Handbook of Industrial Crystallization [3 ed.] 9780521196185

Table of contents :
Contents......Page 6
List of Contributors......Page 9
Preface to the First Edition......Page 10
Preface to the Second Edition......Page 12
Preface to the Third Edition......Page 14
1.3 Solubility of Inorganics......Page 16
1.4 Solubility of Organics......Page 24
1.5 Supersaturation and Metastability......Page 32
1.6 Solution Properties......Page 37
1.7 Thermal Properties......Page 40
References......Page 46
2.2 Basic Concepts of the Solid State......Page 47
2.4 Crystal Growth......Page 56
2.5 Crystal Habit......Page 66
2.6 Crystal Polymorphism......Page 74
2.7 Multicomponent Crystals......Page 79
2.8 Crystal Size......Page 83
2.9 Concluding Remarks......Page 85
References......Page 86
3.1 Introduction......Page 91
3.2 Homogeneous Nucleation......Page 92
3.3 Heterogeneous Nucleation......Page 110
3.4 Secondary Nucleation......Page 111
3.5 Nucleation Kinetics......Page 113
3.6 Control of Nucleation......Page 117
3.7 Nucleation in Polymorphic Systems......Page 119
3.8 Methods to Induce Nucleation......Page 123
References......Page 125
4.2 Retention of Foreign Species in Crystals......Page 130
4.3 Impact of Foreign Species on Growth Rate......Page 137
4.4 Impact of Foreign Species on Crystal Properties......Page 143
4.5 Conclusion......Page 147
References......Page 148
5.2 Crystal Structure Visualization and Analysis Software......Page 151
5.3 Morphology Prediction......Page 153
5.4 Crystal Structure Determination from X-Ray Powder Diffraction Data......Page 160
5.5 Modeling Nucleation and Phase Transitions......Page 162
5.6 Polymorph Searching and Prediction......Page 169
5.7 Solubility Prediction......Page 179
5.8 Chapter Summary and Outlook......Page 183
References......Page 184
Åke C. Rasmuson 6.1 Introduction......Page 187
6.2 Particle Size and Size Distribution......Page 188
6.3 Crystallization Kinetics......Page 192
6.4 Population Balance Modeling......Page 193
6.5 The Idealized MSMPR Concept......Page 196
6.6 Continuous Crystallization and Deviations from the MSMPR Model......Page 199
6.7 Batch Crystallization......Page 201
6.8 Population Balance Modeling of Non-Well- Mixed Processes......Page 203
6.9 Determination of Crystallization Kinetics for Process Modeling......Page 204
6.10 Conclusions......Page 208
References......Page 209
7.1 Introduction......Page 212
7.3 Crystallization Methods......Page 213
7.4 Equipment Design......Page 216
7.5 Instrumentation and Actuation......Page 226
7.6 Case Study: Optimization of a Base-Case Design......Page 227
References......Page 230
8.1 Introduction......Page 231
8.2 Physical and Thermodynamic Properties......Page 232
8.3 Nucleation Kinetics......Page 240
8.4 Crystal Growth Kinetics in Precipitation......Page 243
8.5 Other Processes and Models in Precipitation......Page 246
8.6 Experimental Techniques......Page 260
8.7 Modeling and Control of Crystal Size and Crystal Size Distribution......Page 263
8.8 Scale-Up Rules for Precipitation......Page 268
8.9 Precipitation in Practice......Page 271
8.10 Summary......Page 277
References......Page 278
9.2 Benefits of Melt Crystallization......Page 281
9.3 Phase Diagrams......Page 283
9.4 Crystallization Kinetics......Page 285
9.5 Processes of Melt Crystallization......Page 288
9.6 Post-Crystallization Treatments......Page 290
9.7 Concepts of Commercial Plants......Page 295
9.8 Eutectic Freeze Crystallization......Page 300
References......Page 302
10.2 Crystallizer Flows......Page 305
10.3 Crystallizers......Page 309
10.4 Scale-Up......Page 315
10.5 Modeling......Page 317
10.6 Summary......Page 325
References......Page 326
11.1 Introduction......Page 328
11.2 Crystallization Process Monitoring......Page 331
11.3 Model-Based Optimization and Control of Crystallization Processes......Page 338
11.4 Model-Free (Direct Design) Approaches......Page 352
References......Page 357
12.2 Batch Crystallizers......Page 361
12.3 Batch Crystallization Analysis......Page 364
12.4 Factors Affecting Batch Crystallization......Page 371
12.5 Batch Crystallization Operations......Page 383
12.6 Scale-Up of Batch Crystallization......Page 390
12.7 Summary......Page 391
References......Page 392
13.1 Introduction......Page 395
13.2 Simple Systems......Page 396
13.3 Increasing Complexity......Page 400
13.4 Crystallization Kinetics......Page 406
13.5 Particle Engineering......Page 415
13.6 After the Crystallizer......Page 420
13.7 Intermediates......Page 422
13.8 Other Crystallizations......Page 423
References......Page 426
14.2 Protein and Protein Crystals......Page 429
14.3 The Thermodynamics of Protein Crystallization......Page 432
14.4 Methods of Protein Crystallization......Page 436
14.5 The Role of Nonprotein Solution Components and the Intermolecular Interactions in Solution......Page 439
14.6 Crystal Nucleation......Page 445
14.7 Mechanisms of Crystal Growth......Page 456
14.8 Impurities......Page 463
14.9 Crystal Perfection......Page 464
References......Page 469
15.1 Characteristics of Crystallization in Foods......Page 475
15.2 Controlling Crystallization in Foods......Page 478
15.3 Factors Affecting Control of Crystallization in Foods......Page 489
References......Page 492
16.2 Types of Pigments, Pigment Chemistry, and Pigment Properties......Page 494
16.3 On the Design of Pigment Synthesis Processes......Page 504
16.4 Practical Applications and Special Aspects......Page 517
References......Page 526
Index......Page 528

Citation preview

Handbook of Industrial Crystallization THIRD EDITION Since publication of the first edition of this invaluable resource in 1993 and the second edition in 2001, interest in crystallization science and technology has increased dramatically, and with that interest has come major new developments in the field. This third edition builds on the increased interest in crystallization and incorporates new material in a number of areas, including new chapters on crystal nucleation, molecular modeling applications, and precipitation and crystallization of pigments and dyes as well as completely revised chapters on crystallization of proteins, crystallizer selection and design, control of crystallization processes and process analytical technologies, and an updating of all the other chapters. This book continues to be the perfect reference for industrial and academic scientists and engineers, with this new edition making it even timelier and more important in the field. Allan S. Myerson is Professor of the Practice of Chemical Engineering in the Department of Chemical Engineering at the Massachusetts Institute of Technology. Professor Myerson’s research focuses on separations processes in the chemical and pharmaceutical industry with an emphasis on crystallization from solution, nucleation, polymorphism, and pharmaceutical manufacturing. Professor Myerson has received a number of awards including the American Chemical Society Award in Separations Science and Technology (2008), the AIChE Separations Division Clarence G. Gerhold Award (2015), and the AICHE Process Development Division Excellence in Process Development Research Award (2015). Deniz Erdemir is a Principal Scientist at Bristol-Myers Squibb Company. Dr Erdemir’s research focus lies at the drug substance/drug product interface with emphasis on crystal polymorphism and design of materials via particle engineering to enable robust drug products. She has published twelve papers and she is the inventor on two US patents. Alfred Y. Lee is a Principal Scientist in the Department of Process Research and Development at Merck & Co., Inc. Prior to his current position, Dr Lee was a Principal Scientist at GlaxoSmithKline plc. Dr Lee’s scientific interests are in the area of crystal engineering, crystallization process development, materials characterization, polymorphism, and solid-state chemistry.

Handbook of Industrial Crystallization Third Edition

Edited by

Allan S. Myerson Massachusetts Institute of Technology

Deniz Erdemir Bristol-Myers Squibb Company

Alfred Y. Lee Merck & Co., Inc.

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314–321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9780521196185 DOI: 10.1017/9781139026949 First and Second Editions © Butterworth-Heinemann 1993, 2002 Third Edition © Cambridge University Press 2019 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1993 Printed in the United Kingdom by TJ International, Ltd., Padstow, Cornwall. A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Myerson, Allan S., 1952– editor. | Erdemir, Deniz, editor. | Lee, Alfred Y., editor. Title: Handbook of industrial crystallization / edited by Allan S. Myerson, Deniz Erdemir, Alfred Y. Lee. Description: Third edition. | Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2019. | Includes bibliographical references. Identifiers: LCCN 2018041796 | ISBN 9780521196185 (hardback) Subjects: LCSH: Crystallization – Industrial applications. Classification: LCC TP156.C7 H36 2019 | DDC 660/.284298–dc23 LC record available at https://lccn.loc.gov/2018041796 ISBN 978-0-521-19618-5 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents List of Contributors viii Preface to the First Edition ix Preface to the Second Edition xi Preface to the Third Edition xiii 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7 2

4.5 5

Introduction 76 Homogeneous Nucleation 77 Heterogeneous Nucleation 95 Secondary Nucleation 96 Nucleation Kinetics 98 Control of Nucleation 102 Nucleation in Polymorphic Systems 104 Methods to Induce Nucleation 108 References 110

The Influence of Impurities and Additives on Crystallization 115 Lucrèce H. Nicoud and Allan S. Myerson 4.1

Introduction 115

5.3 5.4 5.5 5.6 5.7 5.8 6

Retention of Foreign Species in Crystals 115 Impact of Foreign Species on Growth Rate 122 Impact of Foreign Species on Crystal Properties 128 Conclusion 132 References 133

Molecular Modeling Applications in Crystallization 136 Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout 5.1 5.2

Introduction 32 Basic Concepts of the Solid State 32 Crystal Nucleation 41 Crystal Growth 41 Crystal Habit 51 Crystal Polymorphism 59 Multicomponent Crystals 64 Crystal Size 68 Concluding Remarks 70 References 71

Crystal Nucleation 76 Deniz Erdemir, Alfred Y. Lee, and Allan S. Myerson 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8

4

Introduction and Motivation 1 Units 1 Solubility of Inorganics 1 Solubility of Organics 9 Supersaturation and Metastability 17 Solution Properties 22 Thermal Properties 25 References 31

Crystals and Crystal Growth 32 Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

3

4.2 4.3 4.4

Solutions and Solution Properties 1 Jennifer Moffitt Schall and Allan S. Myerson

Introduction 136 Crystal Structure Visualization and Analysis Software 136 Morphology Prediction 138 Crystal Structure Determination from X-Ray Powder Diffraction Data 145 Modeling Nucleation and Phase Transitions 147 Polymorph Searching and Prediction 154 Solubility Prediction 164 Chapter Summary and Outlook 168 References 169

Crystallization Process Analysis by Population Balance Modeling 172 Åke C. Rasmuson 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10

Introduction 172 Particle Size and Size Distribution 173 Crystallization Kinetics 177 Population Balance Modeling 178 The Idealized MSMPR Concept 181 Continuous Crystallization and Deviations from the MSMPR Model 184 Batch Crystallization 186 Population Balance Modeling of Non-WellMixed Processes 188 Determination of Crystallization Kinetics for Process Modeling 189 Conclusions 193 References 194

v

Contents

7

Selection and Design of Industrial Crystallizers 197 Herman J. M. Kramer and Richard Lakerveld 7.1 7.2 7.3 7.4 7.5 7.6

8

Precipitation Processes 216 Piotr H. Karpiński and Jerzy Bałdyga 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10

9

Introduction 197 Performance Criteria 198 Crystallization Methods 198 Equipment Design 201 Instrumentation and Actuation 211 Case Study: Optimization of a Base-Case Design 212 References 215

Introduction 216 Physical and Thermodynamic Properties 217 Nucleation Kinetics 225 Crystal Growth Kinetics in Precipitation 228 Other Processes and Models in Precipitation 231 Experimental Techniques 245 Modeling and Control of Crystal Size and Crystal Size Distribution 248 Scale-Up Rules for Precipitation 253 Precipitation in Practice 256 Summary 262 References 263

Melt Crystallization 266 Joachim Ulrich and Torsten Stelzer 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9

Definitions 266 Benefits of Melt Crystallization 266 Phase Diagrams 268 Crystallization Kinetics 270 Processes of Melt Crystallization 273 Post-Crystallization Treatments 275 Concepts of Commercial Plants 280 Eutectic Freeze Crystallization 285 Summary and View to the Future 287 References 287

10 Crystallizer Mixing 290 Understanding and Modeling Crystallizer Mixing and Suspension Flow Daniel A. Green 10.1 10.2 10.3 10.4 10.5 10.6

vi

Introduction 290 Crystallizer Flows 290 Crystallizers 294 Scale-Up 300 Modeling 302 Summary 310 References 311

11 Monitoring and Advanced Control of Crystallization Processes 313 Zoltan K. Nagy, Mitsuko Fujiwara, and Richard D. Braatz 11.1 11.2 11.3 11.4 11.5

Introduction 313 Crystallization Process Monitoring 316 Model-Based Optimization and Control of Crystallization Processes 323 Model-Free (Direct Design) Approaches 337 Summary 342 References 342

12 Batch Crystallization 346 Piotr H. Karpiński and Jerzy Bałdyga 12.1 12.2 12.3 12.4 12.5 12.6 12.7

Introduction 346 Batch Crystallizers 346 Batch Crystallization Analysis 349 Factors Affecting Batch Crystallization 356 Batch Crystallization Operations 368 Scale-Up of Batch Crystallization 375 Summary 376 References 377

13 Crystallization in the Pharmaceutical Industry 380 Simon N. Black 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9

Introduction 380 Simple Systems 381 Increasing Complexity 385 Crystallization Kinetics 391 Particle Engineering 400 After the Crystallizer 405 Intermediates 407 Other Crystallizations 408 Summary 411 References 411

14 Crystallization of Proteins Peter G. Vekilov 14.1 14.2 14.3 14.4 14.5

14.6 14.7 14.8 14.9 14.10

414

Introduction 414 Protein and Protein Crystals 414 The Thermodynamics of Protein Crystallization 417 Methods of Protein Crystallization 421 The Role of Nonprotein Solution Components and the Intermolecular Interactions in Solution 424 Crystal Nucleation 430 Mechanisms of Crystal Growth 441 Impurities 448 Crystal Perfection 449 Concluding Remarks 454 References 454

Contents

15 Crystallization in Foods 460 Richard W. Hartel 15.1 15.2 15.3 15.4

Characteristics of Crystallization in Foods 460 Controlling Crystallization in Foods 463 Factors Affecting Control of Crystallization in Foods 474 Summary 477 References 477

16 Precipitation and Crystallization of Pigments 479 Lars Vicum, Marco Mazzotti, and Martin Iggland 16.1 16.2 16.3 16.4

Index

Introduction 479 Types of Pigments, Pigment Chemistry, and Pigment Properties 479 On the Design of Pigment Synthesis Processes 489 Practical Applications and Special Aspects 502 References 511

513

vii

Contributors

viii

Jerzy Bałdyga Warsaw University of Technology

Alfred Y. Lee Merck & Co., Inc.

Simon N. Black AstraZeneca

Allan S. Myerson Massachusetts Institute of Technology

Richard D. Braatz Massachusetts Institute of Technology

Marco Mazzotti ETH Zürich

Keith Chadwick Massachusetts Institute of Technology

Zoltan K. Nagy Purdue University

Jie Chen Massachusetts Institute of Technology

Lucréce Hèlène Nicoud Massachusetts Institute of Technology

Deniz Erdemir Bristol-Myers Squibb Company

Åke C. Rasmuson Bernal Institute, University of Limerick

Mitsuko Fujiwara Massachusetts Institute of Technology

Erik E. Santiso North Carolina State University

Daniel A. Green GlaxoSmithKline plc

Jennifer Moffit Schall Massachusetts Institute of Technology

Richard W. Hartel University of Wisconsin

Torsten Stelzer University of Puerto Rico

Martin Iggland ETH Zürich

Bernhardt L. Trout Massachusetts Institute of Technology

Piotr H. Karpiński Consultant and expert witness

Joachim Ulrich Martin Luther University Halle-Wittenberg

Herman J. M. Kramer Delft University of Technology

Peter G. Vekilov University of Houston

Richard Lakerveld Hong Kong University of Science and Technology

Lars Vicum BASF SE

Preface to the First Edition

Crystallization is a separation and purification process used in the production of a wide range of materials, from bulk commodity chemicals to specialty chemicals and pharmaceuticals. While the industrial practice of crystallization is quite old, many practitioners still treat it as an art. Many aspects of industrial crystallization have a well-developed scientific basis, and much progress has been made in recent years. Unfortunately, the number of researchers in the field is small, and this information is widely dispersed in the scientific and technical literature. This book will address this gap in the literature by providing a means for scientists or engineers to develop a basic understanding of industrial crystallization and provide the information necessary to begin work in the field, be it in design, research, or plant troubleshooting. Of the eleven chapters in this book, the first two deal with fundamentals such as solubility, supersaturation, and basic concepts in crystallography, nucleation, and crystal growth and are aimed at those with limited exposure in these areas.

The second two chapters provide background in the important area of impurity crystal interactions and an introduction to crystal size distribution measurements and the population balance method for modeling crystallization processes. These four chapters provide the background information that is needed to access and understand the technical literature and are aimed at those individuals who have not been previously exposed to this material or who need a review. The remaining seven chapters deal with individual topics important to industrial practice, such as design, mixing, precipitation, crystallizer control, and batch crystallization. In addition, topics that have become important in recent years, such as melt crystallization and the crystallization of biomolecules, are also included. Each chapter is self-contained but assumes that the reader has knowledge of the fundamentals discussed in the first part of the book. Allan S. Myerson

ix

Preface to the Second Edition

Crystallization from solution and the melt continues to be an important separation and purification process in a wide variety of industries. Since the publication of this volume’s first edition in 1993, interest in crystallization technology, particularly in the pharmaceutical and biotech industries, has increased dramatically. The first edition served as an introduction to the field and provided the information necessary to begin work in crystallization. This new edition incorporates and builds on increased interest in crystallization and incorporates new material in a number of areas. This edition of the book includes a new chapter on crystallization of proteins (Chapter 12), a revised chapter on crystallization of pharmaceuticals (Chapter 11), and a new chapter in an area gaining great importance: crystallization in the food industry (Chapter 13).

Other topics that have become important in crystallization research and technology include molecular modeling applications, which are discussed in Chapters 2 and 3; computational fluid dynamics, which is discussed in Chapter 8; and precipitation, which is discussed in a totally revised Chapter 6. As in the first edition, the first four chapters provide an introduction to newcomers to the field, giving fundamental information and background needed to access and understand the field’s technical literature. The remaining nine chapters deal with individual topics important to industrial crystallization and assume a working knowledge of the fundamentals presented in Chapters 1 to 4. Allan S. Myerson

xi

Preface to the Third Edition

Crystallization science and technology have expanded dramatically since the first edition of this Handbook appeared in 1993. Advances in instrumentation and computation have improved our fundamental understanding of crystallization and have also advanced and improved the practice of crystallization in the chemical, food, and pharmaceutical industries. Interest in crystallization and the development of new crystalline materials has resulted in several new research journals such as Crystal Growth and Design (American Chemical Society) and Crystal Engineering Communications (Royal Society of Chemistry), both established around the same time as the second edition of this Handbook appeared in 2001. This new edition attempts to address the many developments in the field by the addition of a number of new chapters as well as revisions and updates to all the other chapters. This edition includes new chapters on crystal nucleation (Chapter 3), molecular modeling applications in crystallization (Chapter 5),

precipitation and crystallization of pigments and dyes (Chapter 16) and completely revised chapters on crystallizer selection and design (Chapter 7), crystallization process monitoring and control by process analytical technology (Chapter 11), crystallization in the pharmaceutical and bioprocess industries (Chapter 13), and crystallization of proteins (Chapter 14). As in the previous editions, the first five chapters provide an introduction to newcomers to the field, giving fundamental information and background needed to access and understand the field’s technical literature. The remaining eleven chapters focus on individual topics relevant to industrial crystallization and assume a working knowledge of the crystallization fundamentals presented in Chapters 1 to 5. Allan S. Myerson, Deniz Erdemir, and Alfred Y. Lee

xiii

Chapter

1

Solutions and Solution Properties Jennifer Moffitt Schall Massachusetts Institute of Technology Allan S. Myerson Massachusetts Institute of Technology

1.1 Introduction and Motivation Crystallization is a separation and purification technique employed to produce a wide variety of materials. Crystallization may be defined as a phase change in which a crystalline product is obtained from a solution. A solution is a mixture of two or more species that form a homogeneous single phase. Solutions are normally thought of in terms of liquids, but solutions may include solids and even gases. Typically, the term solution has come to mean a liquid solution consisting of a solvent, which is a liquid, and a solute, which is a solid, at the conditions of interest. The term melt is used to describe a material that is solid at normal conditions and is heated until it becomes a molten liquid. Melts may be pure materials, such as molten silicon used for wafers in semiconductors, or they may be mixtures of materials. In that sense, a homogeneous melt with more than one component is also a solution, but it is normally referred to as a melt. A solution can also be gaseous; an example of this is a solution of a solid in a supercritical fluid. Virtually all industrial crystallization processes involve solutions. The development, design, and control of any of these processes involve knowledge of a number of the properties of the solution. This chapter presents and explains solutions and solution properties and relates these properties to industrial crystallization operations.

1.2 Units Solutions are made up of two or more components, of which one is the solvent and the other is the solute(s). There are a variety of ways to express the composition of a solution. If we consider the simple system of a solvent and a solute, its composition may be expressed in terms of mass fraction, mole fraction, or a variety of concentration units, as shown in Table 1.1. The types of units that are commonly used can be divided into those that are ratios of the mass (or moles) of solute to the mass (or moles) of the solvent, those that are ratios of the mass (or moles) of the solute to the mass (or moles) of the solution, and those that are ratios of the mass (or moles) of the solute to the volume of the solution. While all three units are commonly used, it is important to note that use of units of type 3 requires knowledge of the solution density to convert these units into those of the other types. In addition, type 3 units must be defined at a particular temperature because the volume of a solution is a function of temperature. The best units to use for solution preparation are mass of solute per mass of solvent. These units have no temperature dependence, and solutions can be prepared simply by weighing each species.

Table 1.1 Concentration Units

Type 1: Mass (or moles) solute/mass (or moles) solvent grams solute/100 grams solvent moles solute/100 grams solvent Molal: moles solute/1000 grams solvent lbm solute/lbm solvent moles solute/moles solvent Type 2: Mass (or moles) solute/mass (or moles) solution Mass fraction: grams solute/grams total Mole fraction: moles solute/moles total Type 3: Mass (or moles) solute/volume solution Molar: moles solute/liter of solution grams solute/liter of solution lbm solute/gallon solution

Conversion among mass (or mole) -based units is also simple. Example 1.1 demonstrates conversion of units of all three types.

1.3 Solubility of Inorganics 1.3.1 Basic Concepts A solution is formed by the addition of a solid solute to a solvent. The solid dissolves, forming the homogeneous solution. At a given temperature, there is a maximum amount of solute that can dissolve in a given amount of solvent. When this maximum is reached, the solution is said to be saturated. The amount of solute required to make a saturated solution at a given condition is called the solubility. Solubilities of common materials vary widely, even when the materials appear to be similar. Table 1.2 lists the solubility of a number of inorganic species (Mullen 1972; Myerson et al. 1990). The first five species all have calcium as the cation, but their solubilities vary over several orders of magnitude. At 20° C, the solubility of calcium hydroxide is 0.17 g/100 g water, while that of calcium iodide is 204 g/100 g water. The same variation can be seen in the six sulfates listed in Table 1.2. Calcium sulfate has a solubility of 0.2 g/100 g water at 20°C, while ammonium sulfate has a solubility of 75.4 g/100 g water. The solubility of materials depends on temperature. In the majority of cases, the solubility increases with

1

Jennifer Moffitt Schall and Allan S. Myerson

Example 1.1 Conversion of Concentration Units Given: 1 molar solution of NaCl at 25°C

Table 1.2 Solubilities of Inorganics at 20°C

Compound

Chemical formula

Density of solution = 1.042 g/cm3 Molecular weight (MW) NaCl = 58.44 g/mol

Solubility (g anhydrous/ 100 g H2O) 74.5

Calcium chloride

CaCl2

Solution:

Calcium iodide

CaI2

204

1liter 58:44 g NaCl 1 cm3 1 molar ¼ 1 mol NaCl liter of solution 1000 cm3 mol NaCl 1:042 g

Calcium nitrate

Ca(NO3)2

129

Calcium hydroxide

Ca(OH)2

0.17

Calcium sulfate

CaSO4

0.20

Ammonium sulfate

(NH4)2SO4

75.4

Copper sulfate

CuSO4

20.7

Lithium sulfate

Li2SO4

34

Magnesium sulfate

MgSO4

35.5

Silver sulfate

Ag2SO4

0.7

¼

0:056 g NaCl g solution

¼ 0:056 wt fraction NaCl ¼ 5:6 wt% NaCl 0:056 g NaCl 0:056 g NaCl ¼ g solution 0:944 g water þ 0:056 g NaCl ¼ 0:059 g NaCl=g water 0:056 wt fraction NaCl ¼

0:056 g NaCl 0:944 g water þ 0:056 g NaCl

Source: Based on data from Mullen 1972 and Myerson et al. 1990.

0:056 g NaCl 58:44 g=mol NaCl ¼ 0:056 g NaCl 0:944 g water þ 58:44 g=mol NaCl 18 g=mol water ¼ 0:018 mol fraction NaCl

Concentration (g/1000 g H2O)

3000 KNO3

2500 2000 1500

Concentration (g/1000 g H2O)

1.720

1.719

1.718

1.717

1.716

1000

CuSO4

500

0

20

40 60 Temperature (°C)

80

100

Figure 1.2 Solubility of calcium hydroxide in aqueous solution Source: Data from Myerson et al. 1990.

NaCl

0 0

20

40 60 80 Temperature (°C)

100

120

Figure 1.1 Solubility of KNO3, CuSO4, and NaCl in aqueous solution Source: Data from Mullen 1972.

increasing temperature, although the rate of the increase varies widely from compound to compound. The solubilities of several inorganics as a function of temperature are shown in Figure 1.1. Sodium chloride is seen to have a relatively weak temperature dependence, with the solubility changing from 35.7 to 39.8 g/100 g water over a 100°C range. Potassium nitrate, by contrast, changes from 13.4 to 247 g/100 g water over the same temperature range. This kind of information is very important in crystallization processes because it will determine the amount of cooling required to yield a given amount of product and whether cooling will provide a reasonable product yield.

2

In sparingly soluble materials, solubility can also decrease with increasing temperature. A good example of this is the calcium hydroxide–water system shown in Figure 1.2. The solubility of a compound in a particular solvent is part of the system phase behavior and can be described graphically by a phase diagram. In phase diagrams of solid-liquid equilibria, the mass fraction of the solid is usually plotted versus temperature. An example is Figure 1.3, which shows the phase diagram for the magnesium sulfate–water system. This system demonstrates another common property of inorganic solids, the formation of hydrates. A hydrate is a solid formed on crystallization from water that contains water molecules as part of its crystal structure. The chemical formula of a hydrate indicates the number of moles of water present per mole of the solute species by listing a stoichiometric number and water after the dot in the chemical formula. Many compounds that form hydrates form several hydrates with varying amounts of

Solutions and Solution Properties

210

n

f

200 190 MgSO 4 · H2 O

Solution + MgSO4 · H2O

180 170

MgSO 4 · H2O 0.87

160

l

· 6H

Liquid Solution

Solution + MgSO 4 · 6H2O

MgS O

4

140 130

j

d

120

m

MgSO 4 · 6H 2O + MgSO 4

k

110 A

100

MgSO 4 · 7H 2O

Temperature (°F)

2O

150

0.527

MgSO 4 · 6H 2O

e

O · 4 7H 2O

90 B

70

MgS

80

Solution + MgSO4 · 7H2O

60

C

D

E 0.488

50 MgSO 4 · 12H2O

Ice Ice + Solution

c 0.165

a

b

h Solution + MgSO 4 · 12H2O

0.358

40 30

MgSO 4 · 7H2O + MgSO 4

i g

MgSO 4 · 12H2O + MgSO4

20 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

Weight Fraction MgSO4 Figure 1.3 Phase diagram for MgSO4·H2O Source: Reprinted with permission of the publisher from R. M. Felder and R. W. Rousseau (1986), Elementary Principles of Chemical Processes, 2nd ed., p. 259. Copyright © 1986, John Wiley & Sons, Inc.

water. From the phase diagram (Figure 1.3) we can see that MgSO4 forms four stable hydrates ranging from 12 mol of water/mol MgSO4 to 1 mol of water/mol of MgSO4. As is usual with hydrates, as the temperature rises, the number of moles of water in the stable hydrate declines, and at some temperature, the anhydrous material is the stable form. The phase diagram contains much useful information. Referring to Figure 1.3, the line abcdef is the solubility or saturation line that defines a saturated solution at a given temperature. Line ab is the solubility line for the solvent (water) because when a solution in this region is cooled, ice crystallizes out and is in equilibrium with the solution. Point b marks what is known as the eutectic composition. If the solution is cooled at this composition (0.165 wt fraction MgSO4), both ice and MgSO4 will separate as solids. The rest of the curve from b to f represents the solubility of MgSO4 as a function of temperature. If we were to start with a solution at 100°F and 27.5 wt% MgSO4 (point A in Figure

1.3) and cool that solution, the solution would be saturated at the point where a vertical line from A crosses the saturation curve, which is at 80°F. If the solution were cooled to 60°F as shown at point D, the solution will have separated at equilibrium into solid MgSO4·7H2O and a saturated solution of the composition corresponding to point C. The phase diagram also illustrates a general practice concerning hydrate solubility. The solubilities of compounds that form hydrates are usually given in terms of the anhydrous species. This saves much confusion when multiple stable hydrates can exist but requires that care be taken when performing mass balances or preparing solutions. Example 1.2 illustrates these types of calculations. Phase diagrams can be significantly more complex than the example presented in Figure 1.3 and may involve additional stable phases and/or species. A number of references discuss these issues in detail (Gordon 1968; Rosenberger 1981).

3

Jennifer Moffitt Schall and Allan S. Myerson

Example 1.2 Calculations Involving Hydrates Given solid MgSO4·H2O, prepare a saturated solution of MgSO4 at 100°F. (a) Looking at the phase diagram (Figure 1.3), the solubility of MgSO4 at 100°F is 0.31 wt. fraction MgSO4 (anhydrous), and the stable phase is MgSO4·7H2O. First, calculate the amount of MgSO4 (anhydrous) necessary to make a saturated solution at 100°F. 0:31 ¼ xf ¼

weight MgSO4 ðgÞ weight MgSO4 ðgÞ þ weight H2 O ðgÞ

Using a basis: For 1000 g H2O, the weight of MgSO4 (g) needed to make a saturated solution is 449 g MgSO4 (anhydrous). (b) Because the stable form of the MgSO4 available is MgSO4·7H2O, to do this, we need to know the molecular masses of MgSO4, H2O, and MgSO4·7H2O. These are 120.37 g/gmol and 246.48 g/gmol, respectively. xMgSO4 ¼ xH 2 O ¼

wt of MgSO4 in the hydrate 120:37 ¼ 0:488 ¼ wt of MgSO4  7H2 O 246:48 wt of H2 O in the hydrate 126:11 ¼ 0:512 ¼ wt of MgSO4  7H2 O 246:48

1.3.2 Sparingly Soluble Species: Dilute Solutions As we saw in Section 1.3.1, the solubility of materials varies according to their chemical composition and with temperature. Solubility is also affected by the presence of additional species in the solution, by the pH, and by the use of different solvents (or solvent mixtures). When discussing inorganic species, the solvent is usually water, whereas with organics, the solvent can be water or a number of organic solvents or solvent mixtures. If we start with a sparingly soluble inorganic species such as silver chloride and add silver chloride to water in excess of the saturation concentration, we will eventually have equilibrium between solid AgCl and the saturated solution. The AgCl is, like most of the common inorganics, an electrolyte and dissociates into its ionic constituents in solution. The dissociation reaction can be written as AgClðsÞ ↔ Agþ þ Cl

ð1:1Þ

The equilibrium constant for this reaction can be written as K¼

aAgþ aCl aAgCl

ð1:2Þ

where a denotes the activities of the species. If the solid AgCl is in its stable crystal form and at atmospheric pressure, it is at a standard state and will have an activity of 1. The equation can then be written as Ksp ¼ aAgþ aCl ¼ ðγAgþ mAgþ ÞðγCl mClÞ

4

ð1:3Þ

Mass balances: Total weight ¼ wt H2 O þ wt of MgSO4 in the hydrate 0:31ðtotal weightÞ¼ wt of MgSO4 in the hydrate 0:69ðtotal weightÞ¼ wt of H2 O in the hydrate þ wt of H2 O solvent First, we will examine the total mass balance. Because we are using a basis of 1000 g of H2O, and the weight of MgSO4 in the hydrate is equal to the weight of MgSO4 (anhydrous) calculated in (a), the total weight of our system is 1449 g. By substituting the mole fraction expressions into the species material balances, we can solve for the amount of MgSO4·7H2O needed to make a saturated solution at 100°F. 0:31ð1449 gÞ¼ 0:488ðwt of MgSO4  7H2 OÞ wt of MgSO4 . 7H2 Og ¼ 920 g 0:69 ð1449 gÞ ¼ wt H2 O solvent þ 0:512 ðwt of MgSO4  7H2 OÞ 0:69 ð1449 gÞ ¼ wt H2 O solvent þ 0:512 ð920 gÞ wt H2 O solvent ¼ 529 g Therefore, to make a saturated solution of MgSO4 at 100°F starting with MgSO4·7H2O, we need to add 920 g of the hydrate to 529 g of H2O.

where γ is the activity coefficient of the species and m represents the concentration in solution of the ions in molal units. For sparingly soluble species such as AgCl, the activity coefficient can be assumed to be unity (using the asymmetric convention for activity coefficients), so that Equation (1.3) reduces to Ksp ¼ ðmAgþ ÞðmClÞ

ð1:4Þ

This equation represents the solubility product of silver chloride. Solubility products are generally used to describe the solubility and equilibria of sparingly soluble salts in aqueous solutions. Solubility products of a number of substances are given in Table 1.3. It is important to remember that use of solubility product relations based on concentrations assumes that the solution is saturated, in equilibrium, and ideal (the activity coefficient is equal to 1). It is therefore an approximation, except with very dilute solutions of one solute. Equation (1.4) can be used for electrolytes in which there is a 1:1 molar ratio of the anion and cation. For an electrolyte that consists of univalent and bivalent ions, such as silver sulfate, which dissociates into 2 mol of silver ions for each mole of sulfate ions, the solubility product equation would be written as Ksp ¼ ðmAgþ Þ2 ðmSO2 Þ 4

ð1:5Þ

In the dissociation equation, the concentration of the ions of each species is raised to the power of the species’ stoichiometric number. The solubility product principle enables simple calculations to be made of the effect of other species on the solubility of a given substance and may be used to determine the species that

Solutions and Solution Properties Table 1.3 Solubility Products

Substance

Solubility Product at 25°C

Aluminum hydroxide

3.70 × 10−15

Barium carbonate

2.58 × 10−9

Barium chromate

1.17 × 10−10

Barium fluoride

1.84 × 10−7

Barium iodate monohydrate

1.67 × 10−9

Barium sulfate

1.08 × 10−10 −9

Calcium carbonate (calcite)

3.36 × 10

Calcium fluoride

3.45 × 10−11

Calcium iodate hexahydrate

7.10 × 10−7

Calcium oxalate monohydrate

2.32 × 10−9

Calcium sulfate

4.93 × 10−5

Cupric iodate monohydrate

6.94 × 10−8

Cupric oxalate

4.43 × 10−10

Cuprous bromide

6.27 × 10−9

Cuprous chloride

1.72 × 10−7

Cuprous iodide

1.27 × 10−12

Ferric hydroxide

2.79 × 10−39

Ferrous hydroxide

4.87 × 10−17 −14

Lead carbonate

7.40 × 10

Lead sulfate

2.53 × 10−8

Lithium carbonate

8.15 × 10−4

Magnesium carbonate

6.82 × 10−6

Magnesium fluoride

5.16 × 10−11

Magnesium hydroxide

5.61 × 10−12

Magnesium oxalate dehydrate

4.83 × 10−6

Manganese carbonate

2.24 × 10−11

Silver bromate

5.38 × 10−5

Silver iodide

8.52 × 10−17

Zinc hydroxide

3.00 × 10−17

Source: Data from Weast 1975.

will precipitate in an electrolyte mixture. One simple result of applying the solubility product principle is the common ion effect. This is the effect caused by the addition of an ionic species that has an ion in common with the species of interest. Because the solubility of a species is given by the product of the concentration of its ions, when the concentration of one type of ion increases, the concentration of the other must decline, or the overall concentration of that compound must decline. We can illustrate this simply by using our previous example of silver

chloride. The solubility product of silver chloride at 25°C is 1.56 × 10−10. This means that at saturation we can dissolve 1.25 × 10−5 mol of AgCl/1000 g of water. If, however, we were to start with a solution that has a concentration of 1 × 10−5 molal NaCl (hence 1 × 10−5 molal Cl−), the solubility product equation would be written in the form Ksp ¼ ðmAgþ ÞðmCl Þ ¼ ðxAgþ ÞðxCl þ 1  105 Þ 5

Ksp ¼ x þ 10 x 2

ð1:6Þ ð1:7Þ

where x is the amount of AgCl that can dissolve in the solution. Solving Equation (1.7) results in x ¼ 0:725  105 molal. The common ion effect has worked to decrease the solubility of the compound of interest. It is important to remember that this is true only for very dilute solutions. In more concentrated solutions, the activity coefficients are not unity, and more complex electrical effects and complexation may occur. This is discussed in detail in Section 1.3.3. Another use of solubility products is the determination, in a mixture of slightly soluble materials, as to what material is likely to precipitate. This is done by looking at all the ion concentrations and calculating their products in all possible combinations. These are then compared with the solubility products that must already be known. This is useful in situations where scale formation is of interest or in determining the behavior of slightly soluble mixtures.

1.3.3 Concentrated Solutions Unfortunately, like all easy-to-use principles, the solubility product principle is not generally applicable. At higher concentrations, electrical interactions, complex formation, and solution nonideality make the prediction of the effect of ionic species on the solubility of other ionic species much more complicated. In Section 1.3.2 we used the solubility product principle to calculate the effect of a common ion on the solubility of a sparingly soluble species. The common ion effect, however, is completely dominated by a more powerful effect when a large concentration of another electrolyte is present. In fact, the solubility of sparingly soluble materials increases with increasing ion concentration in solution. This is called the salt effect and is illustrated in Figures 1.4 through 1.6, where we see the increase in solubility of AgCl as a function of increasing concentrations of added electrolytes. We see this effect in both added salts with a common ion and without. This effect can also be induced by changing the pH of the solution because this changes the ion content of the solution. The solubility of many inorganics in aqueous solution is available in the book by Linke and Seidell (1965). This reference also contains the solubilities of electrolytes in the presence of other species. For example, Figure 1.7 shows the solubility of NaCl as a function of NaOH concentration. As a general rule, the solubility of most inorganics in water is available as a function of temperature. What is more difficult to find is the effect of other species on the solubility. If several other species are present, the data will usually not be available. Given this situation, there are two alternatives. The first is to measure the solubility at the conditions and composition of interest. Experimental

5

Jennifer Moffitt Schall and Allan S. Myerson

10–3

0.016

g AgCl/1000 g H2O

g AgCl/1000 g H2O

2.2

2.1

2.0

1.9

1.8

0

0.1

0.2

0.3

0.4

0.012

0.008

0.004

0.000 0.000

0.5

0.002

0.004

0.006

0.008

0.010

g NaNO3/1000 g H2O

g CaSO 4/1000 g H2O

Figure 1.5 Solubility of AgCl in aqueous NaNO3 solution at 30°C Source: Data from Linke and Seidell 1965.

Figure 1.4 Solubility of AgCl in aqueous CaSO4 solution at 25°C Source: Data from Linke and Seidell 1965.

400

g NaCl/1000 g H2O

7

g AgCl/1000 g H2O

6 40°C

5 4 3

20°C

200

100

2 10°C

1 0

0

0

100

200

300

400

200

400

600

800

1000 1200 1400

Figure 1.7 Solubility of NaCl in aqueous NaOH solution Source: Data from Linke and Seidell 1965.

Figure 1.6 Solubility of AgCl in aqueous CaCl2 solution Source: Data from Linke and Seidell 1965.

methods for solubility measurement will be discussed in Section 1.4.6. The second alternative is to calculate the solubility. This is a viable alternative when thermodynamic data are available for the pure components (in solution) making up the multicomponent mixture. An excellent reference for calculation techniques in this area is the Handbook of Aqueous Electrolyte Thermodynamics by Zemaitis et al. (1986). A simplified description of calculation techniques is presented later in Section 1.3.3. Solution Thermodynamics. As we have seen previously, for a solution to be saturated, it must be at equilibrium with the solid solute. Thermodynamically, this means that the chemical potential of the solute in the solution is the same as the chemical potential of the species in the solid phase. μisolid ¼ μisolution

0

50°C

g NaOH/1000 g H2O

g CaCl2/1000 g H2O

ð1:8Þ

If the solute is an electrolyte that completely dissociates in solution (strong electrolyte), Equation (1.8) can be rewritten as ð1:9Þ μisolid ¼ vc μc þ va μa where vc and va are the stoichiometric numbers, and μc and μa are the chemical potentials of the cation and anion,

6

300

respectively. The chemical potential of a species is related to the species activity by μi ðTÞ ¼ μ0ðaqÞ ðTÞ þ RTlnðai Þ

ð1:10Þ

where ai is the activity of species i, μ0ðaqÞ is an arbitrary reference state chemical potential, R is the gas constant, and T is temperature. The activity coefficient is defined as γi ¼

ai mi

ð1:11Þ

where mi is the concentration in molal units. In electrolyte solutions, because of the condition of electroneutrality, the charges of the anion and cation will always balance. When a salt dissolves, it will dissociate into its component ions. This has led to the definition of a mean ionic activity coefficient and mean ionic molality defined as γ ¼ ðγvc c γvaa Þ1=v m ¼ ðmvc c mvaa Þ

1=v

ð1:12Þ ð1:13Þ

where vc and va are the stoichiometric number of ions of each type present in a given salt. The chemical potential for a salt can be written as

Solutions and Solution Properties

μsaltðaqÞ ¼ μ0ðaqÞ þ vRTlnðγ m Þv μ0ðaqÞ

where is the sum of the two ionic standard-state chemical potentials, and v is the stoichiometric number of moles of ions in 1 mole of solid. In practice, experimental data are usually reported in terms of mean ionic activity coefficients. As we discussed earlier, various concentration units can be used. We have defined the activity coefficient on a molal scale. On a molar scale, it is ai ðcÞ yi ¼ ci

ð1:15Þ

where yi is the molar activity coefficient, and ci is the molar concentration. We can also define the activity coefficient on a mole fraction scale as fi ¼

ai ðxÞ xi

ð1:16Þ

where fi is the activity coefficient, and xi the mole fraction. Converting activity coefficients from one type of unit to another is neither simple nor obvious. Equations that can be used for this conversion have been developed and include (Zemaitis et al. 1986) f ¼ ð1:0 þ 0:01Ms vmÞγ ρ þ 0:001cðvMs  MÞ y ρ0
 ρ  0:001cM c γ ¼ y ¼ y ρ0 mρ0
 mρ  ρ 0 y ¼ ð1 þ 0:001mMÞ γ ¼ γ ρ0 c f ¼

ð1:17Þ ð1:18Þ ð1:19Þ ð1:20Þ

where v = stoichiometric number = vþ þ v ρ = solution density ρ0 = solvent density M = molecular weight of the solute Ms = molecular weight of the solvent Solubility of a Pure-Component Strong Electrolyte. Calculation of the solubility of a pure-component solid in solution requires that the mean ionic activity coefficient be known along with a thermodynamic solubility product (a solubility product based on activity). Thermodynamic solubility products can be calculated from standard-state Gibbs free energy of formation data. If, for example, we wish to calculate the solubility of KCl in water at 25°C, KCl ↔ Kþ þ Cl

ð1:21Þ

the equilibrium constant is given by Ksp ¼

aKþ aCl ¼ ðγKþ mKþ ÞðγCl mCl Þ ¼ γ2 m2 aKCl


 DGf 0 Ksp ¼ exp RT

ð1:14Þ

ð1:22Þ

The equilibrium constant is related to the Gibbs free energy of formation by the relation

ð1:23Þ

The free energy of formation of KCl can be written as DGf 0 ¼ DGf 0 Kþ þ DGf 0 Cl  DGf 0 KCl

ð1:24Þ

Using data from the literature, one finds (Zemaitis et al. 1986)

so that

DGf 0 ¼ 1282 cal=gmol

ð1:25Þ

Ksp ¼ 8:704

ð1:26Þ

Employing this equilibrium constant and assuming an activity coefficient of 1 yields a solubility concentration of 2.95 molal. This compares with an experimental value of 4.803 molal (Linke and Seidell 1965). Obviously, assuming an activity coefficient of unity is a very poor approximation in this case and results in a large error. The calculation of mean ionic activity coefficients can be complex, and a number of methods are available. Several references describe these various methods (Robinson and Stokes 1970; Guggenheim 1986; Zemaitis et al. 1986). The method of Bromley (1972, 1973, 1974) can be used up to a concentration of 6 molal and can be written as pffiffi Ajzþ z j I ð0:06 þ 0:6BÞjzþ z jI pffiffi þ logγ ¼ þ BI ð1:27Þ  2 1þ I 1 þ jz1:5 þ z j I where γ = activity coefficient A = Debye-Hückel constant z = number of charges on the . cation X or anion I = ionic strength, which is 1 2 i mi zi2 B = constant for ion interaction Values for the constant B are tabulated for a number of systems (Zemaitis et al. 1986). For KCl, B = 0.0240. Employing Equation (1.27), γ can be calculated as a function of m. This must be done until the product γ2 m2 ¼ Ksp . For the KCl-water system at 25°C, γ is given as a function of concentration in Table 1.4 along with γ2 m2 . You can see that the resulting calculated solubility is approximately 5 molal, which compares reasonably well with the experimental value of 4.8 molal. Electrolyte Mixtures. Calculation of the solubility of mixtures of strong electrolytes requires knowledge of the thermodynamic solubility product for all species that can precipitate and requires using an activity coefficient calculation method that takes into account ionic interactions. These techniques are well described by Zemaitis et al. (1986), but we will discuss a simple case in this section. The simplest case would be a calculation involving a single possible precipitating species. A good example is the effect of HCl on the solubility of KCl. The thermodynamic solubility product Ksp for KCl is defined as Ksp ¼ ðγKþ mKþ ÞðγCl mCl Þ ¼ γ2 m2

ð1:28Þ

7

Jennifer Moffitt Schall and Allan S. Myerson Table 1.4 Calculated Activity Coefficients for KCl in Water at 25°C

0.768

5.8 × 10−3

1.0

0.603

0.364

1.5

0.582

0.762

2.0

0.573

1.31

2.5

0.569

2.02

3.0

0.569

2.91

3.5

0.572

4.01

4.0

0.577

5.32

4.5

0.584

6.91

5.0

0.592

8.76

3.00

2.00

1.00

0.00

Experimental Calculated

HCl Molality Figure 1.8 Calculated versus experimental KCl solubility in aqueous HCl solution at 25°C Source: Reproduced from J. F. Zemaitis, Jr., D. M. Clark, M. Rafal, and N. C. Scrivner (1986), Handbook of Aqueous Electrolyte Thermodynamics, p. 284. Used by permission of the American Institute of Chemical Engineers © 1986 AIChE.

Note: Ksp = 8.704 from Gibbs free energy of formation. Source: Data from Zemaitis et al. 1986.

Bij ¼ In the preceding example, we obtained Ksp from the Gibbs free energy data and used this to calculate the solubility of KCl. Normally for a common salt, solubility data are available. Ksp is therefore obtained from the experimental solubility data and activity coefficients. Using the experimental KCl solubility at 25°C (4.8 molal) and the Bromley activity coefficients yields Ksp ¼ 8:01. If we wish to calculate the KCl solubility in a 1 molal HCI solution, we can write the following equations: ðγKþ mKþ ÞðγCl mCl Þ ¼1 Ksp zKþ mKþ þ zHþ mHþ ¼ zCl mCl þ zCl mCl ðfrom KClÞ ðfrom HClÞ ðfrom KClÞ ðfrom HClÞ

ð1:29Þ ð1:30Þ

Equations (1.29) and (1.30) must be satisfied simultaneously for a fixed value of 1 molal HCI. Using Bromley’s method for multicomponent electrolytes, pffiffi Azi2 I pffiffi þ Fi logγi ¼ ð1:31Þ Iþ I where A = Hückel constant I = ionic strength i = any ion present zi = number of charges on ion i Fi = an interaction parameter term X Fi ¼ Bij zij2 mj

ð1:32Þ

where j indicates all ions of opposite charge to i zij ¼ where mj = molality of ion j

8

zi þ zj 2

10.00

0.1

4.00

8.00

8.11 × 10−6

6.00

0.901

4.00

0.01

2.00

γ±m

0.00

γ+

KCl Molality

m (molality)

5.00

ð1:33Þ

ð0:06 þ 0:6BÞjzi zj j ½1 þ ð1:5=jzi zj jÞI2

þB

ð1:34Þ

Employing these equations, the activity coefficient for K+ and Cl are calculated as a function of KCl concentration at a fixed HCI concentration of 1 molar. These values, along with the molalities of the ions, are then substituted into Equation (1.29) until it is an equality (within a desired error). The solubility of KCl in a 1 molal solution of HCl is found to be 3.73 molal, which compares with an experimental value of 3.92 molal. This calculation can then be repeated for other fixed HCl concentrations. Figure 1.8 compares the calculated and experimental values of KCl solubility over a range of HCl concentrations. Unfortunately, many systems of interest include species that form complexes, intermediates, and undissociated aqueous species. This greatly increases the complexity of solubility calculations because of the large number of possible species. In addition, mixtures with many species often include a number of species that may precipitate. These calculations are extremely tedious and time consuming to do by hand or to write a specific computer program for each application. Commercial software is available for calculations in complex electrolyte mixtures. ProChem, part of the OLI Toolkit developed by OLI Systems, Inc. (Cedar Knolls, NJ), is an excellent example. The purpose of the package is to simultaneously consider the effects of the detailed reactions as well as the underlying species interactions that determine the actual activity coefficient values. Only by such a calculation can the solubility be determined. A good example of the complexity of these calculations can be seen when looking at the solubility of CrðOHÞ3 . Simply assuming the dissociation reaction −

CrðOHÞ3 ↔ Cr3þ þ 3OH

ð1:35Þ

and calculating a solubility using the Ksp obtained from Gibbs free energy of formation lead to serious error. This is because a

Solutions and Solution Properties Table 1.5 Calculated Results for Cr(OH)3 Solubility at 25°C

Equilibrium constant

K (mol/kg)

H2O

9.94 × 10−15

CrOH+2

1.30 × 10−10

Cr2(OH)2+4

2.35 × 10−5

Cr3(OH)4+5

2.52 × 10−7

Liquid-phase pH = 10

Ionic strength = 1.01 × 10−2

Species

Moles

H2O

55.5

OH−

1.00 × 10−4

0.902

Cr+3

2.21 × 10−18

0.397

CrOH+2

9.32 × 10−13

0.655

Cr(OH)2+

1.65 × 10−8

0.899

−7

Cr(OH) (aq)

6.56 × 10

Cr(OH)4−

3.95 × 10−6

Cr3(OH)4

+5

Cl− −

Na

2.98 × 10 4.48 × 10

−21 −22

1.00 × 10−2 1.01 × 10

CrOH2+

10

–14

10–16

1.0 −10

1.22 × 10

Cr2(OH)2

10–12 Cr2(OH) 4+ 2

Activity coefficient

H

+4

Molality

6.44 × 10

1.67 × 10−5

Cr3(OH) 5+ 4

10–10

−31

Cr(OH) +2

–8

2.03 × 10−6

−

3

10

2.72 × 10

Cr(OH)3(crystal)

+

Cr(OH) 3(aq) Cr(OH) –4

−9

+

Cr(OH)3(aq)

Cr(OH)4

Total chrome

10–6

−2

0.904

1.0 0.899

Cr3+

10–18

10–20 6.5

7.0

7.5

8.0

8.5

9.0

9.5

10.0 10.5 11.0

pH Figure 1.9 Chrome hydroxide solubility and speciation versus pH at 25°C Source: Reproduced from J. F. Zemaitis, Jr., D. M. Clark, M. Rafal, and N. C. Scrivner (1986), Handbook of Aqueous Electrolyte Thermodynamics, p. 284. Used by permission of the American Institute of Chemical Engineers © 1986 AIChE.

0.185 10–4

0.0725 0.898 0.898

Source: Data from Zemaitis et al. 1986.

number of other dissociation reactions and species are possible. These include CrðOHÞ3 (undissociated molecule in soluþ 2þ 4þ tion), CrðOHÞ 4 , CrðOHÞ2 , CrðOHÞ , Cr2 ðOHÞ2 , and 5þ Cr3 ðOHÞ4 . Calculation of the solubility of CrðOHÞ3 as a function of pH using HCl and NaOH to adjust the pH requires taking into account all species, equilibrium relationships, mass balances, and electroneutrality, as well as calculation of the ionic activity coefficients. The results of such a calculation (employing Prochem) is shown in Table 1.5 and Figures 1.9 and 1.10. Table 1.5 shows the results obtained at a pH of 10. Figure 1.9 gives the solubility results obtained from a series of calculations and also shows the concentration of the various species, whereas Figure 1.10 compares the solubility obtained with that calculated from a solubility product. The solubility results obtained by the simple solubility product calculation are orders of magnitude less than those obtained by the complex calculation, demonstrating the importance of considering all possible species in the calculation.

Chrome Solubility

Cr(OH)2

10–4

Pure Water

10–5

Dilute Brine

10–6 Cr+3 from solubility product

10–7 6

7

8

9

10

pH Figure 1.10 Chrome solubility versus pH Source: Reproduced with permission of OLI Systems, Inc.

1.4 Solubility of Organics In crystallization operations involving inorganic materials, we virtually always employ water as the solvent, thus requiring solubility data on inorganic water systems. Because most inorganic materials are ionic, this means that dissociation reactions, ionic interactions, and pH play a major role in determining the solubility of a particular inorganic species in aqueous solution. When dealing with organic species (or inorganics in nonaqueous solvents), a wide variety of solvents and solvent mixtures usually can be employed. The interaction between the solute and

9

Jennifer Moffitt Schall and Allan S. Myerson

calculate the solubility of an organic solute in different solvents and explain how to assess mixed solvents.

Solubility (mole fraction)

0.10 0.08

Isopropanol + Water

1.4.1 Thermodynamic Concepts and Ideal Solubility

0.06 0.04

As we have shown previously, the condition for equilibrium between a solid solute and a solvent is given by the relation

Ideal

0.02

Methanol

μisolid ¼ μisolution

Ethanol

0.00 10

15

20

25

30

35

40

45

Temperature (°C) Figure 1.11 Solubility of hexamethylenetetramine in different solvents Source: Reprinted with permission from S. Decker, W. P. Fan, and A. S. Myerson, Solvent selection and batch crystallization, Ind. Eng. Chem. Fund. 1986; 25:925. Copyright © 1986, American Chemical Society.

A thermodynamic function known as the fugacity can be defined as
 fi 0 μi  μi ¼ RT ln 0 ð1:37Þ fi Comparing Equation (1.10) with Equation (1.37) shows us that the activity ai ¼ fi =fi0 . Through a series of manipulations, it can be shown that for phases in equilibrium (Prausnitz et al. 1986),

0.20

Solubility (mole fraction)

Ideal

fisolid ¼ fisolution

0.15

In Ethanol

0.05

In IPA In Water

25.0

30.0

35.0

40.0

45.0

Temperature (°C) Figure 1.12 Solubility of adipic acid in different solvents Source: Reprinted with permission from S. Decker, W. P. Fan, and A. S. Myerson, Solvent selection and batch crystallization, Ind. Eng. Chem. Fund. 1986; 25:925. Copyright © 1986, American Chemical Society.

the solvent determines the differences in solubility commonly observed for a given organic species in a number of different solvents. This is illustrated in Figures 1.11 and 1.12 for hexamethylene tetramine and adipic acid in several different solvents. In the development of crystallization processes, this can be a powerful tool. In many cases the solvent chosen for a particular process is an arbitrary choice made in the laboratory with no thought of the downstream processing consequences. Frequently, from a chemical synthesis or reaction point of view, a number of different solvents could be used with no significant change in product yield or quality. This means that the solubility and physical properties of the solvent (solubility as a function of temperature, absolute solubility, and vapor pressure) should be evaluated so that the solvent that provides the best characteristics for the crystallization step is chosen. This requires that the process-development engineers be in contact with the synthetic organic chemists early in process development. In this section we will describe the basic principles required to estimate and

10

ð1:38Þ

Equation (1.38) will be more convenient for us to use in describing the solubility of organic solids in various solvents. The fugacity is often thought of as a “corrected pressure” and reduces to pressure when the solution is ideal. Equation (1.38) can be rewritten as f2solid ¼ γ2 x2 f20 ð1:39Þ

0.10

0.00 20.0

ð1:36Þ

where f2 = fugacity of the solid x2 = mole fraction of the solute in the solution f20 = standard state fugacity γ2 = activity coefficient of the solute or x2 ¼

f2solid γ2 f20

ð1:40Þ

Equation (1.40) is a general equation for the solubility of any solute in any solvent. We can see from this equation that the solubility depends on the activity coefficient and on the fugacity ratio f2 =f20 . The standard state fugacity normally used for solid–liquid equilibrium is the fugacity of the pure solute in a subcooled liquid state below its melting point. We can simplify Equation (1.40) further by assuming that our solid and subcooled liquid have small vapor pressures. We can then substitute vapor pressure for fugacity. If we further assume that the solute and solvent are chemically similar so that γ2 ¼ 1, then we can write x2 ¼

P2s solid solute P2s subcooled liquid solute

ð1:41Þ

Equation (1.41) gives the ideal solubility. Figure 1.13, an example phase diagram for a pure component, illustrates several points. First, we are interested in temperatures below the triple point because we are interested in conditions where the solute is a solid. Second, the subcooled liquid pressure is obtained by extrapolating the liquid–vapor line to the correct temperature.

Solutions and Solution Properties


 DHtp 1 1 1  x2 ¼ exp γ2 Ttp T R

Critical Point

Pressure

SOLID

LIQUID

P Ssubcooled liquid

or because

Triple Point

P Ssolid

VAPOR Solution Temperature

Temperature Figure 1.13 Schematic of a pure-component phase diagram Source: Reprinted by permission of Prentice-Hall from J. M. Prausnitz, R. N. Lichenthaler, and E. Gomes de Azvedo (1986), Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed. Copyright © 1986, Prentice-Hall Publishers.

Equation (1.41) gives us two important pieces of information. The first is that the ideal solubility of the solute does not depend on the solvent chosen; the ideal solubility depends only on the solute properties. Second, it shows that differences in the pure-component phase diagrams that result from structural differences in materials will alter the triple point and hence the ideal solubility. A general equation for the fugacity ratio is !

 DHtp 1 1 DCp Ttp Ttp f2 ln ln  þ 1   ¼ Ttp T f2subcooled liquid solute R R T T 

DV ðP  Ptp Þ RT

ð1:42Þ

where DHtp = enthalpy change for the liquid solute transformation at the triple point Ttp = triple point temperature DCp = difference between the Cp of the liquid and the solid DV = volume change for liquid–solid transformation R = universal gas constant If substituted into Equation (1.40), this yields the solubility equation 

 DHtp 1 DCp Ttp Ttp 1 1 ln  þ1 x2 ¼ exp   γ2 Ttp T R R T T 

DV ðP  Ptp Þ RT

ð1:43Þ

Equation (1.43) is the most general form of the solubility equation. In most situations (though not all), the effect of pressure on solubility is negligible, so the last term on the right-hand side of the equation can be dropped. In addition, the heat-capacity term also usually can be dropped from the equation. This yields

DStp ¼ DHtp =Ttp 
 DStp Ttp 1 1 x2 ¼ exp γ2 R T

ð1:44Þ

ð1:45Þ ð1:46Þ

In many instances, the triple-point temperature of a substance is not known. In those cases, the enthalpy of melting (fusion) and the melting-point temperature are used because they are usually close to the triple-point temperature: 
 1 DHm 1 1 x2 ¼ exp  ð1:47Þ γ2 Tm T R For an ideal solution, when the activity coefficient equation equals 1, this reduces to 
 DHm 1 1 x2 ¼ exp  ð1:48Þ Tm T R Equation (1.48) allows the simple calculation of ideal solubilities and can be used profitably to see the differences in solubility of chemically similar species with different structures. This is illustrated in Table 1.6, where calculated ideal solubilities are shown together with DHm and Tm . Isomers of the same species can have widely different ideal solubilities based on changes in their physical properties, which relate back to their chemical structures. Equation (1.48) also tells us that for an ideal solution, solubility increases with increasing temperature. The rate of increase is approximately proportional to the magnitude of the heat of fusion (melting). For materials with similar melting temperatures, the lower the heat of fusion, the higher is the solubility. For materials with similar heats of fusion, the material with the lower melting temperature has the higher solubility. A good example of this is shown in Table 1.6 when looking at ortho-, meta-, and para-chloronitrobenzene. The lower-melting ortho has an ideal solubility of 79 mol% compared with 25 mol% for the higher-melting para. While Equation (1.48) is useful for comparing relative solubilities of various solutes, it takes no account of the solvent used or solute–solvent interactions. To account for the role of the solvent, activity coefficients must be calculated.

1.4.2 Regular Solution Theory In electrolytic solutions, we were concerned with electrostatic interactions between ions in the solution and with the solvent (water). In solutions of nonelectrolytes, we will be concerned with molecule–solvent interactions due to electrostatic forces, dispersion forces, and chemical forces. Even though a solution contains no ions, electrostatic interactions can still be significant. This is because of a property called polarity. An electrically neutral molecule can have a dipole moment that is due to an asymmetric distribution of its electrical charge. This means that one end of the molecule is positive and the other end is negative. The dipole moment is defined by

11

Jennifer Moffitt Schall and Allan S. Myerson Table 1.6 Melting Temperature, Enthalpy of Melting, and Ideal Solubility of Organic Solutes at 25°C

Table 1.7 Permanent Dipole Moments

Im (K)

ΔHm (cal/mol)

Molecule

µ (Debyes)

Molecule

Ideal solubility (mol%)

µ (Debyes)

CO

0.10

CH3I

1.64

C3H6

0.35

CH3COOCH3

1.67

orthoChloronitrobenzene

307.5

4546

79

C6H5CH3

0.37

C2H5OH

1.70

metaChloronitrobenzene

317.5

4629

62

PH3

0.55

H2O

1.84

HBr

0.80

HF

1.91

paraChloronitrobenzene

356.7

4965

25

CHCl3

1.05

C2H5 F

1.92

(C2H5)2O

1.18

(CH3)2CO

2.87

Naphthalene

353.4

4494

31

NH3

1.47

C6H5COCH3

3.00

Urea

406.0

3472

21

C6H5NH2

1.48

C2H5NO2

3.70

Phenol

314.1

2695

79

C6H5Cl

1.55

CH3CN

3.94

Anthracene

489.7

6898

1

C2H5SH

1.56

CO(NH2)2

4.60

Phenanthrene

369.5

4456

23

SO2

1.61

KBr

9.07

Biphenyl

342.2

4235

39

Substance

Source: Data from Prausnitz et al. 1986.

Source: Based on data from Walas 1985.

μ ¼ el

ð1:49Þ

where e is the magnitude of the electric charge and l is the distance between the two charges. The dipole moment is a measure of how polar a molecule is. As the dipole moment increases, the molecule is less symmetric in terms of its electrical charge. A list of molecules and their dipole moments is given in Table 1.7. As you can see from the table, water is quite polar. There are also molecules with more complex charge distributions called quadrupoles, which also display this asymmetric charge behavior. This shows that even without ions, electrostatic interactions between polar solvent molecules and polar solute molecules will be of importance in activity coefficient calculations and will therefore affect the solubility. Organic solutes and solvents are usually classified as polar or nonpolar, although, of course, there is a range of polarity. Nonpolar solutes and solvents also interact through forms of attraction and repulsion known as dispersion forces. Dispersion forces result from oscillations of electrons around the nucleus and have a rather complex explanation; however, it is sufficient to say that nonidealities can result from molecule–solvent interactions that result in values of the activity coefficient not equal to 1. An excellent reference in this area is the book by Prausnitz et al. (1986). Generally, the activity coefficients are less than 1 when polar interactions are important, with a resulting increase in solubility of compounds compared with the ideal solubility. The opposite is often true in mixed polar–nonpolar systems, with the activity coefficients being greater than 1. Nonpolar solutions typically have activity coefficients around 1 because they do not tend to exhibit the high degree of interactions that exist in polar and hydrogen-bonding systems. A number of methods are used to calculate activity coefficients of solid solutes in solution. A

12

frequently used method is that of Scatchard–Hildebrand, which is also known as regular solution theory (Prausnitz et al. 1986): lnγ2 ¼

V2L ðδ1  δ2 Þ2 ϕ21 RT

ð1:50Þ

where V2L = molar volume of the subcooled liquid solute δ2 = solubility parameter of the subcooled liquid δ1 = solubility parameter of the solvent ϕ1 = volume fraction of the solvent, defined by ϕ1 ¼

x1 V1L x1 V1L þ x2 V2L

ð1:51Þ

The solubility parameters are defined by the relations
v 1=2
v 1=2 Du1 Du2 δ1 ¼ δ2 ¼ ð1:52Þ v1 v2 where Du is the enthalpy of vaporization and v is the molar liquid volume. Solubility parameters for a number of solvents and solutes are given in Table 1.8. This method works moderately well at predicting solubilities in nonpolar materials. Calculated solubility results employing this theory are shown in Table 1.9 along with experimentally determined values. It is apparent that in many cases this theory predicts results very far from the experiment. Regular solution theory has limitations in predicting activity coefficients because it was developed to quantitatively describe solutions with no excess entropy, no change in volume on mixing, and a nonzero enthalpy of mixing. This theory always predicts that all deviations from ideality will be positive, with γ > 1. Because strong polar–polar interactions generally exhibit behavior with γ < 1, regular solution theory should not be used for polar systems without extreme caution.

Solutions and Solution Properties Table 1.8 Solubility Parameters at 25°C

Substance

δ (cal/cm )

3 1/2

Anthracene

9.9

Naphthalene

9.38

Phenanthrene

10.52

Acetic acid

10.05

Acetone

9.51

Aniline

11.46

1-Butanol

11.44

Carbon disulfide

9.86

Carbon tetrachloride

9.34

Chloroform

9.24

Cyclohexane

8.19

Cyclohexanol Diethyl ether Ethanol

11.4 7.54 12.92

n-Hexane

7.27

Methanol

14.51

Phenol

12.11

1-Propanol

12.05

2-Propanol

11.57

Perfluoro-n-heptane

6.0

Neopentane

6.2

Isopentane

6.8

n-Pentane

7.1

1-Hexene

7.3

n-Octane

7.5

n-Hexadecane

8.0

Ethyl benzene

8.8

Toluene

8.9

Benzene

9.2

Styrene

9.3

Tetrachloroethylene

9.3

Bromine

9.5

Sources: Based on data from Prausnitz et al. 1986 and Walas 1985.

Similarly, regular solution theory cannot generally describe solutions with strong hydrogen bonding or acid–base interactions. A number of modifications of the Scatchard–Hildebrand theory as well as other methods are available for activity

coefficient calculations and are described in various other texts (Walas 1985; Prausnitz et al. 1986; Reid et al. 1987). Empirical methods have been developed to mitigate some of the limitations of regular solution theory. For example, Hansen solubility parameters (HSPs) account for hydrogen bonding, dispersion, and polar bonding forces, expanding on the Hildebrand solubility parameters, which divided interactive forces into dispersive and polar interactions (Hansen 2007). Using HSPs, qualitative comparisons of solubility can be made for compounds in various solvents using the relation Ra ¼ ½4ðDsolvent  Dsolute Þ2 þ ðPsolvent  Psolute Þ2 þðHsolvent  Hsolute Þ2 0:5

ð1:53Þ

where Ra is the distance between the solute and the solvent in Hansen phase space, D is the dispersive solubility parameter, P is the polar solubility parameter, and H is the solubility parameter for hydrogen bonding. Tabulated HSPs may be found in Hansen Solubility Parameters: A User’s Handbook (Hansen 2007). Solute– solvent systems with smaller Ra values tend to have higher solubilities. Still, no theoretical method is accurate for activity coefficient calculation of solid solutes in liquids for all types of systems.

1.4.3 Group Contribution Methods As we discussed in Section 1.3 for inorganic materials, industrial crystallization rarely takes place in systems that contain only the solute and solvent. In many situations, additional components are present in the solution that affect the solubility of the species of interest. With an organic solute, data for solubility in a particular solvent are often not available, whereas data for the effect of other species on the solubility are virtually nonexistent. This means that the only option available for determining solubility in a complex mixture of solute, solvent, and other components (impurities or by-products) is through calculation or experimental measurement. While experimental measurement is often necessary, estimation through calculation can be worthwhile. The main methods available for the calculation of activity coefficients in multicomponent mixtures are called group contribution methods. This is because they are based on the idea of treating a molecule as a combination of functional groups and summing the contribution of the groups. This allows the calculation of properties for a large number of components from a limited number of groups. Unlike calculations completed using regular solutions, activity coefficient predictions from group contribution methods can predict both negative and positive deviations from ideality. Two similar methods are used for these types of calculations: analytical solution of groups (ASOG) and universal quasi-chemical (UNIQUAC) functional group activity coefficient (UNIFAC). In the ASOG method, activity coefficients are assumed to be temperature-dependent functions of size, as estimated using the Flory–Huggins equation, and interactions between structural groups. Similarly, the UNIFAC method can be used to estimate activity coefficients in nonelectrolyte mixtures by assuming that activity coefficients are functions of molecular size and shape, as well as intermolecular forces. While UNIFAC makes the limiting

13

Jennifer Moffitt Schall and Allan S. Myerson Table 1.9 Solubility of Naphthalene in Various Solvents by UNIFAC and Scatchard–Hildebrand Theory

Solvent

Solubility (mol%) Experimental

UNIFAC

Scatchard–Hildebrand

Methanol

4.4

4.8

0.64

Ethanol

7.3

5.4

4.9

1-Propanol

9.4

9.3

11.3

2-Propanol

7.6

9.3

16.3

1-Butanol

11.6

11.1

18.8

n-Hexane

22.2

25.9

11.5

Cyclohexanol

22.5

20.5

20.0

Acetic acid

11.7

12.5

40.1

Acetone

37.8

35.8

42.2

Chloroform

47.3

47.0

37.8

Ideal solubility = 4.1 mol% Source: Data from Walas 1985.

assumption that each functional group is independent of other functional groups, it can still be used to predict eutectic compositions and solubilities in binary and higher solvent mixtures with reasonable accuracy in many cases. For very large or complex molecules, though, UNIFAC parameters may not exist to describe all functional groups. Modifications to the original UNIFAC method include defining new functional groups and expanding parameters to be temperature dependent. Further explanations of these and other group contribution methods are detailed in a number of references (Fredenslund et al. 1977; Kojima and Tochigi 1979; Walas 1985; Reid et al. 1987), and a survey of group contribution methods and advancements is provided by Gmehling (2009) and Gmehling et al. (2015). Both ASOG and UNIFAC rely on the use of experimental activity coefficient data to obtain parameters that represent interactions between pairs of structural groups. These parameters are then combined to predict activity coefficients for complex species and mixtures of species made up from a number of these functional groups. An example of this would be calculation of the behavior of a ternary system by employing data on the three possible binary pairs. Lists of parameters and detailed explanations of these calculations can be found in the references mentioned previously. Group contribution methods can also be used to calculate solubility in binary (solute–solvent) systems. A comparison of solubilities calculated employing the UNIFAC method with experimental values and values obtained from the Scatchard– Hildebrand theory is given in Table 1.9.

1.4.4 Quantum Mechanical Predictions In previous sections we discussed ways to estimate solubility using empirical methods or methods that only work for certain classes of molecules. Those methods have limitations and cannot predict

14

solubility for all classes of compounds. Quantum mechanics (QM) and statistical thermodynamic calculations overcome some of these barriers by enabling us to estimate molecular chemical potentials without relying on a database of parametrized functional group contributions. Instead, only universal parameters are used in predictions. Calculated chemical potentials can then be used to estimate thermodynamic equilibrium properties such as solubility, activity coefficients, and partition coefficients. To describe how QM and statistical thermodynamic calculations are completed, we will focus on a commonly used software package: COSMO-RS (Conductor-like Screening Model for Real Solvents; Software for Chemistry & Materials BV, Amsterdam, The Netherlands). First, the molecular electronic structure is optimized using density functional theory (DFT) calculations. Then the charge distribution of the surface of the molecule is calculated using QM, assuming a reference state where the molecule is embedded in a perfect conductor. With this reference state for each molecule of interest, COSMO-RS can use statistical thermodynamics to analyze surface interactions between solvent and solute molecules. From the surface interactions, a charge-density profile can be constructed and used to estimate chemical potential and, subsequently, solubility at infinite dissolution. Calculations can be updated for various temperatures of interest, allowing solubility curves to be constructed. Using this general procedure, solubility can be screened qualitatively in hundreds of solvents rapidly, enabling selection of a solvent list for further experimental evaluation. When using COSMO-RS, it is important to be aware of three major assumptions embedded in the software calculation blocks: 1. The liquid phase is incompressible. 2. The dielectric permittivity of the solvent is infinite.

Solutions and Solution Properties

3. Only pairwise molecular interactions are allowed at the molecular surface, but all parts of the molecular surfaces can contact other molecular surfaces.

1.4.5 Solubility in Mixed Solvents In looking for an appropriate solvent system for a particular solute to allow for the development of a crystallization process, often the desired properties cannot be obtained with the pure solvents that can be used. For a number of economic, safety, or product stability reasons, you may be forced to consider a small group of solvents. The solute might not have the desired solubility in any of these solvents, or if soluble, the solubility may not vary with temperature sufficiently to allow cooling crystallization. In these cases, a possible solution is to use a solvent mixture to obtain the desired solution properties. The solubility of a species in a solvent mixture can significantly exceed the solubility of that species in either pure-component solvent. This is illustrated in Figure 1.14 for the solute phenanthrene in the solvents cyclohexane and methyl iodide. Instead of a linear relation between the solvent composition and the solubility, the solubility has a maximum at a solvent composition of 38 mol% cyclohexane (solutefree basis). The large change in solubility with solvent composition can be very useful in crystallization processes. It provides a method other than temperature change to alter the solubility of the system. The solubility can be easily altered up or down by adding the appropriate solvent to the system. The method of changing solvent composition to induce crystallization will be discussed in more detail in Section 1.5.4.

0.30

Mole Fraction Phenanthrene

These assumptions allow chemical potentials to be calculated, but they also expose some limitations of using COSMO-RS. From assumption 1, we see that COSMO-RS cannot be used to estimate pressure-dependent equilibrium properties, and assumption 2 indicates that we are defining the solvent as a conductor. From assumption 3, we see that COSMO-RS cannot be used to calculate ternary and higher-order interactions. This assumption can be expanded by using associated molecules as the model molecular complex. For example, acetic acid molecules may be modeled as dimers instead of monomers to enable three-body interactions to be estimated and account for internal hydrogen bonding between molecules. Although COSMO-RS can be used to qualitatively compare solubilities of compounds in various solvent systems, COSMORS is limited in its ability to predict experimental data. For example, bond lengths and angles, as well as intermolecular interactions, may be different in bulk fluid than in the small numbers of molecules that can be simulated in COSMO-RS. Therefore, it is useful to use physical property data, such as heat of fusion derived from differential scanning calorimetry (DSC) experiments, to train COSMO-RS and improve the program’s quantitative prediction capabilities. Beyond polarizable continuum models (PCMs) like COSMO-RS, explicit solvation models also may be used to estimate solubility without the use of empirical equations or experimental data. These methods include using Monte Carlo (MC) and molecular dynamics (MD) simulations. Further information on these and other methods is presented in the review article by Skyner et al. (2015).

0.35

320 K 0.25 0.20

310 K

0.15

300 K 290 K

0.10

280 K 0.05 0.00 0.0

CH2I2

0.2

0.4

0.6

x

0.8

1.0

C6H12

Figure 1.14 Solubility of phenanthrene in cyclohexane–methylene iodide mixtures Source: Reprinted with permission from L. J. Gordon and R. L. Scott, Enhanced methylene iodide solubility in solvent mixtures: I. The system phenanthrene– cyclohexane–methylene iodide,” J. Am. Chem. Soc. 1952; 74:4138. Copyright © 1952, American Chemical Society.

Finding an appropriate mixed solvent system should not be done on a strictly trial and error basis. It should be examined systematically based on the binary solubility behavior of the solute in solvents of interest. It is important to remember that the mixed solvent system with the solute present must be miscible at the conditions of interest. Initial estimates may be completed without collecting experimental solubility data. For systems that exhibit positive deviations from ideality, are not highly polar, and are not strongly hydrogen bonding, the observed maximum in the solubility of solutes in mixtures can be predicted by Scatchard–Hildebrand theory. Looking at Equation (1.50), we see that when the solubility parameter of the solvent is the same as that of the subcooled liquid solute, the activity coefficient will be 1. This is the minimum value of the activity coefficient possible employing this relation. When the activity coefficient is equal to 1, the solubility of the solute is at a maximum. This then tells us that by picking two solvents with solubility parameters that are greater than and less than the solubility parameter of the solute, we can prepare a solvent mixture in which the solubility will be a maximum. For example, let us look at the solute anthracene. Its solubility parameter is 9.9 (cal/cm3)1/2. Looking at Table 1.8, which lists solubility parameters for a number of common solvents, we see that ethanol and toluene have solubility parameters that bracket the value of anthracene. If we define a mean solubility parameter by the relation P xi Vi δi δ¼ P ð1:54Þ xi Vi we can then calculate the solvent composition that will have the maximum solubility. This is a useful way to estimate the optimal solvent composition prior to experimental

15

Jennifer Moffitt Schall and Allan S. Myerson

measurement. Examples of these calculations can be found in the text by Walas (1985). For polar systems or systems that exhibit hydrogen bonding, other techniques should be used for estimating solubility in solvent mixtures in the absence of experimental data. These methods include using HSPs or partition coefficients, such as octanol–water partition coefficients. To estimate solubility in a solvent mixture using HSPs, each HSP must be calculated for the mixture using a linear mixing rule: X Dblend ¼ ϕi Di ð1:55Þ i

Pblend ¼

X

ϕi Pi

ð1:56Þ

ϕi Hi

ð1:57Þ

i

Hblend ¼

X i

where ϕi is the solvent volume fraction of component i in the mixture (Hansen 2007). Then the solvent mixture and the solute HSPs can be used to estimate the minimum HSP distance Ra , as shown in Equation (1.53). Another useful method is to employ the group contribution methods described in Section 1.4.3 with data obtained on the binary pairs that make up the system. Finally, COSMO-RS can be used to qualitatively screen solubility in solvent mixtures. Frank et al. (1999) present a good review of these and other calculation-based methods to quickly screen solvents for use in organic solids crystallization processes. For the special case of ternary systems where the binary interactions are characterized and some ternary experimental data have been collected, solubility can be estimated at a single temperature using the reduced three-suffix solubility equation (R3SSE) method presented by Williams and Amidon (1984a, 1984b, 1988) For this method, four types of data are required: 1. Solute solubility in each pure solvent 2. Binary vapor pressure data for the solvents as a function of solvent composition 3. Molar volume of each pure compound 4. One solubility measurement of the solute in the solvent mixture at the temperature of interest First, the binary vapor pressure data are evaluated to determine solvent–solvent interaction constants. Then linear regression is used to determine the last remaining interaction parameter, which accounts for the interaction between the solute and the two solvents, by minimizing the difference between the reported experimental ternary solubility and the estimated solubility. Once all interaction parameters are evaluated, solubility throughout the ternary-phase space can be calculated at the temperature of interest. This method assumes that there are no four-body interactions in the system. If one wishes to also consider four-body interactions, then a second solubility measurement is required in the solvent mixture at the temperature of interest, and the reduced four-suffix solubility equation must be used. Solubility can be estimated in quaternary systems as well using a similar approach (Williams and Amidon 1984c).

16

1.4.6 Measuring Solubility Accurate solubility data are a crucial part of the design, development, and operation of a crystallization process. When confronted with the need for accurate solubility data, it is common to find that the data are not available for the solute at the conditions of interest. This is especially true for mixed solvents, nonaqueous solvents, and systems with more than one solute. In addition, most industrial crystallization processes involve solutions with impurities present. If it is desired to know the solubility of the solute in the actual working solution with all impurities present, it is very unlikely that data will be available in the literature. Methods for the calculation of solubility were discussed previously. These can be quite useful for solvent screening but often are not quantitatively accurate owing to a lack of adequate thermodynamic data. This means that the only method available to determine the needed information is experimental solubility measurement. The measurement of solubility appears to be quite simple. However, these measurements can easily be done incorrectly, resulting in very large errors. Two dominant options exist for collecting solubility data: isothermal methods and nonisothermal methods. Each technique has strengths and limitations. Isothermal solubility measurements always are collected at a constant, controlled temperature while employing agitation. Isothermal methods tend to be more accurate but generally require more raw materials and time than non-isothermal solubility measurement techniques. Typically, a solute concentration calibration curve is required to analyze data, requiring further experimentation and additional raw materials. A generalized procedure for measuring solubility isothermally is as follows: 1. To a jacketed or temperature-controlled vessel (temperature control should be 0.1°C or better), add a known mass of solvent. 2. Heat or cool the solvent to the desired temperature. If the temperature is above room temperature or the solvent is organic, use a condenser to prevent evaporation. 3. Add the solute in excess (having determined the total mass added), and agitate the solution for a period of at least 4 hours. A 24-hour agitation period is preferable. 4. Sample the solution, and analyze for the solute concentration. 5. Repeat step 4 for each temperature of interest to construct the solubility curve. If the solute analysis is not simple or accurate, step 4 can be replaced by filtering the solution, drying the remaining solid, and weighing. The amount of undissolved solute is subtracted from the total initially added to calculate the solubility. The long agitation period is necessary because dissolution rates become very slow near saturation. If a short time period is used (≤1 hour), the solubility will generally be underestimated. If care is taken, data obtained using this procedure will be as accurate as the concentration measurement or weighing accuracy achieved. Generally, non-isothermal solubility measurements take less time than isothermal measurements and, if designed correctly, eliminate the need to sample the solution during experiments and analyze results against a solute concentration calibration curve. In

Solutions and Solution Properties

non-isothermal experiments, excess solute is mixed with a solvent, and the resulting solution is continuously stirred and slowly heated until all solids dissolve. In practice, heating occurs as a series of alternating heating and holding steps, although heating rates are generally specified in experimental procedures. Commercial equipment exists to collect multiple solubility measurements at once. For example, the Crystal16 produced by Technobis Crystallization Systems (Aalkmar, The Netherlands) can screen solubility non-isothermally in up to 16 samples at the same time. Samples are typically small, with volumes on the order of 1 ml solution. A generalized procedure for conducting non-isothermal solubility measurements in a single sample is as follows: 1. To a small vial, add a known mass of solvent and solute. The mass of solute should be selected to dissolve within the temperature range of interest. Insert a stir bar, and cap the vial to seal the contents. 2. Heat or cool the vial to the lowest temperature of interest, and hold it at this temperature for at least 30 minutes. 3. Slowly heat (at ~0.1°C/min) the vial to the highest temperature of interest. Record the temperature at which the last solids dissolve in the vial. 4. Repeat this procedure to obtain the desired number of data points to construct the solubility curve. Typically, the dissolution or recrystallization of a solid during nonisothermal solubility experiments is assessed by measuring the turbidity of the solution. As the last solids dissolve, the transmissivity of laser light inside the solution reaches a maximum. This is known as the clear point temperature and corresponds to the solubility. Two common errors should be avoided if using non-isothermal techniques. In one technique, a solution of known concentration is made at a given temperature above room temperature and cooled until the first crystals appear, assuming that this temperature is the saturation temperature of the solution of the concentration initially prepared. This is incorrect because solutions become supersaturated (exceed their solubility concentration) before they crystallize. The temperature at which crystals appear is likely to be significantly below the saturation temperature for that concentration, so the solubility will be significantly overestimated using this erroneous method. Another method that will result in error is to add a known amount of solute in excess to the solution and quickly raise the temperature until it all dissolves. It is assumed that the temperature at which the last crystal disappears is the solubility temperature at the concentration of solution (total solute added per solvent in system). This is again incorrect because dissolution is not an instantaneous process and, in fact, becomes quite slow as the saturation temperature is approached. This method will underestimate the solubility because the solution will have been heated above the saturation temperature. Instead, slow heating rates should be used to for non-isothermal solubility measurements. If slow dissolution kinetics are a concern, one can increase the temperature of the solution by 1°C every 30–40 minutes until full dissolution is reached, resulting in solubility curves that are generally

accurate within 5 percent (Yi et al. 2005). Regardless, it is important to note that the accuracy of non-isothermal solubility measurements is a function of hold time between temperature ramp steps and the ramp step size. Accurate solubility data are worth the time and trouble it will take to do the experiment correctly. Avoid the common errors discussed, and be suspicious of data where the techniques used in measurement are not known.

1.5 Supersaturation and Metastability As we saw in Section 1.4, solubility provides the concentration at which the solid solute and the liquid solution are at equilibrium. This is important because it allows calculation of the maximum yield of product crystals accompanying a change of state from one concentration to another in which crystals form. For example, if we look at Figure 1.15, which gives us the solubility diagram for KCl, if we start with 1000 kg of a solution at 100° C and a concentration of 567 g/kg of water and cool it to 10°C at equilibrium, we will have 836 kg of solution with a KCl concentration of 310 g/kg of water and 164 kg of solid KCl. While this mass balance is an important part of crystallization process design, development, and experimentation, it tells us nothing about the rate at which the crystals form and the time required to obtain this amount of solid. This is because thermodynamics tells us about equilibrium states but not about rates. Crystallization is a rate process, meaning that the time required for the crystallization depends on some driving force. In the case of crystallization, the driving force is called the supersaturation. Supersaturation can be easily understood by referring to Figure 1.15. If we start at point A and cool the solution of KCl to a temperature of 40°C, the solution is saturated. If we continue to cool a small amount past this point to B, the solution is likely to remain homogeneous. If we allow the solution to sit for a period of time or stir this solution, it will eventually crystallize. A solution in which the solute concentration exceeds the equilibrium (saturation) solute concentration at a given temperature is known as a supersaturated solution. Supersaturated solutions are metastable. We can see what this means by looking at Figure 1.16. A stable solution is represented by Figure 1.16a and appears as a minimum. A large disturbance is needed to change the state in this instance. An unstable solution is represented by Figure 1.16b and is just the opposite, with the solution being represented by a sharp maximum so that a differential change will result in a change in the state of the system. A metastable solution is represented by Figure 1.16c as an inflection point where a small change is needed to change the state of the system, but one that is finite. Metastability is an important concept that we will discuss in greater detail in Section 1.5.3.

1.5.1 Thermodynamics of Supersaturated Solutions Chemical potential, by thermodynamic definition, is the partial molar free energy of a species, which can change due to chemical reaction or phase transition. At constant temperature, pressure, and number of moles of other species in the system, the chemical potential may be defined as the partial molar Gibbs free energy

17

Jennifer Moffitt Schall and Allan S. Myerson

600

g KCl/1000 g H2O

550 500 450 B

A

400 350

Stable (a)

300 250

Unstable (b)

Metastable (c)

Figure 1.16 Stability states

0

20

40

60

80

100

120

Temperature (°C) Figure 1.15 Solubility of KCl in aqueous solution Source: Data from Linke and Seidell 1965.


μi ¼ G i ¼

∂Gi ∂Ni

 ð1:58Þ T;P;Nj≠i

As we saw in Equation (1.37) in Section 1.4.1, chemical potential may be expressed as a function of fugacity
 fi μi ðTÞ ¼ μ0ðaqÞ ðTÞ þ RTln 0 ð1:59Þ fi where fi0 and μ0ðaqÞ are the standard-state fugacity and chemical potential, respectively. The driving force for crystallization or dissolution arises from a difference between the chemical potential of a solute and the chemical potential that solute would exert at equilibrium. At supersaturated conditions, the solute has a chemical potential of μi ðTÞ ¼ μ0ðaqÞ ðTÞ þ RTlnðai Þ ð1:60Þ Similarly, at saturation, denoted by an asterisk (*), the solute has chemical potential μi ðTÞ ¼ μ0ðaqÞ ðTÞ þ RTlnðai Þ

ð1:61Þ

Crystallization can occur when the chemical potential of a species is higher than the chemical potential that species would exert at equilibrium, and this difference in chemical potentials is called the supersaturation. Assuming the same reference state for the solute, the chemical potential difference becomes
 ai  Dμi ¼ μi  μi ¼ RTln  ð1:62Þ ai Simplifying this expression and substituting for the activity yield the thermodynamic expression for supersaturation in dimensionless form
 μ  μi γ xi σ¼ i ¼ ln i  ð1:63Þ RT γi xi In this expression, four quantities are needed to calculate supersaturation in the solution: the mole fraction of the solute in the supersaturated and saturated solutions and the activity

18

coefficient of the solute in the supersaturated and saturated solutions. Experimentally, solution composition can be evaluated for supersaturated and saturated conditions, and the mole fractions of interest can be calculated. Similarly, the activity coefficient at saturation can be calculated from the generalized solubility equation, Equation (1.43), once the solubility and solute physical properties have been determined. Unfortunately, it is extremely tedious and difficult to measure the solute activity coefficient at supersaturated conditions because the system is not at an equilibrium state. For this reason, the thermodynamic expression for supersaturation is not immediately useful, and we must make simplifying assumptions to estimate the true supersaturation of a solution. Binary Solutions. For a system containing a single solute and a single solvent, the ratio of the solute mole fraction in the supersaturated and saturated phases is the same as the ratio of the solute concentration in the supersaturated and saturated phases. This allows us to simplify the supersaturation expression as
 μi  μi γi ci σ¼ ¼ ln   ð1:64Þ RT γi ci For ideal systems, the activity coefficient is 1, and the supersaturation expression can be simplified to
 ci σ ¼ ln  ð1:65Þ ci This simplified expression is also acceptable when γ=γ ¼ 1. For cases where either of these assumptions holds and the supersaturation is also low (σ ≪ 1) such that lnðσ þ 1Þ ¼ σ, the dimensionless chemical potential difference can be approximated by a dimensionless concentration difference σ¼

c  c c

ð1:66Þ

This is generally a poor approximation at σ > 0:1, but it is still normally used because the needed thermodynamic data are usually unavailable (Kim and Myerson 1996). If the supersaturation is high or the system is highly nonideal, the original thermodynamic expression for supersaturation should be used, and the activity coefficient at supersaturated conditions should be approximated using the method proposed by Valavi et al. (2016). In this method, the

Solutions and Solution Properties

activity coefficient in the supersaturated solution is assumed to be the same as the activity coefficient in a saturated solution of the same composition, allowing the activity coefficient to be approximated using only solubility data and the generalized solubility equation. The underlying assumption is that the activity coefficient is a strong function of composition but a weak function of temperature. Ternary and Higher Solutions. For a system containing a single solute in a mixture of solvents, many of the simplifications used for binary solutions do not apply. For example, the ratio of mole fractions may not be interchanged for a concentration ratio if the molecular weights of the two solvents are different. Instead, the thermodynamic supersaturation expression should be used as presented in Equation (1.63). Because the activity coefficient at supersaturation cannot be determined easily through experimentation, the following process is recommended for estimating the activity coefficient at supersaturated conditions: 1. Determine the physical properties and solubility of the active pharmaceutical ingredients (APIs). DSC experiments may be used to determine the heat of fusion, melting temperature, and temperature-dependent heat capacity for the solute. Solubility as a function of both temperature and solvent composition can be measured in the range interest. 2. Regress parameters for the solubility model to account for temperature and solvent dependence. For example, one might assume that the solubility follows a modified Appelblat solubility with respect to temperature and a modified Jouyban–Acree model with respect to solvent composition (Ma et al. 2012). This enables calculation of xi in the supersaturation expression throughout the operating range of interest. 3. Calculate γi , the activity coefficient at saturation, using the general form of the solubility equation, Equation (1.43). 4. Estimate γi , the activity coefficient at supersaturated conditions, in two stages. First, use the solubility model in step 2 to solve for the “effective” temperature corresponding to solute mole fraction xi and solvent fraction at the supersaturated conditions of interest. Then use this effective temperature to solve for activity coefficient in the generalized solubility equation at the supersaturated conditions of interest. This method is an extension of the work by Valavi et al. (2016) and is based on the assumption that the activity coefficient is a strong function of composition but a weak function of temperature. At this point, one can estimate the supersaturation using Equation (1.63). This method works as a reasonable approximation for systems containing a noncharged nonelectrolyte that crystallizes in a pure, single-polymorphic state. Ionic Systems. For a system containing ionic species, the chemical potential for the liquid phase is defined as X μ¼ vi μ i ð1:67Þ i

where vi and μi are the stoichiometric coefficient and chemical potential for species i, respectively. From this, we see that the

dimensionless thermodynamic driving force for crystallization is 0 1 ∏ avi i μ  μ i¼1 A ¼ ln@ ð1:68Þ i RT ∏ avi;eq i¼1

i Looking at the denominator, we see that the quantity ∏i¼1 avi;eq is the solubility product, as defined in Equation (1.22). Therefore, for ionic systems, we can define the supersaturation as 0 1 ∏ avi i i¼1 A ð1:69Þ σ ¼ lnðSÞ ¼ ln@ Ksp

This definition of σ, also known as the growth affinity, is defined on a dimensionless basis (Hartman 1973). To define the supersaturation ratio on a per-ion basis, the following expression is generally used: S

0

¼ S1=ðv

þ

þv Þ

ð1:70Þ

where vþ and v are the stoichiometric species of the positive and negative species, respectively (Peng et al. 2015). From these expressions we see that activities, instead of concentrations, must be used to estimate supersaturation in ionic solutions. Activity coefficients for each ionic species can be calculated using models in the absence of experimental data. Commonly used activity coefficient models for ionic species include the mixed-solvent–electrolyte (MSE) thermodynamic model, the Bromley method, the Pitzer method, and the generalized electrolyte–non-random two-liquid (NRTL) model. Comparisons of activity coefficient models for electrolyte systems are provided by Lin et al. (2010) and Jaworski et al. (2011).

1.5.2 Alternate Supersaturation Units Earlier we defined supersaturation as a dimensionless quantity. However, it is often convenient to make simplifications and express supersaturation in a dimensional form. Supersaturation is often expressed as a concentration difference Dc ¼ c  c

ð1:71Þ

and as a ratio of concentrations S¼

c c

ð1:72Þ

It is important to note that these definitions of supersaturation assume an ideal solution with an activity coefficient of 1. In the literature, it is common practice to ignore activity coefficients in most cases and employ concentrations in expressions of supersaturation, but this can result in large calculation errors. Any supersaturation simplifications should be justified before use. In very nonideal solutions and in precise studies of crystal growth and nucleation, activity coefficients should be used as described in Section 1.5.1. Further caution should be used if these alternate supersaturation units are employed in ternary or

19

Jennifer Moffitt Schall and Allan S. Myerson

1.5.3 Metastability and the Metastable Limit As we have seen previously, supersaturated solutions are metastable. This means that supersaturating a solution some amount will not necessarily result in crystallization. Referring to the solubility diagram in Figure 1.17, if we were to start with a solution at point A and cool to point B just below saturation, the solution would be supersaturated. If we allowed that solution to sit, it might take days before crystals form. If we took another sample, cooled it to point C, and let it sit, this might crystallize in a matter of hours. Eventually we will get to a point where the solution crystallizes rapidly and no longer appears to be stable. As we can see from this experiment, the metastability of a solution decreases as the supersaturation increases. It is important to note, however, that we are referring to homogeneous solutions only. If solute crystals are placed in any supersaturated solution, they will grow, and the solution will eventually reach equilibrium. The obvious question that comes to mind is why supersaturated solutions are metastable. It seems reasonable to think that crystals should form whenever the solubility is exceeded in a solution. To understand why they do not, we will have to discuss something called nucleation. Nucleation is the start of the crystallization process and involves the birth of a new crystal. Classical nucleation theory tells us that when the solubility of a solution is exceeded and it is supersaturated, solute molecules start to associate and form nuclei. If we assume that these nuclei are spherical, we can write an equation for the Gibbs free energy change required to form a nucleus of a given size:
3 4πr 2 DG ¼ 4πr σ  RTlnð1 þ SÞ ð1:73Þ 3Vm where r is the nuclei radius, σ is the solid–liquid interfacial tension, and Vm is the molar volume of a solute molecule in the crystal phase. The first term is the Gibbs free energy change for forming the surface, and the second term is for the volume. For small numbers of molecules, the total Gibbs free energy change is positive. This means that the nuclei are unstable and will dissolve. A plot of DG as a function of nucleus size (Figure

20

5.0

Concentration (molar)

higher-order solutions because differences in solvent composition can affect the calculation of solute concentration in the solution. Supersaturation also may be expressed as a percentage, where 0 percent supersaturation corresponds to a saturated solution and 100 percent supersaturation corresponds to twice the saturation concentration, although this is more common in vapor–liquid applications that in solid–liquid applications. Finally, supersaturation may be characterized in terms of degrees. This refers to the difference between the temperature of the solution and the saturation temperature of the solution at the existing concentration. A simpler way to explain this is that the degrees of supersaturation are simply the number of degrees a saturated solution of the appropriate concentration was cooled to reach its current temperature. This is generally not a good unit to use, yet it is often mentioned in the literature.

4.6 mit

n Li

atio

atur

ers Sup

4.2

C

3.8

A

on

ati

tur

Sa

B

3.4 3.0 0

10

20 30 Temperature (°C)

40

50

Figure 1.17 Metastable zone width from a KCl–water system Source: Data from Chang 1985.

1.18) shows that as the size of the nucleus increases, we reach a point where the Gibbs free energy change is negative, and the nucleus would grow spontaneously. According to classical nucleation theory, nucleation will occur when this happens. The reason that supersaturated solutions are metastable is therefore because of the need for a critical-sized nucleus to form. From Equation (1.73), we can derive an expression for the critical size by setting the derivative dDG=dr ¼ 0 (the minimum in Figure 1.18), which yields rc ¼

2Vm σ RTlnð1 þ SÞ

ð1:74Þ

We can see from this equation that as the supersaturation increases, the critical size decreases. That is why solutions become less and less stable as the supersaturation is increased. Unfortunately, Equations (1.73) and (1.74) are not useful for practical calculations because of the inherent limitations of classical nucleation theory. Classical nucleation theory assumes that all nuclei are spherical, that each nucleus grows by one molecule at a time, and that each nucleus is morphologically identical to a bulk-size crystal; this is not true for most systems. Beyond this, classical nucleation cannot describe many phenomena, including nucleation of complex molecules and systems containing impurities. For this reason, two-step nucleation theory should be used, as will be discussed in greater detail in Chapter 3. Still, every solution has a maximum amount that it can be supersaturated before it becomes unstable. The zone between the saturation curve and this unstable boundary is called the metastable zone and is where all crystallization operations occur. The boundary between the unstable and metastable zones has a thermodynamic definition and is called the spinodal curve. The spinodal is the absolute limit of the metastable region where phase separation must occur immediately. The practical limits of the metastable zone, however, are much smaller and vary as a function of crystallization conditions for a given substance. This is because the presence of dust and dirt, the cooling rate employed, the solution volume, the solution history, and the use of agitation all can affect nuclei formation and change the apparent

Solutions and Solution Properties Table 1.10 Metastable Zone Width

Gs

Free Energy,

G

Substance

Gcrit = 4/3

Equilibrium temperature (°C)

Cooling rate

r2c

rC

Gv

Maximum undercooling before nucleation

G

20°C/h

30.8

1.65

2.17

3.27

CuSO4·5H2O

33.6

5.37

6.82

9.77

60.4

0.93

1.30

2.16

30.0

0.89

1.21

1.93

40.6

0.57

0.83

1.46

30.33

1.62

2.33

4.03

61.0

1.69

2.41

4.11

29.8

1.62

1.86

2.30

59.8

1.02

1.18

1.48

MgSO4·7H2O

32

1.95

2.63

4.15

NH4Al (SO4)2·12H2O

30.2

0.81

1.34

2.88

63

1.19

1.95

4.13

30.6

4.6

6.97

13.08

Figure 1.18 Free energy versus cluster radius Source: Reproduced with permission from Mullin 1972.

KBr

solution metastability. Ultimately, the metastable zone width (MSZW) is a nonisothermal measurement of nucleation induction time. MSZW should not be used quantitatively because it is a qualitative measure of how easily a material nucleates as a function of supersaturation. For quantitative measurements of nucleation, experiments should be completed to assess the nucleation induction time or the nucleation kinetics for a system. Still, MSZW measurements are common in the literature and give extremely preliminary guidance for crystallization process design. In general, there are two methods for qualitatively measuring the metastable limit. In the first method, solutions are cooled to a given temperature rapidly, and the time required for crystallization is measured. When this time becomes short, the effective metastable limit has been approached. A second method is to cool a solution at some rate and observe the temperature where the first crystals form. The temperature at which crystals are first observed will vary with the cooling rate used. Measured metastable limits for a number of materials are given in Table 1.10, and Figure 1.17 gives an estimated metastable zone width for KCl in water. Generally, the MSZW is most narrow for ionic materials, has a greater width for simple organic molecules, and is widest for large or complex organic molecules. Assessing solution metastability through measuring nucleation induction time or nucleation kinetics is important for proper crystallizer design. As we will see in later chapters, formation of small crystals, which are known as fines, is a common problem. Fines cause filtration problems and often are not wanted for various reasons in the final product. When a crystallization occurs at a high supersaturation (near the metastable limit), this usually means that many small crystals will be formed. If the supersaturation is too low, the nucleation and growth rates of crystals may be too small to produce material at acceptable yield within a reasonable processing time. Therefore, crystallizers must be carefully designed to operate at intermediate supersaturations to enable reasonable growth and nucleation rates without producing high quantities of fines.

5°C/h

Ba(NO3)2

FeSO4·7H2O

Size of Nucleus, r

2°C/h

KCl

NaBr·2H2O

Source: Data from Nývlt et al. 1985.

1.5.4 Methods to Create Supersaturation In our discussions of supersaturation and metastability, we have always focused on situations where supersaturation is created by temperature change (cooling). While this is a very common method to generate supersaturation and induce crystallization, it is not the only method available. There are four main methods to generate supersaturation: 1. 2. 3. 4.

Temperature change Evaporation of solvent Chemical reaction Changing the solvent composition

As we have discussed previously, the solubility of most materials declines with declining temperature, so cooling is often used to generate supersaturation. In many cases, however, the solubility of a material remains high even at low temperatures, or the solubility changes very little over the temperature range of interest. In these cases, other methods for the creation of supersaturation must be considered. After cooling, evaporation is the most commonly used method for creating supersaturation. This is especially true when the solvent is nonaqueous and has a relatively high vapor pressure. The principle of using evaporation to create supersaturation is quite simple. Solvent is being removed from the system, thereby increasing the solute concentration. If this is done at a constant temperature, eventually the system will

21

Jennifer Moffitt Schall and Allan S. Myerson

22

20 Terephthalic Acid Solubility (% weight)

become saturated and then supersaturated. After some maximum supersaturation is reached, the system will begin to crystallize. There are a number of common methods used to evaporate solvents and crystallize materials based on the materials’ properties and solubility. One very common method for a material that has a solubility that decreases with decreasing temperature is to cool the system by evaporating solvent. Evaporation causes cooling in any system because of the energy of vaporization. If a system is put under a vacuum at a given temperature, the solvent will evaporate, and the solution will cool. In this case, the concentration of the system is increased while the temperature of the system is decreased. In some cases, the cooling effect of the evaporation slows the evaporation rate by decreasing the system vapor pressure; in these cases, heat is added to the system to maintain the temperature and thereby the evaporation rate. Virtually all evaporations are done under vacuum. As we saw in our discussion of solubility, the mixing of solvents can result in a large change in the solubility of the solute in the solution. This can be used to design a solvent system with specific properties and can also be used as a method to create supersaturation. If we took, for example, a solution of terephthalic acid (TPA) in the solvent dimethylsulfoxide (DMSO) at 25°C, the solubility of the TPA at this temperature is 16.5 wt%. A cooling crystallization starting from some temperature above this to 23°C (about room temperature) would leave far too much product in solution. Imagine that evaporation cannot be used because of the lack of reasonable equipment, because the solvent is not volatile enough, or because the product is heat sensitive. The third option is to add another solvent to the system to create a mixed-solvent system in which the solubility of the solute changes drastically. If the addition of a specific solvent causes the solubility to decrease greatly, we call that solvent an antisolvent. If we were to add water as an antisolvent to the TPA–DMSO system, the solubility changes rapidly from 16.5 wt% to essentially 0 wt% with the addition of 30 percent water (by volume on a solute-free basis). This is shown in Figure 1.19. By controlling the rate of addition, we can control the supersaturation just as we can by cooling or by evaporation. In this case, however, good mixing conditions are important so that we do not have local regions of high supersaturation and other regions of undersaturation. This method of creating supersaturation is often called drowning out or adding a miscible nonsolvent. Normally, you can find an appropriate solvent to add by looking for a material in which the solute is not soluble but is miscible with the solute–solvent system. This is a particularly valuable technique for crystallizing organic materials. As a general rule, water is selected as an antisolvent for organic compounds, whereas organic solvents are selected as antisolvents for inorganic compounds. However, solubility and miscibility should be screened either experimentally or using solubility calculations in multiple-solvent systems before selecting appropriate solvents and antisolvents.

15

10

5

0 60

70

80 % Volume DMSO

90

100

Figure 1.19 Solubility of terephthalic acid in DMSO–water mixtures at 25°C Source: Data from Saska 1984.

The last method of generating supersaturation is through chemical reaction. This is commonly called precipitation and will be discussed in detail in Chapter 8. In this case, two soluble materials are added together in solution and react to form a product with a low solubility. Because the solubility of the product is soon exceeded, the solution becomes supersaturated, and the material crystallizes. This technique is commonly used in the production of inorganic materials. An example of a precipitation is the reaction of Na2SO4 and CaCl2 to form NaCl and CaSO4 (the insoluble product). The solubilities of the reactants and products are shown in Figure 1.20. Again, in this type of process, mixing is crucial to obtaining a homogeneous supersaturation profile. Precipitation is important in the manufacture of a variety of materials. TPA, which is an organic commodity chemical used in the manufacture of polymers, is made from the oxidation of p-xylene in an acetic acid–water mixture. The product has a very low solubility in the solvent system and rapidly precipitates out. Control of the supersaturation in a precipitation process is difficult because it involves control of the mixing of the reactants and/or the reaction rate. In general, you usually have the choice of more than one method to generate supersaturation. You should evaluate the system equipment available, the solubility versus temperature of the material, the selection of solvents available for use, the thermal and chemical stability of the product, and the production rate required before choosing one of the methods we discussed.

1.6 Solution Properties 1.6.1 Density The density of the solution is often needed for mass balance, flow rate, and product yield calculations. Density is also needed to convert from concentration units based on solution volume to units of concentration based on mass or moles of the solution. Density is defined as the mass per unit volume and is commonly reported in grams per cubic centimeter (g/cm3), but

Solutions and Solution Properties Table 1.11 Density and Viscosity of Common Solvents

1000 NaCl

Na 2SO4

100

10

Substance

Density at 20°C (g/cm3)

Viscosity at 20°C (cP)

Water

0.999

1.00

Acetone

0.789

0.322

Benzene

0.879

0.654

Toluene

0.866

0.587

Carbon tetrachloride

1.595

0.975

Methanol

0.791

0.592

Ethanol

0.789

1.19

n-Propanol

0.804

2.56

CaCl 2

CaSO4

1 21 23 25 27 29 31 33 35 37 39 41 43 45 Temperature (°C) Figure 1.20 Solubility of NaCl, Na2SO4, CaSO4, and CaCl2 in water Source: Data from Linke and Seidell 1965.

other units such as pound-mass per cubic foot (lbm/ft3) and kilogram/cubic meter (kg/m3) are often used. When dealing with solutions, density refers to a homogeneous solution (not including any crystals present). Specific volume is the volume per unit mass and is equal to 1=ρ. Densities of pure solvents are available in handbooks such as the CRC Handbook of Chemistry and Physics (Weast 1975). The densities of a number of common solvents are listed in Table 1.11. The densities of solutions as a function of concentration are difficult to find except for some common solutes in aqueous solution. The densities of NaCl and sucrose as a function of concentration are given in Figure 1.21. Densities are a function of temperature and must be reported at a specific temperature. A method for reporting densities uses a ratio known as the specific gravity. Specific gravity is the ratio of the density of the substance of interest to that of a reference substance (usually water) at a particular temperature. To make use of specific gravity data, it is necessary to know the density of the reference material at the correct temperature and to multiply the specific gravity by the reference density. If density data are not available for the solution of interest, the density can be estimated by using the density of the pure solvent and pure solid solute at the temperature of interest and assuming that the volumes are additive: wcrystal wsolvent 1 ¼ þ ρsolution ρcrystal ρsolvent

ð1:75Þ

where w is the mass fraction of crystal or solvent. Calculating the density of a saturated solution of NaCl at 25°C using Equation (1.75) results in a value of 1.17 g/cm3 compared with the experimental value of 1.20 g/cm3. Density can be calculated with more accuracy using thermodynamic techniques, as described in Reid et al. (1987). Density can be measured in the laboratory in a number of different ways depending on the need for accuracy and the number of measurements required. Solution density can be easily estimated with reasonable accuracy by weighing a known volume of solution. Very precise instruments for the measurement of density that work by employing a vibrating quartz

Source: Based on data from Mulin 1972 and Weast 1975.

1.20

1.15

(g/cm3)

Concentration (g/1000 g H2O)

10000

NaCl

1.10 Sucrose

1.05

1.00

0

50

100

150

200

250

Concentration (g/kg water) Figure 1.21 Density of sodium chloride and sucrose aqueous solutions at 20°C Source: Data from Weast 1975.

element in a tube are sold by Mettler-Toledo, LLC (Columbus, OH). The period of vibration of the element is proportional to the density of the material placed in the tube. With careful calibration and temperature control, the accuracy of these instruments ranges from 1 × 10−3 to 2 × 10−5 g/cm3. It is possible to use these instruments for online solution density measurement of fluid in a crystallizer (Cole 1991). Another term typically used to describe solid–liquid mixtures is slurry or magma density. This is usually defined in terms of the mass of solids per unit volume of solution. A 10 percent slurry density therefore would indicate 100 g of solids/ liter of solution. Slurry density is not actually a true density but is a convenient term for indicating the amount of suspended solids in the solution.

1.6.2 Viscosity The design of any equipment that involves the flow or stirring of liquids requires knowledge of the fluid viscosity. Because crystallization operations involve the stirring and movement of suspensions of particles in fluids, the viscosity of suspensions is

23

Jennifer Moffitt Schall and Allan S. Myerson Table 1.12 Viscosity Units and Conversion Factors

Multiply by table value to convert to these units →

Given a quantity in these units ↓

g cm−1 s−1

kg m−1 s−1

lbm ft−1 s−1

lbf ft−2 s−1

cP

lbm ft−1 h−1

g cm−1 s−1

1

10−1

6.7197 × 10−2

2.0886 × 10−3

102

2.4191 × 102

kg m−1 s−1

10

1

6.7197 × 10−1

2.0886 × 10−2

103

2.4191 × 103

lbm ft−1 s−1

1.4882 × 101

1.4882

1

3.1081 × 10−2

1.4882 × 103

−2 −1

lbf ft

s

cP −1

lbm ft

h

−1

2

4.7880 × 10

4.7880 x 10

10−2

10−3 −3

1

−4

4.1338 × 10

4.1338 x 10

32.1740

1

6.7197 × 10−4

2.0886 × 10−5

−4

2.778 × 10

8.6336 × 10

4.7880 × 10

−5

4

1

3600 1.1583 × 105 2.4191

4.1338 × 10

−1

1

Source: Reprinted by permission of the publisher from R. B. Bird, W. E. Stewart, and E. N. Lightfoot (1960), Transport Phenomena. Copyright © 1960, John Wiley & Sons, Inc.

important in crystallization design and operation. Viscosity is a property of a particular material defined as the ratio of the shear stress and the shear rate. Viscosity can be thought of as a measure of the resistance of a fluid to flow. When the relationship between shear stress and shear rate is linear and passes through the origin, the material is said to be Newtonian and the relationship can be represented by τyx ¼ μ

dux dy

ð1:76Þ

where τyx = shear stress dux =dy = shear rate μ = viscosity Most common solvents are Newtonian fluids. Looking at Equation (1.76), we can see that the units of viscosity will be given by the ratio of the shear stress and the shear rate, which is mass/distance × time. Typical units used for viscosity are given in Table 1.12 along with their conversion factors. The ratio of the viscosity and the density is another commonly used term that is known as the kinematic viscosity. The kinematic viscosity has units of length squared per unit time. The viscosity of most common solvents is available in the literature. The viscosity of solutions of solids dissolved in liquids is normally not available at high concentrations except for common solutes in aqueous solution. In crystallization operations, the viscosity of the slurry of solution and crystals is of importance. The viscosity of a slurry of solution and crystals usually does not obey Newton’s law of viscosity but instead follows other, more complex empirical relations that must be obtained from experimental data. Systems that do not obey Newton’s law of viscosity are called non-Newtonian fluids. A discussion of a number of non-Newtonian fluid models can be found in Bird et al. (1960). A commonly used non-Newtonian viscosity model used is the Power law and can be written as τyx ¼ mj

24

dux n1 dux j dy dy

ð1:77Þ

When n ¼ 1, the Power law model reduces to Newton’s law with m ¼ μ. Power law parameters for several different suspensions of particles in a fluid are given in Table 1.13. The viscosity of slurries is a function of the solution and solid involved, as well as the slurry density. The viscosity can also be significantly affected by the particle size, size distribution, and particle shape. As a general rule, as particle shape varies from spheres to needles, the viscosity moves further from Newtonian behavior. A detailed discussion of factors affecting the viscosity of suspensions can be found in Sherman (1970). Instruments used to measure viscosity are called viscometers. A number of techniques and configurations are available for viscosity measurement. In rotational viscometers, some part of the viscometer is rotated, imparting movement to the fluid that is transferred through the fluid to a measuring device. In capillary viscometers, the fluid flows through a capillary under the force of gravity, and the time required for the fluid to flow through the capillary is measured. Some of the more common viscometers are listed in Table 1.14.

1.6.3 Diffusivity If we were to prepare a solution made up of a solute in a solvent at two different concentrations and place them in contact with each other, eventually they would achieve the same concentration through the process of diffusion. The solute molecules would diffuse from the region of high concentration to the region of lower concentration, and the solvent molecules would diffuse in the opposite direction (from higher to lower concentration of solvent). This process is described by Fick’s first law of diffusion, which is JA ¼ DAB

dCA dx

ð1:78Þ

where JA = molar diffusive flux CA = concentration DAB = diffusivity (or diffusion coefficient) The diffusion coefficient is a property of a given solute in a given solvent and tells us the rate at which the solute will

Solutions and Solution Properties Table 1.13 Power-Law Model Parameters

Table 1.14 Viscometers

Fluid composition (wt%)

m (lbf sn ft−2)

n (dimensionless)

23.3% Illinois yellow clay in water

0.116

0.229

0.67% CMC in water

0.00634

0.716

1.5% CMC in water

0.0653

0.554

3.0% CMC in water

0.194

0.566

33% lime in water

0.150

0.171

10% napalm in kerosene

0.0893

0.520

4% paper pulp in water

0.418

54.3% cement rock in water

0.0524

Type

Operation

Rotational Stormer

Stationary center cup, inner rotor

Haake Rotovisko

Fixed outer cup and inner rotor

Epprech Rheomat

Fixed outer cup and inner rotation bob

Brookfield

Measure viscous traction on spindle rotating in sample

Cone Plate

Rotating small-angle cone and stationary lower flat plate

0.575

Weissenberg Rheogoniometer

Cone rigidly fixed while lower flat plate rotates

0.153

Capillary

Source: Reprinted by permission of the publisher from R. B. Bird, W. E. Stewart, and E. N. Lightfoot (1960), Transport Phenomena. Copyright © 1960, John Wiley & Sons, Inc.

diffuse under a concentration gradient. The units of diffusivity are length2/time. Diffusion coefficients vary with temperature and with solute concentration. The diffusion coefficient is important to crystallization operations because it is one of the properties that determines the degree of agitation required. If insufficient agitation is used in a crystallization process, the crystal growth rate can be controlled by the rate of solute transfer from the bulk solution to the crystal–liquid interface. This is called mass transfer controlled crystal growth. Normally, this is undesirable because the crystal growth rate obtained is usually significantly slower than the rate that would be obtained if interfacial attachment kinetics were the rate-controlling step. This will be discussed in more detail in Chapter 2, but the important point is that the diffusion coefficient is a property that must be taken into account when considering mass transfer, mixing, and agitation in crystallization processes. Data on the diffusion coefficients of solid solutes in liquid solvents are difficult to find and, if available, are usually found at low concentrations (or infinite dilution) at only one temperature. The concentration and temperature dependence of diffusion coefficients in the glycine–water system is illustrated in Figure 1.22. The behavior shown in the figure is typical nonelectrolyte behavior, with the diffusivity declining from a maximum value at infinite dilution in an approximately linear fashion. A comparison of the curves at different temperatures shows that the diffusion coefficient increases with increasing temperature.

1.7 Thermal Properties A fundamental aspect in the development and design of any process involves the performance of an energy balance. Crystallization operations involve the transfer of energy in

Ostwald U-tube

Reservoir bulb from which fixed volume of sample flows through capillary to receiver in other arm of U-tube

Common tensile

Reservoir and receiving bulbs in same vertical axis U-tube viscometer with third arm

Bingham

Sample extruded through capillary by air pressure

Source: Data from Sherman 1970.

and out of the system. In addition, because phase changes are involved through the formation of the product and through changes to the solvent system (if evaporation or change in solvent composition are used), data on the thermal properties of the solute–solvent system are important. In a simple cooling crystallizer, for example, it is obvious that a calculation must be done to determine the amount of energy to be removed from the system to cool the solution to the final temperature desired. The calculation could be seriously in error, however, if the heat effects due to the crystallization (heat of crystallization) are ignored. In crystallizations that involve evaporations, mixed solvents, or reactions, the heat effects that accompany each of these phenomena must be known. In addition, any operation involving the dilution of a concentrated solution or the dissolution of a solid into a liquid is accompanied by heat effects that must be taken into account in any energy-balance calculation.

1.7.1 Heat Capacity Imagine that you have a pure liquid and you wish to calculate how much energy is required to heat that liquid from one temperature to another. This calculation is simple, provided that you know a property of the material known as the heat capacity (or specific heat). Two types of heat capacities can be defined. The heat capacity at constant temperature Cp and the heat capacity at constant volume Cv, which are defined as

25

Jennifer Moffitt Schall and Allan S. Myerson

practice is to neglect the solute and use the solvent heat capacity. For a concentrated solution, neglecting the solute can lead to inaccuracies, so the use of enthalpy data is suggested.

Diffusivity (cm2/s) x 105

1.8 1.6

45°C

1.4 1.2

1.7.2 Latent Heat

35°C

1.0

25°C

0.8 0.6

0

1

2 3 Concentration (molar)

4

5

Figure 1.22 Diffusion coefficients of aqueous glycine solutions at 25, 35, and 45°C Source: Data from Chang 1985.


 dH dT P
 dU Cv ðTÞ ¼ dT V Cp ðTÞ ¼

ð1:79Þ ð1:80Þ

where H and U are the enthalpy and internal energy per mole, respectively. The two heat capacities are related to each other as follows: Ideal gases: Cp ¼ Cv þ R

ð1:81Þ

Liquids and solids: Cp ≈ Cv

ð1:82Þ

Heat capacities are given in units of energy per mole (or mass) per unit temperature interval. As with viscosity, typical units used and conversion factors are available for reference in the text Transport Phenomena (Bird et al. 1960). Specific heat is another term used when heat capacity is expressed on a per-mass basis. Heat capacities are a function of temperature and are usually expressed as a polynomial such as Cp ðTÞ ¼ a þ bT þ cT 2 þ dT 3

ð1:83Þ

Tables of heat capacities for a variety of pure substances can be found in Perry’s Chemical Engineers’ Handbook (Green and Perry 2007). The heat capacity of a number of liquids can be found in the textbook Introduction to Chemical Engineering Thermodynamics (Smith et al. 2005). Heat capacity data are available for most liquids, most gases, and many solids. If they are not available, estimation techniques have been developed and are summarized in Reid et al. (1987). Heat capacities of liquid mixtures can be estimated by calculating an average heat capacity from the heat capacities of the components using the relation X Cp;avg ðTÞ ¼ xi Cp;i ðTÞ ð1:84Þ The heat capacity of solutions of solids dissolved in liquids is usually not available. If the solution is dilute, the usual

26

When a pure material undergoes a phase change, the process takes place at a constant temperature and pressure. Even though the temperature remains the same, there is an enthalpy change associated with the phase change that must be taken into account in energy balance calculations. These enthalpy changes associated with phase changes in pure materials are often referred to as latent heat. The latent heat of vaporization refers to the enthalpy change required to vaporize a given amount of a saturated liquid to saturated vapor at a constant temperature and pressure. Other latent heats commonly mentioned are fusion (melting a material) and sublimation (solid to gas). It is important to note that there are also heat effects that accompany phase changes from one solid phase to another. If a material crystallizes in one crystal form and then transforms to another crystal form, an enthalpy change will be involved. Latent heats vary as a function of temperature and, to a smaller extent, pressure. The latent heat for a phase change at 1 atm (101.325 kPa) is often called the standard heat of phase change and is available for many materials in the literature (Green and Perry 2007). A number of techniques have been developed to estimate the latent heat of vaporization and, to a lesser extent, the latent heat of fusion and sublimation. A description of these techniques can be found in Reid et al. (1987). One very useful and commonly used method known as the Watson correlation is   1  ðT2 =Tc Þ 0:38 DH v ðT2 Þ ¼ DH v ðT1 Þ ð1:85Þ 1  ðT1 =Tc Þ where DH v ðT2 Þ and DH v ðT1 Þ are the heats of vaporization at T2 and T1, and Tc is the critical temperature. Equation (1.85) allows calculation of the latent heat of vaporization at any temperature if the heat of vaporization is known at one temperature. If no heat of vaporization data are available and the normal boiling point (boiling point at 1 atm) of the liquid is known, Chen’s equation can be used to estimate the heat of vaporization at the boiling point: DH vb ¼

RTb ð3:978Trb  3:958 þ 1:555lnPc Þ 1:07  Trb

ð1:86Þ

where Trb = reduced boiling point temperature (Tb/Tc) Pc = critical pressure Latent heats of vaporization can be calculated from vapor pressure data employing the Clausius–Clapeyron equation: DH v dðlnP Þ ¼ dð1=TÞ R

ð1:87Þ

where P is the vapor pressure at a particular temperature. By plotting lnP versus 1=T and obtaining the slope at the

Solutions and Solution Properties

temperature of interest, Equation (1.87) can be used to calculate the heat of vaporization. Calculation of latent heats of fusion and sublimation is more difficult and less accurate than calculating heats of vaporization. Reid et al. (1987) summarized available methods. A crude but simple approximation for standard heats of fusion (Felder and Rousseau 1986) is DH m ≈ 0:0092Tm ðKÞ ðmetallic elementsÞ ≈ 0:0025Tm ðKÞ ðinorganic compoundsÞ ≈ 0:050Tm ðKÞ ðorganic compoundsÞ The melting temperatures and latent heats of fusion for a number of organic species are listed in Table 1.15.

1.7.3 Heats of Mixing, Solution, and Crystallization In any process where the concentration of a solution is changed, there will be an enthalpy change accompanying the concentration change. This is true when two liquids are mixed, a concentrated solution is diluted, a solid is dissolved in a liquid,

and a solute crystallizes from solution. These enthalpy changes are known as heats of mixing, solution, and crystallization, respectively, and can be very significant in energy balance calculations. The heat of mixing refers to the enthalpy change accompanying the mixing of two or more pure substances to form a solution at a constant temperature and pressure. When one species being mixed is a gas or solid, this enthalpy change is known as the heat of solution. Heats of solution are often given in terms of the dissolution of 1 mol of solute in moles of solvent at a particular temperature. This is known as an integral heat of solution. Integral heats of solution for HCl, NaOH, and H2SO4 are listed in Table 1.16. The table shows that for each of these substances, the heat of solution is negative, meaning that the solution process results in the evolution of heat. In addition, Table 1.16 shows that the heat of solution is a function of concentration, increasing as the solution becomes less concentrated (n, the moles of water per moles of solute increases) to a limiting maximum value. This maximum value is known as the heat of solution at infinite dilution. Data in Table 1.16 can be used to calculate the enthalpy change that would result from making a solution of desired concentration from its

Table 1.15 Enthalpy of Fusion and Melting Temperature of Some Organics

Compound

Tm (°C)

Molecular weight (g/mol)

ΔHm (cal/g)

(cal/g mol)

Methane

−182.5

16.04

14.03

225

Ethane

−183.3

30.07

22.73

683

Propane

−187.7

44.09

19.10

842

n-Butane

−138.4

58.12

19.17

1114

Isobutane

−159.6

38.12

18.67

1085

n-Pentane

−129.7

72.13

27.81

2006

Isopentane

−159.9

72.13

17.06

1231

Neopentane

−16.6

72.13

10.79

779

n-Hexane

−95.4

86.17

36.14

3114

2-Methylpentane

−153.7

86.17

17.41

1500

2,2-Dimethylbutane

−99.9

86.17

1.61

139

2,3-Dimethylbutane

−128.5

86.17

2.25

194

n-Heptane

−90.6

100.2

33.47

3350

2-Methylhexane

−118.3

100.2

21.91

2195

3-Ethylpentane

−118.6

100.2

22.78

2283

2,2-Dimethylpentane

−123.8

100.2

13.89

1392

2,4-Dimethylpentane

−119.2

100.2

16.32

1635

3,3-Dimethylpentane

−134.5

100.2

16.86

1689

2,2,3-Trimethylbutane

−24.9

100.2

5.39

540

27

Jennifer Moffitt Schall and Allan S. Myerson Table 1.15 (cont.)

Compound

Tm (°C)

Molecular weight (g/mol)

ΔHm (cal/g)

(cal/g mol)

n-Octane

−56.8

114.2

43.40

4956

3-Methylheptane

−120.5

114.2

23.81

2719

4-Methylheptane

−120.9

114.2

22.68

2590

n-Nonane

−53.5

128.2

28.83

3696

n-Decane

−29.6

142.3

48.24

6865

n-Dodecane

−9.6

170.3

51.69

8803

n-Octadecane

28.2

254.4

57.65

14,660

Benzene

5.5

78.1

30.09

2350

Toluene

−94.9

92.1

17.1

1575

Ethylbenzene

−94.9

106.1

20.63

2188

o-Xylene

−25.1

106.1

30.61

3250

m-Xylene

−47.9

106.1

26.04

2760

p-Xylene

13.3

106.1

38.5

4080

n-Propylbenzene

−99.5

120.1

16.97

2040

Isopropylbenzene

−96.0

120.1

14.15

1700

1,2,3-Trimethylbenzene

−25.4

120.1

16.6

1990

1,2,4-Trimethylbenzene

−43.8

120.1

25.54

3070

1,3,5-Trimethylbenzene

−44.7

120.1

19.14

2300

Cyclohexane

6.5

4.1

7.57

637

Methylcyclohexane

−126.5

98.1

16.4

1610

Ethylcyclohexane

−111.3

112.2

17.73

1930

1,1Dimethylcyclohexane

−33.5

112.2

1.32

148

1,cis-2Dimethylcyclohexane

−50.0

112.2

3.50

393

1,trans-2Dimethylcyclohexane

88.1

112.2

22.34

2507

Source: Data from Reid et al. 1987.

components. It also can be used to calculate the enthalpy change that would result from the dilution of a concentrated solution. The substances listed in Table 1.16 all have negative heats of solution, meaning that heat is released when the materials are dissolved or a concentrated solution is diluted. This is not generally true; many substances have positive heats of solution, indicating that heat is absorbed when they are dissolved. The heats of solution of selected organic and inorganic species in water are listed in Table 1.17. Additional data can be found in Perry’s Chemical Engineers’ Handbook (Green and Perry 2007).

28

The data in Table 1.17 illustrates two other points. The first is that for hydrated compounds, the heats of solution vary as a function of the number of waters of hydration present in the solid. Looking at magnesium sulfate, we see that the heat of solution at infinite dilution at 18°C varies from –21.1 kcal/g mol for the anhydrous form to +3.18 kcal/g mol for MgSO4·7H2O (which is the stable form at this temperature). The second point is that organic compounds with the same chemical formula but different structures such as para-, meta-, and ortho-nitrophenol have differing heats of solution. In this case, they are 4.49, 5.21, and 6.3 kcal/g mol, respectively, at infinite dilution and 18°C.

Solutions and Solution Properties Table 1.16 Integral Heats of Solution at 25°C

Table 1.17 Heats of Solution in Water at Infinite Dilution at 18°C

n (mol water/mol solute)

ðDĤs ÞHClðgÞ (kJ/mol HCl)

ðDĤs ÞNaOHðsÞ (kJ/mol NaOH)

ðDĤm ÞH2 SO4 (kJ/mol H2SO4)

0.5

—

—

−15.73

1

−26.22

—

−28.07

1.5

—

—

−36.90

2

−48.82

—

−41.92

3

−56.85

−28.87

−48.99

4

−61.20

−34.43

−54.06

5

−64.05

−37.74

−58.03

10

−69.49

−42.51

−67.03

20

−71.78

−42.84

—

25

—

—

−72.30

30

−72.59

−42.72

—

40

−73.00

−42.59

—

50

−73.26

−42.51

−73.34

100

−73.85

−42.34

−73.97

200

−74.20

−42.26

—

500

−74.52

−42.38

−76.73

1000

−74.68

−42.47

−78.57

2000

−74.82

−42.55

—

5000

−74.93

−42.68

−84.43

10,000

−74.99

−42.72

−87.07

50,000

−75.08

−42.80

—

100,000

−75.10

—

−93.64

500,000

—

—

−95.31

∞

−75.14

−42.89

−96.19

Source: Reprinted by permission of the publisher from R. M. Felder and R. W. Rousseau (1986), Elementary Principles of Chemical Processes, 2nd ed. Copyright © 1986, John Wiley & Sons, Inc.

The enthalpy change that results from the crystallization of a species from solution is called the heat of crystallization. The heat of crystallization is normally assumed to be of the same magnitude but opposite sign as the heat of solution at a concentration near saturation. This is not exactly correct because the supersaturated solution from which the crystallization is occurring has a higher concentration than a saturated solution, but the error should be relatively small. For binary systems of solutes and solvents that are relatively common or have industrial importance, enthalpy concentration diagrams often exist that are quite useful in energy balance

Substance

Formula

ΔHs (kcal/g mol)

Ammonium chloride

NH4Cl

3.82

Ammonium nitrate

NH4NO3

6.47

Ammonium sulfate

(NH4)2SO4

2.75

Barium bromide

BaBr2

−5.3

BaBr2·H2O

0.8

BaBr2·2H2O

3.87

CaCl2

−4.9

CaCl2·H2O

−12.3

CaCl2·2H2O

−12.5

CaCl2·4H2O

−2.4

CaCl2·6H2O

4.11

LiBr

−11.54

LiBr·H2O

−5.30

LiBr·2H2O

−2.05

LiBr·3H2O

1.59

MgSO4

−21.1

MgSO4·H2O

−14.0

MgSO4·2H2O

−11.7

MgSO4·4H2O

−4.9

MgSO4·6H2O

−0.55

MgSO4·7H2O

3.18

Potassium sulfate

K2SO4

6.32

Benzoic acid

C7H8O2

6.5

Citric acid

C6H8O7

5.4

Dextrin

C12H20O10

−0.268

Hexamethylenetetramine

C6H12N4

−4.78

m-Nitrophenol

C6H5NO3

5.2

o-Nitrophenol

C6H5NO3

6.3

p-Nitrophenol

C6H5NO3

4.5

Phthalic acid

C8H6O4

4.87

Urea

CH4N2O

3.61

Calcium chloride

Lithium bromide

Magnesium sulfate

Source: Adapted from data in Green and Perry 2007.

calculations involving solution, dilution, or crystallization. It is important to be aware of the reference state on which the diagram is based when employing these data in energy balance calculations.

29

Jennifer Moffitt Schall and Allan S. Myerson

Nomenclature

30

(cont.)

a

Activity

Equation (1.2)

A

Debye–Hückel constant

B

Constant for ion interaction

c

Concentration of solute Equations (1.15), (1.64), (1.78)

Equation (1.27) Equation (1.27)

P

Vapor pressure

Pc

Critical pressure

P2s

Vapor pressure of component 2 Equation (1.41)

Ptp

Triple-point pressure

R

Gas constant

Ra

Distance in Hansen phase space Equation (1.53)

r

Nuclei radius

S

Supersaturation ratio

Equation (1.87) Equation (1.86)

Equation (1.42)

Equation (1.10)

c

Saturation concentration

Cp

Specific heat at constant pressure (1.79)

T

Temperature

Cv

Specific heat at constant volume Equation (1.80)

Tb

Temperature at boiling point

D

Dispersive solubility parameter

Tm

Melting point temperature

DAB

Diffusion coefficient

Trb

Reduced boiling point

dCA dx

Concentration gradient

Tc

Critical temperature

dux dy

Shear rate

Ttp

Triple-point temperature

Equation (1.42)

e

Magnitude of the electric charge Equation (1.49)

U

Internal energy per mole

Equation (1.80)

Fi

Interaction parameter

V

Molar volume

f

Fugacity, activity coefficient (1.37)

Vm

Molar volume of solute

V2L

G

Gibbs free energy

Molar volume of subcooled liquid solute Equation (1.50)

G

Partial molar Gibbs free energy

w

Mass fraction Equation (1.75)

H

Hydrogen bonding solubility parameter Equation (1.53)

x

Mole fraction Equation (1.39)

y

Molar activity coefficient Equation (1.15)

H

Enthalpy per mole

z

I

Ionic strength

Number of charges on the cation or anion Equation (1.27)

i

Any species or ion present

Dc

j

Ion of opposite charge to i, any non-i species Equations (1.33), (1.58)

Supersaturation, expressed as a concentration difference Equation (1.71)

DCp

Difference in specific heats of a liquid or solid Equation (1.42)

DG

Gibbs free energy change

DGf 0

Gibbs free energy of formation (1.23)

Equation (1.64) Equation

Equation (1.53)

Equation (1.78) Equation (1.78)

Equation (1.76)

Equation (1.31) Equations (1.16),

Equation (1.58) Equation (1.58)

Equation (1.79)

Equation (1.27) Equation (1.8)

Equation (1.73) Equation (1.69)

Equation (1.10) Equation (1.86) Equation (1.47)

Equation (1.86) Equation (1.85)

Equation (1.51) Equation (1.73)

JA

Molar diffusive flux

K

Equilibrium constant

Ksp

Solubility product

l

Distance between two charges

Equation (1.49)

DHm

Latent heat of fusion

M

Molecular weight of the solute

Equation (1.18)

DHtp

Ms

Molecular weight of the solvent Equation (1.17)

Enthalpy change at the triple point (1.42)

m

Molal concentration, empirical viscosity parameter Equations (1.3), (1.77)

DH v

Heat of vaporization

DStp

N

Number of moles Equation (1.58)

Entropy change at the triple point (1.45)

n

Power law exponent

Du

Enthalpy of vaporization

P

Pressure, polar solubility parameter (1.42), (1.53)

DV

Volume change for liquid–solid transformation Equation (1.42)

Equation (1.78) Equation (1.2)

Equation (1.3)

Equation (1.77) Equations

Equation (1.73) Equation

Equation (1.47) Equation

Equation (1.85) Equation

Equation (1.52)

Solutions and Solution Properties

Greek (cont.)

γ

Activity coefficient

δ

Mean solubility parameter

μ

Chemical potential, dipole moment, viscosity Equations (1.8), (1.49), (1.76)

δ1 , δ2

Solubility parameters of the solvent and subcooled liquid, respectively Equation (1.52)

v

Stoichiometric number, molar liquid volume Equations (1.9), (1.52)

ϕ

Solvent volume fraction

ρ

Solution density

Equation (1.18)

σ

Relative supersaturation, solid–liquid interfacial tension Equations (1.63), (1.73)

ρ0

Solution density

Equation (1.18)

τyx

Shear stress

Equation (1.3)

References Bird, R. B., Stewart, W. E., and Lightfoot, E. N. Transport Phenomena (1st edn). Wiley, New York, 1960. Bromley, L. A. J. Chem. Thermodyn. 1972; 4:669–73. Bromley, L. A. AlChE J. 1973; 19(2):313–20. Bromley, L. A., Singh, D., Ray, P., Sridhar, S., and Read, S. AIChE J. 1974; 20(2):326–35. Chang, Y. C. Concentration Dependent Diffusion of Solid-Solute, Liquid-Solvent Systems in the Supersautrated Region, Georgia Institute of Technology, Atlanta, GA, 1985. Cole, L. S. On-Line Measurement of Crystal Size Distribution in a Mixed Suspension Mixed Product Removal Crystallizer, Master’s thesis, Polytechnic University, 1991.

Equation (1.54)

Equation (1.50)

Equation (1.76)

Hansen, C. M. Hansen Solubility Parameters: A User’s Handbook (2nd edn). CRC Press, Boca Raton, FL, 2007.

Robinson, R. A., and Stokes, R. H. Electrolyte Solutions (2nd edn). Butterworths, London, 1970.

Hartman, P. Crystal Growth: An Introduction (1st edn). North Holland, Amsterdam, 1973.

Rosenberger, F. E. Fundamentals of Crystal Growth (Springer Series in Solid-State Sciences vol. 5, 2nd edn). Springer-Verlag, New York, NY, 1981.

Jaworski, Z., Czernuszewicz, M., and Gralla, Ł. Chem. Process Eng. 2011; 32(2):135–54. Kim, S., and Myerson, A. S. Ind. Eng. Chem. Res. 1996; 35:1078–84. Kojima, K., and Tochigi, K. Prediction of Vapor-Liquid Equilibria by the ASOG Method. Elsevier Scientific, New York, NY, 1979. Lin, Y., ten Kate, A., Mooijer, M., et al. AIChE J. 2010; 56(5):1334–51. Linke, W. F., and Seidell, A. Solubilities, Inorganic and Metal-Organic Compounds: A Compilation of Solubility Data from the Periodical Literature (4th edn). American Chemical Society, Washington, DC, 1965.

Saska, M. Crystallization of Terephthalic Acid. Georgia Institute of Technology, Atlanta, GA, 1994. Sherman, P. Industrial Rheology: With Particular Reference to Foods, Pharmaceuticals, and Cosmetics. Academic Press, New York, NY, 1970. Skyner, R. E., McDonagh, J. L., Groom, C. R., van Mourik, T., and Mitchell, J. B. O. Phys. Chem. Chem. Phys. 2015; 17:6174–91.

Ma, H., Qu, Y., Zhou, Z., Wang, S., and Li, L. J. Chem. Eng. Data 2012; 57:2121–27.

Smith, J. M., Van Ness, H. C., and Abbott, M. M. Introduction to Chemical Engineering Thermodynamics (7th edn). McGraw-Hill, New York, NY, 2005.

Frank, T. C., Downey, J. R., and Gupta, S. K. Chem. Eng. Prog. 1999; 95(12):41–61.

Mullin, J. W. Crystallisation (2nd edn). Butterworths, London, 1972.

Valavi, M., Svärd, M., and Rasmuson, A. C. Cryst. Growth Des. 2016; 16:6951–6960.

Fredenslund, A., Gmehling, J., Michelsen, M. L., Rasmussen, P., and Prausnitz, J. M. Ind. Eng. Chem., Process Des. Dev. 1977; 16(4):450–62.

Myerson, A. S., Decker, S. E., and Welplng, F. Ind. Eng. Chem. Process Des. Dev. 1986; 25:925–29.

Walas, S. M. Phase Equilibria in Chemical Engineering; Butterworth-Heinemann, Stoneham, MA, 1985.

Gmehling, J. J. Chem. Thermodyn. 2009; 41:731–47.

Myerson, A. S., Lo, P. Y., Kim, Y. C., and Ginde, R. M. In Proceedings of the 11th Symposium on Industrial Crystallization. European Federation of Chemical Engineers, Garmisch, Germany, 1990, p. 847. Nývlt, J., Söhnel, O., Matachová, M., and Broul, M. The Kinetics of Industrial Crystallization. Elsevier, Amsterdam, 1985.

Weast, R. C. CRC Handbook of Chemistry and Physics (56th edn). CRC Press, Cleveland, OH, 1975.

Felder, R. M., and Rousseau, R. W. Elementary Principles of Chemical Processes (2nd edn). Wiley, New York, NY, 1986.

Gmehling, J., Constantinescu, D., and Schmid, B. Annu. Rev. Chem. Biomol. Eng. 2015; 6:267–92. Gordon, L. J., and Scott, R. L. J. Am. Chem. Soc. 1952; 74:4138–40. Gordon, P. Principles of Phase Diagrams in Materials Systems; McGraw-Hill, New York, NY, 1968. Green, D. W., and Perry, R. H. Perry’s Chemical Engineers’ Handbook (8th edn). McGraw-Hill, New York, NY, 2007. Guggenheim, E. A. Thermodynamics: An Advanced Treatment for Chemists and Physicists (8th edn). North Holland, Amsterdam, 1986.

Peng, Y., Zhu, Z., Braatz, R. D., and Myerson, A. S. Ind. Eng. Chem. Res. 2015; 54:7914–24. Prausnitz, J. M., Lichenthaler, R. N., and Gomes de Azvedo, E. Molecular Thermodynamics of Fluid-Phase Equilibria (2nd edn). Prentice-Hall, Englewood Cliffs, NJ, 1986. Reid, R. C., Prausnitz, J. M., and Poling, B. E. The Properties of Gases and Liquids (4th edn). McGraw-Hill, New York, NY, 1987.

Williams, N., and Amidon, G. J. Pharm. Sci. 1984; 73(1):14–18. Williams, N., and Amidon, G. J. Pharm. Sci. 1984; 73(1):9–13. Williams, N., and Amidon, G. Pharm. Res. 1988; 5(3):193–95. Williams, N., and Amidon, G. J. Pharm. Sci. 1984; 73(1):18–23. Yi, Y., Hatziavramidis, D., Myerson, A. S., et al. Ind. Eng. Chem. Res. 2005; 44:5427–33. Zemaitis, J. F. J., Clark, D. M., Rafal, M., and Scrivner, N. C. Handbook of Aqueous Electrolyte Thermodynamics: Theory & Application. Wiley, New York, NY, 1986.

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Chapter

2

Crystals and Crystal Growth Alfred Y. Lee Merck & Co., Inc. Deniz Erdemir Bristol-Myers Squibb Company Allan S. Myerson Massachusetts Institute of Technology

2.1 Introduction Crystallization can be regarded as a self-assembly process in which randomly organized molecules in a fluid come together to form an ordered three-dimensional molecular array with a periodic repeating pattern. It is vital to many processes occurring in nature and manufacturing. Geologic crystallization is responsible from huge deposits of carbonates, sulfates, and phosphates that often grow in mountains and quarries. This process occurs over long periods of time, often at high temperatures and pressures, and results in large and usually highly ordered crystals such as diamond. In living organisms, crystallization of carbonates, phosphates, oxalates, oxides, silicates, and other inorganic materials is controlled through a biomineralization process that involves application of functionalized macromolecules (Mann 2002). The crystallized biominerals may be beneficial, such as bones, teeth, eggshells, and mussels, or harmful, as in the case of kidney stones (Boistelle 1986) and gout (Martillo et al. 2014). In the chemical, food, and pharmaceutical industries, solution crystallization is regarded as the most important separation and purification method (Garside 1985; Paul et al. 2005). The fact that it combines particle formation and purification within a single process makes crystallization an attractive isolation step during manufacturing (Price 1997). It ranks as the oldest chemical engineering unit operation, because sodium chloride has been manufactured by this process since the dawn of civilization (Mullin 1993). Almost all the products based on fine chemicals, such as dyes, explosives, and photographic materials, require a crystallization of some kind in their manufacture (Carpenter and Wood 2004). According to Melikhov and Kutepov (2001), no fewer than 200,000 various substances are crystallized daily in plants and laboratories all over the world. Over 90 percent of all pharmaceutical products contain bioactive drug substances and excipients in the crystalline solid state for reasons of stability and ease of handling during the various stages of drug development (Shekunov and York 2000; Vippagunta et al. 2001). Crystallization is also exploited for the synthesis of wellordered single crystals of proteins with sufficient size, which are required for accurate structural determination by X-ray diffraction, as well as very small crystals with large surface areas, which are important for the purpose of gas sensing and catalytic applications (Schuth 2001; Garcia-Ruiz 2003). The ability to grow single crystals of precisely controlled composition and structural perfection also plays an essential role in microelectronics (Garside 1985). By contrast, undesired

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crystallization of inorganic salts such as calcium carbonate on pipes or water-handling equipment creates a serious problem known as scaling in many industrial processes (Katz et al. 1993). The significance of crystallization in a wide range of disciplines makes this separation process an active area of research. It is necessary to control the crystallization process in order to obtain products with desired and reproducible properties or prevent the unwanted occurrence of crystals such as kidney stones and scaling. The quality of a crystalline product is commonly judged by four main criteria: size, purity, morphology, and crystal structure. For instance, it is vital in the pharmaceutical industry to manufacture only a particular polymorph to ensure the bioavailability and stability of the drug substance. While the target of many crystallization operations is to produce a crystal that is large enough to be isolated easily on filtration equipment, smaller particle sizes are desired occasionally in pharmaceuticals to enhance the dissolution rate, thus improving bioavailability (Shekunov and York 2000; Carpenter and Wood 2004). Control of crystal habit is also essential in many industries because the particle morphology may have a significant impact on post-crystallization processes such as filtration and drying. Crystal nucleation and the subsequent growth of crystals play a decisive role in determining solid-state properties. This chapter covers the fundamental aspects of crystallization and highlights the importance of controlling the physical properties of the crystalline products and the various strategies undertaken to design and engineer crystals.

2.2 Basic Concepts of the Solid State 2.2.1 Crystals Crystals are solids in which the atoms, ions, or molecules are arranged in a periodic repeating pattern that extends in three dimensions. While all crystals are solids, not all solids are crystals. Materials that have short-range rather than longrange ordering, like glass, are described as non-crystalline or amorphous solids. Many materials can form solids that are crystalline or amorphous depending on the conditions of growth or method of preparation. Some materials can form crystals of the same chemical composition but with different crystal packing arrangements. Other materials can have the same three-dimensional structure but may have different physical appearances or external crystal shapes. To understand the true nature of crystals and how they

Crystals and Crystal Growth

Figure 2.1 A point lattice

are identified requires some knowledge of the internal structure. This area of research is known as crystallography and is described in a number of references (Bunn 1961; Glusker and Trueblood 2010). In this section, the basics of crystal science will be covered.

Lattices and Crystals Systems Crystals are solids in which the atoms, ion, or molecules are arranged in a three-dimensional repeating periodic structure. If we think of crystals in a purely geometric sense and forget about the actual atoms, the concept of a point lattice can be used to represent a crystal. A point lattice is a set of points arranged so that each point has identical surroundings. In addition, the point lattice can be characterized in terms of three spatial dimensions a, b, and c and three angles α, β, and γ. An example of a point lattice is shown in Figure 2.1. Examination of the figure reveals that the lattice consists of repeating units that can be characterized by the three dimensions and three angles mentioned. We can arbitrarily choose any of these units and, by making use of the spatial dimensions and angles, can reproduce the lattice indefinitely. The lengths and angles mentioned are known as lattice parameters, and a single cell constructed employing these parameters is called a unit cell. A unit cell is illustrated in Figure 2.2. Obviously, a number of different lattice arrangements and unit cells can be constructed. It was shown, however, in 1848 by Bravais that only 14 possible point lattices can be constructed. These point lattices can be divided into seven categories (crystal systems) that are shown in Table 2.1. Figure 2.3 shows all 14 of the Bravais lattices. Looking at the crystal systems, we see that they are all characterized by these unitcell dimensions. For example, cubic systems all must have equal lengths (a = b = c) and angles equal to 90 degrees. In addition, lattices can be classified as primitive or non-primitive. A primitive lattice has only one lattice point per unit cell, whereas a non-primitive unit cell has more than one. If we look at the cubic system, a simple cubic unit cell is primitive. This is because each lattice point on a corner is shared by eight other cells so that 1/8 belongs to a single cell. Because there are eight corners, the simple cubic cell has one

lattice point. Looking at a body-centered cubic cell, the point on the interior is not shared with any other cell. A bodycentered cubic (BCC) cell, therefore, has two lattice points. A face-centered cubic (FCC) cell has a lattice point on each face that is shared by two cells. Because there are six faces as well as the eight corners, an FCC cell has four lattice points. Another property of each crystal system that distinguishes one system from another is called symmetry. There are four types of symmetry operations: reflection, rotation, inversion, and rotation-inversion. If a lattice has one of these types of symmetry, it means that after the required operation, the lattice is superimposed on itself. This is easy to see in the cubic system. If we define an axis normal to any face of a cube and rotate the cube about that axis, the cube will superimpose on itself after each 90° of rotation. If we divide the degrees of rotation into 360 degrees, this tells us that a cube has three fourfold rotational symmetry axes (on axes normal to three pairs of parallel faces). Cubes also have threefold rotational symmetry using an axis along each body diagonal (each rotation is 120 degrees) and twofold rotational symmetry using the axis formed by joining the centers of opposite edges. Each lattice system can be defined in terms of the minimum symmetry elements that must be present. Table 2.2 lists the minimum symmetry elements that must be present in a given crystal system. A more complete discussion of lattices and symmetry can be found in Cullity and Stock (2001).

Miller Indices and Lattice Planes If we take any point on a lattice and consider it the origin, we may define vectors from the origin in terms of three coordinates. If, for example, we started with a cubic cell and defined a vector going from the origin and intersecting point 1,1,1, the line would go in the positive direction along the body diagonal of the cube and would also intersect the points 2,2,2 and all multiples. This direction is represented in shorthand by [1,1,1], where the numbers are called the indices of the direction. Negative numbers are indicated by putting a bar over a number, so [1; 1; 1] means that the first index is negative. We can represent a family of direction by using the symbol . This represents all the directions using both positive and negative indices in all combinations. In this case, it represents all the body diagonals of a cube. By convention, all indices are reduced to the smallest set of integers possible either by division or by clearing fractions. An illustration of various indices and the directions they represent is given in Figure 2.4. The representation of planes in a lattice makes use of a convention known as Miller indices. In this convention, each plane is represented by three parameters {hkl}, which are defined as the reciprocals of the intercepts the plane makes with three crystal axes. If a plane is parallel to a given axis, its Miller index is zero. Negative indices are written with bars over them. Miller indices refer not only to one plane but also to a whole set of planes parallel to the plane specified. If we wish to specify all planes that are equivalent, we put the indices in braces. For example, {100} represents all the cube faces. Examples of Miller indices in a cubic system are shown in Figure 2.5.

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson Figure 2.2 A unit cell

Real crystals are often described in terms of the Miller indices of the faces (planes) present. Examples of some common crystals with their faces given in terms of Miller indices are given in Figure 2.6.

Crystal Structure and Bonding In the preceding subsection, a geometric system was described that can be used to represent the structure of an actual crystal. In the simplest actual crystal, the atoms coincide with the points of one of the Bravais lattices. Examples include chromium, molybdenum, and vanadium, which have a BCC crystal structure, and copper and nickel, which have an FCC crystal structure. In a more complex arrangement, more than one atom of the same type can be associated with each lattice point. A structure that a number of metals have, which is an example of this, is the hexagonal close-packed structure. It is called close packed because if the molecules are assumed to be spherical, this arrangement is one of only two possible ways spheres can be packed together to yield the greatest density yet still be in a periodic structure. The packing arrangement in crystals is another part of the information that helps us understand crystal structure. More information on crystal packing can be found in Ruoff (1973) and Cullity and Stock (2001). Many inorganic molecules form ionic crystals. An example of an ionic structure common to a number of molecules is that of sodium chloride, shown in Figure 2.7. Ionic crystals are made up of the individual ionized atoms that make up the species in their stoichiometric proportion. They are held in place by electronic forces. The sodium chloride structure is FCC, and the unit cell contains four sodium ions and four chloride ions. Because the unit cells contain two types of atoms, some additional constraints on the structure exist. For example, a symmetry operation on the crystal must superimpose atoms of the same type. Most organic species form molecular crystals in which discrete molecules are arranged in fixed positions relative to the lattice points. This, of course, means that the individual atoms making up the molecules are each arranged at fixed positions relative to each other, the lattice point, and the other molecules. The forces between molecules in molecular crystals are generally weak compared with the forces within a

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molecule. The structure of molecular crystals is affected by both the intermolecular forces and the intramolecular forces because the shapes of the individual molecules will affect the way the molecules pack together. In addition, the properties of the individual molecules, such as the polarity, will affect the intermolecular forces. The forces between the molecules in molecular crystals include electrostatic interactions between dipoles, dispersion forces, and hydrogen bond interactions. More information about the structure and energetics of molecular crystals can be found in Kitaigorodsky (1973) and Wright (1995). An important tool used to identify crystals and to determine crystal structure is X-ray diffraction. Crystals have atoms spaced in a regular three-dimensional pattern. X-rays are electromagnetic waves with wavelengths of similar size as the distance between the atoms in a crystal. When a monochromatic beam of X-rays is directed at a crystal in certain directions, the scattering of the beam will be strong, and the amplitudes of the scattering will add, creating a pattern of lines on photographic film. The relationship between the wavelength of the X-rays and the spacing between atoms in a crystal is known as Bragg’s law, which is as follows: λ ¼ 2d sin θ

ð2:1Þ

where λ is the wavelength of the incident X-rays, d is the interplanar spacing in the crystal, and Ɵ is the angle of the incident X-rays on the crystal. Bragg’s law shows us that if X-rays of a known wavelength are used and the incident angle of the radiation is measured, determination of the interplanar spacing of a crystal is possible. This is the foundation of X-ray diffraction methods that are used to analyze or determine the structure of crystals. Several different experimental methods making use of X-ray diffraction and Bragg’s law have been developed and are used depending on the type of sample that is available and the information desired. The most powerful method that can be used to determine unknown crystal structures is the rotating crystal technique. In this method, a single crystal of good quality (of at least 0.1 mm in the smallest dimension) is mounted with one of its axes normal to a monochromatic beam of X-rays and rotated about in a particular direction. The crystal is surrounded by

Crystals and Crystal Growth Table 2.1 Crystal Systems and Bravais Lattice

System

Axial lengths and angles

Bravais lattice

Cubic

Three equal axes at right angles a=b=c α = β = γ = 90 degrees

Simple Body centered Face centered

Tetragonal

Three axes at right angles, two equal axes a=b≠c α = β = γ = 90 degrees

Simple Body centered

Orthorhombic

Three unequal axes at right angles a≠b≠c α = β = γ = 90 degrees

Simple Body centered Base centered Face centered

Rhombohedral

Three equal axes, equally inclined a=b=c α = β = γ ≠ 90 degrees

Simple

Hexagonal

Two equal coplanar axes at 120 degrees, third axis at right angles a=b≠c α = β = 90 degrees; γ = 120 degrees

Simple

Monoclinic

Three unequal axes, one pair not at right angles a≠b≠c α = γ = 90 degrees ≠ β

Simple Base centered

Triclinic

Three unequal axes, unequally inclined, and none at right angles a≠b≠c α ≠ β ≠ γ ≠ 90 degrees

Simple

cylindrical film, with the axis of the film being the same as the axis of rotation of the crystal. By repeating this process of rotation in a number of directions, the rotating crystal method can be used to determine an unknown crystal structure. It is unlikely that you will ever need to use the rotating crystal method to determine an unknown structure because most materials you are likely to crystallize have structures that have been determined. This will not be true for a newly developed compound and is rarely true for proteins and other biological macromolecules. An X-ray method more commonly used is the powder X-ray diffraction method. Instead of using a single crystal, a very fine powder of the crystal is used. This is convenient because you do not have to grow a single crystal of the size and quality needed for single-crystal methods. The powder method relies on the fact that the array of tiny crystals randomly arranged will present all possible lattice planes present for reflection of the incident monochromatic X-ray beam. The

powder pattern of a particular substance acts as a signature for that substance, so a powder diffraction pattern can be used for identification, chemical analysis (presence of impurities), and determining whether a material is crystalline or amorphous. Description of powder diffraction methods and analysis can be found in Bunn (1961) and Cullity and Stock (2001). It is now common to determine the structure of molecular crystals from powder diffraction data because significant advances have been made over the past two decades in algorithmic developments and user-friendly computer programs (David and Shankland 2008). The structure solution process commonly involves powder indexing (determination of the unit cell), space group determination, simulated annealing to identify the global minimum that would give the best agreement between the calculated and measured powder diffraction data followed by structure refinement. This is of great value when it is not possible to grow a crystal of sufficient size for single-crystal X-ray crystallography. Another method of X-ray diffraction is the Laue method, which employs a single crystal oriented at a fixed angle to the X-ray beam. The beam, however, contains the entire spectrum produced by the X-ray tube. In this method, therefore, the angle Ɵ in Bragg’s law is fixed but a variety of wavelengths are impinging on the crystal. Each set of planes that will satisfy Bragg’s law with a particular wavelength will diffract and form a pattern known as a Laue pattern. The Laue method is used as a way to assess crystal orientation and to determine crystal quality. Other X-ray techniques are also available, making use of X-ray spectrometers and variations on the methods mentioned previously. A number of references can supply more details on any of the X-ray methods (Bunn 1961; Bertin 1975; Cullity and Stock 2001). It is important to remember that X-ray diffraction is one of the more common analytical techniques employed to determine whether a material is truly crystalline. Furthermore, as will be described in Section 2.6, powder X-ray diffraction is often necessary to determine whether a material is cocrystallized or has crystallized into more than one crystalline structure.

Isomorphism and Solid Solutions It is quite common for a number of different species to have identical atomic structures. This means that the atoms are located in the same relative positions in the lattice. We have seen this previously with the sodium chloride structure. A number of other species have this structure. Obviously, species that have the same structure have atoms present in similar stoichiometric proportion. Crystals that have the same structure are called isostructural. If crystals of different species are isostructural and have the same type of bonding, they also will have very similar unit-cell dimensions and will appear almost identical macroscopically. This is known as isomorphism. Examples of isomorphic materials include ammonium and potassium sulfate, and potassium and ammonium dihydrogen phosphate. In each of these materials, the potassium and ammonium ions can easily substitute for each other in the lattice because they are of almost the same size. This illustrates one of the properties of isomorphous

35

Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson Figure 2.3 Bravais lattices (P, R = primitive cells; F = face centered; I = body centered; C = base centered)

materials; that is, they tend to form solid solutions or mixed crystals. Crystallization from a solution of two isomorphous materials therefore can result in a solid with varying composition of each species with unit-cell dimensions intermediate between the two components. Therefore, the purification of isomorphous substances can be difficult. Solid solutions do not all result from the substitution of isomorphous materials in the lattice sites; other types of solid solutions are possible and are described in Vainshtein (1994). Solid solutions are not only limited to inorganic minerals or salts but also have been encountered with organic materials, metallic compounds, coordination polymers, and metalorganic frameworks (MOFs). In the latter case, the term alloy has been used to describe two distinct MOFs – crystalline solids consisting of metal ions and bridging organic ligands (Panda et al. 2017). Variations in the chemical compositions of the solid solutions can lead to disparities in the electrical, optical, physicochemical, and photochemical properties. Schmidt et al. (2006) demonstrated with mixed crystals of Pigment Red 170 and

36

derivatives (methyl-, fluoro-, chloro-, bromo-, and nitro-substituted analogues) that the light- and weather-fastness improves as a result of the efficiently packed and denser structures. Multicolored photochromic two- and three-component diarylethenes mixed crystals were reported with intermediate color tones available by tuning the component of the solid solutions (Morimoto et al. 2003). In an organometallic molecular salt solid solution, the phase-transitions and melting transitions behave in a linear manner with chemical compositions (Braga et al. 2001). A comprehensive account on the engineering of solid-state properties through solid solutions was recently provided by Lusi (2018).

Imperfections in Crystals In the earlier section “Lattices and Crystals Systems,” we showed that each atom, ion, or molecule has a precise location in a repeating structure. If this structure is disrupted in some way, the crystal is said to have imperfections. A number of different kinds of imperfections can occur. If a foreign atom (or molecule in a molecular

Crystals and Crystal Growth Table 2.2 Symmetry Elements

System

Minimum symmetry elements

Cubic

Four threefold rotation axes

Tetragonal

One fourfold rotation (or rotation-inversion) axis

Rhombohedral

Three perpendicular twofold rotation (or rotation-inversion) axes

Hexagonal

One sixfold rotation (or rotation-inversion) axis

Monoclinic

One twofold rotation (or rotation-inversion) axis

Triclinic

None

Figure 2.4 Indices of direction

Source: Adapted from Cullity and Stock 2001.

crystal) is present in the crystal lattice, this is known as a chemical imperfection. The foreign atom can be present at a lattice site having substituted for an atom in the structure, as we saw in our brief discussion of isomorphism and solid solutions. This is called a substitutional impurity. The foreign atom can also be present in the crystal by fitting between the atoms in the lattice. This is called an interstitial impurity. Both of these types of impurities can cause the atoms in the crystal to be slightly displaced because the impurity atoms do not really fit in the perfect lattice structure. The displacement of the atoms causes a strain in the crystal. Another type of imperfection is due to vacancies in the crystal. A vacancy is simply a lattice site in which there is no atom. A region is called a Frenkel defect, while one in which a vacancy is just an empty lattice site missing an atom is called a Schottky defect. These types of imperfections are very important in semiconductors and microelectronics. Figure 2.8 illustrates the various types of vacancies and chemical imperfections. The imperfections in crystals discussed so far are called point defects because they involve a single unit of the crystal structure that is an atom or a molecule. Another type of imperfection is known as a line defect or dislocation. There are two types of dislocations known as edge dislocations and screw dislocations. An edge dislocation is illustrated in Figure 2.9, which is a cross section of a crystal lattice. Looking at the figure, you can see that half of a vertical row (the bottom half) in the middle of the lattice is missing. This row of atoms is missing in each plane of the lattice parallel to the page. The dislocation is marked at point A. If a line is drawn vertically from the book out, going through all the layers of the crystal, it would represent the edge-dislocation line. Looking at the figure, we can see that the lattice points are displaced in the region of the dislocation and that this displacement gets smaller as we move away from the dislocation until the lattice returns to normal. The dislocation is a weak point in the crystal. Again, looking at Figure 2.9, if we draw a line through the crystal at the dislocation, this defines what is called a slip or cleavage

plane. If a force is exerted in the direction of the arrow on the upper half of the crystal, it will shear at this plane. In molecular crystals, differences in mechanical properties among polymorphic forms can be rationalized through knowledge of the cleavage planes (Sun and Grant 2004). The most common approach to identify slip planes is through interrogation of the crystal structure, where crystallographic planes with high molecular density and large d-spacing would correspond to the slip planes. An alternative method but sometimes less accurate to predict slip planes is based on attachment energy calculations (which is discussed in Section 2.5.3). In this method, the slip plane is assumed to have the smallest (absolute) attachment energy (Sun and Kiang 2007). A more detailed discussion of dislocations and defects can be found in several references (StricklandConstable 1968; Laudise 1970; Chanda 1979). The other type of line defect is called a screw dislocation, which is illustrated in three dimensions in Figure 2.10. Imagine that we cut the crystal along the line BC going toward point A. If we then pushed up at the cut until point B was displaced one lattice unit higher at the edge along the cut, the result would be a ledge running from A to B. The screw dislocation is very important in crystal growth, as we will see in Section 2.4.

2.2.2 Amorphous Solids The antithesis of crystalline material is the amorphous state in which atoms or molecules in this non-crystalline state possess no long-range periodicity, and the atoms or molecules are organized in essentially a random arrangement. This is in complete contrast to a crystal, where the atoms or molecules are in a fixed, welldefined arrangement within the crystal lattice with long-range order as a consequence of the repeating unit cells that are extended in three dimensions along the crystallographic directions. Local or short-range order exists in amorphous solids, and in most cases, the local order is similar to that observed in liquids. In the case of small organic molecules, the short-range order is on the order of 20–25 Å or less in terms of the next nearest neighbor interaction. An experimental approach to quickly confirm the absence or presence of long-range order in the solid material is achieved by collecting the powder X-ray diffraction pattern. Figure 2.11 is an illustration of the powder X-ray diffraction pattern of the amorphous and crystalline phases of the same

37

Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson Figure 2.5 Miller indices of planes in a cubic system. The distance d corresponds to the interplanar spacing.

Figure 2.6 Crystal habits of some common materials with their Miller indices shown

material. For amorphous materials, the diffraction pattern generally would exhibit a broad halo with few or a single maximum. In contrast, the crystalline counterpart would have characteristic diffraction 2-θ peaks that correspond to the different symmetry planes within the crystal lattice (see the section “Crystal Structure and Bonding” above). Due to the inefficient packing and random arrangements of atoms or molecules in the non-crystalline state, the free volume is generally much higher than that with ordered materials. As a consequence, rotational and translational molecular motions are enhanced in disordered solids; in

38

turn, both the chemical and physical reactivity is affected (Prodan 1990). In contrast, molecules are restricted in a crystalline solid and can be considered to be static systems, so crystalline solids are less susceptible to chemical degradation owing to the reduced molecular mobility. This has been reported for a number of systems, including indomethacin and cefoxitin sodium (Oberholtzer and Brenner 1979; Carstensen and Morris 1993). There have also been instances where the reverse is true – a highly ordered crystalline state is chemically more reactive than an amorphous state (Sukenik et al. 1977).

Crystals and Crystal Growth

Figure 2.7 The structure of sodium chloride (NaCl)

In pharmaceutical development, amorphous solids have attracted great interest because of the increased solubility and bioavailability they offer. Several excellent reviews and books devoted to amorphous pharmaceutical solids have appeared in recent years (Shah et al. 2014; Newman 2015; Descamps 2016a, b; Rams-Baron et al. 2018).

The Disordered State Crystalline solids are considered to be perfectly ordered materials, although imperfections exist enough that some level of disorder will always be present (see the section “Imperfections in Crystals” above). By contrast, the complete absence of longrange order in particular translational periodicity is a characteristic of an amorphous material. In this respect, the structure of an amorphous state has often been described as a frozen liquid with the rheologic properties of a solid. The physical properties tends to be quite dissimilar to those of its crystalline counterpart. Hancock and Parks (2000) demonstrated that the solubility ratio between an amorphous form and the most stable crystalline form (of the same material) is estimated to be between 10- and 1600-fold. By contrast, the solubility ratio for different polymorphic forms of the same compound is typically less than 2 (Pudipeddi and Serajuddin 2005). The higher energy state that an amorphous form provides makes it extremely appealing for solubility and dissolution rate enhancement. The tradeoff is that the greater the difference between the amorphous–crystal solubility ratio, the higher is the supersaturation with respect to the crystalline form, and there is a significant thermodynamic driving force for crystallization from solution. This is the primary reason why it is difficult to truly measure the solubility of an amorphous material under true equilibrium conditions. Figure 2.12 is schematic illustration of enthalpy, entropy, or specific volume with temperature. In the case of an enthalpy plot, the slope of each segment corresponds to the heat capacity

of the respective state. With the crystalline state, there is a small change in enthalpy, entropy, or volume relative to temperature. At the melting transition, there is a first-order phase transition to the molten state whereby a discontinuity occurs in the enthalpy, entropy, or specific volume plot. The melting event is generally associated with an increase of free volume as a result of thermal expansion effects. On rapid cooling of the melt (and assuming that a crystallization has not occurred below the melting temperature), a supercooled liquid is obtained. It is often described as the “rubbery state” because its structural characteristics are those of a viscous liquid. On further cooling, the supercooled liquid shifts from a rubbery state to a glassy state at the glass-transition temperature. This is accompanied by a decrease in the heat capacity and a significant reduction in molecular mobility as rotational and translational motions are ‘kinetically frozen.’ This process is also known as vitrification. The glass transition Tg is highly dependent on kinetic factors and may vary with cooling rates and sample history. Slow cooling will generally result in lower Tg values than rapid cooling of the same supercooled liquid. Generally, the glass transition can vary 3–5°C when the rate of cooling is changed by one order of magnitude. In contrast to the melting transition, the glass transition is not a phase transition and does not involve any discontinuous structural change. The structure remains that of a liquid. Based on observations from polymers, organic liquids, and molten oxides, the ratio of the glass transition of the amorphous form and the melting temperature Tm of the crystal is approximately 2/3, or 0.67 (Donth 2001). This rule of thumb is a rapid and convenient way to estimate the glass transition if the Tm of the crystalline material is known. Experimentally, there are many analytical techniques that can be employed to measure the Tg , including calorimetry, dilatometry, inverse gas chromatography (iGC), nuclear magnetic resonance (NMR) spectroscopy, and terahertz (THz) timedomain spectroscopy. Each approach has its own sensitivity and time scale of measurements. In a way, the glass transition should be considered to be a “region” of temperatures instead of a single temperature. In the case of the enthalpy plot in Figure 2.12, extrapolation of the heat capacity of the supercooled liquid beyond the glasstransition temperature to the point where the enthalpy of the supercooled liquid is equivalent to that of the crystalline state yields a critical temperature known as the Kauzmann temperature TK. At this temperature, the configurational entropy of the system vanishes and reaches zero. It is the lower limit of the glass transition. Knowledge of TK as determined by extrapolation may help guide long-term storage stability conditions because physical instability risks no longer exist below this critical temperature. Amorphous solids can be prepared in many different ways – some intentionally and others inadvertently, such as structural disorder induced by processing (e.g., milling, compression, thermal processing, granulation). Figure 2.13 summarizes the various preparative approaches and their time scales to form a disordered, high-energy amorphous solid. The methods to produce amorphous solids can be broken up into two categories: (1)

39

Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

Figure 2.8 Imperfection in crystals (a = perfect crystal; b = substitutional impurity; c = interstitial impurity; d = Schottky defect; e = Frenkel defect)

Figure 2.9 An edge dislocation shown in two dimensions

Figure 2.10 A screw dislocation in a simple cubic crystal. The screw dislocation AD is parallel to BC (D is not visible).

Figure 2.11 Powder X-ray diffraction of an amorphous solid and its corresponding crystalline counterpart

solution-based and (2) solid-state approaches. In the former, all the precipitation methods and spray drying are reliant on the generation of very high supersaturation, whereby amorphous precipitates are formed before nucleation can occur. In the case of lyophilization or freeze drying, amorphous formation more likely occurs during the freezing step as the solution is rapidly quenched to avoid crystal nucleation. The drying step is also critical because residual solvents must be sufficiently removed; otherwise, the solvents that are present can serve as plasticizers and induce crystallization. The most common approach to generate glassy solids, especially for thermally stable material, is through vitrification or melt-quenching. This assumes that decomposition does not

40

Figure 2.12 Schematic illustration of the variation in enthalpy, entropy, or specific volume as a function of temperature. Tg , TK, and Tm correspond to the glass-transition, Kauzmann, and melting temperatures, respectively.

occur on melting. The process involves cooling the molten phase below the glass-transition temperature without crystallization occurring. As described earlier, the cooling rate is a critical parameter in controlling the final physical form. Highenergy processes such as comminution and milling are also

Crystals and Crystal Growth Figure 2.13 Methods of preparation for amorphous solids

effective in generating disordered solids. Desolvation of a crystalline solvate also may yield a non-crystalline material. In most cases when this occurs, the solvent plays a major role in stabilizing the crystal structure, and on removal, the crystal lattice completely collapses and the material is rendered amorphous. Physical vapor deposition is a more recent method to prepare stable glasses (Ediger 2017). Vapor-deposited glasses have been demonstrated to be more kinetically and thermodynamically stable than ordinary liquid-cooled glasses. In the case of indomethacin, these high-density glasses are also reported to have slower photodegradation rates and are less susceptible for water uptake than liquid-cooled glasses (Dawson et al. 2009; Qiu et al. 2018). The properties and density of the stable glasses depend on the deposition rate and substrate temperature during the physical vapor deposition process. For small organic molecules, substrate temperature just below the glass transition formed kinetically stable glasses (Kearns et al. 2007).

Polyamorphism Analogous to polymorphism in molecular crystals (Section 2.6), polyamorphism corresponds to the existence of two or more distinct amorphous states of the same material. Angell (1995) defines true polyamorphic materials as those in which a first-order phase transition occurs between the amorphous states. The most well-known examples are pressure-induced polyamorphism of carbon, chalcogenide, ice, phosphorus, silica, and silicon. This phenomenon is not only limited to inorganics but also is encountered in organic molecules such as triphenyl phosphite and mannitol (Wiedersich et al. 1997; Zhu and Yu 2017). True polyamorphs should be clearly distinguished from amorphous forms that are trapped in a kinetic state (which have been referred to as relaxation polyamorphs) in which a phase transition is not expected. There are numerous examples of these amorphous states, but they are not considered to be true polyamorphisms according to the thermodynamic definition.

2.3 Crystal Nucleation Crystallization from solution can be regarded as a two-step process. The first step is the phase separation, or “birth,” of new crystals. The second step is the growth of these crystals to larger sizes by the addition of solute molecules from the supersaturated solution. These two processes are known as nucleation and crystal growth, respectively. Analysis of industrial crystallization processes requires knowledge of both nucleation and crystal growth. The fundamentals of nucleation are discussed in details in Chapter 3.

2.4 Crystal Growth Crystal growth, along with nucleation, controls the final particle size distribution obtained in the system. In addition, the conditions and rate of crystal growth have a significant impact on the product purity and the crystal habit. An understanding of crystal growth theory and experimental techniques for examining crystal growth from solution is important and very useful in the development of industrial crystallization processes.

2.4.1 Basic Concepts The growth rate of a crystal can be described in a number of different ways that are often used interchangeably in the literature. It is important to understand the definition of each of these terms and their relation to each other. The growth of a crystal is often described by the change in some dimension of the crystal with time. This is called the linear growth rate and has dimensions of length per unit time. It is not enough, however, to indicate that you are interested in a linear growth rate because this can mean a number of different things. As we have seen in previous sections, crystals are made up of a number of faces that can grow at different rates. A fundamental expression of the growth rate, therefore, is the linear growth rate of a particular face. This refers to

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

the rate of growth of the face in the direction normal to the face. While face growth rates are of interest in studies of the fundamentals of crystal growth, they are not normally used to describe the overall growth of a crystal. When a linear growth rate is used to describe the growth of an entire crystal, it is describing the increase in some characteristic dimension of the crystal. If the crystal were a sphere, the characteristic dimension would be the diameter, and we would express the crystal growth rate in terms of the increase in the diameter with time. If the crystal were of another shape, the characteristic dimension used usually would be the second longest dimension. This can be understood by thinking of what happens when you pass crystals through screens or sieves. The dimension of the crystal, which determines if it will pass through a sieve of a given size, is the second largest dimension rather than the largest dimension. This characteristic dimension can be related to the volume and surface area of the crystal through the shape factor. Another way of measuring crystal growth rates is through measurement of the mass change of the crystal. The increase in mass with time is often used and can be directly related to the overall linear growth velocity (G) through the relation RG ¼

1 dm α α dL ¼ 3 ρG ¼ 3 ρ A dt β β dt

ð2:2Þ

where RG = increase of mass per unit time per unit surface area A = surface area of the crystal α, β = volume and area shape features, respectively ρ = crystal density L = characteristic dimension As shown in Equation (2.2), the surface and volume shape factors must be known or estimated to allow linear growth rates to be calculated from mass growth rates. It is quite common for investigators to assume a sphere and thus calculate growth rates based on an equivalent spherical geometry. This can be a reasonable estimate or can be quite poor depending on the crystal’s actual shape.

2.4.2 Theories of Crystal Growth Crystals and their growth and habit have long been of interest to scientists. Much early work on crystal growth centered on explaining the difference in observed crystal habits from an equilibrium or thermodynamic point of view. Of more interest to practitioners of industrial crystallization are theories that deal with the kinetics (rate) of crystal growth. Reviews of crystal growth theories can be found in the work of Ohara and Reid (1973), Strickland-Constable (1968), Nyvlt et al. (1985), Levi and Kotrla (1997), Irisawa (2003), and Rudolph (2015). Many of the theories are mathematically rather complex, but there are certain features of these theories that are worth reviewing and are quite helpful in understanding the nature of the crystal growth process. When discussing crystal growth, it is necessary to focus on a single face or plane and examine the growth of that face. The

42

Figure 2.14 The structure of a surface of a growing crystal

linear growth velocity of a face is usually defined as the rate of growth of a face in the direction normal to the face. Crystals are thought to grow in a layer-by-layer fashion. This can be understood by examining Figure 2.14. A molecule in solution must desolvate and adsorb on the crystal surface. Looking at Figure 2.14, we can see three possible sites for the molecule to incorporate into the crystal surface. Sites A, B, and C can be distinguished by the number of bonds the molecule will form with the crystal. At site A, the molecule will be attached only to the surface of a growing layer, whereas at site B, the molecule is attached to both the surface and a growing step. At site C, the molecule is attached at three surfaces at what is known as a kink site. From an energetic point of view, C is more favorable than B, and B more favorable than A. This can be generalized by saying that molecules tend to bond at locations where they have the maximum number of nearest neighbors. These are the most energetically favorable sites. The general mechanism for incorporation of a molecule into a crystal face is its adsorption onto the surface followed by its diffusion along the surface to a step (B-type) or kink (C-type) site for incorporation. From this explanation you can see why crystals grow in a layer-by-layer fashion because it is easier for molecules to bond to an existing step that is spreading over a surface than to form a new one. The linear growth rate G of a face therefore can be described in terms of the step velocity V∞, the step height h, and the step spacing γ0: G ¼ V∞ h=γ0

ð2:3Þ

Equation (2.3) provides a simple illustration of the growth of a face, but it does not help us with two fundamental questions: (1) where do the steps come from, and (2) what is the ratecontrolling factor in determining the crystal growth rate? The goal of crystal growth theories is to try to answer these two questions.

Two-Dimensional Growth Theories One way to describe the birth of a step is to use what is called two-dimensional nucleation theory. Nucleation (which is discussed in Chapter 3) requires the formation of a critical size cluster of molecules. When this critical size is reached, the Gibbs free energy favors growth of the cluster; at sizes below the critical size, the cluster dissolves. This also extends to the nucleation of a two-dimensional circular nucleus on a flat surface. Molecules will be continually adsorbing on the surface, diffusing, and desorbing. They also will collide with each other and form two-dimensional aggregates, as illustrated in Figure 2.15.

Crystals and Crystal Growth

Figure 2.15 Formation of a two-dimensional critical nucleus on a crystal surface

An application of nucleation theory in two dimensions yields the expression for the critical size: rc ¼ σVm =kT lnS

I ¼ C1 ½lnðSÞ1=2 exp½C2 =T 2 lnðSÞ C2 ¼ πhσ Vm =k 2

2

G ¼ hAI

ð2:8Þ

G ¼ hAC1 ½lnðSÞ1=2 exp½C2 =T 2 lnðSÞ

ð2:9Þ

ð2:4Þ

where rc is the radius of a critical size cluster, σ is the surface energy, k is the Boltzmann constant, T is the temperature, S is the supersaturation, and Vm is the volume of a molecule. Equation (2.4) depends on knowledge of the surface energy, which can be difficult to obtain. Figure 2.16 gives the critical radii as a function of supersaturation at three different values of the surface energy for the KCl–water system. Employing nucleation theory, it is possible to derive an expression for the rate of formation of critical-sized nuclei per unit surface area per unit time. A complete derivation can be found in Ohara and Reid (1973). A simplified version of the rate expression is

C1 ¼ ð2=πÞn2 vðVm =hÞ

Figure 2.16 Two-dimensional critical radii estimated for KCl at 300 K as a function of supersaturation

1=2

ð2:5Þ ð2:6Þ ð2:7Þ

where I is the two-dimensional nucleation rate, n is the equilibrium number of molecules or monomers in the solution, and v is the speed of surface-adsorbed molecules. Because these values, along with the surface energy, are rarely known, C1 and C2 are often used as empirical parameters obtained from experimental data. As you would expect, Equation (2.5) shows that the rate of two-dimensional nuclei formation is a strong function of supersaturation and temperature. Once a surface nucleus is formed, the next question is how does the nucleus spread to form a complete layer. The simplest crystal growth theory assumes that when a surface nucleus is formed, it spreads across the surface at an infinite velocity. The surface must then await the formation of another surface nucleus. Because the rate-determining step in this model is the formation of a surface nucleus, the growth rate of the crystal can be expressed as

or, substituting for I,

Equation (2.9) predicts that the growth rate of a face is proportional to the area of that face. This model is known as the mononuclear model. If we were to assume that the twodimensional nucleus does not spread at all when it forms, a layer would be formed by the formation of enough two-dimensional nuclei of the critical size to cover the layer. The growth rate expression therefore would be G ¼ Iπrc2 h

ð2:10Þ

or G ¼ fC3 =T 2 ½lnðSÞ3=2 gexp½C2 =T 2 lnðSÞ

ð2:11Þ

where C3 is an empirical parameter obtained from experimental data. This model is known as the polynuclear model. Between the extremes of the mononuclear model, in which the spreading velocity is infinite, and the polynuclear model, in which it is zero, is a model referred to as the birth and spread model. The birth and spread model allows the spreading of nuclei at a finite constant rate that is assumed to be independent of size. It also assumes that nuclei can form at any location, including incomplete layers, and that there is no intergrowth between the nuclei. A growth rate expression for this model is derived by Ohara and Reid (1973) and is n o G ¼ C4 ðS  1Þ2=3 ½lnðSÞ1=6 exp½C5 =T 2 lnðSÞ ð2:12Þ where C4 and C5 are empirical parameters obtained from experimental data. A number of modifications of this model have been developed and reported in the literature (Bennema

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

et al. 1973; Garside et al. 1975; Van Rosmalen and Bennema 1975). If we examine these three models, each one makes some prediction about how crystals grow as function of supersaturation, temperature, or face area. In the case of the mononuclear model, Equation (2.8) tells us that the crystal growth rate of a face is directly proportional to the area of that face. This would indicate that large faces grow faster than small faces, which contradicts the observation that the fastest growing faces on a crystal have the smallest areas, whereas the slowest growing faces have the largest areas. This essentially eliminates the mononuclear model from serious consideration as a useful model. The polynuclear model [Equation (2.11)] predicts that the growth rate will increase with increasing nucleation rate, but the growth rate will decline with decreasing critical nuclei size. The critical nuclei size declines as the supersaturation increases, making the growth rate a complex function of the supersaturation ratio S. Equation (2.11) also predicts that the growth rate should display a maximum in G at some supersaturation. This means that the polynuclear model does not predict that crystal growth rate increases continuously with supersaturation; instead, it predicts that at some supersaturation, growth is a maximum that will decline if the supersaturation is increased or decreased. This result seems unlikely and has not been observed. The birth and spread model and its modifications predict that the growth rate G increases with increasing supersaturation and increasing temperature. The dependence of growth rate on these variables is not a simple function of supersaturation, and the model does not have the obvious problems of the other two models. For this reason, some semi-empirical relations obtained from Equation (2.12) are sometimes used to correlate experimental growth data and obtain the needed constants. Another difficulty with all models that depend on twodimensional surface nucleation is that unless a very low value for surface energy is used in Equation (2.5), each of the models fails at low supersaturation, predicting a much lower growth rate than is observed experimentally. The models discussed in this section are useful in developing some understanding of possible crystal growth mechanisms and are quite illustrative in that regard. It is not recommended, however, that these models be used in any predictive sense, although empirical equations obtained by reducing Equations (2.9), (2.11), and (2.12) are sometimes used.

Burton–Cabrera–Frank (BCF) Model The models discussed in the preceding subsection all require two-dimensional nucleation events for a new layer to start. These models fail to account for observed crystal growth rates at low supersaturations and are unsatisfying in the sense that they make crystal growth a non-continuous process with the formation of a critical size two-dimensional nucleus the rate-determining step. A basis for a model in which the steps are self-perpetrating was put forward by Frank (1949). Frank’s idea was that dislocations

44

Figure 2.17 Development of a growth spiral from a screw dislocation

in the crystal are the source of new steps and that a type of dislocation known as a screw dislocation could provide a way for the steps to grow continuously. A simple example of the growth of a screw dislocation is given in Figure 2.17. Molecules adsorb on the crystal surface and diffuse to the top step of the two planes of the screw dislocation. The surface becomes a spiral staircase (which can be right or left handed depending on the screw dislocation). After a layer is complete, the dislocation is still present; it is just a layer higher. The appeal of Frank’s idea was that surface nucleation was not required for growth and that growth could occur at a finite rate at low supersaturations. Burton et al. (1951) formularized this concept in a growth model in which surface diffusion was assumed to be the determining rate. Descriptions and derivations of the Burton–Cabrera– Frank (BCF) growth equation can be found in Nyvlt et al. (1985) and Ohara and Reid (1973). The resulting kinetic expression is G ¼ K1 TðS  1ÞlnðSÞtan h½K2 =TlnðSÞ

ð2:13Þ

While Equation (2.13) has system constants (K1 and K2), which again are difficult to calculate or obtain from experimental data, it does provide significant information. At low supersaturations, the equation reduces to a form in which the growth is proportional to the supersaturation to the second power, whereas at higher supersaturations, growth is linear with supersaturation. The BCF theory tells us that crystal growth rates vary from a parabolic dependence on supersaturation to a linear dependence as the supersaturation increases. The model we have just described is called the BCF surface diffusion model because diffusion on the crystal surface is considered to be the rate-controlling step. While this is true in vapor growth, it is often not true in solution growth, where diffusion from the bulk solution to the crystal–liquid interface can often be the rate-limiting step. Crystal growth models based on the idea of the self-perpetuating growth spiral of the BCF theory have been developed for the situation in which bulk diffusion is limiting. The best known of these models is that of Chernov (1961). In this model, diffusion of solute molecules through a boundary layer is the rate-determining step. Details of the derivation of the model can be found in Ohara and Reid (1973) and Nyvlt et al. (1985). The resulting growth equation is

Crystals and Crystal Growth

G¼

Vm ξC ðS  1Þ2 h   Scr δ 1 þ ðξh=DÞlnfScr δ=½ðS  1Þhsin h½ðS  1Þ=Scr g ð2:14Þ

where ξ is a coefficient that depends on kink density, δ is the boundary layer thickness, and Scr is a parameter defined by Scr ¼ 4Vm σ=kTδ

ð2:15Þ

This parameter (Scr) is a dimensionless group that is sometimes called the critical transition supersaturation because when (S – 1) is much less than Scr, Equation (2.14) reduces to the following relationship: G ∝ ðS  1Þ2 =½1 þ k lnðδ=hÞ

ð2:16Þ

where k ¼ ξh=D. Equation (2.16) shows that at low relative supersaturations, the growth rate follows a parabolic relation with supersaturation. This is the same result as the BCF surface diffusion model. In addition, however, the growth rate G decreases with an increase in δ, the boundary layer thickness. This is an important result because the boundary layer thickness is directly related to hydrodynamic conditions and stirring rates. If (S – 1) is much greater than Scr but much less than 1, Equation (2.15) results in a relationship in which G increases linearly with supersaturation and declines with increasing δ. The Chernov bulk diffusion model provides an important link between crystal growth theory and the practical world of industrial crystallization, where fluid flow and agitation are important. The effect of hydrodynamics on crystal growth will be discussed in the next subsection. We have presented two models that make use of the selfperpetuating growth spiral, the BCF surface diffusion model, and the bulk diffusion model of Chernov. While there are other models of this type, these two are the best known and most useful. In the surface diffusion model, the rate-limiting step is the diffusion of a molecule on the crystal surface to a step. In the bulk diffusion model, it is the diffusion of molecules from the bulk solution to the crystal–liquid interface. A more general model employing the BCF growth mechanism combines surface and bulk diffusion and considers these effects in parallel or series on the crystal growth rate. These models are mathematically complex and are described in detail in the literature (Bennema 1969; Gilmer et al. 1971). One important result that is predicted by these types of models is that as the relative velocity between a crystal and the solution is increased, the growth will increase to a maximum value and then will remain the same. This maximum value is the value obtained when only surface diffusion limits growth. In the literature, this is known as a growth limited by interfacial attachment kinetics. When the crystal growth rate can be changed by changing the hydrodynamic conditions, it is known as a mass transfer limited growth.

Figure 2.18 Schematic representation of the concentration profile near a growing crystal

The Diffusion Layer Model The models of crystal growth discussed to this point have been mathematically complex and focused on the propagation of steps on the crystal surface either by two-dimensional nucleation or by the screw dislocation of the BCF theory. As we have seen in the bulk diffusion model of Chernov, the diffusion of the solute in the boundary layer and the boundary layer thickness can play a significant role in controlling the crystal growth rate. A simple model that focuses on the diffusion of solute through the boundary layer is known as the diffusion layer model. Generally, it is this model that is used in correlating data for industrial crystallization processes. When a crystal is growing from a supersaturated solution, solute is leaving the solution at the crystal–liquid interface and becoming part of the crystal. This will deplete the solute concentration in the region of the crystal–liquid interface. Because the concentration of the solute is greater as you go away from the interface, solute will diffuse toward the crystal surface. The concentration of the solute will continuously increase from the value at the interface to the value in the bulk solution. The region in which the concentration is changing is called the concentration boundary layer (there is also a momentum and thermal boundary layer). The distance from the crystal surface to the region where the concentration is that of the bulk is called the boundary layer thickness. This is illustrated in Figure 2.18. The basis of the diffusion layer model is that solute diffuses through the boundary layer and is then incorporated into the crystal. For the single one-dimensional case shown in Figure 2.18, the rate of mass increase of the crystal can be equated to the diffusion rate through the boundary layer by the expression
 dmc dC ¼ DA ð2:17Þ dx dt where A is the surface area of the crystal, and D is the diffusion coefficient. The concentration versus position through the boundary layer can be written as

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

dC C  Ci ¼ dx δ

ð2:18Þ

where δ is the boundary layer thickness, and C and Ci are the bulk and interfacial concentrations, respectively. Substituting Equation (2.18) into Equation (2.17) yields dmc ¼ kd AðC  Ci Þ dt

ð2:19Þ

where kd ¼ D=δ. The rate of solute integration into the crystal surface can be approximated by the relation dmc ¼ ki AðCi  CÞi dt

ð2:20Þ

where i is between 1 and 2, and ki is a rate constant. If i = 1, then Equations (2.19) and (2.20) can be combined to eliminate the interfacial concentration, which is difficult to obtain. This yields dmc ¼ KG ADC dt where

1 1 1 ¼ þ KG kd ki

ð2:22Þ

ð2:24Þ

where g is between 1 and 2. When i in Equation (2.20) is a value between 1 and 2, an explicit expression such as Equation (2.21) or Equation (2.23) cannot be obtained. However, Equations (2.21) and (2.24) are generally used in industrial crystallization processes. The constants are normally obtained from experimental data. This will be discussed in more detail in Section 2.4.3 on crystal growth kinetics. Garside et al. (1976) defined an effectiveness factor for crystal growth that is a measure of the relative importance of diffusion and surface integration as the rate-controlling factors. The effectiveness factor is defined by the relation measured growth rate RG ¼ ð2:25Þ rate with negligible diffusion resistance ki DCi

Because η → 1, diffusion plays a smaller and smaller role in controlling the rate. Methods for estimation of η as well as

46

The development and operation of industrial crystallization processes can be made significantly easier if some data on the kinetics of crystal growth are available. This information can be incorporated in process models, be used in process and crystallizer design, and can shed light on the observed behavior of the system. In Section 2.4.2 we reviewed a number of different crystal growth theories. These provide a theoretical basis for the correlation of experimental crystal growth data and the determination of kinetic parameters from the data to be used in models of industrial crystallization processes. In general, two basic expressions are used to describe the relationship between supersaturation and crystal growth. These are

and

Equation (2.23) is a rather complex expression relating the crystal growth rate, supersaturation, and the two constants kd and ki. Normally, this equation is approximated by the simple relation

η¼

2.4.3 Crystal Growth Kinetics

ð2:21Þ

Looking at Equations (2.21) and (2.22), we see that when kd ≪ ki, the crystal growth rate will be diffusion controlled and Kg = kd. When ki ≪ kd, the crystal growth rate will be controlled by the rate of solute incorporation into the crystal. If i = 2 in Equation (2.20), combination with Equation (2.19) yields (Nyvlt et al. 1985) "
1=2 # dmc 1 2ki 4ki ¼A 1þ DC  DC þ 1 ð2:23Þ 2ki =k2d dt kd kd

dmc ¼ KG ADC g dt

other methods to determine the relative importance of diffusion and surface integration are discussed in Nyvlt et al. (1985).

G ¼ kg DCg

ð2:26Þ

RG ¼ Kg DCg

ð2:27Þ

Equation (2.26) employs a linear crystal growth velocity (length/time), and Equation (2.27) employs a mass rate of crystal growth (mass/area time). The constants in Equations (2.26) and (2.27) can be related to each other through the expression α Kg ¼ 3 ρkg β

ð2:28Þ

Typical units for the growth constants are m h ig ðkg soluteÞ s ðkg solventÞ for kg and

sm2

h

kg

ig

ðkg soluteÞ ðkg solventÞ

for Kg. The power g in the growth equations does not depend on the form of the equation used and is normally a number between 1 and 2. The constants kg and Kg are temperature dependent and are usually fitted to the Arrhenius equation to obtain a general expression for growth rate as a function of temperature. The Arrhenius equation can be written as kg ¼ AexpðEG =RTÞ

ð2:29Þ

where A is a constant, and EG is an activation energy. The activation energy can be used to obtain information of whether the rate-controlling step is diffusion or surface integration (Lefever 1971; Nyvlt et al. 1985). A complete crystal growth expression that includes both the effect of temperature and supersaturation on the growth rate therefore would be written as

Crystals and Crystal Growth

G ¼ AexpðEG =RTÞDCg

ð2:30Þ

A number of experimental methods can be used to obtain the crystal growth rate data needed to obtain kinetics. Unfortunately, unless great care is taken, these methods can provide very different results for the same system at the same supersaturation. We will first review the measurement techniques and then describe the various problems and challenges that may be encountered.

Measurement of Crystal Growth Rates Techniques used to measure crystal growth rates can be divided into two main groups. The first group is comprised of methods that rely on the growth of a single crystal to obtain the needed data. The second set of methods involves the growth of a suspension of crystals. Discussion of experimental methods can be found in a number of references (Laudise 1970; Nyvlt et al. 1985; Randolph and Larson 1986; Mullin 1993). In experiments in which single crystals are grown, the goal is to allow the crystal to grow at a known supersaturation without any nucleation occurring. This implies, therefore, that the supersaturations used in these experiments must be small. In addition, seed crystals must be prepared that can be used in the experiments. A common method to prepare seed crystals is to make a saturated solution at some temperature above room temperature and then allow the solution to cool without stirring. The resulting crystals are then examined under a microscope, and the ones of the desired size showing discernible faces are used. In some systems this method produces very large crystals. In those cases, gentle agitation will reduce the size obtained. As we have seen in our discussion of the diffusion layer theory of crystal growth, the rate of crystal growth is affected by the boundary layer thickness. The boundary layer thickness is, in turn, affected by the relative velocity between the crystal and the solution. In order to prevent diffusion from controlling the crystal growth rates in single-crystal experiments, all methods employ some means to create a velocity between the growing crystal and the solution. A simple and commonly used experimental apparatus is shown in Figure 2.19. This apparatus consists of a round bottom flask, usually of volume of 500–1000 cm3, to which glass tubing (2.5 cm diameter) has been added to form a loop. If a stirrer is placed in the tube just below the bottom of the flask, it will push the solution down, resulting in flow in a clockwise direction. The rate of flow through the tube will increase with the speed of the impeller. The crystal is attached to a rod and placed in the path of the flow. The entire apparatus is normally placed in a constant-temperature bath so that the temperature can be controlled. Because the volume of the solution is large compared with the mass of solute that will deposit on the single growing crystal, these experiments can be assumed to occur at a constant supersaturation. The crystal is normally allowed to grow for a period of hours. The growth rate of the crystal can be measured in several ways. The simplest method is to weigh the seed crystal at the beginning of the experiment and the product crystal at the end of the experiment. The linear growth rates can also be

Figure 2.19 Recirculation apparatus for single crystal growth

obtained this way by measuring the crystal dimensions before and after the experiment through a microscope. By use of a traveling microscope or cathetometer, the growing crystal can be observed during the experiment, and the growth of a particular dimension or face can be observed. This method has been used by a number of investigators for a variety of substances (Botsaris and Denk 1970; Clontz et al. 1972; Slaminko and Myerson 1981). One of the most difficult aspects of this method is the attachment of the seed crystal to the rod. This can be accomplished by heating a glass rod tip and attaching the crystal to the molten glass. Other methods include using a heated metal rod and inserting the heated metal directly into the crystal. The use of epoxy and silicon rubber to attach the crystal has also been reported. An alternate method that is quite similar to the circulation method and is described in the literature (Myerson and Kirwan 1977) is shown in Figure 2.20. In this method, solution is pumped past a crystal that is suspended from a rod in a chamber. The flow rate can be directly measured by a rotameter, which is not possible in the circulation apparatus. As mentioned previously, the mass or change in dimension of the crystal can be measured at the beginning and end of the experiment or can be observed directly by using a microscope. This method is more difficult to use for several reasons. First, pumping of the solution tends to increase the chance of nucleation, as does the presence of the various tubes and connections. Second, the lines must be kept at the desired temperature or material will crystallize in the lines. A third single-crystal growth method involves rotating the crystal instead of moving the liquid. The simplest way this is done is by attaching a seed crystal to a rod that is inserted into a variable-speed agitator. A more complex sample preparation method known as the rotating-disk method involves fixing the

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

Figure 2.20 Flow apparatus for single crystal growth

Figure 2.21 Effect of flow rate on the growth rate of a single crystal of potassium aluminum sulfate dodecahydrate

crystal in a Teflon epoxy disk with a particular face exposed. The advantages of the rotating disk are the well-defined geometry and hydrodynamics. More details on this method can be found in Nyvlt et al. (1985). Determining the crystal growth kinetics of a particular material using one of the single-crystal growth methods just described involves conducting a series of experiments at different supersaturations (at a given temperature). The resulting data at each supersaturation are used with Equation (2.26) or Equation (2.27) to obtain the parameter kg (or Kg) and g. Normally, this means that the natural log (ln) of G is plotted versus the ln of ΔC, and a least squares fit is performed. The slope of this line will be g and the intercept ln kg. The entire process can then be repeated at a different temperature if desired. It is important to conduct these experiments at welldefined flow or stirring rates. Usually in measurements of this type it is desired to minimize the effect of bulk diffusion so that surface integration is the rate-controlling step. This is normally done in practice by measuring the growth rate as a function of flow rate (or stirring speed) at a given supersaturation. If mass transfer is controlling, the growth rate will increase with increasing flow rate. This is continued until a point is reached

48

where no observable increase in growth rate is noted with increasing flow rate. A flow rate in this range is then used for all additional experiments. An example of such data is shown in Figure 2.21 for potassium aluminum sulfate dodecahydrate (potassium alum). Looking at Figure 2.21, we see that above a flow rate of about 2300 cm3/min, the growth rate does not increase. It is important to note that in our discussion of crystal growth to this point we have assumed that the size of the crystal does not affect its growth rate. This is often known as a McCabe ΔL law (McCabe 1929). This is a good first assumption, but it is not necessarily true. We will discuss size-dependent growth and related issues in Section 2.4.5 of this chapter. The methods for measurement of crystal growth kinetics discussed to this point all involve the growth of a single seed crystal. This is clearly different from the environment in an industrial crystallizer, where many crystals are growing in a suspension. A laboratory method known as a fluidized-bed crystallizer, which attempts to measure the growth rates of a number of crystals, is shown in Figure 2.22. In this method, solution is circulated through a crystallizer in which seed crystals are suspended by the flowing fluid. The flow is such that the crystal will not settle and will not flow out of the crystallization vessel. As in single-crystal growth experiments, the mass of solute depositing on the growing crystals must be small enough so that the system is considered to be at constant supersaturation. By using seed crystals of the same initial size, the final size and mass of the crystals can be used to obtain an average growth rate. More details on the method and the operation can be found in the literature (Mullin et al. 1966; Mullin and Gaska 1969). The most common method for obtaining crystal growth kinetics involves the use of a mixed suspension, mixed product removal continuous crystallizer operating at steady state. By using the population balance concepts developed and described by Randolph and Larson (1986), growth rates can be obtained. The population balance method and use of the crystal size distribution in obtaining kinetic parameters will be discussed in detail in Chapter 6.

Estimation of Crystal Growth Kinetics The techniques for crystal growth measurement discussed in the preceding subsection all involved direct measurement of the change in mass or size of a crystal (or crystals) at a fixed temperature and supersaturation. To obtain kinetic parameters, these experiments are repeated at several different supersaturations at each temperature of interest and then fit to a power law model given by Equation (2.26) or Equation (2.27). In the mixed suspension, mixed product removal (MSMPR) method, which will be described in detail in Chapter 6, a continuous crystallizer must be operated at steady state to obtain the crystal size distribution and the growth rate. To obtain kinetics, this must be repeated at several different supersaturations. All the direct measurement techniques are time consuming and require a significant number of experiments to obtain sufficient data to obtain kinetic parameters. This has led a number of investigators (Garside et al. 1982; Tavare and

Crystals and Crystal Growth

zero. It is assumed in this method that the concentration change is due only to crystal growth with no nucleation occurring. Using this analysis, the power g and constant kg of Equation (2.27) can be obtained from the relations :: 2FDC0 DC0 DC 0 þ ð2:31Þ g¼ 2 3ρL 0 At0 DĊ 0 Kg ¼

Figure 2.22 Fluidized-bed crystallizer: glass tube (1), stock vessel (2), cooler (3), thermostat (4), centrifugal pump (5), infrared lamp (6), power source (7), contact thermometer (8), mercury thermometer (9), and stirrer (10)

Garside 1986; Qiu and Rasmussen 1990) to look at indirect methods for the estimation of both growth and nucleation kinetics. Most of the indirect methods are based on measurement of the solution concentration versus time in a seeded isothermal batch experiment. This is often called the desupersaturation curve because the concentration and solubility can be used to calculate the supersaturation of the system versus time. The procedure using this type of experiment involves preparation of a saturated solution of known concentration in a batch crystallizer or other vessel where the temperature can be controlled. The vessel should be equipped with stirring. With the stirrer on, the temperature of the solution is reduced by several degrees, creating a supersaturated solution. The solution must remain clear with no crystals present at the lower temperature. Seed crystals of the solute of known mass and size (or size distribution) are added to the solution. Because the solution is supersaturated, these crystals will grow, causing a depletion in the concentration of the solution that is measured as a function of time. This is done by taking samples and using one of a number of concentration measurement techniques. An illustration of an apparatus to obtain desupersaturation curves online via density measurements is shown in Figure 2.23. Solution is continuously pumped from the crystallizer to a ceramic cross-flow filter. Clear solution is obtained in a small side stream that flows through the density meter. All solution and crystals are returned to the crystallizer. Readings from the density meter are sent directly to a computer. Typical results obtained from this system for potassium aluminum sulfate dodecahydrate are shown in Figure 2.24. A number of methods can be used to estimate growth kinetics from the de-supersaturation curve obtained during batch-seeded isothermal experiments. The simplest of these methods was developed by Garside et al. (1982) and involves using the derivatives of the de-supersaturation curve at time

ΔĊ 0 g At0 DC0

ð2:32Þ

where ΔC0 = supersaturation at time zero :: 2 DĊ20 and DC 0 = first and second derivatives of the de-supersaturation curve at time zero, respectively F = shape-factor ratio (β/α) α = volume shape factor β = area shape factor L 0 = average size of the seeds At0 = surface area of the seeds at time zero :: The derivatives DĊ0 and DC 0 are usually obtained by fitting the de-supersaturation to a polynomial of the form DC ¼ a0 þ a1 t þ a2 t 2 ð2:33Þ :: so that ΔC0 = a0, DĊ0 = a1, and DC 0 = 2a2. While simple, results obtained from this method are quite sensitive to measurement errors because the method relies on derivatives. More sophisticated techniques for the estimation of growth kinetics involve the use of the entire de-supersaturation curve with parameter estimation techniques (Qiu and Rasmussen 1990). The combination of the de-supersaturation curve and the crystal size distribution can be used to estimate both growth and nucleation kinetics (Tavare and Garside 1986; Tavare 1995). While these methods do not supply very accurate kinetic data, the experiments are fast and the data are easy to obtain, thus allowing for relatively quick estimation of kinetic parameters that can be used for process design and modeling purposes.

2.4.4 Ostwald Ripening As we saw in Chapter 1, the solubility of a solid solute in a liquid solution is an equilibrium property of the solution. When a crystalline solid with a particular crystal size distribution (CSD) is in contact with a saturated solution, however, the CSD can change with time. This is the result of the system trying to minimize its Gibbs free energy. This can be seen by looking at the following equation: X dG ¼ SdT  VdP þ μi dni þ σ s dA ð2:34Þ where μi is the chemical potential of species, and σs is the specific surface energy of the solid particle. Because the system will try to minimize G, the surface area A will try to achieve a minimum. What this means in practice is that particles have different solubilities based on their size when present in a suspension of

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

The equilibrium solubility refers to the solubility that would be obtained with very large particles (or small surface areas). As in all thermodynamic calculations, activity coefficients should be used in nonideal solutions. If the crystal growth process accompanying the aging phenomenon is diffusion controlled, it can be shown (Nyvlt et al. 1985) that the size of particle in equilibrium with the solution can be obtained from the relation rðcÞ ¼

Figure 2.23 Apparatus to measure the de-supersaturation curve on line

βVσ s c ðc  c ÞkT

ð2:36Þ

Particles larger than r(c) will grow and particles smaller than r(c) will dissolve. Ostwald ripening is often important in processes in which crystallization is rapid and crystal sizes are small. This is especially true in a precipitation process and will be discussed in detail in Chapter 8. It is important to remember that the effect of Ostwald ripening is to alter the crystal size distribution with time in a suspension of crystals that is an apparent equilibrium with its saturated solution. Ostwald ripening is an important phenomenon if you are concerned with obtaining fine particles or about changes in the crystal size distribution of your product prior to isolation and drying.

2.4.5 Size-Dependent Growth and Growth Rate Dispersion

Figure 2.24 De-supersaturation curve

particles with a size distribution. This difference in solubility results in the small particles dissolving and depositing on the larger particles, decreasing the surface area A and moving the system toward a minimum in Gibbs free energy. It also increases the average crystal size. This process is called Ostwald ripening. The effect of size on particle solubility can be shown using the Gibbs–Thomson (Ostwald–Freundlich) equation ln

cðrÞ ¼ βVσ s =rkT c

where c(r) = solubility of a particle of size r c* = equilibrium solubility β = surface shape factor r = particle radius V = molecular volume

50

ð2:35Þ

In our discussions of crystal growth theory and kinetics, we have assumed that the crystal growth rate is not a function of crystal size. While a good first assumption, this is not always true. As we saw in Section 2.4.4, there is a difference in solubility between very small particles and larger particles. This differential solubility will result in a difference in supersaturation as a function of particle size. The smaller particles will be at a lower supersaturation than larger particles and will therefore not grow as fast. This is an example of size-dependent growth, but this mechanism is important only when the crystal size is very small (250 µm) tend to be free flowing, whereas fine powder with high surface-area-to-mass ratios becomes cohesive and tends to stick, especially particles that are less than 10 µm, because the interparticulate cohesive forces arising from van der Waals attraction become more dominant and the gravitational forces are less influential. In pharmaceutical solids, the particle size often impacts the performance of the solid dosage forms. For oral administration of water-insoluble drugs, the particle size is closely linked with the dissolution rate and solubility. According to the Noyes– Whitney equation, the dissolution rate of a solute is described by

Electron microscopy

0.001–5

Laser diffraction (Fraunhofer diffraction)

1–1000

Light microscopy

0.5–150

Photon correlation spectroscopy

0.001–1

Sedimentation

0.01–50

Sieving

>20

dw DA  ¼ ðc  cÞ dt δ

ð2:42Þ

where dw/dt is the rate of dissolution, D is the diffusion coefficient of the dissolved solute, δ is the effective boundary layer, A is the specific area of the particle, c* is the saturation solubility of the solute, and c is the transient concentration of the solute. A reduction in the size of the drug particles leads to a dramatic increase in the surface-area-to-volume ratio, in turn enhances the dissolution rate as a result of an increase in the mass transfer rates (Horn and Rieger 2001). For instance, if the size of spherical-shaped particles is reduced from 10 µm to 200 nm, there would be a 50-fold increase in the surfacearea-to-volume ratio, which can significantly improve the performance and bioavailability of a poorly water-soluble drug. Furthermore, a reduction in the particle size can enhance the equilibrium solubility. This is best described by the Ostwald–Freundlich equation (2.35), where the apparent solubility increases exponentially as the particle radius decreases. The effect on solubility is quite significant for submicron particles. As a consequence of the highersurface-area particles, the drug absorption and bioavailability are affected. In the case of danazol, a water-insoluble drug (10 µg/ml), the bioavailability improved when the crystal size was reduced to less than 200 nm (Liversidge and Cundy 1995). The crystal size is typically characterized by a range of sizes best described by a size distribution curve. Most often the distribution is expressed either by the volume or the mass of the crystals. Common methods to measure particle size are described in Table 2.8 (Wedd et al. 1993). A crystal size distribution that is symmetric is a normal distribution in which the mean, median, and mode are equivalent. Often

68

the case, the size distribution is skewed or asymmetric, and the mean, median, and mode are different. Depending on the skewness, there would be either an increased proportion of fines or larger crystals. Given the varying rates of nucleation, growth, and agglomeration, it is not unexpected to encounter a bimodal distribution – appearance of a second mode or maxima in the size distribution. In the end product, the second maxima may also be a consequence of particle attrition that occurred during the isolation and drying processes. To achieve large crystal size, the most common approach is a seeded crystallization process in which a certain amount of uniform seed crystals is introduced to a supersaturated solution. Crystallization ensues at low supersaturation levels within the metastable zone width to avoid unwanted nucleation and for growth of the seed crystals to occur. For cooling and antisolvent addition crystallization processes, cubic cooling and addition profiles are occasionally practiced, respectively (Kim et al. 2005) to ensure that low supersaturations are maintained. Typically, the seeds should be small and have a narrow size distribution. Most often the seeds are milled prior to use. The seed loading and size are important factors that dictate the final crystal size. Assuming that secondary nucleation is suppressed, the size of the grown seed crystals can be described by
 1 þ Cs 1=3 Lsp ¼ Ls Cs

ð2:43Þ

where Lsp is the theoretical final crystal size (of the grown seeds), Ls is the initial size of the seeds, and Cs is the seed mass or concentration. If the seed loading is not sufficient, secondary nucleation may arise, and the overall size distribution may be dispersed. In this event, a large seed mass is needed, which may come at the expense of productivity. The effects of seeding in particular seed loading and seed size or surface area have been investigated in batch cooling crystallization (Kubota et al. 2001; Lung-Somarriba et al. 2004). Control of supersaturation is critical in a seeded crystallization process and must be regulated to eliminate spontaneous nucleation and to maximize crystal growth. Integration

Crystals and Crystal Growth

of crystallization kinetics modeling (as described in Chapter 6) and process analytical technology (as described in Chapter 11) can be valuable to help better understand the role of nucleation and crystal growth over the final crystal size distribution. As described earlier, temperature cycling is an alternative approach to improve particle size, especially if small particles (or fines) are present. Taking advantage of Ostwald ripening, heat-cool cycles effectively remove fines and increase the size of larger crystals as an outcome of the higher stability of the larger particles. For the production of micro- and nanoparticles, crystallization-based methods such as impinging jet crystallization, supercritical fluid crystallization, solvent shifting, ultrasound, spray drying, and confined crystallization are employed on occasion. With the exception of sonocrystallization and confined crystallization, all the bottom-up methods revolve around the creation of high supersaturation, which favors nucleation (over crystal growth) and fine particles formation. Impinging jet crystallization comprises two jet streams opposite one another, where the formation of micron-sized particles is a result of the impingement of these two reactant streams: one contains the solute and solvent, and the other contains the antisolvent. Rapid, intense mixing and high supersaturation lead to the fast precipitation of microcrystals because crystal nucleation is favored over growth. The rapid and homogeneous mixing can be controlled by the flow rate of each stream. Midler et al. (1994) were the first to crystallize pharmaceutical particles in the range of 5–20 µm using miscible organic solvents. It was also reported that crystal purity was enhanced as a result of the process. Later, Lindrud et al. (2001) patented the process where a sonic probe is positioned near the two impinging streams, and sonication at the impingement point led to the production of micro- and nanocrystals. It was also claimed that the mixing process was enhanced by ultrasound. The process of using colloidal stabilizers in the reactant streams of the turbulent jet mixers has also been developed. Johnson and Prud’homme (2003) termed the process flash nanoprecipitation and showed that particle size can be tuned from 80 nm to 1 µm and that the amphiphilic diblock copolymers stabilize the particles as well as alter the nucleation and growth of the organic molecules. Supercritical fluid crystallization (SCF) is another technique that is employed to produce micron and submicron particles with a narrow size distribution. Given its “green” characteristics, carbon dioxide is often used as the fluid. The two most common processes in SFC technology used to generate fine particles are rapid expansion of supercritical solutions (RESS) and supercritical antisolvent (SAS). In the former process, the solute is dissolved in the SCF, and fine particles are created when the solution is depressurized across the capillary nozzle into ambient air (Debenedetti et al. 1993). RESS has been used to prepare small particles of organic and inorganic molecules. As an alternative to ambient air, the

supercritical mixture can be rapidly expanded into a liquid solvent. Pathak et al. (2004) termed this process rapid expansion of a supercritical solution into a liquid solvent (RESOLV) and demonstrated that nanoparticles of water-insoluble drugs, specifically ibuprofen and naproxen, can be generated. In SAS, the SCF serves as an antisolvent as the solute is dissolved in a solvent. This process is also referred to as solution enhancement dispersion by supercritical fluids (SEDS) or an aerosol solvent extraction system (ASES). Fine particles for a variety of materials have been produced, including steroids, peptides, and proteins. A comprehensive review of crystal design using supercritical fluids can be found in Tom and Debenedetti (1991), Jung and Perrut (2001), and York et al. (2004). Solvent changing or solvent shifting is the process by which precipitation occurs as a result of rapid addition of an antisolvent to a saturated solution (Texter 2001). This method requires the solute to have poor solubility in an antisolvent, and the antisolvent must be miscible with the first solvent. A high degree of supersaturation is generated instantaneously with the addition of the antisolvent as the solubility is dramatically reduced, resulting in fast crystallization (Mahajan and Kirwan 1993). Crystal nucleation is favored because there is little or no crystal growth. To avoid agglomeration or aggregation, colloidal stabilizers are sometimes employed to wet the particles and prevent Ostwald’s ripening by providing a barrier. The rapid precipitation of small particles using this approach has been demonstrated for a variety of organic systems (Violanto and Fischer 1989; Brick et al. 2003). Often it is difficult to obtain a narrow particle size distribution because nucleation and crystal growth cannot be controlled. One drawback with this rapid precipitation method is that it can increase the possibility of solvent and impurity entrapment Spray drying is the process by which micronized spherical particles are prepared by atomization of a liquid into a spray of droplets, which are then exposed to a heated atmosphere. Generally, the process produces particles with uniform size distribution. Spray drying can be used to direct polymorph selectivity through control of the drying conditions. Matsuda et al. (1984) demonstrated that for phenylbutazone, all three polymorphs can be formed from methylene chloride by changing the drying temperature. Also, high-energy metastable polymorphs have been reported to be generated from spray drying (Corrigan et al. 1983). Normally, spray drying leads to the creation of noncrystalline or amorphous materials owing to rapid solidification (Elamin et al. 1995). All four of the above-mentioned bottom-up techniques are contingent on the generation of high supersaturation. While effective in fabricating small crystals, the high levels of supersaturation can result in (1) “oiling out,” or liquid– liquid phase separation, (2) formation of amorphous material, (3) the uncontrolled nucleation of a metastable polymorph, or (4) impurity entrapment or solvent occlusion. Other crystallization methods such as the use of ultrasound

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Alfred Y. Lee, Deniz Erdemir, and Allan S. Myerson

or crystallization in confined volumes circumvent these issues. On top of being an effective method for habit modification, ultrasound-assisted crystallization can also be leveraged to create micron- and submicron-sized particles. As described in the section “Sonocrystallization” above, ultrasonic cavitation is particularly effective for inducing nucleation because it has been reported to control polymorphs in melt crystallization as well as influence the crystallization of organic and inorganic molecules (Gatumel et al. 1998; Li et al. 2003). Sonocrystallization also can eliminate the need for seeding because insonation can generate nuclei in a reproducible manner (McCausland et al. 2001). Long sonication time results in an increase in the number of nuclei formed and can be effective in generating small crystals because secondary nucleation is created and crystal growth is inhibited. Moreover, ultrasonic irradiation may result in mechanical disruption and breakage of existing crystals. Dennehy (2003) examined the control of particle size using sonocrystallization or sonofragmentation. It was observed from case studies of three different drug substances that the crystals produced were considerably smaller than those generated from conventional methods. Ultrasound can also be used in the post-crystallization process to reduce particle size. Kim et al. (2003), using a temperature oscillation protocol in combination with ultrasound, were able to decrease the particle size of a pharmaceutically active substance along with producing more uniform particles with a narrow crystal size distribution. Milling was unable to reduce the particle size because the crystal quality could have been compromised as a result of the soft texture of the crystals and the low melting point. Sonication was also reported to modify the crystal habit because needle-like crystals were reduced in size to brick-like crystals. While ultrasound may appear to be more effective than milling in obtaining the desired crystal size, a downside with using ultrasound is cavitation erosion, where surface pitting can occur as a result of cavitation bubbles collapsing unsymmetrically near the sonic probe, causing metal contamination of the crystal slurry (Dennehy 2003). Crystallization in confined volumes, such as emulsified droplets, has shown to be useful in producing nano- and microsized particles. In addition to controlling polymorphic form as described earlier, microemulsions can be used as templates to control particle size through confined crystallization (Sjostrom et al. 1994). Yano et al. (2000) observed a significant decrease in the crystal size of glycine from water–isooctane microemulsions stabilized by sodium di-2-ethylhexyl sulfosuccinate (AOT). The glycine crystals ranged in size from submicron to micron, and the crystal form that appeared from microemulsions was predominately the γ-polymorph, whereas the α-form was observed only in aqueous solution. Also, it was observed for L-phenylalanine that the size of the crystal was reduced due to the confined space. In another case of precipitation in constrained environments, crystal sizes of glycine were significantly diminished when grown in liquid lamellae of foams stabilized by leucine or α-amino octanoic acid.

70

Moreover, the crystals displayed either a pyramidal or plate-like morphology, slightly different than the bipyramidal morphology that is typically observed when nucleated from bulk solution (Chen et al. 1998). A shortcoming with this technique is that for compounds that are insoluble in both water and organic solvents, particle size control cannot be achieved. The most common method to reduce particle size is through milling. Milling involves cutting/shearing, compression, and impact or attrition of particles of a given material. Typically, fluid-energy mills (e.g., jet mills) and impact mills (e.g., hammer mills) are used. In both types of mills, high energy is required to break the large particles to micron- and submicron-sized particles. The process is quite effective in diminishing the size of the particle, but the high energy input can sometimes result in a polymorphic change or a decrease in the crystallinity of the solid crystal. For sulfathiazole, ball milling induces the solid-state phase transformation in which Form III initially converts to an amorphous solid that then eventually transforms to Form I (Shakhtshneider and Boldyrev 1993). In most cases, mechanical stresses applied to the crystals result in defects within the crystal lattice, lattice disorder, or reorganization in the lattice arrangement (Morris et al. 2001). Milling-induced phase transformations are not uncommon because there are numerous reports of the appearance of a different crystalline form or crystallinity loss after milling. Wet milling is an alternative milling approach and overcomes some of the liabilities of dry milling (Harter et al. 2013). However, the lowest limit for particle size is in the range of 10–15 µm. During dry milling, localized heating may be generated, which, in turn, can lead to some thermal degradation. In cases where milling studies are performed on a hydrate form, it is not untypical for partial dehydration to occur. As a consequence, it may lower the degree of crystallinity in the solid and consequently reduce the dehydration temperature, thereby making it easier for the removal of lattice-bound water. The high energy input plus the combination of a potential phase transformation and disruption of the crystal lattice have led to milling often being described as extremely inefficient (Parrott 1990).

2.9 Concluding Remarks Crystallization is one of the oldest unit operations used for industrial separation and chemical purification. The first human-directed crystallization is believed to have involved the crystallization of salt from seawater by evaporation in prehistoric time (Schoen et al. 1956). Since then, it is widely applied in many industries to isolate high-purity crystalline materials. The significance of crystallization science is also reflected in the number of books published on this topic in the past decade (Sangwal 2007; Cőlfen and Antonietti 2008; Tung et al. 2009; Chianese and Kramer 2012; Beckmann 2013; Lewis et al. 2015; Rudolph 2015; Duroudier 2016; Sato 2018) – nine books – which is roughly the same number of publications prior to 2002 (Nyvlt et al. 1985; Mullin 1993; Sőhnel and Garside 1992; Tavare 1995; Hartel 2001; Mersmann 2001; Davey and Garside 2001; Jones 2002; Myerson 2002).

Crystals and Crystal Growth

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Chapter

3

Crystal Nucleation Deniz Erdemir Bristol-Myers Squibb Company Alfred Y. Lee Merck & Co., Inc. Allan S. Myerson Massachusetts Institute of Technology

3.1 Introduction Crystallization from solutions is a complex process completed in several stages. The first stage is the formation of supersaturated solution because the spontaneous appearance of a new phase can occur only when a system is in a nonequilibrium condition. In the next stage, molecules dissolved in solution begin to aggregate to relieve the supersaturation and move the system toward equilibrium. The molecular aggregation process eventually leads to the formation of nuclei that can act as centers of crystallization. A nucleus can be defined as the minimum amount of a new phase capable of independent existence (Khamskii 1969). The nature of nuclei (i.e., whether they are amorphous particles or tiny crystals) is still unknown. The birth of these small nuclei in an initially metastable phase is called nucleation, which is a major mechanism of first-order phase transition. Kashchiev and van Rosmalen (2003) describe nucleation as the process of fluctuational appearance of nanoscopically small clusters of the new crystalline phase, which can grow spontaneously to macroscopic sizes. The growth stage, which immediately follows nucleation, is governed by the diffusion of particles, called growth units, to the surface of the existing nuclei and their incorporation into the structure of the crystal lattice (Khamskii 1969). This stage continues until all the solute in excess of saturation is consumed for the development of mature crystals. The initial stages of crystallization, which can be defined as the period between the achievement of supersaturation and the formation of nuclei, plays a decisive role in determining properties of the resulting solid phase, such as purity, crystal structure, and particle size. Thus higher levels of control over crystallization cannot be achieved without understanding the fundamentals of nucleation. Nucleation may occur spontaneously or be induced artificially by seeding, agitation, mechanical shock, electric and magnetic fields, and other external influences (Mullin 1997). It arises through different mechanisms, which are summarized in Figure 3.1. One of these mechanisms is the generation of nuclei directly from the previously crystal-free solution, referred as primary nucleation. This mechanism involves the formation of prenucleation aggregates in supersaturated solutions. Another mechanism, called secondary nucleation (Section 3.4), takes place in the presence of preexisting parent crystals of the same solute in the solution, which act as catalysts for further nucleation. The secondary nuclei are formed either on the surfaces of the seed crystals, by interaction of crystals with each other or with parts of the crystallization vessel, or from semiordered surface layers removed from the crystal

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surfaces through fluid motion. Primary nucleation can be either homogeneous or heterogeneous depending on the role of foreign bodies. In pure solutions consisting of only solute and solvent molecules, nucleation is triggered by the fluctuations of solute concentration, with the probability of a given fluctuation occurring being identical over the whole volume of the system (Izmailov and Myerson 1999; Garcia-Ruiz 2003). This is called homogeneous nucleation, which is discussed in detail in Section 3.2. Homogeneous nucleation rarely occurs in practice, but it forms the basis of several nucleation theories. Heterogeneous nucleation takes place in solutions that contain surfaces of different composition than the crystallizing solid, such as the crystallizer walls, dust particles, and solid impurities (Boistelle 1986). These foreign surfaces provide active centers for nucleation by decreasing the work required for formation of the nuclei, thus increase locally the probability of nucleation with respect to other locations in the system (Garcia-Ruiz 2003). The mechanism of heterogeneous nucleation is discussed in Section 3.3. The classical nucleation theory (CNT; Section 3.2.1) describes the homogeneous nucleation process in terms of Gibbs free energy and a critical cluster size beyond which the growth occurs instantaneously (Volmer and Weber 1926). It is widely used due to its analytic simplicity. However, whereas the classical theory allows one to estimate the critical size and nucleation rate, it does not offer any information about the structure of prenucleation clusters or pathways leading from

Figure 3.1 Mechanisms of nucleation

Crystal Nucleation

the solution to the solid crystal. The CNT is based on major assumptions, summarized in Section 3.2.2, that simplify the description of the process but at the same time restrict its application to certain cases. Numerous studies, reviewed in Sections 3.2.3, 3.2.4, and 3.2.5, suggest that CNT is both quantitatively and qualitatively incorrect, and nucleation from supersaturated solutions is a two-step process where structure fluctuations follow density fluctuations. Thus, despite the important role of nucleation in determining the solid-phase outcomes, an accurate description of the process is still missing, and design of the crystallization processes is more often than not done on an empirical basis. The essential difficulty of studying nucleation and developing an accurate description of the process arises from the fact that the critical nucleus sizes typically fall in the range of 100–1000 atoms, which is not accessible to most of the current experimental methods (Schuth et al. 2001). Even if they are detected by microscopic techniques, it is hard to distinguish their structure due to small size. Furthermore, they exist for extremely short times and freely move throughout the available volume of solution, reducing the change of their appearance in the volume being examined. Understanding the structure and properties of supersaturated solutions, more specifically addressing the following questions regarding the self-assembly of solute molecules in solution prior to nucleation, is needed to obtain the comprehensive picture of nucleation (Myerson and Izmailov 1999; Davey et al. 2002): • Do the prenucleation clusters actually exist? • What is their role in the nucleation process? • What is their size and time distribution? • Do they contain solvent molecules? • Are their structures influenced by the nature of the solute– solvent interactions? • Are they loose aggregates of solute molecules or ordered in periodic structures that resemble the eventual crystalline solid? In other words, is there a correlation between the prenucleation molecular aggregation and the structural outcome? • Do they exist with packings corresponding to all the potential polymorphs in a system?

certain critical size. The clusters that attain the critical size become stable and continue to grow spontaneously, resulting in the formation of crystal nuclei. The thermodynamic description of this process was originally developed at the end of nineteenth century by Gibbs, who defined the free energy change required for cluster formation (ΔG) as sum of the free energy change for the phase transformation (ΔGv) and the free energy change for the formation of a surface (ΔGs) DG ¼ DGs þ DGv ¼ βL2 σ þ αL3 DGv

ð3:1Þ

where σ is the surface tension, and β and α are the area and volume shape factors (based on the characteristic length L), respectively. For a spherical nucleus, the area factor β = π, and the volume factor α = π/6 based on the diameter d of the nuclei. The shape factors are defined in Section 2.5 and the values for various geometric shapes in Table 2.3. For spherical clusters with radius r, Equation (3.1) becomes 4 DG ¼ 4πr2 σ þ πr3 DGv 3

ð3:2Þ

In terms of crystallization from solutions, the second term (ΔGv) describes the spontaneous tendency of a supersaturated solution to undergo deposition. Because the solid state is more stable than the liquid (i.e., μsolid < μliquid ), ΔGv becomes negative, thus decreasing the Gibbs free energy of the system. By contrast, introduction of a solid–liquid interface increases the free energy by an amount proportional to the surface area of the cluster. As a result, growth of clusters depends on the competition between a decrease in ΔGv, which favors growth, and an increase in ΔGs, which favors dissolution. This concept can be seen more clearly in Figure 3.2, where the two opposing

Numerous studies have reported the existence of prenucleation clusters in supersaturated solutions, and they are reviewed in Sections 3.2.6 and 3.7.

3.2 Homogeneous Nucleation 3.2.1 Classical Nucleation Theory CNT is the simplest and most widely used theory that describes a nucleation process. Even though CNT was originally derived for condensation of a vapor into a liquid, it has also been employed “by analogy” to explain precipitation of crystals from supersaturated solutions and melts (Mullin 1997). According to this theory, density fluctuations in a metastable phase give rise to the formation of prenucleation clusters, which continuously assemble and fall apart until reaching a

Figure 3.2 Free energy diagram showing free energy versus cluster size

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Deniz Erdemir, Alfred Y. Lee, and Allan S. Myerson

terms and the total free energy change required for cluster formation (ΔG) are plotted with respect to the cluster radius r. The positive surface free energy ΔGs term dominates at small radii, which causes an increase in the total free energy change initially. Thus the smallest clusters in solution typically dissolve. As cluster size increases, the competition between the positive ΔGs and negative ΔGv terms becomes significant, and total free energy goes through a maximum at a value of r called the critical size rc. For clusters larger than the critical size, the total free energy decreases continuously, and thus growth becomes energetically favorable. In other words, the critical size represents the minimum size of a stable nucleus that would continue to grow into a macroscopic crystal (Mullin 1997). This size can be evaluated by minimizing the free energy function with respect to the radius dðDGÞ ¼ 8πrc σ þ 4πrc2 DGv ¼ 0 dr or rc ¼ 

2σ DGv

ð3:3Þ

ð3:4Þ

Accordingly, the maximum value of the free energy change, called the activation barrier for nucleation (ΔGcrit), is found by substituting Equation (3.4) into Equation (3.3) DGcrit ¼

4πrc2 σ 3

ð3:5Þ

The growth of clusters is governed by the Gibbs– Thompson equation ln

c ¼ lnS ¼ 2σv=kTr c

ð3:6Þ

where c is the concentration of clusters of size r. Substituting for rc in Equation (3.5) from Equation (3.6), we get DGcrit ¼

16πσ3 v2 3ðkT lnSÞ2

ð3:7Þ

In 1926, Volmer and Weber used Gibbs’s formalism to initiate a kinetic theory for the formation of a nucleus that was further developed by Becker and Doring (1935), Turnbull and Fisher (1949), and Frenkel (1955). In this theory, the steady-state rate of nucleation B0, which is equal to the number of nuclei formed per unit time per unit volume, is expressed in the form of the Arrhenius reaction rate equation as
 DGcrit B0 ¼ Aexp  ð3:8Þ kT where k is Boltzmann’s constant, and A is the pre-exponential factor. Substituting Equation (3.7) into Equation (3.8) gives " # 16πσ 3 v2 ð3:9Þ B0 ¼ Aexp  3k3 T 3 ðlnSÞ2 It is clear that the nucleation rate increases with increasing supersaturation and temperature and decreases with an

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Table 3.1 Induction Time for Nucleation of Water Vapor

Supersaturation S

Time

1.0

Infinity

2.0

1062 years

3.0

103 years

4.0

0.1 second

5.0

10–13 second

increase in surface energy. The importance of the level of supersaturation is evident from the calculation of the induction time (Mullin 1960) for the formation of nuclei in supercooled water (Table 3.1). Although water would nucleate spontaneously at any supersaturation exceeding unity given enough time, instantaneous nucleation is possible only at supersaturation levels of around four. The pre-exponential factor is related to the rate of attachment of molecules to the critical nucleus and thus depends on the molecular mobility (Oxtoby 1994; Granasy and Igloi 1997). Because the molecular mobility changes rapidly with temperature, the temperature dependence of the pre-exponential factor can be quite significant. It is also affected by solubility of solute in the case of solution crystallization because the probability of intermolecular collisions decreases with decreasing solubility. The pre-exponential factor is the product of the equilibrium attachment frequency, concentration of nucleation sites, and Zeldovich factor, which converts the equilibrium expression into a steady-state expression (Oxtoby and Kashchiev 1994; Roelands et al. 2004). In other words, the Zeldovich factor takes care of the fact that not all particles at the top of the nucleation barrier end up in the solid phase but can also recross the barrier and dissolve (Auer 2002). The theoretical value of the pre-exponential factor is given as 1030 cm−3 s−1; however, it is very difficult to measure this factor in practice because homogeneous nucleation is difficult to observe in reality owing to the presence of dissolved impurities and physical features such as crystallizer walls, stirrers, and baffles. This type of nucleation has been studied in several inorganic systems (KNO3, NH4Br, NH4Cl; White and Frost 1959; Melia and Moffit 1964) using the dispersed-phase method proposed by Vonnegut (1948). All predict a pre-exponential factor of 103 to 105, which is much lower than the theoretical value of 1030. However, in a highly supersaturated NaCl precipitation system, a value close to the theoretical was found by using a crystalloluminescence technique (Garten and Head 1966).

3.2.2 Assumptions of Classical Nucleation Theory The CNT is based on major assumptions that simplify the description of the process but at the same time restrict its applications to certain cases. These assumptions, made originally for a vapor–liquid condensation system, are summarized as follows (Schmelzer et al. 1996; Oxtoby 1998; Fokin and Zanotto 2000; Izmailov and Myerson 2000; Bahadur and

Crystal Nucleation

McClurg 2001; Djikaev et al. 2001; Laaksonen and Napari 2001; Moody and Attard 2003; Vekilov 2004; Cacciuto and Frenkel 2005): • The clusters are modeled as spherical droplets having uniform interior densities and sharp interfaces (droplet model). • The density of the droplet is independent of the droplet size and equal to the macroscopic density of the bulk condensed phase. For crystallization from supersaturated solutions, this assumption implies that the building blocks come together in ordered arrays, and thus the molecular arrangement in a crystal’s embryo is identical to that in a large crystal. • The surface tension of a liquid droplet is equal to the respective value of this quantity for a stable coexistence of both phases at an infinite planar interface (capillarity approximation). In other words, the curvature (or size) dependence of the surface tension is neglected. In addition, the surface energy is assumed to be temperature independent. • The growth of clusters takes place by addition of one monomer at a time. Similarly, the preexisting clusters dissolve by detachment of single molecules. The process can be represented by the reaction Cn þ C1 ↔ Cnþ1

•

• • •

where C1 and Cn are monomer and n-mer, respectively. Furthermore, collisions between more than two particles, as well as two preexisting clusters, and break-off of preexisting clusters into two or more smaller clusters are ignored. The stationary distribution of subcritical solute clusters is established instantaneously after the onset of supersaturation. The nucleation rate is time independent (i.e., the features of the process are considered in terms of steady-state kinetics). The clusters are at rest (i.e., they do not undergo translational, vibrational, or rotational motion). The clusters are incompressible, and the vapor surrounding them is an ideal gas with a constant pressure. Thus the formation of clusters does not change the vapor state. The nucleation rate does not depend on the thermal history of the sample (i.e., how it was made and stored).

3.2.3 Shortcomings of Classical Nucleation Theory Although it was reported by a number of researchers that the CNT adequately predicts the nucleation rates for condensation of single-component fluids, this simple model provides only a qualitative description (Wagner and Strey 1984; Granasy and Igloi 1997; Oxtoby 1998; Laaksonen and Napari 2001). Giving that all input parameters of CNT are known with a high accuracy for condensation experiments, even a few orders of magnitude difference between the predicted and measured nucleation rates can be an indication of the inadequacy of the CNT. For condensation of water, the CNT predicts the nucleation rates by 1–2 orders of magnitude higher than the rates inferred from expansion cloud chamber experiments (Sharaf and Dobbins 1982). Expansion chamber experiments showed

that the predicted nucleation rates of toluene and n-nonane by the CNT departed from the experimental data with decreasing temperature (Adams et al. 1984). In addition, the surface tensions inferred from the CNT were physically unrealistic. In a similar kind of experiment, Wagner and Strey (1984) demonstrated that the experimental nucleation rates for n-nonane were higher than the theoretical rates by factors of about 102, 103, and 105 for temperatures 238, 219, and 203 K. The deviations were explained by the fact that in the CNT the critical clusters are assumed to be macroscopic spherical droplets, whereas actually the critical clusters consist of only about 9– 18 molecules. It was concluded that a multiplicative temperature-dependent correction that ranges from 10−5 at 233 K to 103 at 315 K was required to make the CNT agree with experimental findings (Hung et al. 1989). For condensation of n-alcohol vapors, the experimental and theoretical rates were found to differ significantly, with the ratio between the two values ranging from about 10−10 for methanol to 107 for n-hexanol (Strey et al. 1986). Furthermore, even though the dependence of the nucleation rate on the supersaturation was correctly described by the CNT, the predicted temperature dependences significantly disagreed with the experimental findings. It is interesting to note that for each alcohol there was one temperature where experiment and theory agreed quantitatively. This is consistent with the statement by Oxtoby (1998) that the CNT gives too high a nucleation rate at high temperatures and too low a rate at low temperatures. Because of these deviations, it is accepted that the CNT gives qualitatively reasonable but quantitatively incorrect results for the gas–liquid transition of single-component nonpolar fluids, predicting correct supersaturation but incorrect temperature dependences of the nucleation rate. According to Oxtoby (1998), the systematically incorrect temperature dependence of the CNT can be explained by two physical features of nucleation omitted by the CNT: the dependence of surface tension on curvature and the vanishing of the nucleation barrier at spinodal. Because the former lowers the nucleation rate while the latter increases it, a quantitative agreement between experiment and theory is obtained at some temperature where the two errors accidentally cancel each other. It was also argued that the qualitative success of the CNT is “largely fortuitous” because it would disappear completely with 15–20 orders of magnitude error in nucleation rates if nature had supplied a slightly longer range of attractive potentials. For the condensation of single-component highly polar fluids such as acetonitrile, the experimental results were found to be in serious disagreement with the predictions of the CNT, which was explained in terms of dipole–dipole interaction within the curved surface of the embryonic droplets (Wright et al. 1991). Even though the CNT has practical success for single-component nucleation, the classical binary nucleation theory used for the mixed nucleus is oversimplified. In the CNT, the surface of the nucleus is idealized as a region of zero thickness, and the composition is assumed to be uniform throughout the droplet with no allowance for the surface enrichment effects (Wilemski 1984). However, it is widely believed that the surfaces of alcohol–water and acetone–water clusters can have a

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considerably different composition than bulk owing to surface enrichment effects. For example, at low ethanol concentrations, there exists a very significant surface enrichment of ethanol (Schmitt et al. 1990; Laaksonen et al. 1999). For these systems, the surface energy and nucleation rate depend sensitively on the composition of the critical nucleus. Therefore, the CNT fails both quantitatively and qualitatively to describe binary nucleation in water-rich aqueous alcohol or acetone mixtures because the composition of nucleus is incorrectly predicted (Wilemski 1988; Strey and Viisanen 1993). For instance, at certain vapor activities, the theory predicts a negative number of water molecules in the critical nucleus of a water–ethanol system (Laaksonen and Napari 2001). The details of the modified theories that attempt to address this deficiency and properly describe binary homogeneous nucleation can be found in Section 3.2.4. By contrast, Strey and Viisanen (1993) showed that for fairly ideal mixtures such as ethanol–hexanol, the classical binary theory predicts the observed experimental rates and nucleus compositions qualitatively correctly, with some systematic quantitative deviations. At intermediate compositions, this deviation from the experimental results is not larger than with one of the pure components. Thus it seems like the CNT is a failure only when it comes to nonideal, surface-enriching mixtures. However, even for nonideal systems, it can predict the slopes of nucleation rate versus vapor density curves correctly at constant vapor mole fraction, which corresponds to an almost correct prediction of total number of molecules in the nucleus (Laaksonen 1997). Laaksonen showed that the breakdown of the CNT occurs when the classical nucleus mole fraction is the most disproportionate compared with the vapor mole fraction in the equilibrium diagram. This can be explained by the fact that bulk composition of the critical nucleus predicted by CNT stays the same at constant vapor mole fraction regardless of nucleus size. It was also suggested by the same group that the unphysical predictions produced by the classical binary nucleation theory could be a direct consequence of the breakdown of capillary approximation (i.e., the assumption of a curvature-independent surface tension; Laaksonen et al. 1999; Laaksonen and Napari 2001). In principle, the surface tension depends on curvature, but in practice, these corrections are difficult to quantify (Moody and Attard 2003). The capillarity approximation also results in quantitatively and qualitatively incorrect results for large supersaturations in one-component systems. Because the work of critical cluster formation represents the barrier for the transition into the new phase, it has to tend to zero in the immediate vicinity of the spinodal curve. However, the CNT does not fulfill this necessary condition of validity. The analysis shows that the origin of this problem is connected with the fact that the bulk properties of the critical clusters can differ considerably from the properties of the newly evolving macrophases for high supersaturations (Schmelzer 2003). Many studies revealed the fact that clusters of molecules exist even in undersaturated solutions (van Drunen et al. 1993; Minezaki et al. 1996; Niimura et al. 1999). Because the time evolution of prenucleation clusters in a supersaturated

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solution is superimposed on the change that occurs in an undersaturated solution, an exact and full understating of the latter process is required for that of the former. However, because the CNT is valid only for supersaturated solutions, it cannot provide any information about the nature of clusters in undersaturated solutions. The CNT cannot predict the absolute nucleation rates given that the pre-exponential factor in the kinetic equations remains undetermined. Rather, the kinetic factor, as well as the surface free energy, is adjusted to fit the experimental nucleation rate data to the theory (Auer and Frenkel 2001b). For calculation of the kinetic factor at various temperatures, it is necessary to neglect not only the size dependence but also the temperature dependence of the surface energy. Even so, when the thermodynamic driving force of the transformation was calculated as the difference of the chemical potentials in the respective macroscopic phases, large discrepancies were observed between the experimental and predicted kinetic factors (Schmelzer et al. 2004). In some cases, the experimental factor exceeded the theoretical one by ~130 orders of magnitude (Fokin and Zanotto 2000). These significant deviations may be caused by one of the assumptions made in the CNT that neglects the movement of clusters because the pre-exponential factor is related to the molecular mobility (Granasy and Igloi 1997). In reality, the clusters may behave dynamically like large gas molecules, having free rotational and translational motion (Ulbricht et al. 1998). The CNT assumes that the clusters evolve in size by attachment of single molecules and neglects the collision between two clusters. However, based on molecular dynamics (MD) computer experiments, Zurek and Schieve (1980) suggested that cluster–cluster interactions may have a considerable influence on the process of nucleation. They found that large monomer concentrations are not enough to guarantee validity of the monomer addition approximation. In fact, theoretical and experimental studies imply that the nucleation may involve the assembly of preformed clusters. For instance, simulation of crystallization in a solute–solvent system consisting of atoms of two noble gases showed that the solute particles aggregate into small clusters first, and then these small clusters come together to form a single large cluster, which eventually nucleates to the final crystalline phase (Anwar and Boateng 1998). Many crystals are known to have growth units other than monomers, such as α-glycine, whose essential building block is a centrosymmetric dimer (Sazaki et al. 1993; Gidalevitz et al. 1997; Budayova-Spano 2002; Davey et al. 2002). It is highly possible that the formation of prenucleation clusters may proceed through successive aggregation of these preassembled growth units rather than the monomer addition model suggested by the CNT. Mullin (1997) stated that the existence of an induction time period in supersaturated systems contradicts the CNT, which assumes ideal steady-state conditions and predicts immediate nucleation once supersaturation is achieved. In fact, the steady-state nucleation rate in the CNT is calculated for the condition that the size distribution of clusters does not change in time (Ulbricht et al. 1998). This leads to a constant nucleation rate (i.e., a linear increase of the number of nuclei with

Crystal Nucleation

time; Drenth and Haas 1998). This assumption fails at the very beginning of the nucleation process because a certain time, called the transition time, is required to establish the steadystate distribution of subcritical clusters. Furthermore, in some experiments, the relaxation process into steady state takes a much longer time than the characteristic lifetime of the supersaturated system; hence a steady-state nucleation does not exist (Ulbricht et al. 1998). In the CNT frame, the probability of forming critical nuclei is expected to increase rapidly with supersaturation due to a decrease in the nucleation barrier. However, experiments show that the nucleation rate in many systems goes through a maximum as the supersaturation is increased (Auer and Frenkel 2001a). This maximum can be justified by dependence of the nucleation process on temperature, which is ruled by two factors acting against each other: the increasing of supersaturation due to a decrease in temperature is compensated by the decrease in the kinetic coefficients (e.g., viscosity) governing the rates of the aggregation process (Schmelzer 2003). If the increase in supersaturation is a result of an increase in concentration, the peak in the nucleation rate can be explained as a consequence of a decrease in the growth rate of nuclei with increasing concentration. The CNT is based on the liquid droplet model, in which the small droplets are considered to have the same properties as bulk condensed phases and surface energies that are equal to that of an infinite planar surface. However, there has been a long-lasting debate concerning whether a macroscopic thermodynamic description of a liquid drop can be used to model small clusters containing only a few tens of molecules (Strey and Viisanen 1993; Mokross 2003). For example, Oxtoby (1998) stated that this assumption fails for nuclei containing only 20–50 molecules, small enough that the center is not in the thermodynamic limit and the interface is sharply curved, changing its free energy. The failure of the droplet model in describing small particles is evident from the fact that the work of formation of monomers differs from zero (Granasy and Igloi 1997). Because the properties of small clusters cannot be divided into volume and surface characteristics, the concept of surface tension seems to be artificial as applied to these clusters (Anisimov 2003). According to Yau and Vekilov (2001), the surface tension is ill defined for clusters smaller than 100 molecules, and the nucleus shape cannot be approximated with a sphere. They showed, using atomic force microscopy, that the critical nucleus of apoferritin contains between 20 and 50 molecules arranged identically to that found in crystals and consists of planar arrays of one or two monomolecular layers, which makes the nucleus to look like a raft. This disagrees with the droplet model, which states that aspherical molecule, such as apoferritin, should form aspherical nucleus, like a tiny ball cut out from the bulk crystal lattice (Oxtoby 2000). If the critical nuclei are not spherical, the classical nucleation theory can no longer be valid. Georgalis et al. (1995) stated that a critical nucleus cannot be a smooth sphere because cubes or polyhedra can represent better lattice-forming shapes. If one assumes a cubic instead of a spherical shape, the number of molecules present in the critical nucleus would

be expected to be twice as large. Furthermore, ten Wolde et al. (1998) showed that homogeneous gas–liquid nucleation in a model polar fluid is initiated by chainlike clusters instead of spherical ones. Even though these clusters condensed to form compact droplike nuclei on exceeding a certain size, the interface of droplets, with a high degree of chain formation, was noticeably different from the planar interface. Similarly, computer simulations and advanced models showed that the crystal–liquid interface is diffuse (several molecular layers thick), which contradicts the sharp interface hypothesis on which the droplet model is based (Granasy and Igloi 1997). The CNT assumes that the molecular arrangement in nearcritical clusters is identical to that in crystals. Vekilov (2004) observed critical clusters consisting of only 1–10 lysozyme molecules, which contradicts this assumption because clusters so small cannot have the structure of a tetragonal lysozyme crystal. Furthermore, theoretical studies showed that the properties of a critical nucleus can differ drastically from the eventual new phase in composition and structure (Oxtoby 1998). For instance, the precritical nucleus of a Lennard–Jones solid, whose structure is known to be face-centered cubic, was found to be predominantly body-centered cubic ordered (ten Wolde et al. 1995). Additionally, Cacciuto and Frenkel (2005) argued that one of the assumptions made in the CNT that supports the incompressibility of a crystal nucleus is not really necessary because it is not made in the droplet model. In fact, they stated that this assumption leads to a seriously flawed estimate of the pressure inside a critical nucleus, which has consequences for the prediction of possible metastable phases during the nucleation process. A simulation study of crystal nucleation in binary mixtures of hard spherical colloids showed that in the vicinity of the solid–solid critical point where crystallites are highly compressible, small crystal nuclei are less dense than large nuclei (Cacciuto et al. 2004). This phenomenon cannot be accounted for by the CNT and thus illustrates both the qualitative and quantitative breakdown of CNT and the droplet model in the case of crystal nucleation. While the CNT allows one to estimate the size of a critical nucleus and the nucleation rate, it does not provide any information about the structure of aggregates or pathways leading from the solution to the solid crystal (Schuth 2001). Perhaps the most significant shortcoming of the CNT is that size is accepted to be the only criterion determining whether the aggregates become nuclei or not. Because the CNT was developed for condensation of vapor to liquid (transition of one disordered phase into another), it does not distinguish between organized clusters and aggregates in which the orientation of the molecules does not correspond to the orientation in the resulting crystal. Consequently, local density is the only order parameter that differs between the two phases. This picture is not complete for crystallization from solutions, where at least two order parameters (e.g., density and periodic structure) are necessary to sufficiently distinguish between old and new phases (Vekilov 2004, 2005). The CNT assumes simultaneous fluctuations along both density and structure order parameters (i.e., the molecules get together in ordered arrays). In reality, these two parameters for transition need not go together; one

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can dominate the critical nucleation and serve as a prerequisite for the other one (Talanquer and Oxtoby 1998). The CNT is limited in that sense because it cannot identify the different pathways to crystallization when various order parameters do not all change at the same time (Oxtoby 2000). In addition, computer simulations, theories, and experimental studies suggested a mechanism of crystal nucleation in which the structure fluctuation follows and is superimposed on density fluctuations (ten Wolde and Frenkel 1997; Talanquer and Oxtoby 1998; Vekilov 2004). In other words, nucleation proceeds in two steps: the formation of a droplet of dense liquid, metastable with respect to the crystalline state, followed by ordering within this droplet to produce a crystal (Vekilov 2004). The two-step nucleation theory is discussed in more detail in Section 3.2.5. Based on these findings, it becomes clear that nucleation of solids from solution does not proceed via CNT pathways but by much more complex routes.

3.2.4 Modifications of Classical Nucleation Theory and Nonclassical Approaches Over the last 60 years, many extensions and modifications have been made to the CNT in order to eliminate some of its assumptions and address the shortcomings described in the preceding section. In addition, numerous researchers introduced fresh theoretical approaches that avoid the macroscopic capillarity approximation altogether. In this section, a review of these developments is presented. However, it must be noted that despite these advances, a well-proven theory of crystal nucleation is still missing, and further work is needed to develop a general theory that is applicable and reliable for a variety of systems. Because σ3 enters the exponential of Equation 3.9, the nucleation rate depends very strongly on the value of the surface tension. For instance, a change in σ of only 10 percent leads to a change in the nucleation rate by eight orders of magnitude (Ulbricht et al. 1998). However, there is no reliable method to measure the surface tension except by fitting experimental nucleation rate data to theory (Fokin and Zanotto 2000). Moreover, the most serious assumption of the CNT, known as capillarity approximation, neglects the curvature and size dependence of the surface tension, which may lead to quantitatively incorrect results for the calculated nucleation rates (Polak and Sangwal 1995; Schmelzer et al. 1996). It was already noted by Gibbs that curvature corrections to σ are of importance for small drops comparable in size with the width of the inhomogeneous interface between two phases and that the surface tension decreases with decreasing size. Thus one of the extensive modifications of the CNT so far is the use of a size- and curvature-dependent surface tension in order to address this problem. Tolman (1949) suggested that for sufficiently curved surfaces comparable with molecular dimensions, the value of the surface tension σ follows σ 1 ¼ σ 0 1 þ 2τ=r

ð3:10Þ

where σ0 is the surface tension of a planar interface, and τ is the interfacial thickness, which is expected to be on the order of

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10–8 cm. According to this equation, the surface tension monotonically increases from zero to a certain asymptotic value with an increase in the radius of curvature r of the particle (Treivus 2002). However, experiments showed that the surface tension remains constant at radii of curvature lower than 5 nm for liquid water, which puts the usefulness of this equation in doubt (Wu and Nancollas 1996). Furthermore, several researchers showed that the surface tension is a not a monotonic function of the radius; instead, there are two regions in which the surface tension either increases or decreases with increasing radius, with a maximum at some finite value of radius (Schmelzer and Mahnke 1986). Several other equations suggested for the curvature dependence of the surface tension can be found in Schmelzer and Mahnke (1986) and Fokin and Zanotto (2000). It is important to note that because the chemical potential can be considered to be a function of the temperature and density of clusters, the surface tension is also a function of these two parameters (Ulbricht et al. 1998). In fact, it was shown by Schmelzer et al. (1996) that a correct explanation of a possible influence of the curvature dependence of surface tension on the energy of formation of critical clusters requires a simultaneous determination of the size dependence of the density of clusters. Another important point is that the account of curvature dependence of surface tension may result in significant quantitative variations of the nucleation rate, but it does not lead to a qualitatively different behavior compared with the results obtained without this consideration. A phenomenological nonclassical theory, called the diffuse interface theory of nucleation (DIT), was introduced by Granasy (1996) to correct for the size dependence of interfacial energy. This theory was based on parameterization of the interfacial enthalpy and entropy distributions within a characteristic interface thickness, and the predicted size-dependent interfacial energies were found to be consistent with experiments on condensation of vapors and crystal nucleation in molten metals, water, and metallic glasses. Zhukhovitskii (1994) proposed a semiphenomenological approach in which the cluster surface energy is proportional to the number of molecules on the cluster surface rather than the cluster surface area suggested in the CNT. In such way, the surface tension, which has no sense for small clusters, appears only in the limit of large clusters, which extends the validity range of this model from dimers to macroscopic droplets. Another way to avoid the problem of uncertainty in the estimation of surface tension is to describe the clustering process on the basis of molecular interactions. Polak and Sangwal (1995) developed a theory of the formation of clusters in aqueous solutions of ionic salts considering the coulomb interaction between ions, adsorption of water molecules on ions, and cohesion forces between water molecules. The theory was able to predict the existence of clusters in both undersaturated and supersaturated solutions, as well as the size dependence of interfacial tension. As discussed in Section 3.2.3, classical binary theory fails to give correct predictions for water–alcohol systems because it assumes a uniform droplet composition and does not take into account the surface enrichment of alcohol. A number of

Crystal Nucleation

researchers developed modified theories to address this deficiency and properly describe binary homogeneous nucleation. The approach proposed by Flageollet-Daniel et al. (1983) uses a lattice model to describe a microscopic cluster, which allows the determination of the interior and exterior compositions of the critical droplet. It was shown that for mixtures that exhibit strong changes in the macroscopic surface tension with composition, the surface tension itself varies with the size of the cluster. In an alternative cluster model by Laaksonen and Kulmala (1991), the two-component liquid clusters were divided into a unimolecular surface layer and an interior layer, in which the relative fractions of the molecules were allowed to change with cluster size. The predictions from both theories were shown to account qualitatively and almost quantitatively for the observed nucleation rates and cluster compositions (Laaksonen 1992; Viisanen et al. 1994). The revised classical theory developed by Wilemski (1984) relies only on the macroscopic surface tension but implicitly accounts for surface enrichment through the calculation of interior cluster composition using thermodynamically consistent Kelvin equations. The third theory, proposed by Rasmussen (1986), showed that replacement of the macroscopic equilibrium surface tension with the dynamic surface tension of a freshly formed interface yields the surface tension required by classical binary theory. The rate equation predicts exponential growth once a critical supersaturation is attained, but in practice, an optimal temperature exists below which the liquid is too viscous to nucleate and above which molecular motion prevents crystal formation. This was observed by Tamman (1925) for several organic salts. He found that the optimal nucleation temperature was lower than that required for maximal crystal growth. A similar observation was made by Mullin and Leci (1969a) for the spontaneous nucleation of citric acid solutions and is shown in Figure 3.3. The viscous effects can be incorporated into the rate equation by taking into account the viscous free energy (Turnbull and Fisher 1949) " # 16πσ 3 v2 DGvisc B0 ¼ Aexp  þ ð3:11Þ kT 3k3 T 3 ðlnSÞ2 At higher supersaturation values, the first term becomes smaller, whereas the viscous energy term ΔGvisc becomes larger. Thus the nucleation rate declines after a critical supercooling has been attained. In addition, various modifications have been proposed to eliminate the assumption made in the CNT that neglects the motion of molecular clusters. Abraham and Pound et al. (1968) suggested that there are statistical mechanical contributions to the free energy of formation of critical nuclei that arise from consideration of the absolute entropy of the clusters. They found that the inclusion of translational and rotational motion of clusters changes the calculated nucleation rate by a factor of 1017. By contrast, Reiss and Katz (1967) disagreed with this approach, pointing out that the free energy assigned to a drop in the CNT does not correspond to that for a drop at rest but rather for a drop whose center of mass already fluctuates. They

Figure 3.3 Spontaneous nucleation in supercooled citric acid solution: (a) 4.6 kg of citric acid monohydrate per kilogram of “free” water (temperature = 62° C) and (b) 7.0 kg/kg (temperature = 85°C) Source: Reproduced with permission from Nyvlt 1968.

applied a different statistical mechanics approach and obtained a much smaller correction factor. This significant disagreement led to a discussion of the so-called translational-rotation paradox, which was not satisfactorily resolved. Another possible strategy for evaluation of nucleation rates is to use a prescribed cluster model containing unknown parameters that can be chosen to reproduce known vapor properties (Laaksonen et al. 1994). This kind of strategy representing a combination of statistical mechanical treatment of clusters and empirical data has provided promising results (Kalikmanov and Dongen 1995). For instance, the semiphenomenological theory proposed by Dillmann and Meier (1991), referred to as the DM model, is based on a modification of Fisher’s droplet model, which takes into account the translational, rotational, vibrational, and configurational degrees of freedom, as well as the variation in surface tension with cluster size. Unlike Pound and coworkers (Reiss et al. 1968), who calculated the vibrational and rotational contributions from the Boltzmann distribution, the DM model introduced parameters normalized to the experimental values (Anisimov 2003). The predicted nucleation rates showed an excellent agreement with experiment findings for several substances. The model could also predict the correct temperature dependence of nucleation rates, which is hard to achieve by the CNT, as discussed in Section 3.2.3. However, the model contains inconsistencies arising from employing the ideal gas equation for monomers. Because the CNT assumes steady-state conditions, the cluster size distribution is constrained to be in equilibrium with a supersaturated vapor. However, this assumption fails at the very beginning of the nucleation process because a certain time is required to establish the steady-state distribution of clusters (Knezic et al. 2004). Thus an expression for the time-dependent change in nucleation rate to reach the stead state is

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needed. Kashchiev (1969) derived well-established expressions for nonstationary nucleation rates and showed that the phenomena in the critical region play the decisive role during the establishment of the steady state. In a so-called kinetic approach, Katz and Wiedersich (1993) showed that the evaporation rate of molecules from clusters can be related to the equilibrium size distribution of clusters in a saturated vapor. This procedure has an important advantage because the thermodynamic properties of vapor at saturation are well-defined quantities. Kalikmanov and Dongen (1995) combined this kinetic approach with Fisher’s droplet model applied at the saturation point to create a new semiphenomenological theory of homogeneous vapor–liquid nucleation. Girshick and Chiu (1990) extended the kinetic theory to derive a new expression for the rate of nucleation from an ideal supersaturated vapor. This new expression had a different dependence on supersaturation and temperature than the classical expression. The agreement between the new expression and experimental data was found to be remarkably good for some cases. Later, Girshick (1991) showed that the same result could be derived directly from the CNT and that the modification in the nucleation rate was a consequence of writing the expression for the equilibrium cluster distribution self-consistently, regardless of whether the classical or kinetic theory were followed. In this socalled self-consistent theory (SCT), the work for formation of a monomer is equal to zero, in contrast to the CNT, in which the change in free energy of monomers differs from zero (Anisimov 2003). It is interesting to note that a general relation was proved to exist between the nucleation work and nucleus size regardless of the model used for the free energy of nuclei (Kashchiev 1982). The relation reads as follows: the number of molecules in the nucleus equals the derivative with a minus sign of the nucleation work with respect to the supersaturation. This relation was named nucleation theorem (NT) because of its validity for any kind of one-component nucleation. Bowles et al. (2001) showed that the validity of the NT goes beyond the phenomenon of nucleation and extends to all equilibrium systems containing local non-uniform-density distributions stabilized by electric fields. Perhaps the most important among the studies devoted to the development of modern theories of nucleation was density functional theory (DFT). This theory was pioneered by Cahn and Hillard (1959), who determined the free energy for formation of a critical nucleus in a two-component metastable fluid by treating the nucleus as a fluctuation that is in unstable equilibrium with the exterior phase. Thus it was not necessary to make any assumption about its homogeneity or divide the energy of the nucleus into surface and volume terms. The nucleus was simply identified as the saddle point of a free energy expression that was derived for a system of nonuniform density. The authors found that the work required for the formation of a nucleus approaches zero at the spinodal, unlike in the case of CNT, which erroneously predicts a finite barrier to nucleation in the immediate vicinity of the spinodal curve. Taking this approach as a starting point, Oxtoby and coworkers (Harrowell and Oxtoby 1984; Oxtoby and Evans 1988)

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constructed a new nonclassical nucleation theory that is based on a functional expansion in gradients of the density. In this theory, the free energy is not simply a function of the droplet radius but rather a functional of the inhomogeneous spherical density profile ρ(r), which varies from the center of the nucleus outward. The density at the center of the nucleus need not be that of the new bulk phase, nor must the surface behave like a planar interface (Oxtoby 1998). In fact, the curvature dependence of the surface tension appears naturally in this model. It was shown that use of a realistic interaction potential is the key to the success of the DFT (Zeng and Oxtoby 1991). For condensation of single-component fluids, the DFT captures the observed temperature dependence of nucleation, which is not described correctly by the CNT. This theory is also ideal for studying binary condensation because it produces detailed information about the composition of a critical nucleus. For liquid-to-solid transition, which is more complex than condensation owing to the change in structure along with the change in density, application of the DFT provided interesting results that supported the hypothesis of two-step nucleation (Talanquer and Oxtoby 1998). The details of these studies are provided in the next section.

3.2.5 Two-Step Nucleation Theory The CNT assumes that the fluctuations along density and structure order parameters proceed simultaneously (i.e., the molecules get together in ordered arrays; Vekilov 2004). However, in the last 30 years, a line of theoretical and experimental studies has suggested that nucleation from supersaturated solutions might be a two- step process. The first step involves the formation of a liquid-like cluster of solute molecules, whereas the second involves the reorganization of such a cluster into an ordered crystalline structure, as shown in Figure 3.4. One of the first computational works supporting this view was reported by Heerman and Klein (1983), who studied nucleation and growth in the metastable region of the Glauber kinetic Ising model by the Monte Carlo technique. The data indicated that the nucleating droplets are not compact but quickly become so during the initial phase of growth. ten Wolde and Frenkel (1997) studied homogeneous nucleation in a Lennard–Jones system of short-range attraction by the same technique and confirmed that away from the fluid–fluid critical point (T > Tc or T < Tc), fluctuations along density and structure order parameters occur simultaneously, similar to the classical viewpoint. By contrast, large density fluctuations were observed around the critical point, which caused a striking change in the pathway for crystal nucleation: the formation of a highly disordered liquid droplet was followed by the formation of a crystalline nucleus inside the droplet beyond a certain critical size (Figure 3.5). Furthermore, proximity to the critical point decreased the free energy barrier for crystallization and consequently increased the nucleation rate by many orders of magnitude. The metastability of such fluid–fluid coexistence with respect to solidification and thus the presence of an intermediate phase in the form of a high-concentration liquid were shown to be a generic future for substances that interact through a sufficiently short range of interactions, such

Crystal Nucleation Figure 3.4 Alternative pathways leading from solution to solid crystal: (a) supersaturated solution; (b) ordered subcritical cluster of solute molecules, as proposed by the CNT; (c) liquid-like cluster of solute molecules, a dense precursor as proposed by twostep nucleation theory; (d) ordered crystalline nuclei; (e) solid crystal

Figure 3.5 Contour plots of the free energy landscape along the path from the metastable fluid to the critical crystal nucleus for a system of spherical particles with short-range attraction. The curves of constant free energy are drawn as a function of Nρ (number of connected particles that have a significantly denser local environment than particles in the remainder of the system) and Ncrys (number of solid-like particles belonging to a given crystal nucleus). (a) The free energy landscape well below the critical temperature (T/Tc = 0.89). The lowest free energy path to the critical nucleus is indicated by a dashed curve. This curve corresponds to the formation and growth of a highly crystalline cluster. (b) As (a), but for T = Tc. In this case, the free energy valley (dashed curve) first runs parallel to the Nρ axis (formation of a liquid-like droplet) and then moves toward a structure with a higher crystallinity (crystallite embedded in a liquid-like droplet). The free energy barrier for this route is much lower than the one in (a). Source: From ten Wolde and Frenkel 1997. Reprinted with permission from the American Association for the Advancement of Science.

as proteins (Asherie et al. 1996; Nicolis and Nicolis 2003). It was proposed that one can selectively speed up the rate of nucleation in these systems by adjusting the composition of solvent, and thus changing the location of the metastable critical point, instead of increasing the supersaturation, which tends to form aggregates rather than crystals (ten Wolde and Frenkel 1997). Following this work, a molecular dynamics simulation on a Lennard–Jones solute–solvent system consisting of atoms of two noble gases with one dissolved in the other showed that crystallization in highly

supersaturated systems involves liquid–liquid phase separation followed by nucleation of the solute phase (Anwar and Boateng 1998). In another molecular dynamics simulation on a system consisting of 50 widely separated acetic acid (solute) molecules within a box of 1659 CCl4 (solvent) molecules, Gavezzotti (1999) observed the formation of a liquid-like micelle of acetic acid as the concentration of solute was increased by removing solvent molecules from the system. It was suggested that the formation of a microemulsion of liquid-like particles was the first step of crystal nucleation. Soga et al. (1999) used Brownian

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dynamics simulations to study the phase separation of colloidal particles and detected a metastable colloid vapor–liquid phase coexistence region in which the colloid fluid was metastable with respect to the equilibrium crystal phase. The metastable phase separation resulted in regions of high colloid density, out of which nucleation of the crystal phase was observed to proceed rapidly. Additionally, a molecular dynamics study on nucleation of AgBr in water showed that stable prenucleation clusters as large as Ag18Br18 were disordered, which provided further support to the idea that the initial step in nucleation from solution involves the formation of disordered clusters (Shore et al. 2000). By contrast, clusters as small as Ag4Br4 were found to exist in an ordered configuration in vacuo, which indicated that the interaction with solvent was responsible from the disorder within clusters. More recently, the aggregation-volume-bias Monte Carlo–based simulation technique was applied to the study of the nucleation processes encountered by a supersaturated Lennard–Jones vapor at temperatures below the triple point (Chen et al. 2008). It was found that the crystal formation process in these Lennard–Jones clusters proceed via two separate nucleation steps, first a vapor–liquid–like aggregation followed by the nucleation of crystals inside the aggregates, except at very low temperatures, where the crystal nucleation barrier was found to be negligible. Additional theoretical studies have also provided evidence for the two-step nucleation mechanism. Evans et al. (1997), by using Cahn–Hilliard theory, confirmed that the nucleation rate of a solid can be radically altered by the presence of a metastable liquid phase. While computational studies by ten Wolde and Frenkel (1997) showed that the proximity to the fluid– fluid critical point increases the nucleation rate by many orders of magnitude, Evans and coworkers suggested that away from this critical point, the interface between two equilibrium phases can be separated by a slab of metastable liquid phase, hindering formation of the solid crystal. Talanquer and Oxtoby (1998) applied the DFT to the study of crystal nucleation from solution and concluded that the nature of nucleation changes qualitatively near the metastable critical point, with nucleation rates increasing by several orders of magnitude. At temperatures higher than the critical temperature, nucleation was found to proceed through the formation of crystal-like clusters, whereas at lower temperatures, liquid-like clusters with an extended wetting layer were favored. In accordance with the computer simulations of ten Wolde and Frenkel (1997), these results suggest that close to the critical point, the first step toward the critical nucleus is the formation of a liquid-like droplet, followed by nucleation of a crystal in this droplet at a certain critical size. Beyond the critical point, the number of crystalline particles increases proportional to the increase in the excess number of particles. This mechanism was particularly proposed for the nucleation of colloids and globular proteins from solution (Talanquer and Oxtoby 1998). Shiryayev and Gunton (2004) further extended these results to show in detail how the CNT fails not only in the vicinity of the metastable critical point but also close to the liquidus line. DFT calculations by Lutsko and Nicolis (2005) demonstrated the validity of the two-step nucleation theory for simple atomic

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fluids modeled with Lennard–Jones interactions, suggesting that crystallization involving passage through a metastable disordered state may be a generic phenomenon. They stated that the lack of experimental evidence for two-step nucleation in simple fluids can be explained by the short lifetime of the metastable phase in these systems. Treivus (2001), by calculating the parameters that characterize the concentration fluctuations in solutions, showed that the prenucleation clusters for various salts have an amorphous structure with diffuse boundaries. Based on this finding, it was concluded that nucleation is, at least, a two-barrier process in terms of the thermodynamic potential, in which the first barrier necessary for cluster formation is lower than the main barrier necessary for transformation of the already formed cluster into a stable crystalline nucleus. In addition, calculations of the concentration profile and Gibbs free energy of the interface between protein crystal and aqueous solution, reported by Haas and Drenth (2000), confirmed the two-step nucleation mechanism. Such a mechanism with small activation energy for each step was expected to be faster than a one-step mechanism with a larger activation barrier (Haas and Drenth 1999). For lysozymes, the formation of liquid droplets with high protein concentration (i.e., the first step) was found to be the rate-determining step of the nucleation process. The authors also showed that near and below the critical temperature for metastable liquid–liquid phase separation, the protein crystal is covered by a thin liquid film with a high protein concentration, which lowers the surface energy of the crystal, thus enhancing the sticking probability of molecules diffusing from the dilute solution to the crystal surface. A kinetic model developed by Dixit and Zukoski (2000) agreed with these findings, suggesting that the density fluctuations associated with the metastable fluid– fluid phase boundary significantly enhance the nucleation rate. They reported that the enhanced rate of nucleation is moderated by the vanishing of the diffusivity at the critical point; hence the nucleation rate varies nonmonotonically with supersaturation owing to the nonlinear coupling between these two opposing effects. More recently, Vekilov and coworkers (Pan et al. 2005) developed a simple phenomenological model of protein crystallization via an intermediate liquid state and showed that the rate-determining step in the nucleation mechanism is the formation of an ordered cluster within the dense liquid intermediate. This emphasizes the role of viscosity within the dense liquid drop in the kinetics of nucleation of ordered solid phases. A theoretical investigation of the kinetics of two-step nucleation by Kashchiev et al. (2005) revealed a similar behavior. In their proposed kinetic model, the linear part of the time dependence of the number of crystals was determined by formation rate of the intermediate particles, which differed radically from the one-step nucleation mechanism, in which the linear part is determined by the rate of crystal nucleation directly in the old phase. To distinguish between one- and two-step nucleation from experimental data, the authors suggested relying on the different dependencies of the induction time on the supersaturation. Additional support for the two-step nucleation mechanism has been provided by various experimental studies. For

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instance, the spontaneous formation of two liquid phases during a crystallization process was directly observed in macromolecular systems such as proteins and polymers (Taratuta et al. 1990; Berland et al. 1992; Muschol and Rosenberger 1997; Chen et al. 1998). Georgalis et al. (1992, 1993) applied dynamic light scattering to investigate the nucleation of lysozyme crystals and showed that the monomers rapidly aggregate in the diffusion-limited aggregation regime to form fractal clusters in the initial stages of crystallization. Further static light scattering experiments by the same group revealed progressive restructuring of these fractals to compact structures at the later stages of aggregation (Georgalis et al. 1997a, b). Several other light scattering studies that followed provided similar results, supporting the validity of the two-step nucleation model (Tanaka et al. 1999; Pullara et al. 2005). Numerous small-angle scattering studies on nucleation of proteins and colloidal particles also suggested that the first observable nuclei in solution are droplet-like or fractal aggregates that subsequently rearrange to form more compact structures (Dokter et al. 1995; Boukari et al. 1997; Vidal et al. 1998; Pontoni et al. 2002, 2004). Following these studies, Bonnett et al. (2003) reported for the first time the direct observation of liquid phase separation during crystallization of a small organic molecule from solution. An electron microscopy study on the nucleation of ribosomal crystals showed that the process starts with the formation of amorphous precipitates, which later undergo a rearrangement toward the creation of nucleation centers (Yonath et al. 1982). Ueda et al. (1995) observed using the scanning electron microscope that citric acid molecules in supersaturated solutions form solute clusters of ~20 nm, which then gather together in particles of ~60 nm, called clustercysts. With increasing supersaturation, these clustercysts aggregate to several microns in size and finally rearrange into an ordered crystal structure. Even though it is not clear when the formation of crystallographic symmetry starts during these steps, the rearrangement of clustercysts into agglomerates suggests the existence of a two-step mechanism. A recent experimental study of two-dimensional crystallization of colloidal particles with a tunable short-range attractive interaction consisted of monitoring the size of individual clusters via video microscopy while supercooling the samples (Savage et al. 2012). For an extended period of time following supercooling, samples contained a large number of disordered clusters with a wide range of sizes. Because the particles in these clusters were mobile and the clusters were disordered, lacking any discernible crystalline order, the authors interpreted these clusters as droplets of the liquid phase. A relatively small number of clusters continued to grow in size and eventually became stable crystallites, revealing a two-step nucleation mechanism. Differential scanning calorimetry analysis of the supersaturated lysozyme solution revealed that an unstable structure formed just after preparation of the solution transforms into a more structured, probably ordered aggregate, just before the end of the induction period (Igarashi et al. 1999). These structured aggregates, which are formed through hydrophobic interaction, then grow to large crystals. It was stated that the

initial unstable structure can be a three-dimensional network or an aggregate with the size of a tetramer. Knezic et al. (2004) measured the nucleation induction times of lysozyme using electrodynamic levitation of singlesolution droplets and reported that the two-step nucleation model describes the behavior of the experimental data better than an analysis based on the CNT. They found that the second step does not happen instantaneously, which implies that the bulky protein molecules require some degree of orientation prior to forming a stable nucleus capable of further growth. This model is consistent with the observation that nucleation from solution takes a longer time as the complexity of the molecules increases because it is more difficult for more complex molecules to arrange themselves in the appropriate lattice structures. During nonphotochemical laser–induced nucleation (NPLIN) studies, supersaturated solutions of small organic molecules, such as urea and glycine, that were exposed to the laser light were found to nucleate much faster than control solutions (Zaccaro et al. 2001). This was explained by electric field–induced alignment of the molecules in existing prenucleation clusters in the solution and consequent reduction of the entropic barrier for ordered lattice formation. If the nuclei would form by successive aggregation of molecules in an ordered manner, as proposed by the CNT, induced alignment of the molecules would not cause a significant change in the structure of already ordered clusters; hence there would not be any drastic deviation between the induction times of the lased and control solutions. Moreover, freshly prepared supersaturated solutions that were exposed to the laser light did not go through NPLIN, and the solutions had to be aged before the laser could induce nucleation. This implied that the laserinduced organization of molecules within a cluster would lead to nucleation only if that cluster was of sufficient size. This concept supported the two-step nucleation theory because once the laser light encountered sufficiently large clusters, it would reduce the induction time through the organization step by aligning the molecules within the clusters. Providing more support to the NPLIN mechanism and two-step nucleation theory, two different polymorphs of glycine were crystallized from aqueous solutions depending on the laser polarization state, circular polarization producing the α-form and linear polarization generating the γ-form (Garetz et al. 2002). Perhaps the most important among the experimental studies that contributed to the development of the two-step mechanism were those presented by Vekilov and coworkers (Vekilov 2004, 2005). Galkin and Vekilov (2000a, 2000b) developed a novel method for direct determination of homogeneous nucleation rates and found that the rate of nucleation of lysozyme crystals passes through a maximum in the vicinity of the liquid–liquid phase separation boundary. In other words, the enhanced nucleation rates achieved close to the critical point starts to decrease as the system enters the liquid–liquid separation region below the boundary. Based on these observations, the authors suggested that the nucleation rate enhancement at the critical point may be due to the wetting of the nuclei surface by the dense liquid rather than to

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the stronger density fluctuations predicted by theory and simulations (ten Wolde and Frenkel 1997; Talanquer and Oxtoby 1998). In fact, dynamic light scattering and atomic force microscopy measurements on the protein lumazine synthase showed that the short-lifetime dense liquid droplets, which are metastable with respect to both the crystals and the low-density solution, do not promote the nucleation of crystals but rather play a crucial role in the crystal growth stage (Gliko et al. 2005a, b). The association of dense droplets with the surface of the crystal, followed by structuring of them under the influence of the periodic field of the crystal, was found to be the only mechanism for the generation of the new crystalline layers. By contrast, Galkin et al. (2002) demonstrated that dense liquid droplets facilitate the nucleation of deoxyhemoglobin S polymers by serving as centers for nucleation, which was attributed to the higher hemoglobin concentration in the droplets. Based on this finding, Vekilov (2004) attributed the observed suppression of lysozyme nucleation below the liquid–liquid separation boundary to the high viscosity of the long-lifetime liquid droplets appearing in that region. These results allowed Vekilov to present a two-step nucleation mechanism in which a structure fluctuation occurs within a region of higher density of molecules existing for a limited time due to a density fluctuation. They concluded that there exists a density fluctuation with optimal size and density that provides the highest probability of occurrence of structure fluctuation in the droplet. In other words, structure fluctuations do not require large density fluctuations or long-lifetime droplets, such as those existing below the liquid–liquid separation line, to become crystalline nuclei. On the contrary, large density fluctuations may never become crystals because they spend less time in the optimal parameter region due to their high viscosity. Furthermore, Filobelo et al. (2005) showed that the structuring of the dense liquid precursor into an ordered cluster (i.e., the second step) determines the rate of crystal nucleation because the crystal nucleation was found to be 8–10 orders of magnitude slower than the nucleation of dense liquid droplets. The thermodynamics of the two-step nucleation model can be found in detail in Chapter 14.

3.2.6 Studies on Supersaturated Solutions Prior to and En Route to Nucleation Indirect evidence of molecular clusters in supersaturated solutions has appeared in the literature and includes the results of Mullin and Leci (1969b) with vertical columns of supersaturated citric acid solutions published in 1969. They reported that isothermal columns of supersaturated citric acid solutions developed concentration gradients with higher concentrations at the bottom than at the top of the column, whereas there were no gradients in the saturated or undersaturated columns. The authors attributed these results to the settling of closely packed citric acid clusters, which presumably had a density close to that of the solid phase, to the bottom of the column under the influence of gravitational fields. Following this work, Larson and Garside (1986) measured concentration gradients in a vertical column of an aqueous supersaturated solution of citric

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acid, urea, sodium nitrate, and potassium sulfate and observed concentration gradients in the supersaturated solutions. Such gradients could not be detected in saturated and undersaturated solutions. The cluster size was found to increase with supersaturation, and the less soluble salts, such as potassium sulfate, formed only very small clusters. The authors estimated that the clusters typically contained 1000 molecules and had sizes between 4 and 10 nm by assuming that all the solute in excess of the saturation value existed as clusters and that the clusters had the density of the solid state. It was mentioned, however, that these values might be too small because the clusters, in practice, would have an extended lattice and be partially solvated and as a result would have larger specific volumes than the solid state. Later, Ginde and Myerson (1992) stated that the assumption of clustering only among the excess solute molecules forces the average cluster size to an unrealistically high value, and it is more likely that a significant portion of the molecules participate in clustering. They analyzed their column data using the concept of number average solute cluster size (NASCS) and obtained number average cluster sizes on the order of 2–100 molecules. Ohgaki et al. (1991) reported the existence of density gradients in a vertical column of supersaturated citric acid monohydrate solutions and estimated the cluster size to be about 16 nm in diameter by using a Brownian diffusion equation. However, it was not possible to obtain a correlation between the average number of molecules in the cluster and such factors as supersaturation and temperature from any of these measurements. Among the early efforts in the investigation of solution structures were the diffusivity measurements conducted using Gouy interferometry. The measurements in aqueous urea solutions showed a linear decrease in the diffusion coefficient with increasing concentration in the undersaturated region, whereas the results in the supersaturated region exhibited a much more rapid decrease with increasing concentration than in the undersaturated region such that the linear relationship was no longer applicable (Sorell and Myerson 1982). It was speculated that this phenomenon was a result of molecular aggregation of urea in supersaturated solution, which would alter the nature of the diffusion process. The same kind of results were obtained for the diffusivity measurements in supersaturated solutions of potassium chloride, sodium chloride, and glycine, with a rapid decrease of diffusion coefficient toward zero with increasing concentration in the supersaturated region (Chang and Myerson 1986). This rapid decrease was found to be similar to the behavior observed in liquid– liquid systems near the consolute point; in fact, when the concentrations were obtained by extrapolating the data of Sorell and Myerson to the diffusivity of zero, the results were found to be within 5 percent of the predicted location of the spinodal curve for the urea–water system (Myerson and Senol 1984). In addition, the diffusivity coefficient at a fixed concentration in the metastable region was found be declining with “age,” and the effect was attributed to the evolution of molecular clusters in time in metastable solutions (Myerson and Lo 1990). Furthermore, the modified theory of Binder was used to estimate the average cluster size based on these diffusion

Crystal Nucleation

coefficient data. The results showed that potassium and sodium chloride existed mainly as monomers, glycine existed as dimers, and urea as oligomers in supersaturated binary solutions (Ginde and Myerson 1992). Another technique that was used to study supersaturated solutions was microscopy. The microscopic observations of the nucleation process, especially by electron microscopy (EM) and atomic force microscopy (AFM), provided direct imaging on an individual molecular level, which was difficult to achieve by other methods. The structures of aqueous citric acid solutions of various concentrations, including supersaturated solutions, were examined directly by scanning electron microscopy (SEM), and solute clusters were observed in all samples (Ohgaki et al. 1992). The compartments in the scanning electron microphotographs that represented the solute clusters were found to increase in number while decreasing in volume as the solute concentration was increased. Clusters in aqueous solutions of KCl and sugar were detected by a laser scattering ultramicroscope (LSUM), which provided direct evidence for the existence of clusters (Li and Ogawa 2000). In situ AFM was used to observe the surface topology and growth of satellite tobacco mosaic virus (STMV) crystals, and the results showed that the growth occurred through nucleation of 2D islands and layer-by-layer advancement of monomolecular steps (Malkin et al. 1998). Li and coworkers (1999) used AFM technique to measure the dimensions of individual growth units on lysozyme crystal faces as they were being incorporated into the lattice. These authors showed that a variety of unit sizes corresponding to 43 helices were participating in the growth process, with the 43 tetramer being the minimum observed size. Macroscopic mathematical models for lysozyme crystal growth, involving the formation of an aggregate growth unit, mass transport of the growth unit to the crystal interface, and faceted crystal growth by growth unit addition, were developed based on the crystal face growth rate and AFM measurements (Li et al. 1995; Nadarajah et al. 1995; Nadarajah and Pusey 1999; Pusey and Nadarajah 2002). The calculations showed that the best fits were obtained for tetramer or octamer growth units in those models. By contrast, AFM studies on the (110) faces of lysozyme crystals determined growth steps of monomolecular height, which supported the predominance of monomers in lysozyme solutions, rather than oligomers (Konnert et al. 1994). A similar kind of experiment performed on canavalin protein also revealed a monomolecular step growth; however, in addition, a continual formation of a large number of molecular clusters with diameters of fewer than 10 molecules was observed on the terraces of the steps (Land et al. 1995). While some of these clusters dissolved during the imaging, many were stable and incorporated into the advancing steps, thus providing a significant fraction of molecules in the crystal. However, because of their small size, it was not possible to determine their packing arrangement. Michinomae et al. (1999) used electron microscopy to follow the initial process of crystallization in aqueous lysozyme solutions after supersaturating them by adding NaCl. They distinguished two necessary steps for nucleation; the first was the formation of short treads in which four molecules were aligned

Figure 3.6 Schematic illustration of the initial process of lysozyme crystallization based on electron microscopic observation. In the left column, the change in lysozyme molecule aggregation is shown. In the right column, the time needed for the change to occur is shown for three concentrations of lysozyme. Source: Reprinted by permission from Michinomae et al. 1999. Copyright © 1999, Elsevier.

in tandem, and the other was the formation of amorphous riceball-like structures, probably by aggregation of the short threads, which were formed in advance. From these rice balls, larger structures appeared, some of which showed the arrangement of molecules as in a crystalline lattice (Figure 3.6). In other words, characteristic structures were already present in the induction period prior to the packing of molecules into a crystalline lattice. It was not clear whether the rice balls were the essential growth units of the crystals or not because the process occurring prior to the emergence of the crystals of a visible size was difficult to observe. However, the authors stated that monomolecular growth was unlikely because most of the protein molecules had been transformed into rice ball– like structures, even if some might have remained monomolecularly dispersed. Kuznetsov et al. (1998) investigated the growth of canavalin, thaumation, lipase, xylanase, and catalase and satellite tobacco virus crystals from solution by AFM. The mother liquids were found to contain not only developing crystal surfaces but also stable aggregates of a fractile or linear/

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branched character, as well as large metastable liquid aggregates. While the stable aggregates sedimented on crystal surfaces and produced defects, liquid aggregates had the potential of rapidly ordering into multilayered stacks, probably guided by the underlying lattice, and actually contributing to the longrange structure of a growing crystal. By using AFM in situ during the crystallization of apoferritin, Yau and Vekilov (2000, 2001) were able to image for the first time the arrangement of the molecules in the near-critical clusters. The nuclei were found to contain between 20 and 50 molecules arranged identically to that found in crystals. However, they were not compact spherical molecular clusters; instead, they were planar arrays of several rods of four to seven molecules, making the nucleus look like a raft. Even though these studies demonstrated the feasibility of using AFM as an assay for nucleation, one should bear in mind that the crystallizing systems are disturbed by the presence of interfaces required for the AFM measurements, and thus the results should be interpreted with caution (Schuth 2001). Following this work, crystallization of concentrated colloidal suspensions was studied in real space with laser scanning confocal microscopy, which allowed direct imaging of nucleation (Gasser et al. 2001). The structure of the nuclei was identified to the bulk solid phase, whereas their shape appeared rather nonspherical, with rough rather than faceted surfaces, which contradicted the CNT, where the nuclei are assumed to have a spherical shape due to surface tension. More recently, with the combination technique of scanning electron microscopy, electron diffraction, X-ray microscopy, and cryotransmission electron microscopy, coupled with a special quenching technique, it was shown that formation of calcium carbonate is far more complex than the classical model of nucleation and growth at high supersaturation suggests (i.e., during precipitation; Rieger et al. 2007). On mixing CaCl2 and Na2CO3 solutions, first an emulsion-like structure formed, which then decomposed to CaCO3 nanoparticles. These nanoparticles aggregated to form vaterite spheres of some micrometers in diameter, followed by transformation of these spheres via dissolution and recrystallization to calcite rhombohedra. A number of investigators have used spectroscopy to examine the structure of supersaturated solutions and understand the nature of molecular clusters. In one of the earliest attempts, in situ Raman analyses determined contact-ion pairs and ion aggregates in undersaturated and supersaturated aqueous NaNO3 solutions (Hussmann et al. 1984). The relative concentration of these species with respect to the free ions was diminished near the growing crystal, indicating their effective role in the growth process. These results were reproduced by Rusli et al. (1989) five years later, and the solute clusters were found to consist of solute and solvent molecules rather than having a crystalline structure. There were some indications that as the solution became supersaturated at a given concentration, a rearrangement occurred between lower-ordered ionic species and the solvated clusters. As the solute concentration was increased, the spectra shifted toward higher wave numbers, suggesting an increase in the presence of clusters. A similar study on KNO3

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solutions detected a Raman band representing ordered species that increased in magnitude as concentration was increased into the supersaturation region (McMahon et al. 1984). The splitting of a band peak in Raman spectra of undersaturated and supersaturated ammonium dihydrogen phosphate (ADP) aqueous solutions was interpreted in terms of a well-ordered quasi-crystalline solution structure (Cerreta and Berglund 1984). The same group performed similar experiments on aqueous solutions of ammonium, sodium, and potassium dihydrogen orthophosphates and concluded that monomolecular units were accompanied by phosphate polymers of varying size, formed by strong hydrogen bonds in a fashion dissimilar to the crystals. Time-resolved Fouriertransform infrared spectroscopy (FTIR) measurements performed during the crystallization of poly(bisphenol A-codecane ether) revealed a time shift between the appearance of the characteristic peaks for the first conformational changes and the appearance of the characteristic peaks for the crystalline phase, suggesting that the intramolecular changes occurred before the intermolecular packing process (Jiang et al. 2003). Kadima et al. (1990) used photon correlation spectroscopy (PCS) to examine the aggregation processes leading to crystallization of the protein canavalin and reported that the barrier to crystallization of canavalin was the formation of trimers. The concentration dependence of aggregate size revealed that much stronger associative interactions held aggregating species in aggregates on their path to crystallization compared with aggregates on their way to precipitation. Aqueous solutions of citric acid were investigated by PCS, which provided evidence for strong attractive forces between the molecules, resulting in the formation of small (~1 nm) clusters even in undersaturated solutions (van Drunen et al. 1993). The molecules in the clusters were in another phase-like structure compared with the crystalline molecules. In another PCS study, the growth of clusters in two kinds of supersaturated lysozyme solutions, which produced orthorhombic rectangular or needle-like crystals, was investigated to understand whether the cluster formation in the early stages of crystallization depended on the final crystal habit (Tanaka et al. 1996). The increase in the cluster size was commonly explained by diffusion-limited aggregation model, which suggested that the random aggregates were formed in the early stages regardless of the final crystal morphology. Nuclear magnetic resonance (NMR) was another spectroscopic technique employed to understand the molecular association in supersaturated solutions. In situ NMR analysis of the formation of alumino- and gallophosphate crystals proposed that crystallization went through a fast amorphous phase formation followed by its redissolution and then crystal growth (Taulelle et al. 1999). The NMR studies on undersaturated and supersaturated solutions of p-acetanisidide (PAC) showed that PAC molecules formed aggregates in both cases, whereas the growth of the aggregates was slower in the former than the latter (Saito et al. 2002). The interactions between adjacent PAC molecules in the crystal structure were also observed in

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the local structures of the aggregates but not in the total structure, indicating that the aggregates were not the crystal nuclei. More recently, neutron scattering was used to investigate the time-averaged structure of undersaturated and supersaturated aqueous urea solutions (Burton et al. 2008). In moving into the supersaturated state, the solutions showed a significant and unexpected change in the self-aggregation of urea molecules. Urea–urea interactions present in the undersaturated solution became suppressed in the supersaturated region, in favor of interactions between urea and water, so overall the urea became more hydrated. This was a surprising finding because the expectation was self-aggregation of urea to occur on supersaturation. Instead, the results suggested that in its supersaturated state the enhanced hydration of urea enables it to remain in solution, contributing to the observed metastabilty of the system, and desolvation of the urea molecules would be an essential rate-determining process. Light scattering has been used by a number of researchers to understand molecular clustering in crystallizing solutions. Mikol et al. (1989) monitored the crystallization of lysozyme by dynamic light scattering and found that the apparent size of scatterers increased up to maximum as supersaturation proceeded, followed by a decrease to the initial value. These authors ascribed the peak in the radius to the time corresponding to nucleation. Azuma et al. (1989) using quasi-elastic light scattering reported the presence of solute clustering in the supersaturated lysozyme solutions. The cluster size was estimated to be 21–25 nm, but the major growth units controlling the growth rate were monomolecular entities because the clusters were dissolving on arriving at the surface of the growing crystal. By contrast, light scattering intensity measurements by Pusey (1991) suggested that tetragonal lysozyme crystal growth was by addition of aggregates preformed in the bulk solution, not by monomer addition. The concentration of aggregates appreciably increased with increasing total protein concentration. In another light scattering experiment by Skouri et al. (1991), the simultaneous presence of two scatterer populations (~2.5 and 270 nm) was detected in supersaturated lysozyme solutions, which were assigned to individual protein molecules and larger particles. When the solutions were undersaturated, only individual molecules were observed. Similarly, Tanaka et al. (1999) detected the formation of aggregates only in solutions from which crystals grew within a few days. The aggregation mechanism for the clusters fitted well with the diffusion-limited cluster aggregation model and displayed several similarities to the liquid–liquid phase separation process. Georgalis et al. (1992, 1993) applied dynamic light scattering to investigate the nucleation of lysozyme crystals and showed that the monomers rapidly aggregated in the diffusion-limited aggregation regime to form fractal clusters in the initial stages of crystallization. Further static light scattering experiments by the same group revealed progressive restructuring of these fractals to compact structures at the later stages of aggregation (Georgalis et al. 1997a, b). Nearly during the same time period, Malkin and McPherson (1993b) performed quasi-elastic light scattering to investigate the crystallization of satellite tobacco virus (STMV) and found that the aggregation process

leading to crystal formation was distinctly different from that producing amorphous precipitate. The aggregation growth rate for crystallization followed an r ~ tn behavior, with n = 0.33–0.54, while n associated with amorphous precipitate was in the range of 0.1–0.2. Similar results were obtained from experiments performed on apoferrritin. Unlike the fractal aggregates proposed by Georgalis et al., these studies concluded that the structure of nuclei was compact and that nucleation proceeded via the classical pathway, where induction periods with fluctuations of cluster size below a critical size were followed by nucleation and growth periods. The critical nucleus size for STMV was estimated to be in the range of ~30 to 7 virus particles in the supersaturation range of 1–1.4 (Malkin et al. 1993a). In many of the experiments, small aggregates of two to six STMV particles were present in solutions during the induction period prior to nucleation. Based on their dynamic light scattering and scanning electron microscopy measurements on supersaturated thermolysin solutions, Sazaki et al. (1993) proposed a two-step mechanism for crystallization, where the first step was the formation of primary particles with average diameters of 60 nm and the second step was the formation of crystals by highly ordered aggregation of these primary particles. By contrast, Rosenberger et al. (1996) speculated that these aggregation scenarios, at least in lysozyme solutions, were the consequences of the higher-molecular-weight protein impurities in the chemicals used during the experiments. Moreover, they analyzed their static and dynamic light scattering data by taking into consideration the interactions between lysozyme molecules and did not observe any sign of stable aggregates except for nuclei in supersaturated solutions (Muschol and Rosenberger 1996; Rosenberger et al. 1996). They concluded that the changes in the scattering profiles actually corresponded to monomeric attractive interactions rather than the formation of aggregates, and failure to account for these interactions would result in unrealistic prenucleation aggregation scenarios. This picture was supported by dynamic light scattering analyses by Bishop et al. (1992), who stated that lysozyme existed predominantly as monomers in supersaturated and crystallizing solutions. Similarly, quasi-elastic light scattering experiments by Veesler et al. (1994) showed that in supersaturated solutions of porcine pancreatic α-amylase, the protein was strictly monodispersed prior to nucleation. The formation of aggregates and polydispersity were induced only after nucleation. By contrast, polydispersity was increased when the concentration of protein was decreased to values at which the solutions were undersaturated. Based on these findings, it becomes clear that the structure of protein nuclei (i.e., whether they are compact or loose aggregates such as fractals) and the existence of prenucleation aggregates in supersaturated protein solutions are still ambiguous, even though light scattering techniques have been widely employed to understand these phenomena. Moreover, light scattering experiments have mostly focused on the nucleation of proteins, which are relatively large and slow-diffusing compounds, ignoring the nucleation of small organic and inorganic molecules. Following these studies, dynamic light scattering

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Figure 3.7 Model of lysozyme crystallization process based on small-angle neutron scattering studies Source: Reprinted with permission from Niimura et al. 1995. Copyright © 1995 by Elsevier.

experiments on aqueous supersaturated and undersaturated solutions of NaCl, (NH4)2SO4, and Na citrate revealed two predominant components, where the smaller components with radii below 1 nm were attributed to the mixtures of solvated ions, and the larger with radii between 50 and 500 nm were attributed to the ion clusters (Georgalis et al. 2000). These findings implied that simple electrolytes such as NaCl can aggregate in aqueous solutions, even at moderate concentrations, probably due to direct electrostatic interactions among ions of opposite charges and hydrophobic interactions among ions caused by exclusion of ions by water. Another scattering technique that was widely used for study of the early stages of protein crystallization from solution was small-angle scattering of X-rays (SAXS) and neutrons (SANS). The first SANS study on the early stages of lysozyme crystallization was reported by Boue et al. (1993), who demonstrated that the radius of gyration Rg of species in solution increases with increasing supersaturation. In this work, supersaturation was achieved via temperature lowering. In the undersaturated domain, the species were likely mixtures of monomers and dimers, whereas the supersaturated domain was characterized by the presence of tetramers and octomers, which suggested that the aggregation proceeds by successive addition of monomers to form dimers, dimers to form teramers, and tetramers to form octomers. Because higher nucleation densities were achieved at higher Rg values, it was assumed that octomers were the building units required for obtaining critical nuclei, whereas tetramers were the smallest possible growth units. Niimura et al. (1995) took this work one step forward by covering lower scattering vectors as well and observed that in supersaturated solutions, aggregates of polydisperse nature with Rg values between 200 and 600 Å coexist with a much larger number of monodisperse particles with Rg values between 25 and 40 Å. The former did not change with

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time, whereas Rg values of the latter increased with time, went through a maximum after about 14 hours, and then started to decrease. It was suggested that the decrease corresponds to the supersaturation decrease caused by crystal growth; consequently, the maximum of Rg indicates the time at which nucleation occurred. The authors dismissed the possibility that aggregates themselves straightforwardly become crystals because the number of aggregates was too large when compared with the number of crystals grown in the solution (Figure 3.7). Supporting this view, SANS contrast variation studies by the same group (Niimura et al. 1999) showed that the aggregates cannot become precursors of nucleation because the molecular orientation in aggregates differs from that observed in crystals. It was found by Minezaki et al. (1996) that lysozyme molecules aggregate even in undersaturated solutions, with a continuous increase in size through an increase in the salt concentration. As a result, given that the number of lysozyme molecules in solution remains constant, the total number of particles decreases, which results in an increase in the average interparticle distance, indicated by a peak shift toward low q values. It was stated that because the time evolution of aggregates in a supersaturated solution, achieved by increasing the salt content, is superimposed on the change that occurs in an undersaturated solution, an exact and full understating of the latter process is required for a complete understanding of the former. Furthermore, SAXS, along with wide-angle X-ray scattering (WAXS), has been used to understand the early stages of polymer crystallization, because the initial development of microstructures determines the final product properties of the crystallized polymer. Induction time of polymers consisting of linear chain molecules is the stage when randomly entangled polymer chains transform to the regular aligned lattice (Imai et al. 1995). The topologic obstruction of such entanglements

Crystal Nucleation

slows down the crystallization process, which makes it easy to observe the structural change during the induction period. By contrast, the difficulty in probing the early stages of crystallization is the weak signals from the small volume fraction of the developing structure (Somani et al. 2002). Imai et al. (1992) investigated the structural formation in the induction period of the cold crystallization of poly(ethylene terephthalate) (PET) and found that the density fluctuations with long-range ordered structure occur before formation of critical crystal nuclei. The ordering process during induction time can be divided into two stages. In the early stage, the scattering behavior can be described by Cahn’s linearized theory for spinodal decomposition, whereas the scattering profile is in accordance with Furukawa’s scaling theory for the cluster growth regime in the late stage. Thus the growing process of density fluctuations is very similar to the spinodal decomposition type of phase separation process. Such phase separation is caused by orientation fluctuations of rodlike segments of PET molecules, in terms of transformation from the isotropic phase to the ordered phase owing to an increase of the polymer chain rigidity (Imai et al. 1993, 1995). When such parallel, ordered dense domains grow to a certain size, longitudinal adjustment occurs, resulting in the more efficient packing of the parallel-oriented chains to form a crystal. Following these findings, Ezquerra et al. (1996) studied the induction period during cold crystallization of poly (ether ketone ketone) (PEKK), which has a higher chain stiffness than PET, by means of simultaneous real-time SAXS and WAXS techniques. They provided evidence of an ordering process prior to crystallization as well, which is described by the spinodal decomposition formalism. The WAXS measurements confirmed that no crystalline order takes place during the induction period. These results suggested that the prenucleation density fluctuations, consistent with the spinodal decomposition, can be considered to be precursors of crystal nucleation. An alternative scenario is one of classical nucleation and growth, where the crystalline nuclei form first, followed by a growth process (Samon et al. 2002). The early stages of melt crystallization in fractioned isotactic polypropylene (IPP) and polyethylene (PE) by simultaneous SAXS and WAXS techniques showed that the SAXS peaks corresponding to density fluctuations with an average spacing of 20–24 and 40–80 nm, respectively, appear prior to detection of three-dimensional crystal ordering by WAXS (Wang et al. 2000). However, the spacing associated with the SAXS peaks was found to be decreasing with time or remaining constant, which opposes to the behavior of spinodal decomposition. Furthermore, the more sensitive technique of polarized light scattering detected larger objects with dimensions growing from 300 nm prior to the appearance of SAXS peaks during IPP crystallization. Thus the earlier detection of density fluctuations was attributed to the lower detection limit of crystallinity by WAXS. A detailed examination of the data indicated that the early stages of crystallization can be described with the conventional nucleation and growth theories because the data follow a simple Avrami form. Supporting this theory, Hikosaka et al. (2003) provided direct evidence for the

formation of nuclei during the induction period of PE. They stated that nucleation had not been directly observed by SAXS until that time because the number density of nuclei was too low to be detected. This difficulty was overcome by significantly increasing the number density by addition of a nucleation agent into the samples. The induction period was defined as the state where an isolated nucleus, which is equal to the isolated lamella, grows until it changes to an isolated stacked lamella. It was found that the volume-averaged size of nuclei does not change much during the induction time, and lamellae start stacking much later than nuclei start developing. Kozisek et al. (2005) proposed a model to describe the nucleation kinetics in the presence of a nucleation agent, and the results of the numerical computations were found to be in good agreement with the SAXS data obtained for PE. There exists another hypothesis that challenges the conventional view of crystallization via nucleation and growth that suggests that the initial stages of polymer crystallization involve several steps. The first step is the formation of a metastable preordered structure in the supercooled melt, which then undergoes different stages to become a stable lamellae step by step (Wurm et al. 2005). The evidence for this theory comes from SAXS experiments (Heck et al. 2000). Simultaneous dielectric relaxation and SAXS experiments during isothermal crystallization of polycaprolactone strongly supported the idea of precrystalline order in the polymer melt before the formation of crystals. By means of transmission electron microscopy (TEM), differential scanning calorimetry (DSC), and time-resolved SAXS and WAXS, Wang et al. (2001) observed the formation of a mesomorphic phase prior to IPP crystallization, which consisted of clusters of ordered helical chain segments with a random assembly of helical hands. Because the crystalline phase required specific registrations of different helical hands in the unit cell, the wrong chain segments in the mesophase were found to undergo a reorganization process to correct the hands, which was essentially a melting process. By contrast, the correct assemblies of helical chains in the cluster domains served as primary nuclei for crystallization, which were considered to be the true precursors to crystallization. Clearly, the mechanism for the early stages of polymer crystallization is an area of considerable controversy that needs further, more definitive experiments. There are a few reports of the application of SAXS or SANS technique to the nucleation of small organic and inorganic molecules. SAXS study on aqueous glycine solutions revealed that the system showed power-law behavior that indicated the presence of fractals in the system. Prior to crystallization, a transformation from mass fractal to surface fractal arrangement was observed, which was interpreted as the rearrangement of fractal clusters into more compact structures to form crystal nuclei (Chattopadhyay et al. 2005). Simultaneous collection of SAXS and WAXS data during the nucleation of a small organic molecule 2,6-dibromo-4-nitroaniline (DBA) in a plug-flow crystallizer implied that the first step in the phase transition is the separation of an amorphous (solid or liquid) phase, which then transforms to crystalline material within a few tenths of a second (Alison et al. 2003).

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Computer simulations and theoretical calculations support the concept of molecular clustering in solutions prior and en route to nucleation. For instance, a simple Monte Carlo simulation on randomly distributed particles over a field of 100 × 100 units showed that the distribution of cluster sizes was almost independent of the concentration and that clustering took place even in undersaturated solutions (Nyvlt 1996). Molecular dynamics simulation on a system consisting of 729 cells, 20 of which were randomly occupied by acetic acid (solute) with the remainder occupied by CCl4 (solvent) molecules, revealed a very fast condensation of solute molecules into small clusters made of two to five molecules, with geometries ranging from head-to-head dimers to small hydrogenbonded oligomers bound into cyclic structures (Gavezzotti and Flippini 1998). Anwar and Boateng (1998) reported an identical result for simulation of crystallization in a solute–solvent system consisting of atoms of two noble gases, characterized by the Lennard–Jones potential function. It was found that the solute particles aggregated into small clusters first, and then these small clusters came together to form a single large cluster, which eventually nucleated to the final crystalline phase. This diffuse precursor structure appeared to have the characteristics of both body-centered cubic (BCC) and face-centered cubic (FCC) phases, whereas the final crystalline phase corresponded to the FCC phase. The same kind of trend was also observed in simulation of nucleation in a Lennard–Jones system closer to the freezing point, where the precritical nuclei with predominantly BCC and liquid-like structures transformed to the more stable FCC structure in the core as they grew to their critical size (ten Wolde et al. 1996). These observations added strong support to Ostwald’s rule, which states that the phase that is formed from the melt is not the most stable phase but rather the phase that is closest in free energy to the liquid phase. Using a density functional theory for Lennard–Jones fluids, Shen and Oxtoby (1996) found significant BCC character in both the planar interface between the stable FCC crystal and the stable liquid and the interface of the critical FCC nucleus in a metastable liquid, supporting these findings. It was suggested that the metastable phases might have large effects on equilibrium interfaces and rates of first-order transitions. Molecular dynamics (MD) simulations on an undercooled Lennard– Jones liquid revealed that crystal nucleation occurred along many different pathways, in which critical solid nuclei could be small, compact clusters with a large FCC component in the core and a BCC component on the surface but also larger, looser clusters with a core less FCC-like but more BCC-like (Moroni et al. 2005). Hence it was concluded that whether a nucleus is critical or not is determined not only by its size but also by its shape and structure. There are also miscellaneous techniques that do not belong to the above-mentioned categories but were used to understand the structure of solutions and provide indirect evidence for the existence of clusters. Khamskii (1969) reported that light transmission through supersaturated solutions increased with increasing supersaturation, and decreased with time, prior to the onset of crystallization, whereas no changes in solution properties such as viscosity, density, refractive index,

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or electrical conductivity were noted. This behavior was attributed to the formation of clusters. Wilson and Haymet (2005) analyzed a polarized light passing through a supercooled solution during the moment of nucleation to determine whether a stable equilibrium crystal was nucleated or a transient phase of different composition or broken symmetry was formed prior to nucleation. This method was based on the different birefringence of various phases. The results on a supercooled clathrateforming liquid mixture showed that the crystal was nucleated directly, with no transient phase. Sedimentation equilibrium studies on undersaturated lysozyme solutions determined reversible polymerization with increasing pH, whereas dimer was found to be the predominant polymer between pH 5 and 9 (Sophianopoulos and Holde 1964). Behlke and Knespel (1996) suggested that the larger oligomers were more stable than smaller ones in supersaturated lysozyme and papain solutions based on sedimentation velocity measurements. By contrast, in a more recent study, the precrystallization solution of lysozyme was ultracentrifuged using a mode where the sedimentation and diffusion were in equilibrium, and the protein concentration gradient in the centrifugation cell was found to correspond to monomeric units (Yoshizaki et al. 2005). Hence it was concluded that the lysozyme molecule predominantly existed as a monomer in undersaturated and supersaturated solutions. Even though the results were inconsistent, centrifugation was introduced as an effective method to identify the growth units for crystallization. In another approach, the degree of aggregation in solutions was determined by monitoring the rate at which lysozyme traversed a semipermeable membrane (Wilson and Pusey 1992). The measured concentration profiles, when compared with the growth rate data of the (110) face, supported the theory that tetragonal lysozyme crystals grow by addition of preformed aggregates rather than monomer addition. A considerable population of aggregates larger than dimers was observed at high concentrations. It was also argued that the ultracentrifugation and sedimentation velocity studies would be inadequate to directly characterize selfassociating systems because the data analysis must be done with respect to a predefined model. Intermolecular crosslinking was another method employed to investigate aggregation of lysozyme molecules in supersaturated solutions (Hall et al. 2005). In this method, proteins were cross-linked by an agent and then separated with different amounts of time between the addition of a crystallization agent (NaCl) and the start of cross-linking. The results suggested that the solutions were highly aggregated. Jullien and Crosio (1991) used fluorescence signal to follow the prenucleation phase for bovine pancreatic ribonuclease A. The signal was obtained from a label attached to the residues, and the measured fluorescence anisotropy was a reflection of the rotational diffusion of the molecules. As small aggregates were detected prior to nucleation, the results demonstrated the potential of this technique to monitor early stages of crystallization. Similarly, Berglund and coworkers (1996) studied the onset of crystallization in supersaturated aqueous glucose

Crystal Nucleation

Figure 3.8 Schema of the classical and novel views on precipitation (not to scale). Pre-nucleation-stage calcium carbonate clusters provide an early precursor species of different ACC phases, giving rise to an alternative crystallization reaction channel Source: Reprinted from Gebauer et al. 2008, with permission from AAAS.

solutions by using a fluorescent probe molecule that incorporated selectively into precrystalline glucose aggregates. In a review aimed at understanding crystal nucleation, Weissbuch et al. (2003) used the technique of grazing incidence X-ray diffraction to demonstrate that crystal nucleation could be achieved employing tailor-made auxiliaries which were either nucleation inhibitors or promoters. The basis of their work was the hypothesis that supersaturated solutions contained molecular clusters that resembled the structure of the macroscopic crystals into which they eventually grew. It was also suggested that the clusters would exist with packings corresponding to all the potential polymorphs in a system. The details of this work and several other relevant examples are discussed in Section 3.7. In another study, Gebauer et al. (2008) measured Ca2+ concentrations, facilitating a quantitative determination of all species present at the different stages of crystallization of calcium carbonate while the supersaturation slowly evolved. This was achieved by slow addition of dilute calcium chloride solution into dilute carbonate buffer to induce supersaturation, causing nucleation and precipitation of calcium carbonate. The prenucleation-stage clusters were independently detected by means of analytical ultracentrifugation (AUC). It was found that pre-nucleation-stage clusters formed on the basis of pH-dependent equilibrium thermodynamics. The clusters showed an average size of approximately 70 ions (pH = 9.00) and exhibited “solute character,” meaning that not surface tension, which is a characteristic property of phase boundaries and is classically attributed to clusters, but hydration energy taking solvent effects into account was ascribed to clusters. The surface tension characterizing a phase interface was established when the critical stage was reached, and amorphous calcium carbonate (ACC) was precipitated at first. This was consistent with previous studies that identified ACC as a post-nucleation-stage precursor phase in calcium carbonate mineralization. AUC experiments provided evidence that the clusters were the nucleation-relevant species, and nucleation

was probably via cluster aggregation, because small cluster species could not be detected after nucleation (Figure 3.8). Further examples can be found in a recent review by Pienack and Bensch (2011), which gives a survey of the in situ methods available for the study of the early stages of crystallization of solids, with examples of actual research demonstrating the necessity, potentials, and the limitations of in situ monitoring of the formation of crystalline solids. Even though these experimental and theoretical studies provided indirect and direct evidences for the existence of clusters in supersaturated solutions prior to nucleation, the mechanism through which these clusters grow into crystals, their role in the nucleation process, and the relationship between their structure and the final crystalline product still remain ambiguous. In other words, they do not provide enough solid information to address the key questions listed in Section 3.1.

3.3 Heterogeneous Nucleation Homogeneous nucleation is uncommon in practice because the presence of foreign particles, such as dust and dirt, and surfaces, such as container walls, induce heterogeneous nucleation by decreasing the activation barrier necessary for nucleation via a reduction in the surface excess energy of prenucleation clusters. This decrease in the activation barrier depends on the contact angle between the clusters and the foreign substance surface, as given by the equation (Volmer and Weber 1926) DGcrit;het ¼ ϕDGcrit;hom ℤ

ð3:12Þ

where ɸ is the contact angle factor and is given by 1 ϕ ¼ ð2 þ cosθÞð1  cosθÞ2 4

ð3:13Þ

Equation (3.13) shows that the lowering of the activation barrier for nucleation occurs when ɸ is less than 1 (Figure 3.9). Because of the lower activation barrier, heterogeneous nucleation

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Deniz Erdemir, Alfred Y. Lee, and Allan S. Myerson Table 3.2 Surface Tension Factor for Heterogeneous Nucleation

θ

ɸ

0

0

5

1.80 × 10−5

10

1.72 × 10−4

30

1.29 × 10−2

45

5.81 × 10−2

60

1.56 × 10−1

90

0.50

120

0.84

180

1.0

Source: Reproduced with permission from Nyvlt et al. 1985.

Figure 3.9 Illustration of the static equilibrium described by the Young equation. Three interfacial tensions γsl, γns, and γnl are balanced at a contact angle θ between the nucleating phase and the surface. (a) Favorable nucleus– surface interactions (i.e., good wetting) results in θ < 90°. (b) Unfavorable nucleus–surface interactions (i.e., poor wetting) results in 90° < θ < 180°. (c) A complete absence of wetting results in θ = 180°, which is the customary assumption for applications of the Gibbs–Thomson equation. Source: Reprinted with permission from Hamilton et al. 2012. Copyright © 2012, American Chemical Society.

occurs at much lower supersaturations than that necessary for homogeneous nucleation, which causes a significant decrease in metastable zone width. Energy considerations show that spontaneous nucleation would occur for a system with zero contact angle, but no such system exists in practice. Partial attraction is possible in a case where the foreign substance and the crystal have almost identical atomic arrangements. It was shown that the energy for nucleus formation was reduced only if the difference in isomorphism between the crystal and the foreign particle was less than 15 percent (Preckshot and Brown 1952). For differences greater than 15 percent, the energy requirements were similar to those for a homogeneous system. The contact angle factor ɸ is given in Table 3.2. The use of foreign surfaces such as Langmuir monolayers, self-assembled monolayers, and tailor-made additives to promote heterogeneous nucleation and influence the polymorph selectivity is discussed briefly in Section 3.7. The key interaction in most of these cases is molecular recognition, where strong interaction between non–covalently bound molecules directs and controls the nucleation and growth of the crystals (Weissbuch et al. 1991).

3.4 Secondary Nucleation Secondary nucleation results from the presence of parent crystals of the same solute in the supersaturated solution. These

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parent crystals have a catalyzing effect on the nucleation phenomenon, and thus nucleation occurs at a lower supersaturation than needed for spontaneous nucleation. Although several investigations of secondary nucleation exist, the mechanisms and kinetics are poorly understood.

3.4.1 Origin of Secondary Nuclei Several theories have been proposed to explain secondary nucleation. These theories fall into two categories—one traces the origin of the secondary nuclei to the parent crystal—that include (1) initial or dust breeding, (2) needle or polycrystalline breeding, (3) contact nucleation (microabrasion), and (4) collision or attrition breeding (macroabrasion). Secondary nuclei can also originate from the solute in the liquid phase, and the theories that take this into account include (1) impurity concentration gradient nucleation and (2) nucleation due to fluid shear. In initial breeding or dust breeding, secondary nuclei originate from the seed crystals. Tiny crystallites are formed on the crystal surface during growth of the seed crystals or as a result of fragmentation during storage. When introduced into the solution, the crystallites act as nucleation sites (Ting and McCabe 1934; Strickland-Constable and Mason 1963). These crystallites are larger than the critical nucleus size, and as a result, the rate of nucleation is independent of the supersaturation of the solution or the stirring rate. This mechanism is important only in batch crystallization, but it is important to minimize this effect by retreating the seed crystals with a solvent. At high levels of supersaturation, needle-like or dendritic crystals are formed. These crystals fragment in solution and serve as nucleation sites. This phenomenon is referred to as needle breeding. At even higher supersaturations, irregular polycrystalline aggregates are formed. Fragmentation of the polycrystals causes them to serve as nucleation centers. This process is called polycrystalline breeding. Nucleation by this

Crystal Nucleation

mechanism is not considered very likely in industrial crystallization. Contact nucleation is probably the most important source of secondary nuclei in the crystallizer. Three types of contact can be envisioned in a crystallizer: crystal–crystal, crystal–stirrer, and crystal–crystallizer wall. The collision of the crystals with contact material results in microabrasion (damage of crystal surface), which results in the generation of secondary nucleation sites. The solute and the solution near such sites are thought to be in a loosely ordered phase near the crystal surface. Hence the preordered boundary layers that are dislodged from the surface by the contact may also serve as nucleation sites. At high stirring speeds, macroabrasion of crystals results in fragments that serve as nucleation sites. As opposed to contact nucleation, which involves microabrasion of the crystals, this phenomenon results in the rounding of the edges and corners of crystals. This process is referred to as collision or attrition breeding. The rate of nucleation by this mechanism is a function of crystal hardness, the concentration of the suspension, and the retention time. Although important from an industrial point of view, this mechanism can be obscured by other mechanisms of secondary nucleation. The impurity concentration gradient theory assumes that the solution is more structured in the presence of a crystal. This increases the local supersaturation of the fluid near the crystal. It was suggested that if an impurity suppresses primary nucleation, secondary nucleation might occur. This happens specifically if the impurity incorporation into the growing crystal is significant because this causes the seed crystal to create an impurity concentration gradient by itself such that the concentration of impurity near the surface becomes lower than that in the bulk solution. This concentration gradient can enhance secondary nucleation. Experimental evidence of the theory was presented for the nucleation of potassium chloride in the presence of lead impurities. As expected, stirring the solution causes the impurity concentration gradient to disappear and hence lower the nucleation rates (Denk 1972). Powers (1963) proposed fluid shear as a mechanism for the generation of secondary nuclei. This possibility assumes that the boundary layer between the crystal and the solution is the source of crystal nuclei. The shearing action of the fluid is sufficient to remove a layer of the adsorbed molecules into the solution, where they grow into crystals. It is generally very difficult to determine the origin of the secondary nuclei experimentally. Since the presence of an adsorbed layer as a source of secondary nuclei was initially proposed, several investigators (Clontz and McCabe 1971; Strickland-Constable 1972) verified this hypothesis by sliding a crystal along a solid surface and observing the nucleation and crystallization behavior. It was found that a growing crystal was necessary for the generation of nuclei and that the nucleation rate was a function of the contact energy and degree of supersaturation. Denk and Botsaris (1972) devised a clever experiment in which the optical activity of a sodium chlorate crystal was used to determine the origin of the nuclei. The crystals were

Figure 3.10 Differentiation of the secondary nucleation mechanisms using enantiomorphous crystals. The fraction of the crystals with the same modification as that of the parent crystal is denoted as percent. Source: Reproduced with permission from Denk and Botsaris 1972.

grown from aqueous solutions in the presence of seed crystals with and without the presence of impurities. The results for crystals grown from pure stagnant solutions are shown in Figure 3.10. If the solid phase was the source of the secondary nuclei, the resulting crystals would have the same form as the parent crystal. If the secondary nuclei originated in the liquid or transition phase, then the ratio of optically different nuclei should be similar to the ratio obtained during spontaneous nucleation. From Figure 3.10, the change in the source of nuclei is observed around a supercooling of 3°C. At higher supersaturations, the ratio approaches that of spontaneous nucleation. In the presence of impurity (12 ppm borax), the crystals obtained for various supersaturations show a similar trend, but the curve shifts to the right, as expected. It was concluded that no single mechanism could explain secondary nucleation. The possible sources are from (1) the parent crystals (needle breeding), (2) the boundary layer near the growing crystal (impurity concentration gradient), and (3) the solution due to reordering of water molecules near the crystal leading to local supersaturation. Garside et al. (1979) developed an experiment in which a crystal could be contacted with a fixed energy. Their data, displayed in Figure 3.11, showed that the crystal size distribution (CSD) in a potassium alum system was a function of the supersaturation of the solution. Larson and Bendig (1976) used a similar apparatus to show the dependence of nucleation on contact frequency. They found that the adsorbed layer requires time for regeneration. This regeneration time can be of the order of a few seconds and decreases with increasing undercooling or supersaturation. It was found that the nucleation rates decline by several orders of magnitude if proper regeneration times cannot be met.

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Deniz Erdemir, Alfred Y. Lee, and Allan S. Myerson

Figure 3.11 Nuclei size distribution from contact nucleation Source: Reproduced with permission from Garside et al. 1979.

3.4.2 Factors Affecting Secondary Nucleation The rate of secondary nucleation is governed by three processes: (1) the generation of secondary nuclei on or near a solid phase, (2) removal of the clusters, and (3) growth to form a new solid phase. Several factors influence these processes: the supersaturation, the rate of cooling, the degree of agitation, and the presence of impurities. The degree of supersaturation is the critical parameter controlling the rate of nucleation. The effect of supersaturation on the nucleation rate is threefold. At higher supersaturation, the adsorbed layer is thicker and results in a large number of nuclei. The size of the critical nucleus decreases with increasing supersaturation. Thus the probability of the nuclei surviving to form crystals is higher. As the supersaturation is increased, the microroughness of the crystal surface also increases, resulting in a larger nuclei population. The role of temperature in the production of secondary nuclei is not fully understood. For several systems, the nucleation rate declined with increasing temperature for a given supersaturation. This was attributed to the faster rate of incorporation of the adsorbed layer onto the crystal surface at higher temperatures. Because the thickness of the adsorbed layer was reduced, the nucleation rates also declined with increasing temperatures. A few contradictory results exist – Genck and Larson (1972) found a decrease in nucleation rate with increasing temperature for a potassium nitrate system and increasing rates with increasing temperature for a potassium chloride system. It was shown by Nyvlt (1981) and others that the nucleation order is not sensitive to temperature variations.

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Stirring the solution leads to a thinning of the adsorbed layer and hence should lead to lower nucleation rates. However, Sikdar and Randolph (1976) found that the nucleation rate increased with the degree of agitation for smaller crystals of magnesium sulfate (8–10 µm) – the nucleation rates were independent of the degree of agitation for larger crystals. The results of Melia and Moffit (1964) on the secondary nucleation of potassium chloride are shown in Figure 3.12. These authors found that the nucleation rate increases with supersaturation and the degree of supercooling and agitation. The effect of hardness of the contact material and crystal hardness on the generation of secondary nuclei has been investigated by several researchers. In general, it was found that a harder material is more effective at enhancing the nucleation rate. For example, it was found that a polyethylene stirrer reduced the nucleation rate by a factor of 4–10 depending on agitation (Figure 3.13). In situ experiments have indicated that microattrition may be an important source of nuclei in this case. Crystal hardness also affects nucleation behavior – a hard, smooth crystal is less effective. Irregular crystals with some roughness are generally more active. The effects of other factors such as the presence of impurities and intentional seeding will be discussed further later in this chapter.

3.5 Nucleation Kinetics Secondary nucleation is a complex phenomenon and is not well understood. A general theory for the prediction of nucleation rates does not exist. Several correlations based on the power-law model have been found to explain most of the experimental data satisfactorily. The power law (from the Becker–Doering relationship) is given by B ¼ kN DCn

ð3:14Þ

The nucleation rate B is normally given in units of number of nuclei forming per unit volume per second. This form is valid if the adsorption layer mechanism described earlier is the source of nuclei. The nucleation rate in this case is independent of the suspension concentration. In industrial crystallizers, most of the nuclei are generated by contact with the crystallizer environment. The nucleation rate in this case is a function of the degree of agitation, the suspension density, and the supersaturation 0

j

B ¼ kN W i MT ðDCn Þ

ð3:15Þ

where W is the agitation rate (usually in revolutions per minute [rpm] or impeller tip speed), and MT is the suspension density (mass of crystals per volume of solution). It is impor0 tant to remember that the constants kN and k N have different units that are interchangeable. When Equation (3.14) is used in a situation in which secondary nucleation is important, the constant kN will actually vary with the conditions in the crystallizer (i.e., suspension density and agitation rate). In some situations, an equation that does not include the effect of agitation is used

Crystal Nucleation Figure 3.12 Dependence of number of secondary nuclei produced on stirrer speed and supercooling in secondary nucleation of potassium chloride Source: Reproduced with permission from Melia Moffitt 1964. Copyright © 1964, American Chemical Society.

Figure 3.13 Effect of agitator speed on secondary nucleation rate for steel and plastic impellers Source: Reproduced with permission from Ness and White 1976.

B ¼ kN″ MT ðDCn Þ j

ð3:16Þ

In this case, k″N may vary with the agitation rate. Correlations are available for nucleation in flow situations. For a potassium–aluminum sulfate system, the nucleation rate was given by Toyokura et al. (1976) for a fluidized bed B ¼ 10DC3:3 Re2:5

ð3:17Þ

B ¼ 0:85DC3:3 Re2:5

ð3:18Þ

and for a fixed bed

where Re is the Reynolds number based on crystal size. Several other correlations can be found in the literature (Nyvlt et al. 1985). The kinetics for secondary nucleation can be measured either by measuring the metastable zone width and the induction time or by counting the number of nuclei formed. One of the methods for determining nucleation rates is to measure the maximum possible supercooling that can be obtained in a saturated solution when it is cooled at different rates (metastable zone width measurement). This polythermal experiment (proposed by Nyvlt 1968) is carried out in a jacketed crystallizer cooled by a circulating water–ethylene glycol bath accurate to ±0.1°C. The temperature can be increased or decreased at a constant rate by a programmed controller. The crystallizer is fitted with an accurate thermometer (±0.1°C) to read the solution temperatures. Approximately 200 ml of saturated solution is placed in the crystallizer and allowed to equilibrate thermally. The solution is stirred at a constant rate and cooled slowly until a number of small crystals are formed. The temperature of the solution is then raised at a very slow rate until the last crystal disappears. This temperature is denoted as the saturation temperature Ts. The solution is then heated to a temperature 1°C above Ts and maintained for 30 minutes. The solution is then cooled at a constant rate (r1), and the temperature at which the first crystal appears is noted (T1). The difference between this temperature and the saturation temperature is denoted as ΔT1,max for cooling rate r1. The experiment is repeated for two different cooling rates. The measurements also can be carried out in the presence of solid phases, where two to four large crystals are added to the system and the maximum supercooling is observed in a similar fashion. It must be noted that the visual inspection depends on the subjectivity of the observer and the illumination and transparency of the solution, which can result in inconsistencies. To avoid this situation, nucleation temperatures are often determined by using conductometry, ultrasonic devices, or optical

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Deniz Erdemir, Alfred Y. Lee, and Allan S. Myerson

instruments that can measure turbidity (Omar and Ulrich 1999; Hu et al. 2001; Parsons et al. 2003). The nucleation rate at the metastable limit can be approximated as
 dC DC ¼ ð3:19Þ DTmax dT The maximum supersaturation is given by
 dC DC ¼ DTmax dT

ð3:20Þ

Combining Equations (3.19), (3.20), and (3.14), simplifying, and taking logarithms yield logðr1 Þ ¼ logkN mlogðDTmax Þ þ ðm  1Þlog

dC dT

ð3:21Þ

where m is used in place of n to signify an apparent nucleation order. Thus a plot of the cooling rate and maximum possible supercooling will give the apparent order m and the rate constant kN. If the nuclei are visible immediately after they are formed, the true order n and the apparent order m are identical. In all other cases, the true nucleation order can be found by knowing the exponent for crystal growth g. This relationship is given by n ¼ 4m  3g  4

ð3:22Þ

The dependence of the cooling rate on the maximum attainable supercooling for a sodium chromate system (Nyvlt 1968) is shown in Figure 3.14. The presence of a solid phase (filled circles) decreases the maximum supercooling, but the exponent remains the same. A nomograph for estimation of the apparent nucleation order is given in Figure 3.15. The example shows that for a supercooling ratio of 3 and a cooling rate ratio of 9.5, the apparent order of nucleation is 2.07. The nucleation rate also can be determined by observing the time elapsed between the creation of supersaturation and

the formation of a new phase. This time interval is defined as the induction time and is a function of the solution temperature and supersaturation. The formation of a new phase can be detected in several different ways – for example, by the appearance of crystals or by changes in properties (e.g., turbidity, refractive index) of the solution. The induction time tind is the sum of the time needed for reaching steady-state nucleation ttr, the nucleation time tn, and the time required for the critical nucleus to grow to a size large enough to be experimentally detected tg. Thus tind ¼ ttr þ tn þ tg

ð3:23Þ

It can be shown (Sohnel and Mullin 1988) that the transient period is unimportant in aqueous solutions of moderate supersaturations and viscosities. However, it is significant at very low supersaturations and in the case of large solute molecules, such as polymer or protein molecules, which might need a relatively long time to aggregate into stable stationary-on-average subcritical clusters (Izmailov and Myerson 2000). In such a case, the following analysis is not applicable. Kubota et al. (1986) have suggested a method to take into account the transient period. If the transient period can be ignored, the induction time is a function of the nucleation and growth times. Three cases emerge: 1. tn ≫ tg. 2. tn ~ tg. 3. tn ≪ tg. If the nucleation time is much greater than the growth time, the induction time is inversely proportional to the steady-state nucleation rate, and " # Fσ 3 V 2 ϕ tind ¼ Aexp ð3:24Þ ðkTÞ3 ln2 S where ɸ is the wetting angle, which is 1 for homogeneous nucleation and is less than 1 for heterogeneous nucleation [see Equation (3.13)], and F is the shape factor ratio. Figure 3.14 The dependence of cooling rate on the maximum attainable supercooling in aqueous sodium chromate solutions Source: Reproduced with permission from Nyvlt 1968.

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Figure 3.16 Induction period as a function of supersaturation for CaCO3 precipitation at 25°C showing regions of homogeneous and heterogeneous nucleation Source: Reproduced with permission from Sohnel and Mullin 1988.

Figure 3.15 Nomograph for estimation of the apparent nucleation order. The ratio of the cooling rates and the observed widths of the metastable zone are denoted as points on the left and right axes, respectively. The apparent nucleation order is given by the value at which the straight line joining those points intersects the diagonal. Source: Reproduced with permission from Nyvlt et al. 1985.

If the nucleation and growth times are of the same order of magnitude, the induction time for nucleation followed by diffusion growth is given by (Nielsen 1969) " # v2=3 2Fv2 ϕ3 tind ¼ exp ð3:25Þ 2Dx1=5 5ðkTÞ3 ln2 S where x is the solute mole fraction. The induction time for nucleation followed by polynuclear growth is given by Sohnel and Mullin (1988). If the nucleation time is much smaller than the growth time, the induction times for various cases are given by Sohnel and Mullin (1988). For mononuclear growth, the induction time is given by " # d3 Fσ 2 v4=3 tind ¼ exp ð3:26Þ 6DS rc ðkTÞ2 ln2 S where Ds is the surface diffusion coefficient, and rc is the critical radius of the nuclei. In general, the induction time is given by the expression tind ¼ KSn 0

ð3:27Þ

However, Equations (3.23)–(3.26) are helpful in determining the mechanism of nucleation and growth over a wide range of supersaturations. For example, the validity of

Equation (3.24) has been verified for a number of systems from sparingly soluble (e.g., BaSO4, BaCrO4, and SrSO4; Nielsen 1969; Nielsen and Sohnel 1971) to readily soluble (e.g., potassium and potassium–aluminum phosphates; Joshi and Anthony 1979). A plot of log(tind) versus (logS)−2 at a constant temperature is shown for CaCO3 in Figure 3.16. From this figure it is clear that at low supersaturations the nucleation is heterogeneous [f(θ) < 1], whereas homogeneous nucleation occurs at higher supersaturations. It is difficult to distinguish between the mechanisms proposed by Equations (3.24) and (3.25) by constant-temperature experiments. However, experiments at different temperatures can be done to differentiate between the two mechanisms (Mullin and Osman 1973) because the surface tension σ is temperature dependent. Further experimental evidence can be found in Sohnel and Mullin (1988). Nucleation kinetics are often obtained from continuous experiments employing a mixed suspension, mixed product removal (MSMPR) crystallizer and the concept of population balances. This technique is also used to obtain crystal growth kinetics. The details of the population balance and its uses will be discussed in Chapter 6. A review of methods to estimate nucleation kinetics from batch and continuous experiments can be found in Tavare (1995). In this section, we have described two methods to determine the kinetics governing the nucleation process. The first method, which uses the width of the metastable zone, is easy to use and gives an apparent order of nucleation. The second method uses the induction time to predict the mechanism and order of nucleation processes. A third method, which employs population balance techniques and an MSMPR crystallizer, will be described in Chapter 6.

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Figure 3.17 Phase diagram for a solid–liquid system

3.6 Control of Nucleation The concept of a metastable zone plays an important role in understanding and control of nucleation behavior of each system. In supersaturated solutions, exceeding the solubility does not always result in instantaneous nucleation. For instance, distilled water can be undercooled by more than 30° C below its freezing point without solidifying (Oxtoby 1998). In another example, it was reported that samples of salol (phenyl salicylate) can be kept in the metastable state for periods of several years (Dunitz and Bernstein 1995). The metastable phases persist over long periods of time because there are kinetic barriers to first-order phase transitions. When the supersaturation is increased, eventually a supersaturation is reached at which nucleation occurs spontaneously. This is called the metastable limit. A precise thermodynamic definition of the metastable limit is the locus of points where ∂2G/∂x2 = 0. This set of points is known as the spinodal curve. The zone between the saturation curve and the spinodal curve is defined as the metastable zone (Figure 3.17). In industrial crystallization, the effective metastable zone width is much smaller than that defined by the spinodal curve because the presence of dust and dirt, container walls, and other operating conditions such as agitation induce heterogeneous nucleation at lower concentrations. A number of investigators have used electrodynamic levitation of single-solution droplets to avoid these factors and achieve homogeneous nucleation, which provided valuable information about the actual metastable zone limits (Cohen et al. 1987; Knezic et al. 2004). The effective metastable zones widths, by contrast, are obtained by either cooling a solution at a constant rate and recording the temperature at which the first crystals are observed visually or cooling solutions rapidly to a given temperature and measuring the minimum time required for crystallization. It must be emphasized that the measured effective metastable zones are strongly influenced by solution history (how it was made and stored), cooling rate, solution properties (such as viscosity), volume of the system, mechanical effects (such as stirring), and impurities (Maurandi 1981; Nyvlt 1983; Kim and Mersmann 2001). For instance, dispersing a solution

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into collection of small volumes isolates heterogeneous nucleants within a fraction of drops and thus forces spontaneous nucleation to occur at higher supersaturations (RodriguezHornedo and Murphy 1999). The measurement of effective metastable limits and a tabulation of results for several inorganic species can be found in Section 3.5 and Chapter 1. It is important to gain knowledge about effective metastable zone widths in order to control the nucleation. In fact, the effective metastable zone is used to define a working zone for industrial crystallization processes. When the operations are run at close to or greater than the upper limit of the effective metastable zone for a given crystallizer, in other words at high supersaturations, the crystallization is dominated by nucleation. Nucleation-dominated processes are usually avoided because they result in the formation of a large number of small crystals in a rather uncontrolled manner, which can cause filtration problems. Other potential issues encountered on nucleation at high supersaturations can be listed as lack of control of polymorphism, oiling-out, impurity and solvent entrapment, agglomeration, large batch-to-batch variation, and difficulty in scale-up. Difficulty in reproducible scale-up of a nucleation-dominant process can be associated with mixing issues, particularly nonuniform local supersaturations caused by scale-dependent mixing times. It must be noted that there are cases where a nucleationdominated process may be desirable, for example, to produce small crystals to improve the bioavailability of a pharmaceutical product. In such cases, impinging jet crystallization might be a good choice because this setup can achieve rapid mixing of two streams, possibly before the nucleation occurs, reducing the influence of mixing scale-up on the nucleation rate. Seeding the supersaturated solution with the crystalline material of the solute is the frequently employed method in industrial crystallization to prevent nucleation-dominated processes. Seeding provides control over the overall crystallization process by preventing spontaneous nucleation, hence ensuring batch-to-batch reproducibility and improving process robustness. By seeding, the start of the crystallization process is well defined regardless of the variations between batches, such as the presence of foreign particles or seeds from previous batches, differences in impurities, and differences in mixing scale-up. The other benefits of seeding include providing control over the polymorphic form (as discussed in Section 3.7), preventing oiling-out/amorphous formation (as discussed in Section 3.8), manipulating particle size and size distribution, or even, for some cases, manipulating particle shape. The effectiveness of seeding depends on several factors: quality of seeds, quantity of seeds, timing of seed addition, and rate of supersaturation generation following seeding. The important quality attributes of seed crystals are high purity, correct polymorphic form, defined particle size distribution (generally obtained by milling), and for some cases correct morphology. Larger seed crystals yield more secondary nuclei. Smaller seed crystals generally follow the path of the liquid and have a smaller probability of coming in contact with the stirrer or walls to generate secondary nuclei. It is recommended to

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slurry the seeds in the same solvent as used in the crystallizer in order to introduce the seeds to the system as well-dispersed particles because the storage of dry seeds may result in agglomeration. Making a seed slurry also helps to condition the surfaces of the seeds by dissolution of shards, provide surface activation, dissolve very fine materials to make the seed uniform in size, and recrystallize any amorphous surfaces. The removal of tiny crystallites on the surfaces of seed crystals reduces the chances of secondary nucleation owing to the initial breeding mechanism. Typically, 0.1–1% of seeds are sufficient to prevent oiling-out and provide good control of nucleation. In order to suppress the nucleation of unwanted forms, larger amounts of seeds of the desired form may be necessary. To manipulate the shape of the crystals, it may be necessary to add more than 10% of seeds. The seed quantities required to achieve specific particle size can be calculated by a simple equation relating seed ds and product size dp to the mass of the seed added ms and the mass of the product mp: ms/mp = (ds/dp)3. In general, 0.1% of seeds are recommended for a 10-fold increase in particle size. The supersaturation at the addition of seeds is adjusted by using knowledge of the metastable zone width. The seeds must be introduced to the crystallizer after reaching saturation in order to avoid dissolution of seeds but before spontaneous nucleation takes place. This can be sometimes challenging for cooling crystallizations if the solubility dependence on temperature is steep. Typically, for cooling crystallization, seeding is performed at ~5°C below saturation temperature. During antisolvent addition, the seeds can be slurried in the

antisolvent, and this slurry can be added to the crystallizer close to the saturation point. The goal is to be closer to the solubility curve than to the spinodal curve. If the metastable zone is very narrow, the process would need robust temperature control. Online measurement of the solution concentration, for instance, by FTIR, near-infrared (NIR), or Raman spectroscopy, also would be helpful to detect the correct seeding point, as well for monitoring the events after seeding, such as determining whether or not the seeds persist and how their number and size change with time. After the addition of the seeds, it is important to allow enough time for the seeds to relieve supersaturation and solution to reach saturation before generating more supersaturation. At high temperatures, where the seeding generally takes place, a significant amount of solute may crystallize during this hold time. Online monitoring of the solution would be useful to determine the time necessary to reach saturation. The next step is the crystallization of the remaining materials in the solution. At this stage, it is recommended to create a minimum amount of supersaturation throughout the process (i.e., to operate closer to the solubility curve) to reduce the risk of secondary nucleation, which would result in bimodal particle size distributions, and to maintain a constant growth rate. For cooling crystallization, this is achieved by starting the cooling slowly and increasing the rate over time as the overall crystal surface area grows. For the antisolvent addition method, a cubic addition method is recommended. The possible sources of primary and secondary nuclei in an industrial crystallizer are given in Table 3.3 along with

Table 3.3 Sources of Nuclei in Industrial Crystallization

Source of nuclei

Type of nucleation process

Prevention or remedy

Boiling zone

Primary

Reduce specific production rates, increase crystal surface area

Hot feed inlet

Primary

Enhance heat dissipation, reduce degree of superheating, carefully choose inlet position

Inlet of direct coolant

Primary

Enhance heat dissipation, reduce temperature of coolant chosen inlet position

Heat exchangers, chillers, etc.

Primary

Reduce temperature gradients by increasing surface area, increase liquid velocities

Reaction zone

Primary

Enhance mixing and dissipation of supersaturation, increase crystal surface area

Cavitating moving parts

Primary

Adjust tip speed, suppress boiling by sufficient static head

Crystal–crystallizer contacts: collisions with moving parts (impellers, pumps, etc.)

Secondary

Adjust tip speed and design configuration, coat impellers with soft materials, reduce, if possible, magma density and mean crystal size

Crystal–crystal contacts: crystal grinding in small clearance spaces (impeller/draft tube, pump stator/rotor)

Secondary, attrition breakage

Carefully specify all clearances and hydrodynamics of twophase flow therein

Crystal–solution interaction: fluid shear, effect of impurities, etc.

Primary or secondary

Reduce jetting, get to know the effect of impurities for each particular system, prevent incrustation

Source: Data adapted from Jancic and Grootscholten 1983.

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suggestions for preventing excessive nucleation (Jancic and Grootscholten 1983). Primary heterogeneous nucleation is possible at places where high supersaturation is being created – for example, in heat-transfer equipment. This could lead to crystal deposits on the surfaces. Undesirable nucleation can be reduced by choosing proper production rates and by allowing adequate mixing. Contact nucleation is the most important source of secondary nuclei because collisions with stirrers, pumps, impellers, and crystallizer walls lead to fragmentation and generation of secondary nucleation sites. Small clearances between equipment can also lead to breakage of crystals via crystal contact. Nucleation via this mechanism can be controlled by varying the frequency and energy of contact. Padded impellers and nylon or Teflon stirrers are known to reduce the secondary nucleation rate. Allowing for adequate clearances and optimizing fluid flow conditions can also lead to lower nucleation rates.

3.7 Nucleation in Polymorphic Systems Polymorphism is an important solid property determined during the initial stages of crystallization. It is defined as the ability of a chemical to exist in two or more distinct crystalline phases that have different arrangements of molecules in the crystal lattice. There are numerous instances in industrial environments where control of the polymorphic form of a compound is crucial. However, nucleation of polymorphs can be difficult to control, even in systems whose crystallization processes seem well understood. The most straightforward technique to control polymorphism is via seeding, which is exercised during the nucleation phase by adding seeds of the desired form and thus overriding spontaneous nucleation (Doki et al. 2004; Donnet et al. 2005). If the desired form is the thermodynamically stable modification, development of the seeded crystallization process is very simple because there is no danger of losing this form, for example, via a solvent-mediated phase transformation. In addition, this process is self-regulating because any small amounts of an unstable polymorph in the system would eventually transform into the desired stable form. Only the border of the metastable zone should be avoided to prevent spontaneous nucleation of an unstable form (Beckmann et al. 2001). If the desired form is a metastable polymorph, the seeding procedure must be carried out with great precaution because of the risk of transformation into a more stable polymorph. The equipment used must be thoroughly cleaned with solvent between each operation, and the seeds must be free from the stable polymorph (Veesler 2005). Polymorphism can also be controlled by understanding the dependence of crystallization behavior of the polymorphs on the operational factors. For this purpose, Kitamura (2002) categorized the controlling factors in crystallization of polymorphs by including the most influential factors, such as supersaturation, temperature, and stirring rate, in the primary factors while grouping the factors due to external substances, such as solvents and additives, in the secondary factors.

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Polymorphic outcome of a nucleation process can be controlled via manipulation of the supersaturation level because Ostwald’s rule of stages indicates that at low supersaturation, the stable form preferentially crystallizes, and at high supersaturation, the metastable form tends to precipitate (Ostwald 1897). This is because, according to Ostwald’s rule, the phase that is formed from the solution is not the most stable phase but rather the phase that is closest in free energy to the liquid phase – in other words, the phase that can be reached with minimum free energy loss. This form then transforms into the next most soluble form until the least soluble (thermodynamically most stable) form is achieved. For example, L-glutamic acid and 2,6-dihydroxybenzoic acid (DHB) are dimorphic systems that follow Ostwald’s rule of stages (Blagden and Davey 2003). However, despite its success, this rule is not universally valid because crystallization outcome is affected by many other parameters, such as temperature, nature of solvent, and additives. For example, it was reported that the temperature effect on the relative nucleation rates of the L-glutamic acid polymorphs is more remarkable than the effect of the supersaturation ratio (Kitamura 2002). Polymorphs that nucleate simultaneously, which are known as concomitant polymorphs, are also exceptions to Ostwald’s rule because this phenomenon is controlled by both thermodynamic and kinetic factors (Yu 2003). Examples of concomitant polymorphism can be found in Bernstein et al. (1999, 3440–61) and Lee et al. (2005). As described in the preceding paragraph, a high supersaturation ratio is necessary to access metastable forms of a compound. Crystallization from capillaries is an ideal technique for producing an environment with high supersaturation because small volumes of solution isolate heterogeneous nucleants and reduce turbulence and convection (RodriguezSpong et al. 2004). An additional benefit of this technique is that the crystals can be analyzed in situ by using powder X-ray diffraction (PXRD). Successful examples of the application of crystallization in capillaries for metastable polymorph generation are that of 5-methyl-2-[(2-nitrophenyl)amino]-3thiophenecarbonitrile (ROY), the nonsteroidal anti-inflammatory drug nabumetone, and an active pharmaceutical ingredient in the antihyperglycemic drug metformin hydrochloride (Chyall et al. 2002; Hilden et al. 2003; Childs et al. 2004). Using molecularly well-defined surfaces as templates is another route to controlling polymorph nucleation. Several investigations demonstrated the strong influence of Langmuir monolayers on the polymorphic selectivity, nucleation rate, and crystal morphology and orientation by designing monolayers with an ordered two-dimensional interface that structurally mimics a particular crystal plane of the crystallizing substance (Frostman and Ward 1997; Landau et al. 1989; Lee et al. 2002). This created favorable interactions between the monolayer surface and individual molecules of the crystallizing face, which facilitated the formation of nuclei by lowering the surface energy. The technique, called ledgedirected epitaxy (LDE), has been used successfully to selectively nucleate a particular polymorph via a dihedral angle

Crystal Nucleation

Figure 3.18 (a) Schematic representation of ledge-directed epitaxy on a single-crystal substrate and the role of LDE in selective nucleation of polymorphs. Molecules are thought to attach preferentially to the ledge site due to enhanced stability associated with interfacial interactions with two substrate planes instead of only one available from attachment to the terrace. (b) The dihedral angle between two low-energy planes of the aggregate (θagg) corresponding to polymorph A matches the dihedral angle of the substrate (θsub), providing that interfacial contact occurs at both planes of the ledge site. (c) In contrast, a polymorph B in which the dihedral angle between two low-energy planes differs substantially from that of the substrate ledge (θagg ≠ θsub). Hence, the free energy of the prenucleation aggregate for polymorph A and the corresponding activation energy for nucleation are lowered by interaction with the substrate ledge site, selectively nucleating polymorph A over polymorph B. Source: Reprinted with permission from Bonafede and Ward 1995. Copyright © 1995, American Chemical Society.

match between the ledges of an organic crystal substrate and two crystal planes of the growing crystal (Figure 3.18; Carter and Ward 1993; Bonafede and Ward 1995; Mitchell et al. 2001). More recently, selective nucleation of the less stable form of 2-idio-4-nitroaniline was achieved using selfassembled monolayers (SAMs; Hiremath et al. 2004). In addition, Rodriguez-Hornedo et al. (1992) showed that the metastable anhydrous form of theophylline promotes the nucleation of stable monohydrate crystal, which suggests that the surface of a metastable phase can also serve as a template for the nucleation of another form. During the last 20 years, considerable progress has been made in the control of nucleation of polymorphs using “tailor-made auxiliaries” (Weissbuch et al. 1995). The underlying principle is to stabilize or destabilize prenucleation aggregates of a particular polymorph by designing additives that would either promote or inhibit the nucleation of these aggregates based on the assumption that the prenucleation aggregate for a polymorph resembles the mature crystal (Weissbuch 2003). Inhibitors can be designed so that they bind selectively to a face of a particular polymorphic form, thus preventing its growth and permitting unhindered growth of the other crystalline forms. The successful design of these auxiliaries depends on a detailed knowledge of the packing arrangements of all species that might appear in the system and is facilitated by distinguishing differences in their

symmetry, such as one of the polymorphs belonging to a centric and the other to a noncentric space group (Davey et al. 1997; Caira 1998). Weissbuch et al. (1994) demonstrated this concept on glycine using additive racemic hexafluorovaline, which inhibited the growth of centric α-glycine by binding to its fast-growing face, allowing crystallization of the noncentric γ-form instead from aqueous solution. Using rigid additives that mimic the molecular conformation in the stable β-polymorph of L-glutamic acid, Davey et al. (1997) were able to selectively inhibit its appearance and hence crystallize the metastable α-structure. In a more recent attempt, two new stable and unexpected polymorphs of 1,3,5-trinitrobenzene have been crystallized when a trisindane additive has bound to all crystal faces of the metastable form and inhibited its growth (Thallapally et al. 2004). Even though these cases and several other examples in the literature (Addadi et al. 1982; Weissbuch et al. 1987; Davey et al. 1994; Blagden and Davey 2003) demonstrated the potential of tailor-made additives in the control of polymorphism, it must be emphasized that this strategy is rarely applicable in the pharmaceutical industry because it involves the addition of chemical species in the process that remain soluble and do not easily satisfy legal constrains (Veesler et al. 2005). Several investigators showed that a magnetic field also can influence the polymorphic outcome during nucleation.

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Whereas the calcium carbonate crystals formed in the absence of a magnetic field were found to be in calcite phase, the particles nucleated in a magnetic field consisted mainly of aragonite, the more soluble form of calcium carbonate, which could be used as a way of preventing scale as an alternative to chemicals (Kobe et al. 2001). The formation of aragonite rather than calcite in the presence of a magnetic field was explained by the hypothesis that the kinetic energy required by Ca2+ and CO32− ions to achieve the energy of the electronic ground state of aragonite and then to overcome the potential barrier of this much stiffer ground state is provided by the magnetic field (Kobe et al. 2002). The use of an electric field also was demonstrated for the control of polymorphism of calcium carbonate when aragonite was grown under calcite-favorable conditions through epitaxy with inorganic substrates that were oriented by an electric field (Kim et al. 2003). In addition to the experimental evidence, theoretical studies have been reported on the influence of external fields on polymorphic outcome. Even though the freezing of water is extremely difficult to simulate because of its disordered hydrogen-bond network, which permits a large number of possible network configurations, molecular dynamics simulations by Svishchev and Kusalik (1994) showed that supercooled liquids under atmospheric pressures can be completely transformed into a cubic modification of ice within 200 ps when subjected to a homogeneous static electric field. They explained the reason why the cubic form is obtained rather than the expected hexagonal form by the fact that the diamond-like packing of cubic ice can support an ideal parallel arrangement of the molecular dipoles by the electric field better than isomorphous hexagonal packing. The same authors, in later simulation work, found that the application of an electric field to liquid water under pressures of 3–5 kbar produces a new ice polymorph with an open quartzlike structure, which demonstrates the potential to generate new forms of ice via the electrofreezing of water (Svishchev and Kusalik 1996). The crystalline form also can be affected by light, as demonstrated in several studies. Through non–photochemical laser–induced nucleation (NPLIN), two different polymorphs of glycine were crystallized from aqueous solutions depending on the laser polarization state, circular polarization producing the α-form and linear polarization generating the γ-form (Garetz et al. 2002). This was explained by different efficiencies of linearly and circularly polarized lights in aligning the distinct building blocks of the two polymorphs. Liao et al. (2004) found that the exposure of fresh phenalenyl radical solutions to visible light can give rise to the nucleation of a different crystalline form from the one that crystallizes in the absence of light. Because these two forms were shown to differ by the presence or absence of a C—C bond between unpaired electrons of the parent radical, the authors speculated that photoexcitation of the radicals in solution was responsible for the formation of dimer and hence crystallization of a new form. The solvent system is one of the most important factors that influence the polymorphic outcome of a nucleation process.

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For instance, Iitaka (1961) showed that glycine crystallizes in α-form in aqueous solutions while precipitating in γ-form from water solutions containing acetic acid or ammonia. By contrast, addition of methanol or ethanol to aqueous glycine solutions induces the precipitation of the least-stable β-form (Weissbuch et al. 2005). Experiments designed to study the effect of solvent on the nucleation of polymorphs of sulfathiazole revealed that some solvents selectively favor the nucleation of a particular form or forms (Khoshkhoo and Anwar 1993). To explain this behavior, the authors proposed a kinetic mechanism in which the solvent acts by selective adsorption to certain faces of some of the polymorphs and thereby either inhibits their nucleation or retards their growth to the advantage of others. In addition, the formation of clusters in undersaturated or supersaturated solutions is usually governed by the ability of the solvent to disturb or promote particular hydrogen-bonding networks, which was shown by numerous groups to control the polymorphic outcome of the crystallization. Some of those studies are briefly summarized next.

Molecular Self-Association and Polymorphism Davey et al. (2001) used a combination of solubility, spectroscopic (ultraviolet [UV]), crystallization, and molecular modeling techniques to demonstrate that there can be a direct link between the solvent-induced molecular self-association in solution and the crystallized solid form. For their model system, 2,6-dihydoxybenzoic acid (DHB)–toluene solutions rich in hydrogen-bonded dimers were biased toward nucleation of a structure consisting of dimers, whereas chloroform solutions rich in higher-order chainlike aggregates produced the polymorph for which these were the growth units. In other words, solvent-dependent prenucleation aggregation controlled the appearance of the two polymorphic modifications. This lead to the conclusion that, at least at low supersaturations, where kinetic effects were minimized, nucleation processes in polymorphic systems may be used to probe the nature of molecular aggregation in nucleating solutions. Molecular dynamics simulations by Hamad et al. (2006) demonstrated that the nature of the solvent was able to influence significantly the nature of the initial 5-fluorouracil intermolecular interactions and hence the polymorphic outcome of the crystallization. Cyclic dimers were observed to form readily in dry nitromethane, which was consistent with crystallization of the doubly bonded ribbon structure of form 2 from this solvent, whereas the strong binding of water to 5-fluorouracil molecules hindered the formation of cyclic dimers and consequently promoted crystallization of form 1 from aqueous solutions. Molecular dynamics simulations to investigate the possible configurations of two tetrolic acid (TTA) molecules in a solvent box containing 226 CCl4 solvent molecules predicted the existence of chain and more persistent cyclic dimers in solution, which was in agreement with the existence of two polymorphs, one with the cyclic and the other with the chain motif (Gavezzotti et al. 1997). When the solution behavior of TTA was studied via molecular dynamics simulations in four organic solvents, the results suggested that strong interactions

Crystal Nucleation

Figure 3.19 Link hypothesis. Circles highlight the structural synthons in the crystal phase. Source: Reprinted with permission from Chen and Trout 2008. Copyright © 2008, American Chemical Society.

between TTA and solvent molecules (ethanol or dioxane) prevented the formation of carboxylic acid dimers in solution and thus promoted the crystallization of TTA in a catemer-based form or a solvate form. Weak interactions, however, between TTA and solvent molecules (carbon tetrachloride or chloroform) facilitated the formation of carboxylic acid dimers in solution and thus promoted the crystallization of a dimerbased crystal (Chen and Trout 2008; see Figure 3.19). Similarly, application of FTIR spectroscopy to concentrated solutions of TTA showed a direct relationship between molecular association in solution and H-bonded motifs in the subsequently crystallized solid phases (Parveen et al. 2005). It was observed that the solutions rich in dimers nucleated the αpolymorph, which was based on a classic dimer motif, whereas the solutions in which dimer formation was disturbed produced the β-form, which was based on an H-bonded chain. The concentration-dependent changes in 1H NMR chemical shift of a compound in acetonitrile and chloroform solutions were demonstrated to allow making predictions about how these compounds were packed in the crystalline state because the structures of dimer motifs characterized in solution were in agreement with the structures of dimers found in the crystal lattices (Spitaleri et al. 2004). Therefore, the solution

NMR was introduced as a potential methodology for crystal structure prediction and polymorph identification. Most recently, Kulkarni et al. (2012) demonstrated the reproducible effect of solvents on the formation of isonicotinamide (INA) polymorphs via use of Raman and FTIR spectroscopy. In solvents with strong hydrogen bond acceptors, the dominant configuration of the INA molecules with respect to each other is that of amide–pyridine heterosynthons (head-totail chains; Figure 3.20). Similarly, solvents with strong hydrogen bond donors lead to dominancy of amide–amide homosynthons (head-to-head dimers). It was concluded that this self-association in solution controlled the polymorph nucleation of INA by controlling the building unit attaching to the nucleus. As a conclusion, it is evident that there is a connection between the self-association of solute molecules in solution and the final crystal structure. These findings are consistent with the observation made by Margaret Etter that molecules in solution often tend to form different types of hydrogenbonded aggregates and that these aggregate precursors are related to the crystal structures that form from the supersaturated solution (Byrn et al. 1994). Solvent structure plays an important role in determining the relative stability of various

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Figure 3.20 In solutions, INA can be present in a chainlike structure (left) or in a dimer-like structure (right) depending on the solvent system, which controls the polymorph nucleation of INA. Source: Reproduced from Kulkarni et al. 2012, with permission of the Royal Society of Chemistry.

growth synthons in solution. This suggests that a systematic analysis method of the association processes in solutions would be beneficial in control of polymorphism and crystal nucleation, as well as polymorph discovery, by aiding in the rational selection of solvents to obtain desired polymorphs during crystallization.

3.8 Methods to Induce Nucleation The tendency toward crystallization may be significantly reduced by conformational flexibility. Because flexible molecules exist in solutions or melts as mixtures of energetically similar conformers, the process of crystallization must select the “right” conformers from among the “wrong” conformers, which would reduce the nucleation rate (Yu et al. 2000). It is well known that the conformation of proteins in solution is affected by solvent–protein interactions and that appropriate solvent selection can stabilize the protein conformation that can crystallize. In a recent review by Derdour and Skliar (2014), the findings indicated that systems displaying multiple conformers in solution are likely to be more difficult to nucleate if the energy barrier of conformational change in solution is high. The high energy barrier causes the conformers in solution (wrong conformer) to have a relatively long half-life, which results in low intrinsic supersaturation of the conformers that crystallize (right conformer). The authors proposed strategies to identify routes to first crystals from slow interconverting conformers, which are based on designing conditions to obtain optimal supersaturation profiles that favor nucleation by adjusting the total concentration, temperature, and solvent system. These strategies can be generalized for systems that are difficult to nucleate. The tendency toward oiling-out increases at high supersaturation levels, such as on rapid generation of supersaturation by reverse addition of a solution into an antisolvent. This occurs because the high solution concentration forces the solution to separate into a second liquid phase while the nucleation rate is relatively slow. Even though the oil phase is unstable and may convert to amorphous or crystalline material over time, it is undesirable in industrial applications because this type of operation would be difficult to control, and the oil droplets may form gum balls and increase in size by agglomeration into large masses on scale-up. Oiling-out can be

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minimized or prevented by controlled supersaturation and efficient seeding strategies. Another reason for oiling-out or amorphous formation might be the presence of impurities that inhibit nucleation. It is well known that a small amount of an impurity can profoundly affect the nucleation rate, but it is impossible to predict the effect beforehand. The presence of additives can either enhance or inhibit the solubility of a substance. Enhanced solubilities would lead to lower supersaturations and hence lower nucleation rates. If it is postulated that the impurity adsorbs on the crystal surface, then two opposing effects come into play – the presence of an additive would lower the surface tension and lead to higher growth rates, but adsorption of the impurity blocks potential growth sites and lowers nucleation rates. Thus the effect of impurities is complex and unpredictable. Several experimental investigations have been undertaken to clarify the situation. For example, it was shown that Pb2+ acts as a nucleation agent in an NaCl system, whereas Co2+ inhibits nucleation in a KNO3 solution. The effect of the presence of chromium ions on the crystallization of magnesium sulfate is shown in Figure 3.21 (Khambaty and Larson 1978). Their data show a decrease in both the nucleation and growth rates in increasing chromium concentrations. Although it is difficult to generalize, certain trends can be observed from the experimental evidence. The inhibiting effect appears to increase with increasing charge of the cation. The inhibiting effect also appears to decline above a certain critical impurity concentration. This can be seen in the behavior of the nucleation exponent, which is unaffected at low contaminant concentrations (parts per million [ppm]) but changes erratically at higher concentrations (Broul 1978). Therefore, removal of impurities to achieve a very high level of purity can aid in inducing nucleation. It must be noted that there are special cases where nucleation is not desired. For such cases, for example, solid dispersions used in the pharmaceutical industry to improve the bioavailability of drugs, additives that inhibit nucleation are purposely introduced into the systems to deliberately generate amorphous materials. The most effective way to induce nucleation in slow-nucleating systems is via seeding the supersaturated solution with small particles of the material to be crystallized, which is called homogeneous seeding. For compounds that have never been

Crystal Nucleation

Figure 3.21 Effect of Cr3+ on MgSO4·7H2O crystallization. Chromium ion concentrations: circles = 3 ppm; squares = 8 ppm; triangles = 15 ppm; and diamonds = 130 ppm. Source: Reproduced with permission from Khambaty and Larson 1978. Copyright © 1978, American Chemical Society.

crystallized before, foreign-particle seeds can be introduced, which is called heterogeneous seeding. It is common practice to use isomorphous substances as heterogeneous seeds, which is sometimes referred as cross-seeding. For example, phosphates will often nucleate sodium sulfate decahydrate, phenol can nucleate m-cresol, and so on (Mullin 1997). As discussed in Section 3.7, Langmuir monolayers, self-assembled monolayers, and tailor-made additives can be introduced as heterogeneous seeds to promote nucleation. Heterogeneous seeding also can be done unintentionally by introducing dust particles, clothing fibers, lint, and so on. The beneficial effects of these impurities have even led to claims that researchers with beards or long hair are more likely to succeed at crystallizing difficult proteins (Bergfors 1999). In addition to these, there are a few studies that report the use of electric fields, magnetic fields, and light sources to induce nucleation and enhance the nucleation rate. Lundager Madsen (1995) investigated the magnetic field effect on the precipitation of paramagnetic and diamagnetic inorganic salts and concluded that phosphates and carbonates with diamagnetic metal ion are affected through increased nucleation and growth rates. The author explained this faster nucleation in terms of faster proton transfer from hydrogen carbonate and hydrogen phosphate ions in water due to proton spin inversion in the presence of the magnetic field. Taleb et al. (2001) demonstrated that application of an electric field results in nucleation of a smaller number of larger size crystals and suggested that the electric field produces large concentration fluctuations leading to a local supersaturation area in the crystallization solution. An internal electric field has also been found to be a very effective parameter to control the nucleation of protein crystals

because it decreased the nucleation induction time and the number of crystals deposited around the cathode (Mirkin et al. 2003; Moreno and Sazaki 2004). Vekilov and coworkers (Penkova et al. 2005) showed that low electric fields (1 γ aH ð4:22Þ with α¼ l kB TσλA

GF =G0 ¼ 1  αθF for αθF ≤ 1 Kubota–Mullin : GF =G0 ¼ 0 for αθF > 11 with α¼

γl aH kB TσλA

ð4:23Þ

Depending on the value of α, referred to as the effectiveness parameter by Kubota, three different behaviors are observed in the frame of the Kubota–Mullin model: α > 1: The step velocity approaches zero at θ < 1, i.e., at incomplete coverage. α ¼ 1: The step velocity is equal to zero exactly at θF ¼ 1, i.e., at complete coverage. α < 1: The step velocity never approaches zero, even at θF ¼ 1: These three behaviors are also observed in the frame of the Cabrera–Vermilyea model, except that the transition occurs at α ¼ 1=2. It is seen in Table 4.1 that the expression developed by Bliznakov (1965) is analogous to the one of the Kubota–Mullin

126

λF

model if one defines α ¼ 1  G∞ =G0 , where G∞ is the growth rate at complete coverage of the adsorption sites. However, Blizanokov derived this expression assuming a different mechanism based on the blocking of key growth sites rather than step pinning. Bliznakov considered that the crystal growth rate is the sum of two different contributions, the first one resulting from the direct incorporation of growth units at free growth sites, which occurs at the rate G0, and the second one resulting from the incorporation of growth units at occupied growth sites, which occurs at the reduced rate G∞ owing to the energetic barrier necessary to remove the foreign species already adsorbed on the surface. The growth rate can then be formally written as GF ¼ ð1  θF ÞG0 þ θF G∞ , which can be rearranged to give GF =G0 ¼ 1  ð1  G∞ =G0 ÞθF . Importantly, this model does not predict a supersaturation dependence, does not necessarily lead to impurity incorporation, and is not accompanied by step roughening or rounding. Finally, the model by Davey and Mullin (1974) presented in Table 4.1 can be regarded as a particular case of the model of Bliznakov for which G∞ ¼ 0. This would correspond to the case of immobile foreign species, although the mobility of adsorbed foreign species is not discussed in the original work of Davey and Mullin. In the frame of this model, the foreign species are necessarily incorporated in the crystal lattice, and the growth rate decreases linearly with the fraction of free growth sites. It is hardly possible to predict a priori which of the models presented in this section will be the most suitable to describe a specific system. The selection of a proper model therefore requires performing experiments at various concentrations of foreign species and various supersaturations so as to analyze liq the dependence of GF /G0 with cF and σ. To help the reader select an appropriate model, Figure 4.12 shows the dependence liq of GF /G0 with cF predicted by the Cabrera–Vermilyea and Kubota–Mullin models, both in the case of linear and Langmuir adsorption isotherms. Note that the models by Bliznakov and Davey–Mullin predict similar trends as the Kubota–Mullin model provided that α ≤ 1 and α ¼ 1 for the former and the latter, respectively. A value of HF ¼ 10 was liq selected for the predictions shown in Figure 4.12, so θF ¼ 10cF in the linear case. Several values of α ranging from 0.2 to 4 were investigated, and darker lines correspond to higher values of α. Besides the mechanistic models presented in Table 4.1, a spiral pinning model has been proposed in the literature to describe the effect of foreign species in the case of spiral growth at low supersaturation (Sizemore and Doherty 2009; Kuvadia

The Influence of Impurities and Additives on Crystallization

Kubota-Mullin α = 0.2 α = 0.5 α=1 α=2 α=4

Linear

0.8 GF /G0 [-]

(b)

1

0.6 0.4

(c)

0.02

0.04 0.06 xFliq [-]

0.08

(d)

GF /G0 [-]

GF /G0 [-]

0.4

0.02

0.04 0.06 xFliq [-]

0.08

0.1

0.02

0.04 0.06 xFliq [-]

0.08

0.1

1 0.8

0.8 Langmuir

0.6

0 0

0.1

1

0.6 0.4

0.6 0.4 0.2

0.2 0 0

α = 0.2 α = 0.5 α=1 α=2 α=4

Figure 4.12 Predictions of the Kubota–Mullin and Cabrera–Vermilyea models in the case of (a, b) linear and (c, d) Langmuir adsorption isotherms

0.2

0.2 0 0

Cabrera-Vermilyea 1 0.8

GF /G0 [-]

(a)

0.02

0.04 0.06 xFliq [-]

0.08

0.1

0 0

and Doherty 2013). This model considers that the presence of foreign species increases the characteristic time of rotation of the first spiral turn, without affecting the subsequent turns. All the models described in this section predict a decrease in the growth rate in the presence of foreign species. It is sometimes observed that foreign species accelerate crystal growth (Jones and Ogden 2010; Shtukenberg, Ward, and Kahr 2017), but the underlying mechanisms are still poorly understood. This may, for example, be due to a variation in the supersaturation through a change in the solubility or due to more subtle effects, such as assisted desolvation (Piana and Jones 2006).

Comparison Model/Experiments To sum up, a common approach to modeling the effects of foreign species on crystal growth consists of (1) measuring the growth rate of single crystals with various concentrations of foreign species and various supersaturations, (2) selecting an adsorption model (e.g., linear or Langmuir), if possible based on independent adsorption measurements, (3) selecting a growth model (e.g., Cabrera–Vermilyea or Kubota–Mullin), and (4) regressing the data to estimate the unknown parameters. To illustrate this approach, let us consider the example reported in Figure 4.13a, which shows the relative growth rate of the (100) face of sucrose crystals in the presence of raffinose for various supersaturations (Kubota and Mullin 1995). It is observed that the measured growth rates (symbols) decrease quasi-linearly with the concentration of raffinose for each investigated supersaturation and that the impact of raffinose is less pronounced at high supersaturation. According to Figure 4.12, this indicates that a linear adsorption isotherm

and the Kubota–Mullin growth model can be considered. The results of the model are shown with solid lines in Figure 4.13a together with the fitted values of αHF . The decoupling between α and HF would require independent adsorption experiments, but it can be reasonably assumed that the parameter HF is independent of the supersaturation. Therefore, the decrease in the product αHF with the supersaturation can be attributed to an increase in the factor α, which is in line with Equation (4.23). Another example is presented in Figure 4.13b, which shows the relative growth rate of the {100} faces of KBr crystals as a function of the concentration of various aliphatic carboxylic acids at a constant supersaturation (Kubota and Mullin 1995). The experimental data are represented by symbols, whereas the lines correspond to fits with the Kubota– Mullin model assuming a Langmuir adsorption isotherm. It is observed that for all the carboxylic acids, the relative growth rate reaches a plateau at a nonzero value. In addition, the larger the size of the carboxylic acid, the more pronounced is the growth rate reduction. In the pinning step models’ framework (Kubota–Mullin or Cabrera–Vermilyea), an increase in the size of the foreign species can exacerbate the growth rate reduction in two ways: (1) by displacing the adsorption equilibrium, which is reflected by a change in the value of HF, or (2) by reducing the actual distance between the adsorbed foreign species. The latter is not explicitly accounted for in the Kubota–Mullin and Cabrera–Vermilyea models because the adsorbed species are treated as points. Nevertheless, one may think of taking into account size effects in Equation (4.19) by replacing the distance λF /2 by λF /2 − rF, where rF is the radius of the foreign species, which would then predict that the growth rate decreases with increasing rF . In the case

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Lucrèce H. Nicoud and Allan S. Myerson

1

1.2

σ = 0.020 σ = 0.015 σ = 0.013 σ = 0.0065

1 0.8 α

0.6 0.4 0.2

Relative growth rate [-]

Relative growth rate [-]

(a)

(101) face 0.8

(012) face

0.6 0.4 0.2 0 0

0

Relative growth rate [-]

(b)

0

6 8 2 4 Raffinose mole fraction [-] x10–3

1.2 HCOOH CH3COOH C2H5COOH C3H7COOH

1 0.8 0.6

0.4

0.6

0.8

1

x10–3

Glutamic acid mole fraction [-]

Figure 4.14 Relative growth rate of the (101) and (012) faces of L-asparagine monohydrate crystals as a function of L-glutamic acid at constant supersaturation (σ = 0.177). Symbols correspond to experimental data and lines to fittings with the Kubota–Mullin model. The parameter values used for face (101) and face (012) were αHF = 1.93 × 103 and α = 0.86 and HF = 6.43 × 103, respectively. Sources: Experimental data taken from Black, Davey, and Halcrow (1986), with permission from Elsevier. Parameter values from Kubota and Mullin (1995).

0.4 0.2 0

0

0.1

0.2

0.3

Acid concentration [mol/L] Figure 4.13 (a) Relative growth rate of the (100) face of sucrose crystals as a function of the concentration of raffinose. Symbols and lines correspond to experimental data and model fittings, respectively. (b) Relative growth rate of the {100} faces of KBr crystals as a function of the concentration of various aliphatic carboxylic acids at constant supersaturation (σ = 0.046). Source: Adapted from Kubota and Mullin (1995), with permission from Elsevier.

where such a correction is not implemented, it is expected that the fitted value of the parameter α increases with the size of the foreign species. Coming back to the example of KBr crystal growth in the presence of carboxylic acids, the fitted values of α and HF are shown in Figure 4.13b. It is seen that the values of HF are nearly constant, which can be justified if the foreign molecules adsorb on the crystal surface via the carboxyl group. Besides, it is observed that the parameter α increases with the size of the carboxylic acid, which is in line with the preceding discussion. Other studies showed that growth inhibition is enhanced by steric effects. For instance, it was found that the concentration of poly-L-aspartate necessary to stop the growth of calcite decreases with the chain length of the peptide (Elhadj et al. 2006a, b). The two examples reported in Figure 4.13 dealt with the growth rate of only one particular face of the crystals. Figure 4.14 instead shows the growth rate of two different faces of Lasparagine monohydrate crystals as a function of L-glutamic acid mole fraction (Black, Davey, and Halcrow 1986). It is observed that the two faces are affected differently by the presence of glutamic acid: the growth rate of the (101) face decreases linearly and reaches 0 for mole fractions larger than 5 × 10−4, whereas the growth rate of the (012) face does not reach 0 even at high mole fractions of L-glutamic acid. This is reflected by the fitted value of α larger than 1 (see parameter values in the caption of Figure 4.14). Other studies have shown that the growth rates of the different faces of a crystal can be affected differently by the

128

0.2

presence of foreign species (Davey et al. 1992; Shekunov et al. 1997; Kitamura and Ishizu 1998; Jung et al. 2004). This can be attributed to the different structural and chemical properties of the different faces, as discussed in Section 4.3.2.

Combination of Foreign Species In industrial processes, many different impurities and some additives may be present simultaneously in the crystallizing solution. Understanding how such a combination of foreign species affects the crystallization process is quite challenging. Considering only two different foreign species, three types of effects may be encountered (Shtukenberg, Ward, and Kahr 2017): additive (i.e., the inhibiting effect of the mixture can be predicted from the sum of the individual contributions of the two foreign species), synergistic (i.e., the inhibiting effect of the mixture is stronger than that expected on the basis of the individual contributions), and antagonistic (i.e., the inhibiting effect of the mixture is weaker than that expected on the basis on the individual contributions). In light of the mechanisms described in this section, one can expect that an additive effect is observed when the two foreign species bind to similar sites in a noncompetitive manner. By contrast, the competition of foreign species at the crystal surface is likely to lead to an antagonist effect. Besides competitive adsorption, another aspect to take into account is the potential formation of a complex between the two foreign substances. A synergistic or antagonistic effect may result depending on the strength of binding of the complex with the crystal surface. Such considerations were used to rationalize the impact of three growth inhibitors on the crystallization of calcium oxalate monohydrate (Farmanesh et al. 2014).

4.4 Impact of Foreign Species on Crystal Properties We have seen in Section 4.3 that the presence of foreign species can significantly influence the growth rate. Some studies also

The Influence of Impurities and Additives on Crystallization

(a) pure

(b) BSA

10 μm

(c) C4S

10 μm

Figure 4.15 Morphology of calcium oxalate crystals (a) in the absence of additives and (b) in the presence of bovine serum albumin (BSA), (c) chondroitin sulfate (C4S), and (d) citrate Source: Reprinted from Farmanesh et al. 2014, with permission from the American Chemical Society.

(d) citrate

10 μm

10 μm {001}

[001] [010] [100]

{010} – {121}

report that foreign species can influence the induction time (van der Leeden, Kashchiev, and van Rosmalen 1993; Ottens et al. 2004; Ilevbare 2012; Saleena, Onyemelukwe, and Nagy 2013; Peng et al. 2014; Pons Siepermann, Huang, and Myerson 2017), which is generally attributed to a change in the nucleation rate. Nevertheless, discriminating the impact of foreign species on nucleation and growth rates from induction time measurements is challenging because nuclei need to grow to a certain size before they can be detected. Similarly to the growth rate, the nucleation rate can be affected by the presence of foreign species in various ways, including through a change in solubility or interfacial energy. It has also been suggested that the mechanistic models describing growth inhibition through the adsorption of foreign species at the crystal surface can be extended to describe nucleation inhibition, considering that the foreign species also adsorb at the surface of the nuclei (Zhu 2017). Changes in the nucleation and growth kinetics in the presence of foreign species, in turn, result in changes in crystal properties. While the concentration of impurities is imposed by the upstream part of the process, additives can be deliberately added to the solution to be crystallized with a view toward tuning certain crystal properties, in particular, crystal size and shape but also crystal polymorphism or enantiopurity (Shtukenberg, Ward, and Kahr 2017; Jones and Ogden 2010; Weissbuch et al. 1995; Sauter et al. 1999; Song and Colfen 2011; Shtukenberg et al. 2014). An important class of additives is the so-called tailor-made additives, which are designed to interact in specific ways with selected faces of crystals. These compounds contain two different chemical groups: one group that mimic the host species and is thus readily adsorbed at growth sites on the crystal surface and one group that chemically or structurally differs from the host molecule, thereby disrupting growth processes at the affected faces. Although tailor-made additives have proven helpful to manipulate crystal properties, a limitation of this strategy is the potential decrease of product purity owing to the incorporation of the additive in the crystal lattice.

4.4.1 Crystal Shape Crystal shape may have a strong influence on drug bioavailability and drug processability (Banga et al. 2007; Modi et al. 2013; Dandekar, Kuvadia, and Doherty 2013; Pudasaini et al. 2017). Indeed, crystals of different shapes are, for instance, characterized by a different specific surface area, wettability and aspect ratio, what may impact dissolution kinetics, filterability, compactibility, tabletability, and other properties. The shape of a crystal is determined by the relative growth rates of the various faces. The slower the growth rate in a given direction, the larger is the face developed perpendicular to that direction. If the presence of a foreign species reduces the growth rate to a similar extent in all directions, it is expected that the crystal size decreases (for a given harvest/residence time in batch/continuous crystallizers) without changes in crystal shape. Conversely, when a foreign species inhibits growth in a direction perpendicular to a given face, the area of this face is expected to increase with respect to the areas of the other faces of the crystal, thus leading to a change in crystal shape. To illustrate this, Figure 4.15 shows SEM pictures of calcium oxalate crystals obtained in the presence of various additives. The shape of crystals grown in the absence of additive is presented in Figure 4.15a, where it is seen that control crystals exhibit a hexagonal platelet morphology with a large basal (100) surface. In the presence of bovine serum albumin, diamond-shaped crystals are formed owing to the binding of the protein to the {121} faces, as shown in Figure 4.15b. By contrast, chondroitin sulfate binds to the (010) face, thus increasing the length of hexagonal platelets in the [001] direction, as shown in Figure 4.15c. Finally, citrate binds to the steps on the (100) plane, which produces a quasi-rectangular crystal shape, as shown in Figure 4.15d. The shapes of several other crystals have been proven to be affected by the presence of foreign species, and Table 4.2 provides some examples.

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Lucrèce H. Nicoud and Allan S. Myerson Table 4.2 Examples of Foreign Species Affecting Crystal Shape

Crystallizing material

Foreign species

References

Acetaminophen

Acetanilide

Thompson et al. 2004

p-Acetoxyacetanilide

Chow et al. 1985; Hendricksen et al. 1998

p-Acetamidobenzoic acid

Hendricksen et al. 1998

Adipic acid

n-Alkanoic acids

Davey et al. 1992

o-Aminobenzoic acid

Benzoic acid

Simone, Steele, and Nagy 2015

Benzamide

Benzoic acid

Berkovitch et al. 1982

o- and p-Toluamide

Berkovitch et al. 1982

Calcite

Synthetic peptides

DeOliveira and Laursen 1997

Calcium oxalate monohydrate

Citrate

Farmanesh et al. 2014; Wang et al. 2006; Qiu et al. 2005

Osteopontin

Taller et al. 2007

Bovine serum albumin

Farmanesh et al. 2014

Synthetic peptides

Farmanesh et al. 2013

L-Cystine

dimethylester

Poloni et al. 2017

L-Cystine

bismorpholide

Poloni et al. 2017

L-Cystine

Insulin

Anti-insulin antibodies

Markman et al. 1992

Methyl paraben

Acetaminophen

Cai et al. 2017

p-Methyl acetanilide

Cai et al. 2017

Acetanilide

Cai et al. 2017

Sucrose

Saccharides

Sgualdino et al. 2000

Water

Antifreeze proteins

Jia et al. 1996; Garnham, Campbell, and Davies 2011

Note: No change in polymorphism was reported in the references.

Controlling crystal shape with additives has attracted much interest in the literature. Although the selection of suitable operating conditions still requires the screening of additives and solvents, simulation tools are now increasingly developed with a view to reducing the experimental effort (Lovette 2008; Rohl 2003; Schmidt and Ulrich 2012; Kuvadia and Doherty 2013). A review of the computational methods available to predict crystal properties, including crystal shape, will be presented in Chapter 5. As described in Section 4.3.3, fundamental studies aiming at better understanding of the impact of foreign species on the growth rate of selected crystal faces are typically performed with single crystals. However, single-crystal experiments are not representative of the conditions encountered in industrial crystallizers, where nucleation (and potentially crystal aggregation and breakage) occur in addition to crystal growth. In this case, the proper description of the impact of foreign species on crystal properties (such as size and purity) requires one to solve population balance equations (Févotte and Févotte 2009; Alvarez, Singh, and Myerson 2011), as further described in Chapter 6. If one also wishes to describe crystal shape, multidimensional population balance equations should be solved,

130

where the relevant crystal characteristics (such as length and width) are considered (Borsos, Majumder, and Nagy 2016; Zhu 2017; Puel, Marchal, and Klein 1997; Majumder and Nagy 2013; Zhu et al. 2016).

4.4.2 Crystal Polymorphism Polymorphism is the ability of a species to exist in different crystalline structures owing to differences in packing arrangement or in molecular conformation. Different polymorphs may strongly differ in terms of several properties, such as solubility, morphology, color, hygroscopy, mechanical strength, and melting temperature, which ultimately affect the performance of the material (Lee, Erdemir, and Myerson 2011; Lee 2014). Therefore, producing the target polymorph at a high purity is key in the manufacturing of crystalline products. The polymorphic outcome is dictated by the relative rates of nucleation and growth of the different forms and by the kinetics of transformation (Mangin, Puel, and Veesler 2009; Lee 2014; Black et al. 2018). Consequently, the polymorphic outcome is affected by a variety of factors, including the supersaturation, solvent, temperature, and the presence of

The Influence of Impurities and Additives on Crystallization Table 4.3 Examples of Foreign Species Affecting Crystal Polymorphism

Crystallizing material

Foreign species

References

Acetaminophen

Metacetamol

Agnew et al. 2016

4-Bromobenzoic acid

Agnew et al. 2016

4-Fluorobenzoic acid

Agnew et al. 2016

Polyacrylic acid

Agarwal and Berglund 2003

Acusol

Agarwal and Berglund 2003

Glutamic acid

Poornachary, Chow, and Tan 2008

Aspartic acid

Poornachary, Chow, and Tan 2008

Transglutaconic acid

Davey et al. 1997

Trimesic acid

Davey et al. 1997

L-Phenylalanine

Kitamura and Funahara 1994

Mefenamic acid

Flufenamic acid

Lee, Byrn, and Carvajal 2006

Sulfamerazine

N4-acetylsulfamerazine

Gu et al. 2002

Sulfadiazine

Gu et al. 2002

Calcium carbonate

Glycine

L-Glutamic

acid

foreign species, be they dissolved or not (Llina andGoodman 2008; Kitamura 2009). An emblematic example is Ritonavir (Bauer et al. 2001), which was temporarily withdrawn from the market because of the sudden appearance of a more stable, and thus less soluble, polymorphic form. It has been hypothesized that the appearance of the new form in semisolid formulations was triggered by solvent evaporation, which resulted in a high supersaturation in concomitance with a probable heterogeneous nucleation on solid particles of a degradation product. Soon after the stable form started spreading, all attempts to produce the metastable form failed until a new manufacturing process was developed (Chemburkar et al. 2000). Another example is that of an imidazopyridine derivative, for which a by-product was found to inhibit the formation of the stable polymorph (Machiya et al. 2009). After removal of the impurity, the stable form of the drug could be produced. A number of other dissolved foreign species have been shown to influence crystal polymorphism, and some of those are listed in Table 4.3. Although a comprehensive understanding of the underlying physical mechanisms has not been achieved yet, the impact of foreign species on crystal polymorphism is generally attributed to the growth inhibition of specific polymorphic form(s), thus kinetically favoring the crystallization of the other form(s). For illustrative purposes, Figure 4.16 shows experimental results obtained during the crystallization of L-glutamic acid in batches (Kitamura and Funahara 1994). L-Glutamic acid can form two polymorphs, denoted α and β, that are metastable and stable, respectively. Figure 4.16a shows the fraction of crystals of the metastable form as a function of time, whereas Figure 4.16b shows the concentration of L-glutamic acid in the

liquid phase. In the absence of L-phenylalanine, a mixture of both polymorphic forms is produced, and the metastable form transforms into the stable form. In the presence of L-phenylalanine, the nucleation of the stable form and the polymorphic transformation are inhibited, so the fraction of the metastable polymorph dramatically increases. Note that the concentration in the liquid phase reaches a higher plateau value in the presence of L-phenylalanine owing to the higher solubility of the metastable phase. It was shown in a subsequent study that the formation of form α arises from the preferential inhibition of the growth of form β due to adsorption on the (101) face. Interestingly, it was found that the additive effect is highly stereoselective because the presence of D-phenylalanine scarcely influences the polymorphic outcome (Kitamura and Funahara 1994). The observation that foreign species may adsorb differently on different polymorphs implies that their incorporation in the crystal lattice may differ from one polymorph to another. For instance, it was observed that the monoclinic crystals of lysozyme incorporate less bovine serum albumin and ovalbumin than the tetragonal polymorph (Adawy et al. 2015).

4.4.3 Crystal Enantiopurity In 1848, Louis Pasteur established the foundations of stereochemistry by performing the racemic resolution of sodium ammonium tartrate (Pasteur 1848). Nowadays, there is a high interest in producing pure enantiomers in various industrial sectors and in particular in the pharmaceutical industry because enantiomers may have different pharmacologic properties (Lin, You, and Cheng 2011; Lorenz and Seidel Morgenstern 2014). Addadi et al. (1982), Weissbuch et al.

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Lucrèce H. Nicoud and Allan S. Myerson

(b)

(a)

Conc. in liquid [mol/L]

Fraction metastable [-]

0.8 0.6 0.4 0 mM L-Phe 2.6 mM L-Phe 5.2 mM L-Phe

0.2 0

0

10

20

30

0 mM L-Phe 2.6 mM L-Phe 5.2 mM L-Phe

2.5 2 1.5 1 0.5 0

0

10

Time [h]

(1995), and Addadi, Van Mil, and Lahav (2007) realized the resolution of racemic conglomerates using enantiospecific additives. More precisely, they found that the crystallization of a conglomerate is dramatically influenced by the presence of an enantiopure foreign species. This effect was attributed to the selective adsorption of the resolved foreign species on the growing crystals of similar stereoselectivity, which substantially inhibits the growth of these crystals, thus allowing collection of the unaffected enantiomorph. This kinetic resolution was implemented for various amino acids, including glutamic hydrochloride, threonine, and asparagine hydrate.

4.5 Conclusion The presence of impurities and additives, collectively referred to as foreign species, may affect both the thermodynamics and the kinetics of crystallization. A good understanding of the underlying physical phenomena requires examining the effect of foreign species in the liquid phase, in the solid phase, and at the liquid–solid interface. Considering first thermodynamic aspects, foreign species may affect phase equilibria, and in particular the solubility of the host species. In addition, foreign species may decrease crystal purity by forming a solid solution or by adsorbing at the crystal surface. Various considerations come into play for what concerns kinetic aspects. First, if there is a change in solubility, it is necessarily accompanied by a change in the supersaturation and thus in the nucleation and growth rates. This may, in turn, affect various crystal properties such as crystal size and polymorphism. Besides, a change in the growth rate, due either to the presence of foreign species or to a change in the operating conditions, may affect crystal purity. Indeed, high growth rates promote the accumulation of foreign species in the boundary layer and the formation of inclusions. The latter are detrimental not only to crystal purity but also to mechanical properties. Finally, the adsorption of foreign species at the crystal surface decreases the crystal growth rate. If the adsorption is face specific, a change in crystal shape is expected. Interestingly, the preferential growth inhibition of a certain crystal type, such as a polymorphic form or an enantiomer, kinetically favors formation of the unaffected type and thus may be used to control certain crystal properties.

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Figure 4.16 Influence of L-phenylanaline (L-Phe) on the polymorphism of L-glutamic acid. (a) Fraction of the metastable polymorph as a function of time. (b) Concentration of L-glutamic acid in the liquid phase. Source: Adapted from Kitamura and Funahara 1994 with permission.

3

1

20

30

Time [h]

Nomenclature Symbol

Description

Common units

aH

Surface area of growth unit

nm2

cads F

Concentration of adsorbed foreign species

mol/cm2

cmono F

cads F at monolayer coverage

mol/cm2

DF

Molecular diffusivity of foreign species

m2/s

G0

Growth rate in the absence of foreign species

µm/s

GF

Growth rate in the presence of foreign species

µm/s

G∞

GF at complete surface coverage

µm/s

HF

Henry coefficient of foreign species

kB

Boltzmann constant

KF

Partition coefficient of the foreign species

KH

Partition coefficient of the host species

KF=H

Relative partition coefficient

KFeff

Effective value of KF

eq

J/K

KF

Equilibrium value of KF

eq KF=H

Equilibrium value of KF/H

rc2D

radius of the 2D nucleus

μm

R

Gas constant

J/mol/T

T

Temperature

K

TFfus

Fusion temperature of species

K

v0

Step velocity in the absence of foreign species

μm/s

The Influence of Impurities and Additives on Crystallization (cont.)

(cont.)

Symbol vF vmin liq

xF

Description

Common units

Symbol

Description

Common units

Step velocity in the presence of foreign species

μm/s

DHFmix;sol

Heat of mixing in the solid phase

J/mol

Change in vibrational entropy

J/mol/T

Minimum step velocity

μm/s

DSvib;sol F liq γF

Activity coefficient of the foreign species in solution

Mole fraction of foreign species in the liquid phase

liq;sat xF

Solubility of foreign species

xsol F

Mole fraction of foreign species in the solid phase

liq

Mole fraction of host species in the liquid phase

xH

liq;sat

Solubility of host species

xsol H

Mole fraction of host species in the solid phase

α

Effectiveness parameter

δ

Diffusion layer thickness Heat of mixing in the liquid phase

xH

mix;liq

DHF

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μm

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Radius of curvature of a step

θF

Surface coverage by foreign species

μm

σ

Relative supersaturation

J/mol

σc

Critical relative supersaturation

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Chapter

5

Molecular Modeling Applications in Crystallization Keith Chadwick Massachusetts Institute of Technology Jie Chen Massachusetts Institute of Technology Erik E. Santiso North Carolina State University Bernhardt L. Trout Massachusetts Institute of Technology

5.1 Introduction Crystallization is an extremely important process with extensive industrial applications including, but not limited to, the manufacture of electronics, explosives, fine chemicals, and pharmaceuticals. As such, controlling both crystal shape and crystal structure is vital for the production of high-quality products with desirable properties. However, the processes that govern crystallization, crystal growth, and crystal nucleation are not well understood at present. This is due in part to the limitations of experimental techniques in studying such processes because of the small number of molecules, often tens or hundreds, involved. Furthermore, experimental strategies for identifying and analyzing crystal structures (which may have serious implications in terms of intellectual property rights) and controlling crystal shape are not always successful in yielding the optimal product and often can be costly and time consuming. Over the past two decades, molecular modeling has developed into a potent tool for studying crystallization processes and crystal structures because of advances in computing power and the development of sophisticated algorithms and programs. Molecular modeling offers significant benefits in studying crystallization processes, such as being able to study very small systems, perform calculations under conditions that are not available experimentally, and aid in solid-form screening and crystallization process development, which can minimize experimental work and the quantities of chemicals used. In this chapter, we discuss the different areas of crystallization and solid-state chemistry where molecular modeling may be applied. For each topic, the pros and cons of each method are discussed, and the various software packages currently available for use are listed. The applications of molecular modeling to be discussed are as follows: Crystal Structures Visualization and Analysis. Included are building a crystal structure, analysis of intermolecular forces, calculating bond angles and lengths, predicting crystal morphology, visualizing the results of molecular modeling calculations. Morphology Prediction. Included are the Bravais–Friedel–Donnay–Harker (BFDH) model, attachment energy models, kinetic-based models, and Monte Carlo methods, as well as understanding the importance of chemistry at the crystal–solution interface. Crystal Structure Determination from X-Ray Powder Diffraction Data. Included are collecting appropriate

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diffraction data, indexing diffraction data, and determining the unit cell parameters, Pawley–La Bail refinement, and structure solution (Monte Carlo methods), as well as successes and limitations of structure solution from powder diffraction data. Nucleation and Phase Transitions. Included are studying self-assembly in solution (linking solution structure to nucleation and polymorphism), measuring crystalline order in molecular modeling (the use of order parameters), and studying solid-to-solid phase transitions. Polymorph Searching and Prediction. Included is the development of methods for sampling/searching the energy landscape (simulated annealing, Monte Carlo methods, genetic algorithms). Solubility Prediction. Included are discussion of the thermodynamic concepts behind solubility prediction models, the conductor-like screening method (COSMO), the non-random two-liquid method.

5.2 Crystal Structure Visualization and Analysis Software 5.2.1 Introduction The ability to study the crystal structures of organic compounds (such as an active pharmaceutical ingredient) at the molecular level can be extremely useful in developing a crystallization process. The visualization of crystal structures allows one to understand the structural differences between polymorphic forms, explore the important intermolecular interactions between molecules, and study the surface chemistry of different crystal faces. Such studies can aid in the rational design of additives for crystal growth and polymorph control (Davey et al. 1997; Towler et al. 2004; Torbeev et al. 2005; Poornachary et al. 2008; Chen and Trout 2010a; Jones and Ogden 2010), the crystal engineering of multicomponent crystalline phases such as cocrystals and salts (Aakeröy and Salmom 2005; Friščić and Jones 2007; Cooke et al. 2010; Seaton et al. 2010), and the design of heterogeneous surfaces for controlling nucleation (Bonafede and Ward 1995; Campione et al. 2006; Chadwick et al. 2012). Furthermore, visualization software allows the output from molecular modeling studies of crystallization processes (discussed later in this chapter) to be displayed and analyzed efficiently. In this section, we discuss the tools that visualization software packages offer to users and

Molecular Modeling Applications in Crystallization Figure 5.1 (a) The hydrogen-bonded dimer found in the monoclinic and orthorhombic forms of benzamide, (b) crystal packing in monoclinic benzamide as viewed along the b-axis, (c) crystal packing in orthorhombic benzamide as viewed along the c-axis. The packing of the dimers differs between the two polymorphs.

how these applications can be used to assist in the analysis, design, and study of crystallization processes.

5.2.2 Building a Crystal Structure The most basic tool incorporated into all visualization software is the ability to generate a three-dimensional (3D) crystal structure from a file containing the required crystallographic data (unit cell parameters, space group, atomic coordinates). Such crystallographic files include .cif and .pdb file types. The crystal structure is built by generating a superlattice of dimensions x, y, z, where x, y, and z represent multiples of the unit cell axes a, b, and c. Building the 3D crystal structure allows the user to study differences in molecular packing between different polymorphs (Figure 5.1), the chemical environment of solvent molecules present in a solvate structure, and the packing arrangements of the heterosynthons present in cocrystals and salts.

5.2.3 Crystal Structure Analysis In addition to being able to study the overall crystal structure, all visualization programs allow the user to study the molecular detail. This includes being able to study the intra- and intermolecular interactions among molecules in the crystal structure, calculate covalent and hydrogen bond lengths, and determine bond and molecular torsion angles (Figure 5.2). These tools allow researchers to study the differences in intermolecular interactions among polymorphs, study conformational polymorphism, and analyze hydrogen-bonding patterns. Such analyses have been used successfully in the design of additives for the selective nucleation and growth of specific polymorphs (Davey et al. 1997; Towler et al. 2004; Torbeev et al. 2005; Poornachary et al. 2008) and the crystal

engineering of cocrystals with a desired heterosynthon (Aakeröy and Salmon 2005; Friščić and Jones 2007; Seaton et al. 2010). When designing a crystallization process, it is often prudent to understand the chemistry of particular crystal faces. This is especially true when selecting additives or solvents to control crystal morphology and polymorphism (Davey et al. 1997; Towler et al. 2004; Torbeev et al. 2005; Poornachary et al. 2008; Chen and Trout 2010a; Jones and Ogden 2010) or for understanding epitaxial mechanisms on crystalline substrates (Bonafede and Ward 1995; Campione et al. 2006; Chadwick et al. 2012). The face(s) of interest are built by inputting the Miller indices (hkl) of the crystal into the software, and the resulting crystal plane will be displayed along with the crystal structure and can be manipulated thereafter to determine the surface chemistry (Figure 5.3).

5.2.4 Crystal Morphology Prediction Certain software packages allow for the calculation and display of crystal morphologies predicted by the Bravais–Friedel–Donnay–Harker (BFDH) and attachment energy models (refer to Section 5.3 for method details and examples).

5.2.5 Calculation of X-Ray Powder Diffraction Data Programs such as Mercury (Cambridge Crystallographic Data Centre 2013) convert the 3D crystallographic data provided in a .cif file into two-dimensional (2D) powder diffraction data, which can be displayed, saved, and compared with experimental diffraction data to successfully identify a crystalline phase. Figure 5.4 displays the powder diffraction data for

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout Figure 5.2 (a) Intermolecular contacts between the molecules in the unit cell of form I acetaminophen, (b) hydrogen bond distances in the α polymorph of glycine, (c) C—N—C angle of triclinic diphenylamine, and (d) torsion angle C—S—N—C of form II sulfathiazole.

pharmaceutical excipients. Molecular modeling was used to determine the optimal orientation of acetaminophen crystals on excipient surfaces, and by displaying the results of the calculations in Visual Molecular Dynamics (VMD), the authors were able to probe the epitaxial relationship between the two surfaces (Figure 5.5).

5.2.7 Available Software Packages Table 5.1 Available Visualization Software Packages and Their Applications

Software

Functionality

Mercury

Analysis of crystal structures, display of BFDH crystal morphologies, calculation and display of X-ray powder diffraction data

Materials Studio (Biovia 2014)

Analysis of crystal structures, display of calculated crystal morphologies, display of molecular modeling data

Visual Molecular Dynamics

Display of molecular modeling data

Figure 5.3 View along the (100) face of terephthalic acid showing the carboxylic acid functionality present at the surface

indomethacin crystallized from ethanol. It has been compared with powder diffraction data calculated for the two known polymorphs of indomethacin, referred to as α and γ. Comparison of the experimental and calculated patterns shows the γ-form crystallized.

5.2.6 Visualization of Molecular Modeling Calculations As will be discussed in the rest of this chapter, molecular modeling is a powerful tool for studying crystallization processes at the molecular level and providing mechanistic insight that cannot be achieved through experimental means. The software takes the atomic coordinates held within the data files created by the programs that performed the calculations and displays them as a 3D representation from which further analysis on the data can be carried out. An example of the utility of this software is the study of Chadwick et al. (2012), who investigated the epitaxy of acetaminophen on crystalline

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5.2.8 Summary Visualization software is an important tool in analyzing crystal structures and the results obtained from molecular modeling studies of crystallization processes. In this section, the capabilities of currently available software packages and their applications to crystal engineering and crystallization analysis and design have been discussed.

5.3 Morphology Prediction 5.3.1 Introduction Control of crystal morphology is of great importance industrially because many physical properties of crystals are implicitly

Molecular Modeling Applications in Crystallization

all crystal faces are known, the equilibrium morphology of the crystal can be constructed using Wulff theory. Therefore, the greatest challenge in developing morphology prediction models is to accurately predict the relative growth rates of all crystal faces, especially those with low Miller indexes. In this section, we will discuss some of the well-known morphology prediction models, including their theory, application, and limitations. We will also introduce the application of direct molecular simulation, molecular dynamics, and Monte Carlo methodologies in morphology prediction, as well as in understanding solid–liquid interfaces. Figure 5.4 (a) Experimental diffraction data for indomethacin crystallized from ethanol, (b) calculated diffraction data of the γ polymorph, (c) calculated diffraction data of the α polymorph

5.3.2 The Bravais–Friedel–Donnay–Harker (BFDH) Model The BFDH model simulates the morphology of a crystal by relating it to the lattice geometry (Bravais 1866; Friedel 1907; Donnay and Harker 1937; Docherty et al. 1991). It assumes an inversely proportional relationship between the growth rate Rhkl and the interplanar distance dhkl (Figure 5.7). This method is generally inaccurate in predicting the correct crystal morphology because kinetic factors and the role that a solvent or additive might play in crystal growth are not considered by the model. However, the BFDH method is easy to implement and has been proven to be useful in quickly identifying the important faces of a mature crystal, for example, the fastest-growing and the surrounding faces of a needle-like crystal (Chen and Trout 2010a).

5.3.3 Attachment Energy and Modified Attachment Energy Model

Figure 5.5 The most energetically favorable orientation of acetaminophen (100) on D-mannitol (00–1). Visualization of the molecular modeling data in VMD shows the alignment of the acetaminophen chains on the D-mannitol surface and the formation of hydrogen bonds between the two.

The attachment energy (AE) model, developed by Hartman and Perdok (1955), started to capture the anisotropic structural information of crystals using quantifiable energies. This model assumes that the growth rate of a surface is proportional to its attachment energy in a vacuum (absolute value), defined as the energy released by adding a growth slice to the existing crystal surface. It can be calculated by subtracting the slice energy from the lattice energy vac;att Rhkl ∝ jEhkl j vac;att Ehkl

dependent on their shape, and thus there is a need to understand the fundamental processes determining crystal morphology. Substantial effort has been dedicated over the past six decades to studying crystal growth and to predicting the equilibrium morphologies of crystals. The fundamental theory underlying most equilibrium morphology prediction models is the Wullf construction (Wullf 1901). It states that the total surface free energy of a crystal in equilibrium with its surroundings at constant temperature and pressure would be a minimum for a given volume. This eventually leads to the formula that the ratio of the growth rate of a particular crystal face and its perpendicular distance to the origin stay constant for all crystal faces, as shown in Figure 5.6. Once the relative growth rates of

¼

latt Ehkl



Eslhkl

ð5:1Þ ð5:2Þ

sl , Elatt where Evac;att hkl , and Ehkl are the attachment energy in hkl a vacuum, lattice energy, and slice energy, respectively, as shown in Figure 5.8. While overall the predictions have been accurate for sublimation growth crystals, they have not, in general, been accurate for crystals grown from solution because AE fails to consider the influence of the crystallization solvent(s) and additives/impurities, as well as kinetic factors such as the growth mechanism and supersaturation. A modified version of this model, the modified attachment energy (MAE) model (Hammond et al. 2007), accounts for the impact of the solvent by reducing the vacuum attachment energy using solvent–crystal interaction energies. In the MAE model, the

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Figure 5.7 Relationship between the growth rate of a crystal face (hkl) and the interplanar spacing for the BFDH morphology prediction model Figure 5.6 Demonstration of Wulff construction. Ahkl, γhkl, and Rhkl are the area, the specific surface free energy, and growth-rate face (hkl), respectively. hhkl is the perpendicular distance between the origin and face (hkl).

vac,att

E hkl vac,att

E hkl

(a)

crystal slice crystal

(b)

Figure 5.8 Attachment energy model. (a) Schematic demonstration of the AE model, (b) attachment energy calculation in 2,6-dihydroxybenzoic acid form II crystal. Molecules in dark solid lines are in the crystal slice, and the rest in light gray lines are in the crystal bulk.

growth rate of the crystal face (hkl) is proportional to the modified attachment energy, which is defined as vac;att MAEhkl ¼ Ehkl 

solvent Zhkl

Z

solution Uhkl

ð5:3Þ

where Z is the number of crystal molecules in a unit cell in the solution is the specific interaction incoming crystal slice, Uhkl energy between the crystal face (hkl) and the solvent molecules, solvent and Zhkl is the number of solvent molecules in the volume defined by the unit cell in the incoming crystal slice. This is shown schematically in Figure 5.9. The MAE model was reported to be able to predict the aspirin morphology from aqueous ethanolic solutions better than the AE model (shown in Figure 5.10; Hammond et al. 2007). However, the MAE model still exhibits many of the shortfalls of the original AE model. It is still a thermodynamic model and does not reflect any underlying growth kinetics. Other modifications of the classic AE model are the built-in approach (Niehörster 1997) and the surface-docking approach (Lu et al. 2001) to understand the effect of additives on crystal growth, as shown in Figure 5.11. In the build-in approach, an additive molecule substitutes successively into each symmetry position of the host molecules in the crystal unit cell. The morphology calculations using the AE model are done for all possible arrangements of the unit cell in the presence of the additive. The growth rate of a crystal face with the presence of build-in additives is proportional to the attachment energy modified by

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incorporating those additives in the lattice, as shown in Figure 5.12. The main idea of the surface-docking approach is to analyze the effect of additives on individual crystal faces using their binding energies and to combine these values with the AE model to predict the crystal morphology in the presence of those docking additives. If the additive has a strong interaction on one specific face, the growth rate of this face will be slowed down, and the face will become larger relative to the other faces (Myerson and Jang 1995). In this approach, an additive molecule will be placed on the crystal surface, the most stable interaction conformation (global minimum) will be identified using energy optimization, 0 and the binding energy of the additive E b;hkl will be calculated using that conformation. Using the same approach, the binding energy of a crystal molecule itself (Eob;hkl ) will also be calculated. After this, the attachment energy in vacuum will be modified using Equation (5.4) 0

Eatt;b hkl

¼

Eatt;vac hkl



att;vac Ehkl



0 E b;hkl  Eb;hkl

E0b;hkl

ð5:4Þ

att;vac where Eatt;b denote the attachment energy corhkl and Ehkl rected with and without binding energy, respectively.

5.3.4 Kinetics-Based Models The AE model and its modified version, although easy to use, do not provide any mechanistic detail of crystal growth and are often inaccurate. As such, the development of more detailed kinetics-based morphology prediction models is necessary. From a mechanistic point of view, the crystal growth process involves the incorporation of growth units into a crystal lattice. These growth units could be single molecules/atoms or other structural units (e.g., dimers) formed in the crystallization medium. The surface structure of a growing crystal is shown in Figure 5.13. This figure shows three types of sites at which the growth unit can be incorporated into the crystal: site A, a flat surface with one bonding side; site B, a step with two bonding sides; and site C, a kink with three bonding sides. From an energetic perspective, site C is the most favorable site, followed by site B and finally site A. This translates into a crystal growth mechanism whereby molecules are absorbed on the surface and then diffuse along the surface until they are incorporated into the lattice at a step or kink. When a surface is populated with sites such as B and C, it will follow a rough

Molecular Modeling Applications in Crystallization

crystal slice solution

Uhkl

Figure 5.9 Modified attachment model

solvent

solvent

vac,att

Ehkl

crystal

crystal slice crystal

crystal

Figure 5.10 Morphology of aspirin grown from solution in 38% ethanol–water mixed solvent. (a) Predicted using the AE model, (b) predicted using the MAE model, and (c) observed morphology.

Figure 5.11 Description of the build-in approach and the surface-docking approach Source: Lu 2003.

growth mechanism, where molecules can incorporate into the crystal regardless of their position, and growth is typically limited by the transport of solute molecules to the surface. By contrast, if the surface is populated with site A, it is called flat and will follow a layered growth mechanism, which is usually ~10–100 times slower than the rough growth mechanism (Zhang et al. 2006). Layered growth occurs either via 2D nucleation and growth or via a spiral growth mechanism depending on the method of edge/kink generation (Chernov 1984). In a 2D nucleation and growth mechanism, the nucleation of small, isolated islands on the surface is the origin of edges and is then followed by lateral growth, as shown in Figure 5.14a. The free energy per mole of critical regular n-sided nuclei can be calculated according to the equation

Figure 5.12 The MAE for the build-in additives: A = original molecules in the lattice; B = build-in additives

DGc ¼ ntanðπ=nÞ

ϕedge Dμ

ð5:5Þ

where ϕedge is the surface energy of the edges of the 2D nuclei, and Dμ is the difference between the chemical potentials of the

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout

2008). The growth mechanism as a function of supersaturation is shown in Figure 5.15. Both the 2D nucleation and growth and spiral growth models require a good estimation of the surface energies. One of the most notable contributions is from Winn and Doherty (1998, 2000, 2002). In their model, the kink free energy was calculated as ϕkink ¼ γkink Akink

Figure 5.13 Surface structure of a growing crystal

solute in solution and the crystal phase. The growth rate of a crystal is inversely proportional to DGc . The 2D nucleation and growth model only applies when DGc > 3RT, as is the case when supersaturation is not too high and Δµ is not too large. When ΔGc approaches 3RT, the face becomes roughened by the presence of many nuclei, and growth no longer proceeds by a stable and flat growth mechanism (van der Eerden 1993). When DGc ≫ 3RT, as is the case when the supersaturation is too low and Δµ is small, growth by a 2D nucleation mechanism is extremely slow, and a spiral growth mechanism (Burton–Cabrera–Frank [BCF] mechanism) will take over (Burton et al. 1951). The basic idea of the BCF mechanism is that dislocations in the crystals are sources of new steps. A type of dislocation known as a screw dislocation could provide a way for the steps to grow continuously, as shown in Figure 5.14b. In this model, kink integration is the ratelimiting step, and the face-dependent growth rate is proportional to the density of kink sites multiplied by the distance the step is propagated R∝

d ap ½1 þ 0:5 expðϕkink =RTÞ1 : y

ð5:6Þ

where ap is the distance the edge is propagated by adding a monolayer to it, d is the step height, and y is the lateral distance between steps. The term in brackets is the probability of finding a kink in a step at equilibrium, and ϕkink is the work, or free energy, to create a kink along the edge. This expression remains valid only when ϕkink is greater than RT. When it is smaller, kinks are sufficiently numerous that the edges are roughened, and the edge free energy is sufficiently low that 2D nucleation occurs. The general formula for the interstep spacing is y ∝ κϕedge , where κ depends on the geometry of the spiral. With this, the growth rate in the BCF model becomes R∝

d a ½1 þ 0:5 expðϕkink =RTÞ1 edge p κϕ

ð5:7Þ

The mechanism by which a flat face grows depends on the external growth environment, specifically the supersaturation and temperature at the surface. Generally, a spiral growth mechanism is followed at low supersaturation, a 2D nucleation and growth mechanism at medium supersaturation, and rough growth when the supersaturation goes beyond the roughening transition point (Lovette et al.

142

ð5:8Þ

where γkink is the free energy of forming a kink per unit kink area, and Akink is the kink area. To estimate γkink , a geometric approximation was used qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cryst cryst solv γkink ¼ γ þ γ  2 γi þ γ0;solv ð5:9Þ i i where the superscripts denote crystal and solvent phase surface free energies, and the subscript i denotes a kink site on a particular edge on a crystal face. The prime symbol denotes only the dispersive (van der Waals) component of the surface free energy. Solvent surface free energies γsolv have been measured for many solvents in contact with air (Kaelble 1971; Adamson and Gast 1997). The surface free energy of crystal was approximated using the surface energy with the assumption that changes in internal energy dominate the creation of crystal surfaces in contact with a vacuum (Kitaĭgorodskiĭ 1973; Saska and Myerson 1983). The surface energy of a crystal face can be easily calculated using many available software programs, most notably Materials Studio, which requires only the crystal structure and choice of force field as input. From the relative growth rates, a shape can be drawn using the Wulff construction. Winn and Doherty (1998, 2000) successfully predicted the morphology of ibuprofen grown from hexane and methanol using this approach (Figure 5.16). These detailed kinetics-based growth models are in general more accurate and capable of capturing the impact of solvents and additives, and there are many cases demonstrating their success in predicting the shape of organic crystals grown from solutions: urea grown from water (Liu and Bennema 1996), succinic acid grown from water and from isopropanol (Winn and Doherty 2002), biphenyl grown from toluene (Liu and Bennema 1996; Winn and Doherty 1998), and naphthalene grown from toluene (Liu and Bennema 1996). However, one of the major limitations of this approach is that it does not consider the level of supersaturation when determining which growth model to use. In fact, ϕkink , the only criterion by which to determine the growth mechanism, is a function of crystal structure and solvent properties only. Therefore, it is constant throughout the entire growth process. Thus, a face will follow one growth mechanism from beginning to end. This is not entirely accurate for faces whose growth mechanism changes with supersaturation (Land and De Yoreo 2000). Another drawback is that this approach cannot predict the absolute or relative velocities of faces undergoing rough, transport-limited growth, which is very common for needlelike crystals.

Molecular Modeling Applications in Crystallization Figure 5.14 Kinetic-based growth models: (a) 2D nucleation mechanism, (b) spiral growth mechanism

Rough Growth

Growth Rate

Spiral Growth

2D Nucleation and Growth

Supersaturation Figure 5.15 Growth mechanism of a flat (F) face, as a function of supersaturation Source: Reprinted with permission of the publisher from Lovette et al. 2008. Copyright © 2008, American Chemical Society.

5.3.5 Monte Carlo and Kinetic Monte Carlo Methods Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute results. They are especially useful in studying systems with a large number of coupled degrees of freedom. Boerrigter et al. (2004) developed a Monte Carlo algorithm based on existing software called MONTY, a crystal growth simulation program. The experimental crystal structure is the input for the algorithm. It is modeled by a set of molecular interactions, which are obtained from molecular mechanics calculations. The mother phase is parameterized by its bulk thermodynamic properties. In this algorithm, a growth unit only has limited accessible states (possible interactions with the surface). The probability ratio of a growth unit i growing into state j and leaving state j is given by growth

Pi2j

etch Pi2j



 DGi2j DUi2j  TDS þ PDV ¼ exp ¼ exp ð5:10Þ kB T kB T

where i2 j is a shorthand notation for the state j of growth unit i. This Monte Carlo–based method was capable of dealing with the polar character of faces, as well as the influence of supersaturation. Both the 2D nucleation and growth and spiral growth mechanisms are implemented in this program. It was

demonstrated that, using MONTY, the elongated needle morphology of fat crystals (Meekes et al. 2003) and aspartame (Cuppen et al. 2004a) could be predicted with a 2D nucleation and growth mechanism, and the supersaturation-dependent morphology of monoclinic paracetamol could be understood with a spiral growth mechanism (Cuppen et al. 2004b). The kinetic Monte Carlo (KMC) method is a Monte Carlo–based method of computer simulation intended to simulate the time evolution of certain processes occurring in nature. The KMC method for the simulation of crystal growth, developed by Piana and Gale (2005, 2006), allows for the effect of solvation on the growth and dissolution kinetics to be fully included while extending the size of the simulation to the micrometer length scale and millisecond time scale. This method assumes that crystal growth follows either 2D nucleation and growth or a spiral growth mechanism. The growth and dissolution rates of a growth unit into a surface site (flat surface, step, or kink) are derived from atomistic molecular dynamics simulations (at intervals of δt) of the crystal–solution interface by counting the frequency of each event. At each KMC step, each surface site (occupied or unoccupied) is evolved according to the probability of observing a reactive event for that site in the time interval δt. After all the surface sites have been tested, the clock is advanced by δt, and the process is repeated. This scheme can run on parallel architectures with very high efficiency and makes it feasible to perform simulations on much bigger systems for a longer time scale than traditional molecular dynamics can handle. This KMC method accurately reproduced the experimentally observed crystal morphologies of urea from water and methanol, respectively (Figure 5.17; Piana and Gale 2005). Unlike previous models of crystal growth, no assumption is made in this KMC method that the morphology can be constructed from the results for independently growing surfaces or from an a priori specification of surface defect concentration.

5.3.6 Understanding the Chemistry at the Crystal–Solution Interface Current knowledge of the factors influencing the growth of an organic crystal from different solution conditions is very limited, although highly desirable, because the interplay between solute–solvent, surface–solvent, and surface–solute interactions is very subtle and cannot be easily modeled with a static

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout Figure 5.16 Predicted and experimental morphologies of ibuprofen: (a) predicted from hexane, (b) experimental from hexane, (c) predicted from methanol, and (d) experimental from methanol

Figure 5.17 Comparison of the experimental (in situ optical microscope image) and predicted (by KMC) micrometer-scale crystal morphologies of urea from (a) water and (b) methanol

representation of the crystal surface and its environment. Molecular dynamics (MD) is a technique for computing the equilibrium and transport properties of a classic many-body system, where the motion of the constituent atoms/particles obeys Newton’s second law of motion. It offers an attractive way to study the growth of molecular crystals from solution because it is not limited to a single structure but makes it

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feasible to calculate the properties of an ensemble of conformations, therefore fully incorporating entropic contributions. Here we will use two examples, a steroid and 2,6-dihydroxybenzoic acid, to shed some light into the potential of MD simulations to help understand the molecular interactions at/ on crystal surfaces. The steroid form I crystals grown from methanol exhibit a polar habit experimentally, which indicates a difference in growth rates between the {010} and {0–10} faces. Stoica et al. (2004) performed MD simulations of the {010} and {0–10} faces in contact with methanol. Their results revealed that the methanol molecules have a much stronger interaction with the hydroxyl-rich {0–10} face as opposed to the carbonyl-rich {010} face. When listing the three contributions to the total energy of the surface–solvent interactions present in the Dreiding force field – the van der Waals, Coulomb, and hydrogen bond energy – they found that it is not only the hydrogen bonds but also the electrostatic and van der Waals interactions between the surface and solvent that have to be taken into account to explain the polar morphology (Figures 5.18 and 5.19). 2,6-Dihydroxybenzoic acid (DHB) form II crystals exhibit an elongated needle-like morphology when grown from both chloroform and toluene. Chen and Trout (2010a) performed MD simulations of both the {001} and {00–1} faces (the two needle tips) in contact with various solvents, including chloroform, toluene, diethylether, and ethanol. They showed that two functional groups – an aromatic ring and a hydrogen bond acceptor – form strong interactions with the two needle tips, respectively, by analyzing the MD trajectory and visualizing the crystal–solvent interface at an atomic level (Figure 5.20). Using this knowledge, they designed a solvent mixture that has both an aromatic ring and a hydrogen bond acceptor group to crystallize DHB. This solvent mixture was proven experimentally to be able to reduce the aspect ratio of the DHB crystals.

5.3.7 Available Software 5.3.8 Summary and Outlook Over the past few decades, crystal morphology prediction has been a very active field of research, and many significant

Molecular Modeling Applications in Crystallization

advances have occurred. Among the most commonly used morphology prediction methods, the BFDH and AE models are the simplest but least accurate owing to the impact of solvents/additives and supersaturation being neglected. MAE methods are better models because they incorporate the effects of solvents/additives to some extent but neglect supersaturation. Kinetic-based models are in general more accurate owing to the inclusion of the influence of both solvents/additives and supersaturation, although the influence of supersaturation is only qualitatively related to selection of the proper growth mechanism. With advances in computational power, new computer programs and algorithms have been and continue to be developed (e.g., KMC) to simulate the effects of supersaturation and solvent on crystal morphology using a more quantitative approach with greater accuracy. In addition, new Table 5.2 Available Morphology Prediction Models and Software Packages and Their Ease of Use

Methods for morphology modeling

Software

Ease of use

BFDH

Mercury Materials Studio

Easy, ready to use

AE Modified AE (solvent/additive effect)

Materials Studio

Easy, ready to use

Kinetics based models

Materials Studio

Moderate, ready to use

Monte Carlo Kinetic Monte Carlo

MONTY

Difficult, in development

Investigation of interfaces

Various molecular simulation packages (GULP, NAMD, CHARMM, Materials Studio, etc.)

Moderate, in development

evolution models (Zhang et al. 2006) other than the geometric surface evolution model (meaning that the outward normal velocity of a surface depends only on its local orientation) are under active development. The application of these new methods is expected to become commonplace in the near future.

5.4 Crystal Structure Determination from X-Ray Powder Diffraction Data 5.4.1 Introduction Crystal structure determination using X-ray diffraction has been common practice for nearly a century. The most common method is single-crystal X-ray diffraction. This method relies on the investigator successfully growing a suitable single crystal of correct size and sufficient quality for analysis. However there are many cases in which growing such a single crystal is not possible, and the material is only suitable for analysis by

Figure 5.18 Experimental morphology of a steroid crystal grown from methanol

X-ray powder diffraction. This presents a significant problem when attempting to solve a crystal structure because there are considerably fewer data available. As a consequence, finding important crystallographic parameters such as the unit cell dimensions and space group is a much more challenging process (Ladd and Palmer 2003). In order to overcome this data problem, several computational methods have been developed to solve crystal structures from powder data when the molecular connectivity of the compound is known. This section discusses how to collect suitable X-ray powder diffraction (XRPD) data and the various computational methods available for structure solution.

5.4.2 Collecting Suitable X-Ray Powder Diffraction Data As mentioned previously, fewer data are available from powder diffraction techniques than from a single-crystal study. Hence it is important to obtain a high-quality powder diffraction pattern. The data must have been measured using a monochromatic X-ray source (laboratory or synchrotron X-ray data) and exhibit low background noise (Ladd and Palmer 2003; Shankland et al. 2006). Preferred orientation of the crystalline powder, a crystal plane (or planes) whose diffraction intensity is significantly greater than all others, is undesirable and can cause significant problems when attempting to solve the crystal structure. Therefore, to minimize any potential effect of preferred orientation, it is commonplace to rotate the sample during data collection. One such successful method is to pack the sample into a capillary tube, which is then mounted onto a goniometer and rotated during data collection.

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout

structure determination may be achieved is shown in Figure 5.21.

5.4.3 Indexing a Powder Diffraction Pattern

Figure 5.19 Histogram of the various energy contributions for solvent–surface interactions on both the {010} and {0–10} faces using methanol as the solvent Source: Reprinted with permission of the publisher from Stoica et al. 2004. Copyright © 2004, American Chemical Society.

Put simply, indexing is the determination of the unit cell parameters. The first step in indexing a powder diffraction pattern is to obtain accurate peak positions (Shankland et al. 2006). Figure 5.22 is an example of how DASH assigns peak positions to different data sets. Once the peak assignments have been made, the unit cell parameters are determined using an indexing program. Whereas many programs are available, some of the most common and frequently used are TREOR (Werner et al. 1985; Altomare et al. 2009), DICVOL (Boultif 2005), and ITO (Visser 1969). Each program uses different indexing protocols to generate possible unit cells, but all use a figure-of-merit system to identify the correct cell parameters (Altomare et al. 2009). Each crystal structure determination software package incorporates one or many of these indexing programs in order to determine the correct unit cell.

5.4.4 Pawley/Le Bail Refinement Having successfully indexed the powder diffraction pattern and determined the correct unit cell, the next step is to carry out a Pawley refinement on the data (Pawley 1981; Engel et al. 1999; Ladd and Palmer 2003; Shankland et al. 2006; Bruker Corporation 2014). The refinement allows the peaks in the diffraction pattern to be fitted using a least squares method that varies parameters such as peak shape and intensity. This procedure optimizes the cell parameters (without knowledge of the atomic positions) and assists in the determination of the correct space group. The Le Bail refinement method (Le Bail et al. 1988) was developed in order to overcome peakfitting issues in the original Pawley method (Pawley 1981). While not as commonly used as the Pawley refinement, Le Bail can still be used to successfully optimize the unit cell parameters.

5.4.5 Structure Solution Figure 5.20 The two needle tips, {001} (top) and {00–1} (bottom), of DHB form II crystals in contact with a toluene -diethylether mixture. The toluene molecules form strong interactions with the {00–1} face through π–π interactions. The diethylether molecules with the oxygen atoms depicted as dots, form strong interactions with the {001} face through hydrogen bonding.

Once suitable diffraction data have been collected, a structure may be determined using a suitable structure determination software package (DASH, Shankland et al. 2006; PowderSolve, Engel et al. 1999; TOPAS, Bruker Corporation 2014). While each software package uses different computational methods, the overall scheme by which a crystal structure is solved from powder diffraction data is approximately the same. A scheme depicting how a crystal

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Once the experimental data have been fitted and the required parameters have been determined, the next step is to carry out the structure solution. Currently, numerous programs are capable of solving crystal structures, and advances in programming and computational power ensure the successful solution of increasingly complex systems. The vast majority of these programs (Engel et al. 1999; Favre-Nicolin and Černý 2002; Shankland et al. 2006; Bruker Corporation 2014) employ direct-space methods that use global optimization algorithms, such as a Monte Carlo search with simulated annealing (Newsam et al. 1992; Andreev et al. 1997) to find the global minimum structure. These direct-space methods determine the global minimum by ascertaining which crystal structure best matches the experimental data by adjusting molecular position and orientation within the indexed unit cell (FavreNicolin and Černý 2002; Shankland et al. 2006). David et al. noted that in the case of the simulated annealing structure

Molecular Modeling Applications in Crystallization

Indexing the powder pattern

Pawley–Le Bail Refinement

• The ability to differentiate between similar elements, such as Fe, Ni, and Co • To solve the crystal structures of complex compounds (up to 100 atoms in the asymmetric unit)

5.4.8 Available Software for XRPD Structure Solution 5.4.9 Summary

Structure Solution

Rietveld Refinement

Figure 5.21 Schematic of the basic steps involved in structure determination from powder data

solution used in DASH, it is typically the number of degrees of freedom, such as flexible torsion angles, and not the size of the molecule that determines the success of a structure solution (Shankland et al. 2006). In order to ensure that the global minimum is obtained, it is common practice to run multiple annealing simulations owing to the stochastic nature of the method.

5.4.6 Rietveld Refinement After the global minimum structure has been found, the last step in the process is to refine the structure further using a Rietveld refinement (Rietveld 1969; Ladd and Palmer 2003). In this step, a least squares method is carried out to find the best fit between the whole experimental diffraction pattern and one calculated for the solved structure.

5.4.7 Successful Structure Solutions The success of these structure determination methods is often measured by comparing these structures with those determined by single-crystal data. In 2005, Florence et al. solved the known crystal structures of 35 organic compounds from powder diffraction data using DASH (Florence et al. 2005). They determined that for compounds of differing complexity, it was indeed possible to achieve the correct structure solution. Figure 5.23 shows the DASHsolved structure of 2-{[3-(2-phenylethoxy)propyl]sulfonyl} ethyl benzoate (in black) overlaid with the single-crystal structure (in gray). More recently, Lasocha (2008) gave a succinct overview of what computational methods and advancements in powder diffraction techniques offer today; some are listed below: • The refinement of lattice and crystal structure parameters

Solving crystal structures from powder diffraction data has become a viable alternative to single-crystal X-ray diffraction when obtaining a single crystal of the crystalline phase in question is not possible. Advances in the computational methods used to index powder patterns and for structure solution have led to the successful crystal structure determination of molecules of increasing complexity. Furthermore, studies have shown that crystals structures obtained from powder diffraction data can be as accurate as those solved by single-crystal methods.

5.5 Modeling Nucleation and Phase Transitions 5.5.1 Introduction Understanding nucleation is one of the most difficult problems in crystallization – from both experimental and theoretical perspectives. On the one hand, the initial stages of nucleus formation and growth typically involve only a few molecules or ions, and owing to the stochastic nature of nucleation, one cannot predict in advance where or when the critical nucleus will form. This makes experimental investigations exceedingly difficult. On the other hand, the typical time scales for nucleation and growth are on the order of seconds, hours, or even days, whereas most typical atomistic simulation methods can only explicitly model processes lasting up to a few microseconds. This means that sophisticated simulation methods are required to bridge this time-scale gap and understand the molecular processes involved in nucleation. In this section, we will present some of the theoretical approaches that have been used to understand the molecular-level details of nucleation in solid–liquid and solid–solid phase transformations and discuss some of the remaining problems in this challenging area of crystallization.

5.5.2 Molecular Self-Assembly in Solution and Its Implications for Nucleation and Polymorph Selection Direct study of the nucleation of molecular crystals from solution to understand the formation of polymorphs is quite a challenge for current experimental and computational techniques because of the stochastic characteristics of the nucleation event and the extremely small size of the

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(a)

(b)

(c)

Figure 5.22 Image showing DASH assigning peak positions to powder diffraction data for (a) asymmetric, (b) noisy, and (c) partially overlapping peaks. The data in red are the experimental data, the green curves are the fitted peaks, and the blue lines are the assigned peak positions. Source: Reproduced from Shankland et al. 2006, with permission from the International Union of Crystallography.

Table 5.3 Available Complete Structure Solution Software Packages and Diffraction Indexing Programs

Structure solution software packages

Powder diffraction pattern indexing programs

DASH

TREOR

PowderSolve

DICVOL

TOPAS

ITO

critical nuclei. An alternative approach has been to investigate the behavior of molecules in solution prior to nucleation and has been applied to a number of systems with the aim of helping to understand how a particular polymorph nucleates from specific crystallization conditions (e.g., temperature, solvents, and additives). The selfassembly of solute molecules in solution is a much faster process than nucleation, taking place on a time scale of nanoseconds. MD has been demonstrated to be a very useful tool in understanding this self-assembly process in solution. This approach can track the motion of each atom in the system and provide atomic-level understanding of the solute–solute and solute–solvent interactions. Moreover, MD can be implemented to study directly the processes that experimental approaches still find very difficult. Hence, it is a good complementary tool for experimental techniques. MD has been used to study the molecular self-assembly and precursor formation for a number of systems. 5-Fluorouracil is a good example. As will be discussed in Section 5.6, 5-fluorouracil form I is the originally known form that crystallizes from water; form II was first predicted by constructing the energy landscape of this molecule and later on was crystallized from dry nitromethane. Hamad et al. (2006) studied the interactions between 5-fluorouracil and solvent molecules by performing MD simulations of the compound in water, nitromethane, and wet nitromethane. The radial distribution function (RDF) was calculated to investigate the solute–solvent

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and solute–solute interactions. The nature of the hydration of 5-fluorouracil in aqueous solution is indicated by the differences in radial distribution functions with sharp peaks at an O    Hw distance of 1.6 Å (O refers to the oxygen in the solute molecule; Hw refers to the hydrogen in water; Figure 5.24c) and at an O    Ow distance of 2.7 Å (Ow is the oxygen molecule in water; Figure 5.24b). The hydrophobicity of the fluorine region induces F    F contact in a large water-free cavity. The first peak in the F    F RDF (Figure 5.24d) corresponds to this close hydrophobic contact. In nitromethane, the solvation of 5-fluorouracil is very different from that in water. The first peaks in RDFs for both solute–solvent and solute–solute interactions are much broader. The authors concluded that 5-fluorouracil is more evenly solvated and less strongly bound by nitromethane, thus making it easier to displace nitromethane than water from the solvation sphere. Thus it induces the formation of a dimer, which leads to the crystallization of form II. Although not all polymorphic systems can be understood by studying the molecular interactions in the mother solution using MD (exceptions do exist), it can be a very useful tool to help clear some of the controversies that cannot be resolved by experimental techniques alone. Glycine is a widely studied polymorphic system. It has three polymorphic forms, α, β, and γ, with α-glycine (packed in centrosymmetric dimers) obtained from aqueous solutions and β-glycine (packed in an open-chain-like structure) obtained from water–alcohol solutions. There is controversy as to whether glycine molecules exist mainly as dimers in aqueous solution, which leads to the crystallization of the dimer structure–based α form, by measuring different properties of glycine aqueous solutions (Chattopadhyay et al. 2005; Erdemir et al. 2007; Huang et al. 2008). A number of molecular dynamics studies were performed to study the cluster formation of glycine molecules in solutions directly. The results of the MD simulation of Hamad et al. (2008) showed that glycine monomers are the dominating species in aqueous solution. Chen and Trout (2010b) computed the free energy differences between a glycine centrosymmetric dimer and a glycine open-chain

Molecular Modeling Applications in Crystallization

Figure 5.23 An overlay of the best simulated annealing structure from DASH of 2-{[3-(2-phenylethoxy)propyl]sulfonyl}ethyl benzoate (shown in black) with the single-crystal structure (shown in gray). Both structures are in good agreement. Source: Reproduced from Florence et al. 2005, with permission from the International Union of Crystallography.

catemer and two glycine monomers both in pure aqueous and methanol-containing solutions (Figure 5.25) and found that the open-chain catemer (the building block for β-glycine) is always more stable than the centrosymmetric dimer (the building block of α-glycine), and the dimer is the least favorable among the three states being compared in glycine aqueous solutions.

5.5.3 Measuring Crystalline Order When modeling crystallization or solid-solid transformations, one of the practical difficulties is in how to get a computer program to automatically decide the current state of the system (i.e., liquid versus solid or one solid versus another or something in between). Without a way to quantify where in the phase diagram the system is, it is impossible to model how the system moves from one point to another. The variables used for this purpose are often termed order parameters (OPs). An order parameter is any mathematical function that can differentiate one phase from another. In the most general case, OPs can depend on any coordinates of the system, including all the positions and velocities of all the atoms or molecules, and on the previous values of those coordinates. When studying phase transformations, OPs are typically defined so that they take one particular value (e.g., zero or a small value) when the system is in one state (e.g., liquid) and another value (e.g., one or just a large value) when the system is in the other state (e.g., solid). When more than two phases are involved, one would have different characteristic values corresponding to each phase. OPs are of crucial importance in modeling phase transitions because they define the relevant space where the transformation occurs. As such, it is perhaps not surprising that several different types of OPs have been developed to study

crystallization. There are generally four different types of approaches that people have used to define OPs for crystals: 1. Symmetry-Based OPs. In these approaches, OPs are built using mathematical functions with the property that, when summed for a structure with the symmetry corresponding to the crystal, they interfere constructively to give a large value, whereas they interfere destructively for a disordered structure, giving a low value. Examples of this approach are the bond-orientation OPs of Steinhardt et al. (1983), which have been used quite successfully in studies involving atomic crystals or crystals of small, highly symmetric molecules – examples of their use are discussed in Section 5.5.4, and a more extensive survey can be found in Santiso and Trout (2011). It is difficult, however, to generalize these OPs to crystals of more complex molecules, which limits their applicability for systems of practical interest such as pharmaceuticals. 2. Parameters Related to Molecular Mobility. The main idea behind these approaches is that molecules can move much farther away from a given initial position in a liquid than they can in a solid or glassy phase. Examples of quantities related to molecular mobility that have been used as OPs include the self-diffusion coefficient (Broughton et al. 1982; Zahn 2007) and the Lindemann index (Lindemann 1910; Xu and Bartell 1993). These parameters are more generally applicable than the symmetry-based order parameters discussed earlier, but they depend on the time evolution of the system rather than its current state. Because of this, it is very difficult to construct a system with a prespecified value of these OPs or to bias the system toward it. This makes simulations using these order parameters as decision variables very challenging. 3. Structure-Based OPs. In this case, the structure of the crystal is used as a template against which a given structure is compared to determine whether the structure is crystalline or not. A generalized approach of this kind was recently introduced by Santiso and Trout (2011) – the method constructs a model for the probability distribution of distances, bond orientations, relative orientations, and internal degrees of freedom of the molecules in the crystal and uses this model as a measure of the probability that a given, unlabeled structure is crystalline. This approach has the advantage of being completely general and does not depend on the history of the system, as is the case with mobility-based order parameters. However, it is a relatively recent approach, and it has not yet been applied to a large variety of systems. Examples of this approach are discussed in Section 5.5.4. 4. Ad Hoc OPs. In some cases, it is possible to identify a particular geometric feature or set of features that characterizes well the degree of crystalline order in a particular system. For example, one can measure a particular angle or set of angles between certain atoms, which would have a broad distribution in the liquid state but be much more restricted in the solid state. An example of this type of OP is shown in Section 5.5.4. Although these types of OPs can be used successfully to study

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(a)

(b) 1.6 1.4 Hp1

1.2 1.0

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F-Ow O7-Ow O8-Ow H1-Ow H3-Ow H6-Ow

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Figure 5.24 RDF results for the 5-fluorouracil study: (a) molecular structure of 5-fluorouracil; (b) RDFs between atoms in 5-fluorouracil and oxygen in water; (c) RDFs between atoms in 5-fluorouracil and hydrogen in water; (d) RDFs between two fluorine atoms in two different 5-fluorouracil molecules

crystallization, they are typically useful for only one compound or at best a particular family of compounds and cannot be applied in general. In the following section we will discuss a number of methods to study crystal nucleation, which will exemplify the use of several of the OP types just discussed.

5.5.4 Crystal Nucleation The study of the nucleation and growth of crystals from the melt and solution is very important for many applications in the chemical, pharmaceutical, food, construction, and defense industries (Taylor 1997; Hartel 2001; Lee and Myerson 2006; Bernstein 2008; Santiso and Trout 2011). Modeling nucleation at the molecular level is an extremely challenging problem, however (Kelton 1991; Maddox 1995; Bartell 1997; Price 2009; Santiso and Trout 2011). The typical time scales involved in nucleation are many orders of magnitude larger than those accessible to direct MD simulations. In addition, the diffusive nature of the process makes it hard to explicitly follow the time evolution of crystallization paths, even after forming a critical

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nucleus. This severely limits the applicability of methods that rely on sampling transition paths such as transition path sampling (Bolhuis et al. 2002; Santiso and Gubbins 2004) and reactive flux methods (Chandler 1998; Ciccotti and Ferrario 2000; Santiso and Gubbins 2004). Despite these difficulties, however, there have been multiple simulation studies of crystal nucleation in the past. Most of these studies involve simple systems such as monatomic fluids, highly symmetric molecules, or colloids, for which it is easy to define OPs to characterize the degree of crystallinity of the system (Steinhardt et al. 1983; ten Wolde et al. 1996). As described later in this section, more recent studies have begun to tackle the problem of simulating nucleation, in detail, in more complex systems such as molecular crystals. Most methods used to study nucleation in small, roughly spherical molecules use some variation of the OPs defined by Steinhardt et al. (1983) to measure the degree of crystallinity of the system. Using these OPs, one can use some of the techniques available to model rare events and follow the evolution of the system from the liquid to the crystalline state. One example

Molecular Modeling Applications in Crystallization

2. Molecules with more than just a few atoms typically have significant flexibility, and this must also be taken into account when describing the nucleation process. 3. Finally, large molecules with different functional groups are more likely to have one or more specific interactions that need to be modeled accurately.

2 water

G (kcal/mol)

1.5

methanol–water (3:7)

1 0.5

monomer catemer

0 dimer

–0.5 –1

3

3.5

4

4.5

5

5.5

6

6.5

7

OP (Å) Figure 5.25 Free energy profiles for the glycine dimerization reaction in water and methanol–water 3:7 mixtures at 298 K and 1 atm obtained using the MD umbrella sampling method

of this approach is the work of ten Wolde et al. (1996). In this work, the authors used a local version of the order parameters of Steinhardt to study nucleation in the Lennard–Jones fluid (Figure 5.26) – a model for approximately spherical nonpolar molecules such as argon or methane. Using the umbrella sampling method (Torrie and Valleau 1977), the authors obtained the free energy profile as the system progresses from the liquid to the solid state (Figure 5.26a). Then they studied the structure of the critical nucleus (Figure 5.26b) and obtained the nucleation rate using a reactive flux approach (Chandler 1998; Santiso and Gubbins 2004). This study found that although most of the critical nucleus resembled a face-centered cubic structure (which is the stable crystalline form of the Lennard–Jones fluid), the interface between the critical nucleus and the liquid had a structure similar to the metastable bodycentered cubic structure. Moreover, the authors found that although the critical nucleus for this particular system is roughly spherical, classical nucleation theory does a poor job of predicting both the size of the critical nucleus and the nucleation rate. More recently, Beckham and Peters (2011) used a different method, based on the statistical mechanics of MD trajectories, to obtain a better measure of nucleus size for the crystallization of the Lennard–Jones fluid (Figure 5.27). In their work, they found that the product of the number of particles in the crystalline nucleus and the local degree of crystallinity (as defined by ten Wolde et al. 1996) serves as a better metric of nucleus size in this system and can be used as a true measure of progress along the crystallization path. This result, however, is limited to systems that are adequately modeled by the Lennard–Jones potential, i.e., small highly symmetric molecules. The study of nucleation in general molecular systems is much more difficult than the equivalent process for atomic crystals or small, symmetric molecules. There are three main reasons for this: 1. Because molecules are, in general, not spherically symmetric, their orientation plays an important role in describing the crystal structure and thus in measuring the extent of crystallization.

The third point will be discussed in Section 5.6.4. A consequence of the first two points is that it is difficult to define functions to measure the extent of crystallization in general molecular systems [the equivalent of the order parameters of Steinhardt et al. (1983) mentioned earlier]. Often this is done in a trial-and-error fashion by testing different functions of the atomic coordinates or by taking advantage of a particular structural feature of the crystal or molecule under study. An example of the latter is the study by Yi and Rutledge (2009) of crystal nucleation in n-octane melts (Figure 5.28). In this work, the authors used the definition of Esselink et al. (1994) for the nucleus size in chain molecules – two molecules are considered to be in the same cluster if the distance between their centers of mass and the angle between the directors of their chains are below predefined thresholds. The authors used this definition, in combination with MD simulations and umbrella sampling, to analyze nucleation paths of n-octane and characterize the geometry of the critical nucleus. For this system, the authors obtained the free energy profile along the reaction path and found that the critical nucleus shape is modeled more accurately as a cylinder than as a sphere. More recently, Santiso and Trout (2011) have developed a general procedure to define order parameters to characterize crystalline order in molecular systems in a way that does not rely on trial and error or ad hoc choices of geometric features. In this method, an MD simulation is used to build a model for the distribution of center-of-mass distances, orientations, and relative orientations between pairs of molecules in the crystal at the temperature of interest. This model is then used to construct functions that measure the probability that a molecule in an arbitrary configuration is part of a solid-like cluster. The method can be easily adapted to incorporate information on internal degrees of freedom for flexible molecules. An example application of this method to identify solid glycine in solution is shown in Figure 5.29. More recently, Shah et al. (2011) have used the order parameters developed by Santiso and Trout (2011) in combination with transition path sampling to study the nucleation of benzene from the melt. In this work, the authors found that orientational order parameters play a crucial role in describing the nucleation process. Moreover, they found that there is actually a wide ensemble of critical nuclei with varying shapes and sizes (Figure 5.30) – which illustrates the complexity that arises when considering molecules with more than just a few atoms.

5.5.5 Solid–Solid Phase Transitions The study of solid–solid phase transitions is particularly important in materials science as well as in the chemical and pharmaceutical industries. Many compounds exhibit multiple

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Figure 5.26 Nucleation of the Lennard–Jones fluid: (a) free energy as a function of degree of crystallinity; (b) snapshot of a critical nucleus Source: Reprinted from ten Wolde et al. 1996, with the permission of AIP Publishing.

crystalline forms and, given enough time, will convert to the most thermodynamically stable form, which may have different properties than desired. The transformation may also leave defects in the structure of the solid, which may alter its mechanical properties. From the point of view of molecular modeling, solid–solid phase transitions are especially difficult to model. There are several reasons for this. First, just as in the case of nucleation from a melt or solution, solid–solid transitions usually happen on time scales many orders of magnitude larger than those accessible to direct MD simulations. It is also particularly hard to find good potential energy functions that correctly predict the structure of all the polymorphs and their energy orderings – most force fields are fit to reproduce the thermodynamic properties of liquids, not solids. There is also a unique difficulty to the problem of solid transformations: in general, molecular modeling is done in cells with periodic boundary conditions, and depending on the cell parameters of the solid phases under study, it may be necessary to use a very large simulation box for the system to convert from the lattice of one of the solid forms to the other. Alternatively, one can run a simulation of a crystal in vacuum, but this would correspond to a nanometer-sized crystal or cluster, which could have very different properties from the bulk solid. Because most of the difficulties in modeling solid–solid transitions are similar to those that pertain to modeling nucleation from a liquid phase, it is perhaps not surprising that the same techniques have been applied to this problem. However, because of the additional difficulties in finding reliable force fields and setting up the simulation, studies of solid–solid phase transformations are somewhat scarcer.

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Often studies involve small clusters/crystals in a gas phase or systems undergoing phase transitions driven by an external potential. A recent example of the latter is the work of Sandoval and Urbassek (2009) involving the strain-induced solid–solid phase transitions in iron nanowires (Figure 5.31). In this work, the authors used MD simulations to obtain solid–solid transformation pathways under different strains for nanowires with different crystallographic orientations. They found that the phase transformation exhibited a complex dependence on the orientation of the wire, with the new phase growing in a different direction (radial or axial) depending on orientation. They also found that multiple phase transitions and crystal reorientations may be induced by strain. This illustrates that even for such a simple system, the dynamics of solid–solid phase transformation can be quite complex. A recent example involving the more complex problem of solid–solid transformations in molecular crystals is the work of Beckham et al. (2007, 2008), where they studied the form I to form II transformation of terephthalic acid (TPA) using a transition path sampling approach (Figure 5.32). In their work, the authors considered the form I–II transition in TPA clusters of different sizes in vacuum and found that the transformation occurs via the formation of a localized, elongated nucleus along the crystal edge. They also found that a local ratio of lattice parameters serves as the best measure of progress along the transformation path (i.e., the reaction coordinate). However, their results show that the transformation mechanism depends on the size of the cluster studied (Beckham et al. 2008), highlighting the difficulty in modeling solid–solid transformations in bulk systems using molecular models.

Molecular Modeling Applications in Crystallization

Figure 5.27 (Top) Snapshots of a nucleation process in the Lennard -Jones fluid. Light gray represents liquid-like particles, and dark gray represents crystal-like particles. (a) A precritical cluster; (b) a critical nucleus; (c) mostly solid. (Bottom) A plot of effective nucleus size versus number of particles in the nucleus showing the locus of critical nuclei (bold line). Light gray denotes trajectories that end mostly in the liquid state, and dark gray indicates those that end mostly in the solid state. Source: Reprinted with permission of the publisher from Beckham and Peters 2011. Copyright © 2011, American Chemical Society.

5.5.6 Available Software for Molecular Modeling of Crystal Nucleation Many of the methods described in this section are new and not available for general use. Some of the studies mentioned, however, use variations of atomistic MD to gain insight into crystallization and solid–solid phase transformations. Table 5.4 lists some of the MD codes currently available.

5.5.7 Summary and Outlook

Figure 5.28 Nucleation of n-octane from the melt. (Top) Views from the side (a) and from the top (b) of a critical nucleus of n-octane. The shape of the nucleus is roughly cylindrical. (Bottom) The free energy profile along the nucleation path. Source: Reprinted from Yi and Rutledge 2009, with permission from AIP Publishing.

The examples shown in this section illustrate the difficulty of the molecular modeling of solid–liquid and solid–solid phase transformations, especially for flexible molecules. A lack of generally applicable fast models to predict the energies of solid configurations, combined with the long time scales and the complexity of the phase transition dynamics, makes it a formidable challenge to construct atomistically detailed models of phase transitions involving solid phases. Nevertheless, significant progress has been made in recent years both in developing better, more generally applicable potential models and in developing advanced methods to overcome the problem of the separation of time scales. We have now new, general ways to characterize partial or local crystalline order at the molecular scale, as well as methods that can describe the transformation paths without having to follow the time evolution along them. The steady increase in the power of computers, together with the development of more sophisticated simulation methods, may soon bring the accurate modeling of

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and registration purposes. However, one can never be certain that all possible polymorphs have been discovered experimentally. Thus methods that help predict potential stable and metastable crystal structures from just the knowledge of the molecular structure of a compound would be extremely valuable. Crystal structure/polymorph prediction has been an active research area in the last two decades. The Cambridge Crystallographic Data Centre has been organizing blind tests to determine whether crystal structures can be predicted from the molecular diagram. The results (Figure 5.33) of the blind tests in 1999 (Lommerse et al. 2000), 2001 (Motherwell et al. 2002), 2004 (Day et al. 2005a), and 2007 (Day et al. 2009) showed that it was sometimes possible and quickly advanced the understanding of the challenges facing structure prediction. In the 2001 blind test, most participants predicted structures with an alternative hydrogen-bonding motif to the observed structures for compounds IV and VI. Following this, polymorphs of both have been obtained experimentally (Jetti et al. 2003; Hulme et al. 2007), although only one with the anticipated hydrogen-bonding motif. In this section, we discuss the main issues in the computer-based prediction of crystal structures and the approaches that have been used with some success. Even though there is an important practical difference between polymorph searches (where the crystal structures are known before doing the simulations) and predictions (where the method actually discovers previously unknown structures), most methods that are useful for one type of problem can be used for the other, and as such, they will be discussed together in this chapter.

5.6.2 Energy Landscape Figure 5.29 Identification of solid molecules of glycine in solution. The two snapshots on the left (a and c) correspond to a supersaturated glycine solution. The two on the right (b and d) correspond to an α-glycine nanocrystal surrounded by a saturated solution. The molecules are colored light gray if they are identified as liquid, dark gray if they are identified as a bulk solid, and gray if they are in between. The top snapshots (a and b) use functions sensitive to positional ordering, whereas the bottom two (c and d) use functions sensitive to orientational ordering. The latter are better at identifying regions of partial ordering such as the surface of the crystal. Source: Reprinted from Santiso and Trout, 2011, with the permission from AIP.

liquid–solid and solid–solid transformations in complex systems out of the realm of academic research and into the mainstream of applied science.

5.6 Polymorph Searching and Prediction 5.6.1 Introduction The need to identify all the potential polymorphs of a drug candidate and understand the structural aspects of each polymorph will never be overstressed in the pharmaceutical industry. This knowledge is also important for patenting

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Most polymorph prediction methods involve effectively searching for all the thermodynamically feasible structures and constructing the real crystal energy landscape. Ideally, this needs to be the appropriate free energy surface, which is a function of the thermodynamic variables within the conditions of interest (e.g., practically accessible range of temperature and pressure). However, constructing such a free energy surface is quite challenging computationally. Most current prediction methods use the lattice energy instead. They compute the lattice energies of all the feasible structures and plot them against the order parameter(s) (i.e., density, hydrogen bond donor-acceptor distance, etc.) differentiating these structures. Only the lower energy regions of the crystal energy landscape are of interest. The minima on this surface within the energy range of plausible polymorphism are the thermodynamically feasible structures. These structures and their relative energies are the important features of the crystal energy landscape (Price 2008a). Figure 5.34 shows two idealized energy landscapes with the open symbol indicating the only experimentally known structure: (a) clearly monomorphic, because only one structure falls within the energy range of possible polymorphism, and (b) predictive

Molecular Modeling Applications in Crystallization Figure 5.30 A snapshot showing a critical nucleus of solid benzene-I forming in a melt. The dark gray color denotes the molecules whose positions are changing the most along the crystallization path and are thus the most important to describe the progress along the nucleation path. Source: Reprinted with permission of the publisher from Shah et al. 2011. Copyright © 2011, American Chemical Society.

Figure 5.31 Strain-induced solid -solid phase transitions in an iron nanowire. (a) Nanowire oriented in the direction, with strains of 3.8, 8.3, and 27.5 percent from top to bottom. (b) Nanowire oriented in the direction, with strains of 2.5, 6.3, and 13.8 percent from top to bottom. (c) Nanowire oriented in the direction, with strains of 2.5, 3.9, and 6.3 percent from top to bottom. The colors correspond to the body-centered cubic (light gray), face-centered cubic (medium gray), and hexagonal (gray) structures. Dark gray denotes an intermediate structure. Source: Reprinted from Sandoval and Urbassek, 2009, with the permission of AIP Publishing.

polymorphic, because there is an alternative crystal structure that is more thermodynamically stable than the known form. 5-Fluorouracil provides an example of this type of lattice energy landscape leading to the discovery of the polymorph predicted as the most stable structure.

The only previously known structure of this anticancer agent had an unusual hydrogen-bonding motif in which the most striking feature was close F    F contacts (Figure 5.35a). The crystal energy landscape (Figure 5.36) had many different structures based on two different

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout Figure 5.32 Snapshots from the view along two crystallographic axes (left and right) on a transition path between the form I and form II crystals of TPA. The ratio of lattice parameters b:c serves as a measure of progress along the reaction path. The top three snapshots correspond to the formation of a critical nucleus, the fourth one is taken along the growth path, and the bottom one corresponds to the form II structure. Source: Reprinted with permission of the publisher from Beckham et al. 2007. Copyright © 2007, American Chemical Society.

hydrogen-bonded ribbons (Figure 5.35b). An extensive manual screen eventually crystallized the predicted structure (Hulme et al. 2007). To construct the crystal energy landscape, it is recommended to search the seven most probable space groups (P21/c, P212121, P21, P1, C2/c, Pbca, and Pnma) and the number of molecules in asymmetric cell (Z᾽). The computational expense is considerably greater for a Z’ = 2 search; therefore many methods are intrinsically limited to Z᾽ = 1. However, methods that are capable of searching for Z᾽ = 2 can produce many more low-energy

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structures for analysis and are essential for the polymorph search of molecular salts, monohydrates, cocrystals, and any other systems with two independent entities in the asymmetric cell. The lattice energies of the crystals can be calculated using either an ab initio method (when the molecule is small and usually consists of fewer than 50 atoms) or by force fields that assign parametric potentials to various atoms in their different hybridization states. The accuracy of the relative energies of the crystal energy landscape depends on how accurately the intermolecular energies are estimated. A survey (Day et al. 2005b) of the crystal energy

Molecular Modeling Applications in Crystallization Table 5.4 Some of the Available Atomistic MD Software Packages

NAMD (NAnoscale Molecular Dynamics) LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) GROMACS (GROningen MAchine for Chemical Simulations) DL-POLY (Daresbury Laboratory’s molecular dynamics package) Accelrys’ MaterialsStudio CHARMM (Chemistry at HARvard Molecular Mechanics) AMBER (Assisted Model Building with Energy Refinement)

landscapes for 50 small, rigid molecules showed that replacing an atomic point charge model with an atomic multipole model obtained by a distributed multipole analysis (DMA; Stone and Alderton 1985; Stone 1997, 2005) of the same ab initio molecular charge density led to a significant improvement, with half the known crystal structures being within 0.5 kJ/mol of the global minimum. The sensitivity of polymorph prediction to the intermolecular forces urges a careful choice and continuous development of intermolecular potentials.

5.6.3 Energy Minimization and Simulated Annealing Energy minimization–based polymorph prediction methods rely on the assumption that the structure with the least lattice energy is the most plausible. There are two inherent shortcomings of these methods: (1) the omission of the contribution of entropy to the free energy and explicit kinetic effects and (2) the traps around a local minimum. Fortunately, the second problem can be overcome using a simulated annealing algorithm developed by Kirkpatrick et al. (1983). In this method, the temperature of the physical system is elevated, which permits the system to access a range of higher-energy states. The subsequent gradual reduction in temperature shifts the equilibrium until low-energy states dominate. This method introduces the possibility of accepting energy-increasing moves or “uphill” moves in the minimization of a given function and therefore prevents the system from becoming trapped in a local minimum. The relative probabilities of the existence of two states at a given temperature are represented by Boltzmann statistics. A criterion of acceptance of structures generated by simulated annealing is developed on the basis of the statistics of the Boltzmann distribution. Gdanitz (1992) used the simulated annealing technique to predict the crystal structures of hexamethylbenzene and ethene, and the accuracies of the predicted structures were better than 0.02 and 0.1 Å, respectively, when compared with the optimized experimental structures. Simulated annealing techniques combined with a subsequent minimization of the potential structures have been readily implemented in

a number of commercially available polymorph prediction software (Figure 5.37).

5.6.4 Accelerated Sampling Techniques Main Issues in Computational Polymorph Prediction Molecular dynamics (MD) and other molecular simulation methods (Allen and Tildesley 1989; Frenkel and Smit 2002) have been very valuable in the prediction of thermodynamic and transport properties of systems of interest in science and engineering for several reasons. Some of these reasons include that (1) they have a much lower cost compared with experimental measurements, (2) they can easily predict properties of compounds that would otherwise be expensive or dangerous to work with, (3) extreme conditions (e.g., high temperatures and/or pressures) can be simulated easily, and (4) they can be used to predict properties of compounds or materials that have not yet been synthesized. The latter is of particular interest for polymorph prediction because the goal is to find crystalline forms that have not yet been made. Two critical issues determine the success or failure of any attempt to predict polymorphism using molecular simulation methods. The first crucial aspect for computational polymorph prediction is the molecular model, which encompasses the degree of molecular flexibility (and the associated intramolecular energy terms) and the intermolecular interactions. This is a very important issue, especially because crystal structures are typically separated by only a few kilocalories per mole in energy, and small errors in the molecular model may cause it to predict the wrong stable polymorph or even a nonexistent structure. Molecular models used to model crystallization typically fall into two categories: ab initio models (Szabo and Ostlund 1996; Santiso and Gubbins 2004; Jensen 2007), which use the laws of quantum mechanics to calculate the energy of a particular molecular conformation, and force fields (Leach 2001; Santiso and Gubbins 2004), which use empirical equations for the same purpose. When done properly, ab initio approaches have more predictive value because they do not rely on fitting empirical parameters to available data. However, they are much more computationally expensive than force fields and thus are severely limited in the system sizes to which they can be applied. This is particularly important for systems whose properties are mainly determined by weak intermolecular interactions (e.g., dispersion forces) because modeling those interactions accurately with ab initio methods usually requires prohibitively expensive calculations. In practice, the most successful approaches so far to polymorph prediction have involved a combination of ab initio calculations and a force field model, where the ab initio method is used to generate some of the parameters appearing in the force field or to directly obtain the intramolecular contributions to the energy (Karamertzanis and Pantelides 2005, 2007; Neumann 2008; Neumann et al. 2008; Asmadi et al. 2009; Day et al. 2009; Kazantsev et al. 2011a, b). However, these approaches are quite sophisticated and are not yet reliable for predicting the crystal structures of cocrystals, solvates, or salts. They also generally require large investments of computing resources.

157

Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout Figure 5.33 The success rates for predicting the crystal structures of these molecules from the chemical diagram in the Cambridge Crystallographic Data Centre’s blind tests of crystal structure prediction. x/y indicates that there were x correct predictions out of y groups submitting three predictions for the molecule. Source: Reproduced from Price 2008a, with permission of the PCCP Owner Societies.

(a)

energy range of possible polymorphism

(b)

The second critical issue in the computational prediction of polymorphs is the method used to sample the energy (or, better, free energy) landscape for stable crystal structures. This is extremely challenging for two main reasons: 1. The stable structures are often separated by large kinetic barriers (Figure 5.38). In methods such as MD, where the simulation follows the time evolution of the system, this means that prohibitively large simulations would be

158

Figure 5.34 Idealized crystal energy landscape: (a) clearly monomorphic and (b) predictive polymorphic

density

free energy stability

free energy stability

density

required to see a transition between two crystalline forms. The reason for this is that MD involves numerically integrating Newton’s equations of motion for every atom or particle in the system. In order for the integration to be stable, the time steps must be significantly shorter than the characteristic time for the fastest degree of freedom in the system. Typically, the fastest motion is bond vibration, which has typical characteristic times between ~8 and ~35 fs, and thus the time step is usually of the order of 1 fs. On

Molecular Modeling Applications in Crystallization Figure 5.35 Competitive hydrogen-bonding motifs found in the crystal energy landscape of 5-fluorouracil

(a) Form I (known form)

–83 1.22

1.23

1.24

(a) Form II (predicted form)

1.25

1.26

1.27

1.28

1.29

1.3

1.31

Figure 5.36 Scatter plot showing the cell volume per molecule against the lattice energy for all hypothetical structures of 5-fluorouracil within 10 kJ/mol of the global energy minimum. The lattice energy minima corresponding to experimental form I and form II crystals are shown in open symbols. Source: Reprinted with permission of the publisher from Hulme et al. 2007. Copyright © 2007, American Chemical Society.

1.32

Lattice Energy (kJ/mol)

–84

–85

–86 Form 1 Form 2 chain dimers

–87

–88

–89

–90 Density (g/cm3)

the other hand, the characteristic times for polymorph transformations are typically of the order of seconds, which means that one would require several quadrillion time steps to see one such event. With current computational power, one can expect, at best, to run a realistic system for a few billion time steps. Thus standard MD cannot be used to search for polymorphs. Other traditional simulation methods (e.g., Metropolis Monte Carlo) suffer from similar problems – the simulation time required to cross a free energy barrier is generally not accessible. 2. The search space has a very large number of dimensions. There are many variables that define a crystal structure – not only the lattice parameters but also the number of molecules in a unit cell and their relative positions and orientations within the cell, as well as their internal configuration if the molecules are flexible – the more complex the molecules are, the more internal degrees of freedom they have (e.g., rotatable bonds). Even for moderately complex molecules, the number of variables needed to completely describe a crystal structure is large enough that traditional optimization methods (e.g., quasiNewton methods) have a very low probability of finding the

global (free) energy minimum. Furthermore, one cannot rely on carrying out an exhaustive search for minima because evaluating the energy (or, even more so, the free energy) may be computationally expensive. Thus, if one wanted to bypass the time-scale problem described previously by just conducting a large number of optimizations, it is still challenging to find stable polymorphs computationally. In the rest of this section we will discuss some of the approaches that are used to overcome the difficulties just outlined, as well as their strengths and limitations.

Crossing Free Energy Barriers In this subsection, we will describe some of the methods designed to overcome the time-scale problem described previously. The main goal of such methods is to modify MD (or a similar method) in a way that enables the simulation to jump over free energy barriers while retaining a realistic description of the system. There are many methods to study activated processes in the literature [see, e.g., Santiso and Gubbins (2004) and Vanden-Eijnden (2009)], but for this discussion we will limit ourselves to those that do not require knowledge

159

Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout

Figure 5.37 The flowchart for polymorph prediction in MaterialStudio

of the final state of the system, which are those useful for discovering unknown crystal structures. Before attempting to search for crystal structures, it is necessary to define the search space, i.e., the coordinates that would be used by the computer to distinguish between different crystalline structures. There are several possible choices for this. In the case of crystals of small, symmetric molecules or atomic crystals, the order parameters of Steinhardt et al. (1983) have been used successfully. These parameters are built by using spherical harmonics, which interfere constructively when the system has the right symmetry and destructively when it does not. However, the authors do not incorporate any information about the internal structure of the molecules in the crystal (and thus their relative orientation), and therefore, the spherical harmonics are of limited applicability for molecular systems. Extensions of this approach for molecular crystals have been proposed (Briels 1980; Mettes et al. 2004), but often the resulting parameters are quite mathematically complex and useful only for relatively small molecules. Another option is to use the lattice parameters and some subset of the positions and orientations of the molecules within the unit cell, possibly symmetry adapted to a choice of space group for the crystal (Day et al. 2009). More recently, a general set of order parameters based on the finite-temperature dynamics of a known crystal form has been proposed (Santiso and Trout 2011). This method can be applied to more complex molecules

160

but has yet to be fully extended to sample unknown crystal forms. Having chosen the variables defining the search space, we are left with the problem of how to cross the free energy barriers separating different crystal forms while retaining a realistic description of the crystals. One such approach is metadynamics, a method first proposed by Laio and Parrinello (2002). In this method, the MD Hamiltonian is augmented with a history-dependent potential that increases the energy of the system when it spends too much time in the same region of the search space (Figure 5.39). This eventually causes the system to “jump” out of the free energy minimum in which it started (because it reaches a high energy) and move to a different minimum. The process is then repeated until the energy landscape is “filled”; i.e., the system cannot encounter any more stable basins. From such a simulation, one obtains the different stable conformations that the system can attain, and an analysis of the history-dependent potential can be used to reconstruct the free energy surface of the system (Bussi et al. 2006; Raiteri et al. 2006). The latter is useful in gaining information on the mechanisms and barriers for the transitions between the different states. Laio–Parrinello metadynamics has been successfully applied to structural transformations in molecular crystals, in particular, the crystalline structures of benzene (Raiteri et al. 2005; Figure 5.40), Li-ABW zeolite (Figure 5.41), and carbon (Martoňák et al. 2005). The main difficulties in using this method lie in the definition of the functions used to increase the energy when the system remains in the same region of space for too long – the parameters in these functions usually require some modification before the method is able to successfully sample the free energy landscape. This, in turn, increases the computational cost of using the method. Additionally, a poor choice of order parameters used to define the different crystal structures may lead to an incomplete set of crystal structures. Despite these difficulties, however, the method has shown to be a valuable tool in the search for polymorphs and the reconstruction of free energy landscapes in molecular systems. Another promising, more recent approach to sample free energy landscapes is the temperature-accelerated MD method (TAMD) of Maragliano and Vanden-Eijnden (2006) and Vanden-Eijnden (2009). This method is based on a simple idea: when a system is heated to a high temperature, it becomes easier for it to jump over free energy barriers. However, just heating the whole system is often not a good option because the temperature required for the activated process to occur within the time accessible to an MD simulation may be so high that the system would simply vaporize or have properties very different from those of the real system at the lower temperature. Thus, in TAMD, only the degrees of freedom describing the structure of the crystal are heated up, whereas everything else remains at the low temperature. This is achieved by using “thermostats,” which are mathematical devices used in MD to simulate systems at constant temperature. In a standard MD simulation at constant temperature, all the degrees of

Molecular Modeling Applications in Crystallization

2 0

V(X)

–2

320

160 20 10

80

–4 40 –6 –8 –10 –6

Figure 5.38 A depiction of a free energy landscape separating two stable basins (e.g., two stable crystal structures). There are large free energy barriers separating the two stable basins and many possible pathways connecting them. Source: From Bolhuis et al. 2002. Reproduced with permission of the Annual Review of Physical Chemistry, Vol. 53. Copyright © by Annual Reviews (www .annualreviews.org).

freedom are kept at the same temperature by the thermostat. TAMD couples “hot” thermostats to the degrees of freedom that are most likely to describe the transformation from one crystal structure to the other while keeping all the remaining degrees of freedom at the true (colder) temperature. For example, the lattice parameters of the crystal could be set to a much higher temperature than that of the system to allow the system to jump over free energy barriers between different crystal structures. Even though this approach has not yet been applied directly to searching for polymorphs, it has been shown to be successful for similar problems involving barrier crossings (Vanden-Eijnden 2009) and is a promising alternative to Metadynamics.

Sampling Large Spaces As explained at the beginning of this section, another aspect that complicates the search for crystal structures is the fact that the search space has a very high number of dimensions, especially for complex molecules (Oganov 2011). Generally, there are two kinds of approaches that can be used to deal with such a complex optimization problem: (1) exact methods, which are guaranteed to find a solution if run for long enough but have exceedingly large computational costs for high-dimensional problems, and (2) metaheuristic methods (Talbi 2009), which are not guaranteed to find a global optimum but can obtain reasonable solutions in highdimensional spaces. The field of global optimization is too broad to cover in this discussion, but we will focus on a few of these methods that have been used for the computational discovery of crystal structures. The most basic exact methods for finding crystal structures are systematic grid searches (Busing 1981; Price 2008b). These searches typically involve a prior choice of the space group for the crystal under consideration and must be repeated for different orientations of the molecule(s) in the unit cell. Usually each structure generated in this way is

–4

–2

0 X

2

4

6

Figure 5.39 Time evolution of the potential energy in a Laio–Parrinello metadynamics simulation. The numbers next to the curves indicate the simulation time. The system starts in the central minimum (at x = 0). This minimum gets filled up by the history-dependent potential until, at t = 40, the system “jumps” into the minimum on the left (at x = −4). The process is repeated, and the system reaches the minimum on the right at around t = 160. Finally, the whole energy landscape gets filled, and no more stable states can be reached. Source: Reprinted with permission of the publisher from Laio 2002. Copyright © 2002, National Academy of Sciences, USA.

“relaxed” by energy minimization. One limitation of this approach is that the number of positions and orientations one needs to test grows exponentially with the number of molecules in the unit cell, which is why such methods have been used mostly to generate structures with a single molecule per unit cell. Another very simple exact approach used to obtain crystal structures is a random search (Schmidt and Englert 1996; Mooij et al. 1999) – this is usually done by first choosing a space group and then randomly assigning the independent lattice parameters, as well as the positions and orientations of the symmetry-unique molecules in the unit cell. As with grid-based searches, one would then typically relax the randomly generated structure. This procedure has one advantage over grid-based searches: if there are regularities in the molecular model that would cause a grid-based search to miss an optimum, a random search would be able to find it. However, the number of structures that one would expect to test in order to find an optimum also grows exponentially with the number of variables (Zabinsky 2003), and it is possible for a purely random search to miss important regions of the search space. Often random searches are used as a first screening step before applying a more sophisticated heuristic method. A third approach that has the advantages of both uniform grid sampling and random sampling is the use of lowdiscrepancy sequences (Price 2008b). These sequences are similar to random sequences in that they do not fall on any regular grid, but they cover the search space in a more uniform way than purely random searches. This approach has also been used in applications of computational crystal structure prediction (Karamertzanis and Pantelides 2005, 2007; Della Valle et al. 2008).

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout

Figure 5.40 The structures of the polymorphs of benzene as obtained from Laio–Parrinello metadynamics simulations Source: Reprinted with permission of the publisher from Raiteri et al. 2005. Copyright © 2005, Wiley-VCH Verlag GmbH & Co.

Among the many metaheuristic approaches available for high-dimensional optimization problems, the ones that are most often used to predict crystal structures are simulated annealing (Pillardy et al. 2001; Schön and Jansen 2010) and genetic algorithms (Goldberg 1989; Bazterra et al. 2007). Simulated annealing (SA) is an optimization method inspired by the eponymous technique used in metallurgy to obtain crystalline materials with low defect density. The SA protocol involves heating an initial configuration of the system, which may be an approximate crystal structure or even an amorphous structure, to a very high temperature. This allows the system to escape its initial state and rapidly explore its conformational space. The system is then cooled down slowly, allowing it to relax to its minimum energy structure. The cooling speed is important if one is interested in

162

obtaining the global minimum – rapid cooling may cause the system to become prematurely trapped in a local metastable structure. In practice, SA is usually implemented in the context of stochastic sampling of crystal conformations. A typical SA procedure would involve the following steps: 1. Generating an initial configuration and defining the initial (high) temperature 2. Changing the configuration (This can be done by any combination atomic/molecular displacements, changing the size/shape of the unit cell, or other, more complex moves.) 3. Finding the energy of the new configuration 4. Deciding whether to accept the new configuration [There are different criteria that can be used for this, a common

Molecular Modeling Applications in Crystallization

Figure 5.41 Two Li-ABW zeolite structures obtained by Laio–Parrinello metadynamics simulations Source: From Martoňák et al. in De Gruyter, Simulation of Structural Phase Transitions by Metadynamics (Walter De Gruyter, Berlin, 2005). Copyright and all rights reserved. Material from this publication has been used with the permission of Walter De Gruyter GmbH.

one is the Metropolis criterion (Metropolis et al. 1953; Allen and Tildesley 1989; Frenkel and Smit 2002), where new configurations are accepted immediately if they yield a lower energy or accepted with a probability proportional to a Boltzmann factor if they yield a higher energy. This makes higher energy configurations more accessible at higher temperatures. If the new configuration is not accepted, the previous configuration is kept.] 5. Updating the temperature (Typically, one would run a number of steps at high temperature and then slowly reduce the temperature to 0 K, after which one would go back to step 2 and repeat until convergence has been achieved.) The SA approach has been used successfully (MellotDraznieks 2007; Oganov 2011) to predict crystal structures of both inorganic salts (Figure 5.42) and molecular crystals, in some cases predicting structures before they were observed experimentally. However, for molecular crystals, an SA method may in general need to be augmented with additional heuristics or experimental data to be successful. Genetic algorithms (GAs) are a family of optimization methods that finds minima in complex energy surfaces by a procedure that mimics biological evolution. In GAs, one starts with an initial, diverse set of structures that do not need to be good approximations to the correct crystal structure of the system. This initial population of structures is then allowed to “reproduce” by following rules similar to those that govern the transmission of genes in sexual reproduction. Individual structures are then selected based on their “fitness” – e.g., one could choose structures with lower energies with higher probability – and moved on to the next generation. This process is repeated until the best structures do not change significantly. GAs are a powerful technique for searching complex multidimensional spaces and can be combined, with

relative ease, with other heuristics to improve the probabilities of finding good structures. The first step in carrying out an optimization with a GA is mapping the variables defining the system under study onto a “genome” – usually this involves encoding the state of the system (e.g., the positions of the atoms in the crystal unit cell, the cell parameters) as a string of bits. With this representation, the steps in a typical GA are 1. Generating an initial “population,” i.e., an initial set of structures (These can be generated at random, using a grid in position/orientation/cell parameter space, or even using a different search method such as SA. The structure is then encoded as a string of bits [the ”chromosome” of the structure].) 2. Calculating the fitness of each structure (This could be the negative of the energy calculated using a force field or a quantum mechanical method, or it could be a more sophisticated function of the energy – often it is a good idea to scale the energy in some way to prevent premature convergence to a metastable structure.) 3. Selecting pairs of structures for “reproduction” based on their fitness (A basic approach would be to select structures with a probability proportional to their fitness [the socalled roulette wheel method]. However, a better approach would be to select groups of a certain size [e.g., two] at random and choose the best of each group [the “tournament” method].) 4. For a pair of structures chosen in step 3, exchange some subset of the bits in the bit string (For example, one could cut the string at some intermediate point and then mix the left side of one string with the right side of the other – this is a “crossover” move. This generates a new bit string. Following crossover, one can also modify bits at random with a certain, small probability (a “mutation” move). 5. Optionally, one can select one or a few of the best individuals from the previous generation and copy them, with or without mutation, into the next generation. 6. Steps 4 and 5 are carried out until a new population of structures is generated; then go back to step 2 and repeat until convergence is achieved. Provided that one takes measures to prevent premature convergence to local minima, this procedure will often produce good crystal structures. The GA procedure is explained in much more detail in Goldberg (1989), and its use for crystal structure prediction in particular is discussed in Oganov (2011). GAs and modified GAs have been used successfully to predict crystal structures, even for molecular crystals, but just like any other heuristic, they do not work in all cases. A recent paper found that GA-like methods were able to find correctly the structures of 16 of 22 molecular crystals using a generalized force field (Kim et al. 2009; an example is shown in Figure 5.43). As with other heuristics, GA results can be improved by using it in combination with other optimization methods.

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Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout

5.6.5 Available Software for Computational Polymorph Prediction 5.6.6 Summary and Outlook Despite the significant challenges that must be overcome in using molecular simulation methods to predict crystal structures, much progress has been made in recent years. As increasing computer power becomes more widely available and, more importantly, more sophisticated search heuristics and potential energy models are developed, the “holy grail” of reliably predicting the crystal structures of most atomic and molecular systems will become increasingly nearer. This is illustrated by the most recent international blind test of crystal structure prediction hosted by the Cambridge Crystallographic Data Centre (Day et al. 2009). Methods with very high success rates exist even now, although they require very large computing power and are not yet reliable for predicting structures of solvates and cocrystals. This does not mean that there are no challenges left to overcome, and users should be cautious of the ability of these software packages to predict crystal structures for complex materials. In particular, the development of accurate, inexpensive potential energy functions to predict the energies of molecular structures remains challenging. Currently, the most successful approaches often involve computationally costly quantum mechanical models, and even those models can be inaccurate in some cases. The development of methods to successfully obtain optimal structures in complex energy landscapes also remains an active research area, and the combination of multiple heuristics in a single search (Blum et al. 2008) seems promising for many complex optimization problems and, in particular, for the prediction of crystal structures.

5.7 Solubility Prediction 5.7.1 Introduction Predicting the solubility of solids in both pure solvents and mixtures is of prime importance in the development of crystalline products. In this section, we explore molecularmodeling-based approaches to solubility prediction. The thermodynamic framework behind most models is outlined in Section 5.7.2, and one of the most common molecular-modeling-based methods, COSMO, is described in Section 5.7.3. A different approach, the non-random twoliquid (NTRL) method, is briefly discussed in Section 5.7.4. Other methods with a more empirical basis are discussed in Chapter 1 of this Handbook.

5.7.2 The Thermodynamic Cycle and Ideal Solubility Calculation of the solubility of a particular solute in a solvent is equivalent to finding the composition of the liquid phase in equilibrium with the solid phase. From classical

164

thermodynamics, phase equilibrium requires the equality of pressure, temperature, and the chemical potentials or, equivalently, fugacities of all the components in both phases. We consider here the case of a single solute and a single solvent – for this particular case, the phase equilibrium condition is (Prausnitz et al. 1998) ^f s ¼ ^f l

ð5:11Þ

where ^f denotes the fugacity of the solute in the mixture, and the superscript indices s and l denote the solid and the solution. Often the solid phase contains only pure solute, and we can replace the fugacity of the solute in the solid phase with s the fugacity of the pure solid fpure . The fugacity of the solute in the liquid phase is often written in terms of an activity s coefficient γ ≡ ^f =xf 0 , where x is the solubility as a mole fraction, and f 0 is a reference fugacity, usually the fugacity of the pure subcooled liquid at the same temperature as the solution and the vapor pressure of the solute. With this, Equation (5.11) can be rewritten as (Prausnitz et al. 1998; Mullins 2007) s fpure ¼ γxf 0 ) x ¼

s fpure

ð5:12Þ

γf 0

Thus the problem of obtaining the solubility involves finding s the fugacity ratio fpure =f 0 , which is a property of the pure solute (not the solvent), and the activity coefficient of the solute in the solution, which quantifies the influence of the interactions between the solute and the solvent in the solution. The fugacity ratio can be obtained by using the thermodynamic cycle depicted in Figure 5.44. From the definition s of fugacity, fpure =f 0 ¼ expðDgad =RTÞ, where g ¼ h  Ts is the Gibbs free energy (h is the enthalpy and s is the entropy), a denotes the pure solid at the temperature of interest, and d indicates the pure, subcooled liquid at the same temperature and vapor pressure as the solute. From the definition of the Gibbs free energy, we have Dgad ¼ Dhad  TDsad . And from the cycle in Figure 5.44, we can write Dhad ¼ Dhab þ Dhbc þ Dhcd Dsad ¼ Dsab þ Dsbc þ Dscd

ð5:13Þ

where Dhbc and Dsbc are the heat of fusion Dhfus and the entropy of fusion of the pure solute at the triple point, respectively. Because the Gibbs free energies of the liquid and the solid at the triple point are equal (they are at equilibrium at the triple point), we can write Dgbc ¼ 0 ¼ Dhbc  Tt Dsbc ) Dsbc ¼

Dhbc Dhfus ¼ Tt Tt ð5:14Þ

The differences Dhcd ; Dscd ; Dhab , and Dsab are properties of the pure solid (a–b) and the pure liquid (c–d) and can be written in terms of the specific heats of the pure solid and pure liquid

Molecular Modeling Applications in Crystallization

Coordination

Structure type

4 5 5 6 6 6 7 8 mixture mixture

tetrahedral nets (anti-PtS,...) trigonal bipyramid square-based pyramid NaCl NiAs anti-NiAs mono-capped prism CsCl dense structures open structures

(a)

Figure 5.42 The energy landscape of NaCl, as obtained from an SA approach. The table lists different structures obtained. (a) The initial random configuration used. (b) The global minimum, corresponding to the stable NaCl structure. (c) A higher-energy structure found by SA that was previously unknown in ionic systems. This structure was later found experimentally in Li4SeO5. Source: Reproduced from Mellot-Draznieks 2007, with permission from the Royal Society of Chemistry.

Frequency (%)

(b)

5.5 6.8 3.0 37.2 20.0 4.3 0.3 0.1 12.0 9.0

(c)

37.2 %

6.8 %

Table 5.5 Some of the Available Software Packages for Crystal Structure Prediction

Software

Methods used

AMS-GRACE

Dispersion-corrected density functional theory, Monte Carlo parallel tempering

XtalOpt

Genetic algorithm

Accelrys Polymorph Predictor

Simulated annealing

CALYPSO

Particle swarm optimization

CrystalPredictor and CrystalOptimizer

Low-discrepancy sequences, dispersioncorrected density functional theory

FlexCryst

Data mining from a crystal structure database

ðT Dhcd ¼

ðT clp dT

Dscd ¼

T

Tt

Tt

ðT

ðT

Dhab ¼

csp dT Tt

clp

Dsab ¼

csp T

dT ð5:15Þ dT

Tt

Combining Equations (5.13), (5.14), and (5.15), we thus have
 ðT ðT Dcp T dT Dgad ¼Dhad TDsad ¼Dhfus 1 þ Dcp dTT Tt T Tt

Figure 5.43 Matching between the crystal structure of 3-hydroxybiuret as predicted by a GA and the experimental crystal structure. The mean square deviation between the structures is less than 0.5 Å. Source: Reprinted with permission from Kim et al. 2009. Copyright © 2009, Wiley Periodicals, Inc.

Tt

ð5:16Þ where Dcp ≡ clp  csp is the difference between the specific heats of the pure liquid and the pure solid. In many cases, this can, to a very good approximation, be assumed to be constant between

the triple-point temperature Tt and the temperature of interest, and thus

 T T Dgad ¼ Dhfus 1  þ Dcp ðT  Tt Þ  TDcp ln ð5:17Þ Tt Tt Using the definition of fugacity, we have s lnðfpure =f 0 Þ ¼ Dgad =RT, and we get the final expression for the fugacity ratio


 s fpure Dcp Tt Dcp Dhfus T Tt ln 0 ¼ 1  ln 1 þ ð5:18Þ f RT Tt R T R T

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Equation (5.18) allows us to estimate the fugacity ratio in Equation (5.12) in terms of the heat of fusion of the solute at its triple point, the temperature of the triple point, and the specific heats of the solid and the liquid. If the specific heat data are not available, the last two terms in Equation (5.18) often can be neglected because they are small compared with the first term and have opposite signs. Also, for most materials, the normal melting temperature is often quite close to the triplepoint temperature, and thus the heat of fusion at the triple point, as well as the triple-point temperature, can be replaced by the heat of fusion at the normal melting point and the normal melting temperature if the former quantities are not available (Prausnitz et al. 1998). The second contribution to the solubility – the activity coefficient γ of the solute in the solution – is much harder to estimate accurately than the fugacity ratio because it depends on the nontrivial interactions between the solute and the solvent. The simplest assumption one can make is that the solution behaves as an ideal mixture, and γ ¼ 1. In this case, one can combine Equations (5.12) and (5.18) (with γ ¼ 1) to get the ideal solubility


 Dcp Tt Dcp Dhfus T Tt id ln x ¼ 1  ln 1 þ ð5:19Þ RT Tt R T R T This is a good approximation only in cases where the solute and the solvent are very similar in size, shape, and chemical groups. Unfortunately, most mixtures of industrial interest are not of this classification. Therefore, it is crucial to have good methods for predicting activity coefficients in order to obtain good estimates of solubility. There are many different methods to predict activity coefficients, many of them relying on the use of experimental data to fit parameters to a particular functional form. These methods are discussed elsewhere in this Handbook. Because we are mostly concerned with methods involving molecular modeling, in Section 5.7.3 we will discuss a common method to estimate activity coefficients that uses molecular modeling to obtain its parameters rather than experimental data.

5.7.3 Conductor-Like Screening Model (COSMO) Methods As mentioned previously, there are many different approaches to estimating the activity coefficients needed to estimate solubilities. Many of these approaches, however, use parameters fitted to experimental data and therefore cannot be applied to new compounds or functional groups without additional experimental measurements. An alternative is to use solvation thermodynamic models, which are based on quantum mechanical calculations of the compounds of interest and thus do not require experimental data. Two widely used such methods are the conductor-like screening model for realistic solvation (COSMO-RS; Klamt 1995, 2005) and the conductor-like screening model – segment activity

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Figure 5.44 Thermodynamic cycle used to obtain the fugacity ratio of the solute appearing in Equation (5.11) Source: Used with permission from Prausnitz et al. 1998.

coefficient (COSMO-SAC; Lin 2000; Lin and Sandler 2002; Mullins 2007). COSMO-based methods use the solvation process depicted in Figure 5.45: the free energy difference between the molecule in the ideal gas state and the molecule surrounded by the solvent (DGsol i ) can be expressed as the sum of the free energy required to make a cavity of the right shape in the solvent into which the solute molecule will fit (DGcav ) and the free energy i required to remove the fictitious screening charges from the surface of the cavity so that the molecule’s charges are visible to chg the solvent (DGi ). These contributions can be related to quantities that can be obtained from quantum chemical calculations done on the pure solute, regardless of the other components in the mixture. The chemical potential (and hence the activity coefficient) can be related to the solvation free energy DGsol obtained in this way, with an additional combinatorial i contribution – this contribution is different for different COSMO-based models. The resulting expressions for the activity coefficients are complex and will not be discussed here, but details can be found in Lin and Sandler (2002), Klamt (2005), Mullins (2007), and Pye et al. (2009). COSMO-based models are implemented in several process simulators such as ASPEN and COSMOtherm. Databases of the parameters required for many substances are available (Mullins 2007), and several quantum chemistry packages such as GAMESS (Schmidt et al. 1993) and TURBOmole can generate these parameters when they are not available. COSMO-based methods have been used successfully to predict the solubilities of many classes of compounds such drugs, pesticides, and nutraceuticals (Klamt et al. 2002; Guo et al. 2007; Mullins 2007; Kaemmerer et al. 2010; Shu and Lin 2011) in different solvents and solvent mixtures. In a recent paper, Guo et al. (2007) evaluated the ability of the COSMO-RS method to predict the solubility of the flavonoids rutin and esculin in 12 different ionic liquid solvents (Figure 5.46). The agreement between experimental and predicted solubility is quite good for most of the solvents tested, with a root-mean-square deviation (RMSD) of less than 0.25 log units for esculin and less than 1.51 log units for rutin. This illustrates the ability of COSMO-based

Molecular Modeling Applications in Crystallization Figure 5.45 Ideal solvation process used in COSMObased models Source: Reprinted with permission from Mullins 2007.

Solvent Solvation ΔGi*sol

Turn Off Charges

Turn On Charges

ΔGi*chg

Solvent ΔGi*cav Insertion

2,5

M

A.

TF

A

3 tO

EMIM.OctSO4dMIM.dMP

1,5

0,5 BMIM.PF6 BMIM.BF4 tOMA.tf2N

–0,5

(a)

1

EMIM.OTs HMIM.Cl

EMIM.MDEGSO4 EMIM.ES

BMPyi.N(CN)2 BMPyo.N(CN)2

BMPyi.N(CN)2 BMPyo.N(CN)2 EMIM.ES dMIM.dMP EMIM.OctSO4 HMIM.Cl EMIM.MDEGSO4 EMIM.OTs tOMA.TFA

Experimental log_S (mol%)

Experimental log_S (mol%)

(a)

BMIM.BF4 tOMA.tf2N

–1

–3

BMIM.PF6

–5

–7

–1,5 –1,5

–0,5

0,5

1,5

–7

2,5

–5

–3

–1

3

1

Predicted log_S (mol%)

Predicted log_S (mol%) 2,5

0,5 BMIM.BF4

tOMA.tf2N

BMIM.PF6

–0,5

–1,5 –1,5

tO

1

BMPyi.N(CN)2 BMPyo.N(CN)2

Experimental log_S (mol%)

1,5

dMIM.dMP EMIM.OctSO4 EMIM.ES EMIM.OTs EMIM.MDEGSO4 HMIM.Cl

(b) Experimental log_S (mol%)

BMPyi.N(CN)2 EMIM.ES BMPyo.N(CN)2 EMIM.OctSO4 dMIM.dMP tOMA.TFA HMIM.Cl EMIM.MDEGSO4 EMIM.OTs

(b)

M

A.

TF

A

3

BMIM.BF4

tOMA.tf2N

–1 BMIM.PF6

–3

–5 –0,5

0,5

Predicted log_S (mol%)

1,5

2,5

–5

–3

–1

1

3

Predicted log_S (mol%)

Figure 5.46 Experimental versus predicted solubility of esculin (left) and rutin (right) in 12 different ionic liquids at 40℃ (top) and 60℃ (bottom)

167

Keith Chadwick, Jie Chen, Erik E. Santiso, and Bernhardt L. Trout Figure 5.47 The predicted solubilities of lovastatin, simvastatin, etoricoxib, and rofecoxib at 45℃ using NRTL-SAC Source: Reprinted from Tung et al. 2008, with permission from Elsevier (Copyright © 2007 Wiley-Liss, Inc., published by Elsevier, Inc).

Predicted Mass Fraction

1.E+00 1.E–01 1.E–02 1.E–03 1.E–04

Etoricoxib Rofecoxib

1.E–05

Lovastatin Simvastatin

1.E–06 1.E–06

1.E–05

1.E–04

1.E–03

1.E–02

1.E–01 1.E+00

Experimental Mass Fraction

Table 5.6 Some of the Available Software Packages for Solubility Prediction Using COSMO- and NRTL-Based Methods

Software

Methods used

COSMOlogic’s COSMOtherm

COSMO-RS

Aspen Plus

COSMO-SAC, NRTL-SAC

COSMO-SAC-VT

COSMO-SAC

methods to make reasonable predictions for solubilities in highly nonideal systems. In a more recent paper, Kaemmerer et al. (2010) used a COSMO-SAC method to carry out solvent screening for the selective crystallization of a mixture of enantiomers of bicalutamide. In their work, they found that the predictions agree with experimental solubility measurements, even for mixed solvents. Again, this is a good example of the ability of COSMO-based models to predict solubilities in complex systems.

5.7.4 Non-Random Two-Liquid (NRTL) Theory Methods Another often-used and modern approach to solubility prediction is based on the non-random two-liquid (NRTL) theory (Reid et al. 1987), particularly the segment activity coefficient (NTRL-SAC) model (Chen and Song 2004; Chen and Crafts 2006). In this approach, molecules are modeled as collections of conceptual segments, as in the COSMO-SAC approach. However, unlike COSMO, in the NRTL-SAC method the segment parameters for a particular solute are fitted to experimental data for the solubility of that solute in solvents with different characteristic interactions. Typically, one needs experimental data for four to eight different solvents to get good results. The NRTL-SAC method has been used successfully to predict the solubility of electrolytes and pharmaceutical compounds, with accuracy rivaling or exceeding that of COSMO-based methods. In a recent example, Tung et al. (2008) used the NRTL-SAC method to predict the solubility of four different pharmaceutical compounds (lovastatin,

168

simvastatin, etoricoxib, and rofecoxib) in multiple solvents and obtained errors of a tenth of a log unit or less (Figure 5.47). In this particular case, the use of experimental data to fit the NRTL-SAC parameters produced improved results over the predictions of COSMO-SAC.

5.7.5 Available Software for Solubility Prediction Using COSMO- and NRTL-Based Methods 5.7.6 Summary In the past two decades, the predictive ability of theoretical approaches to estimating the solubility of solids in both pure solvents and solvent mixtures has increased dramatically, to the extent that, for many purposes, they can be used in lieu of experimental measurements. Even in cases where experimental measurements are required, the number of experiments needed can be greatly reduced by using the data in combination with theoretical models. Solubility prediction is one of the areas in crystallization where theoretical approaches have proven most successful.

5.8 Chapter Summary and Outlook In this chapter, we have described the different computational methods available for (1) analyzing crystal structures, (2) predicting morphology, polymorphism, and solubility, (3) solving crystal structures from X-ray powder diffraction data, and (4) studying the mechanisms underlying nucleation and phase transitions. Advancements in computational power and the increased sophistication of the methods used to model crystallization processes allow for the solution of problems that would have been either extremely difficult to study experimentally or for which a trial-and-error–based approach was necessary. In the future, we envision that with further development, molecular modeling could be used in industrial crystallization as the initial screening technology, providing a platform for the rational design of experimental strategies that allow investigators to obtain the correct crystal form with the desired properties in a more timely and efficient manner.

Molecular Modeling Applications in Crystallization

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6

Crystallization Process Analysis by Population Balance Modeling A˚ke C. Rasmuson Bernal Institute, University of Limerick

6.1 Introduction In the area of industrial crystallization, population balances are used to model how the number and properties of the crystals in a crystallizer are generated and eventually appear as the solid product. A population balance is a mathematical description of conservation of number of particles, and accounts for how the number of particles having a particular set of properties (e.g., size, shape, density) may change during the process. The population balance has the same format as a mass balance or an energy balance. However, while mass and energy are conserved, particles having specific properties are not, and the population balance aims to account for how various mechanisms lead to changes. Traditionally, a population balance is a number balance, accounting for the number of particles of each particular size. Even though linear size is by far the most common particle characteristic on which the balance is based, other independent variables of the particle phase space can be of interest and modeled. By particle phase space is meant the multidimensional space of various particle properties. The population balance can also be formulated with more than one independent variable describing different particle properties. For example, if two linear dimensions of the particle are used in its characterization and are included in the modeling, we obtain a balance that also contains a description of shape changes, not only size changes. The scientific development of crystallization processes has been lagging behind that of other unit operations such as distillation, absorption, and liquid–liquid extraction. During the 1950s, crystallization processes were still treated very much as an art. However, in the early 1960s, a more scientific treatment was initiated. The basis and catalyst for this new turn in the field of industrial crystallization were two pioneering scientific papers (Randolph and Larsson 1962; Hulburt and Katz 1964). Both of these papers provided a population balance framework for modeling the generation and evolution of crystal size distributions in crystallizers. These groundbreaking publications initiated a strong scientific development of the understanding of industrial crystallization processes. However, the rumor says that the paper by Randolph and Larsson (1962) had significant problems in passing the peer reviewing and that the editor finally decided against a quite unfavorable review and accepted the paper. The population balance provides a mathematical description of how size distributions are created and evolve during a process, but it also clearly establishes the need for determination of crystallization kinetics, i.e., the kinetics of the

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mechanisms that lead to changes in the crystal population. For a description of how size distributions are created and evolve during the process, we need kinetics of particle generation and kinetics describing how particles move along the coordinates of the particle phase space (i.e., change size). In essentially all crystallization processes, we experience nucleation of crystals and crystal growth. In some processes, agglomeration and breakage are also of importance. During the 1970s and 1980s, there was a very strong focus on determination of kinetics, development of methods for determination of kinetics, and population balance modeling of various processes – primarily continuous but also batch. At scientific conferences during late 1980s and early 1990s, one could sometimes sense an exaggerated believe that population balance modeling could provide the solution to all problems in crystallization. During the 1970s and 1980s, industrial crystallization research was dominated by inorganic compounds and continuous crystallization processes. In these processes, the product size distribution is always of significant importance, often together with purity. If the crystals are too small, the downstream separation of the crystals from the mother liquor becomes tedious and expensive, and the washing of the particles to remove the remaining impure mother liquor becomes problematic. Today, research in the area of industrial crystallization is much more devoted to organic compounds and batch processing, very much driven by the challenges of crystallization of pharmaceutical compounds. In the pharmaceutical industry, even though crystal size is rarely unimportant, other properties of the product, in particular, purity and crystal structure (polymorphism), are often more critical. The crystallization of larger, flexible molecules with various functional groups in batch processes offers other challenges than the crystallization of inorganic salts in continuous processes. Because of this, attention has gradually moved on to problems that are more related to the fundamentals of crystallization and to problems that are not treated primarily by population balance modeling (e.g., polymorphism, purity, separation of enantiomers). However, most phenomena in crystallization are strongly dependent on the prevailing supersaturation. The supersaturation in the process cannot be directly controlled but establishes itself as a balance between supersaturation generation and supersaturation consumption. The generation is the result of, for example, cooling or evaporation, and the consumption is governed by crystal growth. The rate of consumption depends on the available crystal surface area, which depends on the number of crystals and their size, and in order

Crystallization Process Analysis by Population Balance Modeling

to get a proper modeling of the supersaturation during the process, a population balance over how crystal size distribution changes is required. Today, population balance modeling is employed as an established tool for analysis of crystallization processes, continuous as well as batch, perhaps more in academia than in industry. It is an active scientific field, taking on gradually more advanced crystallization processes and more elaborated descriptions of solid-phase properties. However, the possibilities and limitations are also fairly well known. For all population balance modeling of crystallization processes, the biggest limitation is the problem of providing the models with kinetics that are relevant for the actual industrial process to be modeled. We are still not able to determine crystallization kinetics by independent experiments that can be used in quantitative modeling of real processes. Already in 1984 (Palwe et al. 1984), it was shown that the crystal growth rate for the same compound could vary by two orders of magnitude in the data from different laboratories. For nucleation data, we have to expect that the variation is even worse. Today, we understand to some extent the background for this variation, and the uncertainty can be reduced, but crystallization kinetics are complex and very sensitive to the conditions and are thus difficult to determine with accuracy. Developments in this area will be made, but rapid, dramatic progress cannot be expected in the near future. The underlying problem of crystallization kinetics relates to the reality of the physics. Crystal nucleation and growth are mechanisms relying on the detailed recognition between molecules in solution and at the solid–liquid interface and the kinetics have a very strong dependence of the driving force. While normal mass transfer and heat transfer have a first-order dependence on the driving force, crystal growth has an order of dependence in the range 1 to 2 for the Burton–Cabrera–Frank (BCF) model, and the order of dependence can be higher for a surface nucleation growth mechanism. For secondary nucleation, we expect that the order of dependence on the driving force normally is in the range 2 to 4 and for primary nucleation the range is 5 to 15. In addition, crystallization kinetics can be very sensitive to impurities at low concentrations as well as minor variations in solvent composition. Population balances provide a framework for understanding and rationalizing how the product crystal size distribution is generated in crystallization processes. As such, it can be a valuable tool in the development and design of industrial processes. However, we are far from being able to determine kinetics in independent laboratory experiments, based on which a quantitative design of an industrial process for a particular compound can be made to the same level of confidence as for distillation and absorption processes. Another problem, especially for continuous crystallization processes, is that the potential to exert control over the product size distribution is much greater in silico than in practice. In the modeling, many different alternatives and modifications to alter the product size distribution can be explored, but in the reality, the possibilities are quite limited. This chapter will not explore all the different models that have been presented in the literature but rather analyze some

basic types and principal features. The chapter starts with an analysis of particle size and shape and how distributions are described and measured. After that follows a brief summary of crystallization kinetics, highlighting aspects that are of prime relevance to our modeling ambitions. After that follows a general description of population balance modeling: the general formulation, the coupling to a supersaturation balance, and methods of solution. Chapter 7 focuses on modeling and the behavior of the mixed suspension, mixed product removal (MSMPR) crystallizer. The MSMPR crystallizer serves as a reference for continuous crystallizers and in the laboratory serves as a method to determine crystallization kinetics. This chapter contains some calculation illustrations to show how population densities are calculated and how kinetics are determined from the data. A section on modeling of batch crystallizers follows and then a paragraph on determination of kinetics from process experiments.

6.2 Particle Size and Size Distribution 6.2.1 Size and Shape Crystals are not spherical objects and hence cannot be fully described by one single linear dimension (Figure 6.1). As a consequence, every specification of size by a linear dimension is directly related to how that dimension is measured. Quite a range of instruments and techniques are used for the determination of particle size. Sieving is normally assumed to fractionate the particles according to their second-largest dimension. Electrosensing zone measurements respond to the volume of the particle, and reported is often the size as the diameter of a sphere having the same volume. Laser diffraction techniques make use of the fact that the angle of diffraction depends on the distance between the edges of the particle. Focused-beam reflectance measurement reports the chord length of a laser beam sweeping over the particle. Regardless of method used, the result is often reported as a distribution in terms of a linear dimension of the particles. Accordingly, the meaning of this linear dimension directly depends on the method of determination. Following this comes an understanding that when the sizes of nonspherical particles are determined by different methods, we should

Figure 6.1 Nonspherical particle seen from two different angles

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6.2.2 Size Distributions In every mass crystallization process (i.e., a process where the solid phase appears as a very large number of individual particles), there is a spread in size among the particles. Accordingly, there is a particle size distribution that can be wide or narrow, can be skewed to different ranges, and can be multimodal (i.e., not having just one single peak but two or more peaks). The size distribution is normally given either as a number distribution or a mass (volume) distribution. The number distribution is more directly linked to the physics of nucleation and to methods of particle size distribution determination where the particles are examined one by one (e.g., electrosensing zone methods). The mass distribution has been favored in particular by Nyvlt et al. (1985) and is more related to the industrial context and to determination of the size distribution by sieving. In some cases, area distributions are also used. The size distribution aims to represent the number of particles within each size range. In the case of sieving, this comes naturally because the mass within the boundaries of a certain sieve fraction and the number of sieve fractions are quite limited for practical reasons. In the case of electrosensing zone analysis, the number of size ranges can be very large (i.e., the instrument is capable of delivering size distribution information with a very high resolution in size, e.g., 128 channels). In laser diffraction analyses, the number of size classes primarily relates to the number of detector “rings.” The size distribution can be described by a histogram or by a frequency distribution. In a histogram, the number or mass of particles of a size fraction is given as a bar having a height corresponding to the number or mass and a width that corresponds to the size fraction range (Figure 6.2). In the frequency distribution representation, the number is replaced by a number density also called the population density (i.e., the number of particles per unit length of particle size) and is calculated as the number of particles in the particular size range divided by the size range nðLi Þ ¼

Ni ðno:=mÞ Liþ1  Li

If the number of particles is expressed as number per unit amount (mass or volume) of suspension or liquid, the population density receives the corresponding units (e.g., no./m, m3). Obviously, the population density will increase if the same num-

174

0,5 0,4 Mass Fraction

expect a difference that relates to the different dimensions of the particles that the methods are actually measuring. A single linear dimension of a particle can never carry information about the shape regardless of the method used for determination. Determination of shape has to be based on more than one independent determination of particle size. Of particular interest for shape determination are, of course, methods that in themselves can measure more than one linear dimension. The prime example is the methods that use actual imaging of the particles. Off-line microscopy has been used widely in the past, but today there has been a development of techniques by which particles can be imaged in situ during crystallization, and the images can be analyzed automatically for various properties.

0,3 0,2 0,1 0 0,1

0,2

0,4 0,6 0,8 1,0 Aperture size, mm

1,2

1,4

1,6

Figure 6.2 Particle mass distribution in the form of a histogram

ber of particles is found in a more narrow size range. If the population density is calculated for all size fractions, the result forms a population density distribution. If the number of size fractions decreases and becomes very low (i.e., each size fraction is very broad), the population density curve becomes more and more smeared out, and important information is lost. By contrast, at very high resolution of the size axis, the population density curve will become noisy because each channel will contain an insufficient number of particles for statistical confidence. The population density can be defined mathematically as n¼

dNc dL

where Nc stands for cumulative undersize number of particles, and L is the linear dimension used to characterize the size of the particles (Figure 6.3). Different mathematical distribution functions are used to represent the population density size distribution. However, there is no size distribution functional form that has a general applicability to crystallization processes. Depending on the process and the process conditions, not only the size distribution can vary but also the functional form of the size distribution. The classical normal distribution is usually not used because its range is from minus to plus infinity. In some cases, though, the negative part can just be neglected. The starting point for the size distribution of a continuous steadystate crystallization process tends to be an exponential distribution (as explained below). Two mathematical functions of applicability for batch processes are the log-normal and the gamma distributions (Figure 6.4).

Log-Normal Distribution

"

2

1 ðlnL  μL Þ nðLÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp  2 2σ2 L 2πσ 
 lnL  μL Nc ðLÞ ¼ 0:5 1 þ erf pffiffiffiffiffiffiffi 2σ2

#

Crystallization Process Analysis by Population Balance Modeling Figure 6.3 Frequency distribution (left), cumulative distributions (right)

0,4

100

0,3

80

Cumulative

Fraction

0,5

0,2 0,1

Oversize

Undersize

60 40 20 0

0 0

0,2 0,4 0,6 0,8

1

1,2 1,4 1,6

0

0,2

Average particle size, mm

0,4

0,6

0,8

1

1,2

1,4

Aperture size, mm

Figure 6.4 Normal, log-normal, and gamma distributions

ð∞

Gamma Distribution nðLÞ ¼ L

k1

Nc ðLÞ ¼

mj ¼

expðL=θÞ ΓðkÞθk

γðk; L=θÞ ΓðkÞ

where γ is the lower incomplete gamma function. From other areas of science and engineering, various functional forms have been adapted, in particular, the Rosin–Rambler distribution, which is a special form of the Weibull distribution with k = 1.

nðLÞLj dL 0

where mj stands for the moment number j. If j = 0, the corresponding moment of the distribution equals the total number of particles in the distribution. The first moment of the distribution (j = 1) corresponds to the total length of particle characteristic dimension. The second moment of the distribution (j = 2) describes the total surface area of the crystals if an area shape factor ka is introduced, and the third moment (j = 3) equals the total volume of particles if a volume shape factor kv is introduced:

Weibull Distribution "
#
 k L k1 L k nðLÞ ¼ exp  λ λ λ "
# L k Nc ðLÞ ¼ 1  exp  λ

ð∞ NT ¼

ð∞ nðLÞdL ¼ m0

LT ¼

0

0

ð∞

ð∞

AT ¼ ka nðLÞL dL ¼ ka m2 2

VT ¼ kv nðLÞL3 dL ¼ kv m3

0

6.2.3 Moments of Distributions Sometimes the full size distribution information is not needed, or the total model is so complex that the detailed solution of the size distribution would be an unrealistic goal from a computational effort point of view. In that case, the partial differential population balance equation can be transformed into a set of ordinary differential equations in terms of moments of the distribution. Moments of the population density distribution are obtained by the general equation

nðLÞLdL ¼ m1

0

The shape factors are defined by the equations Asp ¼ ka L2 Vsp ¼ kv L3 and accordingly will depend on the characteristic linear dimension used. For a sphere, the area shape factor becomes ka ¼ π, and the volume shape factor becomes kv ¼ π=6 if L

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stands for diameter. Of course, it is assumed in the preceding moment definitions that the shape factors are equal for all particles of the distribution. This is not necessarily a reasonable assumption, and we should rather expect a shape distribution even though this is usually neglected.

To characterize the size distribution, various linear dimension average values can be defined (Randolph and Larsson 1971, 1988) by
1=ðjkÞ mj Lj;k ¼ mk One set of mean values is obtained by using k = 0: LT m1 ¼ ¼ L1;0 Average particle linear dimension: LL ¼ NT m0 Linear dimension of particle having average area:

1=2 AT 1=2 m2 La ¼ ¼ ¼ L2;0 k a NT m0 Linear dimension of particle having average volume:

1=3 VT 1=3 m3 Lv ¼ Lm ¼ ¼ ¼ L3;0 kv NT m0 Another set of mean values is obtained from the equation
 miþ1 Liþ1;i ¼ mi m1 LT ¼ m0 NT m2 AT Length weighed mean size: L2;1 ¼ ¼ m1 ka LT m3 VT ka Area weighed mean size: L3;2 ¼ ¼ m2 AT kv m4 Volume weighed mean size: L4;3 ¼ m3 The volume-weighed mean size equals the mean of the volume or of the mass distribution if multiplied by the crystal density. Besides the average value of the distribution, we are usually also interested in the width. The relative width is the standard deviation squared divided by the mean Population weighed mean size: L1;0 ¼

σ2 L

ð∞ ðL  LÞ2 nðLÞdL

σ ≡ 2

0

ð∞ L≡

LnðLÞdL 0

Sometimes we are also interested in the symmetry of the distribution, in example given by the skewness and the kurtosis (Randolph and Larson 1971):

176

using the notation

φ3 σ3

Kurtosis: ψK ≡

φ4 3 φ22

ð∞ φj ≡ ðL  LÞj nðLÞdL 0

6.2.4 Average Values

Coefficient of variation ðCVÞ ≡

Skewness: ψS ≡

If ψS > 0, the distribution is skewed to the right, which is typical for crystal size distributions and is a characteristic of all the size distribution functions given earlier. Kurtosis measures whether the peak is sharper or wider than the normal distribution. If ψK > 0, the distribution has a sharper peak but broader tails. Various features of the normal, gamma, and log-normal distributions are tabulated by Randolph and Larson (1971).

6.2.5 Methods of Determination For a detailed description of various methods to determine particle size and size distributions, the reader is referred to specialized monographs such as those by Allen (1997, 2003), Henk (2008), and Merkus (2009). Here we will give only a brief account in relation to crystals and crystallization. The various techniques differ in terms of the size range of particles that can be determined, the physical state of the sample (e.g., dry or suspension), particle concentration, resolution, and whether the absolute or relative distribution is obtained. For calculation of the population density, the actual number of crystals in a certain particle size range and a certain amount of suspension need to be determined. As discussed later, not all techniques can actually determine this information. For population balance modeling and comparison with experimental data, shape factors also need to be determined. The volume shape factor is used to calculate the mass associated with a certain number of crystals and can be determined by counting and weighing, provided that the crystal density is known. The area shape factor is used to calculate the supersaturation consumption due to crystal growth and is either estimated from an analysis of the geometric shape or, for example, by adsorption methods. The classical method for characterization of size distributions is sieving. A known sample of the suspension is weighed, filtered, washed, and dried, and the dried material is sieved. There is no standardized method of washing the sample, but the purpose is to as far as possible displace remaining mother liquor to reduce agglomeration in the following drying step without dissolving crystalline material. For a proper size distribution determination by sieving, it is important to use an appropriate amount of sample and to allow sufficient time for the sieving operation to reach the end point. This time will depend on the frequency and amplitude of the vibrations, the crystal shape, and the amount of material. At completion, the mass of each fraction is weighed. For each particular fraction, the specification of the sieve just being passed and the specification of the sieve on which the particles are retained are used to characterize the size. However, we receive no information about the size distribution of the particles of that particular sieve fraction, and hence an uncertainty is introduced if a certain mean size is to be used to characterize the particular fraction. One alternative is to just take the arithmetic mean

Crystallization Process Analysis by Population Balance Modeling

value of the upper and the lower sieve apertures. However, perhaps it would be more appropriate to characterize the mass of the sieve fraction by a volumetric average, that is, rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 3 3 L þ L i iþ1 Li;iþ1 ¼ 2 where i stands for sieve number in the stack. It should also be noted that sieves may have circular holes, but often sieves are woven and the hole shape is noncircular. There are many instruments on the market that make use of laser diffraction. This is a method by which the size distribution of particles in suspension can be determined, but the slurry concentration has to be low. Accordingly, systems have been constructed in which a sample of the slurry of the crystallizer is diluted before it is brought to the analyzer. The transport and the dilution of the sample are delicate operations. The original size distribution of the crystals must be maintained, neither allowing the crystals to grow nor to dissolve. It is further important to ensure that the particle aggregation is controlled in accordance with the objectives of the size distribution determination. Furthermore, for determination of particle concentration, the sample volume and the dilution volume must be carefully recorded. The relative size distribution can be turned into an absolute size distribution if the total mass of solid per unit volume of sample is determined. In summary, the laser diffraction instruments usually are simple enough to handle and rapidly give the result. However, on the downside is that the sample often has to be diluted before the measurement, and only a relative size distribution is obtained, not absolute numbers. In addition, laser diffraction is a method that determines the size distribution of all the particles in the slurry simultaneously, as opposed to determining the size of each particle one by one. On the market, there are at least two different methods by which the size distribution is determined by measuring particle by particle. Both techniques work off-line and are limited to fairly dilute slurries and hence often require a sample treatment before the measurement can be done. The electrosensing zone technique determines the volume of each particle that is sucked through a small orifice over which there is an electrical field. An electrically conducting solution is required, and the change in conductivity when a particle is present in the field is recorded; this response is assumed to be proportional to particle volume and fairly independent of shape. The instrument allows for an accurate determination of the suspension volume that is analyzed during the measurement. By diluting a weighed sample of the original slurry in a known amount of an electrically conducting solution, the actual number of crystals of different sizes per unit amount of crystallizer suspension can be determined. The light-blockage technique brings a dilute slurry of particles in front of a light source, and the light intensity obscuration is determined particle by particle; this method can be used in organic nonconducting solutions. A recent development is that instead of measuring the light blockage, a picture of each particle is taken and then analyzed for various size measures. The focused-beam reflectance measurement (FBRM) uses a narrow laser light beam to rapidly traverse over the particles in situ, measuring the chord length. The advantage of this

technique is that it can be used in situ even at fairly high slurry concentrations. In addition, the actual dimensions of an individual particle are determined. However, even a perfectly spherical particle has a multitude of different chord lengths. Hence, for nonspherical particles, the chord length distribution determined for thousands of particles in the slurry being transported in front of the probe window cannot be easily converted into a size distribution. Even if all particles have identical shape, this conversion is not straightforward because the particles may not be randomly oriented when they pass in front of the window. However, by contrast, this problem is not necessarily worse than for other size distribution determination methods. Each method actually records a limited amount of information in a biased way, and the data of each method have to be treated accordingly. Another “problem” with the FBRM technique is that the volume of suspension actually covered in the measurement is not well defined but depends on the suspension flow passing by the window as well as the volume in front of the window actually covered in the measurement.

6.3 Crystallization Kinetics In modeling of crystallizers, often crystallization kinetics are specified as empirical power-law equations, for example: Primary nucleation: Bp ¼ knp Dcnp Secondary nucleation: Bs ¼ ks MTm NSh Dcns Crystal growth: G ¼ kg Dcg where Dc ¼ ðc  cÞ denotes the supersaturation. Alternatively, the supersaturation is expressed as S ¼ c=c. Bp and Bs typically have units of number/s, m3 and G has units of meters per second. Typically, g is in the range of 1–2 but may reach higher values in the case of surface nucleation growth mechanism control. ns is in the range of 2–4, and np is in the range of 5–15, whereas h is typically in the range of 2–4, and m tends to be equal to unity. Accordingly, in particular, the crystal nucleation rate exhibits a very nonlinear dependence on the driving force, and the primary nucleation equation is almost like an on/off function. The driving force is often a small difference between two larger numbers (the concentration and the solubility). For accurate determination of the influence of the supersaturation on the kinetics, obviously, the concentration in the supersaturated solution has to be very carefully recorded, and the solubility must be known with high accuracy. In theory and in careful laboratory work, this can be done, but in the industrial practice, this is a major challenge. In a large-scale crystallizer, there will be spatial variations in concentration and temperature, and the composition and hence the solubility may vary from batch to batch. These variations do not have to be large for the corresponding variations in the nucleation rate to be substantial. Secondary nucleation depends on the hydrodynamics, often described as a function of the agitation rate. However, mechanistically, secondary nucleation is due to particle collisions and fluid shear, and in an agitated tank, these conditions vary substantially from location to location. The agitation rate in the equation cannot be expected to be more than a simple empirical

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parameter just used to correlate the data of the experiments. Accordingly, the particular coefficients in the equation determined in the laboratory cannot be expected to describe the conditions in a much larger crystallizer, perhaps with different geometry. Occasionally, a dependence on crystal size is included in the growth-rate equation. However, it appears as if there is not much experimental support for the idea that crystal growth is actually size dependent. Instead, what is perceived as sizedependent growth rather is growth-rate dispersion; i.e., crystals of the same size in the same environment may have different growth rates. In the random fluctuation (RF) model, growthrate dispersion is described as a variation of the growth rate for each crystal as a function of time (Randolph and White 1977). In the constant crystal growth (CCG) model, it is rather described as each crystal from birth having an intrinsic proclivity for crystal growth, but among the crystals, there is a distribution in this intrinsic growth rate (Janse and deJong 1976), for example, as a result of a variation in concentration of lattice defects. Growth-rate dispersion broadens crystal size distributions and can be mistaken for size-dependent growth. In population balance modeling, it is more complicated to include growth-rate dispersion than to model size-dependent growth. Using the concentration difference as a representation of the supersaturation, as was done earlier, is often a reasonable approximation for low supersaturations, e.g., in cooling and evaporation crystallization of reasonably soluble inorganic compounds. However, for crystallization of organic compounds and, in particular, in case of reaction crystallization, the supersaturation is often higher, and a more correct representation is required. From a thermodynamic point of view, the true driving force, of course, is the difference in chemical potential of the solute in the supersaturated solution and the solute in a corresponding saturated solution at the same temperature: Dμ ¼ μ  μ ¼ RTln

a a

For organic compounds, it is customary to express the activity as a ¼ γx x or as a ¼ γc c, where x is the solute mole fraction, c is concentration, and γ is the activity coefficient in solution. If we assume that the activity coefficient ratio can be approximated by unity, the difference in chemical potential, and hence the true driving force, in concentration terms is approximated by c ¼ RTlnS c c  c When S → 1: lnS → ðS  1Þ ¼  . It should be pointed out c that assuming that the activity coefficient ratio equals unity is not necessarily a good approximation but is commonly used because we usually lack data on how the activity coefficient of the solute changes with concentration, and prediction models are not sufficiently accurate. However, as a simple calculation example, let’s assume that S = 1.1 and that the corresponding activity coefficient ratio is 1.1. Neglect of the activity coefficient ratio would lead to an error of about 100 percent in estimation of the chemical potential driving force. It should be emphasized that the preceding kinetic equations are empirical simple Dμ ≈ RTln

178

representations and cannot be used for extrapolation outside the range of the actual experimental data.

6.4 Population Balance Modeling In most population balance modeling, the particles are characterized by a linear dimension called size, and this is the terminology that will be used here for simplicity, even though a population balance could concern any property characterizing the particle and is not necessarily limited to capturing just one single property. However, so far most population balance modeling in crystallization has been limited to one dimension characterizing particle size. It has been argued that in the case of processes where not only crystal growth but also agglomeration and breakage are of importance, it would be better to formulate the balance using particle volume as the independent variable. However, the rate of crystal growth is usually fairly independent of size when expressed as a change in a linear dimension (McCabe ΔL law). This would not be the case in terms of a volumetric growth rate, for which the growth rate would also approach zero at zero size (Costa et al. 2007). The population balance is almost exclusively formulated using the population density as the dependent variable. The population density is the rate by which the number of crystals changes with crystal size (i.e., along the size scale). It should be noticed that the population density is the average number of particles per unit length of particle size and as such is a continuous variable, and accordingly, this application of the population density assumes that the population of particles is large and that the particle is small compared with the length scale of variation in the continuous phase. This latter assumption is not necessarily fulfilled in reaction crystallizations, and this limit of population balance modeling has been explored by Manjunath et al. (1994). Various units are used for the population density and can be selected in part for convenience. Typical units would be number of crystals per unit size of crystal and per unit volume or mass of suspension. By this definition, the population density can change as a result of changing the actual number of crystals, as a result of the fact that the same number of particles are distributed over a larger size range (e.g., size-dependent growth or growth-rate dispersion), because the volume of suspension changes (e.g., by evaporation of the solvent or by addition of an antisolvent) or due to a combination of these alternatives. To facilitate the modeling of a process, the population density can be defined per unit amount of solvent when that is constant, or in cases where the amount of solution/suspension changes greatly, the population density can be defined on a total basis, i.e. in units of 1/m. It is always advisable to set up the balances from first principles to ensure that all terms are properly accounted for. In general terms, the population density n is defined by n¼

dN dΩ

where N is the number of crystals, and Ω is any particle phase space parameter of interest. The population density is then nðΩÞ and has units of number per unit of Ω.

Crystallization Process Analysis by Population Balance Modeling

6.4.1 The General Formulation The general population balance formulates that Accumulation = input − output + net generation Depending on the process and how the size distribution is characterized experimentally, different units are favorable. The population density varies with crystal size but in general terms may vary with a change in any variable characterizing the solid phase, e.g., shape, agglomeration degree, and porosity. The different variables characterizing different properties of the particles all belong to the internal coordinate system, sometimes called the particle phase space. The population density also may exhibit spatial variations in the crystallizer. The three geometric coordinates defining the position in the crystallizer represent the external coordinate system. Furthermore, the population density at any location in the internal and external coordinate systems may vary with time. The general formulation becomes ∂n þ rðve nÞ þ rðvi nÞ þ D  B ¼ 0 ∂t where r is the divergence operator. The second term on the left-hand side accounts for population flux along the external spatial coordinates ve , and the third term accounts for population flux along the internal phase space coordinates vi . D and B represent “death” and “birth” functions and in general depend on the internal and external coordinates. Most modeling so far has assumed that the crystallizer volume over which the balance is formulated is well mixed and that the particles are characterized by one linear dimension. For this case, we obtain ∂n ∂ðGnÞ dlnV X nk Qk þ þ DðLÞ  BðLÞ þ n þ ¼0 ∂t ∂L dt V k where the population density has units of number per meter, and meter cubed, the fifth term accounts for volume changes, and the sixth term accounts for inflow to and outflow from the mixed volume. As shown, the crystal growth rate appears in the second term and is responsible for population flux along the internal phase space coordinate: size L. The nucleation rate normally appears in a boundary condition: n0 ¼ nðL ¼ 0Þ ¼

B0 GðL ¼ 0Þ

describing that nuclei enter the population at negligible size.

6.4.2 The Supersaturation Balance Because the kinetics of nucleation and crystal growth depend on the supersaturation, the population balance equation must be supplemented by a supersaturation balance. The supersaturation balance is a normal mass balance of the solute in the solution, but using the saturated solution as a reference. If the supersaturation is formulated as a difference between the concentration and the solubility, that is, Dc ¼ c  c, the

supersaturation balance for a well-mixed constant volume V can be written as V

dDc ¼ ðQcÞin  ðQCÞout þ VðNG  NC Þ dt

where NG denotes the rate of generation of supersaturation, and NC denotes the rate of consumption per unit volume. The consumption of supersaturation is normally only by crystal growth ð∞ NC ¼ ρc kv 3L2 GðLÞnðLÞdL 0

when the growth rate is defined as G¼

dL dt

Occasionally, nucleation is also included given a specific size to each nucleus at birth. However, unless the product crystals are very small, the mass associated with nuclei should be negligible. Agglomeration and breakage are normally not associated with changes in the supersaturation. The population balance is a hyperbolic differential equation, i.e., the solution to which is “like a wave.” A disturbance in the initial condition does not lead to changes over the whole range of the independent variables immediately but travels like a wave through the internal phase space coordinate system. Together with the supersaturation balance, the model forms an integro-differential problem, and there are very few analytical solutions besides that of the idealized MSMPR crystallizer discussed later. In addition, the kinetic processes of nucleation and growth exhibit nonlinear or even strongly nonlinear dependence on the driving force. It should also be noted that the key parameter for the process – the supersaturation – cannot be easily controlled. The supersaturation is the driving force for nucleation and crystal growth, the prime mechanisms behind the formation of the size distribution, the total surface area of which consumes supersaturation by growth, and the total mass that influences secondary nucleation. Accordingly, the supersaturation participates in a self-regulating system where a high supersaturation leads to the generation of more crystal surface area, which leads itself to faster consumption of supersaturation. This interplay is illustrated in Figure 6.5.

6.4.3 Methods of Solution A comprehensive summary of solution methods to population balance models is given by Costa et al. (2007), and just a few comments will be given here. Numerical solution methods fall into a few different categories (Gunawan et al. 2004; Majumder et al. 2012): the method of moments, the method of characteristics, the method of weighed residuals/orthogonal collocation, Monte Carlo simulation, and discretized methods. The method of moments performs a moment transformation of the population balance by multiplying the balance by Lj and integrating from zero to infinite size:

179

Åke C. Rasmuson Figure 6.5 Interrelation chart Source: Redrawn with permission of the publisher from Jancic and Grootsholten 1984. Copyright © 1985, Wiley-VCH Verlag GmbH & Co.

# ð∞ " X nk Qk ∂n ∂ðGnÞ dlnV þ þ DðLÞ  BðLÞ þ n þ Lj dL ¼ 0 ∂t ∂L dt V k 0

To link with the supersaturation balance, this operation is performed for at least j = 0 to j = 3, but higher numbers can be included. The result is that the population balance partial differential equation is transformed into a set of at least four ordinary differential equations in terms of the first four moments of the distribution that are to be solved simultaneously. For the case of a well-mixed, constant-volume, transient crystallizer with negligible agglomeration and breakage, size-independent growth, and no crystals in the feed, this set of equations can be written as dmj mj þ ¼ 0j B0 þ jGmj1 dt τ If we consider the corresponding simple batch cooling crystallizer described by the population balance equation ∂n ∂n þG ¼0 ∂t ∂L moment transformation leads to j = 0: j = 1: j = 2: j = 3 dm0 ¼ B0 dt

dm1 ¼ Gm0 dt

dm2 ¼ 2Gm1 dt

dm3 ¼ 3Gm2 dt

where B0 is the nucleation rate. The method of moments will not retrieve the entire size distribution, only the number of moments that are included in the solution. Attempts have been made to estimate the shape of the crystal size distribution by, for example, combining the quadrature method of moments with the method of characteristics (see below; Aamir et al. 2009). For the general case, there is no method that can transform a limited number of moments to a complete description of the size distribution unless a certain shape of the distribution is assumed. This problem may be of particular importance in the case of processes where the distribution is far from unimodal. Another problem

180

with the method of moments arises when the growth rate is assumed to be size dependent, and growth rate dispersion is even more complicated to include in the population balance modeling, especially the CCG model. For more complex processes, the moment equations are not closed, but lower moments will depend on values for higher moments. The moment transformation becomes impossible when agglomeration and breakage are included in the model (Costa et al. 2007). In the method of characteristics the nucleation is discretized into separate pulses in time, and the size change because of crystal growth (or change in any phase space coordinate) of each pulse of nuclei is tracked through the size range (phase space). Mathematically, the population balance is transformed into a set of ordinary differential equations describing the size change of each pulse of nucleated particles. The method of characteristics is of particular use when the process involves only nucleation (of particles of negligible size) and crystal growth and has been applied successfully to batch crystallization processes with sizedependent growth and growth-rate dispersion (e.g., Bohlin and Rasmuson 1992), as well as to non-well-mixed reaction crystallization processes (e.g., Ståhl and Rasmuson 2009). The method of weighed residuals approximates the distribution by a linear combination of functions, resulting in a system of ordinary differential equations that can be solved with standard software. In Monte Carlo simulations, the random behavior of each particle is tracked, and the method is especially competitive for complex cases. In the finite-difference methods/discretized population balances methods, the population balance is solved numerically by various techniques of discretization such as finite-difference methods and finite-volume methods. In finitedifference methods, derivatives are approximated by difference ratios, e.g., the Crank–Nicolson method. In the finite-volume method, a small volume surrounding each node point on a mesh is defined and fluxes at the surfaces of each finite volume evaluated. The method is used in many computational fluid dynamics packages and can be used to solve population balance equations. This approach has been evaluated, for example, by Braatz and coworkers (e.g., Ma et al. 2002; Gunawan et al. 2004), as well as for multidimensional population balance equations over batch processes. Finite-volume methods can be computationally

Crystallization Process Analysis by Population Balance Modeling

expensive. For problems with size-dependent growth, Majumder et al. (2010) propose a variable transformation by which larger time steps can be used without creating numerical instabilities. The method is also applied to a batch cooling crystallizer model with two independent phase space coordinates. For solving the population balance with two independent phase space parameters, Qamar et al. (2007) adopt a finite-volume scheme with a moving-mesh technique. The moving-mesh techniques will allocate the computational effort to the regions where the most dramatic changes occur and is particularly useful for solution of models over physical phenomena containing singular or nearsingular solutions in localized regions. The technique is applied to a model over batch cooling crystallization without aggregation and breakage. For processes dominated by aggregation and breakage, Kumar et al. (2009) provide an analysis of different numerical solution methods. Majumder et al. (2012) develop a lattice Bolzman method to solve multidimensional population balance equations. Orkomi and Shahrokhi (2014) adopt a numerical method called the conservation element and solution element. The method is used to solve one-dimensional population balances in the presence of different types of source terms and two-dimensional population balances to capture crystal shape. The models are used to determine the optimal temperature trajectory for a case with size-dependent growth and agglomeration and for optimizing the aspect ratio when minimizing the process time.

6.4.4 Multidimensional Population Balance Modeling So far most modeling work has been concerned with one independent parameter of the particle phase space. Multidimensional population balance modeling is not only numerically demanding but also requires kinetics for the transport along each independent coordinate of the particle phase space. For determination of the kinetics and for model validation, measuring techniques for each independent “phase space dimension” are required. However, gradually, two-dimensional (2D) and higher-order modeling are emerging, especially to capture crystal shape. Puel, Févotte, and Klein (2003) developed a population balance model with two independent phase space variables for a non-steadystate continuous crystallizer and used the method of classes for the solution. The model is evaluated using data for crystallization of hydroquinone in water. Ma et al. (2007) use the same method of solution to solve a population balance model to describe the changes in crystal shape of L-glutamic acid being produced by batch cooling crystallization. Growth kinetics were derived from in-line shape imaging. Wan et al. (2009) combine a multidimensional population balance formulation with a polytopic geometric description of crystal shape. The model is used to derive optimal cooling profiles to produce a particular crystal shape. In a study devoted to the influence of shape on secondary nucleation, Briesen (2009) developed a 2D population balance model to account for the fact that the rate of attrition will decrease as a crystal gradually becomes more rounded. Another approach to deal with shape changes is adopted by Zhang and Doherty (2004). A one-dimensional (1D) population balance is used and is linked

to a shape-evolution model. The model is applied to succinic acid crystallization in water. Shaikh et al. (2014) develop a polyhedral population balance model for a well-mixed, seeded batch crystallizer where the evolution of crystal shape is due to different growth rates of three independent faces, the kinetics of which are described by power-law equations taken from the literature. Nucleation, agglomeration, and breakage are assumed to be negligible. In the work of Jiang et al. (2014), the shape is described by the crystal’s length and width, and the population balance description includes both a crystal growth equation and a dissolution equation, with the corresponding kinetics for the change in length and width. The model is correlated with experiments where growth and dissolution are cycled for the purpose of changing crystal shape and for determination of the relevant kinetics. Singh and Ramkrishna (2013) develop a framework for the dynamics of single-crystal morphologies and morphology distributions. This approach is combined with a population balance model to cover all possible morphologies based on face-specific growth rates. The morphology is composed of a finite number of low-energy faces, and their growth rates are either determined experimentally or estimated from theory. Crystallization of proteins tends to be more difficult than crystallization of inorganics and small molecules. However, for an industrial protein crystallization process, the basic principles are the same. Liu et al. (2013) use a 2D population balance to describe the shape distribution of lysozyme crystals. A seeded process is simulated, and nucleation is neglected. The model is used to identify the cooling profile required to obtain the desired crystal shape. Similar work, however, also considering aggregation of crystals is presented by Kwon et al. (2013). In many crystallization processes, crystal breakage cannot be neglected. A 2D population balance model was developed by Borsos and Lakatos (2014) that included nucleation and growth of two characteristic faces and random breakage of high-aspect-ratio crystals described by a beta distribution of the broken fractions. The model is solved by moment transformation based on the joint moments of the two different dimensions of the crystals. Szilagyi et al. (2015) investigated different breakage functions in a 2D population balance over a continuous cooling crystallizer. Crystals having a high aspect ratio are considered, and the model includes primary and secondary nucleation, nonlinear size-dependent growth rate of the facets, and size-dependent breakage. Moment transformation is used, and the dynamics of the process are investigated. Sato et al. (2008) include a breakage model in a 2D population balance model in order to simulate the fact that more needle-shaped crystals will break more easily.

6.5 The Idealized MSMPR Concept The mixed suspension, mixed product removal (MSMPR) crystallizer serves as an idealized model for back-mixed continuous crystallizers. It is the crystallization model corresponding to the perfectly back-mixed continuous tank chemical reactor. The MSMPR model assumes that the particle suspension is perfectly mixed in the crystallizer and that the suspension leaving the crystallizer is a perfect representation of the

181

Åke C. Rasmuson

1n n°

–1 Gτ

1n n

Figure 6.7 Isokinetic removal Source: Reprinted with permission from the publisher from Jancic and Grootsholten 1984. Copyright © 1985, Wiley-VCH Verlag GmbH & Co. Figure 6.6 MSMPR size distribution plot

contents of the crystallizer. If we further assume steady state, no crystals in the feed to the crystallizer, and that the only population events are nucleation and crystal growth, the general population balance is simplified to dGn Qout ¼ n dL V which by integration over all particles leads to ln

n Qout L ¼ n0 GV

if the growth rate is independent of size. For the MSMPR crystallizer under the assumptions just given, the size distribution is a straight line in a diagram of ln n versus L (Figure 6.6). The slope of this line is –Qout/GV, and accordingly inversely proportional to the growth rate, and the intercept at L = 0 is ln n0, the population density at zero size. If the size of the nucleus (i.e., the initial size of new particles when they are born) is assumed to be negligible, n0 is usually given as n0 ¼

B G

which is the ratio of the nucleation rate divided by the growth rate. Accordingly, an experiment that fulfills the assumptions of the model allows for determination of the crystal growth rate and the nucleation rate at the prevailing steady-state conditions. For the evaluation, the size distribution of the crystals in the crystallizer or in the suspension leaving the crystallizer at steady state has to be determined. The method used for the size distribution analysis will directly determine the meaning of the size parameter L, and the determined growth rate will be the rate of change of this linear dimension with time. If sieving is used, the size reflects the crystal size in terms of how it appears in a sieving characterization, and the growth rate describes the rate by which the crystal size moves through the sieving stack. The prevailing supersaturation can be determined by sampling and concentration analysis of the liquid phase. Without this determination, however, the experiment does give the relationship between the

182

growth rate and the nucleation rate, which is of prime importance for the size distribution. Unfortunately, while the modeling and evaluation of the experiment are fairly straightforward, it is more difficult to experimentally fulfill the underlying assumption of a perfectly mixed suspension with a perfectly representative suspension withdrawal. In a laboratory-scale experiment, the suspension withdrawal rate (ml/min) becomes very low – too low for a continuous withdrawal of a representative slurry. Crystals will not be withdrawn in a representative manor and will not be kept suspended through the pipes. In laboratory units, sometimes a continuous withdrawal is replaced by intermittent withdrawal – a higher rate of slurry withdrawal for a shorter time interrupted by no withdrawal. But even in this case the withdrawal easily becomes classified. The optimal withdrawal conditions for continuous flow have been called isokinetic removal (Jancic and Grootsholten 1984), which stands for the idea that the liquid flow rate in the withdrawal pipe and the direction at the entrance to the withdrawal pipe in the crystallizer should perfectly reflect the conditions in the crystallizer where the withdrawal point is located (Figure 6.7). The supersaturation balance of the MSMPR cooling crystallizer becomes ð∞ Q ðcin  cout Þ ¼ ρc kv 3L2 GðLÞnðLÞdL V 0

where the left-hand side represents the supersaturation generation rate, and the right-hand side is the supersaturation consumption rate Nc. If the growth rate is independent of size, we obtain 1 NC ¼ AT Gρc 2 In addition, it can be shown that (Tavare 1995) L1;0 ¼ Gτ NT ¼ n0 ðGτÞ LT ¼ n0 ðGτÞ2 AT ¼ 2ka n ðGτÞ 0

L2;1 ¼ 2Gτ 3

MT ¼ 6kv ρc n0 ðGτÞ4

L3;2 ¼ 3Gτ L4;3 ¼ 4Gτ

Crystallization Process Analysis by Population Balance Modeling

where τ is the residence time. The mode is Ld ¼ 3Gτ, and the median is L50 ¼ 3:67Gτ. Obviously, the product crystal size distribution from the MSMPR crystallizer is an exponential function, and the coefficient of variation of the mass distribution function is always 50 percent. The preceding equations may be interpreted as if increasing the residence time would be an efficient method to increase the mean size of the crystals. However, this is normally not the case because at increasing residence time, the supersaturation becomes lower, and hence the growth rate also decreases. The mean size is governed by the relation between the nucleation rate and the growth rate. If the density of the feed and the product solutions are about equal, the residence time can be calculated as τ¼

V Qout

Figure 6.8 Influence of residence time on product mean size in an MSMPR crystallizer

If the nucleation and crystal growth rates are described by: j

BS ¼ kn MT DcnS G ¼ kg Dcg we obtain by elimination of the supersaturation BS ¼ kn kgnS =g MT GnS =g j

which leads to

i1 n0 ¼ kn ki g MT G j

if i = nS/g. For a process where the residence time changes but the magma density remains constant, we obtain MT;1 ¼ 6kv ρc ½n0 ðGτÞ4 1 ¼ MT;2 ¼ 6kv ρc ½n0 ðGτÞ4 2 Figure 6.9 Influence of feed stock concentration of mean size in an MSMPR crystallizer

which leads to n02 ¼ n01



ði1Þ=ðiþ3Þ G1 τ 1 4 Lm;2 τ2 and ¼ G2 τ 2 τ1 Lm;1

These equations are illustrated in Figure 6.8 and show that for i = 1, the mean size is independent of the residence time. For i > 1, the mean size does increase with increasing residence time, but this increase is quite moderate unless the i value is fairly high. If i = 2, doubling of the residence time only leads to a 15 percent increase in the mean size (Randolph and Larsson 1971). An increasing concentration in the feed stock leads to a higher magma density in the crystallizer. For a constant residence time, we obtain MT;1 MT;2 0 4 ¼ 6kv ρc n1 G1 6kv ρc n02 G42 which leads to Lm;2 ¼ Lm;1




MT;2 MT;1

ð1jÞ=iþ3

as illustrated in Figure 6.9. Because j often is approximately unity, this equation shows that the mean size would be approximately independent of production rate if the residence time remains constant. The reason for this is that increased

magma density will reduce the supersaturation in the crystallizer, which tends to lower the nucleation rate, while the increased magma density tends to increase secondary nucleation. The two effects tend to balance out.

Example: Determination of Kinetics by MSMPR Experiments Three MSMPR experiments (Table 1) have been performed for potash alum in a 0.5-liter crystallizer. The suspension volume is 0.45 liter, the rate of agitation is 510 rpm, and the temperature of the crystallizer is 24.9℃. Table 6.1 Experiments

Experiment

1

2

3

Q (ml/min)

30.2

30.2

75.9

cin (g/g)

0.167

0.145

0.172

cout (g/g)

0.123

0.123

0.128

Gram crystalsa

21.1

10.58

21.15

a

Obtained by filtering 0.45 liter of suspension.

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Åke C. Rasmuson

The density of the crystals is 1.760 kg/liter, and the shape factor is 0.5. The solubility data are given by c ¼ 0:057207 þ 3:3573  103 T  8:4857  105 T 2 þ 2:7917  106 T 3 given as grams of potash alum (hydrate) per kilogram of water and temperature as degrees centigrade. The product size distribution has been determined by sieving (Table 2). Table 6.2 Product Size Distributions Obtained by Sieving

Sieve opening (μm)

Weight of sieve + sieve fraction (g)

Weight of sieve (g) 1

2

3

137.14

137.14

137.14

137.14

710

145.20

146.08

145.64

145.74

500

144.89

148.46

146.52

147.42

355

140.71

144.33

143.38

145.55

250

137.83

141.81

140.44

143.12

180

131.35

134.72

132.95

134.85

125

129.03

131.33

129.91

131.06

slope ¼1:0814  104 )

90

126.77

128.30

127.09

127.53

Lmedian ¼ 3; 67Gτ ¼ 339:4 μm

63

124.85

125.99

125.07

125.63

B0 ¼ n0 G ¼ 1:106  107 =m3 ; s

Bottom

126.36

127.09

126.57

127.24

Calculations 1. Calculate the solubility at 24.9℃: c* = 0.131 kg/kg of water = 0.116 kg/kg of solution (for Experiment 1) 2. Calculate the supersaturation: Δc = 1070(0.123 – 0.116) = 7.49 kg/m3 (for Experiment 1) 3. Calculate the magma density: crystalmass ¼ 46.9 g/liter (for Experiment 1) a. MT ¼ suspensionvolume

b. MT ¼ cin  cout ¼ 47.1 g/liter (for Experiment 1) 4. Calculate suspension residence time: τ ¼ VQ ¼ 973.4 s (for Experiment 1) 5. Calculate population densities: L¼ nðLÞ ¼

Liþ1  Li DL ¼ Liþ1  Li 2 Mi 3

kv ρc L DLV

The sieves are numbered from the bottom to the top, and Li is the sieve opening (Table 6.3).

184

Figure 6.10 ln(n) versus L plot for example data Source: Reprinted with permission of the publisher from Granberg and Rasmuson 2005. Copyright © 2005, American Institute of Chemical Engineers.

1000

6. Plot ln(n) versus L (see Figure 6.10), and determine the intercept and the slope and from that the nucleation and growth rates (Table 4): For Experiment 1: Intercept ¼ lnn0 ¼ 32;388 ) n0 ¼ 1:164  1014 =m; m3 G ¼ 9:50  108 m=s

7. Calculate the kinetic orders:   7  7 ln 2:2210 8 ln 3:4310 9:5010 7 1:1110  ¼ 1:58 nS ¼ 12:84 ¼ 2:09 g¼ ln ln 12:84 7:49 7:49   1:11107 ln 5:55106  47 ¼ 1:0 j¼ ln 23:5

6.6 Continuous Crystallization and Deviations from the MSMPR Model If the conditions deviate from the underlying assumptions, the size distribution of the product particles can deviate from the straight MSMPR line. Such deviations can relate to that the population mechanisms deviate from the assumptions. Typical cases that have been explored are size-dependent growth, growth-rate dispersion, agglomeration, and breakage. Such deviations can also relate to that the processing conditions deviate from the assumptions: classification, fines dissolution, etc. A few examples of such deviations are given by Nyvlt et al. (1985, p. 225) and Berglund (2002). When the MSMPR straight line exhibits an upward curvature at decreasing size, this would typically be discussed in terms of size-dependent growth. Mechanistically, however, a more probable explanation is growth-rate dispersion, but to include that in the model would be more complicated. Classified product removal and fines dissolution have been explored extensively by modeling (e.g., Randolph and

Crystallization Process Analysis by Population Balance Modeling Table 6.3 Calculation of population densities

L (μm)

ΔL (μm)

Experiment 1 n(no./m3, m)

Mass (g) 31.5

63

0.73

9.362 ×10

14

Experiment 2 ln n 34.47

14

33.10 32.12

Mass (g)

n (no./m3, m)

4.596 × 10

31.46

0.78

1.629 × 10

32.72

0.32

1.858 × 1013

30.55

0.76

4.414 × 1013

31.42

0.88

1.139 × 10

13

2.03

13

30.90

5.808 × 10

12

13

30.17

12

28.45

5.29

12

4.596 × 10

29.16

27.11

4.84

1.079 × 1012

27.71

2.53

11

25.65

9

22.74

107.5

35

1.53

8.886 × 1013

2.30

2.978 × 10

13

1.223 × 10

13 12

28.87

2.61

2.268 × 10

27.42

2.67

5.952 × 1011

1.63

8.851 × 10

10

6.130 × 10

9

70

3.37

302.5

105

3.98

3.458 × 10

427.5

145

3.62

8.069 × 1011

3.57

1.939 × 10

10

1.226 × 10

10

605 855

210 290

0.88

31.13

25.99 23.23

1.60

0.44

Table 6.4 Calculation of kinetics

Experiment 1

2

3

7.49

7.49

12.84

Method a

46.9

23.5

47.0

Method b

47.1

22.0

44.0

Supersaturation (g/liter) MT (g/liter)

Residence time (s)

973.4

981.9 −8

387.2 −8

ln n

0.22

2.382 × 10

215

n (no./m3, m)

13

1.14

31.02

Mass (g)

2.693 × 10

0.21

27

55

ln n

14

76.5

152.5

Experiment 3

2.22 × 10−7

Growth rate (m/s)

9.50 × 10

9.50 × 10

Nucleation rate (no./m3, s)

1.11 × 107

5.55 × 106

3.43 × 107

Median size (μm)

339,4

342,3

316,1

Larsson 1971; Jancic and Grootscholten 1984). Classified product removal can be achieved deliberately by various means but also often occurs unintentionally as a result of the problem of actually removing a suspension that is truly representative of the suspension in the crystallizer. Classified product removal can produce a more narrow size distribution at the expense of a reduced mean size. Fines dissolution is used to actually reduce the number of particles in the system and leads to larger product crystals. Strong fines dissolution may generate oscillations in the product size distribution. An increased product size and narrower product size distribution can be obtained by combining fines dissolution and classified product removal (Randolph and Larsson 1971, 1988). In recent work, Quon et al. (2012) used a two-stage MSMPR crystallization process for reactive crystallization of a drug molecule: aliskiren hemifumarate. The process was modeled by a population balance approach and the impurity incorporation described by a distribution coefficient. The model was fitted to the crystal size distribution of each stage

33.23

30.06 29.39

25.21 22.54

0.88

3.50

0.54

15

34.66

14

1.129 × 10

2.268 × 10 1.270 × 10

1.374 × 10 7.523 × 10

for determination of five parameters of the kinetics of nucleation and growth. Experimental crystal size distributions were determined by FBRM. The model was then used to evaluate the influence of processing conditions on the yield, purity, and crystal size distribution. Kwon et al. (2014) develop a dynamic population balance approach to control a seeded plug-flow crystallizer for production of hen egg white (HEW) lysozyme crystals. The model is solved by the method of moments and is used in a control scheme where the temperature profile along the crystallizer and the superficial flow velocity are adjusted in response to disturbances in order to generate the desired product crystal size and shape. The analysis leads to a multivariable optimization problem in which the objective function is defined by the sum of squares of size and shape deviations from the setpoint values. Ridder et al. (2014) investigate the corresponding antisolvent process with multiple injection ports for finer control of the supersaturation. A multiobjective optimization is used to determine the feed rate into each segment for maximizing the mean crystal size and minimizing the distribution width. The population balance was solved by either the method of moments or a finitevolume approach. Zhao et al. (2015) applied population balance modeling to a continuous plug flow drowning out crystallization of benzoic acid and examined the relevance of different growth-rate representations. In the work of Zhou et al. (2014), such a model is combined with a reactant mixing model to describe the production of nanoparticles. The work includes analysis of the mixing performance in various mixing devices. A framework for comparison of different continuously operating modes and the region of attainable product crystal mean size at given crystallization kinetics and process starting/ending points are explored by Vetter et al. (2014) using population balance modeling. In continuous crystallization processes, encrustation is regularly a serious problem. Majumder and Nagy (2015) develop a model that combines a representation of the encrustation in a plug-flow crystallizer with a population

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Åke C. Rasmuson

balance model over the crystal size distribution, including both primary and secondary nucleation. The model is solved by a finite-volume method. In a segmented-flow crystallizer, a continuous plug flow is segmented into batches of liquid, e.g., by air bubbles, that travel along the tube. A seeded segmented-flow crystallizer was modeled by Besenhard et al. (2014) like a train of batch reactors having equal supersaturation profiles and residence times. The model includes crystal growth and agglomeration.

6.7 Batch Crystallization From population balance modeling point of view, a batch process is automatically more complex because conditions change with time. At a minimum, the balance will contain two independent variables, i.e., time and a particle characteristic dimension. For the simplest case, the population balance takes the form ∂n ∂n þG ¼0 ∂t ∂L in which it is assumed that • The crystallizer is well mixed. • The mass or the volume (depending on the units of the population density) in the crystallizer is constant, e.g., a batch cooling process. • The crystal growth rate is independent of crystal size. • Nuclei are born having a size that is negligible. The equation is a hyperbolic partial differential equation, which means that simple, straightforward numerical solutions easily run into problems. Historically, this equation was solved by the method of characteristics, in which the nucleation is discretized into separate pulses in time, and the size of each pulse of nuclei is tracked through the size range. Today, more efficient algorithms and computers have essentially eliminated the problems with the numerical solution. The supersaturation generation in a constant-volume batch cooling process is written NG ¼ 

dc dc dT ¼ dt dT dt

In an evaporation process at constant temperature, the volume is decreasing by evaporation of the solvent. This evaporation generates supersaturation according to NG ¼ c

dlnV dt

but will also increase the population density if the units are (m, m3)−1, i.e., numbers per unit characteristic length and unit volume of suspension. Accordingly, the population balance either has to be complemented by a volume-change term ∂n ∂lnV ∂n þn þG ¼0 ∂t ∂t ∂L

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or the balance can instead be expressed in terms of the total number in the crystallizer, i.e., in numbers per unit characteristic length (m)−1 only. Lindenberg et al. (2009) developed a population balance model for a batch antisolvent crystallization. The solubility c* decreases because of the changing composition Y of the solution as a result of feeding the antisolvent. The supersaturation balance has one term describing the decay because of crystal growth, one term accounting for dilution, and one term describing the change in the solubility because of the changing solvent composition Y: ð∞ dDc 3ρc kv G 2 cQ dc dY ¼  L nðLÞdL  dt Mv V dY dt 0

Population balance modeling has been used quite extensively to increase the understanding of batch crystallization processes, the work of Mullin and Nyvlt (1971) showing the potential of controlled cooling perhaps being the first example. Most work relies on the fact that crystallization kinetics for the process are known. This may perhaps be true for a small laboratory process, but kinetics would very rarely be available for a full-scale production process. Bohlin and Rasmuson (1992a) used modeling to explore how seeding and controlled cooling would apply in the industrial practice of batch-cooling crystallization. Because the metastable region is often fairly narrow – from a fraction of a degree to a few degrees centigrade – the right moment to add the seeds and/or begin the cooling is quite difficult to identify. The reason is that in the industrial practice, the temperature may vary somewhat in the tank, the temperature measurement may lack sufficient precision, and the concentration of the crystallizing compound, as well as the overall composition of the solution, is not exactly known. This leads to uncertainty in the actual state of the solution in relation to saturation. Taking kinetics for nucleation and growth for potassium sulfate and citric acid monohydrate, it was shown that while seeding and controlled cooling are theoretically perfectly sound for product crystal size distribution control, in practice, the application becomes very much a trial-and-error gamble. It was also shown that selecting the right amount of seeds is equally difficult because data on the secondary nucleation rate are normally unknown for the plant conditions. Most modeling work assumes that the batch-cooling crystallizer is well mixed, which appears to be a good approximation (Bohlin and Rasmuson 1996). In more recent work, methods to determine the cooling profile that would maximize the size of the seeded crystals or minimize the amount of fines have been developed. Hu et al. (2005a, b) and Kim et al. (2009) used a population balance method to find the optimal temperature profile in batchcooling crystallization to meet product size distribution requirements. Paengjuntuek et al. (2008) used population balance modeling to improve a seeded batch-cooling crystallization by updating the model kinetics after each experimental batch performed. Two kinetic parameters were updated by fitting a moment equation model to the product

Crystallization Process Analysis by Population Balance Modeling

size distribution determined off-line. Lindenberg et al. (2009) applied a holistic approach to the design and optimization of a combined cooling/antisolvent crystallization process. A population balance was formulated that accounts for the volume change at antisolvent addition, and the solute concentration balance accounts for cooling as well as antisolvent addition, where the solubility is allowed to depend on temperature and solvent composition. Experiments are conducted where solute concentration is recorded by ATR-FTIR and the particle size distribution by FBRM. Growth kinetics are determined by seeded-batch desupersaturation experiments at different temperatures and solvent compositions. Nucleation kinetics were determined by unseeded crashcooling experiments recorded by FBRM. The model and the kinetics are used to determine an optimal trajectory for cooling and antisolvent addition of a seeded process, and this trajectory is implemented experimentally. The process was optimized with respect to minimum process time for a specified crystal size distribution. The cooling and the antisolvent addition profiles were discretized into six time steps characterized by in total 13 parameters to be determined by the optimization. The resulting trajectory differs from standard industrial procedure and leads to a process of growing seeds without nucleation. Aamir et al. (2010a) analyzed the influence of seed quality on the shape of the product size distribution in seeded-batch cooling crystallization. The population balance equation is solved by a combination of quadrature method of moments and method of characteristics. Growth kinetics are determined by fitting to seeded-batch crystallization experiments at negligible nucleation supervised by in situ ultraviolet/ visual (UV/vis) spectroscopy. Aamir et al. (2010b) used the same approach to optimize the seeding strategy in batch cooling crystallization. In the model on the precipitation of L-glutamic acid by addition of hydrochloric acid to sodium glutamate (Borissova et al. 2005), a full account of the chemistry is included. A similar study on precipitation of barium sulfate is presented by Steyer et al. (2010), and in both cases the volume is assumed to be well mixed. However, this assumption is not likely to be valid because most precipitations/reaction crystallizations occur under conditions of partial segregation, and the product size (and shape) distribution will depend on the mixing conditions, especially at the feed point. Fines dissolution is used not only for continuous crystallization but also for batch crystallization as a method of increasing the product crystal size. Experimentally, this was originally evaluated by Jones and Chianese (1987), and recently, a cooling crystallizer with fines dissolving has been modeled by Qamar, Mukhtar, and Seidel-Morgenstern (2011). The model is solved by a combination of moment transformation and the method of characteristics. In a follow-up paper (Qamar et al. 2011), this work is extended to a population balance model with two independent phase space coordinates. Nagy et al. (2011) propose controlled crystal dissolution as a method of controlling the final crystal size distribution in batch-cooling crystallization. In a population balance model,

nucleation, crystal growth, and dissolution are accounted for in two separate balance equations. The model is solved by the quadrature method of moments in conjunction with the method of characteristics. Optimal temperature trajectories are determined to obtain the desired crystal size distribution by exploiting in situ fines removal policies. Partial dissolution is also proposed as a method to change the shape distribution in nonseeded batch cooling crystallization (Shoji and Takiyama 2012). The population balance includes primary and secondary nucleation and crystal growth in two directions and is solved by moment transformation based on the double integral joint or cross-moments of the population density with the two internal coordinates corresponding to the two dimensions of the crystal characterizing the shape. Schöll et al. (2006) modeled the polymorphic transformation of L-glutamic acid. For each polymorph, a population balance was formulated, and individual kinetics were assigned. Ten parameters of crystallization kinetics were determined by fitting the model to experimental data from batch experiments. In particular, solute concentration data by ATR-FTIR, relative polymorphic composition of the solid phase by Raman spectroscopy, and average particle size measured by FBRM and particle vision and measurement (PVM) were used. In further studies on this system (Cornel et al. 2009), a fully descriptive model of the polymorphic transformation is developed. In additional work (Lindenberg et al. 2008), the agglomeration of the α-form during precipitation by a pH shift is investigated, and a population balance model accounting for nucleation, growth, and agglomeration is used to predict particle size distributions. The agglomeration of the α-form crystals is examined. Fevotte et al. (2007) investigate the transformation of anhydrous citric acid into the monohydrate. The population balance model has one balance for the dissolving phase and one balance for the growing phase and was solved by the software FEMLAB. Kinetics for dissolution of the anhydrate and for the nucleation and growth of the monohydrate are determined by fitting the model to experiments using Raman spectroscopy for determination of solution concentration and solid phase composition and occasional off-line size distribution determination by laser diffraction. A crystallization process can depend strongly on the presence of impurities and additives. Majumder and Nagy (2013) developed a population balance model to describe the influence on size and shape. The model is applied to a case where the shape is shown by two internal coordinates, and the influence of the growth modifier on the face growth rate is given by the Kubota–Mullin model. The influence of the additive depends on the time of contact between the crystal and the solution and hence will differ for crystals depending on when they appear in the solution, by seeding or by nucleation. The model is used to examine the shape changes of seeds during a cooling crystallization process and to develop a scheme for controlled additive concentration in an operation mode where the solid phase is retained in the crystallizer while the solution with a variable additive concentration is flowing through continuously. Zhang et al. (2015) investigate

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the influence of impurities in cooling crystallization of glutamic acid. A 2D population balance was furnished with growth kinetics in the length and width directions from independent experiments. Agglomeration, breakage, and nucleation were neglected. In the experimental work, the size and shape distributions of dry powders were determined by image analysis.

6.8 Population Balance Modeling of Non-Well-Mixed Processes Often the crystallization process is assumed to be well mixed. However, this is not necessarily a reasonable assumption, and in particular, two different cases have been examined by population balance modeling: full-scale cooling and evaporation crystallizers and precipitation/reaction crystallization processes. In an industrial-scale cooling or evaporation crystallization, the supersaturation is generated locally, and the crystals are not evenly distributed in the slurry. Thus the rate of supersaturation consumption varies in space. To this complexity we can also add that the suspension flow rate and the level of turbulence vary substantially in the crystallizer and hence also the crystal growth rate and the rate of secondary nucleation. Such processes have been modeled using compartment modeling. The entire unit is divided into a set of well-mixed subunits, where different conditions may prevail. Slurry is flowing between the subunits as defined in the model. In a reaction crystallization process, the solubility of the product to be crystallized is often much lower than the concentrations of the reactant solutions that are mixed to form the product. This leads to the supersaturation at the reactant feeding point potentially reaching very high values, and nucleation and growth will proceed under conditions of partial segregation. Studies of mixing effects on precipitation or fast reactions have often focused on the micromixingcontrolled regime. The corresponding experiments are specifically designed to eliminate the influence of mesomixing by using a low feeding rate (Phillips et al. 1999; UeharaNagamine 2001; Akiti and Armenante 2004). Micromixing can be described by phenomenological models with parameters that must be determined either from experimental data or from interpretation of the underlying physics, e.g., the “interaction by exchange with the mean” model and the “segregated feed model” (Villermaux and Falk 1994). The segregated feed model includes both micro- and mesomixing. Zauner and Jones (2000a, b, 2002) claimed that the model gave good results for precipitation of calcium oxalate, but the model was not compared with experimental data on the influence of the feed time and feed pipe diameter. A different approach is to describe the most relevant flow and diffusion processes in a physical model. The parameters can then be determined from fluid dynamics theory or measurements. Examples include the engulfment–deformation–diffusion (EDD) model and the engulfment model (E model; Baldyga and Bourne 1999), as well as the diffusion model by Lindberg and Rasmuson (1999).

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Nowadays, computational fluid dynamics (CFD) is often used to calculate the flow field. The population balance equations can be solved either together with the CFD equations (Piton et al. 2000; Baldyga and Orciuch 2001; Marchisio and Barresi 2003; Vicum et al. 2004; Baldyga et al. 2005; Woo et al. 2006) or the CFD simulation can be done separately to calculate energy dissipation rate values, which are then used in a mechanistic mixing model (Zauner and Jones 2000a; UeharaNagamine 2001; Akiti and Armenante 2004; Cheng et al. 2009). Another, simpler approach was used by Phillips et al. (1999), in which the reactor was divided into zones with different hydrodynamic properties, and the flow map was used to calculate the corresponding variation in the mixing parameters as mixing proceeded. Multiscale Eulerian mixing models that account for all levels of mixing are usually combined with a CFD code to calculate the flow field. Baldyga et al. (2005) used the turbulent mixer model (Baldyga and Bourne 1999) and a beta distribution to describe micromixing for a single-feed semibatch process. The variation of selectivity with varying mean energy dissipation rate and feed concentration was well described, but the influence of feed time on the selectivity was less pronounced than in the experiments. The authors also applied the model to precipitation, but it is difficult to judge the model’s capability from the results. Wei et al. (2001) modeled the precipitation of BaSO4 in a semibatch reactor and compared their results with the experimental results of Chen et al. (1996) concerning the variation of mean size with agitation rate. A correct description of the trend in mean size was obtained, even though the model simply ignored possible fluctuations in concentrations on a scale smaller than the grid in the CFD simulation. However, the product size predicted by the model was about four times larger than the experimental result. Ståhl and Rasmuson (2009) developed a population balance model over single-feed semibatch reaction crystallization of benzoic acid. The model accounts for chemical reaction, micro- and mesomixing, primary nucleation, crystal growth, and growth-rate dispersion. The model is evaluated by comparison with experimental data over benzoic acid precipitation from previous studies (Åslund and Rasmuson 1992; Torbacke and Rasmuson 2001; Torbacke and Rasmuson 2004), in which it was shown that mesomixing has a decisive role for the product size distribution. When the mixing is described by the E-model and growth rate dispersion is accounted for, the model captures the influence of reactant concentrations, agitation rate, feed point location, feed pipe diameter, total feeding time, and crystallizer volume quite well on product weight mean size. The kinetics of nucleation and crystal growth have a great impact on the results of the simulations, influencing the product weight mean size as well as the response to changes in the processing conditions. Cheng et al. (2009) combine CFD computations with population balance modeling of a continuously stirred tank crystallization of barium sulfate. The flow field is assumed to be independent of the solid phase, and the particles are assumed

Crystallization Process Analysis by Population Balance Modeling

to follow the flow. The flow field is mapped into concentrations and population balance models with three independent external space coordinates. The population balance is solved by the standard and quadrature method of moments. The model accounts for nucleation, crystal growth, agglomeration, and breakage. Cheng et al. (2012) extend the model to account for specific feed mixing conditions and compare model results with experimental data.

6.9 Determination of Crystallization Kinetics for Process Modeling The biggest obstacle to a more widespread use of population balancing for quantitative analysis and design of real crystallizers is the difficulty of providing the models with kinetics that are relevant to the actual process. Kinetics of crystallization processes are often clearly nonlinear with respect to the driving force, and in the case of nucleation, the dependence can be strongly nonlinear. Furthermore, accurate determination of the supersaturation is difficult. In inorganic continuous crystallization, the supersaturation concentration difference is often only a few percent of the solubility. Accordingly, an accurate determination of supersaturation obviously requires a very accurate determination of concentration, on the one hand, and accurate knowledge about the solubility, on the other hand. In crystallization of organic compounds, the supersaturation is often higher, but the solution composition can be quite complex. In a full-scale process, conditions such as temperature and composition may vary somewhat in the crystallizer, leading to uncertainties in the accuracy of the solubility. In addition, accurate determination of concentration in complex mixtures is not always trivial. If the compositions of impurities and by-products vary, this may have a significant influence in particular on nucleation kinetics. Accordingly, the conditions at which the crystallization kinetics are determined should be as close as possible to the real conditions, and great care should be taken to reach high accuracy in the data determination. The rates of secondary nucleation and crystal growth depend on the hydrodynamics. In most experiments on determination of secondary nucleation, the measurements are done in a small-scale agitated crystallizer, and the kinetics are obtained in units of number/m3, s, where the volume refers to slurry volume. However, secondary nucleation relates to fluid shear and, in particular, to collisions between crystals and against equipment surfaces. In a small laboratory unit, collisions between crystals and equipment dominate, whereas in large industrial units, it is likely that crystal– crystal collisions dominate. Hence the governing mechanism of nucleation may actually depend on scale. The rate of secondary nucleation varies in the crystallizer because the fluid dynamics vary substantially, and hence the kinetics determined represent an average over the entire volume, in spite of the fact that most nucleation should occur in quite local regions of high-intensity flow and mixing. Accordingly,

Figure 6.11 Objective function surface in determining growth-rate parameters Source: Reprinted from Granberg and Rasmuson 2005 with permission from John Wiley and Sons. Copyright © 2005, American Institute of Chemical Engineers.

there is low scalability of the kinetics unless this fact is accounted for. Finally, it is difficult to determine rate constants and order of dependence with high confidence. As is sometimes the case in determination of chemical reaction kinetics, there tends to be a bias between the value of the rate constant and the value of the exponent in the power-law equation (Qiu and Rasmuson 1991). The objective function exhibits a valley shape, as shown in Figure 6.11 (Granberg and Rasmuson 2005) and various combinations of the rate constant and the exponent values along the bottom of the valley give about the same sum of squares deviation. The MSMPR experiment provides a method to determine the nucleation rate as well as the growth rate at a specific supersaturation in a steady-state experiment. For a wider characterization of the kinetics, the experiment has to be operated at (1) different supersaturations (e.g., by changing the residence time), (2) different magma densities by changing the feed concentration, (3) different temperatures, and (4) different hydrodynamic conditions. This method of determination of kinetics, like most other methods, assumes that the suspension is well mixed. Occasionally, the MSMPR evaluation is also adopted for steady-state continuously agitated tank experiments that are obviously not well mixed (e.g., reaction crystallization processes). Such kinetics can only provide an empirical average representation of the kinetics of that particular experiment, having a very low validity for other conditions. Besides the MSMPR method, there is a variety of more or less refined methods for determination of crystallization kinetics, including single-crystal experiments and fluidized-bed experiments, especially used for determination of crystal growth kinetics. A more efficient route to determination of kinetics for process modeling is to use non-steady-state experiments. A batch process normally travels through a whole range of supersaturation and magma density conditions. The supersaturation

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dependence of crystal growth and nucleation can be determined from one single or a few experiments depending on the extent to which the evolution of the process conditions is recorded. Perhaps the first application of this principle was adopted in the so-called seeded isothermal desupersaturation experiment developed by Tanimoto et al. (1964) and later adopted by Mullin and coworkers (e.g., Jones and Mullin 1973). To an agitated isothermal supersaturated solution, seeds are added, and the solute concentration decay by crystal growth is recorded. The evaluation permits determination of the growth rate versus supersaturation dependence at that temperature and condition of agitation. A well-defined sieve fraction of seeds should be used. By adjusting the mass of seeds to the liquid volume and the initial supersaturation, the extent of growth of each crystal can be limited to avoid, e.g., substantial shape changes. Assumptions underlying the standard evaluation are that nucleation is negligible and that the crystal shape can be approximated as being constant and equal for all crystals. A more complete determination of kinetics from batchwise operated experiments (e.g., a cooling crystallization) can be obtained by fitting a population balance model of the experiment to experimental data (e.g., Qiu and Rasmuson 1994; Ståhl and Rasmuson 2009). It is quite clear that one single experiment can provide rich information depending on what is measured during the experiment. It should be recognized that determination of kinetics involves two different aspects: the actual rate of nucleation and growth, respectively, and the supersaturation at which these rates are observed. Solution concentration and size distribution (number of crystals or amount and size of seeds) at the beginning of the experiment are often known from how the experiment is set up. After termination of the experiment, the product size distribution can be determined, and it is often reasonable to assume that from a mass balance point of view the supersaturation at the end is negligible. In a more advanced approach, the supersaturation can be measured either by sampling or by in situ methods during the course of the experiment, and the same is also possible, although with some difficulty, with respect to the size distribution. As is the case in seeded isothermal desupersaturation experiments, if the crystal population is known, the crystal growth kinetics can be determined by recording the solution concentration only. Alternatively, recording the increase in crystal size also allows for determination of the growth rate. The prevailing supersaturation at each moment can be determined by a mass balance calculation. For determination of nucleation kinetics, there has to be some determination of the actual change in the number of crystals in the experiment. Without number data, nucleation cannot be distinguished from the crystal growth, besides the fact that the mass consumption on nucleation is normally completely negligible compared with that of crystal growth. In order for the supersaturation dependence of the kinetics of nucleation and growth to be established, solution concentrations must be determined, either by

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measurement of by a mass balance using information only on the solid phase, i.e., the amount of solid crystallized combined with solubility information. However, the latter is likely to lead to insufficient accuracy, which is why it is recommended to actually measure the concentration. Of course, the development of advanced process analytical technologies has significantly facilitated collection of information for evaluation of kinetics as long as the need for accuracy is acknowledged. The parameter determination should be done by fitting a mathematical model of the experiment to the actually measured experimental data by direct, nonlinear parameter estimation methods. The experimental information determined is compared with the corresponding data calculated by the model in an objective function X F¼ ðR2i Þ i

Ri ¼ ðXmod  Xexp Þ The optimization procedure determines the parameters in the kinetic equations such that the difference between the model and the experimental data is minimized (i.e., minimize the value of F). The objective function should be defined as far as possible in terms of the data that are actually determined experimentally (i.e., without correlation and averaging of the actual experimental data by, e.g., polynomials). For example, if the product size distribution is determined by sieving, the experimental sieve fraction weights should be compared with the corresponding data calculated by the model. The procedure should not be that the experimental size distribution is fitted to a size distribution function, and this function is then used in the objective function. The same can be said regarding concentration measurements. The value of each term Ri ¼ ðXmod  Xexp Þ is called a residual, and the objective function is the sum of these residuals squared. A plot of the different residuals against, e.g., process time or supersaturation can reveal whether deviations between the model and the experimental data are random or systematic in some way. In the latter case, this would suggest that the model is not completely capable of describing the experiments. If there are systematic deficiencies of the model, this will, of course, reduce its validity for extrapolations. The residual terms can be given different form, e.g., absolute difference, relative difference, and absolute difference between logarithmic values: R ¼ ðXmod  Xexp Þ R¼

ðXmod  Xexp Þ Xexp

R ¼ ðlnXmod  lnXexp Þ It is also possible to apply weighing factors to tune the influence of different terms, for example, if the data vary substantially in magnitude or if data of different character are

Crystallization Process Analysis by Population Balance Modeling

included, e.g., solution concentration and size distribution data. Ståhl et al. (2001) investigated the crystallization kinetics of benzoic acid. Benzoic acid was crystallized in a T-mixer by the mixing of hydrochloric acid and an aqueous sodium benzoate solution. A population balance model was developed over the experiment, and parameters were estimated by nonlinear optimization. The actual conditions of the experiments could be analyzed by fitting different process models and evaluating the parameter values obtained as well as the quality of fit. It was found that the objective function is highly nonconvex, and the results have to be scrutinized in order to identify the appropriate optimum. Six parameters of nucleation and growth kinetics were determined simultaneously in an optimization that included experimental product size distributions from 14 experiments at eight different initial supersaturations. Size distributions where determined by an electrosensing zone technique. The final model describes experimental data well, and the estimated parameters are physically reasonable and provide a physically reasonable description of an entire individual experiment. The objective function was defined as F¼

kmax X lmax X k¼1 l¼1

R2kl .

where k refers to different experiments, and l refers to different size fractions in a particular product size distribution. Residuals were defined as relative deviations Rkl ¼

nsim  nexp . nexp

which reduces the influence of the size intervals with large numbers of crystals, typically in the low size region. However, if the starting point of the optimization is located far from the optimum, the preceding definition of the residuals had to be replaced by logarithmic residuals Rkl ¼ lnðnsim Þ  lnðnexp Þ The objective function can contain different kinds of experimental information simultaneously, e.g., size distribution information and solution concentration measurements. However, these terms have to be balanced or scaled because they would normally be of different numerical magnitude. Size distribution data (population densities) may very well be of order 10 raised to power 3–15 and vary substantially from size range to size range and over time in a nonstationary process, whereas, of course, concentration values are usually much lower and vary less. The balancing may also account for the fact that the number of terms (residuals) related to concentration values and size distribution values can be substantially different. Even in the case of batch experiments, one would normally perform a number of experiments at different conditions, partly to increase the experimental basis for determination of kinetics, but also to resolve different biases in the experimental information (e.g., temperature is decreasing

coupled with magma density increasing). As far as possible all data from all experiments where the kinetic parameters to be determined are supposed to be the same should be used in one single optimization for determination of kinetic parameters, and in this case, it is of particular value to analyze residual distributions. In the work of Qiu and Rasmuson (1994), secondary nucleation and crystal growth rates for succinic acid were evaluated from seeded batch-cooling crystallization experiments. The experimental data recorded and used include temperature and solute concentration as a function of time and the seed and final product size distributions. Parameters were determined by nonlinear optimization of a dynamic model of the experiment, and intermediate approximations of experimental data are avoided. Data from several different cooling crystallization experiments are simultaneously supplied into one single optimization for determination of seven different parameters, four of secondary nucleation and three of crystal growth. The optimization objective function includes both solution concentration data and product mass distribution data   m  s1  PX cðti Þ  ci 2 1  P X MðLi Þ  Mi 2 F¼ þ m i s c0  cf MT i In the concentration part, each term is normalized by the total concentration change, and in the size distribution part, each term is normalized by the total mass of crystals at the end MT. Also, m is the total number of data points in the concentration determination, and s is the total number of sieve fractions – 1 (the upper sieve fraction does not have an upper limit). In the equation, m and s provide for a balancing of the two parts with respect to differences in the number of data points within each set of data, and the parameter P allows for a balancing between concentration data and size distribution data. The output data allow for estimation of the Hessian matrix, from which the covariance matrix can be calculated, providing information on the statistical confidence of each parameter value. A similar approach is taken by Hu et al. (2004). If the model gives a perfect description of the experiment, and when the scatter in experimental measurements is low, the balancing and actual design of the objective function have little influence on the outcome. However, normally, the experimental data contain both random variations and systematic errors/ uncertainties, and the model is not a perfect representation of reality. In this case, the detailed formulation of the objective function will influence the outcome. However, the detailed formulation of the objective function in this case is not primarily a matter of right or wrong but is rather a decision about how to distribute the deviation between the model and the experimental data. Weighing factors can be used to reduce the influence of experimental data that are believed to be more uncertain. Weighing factors can also be used to give different weights to different experiments, to different parts of each experiment, and to different types of information based on the experimental work.

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To carry out the actual optimization calculations, various subroutines are available, and the details are beyond the scope of this presentation. In general, it is advisable to scale the problem such that residuals and parameter values to be determined are of the order of unity. There is a wide range of different optimization methods, different in the techniques for finding an efficient route towards the optimum, and in their robustness. Formerly, the computation speed was an important issue, but for complex problems today, perhaps robustness is more critical. The outcome of the optimization should be carefully analyzed, for example: • Has the computation reached an optimum or just stopped halfway? • Are the same parameter values produced when different starting values are used? • Are residuals evenly distributed between different parts of the experimental data? • Are parameter values physically realistic? • Is the range of confidence of each parameter reasonable? There are in principle two different applications of the kinetics determined. One application is to use the kinetics for modeling the process. The second is to use kinetic parameter values for analysis of mechanisms. For the first application, of course, the main aspect is that the parameter values and their associated functions do provide for an accurate description of the experimental product crystal size distribution. In this case, empirical equations representing the kinetics can be used, and whether there is a bias between different parameters is not of significant importance as long as an adequate numerical representation of the rate is obtained. For the second objective, it is more critical that the values do represent the actual physics, and this will require equations that are physically realistic, and it is important that the optimization delivers a value of each parameter that is physically appropriate and statistically valid. The parameterization of the model in population balance modeling may involve a substantial number of parameters that need to be determined by the optimization. Of course, the higher the number, the better is the fit to the experimental data, but the physical meaning of each parameter may deteriorate, and the statistical validity will decrease, as will be reflected in wider ranges of confidence of the parameters. Interesting progress in this area is presented in two papers (Hermanto et al. 2008; Su et al. 2014) where kinetic parameters are estimated by Bayesian interference from batch crystallization data, and probability distributions of the parameter estimates are obtained by Markow chain Monte Carlo simulations. Hermanto et al. (2008) apply these methods to a batch polymorphic crystallization of L-glutamic acid, during which the solution concentration is determined by ATR-IR and the size distribution is recorded by FBRM. The population balance is solved by the method of moments. The kinetic model includes secondary nucleation, growth, and dissolution kinetics of the α-form and nucleation and growth kinetics of the β-form. Growth kinetics take temperature

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dependence into account. The nucleation kinetics of the βform include secondary nucleation and nucleation on the surface of α-form crystals. By regression, the values of 10 parameters are determined based on six seeded crystallization experiments. Su et al. (2014) apply the same parameter estimation methods to a semibatch reactive crystallization process. The model includes protonation changes with pH and nonideality in the specie balance, polymorphic transformation, and mixing effects in terms of different average mixing intensity due to different agitation rate/agitator design. Primary and secondary nucleation of both forms is accounted for, as well as the influence of the α-form on the nucleation of the β-form. For the α-form, dissolution kinetics are included. The model is solved by a finite-volume method, and 29 parameters are estimated based on six different experiments at different reagent concentrations, agitation rates, and agitator shape. Schöll et al. (2007) used a combination of seeded desupersaturation experiments and nucleation induction time experiments to determine crystal growth and nucleation parameters for a pharmaceutical compound. Besenhard et al. (2014) investigated the parameter determination from seeded batch-cooling experiments with crystal growth and agglomeration. A population balance is developed and solved by the method of classes. The crystal growth rate is evaluated from solute concentration versus time data, whereas the agglomeration kinetics are extracted from size distribution data. The choice of objective function and optimization algorithm is investigated. Oucherif et al. (2013) combined seeded and nonseeded desupersaturation experiments to determine the influence of a polymer additive on the nucleation and growth kinetics of felodipine. Growth kinetics were determined from the seeded experiments and then inserted into a population balance model over the nonseeded experiments for determination of the nucleation kinetics. The influence of the polymer on the nucleation rate was much more pronounced than on the growth rate. A very similar approach has been used for determination of the nucleation and growth kinetics of mefenamic acid (Cesur and Yaylaci 2012). Gherras and Fevotte (2012) perform batch-cooling crystallization during which the solute concentration is determined by in situ infrared spectroscopy, and the product crystal size distribution is determined by image analysis. The population balance model includes primary and secondary nucleation and crystal growth, and parameters are determined by fitting to experimental data by nonlinear regression. Pohar and Likozar (2014) used laboratory experiments on batch-cooling crystallization of amlodipine to determine nucleation, growth, and agglomeration kinetics by fitting a population balance model. Nucleation and growth kinetics were described by power-law expressions and agglomeration by the Abrahamson model. The model was then used to analyze an industrial-scale crystallizer and to help explain the unexpected formation of bimodal product distributions. A general discussion of the parametrization of population balance models for solution-mediated polymorphic

Crystallization Process Analysis by Population Balance Modeling

transformations is given by Flood and Wantha (2013) highlighting the need for better mechanistic models over the underlying kinetics of nucleation, growth, and dissolution. In the population balance model of Kobari et al. (2014), primary as well as secondary nucleation of the stable form is accounted for, whereas agglomeration, breakage, and growth-rate dispersion are neglected, and the growth rate/dissolution rate is assumed to be independent of size. Particular attention is given to the secondary nucleation of the stable phase on the transformation time and the scale-up. For antisolvent nonisothermal batch crystallization, Cogoni et al. (2014) compare a population balance approach to a stochastic modeling approach, without finding a clear advantage of adopting one over the other. Both models were capable of quantitatively describing the development of the population density distribution. In the population balance approach, the kinetics were determined by fitting the model to nine experiments simultaneously for the determination of two nucleation parameters and 11 growth-rate parameters. The population balance was solved by a first-order finitedifference scheme. Models describing the evolution of crystal shape distributions in a process require growth kinetics of individual faces or at least in more than one overall direction of the crystal. Schorsch et al. (2012, 2014) developed a system with two orthogonal cameras recording the crystals in suspension being circulating from the crystallizer through a flow cell. The procedure allows for determination of multidimensional size distributions during crystallization. The setup is used (Ochsenbein et al. 2014) in the determination of 2D crystal growth kinetics for β-L-glutamic acid from seeded desupersaturation experiments. Ma and Wang (2012a) recorded size and shape distributions by image analysis and determined kinetics of growth in two directions by fitting a 2D population balance to the measurements. This work is extended into 3D kinetics by Ma et al. (2012b).

6.10 Conclusions There has been a very strong development in the use of population balances for analysis of crystallization processes, especially during the last 5–10 years, focused on various types of batch and fed-batch processes. The development of faster computers allows one to solve gradually more complicated problems. Systems with increasing kinetics complexity, including agglomeration and crystal breakage, are routinely handled, and kinetic parameters are determined by nonlinear optimization from a combination of different experiments targeting different regions of the parameter space. Population balances over several phase space coordinates and models over non-well-mixed systems where the population balance is combined with computational fluid dynamics are in strong development. However, the progress in understanding the underlying kinetic mechanisms is slower, and kinetics determined under one set of conditions can normally not be transferred to other conditions. Hence population balance process design from first principles based on kinetic parameters determined by independent experiments and

model based scaling up of a real industrial process still lie in the future.

Nomenclature a

activity

A

area (m2)

Asp

surface area of crystal (m2)

AT

total surface area of crystal population (m2; m2/ kg; m2/m3)

B

population birth function (1/m3, s)

Bp

primary nucleation rate (m−3, s−1)

Bs

secondary nucleation rate (m−3, s−1)

c

solution concentration

D

population death function (1/m3, s)

F

objective function

g

order of growth-rate dependence on supersaturation

h

order of nucleation rate dependence on agitation rate

j

order of nucleation rate dependence on magma density

k

distribution parameter

ka

area shape factor

knp

rate-constant primary nucleation

ks

rate-constant secondary nucleation

kv

volume shape factor

L

linear characteristic dimension of crystal (m)

LT

total length of crystals (m; m/kg; m/m3)

L

mean crystal size (m)

Lm

median crystal size (m)

mj

moment j of distribution

M

mass of crystals per unit volume or mass of suspension (kg/m3; kg/kg)

MT

total mass of crystals (kg; kg/kg; kg/m3)

n

population density (1/m; 1/m, m3; 1/m, kg)

np

order of primary nucleation rate dependence on supersaturation

ns

order of secondary nucleation rate dependence on supersaturation

Mv

molecular mass

N

number of crystals (–; 1/m3; 1/kg)

Nc

cumulative number of crystals (–; 1/m3; 1/kg)

NC

consumption by growth (kg/m3, s; kg/kg, s)

193

Åke C. Rasmuson

(cont.)

(cont.)

supersaturation generation (kg/m3, s; kg/kg, s)

NG

−1

NS

agitation rate (s )

chemical potential (J/mol)

μL

mean of distribution

Nt

total number of crystals (-; 1/m ; 1/kg)

ψK

kurtosis

Q

volume flow rate (m3/s)

ψS

skewness

R

universal gas constant (J/K, mol)

νe

velocity vector along external coordinates

Ri

residual

νi

velocity vector along internal coordinates

S

supersaturation ratio (=c/c*)

ρ

density (kg/m3)

t

time (s)

τ

residence time

T

temperature (K)

θ

distribution parameter

λ

distribution parameter

3

3

V

volume (m )

Vsp

volume of crystal (m3)

VT

total volume of crystal population (m3; m3/kg; m3/m3)

x

mole fraction

X

parameter value in objective function

Y

solvent mixture composition parameter (m3/ m3; kg/kg)

Greek

Upper index in

in inlet stream

out

in outlet stream

*

at equilibrium

0

at zero size, nuclei

Lower index

γc

activity coefficient based on concentration

c

crystal

γx

activity coefficient based on mole fraction

i

particle size i

σ

standard deviation of distribution

j

moment j

T

total

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Chapter

7

Selection and Design of Industrial Crystallizers Herman J. M. Kramer Delft University of Technology Richard Lakerveld Hong Kong University of Science and Technology

7.1 Introduction Crystallization is one of the most important separation and product-formation technologies in the chemical industry. Typical advantages of crystallization are the low energy consumption, mild process conditions, and high product purity that can be obtained in a single separation step. The future impact of crystallization is even expected to increase further because many new high-added-value products are often in crystalline form. However, future crystalline products are also subject to increasingly stringent product quality requirements related to, for example, flowability, filterability, bioavailability, stability, and dissolution behavior. Product quality requirements for crystalline products typically vary strongly depending on the field of application. In the last 50 years, we have seen technical advancements in the design and operation of industrial crystallizers that have enabled the production of a large variety of products via industrial crystallization processes in completely different quantities and with varying product specifications. Examples of such products include • • •

Bulk products such as sugar, sodium chloride, ammonium sulfate, urea, and adipic acid Products that require ultrahigh purity, such as para-xylene and naphthalene A wide range of fine chemicals and pharmaceutical products such as pigments and numerous active pharmaceutical ingredients

The design of crystallization processes traditionally has been considered “more an art than a science” by relying on heuristic knowledge of experts in the field rather than on a fundamental understanding of the phenomena that are part of crystallization processes. However, advancements in various fields allow for a more scientific approach to the design of industrial crystallizers. In particular, knowledge in the following fields enhances the rational selection and design of industrial crystallizers: •

•

Thermodynamics. Because crystallization processes are typically operated close to equilibrium, understanding of the phase behavior of the system at hand allows for identification of the optimal separation route. Furthermore, thermodynamics dictate the maximum attainable yield of the crystallizer design, which is an important performance criterion. Kinetic Phenomena. The deviation from equilibrium drives various kinetic phenomena inside an industrial crystallizer,

•

•

such as nucleation, growth, agglomeration, and attrition. The interplay between these kinetic phenomena determines the final product quality attributes related to, for example, crystal size distribution (CSD), purity, and shape. Measurement and Control. The availability of advanced process analytical tools (PATs) has enabled various opportunities to improve the design of industrial crystallizers. In an off-line setting, PATs can be used to obtain reliably and, sometimes with high-throughput, fundamental understanding of the crystallization process at hand. In an online setting, PATs can be used to improve the control of industrial crystallizers. Modeling and Optimization. Process models of industrial crystallizers have received significant academic interest in recent decades, which has led to validated model structures and computational tools for simulation and optimization that can be used for the design of industrial crystallizers.

The aim of this chapter is to present a systematic approach to the design of industrial crystallizers that combines both advancements in fundamental understanding of crystallization and useful heuristics for design. Each of the five sections represents a step in the sequential design procedure, as illustrated in Figure 7.1. 1: The typical performance criteria for industrial crystallizers will be discussed. A key challenge is to translate the functional requirements of the crystalline product into physical requirements, which sets objectives for crystallizer design (Kramer and van Rosmalen 2000; Bermingham et al. 2003; Kramer and Jansens 2003). Furthermore, performance criteria related to process operation can be specified. The overall design objective is dictated by the economics of the process, which often requires tradeoffs between various performance criteria. 2: The selection of the crystallization methods will be discussed, which results in a process block diagram indicating the material and energy flows in the design. Selection of the crystallization methods and corresponding material flows aims to optimize the overall separation route in which equilibrium between solid and liquid phase is typically assumed (Wibowo and Ng 2000; Mersmann 2001; Bennett 2002). 3: The block diagram will be extended by selecting suitable equipment to perform the individual crystallization steps. Typically, a certain type of crystallizer is selected from a set of well-known crystallizers based on heuristics, which is subsequently adjusted for the specific application by design of equipment dimensions. 4: The instrumentation and actuation will be discussed, which involve evaluation of PATs and design of additional actuation such as fines removal or seeding. 5: A final optimization step is recommended to

197

Herman J. M. Kramer and Richard Lakerveld

physical properties, and more important, these properties are directly related to the thermodynamics and kinetic models that are used to describe the crystallization processes (Kramer and van Rosmalen 2000; Bermingham et al. 2003). Therefore, in the design of crystallization processes, product quality is often expressed in terms of well-defined physical properties. The main performance criteria with respect to the properties of the product are • • •

Capacity. This is the production per unit of time. Yield. This is defined as the production of the solute in the crystal phase divided by the amount of solute entering the process via the feed. Crystal size distribution. In most cases, only a few characteristics of the size distribution are of interest: •

•

•

Figure 7.1 Key recommended steps for the design of industrial crystallizers

•

improve the efficiency of the overall design. Process modeling can be used to optimize, for example, a rate of fines destruction in the case of continuous crystallization or the dynamic profile of supersaturation generation and the properties of the seeding procedure in the case of batch crystallization. The chapter ends with a case study from the literature to illustrate the systematic design approach.

•

7.2 Performance Criteria The product specifications in crystallization processes depend on whether the process aims to the purification of the the liquid phase or the production of a solid phase (Bermingham et al. 2003). Whereas the liquid phase can be uniquely specified in terms of pressure, temperature, and composition, specification of a solid product is difficult to quantify and to measure. In this respect, a distinction can be made between the primary and secondary properties of the product. The former are expressed in terms of the fundamental properties of the solid product, such as purity, size distribution, shape, and crystal structure. The secondary properties of a solid product specify the composite properties of the powder made up from crystals with the specific primary properties. Examples of secondary properties include free-flowing behavior, filterability, caking behavior on storage, rate of dissolution, and bioavailability. The main problem with these secondary properties is that they are in most cases ambiguously defined and specific for the typical application (Kramer and van Rosmalen 2000). This complicates the translation of product performance specifications into well-defined physical properties of crystalline products. The primary properties of the product, by contrast, can be expressed in terms of well-defined

198

•

The average size expressed as the mean, median, or mode of the distribution The width of the distribution expressed by, for example, the coefficient of variation or the quartile ratio Quantiles expressed as the mass fraction of crystals below or above a specified size (Quantiles are often used in practice and express the wish to avoid a large amount of fines or a long tail of large crystals in the product.)

Shape. This property is related to crystal structure and can be strongly affected by the presence of impurities, the level of supersaturation, solvent, polymorphic form, and various kinetic processes such as attrition and agglomeration. Crystal internal structure. This typically influences product performance criteria such as shape, dissolution behavior, flowability, and bioavailability. Purity.

In addition, a number of properties related to the process are important to characterize the performance of the crystallization process, such as the energy duty, flexibility, stability, and controllability (Mersmann 2001; Bennett 2002; Kramer and Jansens 2003; Lakerveld et al. 2010).

7.3 Crystallization Methods The first step in the design of an industrial crystallizer is to estimate the mass and energy flows around the crystallizer. Crystallization processes are typically operated close to thermodynamic equilibrium. Therefore, phase behavior will dictate the optimal separation pathway and the associated methods for crystallization. In general, four methods can be used to invoke crystallization: 1. Change in temperature (e.g., cooling crystallization) 2. Selective removal of one of the components (e.g., evaporative crystallization) 3. Addition of a component (e.g., antisolvent crystallization) 4. Change of chemical species (e.g., reactive crystallization or precipitation) Often crystallization from the melt is considered to be a separate method, although from a thermodynamic point of view this is in most cases a cooling process. However, because of the different objective, i.e., the production of an ultrapure

Selection and Design of Industrial Crystallizers

product, the absence of a solvent, and the difference in growth mechanism (heat transfer–limited versus mass transfer–limited growth), melt crystallization can be considered to be a completely different crystallization method from the four basic methods just mentioned. The thermodynamic behavior of a crystallization system is a function of temperature, pressure, and composition. The effect of pressure normally can be neglected. The points of interest of a phase diagram are boundaries for separation such as eutectic points and solubility lines, which shape regions in thermodynamic space where a desired component can be crystallized (Wibowo and Ng 2000). The complexity of selecting optimal crystallization methods depends on the number of components involved because the thermodynamic space grows exponentially with the number of components. For binary systems, a solubility diagram of the system is sufficient to select the optimal crystallization method. A decision flowchart, as shown in Figure 7.2, can be used to select the optimal crystallization method based on key characteristics of the solubility diagram. Evaporative crystallization and cooling crystallization are the most frequently applied methods in large-scale industrial plants and are the first choice for solutes with a relative high solubility. When the solubility curve is relatively steep, such as, for example, in the case of glauber salt or acidic acid in water, cooling crystallization is preferred, which enables a high yield at low energy consumption. For compounds with a flat or reversed solubility curve, evaporative crystallization is the preferred method. Intermediate cases in which either cooling or evaporative crystallization can be used are also possible. Energy consumption in cooling crystallization is often lower than in evaporative crystallization because cooling only requires the removal of the heat of crystallization

and the sensible heat of the mother liquor, which are much lower than the heat of evaporation that has to be provided in the case of evaporative crystallization. The relative energy consumption of evaporative crystallization and cooling crystallization can be estimated with
 dweq Eevap DHevap 1 ¼ 1 weq Ecool Cp dT

where E is the required duty for the evaporation or cooling, weq is the saturation concentration expressed in mass fraction (kg/ kg of liquid) at the equilibrium temperature, Cp is the specific heat of the saturated solution, and ΔHevap is the heat of evaporation. From Equation (7.1), it is clear that, at least energetically, cooling crystallization is often more favorable than evaporative crystallization, except for highly soluble substances with flat solubility curves. Flash cooling is often used to avoid scaling on the cold heat exchanger surface, which is a well-known problem in cooling crystallization, especially when the solubility curve is steep. Melt crystallization is used for the ultrapurification of materials with relatively low melting points and viscosities. Drowning out with an antisolvent has the disadvantage that it requires an auxiliary (usually organic) phase, which has to be recovered afterwards. Nevertheless, addition of a third component to invoke crystallization is a widely used method, for example, in the production of proteins or in cases where very precious or harmful components have to be separated. Precipitation is only applied for the production of slightly soluble compounds. In the case of precipitation and drowning out, two feed streams have to be mixed. Well-controlled mixing is crucial to avoid locally high supersaturation, which would trigger abundant

Design information - feed composition - thermodynamics - physical properties

Specifications purity yield average size 0 < Tmelt < 100 ? High purity?

yes

ð7:1Þ

Figure 7.2 Decision diagram to choose the method of crystallization Source: Reproduced by permission of the publisher from Lewis et al. 2015. Copyright © Cambridge University Press.

Melt crystallization (not at high viscosity)

no dWeq/dT > 0.005? No severe fouling?

yes Cooling crystallization

no Weq. > 0.2 ? Thermal stable?

yes Evaporative crystallization

no Weq. > 0.02 ?

yes Anti-solvent crystallization

no Precipitation

199

Herman J. M. Kramer and Richard Lakerveld Figure 7.3 Example of the selection of various crystallization methods to separate a ternary mixture. The left-hand side shows a polythermal projection of the ternary phase diagram. A process pathway is drawn in the phase diagram, which uses seven process manipulations to move through thermodynamic space. The process pathway is translated to a process flow diagram in the righthand side of the figure.

primary nucleation and agglomeration. Several researchers have developed structured approaches to select an optimal sequence of crystallization methods for multicomponent systems. The basic idea is to use the special features of the phase diagram to divide the thermodynamic space into operating regions, which are subspaces of thermodynamic space in which only a certain component can be crystallized. The operating regions are bounded by various manifolds, which include (Wibowo and Ng 2000) 1. Solubility Manifold. A single component is crystallized on passing the solubility manifold. The number of these manifolds is equal to the number of components. 2. Eutectic Manifold. Several components are crystallized at a eutectic manifold. The number of different types and dimensionality of eutectic manifolds depends on the number of components involved. A system has (n − 1) binary eutectic manifolds in which two components are crystallized simultaneously, (n − 2) ternary eutectic manifolds in which three components are crystallized simultaneously, etc. Note that a manifold itself is a multidimensional subspace depending on the number of components. 3. Decomposition Temperatures. The operating temperatures of the streams and process manipulations are limited by the decomposition temperature of the least stable component. 4. Compound Formation. The formation of compounds (including solvates) is common for a large number of systems. A compound is a stable crystalline structure formed from different components in the mixture. The manifold describing the formation of compounds limits free movement from a neighboring region. Compounds can also be used as intermediate or final products. The experimental effort involved in determining a full phase diagram quickly explodes on increasing the number of components. However, the experimental work can be targeted toward finding the most crucial states (mentioned earlier), which determine the boundaries of a certain region. Visualization of the phase diagram becomes impractical once more than three components are involved. Therefore, several tools have been developed to organize the information for process design obtained from experimental data and model

200

predictions (Cisternas and Rudd 1993; Samant et al. 2000; Samant and Ng 2001; Wibowo and Samant 2002). In each bounded region of the phase diagram, only a single component can be crystallized in pure form. Crystallization of more than one component is possible for components sharing a eutectic manifold. Each bounded region in which a component with tight product requirements for purity can be crystallized should be visited at least once. The eutectic manifolds need to be crossed to move to a different operating region. The optimal process manipulation for passing eutectic manifolds depends on the special features of the phase diagram and has been addressed extensively by various researchers, which has resulted in the availability of heuristics and optimization methods to support decision making (Thomson et al. 1998 Takano et al. 2000; Wibowo and Ng 2000; Cisternas et al. 2006). Figure 7.3 is an example of how to select a sequence of crystallization methods to separate three components based on a phase diagram that is based on work by Wibowo and Ng (2000). The diagram is given as a polythermal projection with the eutectic manifolds projected on the ternary phase diagram. Three binary eutectic manifolds are present, which are lines in the full temperature–composition (Tx) space. The lines coincide in the ternary eutectic point, where all components crystallize simultaneously. The solubility manifolds are twodimensional (2D) surfaces bounding the upper temperature of each operating region and are connected by the eutectic manifolds. Note that the solubility manifolds are not shown in the diagram owing to the chosen polythermal projection. The challenge is to select an optimal sequence of process manipulations to move through this thermodynamic space. Three operating regions are identified bounded by the eutectic manifolds. In each operating region, one component can be crystallized, as indicated in the figure. The objective for the presented case is to obtain the components A and B in pure form, which implies that both operating regions A and B need to be visited. The composition of the undersaturated feed is located in the operating region for component A, as shown in the figure. Component A can be crystallized directly by cooling the feed. However, this would move the system in the direction of the eutectic manifold of components A and S from which operation region B is difficult to access. Therefore, mixing is applied first to create a mixture with thermodynamic state S1. Subsequently, cooling is applied to move the process along

Selection and Design of Industrial Crystallizers

the solubility surface, which provokes crystallization of component A. Cooling is stopped just before the eutectic manifold AB is reached (thermodynamic state S2) to prevent simultaneous crystallization of components A and B, which will be difficult to separate once crystallized together. A solid–liquid separation is used to remove the crystals from stream S2. The liquid stream with thermodynamic state S3 has to be transferred to operating region B to selectively remove component B. In the example, two process manipulations are used to affect this movement. First, heating is applied to create an undersaturated solution (S4), and subsequently, solvent is removed to move the operating point into region B (S5). In this region, cooling is used to move the composition along the solubility surface of component B, which causes component B to crystallize. Once more, cooling is aborted just before the eutectic manifold AB is reached to prevent simultaneous crystallization of the components A and B. Crystals are removed from stream S6 with a solid–liquid separation, and the liquid stream (S7) is mixed with the feed to move the operating point across the eutectic manifold back into region A, which completes the cycle. The next decision to take in the design process is to choose between continuous and batch-wise operation. By far, most crystallization processes are operated batch-wise. Especially in the pharmaceutical and fine chemical industry, a massive number of small-scale batch crystallization processes are carried out. Nevertheless, there is a growing interest in the pharmaceutical industry to move from traditional batch-wise processing to continuous processing, which includes crystallization processes (Plumb 2005; Myerson 2010; Mascia 2013; Zhang 2014). Therefore, a growing number of future pharmaceutical crystallization processes may very well be operated in continuous-flow mode. The most important considerations to choose between batch and continuous operation include •

•

•

Production Capacity. For small production levels up to 5 kt/year, batch-wise production in simple crystallizers is common practice. This is mostly the case when expensive materials such as fine chemicals, food additives, or pharmaceuticals are being handled, where flexibility in production and losses of the precious solute are more important than the somewhat higher operating costs. Above 20 kt/year, operation is in general continuous. Product Quality. The well-defined residence time of the crystals can lead to much narrower size distributions with batch-operated processes because each crystal has the same processing history in the absence of secondary nucleation. By contrast, continuous crystallizers are simple to operate and deliver a constant product quality, whereas batch-tobatch reproducibility can be problematic in the case of batch-wise operation. Operating Costs. The operating costs of batch-operated crystallizers are usually higher because of the continuous human intervention that is needed, especially at the startup and final stages. The complicated startup procedure may include seeding, the less efficient use of the utilities, and the consequences of the solid–liquid separation and solid handling steps.

•

Cleaning/Descaling. When the crystallization system has a strong tendency for scaling or when regular cleaning is required, for instance, for food production, batch operations might be preferred. As a result, scale removal and equipment cleaning can be easily incorporated into the batch schedule, whereas a full production stop or parallel equipment would be required to realize a descaling or cleaning procedure during continuous operation.

7.4 Equipment Design 7.4.1 Introduction In this section, the design of the equipment to produce a specified crystalline product is discussed. Based on heuristic knowledge, a selection is made from a list of documented crystallizer types, which is then adapted and dimensioned according to the design specifications. In addition, the circulation device, the heat exchanger, and possibly the classification devices are designed. The basis for almost all the available crystallizer types for solution crystallization processes is a well-mixed vessel. The choice for a well-mixed vessel is driven by the need for long residence times and turbulent conditions to enhance mass and heat transfer and minimize gradients in supersaturation and solid fraction. The various designs differ in the features they offer to manipulate the product quality either by influencing the balance between the (secondary) nucleation and size-increasing phenomena such as growth and agglomeration or by changing the residence time of either the large or small crystals in the crystallizer. Figure 7.4 illustrates the expected mean crystal size for a so-called Oslo (or suspension) crystallizer, a draft-tube baffle (DTB) crystallizer, and a forced-circulation (FC) crystallizer. The graph shows that these crystallizers deliver a decreasing mean crystal size in the order just listed. Before we describe

Figure 7.4 Expected median crystal size range for three basic crystallizer types as a function of residence time and nucleation rate Source: Reproduced with permission of the publisher from Wöhlk et al. 1991. Copyright © 1991, Wiley-VCH Verlag GmbH & Co.

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Herman J. M. Kramer and Richard Lakerveld

the different types of crystallizers in more detail, some theoretical work done by Randolph and Larson (1988) will be summarized that describes the effects of the residence time of part of the crystal population in a crystallizer on the product crystal size distribution. With this theoretical analysis, the explanation of the effects of the different features of the various types of crystallizers will become clearer. The efficiency of the various features in the different crystallizer designs is not always clear because they will depend not only on how well the feature is designed but also on the crystallization system or scale of operation. The most used selection criteria for the selection of a crystallizer type are aspects of the product quality, such as the coarseness and width of size distribution and the required flexibility in production rate. The attainable yield and energy duty are mainly determined by the choice of the crystallization method, as discussed earlier.

where B0 is the particle flux into the domain by nucleation, and G(0) is the growth rate at the boundary of the crystal size domain. In addition, an initial distribution is needed that defines the population density at the start of the process formed either by primary nucleation or an added seed distribution. The model is completed with material and energy balances and with the kinetic expressions for the growth, nucleation, and agglomeration as a function of the process conditions. For the MSMPR concept, mostly power-law relations are used to describe the kinetics as a function of the process conditions, although more detailed kinetic models are available from the literature. The growth and nucleation rates are given in a power-law form, for example, as

7.4.2 Influence of the Crystallizer Configuration and Process Conditions on the Product Quality

where B0 is the secondary nucleation rate, σ is the relative supersaturation, ε is the mean power input by the impeller, and MT is the total crystal mass in the crystallizer per unit volume of crystal suspension. The parameters kN, b, k, j, m, and i are the system- and crystallizer-specific parameters for the model equations. In continuously operated processes, the secondary nucleation is mainly due to attrition of the larger crystals induced by mechanical contact with the impeller, baffles, pump internals, or heat exchangers or by particle– particle collisions (Mersmann 2001). Other secondary nucleation mechanisms might play a role as well, such as initial and surface breeding. For an MSMPR crystallizer, the classification function is defined as

To get a better idea about the influence of the crystallizer configuration and the process conditions on product quality in crystallization processes, a model is needed that describes the evolution of the crystal size distribution in time. For continuous crystallization processes, the widely used mixed suspension, mixed product removal (MSMPR) concept was introduced by Randolph and Larson (1988), and it provides a simple solution of model equations based on energy, material, and population balances. An MSMPR crystallizer is a wellmixed continuous crystallizer in which the feed is crystal free and the product composition is equal to the crystallizer content. Additional concepts were introduced to simulate a crystallizer with fines removal (R model), clear liquor advance, selective product removal (Z model), and a crystallizer with fines and selective product removal (R-Z model). The main advantage of the MSMPR concept is that it gives quick insight into the effects of process conditions and crystallizer configuration on the CSD of the product. However, because real crystallizers do not behave as idealized models, the shortcut calculations discussed here are mainly used to predict certain trends. For the design and optimization of a specific design, detailed modeling is needed, as is discussed in Section 7.6. The size distribution in an MSMPR crystallizer can be described by the population balance equation ∂nðL; tÞ ∂nðL; tÞ ϕp ¼ G  hðLÞnðL; tÞ ∂t ∂L V

ð7:2Þ

where n is the population density (number/m4), G is the growth rate (which is assumed to be size independent), ϕp is the product flow rate (m3/s), and L is the size of the crystals. The variable h represents the classification function, which is a size-dependent function and describes the effect of selective fines or product removal. The boundary conditions for this partial differential equation are nð0; tÞ ¼

202

B0 ; nðL; t ¼ 0Þ ¼ n0 Gð0Þ

ð7:3Þ

G ¼ kG σ m

ð7:4Þ j

0

j

B0 ¼ kN σ b εk MT ¼ k N Gi εk MT

hðLÞ ¼ 1; 0 ≤ L ≤ Lmax

ð7:5Þ

ð7:6Þ

which leads to the following steady-state solution of the population balance: n¼

B0 L=τG e ¼ n0 eL=τG G

ð7:7Þ

qith τ = V/ϕp and B0 as the steady-state nucleation rate (number/m3). When the logarithm of the size distribution is plotted as function of crystal size, a straight line is found with a slope equal to −1/Gτ. The number of crystals at the lowest size n0 is equal to the ratio B/G. Furthermore, the mean crystal size is given by Lm ¼ 4Gτ

ð7:8Þ

The total crystal mass in the crystallizer is given by MT ¼ 6kv ρc n0 vG4 τ4

ð7:9Þ

From these equations, it is clear that in an MSMPR crystallizer, the crystal quality is determined by the residence time in the crystallizer and the process conditions that influence the kinetic phenomena such as the supersaturation, the specific power input, and the crystal concentration in the crystallizer. The dependency of crystal size on the process conditions is not directly clear from the preceding equations but can be derived

Selection and Design of Industrial Crystallizers

easily using the moment method (Randolph and Larson 1988; Mullin 2001). For example, the ratio of mean crystal size Lm for two different values of residence time τ1 and τ2 is given by
ði1Þ=ðiþ3Þ Lm;1 τ1 ¼ ð7:10Þ Lm;2 τ2 Because the parameter i is on the order of 1–3, we observe a weak dependency of the mean size on the residence time. The CV value does not depend on the residence time. The influence of the crystal mass is given by
 Lm;1 MT;1 ð1jÞ=ð3þiÞ ¼ ð7:11Þ Lm;2 MT;2 Because j is mostly assumed to be on the order of 1, only a very weak dependency on the crystal mass is expected. Another possibility to influence crystal size is the application of a finesremoval system, as is found, for instance, in a DTB crystallizer, which can be modeled by the so-called R-crystallizer with the following classification function: hðLÞ ¼ R if 0 ≤ L ≤ LF hðLÞ ¼ 1 if LF ≤ L ≤ Lmax

ð7:12Þ

The solution of the population balance is then given by nðLÞ ¼ n0 eRL=τG nðLÞ ¼ n0 eðR1ÞLF =Gτ  eL=GL τ

if 0 ≤ L ≤ LF if LF < L ≤ Lmax

ð7:13Þ

The Z-crystallizer model describes an MSMPR crystallizer in which the residence time of the large crystals above a size Lp is reduced with a factor z, maintaining the residence time of the liquid phase at the same value, resulting in the following classification function: hðLÞ ¼ 1 if 0 ≤ L ≤ Lp hðLÞ ¼ z if Lp ≤ L ≤ Lmax

ð7:14Þ

Finally, the fines removal and product classification models can be combined into the so-called R-Z model hðLÞ ¼ R if 0 ≤ L ≤ LF hðLÞ ¼ 1 if LF ≤ L ≤ Lp hðLÞ ¼ z if Lp ≤ L ≤ Lmax

ð7:15Þ

The effects of the selective crystal removal are shown in Figure 7.5 and Table 7.1. Figure 7.5 shows the logarithm of the

steady-state distributions for the different crystallizer configurations. Compared with the MSMPR method, the crystallizer with fines removal shows a higher growth rate, large mean crystal sizes, and a slightly narrower distribution. If the R factor is increased, large crystals can be obtained. However, care should be taken to avoid instability of the crystallizer, which in extreme cases can lead to cyclic behavior. Product classification does not have much influence on the CSD, whereas a combination of fines removal and product classification leads to a much narrower size distribution.

7.4.3 Crystallizer Selection This section provides an overview of the main crystallizer types together with a short discussion on the characteristics of these crystallizers. The list of crystallizers discussed here has no pretense to being complete and only serves as a means to highlight important features and differences between the various types of crystallizers.

Stirred Draft-Tube Crystallizer The stirred draft-tube (DT) crystallizer represents the most basic crystallizer type and is the one that approaches most closely an MSMPR crystallizer. Figure 7.6 shows an example of a DT crystallizer with an internal heat exchanger installed. The internal body is well mixed through the action of the marine-type impeller. The internal heat exchanger has a number of disadvantages. The most important ones are the limitations on the heat-transfer area, which make it only feasible for low-heat duties and relatively small volumes, and the lack of flexibility, because the heat exchanger is built into the crystallizer. Second, the presence of the heat exchanger will cause considerable attrition as a result of the collisions of crystals with the tubes of the heat exchanger. For these reasons, often the tube bundle is placed outside the crystallizer body using an external heat exchanger, which makes the crystallizer more flexible and robust. The external loop through the heat exchanger requires an additional pump, which gives rise to some additional attrition in the pump, piping, and heat exchanger. The DT crystallizer can be used for both evaporative and cooling crystallization. For cooling applications with low cooling duties, alternative designs with a jacketed heat exchanger exist. The DT crystallizer has the advantage of a well-defined directed flow for optimal mixing at low power input compared with a mixed-tank crystallizer with a pitched blade or a Ruston turbine, at least when large crystals are desired. Reasonably

Figure 7.5 Steady-state distribution of an MSMPR and R-crystallizer (left), an MSMPR and Z-crystallizer (center), and an R-Z crystallizer (right)

203

Herman J. M. Kramer and Richard Lakerveld Table 7.1 Mean Crystal Size Lm, Steady-State Growth Rate G, and Width of the Size Distribution CV for Different Crystallizer Configurations and Different R and Z Values Calculated for an Ammonium Sulfate–Water System with a Residence Time of 4500 s

Type

MSMPR

R

Z

R-Z

Dim

R

1

5

20

1

1

5

–

Z

1

1

1

5

20

5

–

Lm

0.7

1.0

1.5

0.6

0.6

0.8

mm

G

38

53

94

40

40

58

nm/s

CV

0.5

0.49

0.41

0.4

0.39

0.32

–

vapor barometric condensor demister vapor

boiling zone

steam

swirl breaker

draft tube steam

condensate expansion joint

product

Figure 7.7 FC crystallizer with a tangential inlet, Swenson type Source: Adapted from Bennett 2002. Taken with permission from the publisher from Lewis et al. 2015. Copyright © 2015, Cambridge University Press.

condensate product

feed Figure 7.6 Stirred DT crystallizer with an internal heat exchanger Source: Reproduced with permission of the publisher from Lewis et al. 2015. Copyright © 2015, Cambridge University Press.

coarse crystals can be produced, but this crystal size can hardly be manipulated without external devices owing to the simplicity of the design.

Forced-Circulation Crystallizer The forced-circulation (FC) crystallizer is the most widely used crystallizer. Its most common use is in vacuum evaporative crystallization of substances with a flat solubility curve such as sodium chloride and salts with an inverted solubility curve. The FC crystallizer is the least expensive vacuum crystallizer, especially when substantial evaporation is required.

204

feed

The crystallizer comprises two separate bodies (Figure 7.7), which can be designed independently for crystallization and heat input. The crystallizer body should be large enough for vapor release, for accommodation of the boiling zone, and to maintain a sufficiently large liquid volume for retaining the growing crystals until all the generated supersaturation is consumed. A slurry pump circulates the crystal slurry between the heat exchanger and the boiling zone via a tangential, a vertical, or a radial inlet. The heat exchanger is designed for relatively low temperature rises of the slurry per pass to prevent scaling by boiling liquid at the walls of the tube. Each type of inlet affects the performance of the crystallizer in several ways. A tangential inlet, for example, causes a toroidal circulation, and vortexing occurs. A swirl breaker is usually installed at the bottom of the crystallizer body. The tangential inlet enters the crystallizer body where the static height of the liquid above the inlet prevents boiling of the mother liquor in the heat exchanger pipes. Its height is thus directly coupled with the temperature difference over the heat exchanger, and the liquid starts to boil directly at the outlet.

Selection and Design of Industrial Crystallizers

The tangential stream not only causes a macroscopic swirl around the crystallizer, but on top of this circulation a swirl is formed from the outside wall to the inside of the crystallizer, which causes the liquid to mix. The flow pattern induced by the incoming flow of slurry affects the residence time in the boiling zone for crystals of various sizes. For a tangential inlet, the larger crystals almost directly leave the boiling zone. This classification effect owing to the hydrodynamic characteristics results in an apparently slower growth rate of the larger crystals and thus in a narrower CSD. By contrast, circulation of the total slurry by a circulation pump can cause more attrition than a stirrer in a DT or DTB crystallizer, and special pumps must be installed to minimize the negative effect of enhanced attrition on mean crystal size. Usually an (axial-type) centrifugal pump is used. Another often observed effect of a tangential inlet is so-called thermal short circuiting. Fluid entrapped between larger crystals leaves the boiling zone soon after reentering the crystallizer through the recirculation inlet nozzle such that the heat uptake by these trapped fluid packages is not released in the boiling zone, which results in a temperature difference between the top and bottom of the crystallizer body. Sophisticated designs of nozzles try to overcome this problem. With an axial inlet, the temperature short circuiting is much lower than for a tangential inlet. The FC crystallizer with an axial inlet is therefore especially important when the FC crystallizer is operated in combination with mechanical vapor recompression. The FC crystallizer is considered to closely approximate an MSMPR crystallizer because the forced circulation causes a reasonably good mixing of the crystallizer contents. However, in practice, the circulation time can become much larger than the depletion time of the supersaturation. It has therefore been estimated that growth in an FC crystallizer mainly takes place in the boiling zone and that the effective volume for crystal growth is only a fraction of the total volume (Bermingham et al. 1998; Kramer et al. 2000). Compared with the DT crystallizer, the FC crystallizer suffers from enhanced attrition owing to the large circulation flow of the full suspension through the external heat exchanger, which results in smaller crystals. The FC crystallizer can be used as a cooling crystallizer when flash cooling suffices to produce crystals. In this case, the heat exchanger is not needed. The FC crystallizer lacks the flexibility to adapt the mean size of the CSD because of the lack of an option to install a fines-removal loop. Addition of a classifier in the product slurry discharge is, of course, still possible. Often an elutriation leg beneath the crystallizer body is used to classify and wash the product crystals.

The DTB Crystallizer To obtain a coarser product, a DT crystallizer can be extended with a fines-removal and fines-destruction mechanism. An annular zone (Figure 7.8) facilitates the separation of small crystals, which can be dissolved subsequently in an external heat exchanger through a temperature increase or by mixing with a solution that has a concentration below saturation. Such a crystallizer, with a skirt or baffle in order to obtain an annular zone, is called a draft-tube baffle (DTB) crystallizer. In this

vapor

barometric condensor

annular zone fines loop skirt baffle

feed elutriation leg

product

Figure 7.8 Stirred DTB crystallizer with an external heat exchanger and fines destruction Source: Adapted from Bennett 2002. Taken with permission of the publisher from Lewis et al. 2015. Copyright © 2015, Cambridge University Press.

design, the flow through the draft tube has to be directed upward. The feed stream is typically introduced into the draft tube just below the impeller blades. In the DTB design, the total crystal slurry is circulated through the impeller zone, which can cause considerable attrition induced by crystal–impeller collisions. Special attention therefore must be paid to the impeller design to minimize attrition. Usually marine-type propellers with three or five blades are used, although pitched-blade turbine propellers are also encountered depending on the viscosity of the suspension and the crystalline material. To avoid crushing of in particular larger crystals, the clearance between the tips of the impeller blades and the draft tube has to be at least three times larger than the largest crystals. Compared with the DT crystallizer with an external heat exchanger and the FC crystallizers, the DTB crystallizer has the advantage that the flow though the heat exchanger only contains fines and no large crystals, which minimizes the attrition in the external loop. The draft tube is often tapered with the wide end near the top. This reduces the fluid circulation velocity near the liquid surface (and thus reduces air entrainment) and increases the fluid velocity at the bottom of the draft tube (and thus gives a better suspension of crystals at the bottom). The ratio of the diameter of the draft tube and the crystallizer is in the range of 0.4–0.7 depending on the viscosity of the suspension and on the density difference between the crystals and the liquid phase.

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Herman J. M. Kramer and Richard Lakerveld

The design of the fines-removal loop is often quite delicate. An increase in the fines flow increases the number of fines that are removed from the crystallizer but also increases the cut size of the fines. Therefore, care has to be taken that sufficient fines are indeed destroyed before the stream reenters the crystallizer in order to increase the mean size of the product, as predicted in the Z crystallizer model. In practice, the fines removal is often not very efficient, which can be due to incomplete classification in the annular zone or incomplete dissolution of the fines in the heat exchanger. The addition of an elutriation leg at the bottom of the crystallizer (Figure 7.8) allows for some classification of the product. Part of the clear liquid feed flow enters the elutriation leg at the bottom and flows countercurrently with respect to the crystal mass. The smaller crystals are washed back into the crystallizer, and coarser product slurry can be obtained from the side of the elutriation leg. Instead of an elutriation leg, other classifiers with a sharper classification function also can be installed. DTB crystallizers also can be applied as cooling crystallizers by using the heat exchangers as a cooling system or by applying flash cooling. In the former case, the fines in the stream from the annular settling zone are destroyed by heating before the stream is returned to the crystallizer or to a cooler. In the latter case, the stream is mostly first combined with the feed stream to avoid locally induced nucleation that may be caused by evaporation of solvent. The DTB crystallizer is a versatile crystallizer (van Esch et al. 2008), which can be used to produce a broad range of products. The fines-removal option allows for the production of larger crystals and reduces the width of the crystal size distribution, which is often of critical importance in wellmixed crystallizers. Therefore, the DTB crystallizer is frequently used for the production of course products because optimization of the fines removal and dissolution can yield large product sizes (Hofmann et al. 2010). Especially in such cases, care should be taken to avoid cyclic behavior induced by the complex interplay between fines removal and the kinetics of secondary nucleation and growth. In recent patents, it is claimed that the temporary shortage of nuclei, which seems to cause this cyclic behavior, can be avoided in the production of very course crystals in a DTB crystallizer by continuous seeding with seed crystals from a small precrystallizer (Hofmann et al. 2010).

Fluidized-Bed Crystallizer A fluidized-bed or fluidized-suspension crystallizer (Figure 7.9), also called an Oslo crystallizer, is especially designed to produce large and uniformly sized crystals. A hot circulating stream enters the vaporizer chamber, where the solvent is evaporated adiabatically. The supersaturated solution leaves the vaporizer through the downcomer and enters a densely packed fluidized bed of crystals at the bottom of the suspension chamber. The supersaturation is subsequently consumed when the mother liquor flows upward through the fluidized bed. A coarse crystalline product is removed continuously from the bottom of the classified bed. The crystals settle at the top of the bed, and

206

fines loop vapor

down comer steam feed

product

condensate

Figure 7.9 Fluidized-bed crystallizer Source: Reproduced with permission of the publisher from Lewis et al. 2015. Copyright © 2015 Cambridge University Press.

consequently, only the fines leave the suspension chamber together with the depleted mother liquor. The circulating stream is mixed with the feed stream and reheated in a heat exchanger. The fines from the suspension chamber are dissolved to a large extent in the heat exchanger. The fraction of the fines that did not dissolve can still contribute to the growth of crystals in the fluidized bed by agglomeration. In case too many fines are circulated, a draw-off of fines from the settling zone could be installed. These fines could be destroyed, for example, by dilution with solvent before returning this stream to the recirculation line. A major problem in the design of the Oslo crystallizer is the close clearance through which the supersaturated liquid enters the fluidized bed at the bottom of the crystallizer. Such close clearance is needed to generate sufficient liquid velocity to prevent settling of the large crystals at the bottom of the crystallizer. Together with the recirculation rate, this clearance determines the superficial velocity in the bed. Partial blockage of the clearance owing to local scaling of the supersaturated mother liquor (especially in the case of slightly eccentric positioning of the downcomer pipe) or as a result of scaling fragments falling down from the walls will disturb the fluidization of the bed and consequently disrupt the performance of the crystallizer. To minimize occasional production stops for cleaning of the crystallizer because of such blockages, the velocity of the fluid all over the clearance has to be maintained above a certain value. The mean residence time of the crystals can be increased considerably in an Oslo crystallizer versus a DTB or FC crystallizer. Furthermore, attrition of crystals induced by a stirrer or pump is avoided because no large crystals are circulated through the pump. This pump is usually a (axial-type) centrifugal pump. Crystal–crystal attrition in a fluidized bed could, however, be considerable, and in case of, for example, sodium chloride, the crystals from an Oslo crystallizer are clearly rounded off. Nevertheless, the Oslo crystallizer yields a coarse product with a mean size of a few millimeters. For this reason, the Oslo crystallizer is sometimes also referred to as a classifying growth type of crystallizer.

Selection and Design of Industrial Crystallizers

The capacity is in general rather low compared with a DTB or FC crystallizer. One of the reasons is the restriction of the circulation velocity by the requirement that a fluidized bed has to be maintained. In addition, the supersaturation in the downcomer has to be kept low to avoid scaling. A clear liquor advance can be applied for systems with a low natural solid content, removing clear liquor from the settling zone. An Oslo crystallizer can in principle also be used as a flash cooling or cooling crystallizer. In the latter case, the heat exchanger is used to cool the circulating stream, although the low circulation rates could cause severe scaling in the heat exchangers. Because no vapor head is needed, the recirculation pipe is directly connected with the downcomer.

Growth Crystallizer The so-called growth crystallizer operates in the same way as a fluidized-bed crystallizer but allows for a higher capacity. Much higher supersaturation can be maintained in the downcomer compared with the Oslo crystallizer without abundant scaling by recycling a large fraction of the smaller crystals. The supersaturation is partly depleted in the downcomer by the circulating crystals, and the capacity is claimed to be even higher than that of a DTB crystallizer. The circulation loop typically contains 10–60 percent of the total crystal mass, which equals in most cases half the mass of the lower crystal bed. High circulation flows of 1 m/s are generally applied. The lowest part of the crystal bed acts as a so-called spouted bed where the velocities of the crystals decrease with the bed height. This suspended crystal bed extends to the baffle of the outlet for the circulation loop. The crystals are allowed to settle in the section above this outlet, which opens the potential to create a clear zone at the top of the crystallizer body where liquor can be withdrawn for either fines removal or clear liquor advance. The growth crystallizer can easily be used as a cooling crystallizer because of the high circulation velocities.

Spray Evaporative Crystallizer Spray evaporation is often used when waste heat at a low pressure is available and when product quality demands are not high (Figure 7.10). A product is obtained by cooling and evaporation of mostly an aqueous solvent from fine droplets obtained by spraying the slurry into an air stream. Agglomeration is sometimes difficult to avoid because nucleation at the surface of the droplets might be difficult to control. Spray evaporation is a cheap method for the recovery of solids from waste streams.

humidified air

feed l.p.steam air

product

Figure 7.10 Spray evaporative crystallizer

refrigerant vapor

fines loop skirt baffle

settling zone

refrigerant feed product

Direct-Cooling Crystallizer In a direct-cooling or direct-contact refrigeration crystallizer, no heat is withdrawn from the suspension through a cooling surface of a heat exchanger, but a refrigerant is introduced directly into the solution (Figure 7.11). Direct cooling is applied in cases where cooling crystallization has to occur at a very low temperature or when the solute has a strong tendency to scale on the heat exchanger tubes. The refrigerant has to be insoluble in the solvent. Liquid propane and liquid

Figure 7.11 Direct-cooling crystallizerSource: Reproduced with permission from Bennett 2002.

carbon dioxide are examples of refrigerants. The refrigerant absorbs heat from the suspension via evaporation. The vapor leaves the crystallizer similarly as the vapor in a conventional evaporative crystallizer. Therefore, a slightly modified version

207

Herman J. M. Kramer and Richard Lakerveld

overflow. The suspension flows countercurrently to the coolant to maximize the average driving force for cooling.

fines loop

Cascade of Crystallizers

skirt baffle

feed product

coolant

Figure 7.12 Surface-cooling crystallizerSource: Reproduced with permission from Bennett 2002.

of either a DTB or an FC crystallizer can be used for directcooling crystallization. The refrigerant vapor has to be recycled by condensation by recompression and cooling. Several examples exist of organics with a low melting point that are separated from their mother liquor or from an impure melt by this technique.

Surface-Cooling Crystallizers Besides the DTB and the Oslo crystallizer, two other types of surface-cooling crystallizers are often encountered. In one type, a crystallizer body provides the retention time for the growing crystals, and either a slurry or an overflow is circulated through a heat exchanger (Figure 7.12). For crystallization from solution, the slurry is pumped through a tube and shell heat exchanger, with a temperature difference between the tube and the wall in the range of 5–10°C. For melt crystallization, the “clear” liquor overflow is mostly cooled down in a scraped heat exchanger to prevent scaling on the cooled surfaces. Another often applied type of surface-cooling crystallizer is the cooling-disk crystallizer, which is typically applied for crystallization from solution. A cooling-disk crystallizer can be seen as a compact cascade of cooling crystallizers. The crystallizer consists of a trough, which is divided by fixed hollow cooling elements into several compartments (Figure 7.13). Each compartment acts as a separate crystallizer and contains a disk equipped with wipers (typically made from Teflon) and mixing blades. The disks are mounted on a longitudinal axis, which rotates slowly (5–45 rpm). The wipers keep the cooling surfaces clean from scaling. Internal slurry transport proceeds without the need of a pump because both the crystals and the mother liquor flow freely from one compartment to the next via openings at the bottom of the cooling elements. The crystal content increases in the successive compartments and may reach up to 45 vol% in the last compartment. The slow rotation of the disks provides a gentle agitation of the slurry in the compartments. The product leaves the last compartment via an

208

A cascade of crystallizers other than the cooling-disc crystallizer often involves evaporative crystallizers. The purpose is mainly to minimize operating (mainly energy consumption) and investment costs. Mostly, DTB and FC vacuum crystallizers are operated in a cascade of typically three to five crystallizers. Each crystallizer is controlled at a lower pressure and temperature than the crystallizer upstream. In case of, for example, sodium chloride, where crystal production follows from flash evaporation of the aqueous solvent, the heat of condensation of the vapor is partly reused to evaporate the water in the next stage and partly for countercurrent heating of the feed (Figure 7.14). In the case of potassium chloride, often a cascade of five DTB crystallizers is used, and the vapor is only used to reheat the recycled mother liquor. Finally, water can be added for fines destruction. Also, for precipitation, often cascades of simple stirred vessels are used. In this case, a cascade is mainly applied to improve the product quality by dividing the feed stream via more than one inlet stream over the various vessels and by recycling part of the slurry stream. In this way, a higher solid content in the first vessel(s) is maintained, and the supersaturation is suppressed.

7.4.4 Shortcut Design Calculations After selecting the crystallizer type, shortcut calculations are performed to estimate 1. The dimensions of the crystallizer 2. The duty and dimension of the heat exchanger 3. The required circulation flows for the equipment with external heat exchangers 4. The required impellers or pumping duties to establish sufficient mixing and a homogeneous suspension in the crystallizer

Crystallizer Dimensions Crystallizer dimensions are directly related to the residence time of the crystals in the crystallizer and the average growth rate. For a well-mixed crystallizer, the mean crystal size is given by Lm ¼

Gτ 4

ð7:16Þ

At this stage of the design, it is difficult to predict the growth rate in the crystallizer. Usually, a constant growth rate is assumed, for example, from a correlation given by Kind and Mersmann (1989) that gives the growth rate and the relative supersaturation in industrial crystallizers of a number of inorganic salts (Figure 7.15; Mersmann 2001). Alternatively, laboratory experiments can be used to estimate the growth rate of the crystals. However, such data, mostly obtained from small-scale crystallization experiments, must be used with caution because the growth rate itself is affected by

Selection and Design of Industrial Crystallizers Figure 7.13 Cooling-cisc crystallizer with a detail of a cooling element Source: Reproduced with permission of the publisher from Lewis et al. 2015. Copyright © 2015, Cambridge University Press.

product out coolant out coolant in feed

trough rotating disk cooling element coolant inlet

coolant outlet

axis axis cooling element

wipers

rotating disk with wipers

feed

vapor

Figure 7.14 Cascade of four FC crystallizers for NaCl crystallization Source: Reproduced with permission of the publisher from Lewis et al. 2015. Copyright © 2015, Cambridge University Press.

steam

product

the nucleation rate and thus by the residence time and the geometry and the scale of the equipment, as explained in Section 7.4. Using Equation (7.16), the residence time can be calculated from the desired mean crystal size and the assumed growth rate. The volumetric flow rate is determined by the production capacity and the fraction of clear liquid given by ϕv;p ¼

P ð1  εÞρc

ð7:17Þ

The volume fraction of clear liquid or the crystal fraction is a design variable that can be used to optimize the design, making a tradeoff between economics (crystallizer volume), ease of pumping the slurry, and product quality owing to enhanced attrition or breakage at high crystal concentrations. Finally, the slurry volume of the crystallizer is determined by the residence time and the volumetric flow rate of the product stream. Based on the calculated suspension volume for the crystallizer, the basic dimensions of the crystallizer can be

approximated using a simplified model. Assuming a cylindrical crystallizer vessel with a suspension volume V and a fixed liquid height-to-diameter ratio of RHD, the diameter of the crystallizer vessel becomes rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 4V  RHD Dvolume ¼ ð7:18Þ π Especially for evaporative crystallizers, the total vessel height will be higher. For DTB crystallizers, the dimensions of the vessel are enlarged by the presence of a skirt baffle, in which the fine crystals are separated from the large ones (see Section 7.4.3). The diameter of the DTB crystallizer then becomes rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 AAZ þ D2crystallizer DDTB−crystallizer ¼ ð7:19Þ π where AAZ is the cross-sectional area of the skirt baffle or annular zone (Figure 7.15).

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Herman J. M. Kramer and Richard Lakerveld

to the crystallizer when an external heat exchanger is used. In general, boiling can be avoided if the heat exchanger and the inlet of the superheated suspension are placed well below the liquid level in the crystallizer. The static pressure of the liquid should prevent boiling in the tubes. In forced-circulation and fluidized-bed crystallizers, however, the inlet of the circulation flow is situated just below the boiling zone. This poses a constraint on the temperature increase of the suspension in the heat exchanger. For a given heat duty, a minimal circulation flow through the heat exchanger ϕv,circ,min thus needed, given by

Growth rate [m/s] PRECIPITATION (primary nucleation)

10–7

F I

D A

B

C

E

G

H

J

10–8 CRYSTALLIZATION (secondary nucleation)

N Q

ϕv;circ;min ¼

O 10–9

10–2

100

ð7:23Þ

102

Relative Supersaturation Figure 7.15 Growth rate of a number of inorganic salts versus relative supersaturation: (a) KCl, (b) NaCl, (c) (NH2)2CS, (d) (NH4)2SO4, (e) KNO3, (f) Na2SO4, (g) K2SO4, (h) (NH4)Al(SO4)2, (i) K2Cr2O7, (j) KAl(SO4)2, (k) KClO3, (l) NiSO4, (m) BaF2, (n) CaCO3, (o) TiO2, (p) CaF2, (q) BaSO4 Source: Adapted with permission from Mersmann and Kind 1988.

Cross-Sectional Area for Evaporation For evaporative crystallizers, the main constraint is in general the cross-sectional area for evaporation. In order to avoid entrainment of liquid droplets into the condenser, the linear velocity of the vapor flow should not exceed certain limits. A conservative relation is derived for distillation columns as !1=2 ρliquid  ρvapor vmax ¼ Cv ð7:20Þ ρvapor For vapor heads, a value of 0.0244 m/s is recommended for Cv. This then leads to a minimal diameter of the evaporation zone of sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4ϕv;vapor Dvaporhead ¼ ð7:21Þ πvmax The diameter of the crystallizer should be equal to the maximum of the values determined by the Equations (7.18) and (7.21): Dcrystallizer ¼ maxðDvolume ; Dvaporhead Þ

ð7:22Þ

Note that this value will also affect the volume of the crystallizer and thus the residence time. Alternatively, the diameter-toheight ratio can be changed to adapt the crystallizer diameter without changing the volume.

Heat Exchanger Area In evaporative crystallizers, the suspension in the crystallizer is heated. Because the solubility in general will increase with temperature, scaling on the heat exchanger walls is not likely to occur, and the design of the heat exchanger will not be very critical. The only constraint is that boiling should be avoided in the heat exchanger and in the tubing from the heat exchanger

210

Qheat : DThex;max ½ερliquid Cpliquid þ ð1  εÞρcrystal Cpcrystal 

where ΔThex,max is the maximal temperature difference of the process stream between the inlet and outlet of the heat exchanger. In general, a maximal value of 1–2℃ is used for FC crystallizers. A major problem in a cooling crystallizer is the design of the heat exchanger in such a way that encrustation of the heat exchanger wall is avoided. The simplest way to achieve this is to limit the heat flux through the heat exchanger area. For a conventional heat exchanger, a maximal heat flux of 200– 400 W/m2 should be used. Note that this leads to large heat exchanger areas. When wipers or scrapers are used, maximal heat fluxes of up to 1000 W/m2 are allowed. These devices not only prevent encrustation on the wall but also improve the heat transfer coefficient by creating turbulence near the wall. The required heat exchanger area to be used is given by Ahex ¼

Qheat

ϕ

00

ð7:24Þ

hex;max

where Ahex is the heat exchanger area (m2), ϕ00 hex;max is the maximal allowable heat flux (W/m2) [see, e.g., Sinnott (2009) for further reading].

Cross-Sectional Area of the Skirt Baffle The cross section of the skirt baffle must be designed on the basis of the settling velocity of the crystals. A fines cut size can be defined as the size at which the settling velocity is equal to the average superficial velocity of the liquid in the baffle zone. In most cases, compromises must be made for the cut size of the fines to keep the area of the annular zone within acceptable limits. To calculate the settling velocity of the particles, several approaches are available (Barnea and Mizrahi 1973). In the model of Barnea and Mizrahi, the settling velocity is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi32 2 BðxÞ þ BðxÞ2 þ ACðxÞ 5 vss ¼ 4 ð7:25Þ CðcÞ where the parameters of this equation are given in Table 7.2. To calculate the settling velocity, the final crystal concentration in the circulation loop must be known and must be

Selection and Design of Industrial Crystallizers Table 7.2 Parameters for Settling Velocity Equation of Barnea and Mizrahi

A¼

pffiffiffiffiffi α1

ρ

7.5 Instrumentation and Actuation

ρ

α1 ¼ g ρcrystal þρliquid crystal

pffiffiffiffiffi BðxÞ ¼ 0:5α4 α2

α2 ¼ 34 ðρ

pffiffiffiffiffi CðxÞ ¼ α3 α2

liquid

1þð1εÞ1=3 2 crystal þρliquid Þx ε

pffiffiffiffiffiffiffiffiffiffiffiffiffi α3 ¼ 0:63 xρliquid i pffiffiffiffi h α4 ¼ 4:80 ηl exp 56 ð1εÞ ε

estimated. In a fines loop, this concentration is rather low, and a crystal fraction of around 0.01 can be used. The cross-sectional area for the annular zone then can be calculated as follows: AAZ ¼

ϕv;circ vss

ð7:26Þ

Suspension Criterion To meet a suspension criterion, the volumetric flow rate of the slurry that is generated by a stirrer or a pump must be sufficient to keep all the crystals in the crystallizer in a suspended state. This implies that the superficial flow velocity of the solution vsup must exceed the settling velocity vss of a swarm of particles. Settling velocities of suspensions of particles can be estimated using, for example, the model of Barnea and Mizrahi (1973), which is given in Equation (7.25) and Table 7.2. The flow velocity depends on the impeller geometry, crystallizer geometry, and specific power input. Depending on the flow direction, the ratio by which vsup must exceed vss depends on the flow conditions of the suspension, i.e., flow direction and geometry. Mersmann (2001) recommends the following criteria: vsup > 5vss (vertical pipe), vsup > 10vss (horizontal pipe), and vsup ≈ vss (fluidized beds). For large stirred vessels (Dcrystallizer/Lxm > 103), Psp;min ¼ 5ð1  εÞvss g

ρcrystal  ρl ρsl

ð7:27Þ

where Psp,min is the minimal specific power input, 1 − ε is the volumetric holdup of the crystals, g is the gravitational constant, ρcrystal is the crystal density, and ρsl is the slurry density, which is defined as ρsl ¼ ερliquid þ ð1  εÞρcrystal

ð7:28Þ

A criterion for scale-up to large stirred vessels is to maintain a constant stirrer tip speed, that is: vt ¼ πNDimp ¼ constant

ð7:29Þ

where Dimp is the stirrer diameter. For small-stirred vessels (Dcrystallizer/Lm < 500), a constant specific power input Psp,min is recommended as the suspension criterion (Mersmann 2001). The difference between large and small vessels is a result of enhanced particle settlement in narrow geometries. Consequently, it can be said that the larger the vessel, the easier the crystals are kept suspended.

Manipulation of product quality requires the availability of process actuators. The most well-known actuators are product classification, fines removal and destruction loops, heat input or output, and residence time. These process actuators form the handles that can be used to optimize and control the product specifications. Some of these actuators, such as the impeller frequency and fines removal, can make up an integral part of specific crystallizer configurations. Active use of the process actuators to control the quality of the crystals flow produced can therefore influence the choice and design of the crystallizer equipment. An additional iteration might be needed in such a case. Alternatively, process actuation can be achieved in external devices such as external screens and settlers. Earlier, the effects of product classification, fines removal, heat in or output, and residence time were discussed on the basis of the MSMPR model. The impeller frequency is not often used as a process actuator because of the strong and complex interaction between the impeller frequency and process behavior, which makes optimization of certain product specifications almost impossible. The impeller frequency has a direct impact on the mixing, affecting temperature and concentration profiles in the crystallizer. However, it also determines the suspension behavior of the crystals in the mother liquor and the attrition and agglomeration kinetics. Design of the process actuators for the active control requires the prediction of the effects on product properties, and therefore, a detailed process model with a state-of-the-art kinetic model is needed. The design of the instrumentation is closely connected to the ability to monitor the performance of the plant and to control the production rate and quality. These topics are discussed in detail in Chapter 11 of this book. Here only a short discussion is given on the relevant process variables and some indication of measurement techniques. The crucial process variables for crystallization processes are the temperature and concentration of the solute in the liquid phase and the properties of the crystalline phase. Whereas temperature measurement is a standard technique, robust determination of the solute concentration in situ (in the presence of crystals) is still problematic. Most techniques use refractive index, near- or middle-infrared absorption with attenuated total reflection (ATR), or conductivity measurements. The position of the sensor in the equipment, the calibration, and the robustness have to be considered as well. Measurement of particle properties such as crystal size distribution, particle shape, and number of fine particles is also important. Unfortunately, there is no in situ technique that easily allows the robust monitoring of all these properties. Most common techniques include laser diffraction (offline or online with automatic dilution), in situ imaging, focused-beam reflection measurement (FBRM), and ultrasound attenuation.

211

Herman J. M. Kramer and Richard Lakerveld Figure 7.16 Process flow diagram case study Source: Adapted with permission from Bermingham 2003.

7.6 Case Study: Optimization of a Base-Case Design Model-based optimization can be used to further improve the performance of a crystallizer design once a base-case design has been established. In this section, an example based on a case study from the literature is presented (Bermingham 2003). The aim is to give a flavor of typical models and constraints that can be used to obtain more detailed design information and to optimize a base case. The case study involves a 360-m3 DTB crystallizer, which produces 94 kt of ammonium sulfate per year by evaporative crystallization from water (Bermingham 2003). A typical model that is used for modelbased optimization of industrial crystallizers involves component balances, energy balances, a population balance, and possibly a CFD model or compartment model (Kramer 1999) to account for the influence of hydrodynamics. A schematic drawing of the mass and energy flows for the case study is given in Figure 7.16. The mass flow rates around the crystallizer can be estimated based on the approximation of a saturated product flow rate out of the crystallizer and by imposing a maximum solid concentration in the system. The latter is often determined by practical considerations such as the effort needed for pumping or the risk of blockage of piping. These constraints fix the outlet flow rate of the crystallizer at 13.5 kg/s (9.9 kg/s saturated solution + 3.6 kg/s solids). Based on the conditions of the feed flow rate at the beginning of the process (saturated at the temperature of the crystallizer for our case), the inlet flow rate of the crystallizer can be determined (17.8 kg/s) as well as the required heat duty (11 MW) to evaporate sufficient solvent (4.3 kg/s) to achieve the targeted solid production. Finally, the required feed flow rate to the process can be determined by subtracting the feed flow rate to the crystallizer from the recycle flow rate. Note that typically a part of the recycle stream has to be purged to prevent the buildup of impurities in the system, which has been omitted here for the sake of simplicity. The crystallizer has been divided into three compartments, which represent the crystallizer body, annular zone, and fines dissolution compartment. The choice for the number of compartments is a model assumption and depends primarily on the

212

expected internal classification, which can be guided by comparing the half-time for supersaturation decay with the turnover time of the crystallizer (Bermingham et al. 1998). Classification of the crystallizer in various compartments is expected to yield limited improvements compared with a model with a single well-mixed compartment if half-time for supersaturation decay is an order of magnitude larger than the turnover time because process conditions in each compartment will be similar, which is typically the case for operation on a pilot-plant scale. The turnover time inside a crystallizer increases with increasing scale. For our case study, both halftime for supersaturation decay and turnover time are of the same order of magnitude (60s), and improved performance of the model can be expected by adding more compartments. Each compartment is described by component and energy balances as follows: nout nin X dmi X ¼ ϕk wk;i  ϕp wp;i  ϕvap wvap;i ; dt p¼1 k¼1

8t2ðt0 ; tf ; mi ðt0 Þ ¼ mi;0 ; i ¼ 1; 2; …; nC dH ¼ dt

nIN X k¼1

ϕk h k 

nout X

ð7:30Þ

ϕp hp  ϕvap hvap þ Q;

p¼1

8t2ðt0 ; tf ; Hðt0 Þ ¼ H0

ð7:31Þ

where mi is the mass of component i in a compartment of the crystallizer, ϕk are the inlet flow rates, ϕp are the outlet flow rates, wi is the weight fraction of component i in a compartment, H is the total enthalpy, h is the specific enthalpy of the inlet and outlet flow rates, and Q is the heat input into the crystallizer. Note that the component balance can be written for both the liquid and the solid phases. Also note that for determination of the mass flow rates around the crystallizer, as described in Section 7.5, equilibrium between the solid and liquid phases was assumed. However, the derivation from equilibrium drives the kinetic processes that shape the crystal size distribution, which cannot be neglected when kinetic models are used. The dynamic development of the crystal size distribution in each compartment is described by a population balance, which is given by

Selection and Design of Industrial Crystallizers Table 7.3 Design Variables and Decision Variables for the Case Study

Fixed design variable

Value

Crystallizer and feed temperature (°C)

50

Specific heat input (kW/m3)

30

Target mean crystal size (μm)

2000

Decision variables

Lower bound : upper bound

Product residence time (h)

Table 7.4 Optimal Values for the Decision Variables, Constraint Variables, and Equipment Dimensions for the Case Study

Decision variables

Value

Product residence time (h)

9.1

Axial velocity induced by impeller (m/s)

0.50

Fines flow rate over product flow rate (–)

14.7

Superficial upward velocity in annular zone (m/s)

0.50

1.4:14

Residence time in fines dissolver (s)

5.5

Axial velocity induced by impeller (m/s)

0.5:1.5

Constraint variables

Value

Fines flow rate over product flow rate (–)

5:100

Crystal growth rate G (nm/s)

16

Superficial upward velocity in annular zone (m/s)

0.002:0.5

Temperature increase fines dissolution ΔTfines (°C)

20.2

Superficial vapor velocity vvap (m/s)

3.03

Residence time fines dissolver (s)

5.0:1000

Selected equipment dimensions

Value

Volume main body (m3)

328

Source: Adapted with permission from Bermingham et al. 2003.

3

∂ ∂ ½VðtÞnðL; tÞ þ V ½nðL; tÞVðtÞ ¼ BðL; tÞ  DðL; tÞ ∂t ∂t nout nin X X þ ϕk ðtÞnk ðL; tÞ  ϕp ðtÞnp ðL; tÞ; k¼1

p¼1

B0 ðtÞ or LðLmax ; tÞ ¼ 0 GðLmin ; tÞ

2.3 3

Volume of fines dissolution loop (m )

0.81

Diameter of main body (m)

4.67

Diameter of annular zone (m)

4.71

Source: Adapted with permission from Bermingham et al. 2003.

8t2ðt0 ; tf ; 8L2½Lmin ; Lmax ; nðL; t0 Þ ¼ n0 ðLÞ; LðLmin ; tÞ ¼

Volume annular zone (m )

ð7:32Þ

where n is the crystal size distribution, B and D represent, respectively, the birth and death rates of crystals, ϕ represents a volumetric flow rate, B0 is the nucleation rate at the smallest size class, and G represents the crystal growth rate. Note that the boundary condition in Equation (7.32) changes when the growth rate switches sign. The birth and death rate of crystals are typically determined by the net result of various kinetic processes such attrition, agglomeration, dissolution, and breakage. Attrition and dissolution are considered for the case study in this section. Various kinetic models for attrition can be implemented ranging from simple power laws to more detailed mechanistic models such as the Gahn kinetic model (Gahn and Mersmann1999a, b). For the case presented here, the Gahn kinetic model has been implemented to account for attrition as a result of crystal– impellor collisions in the pump compartment (Figure 7.16) and a diffusion-limited dissolution model. A common assumption is complete dissolution of the crystals in the fines-dissolution system, which can be validated by comparing the order of magnitude of the half-time for size reduction and residence time in the fines dissolver. The terms for agglomeration, breakage, and primary nucleation from Equation (7.32) are not taken into account because these kinetic processes are of limited influence for crystallization of ammonium sulfate from water. In daily practice, detailed kinetic models for crystallization are often not readily available. Therefore, care has to be taken when interpreting the

results from model-based simulations and optimizations. However, based on approximate correlations obtained from laboratory or pilot-plant experiments, model-based optimizations often can be used to evaluate, for example, the optimal residence time, the usefulness of fines destruction, and the optimal number of crystallization stages. The model equations that describe the behavior of the crystallizer consist of partial differential algebraic equations (PDAEs), which are often reduced to a set of differential algebraic equations (DAEs) by applying a suitable discretization scheme to the population balance or by applying the moment transformation to the population balance. The latter method yields only a selected number of moments from the crystal size distribution. The fixed design variables and decision variables for the case illustrated in Figure 7.16 are given in Table 7.2. The aim is to produce crystals with a median size of 2000 μm. The objective function to be minimized includes terms for the capital cost of the equipment and operating costs for the stirrer and the circulation pump. Note that the heat duty for evaporation is not relevant for optimization as a result of the fixed production rate. The following constraints are added to the optimization problem: 1. Gmax ≤ 50 nm=s 2. DTfines;max ≤ 20°C ffi qffiffiffiffiffiffiffiffiffiffiffiffi ρ ρ 3. vvap;max ¼ 0:0244 liqρ vap vap

213

Herman J. M. Kramer and Richard Lakerveld

The first constraint represents an implicit constraint on crystal purity because impurity uptake typically increases with crystal growth rate. The second constraint aims to prevent boiling conditions in the fines-removal loop and is determined by operating conditions and the hydrostatic pressure in the fines-removal loop. The third constraint prevents excessive vapor flow inside the crystallizer, which aims to avoid entrainment of mother liquor in the condenser. The results of the optimization of the case study are summarized in Tables 7.3 and 7.4 (Bermingham 2003). The results demonstrate that the fines-dissolution system is not useful for the studied case because two decision variables related to fines dissolution – the superficial upward velocity in the annular zone and the residence time for fines dissolution – move to the boundary of the domain corresponding to minimum fines dissolution (i.e., high cut size in the annular zone and minimum dissolution). On further reducing the targeted mean crystal size to 1000 μm, the model suggests that the impeller would have to be used as an actuator to reduce mean size by generating attrition fragments. For a larger mean size compared with the base case of 2000 μm, fines dissolution has to be used to meet specifications on product size. Furthermore, the optimization study reveals that the vapor velocity inside the crystallizer is at the maximum value, which indicates that the residence time is also constrained (i.e., reducing the residence time would increase the vapor velocity, which would result in entrainment of liquid droplets in the condenser). The constraint on vapor velocity can be relaxed by implementing a number of crystallizers in parallel, which potentially leads to an overall design with reduced cost. Introducing the number of crystallizers in parallel as an additional degree of freedom revealed for the presented case that an optimal number of crystallizers in parallel lies at between four and nine crystallizers. The model and case study briefly presented in this section demonstrate the importance of constraints. Often, model-based optimization will result in constrained values for at least some of the decision variables, and some of the imposed constraints on process variables will be active. Insights into active constraints by themselves is useful for debottlenecking an existing design of an industrial crystallizer.

Nomenclature

214

(cont.)

G

linear growth rate (m s−1)

H

enthalpy (J)

ΔH

enthalpy change for the phase transition (J kg−1)

L

crystal size (m)

Lm

mean crystal size (m)

MT

total crystal mass per unit volume of suspension (kg−3)

P

production rate (kg s−1)

Psp

specific power input (W/kg)

Q

heat flow rate (W)

R

ratio of the fines and product-removal flows (–)

RHD

height–diameter ratio (–)

T

temperature (K)

V

suspension volume crystallizer (m3)

h

classification function (–)

h

specific enthalpy (J kg−1)

kN

nucleation rate constant (kg−1 s−1)

kv

volume shape factor of the crystals (–)

mi

mass of component i (kg)

n

population density (no. m−4)

v

linear velocity (m)

w

mass fraction

z

ratio of the removal rates of large crystals (>Lz) over the other crystals in the crystallizer (–)

Subscripts 0

boundary value

circ

circulation flow

cool

cooling

crystal

crystalline phase

evap

evaporation

eq

at equilibrium

A

area (m2)

hex

heat exchanger

A, B, C

variables in settling velocity equation (Barnea and Mizrahi 1973)

liquid

liquid phase

AAZ

cross-sectional area of annular zone (m2)

p

product stream

B0

nucleation rate (no. m−3 s−1)

ss

of swarm settling

Cp

specific heat (J kg−1 K−1)

sup

superficial

D

diameter (m)

v

volume

E

duty of the process (W)

vapor

vapor phase

Selection and Design of Industrial Crystallizers

Greek

(cont.)

α1, α2, α3, α4

variables in settling velocity equation (Barnea and Mizrahi 1973)

ϕv

volume flow rate (m3 s−1)

ϕ00hex

heat flux (W m−2)

References Alvarez, A., and Myerson, A. S. Crystal Growth Des 2010; 10:2219–28. Barnea, E., and Mizrahi, J. Chem. Eng. J. 1973; 5(2):171–89.

ε

volume fraction of clear liquid (–)

σ

relative supersaturation (–)

ρ

density (kg m−3)

τ

residence time (s)

Kind, M., and Mersmann, A. Ind. Cryst. 1989; 87:101–5.

Plumb, K. Chem. Eng. Res. Des. 2005; 83 (6):730–38.

Kramer, H. J. M., Bermingham, S. K., and van Rosmalen, G. M. J. Crystal Growth 1999; 198:729–37.

Randolph, A. D., and Larson, M. A. Theory of Particulate Processes: Analysis and Techniques of Continuous Crystallization (2nd edn). Academic Press, San Diego, CA, 1988.

Kramer, H. J. M., Dijkstra, J. W., Verheijen, P. J. T., and Van Rosmalen, G. M. Powder Technol. 2000; 108(2–3):185–191.

Bennett, R.C. In Crystallizer Selection and Design: Handbook of Industrial Crystallization, ed. Myerson, A. S. Butterworth-Heinemann, Amsterdam, 2002, pp. 115–140.

Kramer, H. J. M., and Jansens, P. J. Chem. Eng. Technol. 2003; 26(3):247–55.

Bermingham, S. K., Neumann, A. M., Muusze, J. P., Kramer, H. J. M., and Verheijen, P. J. T. Particle 1998; 15(1): 56–61.

Kramer H. J. M., and van Rosmalen, G. M. In Encyclopedia of Separation Science, ed. Wilson, I. D. (vol. 1). Academic Press, New York, NY, 2000, pp. 64–84.

Bermingham, S. K., Verheijen, P. J. T., and Kramer, H. J. M. Chem. Eng. Res. Des. 2003; 81(A8):893–903.

Lakerveld, R.; Kramer, H. J. M.; Stankiewicz, A. I.; Grievink, J. Application of generic principles of process intensification to solution crystallization enabled by a taskbased design approach. Chemical Engineering and Processing 2010, 49 (9), 979–991.

Cisternas, L. A., and Rudd, D. F. Ind. Eng. Chem. Res. 1993; 32(9):1993–2005. Cisternas, L. A., Vasquez, C. M., and Swaney, R. E. AIChE J. 2006; 52(5):1754–69. Gahn, C., and Mersmann, A. Chem. Eng. Sci. 1999a; 54(9):1273–82. Gahn, C., and Mersmann, A. Chem. Eng. Sci. 1999b; 54(9):1283–92. Hofmann, G., Scholz, R., and Widua, J. Producing coarsely grained ammonium sulfate product, useful as fertilizer, comprises, e.g., circulating mother liquor and ammonium sulfate suspension in internal circuit and removing clarified partial flow of solution and dissolving solids, Patent DE102008059754-A1, sWO2010063584-A1, 2010.

Lewis, A., Seckler, M., Kramer, H. J. M., and van Rosmalen, G. M. In Industrial Crystallization: Fundamentals and Applications. Cambridge University Press, Cambridge, 2015. Mascia, S., Heider, P. L., Zhang, H., et al. Angew. Chem. Int. Ed. 2013; 52(47):12359–63. Mersmann, A. Crystallization Technology Handbook (2nd edn). Marcel Dekker, New York, NY, 2001.

van Rosmalen, G. M., and Kramer, H. J. M. Solids Production by Crystallization and Precipitation Processes (Paon Course). Delft University Press, Delft, 2007. Samant, K. D., Berry, D. A., and Ng, K. M. AIChE J. 2000; 46(12):2435–55. Samant, K. D., and Ng, K. M. AIChE J. 2001; 47(4):861–79. Sinnott, R. K. Chem. Eng. Des. 2009. Takano, K., Gani, R., Ishikawa, T., and Kolar, P. Chem. Eng. Res. Des. 2000; 78 (A5):763–72. Thomsen, K., Rasmussen, P., and Gani, R. Chem. Eng. Sci. 1998; 53(8):1551–64. Van Esch, J., Fakatselis, T. E., Paroli, F., Scholz, R., and Hofmann, G. In Proceedings of 18th Symposium on Industrial Crystallization (vol. 2). 2008, pp. 14–17. Wibowo, C., and Ng, K. M. AIChE J. 2000; 46 (7):1400–21. Wibowo, C., Samant, K. D., and Ng, K. M. AIChE J. 2002; 48(10):2179–92.

Mersmann, A., and Kind, M. Chem. Eng. Technol. 1988; 1:264–76.

Wöhlk, W., Hofmann, G., and de Jong E. J. Chem. Ing. Technik. 1991; 63:293–97.

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Chapter

8

Precipitation Processes Piotr H. Karpin´ski Consultant and expert witness Jerzy Bałdyga Warsaw University of Technology

8.1 Introduction Precipitation generally refers to a relatively rapid formation of a sparingly soluble crystalline – or sometimes amorphous – solid phase from a liquid solution phase. Precipitation is rather poorly understood when compared with crystallization of more soluble materials. It generally involves the simultaneous and rapid occurrence of nucleation and growth together with the so-called secondary processes, such as Ostwald ripening and agglomeration. In many cases, these processes are difficult to separate and investigate independently and mechanistically. Precipitation has several important characteristics. First, the “precipitates” are usually sparingly soluble, and their formation occurs under relatively high supersaturation conditions. In this case, precipitation does not depend on the presence of solute crystalline material (i.e., it does not involve secondary nucleation). Rather, it results from homogeneous or heterogeneous nucleation processes. Second, because of the presence of high supersaturation, nucleation plays a major role in the precipitation processes. As a result, a large number of crystals with relatively small sizes are produced. The particle concentration is usually between 1011 and 1016 particles per cubic centimeter, and the crystal size is typically between 0.1 and 100 µm. Third, because of the high particle concentration and small crystal size, the aforementioned secondary processes, such as Ostwald ripening and aggregation, may occur and can greatly affect the properties of the resulting precipitates. Thus the development of colloidal systems must be considered in order to manage these secondary processes to achieve the desirable precipitate quality. Fourth, the supersaturation necessary for precipitation frequently results from a chemical reaction; indeed, precipitation is sometimes referred to as reactive crystallization. Chemical reactions are generally very fast and may involve rapid mixing of concentrated chemical reagents. Thus the role of mixing (both macro and micro) is frequently important in precipitation processes. The mixing effects may be even more significant in the case of an antisolvent precipitation or precipitation based on pH adjustment. Many of these key precipitation characteristics will be discussed further in later sections. Precipitation processes are of great importance in the chemical and process industries. The increasing emphasis on high-added-value specialty chemicals (e.g., dyes, pigments, paints, printing inks, and pharmaceuticals) has also highlighted the important role of precipitation. Many consumer products (e.g., pharmaceuticals, pigments, metal oxides, catalysts, ceramics, and various nanoparticulate materials) are

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produced via precipitation processes. In addition, precipitation has also played an important role in wastewater treatment (e.g., removal of calcium salts and heavy metals) and in reducing emissions to the environment. In most practical applications, the properties of precipitated particles are of key importance, and great attention must be paid to “particle design.” Many crucial properties are physical in nature. The crystal size distribution and crystal habit or morphology can have a major impact on successive downstream processes such as filtration, centrifugation, and drying. Other key particle properties are chemical in nature. Chemical composition, as represented by the chemical purity and impurity levels, must be controlled. Many precipitated products, notably pharmaceutical small organic molecules, can form a range of polymorphs or solvatomorphs (Karpinski 2006). Physical properties of precipitate, along with its polymorphic or solvatomorphic form and chemical purity level, greatly influence postprecipitation particle processing leading to the ultimate quality of consumer products. In most cases, only one specific form is acceptable, and therefore, precipitation conditions need to be carefully investigated and implemented in a controlled manner in order to obtain the desired polymorphic form or hydrate. In addition to particle design, one needs to ensure reproducibility of particle properties from batch to batch. This is of particular importance if the product is subject to statutory or regulatory requirements, as is the case with pharmaceuticals. In industrial applications, significant challenges still exist in scaling up the precipitation processes that involve fast chemical reactions of concentrated reagents or other means of creating very high supersaturation levels. Some of these practical aspects of precipitation will be discussed further in later sections. Up to the present, major attempts were directed at controlling the product particle size distribution (PSD), and a good number of research efforts have been devoted to the development of mathematical models for PSD prediction. As in most precipitation devices, the flow and mixing are turbulent, and there is a real challenge to predicting interactions of turbulence with particle-formation phenomena, which define the closure problem for precipitation. Despite its practical importance and the development of several theoretical approaches for process interpretation and related modeling methods, precipitation still remains a relatively underdeveloped field. This is reflected in the small number of books devoted specifically to precipitation. Two specific books were published in the 1960s – one by A. E. Nielsen

Precipitation Processes

(Kinetics of Precipitation, 1964) and the other by A. G. Walton (The Formation and Properties of Precipitates, 1967). Walton’s book tends to focus on the materials science aspects of precipitation and precipitates, whereas Nielsen’s book focuses on the mechanistic and more process-oriented aspects. Nielsen also covers some of the mass-transport aspects of fluid mechanical mixing of reactive solutions. Much of the content of these books is now outdated, but these books still serve as useful introductions to the topics covered. A more recent addition to the literature of precipitation is the book by O. Söhnel and J. Garside (Precipitation, 1992). It places considerable emphasis on developing a rigorous theoretical background to the kinetics of precipitation as well as attempting to show how this might be applied to industrial operations. The complex interrelationships between mixing and precipitation product quality are discussed in the Crystallization Technology Handbook, 2nd edition, edited by A. Mersmann (2001). The aforementioned closure problem has been discussed by J. Bałdyga and J. R. Bourne (Turbulent Mixing and Chemical Reactions, 1999). Precipitation with supercritical fluids (SCFs) used as either solvents or antisolvents is presented in Supercritical Fluid Technology for Drug Product Development, edited by P. York, U. B. Kompella, and B. Y. Shekunov (2004). Handbook of Industrial Mixing, Science and Practice, by E. L. Paul, V. A. Atiemo-Obeng, and S. Kresta (2003), touches on the industrial aspects of precipitation. Applications of precipitation technologies to active pharmaceutical ingredient (API) nanoparticle production are reviewed by J. M. Rowe and K. P. Johnston [chapter 12 in Formulating Poorly Water Soluble Drugs, edited by O. O. Robert et al. (2012)].

8.2 Physical and Thermodynamic Properties 8.2.1 Driving Force for Precipitation To begin consideration of the driving force for precipitation, one should start from the thermodynamic Gibbs phase rule and the application of phase diagrams. Phase diagrams are constructed based on measured or predicted solubility data, including information on number and nature of the possible crystallizing phases. They provide information on the possible process yield and help to define variables that can be used to control the process. A more detailed description of the phase rule and phase diagrams, using a specific example of antisolvent precipitation of solutes with CO2, is discussed in Section 8.2.5 and in Bałdyga et al. (2010b). Phase diagrams are maps that can show the process starting point and the final point after an infinitely long time but cannot show how fast the process is. Precipitation is a kinetic process, and the rate of this process depends on the driving force, which is an appropriately defined difference between properly defined variables. Davey and Garside (2000) define the driving force for crystallization/precipitation by comparing the actual and equilibrium compositions or supersaturations on a molecular scale. Thus the supersaturation σ can be defined as the dimensionless difference in the chemical potential between a molecule in the equilibrium (saturated) state μeq and a molecule in its supersaturated state μss

σ¼

μss  μeq

ð8:1Þ

kB T

Replacing the chemical potentials of the molecule with molar potentials and thus the Boltzmann constant kB by the universal gas constant R, a change in the molar chemical potential (the molar Gibbs free energy) from µ1 in supersaturated solution to µ2 in the crystalline state (equal to the potential in saturated solution) can be expressed using solute activities ai based on the activity-defining equation dμi ¼ RT d ln ai

ð8:2Þ

as follows: σ¼

μ  μeq RT

¼ ln

a aeq

ð8:3Þ

where µ and a are the actual molar chemical potential and the actual activity of the solute in solution, respectively, and T is the absolute temperature. Thus Dμ ¼ μ  μeq ¼ 0 defines equilibrium: for Δµ > 0, spontaneous crystallization may occur, and Δµ < 0 means that an opposite transformation (i.e., dissolution) becomes spontaneous. The sign of Δµ determines the direction of the possible phase transformation, whereas the absolute value of Δµ signifies the distance from the equilibrium. Δµ/RT represents the thermodynamic driving force for crystallization, and both the rate of nucleation and the rate of crystal growth depend on Δµ/RT. Because the chemical potential of the dissolved solute is directly related to its activity, the driving force can be expressed using the activity coefficients in a simpler form than Equation (8.3):
 a  aeq x  xeq a σ ¼ ln ≈S  1 ¼ ð8:4Þ ≈ Sa  1 ¼ aeq aeq xeq Equation (8.4) defines the activity-based and concentration (x)–based supersaturation ratios Sa and S, respectively, and is valid for small (a  aeq ) difference and – in the case of ideal solutions – small (x  xeq ) difference. In chemical engineering applications, concentration is often expressed in the units appropriate to the specific application, e.g., molar concentration c: Sa ¼

ac c ≈S ¼ ceq ac;eq

ð8:5Þ

where ac represents the activity of the dissolved component of concentration c, ceq is the equilibrium solubility, and Sa ≈ S is the supersaturation ratio, which is also known as relative supersaturation or saturation ratio. Supersaturation is also sometimes expressed as a concentration difference Dc ¼ c  ceq

ð8:6Þ

The solution is supersaturated when Sa > 1, S > 1, Δc > 0, or σ > 0. For electrolyte, which dissociates giving νþ cations and ν anions, ν ¼ νþ þ ν , and

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Piotr H. Karpiński and Jerzy Bałdyga

" #1=ν ν ν ðaνþ 1 ðaνþ þ a Þ þ a Þ ¼ ln νþ ν ¼ ln νþ ν σ¼ ¼ lnSa ν ðaþ a Þeq νRT ðaþ a Þeq μ  μeq

ð8:7Þ

Sa ¼

P Ps

1=ν

ð8:8Þ

to define both the ionic activity product Π and the thermodynamic solubility product Πs. For dilute solutions, activities are often replaced by concentrations in both Π and Πs. Equations (8.7) and (8.8) are valid for stoichiometric ion concentrations in the solution as well as when there are inert electrolytes having a common ion with the crystallizing compound. Again, in practice, supersaturation is sometimes expressed as σ = Sa – 1 or as a “concentration” difference Dc ¼ ½P1=ν  P1=ν s 

ð8:9Þ

Notice that Equations (8.7) to (8.9) should be combined with the relations describing equilibrium in the solution to account for the effect of solution composition on both solubility and driving force. The following two examples will illustrate these effects. At first, let us consider the effect of solution composition on the solubility and supersaturation of titanium hydrous oxide TiO(OH)2. At saturation, the following equilibria should be considered:

þ TiO2þ eq þ 4H2 O ⇄ TiOðOHÞ2;eq þ 2H3 O

ð8:10Þ

with ½TiOðOHÞ2 eq ½H3 Oþ 2 ½TiO2þ eq

¼

8 K1;0 Ks 4pOH 8Ks K8;12 þ 10 Kw2 Kw12

Defining the supersaturation ratio by ! ½TiO2þ ½OH 2 S¼ Ks

ð8:17Þ

ð8:18Þ

the effect of solution composition [expressed by the concentration of titanium(IV) and pOH] on the solubility and supersaturation can be depicted as in Figure 8.1. There are regions of strong dependence of these composition variables on pH and regions of their strong dependence on titanium concentration, which is typical for weakly dissociated species. The second example is concerned with the application of a thermodynamic model to precipitation of BaSO4, which is characterized by a much higher level of dissociation than TiO (OH)2. Vicum et al. (2003) evaluated the available thermodynamic models for aqueous Ba2+–SO42−–Na+–Cl− solutions at different levels of complexity. One should remember that the model choice depends mainly on the value of the ionic strength I, that is, X I ¼ 0:5 mi zi2 ð8:19Þ i

For the formation of ion triplets:

K1;0 ¼

 ½TiðIVÞsol ¼ ½TiO2þ eq þ 8½ðTiOÞ8 ðOHÞ4þ 12 eq þ ½TiOðOH Þ2 eq

¼ 102pOH Ks þ

which can be rearranged as


The total concentration of titanium(IV) in the solution can now be expressed as

½TiOðOHÞ2 eq Kw2 ½TiO2þ eq ½OH 2

where mi and zi represent the molality of ion i and the charge on ion i, respectively. The dependence of solubility on composition can be expressed using the thermodynamic solubility product

ð8:11Þ

where Kw ¼ ½H3 Oþ ½OH 

ð8:12Þ

For the formation of complex ions ½ðTiOÞ8 ðOHÞ4þ 12  (Einaga and Komatsu 1981): 4þ þ 8TiO2þ eq þ 24H2 O ⇌ ðTiOÞ8 ðOHÞ12;eq þ 12H3 O

ð8:13Þ

with K8;12 ¼

þ 12 ½ðTiOÞ8 ðOHÞ4þ 12 eq ½H3 O 

½TiO2þ 8eq

¼

12 ½ðTiOÞ8 ðOHÞ4þ 12 eq Kw

½TiO2þ 8eq ½OH 12 ð8:14Þ

For precipitation equilibrium: TiOðOHÞ2;eq ⇌ TiOðOHÞ2;ppt with

218

 2

þ 2

Ks ¼ ½TiO s ½OH  ¼ ½TiO s ½H3 O  2þ

2þ

ð8:15Þ Kw2

ð8:16Þ

Figure 8.1 Precipitation diagram for titanium(IV) (titanium hydrous oxide) at 20°C Source: Based on Bałdyga and Jasińska 2005.

Precipitation Processes

KSP ¼ aBa2þ ðaqÞ aSO4 2 ðaqÞ ¼ mBa2þ mSO4 2 γBa2þ γSO4 2 ¼ mBa2þ mSO4 2 γ2

ð8:20Þ

where γi is the activity coefficient of the ionic species i, and γ± represents the mean ionic activity coefficient. The mean ionic activity coefficient γ± can be calculated using semiempirical forms of the Debye–Hückel law, such as the model by Bromley (1973), applicable for ionic strengths up to 6 M, and the Pitzer (1991) model, which includes ion–pair and ion–triplet interactions and is also valid for higher values of ionic strength. Dissociation is not complete, so the activity coefficient models should be used together with the equilibrium between ions and undissociated ion pairs (complexes), as expressed below by the corresponding dissociation (ionization) constant: Ka ¼

mBa2þ mSO4 2 γ2 mBaSO4 ðaqÞ γBaSO4 ðaqÞ

ð8:21Þ

The corresponding driving force SA can be expressed, based on the definition [Equation (8.8)], by
 mBa2þ mSO4 2 0:5 SA ¼ γ ð8:22Þ KSP The impact of the composition, including its effect on complex formation, is presented in Figure 8.2. Close to the stoichiometric conditions, complex formation has a significant impact on the supersaturation ratio only at high concentrations. For nonstoichiometric conditions, a major decrease in supersaturation is caused by the complex formation, particularly at high solution concentrations.

8.2.2 The Gibbs–Thomson Equation and Surface Energy The Gibbs–Thomson equation links crystal size with the equilibrium solubility. For crystal–solution equilibrium, the solur bility Ceq of spherical crystals of radius r is related to the ∞ solubility Ceq of crystals of very large sizes by the following equation (Enustun and Turkevich 1960): ! r Ceq 2σVm ð8:23Þ ln ∞ ¼ Ceq RTr where σ is the surface energy of the solid phase, and Vm is the crystal molar volume. According to Equation (8.23), the solubility of a substance increases with decreasing crystal size. Principally, the surface tension forces between the crystal and the solution cause this larger solubility of small crystals. There is little unambiguous experimental evidence confirming the validity of the Gibbs–Thomson equation for crystal–solution equilibrium. Observations carried out for some sparingly soluble substances represent one piece of evidence. For example, the solubility of AgBr crystals of 0.05 µm size is about 15 percent higher than that of large crystals (Berry 1976). In general, a significant increase in solubility is only apparent for crystals smaller than about 1 µm.

Figure 8.2 Precipitation diagram for BaCl2 and Na2SO4 in aqueous solution at 25°C Source: Based on Vicum et al. 2003.

The concept of the surface energy of a solid phase is somewhat controversial. The experimental methods that have been suggested for determination of the solid-phase surface energy are frequently ambiguous. Söhnel and Garside (1992) gave a summary of the energy parameters of the surface under consideration. They also showed the values of the experimentally determined surface energy as a function of equilibrium solubility for many solid substances that covered a solubility range of approximately eight orders of magnitude. The influence of the Gibbs–Thomson equation and the surface energy can be important for sparingly soluble substances. For example, on Ostwald ripening, very small crystals (which are more soluble than larger ones) dissolve preferentially, and the dissolved material, which forms a supersaturated solution with reference to the coarse crystals, is consumed on the growth of the latter. Thus the large crystals grow at the expense of the small ones. The Ostwald ripening process will be discussed further later.

8.2.3 Precipitation Diagrams As already indicated, practically important information on the type of precipitate that results from given reaction conditions can be found from precipitation diagrams. The precipitation diagram of a sparingly soluble solid depicts the regions of concentration, pH, temperature, or other property where exists a solid phase of uniform composition, morphology, color, or crystal habit (Söhnel 1991). For example, Figure 8.3a shows an idealized precipitation diagram in p[Me+] = – log[Me+] and p[X−] = −log[X−] coordinates, where [Me+] and [X−] are the cation and anion concentrations. A straight line bisecting the diagram, a so-called equivalence line, represents

219

Piotr H. Karpiński and Jerzy Bałdyga

(a)

Figure 8.3 (a) Idealized precipitation diagram. Zones: 1 = undersaturated solution; 2 = metastable supersaturated solution; 3 and 4 = heterogeneous nucleation; 5 = homogeneous nucleation. Lines: a = equivalence line (stoichiometric composition); b = solubility line; c = metastable zone boundary; d and e = zone boundaries separating zones 3, 4, and 5. (b) Precipitation diagram with nonlinear boundaries separating the regions delineated in part a. Zones: 1–4 as in part a; 5 = transient precursor formation; 6 = positively charged colloids; 7 = negatively charged colloids. Source: After Söhnel 1991.

(b)

stoichiometric compositions for the compound MeX, that is, [Me+] = [X−]. Equations for a constant ionic product [Me+] [X−] = constant or p[Me+] + p[X−] = constant are satisfied along any line perpendicular to the equivalence line. However, the ratio [Me+]:[X−] changes along the former. Thus, if line a represents the equilibrium solubility, lines b and c represent the boundary of the metastable region and the boundary between heterogeneous and homogeneous nucleation. In many cases, precipitation does not follow a simple stoichiometric reaction such as Me+ + X− → MeX

(8.24)

but proceeds through formation of polymeric species that transform to one another with various rates. The actual solidphase composition is then time dependent and changes during the initial part or even the entire system lifetime. If the ion complexation occurs, that is, if the total analytical concentration of the anion is expressed by [X] = [X−] + [MeX] + [MeX2−] + …

(8.25)

then the boundaries separating the regions take on the form of curves, as depicted schematically in Figure 8.3b. Area 5 may represent the region of precursor formation (Tanaka and Iwasaki 1983, 1985), whereas islands 6 and 7 may represent the regions of stable negatively and positively charged colloidal suspensions. Regions 5, 6, and 7 may or may not appear in the precipitation diagram of a specific compound.

8.2.4 Surface Chemistry and Colloidal Stability Electrostatic interactions between colloidal particles may affect or even control the stability of colloidal suspensions and particle aggregation. Particles dispersed in aqueous media are usually charged and interact with ions present in the aqueous solution. The attraction forces tend to place the oppositely charged ions (i.e., counterions) close to the charged surfaces, whereas the thermal motion of counterions tends to randomize them, which constitutes the electrical double layer consisting of

220

the charged surface and the counterion-rich charged liquid layer. In many applications, control of the surface charge is needed to induce aggregation or deposition or to stabilize the suspension. This is not possible without identifying and understanding the mechanism of creation of the surface charge. The first important mechanisms to be considered here is nonstoichiometric dissolution of potential-determining ions – resulting from a difference in affinity of ions for the two phases – and preferential adsorption of ions from the solution. A simple relation between the concentration of the potentialdetermining ions and the surface potential Ψ can be expressed, using the Nernst equation, by Ψ ¼ Ψ0 þ

kB T RT lnðai Þ ¼ Ψ0 þ lnðai Þ zi e zi F

ð8:26Þ

where the following two substitutions were used: R = kBNA and F = eNA; and where e is the elementary charge, zi is the charge on the ion (valence), kB is the Boltzmann constant, and F is the Faraday constant. For sparingly soluble materials, and thus low electrolyte concentrations, the ion activity ai can be replaced by the ion concentration ci. At point of zero charge (PZC), or isoelectric point, when ci ¼ ci;iso , Ψ= 0 and the constant Ψ0 in the preceding equation can be expressed in terms of ci;iso . Equation (8.26) thus becomes

 kB T ci RT ci ln ln Ψ¼ ¼ ð8:27Þ zi e zi F ci;iso ci;iso For a simple ionic reaction Aþ þ B → Csolid , with the solubility product Ks ¼ cA cB and zA ¼ 1, zB ¼ 1,

 kB T cA kB T cB ln ln Ψ¼ ¼ ð8:28Þ e e cA;iso cB;iso At the isoelectric point, the surface concentrations of both ions are equal to each other, whereas the concentrations cA,iso and cB,iso usually differ significantly. For example, for AgI at 25°C,

Precipitation Processes

cI ;iso = 2.5 × 10−11 mol/dm3 and cAgþ ;iso = 3.0 × 10−6 mol/dm3, which demonstrates that I− anions adsorb on the solid surface much more easily than do the Ag+ cations, and KS = 7.5 × 10−17 mol2/dm6 (Stokes and Evans 1997). Equation (8.26) is the first step in determining how the ion concentration and the electrical potential vary with distance from the solid surface and how the surface charge is related to the surface potential. This can be resolved using the doublelayer model (DLM), which has been developed for an infinite, flat, solid interface with uniformly distributed surface charge and potential contacting a liquid medium with uniform properties. The local number concentration ni of ion i is related to the local potential by the Boltzmann equation
 zi eΨ ni ¼ ni0 exp  ð8:29Þ kB T For Ψ ¼ 0, ni is equal to its bulk concentration ni0 . The spatial distribution of the potential depends on the distance x from the surface according to the Poisson–Boltzmann equation, presented here for symmetrical z–z electrolytes
 d2 Ψ 2zen0 zi eΨ sinh ¼ ð8:30Þ dx2 kB T χ

where χ represents the dielectric permittivity, and n0 is the number concentration of each ion. Solving Equation (8.30) gives distribution of both Ψ and ni from Equation (8.29). The complete solutions of Equation (8.30) can be found in the book by Elimelech et al. (1995); a useful approximation, valid for zeΨ Gcrit, then renucleation is inevitable, the population of crystals may increase even by orders of magnitude during the course of the growth stage due to renucleation, and a bimodal (or polymodal) CSD could result. The simplified overall (total, integral) material balance of batch precipitation states that the mass increase due to the growth of precipitate crystals of molecular weight M from initial size L0 to an arbitrary size L in an arbitrary time t is equal to the mass of solute in the solution of volume V delivered by the equimolar double feed whose molar concentration is CR NkV ρðL3  L30 Þ ¼ VMCR

ð8:145Þ

where kV is the volumetric shape factor, constant for a given morphology, and ρ is the crystal density. The number of individual crystals N fixed in the nucleation stage or supplied by seeding remains constant throughout the growth of the precipitate. The material balance equation [Equation (8.145)] may be differentiated with respect to time to give 3NkV ρGL2 ¼ QMCR

ð8:146Þ

where Q ¼ dV=dt denotes the equimolar volumetric flow rate of reagents. One can proceed with the differentiation of Equation (8.146) until all the quantities but Q ¼ QðtÞ become constant. After the two subsequent differentiations, d2 Q 6NkV ρG3  ¼0 dt 2 MCR

ð8:147Þ

Equation (8.147), whose second term on the left side is now constant, is a second-order ordinary differential equation (ODE). In order to solve this ODE for Q(t), two initial conditions are necessary. They come in naturally by realizing that at time t = 0, the size of crystals is L = L0 (seeds or nuclei size). The first initial condition is obtained from Equation (8.146) as Qð0Þ ¼ Q0 ¼

3NkV ρGL20 MCR

ð8:148Þ

and the second initial condition is dQð0Þ 6NkV ρL0 G2 ¼ dt MCR

ð8:149Þ

The solution to the ODE [Equation (8.147)] subject to the initial conditions (8.148) and (8.149), yields the sought-after growth versus time equation (growth ramp or growth profile) Q¼

3NkV ρ 2 ðL G þ 2L0 G2 t þ G3 t 2 Þ MCR 0

ð8:150Þ

Equation (8.150) is parabolic in t. For a nonzero initial size, it does not reduce to Q = 0 at t = 0. This is quite understandable because the initial crystals whose size is L0 are able to accept the

growth material at a volumetric rate equal to that given by Equation (8.148). The latter equation determines the magnitude of the initial flow for the first moment of the growth stage of batch precipitation needed to ensure G = constant. The significance of Equation (8.150) lies in the fact that it prescribes the reagent flow rate at any moment of the growth of precipitate crystals, provided that the number and size of the seeds used are known or, alternatively, that the number and size of the nuclei population at the beginning of the growth stage of precipitation are known from the previous experiments. In the latter case, by knowing the size of the final precipitate and the number of moles precipitated, one can determine the number of individual crystals that were formed as a result of the particular nucleation procedure employed in the process. Equation (8.150) may be rearranged to the form (one may find it more convenient to use)
 nn G 2G2 t G3 t2 þ 2 þ 3 Q¼ ð8:151Þ L0 CR L0 L0 where nn denotes the number of moles of solute used in the nucleation stage. In order to use Equation (8.151) in programming Q versus t, it is necessary to substitute the moles of solute used in the nucleation stage nn and the size of crystals after nucleation L0 = Ln. The former is known for a given process, and the latter can be measured by an online or offline particle sizer. The manipulative variables in Equations (8.150) and (8.151) are the growth rate G, the reagent concentration CR , the total growth time tf , and the final size of the precipitate crystals Lf . There are always limiting values for these variables. The maximal reagent concentration is limited by the solubility of the least soluble reagent. In addition, the total growth time tf and the minimum value for CR should, for practical reasons, be such that the product cycle time is minimized. G, tf , and Lf are related as follows: G ¼ ðLf  L0 Þ=tf < Gcrit. This relationship and the preceding discussion impose certain limitations on the achievable final precipitate size Lf . If the dependence of the crystal size on Gcrit is known, the growth time may be further optimized (shortened) by the use of two or more growth segments, each having its own constant growth rate Gi < Gcrit,i.

8.8 Scale-Up Rules for Precipitation There is a general agreement that the history of supersaturation and the history of supersaturation structure determine the PSD and the crystal morphology in precipitation [see the discussion in Bałdyga and Bourne (1999, chap. 15) and here in Sections 8.5.3 and 8.7]. The prediction of supersaturation is possible with the use of suitable micromixing models, a CFD technique, and closure schemes. The design of the industrial plant and equipment is often based on scale-up procedures that are not obvious in the case of precipitation processes – posing an important and challenging task for the chemical engineers. The processes of mass, momentum, and heat transfer – and the resulting concentration, stress, and temperature distributions, as well as the properties of populations of particles present in the system – are scale dependent and portrayed with very complicated models, which underscores the scale-up problem.

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Piotr H. Karpiński and Jerzy Bałdyga

To scale up any process, precipitation included, one can investigate the process in stages, at several increasing scales, by performing laboratory and pilot-scale experiments and in this way obtain information that enables process extrapolation to a larger scale using the principles of similarity. In practice, an industrial-scale crystallizer–precipitator is not geometrically similar to the laboratory equipment. Also, the flow and property distributions are not expected to be similar at both scales, but instead, the process has to be designed in such a way that the most important product properties obtained at either scale are almost identical. Extrapolation steps up to and including the industrial scale are usually based on the principles of similarity that were developed using the procedures of dimensional analysis. According to Zlokarnik (2002, p. 43), such procedures are “expensive and basically unreliable.” Dimensionless similarity criteria can be derived using the available governing equations for mass, momentum, species, energy, and population balances; when they are not available, the Buckingham PI THEOREM can be used instead (Zlokarnik 2002). In geometrically similar systems, a complete similarity occurs if all necessary dimensionless criteria either derived from the differential equations or obtained using the pi theorem are equal. In complex precipitation processes, such complete similarity is impossible. We usually want to reproduce product quality (particle size, particle morphology), mixture composition, and the structure of the suspension at the larger scale because we want to obtain identical products from the systems differing in scale. Thus we settle with a partial similarity, which means that we lose several degrees of freedom (because we cannot simultaneously manipulate particle size, solution composition, viscosity, and diffusivity), and we obtain in this way a reduced number of the most important similarity criteria for the process (York et al. 2004). Furthermore, one needs to abandon deliberately some similarity criteria that are in conflict with other, more important ones; the decision as to the choice of a less important criterion to be neglected is based on our experience, intuition, modeling, or – preferably – welldesigned experimental procedures. In what follows, our considerations are based on the scaleup rules, the CFD simulations, and the experimental data for a selected precipitation process. The scale-up rules are developed based on time-constant analysis. The relevant time constants are presented in Section 8.5.3 [see Table 8.1 and Equations (8.105)–(8.114)]. To scale up or down a process, one needs invariability of all characteristic times: τ ¼ idem, τ C ¼ idem, τ S ¼ idem, τ D ¼ idem, τ E ¼ idem, τ G ¼ idem, τ R ¼ idem, τ N ¼ idem, and τ Ccr ¼ idem, which is practically impossible in the case of complex processes. Let us consider, as an example, the MSMPR crystallizer. There is no reaction, and mixing is ideal, so we need only identical values at scales 1 and 2 for the mean residence times τ1 ¼ τ2 and identical time scales for nucleation and growth, respectively, τ N1 ¼ τN2 , τCcr1 ¼ τCcr2 . For the growth and nucleation rates, which depend only on supersaturation, the scale-up rule reduces to τ1 ¼ τ2 and cP01 ¼ cP02 , where cP0 denotes the feed concentration of the precipitating substance P.

254

Figure 8.37 The structure of the flow in the SEDS particle formation vessel, represented by the stream lines Source: Reprinted with permission of the publisher from Baldyga et al. 2010a. Copyright © 2010, Elsevier.

In the case of the more complex SEDS process [presented in Section 8.7.2 and described in detail by Bałdyga et al. (2010a)], the scale-up problem is more complicated. The precipitation vessel described in Section 8.7.2 can be interpreted as a pipe-inpipe system, with the outer tube fluid velocity equal to zero. Instead of the outer tube, we have in fact the precipitation vessel. Similarity of the flow pattern is characterized by the Curtet number (Becker et al. 1963; Craya and Curtet 1955) d0 =DV Ct ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  0:5ðd0 =DV Þ2

ð8:152Þ

where d0 is the nozzle orifice diameter, and DV is the diameter of the precipitation vessel. The Curtet number Ct characterizes turbulent self-entrainment. Flow similarity is ensured when Ct is maintained during scaling-up, which means that d0 =DV is kept constant at all scales. The flow induces a recirculation eddy, as shown in Figure 8.37. The eddy induces backmixing, and the exhausted residual fluids dilute the fresh supersaturated solution. This significantly reduces the nucleation rate, and the supersaturation is then unloaded mainly owing to particle growth. To characterize the flow and mixing in geometrically similar systems (precipitation vessels with coaxial nozzles) with the identical physical properties, one can use two characteristic scales, the first for the characteristic length scale and the second for the characteristic velocity in the system. The proper length scale is the jet nozzle diameter d0, and the proper velocity scale in the system under consideration is the nozzle velocity u0. Then both the mean residence time τ and the characteristic convection time τC can be expressed by τ ∝ τC ¼

d0 u0

ð8:153Þ →

Hence the age of a fluid element at position x scales with the characteristic convection time τC ¼ d0 =u0 , and the fluid → elements should have the same dimensionless positions x =d0 after the same dimensionless time period t=τ C . At very high values of the Reynolds number, Re ¼ d0 u0 =ν, the rate of energy dissipation per unit mass scales as →

εð x =d0 Þ ¼

u30 → → f ð x =d0 Þ ¼ ε0 f ð x =d0 Þ d0

ð8:154Þ

whereas the kinetic energy of turbulence scales as →

k ¼ u20 gð x =d0 Þ

ð8:155Þ

Precipitation Processes →

→

where functions f ðx =d0 Þ and gðx =d0 Þ represent the dimensionless distributions of the rate of energy dissipation and the kinetic energy of turbulence, respectively. To estimate the time constants in Equations (8.108) and (8.109), one assumes that the velocity fluctuations scale as u0 and the integral scales as d0. The time constant for mesoscale mixing by turbulent diffusion can be expressed by τD ¼

Q u0 d2 d0 ∝ 1=3 0 4=3 ¼ uDt u0 ε d u0 0

Table 8.2 Physical Dimensions of SEDS Equipment

Scale

Laboratory

Pilot

SMP

V (m3)

5.0 × 10–5

2.0 × 10–3

1.0 × 10–2

d0 (m)

2.0 × 10–4

4.0 × 10–4

9.0 × 10–4

L (m)

0.32

0.40

1.05

ð8:156Þ

0

0

0

with Dt ∝ u L ∝ ðεd0 Þ1=3 d0 and ε ∝ ðu Þ3 =L ∝ u30 =d0 . Similarly, the time constant for mesoscale mixing by the inertial-convective fluctuations is 2=3

τS ∝

L2=3 d0 d0 ∝ ∝ 1=3 1=3 3 u0 ε ðu0 =d0 Þ

ð8:157Þ

The time constants for the viscous-convective and viscousdiffusive micromixing processes scale as ε1=2 [see Equations 1=2 3=2 (8.79) and (8.105)}, which means that τE ¼ 1=E ∝ d0 =u0 1=2

3=2

and τG ¼ 1=Gm ∝ d0 =u0 when the viscosity ν and the molecular diffusivity Dm (Sc ¼ ν=Dm ) have the same value in both scales considered. Hence the ratio of any mesomixing and micro1=2 mixing time-constant scales as Re0 , where Re0 ¼ u0 d0 =ν, for 1=2

example, τS =τE ¼ u1=2 d0 =ν1=2 , and this means that at large Reynolds number values, τS ≫ τE , with mesomixing controlling the process and negligible micromixing effects. The convection time constant and both mesomixing time constants all scale as d0 =u0 , so the spatial profiles of the passive tracer concentration and the intensity of segregation in any apparatus cannot be altered just by changing the flow rate at high Re0. The analysis of the time constants leads to the following conclusions (Bałdyga et al. 2010a): 1. For developed turbulence, characterized by the high Re0 values, when micromixing effects are negligible, the scaleup criterion is ðd0 =u0 Þ1 ¼ ðd0 =u0 Þ2 ; introducing this criterion to the throughput ratio Q2 =Q1 ¼ ðuo d02 Þ2 =ðuo d02 Þ1 ¼ X , one gets ðu0 Þ2 =ðu0 Þ1 ¼ X 1=3 and ðd0 Þ2 =ðd0 Þ1 ¼ X 1=3 . 2. For weakly turbulent flows, when micromixing dominates, the scale-up criterion is τ E1 ¼ τE2 or ε1 ¼ ε2 , or ðu30 =d0 Þ1 ¼ ðu30 =d0 Þ2 , which leads to ðu0 Þ2 =ðu0 Þ1 ¼ X 1=7 and ðd0 Þ2 =ðd0 Þ1 ¼ X 3=7 . Because scaling up is related to an increase in the Reynolds number, ðu0 d0 Þ2 =ðu0 d0 Þ1 ¼ X 2=3 , a safe scale-up method is to use high Reynolds number flows to avoid the possible effect of micromixing. The Curtet number then should be kept constant. To validate this criterion, CFD simulations and experimental investigations for acetaminophen precipitation were carried out for three equipment scales: laboratory scale, pilot scale, and a small manufacturing plant (SMP), as characterized in Table 8.2 (Bałdyga et al. 2010a). Generally, good agreement between the model prediction and the experimental data was observed (Bałdyga et al. 2010a).

Figure 8.38 Particle size distribution of the product obtained in vessels of different diameters Dv with a nozzle diameter d0 = 9 × 10–4 m; Ct = 0.036 for Dv = 0.025 m, Ct = 0.018 for Dv =0.050 m, Ct = 0.0082 for Dv = 0.11 m (p = 120 bar, T = 323 K) Source: Reprinted with permission of the publisher from Baldyga et al. 2010a. Copyright © 2010, Elsevier.

The experimental results illustrating the scale-up problems are presented below. Figure 8.38 underscores how important in this case is the geometric similarity, expressed by the Curtet number. The particle size distributions obtained for the same process conditions but different vessel diameters and thus different d0 =DV values are shown in Figure 8.38. Figures 8.39 and 8.40 show the results of application of the scale-up criterion ðd0 =u0 Þ1 ¼ ðd0 =u0 Þ2 . The results presented in Figure 8.40 agree with the CFD predictions presented in Figure 8.41; clearly, in this case, a slightly higher supersaturation in the laboratory equipment causes slightly faster nucleation, which results in a slightly larger number of slightly smaller crystals. The micromixing effects decrease with increasing system scale [see the relation Re2 =Re1 ¼ ðd2 =d1 Þ2 ] and are negligible in the case of both the pilot plant and the SMP. Therefore, the results of scale-up are much better in these cases, as shown in Figure 8.40. More results of computations and experimental data for validation of scale-up procedures can be found in Bałdyga et al. (2010a) and Czarnocki (2007). The importance of the time constants, either for the direct modeling–based scaling up or for the formulation of scalingup rules, has been emphasized in several publications. For scale-up in a range from 0.3 to 25 dm3, Zauner and Jones (2000) applied the so-called segregated feed model [see also

255

Piotr H. Karpiński and Jerzy Bałdyga

Figure 8.39 Scale-up from laboratory scale to small production scale: application of scale-up criterion ðu0 =d0 Þ2 ¼ ðu0 =d0 Þ1 (log PSD; p = 120 bar; T = 323 K; Ct ≈ 0.015) Source: Reprinted with permission of the publisher from Baldyga et al. 2010a. Copyright © 2010, Elsevier.

Zauner and Jones (2002) for details of model application]. The segregated feed model can be characterized as a simple compartmental mixing model or, better, as a three-zone model with two feeding zones and one bulk zone. The model is based on the population balance and supplemented with the precipitation kinetics. The time constants for micro- and mesomixing were applied to model the mass transfer between zones by diffusion and convection, respectively. Using the resulting precipitation-mixing model, Zauner and Jones (2000) were able to predict accurately the mixing effects observed during the continuous mode of operation of calcium oxalate precipitation, including maxima in mean particle size and the coefficient of variation observed with an increasing power input. Gavi et al. (2007) considered the scale-up of confined impinging jet reactors (CIJRs) and have shown that the Damköhler number, defined as a ratio of the time constant for mixing (a sum of macro-, meso-, and micromixing time scales) and the time constant for the second, slower reaction of the system of two parallel reactions, unequivocally determines the conversion of the second reaction. Gavi et al. (2007) argue that similar methodology can be extended to other chemical systems with two competing processes (e.g., particle nucleation and molecular growth) and show as an example the case of nanoparticle precipitation in the CIJR (Marchisio et al. 2006), where for Da ≫ 1 (slow mixing), the particles are big, and for Da < 1 (fast mixing), the particles are small.

8.9 Precipitation in Practice Requirements regarding the morphology, purity, yield, size range, and CSD of precipitate are case specific. Therefore, only brief, qualitative discussion will be devoted to the practical aspects of the precipitation process. Precipitation, as much as typical crystallization, is seen now not just as a separation and purification operation but rather as a process capable

256

Figure 8.40 Scale-up from a pilot plant to a small production scale (SMP) with scale-up criterion ðu0 =d0 Þ2 ¼ ðu0 =d0 Þ1 (log PSD; p = 120 bar; T = 323 K; Ct ≈ 8.5 × 10−3) Source: Reprinted with permission of the publisher from Baldyga et al. 2010a. Copyright © 2010, Elsevier.

of delivering a range of value-added materials that meet certain specifications. Furthermore, the circumstances of the competitive global economy add factors outside the specification, such as speed in developing new materials or meeting specific customer needs. Very frequently, the manner by which the precipitation process is carried out predetermines the final properties of the material manufactured.

8.9.1 Batch Precipitation Batch (Semibatch) Precipitator This is the most commonly used precipitator. The batch volume may range from 100 to 12,000 liters (Rusli 1991). There are two basic types of batch (semibatch) precipitation operations: single feed (single jet) and double feed (double jet), illustrated schematically in Figure 8.42. In a single-feed operation, one of the reagents is placed in an agitated vessel, and the other is fed throughout the batch time or its part. In a doublefeed operation, the reagents are fed to the vessel, which initially contains a certain amount of a solvent at preset conditions. The addition rate of reactant may be constant or may be accelerated with time (Karpinski 1996). In a double-feed operation, the reagent concentration usually reflects the stoichiometry of the compound being precipitated, and so do the reagent flow rates. One of the feeds may be controlled in order to maintain a certain property found to be important for successful operation at a constant level. An excess concentration of one or more ionic species above the stoichiometric value may be employed (and controlled) in order to ensure formation of a specific morphology of the precipitate. It has been well documented in the literature (Söhnel and Garside 1992; Marcant and David 1993; Bałdyga et al. 1995) that the feed position in double-feed precipitation can significantly influence such properties as the size and the CSD of the product.

Precipitation Processes Figure 8.41 Supersaturation ratio distribution in the nozzle and the jet: (a) the laboratory scale system, (b) SMP Source: Reprinted with permission of the publisher from Baldyga et al. 2010a. Copyright © 2010, Elsevier.

Marcant and David (1993) studied the precipitation of calcium oxalate using several precipitator configurations. They observed that in a single-feed precipitation, the largest average size of precipitate was obtained with a feed point next to the tip of the impeller, followed by a feed point above the impeller and close to the impeller shaft. The smallest average size was obtained when the feed point was located near the liquid surface. Bałdyga et al. (1995) obtained qualitatively similar results for barium sulfate. Double-feed precipitation gave a much larger average size of precipitate and appeared to be very sensitive to impeller stirring speed (Marcant and David 1993; Bałdyga et al. 1995). In batch precipitation, initial contacting and mixing of fluids play a major role in ensuring product precipitate quality [see Bałdyga et al. (1995), Bałdyga and Bourne (1999), and Figures 8.22, 8.28, and 8.35]. Therefore, significant efforts have been focused on perfecting

both the contact of the two liquids (either reagent feeds or solute-rich solution and antisolvent) and the rate of homogenization of the concentration profile in the reactor– precipitator. A classic example here is the silver halide precipitation technology that since the early 1980s used a special rotor–stator device (e.g., Kodak’s PEPA mixer, US Patent No. 6136523) to ensure atomization and intimate (micro)mixing of the reagent feeds. Technically, such a device has separated two domains: a high-supersaturation zone, where rapid nucleation and formation of intermediate precursors take place, and a low-supersaturation zone, where the growth of nuclei, associated with the Ostwald ripening and agglomeration, takes place. A similar underlying principle, although for a different configuration of the reactor–precipitator, is used, for example, in the sliding-surface mixing device and the vortex reactor (Bénet et al. 2002).

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Piotr H. Karpiński and Jerzy Bałdyga

Sliding-Surface Mixing Device A flat-bottom baffled vessel with height equal to diameter is equipped with a disk mixer of large diameter (80–90 percent of the vessel diameter), which rotates at high speed (1200– 3000 s−1) just few millimeters from the bottom and separates the precipitator in two zones: • A confined, high-intensity mixed zone under the disk, where the injected and rapidly mixed reactants create high supersaturation inducing rapid massive nucleation • The upper low-intensity mixing zone above the disk with lower supersaturation where growth, Ostwald ripening, and agglomeration occur

Vortex Reactor This is a high-aspect-ratio (>2) cylindrical vessel without baffles equipped with a bottom-mounted Rushton turbine that generates a forced central cylindroconical vortex. The reacting liquids are injected onto the free surface of the vortex, where very high supersaturation and rapid nucleation occur. There is no shaft, baffles, or walls in the proximity of the vortex, so there is no incrustation problem. In the vortex-free zone, where supersaturation is lower, the growth of the nuclei with possible agglomeration and Ostwald ripening takes place. In any batch precipitation scheme, a selection must be made between a high-yield, high-supersaturation (without exceeding the critical supersaturation), short-operation process and longer-batch-time, lower-supersaturation process. Such a decision must weigh both the product properties desired and the cost factor associated with implementation of either scheme. Often additional considerations may need to be taken into account. For example, as reported by Jiang et al. (2009), different polymorphic forms of precipitate can be obtained by controlling the supersaturation level in a doublejet precipitation.

A semibatch precipitation process of barium sulfate formation from aqueous solutions of Na2SO4 (A) and BaCl2 (B) was experimentally investigated by Podgórska (1993). The experiments were carried out in a Rushton-type single- and doublefeed stirred-tank reactor of diameter T = 0.242 m and impeller diameter of dimp = 0.075 m, as shown in Figure 8.42. Two addition modes were investigated: a single vertical feed tube with an outlet located 0.05 m below the initial liquid Na2SO4

BaCl2 single feed mode

BaCl2

water double feed mode

Figure 8.42 Schematic of the semibatch precipitation processes investigated by Podgórska (1993)

258

where Ahe = 43.13, Rmax,he = 5.36 × 1011 m−3s−1 for heterogeneous nucleation, and Aho = 3137, Rmax,ho = 1.24 × 1049 m−3s−1 for homogeneous nucleation, as given by Vicum et al. (2002). The saturation ratio SA is
0:5 cBa2þ cSO2 4 SA ¼ γ ð8:159Þ KSP where the activity coefficient γ± is calculated from the formulas derived by Bromley (1973). The calculations take into account that the barium sulfate does not dissociate completely in aqu2 eous solutions forming ion pairs, Ba2þ ðaqÞ þ SO4ðaqÞ ⇌ BaSO4ðaqÞ , which decreases the saturation ratio compared with the case of complete dissociation observed with very strong electrolytes "
#2 0:5 cBa2þ ;int cSO2 4 ;int G¼kr γ;int 1 ¼kD ðcBa2þ ;bulk cBa2þ ;int Þ KSP cSO2 Þ ¼kD ðcSO2 4 ;bulk 4 ;int

Example of Semibatch Precipitation

Na2SO4

level (distance from the axis r = 0.06 m, distance from the tank bottom z = 0.22 m) and two feed tubes of internal diameter equal to 0.0015 m placed in the plane defined by both tubes and the reactor axis symmetrically to the axis. For the single-feed mode, 0.01 m3 of the barium chloride solution was placed in the vessel initially, and the sodium sulfate solution was fed by the feed tube. For the double-feed mode, 0.01 m3 of water was placed in the tank initially, and equimolar amounts of the reactant solution were fed simultaneously through the feed tubes with the same flow rate. The volume ratio αV is defined in both cases as a ratio between the initial tank volume and the volume of the added sodium sulfate solution. The nucleation rate was calculated using the following equation, representing the sum of homogeneous and heterogeneous nucleation rates:

 Ahe Aho RN ¼ Rmax;he exp exp þ R ð8:158Þ max;ho ln2 SA ln2 SA

ð8:160Þ

with kr = 9.1 × 10−12 m/s (Vicum et al. 2003) and kD = 4.0 × 10−5 m4/s/kmol (Bałdyga et al. 1995). In Equation (8.160), the subscript “int” describes the concentration at the solid–liquid interface and the subscript “bulk” indicates the concentration in liquid far from the crystal surface. The average equivalent particle size was recalculated from the moments of the particle size distribution using the following equation: d43 ¼ ðm4 =m3 Þ  ðπ=ka Þ1=3

ð8:161Þ

Figure 8.43 presents a visualization of the mixing zone for the single-feed simulations, which is the zone of the intensity of segregation, IS ¼ ðf 0 Þ2 = ½f ð1  f Þ larger than 0.01. The micromixing is not instantaneous and not localized at the feed point. Figure 8.44 compares the measured and predicted effects of the initial concentration on the particle size for the single-feed mode. The effect of concentration is predicted well, especially at the higher concentration values. In the case of double-feed precipitation, the mixing effects are expected to be very important. The initial segregation

Precipitation Processes

Figure 8.44 Comparison of the model predictions with the experimental results for single-feed mode with the feed introduced above the impeller. The effect of the initial concentration on mean particle size. N = 5 s−1, αV = 50, tf = 1230 s. Model II accounts for the micromixing effect. Source: From Bałdyga et al. 2005.

Figure 8.43 Visualization of the mixing zone for a single-feed mode for N = 5 s−1 and tf = 1230 s. Other conditions are found in Podgórska (1993).

Figure 8.46 The effect of stirrer speed on mean crystal size for qf = 0.12 × 10−6 m3/ s, c 0 = 0.0045 kmol/m3, αV = 50, and feed concentrations cA0 = cB0 = 0.234 kmol/m3. Solid line = predicted by model II accounting for micromixing effects; dashed line = predicted by model I neglecting micromixing effects. Source: From Bałdyga et al. 2005.

Figure 8.45 Mixing in the feed zone of a double-feed precipitator operating at N = 3 s−1 and qf = 0.12 × 10−6 m3/ s: 3D representation of the feed zone for the intensity of segregation IS ≥ 0.01 Source: From Bałdyga et al. 2005.

between the fresh feeds A and B prevents fast nucleation, and thus, close to the feed pipe, micromixing increases nucleation rate. Afterwards, at a longer distance from the feed point, micromixing with the bulk liquid decreases supersaturation and limits nucleation. The local segregation intensity for fresh fluids A and B is portrayed in Figure 8.45. The segregation in the region between the feed points and the impeller defines the zone where mixing can affect the precipitation process.

Figure 8.46 shows a satisfactory agreement of the experimental data with the model predictions – when the micromixing effects are included (model II). Neglecting the micromixing affects (model I) results here in the production of the large amount of very small particles, about two orders of magnitude smaller than those obtained in the experiment. This is due to the fact that when there is no segregation close to the feed, very high supersaturation values are created, which – due to the nonlinearity of Equations (8.45) and (8.47) – result in a very large amount of small particles. The feed concentrations are in this case many times larger than the average value, which results in very small values of the time constant for nucleation (about 10−5 s), which is much smaller than the time constant for micromixing, which – close to the impeller – is on the order of milliseconds. At high supersaturation, the micromixing effects need to be included to match the simulation results

259

Piotr H. Karpiński and Jerzy Bałdyga Table 8.3 Precipitation of Cubic AgBr Crystals Grown from In Situ Seeds According to the Growth Profile [Equation (8.150)]

Final size, μm

G × 1010

tbatch

Run ID

(m/s)

(min)

Calculated

Obtained

Size ratio

CSD width index

PK201(a)

0.50

60

0.266

0.259

0.97

1.09

PK202

0.50

60

0.266

0.29

1.09

1.10

PK203

0.25

60

0.176

0.195

1.11

1.09

PK204

0.75

30

0.221

0.225

1.02

1.08

60

0.356

0.358

1.01

1.09

1.00

50

0.386

0.378

0.98

1.07

1.25

42

0.416

1.04

1.12

PK205 PK206(b)

3

0.401 15

3

Note: V = 6 dm , T = 70℃, L0= 86.1 nm, N = 2.85 × 10 , CR = 3.0 mol/dm .

with the experimental data. The model based on the concept of mixture fraction predicts all trends observed in the experiments without any fitting.

Example of Application of the Growth versus Time Profile (Growth Ramp) The experimental verification of the growth versus time profile [Equation (8.150)] was carried out by Karpinski (1996) for octahedral and cubic AgBr submicron crystals in a 6.0-liter batch reactor–precipitator. In the preliminary trials, the number and size of the effective (stable) nuclei or in situ seeds was determined. To this end, test precipitations were performed with samples for sizing taken just prior to the growth stage and at the end of precipitation. Crystal size was measured by a disk-centrifuge sizer, and the number of crystals was determined from the material balance. With these data, the growth profiles Q versus t were calculated from Equation (8.150) for linear growth rates G = 0.25, 0.50, 0.75, 1.00, and 1.25 × 10−10 m/s. Computer-controlled precise peristaltic pumps were used to deliver the AgNO3 and NaBr solutions in a double-feed reactor–precipitator configuration to the stator of a high-shear rotor–stator mixing device. The results of the experiments are shown in Table 8.3 for cubic AgBr crystals, whereas Figure 8.47 presents a scanning electron microscope (SEM) photomicrograph of the typical crystals of the nearly monodisperse final product. Very good agreement between the calculated and measured final sizes of precipitate was obtained. The author concluded that the proposed growth versus time profile [Equation (8.150) or Equation (8.151)], which assumes a constant growth rate throughout the segment of a growth stage or the entire growth stage, provided a successful growth strategy for the precipitation of AgX (X = Cl, Br, I) photosensitive materials. The constant-growth-rate strategy just presented allows one to predictably (i.e., quantitatively) manipulate the final size of the precipitate by adjusting the growth rate and/or the batch time of the batch precipitation process, for which known data exist regarding the nucleation stage. By setting the supersaturation – and thus the growth rate – at a constant level, an additional means of controlling crystal

260

Figure 8.47 SEM photomicrograph of typical cubic crystals of AgBr obtained in the experiments presented in Table 8.3.

morphology, polymorphism, and/or properties of the crystal surface may be achieved. The use of any growth rate that does not exceed the critical (maximum) one ensures a renucleation-free process and a narrow CSD with limited fines contribution.

pH Swing Precipitation The pH swing precipitation method is used to obtain a desired microstructure of the catalysts; the microstructure is characterized by the specific surface area or pore size distribution. These can be related either to the catalyst itself or to the catalyst support. As catalyst support, such materials as MgO, Al2O3, TiO2, and SiO2 (Hoskawa 2007) are often used. Consider as a first example a pH swing method for controlling the size of micropores in alumina (Hoskawa 2007). Two solutions are prepared initially – aluminum nitrate of pH = 2 and sodium aluminate of pH = 10 – and used as aluminum sources. The pH of the solution is then swung by carrying out a sequence of alternate additions of both solutions to the reactor. The

Precipitation Processes

pseudobohemite precipitates close to neutral pH. The small pseudobohemite particles are dissolved at pH = 2 and pH = 10, and the large ones survive without dissolving completely. Hence only a limited number of particles survive, and they can grow up to obtain their desired size and the resulting target pore size of the catalyst. The next example is related to TiO2 precipitation to obtain either anatase or rutile phases. Neppolian et al. (2005, 2008) applied the pH swing precipitation between pH = 8 (creation of particles) and pH = 2 (dissolution of small particles), which can be explained by the structure of the precipitation diagram represented in Figure 8.1. As demonstrated by Neppolian et al. (2005), depending on the number of swings – between 1 and 30 – and the swing duration, different particle sizes, surface areas, crystallinity, pore volume, and diameters, as well as anatase and rutile phase composition (after calcination), were obtained, which may enable process control. In particular, a potential to prevent the phase transition from anatase to rutile was indicated. Still better product control was possible with ultrasonic irradiation (Neppolian et al. 2008).

8.9.2 Continuous Precipitation A basic criterion used to select a continuous-precipitation process is the amount or the mass rate of product to be manufactured. Bulk inorganic chemical, catalyst, fertilizer, and food-processing industries are typical examples of such high-capacity precipitation operations. Continuous precipitation of crystalline and amorphous particles has the following advantages over a batch process (Laird 2011): >90 percent reduction in footprint, lead time, and waste/ toxic emissions and around 50 percent reduction in capital cost. Moreover, consistency of product quality is ensured because all material experiences the same history; process control and scaling up rules are simpler, and the CSD and polymorph control is improved. Integration of the continuous-precipitation process with upstream (e.g., manufacture of substrates) and downstream operations (e.g., filtration, centrifugation, drying, and milling) is necessary and may not be an easy task. Recently, there has been a new technological initiative – continuous manufacturing – aimed at exploration of advanced approaches to integrated continuous manufacturing processes that can deliver efficiency gains and cost reductions in the production of medicines, foodstuffs, dyes, pigments, and nanomaterials. For example, continuous manufacturing has generated significant interest in the pharmaceutical industry, which sees it as s a potential “paradigm shift” that could make large-scale drug manufacturing more economically sustainable, cleaner, leaner, and more energy efficient; cut down development time; and offer greater potential to leverage new technological advances.

CMSMPR Precipitator In this classical continuous-precipitation technique, continuous streams of the reactants are fed to a stirred vessel while product is simultaneously removed to maintain a constant reaction volume. Following a transient period, a steady state is reached, during which the CSD and morphology of

crystals removed from the precipitator remain unchanged. Uniform morphology and rather polydisperse precipitate crystals are typically obtained. Such a continuous operation may exhibit more or less pronounced oscillations. The results of application of a CMSMPR precipitator to an inorganic material such as AgBr were summarized in Section 8.7 (Wey et al. 1980). Raphael et al. (1997) reported that application of the CMSMPR precipitator to protein precipitation may result in unimodal or bimodal CSD depending on the precipitation conditions and the tendency of the material to aggregate. They found that at a low-protein feed concentration, the growth process is dominant, leading to the bimodal CSD at longer mean residence times. At high-protein feed concentrations, smaller particles with a unimodal CSD were obtained after longer mean residence times.

Impinging-Jets Mixing Devices Various mixing devices, such as rapid or very rapid mixers, have been tried to ensure better control of the contacting mode of reactants and mixing effects. Examples include simple T-tube or Y-tube devices or more sophisticated ones such as the Hartridge–Roughton chamber, two impinging jets, or two impinging sheets. As a general principle, the reactive solutions are pumped with high flow rates in feed tubes, and mixing takes place rapidly in a very small volume confined by the tube walls or by the environmental fluid. Variable parameters of such devices are the jet velocities, the diameter of the jet orifice, and the distance between the two injection points. Typically, very small particles are obtained with a narrow CSD. The major problems are plugging of the tubes and difficulty in obtaining a perfect impinging point. A promising alternative consists of impinging the reactive solution in the host solvent or solution. The mixing zone is no longer confined by the tubes but by the environmental fluid. Thus plugging cannot occur. For this technology, called a two-impinging-jets mixing device (Bénet et al. 2002), the variable parameters are the jet velocities, the diameter of the jet orifice, and the distance between the two injection points. The two-impinging-sheets mixing device is derived from the two-impinging-jets mixing device.

Tubular Precipitator This type of continuous operation may be employed to reduce polydispersity of precipitates (Raphael et al. 1997; Raphael and Rohani 1999). The tubular precipitator may operate under either turbulent flow or laminar flow. The reactants are added in the inlet section equipped with static mixers and may also enter as a multiport feed along the length of the tubular precipitator. If the reactant feeding streams are too concentrated, or if excessive formation of precipitate occurs in the inlet section of the precipitator, a third stream of solvent is also fed to dilute the flowing suspension. The latter may contain a protective colloid or surfactant that prevents agglomeration of precipitate.

Continuous Oscillatory Baffled Crystallizer (COBC) The COBC is a tubular crystallizer containing periodically spaced orifice baffles with oscillatory motion superimposed on the net flow (Lawton et al. 2009). The mixing in a COBC

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Figure 8.48 Continuous oscillatory baffled crystallizer (COBC) Source: Reprinted with permission from Lawton et al. 2009. Copyright © 2009, American Chemical Society.

is provided by the generation and cessation of eddies when flow interacts with baffles and is thus decoupled from the net flow-driven turbulence. With the repeating cycles of vortices, strong radial motions are created, giving uniform mixing in each interbaffle zone and cumulatively plug flow conditions along the length of the column at flows that would otherwise result in laminar flow regimes. The fluid mechanical conditions in a COBC are governed by the oscillatory Reynolds number and the Strouhal number, defined as 2πfxo ρD μ D St ¼ 4πxo

Reo ¼

ð8:162Þ ð8:163Þ

where D is the column diameter, ρ is the liquid density, μ is the liquid viscosity, xo is the oscillation amplitude, and f is the oscillation frequency. The oscillatory Reynolds number describes the intensity of mixing applied, whereas the Strouhal number is the ratio of column diameter to stroke length, measuring the effective eddy propagation. According to Lawton et al. (2009), the combination of these aspects in COBC offered more control over the crystallization process and shortened the crystallization process of a model pharmaceutical to 12 minutes compared with the 9 hours and 40 minutes for the reference batch process.

8.10 Summary Precipitation or reactive crystallization is very common in industrial applications and laboratory practice. A large number of high-value-added products and intermediate materials are produced via precipitation. The precipitation process is very complex, and the properties of precipitate strongly depend on the kinetics of the component subprocesses and their conditions. All these factors, as well as the fact that the typical size of precipitate is in the 10-nm to 100-µm range, make the precipitation process very unique. Frequently, different theoretical and experimental approaches than those used for typical crystallization processes are required.

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In this expanded, revised, and updated chapter, a strong emphasis has been put on sparingly soluble, rapid kinetics crystalline materials because the authors felt that this domain of the precipitation processes spectrum is the most representative of precipitation as well as distinct from other crystallization processes. In addition to considering a typical sequence – generation of supersaturation, nucleation, and crystal growth – one may be concerned with the chemistry of precursors, colloidal properties, mixing effects, Ostwald ripening, renucleation leading to bimodal or polymodal distribution, aggregation, and aging of precipitates. All these physicochemical phenomena and their underlying principles are discussed, with emphasis on surface chemistry and colloidal stability and aggregation. Precipitation-related properties of supercritical fluids and precipitation processes that are induced by supercritical fluids, such as SEDS, are discussed. At high supersaturation levels characteristic of precipitation, particle agglomeration may dominate the entire precipitation process. Therefore, both Lagrangian and Eulerian frames of reference are used to estimate agglomeration rates. Particular attention is also devoted in this chapter to the effect of mixing on precipitation. Owing to the milli- to microsecond-range time scale of nucleation and chemical reaction stages of precipitation, micromixing phenomena occurring on a molecular scale become critically important. Although the incorporation of micromixing adds to the complexity of the theoretical description, it must not be neglected in any rigorous approach to precipitation. Mixing–precipitation models combined with CFD and population balances are presented to illustrate the significance of mixing. Also discussed are experimental techniques employed to study precipitation, such as supersaturation measurements, the constant-composition (CC) method, instantaneous mixing devices, maximum (critical) growth rate experiments, and size measurement. Given the intrinsic difficulties with direct supersaturation measurements, the CC method is used to study precipitation kinetics. For the same reasons, the critical growth experiments are used to delineate the domain of the reactant feed rate that ensures a renucleation-free process and the unimodal CSD. Despite the complexity of precipitation, the population balance modeling tools for CMSMPR and semibatch crystallizers have been adopted to predict the CSD. They are reviewed in this chapter. Once the outcome of high-rate prenucleation and nucleation processes has been established as a stable nuclei population (or seeding is possible), the growth part of a batch precipitation process can be conveniently carried out thanks to the appropriate time profiling of the reactant feed rate. This concept is described in detail and illustrated with an example for AgBr submicron-sized crystals. Scale-up rules for precipitation are reviewed and illustrated using a SEDS process. Finally, we touch on selected practical aspects of typical precipitation operations using continuous and semibatch precipitators as illustrative examples.

Precipitation Processes

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Chapter

9

Melt Crystallization Joachim Ulrich Martin Luther University Halle-Wittenberg Torsten Stelzer University of Puerto Rico

Melt crystallization is an important separation, purification, and concentration technique used in the chemical, pharmaceutical, and food industries. Crystallization from melt is a very powerful separation process for the purification of organic compounds up to very high purities of 99.99 percent. Therefore, the objectives of melt crystallization (i.e., purity, separation, or concentration) are quite often different from crystallization from solution (i.e., purity and defined crystal size distribution). Good background information about the theory of melt crystallization can be found, for example, in Arkenbout (1995), Atwood (1972), Jansens and van Rosmalen (1994), Matsuoka (1991), Matz (1969), Molinari (1967), Mullin (2001), Özoğuz (1992), Rittner and Steiner (1985), Sloan and McGhie (1988), Toyokura and Hirasawa (2001), Ulrich and Bierwirth (1995), Ulrich and Kallies (1994), Ulrich and Nordhoff (2006), Ulrich and Stelzer (2011), Verdoes et al. (1997), and Wintermantel and Wellinghoff (2001). In the following sections, the basics and design examples of plants for melt crystallization will be given.

9.1 Definitions In crystallization processes, two expressions are often used: crystallization from solution and crystallization from melts. A solution is a homogeneous mixture of more than one species. A melt most correctly refers to a pure molten solid such as molten silicon. Unfortunately, the term melt is used in a more general way to describe solutions of materials that are usually solid at room temperature. From a thermodynamic point of view, however, there is in the case of a two-component system no difference between a solution and a melt when looking at the phase diagram. Phase diagrams, as shown in Figure 9.1, give the meltingpoint curve for the full range of concentrations possible for a binary system. Solubility diagrams, which are mostly used in solution crystallization, however, are usually plotted with exchanged axes and give only a limited section of concentrations (an example would be Figure 1.1). In order to differentiate between the two expressions melt and solution, it was therefore suggested by Ulrich, Özoğuz, and Stepanski (1988) that whenever the expression solution is used, the mass-transfer effects should dominate a process, and whenever heat transfer is dominating a process of liquid–solid phase change, it should be called melt crystallization. An additional explanation can be found in the different growth rates that have their origin in the rate-domination effect. Crystal growth rates within crystallization processes from solutions in most cases are in the range from 10−7 to

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10−9 m/s. Growth rates in melt crystallization, however, are quite often in the range of approximately 10−6 m/s and in extreme cases as high as 10−4 m/s and still lead to high-purity products. This allows the economical use of technologies such as solid-layer melt crystallization (see Section 9.5.1). Another expression that needs a definition is fractional crystallization. Fractional crystallization is a repetition of a crystallization process performed in order to receive a further purified product. A fractional crystallization therefore can be conducted from a melt as well as from a solution but does not differentiate between the two.

9.2 Benefits of Melt Crystallization Melt crystallization as a unit operation can have advantages and features that establishing it as an alternative separation technique to other thermal unit operations in chemical engineering (e.g., distillation or extraction). Melt crystallization may be a reasonable alternative to other separation and purification technologies when comparing the energy required for the phase change [melt crystallization (solid–liquid) and distillation or evaporation crystallization (liquid–vapor)]. Because the heat of vaporization (liquid–vapor) of most volatile organic materials is between two and five times their heat of fusion (solid–liquid), and in case of water even seven times, it is required for evaporation crystallization out of solution or distillation. A further point in this context is that most known organic chemicals have melting points in a range where low-level waste heat from other processes can be used. This last point becomes clearer when looking at Figure 9.2. The data shown from Matsuoka and Fukushima (1986) are based on the data available in the CRC Handbook of Chemistry and Physics (Weast 1976). The numbers of substances are plotted versus their melting points. More than 71 percent of the substances examined (10,600 organics) have melting points between 0 and 200°C and more than 86 percent between 0 and 300°C. The tendency could be verified by Ulrich (2003) based on the commercially available chemistry database of MERCK (Germany) in 1999. On this account, the much lower temperature level and the much smaller difference in latent heats are the benefits compared with evaporative processes. The moderate temperature level shows obvious advantages when heat-sensible substances (e.g., foods, polymers, or pharmaceuticals) are processed, as described by Ryu, Jones, and Ulrich (2010), as well as Tiedtke, Ulrich, and Hartel (1996). The requirement is thermal stability (no thermal degradation) or at least a slow enough decomposition at the melting point. In the case of materials

Melt Crystallization Figure 9.1 Frequency of types of phase diagrams of 1486 binary organic systems Sources: According to Matsuoka 1977, as well as Matsuoka and Fukushima 1986.

Figure 9.2 Distributions of melting points of 10,600 organic materials Source: According to Matsuoka and Fukushima 1986.

such as waste waters from heat-treating processes, a temperature above a certain level can lead to chemical reactions and accelerated corrosion. Hence a treatment with crystallization (low temperature level) instead of distillation would be appropriate. Another example for heat-sensible substances, aside purification and separation, is the concentration of fresh juice, beer, or wine in order to reduce the volume by freezing out the water (called freeze concentration). The advantages of freeze-concentration technology compared with evaporation technology (the most common method for liquid food concentration) is, among others, the nondegrading of thermally fragile components such as aroma as well as the vitamin content due to the low process temperatures (Hartel 2001). If the advantages are not that unambiguous, as in case of freeze concentration of fruit juice, it has to be stated that an analysis of the energy requirements in melt processes has shown that such processes can compete with other thermal separation techniques only if the plant is well designed and the process precisely controlled (Sloan and McGhie 1988). The aforementioned advantages of melt crystallization processes concerning temperature level of the product component

do not exist if the comparison is made with crystallization from solutions (see Chapter 1) because the product is crystallized in solution at temperatures lower than its melting point. However, the solvent has to be evaporated, usually in larger amounts, and is often treated as an impurity. This fact makes up for the disadvantages in many cases. Hence melt crystallization does not need any additional substances compared with extraction or solution crystallization. Therefore, no waste water will be produced, and no other chemicals (solvents) have to be reprocessed. The capital and energy costs for solvent recycling can represent a major portion of a product isolation process using solution crystallization. A further very important point about melt crystallization processes is the high selectivity, which can lead to an almost pure product within one separation step if the mixture to be separated is a eutectic one. If a system is eutectic, then product recovery is limited, although product purity is not. Here again, it can be seen in Figure 9.1 that 54.3 percent of all organic mixtures for which phase diagrams can be found [there are 1496 in International Critical Tables (Washburn 1928)] are eutectic (Matsuoka 1977). Further, 31.6 percent belong to the

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group of molecular-forming compounds and systems with peritectic and eutectic points. Only 14.1 percent of all the investigated substances belong to the group of solid solutionforming substances (over a part or the total range of composition). In solid solutions, the separation has to be done step by step, as in distillation processes. The number of steps is determined by the phase diagram and the required purity of the product. As a result, complete product recovery is more complicated but possible. The number of steps required, however, increases rapidly as recovery approaches unity. Known solid-solution-forming systems are naphthalene– theonaphthalene, thiophene–benzene, hexadecane–octadecane, and m-chloronitrobenzene–m-fluoronitrobenzene. Further advantage of melt crystallization is the smaller volume of the liquid phase compared with the vapor phase of a substance. A smaller volume leads to less space or less construction work, which means less capital costs. These advantages are sometimes lost if the process of crystallization and remelting is very slow. Therefore, the retention time in the apparatus is high. The nonexistence of a vapor phase, however, also leads to better control in terms of safety (e.g., leakage and collection of lost product). Totally closed equipment leads to high environmental safety. Another advantage is reflected in investment and running costs. Tiedtke (1997), for example, compared these two types of costs for a melt-suspension crystallization and a highpressure extraction process applied in the fractionation of milk fats. For the same capacity, the extraction plant has investment costs that are three times higher than those of the crystallization process and running costs exceeding those of the crystallization process by a factor of eleven.

extreme in phase diagrams, show that only an enrichment of one component is possible. Therefore, a number of steps would be necessary to achieve the desired purity. In chemical engineering practice, however, no separation process yields a perfect separation. Therefore, the effectiveness of the process has to be considered. This means that more steps are necessary than indicated by the equilibrium phase diagram. If the melting points of the components are too close, the separation will be difficult for solid solutions. If the feed composition of a eutectic-forming mixture is at or near the eutectic point, then little or no product can be recovered. This is true for solution crystallization as well unless the solvent–solute interaction is far from ideal, resulting in different activity coefficients for the components. In such cases, other separation technologies should be used. All that has been said thus far applies only to constant pressure. However, phase diagrams are affected by a change in pressure if the pressure is as high as 10–100 MPa. This is demonstrated for p- and m-cresol, in the work of Moritoki and Fujikawa (1984) and for benzene–cyclohexane in the work of Moritoki et al. (1989), as shown in Figure 9.3. The dependence of the phase diagram on the pressure can be exploited to improve both crystallization and post-crystallization treatment, as demonstrated for sweating by Freund, König, and Steiner (1997). One problem is that there are hardly any phase diagrams available giving the pressure dependence, except those published in the works of Moritoki, Wakabayashi, and Fujikawa (1979) as well as Moritoki et al. (1989). In addition to the aforementioned substances, this includes p- and m-xylene and benzene from benzene–cyclohexane mixtures.

9.3 Phase Diagrams 9.3.1 What to Learn from Phase Diagrams Phase diagrams provide fundamental information about the physicochemical interaction between the substances in a mixture to be separated by the crystallization process. Knowledge of the phase diagram provides the information about temperature level at which the process must be operated as well as the change in composition. This is important to ascertain other physical properties, such as density, viscosity, and diffusion coefficient at the right temperature for the mixture. These physical properties are important not only for the separation process itself but also for calculation of the flow regime (laminar or turbulent), which is important for the pump design and so on. The most important information from phase diagrams is knowledge of the fundamental type of the mixture to be separated (see Figure 9.1; see also, e.g., König 2003). For instance, eutectic-forming mixtures provide a theoretical opportunity to achieve one almost pure component by crystallization. This is possible, of course, only if, on the one hand, the crystal growth rates are so extremely slow that perfect crystals are created and, on the other hand, the necessary solid–liquid separation after the growth process is also perfect. Solid solutions, which represent the other

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Figure 9.3 Phase diagram of benzene–cyclohexane Source: According to Moritoki et al. 1989.

Melt Crystallization

9.3.2 How to Obtain Phase Diagrams There are three ways to obtain phase diagrams: 1. Literature or databases 2. Experiments 3. Calculations The simplest and probably the fastest way is to find the required data in the literature or databases on physical properties. Some examples include Beilstein’s Handbook of Organic Chemistry (Luckenbach 1992), the CRC Handbook of Chemistry and Physics (Hynes 2013), Taschenbuch für Chemiker und Physiker (Lechner 1992), International Critical Tables (Washburn 2003), Kirk-Othmer’s Encyclopedia of Chemical Technology (KirkOthmer 2014), the Landolt–Börnstein Database (Springer 2014), Ullmann’s Encyclopedia of Industrial Chemistry (WileyVCH 2011), and VDI Heat Atlas (VDI 2010). If the data required to construct a phase diagram are not available in the literature for the mixture of interest, the data will have to be obtained experimentally. In this case, determination of the melting points of known compositions of the mixture has to be carried out. Several experimental techniques are described in detail by Sloan and McGhie (1988). Melting points can be measured with commercially available melting-point measuring devices or with highly sensitive temperature-measuring instruments. The systems work by detecting the temperature (the melting temperature) at that moment when the optical appearance of the material is changing due to melting. This optical method can be conducted by the human eye or by the change in light absorption, for which laser light is usually used. Another method is to record the temperature curve as a function of time while the temperature increases with a constant temperature gradient. There will be an inflection in the temperature–time curve at the melting temperature because energy supplied will be used for the phase change, particularly before the sample can continue its temperature increase. A differential scanning calorimeter (DSC) is in widespread use for these measurements. Many mixtures, however, have a melting zone rather than a melting point. This is due to a number of reasons. One is the

error in the detection; another is the quality of the specimen. This second point can be the result of an unrepresentative specimen or because of a less than ideal mixture within the specimen. Furthermore, something that has to be mentioned in this context of melting temperatures is that the melting point also can be used to specify the purity of a product. This means that the purification effect achieved by a crystallization process can be evaluated by comparing melting points because the melting point is depressed as the impurity content of the product increases. In addition to melting-point measurements, however, there are other methods to evaluate or prove the achieved progress in purification. These are gas chromatography, density measurements, and the different types of color tests. As we tried to describe earlier, there is no one, single method to determine phase diagrams experimentally. For each system, the method has to be chosen based on several criteria or even pre-experiments. One very important criterion, e.g., is to achieve the desired information in an appropriate time frame. A good overview of this topic is given, e.g., by Nývlt et al. (1985). In some cases, if the system shows ideal behavior (activity coefficient = 1), the phase diagram can be simply calculated by means of the known melting points and the heats of fusion of the pure components of a mixture. Hence experimental time can be saved by this method. Several equations exist to calculate a phase diagram (Gmehling et al. 2012). The most famous one is probably the van’t Hoff equation lnxil ¼ 


 Dhm;i T 1 Tm R T

ð9:1Þ

where xil is the liquid concentration of component i, Δhm,i is the heat of fusion of component i, R is the universal gas constant, T is temperature, and Tm,i is the melting temperature of component i. By means of Equation (9.1), the liquidus and solidus line can be calculated, as demonstrated in Figure 9.4 for a eutectic phase diagram (panel a) and a solid solution type (panel b). In addition, systems with nonideal behavior (activity coefficient ≠ 1) can be calculated. In this case, the activity

Figure 9.4 Phase diagrams of (a) benzene–naphthalene and (b) anthracene–phenanthrene; lines are calculated values, and points are experimental values Source: From Gmehling et al. 2012.

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coefficient values are required (Gmehling et al. 2012). These values, however, are not that easy to measure compared with melting temperature and heat of fusion, as well as not that good in tabulated databases.

9.4 Crystallization Kinetics 9.4.1 Importance of the Crystallization Kinetics to Melt Crystallization Kinetics in crystallization means nucleation and crystal growth. Both effects are important in melt crystallization. In melt crystallization, the process is mainly controlled by the rate of heat transfer and not by mass transfer, as it often is in solution crystallization. All crystallization processes need a nucleation step. Nuclei are the smallest solid units to survive and are able to grow as the crystalline substance (see Chapter 3). The nucleation could be heterogeneous, meaning that particles other than the crystallizing substance itself (e.g., dust or roughness of walls) are the nuclei. The nuclei also could be homogeneous, which needs considerably higher driving forces to be created. The most common way to start and maintain crystallization, however, is to have a seeding procedure or a secondary nucleation effect. Secondary nuclei are nuclei of material from the crystalline substance that already exist within the crystallizer apparatus. The nature of the nuclei and the place where they form inside the crystallizer both influence the following growth process. The nucleation process therefore should be controlled to ensure reproducibility. The nucleation details depend strongly on the process and the equipment. Therefore, the growth of crystals from the melt is influenced by at least four steps. 1. Transport of crystallizing material from the bulk of the melt to the vicinity of the crystal surface 2. A surface integration process, which means an integration of molecules or ions into the crystal lattice 3. A transport of the noncrystallizing substance (impurity) away from the vicinity of the crystal surface into the bulk of the melt 4. A transport away from the crystal and its vicinity of the dissipated heat resulting from the solidification process The mass transfer can be influenced by convection, whereas surface integration cannot. Surface integration is affected by the impurity content. Both effects depend on the temperature because physical properties such as diffusion coefficient and viscosity are temperature dependent. Melt crystallization processes are dominated by heat transfer. In the case of melt crystallization in suspension processes, the crystals are surrounded by melt, and the system is almost isothermal at the crystallization temperature. The heat of the solidification process is only transported away through the melt. Such a system is often referred to as adiabatic growth, which results in moderate growth rates with rather pure crystals. The so-called solid-layer processes (see Section 9.5.1), where the crystals are formed on a cooled surface, show nonadiabatic conditions. The heat resulting from the phase change

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is directly transferred through the crystals themselves and the cooled surface to the outside. In contrast to suspension processes (see Section 9.5.2), the supercooling at the solid–liquid interface is not limited by the requirement to avoid encrustations at the heat exchanger surfaces because the encrustation is the desired product. Consequently, high supercoolings and, hence, rather fast growth rates are possible. Furthermore, in suspension crystallization, the maximum achievable supercooling of the process is limited by the metastable limit (width of metastable zone). Otherwise, the whole melt will be solidified, and no separation as well as purification takes place. Unfortunately, these fast growth rates in solid-layer processes are in most cases accompanied by increased impurity concentrations of the crystallized material. Therefore, it can be stated that in the ideal case, very slow, indefinitely slow, crystal growth leads to perfect crystals, which also means perfect purity of the crystals. However, indefinite slow growth is not possible in industrial applications. Faster growth leads from the perfect, flat, planar crystals to a more rough interface between the melt and crystal or melt and crystal layers. An even faster growth rate, achieved by higher driving forces and expressed by higher temperature gradients, will lead to a dendritic type of growth. The dendrites, which look like Christmas trees, are still pure crystals, but they have entrapped a lot of melt between the dendrites. This melt is, of course, highly contaminated with the unwanted impurities. Owing to the fast growth in these cases, only a local separation of the compounds occurs. This means the pure compound has formed the dendrite, and the impurity is enriched in the remaining, entrapped melt. In order to quantify the achieved purity after a crystallization process, the effective distribution coefficient is quite often used and plotted in diagrams in correlation with, for example, the crystal growth rate (kinetic). The effective distribution coefficient has been defined by Burton, Prim, and Slichter (1953), as well as Wintermantel (1986), as the ratio of the impurity content in the crystals to the impurity content in the feed melt.

9.4.2 Theoretical Approach to Crystallization Kinetics Impurities can be incorporated in crystals or crystal layers as a result of a kinetic process, as mentioned earlier. On this account, it is unavoidable and essential to have a complete understanding of the crystallization kinetics for the design and optimization of melt crystallization processes. Nevertheless, there is only little theory available to describe melt crystallization processes mathematically or to predict their separation efficiency. This is mainly a consequence of the complex heat and mass transfer processes prevailing in the crystallizers, which lead to a nonlinear system of differential equations for the transfer processes. These equations can only be solved numerically and even then require a considerable number of simplifying assumptions and boundary conditions. Nevertheless, for pure substances, the prediction can be carried out. However, in real processes, multicompound mixtures have to be purified, and the existing models are not able so far to predict either the

Melt Crystallization

thermodynamic or the kinetic owing to interaction of the substances within the mixtures. The first important part of every crystallization operation, namely the nucleation step, reveals the same mechanisms underlying the kinetics in the fields of melt and solution crystallization. Therefore, the reader is referred to Chapter 3, which gives a detailed introduction to the corresponding theoretical background. Crystal growth kinetics, by contrast, differ in that the heat transfer is the dominant mechanism controlling the process in case of melt crystallization, whereas in solution crystallization it is the mass transfer. This, of course, is also reflected in the equations describing the process. In both layer and suspension crystallization, solid material forms from the melt starting with a nucleus through which a solid–liquid interface is created. As crystallization proceeds, the mass of solidified substance steadily increases, which causes the interface to move. The impurity components remaining in the melt thereby enrich in front of the solid–liquid interface, forming a concentration boundary layer. Furthermore, the concentration profile in this boundary layer changes as the interface advances, which in the literature is referred to as the moving-boundary problem. Based on early publications, Wintermantel and Kast (1973) developed the v/k criterion, which enables prediction of the maximum allowable growth rate v at which the layers still grow essentially pure (e.g., without incorporating liquid inclusions as a function of the mass transfer coefficient k). In a later publication, Wintermantel (1986) introduced an equation that is founded on the v/k criterion and describes the relationship between physical properties of the substance, the parameters of the process, and the separation efficiency more successfully for the case of eutectic-forming aqueous and organic mixtures

keff ¼ 1  with

ci  c∞   c∞ eðvcr =kÞðρs =ρl Þ  1 keff ¼

ccr c∞

ð9:2Þ

ð9:3Þ

where keff is the aforementioned effective distribution coefficient, ci is the concentration at the solid–liquid interface, ccr is the average concentration in the layer, c∞ is the concentration in the bulk, vcr is the velocity of the interface normal to the crystal surface, k is the mass transfer coefficient, ρs is the solid density, and ρl is the liquid density. Unfortunately, the concentration at the solid–liquid interface ci changes with time and usually is unknown, which makes calculation of keff impossible. However, Wintermantel (1986) was able to show that Equation (9.2) provides a valuable means to uniformly describe the relationship between the keff and the process parameters if it can be shown experimentally that ci is only a function of vcr, k, ρs, and ρl. The applicability of this approach could be successfully shown by a number of authors (e.g., Özoğuz 1992; Chianese and Santilli 1998). Its advantage is that the user does not have to worry about the special process and its operating conditions because these parameters are not contained in the equation. As mentioned earlier, however, this approach also neglects the coupling of heat and mass transfer in crystallization processes. For this reason, the prediction potential of the approach is limited to systems that exhibit a rather weak dependence between heat and mass transfer. A second kind of approaches avoids this disadvantage by taking the mutual dependence of heat and mass transfer into account. This approach was introduced by Rutter and Chalmers (1953) and is referred to as the gradient criterion. Their theory is based on the conditions shown in Figure 9.5. Figure 9.5 Temperature and concentration profile in front of advancing phase boundary as reason for constitutional supercooling Source: According to Scholz 1993.

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As can be seen in the figure, the solidification of material onto the growing solid layer leads to the rejection of impurities, which, in turn, are enriched in front of the solid–liquid interface. These impurities form a concentration boundary layer because the diffusion of the impurities back into the mother liquor usually is slower than the actual crystallization. To this concentration profile, a profile of equilibrium temperatures corresponds, resulting from the phase diagram. Depending on the specific process conditions, the real temperature can be lower locally than the equilibrium temperature, leading to an additional supercooling in front of the solid–liquid interface. This effect is called constitutional supercooling (see shaded area in Figure 9.5) and leads to dendritic growth of the layer, which, in turn, causes the incorporation of impurities into the crystalline layer, thus resulting in an impure product. To quantify the conditions under which no constitutional supercooling occurs (i.e., pure layers are produced), Rutter and Chalmers (1953) demand the gradient of the real temperature at the solid–liquid interface to be greater than or equal to that of the equilibrium temperature ∂Teq ∂T ≤ ∂x ∂x

ð9:4Þ

with ∂Teq vcr ρs ðvcr =kÞðρs =ρl Þ ¼ m e ∂x D ρl

ð9:5Þ

∂T α ¼ ðTi  T∞ Þ ∂x λ

ð9:6Þ

and

where Teq is the equilibrium temperature, T is the actual temperature, Ti is the temperature at the solid–liquid interface, x is the distance from the cooled surface, m is the linear gradient in the phase diagram, D is the diffusion coefficient, α is the heat transfer coefficient, and λ is the heat conductivity. The strictness of this criterion could be reduced by subsequent investigations, such as, e.g., that of Scholz (1993). The experiments revealed that the growth of pure layers is also possible under constitutional supercooling conditions (gradient criterion) as long as a certain maximum level of supercooling is not exceeded. However, even applying this weaker criterion often leads to uneconomical growth rates in the industrial process. For this reason, the optimum between an economical growth rate and a still acceptable product purity has to be searched for in every process and substance mixture. A modified version of this criterion was developed by Mullins and Sekerka (1964). The authors attribute the incorporation of impurities essentially to the nonplanar growth of the crystalline layer and analytically describe this by superposing a planar layer with a sinusoidal disturbance. Afterwards, they determine the conditions under which this disturbance is damped and derive a criterion that generally guarantees such conditions during the process. The applicability of this stability criterion was proven for a variety of metallic and organic compounds.

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The disadvantages of the approaches just described, namely the v/k criterion and the gradient criterion, are that they both do not provide an analytical relationship between the separation efficiency and the process parameters. For instance, they still rely on experiments. Furthermore, they assume planar growth of the crystalline layer, which often contradicts reality. This nonplanar growth of the crystalline layer has liquid inclusion of impure mother liquor as a consequence and hence leads to an impure layer. However, these inclusions do not necessarily remain in the layer because the temperature gradient always present in the layer causes a migration of the inclusion toward the warm side of the layer. The migration rate of these inclusions is of special interest because it determines the time necessary to remove the inclusions from the layer. For this reason, researchers in the field developed correlations to predict the migration rate as a function of process parameters. Assuming that diffusion in the inclusion is the determining factor in the migration of liquid inclusions and that the influence of convection is negligible, Wilcox (1968) gives the following relationship for the influencing parameters: D

∂2 c ρ ∂c þ vmig s ¼ 0 2 ∂x ρl ∂x

ð9:7Þ

where vmig is the migration rate. By integrating Equation (9.7) twice with the corresponding boundary conditions and further introducing the experimentally found restriction, that all inclusions are smaller than 100 µm, Scholz (1993) finally derived the following correlation for the migration rate:
 ρl D dT vmig ¼ ð9:8Þ ρs mccry dx where ccry is the concentration at the crystallizing side of the inclusion. Equation (9.8) is based on experiments with different water–sodium chloride mixtures and yields good agreement between experiments and calculations. Henning and Ulrich (1997) and Henning, Ulrich, and Niehörster (1996), however, conducted detailed experiments regarding the migration of liquid inclusions for various caprolactam–water mixtures and found not only that the inclusions can be considerably larger than 100 µm but also that they grow and change shape during migration. The authors therefore indicate a growing demand for a correlation that accounts for these factors. The effect of the migration of liquid inclusions theoretically enables the creation of completely pure crystalline layers, namely if the migration rate is of the same order of magnitude as the crystal growth rate. If the migration rate of the inclusions is considerably lower than the crystal growth rate, crystallization will yield a layer still containing inclusions. In order to remove them, additional purification steps, so-called post-crystallization treatments, such as sweating and/or washing will be necessary (see Section 9.6). Figure 9.6 shows the different possible origins of inclusions in the layer as well as measures to reduce or even prevent them. Along with this information in Figure 9.6, the potential

Melt Crystallization Figure 9.6 Mechanisms of impurity incorporation in a crystal layer as well as measures to prevent them and additional purification measures (post-crystallization treatments) Source: According to Scholz 1993.

of the postpurification steps to partially or fully remove the inclusions is listed.

9.5 Processes of Melt Crystallization Concepts of processes of melt crystallization can be divided into two lines of technology: solid layer and suspension crystallization. In Sections 9.5.1 and 9.5.2, the basics of solid layer as well as suspension crystallization will be given. Detailed overviews of the different designs of existing and commercially available plants are discussed in Section 9.7

9.5.1 Solid-Layer Crystallization A solid-layer type of crystallization from the melt is often called progressive freezing (see, e.g., Jančić 1989; Ulrich and Özoğuz 1990) or directed crystallization (Ulrich 1988) and directed solidification (Smith 1988). All expressions describe a crystal layer growing perpendicular to a cooled wall into the bulk of a melt (so-called mother liquid). Thereby, the phase change is used as the basis for separation of the feed mixture. This is possible due to different equilibrium concentrations of the solid and liquid phases of a mixture (see Section 9.3). The driving force for crystal growth is the temperature difference between the equilibrium temperature of the melt (the bulk) in front of the solid layer and the temperature of the cooled surface (see Figure 9.7). The latent heat of crystallization is transported through the solid crystal layer and the cooled surface. Therefore, extremely fast (up to 10−4 m/s) crystal growth rates are achievable. In general, solid-layer crystallization processes are carried out in more than one purification stage (so-called multistage). Furthermore, one purification stage is a stringing together of three single steps (phases):

Figure 9.7 Temperature and concentration profile in front of the solid layer Source: According to Neumann 1996.

1. Crystallization 2. Post-crystallization treatment (see Section 9.6) 3. Melting The stringing together of these three phases in one purification stage is illustrated in Figure 9.8, which shows the sweating (see Section 9.6.1) as post-crystallization treatment. As can be seen in Figure 9.8, solid-layer crystallization is an alternate cooling and heating process. For example, in phase 1, the melt is cooled by the heat exchange surface in order to initiate the crystallization as well as to maintain the supercooling (driving force). In phase 2, however, the surface is slightly heated to initiate the sweating, and finally, in phase 3, the surface completely heated for a total melting of the crystallized material (in most cases the desired product). Besides sweating as post-crystallization treatment (shown in Figure 9.8), further postpurification steps such as washing can be and are conducted in a solid-layer crystallization process.

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•

•

•

Figure 9.8 The principle of the purification stage of solid-layer crystallization Source: Copyright © 2002, Sulzer Chemtech. Ltd.

Therefore, Figure 9.9 shows a flow diagram indicating all liquid streams and all possible operational steps in a solid-layer process. The main advantages of solid-layer crystallization techniques are as follows: •

• • •

• • •

No incrustation problems because the incrustations are the solid layers, which are the product; respectively, the waste and the concentrated residual melt are in this case the product. The formation and removal are controlled by equipment operating strategies. Good, controllable crystal growth rates (layer growth rates) due to a good, controllable driving force (the temperature difference through the cooled surface). Hardly any solid–liquid separation problems due to draining of the liquid by gravity and separately discharging the liquid residue and the remolten solid layer. Easy possibilities for post-crystallization treatment of the crystal layer on the cooled surface. Treatments such as sweating and washing should lead to a further purification of the product (see Section 9.6). The amount of impure liquid entrapped in the layer or sticking on the surface of the crystal coat should be reduced by these treatments. Easy to operate equipment due to no slurry handling in the equipment. This means that other than pumps, there are no moving parts in the process. Easy staging opportunities because the product leaving the process needs no treatment in order to be ready for the next production step. Easy scale-up because just cooling surface (additional tubes or plates) needs to be added.

The limitations are to be found mainly in the following points: •

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The limited surface area of the cooled surface and thereafter the surface area of the solid layer because the surface area is a limiting factor for mass transfer processes. This results in bigger plant sizes (investment costs).

•

Solid-layer crystallization is in general a multistage process to gain the desired purity (owing to constitutional supercooling because of fast growth rates). For this reason, the efforts at crystallization, the energy consumption, and the required plant size are increased. The solid-layer formation by the product on the heat transfer surface requires either an increase in the temperature driving force to maintain the same growth rate (therefore production rate) or a reduction in yield and speed with increasing crystal layer thickness. It is a limiting point that the product sticks like an encrustation on the cooled surfaces of the apparatus and has to be remelted to be discharged. This requires not only energy for melting the crystal coat but also an additional partial heating up of the whole apparatus. By contrast, this remelting makes it necessary to have batch processes. In some cases it can also be a disadvantage for the product to leave the apparatus in liquid form and to be solidified again.

The last two points could well be avoided someday if it becomes possible to build continuously operating processes in one plant for solid-layer melt crystallization processes (see Section 9.7.1).

9.5.2 Suspension Crystallization Contrary to solid-layer crystallization, the solid product in suspension crystallization is present as crystals freely suspended in the melt. This type of melt crystallization is characterized by the generation of suspension as well as the separation of the crystals from the liquid residue (solid–liquid separation). For the generation of suspension, the crystallization is often initiated on cooled surfaces, as in solid-layer crystallization. Subsequently, however, the crystals are periodically scraped off. Hence most of the crystal growth occurs on the crystals suspended in the melt (see Section 9.7.2). The advantages of suspension crystallization are as follows: • •

• •

Because of moderate crystal growth rates and the large surface areas, very highly pure crystals can be created at the best in just one step. The crystals exist as crystalline particulate solids as the final product. Hence the remelting of the solid phase to clear the apparatus, as necessary in solid-layer crystallization, is not necessary here. This saves energy and also the potential process step of resolidification in pastilles or other forms as requested by customers. The plants are more compact. The operating mode is continuous/semicontinuous (see Section 9.7.2).

The limitations in suspension crystallization are to be found mainly in the following points: •

Problems occur in slurry handling and in dealing with the moving parts of the apparatus and encrustation. All the aforementioned limitations are connected in one way or

Melt Crystallization Figure 9.9 Flow diagram of possible process steps in solid-layer crystallization Source: According to Özoğuz 1992.

•

•

•

another with each other. In other words, special pumps to handle the suspensions, stirrers to avoid sedimentation, and discharge units are necessary and might lead to problems. The second set of problems occurs when preparing the final product. The very pure crystals must be separated from the remaining highly contaminated liquid (solid–liquid separation). This requires unit operations such as filtration, centrifugation, and washing by either a solvent (not wanted in melt crystallization) or the melt, which is in most cases purified product or feed material. Afterwards, a drying step is required, during which the remaining impure melt or the washing liquid, respectively, may evaporate or solidify on the surfaces of the pure crystals. During this last step, caking is often a problem that has to be addressed. The additional operations are costly, need energy, contaminate already purified products, or reduce the production capacity from reflux of pure product as wash. Only moderate growth rates can be achieved compared with the very high rates in solid-layer crystallization. This is due to isothermal growth, whereas in the solid-layer technique, high thermal gradients can be forced on the crystals within the layer. Finally, but very important, are the limitations in many of the suspension techniques because of the high viscosities of melts and the very small differences in densities between the melt and the crystals. These difficulties always occur when the process depends on the natural settling velocities of the solids within the melts.

9.5.3 Choice The choice of one of the aforementioned two processes depends primarily on the system to be purified as well as the throughput. In this context, of course, economical aspects have to be considered carefully, too. The main features of the solid-layer and suspension crystallization techniques are summarized in Table 9.1. As can be seen in the table, both techniques have positive and limiting features as well. In general, it can be stated that an

advantage or disadvantage of one of the two types of melt crystallization processes is the disadvantage or advantage of the other. However, solid-layer and suspension-based crystallization processes are used for the treatment of many different substances in the laboratory and at industrial scale. Therefore, there is no one best technology for all separation problems in melt crystallization, but in each case, based on the boundary conditions and product specifications, one or the other technique could be the better choice. As a consequence, the arguments have to be weighed in each individual case.

9.6 Post-Crystallization Treatments In order to obtain the desired purity of the product, additional purification steps are necessary as mentioned earlier owing to adhering or captured liquid inclusions of the residual melt in the crystals after the solid–liquid separation. These purification steps are either a repetition of the first crystallization step or additional so-called post-crystallization treatments. Thus final product purity is the result of different procedural steps, including crystallization in one or more stages, the solid–liquid separation, and/or additional post-crystallization treatments such as sweating and washing. Therefore, it is important to have information available on the purification potential (separation coefficient) of the integral processes as well as of the different procedural steps. In solid-layer crystallization, for example, the melt is seen only in front of the layer. In suspension crystallization, however, the crystals are always surrounded by impure melt. It is therefore easier to explain the three mechanisms of impurity incorporation using the solid-layer process, but the same mechanisms also exist in suspension crystallization as well (see Figure 9.6). These three mechanisms are 1. Nucleation (initiation step) on cooled surface 2. Crystallization (solid-layer growth) perpendicular to the cooled surface 3. End of crystallization (draining of the residual melt) Figure 9.6 showed that impurities caused by nucleation and crystal growth can be reduced to a minimum by controlling the

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Joachim Ulrich and Torsten Stelzer Table 9.1 Comparison of Solid-Layer and Suspension Crystallization

Feature

Solid-layer crystallization

Suspension crystallization

Apparatus

No moving parts besides pumps

Moving parts, especially scrapers

Operating mode

Batchwise

Continuous (predominantly)

Temperature of melt

Above but close to solidification temperature

Below solidification temperature

Melt flow rate

Large

Small

Heat withdraw

Through crystal layer

Through the melt

Crystal growth rate

Large, 10−5 to 10−7 m/s

Small, 10−7 to 10−8 m/s

Relative interface melt–crystal

Small, 10–102 m2/m3

Large, 104 m2/m3

Transportation of product

No problems, all liquid

Problems due to suspensions

Solid–liquid separation

Easy, just draining

Difficult

Encrustation problems

No

Yes

Scale-up

Easy

Difficult

Source: According to Ulrich and Bierwirth 1995.

process parameters of crystallization. Additional postcrystallization treatments such as sweating and washing can improve the purity to the same extent as can be achieved just by a crystallization step. However, an additional crystallization step is more costly than postcrystallization treatment. Therefore, postcrystallization treatment can be the better alternative compared with an additional crystallization step to gain the desired purity.

9.6.1 Sweating Sweating is defined by Ulrich and Bierwirth (1995) as a temperature-induced purification step based on a partial melting of crystals or crystal layers by means of a warm gas or heating of the cooled surface up to close to the melting point of the pure component (about 1–2 K below). As a consequence, the impurities adhering to the crystal surface and those contained in pores of the crystalline material remelt (partially diluted with pure material) and then drain under the influence of gravity. The temperature rise along with the sweating simultaneously reduces the viscosity of the impurities and thus further eases the draining off. A process step of this kind is used with the knowledge that it leads to about 10 percent product loss (according to Jančić 1989). The liquid film sticking to the crystal surface or crystal

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layer surface, on the one hand, and the liquid inclusions, on the other hand, are highly contaminated and hence reduce the overall purity of the product considerably. Therefore, from an economical point of view, it is worthwhile to reduce the amount of product by the sweating step and raise the purity at the same time. However, it is not always recommended to conduct a sweating operation in the fastest possible way, according to Ulrich and Özoğuz (1990). Nevertheless, the sweating step (about 5 minutes) still remains much faster than a crystallization step (2–3 hours). Furthermore, it does not require another phase transformation, as does the latter. Hence it saves a corresponding amount of energy, especially if layer processes are considered, where a final heating is necessary in any case to remove the crystal layer from the cooled surfaces. In addition to the advantage of draining off the attached liquid film, the important part of the purification effect with sweating is that the liquid inclusions have a lower melting temperature, respectively, solidification temperature, than the purer material. At temperatures where the product compound is still solid, the impurity is already liquid or becomes liquid and therefore will drain out of the pores and crystal coats. This process is supported by the decrease in viscosity due to the higher temperature. According to Poschmann and Ulrich (1996), sweating is also possible with single crystals. The purification effect, however, is much stronger in the case of solid-layer crystallization. According to Ulrich and Özoğuz (1989) as well as Delannoy, Ulrich, and Fauconet (1993), it can even reach the same extent as purification by a crystallization step, as shown in Figure 9.10. A distribution coefficient of zero here means a perfect purification, whereas a coefficient of one means no purification at all. In the figure, the effect of sweating can be seen by a decrease in the distribution coefficient from about 0.4 to 0.2 at a growth rate of about 0.3 × 10−6 m/s, as well as from about 0.6 to 0.3 at a growth rate of about 1.7 × 10−6 m/s. The results shown in Figure 9.10 were obtained for the static mode (stagnant melt) of solid-layer crystallization. The effect of sweating for the dynamic mode (flowing melt) of solid-layer crystallization, however, is in general less pronounced than that of the static mode. Furthermore, the sweating times are also rather different with respect to the processes. Sweating times between 10 minutes in the dynamic Sulzer falling-film process (see Jančić 1989) and 30 hours in the static Hoechst process (see Rittner and Steiner 1985) for the purification of monochlorous acetic acid are examples. As rule of thumb, it can be stated that purification-step sweating is more suitable for the static mode than the dynamic mode because the improvements achieved in distribution coefficient are less than for a static-mode crystallization process. This rule depends, of course, on the mixture (the substances) that is used and therefore has to be considered in individual cases. In summary, the possible advantages by sweating are •

Additional purification can be reached in the same range as in a crystallization step.

Melt Crystallization

Figure 9.10 Purification effect of static (stagnant melt) crystallization and sweating of a crystal layer process considering crystal growth rates (feed mixture of methacrylic acid–water) Source: According to Delannoy et al. 1993.

• •

•

Much shorter retention times of the process (about onethird to one-ninth of a crystallization step) are achievable. There is hardly any additional energy consumption compared with a further crystallization step, which means less energy consumption because no phase transition energy is required, and the product has to be remelted anyway. There is a product loss of about 10 percent of the crystal coat, which should be compared, however, with a yield in a crystallization step that in many cases is about 80 percent of the maximum achievable yield.

The importance of the sweating step is discussed in detail by Jančić (1989), Saxer, Stadler, and Ignjatovic (1993), and Ulrich and Nordhoff (2006), the former of which also gives examples of overall product recovery and temperature curves for a multicompound mixture of impurities for the purification of benzoic acid. A theoretical treatment of the sweating process is presented by Wangnick (1994) and Kim and Ulrich (2002) for layer melt processes. Wangnick (1994) derives a correlation for the prediction of additional purification that can be achieved by sweating. The equation accounts for the physical properties of the mixture of the substances under consideration as well as for the history of the layer and reads as follows: keff;sw ¼ 0:028  0:277Psw þ 1:235P2sw

ð9:9Þ

Dkeff ¼ keff;before  keff;after

ð9:10Þ

with

and Psw ¼ keff;before

Tsw Teq

ð9:11Þ

where Δkeff is the difference of the effective distribution coefficients before and after sweating, Пsw is the characteristic dimensionless number of sweating (sweating number), keff,before is the effective distribution coefficient before sweating, corresponding

to the effective distribution coefficient of the preceding crystallization step, keff,after is the effective distribution coefficient after sweating, Tsw is the sweating temperature, and Teq is the equilibrium temperature. The application of this equation to data gained from experiments with p- and o-dichlorobenzene in different plant types (static/dynamic) and at different sweating temperatures leads to a uniform representation for all setups. Some exemplary results are shown in Figure 9.11 and demonstrate the usefulness of this approach using a dimensionless number based on the physical background of the sweating process.

9.6.2 Washing When talking about washing, a distinction must be made between rinsing and diffusion washing as defined, for example, in Ulrich and Bierwirth (1995). Rinsing occurs when the highly contaminated residual melt that is attached to the crystal or crystal-layer surface is substituted with a film of the purer rinsing or washing liquid. The washing liquid consists of pure desired product or a purer mixture (e.g., feed mixture) than the residual melt. When pure product is used as washing liquid, a certain amount of purified product will be contaminated again in order to purify the crystal coat. However, the amount of contaminated deployed pure product as washing liquid is small (~10 percent) compared with the yield of pure product. The post-crystallization rinsing typically requires residence times in the range of only seconds and is driven by mechanical forces (pushing away impure product and substituting it with pure product). The second washing process, however, diffusion washing, effectively operates at residence times longer than 15 minutes. Here the washing liquid causes a liquid–liquid diffusion of impurities from the pores of the crystalline material into the washing liquid. To prevent solidification of pure material onto the crystals or the crystal layer, the washing liquid must be superheated. For this reason, diffusion washing is always, at least to some extent, accompanied by sweating, which further increases the efficiency of the purification effect. During diffusion washing, all three phenomena are involved: rinsing (which takes place when draining the washing liquid), diffusion, and sweating (see above), which increases the purification effect of this post-crystallization treatment. Furthermore, diffusion washing, compared with sweating and rinsing, is a concentration-driven process. However, all three additional purification steps have one thing in common as well: about 10 percent loss of product, which has to be repurified afterwards. Parameters that have to be considered to evaluate the purification efficiency are the purity, the temperature, the amount, and the flow regime of the washing liquid. Therefore, optimization of all these parameters is always necessary and has to be done for each compound in the laboratory (see, e.g., Lüdecke, Brendler, and Ulrich 2003; Neumann 1996; Poschmann and Ulrich 1996; Ulrich, Bierwirth, and Henning 1996; Ulrich and Neumann 1997; Wangnick 1994; Wangnick and Ulrich 1994).

277

Joachim Ulrich and Torsten Stelzer Figure 9.11 Difference in distribution coefficients of a sweating step as a function of sweating number Source: According to Wangnick 1994.

The efficiency of a washing operation depends, of course, on the quality, which means purity as well as smoothness, of the crystal layer. In summary, the possible advantages of washing (rinsing and diffusion washing) include • • • • •

Additional purification Much shorter retention times (rinsing: few seconds; diffusion washing: 15–20 minutes) compared with a further crystallization step (several hours) Less energy consumption than a crystallization step (just pumps and temperature control) No additional contamination by solvents (because pure product or feed is used, which has to be purified again afterwards) About 10 percent product loss, which should be compared with a crystallization step yield of about only 80 percent

In order to predict the effect of purification by rinsing or by diffusion washing, Wangnick (1994) introduced, similar to the sweating number, characteristic dimensionless numbers for rinsing (rinsing number) and for diffusion washing (diffusion washing number). The dimensionless rinsing number is Prins ¼

A cml s2 c∞

ð9:12Þ

where Пrins is the characteristic dimensionless number of rinsing (rinsing number), A is the surface area of the crystalline layer, s is the thickness of the crystalline layer, cml is the concentration of the adhering mother liquor, and c∞ is the initial concentration of the melt. The dimensionless diffusion washing number is Pdiff ¼

Dt ðcinc;0  cwl Þ keff;before Sc s2 c∞

ð9:13Þ

where Пdiff is the characteristic dimensionless number of diffusion washing, D is the diffusion coefficient, t is the diffusion

278

time, cinc,0 is the concentration in the liquid inclusions, cwl is the concentration of the washing liquid, and Sc is the Schmidt number. These numbers were gained, of course, from a dimension analysis of all influencing parameters (as similarly carried out for the sweating step). More details are given by Wangnick (1994). Wellinghoff, Holzknecht, and Kind (1995) derived a correlation directly from a pore diffusion model based on Fick’s law and obtained an equation that predicts the effective distribution coefficient of diffusion washing as a function of four dimensionless numbers. The apparent differences between the two approaches vanish on closer examination; that is, when neglecting the factors that cannot be arbitrarily influenced owing to inherent process restrictions, the two approaches are essentially alike. They only differ in that Wellinghoff et al. (1995) additionally considered a dimensionless area that corresponds to the ratio of the pore surface to the surface of the crystal layer. This factor can only be determined experimentally and yet is fairly complicated and thus limits the applicability of this approach. In order to predict the separation potential of a crystallization step followed by one or more of the aforementioned postpurification processes, Wangnick (1994) proposed to superpose the corresponding equations using the v/ k criterion to model the crystallization step.

9.6.3 Wash Columns By means of experiments, Poschmann and Ulrich (1996) were even able to show quantitatively that the aforementioned postcrystallization treatments are as efficient for suspension crystallization as they are for layer processes. This is also expressed by the BASF patent (Holzknecht et al. 1988) dealing exclusively with the washing step. In the case of crystal suspensions, the final purity of the product strongly depends on the performance of the solid–liquid separation, which is easy to understand. As a consequence, additional washing (rinsing) of the crystals

Melt Crystallization

with a relatively pure washing liquid is required to achieve a highly purified product. Washing on filters or in centrifuges is also possible but difficult because the temperature of the washing liquid needs to be controlled precisely. Otherwise, the washing liquid will crystallize on the crystals or lead to caking of the crystals (filter/centrifuge cake) or partially melts the crystals and hence remelts product. Furthermore, the washing liquid gets contaminated with impurities and needs to be reprocessed. Therefore, wash columns combine solid–liquid separation and washing in one apparatus in which the temperature can be controlled rather precisely and some of the aforementioned problems do not explicitly arise. The higher quality of wash column as post-crystallization treatment compared with conventional solid–liquid separation (mechanical separation) by centrifuge or by filter press with and without wash streams is illustrated in Figure 9.12. In this diagram, product purity is plotted as a function of impurity concentration within the crystallizer. Furthermore, the product purity gained by washing is plotted on the left ordinate and the product purity achieved without washing is plotted on the right ordinate. Figure 9.12 schematically illustrates how product purity is influenced by the residual moisture of the filter cake for different impurity concentrations in the liquid. The residual moisture is the adhered residue mother liquid after mechanical solid–liquid separation. By means of a filter press, e.g., about 20 percent of the achieved filter cake is residual moisture. A centrifuge typically produces a filter cake with a content of 3–10 percent residual moisture. As can be seen in Figure 9.12, the product purity gained decreases with increasing residual moisture. Thus the limitations of the centrifuge and filter press (mechanical separation) compared with wash columns is shown. Furthermore, the difference between a filter press and especially a centrifuge versus a wash column increases with increasing impurity concentration in the crystallizer. This effect might be explained by the fact that less pure melts lead to higher temperature differences over the column. This results in an increased quantity of wash liquid being forced back into the column.

Moreover, the advantage of using a washing liquid is demonstrated in Figure 9.12. For example, at 20 wt% (typical impurity concentration in the final crystallizer stage) and a residual moisture of 5 percent, the final product purity would only be 99.0 wt% in the case of using a centrifuge without washing. This explains why conventional mechanical separations require intensive washing with melted product to achieve even moderately pure product. As a rule of thumb, about 10–20 percent of the crystal product is needed as washing liquid, which, in turn, can reduce the impurity content by up to two-thirds. For the aforementioned example a final product purity of 99.6 wt% is expected to be achieved in the case of washing. In principle, two types of wash columns can be distinguished: gravity and forced transport columns. However, problems with scale-up and the effect of backmixing on product purities have severely limited their commercial application (Scholz and Ruemekorf 2003). Furthermore, the latter can be differentiated in mechanical and hydraulic operating mode. In Figure 9.13, for example, a mechanical wash column is shown. Subsequent to the suspension crystallization process, the slurry of crystals and melt is fed into the wash column. The residual melt leaves through a filter, which causes a compaction of crystals into a so-called packed bed. This bed is forced through the wash column by solid floating (if the density is lower than that of the residual melt, e.g., ice) or settling (if the density is higher than residual melt in general), as well as by pressure, as transport mechanisms. The packed bed is scraped by rotating scrapers at the end of the column (here at the top). Thereafter, the scraped-off crystals are fluidized in a reslurry section by recirculated molten product and fed in an external heat exchanger. The resulting melt of molten crystals is subsequently split into two streams. The largest portion is removed as product, and the other portion of pure molten product is forced back into the porous packed crystal bed. At this point, it has to be stated that no additional energy is required for the wash column because the crystals have to be molten anyway in order to obtain the desired product. This can be compared with the necessary condensation of

Figure 9.12 Product purities achieved for different solid–liquid separation technologies with (left ordinate) and without (right ordinate) washing Source: Reproduced courtesy of GEA Messo PT.

279

Joachim Ulrich and Torsten Stelzer

Figure 9.13 Principle of a mechanical forced wash column Source: Reproduced courtesy of GEA Messo PT.

Sweating, rinsing, and diffusion washing are also as efficient for suspension crystallization as they are for layer processes. In suspension crystallization processes, however, the application of wash columns is more appropriate. In summary, post-crystallization treatments take less time and, if efficient, have higher yields and less energy consumption than an additional costly crystallization step. It is therefore of the utmost importance to know the potential of the post-crystallization treatments before evaluating melt crystallization as an alternative to other thermal separation, purification, or concentration processes. The efficiency of a purification step can be measured experimentally in the laboratory by determining the effective distribution coefficient before and after the post-crystallization treatment or can be predicted by dimensionless numbers based on process conditions.

9.7 Concepts of Commercial Plants the high-boiling compound in distillation to achieve the desired product. The crystals will be washed within the wash column countercurrently with the molten pure product during their passage to the reslurry section. The washing liquid (pure molten product) crystallizes on the surfaces of the crystals as soon as it contacts the colder crystals (due to impurities, generally 5–20 K colder) from the unwashed part of the bed. Therefore, only minor amounts of pure product are lost from the washing liquid to the filtrate. This horizontally extending zone across the crystal bed is called a wash front. The position of the wash front, which indicates the occurrence of purification, is in some cases visible by a color shift (e.g., freeze concentration of fruit juice). Alternatively, the position can be determined by measuring the temperature in the column. A sharp temperature difference is the indicator. Commercial providers of wash columns are GEA Messo PT and Sulzer Chemtech, Ltd. (mechanically forced), as well as TNO (hydraulically forced) nowadays SoliQz in close cooperation with Armstrong-Chemtec. A more detailed overview on wash columns is given by Arkenbout (1995).

9.6.4 Choices The choice for the user as to which of the aforementioned post-crystallization treatments (sweating, rinsing, diffusion washing, or wash column) to use depends on the system to be purified and has to be made in each individual case, on the one hand. On the other hand, it depends, of course, on the type of crystallization process used (layer or suspension). However, sweating is easy to operate and needs no additional equipment. Therefore, it is always worthwhile to try it if the crystalline coat is not slipping off by its own weight when heated. In contrast, washing is probably more efficient but needs modification of the equipment. Furthermore, in the case of diffusion washing, extreme precise temperature control is needed. Otherwise, too much remelting or a crystallization of the washing liquid takes place because both are only wanted in small amounts.

280

The process basics in melt crystallization were given in Section 9.5. Here, in the following subsections, the concepts of existing and commercially available plants will be addressed. Because plants can be divided into solid-layer and suspension crystallization processes, in industrial applications, these two techniques will be split in continuous and batchwise modes as well as in static (stagnant melt) and dynamic (flowing melt) operating modes. Dynamic or flowing means that the melt is forced to convection.

9.7.1 Solid-Layer Crystallization The most discussed technologies in the literature addressing solid-layer crystallization are illustrated in Figure 9.14. However, not all of these listed techniques are really commercially available. Therefore, the underlined equipment examples in this figure will be discussed in the following subsections in more detail because they are established on the market. The first group of commercially available equipment to be discussed is the batch type of solid-layer processes with stagnant melts (only natural convection). This type of equipment is based on the Hoechst Tropfapparat (German) or, in English, dripping-off apparatus (sweating apparatus) patented more than 100 years ago (see Rittner and Steiner 1985). These crystallizers are commercially available, e.g., as the ProABD type, nowadays Sulzer Chemtech, Ltd. (Ab-derHalden 1960; Ab-der-Halden and Thomas 1962; Genin 2003; ProABD 1972; Schwartz 1982), using tubes, and as the staticplate crystallizer from Sulzer Chemtech, Ltd. (see Figure 9.15). Both processes feature cooled surfaces for crystallization of the melt. The cooled surface for the crystal growth of the solid layer is provided in the static-plate crystallizer by plates. These plates are located in the stagnant melt, as can be seen in Figure 9.15. The melt feedstock is progressively crystallized by cooling the heat-transfer surface area. As the crystallization proceeds, the remaining melt becomes more and more enriched with impurities. The crystallization process in such a crystallizer needs about 2–30 hours. The remaining residual melt is allowed to

Melt Crystallization

Figure 9.14 Solid-layer crystallization equipment subdivided based on operating modes Source: According to Özoğuz 1992.

Figure 9.15 Static-plate crystallizer Source: Courtesy of Sulzer Chemtech, Ltd.

drain after the crystallization is stopped by an opening at the bottom of the apparatus. Unfortunately, a film of highly contaminated melt of residue composition will be held back afterwards on the crystal coat. However, this film, or at least most of it, can be removed by post-crystallization treatments such as sweating (hence the Hoechst dripping off apparatus) or

washing (see Section 9.6). The final product is achieved after carrying out these post-crystallization treatments by melting the crystal coat and collecting it in a different tank than the residue. The Hoechst dripping-off apparatus as well as the ProABD process, nowadays Sulzer Chemtech, Ltd., (see Figure 9.16), both are tube-bundle crystallization equipment that runs in the same way. In principle, it can be stated that every plate or tubebundle heat exchanger can be used as static solid-layer crystallizer, but a few special geometric constructions have to be obeyed. An advantage of this solid-layer technique is the high operating safety resulting from the simple assembly (no moving parts) and the fact that no additional solid–liquid separation is required. The space-time yield of this technique, however, is relatively small. The reason for this is that the heat and mass transfer is forced just by conduction, respectively, diffusion, and is supported only by natural convection. Therefore, only very slow crystal growth rates lead to high purities; otherwise, liquid inclusion of impure mother liquor will be entrapped, thus lowering the purity (see Section 9.4). Because of this, large crystallizers are required to achieve high yields with the batchwise technique of stagnant melt. The efficiency of such solid-layer crystallization processes, of course, can be enhanced by mechanisms designed to improve the heat and mass transfer, on the

281

Joachim Ulrich and Torsten Stelzer

Figure 9.16 ProABD process, nowadays Sulzer Chemtech, Ltd. Source: Reproduced with permission from Rittner and Steiner 1985.

one hand, and by a dynamic operating mode (flowing melt), on the other hand. The second group of batchwise solid-layer techniques consists of those with flowing (forced convection) melts (dynamic) because, owing to the circulated (flowing) melt, the heat and mass transfer (reduction of boundary-layer thickness and, therefore, reduction of the potential for constitutional supercooling) is improved. Here the MWB-Sulzer process, nowadays called the Sulzer falling-film process (see Figure 9.17), must be named (Saxer 1971; Stepanski and Schäfer 2003). The crystallization takes place in this process on the inside of tubes (tube bundle) that are cooled from the outside. This process features, as indicated by its name, a falling film on the inside of the tubes for the melt and outside for the cooling fluid. The melt coming from a feed tank is continuously circulated (pumped) through the tubes until the crystal coat at the walls is thick enough – this means, for example, until the desired percentage of product from the feed is reached (crystallized). Afterwards, as mentioned earlier, the residual melt is drained, and post-crystallization treatments such as sweating and washing can be conducted (see Section 9.6). In this context, two more processes – the Imperial Chemical Industries (ICI) process (Barton 1967) and the BASF process (Wintermantel, Stockburger, and Fuchs 1988) – have to be mentioned besides the Sulzer falling-film crystallizer. The important difference is that the ICI and BASF processes have no falling film, but rather the spaces in the tubes are completely filled by the melt. In addition, the cooling fluid is not running on the outside of the tubes, as in the falling-film process. Hence the ICI and BASF processes are in the subdivision of tube-flow processes. References for

282

currently commercially available dynamic solid-layer crystallization technologies are only published by Sulzer Chemtech, Ltd. (e.g., Stepanski and Schäfer 2003). The stated improved purification efficiency of the dynamic operating mode of solid-layer crystallization compared with the static mode for a single stage is demonstrated in Figure 9.18. As can be seen in the figure, the dynamic process (solid lines) has lower distribution coefficients than the static process (dotted lines) almost over the whole range of growth rates. Only for quite slow growth rates are the static processes able to achieve the same purification (distribution coefficient) as the dynamic operating mode. Plants with flowing melt, however, have the same purification efficiency at much higher growth rates (Neumann 1996). Furthermore, if the resulting purity after one cycle does not reach the desired level, both types of batchwise solid-layer crystallization processes are capable of staging. This is expected and part of the strategy when using such techniques. Staging is quite easy because each stage after another can be conducted in the same crystallizer just by storing the products of different purities in different tanks to increase the purification efficiency. Moreover, a modified version of the Sulzer falling-film process (see Bischof 1997) does not realize the cooling as a falling film but condenses the refrigerant on the outside of the tubes instead. Thus a constant temperature along the length of the tubes is enabled. This, in turn, leads to more homogeneous thicknesses of the crystalline layer and, consequently, improves the efficiency of postpurification steps. These plants with the dynamic operating mode have, aside from pumps, no moving parts and produce a product in liquid form. A further advantage of these plants is their quite easy scale-up, just by adding tubes or a new apparatus of tubes. Some examples of a numerous list of purified materials from the Sulzer falling-film process are acrylic acid, benzoic acid, bisphenol A, methacrylic acid, naphthalene, and xylenol (Stepanski and Schäfer 2003). However, the limitation of the processes discussed so far is that these plants run only in batchwise or semibatchwise operating mode. Therefore, the continuously operating ones, the so-called belt crystallizers, will be considered as a third group of processes. Here the Bremband process of Sulzer Chemtech, Ltd., and the Sandvik process systems (see Figure 9.19) have to be named (Ulrich, Stepanski, and Özoğuz 1992). Given that the position of the conveyer is at a countercurrent angle, solid-layer crystallization is possible. Furthermore, the purification efficiency is also proven by Hünken and Ulrich (1993) as well as Ulrich, Stepanski, and Özoğuz (1992). The advantages of these processes are no dead time through filling and draining of the equipment, as well as no waste of energy by cooling and heating of the heat transfer medium and the continuously countercurrent flow (forcedconvection) operating mode. The reason for not explaining all the details of this innovative concept of solid-layer crystallization technology is that this technology, so far, has not been established on the market.

Melt Crystallization Figure 9.17 Flow diagram for a typical falling-film crystallizer Source: Courtesy of Sulzer Chemtech, Ltd.

Figure 9.18 Correlation between distribution coefficient of solid layer crystallization process in static (dotted lines) as well as dynamic (solid line) operating modes (single stage) and crystal growth rate (feed mixture of caprolactam–cyclohexanone) Source: According to Neumann 1996.

However, the belt equipment is established on the market as solidification technology in order to improve the product handling as solid instead of liquid. Prominent providers are Sandvik Materials Technology (Processing Division), Kaiser Steel Belt Systems, and Berndorf Band GmbH.

9.7.2 Suspension Crystallization Equipment or concepts in suspension crystallization that are discussed the most in the literature are illustrated in Figure 9.20. However, not all of these techniques are commercially available. The underlined examples of equipment in the figure will be discussed in greater detail because they are established on the market. In the group of suspension processes, there are the directly cooled (e.g., by inert gases or immiscible coolant) and the jacket-cooled processes. For directly cooled processes, e.g., the

Figure 9.19 Bremband process, continuous solid-layer process

Newton–Chambers process (Molinari 1967) should be mentioned here. The advantage of these directly cooled processes is the energy efficiency. The disadvantages, however, are problems with separating the coolant gas completely from the crystals and that the gas bubbles hamper crystal growth by occupying the crystal surface. Plants were built in 1964 for the purification of benzene (according to Molinari 1967). Despite the directly cooled processes, the crystallization in suspension processes is usually initiated on cooled surfaces, as in solid-layer crystallization. Afterwards, the crystals are

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Figure 9.20 Equipment for suspension crystallization subdivided based on operating modes Source: According to Özoğuz 1992.

periodically scraped off, and most of the crystal growth occurs on the crystals suspended in the melt. Therefore, these socalled scraped-surface heat exchangers (SSHEs) are in fact solid-layer plants, but because of the handling of the products – crystals in suspension – they have to be treated like suspension processes. In this technique, the melt gets supercooled at the surface of the SSHE. Subsequently, the supercooled melt is mixed by the scraper into the bulk of the melt and provides the growth potential. Thus the crystal surface area available for growth is the total surface area of a very large number of crystals (up to 104 m2/m3) and is not limited to the cooled surface area, as in solid-layer crystallization (up to 102 m2/m3). Therefore, the crystal growth rates can be much lower and still have the same yield. Furthermore, the lower growth rates lead to much purer crystals than the solid-layer crystallization technology if, however, just a single-stage (non-multistage) process is considered (see Figure 9.21). At this point, it has to be clarified that given the results shown schematically in Figure 9.21, solid-layer processes are designed as multistage plants in order to gain almost the same purification efficiency as the suspension technology (see also Sections 9.5.1 and 9.7.1). Of course, if only a singlestage process is considered, the product purity achieved by a suspension process (with all the advantages and

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Figure 9.21 Schematic comparison of purification ratio of suspension and solid-layer crystallization technologies versus impurity concentration in a singlestage crystallization process Source: Reprinted courtesy of GEA Messo PT.

disadvantages) is much better than that of solid-layer plants (with all the advantages and disadvantages). Nevertheless, the only criteria for choosing between these technologies are those listed in Table 9.1. Commercially available SSHEs can be obtained, for example, from Armstrong Engineering, Inc. (see Figure 9.22),

Melt Crystallization

Figure 9.22 SSHE with rotating disks Source: Reproduced with permission of Armstrong Engineering, Inc.

Brogtec Mischtechnik GmbH, Fluitec Georg AG, GEA Messo PT, HRS-Spiratube S.L., Royal GMF Gouda, and Sulzer Chemtech, Ltd. (Stelzer and Ulrich 2009). However, the solid–liquid separation of these apparatuses is the limitation. Despite this flaw, the simple construction and operation of the technology feature their application. Given the aforementioned limitations of solid–liquid separation, these technologies are more frequently used to generate the initial crystals in a combination of process steps. The simplest type of combination is that provided by GEA Messo PT (the former Grenco, later Niro PT), shown in Figure 9.23. The principle of the former Grenco piston wash column is based on the generation of crystals on the cooled wall of the SSHE (in Figure 9.23, just named crystallizer). Subsequently, the crystals are scraped off and grow during their suspension within the SSHE. If the crystals have the desired size, a mature crystal suspension (consists of suspension with desired crystal size) is almost continuously (semicontinuous) discharged into the mechanical forced-wash column. Here the post-crystallization treatment takes place, as discussed in Section 9.6.3. Because of the continuous generation and semicontinuous discharge of crystals, the suspension density is kept almost constant. A similar design of suspension crystallization by GEA Messo PT is provided by Sulzer Chemtech, Ltd. (the former Freeze Tec B.V.) and SoliQz. In this concept, an additional so-called growth vessel is coupled with the SSHE (see Figure 9.24). As the name of the growth vessel suggests, the created crystals (in the SSHE or just crystallizer) have time to increase in size (grow) in the vessel. Furthermore, the crystals are kept evenly suspended within the vessel by an agitator. Therefore, it is called suspension crystallization. As can be seen in Figure 9.24, the crystal suspension is continuously circulated through the crystallizer and the growth vessel. Hence new crystals are generated continuously, and the mature crystal suspension (suspension with desired crystal size) is semicontinuously discharged out of the growth vessel to undergo further post-crystallization treatment, here by means of a wash column (see Section 9.6.3). The two suspension crystallization concepts just discussed count for the continuous packed columns with mechanical forced transport. There are numerous examples of purified

Figure 9.23 The principle of process flow of a suspension crystallization process with SSHE and wash column Source: Reproduced courtesy of GEA Messo PT.

materials produced by these two types of suspension crystallization, some of which are, for example, acrylic acid, caprolactam, naphthalene, para-nitrochlorbenzene, para-xylene, and 4,4’-diphenylmethane diisocyanate (Dette 2010; Scholz and Ruemekorf 2003). In a final group of commercially available design concepts of suspension crystallization, the so-called unpacked column has to be introduced. The unpacked columns also count as continuous suspension crystallization technologies. Many other patents, for example, the Brodie purifier (Brodie 1971), and papers (see, e.g., Ulrich et al. 1996) on suspension melt crystallization columns exist on the market today. However, only the double-screw column, nowadays called the Kureha double-screw purifier or the KCP column (see Figure 9.25), has a significant market share and therefore must be named (Yamada, Shimizu, and Saitoh 1982). The principle of a KCP column is that it works with a crystal slurry introduced at the bottom of a column. The crystals are transported from the bottom to the top of the column (mechanical forced transport) by a double-screw conveyer. At the top of the column, the crystals are melted (molten product) by a melting device. Some part of the molten product is used as reflux. The reflux washes the crystals while moving in countercurrently. Therefore, a KCP column is more or less a wash-melt column. Some highly concentrated residue is taken out at the bottom. The KCP column is a proven design concept in Japan with about 20 applications, e.g., p-dichlorobenzene and carboxylic acid. The process is described in more detail by Rittner and Steiner (1985).

9.8 Eutectic Freeze Crystallization The eutectic freeze crystallization process (EFC process) is a novel crystallization technology designed to recover valuable dissolved salts or acids and water from aqueous streams. The benefit of the EFC process compared with the conventional separation technologies, e.g., reverse osmosis, evaporative or cooling crystallization, and freeze concentration, is that saline water can be recovered in an ecologically and economically sustainable way. With use of the EFC process instead of a three-stage evaporative crystallization, costs (e.g., the energy) can be reduced by up to 70 percent (Vaessen, van der Ham, and

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Figure 9.24 Process flow diagram of a suspension crystallization process with SSHE, growth vessel, and wash column Source: Courtesy of Sulzer Chemtech, Ltd.

Figure 9.26 Principle of the EFC process shown in a binary eutectic phase diagram of a brine Source: According to Genceli 2008.

Figure 9.25 Kureha double-screw purifier Source: Reproduced courtesy of Kureha Engineering Co., Ltd.

Witkamp 2000). Furthermore, the yield increases up to almost 100 percent with two separated pure phases, usually a salt and water (ice). On this account, the EFC process changes the focus from the costs to the values by converting waste into raw materials in an energy-efficient process and in accordance

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with physical calculations (Genceli 2008). The principle of the EFC process is illustrated in Figure 9.26. The principle of the EFC process is cooling an unsaturated solution (point A in Figure 9.26) below its freezing point because, for crystal growth, a driving force (supercooling) is required. This means that the crystallization takes place within the metastable zone (not shown in Figure 9.26). At the liquid line itself, no crystallization will occur because of thermodynamic equilibrium. However, at point B, (supercooled) ice

Melt Crystallization

crystals start to grow. Further cooling decreases the temperature and increases the concentration of the saturated solution by growing ice crystals along the line between points B and C (also supercooled) because a driving force is still required. If point C is reached, salt crystals start to grow simultaneously with the ice crystals. The EFC process also can be started if the solution has a higher initial concentration than the eutectic composition (mirror inverted). In this case, however, salt crystals will be formed first. The ice crystals start to grow if point C is reached. Separation of the two generated phases (ice and salt) takes place because of differences in density. Whereas the salt is settling, the ice crystals are floating on the saturated solution. More details on the EFC process can be found in Stepakoff et al. (1974), Chowdhury (1988), Vaessen et al. (2000), Genceli, Gärtner, and Witkamp (2005), and Genceli (2008). The references are listed in chronological order to indicate the development of the EFC process.

9.9 Summary and View to the Future One benefit of melt crystallization is that no additional substances compared with extraction or solution crystallization is needed. To enhance the yield, however, a shift of the eutectic point by adding a further substance (so-called additive) can be appropriate. Such a method is patented by Standard Oil Company (Paspek and Every 1980) for the acrylic acid–water mixture. The continuous/semicontinuous operating mode is the state of the art in the concept of suspension crystallization of melts because this operating mode ensures high throughputs as well as yields of the desired pure products in a more economical manner than the batchwise operating mode. Economical

References Ab-der-Halden, C. (1960). Great Britain Patent No. 837,295, London. Ab-der-Halden, C., and Thomas, B. E. A. (1962). Great Britain Patent No. 899,799, London. Arkenbout, G. J. (ed.). (1995). Melt Crystallization Technology. Lancaster, PA: Technomic Publishing Company. Atwood, G. R. (1972). In N. I. Li (ed.), Recent Developments in Separation Science (vol. 1, pp. 1–33). Boca Raton, FL: CRC Press. Barton, E. (1967). Great Britain Patent No. 1,083,850, London. Bischof, R. (1997). European Patent No. 0811410 A1, Brussels. Brodie, J. A. (1971). Mech. Chem. Eng. Trans. 7(1), 37–44. Burton, J. A., Prim, R. C., and Slichter, W. P. (1953). J. Chem. Phys. 21(11), 1987–91. Chianese, A., and Santilli, N. (1998). Chem. Eng. Sci. 53(1), 107–11. Chowdhury, J. (1988). Chem. Eng. 25, 24–31.

means less investment as well as operating costs with a high-aspossible space-time yield. Especially, in the pharmaceutical industry, a shift away from batchwise toward continuous processes is expected in the future (Pavlou et al. 2010). In solidlayer crystallization, however, the continuous operating mode did not achieve significant market share, although concepts of continuous processes exist in the literature, e.g., the Bremband process (Ulrich et al. 1992). Nevertheless, the number of applications of melt crystallization (solid layer as well as suspension) is growing. However, in the future, a paradigm shift has begun in the design of chemical, pharmaceutical, and food processes away from single plants toward so-called hybrid processes. Hybrid processes combine several separation techniques (e.g., distillation and melt crystallization) to enhance throughput, heat and mass transfers, and reaction rates. Furthermore, such hybrid processes are also used to separate or purify over the full range of concentrations because sometimes this is not possible with only one technique (e.g., melt crystallization or distillation), namely in case of azeotropic or eutectic mixtures. Therefore, hybrid processes are able to solve such problems. Hybrid processes are a subset of the so-called process-intensification techniques. Process intensification paves the way for reducing the size of plants. Therefore, the paradigm shift leads to a “greener” chemistry and engineering that enables sustainable development. Furthermore, process intensification results in lower investment and operating costs. The important thing, however, is that melt crystallization is part of process intensification. An example of a hybrid process consisting of distillation and melt-suspension crystallization is the purification of 4,4’-diphenylmethane diisocyanate (MDI; Dette 2010).

Delannoy, C., Ulrich, J., and Fauconet, M. (1993). In Z. H. Rojkowski (ed.), Proceedings from 12th International Symposium on Industrial Crystallization (pp. 49–54). Amsterdam: Elsevier. Dette, S. S., (2010). Chem. Technik 39(3), 18–19. Freund, H., König, A., and Steiner, R. (1997). In J. Ulrich (ed.), Proceedings of Crystal Growth of Organic Materials (vol. 4, pp. 114–22). Aachen: Shaker Verlag. Genceli, F. E., Gärtner, R., and Witkamp, G. J. (2005). J. Crystal Growth, 275 (1–2), e1369–72.

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Kim, K.-J., and Ulrich, J. (2002). Sep. Sci. Technol. 37(11), 2717–37. Kirk-Othmer (ed.) (2014). Kirk-Othmer Encyclopedia of Chemical Technology. Available at http://onlinelibrary.wiley.com/ book/10.1002/0471238961 (accessed May 6, 2014). König, A. (2003). In J. Ulrich and H. Glade (eds.), Melt Crystallization: Fundamentals, Equipment and Applications (pp. 7–40). Aachen: Shaker Verlag Kureha Engineering Co. Lechner, M. D. (ed.). (1983). Taschenbuch für Chemiker und Physiker (4th edn). Berlin: Springer-Verlag. Luckenbach, R. (ed.). (1992). Beilstein’s Handbook of Organic Chemistry (4th edn). Berlin: Springer-Verlag. Lüdecke, U., Brendler, L., and Ulrich, J. (2003). Eng. Life Sci. 3(3), 154–58. Matsuoka, M. (1977). Bunri Gijutsu 7, 245–49. Matsuoka, M., and Fukushima, H. (1986). Bunri Gijutsu 16, 4–10. Matsuoka, M. (1991). In J. Garside, R. J. Davey, and A. G. Jones (eds.), Advances in Industrial Crystallization (pp. 229–44). Oxford: Butterworth-Heinemann. Matz, G. (ed.) (1969). Kristallisation; Grundlagen und Technik (2nd edn). Berlin: Springer-Verlag. Molinari, J. G. D. (1967). In M. Zief and W. R. Wilcox (eds.), Fractional Solidification (pp. 393–400). New York, NY: Marcel Dekker. Moritoki, M., and Fujikawa, T. (1984). In S. J. Jančić and E. J. de Jong (eds.), Proceedings of Industrial Crystallization (vol. 84, pp. 369–72). Amsterdam: Elsevier Science. Moritoki, M., Ito, M., Sawada, T., et al. (1989). In J. Nývlt, and S. Žáček (eds.), Proceedings of Industrial Crystallization (vol. 87, pp. 485–88). Amsterdam: Elsevier Science. Moritoki, M., Wakabayashi, M., and Fujikawa, T. (1979). In E. J. de Jong and S. J. Jančić (eds.), Proceedings of Industrial Crystallization (vol. 78, pp. 583–84). Amsterdam: Elsevier Science.

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Nývlt, J., Söhnel, O., Matuchová, M., and Broul, M. (eds.) (1985). The Kinetics of Industrial Crystallization. Amsterdam: Elsevier. Özoğuz, Y. (1992). Zur Schichtkristallisation als Schmelzkristallisationsverfahren. Ph.D. dissertation, Universiẗ at Bremen, Bremen, Germany. Paspek, S. C., and Every, W. A. (1980). US Patent No. 4,230,888, Washington, DC. Pavlou, F., Nepveux, K., Schoeters, K., Weiler, A., and Whitfield, S. (2010). Pharm. Technol. Eur. 22, 44–50. Poschmann, M., and Ulrich, J. (1996). J. Crystal Growth 167(1–2), 248–52. Proabd S. A. (1972). DE Patent No. 1793345, Berlin. Rittner, S., and Steiner, R. (1985). Chem. Ingenieur Technik 57(2), 91–102. Rutter, J. W., and Chalmers, B. (1953). Can. J. Phys. 31(1), 15–39. Ryu, B., Jones, M. J., and Ulrich, J. (2010). Chem. Eng. Technol. 33(10), 1695–98. Saxer, K. (1971). CH Patent No. 501421, Bern. Saxer, K., Stadler, R., and Ignjatovic, M. (1993). In Z. H. Rojkowski (ed.), Proceedings of the 12th International Symposium on Industrial Crystallization (vol. 1, pp. 13–18). Amsterdam: Elsevier. Scholz, R. (1993). Die Schichtkristallisation als thermisches Trennverfahren. Ph.D. dissertation, Universiẗ at Bremen, Bremen, Germany. Scholz, R., and Ruemekorf, R. (2003). In J. Ulrich and H. Glade (eds.), Melt Crystallization: Fundamentals, Equipment and Applications (pp. 191–212). Aachen: Shaker-Verlag.

Crystallization: Fundamentals, Equipment and Applications (pp. 167–89). Aachen: Shaker-Verlag. Tiedtke, M., Ulrich, J., and Hartel R. W. (1996). In A. S. Myerson, D. A. Green, and P. Meenan (eds.), Crystal Growth of Organic Materials (pp. 137–44). Washington, DC: American Chemical Society. Tiedtke, M. (1997). Die Fraktionierung von Milchfett – ein neues Einsatzgebiet für die Schichtkristallisation. Ph.D. dissertation, Universiẗ at Bremen, Bremen, Germany. Toyokura, K., and Hirasawa, I. (2001). In A. Mersmann (ed.), Crystallization Technology Handbook (2nd edn, pp. 617–62), New York, NY: Marcel Dekker. Ulrich, J. (1988). Chem. Eng. Symp. Series Jpn 18, 172–75. Ulrich, J. (2003). In J. Ulrich and H. Glade (eds.), Melt Crystallization: Fundamentals, Equipment and Applications (pp. 1–6). Aachen: Shaker-Verlag. Ulrich, J., and Bierwirth, J. (1995). In J. P. von der Erden and O. S. L. Bruinsma (eds.), Science and Technology of Crystal Growth (pp. 245–58). Dordrecht: Kluwer Academic. Ulrich, J., Bierwirth, J., and Henning, S. (1996). Sep. Purific. Methods 25(1), 1–45. Ulrich, J., and Kallies, B. (1994). Curr. Top. Crystal Growth Res. 1, 1–14. Ulrich, J., and Neumann, M. (1997). J. Thermal Anal. 48(3), 527–33. Ulrich, J., and Nordhoff, S. (2006). In R. Goedecke (ed.), Fluidverfahrenstechnik: Grundlagen, Methodik, Technik, Praxis (vol. 2, pp. 1131–96). Weinheim: WileyVCH. Ulrich, J., and Özoğuz, Y. (1989). Chem. Ingen. Tech. 61(1), 76–77.

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Chapter

10

Crystallizer Mixing Understanding and Modeling Crystallizer Mixing and Suspension Flow Daniel A. Green GlaxoSmithKline plc

10.1 Introduction Mixing determines the environment in which crystals nucleate and grow and is therefore intrinsic to industrial crystallization. Individual nucleating and growing crystals respond directly to their microenvironment and not in a simple way to the macroenvironment, often thought of as the bulk or average environment. Because the growing crystal removes solute from solution and the dissolving crystal releases it, the solute concentration and therefore the supersaturation is in general different at the crystal surface than in the bulk. Crystals grow when the microenvironment is supersaturated, stop when it is just saturated, and dissolve when it is undersaturated. In most cases, impurities are rejected by growing crystals; therefore, each growing crystal face creates a zone of locally higher impurity concentration immediately adjacent to it. The growth rate and amount of impurity taken up by the growing crystal are functions of the impurity concentration where growth is occurring – at the crystal face itself. Mixing is the family of processes that links this local microenvironment to the macroscopic scale of the crystallizer by affecting the mass transfer between crystal and the larger environment and the dynamics of crystal suspension flow in the crystallizer. Mixing, therefore, to a large extent creates the crystal microenvironments. Furthermore, it determines the homogeneity of the macroenvironment, both temporally and spatially. Inhomogeneity in the macroenvironment affects the microenvironments around crystals, causing temporal variations as the crystals circulate from one zone to another inside the crystallizer. This is particularly important because local values of key variables such as supersaturation and solids concentration are often much more important in crystallization than the bulk or global averages of these quantities, as discussed below. For example, in practice, it is often found that the feed location plays a large role in determining particle size distribution and other crystal characteristics. This is because the local environment near the feed point is strongly affected by position relative to the agitation system and the rest of the vessel. This is where the feed solution begins the transition from the conditions in the feed line, typically higher concentration but undersaturated, to the supersaturated conditions in the vessel. Therefore, here is where nucleation frequently first occurs. The intent of this chapter is to make the reader aware of mixing and related phenomena as encountered in typical industrial crystallization processes. The effect of mixing on crystal characteristics is stressed. The fundamentals of the flow of suspension crystallizers are presented. The information presented here should enable practitioners to troubleshoot simple mixing

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problems in their processes. The principles of vessel and agitator design, from the standpoint of providing adequate mixing (and not mechanical design), are discussed. The issues and difficulties of scaling up are discussed from the perspective of suspension mixing. Finally, methods of investigation are presented, including both experimental and mathematical modeling.

10.1.1 Progress Since Previous Handbook Editions Crystallizers are rather simple vessels. Arguably, the last major change in the design of crystallizers, particularly as it affects mixing and flow, was the development of foil-type agitators used in some crystallizers. Other than that, there have been no major changes in crystallizers themselves in the last several decades. Our ability to monitor and model crystallization processes, however, has changed dramatically, and this is changing the way crystallizers are operated. Instrumentation has made significant strides. For example, it is now possible to routinely measure particle size distribution and often solute concentration during crystallizer operation. While it is still not in general practical to do so in operating crystallizers, particle image velocimetry and laser-Doppler anemometry are used to measure flows in model crystallizers. The data obtained can be used to refine agitator and vessel designs and verify mathematical flow predictions. The area of the most progress since the publication of the first edition of this Handbook in 1993 is mathematical modeling. Crystallizer flows are complicated: multiphase and turbulent with coupled with heat and mass transport. Analytical solution of the governing nonlinear equations of fluid and suspension flow is impossible, and we must turn to computational fluid dynamics (CFD). Progress in CFD is occurring on several fronts. Computational power, of course, continues to increase and become more affordable. Improved algorithms make solution of the equations of interest far more efficient than in the past. Increased power and efficiency make it possible to improve the models used. For example, it is no longer necessary to settle for spatially averaged turbulence approximations. The simulation of larger turbulent structures and in specific, limited cases even resolution of the complete range of turbulence length scales are possible. CFD modeling of crystallizers is covered in more detail later.

10.2 Crystallizer Flows Crystallizers are typically, although not exclusively, agitated tanks. In many respects, they are similar to vessels used as chemical reactors and mixing vessels in many other processes.

Crystallizer Mixing

Mechanical energy physically mixes the contents. Feed streams are incorporated into the bulk by the turbulence created by large turbines or impellers. Relatively large flows sweep the vessel from top to bottom and from center to sides in an effort to homogenize the contents. One key difference, of course, is that crystallizers need to suspend and distribute the solid crystals that are being formed within them. It is important, then, to recognize that the flow in crystallizers is of a suspension and not a single-phase fluid. There are obvious differences. The effective viscosity of the suspension is larger than that of the solution alone. The flow velocities everywhere should be large enough that the particles do not settle appreciably. Then there are more subtle differences. The presence of particles blunts velocity profiles and affects turbulence. The particles are not in general uniformly suspended but are distributed, often in unexpected ways. Crystallizers do not produce a single particle size but rather a distribution of sizes. This particle size distribution (PSD) is also not uniform throughout the vessel. For example, the fraction of larger particles in the distribution may be higher lower in the crystallizer. The net result is that transport properties and the variables that affect crystallization most, such as supersaturation, are affected. Because crystallizer flows are in general turbulent, we must first distinguish between the instantaneous and time-averaged flows in our descriptions. Consider the flow in a simple crystallizer: a baffled tank with a single axial-flow impeller. If we were to sketch the tracks of individual very small fluid elements, known as streamlines, associated with this configuration, we would be tempted to draw smooth lines representing the overall flow, such as in Figure 10.1. In fact, these lines represent only the time-averaged flow. Superimposed on this timeaveraged flow are turbulent velocity fluctuations associated with swirling eddies of various sizes and strengths. Although we tend to understand the flow in a vessel in terms of the timeaverage flow, it is important to keep in mind the nature of the turbulence because it has a great effect on the mixing and therefore on the crystallization process itself. Let’s consider several characteristics of turbulence. Turbulent fluctuations are associated with eddies of various sizes. In fact, turbulent eddies are characterized by a cascade of length scales. In the case of an agitated crystallizer, turbulence is generated by the agitator impeller, which creates large-scale turbulent eddies on the order of the vessel dimensions, say, impeller blade height, in size. These eddies, in turn, create smaller eddies. Each eddy length scale breeds smaller eddies. Energy input into turbulence by the agitator at the largest turbulent length scales flows down the length-scale cascade until a lower limit is reached, the Kolmogorov length scale (Deen 1998). Turbulence is a highReynolds-number phenomenon, meaning that the ratio of inertial to viscous forces in the flow is very large, and inertial forces dominate. However, as the length scale of the turbulent structures decreases, a limit is reached at which viscous forces become important and finally dominate.1 At this point, viscosity

Figure 10.1 Schematic suggesting streamlines associated with an axial-flow impeller in a baffled tank

dissipates the turbulent energy. The Kolmogorov scale for a typical crystallizer is o(10 μm). Therefore, the cascade of turbulent length scales is huge, say, from o(1) to o(10–5) m in a large crystallizer. The importance of this range of length scales will become obvious later when mixing mechanisms are discussed. Along with the cascade of length scales, there is a cascade of time scales associated with turbulence. As one would expect, the time scale associated with large length scales is longer than that associated with small length scales.

10.2.1 Distribution of Key Variables in Crystallizers Depending on the means of generating supersaturation (see, e.g., Chapter 7), the details of how mixing determines the distribution of process variables change. For example, in reacting systems in which two reactants mix to generate a dissolved product present above saturation concentration, it is the mixing of the two reactants, coupled with their reaction rate, that determines the distribution of the product concentration and subsequently the supersaturation. In cooling crystallization, mixing determines both the dispersion of the hot feed stream into the cooler bulk of the crystallizer and the heat transfer to the heat transfer surfaces. In evaporative crystallizers, the boiling zone, with higher supersaturation because of both cooling and solvent removal, must be mixed into the bulk. Mixing determines the distribution of both solute and suspended solids and the temperature profile in the vessel. These three quantities are coupled, and the net result is the distribution of supersaturation. Because supersaturation is the difference between the actual solute concentration and the concentration

In the Navier–Stokes equations, viscous dissipation is represented by a term proportional to μΔ2 v, where μ is viscosity, v is the fluid velocity ∂2 ∂2 ∂2 vector, and Δ2 is the operator ∂x 2 þ ∂y2 þ ∂z2 . At the smallest length scales, the partial derivatives of the velocity become very large; therefore, this viscous dissipation term, negligible at the larger length scales, becomes dominant. 1

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at saturation, supersaturation is highest where solute concentration is highest and temperature lowest. The distribution of solids affects supersaturation because high solids concentration means a large specific crystal area for crystal growth, which decreases solute concentration. Other variables with a potentially significant impact on crystallization are also affected by mixing: the distribution of impurities and of reactants in chemically reacting systems, for example. The distributions of these key variables that control crystallization are determined by several factors, including • Flow field in the vessel v(r) • Distribution of the turbulent kinetic energy and dissipation in the vessel [k(r), ε(r)] • Location of the feed point, and • Location of the product withdrawal point. Here r is the position vector, v is the velocity vector, and k and ε are the local turbulence kinetic energy and rate of dissipation, respectively. The flow and dissipation fields are determined in turn by the agitator(s), its(their) placement and rotational speed(s), the placement and dimensions of baffles and draft tubes, and the overall geometry of the vessel. It turns out that the distributions of a crystallizer’s key variables are often not uniform. While it is often possible to approach uniformity at laboratory scale, it is almost never achieved at industrial scale. Therefore, the effect of inhomogeneity in crystallizers cannot be underestimated. Let us now consider the effects of nonuniform distribution of supersaturation.Nucleation is a strong function of supersaturation, and this dependence is frequently approximated as B ∝ σn where B is the nucleation or “birth” rate of crystals, σ is the dimensionless supersaturation, defined as σ = (c – csat)/csat, where c is the solute concentration and csat is the concentration at saturation; n is determined by data correlation and typically falls in the range of 1 ≤ n ≤ 3 for secondary nucleation and is often ≥2 (Randolph and Larson 1988). (For primary nucleation, which is encountered much less frequently in industrial crystallization, n is typically much higher.) Because of this strong dependence on supersaturation, local regions of sustained supersaturation that are higher than the vessel average can dominate the nucleation rate of the entire vessel. In these cases, local supersaturation maxima and not the vessel average may effectively control the nucleation rate of the entire process. Regions of locally higher supersaturation will result, for example, where a concentrated feed is injected into a poorly mixed region or if the evaporation zone of an evaporative crystallizer is poorly mixed into the bulk.The growth rate of crystals also depends on supersaturation, frequently estimated as G ∝ σm where G is the growth rate, and 1 ≤ m ≤ n. Then, similar to the case with nucleation, growth of individual crystals may be nonuniform, growing more rapidly in regions of high

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supersaturation. This may alter crystal morphology and possibly induce defects and strain. Controlling the effect of locally higher supersaturation may be particularly important in cases where the goal is to minimize fines generation. Efforts to decrease the average supersaturation in the vessel may fail to reduce the number of fines produced. Often the key is rather to eliminate or moderate a local maximum supersaturation, say, in the vicinity of a feed point or in the boiling zone of an evaporative vessel. Inhomogeneity in supersaturation has a similar effect on encrustation, which is also a strong function of supersaturation. Small high-supersaturation zones may be the trigger to initiate encrustation, which then may spread to cover a much broader area of the crystallizer wall. In this case, a local cold spot on a wall or at the entrance of a heat transfer coil may be where the supersaturation becomes higher locally. It must be noted that while it is almost always beneficial to have a homogeneous suspension, this is not always true. In some cases, there are benefits to having an inhomogeneous suspension. This is the case when there is an advantage to separating nucleation and growth phenomena, for example, when polycrystalline particles are formed by nucleating new crystallites on existing particles in a zone of very high supersaturation and then circulating these particles through a lowersupersaturation zone in which growth of the crystallites occurs without additional nucleation (Green et al. 1995). Similar to the dependence of nucleation and growth on supersaturation, the mechanisms of attrition and aggregation are sensitive functions of particle collision rate. Because particle collision rate scales with solids concentration to the second power (Smoluchowski 1916), these processes are exceptionally sensitive to locally higher particle concentration. Therefore, if particles are not suspended uniformly, much of the attrition and/or aggregation in the system can occur in a relatively small zone of higher particle concentration. Again, the solution to an excessive attrition condition may be to make the solids suspension more uniform, minimizing or moderating the concentration in any small zone with higher solids concentration rather than decreasing the average solids concentration for the entire vessel. Therefore, in these cases, we should strive to achieve greater uniformity in solids suspension, not the “just suspended” condition often used as a criterion for solids suspension (Nienow 1997). Let us now examine what determines the uniformity of these fields. First, we will consider solute concentration in a simple cooling crystallizer fed at a single point. Mixing occurs at all length scales. Many people speak of mixing occurring at small length scales as micromixing and that at larger length scales as macromixing (Bourne 1997). Micromixing is the intimate mixing that occurs at the smallest scales of turbulence. Macromixing is the overall blending that occurs by convection and is associated with length scales that are roughly equivalent to vessel dimensions. The term mesomixing describes mixing occurring at intermediate length scales, say, on the order of the envelope of the feed zone in size (Baldyga and Bourne 1992). Mesomixing is associated with the turbulent diffusion of the feed into the

Crystallizer Mixing Figure 10.2 Dimensionless particle concentration versus dimensionless height in a tank mixed with an axial-flow agitator. Co = 3 wt%. Source: Reprinted with permission of the publisher from Shamlou and Koutsakos 1989. Copyright © 1989, Elsevier.

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surrounding fluid and occurs at length scales above that of micromixing but still quite small relative to vessel dimensions. Mixing in a crystallizer occurs by all these mechanisms. In the immediate vicinity of the feed point, mixing of the concentrated feed into the fluid streaming by is by meso- and then micromixing. Turbulent eddies capture packets of the feed solution and mix it with the bulk fluid. Molecular diffusion takes over below the Kolmogorov scale. Convection carries the fluid in the vicinity of the feed point away and into the rest of the vessel. Mixing at all length scales is important in the crystallizer. In the absence of macromixing, the concentration would build around the feed location and not be blended with the rest of the vessel. Micromixing is required for the final homogenization of the solution. The distribution of solids depends on the velocity field. There is also a feedback effect because the solids themselves affect the flow, altering velocity profiles and the turbulence. The spatial distribution of the solids can be quite surprising. Figure 10.2 shows data from Shamlou and Koutsakos (1989) for solids suspended in a baffled tank agitated with an axial-flow impeller. The distribution is unexpected and complicated. Note several features: generally, the particle concentration is higher lower in the tank, the particles are concentrated immediately above the impeller, and the profiles depend rather strongly on agitator speed (rpm). Because this configuration would commonly be found in a batch crystallizer, these data indicate a range of behavior possible in actual crystallizers. Continuous crystallizers frequently employ draft tubes to enforce mixing up and down the vertical axis. Even with draft tubes, however, the solids will not be uniformly distributed in general. Figure 10.3 shows solids distribution in a laboratory model of a draft-tube crystallizer (Green and Robertson 1993). Notice that there are variations in vertical position inside the crystallizer. In addition, the average concentration inside the draft tube is different from that outside because the net particle velocity and therefore particle holdup is different where the

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particle settling velocity is opposite the net flow (i.e., in upflow) from than where they settle in the direction of the net flow (i.e., downflow). This is discussed in more detail in Mullin (2004), Jones and Mullin (1973), and Section 10.5.1. Temperature also varies spatially in crystallizers, being higher near feed points where hot feed is added (and exothermic reactions are preferentially occurring) and near any heating surfaces. Even zones where crystallization is happening more rapidly could conceivably have higher temperatures because crystallization is normally exothermic. Temperature will be lower near cooling surfaces and in and near the boiling zone of an evaporative crystallizer. Because temperature is in general easier to measure than either solute or solids concentration, one would expect more data on temperature distribution in crystallizers than seem to exist. Chiampo et al. (1996) measured the temperature for cooling crystallization in a drafttube baffled (DTB) crystallizer (see Chapter 7) at five points. They found significant temperature variation, particularly at low agitation rates. Warmer temperatures, varying with time because of the formation of eddies, were found near the feed point. Temperatures near the solid-surface heat transfer surfaces were lower than the bulk, but not greatly so. The temperature just under the surface of the suspension was found to be low in most cases studied. These authors found good thermal homogeneity when the ratio of agitator pumping flow rate to the feed flow rate was 450 or greater for upwardpumping and 700 or greater for downward-pumping agitators. Of course, the combined effect of the distributions of these three variables – solute concentration, solids concentration, and temperature – is to determine the spatial distribution of supersaturation, the dominant variable in determining crystal nucleation and growth rates. To reiterate, it is likely that supersaturation is higher in zones such as the vicinity of feed points and in and near the boiling zone in evaporative crystallizers. Supersaturation can be expected to be lower in the bulk of the crystallizer and anywhere temperature is higher.

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Figure 10.3 Particle concentration as a function of height and agitator rpm (a) outside and (b) inside the draft tube of a laboratory-scale DTB model crystallizer

It should be noted that the situation could be more complicated when a hot undersaturated feed is introduced into a crystallizer. Let’s assume that there is a plume associated with the feed point. The center of the plume may well be hot and undersaturated, indeed similar to the conditions in the feed stream itself. Surrounding this zone, however, will be a zone where the temperature has begun to fall, yet the concentration may still be high enough to create significant supersaturation. Therefore, the picture is of a plume with an undersaturated core surrounded by a shell in which the supersaturation will be higher, perhaps significantly higher, than that in the bulk of the crystallizer. The degree to which this happens and the spatial extent of such a plume, of course, depend strongly on the local mixing conditions near the feed location and on the diameter and velocity of the feed tube or nozzle.

10.3 Crystallizers The design of the crystallizer and the means to agitate the suspension therein are the physical implementation of mixing principles, the way to achieve mixing sufficient to create uniform crystals with desired properties.

10.3.1 Providing Agitation There is always much discussion as to what type of agitator is best for crystallization. There are many considerations. First, one must decide whether a radial- or axial-flow impeller, or some combination, is needed. Figure 10.4 shows some common impellers. Radial-flow impellers produce a high-shear, highturbulence region. They do a good job producing flow near the impeller. When used, they are normally placed near the bottom of the vessel so that the strong radial flow and not the weaker axial flow they induce sweeps the bottom of the vessel and suspends particles. Paradoxically, they should also be placed nearer the center of the tank vertically because typically they do not produce nearly as much vertical mixing as axialflow impellers (oriented to produce vertical flow). Because of the limited axial flow, they are usually a poor choice for vessels

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with high height-to-diameter ratios, unless paired with an axial-flow impeller or impellers or a draft tube, as discussed later. One shortcoming, particularly for radial-flow impellers such as the Rushton turbine, which has a disk as part of its construction, is that they tend to divide the tank into upper and lower sections, with less mixing between regions. (This feature, however, may be beneficial in certain special cases, such as precipitation of the product of two reactants, one fed above the impeller and one below.) There are various forms of radial-flow impellers. The most basic are a number of flat, vertical blades attached to a hub or a horizontal disk (Rushton turbine). The blades can also be swept back (but still oriented vertically), which reduces some of the turbulence associated with blade passage. Another variation is the squirrel-cage impeller, which captures vertical blades between top and bottom plates, one or both of which are open in the center to allow flow to enter from the center and be propelled outward by the blades. A particular type of radial-flow impeller that is becoming less common is the crow’s foot agitator. It is comprised of curved pipe sections and not flat blades. These are generally poor, inefficient agitators. They are often used in glass-lined vessels because they can be readily glass coated. However, techniques have been developed to effectively coat more efficient agitator geometries, so the crow’s foot is becoming obsolete. As the name suggests, axial-flow impellers produce less radial flow and much more axial flow for similar power input. They therefore produce much more vertical mixing when so oriented. They also do a much better job mixing between the regions above and below the impeller. The simplest axial-flow impeller is the pitched-blade turbine. Flat blades inclined from the vertical are attached to a central hub. As the impeller turns, the blades force fluid up or down depending on the direction of rotation and the blade pitch. The pitch of the blades determines how much fluid is moved for each revolution of the impeller. There are many improvements that are generally more efficient. That is, they provide more axial flow for the same power. Because the tips of the blades move farther each revolution than the roots, frequently the blades are twisted so that the

Crystallizer Mixing

Figure 10.4 Impellers from the Lightnin catalog, typical of those often used in crystallization: (a) Rushton turbine, (b) marine propeller, (c) pitched-blade turbine, (d) foil type. The Rushton turbine (a) is radial flow, whereas the others (b–d) are axial flow. Source: Photographs copyrighted and owned by SPX Flow, Inc., and used under license and permission from SPX Flow, Inc. Copyright © 2018, SPX Flow, Inc.

angle of attack of the root of the blade is higher than the tip. This is done in an effort to make the flow from the impeller more uniform rather than disproportionately more flow from the tip region relative to the core. The common marine propeller type of agitator is an example. Much work in recent years has gone into designing more efficient axial-flow impellers. The result is the family of fluidfoil designs, which use airfoil-like blade cross sections to produce more axial flow and less blade drag and turbulence at the blade. Often they also use winglets at the tips of the blades. These reduce the blade tip vortices that form because fluid rushes from the high- to the low-pressure side of the blade, creating a vortical flow trailing from the blade tip. This flow serves little useful purpose, so to increase efficiency, it is damped with these additions. It should be noted that these highly developed designs suffer from changes to their shape caused by encrustation and erosion much more than simpler but more robust designs.

Radial- and axial-flow agitators each have their benefits. One way to achieve the benefits of both and/or to more effectively mix crystallizers with higher height-to-diameter ratios is to use two or more impellers, typically mounted on a common shaft. Using two or more impellers is often especially beneficial (as discussed in more detail later) for batch processes where the use of a draft tube is impractical. Multiple axial-flow impellers can be stacked on a single shaft to enforce top to bottom vessel mixing, or a radial-flow impeller low in the vessel can be used with an axial-flow impeller or impellers in the middle to upper vessel. Because radial-flow impellers generally produce a higher turbulence zone, they make a good location for feed addition, plus the radial flow can be used to sweep the vessel bottom, whereas the axial-flow impeller(s) produces good macromixing involving the rest of the vessel, particularly through the otherwise partially segregated region under the radial-flow impeller. It should be noted that all the energy input to an impeller or system of impellers is dissipated by the flow in the vessel,

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regardless of the type of impeller(s). What does change with the type of impeller(s) is the overall flow field in the vessel and the spatial dissipation of the energy. Turbulent energy is dissipated, as noted earlier, by the smallest scales of turbulence, so, of course, the spatial distribution of dissipation and turbulence intensity are closely related.

10.3.2 Crystallizer Types Our primary focus here is crystallization from solution in agitated suspensions. We will, however, also briefly consider crystallization from solution in fluidized-bed crystallizers and melt crystallization in agitated suspensions and static layers. This subsection discusses crystallizer design from the perspective of mixing and solids suspension. See Chapters 7 and 9 for more information.

Agitated Suspension Let us consider the agitated tank used as a crystallizer. Most agitated tanks are employed for liquid–liquid mixing, but when used as a crystallizer, there are additional considerations. The goal, recall, is generally to achieve as nearly as homogeneous a system as possible. Solids must be suspended. (For this discussion, we will assume that the crystals settle.) Feed streams must be rapidly mixed into the bulk of the suspension. The same is true for any other zones where the solution is concentrated or cooled, such as evaporation or heat transfer zones. Sufficient heat transfer surface and flow must be provided for efficient heat transfer. Finally, all of this must be accomplished in such a way that the shear imposed by the agitation system does not cause undue damage to the growing crystals or cause too much secondary nucleation. Draft tubes are often used in crystallizers because they are particularly effective and efficient at imparting a good degree of vertical mixing. They are discussed in detail later. However, they are usually employed only for continuous crystallizers because they do not operate well with the suspension level below the top of the draft tube, except as noted below. Let us first focus on crystallizer geometries suitable for batch crystallization. Because these usually do not use draft tubes, other arrangements must be made to suspend particles over the entire height of the vessel. A low crystallizer height-todiameter ratio can be used so that a single agitator will keep the solids in the entire tank suspended or multiple impellers on a single shaft can be used in various combinations, as discussed earlier. Ideally, all the impellers in the vessel act in concert. The flow from one impeller needs to feed smoothly into the zone of the next to keep the suspension smoothly flowing vertically in the vessel. Therefore, the spacing between impellers is crucial. Spaced too far apart, the effect is to create individual mixing zones around each impeller, with relatively poor mixing between zones. Figure 10.5 shows the two situations schematically. The use of vertical baffles to provide adequate mixing and suspension is nearly a requirement. Impellers impart axial, radial, and tangential (swirl) flow to the suspension. Vertical baffles convert much of the tangential flow from the impeller to

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Figure 10.5 Schematic suggesting the flow produced by multiple impellers: (a) appropriately spaced with flow sweeping entire tank height, (b) too widely spaced with separate circulations formed about each impeller

axial flow. Because axial flow is needed both for crystal suspension and macromixing, baffles are generally required for efficient mixing and suspension. Nonbaffled tanks are occasionally found in crystallizer service, perhaps in glasslined or other specialty configurations, but on closer examination, one will usually find that mixing can be improved with baffles. Draft tubes function very much like multiple axial-flow impellers in an open tank. They provide efficient vertical mixing because the average flow is forced to go from the impeller up and over the top and bottom of the draft tube. This, plus the fact that there is typically a single and not multiple impellers, each with its own potentially particle-damaging high-shear zone, means that draft tubes are generally the geometry of choice for crystallizers where the suspension level can be maintained within a limited range. Continuous crystallizers usually fit this description, as well as some batch crystallizers, say, when cooling begins only after the vessel is fully filled or when an antisolvent or salting-out component is a small fraction of the overall volume. The reason that draft tubes work only for a limited range of suspension heights is first that there is no flow over the top of the draft tube when the suspension height is significantly lower than the top of the draft tube. When this is the case, the suspension level will be higher in the region of upflow because the impeller imparts a pressure differential. The hydrostatic pressure difference created by the difference in heights outside and inside the draft tube equals the pressure difference produced by the agitator. When the height of the suspension in the upflow region is less than the height of the draft tube, there is no flow up and over the top and therefore no net circulation through the draft tube. Second, when the suspension height is greater than some critical height over the top of the draft tube, a clarified solution layer forms that is very poorly mixed into the rest of the vessel. This effect is often dramatic and is shown

Crystallizer Mixing Figure 10.6 Photograph showing clarified liquid well above the draft tube in a 20-liter laboratory crystallizer

in Figure 10.6 for a 20-liter laboratory crystallizer. The critical height depends in part on the upflow velocity and the vessel geometry. Therefore, it is vital to maintain the suspension level in the crystallizer within these limits. In many cases when the suspension level must vary and be lower than the top of the draft tube for part of the crystallization, a draft tube can still be used by cutting slots or windows in the draft tube that allow the suspension to “short circuit” through the openings when the top of the draft tube is not immersed (Oldschue 1983). Most of the benefit of the draft tube is retained when the suspension level is above the top of the tube (with some, usually minor, losses due to shortcircuiting). Difficulty resuspending solids is sometimes encountered when starting the flow in a draft-tube vessel after the solids have slumped to the bottom, say, after an interruption in agitation. An initial flow rate large enough to disperse the particles and carry them over the top of the draft tube is needed to initiate complete suspension. This flow rate may be considerably higher than the steady-state value needed once flow has been established. If the initial flow rate is insufficient, a situation can develop where liquid flows through the bed of particles that are now fluidized (see below) in the upflow region but not conveyed over the top of the draft tube. Sufficient additional flow must be provided to expand the fluidized bed until the height of the bed reaches the top of the draft tube. At this point, the solids spill over into the downflow region, and the suspension quickly disperses throughout the vessel. There are two schools of thought as to draft-tube diameter. To minimize damage to particles by minimizing the energy and velocities created by the agitation, the first school maintains that because a velocity significantly higher than the particle settling velocity is necessary only in upward flow, it makes sense to have a higher velocity in upward flow. This is created by choosing a draft-tube diameter so that the cross-sectional area of the upflow region is smaller than that of the downflow region. The second school of thought is that because of the dependence of the hindered particle settling velocity on particle size distribution, particle concentration, and fluid

Figure 10.7 Velocity magnitude predicted by CFD model of liquid flow over top of a 0.7Dt draft tube. Uniform upflow in the annulus at the bottom of image is specified as the boundary condition. Note that the core of fluid in the downflow region moves at greater than twice the average speed.

properties, any attempt to optimize the choice of draft-tube diameter is futile because these values change frequently. Therefore, one may just as well choose approximately equal cross-sectional areas inside and outside the draft tube, meaning, of course, that the average axial velocities will also be equal in each zone. A draft-tube diameter approximately 0.7Dt creates these equal cross-sectional areas. Dt is the tank diameter of the crystallizer. It should be noted here that the choice of draft-tube diameter affects mainly the ratio of the average suspension velocities inside and outside the draft tube. The velocity profiles in both regions often show marked deviation from uniform velocity profiles near the average value. Large recirculating zones are common, particularly next to the draft tube and associated with the turn the suspension makes around the top or bottom of the tube. The result is that there will be a higher-speed flow adjacent to a low-speed or recirculating zone. The velocity maximum may be several times the average velocity. Therefore, the average velocity may have little importance to overall performance. A very simple computational fluiddynamic model illustrates this point. Figure 10.7 shows the prediction of flow around the top of a draft tube. A uniform upflow in the annulus outside a 0.7Dt draft tube is assumed. A large low-speed region forms inside and adjacent to the draft

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tube where the flow turns over the top of the tube, whereas a high-speed downflow travels down the center. The velocity of this high-speed core is approximately twice that of the assumed uniform upflow. Empirically, a tracer study in a 50,000-gallon industrial crystallizer found that the tracer velocity was approximately twice the computed average circulation velocity (R. E. Kendall, personal communication, 1997). The region outside the high-speed core but inside the draft tube is typically a recirculating vortex. Recirculating suspension instead of having it move more uniformly through the tube can be detrimental to performance, particularly if the vortex is in or near a high-supersaturation zone such as a feed zone or the boiling zone of an evaporative crystallizer. To counter this tendency, the end of the draft tube can be either flared with a conical section or rounded in an effort to streamline. The benefits of adding these features must be weighed against their additional cost. The most well-known example of the higher-upflowvelocity school is the draft-tube baffled (DTB) design (see Chapter 7). In addition to a 0.5Dt draft tube with upflow inside, an annular zone outside the main body of the crystallizer is provided to serve as a nearly quiescent fines settling zone to disengage mother liquor and fines from the rest of the suspension. The fines are separated for separate treatment such as fines destruction or clear liquor advance (also known as clear liquid overflow; Mullin 2004). The use of this annular zone is possible because the flow outside the draft tube is downward, and the relatively small flow of mother liquor and fines removed from the annulus must be upward to effect crystal segregation based on differing settling velocities. An axial-flow impeller, now shrouded by the draft tube, provides upflow inside the draft tube. This impeller is often placed at such a height inside the draft tube that in the event of an interruption to agitation and subsequent settling of the entire charge, the impeller will be above the sedimented particles in clear fluid and therefore better able to reinitiate flow on restarting. One drawback of this design is the tendency of particles to sediment to the bottom of the vessel and remain. Aeschbach and Bourne (1972) studied vessel bottom configurations of model draft-tube crystallizers to improve particle suspension. They found that a shaped bottom can significantly improve the uniformity of particle suspension. The design found to be best has rounded tank “corners” and a center peak under the agitator. The rounding of the corners means that there is less settling caused by low velocities associated with sharp corners, and the center peak virtually eliminates the stagnation point that would otherwise be present (in the time-averaged flow) at the bottom center of the tank under the agitator. Several features can be added to combat stagnant zones and particle settling in DTB (or similar) crystallizers (Swenson Process Equipment, Harvey, IL, USA). The bottom of the vessel can be made conical and the bottom of the draft tube slightly flared to constrict the clearance between the end of the draft tube and the lower part of the tank, thereby accelerating the flow for better crystal suspension. An upward-pointing conical section in the very center of the bottom of the vessel is there for the same reason, as well as to fill a region

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Figure 10.8 DTB crystallizer. Note features to combat crystal settling: flared draft tube and center upward-pointing conical section beneath agitator.

particularly subject to sedimentation because of the stagnation point in the center of the vessel bottom (Figure 10.8). These additions approximate the slightly more complicated geometry recommended by Aeschbach and Bourne. Accepted designs using equal cross-sectional areas inside and outside the DT include the TSK DP crystallizer (TSK, Tsukishima Kikai Co., Ltd., Tokyo, Japan) and a proprietary design by DuPont (Randolph et al. 1990). The former uses a proprietary “double-propeller” impeller that penetrates the draft tube and imparts momentum to the suspension both inside and outside the draft tube. The latter uses a radial-flow impeller at the bottom of the draft tube, just above the vessel bottom. The principal advantage of these designs is that the impeller diameter can be made large, and therefore, it turns relatively slowly. Circulation velocity nominally scales with the tip velocity of the impeller, whereas specific power input, the agitator power divided by either the swept volume of the agitator or the total suspension volume, scales with the rotational velocity cubed. Therefore, a large-diameter impeller creates relatively high circulation velocities in the suspension with relatively low specific power input. Minimizing the rotational velocity of the impeller and therefore the specific power input for a given circulation velocity is often desired, particularly when crystallizing fragile materials, because particle damage generally increases with increasing specific power input: increasing strongly with impeller tip speed and decreasing with increasing agitator diameter (Offermann and Ulrich 1982). Not all crystallizers are agitated with impellers in the vessel itself in agitated-tank configuration. A common arrangement is to use a pump (or an axial-flow impeller) in an external pipe loop to draw suspension from and discharge it back to the main vessel. Such crystallizers are known as forced-circulation crystallizers. They are widely used, for example, in sodium chloride crystallization and can be quite effective (see Chapter 7). Advantages are that the crystallizer feed can be

Crystallizer Mixing

introduced into the external flow loop, perhaps immediately upstream of the pump, so that local turbulence can be very high, leading to very effective feed mixing. A conventional heat exchanger can be used on the external loop to add or remove heat from the crystallizer, which may be less expensive than a jacketed vessel and/or an internal vessel coil, or it can be used to supplement other heat transfer equipment. Also, product crystal suspension can be removed directly from the circulation loop, eliminating the need for a separate pumping system for product removal. A significant disadvantage is that the pump impeller or in-tube axial-flow impeller is necessarily of a much smaller diameter than one rotating in the tank itself; therefore, as discussed earlier, the specific power input for forced-circulation crystallizers is much larger than for properly designed conventional crystallizers. This, of course, means that they are generally a poor choice for crystallizing fragile materials. Crystallizers need not be mechanically agitated. Air injection inside a draft tube drives the circulation in the older Pachuca (Oldschue 1983) design. Because of the cost of compressing air, this design is no longer widely used. A more recent variation, however, has been developed by Praxair (Praxair, Inc., Tarrytown, NY, USA). A cryogenic liquid is introduced into the bottom of a crystallizer, either with or without a draft tube. The liquid then boils, cooling the crystallizer and simultaneously driving the circulation of suspension with the resulting vapor stream.

Fluidized Bed There is another class of crystallizers that are not mechanically agitated tanks. Rather, a bed of growing crystals is fluidized by a recirculating stream of mother liquor. Fluidization is the state where the drag force on a bed of particles exerted by upwardflowing fluid equals or exceeds their weight. The bed expands and takes on fluid-like properties: it can be stirred; if a hole is punched in the side of the bed, the solids pour out; etc. (Kunii and Levenspiel 1969). Fluidized beds are good solid–fluid contacting devices; hence their use as crystallizers where the fluid transports solute to the bed of growing crystals. One can envision a stable state of fluidization with a nearly rectilinear vertical flow of fluid upward through a bed of largely motionless particles. However, at essentially all fluid flows above the minimum necessary to fluidize the solids, the flow is unstable (Anderson and Jackson 1968). The instability produces voidage fluctuations in the dispersed solid phase. The usual state is that the bed is “bubbling”; that is, particle-lean fluid-filled voids very similar in appearance to bubbles in a gas-liquid system propagate upward through the bed. These “bubbles” drag a particulate wake behind them as they rise through the bed and very efficiently backmix the solids. Therefore, the composition of the solid phase tends to be well mixed. In fluidized bed crystallizers, the larger particles remain in the bed and are not circulated with the mother liquor (and smaller crystals). Therefore, they undergo considerably less attrition and breakage than in circulating suspension crystallizers. This type crystallizer is thus particularly well suited for the crystallization of large crystals (Mersmann and Rennie

1995). In the typical fluidized-bed crystallizer design, supersaturation is generated separately from the bed of particles; therefore, the supersaturation can become higher than it would be in an agitated suspension crystallizer. Consequently, fluidized-bed designs are subject to fouling for systems with narrow metastable zones.

Melt Crystallizers As mentioned earlier, the primary focus of this chapter is mixing in suspension crystallizers, most of which crystallize a solute from solution, but we will consider the mixing of melt crystallizers briefly here. (Chapter 9 provides a more complete review of mixing considerations and equipment options for melt crystallization.) Melt crystallization can be broadly divided into suspension and static-layer processes. Mixing in melt and solution crystallization processes using agitated suspensions is quite similar, although the considerations are slightly different. In melt crystallization, the liquid is predominantly a melt of the substance being crystallized. Mass transfer to the growing crystals is therefore not the rate-limiting step, as it can be in crystallization from solution. Instead, it is frequently the rate of heat transfer from growing crystals in the melt that sets the average crystal growth rate. Also of crucial importance is the mass transfer of impurity away from the growing crystals, where it is usually at least partially rejected (as it also is for solution crystallization). Mixing must be sufficient to maintain enough heat transfer to keep the heat transfer surfaces from becoming enough colder than the melt that unacceptable encrustation occurs. For these reasons, adequate mixing of the crystallizing suspension is still very important in melt crystallization, and many of the same considerations apply, except perhaps for the concern about distributing the crystallizer feed (see below). Encrustation of the heat transfer surfaces limits the application of conventional suspension crystallization to melt crystallization. Encrusted crystals remain attached to the wall and do not suspend. The heat transfer from the heat transfer surfaces to the melt is severely curtailed. One solution is to mechanically scrape the heat transfer surfaces to remove encrusted product. This can be done in shell and tube heat exchangers with mechanical scraping in each tube or in tanks incorporating scraper arms, typically in conjunction with agitation of the tank. For example, Armstrong (Armstrong Engineering, West Chester, PA, USA) manufactures the former and TSK (Tsukishima Kikai, Tokyo, Japan) and others the latter. Besides keeping the heat transfer surfaces relatively clean, the action of the scrapers in these systems provides mixing, distributing the impurities more uniformly through the melt and making the temperature more uniform. Crystal nucleation also may be increased. Perhaps the most common approach to melt crystallization is to crystallize a layer of crystals directly on the heat transfer surfaces, the static-layer process. The ratio of heat transfer surface area to vessel volume is typically much greater than for an equivalent suspension process to make up for the reduced heat transfer from the encrusted surfaces.

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Often there is no internal agitation. However, if agitation is provided, it can be beneficial by reducing the impurity concentration at the growing crystal surfaces. For example, Sulzer (Sulzer Chemtech, Winterthur, Switzerland) claims product quality improvements when their “falling-film” process is used. In this process, melt is flowed down along the interior surfaces of vertical cooled tubes. A static solid layer forms as the melt crystallizes, but the melt is mixed as it flows down the walls and is recirculated.

10.3.3 Feed Strategies No matter what type of crystallizer is chosen, it must be fed. For continuous or semicontinuous operation, feed strategy can be very important. Some designs have preferred feed locations, such as just before the pump or impeller in a forced-circulation design’s external loop. Conventional open-tank designs have much more leeway. In any event, the location and arrangement of the crystallizer feed determine, to a large extent, how well the feed mixes into the bulk. As discussed earlier, this will have a great impact on overall nucleation and growth rates, as well as on the uniformity of crystal growth of the entire process. Therefore, it behooves us to choose feed configurations with care! To generalize, the feed stream should be introduced below the surface of the crystallizer into a highly turbulent region to maximize micromixing. The region should also be subject to strong bulk flow to convect the newly mixed solution away from the feed zone into the bulk of the vessel to avoid concentrating the micro- and mesomixing region, i.e., to create good macromixing. Often in the laboratory and even in some full-scale crystallizers, the feed is dribbled onto the top suspension surface of the crystallizer. This is usually a poor way to feed because the top surface is often not very well mixed into the bulk. The result is that the feed solution pools on the surface, and a high supersaturation envelope forms around it, leading to high nucleation and encrustation rates. As a general rule, it is usually far preferable to introduce the feed well beneath the top surface of the suspension. To satisfy the micro- and macromixing requirements, it is usually preferred to introduce the feed near the agitator: near the tips of the impeller and just above or below an axial-flow impeller or just to the outside of a radial-flow impeller. The reasons for this are that the region adjacent to the agitator generally has the highest turbulence and therefore the best micromixing to be had in the vessel. Adding just above or below an axial-flow impeller ensures a good bulk flow through the micromixing region and therefore good macromixing. Feeding just to the outside of a radial-flow impeller, where the flow is leaving the impeller and washing through the injection zone, accomplishes the same thing. To enhance the turbulence at the very point of injection, it is often helpful to narrow the feed stream into a jet. This has two positive effects. Additional turbulence is generated by the jet action that enhances mixing. Also, the increased momentum of the jet carries the feed farther into the bulk, enlarging the feed mixing zone and therefore decreasing the local maximum supersaturation occurring there.

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Another often fruitful approach to dispersing crystallizer feeds is simply to distribute them. Instead of feeding the vessel at a single point, use multiple injection points. This can be done with discrete feed pipes, or a tube or ring with multiple orifices can be used. It must be recognized, however, that the performance of such a system can be significantly degraded if one or more of the feed points becomes clogged or encrusted or if a distributor ring corrodes so that the feed preferentially flows through fewer orifices. Instrumentation to detect this possibility is helpful. The preceding discussion of feed strategies has been concerned only with the feed of one component. This is typical of many crystallization process configurations, such as cooling or antisolvent addition crystallizations, but not all. In these others, such as reaction precipitation in which two-reactant feed streams are fed, there are two separate feed streams, or one reactant can be fed in semibatch fashion to another reactant already present in the vessel. The strategy of feeding the streams is important, particularly for fast reactions. See, for example, Tosun (1988). Here the position of each of the streams has a strong influence on particle size distribution and even on morphology of the product. To generalize, it is best to separate the streams and to feed them into high-shear regions as in the single-feed stream case.

10.4 Scale-Up Industrial crystallization has the (at least partially deserved) reputation of being difficult to scale up. If one considers that, as stated at the outset of this chapter, individual crystals respond only to their microenvironment and that mixing determines how the micro- and macroenvironments in the crystallizer interact as well as their spatial and temporal homogeneity, it is clear that mixing determines the ease or difficulty of scaling this process. A rapid examination of dimensional analysis is sufficient to convince us that, under most circumstances, the detailed dynamic behavior of a full-scale crystallizer cannot be fully captured with a geometrically similar laboratory-scale experimental model. It is well known that it is impossible in general to achieve dynamic similarity between model and full scale of even liquid-only mixing tanks and still use liquids of similar viscosities and densities (Bird, Stewart, and Lightfoot 1960). In crystallizing systems, of course, we are not free to change the solvent and solute, so we must use the same crystallizing system at both laboratory and full scale. Complicating the situation is the fact that the particle size itself does not scale with the vessel dimensions. Neither, of course, do the transport properties associated with the particles and the suspension. Therefore, the settling velocity of the particles and consequently the necessary suspension velocities to suspend the particles also do not scale. The crux of the scale-up problem is really that it is usually easy to achieve homogeneity at the bench scale but almost always difficult or impossible to achieve at large scale. It is obviously easier to mix a smaller vessel. Length scales are smaller, and circulation times are therefore less. The laboratory unit geometry is inevitably simpler than that of the full-scale unit.

Crystallizer Mixing

We expect overall mass transfer in the system associated with liquid–liquid mixing, that is, the mass transfer between solution in the macro- and microenvironments, to scale with the turbulent energy dissipation rate (Nienow, Harnby, and Edwards 1997). Therefore, let’s assume that the mass transfer scales with the specific power intensity input to the crystallizer, which should roughly equal the turbulence dissipation rate. [Specific power intensity P is frequently defined as the power input by the vessel agitator divided by the volume of the suspension (Oldschue 1983).] Another choice that is often preferred because the influence of the turbulent energy may be localized around the agitator is to divide the power input by the swept volume of the impeller. While it is tempting to simply scale up by keeping the specific power intensity equal between scales, this is usually impractical and often undesirable. The amount of power required is often simply too large. Also, if such power can be applied, it will often cause unacceptable damage to the growing crystals and unwanted secondary nucleation. The ramification of the difference in suspension homogeneity between small and large scale is that mixing-related problems are often encountered on scale-up. Conversely, experimentally modeling full-scale equipment in the laboratory often does not capture mixing-related phenomena observed at the larger scale. The unavoidable compromise of scaling crystallizer mixing is that high specific power input improves mixing and crystallizer homogeneity, which leads to the choice of high agitator speeds. By contrast, particle damage and the control of secondary nucleation (which are strong functions of agitator shear rate) argue for the use of low agitator speeds. Therefore, the appropriate tradeoffs must be made to achieve sufficiently good mixing while minimizing particle damage and managing secondary crystal nucleation. Many “rules” for scaling up mixing in crystallizers (and/or suspending particles in tanks) have been proposed. Geisler and colleagues compiled the list in Table 10.1 and reviewed the many options suggested for scaling the power input to vessels containing suspensions (Geisler, Buurman, and Mersmann 1993). The range of the ratio of specific power intensity from laboratory to full scale is presented as a function of the scale-up ratio, expressed as the ratio of vessel diameters in Figure 10.9. Remarkably, previous workers have suggested a range of specific power input scalings: ðDt;lab =Dt;fs Þ0:5 ≤ Plab =Pfs ≤ ðDt;lab =Dt;fs Þ1 . To more fully appreciate the inherent conflicts in these mixing scale-up recommendations, let us consider solids suspension, fluid mixing, and crystal damage separately. First, we’ll consider solids suspension. Zwietering (1958) found empirically that the angular velocity necessary to just suspend (but not uniformly suspend) similar particles in agitated tanks scales as ωDKa = constant, where Da is the diameter of the agitator, and K is an empirically determined exponent. The criterion of “just suspended” means that all particles are in motion but implies nothing about the state of the suspension. Therefore, this correlation generally should be regarded as the minimum requirement for solids suspension, not the desired

operating point for processes sensitive to particle distribution, particularly crystallization. To suspend particles, the linear velocity developed in the fluid by the impeller must exceed the settling velocity of the particles, that is, V > Vs, where V is the linear velocity produced in the fluid by the impeller, and Vs is the settling velocity of the particles. To first approximation, V scales with the agitator tip speed ωDa , that is, V ∼ ωDa . Assuming that the settling velocity of the particles doesn’t change with the scale of the apparatus and that V = o(Vs), ðωDa Þlab ∼ ðωDa Þfs . Therefore, ωlab =ωfs ≈ Da;fs =Da;lab . This is equivalent to keeping the tip velocity of the impeller constant, as proposed by the more complete analysis of Nienow (1976). [The linear velocity generated by the impeller is related to the tip velocity of the agitator by the flow number NQ (Oldschue 1983). For an axialflow impeller, the average velocity 〈V〉 ≈ 4Q=πD2a , where Q is the volumetric flow rate discharged by the agitator. Because Q ¼ NQ ωD3a , 〈V〉 ≈ 4NQ ωDa =π. Therefore, the tip velocity equals 〈V〉 when NQ ¼ π2 =4.] Let us now consider the fluid mixing requirements. Typically, the first attempt at scale-up involves keeping the specific power input constant between scales (Oldschue 1983). The specific power input scales with the cube of the angular velocity ω3 . Therefore, while the ratio of angular velocities is proportional to the inverse of the ratio of the radii, the specific power intensity is proportional to the inverse cube of the ratio of the radii. Consequently, we cannot maintain both the tip velocity and the specific power input constant on scale-up. Either the tip velocity of the full-scale agitator must be very much greater than in the laboratory if specific power intensity is kept constant, or the specific power intensity of laboratory models is very much greater than that found in the full-scale vessel if the tip velocities are kept constant. Particle damage is a function of the shear produced by the agitator, which is nominally proportional to the agitator tip velocity (at least in the vicinity of the agitator). Particle damage is evidenced in different ways, such as collision-induced defects and crystal attrition. Crystal attrition fragments become additional growing crystals, effectively new nuclei. Additional secondary nuclei are produced by crystal collisions via mechanisms that are still not fully understood but thought to be either disruption of the boundary layer around the crystals or dislodging of microscopic surface nuclei (see the discussion of secondary nucleation in Chapter 3). At any rate, the result is a clear increase in the rate of nuclei produced with increasing agitator-induced shear. Offermann and Ulrich (1982) showed that the number of crystal nuclei produced by an impeller is linearly related to the impeller tip velocity. A commonly used model of the nucleation rate includes B ~ωl, where the exponent l is determined experimentally (Randolph and Larson 1988). This suggests that we minimize tip velocity when possible on scale-up (while keeping it large enough to uniformly suspend particles). This is why one should not use a small-diameter impeller in a large crystallizer: the tip velocity to produce both the required flow to suspend particles and the specific power intensity to mix would be far higher than necessary and

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Daniel A. Green Figure 10.9 Comparison of proposed scale-up rules for agitated tanks with suspended solids. Numbers in brackets indicate references to the original work, as listed in Table 10.1. Source: Reprinted with permission from the publisher from Geisler, Buurman, and Mersmann 1993. Copyright © 1993, Elsevier.

probably would cause excessive crystal damage and secondary nucleation. We must also consider that the typical particle path for a complete circuit of the vessel will be much longer at full scale. Therefore, if the average linear velocities are comparable, the particles in the laboratory-scale vessel pass through the highshear region of the agitator much more frequently than they do in the large-scale vessel. For these reasons, it is often possible to reduce particle damage on scale-up. So the various mixing considerations put conflicting requirements on the parameters involved in scale-up, and there is no clear choice of scale-up rule. Geisler et al. argue that “there is no constant scale-up rule possible . . . scale-up prediction always depends on the particle and fluid properties, as well as on the diameter ratios” (Geisler, Buurman, and Mersmann 1993). The appropriate choice of scale-up variables may also depend on both the nature of the material being crystallized and the equipment configuration. Fragile, damageprone crystals may well have a greater dependence on agitator shear than a more robust crystal. Intelligent, robust vessel design is an asset. For example, a system with a well-designed crystallizer feed, say taking advantage of a highly turbulent high-shear zone near an agitator, may reduce the sensitivity to specific power input. The individual merits of the system being crystallized and the chosen equipment must be carefully evaluated and tested to determine suitability. Certainly, practical measures to enable mixing conditions to be varied at full scale, such as variable-speed agitator drives, should be considered because the scale-up cannot be reliably predicted from simple scaling considerations.

10.5 Modeling By now you should be convinced that good mixing and the ability to achieve it at full scale are vital to good, wellcontrolled crystallization. You should also have an

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understanding of some of the considerations, as well as the overall complexity, of the problem. Compounding these concerns are the nearly myriad possible configurations possible for a proposed process or even the more constrained but still very broad choice of conditions and configurations available to one trying to improve an existing process. A means of investigating and predicting the effects proposed changes in mixing conditions and configurations have is necessary. For these reasons, we turn to modeling. Because, as discussed earlier, it is impossible to achieve dynamic similarity between laboratory and full scale, the predictive capability of empirical modeling of crystallization is limited. Mathematical modeling also has its shortcomings. Suspension flows in crystallizers are turbulent, two- and perhaps even three-phase flows (e.g., boiling crystallizers), particle size is distributed, and the geometry is complicated with perhaps multiple moving parts (impellers). This is, of course, beyond the possibility of analytical solution of the equations of motion, so we must turn to computational fluid dynamics (CFD). However, even CFD is not capable of successfully dealing with all these features. Successful computational models of crystallizers to date are limited, and a comprehensive crystallizer model cannot be realized. Is modeling, then, still worthwhile? Yes. It must, however, be applied to answer specific questions, not yet in general the global prediction of crystallizer operation. Many (perhaps most) of the results are qualitative and not quantitative. Nevertheless, even such limited information can be very enlightening. The knowledge accumulated from model results, perhaps both experimental and computational, combined with knowledge and experience of what occurs in the actual process, frequently allows the crystallization practitioner to piece together a reasonable picture of the entire crystallization process, at least to the extent that specific problems can be solved or avoided.

Crystallizer Mixing Table 10.1 Proposed Scale-Up Rules

1

Zwietering 1958

2

Rieger and Ditl 1982

3

Voit and Mersmann 1986

4

Kneule and Weinspach 1967

5

Zlokarnik and Judat 1988

6

Hemrajani et al. 1988

7

Buurman, Resoort, and Plaschkes 1985

8

Rieger, Ditl, and Havelkova 1988

9

Davies 1986

10

Herndl 1982

11

Molerus and Latzel 1987

12

Chopey and Hicks 1984

13

Gates, Morton, and Fondy 1976

14

Einenkel and Mersmann 1977

15

Nienow 1968

16

Baldi, Conti, and Alaria 1978

17

Mueller and Todtenhaupt 1972

18

Zehner 1986

19

Todtenhaupt et al. 1986

20

Niesmak 1982

Note: References for proposed scale-up laws as reported in Geisler, Buurman, and Mersmann (1993). Numbers refer to references in Figure 10.9.

10.5.1 Experimental Modeling Recognizing that dynamic similarity cannot be achieved, general quantitative predictions from laboratory models are not possible. It has often been found, however, that laboratory modeling is still very valuable. The qualitative results obtained are often quite instructive. Additionally, many of the techniques applied in the laboratory can be adapted and applied to full-scale mixing tests in actual industrial equipment. By far the simplest approach is single-phase (liquid-only) experimentation in a laboratory-model crystallizer. It is of very great value to use a transparent scale model so that the flow can be seen and photographed. Single-phase experiments can be quite valuable as long as their very great limitations are understood, primarily the scaling problems discussed earlier and the fact that the very important effects of particles on the flow are not included. Nonetheless, one can get a qualitative sense of overall flow patterns, areas of recirculation, and often identify poor mixing zones with these experiments. Standard techniques of flow visualization can be used, e.g., dye slowly injected at a point or following the progress of a few tracer particles. One method that is particularly valuable for studying mixing, which can also be applied when particles are present, is to put

a pH indicator in the vessel, which is made either slightly acidic or basic, followed by the addition of the opposite, either base or acid, mixed with the feed stream at the usual feed point. Either a pulse or a continuous feed can be used. Mixing can then be evaluated by following the plume of indicator and noting its position and duration – how long until the plume either dissipates or the entire vessel turns color. We have been successful using phenolphthalein in a slightly acidic model crystallizer and adding a pulse of concentrated base to a continuous feed (Jacobs 1996; see Figure 10.10). Thimol blue, which shows up better in video photography, is another good choice of indicator. These experiments show how fast tracer mixes with the bulk. By contrast, we are primarily interested in the steadystate spatial distribution of solute concentration. With a 1-m-diameter model crystallizer and a 0.7-m draft tube, some regions receive tracer almost immediately, whereas others do not receive the tracer for several seconds or more. This indicates that when there is a continuous flow of solute containing feed into a crystallizer with growing crystals in suspension there will be spatial variation of the solute concentration and therefore supersaturation. This is because nucleating and growing crystals are a solute “sink,” relieving supersaturation everywhere in the suspension. Therefore, areas that receive feed/tracer more slowly will have lower supersaturation than regions that receive it sooner because the longer it takes for feed solute to reach a region, the more time is available for solute removal from solution by crystallization. To fully convert the temporal information from tracer experiments to the spatial distribution of continuous feed injection requires knowledge of the entire flow field in the vessel, which, of course, we do not have a priori. It is necessary, then, to either measure or model the flow field before these results can be fully interpreted. In the interim, tracer experiments are useful in developing mathematical models, in model verification, and in estimating model adjustable parameters. To quantify the quality of feed mixing, a technique involving consecutive fast chemical reactions pioneered by Bourne can be used (Bourne, Kozicki, and Rys 1981). Sequential reactions A + B → R and R + B → S are run in the mixing tank or crystallizer, with A continuously fed into the tank and B already present. In the version proposed by Bourne, A is 1-napthol and B is diazotized sulfanilic acid in dilute alkaline solution. B is the limiting reagent. By examining the ratio of the intermediate and final products, R and S, we can determine whether the reaction occurs uniformly throughout the vessel, in a relatively small zone near the injection point, or at some intermediate point. To clarify, let’s examine the limiting cases. First, the concentration ratio X, the fraction of B that is finally present as S, is defined: X = 2[S]/([R] + 2[S]). (Square brackets indicate the concentration of the enclosed species.) Given the stoichiometry and the fact that B is the limiting reagent, [B0] = [B] + [R] + 2[S], where [B0] is the initial concentration of B in the vessel, and [B0] = [R] + 2[S], assuming the reactions go to completion. Therefore, X = 2[S]/[B0].

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Figure 10.10 Injecting pulse of base into acid solution in model crystallizer with phenolphthalein as an indicator. Net flow is up in the annulus between the 0.7Dt draft tube and the (transparent) tank wall. Agitator is rotating clockwise when viewed from above. Note that the shaded plume from the feed indicates a significant recirculation zone behind the vertical baffle. (a) Frame from video with injection point indicated on south side of vessel. (b) Schematic of image showing approximate location of draft tube, impeller, and baffles with respect to the feed.

Conveniently, for the compounds used by Bourne, the final reaction product composition can be determined simply by spectrophotometry because both R and S are dyes. If the mixing is slow compared with the reaction rate, the reaction occurs in a thin layer at the envelope of the feed plume, as shown schematically in Figure 10.11a. The rate of reaction is controlled by the rate of diffusion of A across the envelope boundary. In this case, all A is immediately consumed by B, and B immediately reacts in the thin layer with the intermediate R, and only S results. X in this case, therefore, equals 1. If, however, the vessel is completely well mixed, the reaction occurs throughout the vessel volume, as shown schematically in Figure 10.11c, because B has equal access to both A and R. Now both R and S are produced. The value of X obtained is determined by the kinetics of the individual reactions but is the minimum value for this system. X is therefore bounded, 1≥ X ≥ Xmin. The intermediate case, where the reaction is neither fully diffusion nor kinetically controlled is shown in Figure 10.11b. The reactions proposed by Bourne are fast enough to probe most mixing applications but may not be fast enough to probe very rapid mixing devices such as impinging jets and grid mixers. Mahajan and Kirwan (1996) have developed an alternate pair of reactions with much shorter time scales to probe these situations. The above-mentioned technique, of course, can be applied quite easily to characterize mixing conditions in laboratoryscale equipment but can also in principle be applied to characterize the mixing performance of full-scale equipment. Laser-Doppler anemometry (LDA; Durst, Whitelaw, and Melling 1981) is a method of measuring fluid and/or particle velocity. It has improved considerably since it was first introduced. Advances such as the use of fiber-optics and the ability to coordinate measurements with agitator passage make it much more accessible. It has been applied to crystallizer geometries and to mixing tank and agitator geometries similar to crystallizers (e.g., Stoots and Calabrese 1995) and has greatly advanced our understanding of these flows. LDA can be used

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Figure 10.11 Schematic showing range of concentration ratio X (see text) for diazo coupling reactions developed by Bourne, Kozicki, and Rys 1981). Reactant A is fed as shown by arrow at left of reaction volume; reactant B is distributed throughout the vessel. Reaction occurs (a) in a narrow zone around the envelop of the feed (indicated by heavy black line), (b) inhomogeneously (indicated by shaded gradient in vessel), and (c) homogeneously throughout the vessel volume (uniform shading).

to simultaneously measure all three spatial components of velocity by using and resolving three colors of laser light simultaneously. LDA is inherently a pointwise measurement. Mapping out an entire time-varying flow field is very cumbersome, and the resolution is necessarily coarse. Particle image velocimetry (PIV) uses images of particles illuminated with a sheet of light that are taken a short, known interval apart and are cross-correlated region by region to determine particle velocity as a function of position in the image plane to measure the velocity over a two-dimensional (2D) slice of the flow field. Both LDA and PIV actually measure particle velocities and not fluid velocities directly. To measure fluid velocity, a low concentration of particles is dispersed in the flow. These particles are chosen so that they very closely follow the fluid; i.e., they are small and with a small density difference relative to the fluid. Both techniques can in principle be used to directly measure crystal velocities, but crystal concentrations are usually much too high to allow the necessary optical access for either of these techniques, except in limited cases with low particle concentration (Yu and Rasmuson 1999). For a more complete understanding, it is necessary to include the effect of prototypic particle concentrations on flow. For experimental flow modeling, it is generally best to employ a noncrystallizing, attrition-resistant solid in

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a convenient liquid. Otherwise, the particle size distribution (PSD) and potentially the morphology of the particles will constantly change over the course of the experiments. Various such solids have been used, such as glass and plastic spheres, sand, and polymer resin pellets. Some thought should be given to the density difference between the solid and liquid, as well as the PSD. It usually makes sense to select a system with a settling velocity approximately equal to that found in the actual process. There are several approaches to obtaining information about flow patterns and mixing. Dye injection and acid/base indicator neutralization experiments used for single-phase experiments have some value but less than for single phase because the particles obstruct the view, and one can only see at most a few centimeters into the flow. Often, however, enough information can be extracted to make this a useful exercise. In particular, it is valuable to follow single-phase experiments with selected two-phase experiments used as verification of the single-phase results. We have used a simple fiber-optic reflectometer (MTI Photonic Reflectometer, Mechanical Technologies, Inc., Latham, NY, USA) to measure local particle concentration (Figure 10.12). For suspensions of particles with a narrow size distribution, this device can be calibrated to give particle concentration. By traversing the probe around the vessel or by using an array of sensors, the spatial uniformity of the particle concentration can be estimated. For suspensions of glass spheres in water in a laboratory model of a DTB crystallizer (average particle concentration 10 vol%; 128-μm average diameter), we find particle concentration gradients vertically that are complicated functions of the agitation rate (Figures 10.3 a and b). There is also a clear difference between the average concentration inside and outside the draft tube (comparing the average particle concentrations of Figures 10.3) caused by the different particle residence times in each zone resulting from the difference in net particle velocities, the vector sum of the particle settling velocity and the local average fluid velocity. This can be crudely modeled by considering hindered settling. The Richardson–Zaki correlation (Richardson and Zaki 1954) can be used along with an estimate of the fluid velocity in each

region: the pumping rate of the agitator divided by the crosssectional area. The predicted particle concentrations are 13 and 9.1 vol% inside and outside the draft tube, respectively, significantly overpredicting the actual difference of approximately 2 vol%. The residence time distribution of both solution and crystals can be measured, although this information typically shows only gross differences in mixing quality. We have injected spikes of sodium chloride into an 8-liter laboratory DTB crystallizer model filled with a suspension of water and glass spheres (Green and Robertson 1993). Water was continuously fed and solution withdrawn through a screen so that particles were retained in the vessel. A spike of sodium chloride solution was added to the feed and the chloride ion concentration of the withdrawal stream monitored as a function of time. As can be seen in Figure 10.13, there is a considerable difference between feed injected on the centerline above the agitator and that injected at the side of the vessel for a low rpm with incomplete suspension. However, the difference for larger rpm and complete suspension is much less. The residence time distribution for the crystals can be characterized as outlined by Nienow (1997). Bourne and Zabelka (1980) used this technique to characterize the average residence time of crystals in a laboratory crystallizer as a function of crystal size. Process tomography has recently made dramatic advances and is a very good candidate for application to crystallizers. McKee (1994) made measurements of particle concentration in an agitated tank, clearly showing gradients in particle concentration. Either electrical resistance or capacitance can be used. Typically, sensors are attached to or embedded in the vessel walls (see Hoyle, McCann, and Scott 2005). Use with draft tubes and other vessel internals can be tricky. Recently, immersed linear sensor arrays have been developed, along with the algorithms to interpret the signals. These are now commercially available (Industrial Tomography Systems, Manchester, UK). This is particularly promising because these probes can be practically deployed in large process vessels, both inside and outside the draft tube if so equipped. Only access at the top of the vessel is required.

Figure 10.12 Schematic of a simple fiber-optic reflectometer used to measure particle concentration in a model crystallizer. For a given PSD, intensity of backscattered light is a function of local particle concentration.

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Daniel A. Green Figure 10.13 Solute residence time distribution in a 19-cm model DTB crystallizer in which 10 vol% 128-μm glass spheres are suspended. On-axis feed injection results in a significantly narrower residence time distribution.

For full-flow visualization, a transparent slurry must be obtained. Transparent particles, such as clear plastic or glass, are used with a liquid formulated to match the index of refraction of the solids (Karnis, Goldsmith, and Mason 1966). The problem encountered is that liquids with a high enough index of refraction to equal that of the transparent solids tend to be either very viscous, toxic, flammable, or likely to dissolve plastics. Abbott et al. (1993) report using polymethylmethacrylate (PMMA) spheres in a solution of tetrabromomethane and a polyalkylene glycol. We have used glass spheres in xylene with a small amount of diiodomethane, but this is only partially satisfactory. Although suspension transparency increased dramatically over the same spheres in water, it was never fully transparent. We suspect that there is some dispersion of the index of refraction of the glass. Merzkirch (1987) has tabulated data from Donnely (1981) on liquids having refractive indices close to PMMA. Opaque tracer particles are used to follow the flow. The particle flow is tracked by marked particles similar in size, shape, and density to the transparent particles, whereas the fluid flow can be resolved by tracking very small [o(1 μm)] marked particles (Bachalo 1994). Once the model suspension has been rendered transparent, a whole range of techniques that rely on optical access to the flow field becomes available. LDA, for example, can measure the velocity at a particular point. Sheets of laser light can illuminate slices of the flow, and PIV can in principle be applied (Liu and Adrian 1993). Generally, these techniques require specialized equipment, but practitioners should be encouraged to investigate crystallizer flows.

10.5.2 Computational Modeling Although instrumentation capabilities are continually improving, the limitation that small-scale experimental modeling cannot be dynamically similar to full-scale processes is absolute, 2

imposed by the physics of the system. However, computational modeling is continually improving and is indeed the great hope for future comprehensive crystallizer mixing models. Particle size distribution (PSD) is the characteristic of particles produced by industrial crystallization processes that is most often modeled. Population balances predict the effect of fundamental processes nucleation and growth, as well as perhaps attrition, breakage, and agglomeration, on PSD (Randolph and Larson 1988). Most applications of crystal population balance modeling have assumed that the solution and suspension in the crystallizer are homogeneous, i.e., the mixed-suspension, mixedproduct removal (MSMPR) approximation.2 This approximation is clearly not justified when there is significant inhomogeneity in the crystallizer solution and/or suspension. For example, it is well known that nucleation kinetics measured at laboratory scale are not very accurate when applied to full scale. It is very likely that the reason for this inaccuracy is because MSMPR models used to define the kinetic parameters may apply fairly well to relatively uniform laboratory crystallizers but do considerably worse for full-scale, comparatively inhomogeneous crystallizers. In cases where significant inhomogeneity exists, a distributed parameter model is needed that allows for and predicts the spatial distribution of key process parameters. The ultimate goal is a comprehensive model that predicts characteristics of the crystals produced, taking into account, e.g., flow fields, mass transfer, dynamic particle population balance, and crystal growth. Of course, developing and solving such a model are vastly more difficult than for the MSMPR counterpart. Therefore, this comprehensive model remains significantly beyond reach for the general case of crystallization. Let’s examine some of the difficulties from the standpoint of crystallizer mixing. Realistic treatment of most crystallizers requires modeling turbulent high-concentration two-phase flow, which is at the cutting edge of CFD development. Improvements both in constitutive relations and in computer processor speed have

This is simply the analogue of the continuous-stirred-tank (CSTR) approximation (Levenspiel 1999) for systems containing particles. It means that the system is well mixed from the standpoint of solute concentration, particle concentration, and PSD. In addition, the effluent is assumed to have the same solute concentration, particle concentration, and PSD as the tank.

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advanced this field, although there are still significant limitations. New developments in user interfaces and the general decrease in cost of computational power are making this type CFD modeling available to more and more crystallization practitioners. Modeling the feed zone, with its attendant micromixing, is difficult because the small length scales of the micromixing [o (Kolmogorov scale)] are below the grid size of the typical vessel-spanning computational mesh. The overall mesh cannot be made fine enough to model these scales because the number of computations necessary will overwhelm even today’s fastest computers. Therefore, micromixing models for the subgrid scales are usually necessary. These are discussed later. It is often useful to increase the computational mesh resolution locally where gradients are high, e.g., around the injection point. This is implemented in several commercial codes, which are capable of generating nonstructured, adaptive meshes. However, the model is not usually changed as the grid size changes. In other words, the same closure models are applied across the entire mesh. Examining some of the successful CFD work on crystallization that has been reported, each model has limitations to its applicability. For example, reasonably accurate models of precipitation processes can be formulated using the assumption that particles present at small size and low concentration do not significantly affect fluid flow. Therefore, single-phase flow prediction coupled with mixing and population balance models can be applied (Wei and Garside 1997; van Leeuwen, Bruinsma, and Rosmalen 1996). The formation of nanoparticles, an area of considerable current interest, is particularly amenable to this approach. For example, Cheng and Fox (2010) combine a sophisticated mixing model with population balance and single-phase CFD models to accurately predict the PSD of nanoparticles produced by rapid precipitation in a small mixing device, a multi-inlet vortex reactor. Others have had a more limited initial goal, such as predicting only the flow field (ten Cate et al. 2001; Derksen, Kontomaris, and Akker 2007) or the flow fields and distribution of solids (Green, Kontomaris, and Kendall 1998). Still others have used greatly simplified hydrodynamic models to examine the effect on other aspects of the process, such as using Poiseuille flow and examining the effect on aggregation (Mumtaz and Hounslow 2000). Alternately, a small volume of the crystallizer can be modeled with a very fine mesh capable of capturing the phenomena whose length scales normally fall below the grid scale. For example, Rodriguez Pascual et al. (2009) have modeled in great detail a small region immediately adjacent to the scraper-blade of a scraped-surface melt crystallizer to better understand what happens in this very important region. There is another, more pragmatic approach to crystallizer modeling. By dividing the crystallizer into individual zones or “compartments,” which are then generally treated as individual MSMPRs, spatial variation in the actual crystallizer can be modeled by allowing the parameters to vary between the individual compartments (Neumann et al. 1999). The relative simplicity of this approach, while still capturing some of the effects of the spatial distribution of variables in the vessel, is encouraging. There are, of course, drawbacks: deciding how to divide

the crystallizer into compartments is vital to the success of the approach. Typical choices of compartments are one for the feed zone, the evaporation zone (if there is one), and several for the bulk of the crystallizer, say at least one inside and one outside of a draft tube. The exchange flow rates between compartments must be specified. Because these are not known in general, they are often treated as adjustable parameters. Alternately, CFD flow modeling can be used to determine appropriate values for these exchange velocities (Zauner and Jones 1999). Coupling CFD with a multiple-zone model is potentially very valuable. It is difficult and time consuming to add the additional complexity of mixing, heat and mass transfer, and dynamic population balance modeling to the CFD model, plus it makes the CFD model very slow. Sufficient accuracy may be achievable for many applications by applying population balance modeling to the simpler zone models, although accuracy will be limited because the effects of particle concentration, distribution and PSD will not be fed back to the transport models. Alternately, comprehensive CFD models can be used to understand the flow and its variation for a limited number of conditions, but simpler zone models may be used for application of the model where speed and convenience are important and detailed accuracy is not (e.g., process control). For example, the commercial crystallizer population balance modeling package gCrystal (Process Systems Enterprise, Ltd., London, UK) is now offering this capability. gCrystal allows the user to define zones. A link is then provided to the commercial CFD package Fluent (ANSYS, Canonsburg, PA, USA) to predict overall crystallizer flow and specify the zone exchange flows. A logical extension of this approach is to divide the crystallizer into a myriad of compartments. This is known as the network-of-zones approach, pioneered by Mann (1993). Here the exchange flows between compartments are not modeled but applied separately, either from knowledge of the overall flow pattern achieved from experimental measurements or from a CFD flow model. Additionally, rules are applied that describe the turbulent transport of particles and species between zones.

Details of Computational Modeling Let us now take a closer look at some of the detailed considerations of modeling crystallizer flows. Impeller Treatment The standard method of CFD is to set up a geometric grid on which to perform the finite difference/finite element analysis of the equations of motion. This mesh is fixed to the boundaries of the problem, and the fluid moves through it. For a tank with an agitator (i.e., the configuration of most crystallizers), the agitator/impeller represents a portion of the boundary that moves relative to the rest of the boundary (i.e., the tank and baffles). This is a complicated problem to treat rigorously. The flow near the impeller is strongly time dependent, whereas the time-average flow in the rest of the vessel is more nearly steady state. Of course, the agitator cannot be treated explicitly

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with a fixed grid, except as noted below. There are several different options to treat the impeller problem; each offers a particular combination of fidelity to the problem and computational complexity. The simplest approach, which is used much less frequently now that more accurate representations have become accessible, is to apply a specified momentum source at or near the impeller location and not model the detailed impeller dynamics. A series of mesh points is selected where a momentum source condition is applied. For example, the fluid velocity can be specified at a plane just above the impeller. Because the impeller is the primary source of turbulent energy in the system, a turbulence source is also included. Modeling in two dimensions is used much less frequently today because computational power has made 3D modeling accessible, but the simple impeller momentum source is the only choice for 2D models and can also be used for 3D models. A disadvantage is that the momentum source must be specified. Frequently, experimental data obtained at the desired conditions are the basis of the values chosen, but for a nonstandard agitator or a standard agitator working under nonstandard conditions, an experimental program to measure the velocities and turbulence in the vicinity of the agitator under process conditions is necessary. An alternate approach may be to perform one of the more detailed calculations discussed below for a single case and then to use this information, perhaps scaled appropriately, to specify the momentum and turbulence sources for production calculations. Another disadvantage is that none of the dynamics of the rotating impeller are captured. In cases where it is important to capture these dynamic interactions, this approach will be inadequate. The next, more complicated treatment is the use of “multiple reference frames” (Ranade 1997). Here a mesh is generated that is stationary with respect to the vessel. Inside this mesh, meeting it in a cylindrical boundary, is another mesh that is stationary with respect to the impeller; in other words, it is “attached” to the impeller and moving with respect to the tank and the other mesh. A quasi-steady-state flow problem is then solved in each mesh, with the two solutions being matched where the two mesh boundaries meet. The problem, as you might imagine, is in making the numerical solutions match at the interface. For this reason, this approach is limited to problems where the inner and outer flows are not strongly coupled. It has been applied with good success for singlephase modeling of agitated, baffled tanks. The advantage is, for an approach that is slightly more computationally intensive, that the impeller itself is actually modeled, which may be important for cases where impeller–fluid or impeller–solids interactions are important. Its disadvantages include that it does require more computational time and the limitation to problems with weakly interacting inner and outer flows. Because the impeller model produces a steady-state solution, the multiple reference frame approach is compatible only with time-averaged and not time-dependent models of turbulence (see below). The third “standard” for modeling rotating impellers is the sliding-mesh approach (Rai 1985). Here a grid is attached to the

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impeller that does not extend much beyond the outer radius of the impeller. This is the most computationally intensive of the three standard techniques but also potentially the most accurate. In principle, it can fully capture the effect of the agitator on flow, including time dependence. It is compatible with time-dependent turbulence models. Advances in impeller treatment are continuing, both in developing new treatments and in making the established treatments more accurate and more efficient computationally. For example, Derksen and Van den Akker (1999) have applied an adaptive force field technique in a model of Rushton turbine flow. Vanderheyden at Los Alamos has developed a technique of switching a momentum source term at certain mesh node points on and off in a pattern that simulates the effect of a rotating impeller (W. B. Vanderheyden, personal communication 1998). These and other approaches promise increased fidelity and/or greater computational efficiency. Treatment of Turbulence Turbulence is inherently a time-dependent phenomenon, but there is value in describing the time-averaged flows. The wellknown Reynolds-averaged Navier–Stokes (RANS) formulation allows us to capture the effects of turbulence in a time-averaged sense and be able to predict time-averaged flows and even to use steady-state models. There are significant limitations, chief among them that the turbulence is assumed to be independent of time. Modeling turbulence is difficult because it is usually impossible to use a mesh fine enough to capture the fine scales of the turbulence for realistically sized vessels. This poses a significant problem for modeling. To rigorously capture turbulent behavior computationally, the numerical grid must resolve the smallest length scales and still be large enough to fit the entire vessel. In practice, in all but some very specific cases, the necessary number of grid points is too large. Even with the fastest modern computers, the computational load is too high. Therefore, several approaches have been taken. One, direct numerical simulation (DNS), resolves the smallest turbulence scales but attempts to model only a very small volume of the flow domain. Boundary conditions are necessarily highly idealized. Because the volume is often too small to allow realistic representations of turbulencegenerating elements, a spatially uniform source of turbulent energy is frequently used (Moin and Mahesh 1998). Despite the obvious limitations, we rely on this type modeling to investigate features of turbulence that we can then incorporate into other, less rigorous, but more realistic models (ten Cate et al. 2004). The usual approach to modeling turbulence in realistic geometries such as process vessels is to employ “closure models.” Because the computational grid cannot be made fine enough to capture the fine scales of the turbulence, the turbulence itself is modeled. For RANS simulations, the familiar k − ε model (Hinze 1975) is often used. The dissipation of the entire cascade of turbulent eddies is captured in the eddy viscosity, which is a function of the turbulent kinetic energy k and the local turbulent energy dissipation rate ε. This

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approximation assumes that turbulence is isotropic, when clearly it is not. Models of turbulent flow continue to evolve [see the review on modeling turbulent mixing in VanDenAkker (2006)]. A significant advance is large eddy simulation (LES; Galperin and Orzag 1993). The larger scales of turbulence, created as they are by inherently anisotropic vessel elements and highly influenced by the vessel envelope, are not isotropic. LES then models the large-scale turbulent structures as time dependent and anisotropic, relying on a closure model that assumes isotropic properties to capture only the subgrid scales. Because there is a very much greater chance that the subgrid scales are approximately isotropic, this is potentially a much more accurate approach. Techniques such as the Lattice–Boltzmann method (Chen and Doolen 1998) have also been developed that significantly speed the computations. This, in turn, allows the use of finer computational grids and therefore more accurate rendering of a broader spectrum of turbulence length scales. RANS models are still widely used, particularly because they are available in commercial CFD packages, but better options are available, although at the cost of increased computation and perhaps greater effort. Also, simpler turbulence models are often chosen for crystallizer flow modeling because of the complexity of the rest of the model system. Some variant of the k − ε model is typically used for the liquid phase. Modeling Multiphase Flow Modeling dense multiphase flows is difficult. Consider some of the complexity of the problem. A full solution to the equations of motion for a solid–liquid suspension flow would require that the flow field around each individual particle be determined, as well as all the interactions among the particles, including their effect on the liquid. The complexity of modeling suspensions of anything much over a few volume percent of solids is daunting and has not been successful. Therefore, we need to approximate. Two fundamental approaches are taken. The first is Lagrangian particle tracking, where computationally we predict the trajectories of a finite number of particles in a flow field and then use this information to extrapolate and infer some properties of more concentrated suspensions. The second approach is to view the particles as their own “continuous” phase. This requires resolving the flow on a large enough length scale that the flow and motion of individual particles are not resolved; rather, the flow of the overall solids phase is. (This seems surprising at first but is in fact analogous to using the continuum approximation for the description of the flow of discrete fluid molecules.) There are then two continuous phases modeled that interpenetrate each other and exchange momentum: the liquid and the solids – hence the name for this technique: the interpenetrating continua (IC) model, also known as the two-fluid or Eulerian–Eulerian model. The principal drawback of Lagrangian particle tracking is just too many particles have to be tracked to model dense suspensions. Because the computational time is at least proportional to the number of particles, the time required for

computation increases very quickly with particle number. Different degrees of realism are possible with this approach. Particle collisions may or may not be allowed, as may the interparticle effects associated with perturbations to the liquid flow field. Of course, each level of additional complexity requires more computation. Part of the method’s strength is that because particles are essentially treated individually, accounting for particle size distribution among particles is straightforward. Particle growth, breakage, and agglomeration can all be modeled. The underpinning of the IC approach is volume averaging of the equations of motion (Drew 1983; Anderson and Jackson 1967). In this approach, the equations of motion are averaged over volumes sufficiently small as to be small relative to the domain being studied but large relative to the individual particle diameter. [The same equations can also be derived from mixture theory (Homsy, El-Kaissy, and Didwania 1980).] The IC model has particle concentration as a parameter, but higher particle concentrations do not automatically require more computations, so it becomes more efficient for dense suspensions. Some of the drawbacks of this technique, however, lie in the empiricism necessary to close the model. Treating the particles as a continuous phase, while of considerable merit, is obviously not completely accurate. Terms arise in the model that have a somewhat unclear meaning and whose values, although estimatable, cannot be rigorously measured. One of these is the viscosity of the dispersed solid phase. This arises because momentum is clearly dissipated by the solids as the suspension is sheared. Think of the flow around each individual particle: each creates shear and has a wake, dissipating momentum. The way in which this is captured in the model is to include a term in which the solids phase is assigned a viscosity. Momentum exchange between the fluid and solids continua also clearly occurs and is included in the model but cannot be rigorously measured. The values for these terms are inferred from measured suspension properties and other relationships such as drag laws. The treatment of particle size distribution, particularly time-varying particle size distribution, is problematic for IC modeling. Particle size distribution figures in the model only in the values of empirical parameters chosen to represent the solids phase. These would be averaged quantities for a given particle size distribution and set of particles. In crystallization, of course, this particle size distribution is in general evolving with time. Because there are no satisfactory models to predict the dependence of all the empirical parameters on the details of the particle size distribution, it is not now possible to capture an evolving particle size distribution with a simple liquid–solids IC model. One possible approach, although prohibitive in computational cost for all but very limited cases, would be to divide the particle size distribution into a number of size ranges (of number n) and then to define each as its own “continuum” and incorporate it into the interpenetrating continuum model. Of course, another drawback would be all the exchange of momentum terms needed to express momentum exchange between what are then n + 1 continua. Although

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limited, this approach would be valuable when the PSD can be divided into a very small number of ranges (say fine and coarse particles). In fluidization, Sinclair (Curtis) has incorporated the effects of bimodal PSDs into IC models (van Wachem et al. 2001). Another approach is to use the modes of the size distribution to account for the effect of particle size distribution on flow. Brown et al. (2006) have shown that this modal approach can be applied to polydisperse aerosol particles. Here moments of the PSD are used to couple the evolving PSD with the IC model. The additional computational load imposed is relatively low. Although less well defined than turbulence of the continuous phase, the dispersed particle phase also experiences turbulence. Turbulence in the solids is usually either treated with a simple RANS model, such as k −ε, or ignored. There are other considerations, such as the influence turbulence in one phase has on turbulence in the other phase. These effects are not captured in the momentum transfer terms contained in the time-averaged equations and must be considered separately, if included. Multiscale Approach The importance of phenomena at various length and time scales in turbulent flows is now broadly recognized. As discussed earlier, full simulation of all scales across the full extent of a typical industrial-size vessel is impossible. Van den Akker (2010) has recently posed an interesting possibility: the use of DNS at various locations within the vessel; then feeding the results into a vessel-spanning LES simulation. In essence, the DNS results would substitute for, or inform, the subgrid closure model of the LES simulation. The importance of the large-scale time-averaged flows, as well as the small scales of turbulence in mixing, has been recognized for some time, and these actions are being modeled. Linking these large and small scales in an efficient, accurate way is the emerging area of mesoscale modeling of flow and turbulence (see, e.g., Derksen 2010). Mixing Models Similar to turbulence itself, mixing occurs over a range of length scales. The treatment of macromixing is straightforward, coupling mass transfer with the equations of motion. However, in regions such as the feed point where micromixing is important, a closure model is needed because the length scales of micromixing are below the scale of the computational grid. This is analogous to the need of a closure model for the smallest turbulent length scales. The engulfment model proposed by Baldyga and Bourne (1989) is commonly used [see also the discussion in Marshall and Bakker (2004)]. The capabilities of mixing models are improving. It is now possible to predict blend times, the time needed for a mixing tank to reach a nearly homogeneous state after the injection of a pulse of tracer. Efficient Lagrangian methods are making it possible to follow both fluid elements and particles to determine fates and develop probability density functions for phenomena in mixing.

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Comprehensive Models The difficulty of obtaining a comprehensive model becomes apparent when we consider that mixing models alone are insufficient. The CFD flow model predicts the flow fields for both solution and crystals, as well as the spatial crystal distribution (as a function of time, in general). The mixing model and coupled mass transport modeling then can predict the solute concentration as a function of position and time. The energy equation must be coupled so that the temperature distribution can be predicted. Once the concentration and temperature fields are solved, the supersaturation can be predicted given knowledge of the saturation of the solute as a function of temperature. Then the dynamic population balance must be applied. Supersaturation is the dominant variable, but a complete representation of the flow opens the possibility of using more realistic models of the fundamental crystallization mechanisms. For example, the dependence of crystal nucleation on local shear rate and particle concentration could be included. The particle population properties – particle concentration and PSD – then must be fed back into the transport models because of the coupling of the particle field with supersaturation and transport properties. Each step in this process is significant, creating great computational demands. When coupled, the ability to achieve this envisioned comprehensive model remains elusive. The computational demands are simply too great. Even though this comprehensive model for the general case of high-solids-loading industrial crystallization remains unachieved, the success of modeling the limited cases of lowconcentration precipitation (to which I’ve previously referred) suggests that this approach should be quite successful when advances in computing power and model efficiency make the extension to conventional crystallization possible.

10.6 Summary Understanding mixing is vital to understanding and controlling the basic phenomena associated with industrial crystallization. Our limited understanding of and ability to predict the effects of mixing have caused many of the problems encountered in controlling and scaling up industrial crystallization. Our hope is that this situation can be improved, particularly as our ability to characterize and predict crystallizer mixing improves. While we do not yet have complete predictive capability applicable to the vast majority of industrial crystallization processes, our ability to model these systems is steadily improving. Computational modeling is advancing. Special cases, such as precipitation, can now be treated that would have been impossible only a few years ago. The fidelity of our models and the range of applicability will steadily improve as both constitutive models and computational power increase. We also have many tools to characterize mixing both in the laboratory and, to a more limited extent, at full scale. While we must constantly remember that experimental modeling is in general limited to a qualitative understanding, we should gather whatever information is possible and use it to synthesize

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a much better understanding of our crystallization processes. We should employ modern characterization techniques to fullscale crystallizers as often as possible. First, by doing so, we will obtain unambiguous data on full-scale crystallizer mixing. Second, it will only be with such data that we will be able to verify the predictions of our computational and laboratory models. In fact, recently, the progress in obtaining such data has not kept pace with the advances being made in modeling, so there is now a paucity of data with which to compare developing computational models. Even though quantitative predictions are not in general available for crystallizers, the importance of good qualitative understanding of the mixing phenomena occurring in operating crystallizers should not be underestimated. Both laboratory and computational modeling now can give us qualitative insight into

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the mixing processes. Ideally, both experimental and mathematical models can be applied. Mathematical models can help us extrapolate and scale up the results we find at the laboratory scale. Tools are also available to investigate specific details of our processes, which is a great help in designing, operating, and troubleshooting. The level of understanding our improving capabilities now provide allows us to intelligently design crystallization vessels and anticipate many of the problems that will be encountered in many, if not all, cases.

Acknowledgment I gratefully acknowledge the input provided by colleagues Ross Kendall, Kostas Kontomaris, Richard Grenville, Minye Liu, David Scott, and Art Etchells.

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Einenkel, W.-D., and Mersmann, A. (1977). Verfahrenstechnik (Mainz) 11(2), 90–94. Galperin, B., and Orzag, S. A. (1993). Large Eddy Simulation of Complex Engineering and Geophysical Flows. Cambridge: Cambridge University Press. Gates, L. E., Morton, J. R., and Fondy, P. L. (1976). Chem. Eng. 83, 144–50.

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Karnis, A., Goldsmith, H. I., and Mason, S. G. (1966). J. Colloid Interfac. Sci. 22, 531–53. Kneule, F., and Weinspach, P. M. (1967). Verfahrenstechnik (Mainz) 1(12), 531–40. Kunii, D., and Levenspiel, O. (1969). Fluidization Engineering. New York, NY: Wiley. Levenspiel, O. (1999). Chemical Reaction Engineering (3rd edn). New York, NJ: Wiley. Liu, Z. C., and Adrian, R. (1993). In M. C. Rocco (ed.), Particulate Two Phase Flow. Boston, MA: Butterworth-Heinemann. Mahajan, A., and Kirwan, D. J. (1996). AIChE J. 42, 1801–14. Mann, R. (1993). Computational fluid dynamics (CFD) of mixing in batch stirred vessels used as crystallizers. Paper presented at the 12th Symposium on Industrial Crystallization, Warsaw. Marshall, E. M., and Bakker, A. (2004). In E. L. Paul, V. A. Atiemo-Obeng and S. M. Kresta (eds.), Handbook of Industrial Mixing: Science and Practice. New York, NY: Wiley. McKee, S. (1994). Tomographic measurement of solid liquid mixing. Ph.D. dissertation, Manchester Institute of Science and Technology, Manchester, UK. Mersmann, A., and Rennie, F. W. (1995). In A. Mersmann (ed.), Crystallization Technology Handbook. New York, NY: Marcel Dekker. Merzkirch, W. (1987). Flow Visualization (2nd edn). Orlando, FL: Academic Press. Moin, P., and Mahesh, K. (1998). Ann. Rev. Fluid Mech. 30, 539–78. Molerus, O., and Latzel, W. (1987). Chem. Eng. Sci. 42(6), 1423–37. Mueller, W., and Todtenhaupt, E. K. (1972). Aufbereit. Tech. 1, 38–42. Mullin, J. W. (2004). Crystallization (4th edn). Oxford: Elsevier Butterworth-Heinemann. Mumtaz, H. S., and Hounslow, M. J. (2000). Chem. Eng. Sci. 55(23), 5671–81. Neumann, A. M., Bermingham, S. K., Kramer, H. J. M., and van Rosmalen, G. (1999). Modeling industrial crystallizers of different scale and type. Paper presented at Industrial Crystallization 1999, 14th International Symposium on Industrial Crystallization, Cambridge, UK. Nienow, A. W. (1968). Chem. Eng. Sci. 23, 1453–59.

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Todtenhaupt, P., Forschner, P., Grimsley, D., and Lases, A. B. M. (1986). Design of agitators for the ERGO carbon-in-leach plant. Paper presented at the International Conference on Gold. Tosun, Guray. (1988). An experimental study of mixing on the particle size distribution in BaSO4 precipitation reaction. Paper presented at the Sixth European Conference on Mixing, Cranfield, UK. van Wachem, B. G. M., Schouten, J. C., van den Bleek, C. M., Krishna, R., and Sinclair, J. L. (2001). AIChE J. 47(6), 1292–302. van den Akker, H. E. A. (2006). Adv. Chem. Eng. 31, 151–229. van den Akker, H. E. A. (2010). Ind. Eng. Chem. Res. 49, 10780–97. Van Leeuwen, M. L. J., Bruinsma, O. S. L., and van Rosmalen, G. M. (1996). Chem. Eng. Sci. 51(11), 2595–600. Voit, H., and Mersmann, A. (1986). German Chem. Eng. 9, 101–6.

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Chapter

11

Monitoring and Advanced Control of Crystallization Processes Zoltan K. Nagy Loughborough University, Purdue University Mitsuko Fujiwara University of Illinois at Urbana–Champaign Richard D. Braatz Massachusetts Institute of Technology

11.1 Introduction Crystallization is one of the main separation and purification processes in the pharmaceutical, biotechnology, food, microelectronics, fine and bulk chemicals industries. The production of more than 70 percent of all solid products involves at least one crystallization as a key processing step, which can have a significant effect on the overall performance of the entire production process and the properties of the final product. The control of crystallization processes is challenging because of the highly nonlinear dynamics, large variations in length and time scales at which the various simultaneous mechanisms occur, variations in crystallization rates over time owing to variations in the impurity profiles of chemical feedstocks, unexpected polymorphic transformations, and nonideal mixing conditions. This chapter reviews the main instrumentation applied for crystallization monitoring and approaches for crystallization process control. The main emphasis is on batch crystallization control, but most of the approaches presented can be easily extended to continuous processes. An overview of the typical basic control systems for industrial crystallization processes is provided by Rawlings et al. (2002). This chapter focuses on the advanced process control approaches that would be implemented at the supervisory level in a hierarchical/cascade structure. To achieve the desired control performance, the basic control systems (e.g., temperature, flow) should be properly designed and tuned before the implementation of any of the advanced control approaches presented in this chapter. The main phenomena that can occur during crystallization are nucleation, growth, agglomeration, and breakage. The main control objectives typically concern (1) the crystal size distribution (CSD), (2) morphology, and (3) molecular purity, which collectively have a strong influence on the efficiency of downstream processes (e.g., filtration, drying), the formulation properties (e.g., flowability, compressibility, friability), and efficacy of the final product (e.g., dissolution rate, shelf life, bioavailability). The final product CSD is determined by the interplay between nucleation and growth, as well as by the presence of agglomeration or breakage of crystals. The complete morphology of the product is described by the polymorphic form, crystal habit, and other variables such as the amorphous fraction or crystallographic defects. Morphology is also determined by the nucleation and growth rates and the presence of breakage. Molecular purity is a key product quality indicator, and the presence of

impurities can also have a significant effect on the final morphology of the product. Agglomeration also can strongly influence molecular purity, for example, by inducing the inclusion of solvent molecules or impurities. The main property indicators are influenced by the initial conditions (e.g., seed crystal characteristics and solvent) and the dynamic operating policies (e.g., cooling, antisolvent addition, evaporation rates). The aim of crystallization control is to design the initial conditions and operating policies that lead to the desired product properties while managing tradeoffs between the various physicochemical phenomena. The control objective can be expressed directly in terms of a desired CSD, morphology, and molecular purity or in terms of quality indicators of the final product. The intensive use of advanced measurement and analysis technologies for the design and operation of processes to manufacture products with the desired properties is referred to as quality-by-design (QbD) in the pharmaceutical industries. Real-time monitoring of the crystallization processes can augment understanding of the crystallization mechanisms, identify potential problems, and provide the support and information needed for the design and control of robust crystallization processes. The large number of available instrumentation and sensor technologies enables a comprehensive monitoring of crystallization processes and provides the measurements needed for the implementation of feedback control strategies. Section 11.2 provides an overview of the most important process analytical technologies (PATs) used for crystallization monitoring. The fundamental driving force for crystallization from solution or melt is the difference in the chemical potential between the solution/melt and the solid phase. For crystallization from solution, the difference in the chemical potential is usually expressed in terms of the closely related supersaturation, which is the difference between the solution concentration and the saturation concentration or some related expression. Supersaturation strongly influences nearly all phenomena during a crystallization, so the product properties depend on the supersaturation profile achieved during the crystallization process. Consequently, supersaturation is typically the main manipulating input employed in crystallization control. The method in which supersaturation is generated also can have a significant influence on control performance and the achievable control objectives and has to be considered at the design stage of the crystallization process. Most often supersaturation is created by

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Antisolvent/Reactant

Narrow PSD

Broad PSD

Different polymorph

Figure 11.1 Schematic representation of a batch crystallizer with the main operating modes – cooling, antisolvent/reactive, and evaporation – used to control supersaturation to achieve desired product properties such as shape, size distribution, or polymorphic form.

Vacuum

Seed

Slurry with supersaturated solution Heating/ Cooling medium

cooling, evaporation, or antisolvent addition (Figure 11.1). Less frequently, other methodologies, such as precipitation (e.g., pH swing) or freezing can also be applied. Depending on the initial conditions, mode of supersaturation generation, and dynamic variation of the supersaturation profile, very different product properties can be achieved (see Figure 11.1). For batch and semibatch crystallization, often applied in the production of pharmaceuticals, cooling and antisolvent addition are used most often for supersaturation generation, whereas in the bulk chemical industries and continuous crystallization systems, evaporation (controlled generally by manipulating the heat input and pressure in the system) is commonly used. This chapter considers mainly batch and semibatch crystallization processes, but most of the monitoring and control approaches discussed apply directly or can be extended easily to continuous processes. The main features of the different methods of supersaturation generation that can be applied in crystallization control are reviewed briefly next. Cooling Crystallization. In cooling crystallization, the rate at which the temperature is decreased influences the level of supersaturation and can be used as a control variable to achieve desired solid-state properties. To achieve a high yield, this method requires a steep decrease in solubility with temperature. An advantage of cooling crystallization is that no additional raw material (e.g., antisolvent) is needed, which could create additional problems in product purity and increase operating and capital costs. Antisolvent Crystallization. Crystallization from solution using an antisolvent is commonly applied in the pharmaceutical industry, in which solute is crystallized from a primary solvent by the addition of a second solvent (antisolvent) in which the solute is relatively insoluble. Typically, a solution of the solute in a solvent, which is often saturated or close to saturation, is initially formed. Then an antisolvent that is miscible with the primary solvent is added. Sometimes multiple solvents or antisolvents are used to achieve a sharper

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solubility curve or to produce a more desirable crystal shape or polymorphic form (crystal structure). When the antisolvent is added to the solution, the solute crystallizes owing to the reduction in solubility. Antisolvent crystallization can be conducted via direct or reverse addition. In direct addition, the antisolvent is added to the saturated solution, whereas in inverse addition, the saturated solution is added to the antisolvent. Direct addition is by far the most common operating mode, whereas reverse addition is employed when the generation of small particles is the objective (Woo et al. 2006; Kim et al. 2009). An advantage of antisolvent crystallization is that it can typically be operated at low temperatures throughout the crystallization, which is important for thermally sensitive products. The solvent activity also changes significantly with its composition during the antisolvent addition, so this approach can have a more profound effect on the crystal morphology or polymorphic form than in cooling crystallization. Such a strong effect on solvent activity can be exploited for increased control of morphology, but in certain cases this effect may be considered to be a disadvantage because the potential formation of a higher number of possible polymorphs or pseudopolymorphs (i.e., hydrates, solvates) may be undesirable. Antisolvent crystallization is characterized by higher gradients in the local supersaturation within the solution, especially in the case of larger-scale industrial crystallizers. To minimize primary nucleation due to the formation of local regions with high supersaturation, the antisolvent should be introduced in a region with high turbulence for fast dispersion. The increase in volume during antisolvent addition may also require progressively increasing agitation speed to achieve proper suspension of the particles. An approach to avoid the formation of highsupersaturation regions is to add antisolvent to the system by diffusion through a membrane over a large surface area (so-called membrane crystallization). Antisolvent crystallizations may have additional costs associated with the solvent separation and require larger capital costs because of the higher operating volumes. By contrast, the solvent/

Monitoring and Advanced Control of Crystallization Processes

antisolvent addition rate can be manipulated very quickly and has a much faster dynamic effect on the crystallization than controlling the temperature of the suspension, as is done in cooling crystallization. Reactive Crystallization. In reactive crystallization, supersaturation is generated by employing a chemical reaction in solution to form a compound at a concentration that is higher than its solubility in the solution. Owing to the rapid generation of high supersaturation, reactive crystallization is mostly used for the formation of small particles (nanoparticles). For some systems, the stabilization and isolation of these particles can be a challenge. Variations in pH (pH swing) to produce a less soluble acid or base from a salt, or vice versa, also can generate supersaturation and is often used for the production of pharmaceuticals. Several control approaches have been applied to various reactive crystallization systems (e.g., see Alatalo et al. 2010). Evaporative Crystallization. In this approach, supersaturation is generated by the evaporation of the solvent from the solution to increase the solute concentration. Both the heating rate and pressure in the system are controlled in these crystallizers. Generally, the level of vacuum is manipulated to achieve the desired evaporation rate at relatively low temperatures. The heat is typically removed by adiabatic evaporation of the solvent. These systems may allow faster control in the case of large operating volumes because generally the heat input and pressure can be manipulated faster than cooling the entire suspension (Bolanos-Reynoso et al. 2008). Freezing Crystallization. Freezing of the solvent from a solution increases the concentration of solute in the solution, which results in supersaturation. The level of supersaturation in this case is controlled by the amount of frozen solvent generated, which can then be removed from a solution by sublimation under vacuum (Connolly et al. 1996). When water is used as the solvent, separation of the two solid phases (product and ice) can be achieved relatively simply based on the density differences. Because the density of ice is lower than that of water, whereas generally the density for the solid product is higher, in a nonagitated slurry the crystals will sediment to the bottom of the reactor, whereas the ice can be collected from the surface of the water. Freezing crystallization can be used for the separation of heat-sensitive products (various active pharmaceutical ingredients [APIs], explosives) or compounds with limited stability in the solution (Connolly et al. 1996). Freezing crystallizaters require special design, and control of the temperature gradient between the jacket and reactor is important to avoid solvent freezing on the walls. The choice of the method of supersaturation generation depends on a series of factors. The level of solubility and the shape (sensitivity) of the solubility curve with respect to temperature or solvent/antisolvent ratio should be examined. If the temperature sensitivity of the solubility is low (flat solubility curve), evaporative or antisolvent crystallization will provide better control of the supersaturation. For

Figure 11.2 Crystallization phase diagram with typical seeded and unseeded operating curves.

thermally sensitive products, antisolvent or freezing crystallization can be an option. Combinations of the different supersaturation-generation techniques such as combined cooling and antisolvent (see Section 11.3.5) and cooling and evaporative crystallization are also commonly applied for better control and increased yield; these combinations typically employ somewhat more complicated control systems. The operating profiles, which correspond to a desired supersaturation trajectory, can be determined and/or implemented in the time domain (e.g., temperature or antisolvent addition profile versus time) or in the phase diagram. Figure 11.2 illustrates two typical operating profiles in the phase diagram in the case of unseeded or seeded batch cooling or antisolvent crystallization processes. In the case of unseeded operation, supersaturation is progressively increased (e.g., via cooling of a saturated solution or antisolvent addition) until spontaneous nucleation occurs, followed by crystal growth. The number of nuclei and the extent of the subsequent growth are determined by the dynamic operating profile. Because spontaneous nucleation is very difficult to control, instead, seeded operation is usually employed by adding initial crystals (seed) with well-defined mass and size ranges at a small but positive supersaturation and operating at a low enough supersaturation that the crystallization is growth dominated. Seeded operation is also preferred when the metastable zone width is very large because nucleation is difficult to initiate or produce highly irreproducible nuclei; often an immiscible additional liquid phase forms (known as oiling out) that can suppress nucleation (Deneau and Steele 2005). When nucleation occurs at high supersaturation, the rate of nucleation is very high and cannot be controlled, generating very small and highly agglomerated particles. The operating curves are similar conceptually independent of whether cooling, evaporation, or antisolvent addition is used to generate the supersaturation, but the boundaries of the metastable zone change, and the shape of the optimal supersaturation trajectory can vary significantly. In unseeded operation, control of the initial nucleation event to provide a consistent in situ seed generation at the production scale is the main challenge, whereas in seeded operation, it is desirable to produce highly consistent seed mass, size,

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and other characteristics to minimize batch-to-batch variability. In continuous crystallization processes, a saturated feed stream with or without seed crystals is continuously added in the crystallizer, a stream of slurry with product is withdrawn, and the control system is designed to maintain a fixed operating point in the phase diagram, with the objective of maintaining constant growth conditions.

11.2 Crystallization Process Monitoring 11.2.1 Classification of Measurement Techniques

•

The measurement and monitoring of crystallization processes are generally differentiated based on whether the information obtained is related to the properties of the liquid or the solid phase. The information about the solution phase is provided by the solution temperature, supersaturation, and concentration, whereas particle size distribution (PSD) and morphology provide information about the dispersed solid phase. The monitoring of the liquid-phase concentration provides information about the degree of supersaturation and the potential yield. The monitoring of the solid phase indicates the existence of agglomeration or breakage and provides a direct indication of how the operating conditions affect the system and the solid product. Measurement of the liquid- and solid-phase variables is required to • Obtain insight into limits to processing and the main governing phenomena; • Design the experiments and obtain data for identification of the crystallization kinetics; • Design controllers to maximize product quality and minimize operating costs; and • Drive the process to its desired state via suitable feedback control. The measurement techniques can be classified into three main categories based on how the sampling is performed: • In Situ Measurements. These techniques consist of measurements made directly in the process medium. Instruments used for in situ crystallization monitoring include attenuated total reflectance (ATR) and Fourier transform infrared (FTIR) or ultraviolet (UV)–visible spectroscopy, ultrasonic attenuated spectroscopy (USS), Raman spectroscopy with immersion probes, focusedbeam reflectance measurement (FBRM), conductivity, pH, turbidity, reflective index (RI), and imaging devices such as a particle vision and measurement (PVM) probe and internal and external bulk video imaging (i/eBVI; Simon et al. 2009a, b, 2010a, b) coupled with multivariate image analysis methods of various types (De Anda et al. 2005; Larsen et al. 2006, 2007; Eggers et al. 2008; Larsen and Rawlings 2009; Sarkar et al. 2009; Darakis et al. 2010). Some of these measurement devices (e.g., PVM) are inserted directly into the crystal suspension, whereas others employ optical images

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obtained through an external observation window in the crystallizer. An advantage of such methods is that they avoid artifacts resulting from sampling and sample preparation. These techniques sample the solid or liquid phase in close proximity to a probe or a window; therefore, good mixing and careful choice of the motion of the suspension flow relative to the window or probe tip are required to obtain representative measurements. In industrial-scale crystallizers, physical limitations such as a lack of available ports and difficulty reaching representative zones of the slurry often preclude the use of these in situ devices. Online Measurements. These technologies provide information during the course of the batch by automatically sampling the slurry and performing measurements on the samples. Commonly used equipment includes laser diffraction and flow cell–based imaging equipment for PSD measurement. Other equipment, such as reflective index measurement in the crystallization of enantiomers and X-ray diffraction in polymorphic crystallization, also has been applied (Hammod et al. 2004). Online measurements can be used for real-time feedback control. The disadvantage of this approach is that it is difficult to guarantee that a representative sample will reach the measurement device because nucleation, growth, and breakage can occur during sampling and transport. Online sampling equipment can be combined with any offline measurement equipment to provide information currently not available with in situ devices. For example, smaller particle sizes can be measured, or with the design of suitable dilution loops, measurement of solid properties in slurries with high solids concentrations can be achieved. By employing online sampling, the same equipment can be used for online and offline characterization of the final products, which reduces the errors and misinterpretation of results that can occur when using different measurement techniques for monitoring and final product characterization. Because the installation of a sampling loop is possible for most systems, these approaches are also suitable for industrial crystallizers when the lack of availability of ports and geometric and safety limitations prevent the use of in situ devices. Offline Measurements. These techniques are used to characterize the properties based on samples taken from the process and analyzed after the crystallization has completed. These techniques can include a large range of solid-state characterization devices, such as optical microscopy, scanning electron microscopy (SEM), atomic force microscopy (AFM), hot-stage microscopy (HSM), differential scanning microscopy (DSC), dynamic vapor sorption (DVS), thermogravimetry (TG), X-ray diffraction, and solid-state Fourier transform infrared (FTIR), Raman, and nuclear magnetic resonance (ssNMR) spectroscopy. These techniques generally cannot be used for real-time monitoring and control, although the online

Monitoring and Advanced Control of Crystallization Processes

PVM

FBRM

Pump Laser diffraction or imaging unit Thermocouple

Comm. interface (serial, modbus, OPc,...)

Figure 11.3 Schematic representation of a crystallization setup with typical in situ and online process analytical technologies connected to an (industrial) supervisory system.

ATR Probe UV-VIS/ FTIR

Supervisory & control computer Thermostat

application of X-ray powder diffraction (XRPD) has been reported (Hammod et al. 2004). These solid-state analytical approaches are generally used in combination with the in situ tools, which are very efficient in signaling potential problems but often do not provide enough information to identify the exact causes (Howard et al. 2009a). Offline measurements are also often used for model identification. For understanding and characterizing crystallization processes and for the design and validation of suitable control strategies, the combined use of solid-state and in situ analytical tools is the most efficient approach. The offline within-batch or end-ofbatch analyses of product using the available comprehensive set of solid-state analytical approaches are the key measurements in the design and application of batch-to-batch iterative learning control (ILC) approaches (see Section 11.3.7) and allow the use of more sophisticated models and various final product quality indicators in the control design. Figure 11.3 illustrates a typical setup, which uses a combination of in situ and online measurement tools for the comprehensive monitoring of crystallization processes, coordinated by a supervisory computer. Figure 11.4 provides an example of a laboratory crystallizer with the simultaneous use of an FBRM probe for monitoring the properties of the solid phase and a UV-visible ATR probe for monitoring solution concentration (with sample information provided by the tools for illustration).

11.2.2 Liquid-Phase Monitoring Many different methods have been developed to measure liquid-phase concentrations in crystallization processes, both in terms of the specific property that is directly measured and in terms of the methods used to relate the measured property to the liquid-phase concentration (Loffelmann and Mersmann 2002). The most commonly used techniques for measurement

of a solute concentration during a crystallization are described next.

Spectroscopy-Based Techniques Most methods for concentration measurement used in crystallization monitoring actually measure a spectrum, which is a plot of the interaction between electromagnetic radiation and the absorbing species of a substance (Atkins and de Paula 2005). The liquid concentrations can be calculated from the spectral data using the Beer–Lambert law for the vast majority of crystallization applications. For the spectrometer to produce a spectrum of the liquid phase in the suspension without being significantly affected by the solid phase, an attenuated total reflection (ATR) accessory is fitted on an immersion probe (Lewiner et al. 2001). Many crystallizations have high concentration levels during at least part of the crystallization, in which case the ATR probe also has the role of attenuating the measured signal into the measurement range of the detector. The penetration length of the light into the solution through an ATR probe is on the order of microns, which minimizes the interaction with the solid, and the desired signal amplitude is achieved (Doyle and Tran 1999). A key advantage of spectroscopy-based techniques is that the multiple data points collected during each measurement enable the simultaneous monitoring of multiple components in a system, provided that suitable calibration methods are applied. ATR-FTIR Spectroscopy In crystallization applications, ATR spectroscopy is employed either in the ultraviolet/visible (UV-visible ATR) or infrared region of the spectrum (ATR-FTIR). The IR regions include the near-IR (NIR), 12,800–4000 cm−1; mid-IR (MIR), 4000– 200 cm−1; and far-IR, 200–10 cm−1. Because most compounds absorb radiation in the mid-IR range, its applicability to different systems is widespread. The IR spectroscope operates by imposing the radiation beam on a sample and measuring

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0.2

counts/sec

600 ATR Probe 400 103

200 0 20 15 tim 10 e (h )

5

0 100

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102 m) (μ th 1 10 eng l d or ch

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2 300 nm

)

1 350 0

h)

e(

tim

Crystallizer

Dynamic monitoring of the chord length distribution using focused beam reflectance measurement

In situ concentration monitoring using ATR-UV/Vis spectroscopy coupled with chemometrics

Figure 11.4 Photograph of a bench-scale crystallizer with jacket for temperature control and FBRM and UV-visible ATR probes for real-time and in situ monitoring of properties of the solid and liquid phases. Typical measurement outputs from each sensor are shown for illustration.

the amount of IR light absorbed at different frequencies. The frequencies at which absorption occurs indicate the identity of the chemical species present, whereas the absorption magnitudes indicate the concentrations of those species. A mathematical treatment, the Fourier transformation, is used to interpret the data and produce a spectrum. ATR-FTIR has been widely used and proven to be a reliable technique for liquid-phase monitoring in crystallization processes (Dunuwila and Berglund 1997; Lewiner et al. 2001; Feng and Berglund 2002; Togkalidou et al. 2001a, 2002; Fujiwara et al. 2002; Liotta and Sabesan 2004; Barrett et al. 2005; Pollanen et al. 2006; Yu et al. 2006a, b; Zhou et al. 2006; Nonoyama et al. 2006; Schöll et al. 2006; Kee et al. 2009a, b). Figure 11.5 shows a sample IR spectrum of an acetaminophen solution. The ATR-FTIR spectral data measured from the crystallization process are almost always linearly related to the solute concentrations. The simplest way of converting the information obtained from ATR-FTIR spectroscopy into concentration is by correlating the height or area of the spectrum corresponding to a specific band of wave numbers with the solute concentration by regular linear regression (Dunuwila and Berglund 1997; Feng and Berglund 2002; Lewiner et al. 2001). Another approach to analyzing the spectral data is through a chemometrics approach, which takes into account the spectra over a wide range of wave numbers using either principal component regression, partial least squares, or variants of these methods (e. g., see Togkalidou et al. 2002; Fujiwara et al. 2002; Liotta and Sabesan 2004; Zhou et al. 2006a; Chen et al. 2009; Borissova et al. 2009). When correctly applied to welldesigned experimental calibration data, chemometrics was shown to produce concentration measurements of very high accuracy (e.g., Togkalidou et al. 2002; Fujiwara et al. 2002) for several reasons, including the averaging of the effects of noise over more data points and the ability to automatically incorporate any additional information

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content in the spectra associated with peak shifts. For crystallization in which the temperature changes, the dependency of the measured spectra on the temperature must be taken into account in the correlation. The usual calibration equation is C ¼ w0 þ

N X

wi Ai þ wNþ1 T

ð11:1Þ

i¼1

where C is the concentration, Ai are the absorbance values measured at different wavelengths (or wave numbers) with i as an index, T is the temperature, and wi are the weighting coefficients in the calibration equation, which are determined by measuring the spectra of solutions with known concentrations at different temperatures and applying chemometrics approaches using all absorbance values over a range of wavelengths or carefully selecting a limited number of absorbance values at characteristic wavelengths. To eliminate the effect of signal drift owing to changes in the intensity of the light source, it is common to use the first derivative of the absorbance in the calibration equation rather than the raw absorbance values. ATR-FTIR spectroscopy has been used to detect metastable limits by detecting a shift in peak absorbance associated with a reduction in solute concentration (Fujiwara et al. 2002) and has been widely applied to monitor normal crystallization operations. However, it is more expensive than other techniques of in situ concentration measurement. Another disadvantage of ATR-FTIR spectroscopy is that the optical material of the probe tip can chemically react or foul for some combinations of probe tips and chemical compounds. UV-Visible ATR Spectroscopy UV-visible spectroscopy uses a radiation beam in the UVvisible region (200–900 nm). Figure 11.5 shows sample IR

Monitoring and Advanced Control of Crystallization Processes

UV Spectra: acetaminophen in Water

IR Spectra: acetaminophen in Water

0.5 absorbance

absorbance

0.15

0.1

Figure 11.5 Typical IR and UV spectra of acetaminophen in water.

0.4 0.3 0.2 0.1

0.05 1800

1600

1400

1200

0

200

wavenumber (cm–1)

250 300 350 wavelength (nm)

and UV spectra of acetaminophen in water as given by ATRFTIR and UV-visible ATR spectroscopy, respectively. The absorbance AðλÞ at a certain wavelength λ obtained from a UV-visible absorption spectrum is directly proportional to the concentration C of the absorbing species in solution according to the Beer–Lambert law: AðλÞ ¼ εðλÞ ℓ C

ð11:2Þ

where εðλÞ is the molar absorption coefficient, and ℓ is the path length of the absorbing solution. The measured absorbance, however, is temperature dependent because the change in temperature influences the refractive index and hence the path length for the ATR element. A temperature correction has to be applied to the UV absorption spectra before or during the development of a calibration model (Thompson et al. 2005), similarly to what is shown in Equation (11.1) for the FTIR spectra. For monitoring crystallizations that operate under very high solute concentrations where the linear Beer– Lambert law does not always apply, nonlinear correlations are used in the calibration. Simple nonlinear forms for the calibration model (Abu Bakar et al. 2009a) have been used successfully, such as C ¼ w0 þ w1 A þ w2 T þ w3 AT

ð11:3Þ

which includes a nonlinear interaction term between absorbance and temperature. Additional higher-order terms for the selected absorbance and/or temperature may be required in particular applications, and use of the first derivative instead of the raw absorbance is common, similar to the FTIR spectrum case. UV-visible ATR spectroscopy has been used for the in situ monitoring and detection of the nucleation event (Simon et al. 2009; Anderson et al. 2001) and polymorphic transformation (Howard et al. 2009a, b). It has also been used quantitatively in the monitoring and measurement of supersaturation levels during crystallization processes (Abu Bakar et al. 2009a; Thompson et al. 2005; Aamir et al. 2010a, b). The use of UVvisible ATR spectroscopy in crystallization monitoring, however, is not as widespread as ATR-FTIR. This is probably due to the limitation of its application only to compounds that contain a chromophore, which is a functional group that causes light absorption in the UV wavelength range. The technique is expected to gain more application in industry because it is less expensive than ATR-FTIR and Raman spectroscopy techniques.

400

Raman Spectroscopy Raman spectroscopy differs from the above=mentioned spectroscopic techniques in that the scattering of the radiation by the sample, rather than an absorption process, is employed. Raman spectroscopy imposes a monochromatic radiation beam on the sample and measures the amount of light scattered at different wavelengths. The difference in wavelength between the incident light and the scattered light is a fingerprint for the types of chemical bonds in the sample. Raman spectroscopy can be used for monitoring both liquid and solid phases. Falcon and Berglund (2003) and Hu et al. (2005) have successfully monitored and measured liquid concentrations using this instrument during crystallization. The Raman probe is less chemically sensitive than an ATR-FTIR probe and can be applied noninvasively.

Other Liquid-Phase Monitoring Approaches Densitometry Density measurement has long been used successfully in the monitoring of the liquid-phase concentration during singlecomponent (Gutwald and Mersmann 1990) and two-component (Zhu et al. 2004) crystallizations. Because the solution to be measured needs to be free from crystals to employ this method, the sample is passed to an external density-measuring cell through a sampling loop (e.g., an oscillating U-tube). The filter can clog unless special care is taken to reduce temperature fluctuations to a very low level. The technique has been used successfully for the online monitoring of a potassium nitrate– water system (Miller and Rawlings 1994). Electrical Conductivity The liquid-phase concentration during a crystallization can also be followed by measuring the electrical conductivity of the liquid phase. This technique has been applied for the crystallization of many inorganic salts (Sessiecq et al. 2000; Hlonzy et al. 1992; Nyvlt et al. 1994). The measurement can be influenced by the solids content, making the correlation of the conductivity signal with the liquid-phase concentration difficult. The need for frequent recalibration of the probe limits its usefulness in long-term industrial crystallization applications. It can be difficult to apply this technique to batch cooling crystallization processes because conductivity is also strongly (and generally nonlinearly) affected by the temperature. This technique has limited applicability to organic systems because

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Zoltan K. Nagy, Mitsuko Fujiwara, and Richard D. Braatz

of the smaller changes in electrical conductivity, and the results are temperature dependent and sensitive to impurities (Mullin 2001). Refractive Index Another technique to measure concentration is the refractive index, which is well correlated with the concentration for many solutions (Zhou et al. 2006b). The technique can work if there is a significant change in refractive index with change in liquidphase concentration, but the technique is sensitive to ambient light and air bubbles. The refractive index also has to be measured online rather than in situ, and the effect of temperature has to be taken into account. A general weakness of nonspectroscopic methods is that a single data point does not provide enough information to measure the liquid-phase concentrations in crystallizations involving more than two chemical species (e.g., solute and solvent), which are very common in industrial applications. In contrast, ATR-FTIR spectroscopy with its information-rich spectra has been employed in crystallization with more than four chemical species (Togkalidou et al. 2002), and there is no fundamental limitations with its application to systems with more than four species.

11.2.3 Solid-Phase Monitoring Many techniques have been used for the in situ monitoring of the solid phase for crystallization processes. The choice of technique depends mainly on the product characteristic that is to be monitored. The measurable properties or events of the solid phase include nuclei formation and crystal size, shape, and polymorphic form.

Image Analysis Image analysis is a simplest technique to monitor crystal size and shape in crystallization processes. This direct observation technique does not require any assumptions for the size or shape of the crystals. In recent years, many applications of online, in situ, and offline image-analysis techniques have been reported for monitoring the shape and size of the particles. Image analysis can be used for the classification of crystals based on their size or morphology. Online imaging and image analysis have been used for classification of polymorphic forms in real time (De Anda et al. 2005). 2D information can be obtained using in situ microscopy, which provides pictures of the crystals in solution using a probe inserted directly into the crystal slurry. Various in situ microscopes are available. The Lasentec Particle Vision Measurement (PVM) system has been used extensively (Barrett and Glennon 2002; Fevotte 2002; Kempkes et al. 2008; Fujiwara et al. 2002; Schöll et al. 2006). PVM is a probe-based high-resolution in situ video microscope that uses laser light to illuminate an area within the slurry; the light scattered back toward the probe is used together with a CCD element to generate an image. This video microscope can collect 10–30 pictures per second, providing 2D snapshots of the crystals in real time. In situ video microscopy can

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measure crystals as small as 3–15 µm, which is not as small as those measured by laser scattering instruments. PVM has been used mainly to validate the result of other in situ solid characterization techniques because the quality of images, especially in dense slurries, limits the ability of the imaging software to automatically identify individual particles and quantify their characteristics. PVM is suitable for use in industrial crystallizers (Braatz 2002), and process video microscopy is becoming increasingly used to image the crystal nucleation and growth in solution to visualize the extent of agglomeration and changes in crystal size and shape. A much less expensive method, bulk video imaging (BVI), has been introduced for nucleation detection (Simon et al. 2009a, b, 2010a, b) using low-cost video hardware (simple camera for eBVI and process endoscope for iBVI). This method samples a larger region of the volume covered by the image and so is less affected by nonuniform mixing. Despite its lowresolution images, BVI has been reported to provide earlier detection of nucleation than spectroscopy or focused-beam reflectance measurement (Simon et al. 2009a, b, 2010a, b). The method is also able in some cases to detect polymorphic transformation, but given the low resolution of the acquired images, crystal size and shape have not been accurately measured using this approach. An online high-speed imaging system developed by GlaxoSmithKline has been used for monitoring the size and shape of crystals during batch cooling crystallization (Dharmayat et al. 2006). The EZProbe sensor collects images from light shone through an orifice in an in situ probe (Presles et al. 2010). Automated image analysis using online measurement approaches with a sampling loop, as indicated in Figure 11.3, also can be implemented to enable the use of better imaging equipment. Whereas in situ and online imaging provide 2D information related to the size and shape of particles, newer methodologies based on digital holography aim to extract 3D information from specially acquired images (Eggers et al. 2008; Darakis et al. 2010). The use of in situ particle imaging for feedback control has been demonstrated in which the concentration of an additive during a cooling crystallization was adjusted to control the measured crystal habit (Patience and Rawlings 2001).

Raman Spectroscopy Raman spectroscopy has been employed for monitoring solid phases during crystallization, particularly during polymorphic transformations (Starbuck et al. 2002; Wang et al. 2000; Schöll et al. 2006; Ono et al. 2004; Fevotte 2007). Raman spectroscopy was used to simultaneously and continuously monitor both the overall solids concentration and the composition of the solid phase in the solvent-mediated phase transformation of citric acid (Caillet et al. 2007) within a measurement accuracy of about 5 percent. Raman spectroscopy has also been used to monitor the dehydration of crystal hydrates over time (Falcon and Berglund 2003). Calibration of the Raman signal for quantitative polymorphic composition measurements can be difficult because the signal intensity is affected by the particle size distribution (O’Sullivan et al. 2003).

Monitoring and Advanced Control of Crystallization Processes

Ultrasonic Spectroscopy Ultrasonic spectroscopy, which is a comparatively new solids characterization technique, has been applied successfully for online crystal size measurement (e.g., see Mougin et al. 2003). The technique, which is based on measurement of the attenuation of ultrasonic waves through the suspension, is capable of measuring crystal sizes within the range of 0.01–1000 μm for solids concentrations in the range of 0.1–50 vol%. In addition to crystal size determination, the technique is also able to detect nucleation, crystal growth, and crystal breakage online and in real time.

Turbidimetry A turbidity probe transmits near-IR light into the slurry. The light scattered by the particles returns to the analyzer, which provides a relative measure of the liquid turbidity. This is a very cost-effective and robust measurement that is suitable for the detection of nucleation and dissolution and so is often used for metastable zone determination (Parsons et al. 2003; Harner et al. 2009). A turbidity probe has been used to detect nucleation and polymorphic transformation during the antisolvent crystallization of sodium benzoate from an IPA–water mixture (Howard et al. 2009b). The turbidity meter, however, is unable to provide quantitative information on the size or number of particles.

Laser Diffraction A laser diffraction instrument works by passing a beam of monochromatic laser light to illuminate the particles. The scattered and unscattered light then passes through a lens to a detector located in the focal plane of the lens. Then, by processing the signals, a spectrum can be obtained to represent either particle size or shape. Laser diffraction technique has been applied to measure both crystal particle size and shape (Ma et al. 2001) by using 2D pixel arrays as the detector so that diffraction pattern for both size and shape can be obtained. Laser diffraction can be used for the online measurement of CSD through a sampling loop, as shown in Figure 11.3 (Aamir et al. 2010a, b). Usually, an additional dilution loop using filtered mother liquor is needed to keep the solids concentration within the measurement limits of the instrument. Careful control of the dilution flow rate may be necessary during the crystallization because as the solids content increases, the dilution flow rate must be increase to keep the solids content within the measurement limit of the equipment.

Laser Backscattering Another solid-phase measurement technique based on the optical properties of the system being studied is laser backscattering, such as the Lasentec focused-beam reflectance measurement (FBRM). FBRM sends a laser beam through fiber-optics to an immersion probe tip where it is finely focused by a rotating lens, which causes the beam to scan in a circular path through a sapphire window at a fixed high speed. The light scattered back from the particles when the laser beam crosses them is collected, and the duration of

Figure 11.6 FBRM and temperature data recorded during an automatic MSZW determination experiment, with the clear and cloud points that delimit the metastable zone.

the backscattering is transformed into chord length. Typically, a large number of chords of different sizes can be obtained from any given particle depending on how the beam intersects the passing particle. These chord lengths are classified into a series of size ranges, known as channels, to produce a chord length distribution (CLD). The CLD is related to the solids concentration, particle size, and particle shape. FBRM also allows monitoring of the change in CLD for different chord size classes (fine, intermediate, and coarse) of the crystals as a function of time, hence providing useful information related to nucleation, growth, or dissolution as well as breakage and agglomeration. FBRM has been used extensively in the monitoring of crystallizations to provide qualitative as well as quantitative information on nucleation and crystal growth. Because a sudden increase in the chord counts per second indicates the onset of nucleation, whereas a sudden decrease in the counts per second indicates dissolution, FBRM has become a standard tool for metastable zone width (MSZW) determination for crystallization processes in highly automated procedures (Fujiwara et al. 2002; Barrett and Glennon 2002; Liotta and Sabesan 2004; Zhou et al. 2006a; Kee et al. 2011a). Figure 11.6 shows the data points from typical MSZW determination experiments where the cloud (nucleation) points and clear (dissolution) points are determined with automated heating/ cooling experiments, with programmed dilutions between each cycle used to alter the concentration. The zone between the clear and cloud points corresponds to the metastable zone in the phase diagram. FBRM was found to detect the MSZW earlier than ATRFTIR spectroscopy and visual observation (Fujiwara et al. 2002). Because the dynamic evolution of the CLD data measured by FBRM is correlated with variation in the CSD, many research groups and industries have used FBRM to monitor the CSD in situ (e.g., Barrett et al. 2005; Barthe and Rousseau 2006). The effects of process conditions on the CSD can also be monitored using FBRM. For example, FBRM has been

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Zoltan K. Nagy, Mitsuko Fujiwara, and Richard D. Braatz

used to monitor the effects of seed loading and size distribution on the average crystal size of a system operated at constant supersaturation (Yu et al. 2006b), study the role of vessel size on the size of the product crystals (Yi and Myerson 2006), and monitor polymorphic transformation (O’Sullivan et al. 2003; Schöll et al. 2006; Barthe et al. 2008; Howard et al. 2009a). The successful monitoring of polymorphic transformation by FBRM requires the crystals to have a distinct change of shape on transformation (Mangin et al. 2009). Process control strategies have used FBRM to selectively crystallize metastable polymorphs (Doki et al. 2004a; Abu Bakar et al. 2009b). Several approaches have been proposed to convert the CLD provided by the FBRM into PSD (e.g., Tadayyon and Rohani 1998; Ruf et al. 2000; Hukkanen et al. 2003; Worlitschek et al. 2005; Li and Wilkinson 2005; Kail et al. 2007, 2008, 2009a, b); however, these methods generally depend on assumptions about the geometry of the particles. Several empirical studies that compared the FBRM outputs with those obtained from more conventional particle sizing techniques, suggest that the square-weighted mean chord length (SWMCL), for a variety of systems, can be correlated with the Sauter mean diameter measured using laser diffraction instrument (Heath et al. 2002; Yu and Erickson 2008) and optical microscopy (Yu and Erickson 2008). Because the instruments are based on different measurement principles, no correlation can be derived that universally quantitatively relates FBRM data to laser diffraction or optical microscopy data (Braatz et al. 2006). The comparison should only be used to evaluate the trends of the dynamic changes related to the CSD. Using a suitable modeling approach and proper processing, FBRM data can be used for the quantitative estimation of the kinetic parameters (e.g., Nagy et al. 2008c 2008; Kee et al. 2011b) for crystallization processes. FRBM measures particles in the range of 0.5–1 μm, and the reliability of the data depends significantly on proper installation of the probe. The probe has to be positioned in a well-mixed zone of the crystallization vessel with a 30- to 60-degree angle into the flow direction. These requirements limit the in situ installation of the probe in many industrial-scale crystallizers, so it is common in industrial practice to place the probe in an external circulation pipe. The PAT tools and measurement approaches described in this section benefit the development of crystallization monitoring and control approaches because they allow the in situ (or online) and real-time measurement of key process variables (e.g., liquid-phase concentrations and CSD), which can be used as the measurement input signal for various feedback-control strategies.

11.2.4 Basic Concepts of Process Analytical Technology Process analytical technology (PAT) is a term used mainly in the context of pharmaceutical crystallization processes, but the associated terms and methods apply to practically all

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crystallization systems. PAT is defined by the US Food and Drug Administration (FDA) as “a system for designing, analyzing, and controlling manufacturing through timely measurements (i.e., during processing) of critical quality and performance attributes of raw and in-process materials and processes, with the goal of ensuring final product quality” (FDA 2004). PAT is often used in the context of quality-bydesign (QbD). The concept of QbD was introduced to fill the need of the pharmaceutical industry to have a more efficient and reliable manufacturing approach. QbD is based on a notion that “quality cannot be tested into products, i.e., quality should be built in by design” (FDA 2006). The application of PAT is the key enabler for the implementation of the QbD concepts and for advanced crystallization control approaches. PAT focuses on the use of real-time measurement tools as a primary means of in-process monitoring. The term analytical in PAT includes chemical, physical, microbiological, and statistical analyses such as chemometrics conducted in an integrated manner (Yu et al. 2003). Chemometrics is typically applied in the interpretation of the multivariate data provided by in situ process monitoring devices. In addition to constructing calibration models for sensors, chemometrics can be used to derive mathematical relationships between the desired product properties, such as the CSD, and the various contributing variables, such as solute concentration. The mathematical relationships can be constructed using multivariate data analysis methods such as multiple linear regression (MLR), principal components analysis (PCA), principal components regression (PCR), partial least squares (PLS), and variations of these approaches. MLR describes its output as a linear combination of dependent variables. Instead of using the original variables of the model, PCA uses linear algebra to derive a new, smaller set of variables as a linear combination of the original variables. The derived variables are determined based on the variability in the data. PCR and PLS are combinations of the MLR and PCA and are different in the way the derived variables are selected. Several reviews of applications of PAT to crystallization processes have been published (e.g., Yu et al. 2003; Barrett et al. 2005; Braatz et al. 2006). PAT includes sensors that can measure critical process properties, such as supersaturation, and desired product qualities, such as size, shape, and polymorphic form. This real-time process information can be integrated into the control algorithm to provide an optimal control strategy. The use of PAT helps to identify the critical parameters that affect the crystallization, which results in a more in-depth understanding of the process but is not a substitute for traditional solid-state characterization techniques.

11.2.5 Toward a Crystallization Process Informatics System While the aforementioned measurement techniques and instruments are often used simultaneously to monitor

Monitoring and Advanced Control of Crystallization Processes Figure 11.7 Architecture of the Crystallization Process Informatics System (CryPRINS) with the industry standard communication interface and composite sensor array (CSA) and distributed control system (DCS).

Model-based design approach Design of experiment

Model identification

Model-based optimization

Nominal optimal recipe

Robust optimization

Figure 11.8 Schematic representation of the modelbased and model-free (direct design) approaches for cooling, antisolvent, or combined crystallization systems.

No Metastable zone determination by in situ monitoring tools

Implement trajectory in time domain (direct operation) or in phase diagram (supersaturation control)

Optimal Yes Robust optimal performance operating achieved ? trajectory No

Select operating trajectory in the phase diagram or apply adaptive direct nucleation control (DNC)

Suboptimal operating trajectory

Model-free (direct design) approach

crystallization processes, the resulting information is not always combined and applied in real time. A crystallization process informatics systems (Figure 11.7) is based on the use of a composite sensor array (CSA) for monitoring multiple process and quality properties at the same time rather than one at a time using simultaneously all signals from all measurement devices for automated decision support and feedback control of the crystallization process. The systems must provide a generic communication interface based on industry standard protocols such as OPC (Object Linking and Embedding [OLE] for Process Control) or Modbus to allow the communication between various PAT technologies provided by different vendors and the distributed control system (DCS) to implement the proposed control actions. The complementary and redundancy in the acquired information allow the implementation of robust crystallization control strategies. The combination of the signals can be performed using modelbased approaches or by applying chemometrics using simultaneously all signals from all measurement devices, with a system that automatically determines the most relevant combination of signals for a particular process or control objective.

Measurement based optimization, (iterative learning control)

11.3 Model-Based Optimization and Control of Crystallization Processes In the majority of industrial crystallizations, typical feedbackcontrol strategies (e.g., PID, cascade) are designed to follow predetermined operating policies (e.g., a temperature profile or an antisolvent addition or evaporation rate; Rawlings et al. 2002). The emergence of modern sensor technologies and advances in crystallization modeling and control have enabled more advanced control strategies to be increasingly applied (e.g., Braatz 2002; Braatz and Hasebe 2002; Fujiwara et al. 2005; Nagy and Braatz 2008b; Mesbah et al. 2011b; Nagy 2009a). Approaches for the design of operating conditions and advanced control of crystallization processes can be divided in two main categories: (1) model-based design and control and (2) model-free (direct design) approaches, which are depicted schematically in Figure 11.8, and discussed in the following sections. The model-based design and control approaches are based on developing a detailed model that is used with optimization techniques to determine a dynamic supersaturation profile and/or seed recipe and addition protocol to achieve desired product properties (Figure 11.9). The supersaturation profile

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Zoltan K. Nagy, Mitsuko Fujiwara, and Richard D. Braatz

∂f ðL; tÞ ∂fGðS; L; θg Þf ðL; tÞg þ ¼ BðS; θn ÞδðL  L0 Þ ð11:4Þ ∂t dL

Figure 11.9 Generic formulation of the model-based control approaches as an optimization problem.

is optimized indirectly by determining temperature versus time and/or antisolvent addition rate versus time or evaporation rate trajectories (Rawlings et al. 1993). The optimization is subject to the model equations and a set of constraints owing to equipment limitations (e.g., maximum and minimum temperature values, maximum and minimum cooling rates, maximum volume, limits on antisolvent addition rate). The crystallization also must satisfy a productivity constraint that ensures a desired yield at the end of the batch. Quality constraints also can be included in the optimization (e.g., Worlitschek et al. 2005; Corriou and Rohani 2008; Sarkar et al. 2006). Advantages in the model-based approaches include their ability to obtain a theoretically optimal recipe, much smaller number of experiments than for statistical experimental design of batches (Togkalidou et al. 2001a, b), increased process understanding, and the possibility of incorporating the effects of nonideal mixing via computational fluid dynamics (e.g., Woo et al. 2006; Woo, Tan, and Braatz 2009). Model-based approaches also enable the design and control of crystallization processes as integrated unit operations, taking into account downstream processing units and considering final product properties as control objectives. Disadvantages associated with model-based approaches are mainly related to the increased difficulty in modeling some of the practical objectives (e.g., filterability, purity, tablet stability) and the significant time and engineering effort required for model development. Additionally, the performance of the modelbased approach depends on model accuracy. The robustness of the approach can be improved by incorporating linear or nonlinear robustness analysis into the optimization (Nagy and Braatz 2003a, b, 2004).

11.3.1 Modeling and Parameter Estimation Model Formulation and Solution Model-based design and control approaches are generally based on the population balance model (PBM) for the crystallizer. For example, a batch cooling crystallization with perfect mixing, one characteristic size L, and growth and nucleation as the main crystallization phenomena has the population balance equation (PBE)

324

where f ðL; tÞ is the crystal size distribution (CSD) expressed in a per-mass or volume of solvent or slurry basis, t is time, GðS; L; θg Þ is the rate of crystal growth, BðS; θn Þ is the nucleation rate (as a result of primary or secondary mechanisms or both), S is the supersaturation, θg and θn are vectors of growth and nucleation kinetic parameters, respectively, L0 is the size of the nuclei, and δ is the Dirac delta function. The necessary initial and boundary conditions are f ðL; 0Þ ¼ fseed ðLÞ

ð11:5Þ

f ð0; tÞ ¼ B=G

ð11:6Þ

where f ðL; 0Þ ¼ fseed ðLÞ is the seed CSD. The supersaturation S can be expressed as an absolute supersaturation S ¼ C  Csat , relative supersaturation S ¼ ðC  Csat Þ=Csat , or supersaturation ratio S ¼ C=Csat , where C is the solute concentration, and Csat ¼ Csat ðTÞ is the saturation concentration (solubility), which is a function of the temperature. The temperature trajectory influences the CSD and hence the optimization objectives through Equation (11.4). The temperature can also have an additional effect on the kinetic parameters in the growth and nucleation rates, which appear in both Equations (11.4) and (11.6). This latter temperature effect is usually neglected in comparison with the generally relatively stronger influence of temperature on the growth and nucleation rates through the supersaturation. Various expressions have been proposed for growth and nucleation rates, but nearly all optimization and control studies have successfully used rather simple power-law relationships. The power law for size-independent growth is G ¼ kg Sg

ð11:7Þ

with the corresponding expression for apparent size-dependent growth being GðS; L; θg Þ ¼ kg Sg ð1 þ γLÞp

ð11:8Þ

which introduces additional parameters to be estimated. The size-dependent growth rate expressed by Equation (11.8) corresponds to an apparent size-dependent mechanism in that it is an empirical expression fit to experimental observations. These observations may be caused by the actual growth kinetics being size dependent, by diffusion-controlled growth (Mullin 2001), by growth rate dispersion (i.e., different crystals of the same size having a variation in growth rates), nonideal mixing (Ma et al. 2002a), or a combination of these causes (Zumstein and Rousseau 1987; Randolph and White 1977). Primary nucleation Bp (considered for unseeded systems) is given by Bp ¼ kb Sb

ð11:9Þ

whereas secondary nucleation Bs (considered in the case of seeded systems) also depends on the total crystal surface area

Monitoring and Advanced Control of Crystallization Processes

or volume, expressed by the second μ2 or third moments μ3 of the CSD, respectively, as Bs ¼ kb Sb μ2

ð11:10Þ

Bs ¼ kb S μ3 :

ð11:11Þ

b

In past studies, fits to experimental data have not been statistically different for these two models (Miller 1993; Gunawan et al. 2002; Togkalidou et al. 2004). The jth moment (j ¼ 0; 1; …) of the CSD is defined by ð∞ μj ¼

Lj f ðL; tÞdL

ð11:12Þ

0

The PBE [Equation (11.4)] is solved together with the overall mass balance, which is also calculated using the third moment μ3 ðtÞ of the CSD (related to the total volume of the crystals); thus CðtÞ ¼ Cð0Þ  kv ρc ½μ3 ðtÞ  μ3 ð0Þ

ð11:13Þ

where ρc is the density of crystals, and kv is the volumetric shape factor. The PBE [Equation (11.4)] can be solved using a variety of different approaches, which include the standard method of moments (SMM; Hulbert and Katz 1964), the quadrature method of moments (QMOM; McGraw 1997; Rosner et al. 2003; Marchisio et al. 2003), the method of characteristics (MOCH; e.g., Hounslow and Reynolds 2006), the method of classes (MOC; Marchal et al. 1988; Puel et al. 2003), direct numerical solution (DNS) techniques such as finite volume and finite difference schemes (e.g., Ma et al. 2002a, b; Gunawan et al. 2004, 2008; Zhang et al. 2008; Hermanto et al. 2009a), and dynamic Monte Carlo (DMC) simulation approaches (Rosner et al. 2003; Haseltine et al. 2005). The MOM techniques are very computationally efficient but provide only a coarse description of the CSD (e.g., mean crystal size, width of the distribution, total area of crystals) and have been widely used for optimization and control purposes (e.g., Rawlings et al. 1993; Fujiwara et al. 2005; Nagy and Braatz 2004). A drawback of MOM techniques is that the entire distribution is not provided, and methods to construct the distribution from the moments require additional assumptions on the distribution (Hulbert and Katz 1964). A combination of QMOM and MOCH can be used to efficiently construct the distribution (Aamir et al. 2009a, 2010b; Qamar et al. 2010) and can be used in real-time modelbased control. For size-independent growth, applying the method of moments to the PBE [Equation (11.4)] gives the infinite set of ordinary differential equations dμ0 ¼ B; dt dμj j ¼ jGμj1 þ BL0 ; j ¼ 1; 2; … dt

ð11:14Þ

The size of the nuclei usually can be neglected (L0 ≈ 0) in Equation (11.14) without significantly affecting the variables that are fit to the experimental data. The initial conditions for unseeded crystallizations are μj ð0Þ ¼ 0; j ¼ 0; 1; …, and for seeded crystallizers are the moments of the seed distribution (μj;s ; j ¼ 0; 1; …). Although the number of Equations (11.14) is infinite, most practical control objectives can be computed using only the first few moments (generally up to four).

Parameter Estimation Model identification requires estimation of the parameter vector θ ¼ ½θg ; θn  from the kinetic expressions, with θg ¼ ½g; kg  (or θg ¼ ½g; kg ; γ; p for size-dependent growth) and θn ¼ ½b; kb . These parameters can be obtained by least-squares optimization, which minimizes an objective function that represents the overall error between the experimentally measured and model-predicted states (Rawlings et al. 1993). Estimation of the model parameters requires the solution of a nonconvex nonlinear program (NLP), which can be solved using various numerical optimization algorithms readily available in several commercial software packages (e.g., Matlab, gPROMS). The objective function for parameter estimation is calculated at the discrete time points t0;l < t1;l <    < tKl ;l with l ¼ 1; …; Nex (Nex being the number of experiments) and Kl being the number of discrete time points in experiment l minfJest ¼ θ

Ny Kl X Nex X X

exp

½yi ðtk;l ; θÞ  yi ðtk;l Þ2 g

ð11:15Þ

l¼1 k¼0 i¼1

subject to the model equations [e.g., Equations (11.4)–(11.13)] and constraints θmin ≤ θ ≤ θmax

ð11:16Þ exp

where Ny is the number of measured model outputs (y), yi are the experimental values, and θ is the vector of model parameters with Nθ elements and bounds θmin and θmax , respectively. When the process is simulated using the moment equations only, the model outputs and experimental measurements are the moments or their ratios together with the concentration (e.g., y ¼ ½C; μ1 =μ0 ; μ2 =μ0 ; μ3 =μ0 ) and θ ¼ ½kb ; b; kg ; g for nucleation and size-independent growth. When the entire CSD is available from the model and measurement, then typically y ¼ ½C; f1 ; f2 ; …; fNy , with fk , k ¼ 1; …; Ny being the values (simulated and experimental) of the discretized particle density functions, corresponding to the experimental size bins. An alternative approach employs y ¼ ½C; F1 ; F2 ; …; FNy , with Fk , k ¼ 1; …; Ny , being values of the cumulative distribution function at discrete values for the size L, which has the advantage of not introducing any binning errors. The confidence intervals of the estimated parameters should always be calculated, which can then be used for quantitative assessment of the uncertainties in the model predictions and ultimately can be incorporated in robust optimization and control approaches. Several approaches are

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available to calculate the confidence intervals (e.g., Beck and Arnold 1977; Rawlings et al. 1993; Hermanto et al. 2008; Nagy et al. 2008c). An approach that can be easily implemented is based on the measurement matrix 3 2 Mθ0 7 6 6 Mθ1 7 7 6 ð11:17Þ Mθ ¼ 6 .. 7 6 . 7 5 4 k MθNex XNex which consists of kNex ¼ K number of ðNy  Nθ Þ sensil¼1 l tivity matrices, calculated in each discrete time step (tk;l , k ¼ 0; …; Kl ) corresponding to all experiments, l ¼ 1; …; Nex , Mθkl ¼

dyðtk;l ; θÞ dθ

ð11:18Þ

with kl ¼ 0; …; kNex . The sensitivity matrices are calculated by concatenating the data from the time steps of different experiments into combined vectors of time points (t0;1 <    < tK1 ;1 < t0;2 <    < tK2 ;2 <    < t0;Nex <    < tKl ;Nex ), with tKl ;l > t0;lþ1 and corresponding model inputs, model outputs, and experimental measurement vectors. The precision matrix Pθ and covariance matrix Vθ are given by Pθ ¼ ðMθT Mθ Þ1

ð11:19Þ

Vθ ¼ s2R P:

ð11:20Þ

where the residual variance (assumed to be equal to the measurement variance) is given by s2R ¼ Jest =Ndf , with Ndf ¼ Ny ðkNex þ 1Þ  Nθ  1 being the number of degrees of freedom. The confidence intervals are calculated using the t-test pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi θ ¼ θ^  tα=2;Ndf diagðVθ Þ ð11:21Þ where ^ θ is the nominal parameter vector, and tα=2;Ndf is the tdistribution with Ndf degrees of freedom. The 95 percent confidence intervals are obtained for α ¼ 0:05. Identification of the model parameters with small confidence intervals requires a large number of experiments unless the experimental are very well designed. The efficiency of the parameter estimation process and the uncertainty on the parameters can be greatly improved by applying robust experimental design approaches (Matthews and Rawlings 1998; Chung et al. 2000; Ma and Braatz 2003; Togkalidou et al. 2001b). High-throughput microfluidic crystallization techniques can be applied to produce a large number of parallel experiments with a very small amount of compounds (especially important in pharmaceutical applications) for the identification of robust crystallization kinetics (Goh et al. 2010).

Aspects of Imperfect Mixing in Crystallization Modeling The mathematical model described in this section and used in the subsequent model-based control designs assumes perfect

326

mixing. Although this is rarely the case in industrial crystallization systems, good mixing can be achieved by proper design of the crystallization vessel and impeller type, as well as by suitable selection of the agitation speed to achieve well-suspended particles. For antisolvent (or reactive) crystallization processes, selection of the addition rate and feed position is also important for achieving quick dispersion of the antisolvent to avoid the formation of local high-supersaturation regions (O’Grady et al. 2007; Woo et al. 2006). In the case of the model-based control approaches described in this chapter, the effects of imperfect mixing are addressed indirectly by ensuring that the designs are robust to uncertainties. A rigorous incorporation of nonideal mixing effects on crystallization requires the coupling of computational fluid dynamics (CFD) with a spatially varying population balance model (Woo et al. 2006, 2009). While the fully coupled multiscale CFD-PBM simulation of industrial stirred vessels is currently computationally too intensive to be used in model-based optimization and control, such simulations can help us to understand scale-up problems, design better crystallization equipment (geometries of vessels, impellers, or baffles), optimize operating conditions (such as antisolvent addition rate and location, seed recipe), and improve the performance of the control approaches described in this chapter. To reduce the large computational requirements of the CFD-PBM approach, multicompartmental approaches can be employed, which only use a small number of compartments to simulate the effects of imperfect mixing (Ma et al. 2002a; Li et al. 2003). These approaches can approximate the effect of spatial variation on crystallization but are unable to accurately model the influence of localized turbulence on crystal nucleation and growth. A simplified approach to include some of the effects of nonideal mixing in a model is to incorporate a volume-averaged mass transfer coefficient into the growth rate. Empirical nucleation rate expressions have also been proposed (Matthews and Rawlings 1996). The mechanisms of agglomeration and breakage and corresponding kernels generally incorporate mixingrelated parameters (Chen et al. 2004). Considering a single compartment and volume-averaged parameters will not allow detailed modeling of the effects of micro- and mesomixing on the PSD; nevertheless, they can decrease parameter uncertainties in the model caused by variations in the crystallization kinetics owing to scale-up. If these effects are incorporated in the model, mixing can be considered to be an additional manipulating input in the control strategy. For example, Mesbah (2010) proposed a multi-input nonlinear model predictive control strategy for an industrial batch evaporative crystallization process with fines-removal system that controlled a constant optimal growth rate and maximized the mean crystal size by manipulating three process inputs: evaporation rate (through the heat input to the system), fines-removal flow rate, and impeller frequency.

11.3.2 Optimal Nominal Open-Loop Control of Batch Cooling Crystallization The identified model with the nominal parameters can be used to determine the open-loop optimal control policy, such as the temperature trajectory versus time, by solving a

Monitoring and Advanced Control of Crystallization Processes

nonlinear optimization problem. A convenient way to describe the optimal trajectories is to discretize the batch time horizon ½0; tf  into Nb equally spaced time intervals of duration Dt (stages). The temperature trajectory is approximated by a piecewise linear function, and the temperature values TðkÞ at every discrete time k ¼ 0; …; Nb are considered to be the optimization variables. The open-loop nominal optimal control problem in this case can be written as optimize Jðtf ; ^ θÞ

ð11:22Þ

TðkÞ

subject to the model equations [e.g., Equation (11.4) or Equation (11.14)] Tmin ðkÞ ≤ TðkÞ ≤ Tmax ðkÞ dTðkÞ ≤ Rmax ðkÞ Rmin ðkÞ ≤ dt Cðtf Þ ≤ Cf ;max

ð11:23Þ

where tf is duration of the batch, and Tmin , Tmax , Rmin , and Rmax are the minimum and maximum temperatures and temperature ramp rates, respectively, during the batch. The first two constraints ensure that the temperature profile stays within the operating range of the crystallizer. The final constraint ensures that the solute concentration at the end of the batch Cðtf Þ is smaller than a certain maximum value Cf ;max set by the minimum yield required by economic considerations. θÞ is some desired characteristic of the The objective Jðtf ; ^ crystalline product at the end of the batch, calculated using the model with the nominal parameters ^ θ. Different objectives have been used in the literature [for an overview, see Ward et al. (2006)]. Typical CSD properties considered for optimization are the number-average crystal size d10 ¼ μ1 =μ0 , the coefficient of variation CV ¼ ðμ2 μ0 =μ21  1Þ1=2 , the nucleated-toseed-mass ratio nucleated=seed ¼ ðμ3  μ3;s Þ=μ3;s (where μ3;s is the third moment corresponding to seed crystals only), and weight-mean size d43 ¼ μ4 =μ3 . The optimizations for CV and nucleated/seed ratio are minimization problems whereas for d10 and d43 are maximizations. For calculation of the nucleated/seed performance index, an additional PBE with a growth-only mechanism (corresponding to the dynamic

evolution of the seed CSD only) has to be solved in parallel with the overall PBE, which gives the total moments (corresponding to both seed and newly nucleated crystals). In the case of the method of moments, the second PBE corresponding only to the seed crystals yield an additional set of moment ordinary differential equations (ODEs). The initial conditions for both the PBEs and the two sets of moment ODEs are identical and result from the seed distribution. Although the performance objectives CV and d43 have been used extensively in the literature for many decades, these objectives tend to produce cooling profiles that generate a large number of fine crystals, with the main weakness of the CV being that the mean size only appears as a ratio and the main weakness of d43 being its very low sensitivity to very small crystals (Chung et al. 1999; Ma et al. 2002c). A more detailed discussion of selection of the optimization objective for industrial crystallizations is available (Ward and Doherty 2006). Considering the particular system of KNO3 in aqueous solution (Miller and Rawlings 1994), the kinetic parameters are g ¼ 1:31, kg ¼ 6:568  103 µm/min, b ¼ 1:84, and kb ¼ 3:532  107 particles/cm3 per minute, relative supersaturation is used in the growth and nucleation expressions [Equations (11.7) and (11.11)], respectively, the crystal density is ρc ¼ 2:11 g/cm3, the shape factor is kv ¼ 1, and the solubility is given by Csat ðTÞ ¼ 0:1286 þ 5:88  103 T þ 1:721  104 T 2 ð11:24Þ with C in grams per gram of water and T in degrees Celsius. Considering the initial values μ(0) = [121.57 /g, 372.32 μm /g, 73072.2 μm2/g, 1.436 × 107 μm3/g, 2.82 × 109 μm4/g], the optimal control results for minimization of the CV and the nucleated/seed mass ratio are shown in Figure 11.10, in comparison with uncontrolled operation using a linear cooling profile. Using different control objectives results in significantly different optimal temperature trajectories and dynamic evolutions of the two CSD properties considered. In the case of CV minimization, the optimal temperature profile leads to a 17 percent improvement compared with linear cooling, whereas the optimal trajectory corresponding to nucleated/seed ratio

Figure 11.10 Sample optimal temperature profiles for the optimization of nucleated/seed ratio and CV indicating the improvements in these performance indexes compared with linear cooling (left). Dynamic evolution of the CV and nucleated/seed ratio corresponding to each optimal temperature profile (right).

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Zoltan K. Nagy, Mitsuko Fujiwara, and Richard D. Braatz

minimization yields a 21 percent reduction of the nucleated mass to seed mass compared with linear cooling. Following the optimal temperature trajectory resulting from CV minimization, the CV is generally smaller during the entire batch than for the optimal trajectory corresponding to nucleated/seed ratio minimization. Similarly, applying the optimal temperature profile corresponding to nucleated/seed ratio minimization, this CSD property used in the objective function (nucleated/seed) is smaller for the entire duration of the batch. While the CSD properties are related, the optimal operating trajectory may differ significantly based on the CSD property used in the optimization. The optimal temperature (or antisolvent addition rate) profiles can be implemented using standard temperature and flow control systems. When a solute concentration measurement is available, then a concentration-versus-time profile can be implemented instead. This so-called concentration control approach can provide greatly increased robustness to variations in crystallization rates and reduced sensitivity to disturbances by judicious use of the additional feedback obtained through the concentration measurement and the extra degree of freedom by allowing the batch time to vary in response to the crystallization rates (Nagy et al. 2008b).

11.3.3 Robust Optimal Control Strategies The optimal trajectories resulting from nominal optimization generally drive the system close to its operating and product quality constraints. Therefore, all the benefits of using optimal control can be lost with subtle variations in either the model parameters or the performance of the feedback controller to track the optimal set-point trajectory. Consequently, the initial steps before considering the implementation of advanced control approaches to industrial crystallization processes should include the tight tuning of the basic control systems and the careful model identification and assessment of model/parameter uncertainties. Comprehensive quantitative uncertainty analysis approaches have been proposed in the literature and applied to crystallization processes (e.g., Matthews 1997; Ma et al. 1999). For the example presented in Section 11.3.2, Nagy and Braatz (2003a) showed that implementing the optimal temperature trajectory for nucleated/seed ratio minimization and considering realistic uncertainties in the growth and nucleation kinetics (characterized by the covariance matrix Vθ resulting from parameter estimation) can lead to an increase in the nucleated/seed ratio by as much as 36 percent compared with the value obtained using the linear cooling profile. This result indicates that applying open-loop optimal control without considering potential uncertainties can result in worse performance than doing no control at all. While the CV also exhibits variations around the nominal value when the trajectory that optimizes the nucleated/seed ratio is implemented, these variations were significantly smaller than the variations in the nucleated/seed ratio. Similarly, when the trajectory that optimizes the CV was implemented, the sensitivity of the CV to

328

parameter and implementation uncertainties was much higher than for the nucleated/seed ratio. These results indicate that control trajectories that optimize a particular nominal performance objective can be much more sensitive to parameter and implementation uncertainties than other control trajectories. That is, there can be a significant tradeoff between nominal performance and robustness. The robust optimal control approaches for batch crystallization are based on determining the best tradeoff by formulating the robust counterpart of the nominal optimization problem. One approach commonly used is the mean-variance approach, according to which the objective function in Equation (11.22) is replaced with a weighted sum of the nominal performance index and its variance owing to uncertainties minfð1  wÞℰ½Jðtf ; θÞ þ wVJ ðtf Þg TðkÞ

ð11:25Þ

where ℰ and VJ 2 ℝ are the expected value and variance, respectively, of the CSD property (J) at the end of the batch, and w 2 ½0; 1 is a weighting coefficient that quantifies the tradeoff between nominal and robust performance. To reduce computational cost, it is common to replace ℰ½Jðtf ; θÞ with Jðtf ; ^θÞ in the optimization [Equation (11.26)]. An advantage of this approach compared with the classical minmax optimization is that the tradeoff between nominal and robust performance can be tailored by selecting the weight w. The robust optimization approach can be applied to either open- or closed-loop control strategies and can yield significant robust performance improvements, as demonstrated for the control of the CSD (Nagy and Braatz 2003b, 2004; Nagy 2009a) and polymorphic transformation (Hermanto et al. 2007).

11.3.4 Distribution Shaping Control with In Situ Fines Removal Using Controlled Dissolution The classical control objectives presented in Section 11.3.3, expressed in characteristics of the size distribution (e.g., average size, coefficient of variation), can sometime lead to conservative and economically inefficient designs of the crystallization systems (e.g., excessively long batches to increase average particle size much above requirements just to shift the fines tail in the CSD, which could cause problems at filtration, and then followed by more intensive size reduction to achieve specifications). Distribution shaping control (DSC), by contrast, can be used to directly design the shape of the CSD to achieve the desired product properties. DSC provides a control framework for novel product engineering approaches and an ideal means for the implementation of QbD technologies. For example, crystallization operation can be designed to minimize filtration time without generating unnecessarily large crystals. Because the dissolution rate depends on the shape of the CSD, when the resulting crystals represent the final product (e.g., drugs for inhalers or capsulated pharmaceuticals), controlling the shape of the CSD can provide novel applications in the area of drug delivery or environmentally friendly dosages of pesticides, where

Monitoring and Advanced Control of Crystallization Processes

particular multimodal distributions can be designed to achieve the desired concentration level of the active ingredient. Designing tailored multimodal distributions can also enhance mechanical properties of the final product such as compactability, packing property, and friability. Tailoring the final product CSD so that an optimal dissolution rate (heat release rate) is achieved for two-phase exothermic reactive processes, in which the dissolution is followed by reactions, can improve the safe operation of these systems. The same strategy also can be applied for three-phase reactions, which produce a significant amount of gas and induce excessive liquid rise (swelling), which can lead to the explosion of reactors (owing to the two-phase flow forming in the connection pipe and pressure buildup). In distribution shaping control, typically the objective function in the optimization [Equation (11.22)] is set to the sum-of-squared errors of the size bins between the target CSD and the predicted CSD at the end of the batch. The batch time can be optimized together with the operating profile, and all relevant crystallization mechanisms should be considered in the model. In addition to the typical crystallization mechanisms (i.e., growth, nucleation, agglomeration, breakage), dissolution also can be modeled to provide an additional freedom for shaping the distribution or responding to disturbances. As an alternative to optimizing the temperature values at the discretized time steps (as illustrated in Section 11.3.3), the slopes (αT ðjÞ, j ¼ 0; 1; …; Nb ) of the piecewise linear temperature trajectory in each discretization period Dt can be optimized. This formulation allows easy incorporation of the temperature-rate constraints as bounds on the decision variables αT ðjÞ, which are important to obtain a practically implementable temperature trajectory. The typical DSC optimization problem is formulated as min

αT ðjÞ; tf

Nd X

½f ðLl ; tf Þ  f target ðLl Þ2

ð11:26Þ

l¼1

subject to the model equations [e.g., Equations (11.4)–(11.13)] αT;min ≤ αT ðjÞ ≤ αT; max

j ¼ 0; 1; …; Nb :

ð11:27Þ

0 ≤ tf ≤ tf ;max

ð11:28Þ

Cðtf Þ ≤ Cf ;max

ð11:29Þ

where αT ðjÞ are the elements of the vector containing the slopes (dT=dt) for the temperature trajectories depending on the implementable heating and cooling capacity of the system, tf is the total batch time, and f ðLl ; tf Þ and f target ðLl Þ are the simulated (at the end of the batch) and target probability distribution functions evaluated in the discretized sizes Ll , l ¼ 1; …; Nd , with Nd being the number of bins. A similar formulation replaces the crystal size distribution in Equation (11.27) with its cumulative distribution, which has the advantage of not introducing the imprecision resulting from binning. Although the beneficial effect of controlled dissolution has been widely recognized in industry, the vast majority of

the model-based crystallization control approaches reported in the literature restrict the operating curve within the metastable zone. The suitable design of operating policy (e.g., application of seed and use of low supersaturation) can in most cases avoid the formation of undesired fines. However, when fines are generated in the crystallizer (e.g., owing to accidental seeding from crust, initial breeding, or attrition, all of which are common phenomena in industrial-scale crystallizers), operating only under supersaturated conditions does not provide enough degrees of freedom to eliminate the effect of fine particles on the product CSD. The dissolution of fines (known as ripening) has been shown to be effective for the production of large crystals with a narrow CSD (Mangin et al. 2006; Garcia et al. 2002). Dissolution can be used to dissolve small crystals and hence can counteract the effects of secondary nucleation, which can be a dominant phenomenon in industrial crystallization, for example, due to attrition (Lewiner et al. 2002). A generic approach to enhance CSD control via tailored dissolution, which is applied in industrial crystallization processes, is to use external fines removal equipment (Zipp and Randolph, 1989). Such equipment consists of a draft pipe, pump, and external heat exchanger through which the slurry is recycled. The maximum fines size to be removed through the vertical draft pipe is given by the upward velocity through the tube determined by Stoke’s law of particle settling. This upward velocity is controlled by the recycle flow rate so that crystals above a certain size sediment in an internal draft pipe, and only fine crystals are eliminated from the system. The slurry with the fine crystals is then recycled through a heat exchanger where fines are dissolved, and the supersaturated solution is reintroduced in the crystallizer. This approach requires additional equipment with associated increased capital and operating costs. When the fines-removal system is available, the additional control input can be used to improve the distribution shaping control approach by optimizing two manipulating inputs: (1) temperature and (2) recycle flow. Heuristic temperature cycling is often used in industrial batch crystallization, in the case of unseeded systems, or for seeded processes as an alternative when no external finesremoval equipment is available. Short time periods of undersaturation dissolve as many fine crystals as long time periods of Ostwald ripening occur at saturated conditions. Temperature cycling can have a significant beneficial effect on the quality of CSD, particularly during the unseeded crystallization of larger organic molecules (e.g., pharmaceuticals) with broad primary nucleation zones. For these systems, primary nucleation occurs at very high supersaturation, typically resulting in a large number of small, highly agglomerated particles. Applying several temperature cycles after an initial nucleation event can greatly improve the uniformity of the CSD and quality of crystals by eliminating fines and minimizing agglomeration. The quality of crystals and productivity of the crystallization process depend on the design of the temperature cycles. A model-based control approach can be used for the systematic design of the operating curve (temperature profile) to

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Supersaturation (g/g)

6 4

x10–3 Supersaturated Region Optimal Temp Profile

40 35 30

2 0

25

Supersaturation

–2 Under-saturated Region –4 0

Temperature (ºC)

8

20

50

15 100

time (min)

Figure 11.11 Target and product CSDs obtained using distribution shaping control with and without considering dissolution mechanism in the model. Both approaches achieve the target distribution, but the secondary fines peak developed during the process due to secondary nucleation can only be eliminated via controlled dissolution.

Figure 11.13 Operation curve in the phase diagram corresponding to the optimal temperature trajectory resulting from distribution shaping control with dissolution. The trajectory exhibits a single in situ fines-dissolution loop corresponding to the heating phase in the temperature profile.

provide in situ fines dissolution. To incorporate dissolution into the optimal control approach, two different PBEs are solved based on whether the system is supersaturated (S > 0) or undersaturated (S < 0). For S > 0, the evolution of the CSD is governed by the PBE [Equation (11.4)] with growth and nucleation, whereas for S < 0, the PBE is characterized by dissolution only, described by ∂f ðL; tÞ ∂½DðS; L; θd Þf ðL; tÞ þ ¼0 ∂t ∂L

S 0 ð12:5Þ dt where the ij cross-moment is defined by ð∞ ð∞ mij ¼

j

Li1 L2 nðL1 ; L2 ; tÞdL1 dL2

ð12:6Þ

0 0

Transformation of Equation (12.3) results in dm0 ¼ B0 dt

ð12:7Þ

dmj j ¼ jGmj1 þ B0 LN dt

ð12:8Þ

with ð∞ mj ¼

nðL; tÞLj dL

ð12:9Þ

0

The changes in concentration in time can be followed using the solute mass balance; for example, for crystals of length L1, width and depth L2, and shape factor defined by V ¼ kV L1 L22 ; dc 0 ¼ ρc k V ðG1 m02 þ 2G2 m11 Þ dt

Figure 12.8 Bulovak draft-tube crystallizer Source: From Bamforth 1965.

→

3 ∂½upi ðx ;tÞn ∂ðG1 nÞ ∂ðG2 nÞ ∂n X → → þ þ þ ¼ Bð x ;tÞ  Dð x ;tÞ ∂t i¼1 ∂L1 ∂L2 ∂xi

ð12:1Þ which is averaged over a macroscopic region of the crystallizer under the assumption of ideal macromixing of the crystallizer content. Assuming further the size-independent growth rates G1 and G2 along each crystal dimension of nucleated crystals of the size (L1N, L2N) and neglecting aggre→ → gation and breakage terms Bðx ; tÞ and Dðx ; tÞ – for the system of constant volume with no feed or outflow – Equation (12.1) is simplified to ∂n ∂n ∂n þ G1 þ G2 ¼ B0 δðL1  L1N ÞδðL2  L2N Þ ð12:2Þ ∂t ∂L1 ∂L2 Assuming direct proportionality of the growth rates G1 and G2 and the nuclei dimensions, and thus a constant value of the shape factor, Equation (12.2) is further reduced to ∂n ∂n þ G ¼ B0 δðL  LN Þ ∂t ∂L

ð12:3Þ

Equations (12.2) and (12.3) are used after the moment transformation. In the case of Equation (12.2),

ð12:10Þ

whereas for size defined by a single dimension L and the shape factor defined by V ¼ kV L3 ; dc ¼ 3ρc kV Gm2 ð12:11Þ dt The energy balances for a crystallizer and a cooling jacket, assuming that the cooling water is well mixed in the jacket, can be represented as ðML CPL þ MC CPC Þ

ρj CPj Vj

dT dc ¼  ðDHC ÞML  UA½TðtÞ  Tj ðtÞ dt dt ð12:12Þ

dTj ¼ UA½Tj ðtÞ  TðtÞ þ Fj ρj CPj ½Tcool ðtÞ  Tj ðtÞ dt ð12:13Þ

respectively, where ML is the mass of solvent and MC the mass of crystals in the crystallizer, Vj represents the volume of cooling water in the jacket, CPL, CPC, and CPj represent their respective heat capacities, U is the overall heat transfer coefficient, A is the heat transfer surface, T is the temperature in the crystallizer, Tj represents the cooling jacket temperature, Tcool is the fresh coolant temperature, DHC is the heat of crystallization, Fj represents the cooling water flow rate, and the concentration c is expressed in kilograms of solute per kilogram of solvent (water). The control objective in batch crystallization is most often to obtain uniform crystals of the desired size and size

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distribution within the shortest time possible. Well-controlled crystallization processes are usually operated within the metastable zone. For seeded processes, the seed is added after crossing the solubility curve by the operation line (see the trajectory AGEFC in Figure 12.25), and afterwards, the operation trajectory remains within the metastable zone. For unseeded process, the seeds are generated in the crystallizer, so the operation trajectory first crosses the metastable limit to generate seeds, and then the supersaturation is kept within the metastable zone, similarly to the seeded process (see the trajectory ADEFC in Figure 12.25). Once the model parameters are identified, the control can be implemented. The temperature control method is the most widely used approach because of the simplicity of the temperature measurement in the crystallizer T and in the jacket Tj. Then open-loop optimal control (Rippin 1983) can be used with the set-point temperature resulting from the model. As pointed out by Vega et al. (1995), the difficulty in this approach is that no compensation is made for modeling errors, and there is no feedback from the process through the measurements. In online optimal control, a feedback scheme can be used to implement an optimal profile. For example, Zhang and Rohani (2003) applied a proportional-integral (PI) controller in the feedback controller system. They have chosen the jacket temperature Tj as the manipulated variable and the crystallizer temperature T as the controlled variable. The calculated optimal trajectory was selected as the set point Topt. Vega et al. (1995) applied feedback PI control with the controlled cooling water flow rate Fj ¼ Kc ðTset  TÞ þ

ðt Kc ðTset  TÞdt τi

where Tset is the crystallizer temperature set point, which changes according to the chosen trajectory, and Kc and τi are the controller constants. Generic model control (GMC) represents a nonlinear model-based algorithm, proposed by Lee and Sullivan (1988). Vega et al. (1995) applied this algorithm to express the manipulated variable Fj as a function of measured temperatures and known model parameters. Paengjuntuek et al. (2008) applied the GMC control algorithm to obtain a cooling profile ð12:15Þ

0

where K1 and K2 are the GMC tuning parameters. Combining Equation (12.15) with Equation (12.12), one can express the jacket temperature as a function of the measured temperatures and known model parameters. The application of batch process control significantly improves the product quality; moreover, as shown by Paengjuntuek et al. (2008), a batch-to-batch optimization strategy can be used to update the kinetic parameters and modify an optimal temperature trajectory for the subsequent operations.

352

For a batch or a semibatch crystallizer with no net inflow or outflow of crystals, whose size may be characterized by one characteristic dimension L, the population balance equation can be written as (Randolph and Larson 1988) ∂ðnVÞ ∂ðGnVÞ þ ¼0 ∂t ∂L

ð12:16Þ

where n is the population density per unit suspension volume, and V is the total suspension volume. Because the working volume of a batch or semibatch system may vary with time, it is convenient to redefine the population density and other volume-sensitive quantities on the basis of the total operating volume of the crystallizer; thus e n ¼ nV

ð12:17Þ

With this substitution, Equation (12.16) becomes ∂e n ∂ðGe nÞ þ ¼0 ∂t ∂L

ð12:18Þ

which is in line with a common way of expressing the population density [see, e.g., Equation (12.3)]. Equation (12.18) requires a boundary condition and an initial condition. The boundary condition e n ð0; tÞ is the nuclei population density e n 0 , which can be related to the nucleation e 0 by rate B e 0 ðtÞ=Gð0; tÞ e n ð0; tÞ ¼ e n 0 ðtÞ ¼ B

ð12:14Þ

0

ðt dT ¼ K1 ðTset  TÞ þ K2 ðTset  TÞdt dt

12.3.2 Characterization of CSD for Batch Crystallization via Population Balance

ð12:19Þ

The initial population density e n ðL; 0Þ for a batch crystallizer is not well defined. For a crystallizer seeded externally, e n ðL; 0Þ may be denoted by an initial seed distribution function e n s ðLÞ. However, in an unseeded system, initial nucleation can occur by several mechanisms, and one cannot realistically use a zero initial condition for the size distribution. To overcome this difficulty, Baliga (1970) suggested using the size distribution of crystals in suspension at the time of the first appearance of crystals as the initial population density. In addition to the population balance equation, appropriate kinetic equations are needed for evaluation of the nucleae 0 and the growth rate G. Empirical power-law tion rate B e0 expressions are frequently used to correlate G and B (Randolph and Larson 1988). For the growth rate, G ¼ kG Dcg

ð12:20Þ

where Dc is the supersaturation, equal to the difference between the bulk solute concentration c and the equilibrium solute concentration c*. The growth kinetic order g is usually 1–2 (Karpinski 1980, 1981). The growth rate constant kG, which may be influenced by temperature and degree of agitation, is a function of size if crystal growth is affected by bulk diffusion or the Gibbs–Thomson effect (Wey and Strong 1977b). The nucleation rate may be represented by

Batch Crystallization

e 0 ¼ kN Dcb mj B T

ð12:21Þ

where mT is the crystal suspension density. The nucleation kinetic order b is usually 0.5–2.5 for secondary nucleation and higher for primary nucleation. The exponent j may be taken as unity if secondary nucleation is the predominant nucleation mechanism. The nucleation rate constant kN is likely to depend on temperature and the degree of agitation. The bulk solute concentration c can be determined from the mass balance for batch system (Baliga 1970) d dmT ðV  mT =ρc Þc þ ¼0 dt dt

ð12:22Þ

where ρc is the crystal density. In general, the volume occupied by the crystals is much smaller than the volume of the mother liquor (mT =ρc ≪ V). Under this assumption, Equation (12.7) reduces to dðVcÞ dmT þ ¼0 dt dt

ð12:23Þ

The mass balance for a semibatch system would contain an additional term in Equation (12.23) for solute input. The crystal suspension density mT can be related to e n by ð∞ mT ¼ kV ρc e n L3 dL

ð12:24Þ

0

where kV is the crystal volume shape factor. The equilibrium solute concentration c* is usually influenced by the suspension temperature T through the solubility– temperature relationship. In a nonisothermal batch crystallizer (e.g., batch cooling crystallizer or when the heat of crystallization is significant), the suspension temperature may change with time. In this case, an energy balance is needed to determine the suspension temperature (and the associated c*) as a function of time. The energy balance can be obtained by applying the first law of thermodynamics to the batch crystallizer. The expression of this energy balance, however, varies according to the specific configuration of the batch crystallization system. The structure and interrelationship of the batch conservation equations (population, mass, and energy balances) and the nucleation and growth kinetic equations are illustrated in an information flow diagram shown in Figure 12.9. To determine the CSD in a batch crystallizer, all the preceding equations must be solved simultaneously. The batch conservation equations are difficult to solve, even numerically. The population balance [Equation (12.18)] is a nonlinear first-order partial differential equation, and the nucleation and growth kinetic expressions are included in Equation (12.18) as well as in the boundary conditions. One solution method involves the introduction of moments of the CSD, as defined by Equation (12.9), for j = 0, 1, 2, 3. The population balance [Equation (12.18)] can be multiplied through by Li and integrated to yield equations in terms e i of e of moments m n:

ð∞ ei dm ∂ n ÞdL ¼ 0 þ Li ðGe ∂L dt

ð12:25Þ

0

For size-independent growth [i.e., kG ≠ kG(L)], Equation (12.25) can be simplified to the following moment equations: e0 dm e0 ¼e nG ¼ B dt

ð12:26Þ

e1 dm e 0G ¼m dt

ð12:27Þ

e2 dm e 1G ¼ 2m dt

ð12:28Þ

e3 dm e 2G ¼ 3m dt

ð12:29Þ

The mass balance [Equation (12.23)] also can be expressed in e 3: terms of the third moment m e3 dðVcÞ dm þ kV ρ c ¼0 dt dt

ð12:30Þ

The initial conditions of the moment equations are derived directly from the initial population density e n ðL; 0Þ. Equations (12.26)–(12.30), together with the nucleation and growth kinetic expressions [Equations (12.20) and (12.21)], can be solved numerically to give the moments of the CSD for a batch or semibatch system as a function of time. The CSD can be reconstructed from its moments by the methods described by, for example, Hulburt and Katz (1964), Randolph and Larson (1988), Bałdyga and Orciuch (2001), and Diemer and Olson (2002). For size-dependent growth [i.e., kG = kG(L)], the mathematical treatment of Equation (12.25) becomes more complicated. One solution method, which included the use of moments and an orthogonal polynomial to simulate the population density function, was discussed by Wey (1985). Tavare et al. (1980) developed a transformation technique in which the time domain t was reduced into the size domain y as given by ðt y ¼ Gdt

ð12:31Þ

0

The population balance [Equation (12.18)] was then expressed in terms of the new variable y and crystal size L. Analytical solutions of the population balance equation were obtained for both size-independent growth and linear sizedependent growth. This technique was applied to the analyses of CSDs in different types of batch crystallizers (Tavare et al. 1980). Other solution methods for the population balance equation were discussed by Tavare (1987). While the classical method of moments (MOM) is focused on tracking lower-order moments of a particle distribution

353

Piotr H. Karpiński and Jerzy Bałdyga Figure 12.9 Flow diagram illustrating interrelationship of batch conservation equations, nucleation and growth kinetic equations, and the resulting CSD in batch suspension crystallizers Source: From Wey 1985.

function rather than the distribution itself, a newer method – the quadrature method of moments (QMOM) – aims at closing moment evolution equations by tracking a small number of quadrature points located so as to reproduce the moments of interest (Yoon and McGraw 2004). QMOM can be used to model processes with nonlinear crystallization/precipitation kinetics, such as size-dependent growth or agglomeration, where MOM is not applicable. Analytical solutions for the CSD for a batch or semibatch crystallizer are difficult to obtain unless both the initial condition for the CSD and appropriate kinetic models for nucleation and growth are known. An example of such an analytical solution – simple yet not overly restrictive – was given by Nývlt (1991). It is assumed that the process, in which both external seeding and nucleation take place, occurs at constant e 0 = const.) in an ideally mixed supersaturation (G = const., B crystallizer. An additional assumption of size-independent growth allows one to rewrite the time-dependent moments [Equations (12.26)–(12.29)] in terms of the physical properties, e and mass e , length e such as the total number N L, surface area A, e as follows: of crystals m e =dt ¼ B e0 dN

e Lð0Þ ¼ 0

ð12:37Þ

e Að0Þ ¼0

ð12:38Þ

e mð0Þ ¼0

ð12:39Þ

Integrating the moment equations [Equations (12.32)–(12.35)] with the initial conditions [Equations (12.36)–(12.39)] from t = 0 to t gives e 0 t 4 =4 þ N e s t3 Þ e ¼ kV ρc G3 ðB m

ð12:40Þ

The total suspension density in the batch crystallizer increases rapidly with time according to the biquadratic function [Equation (12.40)]. This equation is applicable to cooling or evaporative crystallization and illustrates a generalization of more restrictive approaches employed to derive cooling and evaporation profiles in Sections 12.5.1 and 12.5.2.

ð12:32Þ

12.3.3 CSD Analysis and Kinetic Studies

eG de L=dt ¼ N

ð12:33Þ

e LG dA=dt ¼ 2kA e

ð12:34Þ

Several experimental techniques and data-analysis methods can be employed to study the nucleation/growth kinetics and CSD in batch crystallizers.

e e dm=dt ¼ 3kV ρc AG=k A

ð12:35Þ

where kA is the crystal surface area shape factor. All the abovementioned time-dependent physical properties of crystals are defined on the basis of the total operating volume of the crystallizer. At time t = 0, the total number of introduced seeds per unit of the total operating volume of the crystallizer is given as e ð0Þ ¼ N es N

ð12:36Þ

For most practical batch crystallization processes, the total length, surface area, and mass of seeds are negligible compared

354

with the corresponding quantities characterizing the final product. Thus one can safely assume that

Thermal Response Technique. Thermal response is a convenient method for studying a crystallization process in which the heat of crystallization is significant enough to cause a temperature change following the onset of nucleation. Omran and King (1974) applied the thermal response technique to follow ice crystallization in a batch cooling crystallizer in which solution subcooling was closely controlled. Figure 12.10 shows a typical temperature–time curve for an experimental run in which the solution subcooling is held constant before crystallization is started. After the introduction of a seed crystal at t = 0, the temperature of the solution remained constant for a period of time until t = t0. During this period, some nuclei became visible, and other nuclei were being formed in the solution. This was

Batch Crystallization

Garside et al. (1982) developed an elegant technique to evaluate crystal growth kinetics from an integral mode of batch experiments. For size-independent growth, the crystal mass deposition rate RG can be given by RG ¼ kG ρc Dcg

ð12:41Þ

The kinetic parameters kG and g in Equation (12.41) can be related to the mean size L and total surface area AT of the seed crystals and zero-time values of the supersaturation Dc0 and its time derivatives Dc01 and Dc02 : g¼

2kA Dc0 Dc0 Dc02 þ 3kV ρc LAT ðDc01 Þ2

kG ¼ 

Dc01 ρc AT ðDc0 Þg

ð12:42Þ ð12:43Þ

If the initial portion of a measured desupersaturation curve is approximated by a second-order polynomial Figure 12.10 Temperature–time response curve for exothermic nucleation event Source: From Omran and King 1974.

followed by a rise in temperature of the solution (end of stage I). After the early rise (stage II), the temperature rises in a monotonic fashion (stage III) until the equilibrium temperature of the solution is reached. Omran and King (1974) considered mass and energy balances for this system and derived an equation to relate the change of subcooling in the crystallizer to the nucleation and growth kinetics. For a specified small temperature change at the end of stage I, the nucleation kinetic order b can be determined from the slope of a logarithmic plot of t0 versus initial subcooling DT0 . For the crystallization of ice from sugar solutions, the values of b were about 1.25 and independent of the sugar concentration (Omran and King 1974). Stocking and King (1976) improved the experimental technique by using quartz thermometry to achieve high-resolution measurement of the initial thermal response. Kane et al. (1974) also used the thermal response method to determine the crystallization kinetics of ice crystals from saline, emphasizing a later portion of the response curve. Desupersaturation Curve Technique. Two different modes of experiment under the condition of negligible solute consumption by nucleation have often been used to determine crystal growth kinetics. In the differential mode, a small quantity of crystals with relatively uniform size is allowed to grow for a given period of time. In this type of operation, the change in solution supersaturation is relatively small, and the crystals grow at essentially constant supersaturation. Direct measurement of the changes in crystal weight enables determination of the overall growth rate. In the integral mode, however, a relatively large seed loading of closely sized crystals is charged to the crystallizer. The growth process consumes appreciable solute, thus resulting in decay in supersaturation during the experiment. In this case, the growth rate is a time-varying quantity, so the analysis of growth kinetics is somewhat complicated.

DcðtÞ ¼ a0 þ a1 t þ a2 t2

ð12:44Þ

then the zero-time derivatives are given by Dc0 ¼ a0 Dc01 ¼ a1 Dc02 ¼ 2a2

ð12:45Þ

Garside et al. (1982) used this technique to study the growth kinetics of potassium sulfate crystals in a seeded batch crystallizer. The first few Dc values of the experimental desupersaturation curve were fitted to the second-order polynomial [Equation (12.44)] from which the values of Dc0 , Dc01 , and Dc02 were determined according to Equation (12.45). The values of the growth kinetic parameters g and kG were then calculated from Equations (12.42) and (12.43), respectively. Tavare (1985) extended this technique to study the growth kinetics of ammonium sulfate crystals in a batch cooling crystallizer. The initial derivatives of supersaturation and the temperature profiles obtained in a series of integral batch experiments were used to directly evaluate the kinetic parameters in the crystal growth correlations. Cumulative CSD Method. Misra and White (1971) used a cumulative CSD method to study the crystallization kinetics of aluminum trihydroxide from a seeded batch crystallizer. The population balance equation corresponding to Equation (12.18) can be written as ∂ðNFÞ ∂F þ NG ¼ 0 ∂t ∂L

ð12:46Þ

where N is the total number of crystals, and F is the cumulative size distribution on a crystal number basis. The growth rate G represents the rate of change of crystal size at a constant value of F and is given by

 ∂L DL G¼ ≈ ð12:47Þ ∂t F Dt F The sizing methods for the cumulative CSDs are unfortunately limited to a minimum crystal size below which accurate

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Piotr H. Karpiński and Jerzy Bałdyga

counts cannot be made. Thus the total number of crystals N and the number of crystals smaller than a particular size NF are unknown. However, the experimental data can give N(1 – F), the number of crystals with size greater than any prescribed size L. Because a constant value of F corresponds to a constant value of N(1 – F) for the chosen conditions, the concept given by Equation (12.47) is still valid and is illustrated in Figure 12.11. Misra and White (1971) obtained substantially parallel cumulative size distributions of aluminum trihydroxide from caustic aluminate solutions. Their results suggest a size-independent growth behavior because the smaller crystals increased in size by the same amount as the larger ones. Garside and Jančić (1976) applied the same method to study the growth and dissolution of potash alum crystals in the subsieve size range and obtained a strongly size-dependent behavior. Characterization of CSD Maximum. The changes in the magnitude and location of the CSD maximum with time are useful in revealing the growth mechanism involved in batch crystallization and can be given by (Wey and Estrin 1974)
 de n ∂G ¼ e n ð12:48Þ dt ∂L Lp
2  dLp ∂ G=∂L2 ¼ ðGÞLp þ e n 2 dt ∂e n =∂L2 Lp

ð12:49Þ

batch crystallizer. Some of the factors considered here include batch cycle time, supersaturation profile, external seeding, fouling control, CSD control, growth-rate dispersions, and mixing.

12.4.1 Batch Cycle Time

where Lp is the size at which the peak is located. Equation (12.48) reveals how the magnitude of e n at Lp changes with time. For size-independent growth, the peak will remain unchanged in value. For size-dependent growth, the magnitude of the peak will increase with time if ∂G=∂L < 0 and decrease with time if ∂G=∂L > 0. (The same is true for the minimum of the distribution.) Thus, if one can follow the values of the CSD maximum (or minimum) during a batch crystallization experiment, the growth characteristics of the crystals may be inferred. Figure 12.12 shows short-time CSD data obtained from a Couette flow batch crystallizer for the ice–brine system (Wey and Estrin 1974). The experimental data qualitatively show that de n =dt > 0 for Lp < 0.06 cm and de n =dt ≈ 0 for Lp > 0.06 cm. The results suggest that ∂G=∂L < 0 for Lp < 0.06 cm and ∂G=∂L ≃ 0 for Lp > 0.06 cm. This growth behavior is qualitatively consistent with a size-dependent growth model based on a heat and mass diffusion mechanism (Wey and Estrin 1974). Equation (12.49) reveals how the location of the peak changes with time. For size-independent growth, the peak remains distinct and moves toward larger sizes at a velocity (rate) equal to G. For the growth function of ice crystals, the second term on the right is negative, and the minimum moves to the right more rapidly than the maximum. This may be demonstrated by the skewness of the CSD data shown in Figure 12.12.

In the operation of a batch crystallizer, several steps need to be performed in succession to complete a batch cycle (Nývlt 1978). For a cooling crystallizer, the various periods of a batch cycle are shown in a plot of temperature versus time (Figure 12.13). These periods include (1) filling the crystallizer, (2) cooling to the saturation temperature T*, (3) crystallization period, (4) removing the suspension, and (5) cleaning the crystallizer. The time needed to complete a period is determined by the attainable rate of the rate-limiting process associated with each period. For example, the time needed for period 2 is determined by the attainable heat transfer rate for cooling. In period 3, however, the heat transfer rate may need to be restricted in order to control the nucleation and growth rates or to minimize fouling of the cooling surfaces. The time required for period 4 may be determined by the capacity of the filter if filtration is needed during transfer of the crystallizer suspension to a storage vessel. The required total batch cycle time can be calculated if all times required to complete the individual periods are known. Because the size of the crystallizer may affect the duration of all the stages of a batch cycle, the batch cycle time is expected to be a function of crystallizer volume. Thus the batch cycle time information can be used to determine the crystallizer volume required for meeting a production rate specification.

12.4 Factors Affecting Batch Crystallization

12.4.2 Supersaturation Profile

The quality, productivity, and batch-to-batch consistency of the final crystal product can be affected by the conditions of the

356

Figure 12.11 Determination of growth rate from cumulative size distribution. Misra and White (1971) also developed a procedure to determine the nucleation rate based on the cumulative size distribution data. Source: From Misra and White 1971.

The supersaturation profile in a batch crystallizer has a profound effect on the nucleation and growth processes and the

Batch Crystallization

Figure 12.13 Schematic representation of a cooling batch crystallization cycle Source: From Nývlt 1978.

Figure 12.12 Short-time CSD data obtained from a batch crystallizer for an ice–brine system Source: From Wey and Estrin 1974.

resulting CSD. It also can affect other factors (e.g., batch cycle time) related to the batch crystallization operation. Figure 12.14 shows schematically a supersaturation profile in a batch crystallization experiment (Nývlt et al. 1985). At t = 0, the batch crystallizer is filled with a just-saturated solution that contains crystals with a negligible surface area. The solution begins to be supersaturated at a constant rate, and the supersaturation increases until it reaches the limit of the metastable zone Dc max . At this point, nucleation starts, and crystals with a certain surface area begin to form. The increase in total crystal surface area owing to nucleation and crystal growth would remove more and more of this supersaturation until, at the end of the batch (t = tf), the supersaturation decreases to the value Dcf . The supersaturation profile shown in Figure 12.14 represents a special case in which the supersaturation is increased at a constant rate. In practice, the supersaturation profile in a batch crystallizer may deviate from this profile. For example, if a constant cooling rate is imposed throughout the batch cooling crystallization operation, significantly high supersaturation will be generated at the beginning of the run and very low supersaturation will be generated near the end of the run. As a consequence, excessive nucleation will occur at the beginning, and low growth rates will be obtained near the end. Both results will contribute to the production of smaller crystals and lengthen the batch cycle time required to achieve the desired crystal size.

Significant improvement in CSD and operation can be achieved by controlling the supersaturation levels during the batch crystallization process. Mullin and Nývlt (1971) and Larson and Garside (1973) showed how programmed cooling or programmed evaporation could be designed so that the optimal rate of supersaturation generation could be achieved at all stages of the batch run. Their results demonstrated that supersaturation control during a batch run was beneficial for increasing crystal size and reducing batch cycle time. Jones and Mullin (1974) showed that programmed cooling also markedly narrowed the CSD obtained from batch potassium sulfate crystallization. Programmed cooling will be addressed further in Section 12.5.1.

12.4.3 External Seeding One of the main challenges in batch crystallization is to control the supersaturation and nucleation during the initial stage of the batch run. During this period, very little crystal suspension is present on which solute can crystallize, so high supersaturation and excessive nucleation often occur. Another difficulty associated with batch crystallization is in determining the initial condition for the population density function. In an unseeded batch crystallizer, initial nucleation can occur by several mechanisms and usually occurs as an initial “shower” of nuclei followed by a reduced nucleation rate. Thus an initial size distribution exists, and one cannot realistically use the zero initial condition for the size distribution. One method for overcoming these difficulties is the use of external seeds so that initial nucleation can be controlled, minimized, or eliminated altogether. Most crystallization systems exhibit a metastable supersaturation zone where crystal growth continues but the supersaturation is too low for nucleation to take place. Thus, if the supersaturation can be

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Piotr H. Karpiński and Jerzy Bałdyga

Figure 12.14 Evolution of supersaturation profile in an unseeded batch crystallization experiment Source: From Nývlt 1985.

maintained below the metastable zone upper limit after seeding, then only growth on the seed crystals will occur. In this case, the initial condition for the population density will be the initial seed distribution. The external seeding technique also can be used to determine the maximum allowable growth rate without nucleation at the limit of the metastable zone. Larson and Garside (1973) suggested an experimental technique to determine this maximum growth rate for a low-yield (class I) crystallization system in which the supersaturation can be measured. Seed crystals, which are closely sized between two adjacent sieve sizes, are placed in a supersaturated solution of known concentration and allowed to grow until they have about doubled in weight. The final solution concentration is measured, and the crystal product is dried. From the final mass of crystals mf, the growth rate G can be calculated from the following equation: Ls G¼ tf

"


mf ms

#

1=3 1

ð12:50Þ

where Ls and ms are the size and mass of the seed crystals, respectively. Results at different supersaturation may be correlated by the growth kinetic equation [Equation (12.20)]. As the supersaturation is increased, the point at which nucleation just begins to occur is noted. This corresponds to the maximum metastable supersaturation (or the maximum growth rate) allowable without nucleation in the crystallizer. This technique will serve to give a maximum allowable growth rate that is accurate to within about 50 percent. For a high-yield (class II) crystallization system, where the supersaturation generally cannot be measured accurately, determination of the maximum metastable supersaturation and the corresponding maximum growth rate becomes more difficult. Wey and Strong (1977a) presented an experimental technique that involved external seeding to examine the maximum growth rate of AgBr crystals (class II system). Their approach involves a balanced double-feed precipitation in

358

which aqueous silver nitrate and sodium bromide solutions are added simultaneously to an agitated aqueous gelatin solution. For each experiment, the reaction vessel is charged initially with monodisperse AgBr seed crystals. When the number of seed crystals is decreased or the reactant addition rate is increased from one experiment to the next, nucleation eventually occurs at the point where the solution reaches the maximum metastable supersaturation. The corresponding maximum growth rate is determined from a simple mass balance for the class II system. Wey and Strong (1977a) followed this approach for each of five different monodisperse AgBr seed sizes ranging from 0.047 to 0.64 µm. The temperature and bromide ion concentration in suspension were maintained constant for all the experiments. Because the maximum growth rates of different seed sizes were obtained at the same maximum metastable supersaturation condition, the effect of supersaturation can be eliminated from the correlation between growth rate and crystal size. Furthermore, the maximum metastable supersaturation can be estimated when comparing the experimental results with the growth models. The maximum growth rate of AgBr crystals obtained from this study shows a strong size-dependent behavior that can be explained by a growth model based on bulk diffusion and surface integration mechanisms. The supersaturation ratio c/c* at this maximum metastable condition was estimated to be about 1.5. As mentioned earlier, in an ideal case, the supersaturation level during the externally seeded batch growth process stays below the critical (maximum) supersaturation, so the number of uniformly sized seeds introduced at the onset of the batch crystallization remains unchanged throughout the growth. As a result, a fines-free product of uniform size can be obtained. Aside from special cases (e.g., high-value product), the preparation of uniformly sized seeds is considered to be noneconomic and troublesome. Therefore, a common industrial practice is the use of a narrowly sized fraction of the seeds, usually taken from the fines fraction of the product from the preceding batches. Typically, no particular attention is paid to ensure that no primary nucleation (renucleation) occurs during the course of the seeded process. The equipment employed is either not capable or not set to maintain a desirable constant, subcritical supersaturation throughout the run. Nonetheless, it has been a common belief that even bad seeding is better than no seeding at all, for it allows one to harness a massive spontaneous nucleation in the initial part of the run and to prevent excessive supersaturation excursions along the course of the batch crystallization process. The benefits of external seeding are less dramatic for solutions with a narrow width of the metastable zone and for crystalline materials that are prone to secondary nucleation. The most frequently studied effect for externally seeded batch crystallization is that of the number or amount of seeds and their effect on the CSD. In theory, the number of seeds introduced into a well-mixed and well-controlled batch crystallizer should be equal to that of the product. In practice, inadequate seed population, nonuniform mixing, and

Batch Crystallization

excessive cooling or evaporation rates may contribute to the occurrence of spontaneous nucleation despite the presence of seeds. Furthermore, unavoidable secondary nucleation may be an additional – and often rather significant – source of nuclei. As a result, only a portion of the product crystals originates from the seeds. Nevertheless, the benefits of seeding for subsiding the primary nucleation and for coarsening the mean product size are apparent. Figure 12.15 shows an example of computer simulations investigating the influence of the number of seeds on the course of the primary nucleation rate during batch crystallization (Bohlin and Rasmuson 1992a). As expected, the nucleation rate decreased very significantly with a larger number of seeds introduced. In experimental studies, the amount of seeds ranging from a fraction of 1.0–10 wt% of the theoretical yield (e.g., Karpinski et al. 1980a; Johnson et al. 1997) has been used successfully to improve the CSD of the product crystals over an unseeded batch crystallization. It is generally recommended that the seeding should take place at the moment when the initially undersaturated solution is brought to the saturation condition (i.e., when the solubility concentration is achieved). An excessive buildup of supersaturation prior to the seeding should be avoided because it may lead to either spontaneous or seed-induced nucleation, which would undermine the benefits of seeding. Karpinski et al. (1980a) investigated the effect of the “point of seeding” within the subcooling metastable zone on the CSD in cooling crystallization. Two inorganic salts with a wide width of the metastable zone – MgSO4·7H2O and K-alum – were studied. Figure 12.16 shows the effect of the point of seeding – measured by the subcooling below the supersaturation temperature DT– on the average size of the product. The authors concluded that seeding at any point up to 40 percent of the maximum subcooling metastable zone width DTmax had little effect on product size. However, progressively more and more fines were observed in the product as a result of “late” seeding points at the temperatures corresponding to the subcooling within the range 40–100 percent DTmax .

12.4.4 Fouling Control One of the major operational problems in batch (or continuous) crystallization is the fouling (or incrustation) of crystals on the heat transfer surfaces. This is because the supersaturation is normally highest near these surfaces. Once the deposition of crystals takes place on these surfaces, the accumulation of crystal deposits can increase rapidly, resulting in a severe fouling problem of the heat transfer equipment. To minimize this fouling problem, one needs to maintain sufficient agitation and a low temperature gradient on the heat transfer surface. For batch crystallization, the external seeding technique mentioned earlier can also be used to suppress nucleation and fouling during the initial period of the batch run (Randolph and Larson 1988). Other techniques may be used to reduce the fouling problem. For example, dual cooling elements may be operated alternately in a batch cooling crystallizer (Bamforth 1965). When fouling of the first cooling element occurs, the cooling

Figure 12.15 Effect of the number of seeds on primary nucleation rate in a controlled batch cooling crystallization: computer simulation results for citric acid Source: From Bohlin and Rasmuson 1992a.

fluid is then directed to the second cooling element, while the first is heated to remove the fouling deposits. Later, fouling may occur on the second cooling element, and the cycle is repeated. Also, heat transfer equipment with some type of scraper mechanism may be used to prevent excessive buildup of deposits (Bamforth 1965). In addition, fouling can be completely eliminated by using a direct-contact refrigeration technique. In this case, the refrigerant is mixed directly with the crystal suspension, where it absorbs heat and is vaporized. Refrigerant vapor leaves the surface of the crystallizer and is compressed, condensed, and recirculated to the crystallizer. Margolis et al. (1971) successfully used liquid isobutylene as a direct-contact refrigerant to perform ice crystallization in brine solutions.

12.4.5 CSD Control CSD control in batch crystallization can provide a significant improvement in the product quality and properties and in downstream processing efficiency and economics. Two important technologies considered here include fines destruction and preparation of narrow CSD. Fines Destruction. In the operation of industrial crystallizers, one would usually want to avoid the fines (i.e., small crystals) because they may cause difficulties in downstream processing equipment (e.g., filtration) and affect both product quality and process economics. Excessive fines may also require a relatively long batch run time to achieve the desired final size of the product crystals. Karpinski (1981) proposed a controlled dissolution of secondary nuclei to improve CSD from fluidized-bed crystallizers. Jones et al. (1984) first described the application of fines destruction in batch crystallization of potassium sulfate solutions. Their study demonstrated the experimental feasibility of this technology to dramatically reduce the amount of fines in the

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Piotr H. Karpiński and Jerzy Bałdyga

Figure 12.16 The effect of “point of seeding” within the metastable zone on crystal size in batch cooling crystallization. Seed size Ls = 0.4075 mm, linear cooling rate = 7.5 K/h. The saturation temperature and maximum (critical) undercooling were as follows: T* = 298 K and DT max = 6 K for MgSO4·7H2O, and T* = 293 K and DT max = 7 K for potassium alum. Source: From Karpinski et al. 1980a.

final product CSD. Their theoretical predictions, obtained from population balance models, agreed with the experimental results. Zipp and Randolph (1989) studied the crystallization of pentaerythritol, with and without fines removal and dissolution, in a batch cooling crystallizer. The fines-segregation device used a tube tapered at the top through which fines flow was removed. The maximum fines size removed was determined by Stoke’s law of settling by the upward velocity through the tube. Larger crystals settled back into the agitated bulk suspension. Fines were pumped from the crystallizer via the fines-segregation device, passed through a coil immersed in a hot-water bath, and returned to the crystallizer. Figure 12.17 shows the experimental cumulative number distributions of the initial and final samples obtained with and without fines destruction. The results show that fines destruction causes a decrease in the number of small crystals and an increase in the number of large crystals. Fines destruction also gives a larger average crystal size and a smaller coefficient of variation in CSD. Thus selective fines destruction is a viable technology for batch-operated crystallizers that could allow significant manipulation of the CSD. Preparation of Narrow CSD. In batch crystallization, the withdrawal of crystal product is done only at the end of the batch, so no residence time distribution of individual crystals is associated with the product withdrawal. This is different from continuous crystallization, in which the product is continuously withdrawn from the crystallizer and there is a residence time distribution associated with the individual crystals being withdrawn. Thus batch crystallizers are inherently more

360

Figure 12.17 Effect of fines destruction on cumulative number distribution in a batch crystallizer Source: From Zipp and Randolph 1989.

capable of making products with a narrow CSD than continuous crystallizers owing to this difference in product withdrawal. An effective strategy to achieve a narrow CSD in batch crystallization is to separate nucleation and growth. That is, nucleation is controlled so that nuclei are generated only at a very early stage of the batch run. After the initial nucleation, the supersaturation is maintained within the metastable zone, where the growth of these nuclei will take place without the formation of additional nuclei. A population balance analysis may be used to derive cooling or evaporation programs to accomplish this (Randolph and Larson 1988). The external seeding technique mentioned earlier also can be used to control or eliminate the initial nucleation step. For certain batch crystallization systems, some fines may still be formed as a result of nucleation occurring even at low levels of supersaturation. In this case, the above-mentioned fines destruction technique can be used to help achieve a narrow CSD. The strategy of separating nucleation and growth was applied successfully to the preparation of monodisperse AgBr crystals by a double-feed precipitation technique (i.e., a semibatch system), as discussed by Leubner et al. (1980). In the double-feed technique, in which silver and halide ion solutions are simultaneously added to an agitated aqueous gelatin solution, observable nucleation occurs only at the very early stage of precipitation (within 1 minute). The number of stable nuclei quickly reaches a constant value, and further addition of reactants causes only growth of these stable nuclei. Thus relatively monodisperse AgBr crystals can be obtained by this method (see also Section 12.5.7). Sugimoto (1987) provided a comprehensive review of the requirements and general conditions for

Batch Crystallization

obtaining monodisperse particles from many typical colloidal systems.

12.4.6 Growth-Rate Dispersions The phenomenon of growth-rate dispersion (GRD) refers to the fact that individual crystals in suspension, even having the same size and subjected to identical growth conditions, may grow at different rates. The usual consequence of GRD in batch crystallization is widening of the CSD as the process progresses. This is mainly the result of accumulation of small, slow-growing crystals, which may account for as much as 85–90 percent of the total number of product crystals in the case of highly soluble materials (Randolph and Larson 1988). By analogy to the one-dimensional axial dispersion plug flow model, Randolph and White (1977) proposed the random fluctuation (RF) model, in which growth-rate fluctuations were represented by the growth-rate diffusivity DG, which interacts with the average linear growth rate G as follows: ∂n ∂n ∂n þ G ¼ DG 2 ∂t ∂L ∂L 2

ð12:51Þ

Tavare and Garside (1982) used the solution of Equation (12.51) to derive its parameters in terms of the differences in the initial and final mean size DL ¼ Lf  L0 and variance Dσ 2L : G ¼ DL=tf

ð12:52Þ

Dσ 2L 2tf

ð12:53Þ

DG ¼

The RF model was found applicable to sugar and Al2O3 systems (Randolph and White 1977) and some slow-growing systems (Garside 1985). Another concept of GRD, a constant crystal growth (CCG) model, was introduced by Ramanarayanan et al. (1985). They assumed that individual crystals have inherent constant growth rates but that different crystals have different inherent growth rates. The CCG model was found to be applicable to the citric acid monohydrate system and to a number of systems for periods of a few hours (Garside 1985; Shiau and Berglund 1990). A detailed treatment of the available experimental data based on the CCG model that accounts for primary and secondary nucleation and unseeded and seeded processes was given by Bohlin and Rasmuson (1992b). They concluded that even moderate growth rate dispersions can significantly influence CSDs in a batch cooling crystallization. Zumstein and Rousseau (1987) introduced a dual-mechanism model and demonstrated that both RF and CCG mechanisms may be present during batch crystallization and can contribute to the increase in variance of the CSD during growth. According to Lacmann et al. (1996), the growth rate of each individual crystal may be a consequence of the following factors: the amount of the lattice strain, the macro- and

microsurface structures of the growing crystal surface, the adsorption of foreign ionic and molecular species, and the stress introduced by the incorporation of foreign molecules. In order to account for GRD, the population balance needs to be modified, and its solution becomes a very difficult task, in particular for batch crystallization. Recently, Butler et al. (1997) concluded that because different batches of seeds of the same material have different growth rates, it is impossible to assign a “growth-rate dispersion” property for a given material. Instead, they advocated a common-history (CH) seed concept. As the name suggests, CH seed comprises crystals of a common history. Furthermore, CH seed has the following properties: 1. The size of the crystals is directly related to the growth rate and the time of growth. 2. The CSD reflects the distribution of growth rates inherent in the crystals. For such characterized CH seed, the authors were able to obtain a simple solution to the batch crystallization population balance and offered a relatively simple technique to model the behavior of a batch crystallizer.

12.4.7 Mixing Mixing is an important factor affecting batch crystallization. On the one hand, sufficient mixing is required to maintain crystals in suspension to ensure an adequate rate of energy transfer and to achieve uniformity of suspension properties throughout the crystallizer. On the other hand, the effect of mixing on batch crystallization is largely system dependent. In a somewhat idealized case, when crystals do not agglomerate and are not prone to an excessive contact secondary nucleation, use of the general diffusion mass transfer correlation of the form Sh = 0.29 Re0.6Sc0.33

(12.54)

has been proposed (Karpinski and Koch 1979; Karpinski 1981). In this correlation, valid for both stirred tank and fluidized-bed types of crystallization, as well as for single crystals growing in a flow cell, the applicable definitions of the dimensionless groups are as follows: Sh ¼

kD L ρl D

ð12:55Þ

Sc ¼

μ ρl D

ð12:56Þ

the Sherwood number,

the Schmidt number, 1=3

Re ¼

L4=3 ρl Em μ

ð12:57Þ

the particle Reynolds number for stirred vessels, and

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Piotr H. Karpiński and Jerzy Bałdyga

Re ¼

uLρl εf μ

ð12:58Þ

the particle Reynolds number for a fluidized bed. In the preceding equations, L is the average crystal size, kD is a mass transfer coefficient (expressed here in kg/m2/s, as based on the mass growth rate and not on the size growth rate), ρl is the average density of saturated solution, Em is the energy dissipated per unit mass of suspension, u is the relative crystal–solution velocity, and εf is porosity of the fluidized bed. In the case of small particles, smaller than the Kolmogorov microscale λK ¼ ν3=4 =ε1=4 , the model of Batchelor (1980) can be used:
2 1=2 1=3 kd L Lε ¼ 1 þ 0:689 Sh ¼ D Dν1=2

ð12:59Þ

where φ represents the volume fraction of crystals, and ˙γ is the shear rate, with ˙γ ¼ ðε=νÞ1=2 in turbulent flow. The number of binary collisions per unit time per unit volume (Gidaspow 1994) is given by "
1=3 # pffiffiffi pffiffiffi 2 2 φ θ ð12:66Þ Γ ¼ 4 πβnc d 1  φm

In both cases, for particles larger and smaller than the Kolmogorov microscale, the mass transfer coefficient kd, as defined by Equation (12.60), depends on the power input that is expressed here by the rate of energy dissipation ε. The power on the energy dissipation rate is between 0.167 and 0.2, so to increase the mass transfer coefficient twice, the power input should be increased by a factor of 32–64, which may result in excessive secondary nucleation. There are several aspects relating the rate of energy dissipation to the crystallization process. First, according to a two-step growth model (e.g., Karpinski 1980), mass transfer and surface integration take place consecutively and under the assumption of quasi–steady state

Where nc is the crystal number concentration, d represents the equivalent crystal diameter, φm is maximum concentration of crystals, and θ is the granular temperature, and β represents the efficiency of collisions that – based on the results of You et al. (2004) – was found to be proportional to n0:75 . c To calculate the granular temperature and related rate of energy dissipation at microscale, one can use the model proposed by Eskin et al. (2005). The total rate of energy dissipation Pw consists of the viscous dissipation εvisc caused by viscous friction between the bead and the fluid and lubrication and the energy dissipation rate per unit mass εcoll caused by inelastic component of crystal–crystal collisions:

G ¼ kr ðci  ceq Þr ¼ kd ðc∞  ci Þ

Pw ¼ εvisc þ εcoll

ð12:60Þ

Eliminating the interfacial concentration ci, one obtains G ¼ kr ½ðc∞  ceq Þ  G=kd r

ð12:61Þ

Davey and Garside (2000) introduced the surface integration effectiveness factor ηi , defined as the ratio of the growth rate at the interface conditions (i.e., the observed growth rate) and the growth rate if the interface were exposed to the bulk conditions. Defining the Damköhler number by Da ¼ kr ðc∞  ceq Þr1 =kd

ð12:62Þ

ηi ¼ ð1  ηi DaÞr

ð12:63Þ

they obtained

This can be presented as ηi =ð1  ηi Þ ¼ Da1 1=r

ð12:64Þ

For Da → 0, ηi → 1, which defines the kinetic regime, and for Da ≫ 1, ηi ffi Da1 ¼

362

and the diffusive regime takes place. This shows that to increase the surface integration effectiveness factor by changing hydrodynamic conditions, an enormous increase in power input is necessary. Using the theory of Batchelor and Green (1972), one can estimate maximum stresses created during the contacts of the spherical particles:
 5 τmax ¼ 4:9μ0 ˙γ þ 7:6φ ð12:65Þ 2

kd kr ðc∞  ceq Þr1

(12.67)

Application of the method proposed by Wylie (2003) leads to the following relation: Pw ¼ Aθ þ Bθ3=2

ð12:68Þ

where A and B are constants that depend on process conditions and fluid and crystal properties. The problem becomes complicated and essentially escapes a rigorous theoretical description when the crystals in a stirred suspension are susceptible to secondary nucleation or grow via an agglomeration mechanism. In order to suppress the formation of contact secondary nuclei, the impeller may be covered with a plastic coating and its speed reduced to the lowest acceptable level. Sometimes a fines-destruction loop is the only possibility to control an excessive secondary nucleation (Karpinski 1981). Figure 12.18 compares experimental RG versus Dc data for the growth of potassium hydrogen tartrate (KHT) occurring in a fluidized bed and in a stirred vessel (Ratsimba and Laguerie 1991). Because the mixing intensity of the fluidized bed is relatively low as limited by the maximum upward solution velocity, the growth rates are much slower than those occurring in the stirred vessel. The mixing intensity for the stirred vessel experiments was sufficiently high that the diffusional resistance – significant in the case of the fluidized bed – was no

Batch Crystallization

5.0

100 90

fluidized bed

Cumulative undersize fraction (wt. %)

Mass growth rate, RGx106 (kg m–2 s–1)

stirred tank 4.0

3.0

2.0

1.0

80 500 rpm 70

200 rpm

60 650 rpm

350 rpm

50 40 30 20 10 0

0.0 0.0

1.0

2.0

3.0

4.0

Supersaturation, ΔCx103 (mol/kg solvent) Figure 12.18 Comparison of the growth rates of potassium hydrogen tartrate (KHT) occurring in a fluidized bed and in a stirred vessel. Mean seed size Ls = 1.125 × 10–4 m, ethanol concentration 10 vol%, crystallization temperature T = 273 K. Source: From Ratsimba and Laguerie 1991.

longer important. An example of the case of crystal growth occurring via an agglomeration mechanism is illustrated in Figure 12.19, where the mean crystal size dramatically decreases with an increase in mixer speed (Mullin et al. 1990). Needless to say, there are many examples where the mean crystal size initially increases with the intensity of mixing owing to an improved convective mass transfer and then decreases as vigorous mixing produces more and more contact secondary nuclei, and the breakage phenomena may become significant. The inherent scale dependency of mixing and liquid–solid suspension dynamics can cause significant differences between laboratory, pilot, and industrial manufacturing scales (Green et al. 1996). In fact, scale-up usually amplifies inhomogeneity resulting from inadequate mixing. Single-phase-flow theoretical models are of limited value because the complicated flows of mother liquor and crystals in a crystallizer significantly affect viscous dissipation and alter turbulence. Physical models of mixing and suspension flow via application of specialized techniques for flow pattern visualization (e.g., dye, tracer particles) and characterization have been more successful. Nevertheless, the results of physical modeling are usually qualitative because of the nearly impossible scaling of the results from one vessel size to another. Significant progress is being made in fundamental approaches. The current powerful computational fluid dynamics (CFD) tools (e.g., Fluent and CFX software) – based on the solution of differential mass and momentum balances – have made it possible to allow simulations of the flow patterns within the crystallizer. Both physical and mathematical modeling adds to our knowledge and understanding of the nature of high-concentration suspension flows.

1

10

100

1000

6)

Crystal size, L (m x 10

Figure 12.19 Cumulative undersize CSD in a suspension containing SrMoO4 after 1 hour of stirring time as a function of mixer revolutions. Growth occurred via an agglomeration mechanism. Source: From Mullin et al. 1990.

For large crystals and high crystal concentrations, the crystals can be segregated, representing nonuniform distribution in the tank. They can affect the flow structure as well. In what follows, the multiphase Mixture Model by Fluent ANSYS is applied to spherical glass particles as a model of crystalline suspension. The particle concentration distribution predicted by this model is presented in Figure 12.20. For 100-µm particles, the particle distribution is uniform; however, for bigger particles, gravity force plays an increasingly significant role as the particle size gets bigger and the distribution becomes heterogeneous. Because the particles influence the fluid flow, it is important to take into account their distribution, especially when large particles are considered. The effect of the particles on the local velocity distribution is presented in Figure 12.21. One can see only a small influence of the particles on the velocity distribution for small particles and a stronger influence when larger particles are present. The remarks made so far were mainly concerned with bulk or macromixing. For batch reactive crystallization/precipitation, micromixing must be taken into account in order to properly describe the rapid chemical reaction at a molecular level (see Section 8.5.4).

12.4.8 Polymorph Control Polymorphs are crystalline materials that can exist in forms defined by different unit cells but where each of the forms has exactly the same elemental composition (Brittain 1999). Solvatomorphs are crystalline materials that can exist in forms defined by different unit cells but where these unit cells differ in their elemental composition owing to the inclusion of one or more molecules of solvent (e.g., hydrates).

363

Piotr H. Karpiński and Jerzy Bałdyga Figure 12.20 Effect of particle size on particle concentration expressed by the particle volume fraction. Stirrer speed N = 700 rpm; feeding rate through the feeding pipe Q = 90 cm3/min; average volume fraction of particles X V = 0.041. Source: From Malik 2012.

Polymorphs and solvatomorphs differ in their thermodynamic stability and have different physicochemical properties. As a rule, single-polymorph crystalline material (versus mixtures of polymorphs) and the most stable polymorph are targeted. Polymorphs and hydrates add to the complexity of batch crystallization. Figure 12.22a and b show the metastable zones for two polymorphs A and B. In Figure 12.22a, polymorphs A and B are monotropically related, and polymorph A is the most stable of the pair, regardless of temperature. In Figure 12.22b, polymorphs A and B are enantiotropically related, and polymorph A is more stable below the transition temperature, whereas polymorph B is stable above the transition temperature. Let us consider a batch cooling crystallization. For the monotropic system depicted in Figure 12.22a, it would be sufficient to conduct cooling in such a way that the concentration of solute is always below the solubility curve of polymorph B. As Figure 12.22b illustrates, in order to target a more stable polymorph A in the case of enantiotropically related polymorphs, the crystallization process must be carried out in a carefully designed fashion so as to stay in the zone within which nucleation of the less stable polymorph B is not possible. The same general principle applies to hydrated

364

and anhydrous forms or two hydrates of different hydration levels. It is sufficient to substitute the polymorphs A and B in Figure 12.22a and b with the anhydrous form AF and hydrate H or with hydrates HA and HB of different hydration levels, respectively. In practice at a manufacturing scale, targeting a desired polymorph in an enantiotropic system can be difficult. Usually, the targeted form is polymorph stable at ambient temperature, denoted here as polymorph A. As is clear from Figure 12.22b, above the transition temperature Tt, only polymorph B will preferentially nucleate at T > Tt. If the seed crystals of polymorph A are introduced into the solution supersaturated with respect to polymorph B at temperature TsA, they would dissolve. The only safe approach to obtain polymorph A is by seeding with polymorph A seeds at temperature Ts, below the transition temperature and at low supersaturation driving force DC. If a constant-growth-rate strategy is implemented, the trace of crystallization conditions will follow the line depicted in Figure 12.22b. Because the starting concentration of the solution is relatively low, the yield of the crystalline product may suffer. To improve the yield, cooling to very low temperatures and/or addition of antisolvent is advisable.

Batch Crystallization Figure 12.21 Effect of particle size on velocity distribution in a stirred tank. Stirrer speed N = 700 rpm; feeding rate through the feeding pipe Q = 90 cm3/min; average volume fraction of particles X V = 0.041. Source: From Malik 2012.

Figure 12.22 The metastable zones for two polymorphs A and B. Tt = transition temperature; T0 = initial temperature; Tf = final temperature; Ts = seeding temperature. (a) Polymorphs A and B are monotropically related: polymorph A is more stable of the pair, regardless of the temperature. (b) Polymorphs A and B are enantiotropically related: polymorph A is stable below Tt, whereas polymorph B is stable above Tt. Source: Republished with permission of the publisher from Karpinski 2011. Copyright © 2011 by Wiley-VCH, permission conveyed through Copyright Clearance Center, Inc.

Another approach to improve the yield in this situation would be to add very slowly into the crystallizer a hot, clear solution of solute in such a fashion that the conditions of the solution in the crystallizer at any moment remain in the shaded area depicted in Figure 12.22b. Desupersaturation of the added hot solution would result in growth of the crystals of polymorph A. The factors to consider at a large scale are local variations in the concentration and temperature profiles within the crystallization vessel, which are different from bulk conditions. These variations may lead to the formation of an undesired form, dissolution of the desired form, or interconversion into an undesired polymorphic form.

Sometimes different methods chosen for the preparation of a specific polymorphic form may produce crystals of different morphology that, in turn, may critically affect properties and physical stability of the final material. For example, different batches of diclofenac sodium trihydrate can exhibit different characteristics, particularly with respect to stability, under thermal stress (Rodomonte et al. 2008). This further underscores the importance of proper selection of processing parameters to guarantee a faultless batch crystallization process. Hydrates. Hydrates introduce an additional challenge: the final material must be manufactured as a specific hydrate of

365

Piotr H. Karpiński and Jerzy Bałdyga

stoichiometric hydration level [unless the material in question forms nonstoichiometric channel hydrates (Brittain 1999)]. In general, from the manufacturing point of view, the following challenges may present themselves: • • •

Targeting of the desirable hydrate among other hydrate forms and polymorphs of anhydrous form Maintenance of the stoichiometric hydration level throughout the drying and milling operations Reconstitution of the water of crystallization lost in the drying operation, necessary for the maintenance of crystalline properties

In some instances, formation of the hydrate is preceded by formation of a metastable solvate, which subsequently converts to hydrate.

12.4.9 Agglomeration The agglomeration process was considered in Chapter 8, devoted to precipitation processes (Section 8.5.3), because agglomeration typically occurs in crystallizing systems with the small crystal size that is characteristic of precipitation. Agglomeration is defined (Söhnel and Garside 1992) as such clustering of the particles that were formed earlier that the primary particles are usually held together by crystalline bridges, as well as, perhaps, physical forces. In Chapter 8, the efficiency of agglomeration for very small particles PS was defined by the exponential function (Bałdyga et al. 2001, 2003) PS ¼ exp ðtc =ti Þ

ð12:69Þ

where tc represents the average time necessary to build a strong enough bridge of diameter Db between the particles growing with the local rate of crystal growth G Db f ðλÞG

ð12:70Þ

rffiffiffiffiffiffi τ deq Ap

ð12:71Þ

tc ¼ where Db ¼

deq ¼ 2ai aj =ða2i þ a2j  ai aj Þ1=2 , Ap represents the yield strength, and the form of equation describing the stresses τ depends on the relation between the size of particles and the Kolmogorov microscale  ε1=2 τffiμ for Li þ Lj < λk ð12:72Þ ν τffiρ ε

2=3

ðLi þ Lj Þ

2=3

for Li þ Lj > λk

ð12:73Þ

and f(λ) is the shape function introduced by David et al. (1991) pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 ð1 þ λ  λ2  1Þ f ðλÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi! pffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffi2 2 λ λ2  1 2 2 1 þ þ λ  λ  1  λ  λ  1 3 3 3 ð12:74Þ

366

where λ represents the particle size ratio λ ¼ L2 =L1 .

The contact time ti is the average interaction time. For particles of collision radius smaller than the Kolmogorov microscale (i.e., with Li + Lj < γK), one should assume that the average interaction time is equal to the time Kolmogorov microscale (ti ¼ τk ). Thus ν1=2 Li þ Lj Li þ Lj ¼ ¼ ti ¼ ε Dv γ˙ðLi þ Lj Þ for Li þ Lj < λk and ˙γ ¼ ðε=νÞ1=2

ð12:75Þ

This relation can be used for Li + Lj that it is slightly larger than λk , provided that the contact time will be smaller than the 1=2 Lagrangian time microscale τL ffi τK Reλg , where Reλg is the Reynolds number based on the Taylor microscale λg . For the collision radius Li + Lj in the inertial subrange of turbulent scales Li þ Lj >> λk , and the interaction time can be expressed as ti ¼

Li þ Lj Li þ Lj ðLi þ Lj Þ2=3 ¼ ¼ for Li þ Lj > λk Dv ε1=3 ε1=3 ðLi þ Lj Þ1=3 ð12:76Þ

The agglomeration kernel βðL; L1 Þ can be expressed as the product of the collision kernel βc and the agglomeration probability PS : β ¼ βc PS

ð12:77Þ

Note that this approach can be used for very small particles in precipitation applications (see Section 8.5.2), as well as for the large crystals, much larger than the Kolmogorov microscale considered in this chapter. Figure 12.23 shows that at constant supersaturation, an increase in the energy and the increase in particle size decrease the probability of agglomeration during collision. Hence agglomeration during precipitation is more probable than agglomeration of large crystals. A related approach, based on an equation similar to Equation (12.77), was put forward by Hounslow et al. (2001), who proposed the so-called Mumtaz model. This model is based on the collision kernel introduced by Saffman and Turner (1956) in its averaged form rffiffiffiffiffiffiffiffi 8πε 3 d βc ¼ ð12:78Þ 15ν 3;0 and an averaged efficiency of collisions Ps ¼

ðM=M50 Þλ 1 þ ðM=M50 Þλ

ð12:79Þ

where M50 and λ are parameters of the logistic curve, and M represents the ratio of the strength of the neck to the force per area of the neck. As shown by Liew et al. (2003), based on experimental observations, one should use λ = 1, whereas M can be expressed by M¼

Lσ  G 2 ερd3;0

ð12:80Þ

Batch Crystallization Figure 12.23 The effects of the rate of energy dissipation and particle size on the probability of agglomeration for equal-size particles ai ¼ aj ¼ a [black circle symbols trace the probability of agglomeration for a ¼ λK ðεÞ] and a constant crystal growth rate G = 10–6 m/s Source: From Krasiński 2004.

where L represents the characteristic contact length, and σ  is the apparent yield stress. In this case, β ¼ β cP S

ð12:81Þ

In comparing the two approaches, one can conclude that the Mumtaz model is limited to small particles, smaller than the Kolmogorov microscale; this results from application of the simplified Saffman and Turner model – Equation (12.78) and the derivation of Equation (12.81) – where the Kolmogorov deformation rate ðε=νÞ1=2 was applied. The model of Bałdyga et al. (2001, 2003) can be applied for either small or large crystals. To model the course of the crystallization/precipitation processes, the Reynolds-averaged transformation of the population balance, with the birth and death terms for the agglomeration process, can be applied L2 BD¼ 2

ðL

β ½ðL3  L31 Þ1=3 ; L1  n ½ðL3  L31 Þ1=3  nðL1 Þ ðL3  L31 Þ2=3

Lo

dL1

ð∞ nðLÞ βðL; L1 ÞnðL1 Þ dL1

ð12:82Þ

Lo

where βðL; L1 Þ represents the agglomeration kernel, given by either Equation (12.77) or Equation (12.81). Now, using the population balance equation in the form of the moment balance ∂mj ∂ðupi mj Þ þ ¼ 0j RN þ jG mj1 þ Bj  Dj ∂t ∂xi

ð12:83Þ

obtained for i = 1, 2, 3 and j = 0, 1, 2,… and assuming ∂G=∂L ¼ 0, with →

ð∞

mj ð x ; tÞ ¼

nðLÞLj dL



∂mj ∂mj ∂mj ∂upi ∂ þ upi þ mj ¼ DpT þ 0j RN þ jGmj ∂xi ∂t ∂xi ∂xi ∂xi

ð12:84Þ

þBj  Dj

ð12:85Þ

As discussed in Chapter 8, Equation (12.85) should be solved together with the Navier–Stokes equation and the species balances   ∂cα ∂cα ∂ ∂cα ka ρp þ ui ¼ ðDα þ DT Þ G m2 for α ¼ A; B  ∂t ∂xi ∂xi ∂xi 2 Mc ð12:86Þ In practice, the concept of combining the collision rate with some kind of efficiency was used by many researchers. David et al. (2003) applied a three-level agglomeration model that accounts for Brownian, laminar (below the Kolmogorov microscale), and turbulent agglomeration using separate equations for each regime. They additionally assumed that above the Taylor microscale, the agglomeration rate decreases to zero. For laminar and Brownian collisions, they assumed that the agglomeration efficiency is proportional to the growth rate G. This simple approach includes the effects of all important phenomena involved in agglomeration. Ilievski and Livk (2006) considered several efficiency models and decided to use the variables appearing in the efficiency expressions in the following form: Ps ¼

0:00042 1 þ 6:25 

109 μ1:5 ðγ2 =U

tip Þ

2:25

=ðG=L10 Þ3

ð12:87Þ

where γ is the average shear rate, and Utip represents impeller tip speed. Equation (12.87) assumes a maximum efficiency equal to 0.00042, which should be system dependent, and reveals a strong effect of viscosity not predicted by Equation (12.79) with Equation (12.80). In their next paper (Livk and Iliewski 2007), the following agglomeration kernel was proposed:

0

after Reynolds averaging and using the concept of gradient diffusion, one obtains

β¼

GS3ij β1 Sij þ β4 S4ij

ð12:88Þ

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Piotr H. Karpiński and Jerzy Bałdyga

where Sij ¼ Li þ Lj and β1 and β4 depend on the rupture rate and collision–rupture parameters. Faria et al. (2008) modeled agglomeration of sucrose crystals. Based on the earlier observations (e.g., Kuijvenhoven et al. 1983), they assumed that there is a minimum and a maximum size beyond which there is no agglomeration and a critical size Lcrit for which the agglomeration probability is the highest. They considered a minimum size of 10 µm, a maximum size of 250 µm, and a critical size of 100 µm. The agglomeration kernel for sucrose crystals of size Li and Lj is as follows:
 ðLcrit Li Lj Þ2 wc ρm   β ¼ Kag  ð12:89Þ G 1 3 1 3 3 3 kv ρc 2 Lcrit þ Li 2 Lcrit þ Lj where wc is the mass fraction of crystals in suspension, kv is the crystal volume shape factor, and ρc and ρm represent the crystal and massecuite density, respectively. With the fitted Kag value, a good agreement between the observed and predicted results was obtained. One can conclude that there are theoretical models describing both collision frequencies and efficiency of collisions, but for modeling of agglomeration, the simplified models are commonly used, often with the parameters defined by the theoretical approach.

12.5 Batch Crystallization Operations Classification of batch crystallizers and batch crystallization operations according to the means by which supersaturation is created is still a widely accepted method. Therefore, the discussion of such operations may include cooling crystallization, evaporative crystallization, vacuum crystallization, antisolvent crystallization, reaction (reactive) crystallization, sonocrystallization, and so on. The vacuum crystallization operation can be considered as a combination of evaporative and cooling crystallization and thus will not be discussed separately. Reaction crystallization (precipitation) is discussed in detail in Chapter 8. The course of batch crystallization processes may be considered in relation to the phenomenon of metastable state of supersaturated solutions. The metastable regions of concentration are characterized by the solution containing more solute than in equilibrium at the given temperature but with nucleation of the solid phase not occurring until a critical value of supersaturation is reached. Regions where no nucleation occurs and regions where nucleation is very fast are separated by the critical supersaturation Sc. Moreover, heterogeneous nucleation takes place at lower critical supersaturation than homogeneous nucleation (i.e., Sc,hom > Sc,het, as shown in Figure 12.24; see curves II and III and the regions above those curves). As suggested by Davey and Garside (2000), a simple mass balance can be used for determination of the factors that control the rate of crystallization, which is related to the rate of creation of supersaturation. The mass balance of the solute present in the batch crystallizer is as follows:

368

dM dðVcÞ dM dV dc þ ¼ þc þV ¼0 dt dt dt dt dt

ð12:90Þ

where M denotes the mass of crystals in the crystallizer, c is the solute concentration, and V is the volume of the solution. In cooling crystallization, the volume changes are negligible, and the concentration remains close to the equilibrium concentration (saturation) ceq. Hence
 dceq dceq dT dM ¼ V ¼ V ð12:91Þ dt dt dT dt which means that the rate of crystallization depends on the slope of the solubility curve (Figure 12.24) and the rate of cooling. Therefore, for a given system, the rate of cooling determines the rate of crystallization. However, when the saturation curve does not increase much with the temperature, then the process is very slow, and evaporative or drowning-out (antisolvent) crystallization is recommended. In evaporative crystallization, the concentration is approximately constant, but the volume changes in time, which – based on Equation (12.90) – results in dM dV ¼ ceq dt dt

ð12:92Þ

and the evaporation rate −dV/dt, which depends on solubility, determines the rate of crystallization. In drowning-out (antisolvent) crystallization, assuming that c is approximately equal to ceq, Equation (12.90) takes on the following form: dceq dM dV ¼ ceq V ¼0 dt dt dt

ð12:93Þ

because the addition of antisolvent of concentration c0,ant changes both the concentration of the antisolvent cant and the solution volume. Using a similar balance for the antisolvent, Equation (12.90) can be transformed to   dceq dM dV dV ¼ ceq þ  ðc0;ant  cant Þ ð12:94Þ dt dt dt dcant which shows that the rate of antisolvent addition dV/dt determines the rate of the process for a given slope of decrease in solubility with increasing antisolvent concentration dceq/dcant. For small values of dceq/dcant, the process would be inefficient. The first term on the right-hand side represents dilution of the solute with the antisolvent and is negligible compared with the second term in the case of a properly designed process.

12.5.1 Cooling Crystallization Cooling is probably the most common way of creating supersaturation in batch crystallization. It is applicable to aqueous and nonaqueous systems in which the solute solubility sharply increases with temperature. Its yield and economy depend on the temperature and concentration of the feed solution to be crystallized and on whether cooling water or another cooling

Batch Crystallization

Figure 12.24 Boundaries of the metastable zone: I = solubility boundary; II = primary heterogeneous nucleation boundary of the metastable zone; III = primary homogeneous nucleation boundary of the metastable zone

agent is employed. Typically, a high-temperature concentrated solution is delivered into a jacketed crystallizer equipped with a stirrer and often also a cooling coil to increase the cooling surface area. The solution is stirred, and cold water or cooling agent is pumped through the jacket and – if so equipped – the cooling coil. Such a “natural” cooling is continued until the temperature of the solution in the crystallizer is near that of the cooling medium. As shown in Figure 12.10, at the beginning of this process, the temperature difference between the cooling surface and the solution is at a maximum. This huge subcooling quickly brings the solution to the maximum (critical) supersaturation level. Moreover, the excessive subcooling persists for a significant portion of the batch operation. As a result, massive nucleation of the solute can occur more than once (renucleation). By contrast, nuclei subjected to the maximum supersaturation conditions grow at a high rate that may cause dendritic growth and occlusions. Furthermore, the cooling surfaces become quickly covered with layers of crystallizing solute (fouling, scaling, or incrustation). This phenomenon is detrimental to the heat transfer efficiency and may bring about a decrease in the rate change of supersaturation. In addition, as the process progresses, a large surface area of the crystal population created thus far is able to harness the excessive supersaturation level, so no further nuclei are formed spontaneously. The crystals that originate from the spontaneous nucleation and the secondary nuclei that are formed as a result of contact secondary nucleation now may grow at a somewhat slower rate. As the temperature in the bulk approaches that of the cooling medium, the growth of the crystal population becomes much slower. There are two basic practical approaches that can help control CSD in batch cooling crystallization. The first is concerned with the use of seed crystals (external seeding), and the second is concerned with limiting the occurrence of spontaneous nucleation only to the initial stage of the process. Both require control of the cooling rate via control of the solution

Figure 12.25 Schematic of the cooling crystallization process

temperature in the crystallizer and manipulations of the flow rate of the cooling agent. The principle of size manipulation in cooling crystallization is illustrated in Figure 12.25. In the cooling crystallization process, an unsaturated solution of temperature T1 (point A) is cooled in the crystallizer to temperature T2. The final state in the crystallizer is denoted by point C. For very fast cooling, point B is characterized by a very high supersaturation, and thus a very high nucleation rate may be reached, which may result in a fine crystal size at point C. The trajectory BC represents the process of solute depletion due to crystallization. For slower cooling, crystallization may start at a higher temperature (point D); crystallization then competes with cooling, and the resulting trajectory DEFC yields larger product crystals. When the solution is seeded in the metastable region (point G), product size can be controlled using trajectory AGEFC, as shown in Figure 12.25. Note that if this trajectory is parallel to the solubility line, crystallization occurs at constant supersaturation, which – based on Equation (12.20) – implies a constant growth rate. The mass m of uniform 3D crystals of an average size L can be expressed as m ¼ NkV ρc L3

ð12:95Þ

If secondary nucleation does not occur and the concentration of solution C (expressed as mass of solute per mass of solvent) is higher than the solubility, the mass rate of the depletion of solute from solution caused by cooling is equal to the sum of the mass rates owing to spontaneous nucleation and growth msol

dC dmN dmG ¼ þ dt dt dt

ð12:96Þ

where msol is the mass of the solvent part of the solution in the crystallizer. Seeded Cooling Crystallization. It is assumed that a certain number N of seed crystals of a uniform size Ls is introduced to the crystallizer at the saturation temperature T*. No nuclei are

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formed spontaneously, and the seeds are allowed to grow, at a constant growth rate G, until the final size Lf has been reached after batch time tf. In the absence of spontaneous nucleation, Equation (12.96) simplifies to msol

dC dmG þ ¼0 dt dt

ð12:97Þ

By denoting the linear growth rate as G = dL/dt = constant, one can obtain the following ordinary differential equation (ODE) from Equations (12.95) and (12.97): msol

d3 C þ 6NkV ρc G3 ¼ 0 dt 3

ð12:98Þ

with three initial conditions: C(0) = C*

(12.99)

dCð0Þ 3NkV ρc L2s G ¼ dt msol

ð12:100Þ

d2 Cð0Þ 6NkV ρc Ls G2 ¼ dt2 msol

ð12:101Þ

Integration of the preceding ODE [Equation (12.98)] subject to the initial conditions [Equations (12.99)–(12.101)] yields msol  G3 t3 G2 t 2 Gt ðC  CÞ ¼ 3 þ 2 þ Ls 3ms 3Ls Ls

ð12:102Þ

a concentration profile that is cubic in time. In order to obtain a cooling profile T(t), it is necessary to use the appropriate expression for the dependence of concentration on temperature. In a simple case, the concentration may be expressed as a linear function of T: C = aT + q

(12.103)

where a and q are constants. From Equations (12.102) and (12.103), one can obtain the following cooling profile:
 3ms G3 t3 G2 t 2 Gt T ¼ T  þ þ ð12:104Þ Ls amsol 3L3s L2s Because this cooling profile is cubic in time, the cooling rate
 dT 3ms G3 t 2 2G2 t G ¼ þ 2 þ ð12:105Þ dt Ls Ls amsol L3s is parabolic in time. Therefore, as the crystal growth progresses, higher and higher cooling rates are required to maintain the assumed constant growth rate (and supersaturation). As will be shown for evaporative crystallization, a population balance approach subject to the same assumptions will yield similar results as far as the mathematical form of the solution is concerned. Unseeded Cooling Crystallization. In a similar fashion, a cooling profile can be derived for the unseeded case in which spontaneous nucleation and growth are allowed to occur at

370

constant rates. The actual solutions to the resulting third-order differential equations found in the literature differ because of the different sets of the four initial conditions used by various authors (Karpinski et al. 1980b; Nývlt 1991; Randolph and Larson 1988). Understandably, all of them result in a cooling profile of the general form e 0 ; LN ; aÞ T ¼ T   f ðt; G; B

ð12:106Þ

which is biquadratic in time, and a cooling rate profile of the general form dT e 0 ; LN ; aÞ ¼ φðt; G; B dt

ð12:107Þ

which is cubic in time. Karpinski et al. (1980b) investigated the effect of different cooling profiles on the CSD for batch cooling crystallization of MgSO4·7H20. The four cooling profiles studied are shown in Figure 12.26. Curves (1) and (2) represent the cooling profiles calculated from Equations (12.104) and (12.106), respectively. A good agreement between the theoretical and actual product size and a narrow CSD were obtained for these two profiles. The arbitrary linear cooling profile (3) resulted in a product of smaller mean size and much wider CSD. Application of the natural cooling profile (4) produced a very polydisperse product. It is worth noticing that the shape of the natural cooling profile is “opposite” that resulting from Equations (12.104) and (12.106). Based on the two cases discussed so far, one can consider yet another case of interest, in which nucleation is allowed to proceed at a constant rate from t = 0 until a certain time t1, after which no new nuclei are generated. For such a cooling crystallization, a biquadratic cooling profile applies until t = t1 and Equation (12.104) for the time t > t1. This is the preferred method of practical unseeded batch cooling crystallization. In order to calculate these cooling profiles, one needs to know the values of the growth rate G, the nuclei size LN, and the e 0 . For the seeded run, the value of the growth nucleation rate B rate G must be lower than the critical growth rate, which can be determined experimentally in growth kinetics experiments (Karpinski 1981): G = (Lf – Ls)/tf < Gcrit

(12.108)

Again, the critical growth rate is a maximum growth rate attainable without the occurrence of spontaneous nucleation. The average size of effective nuclei LN can be determined from microscopic observations, sieve analysis, or any applicable instrumental sizing technique. If the cooling strategy allows for spontaneous nucleation, the magnitude of the nucleation e 0 must be estimated, which usually requires a population rate B balance approach (Randolph and Larson 1988). Once the value of the parameter n0 is determined from the semilog plot of population density n versus size L via continuous crystallization experiments, the nucleation rate can be calculated as e 0 ¼ n0 VG B

ð12:109Þ

Batch Crystallization

The type and area of the energy-exchange surface, as well as the process control, must be adequate to accommodate both low evaporation rates at the beginning of the batch evaporation and elevated evaporation rates in the final stage of the run. These rates may differ by two orders of magnitude.

12.5.3 Antisolvent Crystallization

Figure 12.26 Cooling profiles for crystallization of MgSO4·7H2O in a 21-liter batch cooling crystallizer. Saturation temperature T* = 308 K. Seeded crystallization: (1) cooling profile calculated from Equation (12.103), mean seed size Ls = 0.4075 mm. Unseeded crystallization: (2) cooling profile calculated from Equation (12.105); (3) linear cooling profile; (4) natural (uncontrolled) cooling profile. Source: From Karpinski et al. 1980b.

12.5.2 Evaporative Crystallization For substances whose solubility is weakly dependent on temperature (e.g., NaCl) or for those with an inverse dependence of the solubility on temperature (e.g., Na2SO4), a method of choice to create supersaturation is evaporation of the solvent. In practice, evaporative crystallizers usually operate at constant temperature and reduced pressure. Larson (1978) considered the operation of a batch, externally seeded evaporative crystallizer with a constant growth rate G in which no spontaneous nucleation was allowed. Applying the population balance [Equation (12.16)] to this system gave the following equations for the solvent profile CV G3 t3 G2 t2 Gt ðVsol;0  Vsol Þ ¼ 3 þ 2 þ Ls 3ms 3Ls Ls and for the solvent evaporation rate
 dVsol 3ms G3 t2 2G2 t G ¼ þ þ L2s Ls dt CV L3s

ð12:110Þ

ð12:111Þ

It is important to recognize the similarity of the mathematical form of Equations (12.104) and (12.110) and Equations (12.105) and (12.111). One may conclude that all batch crystallization processes may be controlled by a properly designed time profile of the supersaturation-inducing quantity (e.g., the cooling or evaporation rate in the cases discussed in this chapter and the reagent addition rate discussed in Chapter 8 on precipitation).

In this type of batch crystallization, a solute is crystallized from a primary solvent by the addition of a second solvent (antisolvent) in which the solute is relatively insoluble. The antisolvent is miscible with the primary solvent and brings about a solubility decrease of the solute in the resulting binary solvent mixture. If an organic solvent or solution of another solute (such as salt solution) is added to an aqueous solution containing the substance to be crystallized (historically, an inorganic salt), the process is termed salting out. The inverse procedure, common in pharmaceutical and organic chemicals technology, when water is added to the organic solvent in order to crystallize a dissolved organic substance, is known as drowning out. The fundamental principle is the same in both cases. The principle of drowning-out or salting-out crystallization is presented schematically in Figure 12.27. The unsaturated solution of the solute (point A) is mixed with the antisolvent, as shown in triangular diagram. The solution composition then varies along a straight line AB–antisolvent; the solution becomes saturated at point B, and further addition of the antisolvent increases supersaturation. The rate of supersaturation creation then depends on the amount of antisolvent added, the addition rate, and the intensity of mixing. Drowning-out precipitation usually produces crystals having a smaller size than those produced by cooling or evaporation at the same supersaturation (Söhnel and Garside 1992). The major advantage of antisolvent crystallization lies in the fact that the process can be carried out at the ambient temperature, which – aside from the convenience and economic aspects – is of paramount importance for heatsensitive substances. The disadvantage of this process is that the binary solvent mixture must be separated subsequently to recover and recycle one or both solvents. Frequently, however, the added cost of the separation operation is fully absorbed by the valuable and expensive products produced, such as pharmaceuticals. Using an analogy with the programmed cooling crystallization, Karpinski and Nývlt (1983) suggested that the quality of the product obtained by means of salting out can be improved by a programmed addition rate of the antisolvent that is proportional to the instantaneous crystal surface area. Typical operational parameters affecting batch antisolvent crystallization are as follows (Budz et al. 1986; Johnson et al. 1997): the solubility and actual concentration of a solute in both primary solvent and binary mixture (primary solvent– antisolvent), the antisolvent addition rate, the use of seeds, the form of seeds (dried versus slurry), the amount and properties of seed crystals, intensity of mixing, the crystallization temperature, and the batch time.

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Piotr H. Karpiński and Jerzy Bałdyga

Figure 12.27 Schematic illustrating the principle of antisolvent or drowningout crystallization

Quite often the addition of antisolvent even at low addition rates may create extremely high supersaturation levels leading to an excessive amount of fines in the final product and subsequent problems in the downstream processes. One way of avoiding this problem, suggested by Karpinski and Nývlt (1983), is a dilution of the antisolvent with the primary solvent or the use of a diluted solute–primary solvent solution, as advocated by Budz at al. (1986). Such measures help with the CSD control but owing to increases in the operating volume also have an adverse effect on the process yield. An interesting approach to maintain a high yield and to operate within a reasonable supersaturation range was proposed by Mydlarz and Budz (1986). Namely, a low-volatility antisolvent can be introduced to the concentrated primary solvent solution in the form of a gaseous mixture, such as a carrier gas–antisolvent vapor mixture, rather than as a liquid. The supersaturation level can be controlled by the ratio between the carrier gas (e.g., air or nitrogen) and the antisolvent vapor. The volume increase from antisolvent vapor condensation is rather small, and therefore, the attainable yield may be increased significantly.

12.5.4 Crystallization from Supercritical Solvents The properties of supercritical fluids (SCFs) and the method of antisolvent precipitation with the SCF as an antisolvent were described in Chapter 8. Here the rapid expansion of supercritical solutions (RESS) process, depicted schematically in Figure 12.28, will be presented. It consists of dissolving the substrate in an SCF and subsequent rapid expansion of this solution through a heated nozzle. As discussed in Section 8.2.5, the dissolving power of SCF depends on its density, which means that solubility in the SCF can be tuned continuously by varying either pressure or temperature or both. When the saturated solution is depressurized into a lowerpressure chamber, solubility decreases immediately, and a very high supersaturation is created. At high initial supersaturation, very fine particles can be created (crystalline or amorphous) that usually form small agglomerates. As a result, in the case of pure substrate components, fine particles of the range 0.5– 20 μm are produced with a narrow size distribution. Composite particles (microspheres, microcapsules) can also be obtained by processing a mixture of substrates in the RESS process. Morphology of the particles depends on the nozzle

372

Figure 12.28 Schematic of the rapid expansion of supercritical solution (RESS) process

design and geometry of the expansion vessel, RESS parameters (temperature, pressure, and the pressure drop), and the physical and chemical properties of the solute. The method can be compared with the classic method of evaporative crystallization, especially spray crystallization. The main difference is that in the classic method, the mass and energy transfer between liquid and vapor controls creation of supersaturation, whereas depressurizing does not create any energy or mass transfer barrier. The RESS method can be used only for substances that are well soluble in SCF, and the process often requires high pressure ranges. A more complete description of the RESS process can be found in the book by York et al. (2004) and review articles (Jung and Perrut 2001; Marr and Gamse 2000; Subra and Jestin 1999).

12.5.5 pH Swing Crystallization The basic strategy in crystallization is to reduce the solute solubility and create supersaturation by cooling, evaporation, addition of antisolvent, or use of the strong effect of process conditions on solubility in supercritical solvents, as presented in Sections 12.5.1 to 12.5.4. Another method of creating supersaturation is based on pH swing, that is, a dramatic variation of the solubility by changing pH, which is, of course, only possible when the solubility depends on pH. This may happen when there are large solubility differences between protonated and deprotonated species. As an example of such dependence, let us observe the effect of pH on the solubility of titanium hydrous oxide TiO(OH)2. To this end, we can use Figure 12.29 to demonstrate that increasing pH from 1 to 6.7 or decreasing pOH from 13 to 7.3 creates high supersaturation, in this case to a saturation ratio as high as 512. The process of pH variation may be repeated, which can result in dissolving small crystals or preventing formation of crystals of unwanted morphology. Neppolian et al. (2005) used as many as 30 pH swing cycles. This has practical implications; for example, Neppolian et al. (2005) presented a multigelation method based on pH swing for the preparation of TiO2 nanoparticle photocatalysts that showed good performance in controlling particle size,

Batch Crystallization

Figure 12.29 Precipitation diagram for titanium IV (titanium hydrous oxide) at 20°C and the effect of decreasing pOH on the saturation ratio and structure of solution; the arrow shows direction of the pOH variation.

crystallinity, pore volume, and pore diameter, as well as the anatase and rutile phase composition of the catalysts. What is important from the viewpoint of this section is that by using a higher number of pH swing cycles, the phase transition from anatase to rutile may be prevented. This means that the catalysts have a higher photocatalytic activity. In their next paper, Neppolian et al. (2008) proposed to prepare nanocrystalline TiO2 particles based on the pH swing method assisted by ultrasonic irradiation in the presence of a surfactant (Pluronic P-123). The prepared TiO2 material was calcined. The enhancement in the particle size of TiO2 by the pH swing method was controlled by combining the pH swing with ultrasonic irradiation (see Section 12.5.6, which addresses ultrasonic crystallization). TiO2 prepared with 15 pH swing cycles and calcined at 700°C was found to show the highest catalytic performance. Creation of high supersaturation by decreasing pH (pH swing) was applied in the case of crystallization/precipitation of benzoic acid by mixing the solution of sodium benzoate with hydrochloric acid; this problem was discussed in relation to aggregation in the semibatch stirred reactor in Section 8.5.2 (see also Bałdyga and Krasiński 2005). The same system was used by Torbacke and Rasmuson (2001) for studying the effects of different mixing scales (from macroscales through mesoscales to microscales) on the reactive crystallization of benzoic acid. This shows that the pH swing method (with a single swing) can be used as a test method to study the mixing and agitation effects observed in crystallization processes. Crystallization of nicotinic acid by mixing an aqueous solution of sodium nicotinate with hydrochloric acid was investigated by Wang and Berglund (2000) in the semibatch stirred tank reactor with HCl added to the sodium nicotinate. They showed that the progress of crystallization can be

monitored using attenuated total reflection Fourier transform infrared spectrometry (ATR-FTIR). Finally, it is worth mentioning a phenomenon of pH swing observed in frozen solutions. Freezing is an important step in the method of lyophilization (or freeze drying). As shown by Sundaramurthi et al. (2010a, b), the pH swing phenomenon can be observed in this process. Namely, in the freeze-drying process, one needs at first to cool the prelyophilized solution and afterwards to sublimate the ice under low pressure; finally, some elevation of temperature is necessary to remove absorbed water. A common practice is to buffer the prelyophilized solutions. However, for multicomponent buffers, selective crystallization of buffering components can strongly affect pH; Sundaramurthi et al. (2010a, b) observed a decrease of pH up to 4 pH units. Because of the high sensitivity of many pharmaceutical species, especially proteins, to pH, one should choose the buffer and its concentration with particular care. The examples presented herein demonstrate both the usefulness of the pH swing phenomenon in controlling crystallization processes and the potential of significant variation of the product quality for pH-dependent materials caused by uncontrolled pH swing.

12.5.6 Ultrasound-Assisted Crystallization It is known from a large body of experimental work that ultrasound affects crystallization processes. The literature reports that ultrasonic irradiation reduces the induction time and accelerates nucleation, reduces crystal size, increases the crystal growth rate, reduces the width of the metastable zone, increases product yield, reduces (or promotes) agglomeration, affects crystal habit, and may narrow the crystal size distribution. Shock waves can induce rapid local cooling, and the cavitation events are expected to allow the excitation of nucleation energy barriers; see, e.g., papers by Nalajala and Moholkar (2011) and de Castro and Priego-Capot (2007) for details. Mechanisms that cause the above-mentioned effects are not fully understood, and there are different opinions and explanations regarding the importance of the effects resulting from the different phenomena involved. Based on an analysis of their experimental results, Nalajala and Moholkar (2011) concluded that the crucial factor that affects ultrasound-assisted crystallization is convection induced in the medium. Two types of convection can be considered: microturbulence, induced in the liquid by oscillatory motions of cavitation bubbles in radial direction, and acoustic shock waves, induced by collapsing bubbles “at the instance of minimum radius.” In the case of microturbulence, the velocity induced in the liquid Vturb decreases significantly with distance r from the bubble center (Leighton 1994)
 R2 dR Vturb ðr; tÞ ¼ 2 ð12:112Þ dt r The shock or acoustic waves are induced when the converging liquid elements forming the bubble walls are suddenly stopped. The amplitude of the resulting wave is given by (Grossmann et al. 1997)

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Piotr H. Karpiński and Jerzy Bałdyga

"
 # ρ d 2 Vb R dR 2 d2 R 2 PAW ðr; tÞ ¼ ¼ρ þR 2 4πr dt 2 r dt dt

ð12:113Þ

where Vb denotes bubble volume. One can see that the magnitude of PAW varies inversely with r. To find dR/dt and d2R/dt2, an adequate form of the Rayleigh–Plesset equation for bubble dynamics should be solved. The role of both local convection phenomena in ultrasound-assisted crystallization is – according to Nalajala and Moholkar (2011) – different; namely, the shock waves increase the nucleation rate significantly, while microturbulence accelerates the rate of crystal growth by increasing the rate of transfer of the solute molecules to the crystal surface. Typically, the effect of shock waves is dominant, and smaller crystals are produced. Another explanation for nucleation induced by ultrasound was given by Kordylla et al. (2008, 2009). They assumed that the observed acceleration of nucleation owing to ultrasound irradiation results from a heterogeneous mechanism of primary nucleation on the cavitation bubble surfaces as well as adsorption of newly formed crystals onto the bubble surface. By contrast, they found that enhancement of the diffusional mass transfer as a result of ultrasounds has a minor effect on the nucleation mechanism. Because the properties of the liquid solvent seem not to affect the width of the metastable zone (Kordylla et al. 2008) and the properties of the shock waves depend on the liquid properties (i.e., sound speed, compressibility, viscosity), it is difficult to judge which mechanism dominates. Cavitation induced by ultrasound and ultrasonic waves themselves can cause deagglomeration or prevent agglomeration (Bałdyga et al. 2008). To estimate the stresses generated by bubble collapse, the Rayleigh–Plesset equation can be used to approximate, first, the interface velocity during collapse Vj ¼

ðp  pV Þ1=2 0:915ρL 1=2

ð12:114Þ

where p is external pressure infinitely far from the bubble, and pV is the vapor pressure within the bubble, and then the localized pressure generated by microjets τp ¼ αρL cVj

ð12:115Þ

where c is the velocity of the compressional wave in the liquid, Vj is microjet velocity, and α is a constant that varies from 0.41 to 3.0 (Crum 1988). Figure 12.30 illustrates the effect of local pressure p on the stresses τp generated by bubble implosion (Bałdyga et al. 2008). The sound wave induces unsteady motions of particles in the fluid, which also generate stresses, but they are much smaller than those presented in Figure 12.30.

12.5.7 Concept of Seeding In Situ The concept of in situ seeding (Karpinski 2004, 2008, 2011) acknowledges that in many practical applications, nucleation is very difficult to control owing to its extremely high rate.

374

Figure 12.30 Approximate values of stresses generated by the collapse of cavitation bubbles Source: Reprinted with permission of the publisher from Baldyga et al. 2008. Copyright © 2008, Institution of Chemical Engineers and Elsevier.

Different process conditions – frequently unavoidable in large-scale operations – may result in very different populations of nuclei that inevitably lead to different final products. According to the model, illustrated in Figure 12.31, instead of trying to control rapid nucleation, the latter is allowed to occur at its own pace determined by the process conditions. Subsequently, the nucleation step is separated from the consecutive growth step by a distinct holding step, during the course of which no supersaturation is generated. Thus – depending on mode of operation – reagent addition is stopped, no further cooling takes place, or the addition of antisolvent is halted. During the holding step, owing to Ostwald ripening, nuclei population is evolving and establishes itself at the end as a quasi-stable set of “effective nuclei.” Certain measures such as • • • •

Dilution (addition of primary or secondary solvent), Solubility increase (temperature rise, addition of common ion), Addition of a ripening agent, and Other (e.g., temperature cycling)

aimed at manipulation of the in situ seed also can be applied during the holding step. At the end of the holding step, the CSD of the effective nuclei population is measured in-line and used in the same fashion as would be the CSD of external seed crystals to design, online, the course of the profile of a supersaturationgenerating property rate, in time t, for the control of the successive growth step. The supersaturation-generating property rate is the reagent addition rate, cooling rate, evaporation rate, and so on and depends on the properties of effective nuclei, their CSD, the growth rate, and time. As soon as this growth profile – see, e.g., Equations (12.116)–(12.118) – has been calculated, it is fed to the control unit of the crystallizer. Now generation of supersaturation resumes according to the growth profile just designed. The in situ seeding approach eliminates external

Batch Crystallization

Figure 12.31 Pictorial representation of the concept of in situ seeding Source: From Karpinski 2011.

seeding while removing the uncertainty of nucleation and significantly reducing batch-to-batch variability. In a simplified example, assuming that in situ seed population has Niss individual crystals with a monodisperse CSD and uniform size Liss, expressions for the cooling rate (in cooling crystallization), the evaporation rate (in evaporative crystallization), and the volumetric flow rate (in reactive crystallization) will now have the following form with regard to time, respectively: dT 3Niss kV ρc 3 2 ¼ ðG t þ 2G2 Liss t þ GL2iss Þ dt amsol

ð12:116Þ

dVsol 3Niss kV ρc 3 2 ¼ ðG t þ 2G2 Liss t þ GL2iss Þ dt CV

ð12:117Þ

Q¼

3Niss kV ρc 3 2 ðG t þ 2G2 Liss t þ GL2iss Þ MCR

ð12:118Þ

In all three of these equations, G = const < Gcrit. For CSD that is not monodisperse, this approach can still be useful if the average size of the in situ seed population is substituted for Liss. A similar concept was recently described by Abu Bakar et al. (2008).

12.6 Scale-Up of Batch Crystallization The problem of scale-up was considered in the context of precipitation in Section 8.8. It was argued that in such a complex process as precipitation, a complete scaling similarity is impossible. This applies to batch crystallization as well. The objective is to repeat the product quality (i.e., particle size, particle morphology), mixture composition, and structure of the suspension at a larger scale with the aim to obtain identical and not just similar products. Again, some similarity criteria that are in conflict with the more important ones have to be abandoned deliberately. Although successful scale-up ratios on the order of 1:100 are typically achievable in practice, it is not possible to maintain all the different scaling-up criteria, which assume geometric, kinematic, and dynamic similarities

(Mullin 2001). The exact scale-up of crystallizers is therefore impossible. This is best illustrated for the kinematic similarity criterion: mixing – whose principal role is to maintain solids in nearly homogeneous suspension and facilitate momentum, heat, and mass transfer without solids attrition – is considered to be intrinsically nonscalable. Historically, scale-up criteria for mixing were based on the constancy of stirrer speed, tip speed, or specific power input. The latter criterion is believed to ensure constant mass transfer throughout the scale-up to avoid excessive stirrer revolutions at larger scales. Maintaining a constant tip speed throughout the scale-up process may result in unacceptably low stirrer revolutions at a large scale. In contrast, maintaining a constant stirrer speed would lead to excessive stirrer revolutions at an industrial scale, excessive power consumption, and extreme solids attrition. For constant power input, assuming turbulent flow and similar geometry of the crystallizer, the stirrer revolutions Nsr2 at a larger scale characterized by volume V2 can be estimated based on Nsr1 and V1 of a small-scale crystallizer as follows (Brozio 2007):
2=9 V1 ð12:119Þ Nsr2 ¼ Nsr1 V2 According to Oldshue (1983), the three dimensionless numbers can then be considered for dynamic similarity of batch crystallization in the stirred tank: the Reynolds number, the Froude number, and the Weber number, defined as follows: Re ¼

Nd2 ρ N2d N 2 d3 ρ ; Fr ¼ ; and We ¼ μ g σ

ð12:120Þ

where Re represents the ratio of the inertial and viscous stresses, Fr is the ratio of the inertial to gravitational stresses, and We is the ratio of the inertial to surface tension stresses. This means that for fluids of the same properties to have hydrodynamic similarity at different scales, we need to have simultaneously Nd2 ¼ const:, N 2 d ¼ const:, and N 2 d3 ¼ const:, which is, of course, impossible. Moreover, to have the same mass transfer coefficients between the fluid and the particles or the same turbulent stresses acting on the particles, the power input per unit volume (P/V) should be kept the same in the systems of different scale. The agitation power can be expressed by the power number Po ¼

P N 3 d5 ρ

ð12:121Þ

which for developed turbulent agitation becomes independent of the Reynolds number. Thus, for geometrically similar agitated tanks, in the turbulent regime, P=V∝N 3 d2 , and the similarity criterion requires N 3 d2 ¼ const: Furthermore, because the impeller tip speed πNd determines the maximum shear rate, in order to maintain the same maximum shear rate at different scales, Nd ¼ const: Finally, to have the same values of the internal circulation time in the turbulent regime of agitation V=ðNd3 Þ, where Nd3 is proportional to the impeller pumping capacity, one needs to use an N = const. rule when scaling up. Because blending requires several circulations, the

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blending time is inversely proportional to N, and N = const. guarantees the same values of the blending time for both scales. Mersmann and Rennie (1995) suggested that for particles in suspension in a fluidized bed, the mean fluid velocity must be as high as the settling velocity of crystals in the swarm, which calls for constant tip speed of the impeller. They recommended the following scale-up criteria: for tank diameter to crystal size ratio T/L 1000, maintain the peripheral speed of the impeller πNd. Based on these rules, one can conclude that scaling up requires N1 d1α ¼ N2 d2α

ð12:122Þ

where 0 ≤ α ≤ 2 depending on the mechanism controlling the crystallization process. The value of the exponent α should be chosen for the most important mechanism. Needless to say, the secondary mechanisms will still affect – to some extent – the process and the resulting product quality. This is the price one pays for incomplete similarity; complete similarity nonetheless is impossible. In one industrial example (Brozio 2007), typical values of mixing characteristics were as follows: • • •

Specific power input: 0.4–2.5 kW/m3 Tip speed: 1–10 m/s (1–4 m/s for anchor mixer) Stirrer revolutions: 10–70 rpm (INTERMIG impeller in a 6-m3 crystallizer)

As far as heat transfer is concerned, the same source recommends a maximum cooling rate of 5–10 K/h. For mixed-suspension crystallizers, when agglomeration is less pronounced than attrition, an approach proposed by Gahn and Mersmann (1999a) – a growth and attrition model – is often applied. This approach is based on hydrodynamics represented by the local values of the rate of energy dissipation and local liquid velocity. The crystallization kinetics is represented by the integration rate constant kr, mass transfer coefficient kd, and the surface-related energy increase Γs that affects crystal solubility. The model assumes that the surface energy accumulated in crystals increases their solubility creal

 ΓS ¼ c exp RTL 




ð12:123Þ

where c* is the solubility of perfect crystals. This leads to a decrease in the driving force for crystal growth and, if true, can cause dissolution of small fragments produced by attrition. Thus secondary nucleation resulting from attrition controls the process, and the frequency and energy of crystal–impeller collisions can be calculated. Then the attrition volume and number and size distribution of attrition fragments produced on a single collision of a crystal with a flat surface of the impeller can be estimated. The model’s growth parameters kr and kd depend on temperature, whereas Γs is assumed to be constant for a given substance. Gahn and Mersmann (1999a, b) have shown that the two unknown growth parameters can be obtained from experimental data from the MSMPR draft-tube

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baffled (DTB ) crystallizer and used to predict the crystal size of potassium nitrate in a 580-dm3 forced-circulation (FC) crystallizer. The same model and similar procedures were applied afterward by many researchers, recently by Kalbasenka et al. (2011). A 75-dm3 draft-tube (DT) crystallizer was used to identify experimental parameters. The two-compartment model was used, with the first compartment representing the well-mixed volume of the crystallizer, and the second compartment is a zero-volume impeller compartment that acts as a source of secondary nuclei. The experimental design allowed a reduction in uncertainty in the initial conditions, namely, the mass and crystal size distribution of the initial population of crystals and the initial supersaturation. It was shown that the main reason for the discrepancy between model predictions and experimental data was the origin of nuclei, which shows that the phenomena considered are more complex than initially expected. The differences in the crystal surface properties, shape, and strain content – in batches that were initiated either by primary nucleation, seeding with small ground seeds, or seeding with coarse crystals – could be responsible for a divergent nucleation and growth behavior. Very recently, Westhoff and Kramer (2011) reported limitations in application of the crystallizer model framework of Neumann (2001) with the above-mentioned growth and attrition model. In particular, their experiments on potassium nitrate at 50°C in a 2-liter batch cooling crystallizer suggested that the model parameter Γs is not an independent constant for a given substance.

12.7 Summary Batch crystallization has several desirable features and advantages in both laboratory and industrial applications. Industrial batch crystallizers are commonly used to manufacture a wide variety of crystalline materials with desirable product features and quality. Laboratory batch crystallizers are often used to characterize crystallization kinetics and CSD and to determine the effects of process conditions on the kinetics and CSD. Over the past decade, off-the-shelf automated laboratory crystallizers with volumes from 50 ml to 5 liters, equipped with process analytical technology (PAT) tools, have become available. The analysis of batch crystallizers normally requires consideration of the time-dependent, batch conservation equations (e.g., population, mass, and energy balances) together with the appropriate nucleation and growth kinetic equations. The solution of these nonlinear partial differential equations is relatively difficult. Under certain conditions, these batch conservation equations can be solved numerically by a moment technique. Several simple and useful techniques to study the crystallization kinetics and CSD are discussed. These include the thermal response technique, the desupersaturation curve technique, the cumulative CSD method, and the characterization of CSD maxima. Several factors affecting batch crystallization are also discussed. These include batch cycle time, supersaturation profile, external seeding, fouling control, CSD control, growth-rate dispersions, mixing, polymorph control, and

Batch Crystallization

agglomeration. The batch cycle time information is needed to determine the crystallizer volume required for meeting the production-rate requirements. The supersaturation profile has a profound effect on the nucleation and growth processes and the resulting CSD. The supersaturation control during batch crystallization is beneficial for both manipulating crystal size and reducing the batch cycle time. The external seeding technique can be used to control the supersaturation and nucleation during the initial stage of the batch crystallization. In fact, external seeding can be used to eliminate or significantly reduce nucleation and control the growth stage of the process. External seeding can also be used to determine the maximum metastable supersaturation (or the maximum growth rate) allowable without causing nucleation in the batch crystallization process. Several techniques are used to minimize the fouling of crystals on the heat transfer surfaces during the batch run. Fines destruction and a strategy to separate nucleation and growth during the batch crystallization operation can be used to prepare crystal products with a narrow CSD. Growth-rate dispersions resulting from inherently different growth rates of individual crystals may significantly influence the CSD in batch crystallization. Several growth-rate-dispersion models were proposed to explain the widening of the CSD during the course of batch crystallization. Good mixing is required in a crystallizer not only to maintain crystals in suspension but also to ensure adequate rates of mass and energy transfer. Crystallizationspecific phenomena, such as secondary nucleation and

References Baliga, J. B. (1970). Crystal nucleation and growth kinetics in batch evaporative crystallization, Ph.D. thesis, Iowa State University, Ames, IA. Bałdyga, J., Jasińska, M., Krasiński, A., et al. (2001). In Proceedings of the 4th International Symposium on Mixing in Industrial Processes. Rugby: Institute of Chemical Engineers, pp. 61–69. Bałdyga, J., Jasińska, M., and Orciuch, W. (2003). Chem. Eng. Technol. 26(3): 334–40. Bałdyga, J., and Krasiński, A. (2005). In Proceedings of 16th International Symposium on Industrial Crystallization, Ulrich, J. (ed.). Düsseldorf: VDI-Verlag, pp. 411–16. Bałdyga, J., Makowski, Ł., Orciuch, W., Sauter, C., and Schuchmann, H. P. (2008). Chem. Eng. Res. Des. 86:1369–81. Bałdyga, J., and Orciuch, W. (2001). Chem. Eng. Sci. 56:2435–44. Bakar, M. R. A., Saleemi, A. N., Rielly, C. D., and Nagy, Z. K. (2008). In Proceedings of 17th International Symposium on Industrial Crystallization, Jansens, J. P., and Ulrich, J. (eds.), vol. 1. Maastricht: European

agglomeration, are also strongly affected by mixing. The inherent scale dependency of mixing and liquid–solid suspension dynamics further complicate rigorous description of the mixing effect on batch crystallization. Many substances, pharmaceuticals in particular, exhibit polymorphism, and polymorph control needs to be implemented. This requires good understanding of the polymorphic landscape and significant upfront research and development activities. The control of supersaturation must be employed in batch crystallization in order to obtain a desired CSD of the product. In the three common batch crystallization operations (i.e., cooling crystallization, evaporative crystallization, and antisolvent crystallization), such control may be realized by employing the appropriate time profiles for cooling rate, evaporation rate, and antisolvent addition rate, respectively. When external seeding is used and crystal growth occurs at a constant rate, mathematically similar expressions for cooling and evaporation-rate profiles can be derived. For an unseeded operation, the knowledge of growth and nucleation rates and the nuclei size distribution is necessary in order to obtain the time profiles via a population balance approach. Less common modes of batch crystallization, such as crystallization from supercritical solvents, pH swing crystallization, and sonocrystallization, are briefly discussed as well. Separation of nucleation and crystal growth is one of the means to control batch crystallization. How this can be done is illustrated through the concept of seeding in situ. Finally, historical and newer scale-up models for batch crystallization are discussed.

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Chapter

13

Crystallization in the Pharmaceutical Industry Simon N. Black AstraZeneca

13.1 Introduction Most crystallizations in the pharmaceutical industry are not carried out by crystallization scientists. The Cambridge Structure Database (CSD) contains over 850,000 crystal structures, and the number of organic molecules that have been isolated as solids is much larger. In many cases these isolations have not been repeated or scaled up. Yet this task, namely the development of robust, reproducible crystallization processes, is the main topic of this chapter. By way of introduction, meet the types of molecules, the types of people, and the nature of the industry. Typically, the compounds of interest have molecular weights (MWs) in the range of 200–700, are solids at room temperature, and are organic compounds containing C, H, O, and N. They may be salts or zwitterions and may contain other elements, most commonly F, Cl, or S. There is a great range of molecular properties such as flexibility, saturation, polarity, and ability to donate or accept hydrogen bonds. Of particular interest to the industry are active pharmaceutical ingredients (APIs), also referred to as the drug substance. APIs are the subject of much regulatory focus because the chemical and physical purity of the drug is easier to assess as a pure API than in the formulated medicine. The physical and chemical properties of an API also have a major influence on downstream processing and performance of the medicine. A patent typically offers an innovator pharmaceutical company exclusivity for 20 years. A successful medicine may have sales of over $10 million per day at the end of this period, when exclusivity ceases and other companies are free to make and sell generic versions of the medicine. For innovator pharmaceutical companies, it is important to proceed from patent to launch as swiftly as possible. However, in order to launch a product, it is necessary to show that it is safe and effective and offers an improvement over other medicines that are already available. This can only be achieved by a series of clinical trials, with each trial requiring increasing amounts of material. The development program is dictated by the timetable for clinical trials, with manufacturing “campaigns” timed so that they are not on the critical path. Hence, for an individual API, process development is not a smooth continuum but a series of campaigns – bursts of activity at increasing levels interspersed with quiet periods. Consistency of product through these campaigns is paramount, to ensure that the data acquired on safety, efficacy and benefit all refer to the same material. As an API proceeds through development, the probability that it will be successful increases, as does the investment to ensure that the manufacturing processes are as efficient and robust as possible.

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This sets the scene for process development in the pharmaceutical industry, through a series of campaigns at increasing scales, with increasing attention to process efficiency, robustness, and understanding. Historically, crystallization has been considered something of an art within the pharmaceutical industry. The crystallization scientist will be familiar with the extensive literature and detailed scientific understanding available from studies on the crystallization of metals, inorganic salts, dyes, energetic materials, and minerals (as discussed in detail elsewhere in this Handbook). The challenge for the crystallization scientist is to present the tenets of crystallization science in ways that are attractive and useful to medicinal chemists, process chemists, process engineers, solid-state analysts, materials scientists, formulation scientists, kilo lab and pilot plant scientists, patent agents, team managers, quality managers, biopharmacists, and project managers. The aim of this chapter is to increase mutual understanding between crystallization scientists and these various disciplines. Hence the chapter focuses on crystallization science that works in practice.

13.1.1 Why Perform a Crystallization Experiment? The aims of crystallization experiments vary, and a failure to appreciate this is a common reason for inappropriate advice being given or followed. The most difficult crystallizations are attempts to persuade a material to crystallize for the first time. Important features of such crystallizations are that the yield does not matter, and the process does not need to be scalable – allowing a wide range of techniques. Yield is also unimportant for screening experiments to find new polymorphs, solvates, or salts, but these are usually constrained by the need for high throughput, often via automation. Experiments to grow single crystals suitable for structure determination are characterized by low yields – one suitable crystal is a success – and long times. These topics are revisited in Section 13.8. Crystallization process development is quite different. The initial focus is usually on yield and chemical purity, with productivity and robustness of increasing importance during latter stages of development. The influence of crystallization on subsequent process steps – filtration, washing, drying, storage, and dissolution – can be profound. For APIs, the influence may extend to size reduction and formulation as well as product performance in vitro and in vivo. During the development of a new medicine, the key is consistency. Given changes in synthetic route and scale, this is a challenging objective. The following example is not unusual.

Crystallization in the Pharmaceutical Industry

A new development compound was initially isolated as the free base, but this failed to crystallize. A salt was identified by screening, and approximately 300 g were manufactured for initial toxicology studies. The process was repeated at 10 times the scale to prepare material for the first clinical studies. A new polymorph of the salt appeared at this stage, which was converted to the desired form. Subsequent work showed that this new form was metastable, so the final crystallization had to be redesigned to avoid it. Subsequent changes to the route and the discovery of crystalline forms of the free base necessitated further changes to the final crystallization. Later still, a solvate of the salt in the selected process solvent was identified, and the process was modified accordingly to avoid that. Crystallization science answered most of the questions raised during this development. The best return on investment in crystallization science is in the development of robust processes for APIs. Improved processes for the isolation of intermediates in the synthetic route inevitably follow once these skills are in place. In this chapter, the initial focus is on recrystallization of an API from a single solvent. The principles developed for this simple system are then applied to progressively more complicated situations.

13.2 Simple Systems One requirement for the final crystallization of an API is that all the product must dissolve so that the solution can be passed through a filter (sometimes called a polish filter or a screen) to remove any extraneous solids before crystallization is initiated. This is a regulatory requirement and places an important constraint on crystallization process development. In the early stages of drug development, a typical challenge is to develop a recrystallization process for an API that includes such a screening step.

Inspection of this solubility curve shows that 250 g can be dissolved in 1 liter of solvent at 77°C, and 90 percent of this can be recovered after cooling and equilibrating at 5°C. This process has high productivity and good yield. The yield could be improved still further by cooling to lower temperatures or by adding antisolvent. In both cases, the risk that impurities will crystallize out with the product is increased. The addition of antisolvent will decrease the productivity. In general, both productivity and yield will be increased by increasing the dissolution temperature. This introduces risks associated with crystallization on the screen and difficulties in agitating concentrated slurries. In this particular case, a more significant restriction is the boiling point of the solvent (81°C), which could be increased by operating at increased pressure. An additional merit of solubility data is that they quantify the supersaturation – the distance from equilibrium (Davey and Garside 2000). In this chapter, supersaturation is defined as

S = (Css – Cs)/Cs where Css is the concentration of the supersaturated solution, and Cs is the equilibrium solubility at the temperature of the supersaturated solution. As can be seen from Figure 13.1, it is difficult to access supersaturations greater than 10 by cooling. The free energy that drives crystallization is given by RT ln (S + 1), where R is the gas constant and T is the absolute temperature. The power of solubility data is that they are thermodynamic and hence independent of scale. The data shown here were determined using a few grams of material, and the process was operated to produce over 200 kg of material at a yield of 86 percent, close to the expected yield. This illustrates the first fundamental tenet of crystallization process development – measure solubility data.

13.2.1 Yield and Productivity Yield is of paramount importance in process development. Consider a 10-step synthesis in which the yield of each step is 80 percent. The total yield is 11 percent. Increasing the yield of each step to 90 percent increases the overall yield to 35 percent. Yield is particularly important in the final step of the synthesis. The major cost of this step is the cost of making the input material that is not recovered. Typical yields are between 80 and 95 percent for final crystallizations, sometimes referred to as pures processes. Imagine a process in which 10 g of solute is dissolved in 1 liter of solvent and 9 g is crystallized out. The yield, at 90 percent, is acceptable. However, the productivity of this process is low because 100 liters of reactor are required for each 1 kg of product. A process in which 100 g of solute is dissolved in 1 liter of solvent and 90 g is recovered is preferred, with acceptable yield and productivity. Yield and productivity are determined by the solubility curve and the temperature of operation. This is illustrated in Figure 13.1, which shows solubility data for an API in acetonitrile (Black et al. 2006).

Figure 13.1 Solubility data for an API in acetonitrile Source: Adapted with permission of the publisher from Black et al. 2006. Copyright © 2006, American Chemical Society.

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purposes, for example, in the selection of solvents for reactions, cleaning solvents, solvents for analytical methods, and solvents for automated screening protocols. The initial requirement in all these cases is to find solvent(s) in which the compound will dissolve. For recrystallization, a second requirement is for suitable variation of solubility with temperature. Before proceeding with methods for solvent selection, it is worth surveying the solubility landscape. Figure 13.2 Solubility equilibrium Source: Adapted with permission of the publisher from Black et al. 2013. Copyright © 2013, American Chemical Society.

13.2.2 Solubility Data There are three golden rules for solubility measurement: 1. Measure the temperature. 2. Achieve equilibrium. 3. Check the solid phase. Two different types of solubility measurement are in common use. The first, preferred by this author, is the excess solid or slurry method (Figure 13.2), in which the solution is sampled and analyzed by nuclear magnetic resonance (NMR), highpressure liquid chromatography (HPLC), or gravimetric analysis, and the solid is checked by powder X-ray diffraction. The major disadvantage of this method is the need to separate solid and liquid phases, normally by filtration. This is problematic at elevated temperatures. An alternative excess liquid method is to use visual inspection or light scattering to detect a clear point on heating a slurry of known composition. This method assumes that the solid phase dissolving at the clear point is that same as the one that was added to the vessel at the start, as well as that the heating rate is sufficiently slow to prevent any overshoot. Trace amounts of highly insoluble impurities (brick dust) can cause this method to give misleading results (Black et al. 2013). The need for solubility data as a first step in designing crystallization processes may seem obvious, but it rests on an important assumption that is not always accepted. This is that solubility is not a function of purity. The alternative view – that solubility varies with purity – implies that solubility varies from campaign to campaign. Inevitably, there are upstream changes in starting materials, synthetic routes, and process details that will lead to changes in the impurity profile of the material to be crystallized. This justifies a linear development timetable, where work on the final crystallization will not start until “typical” input material is available from the preceding step. It is common for timelines to become compressed owing to difficulties in the synthetic route, with the result that there may only be weeks available to develop the final step. It is then difficult to ensure that the correct emphasis is given to obtaining the desired polymorph and particle properties. The recommended way of working is to accumulate solubility data for each API as soon as possible and treat these data as valid for all campaigns. This allows work to start on the final crystallization step in parallel with work on earlier steps in the synthetic route. Solubility data are also required for other

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13.2.3 Solvent Ranking A huge selection of solvents is available for use by the synthetic chemist. Table 13.1 lists common solvents used in final crystallizations of APIs, along with useful physical properties (Reichardt 2004). As can be seen from the table, most solvents have molecular weights in the range of 40–120 and densities of 0.75–1.1 g/cm3 and contain at least one oxygen atom and one carbon atom. There are considerable variations in polarity and ability to donate and accept hydrogen bonds. There are many different schemes in the literature for classifying solvent into different types. In the table, the solvents are ranked according to their dielectric constant, which correlates roughly with the number of moles per liter. Solubility may be considered as the transfer of solute from the solid to the liquid state, followed by the mixing of that liquid with the solvent. Thus solubility is determined by the balance of two factors – the free energy change when the solid melts (which is independent of solvent) and the free energy of mixing of solute and solvent molecules. As pointed out by Atkins and de Paula (2002), this is the phenomenon that accounts for the lowering of melting point by impurities. A common piece of advice for solvent selection is “like dissolves like” (Leonard et al. 2013). This implies that solute and solvent molecules are equally content in their own company or each others’, so the free energy of mixing depends only on the mole fraction. As stated earlier, solubility is usually quoted in moles or grams per liter of solvent. Hence, in the absence of any specific solute–solvent interactions, the solubility of a given solute will vary with the number of moles per liter of solvent. This is one contribution to the efficacy of solvents, setting expectations that, for example, solubilities in ethanol will generally be higher than solubilities in 1-butanol. Most organic molecules encountered during pharmaceutical process development are polar. One interpretation of “like dissolves like” is that such molecules will be more soluble in polar solvents than in nonpolar ones. The dielectric constant of a solvent is a macroscopic measure that correlates with polarity. High dielectric constants would also be expected to be advantageous for solubilizing charged species such as zwitterions and dissociated salts. This sets a general expectation that solubilities will generally be higher in some solvents than in others for polar pharmaceutical development molecules, following the ranking given in Table 13.1. Figure 13.3 shows solubility data for a polar API in 10 common solvents. Qualitatively, this solvent ranking correlates roughly with the preceding expectations. The glaring exception is water, in which this API is less soluble than in all organic

Crystallization in the Pharmaceutical Industry Table 13.1 Common Recrystallization Solvents

Solvent

Molecular weight

Density, g/cm3

Boiling point, °C

Dielectric constant

Moles per liter

ICH class

Water

18.02

1.00

100

78.3

55.5

3

Dimethyl sulfoxide (DMSO)

78.14

1.10

189

46.5

14.1

3

Acetonitrile

41.05

0.78

82

35.9

19.0

2

Methanol

32.04

0.79

65

32.7

24.7

2

N-Methyl pyrrolidone (NMP)

99.13

1.03

202

32.2

10.4

2

Ethanol

46.07

0.79

78

24.6

17.2

3

Acetone

58.08

0.79

56

20.6

13.6

3

1-Propanol

60.1

0.80

97

20.5

13.3

3

2-Propanol (IPA)

60.1

0.78

82

19.9

13.0

3

2-Butanone (MEK)

72.1

0.80

80

18.5

11.1

3

1-Butanol

74.12

0.81

118

17.5

10.9

3

2-Butanol (racemic)

74.12

0.8

99

16.56

10.8

3

Methylisobutylketone

100.16

0.8

116

13.1

8.0

3

Tetrahydrofuran (THF)

72.11

0.89

67

7.6

12.3

3

Ethyl acetate

88.11

0.89

77

6.0

10.1

3

Isopropyl acetate

102.13

0.87

89

6.0

8.5

3

Butyl acetate

116.16

0.88

126

5.01

7.6

3

Anisole

108.14

1.00

152

4.33

9.3

3

Methyl-t-butyl ether (MTBE)

88.15

0.74

55

4.2

8.4

3

Toluene

92.14

0.86

111

2.38

9.3

2

Heptane

100.2

0.68

98

1.92

6.8

3

solvents other than alkanes. This is probably explained by the unusually strong affinity of water for itself. The data in Figure 13.3 are extremely useful for process design. For this API, DMSO and MEK will be suitable cleaning solvents, but the solubilities are too high for crystallizations with high yields. Recrystallizations from anisole, MTBE, water, or heptane will require large solvent volumes and have very low productivities. The other four solvents may be suitable recrystallization solvents. Solubility may be modified by the addition of a second solvent. Figure 13.4 shows two examples of how solubility may vary as the ratio of solvents changes. If the solvents are chemically similar (e.g., ethanol and 2-propanol), then solubility often varies linearly with solvent ratio (dotted line). If the solvents are chemically very different (e.g., ethanol and water), the variation in solubility may be nonlinear and may pass through a maximum (dashed line).

13.2.4 Solubility and Temperature The yield from a cooling crystallization is closely linked to the variation in solubility with temperature. A full account of the

relevant theory (Muller et al. 2009) shows that if the enthalpy and entropy of dissolution are constant over the relevant temperature range, then the variation in solubility can be described by S = c exp(0.693/ΔTsd) where S is the solubility in grams per liter, T is the temperature in °C, and c is the solubility at 0°C. ΔTsd is the solubility doubling temperature, which is related to the enthalpy of dissolution ΔTsd × ΔHd ≈ 500 K · kJ/mol ΔTsd is a convenient way to link solubility data to yield. For example, in Figure 13.1, the curve is an exponential fit to the four experimental data points, with R2 = 0.996 and ΔTsd = 21.2°C. It follows that the yield from a cooling crystallization over a temperature range of 3 × ΔTsd (= 63.6°C) should be 87.5 percent. In recent studies of 160 data sets, an average value for the solubility doubling temperature of 20°C was reported (Muller

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Simon N. Black Figure 13.3 Solubility (grams per liter) of a polar API in some organic solvents at 25°C

Figure 13.4 How solubility varies with solvent ratio (schematic)

et al. 2009; Black and Muller 2010). However, ΔTsd varies considerably, even for the same solute (e.g., from 16–27°C for succinic acid [Yu, et al. 2009] and 16–33°C for fumaric acid [Dang et al. 2009]). The author has not encountered any examples of inverse solubility (decreasing with increasing temperature) for pharmaceutical development compounds, although a few examples are known for inorganic salts (see table A4 in Mullin 2001). The available temperature range will be restricted by the melting and boiling points of the solvent. For the solute in Figure 13.3, the solubility in ethanol is double the solubility in 1-butanol. However, the higher boiling point of n-butanol (118°C compared with 78°C) means that solubility at reflux may be considerably higher in n-butanol than in ethanol. Conversely, acetic acid and DMSO are examples of solvents that may freeze in winter (or at other times in some parts of the world).

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13.2.5 Solvent Selection Solvents perform several useful functions, including heat and mass transfer, removal of impurities, promotion of desired reactions, and inhibition of undesired reactions. Safety, environmental, and cost concerns will also influence solvent selection. Chemical knowledge is essential to avoid choosing solvents that react with solutes, for example, primary alcohols with carboxylic acids or ketones with amides. Personal preferences of the development scientist may also influence solvent selection. For final crystallizations of an API, an additional concern is the need for acceptable levels of solvent in the product. This is ensured by the International Conference on Harmonisation of Technical Requirements for Registration of Pharmaceuticals for Human Use (ICH) classification scheme (ICH 2011), which lists 30 Class 3 solvents that can be present in APIs at levels up

Crystallization in the Pharmaceutical Industry

to 0.5 percent and for which standard analytical methods are available. The use of Class 2 solvents is by exception, owing to the additional analytical burden and the increased technical difficulty in drying to the lower levels required. Fortunately, it is rarely absolutely necessary to go beyond Class 3 solvents. Solvent classifications change over time, so it is important to consult current ICH guidance. The yield and productivity of a cooling crystallization are both dictated by the temperature–solubility curve. The link between yield and ΔTsd has already been exemplified – a temperature drop of ΔTsd × 3 corresponds to a yield of 87.5 percent. The productivity is determined by how much solvent is required to dissolve the solute at elevated temperatures. This is often quoted in “vols” or “rel vols,” this being the amount of solvent (in liters) required to dissolve 1 kg of solid. Productivities are typically 5–20 rel vols, and there may be considerable cost benefits (batch size and number) in reducing the number of rel vols. A useful procedure is to screen 10 Class 3 solvents of diverse types at 20°C (as in Figure 13.3). The slurry method is preferred at this stage because it also allows for a check for polymorph/solvate formation. Solvents that give solubilities in the range of 5–20 mg/ml are taken forward to the next stage. If no suitable solvents are found, the screen may be repeated with other solvents or with mixtures of solvents. In many cases, interpolation from plots such as Figure 13.4 may be used to suggest solvent mixtures that will have solubilities in the desired range. For each suitable solvent with solubility of 5–20 mg/ml, a cooling crystallization is attempted: 10 rel vols of solvent and heat until dissolution is complete, adding measured extra amounts of solvent if necessary. The solution is cooled to 10° C below the dissolution point, seeded, and allowed to equilibrate. After sampling the liquid phase, the suspension is cooled to 0°C. The solute levels in the solvent (losses to liquors) are determined, and the solid phase is characterized for purity and polymorph. The use of in-line technology (turbidity, Lasentec focused-beam reflectance measurement [FBRM]; see below) to track the behavior of suspended solids in this experiment is extremely helpful. This experiment reveals a wealth of information about each solute–solvent system. Yield and productivity data are obtained directly. At this scale, information about yield is best deduced from the amount of solute remaining in solution rather than the weight of solid recovered. From the four solubility–temperature data points, the solubility curve may be plotted and the data fitted to the exponential equation. This allows the dissolution and isolation temperatures to be optimized. The ability of the seed to control the final polymorph and the particle size is tested, as is the chemical purification in the process. The product may be forwarded to formulation colleagues for a preliminary assessment. If none of the solvents gives acceptable product, then the solubility screen may be attempted again. Antisolvent crystallizations are popular because they are quick and often give high yields. They suffer from many disadvantages, as discussed in more detail later. Hence cooling

crystallizations are recommended wherever possible, but particularly when control of polymorph and particle size is required. Hence the above-mentioned screening protocols are designed to identify cooling crystallizations first. Fortunately for the crystallization scientist, life is not always this simple. So far we have considered systems containing an API and one solvent. Some systems are more complicated, and the most common complication is polymorphism.

13.3 Increasing Complexity So far the discussion has focused on single solute and a single solvent. The phase rule of Gibbs offers a convenient way to characterize and group more complex systems (Ricci 1966). In the formal language of the phase rule, Figure 13.1 shows a solubility curve for a system containing two components – the solute and the solvent – and two phases – liquid and solid. The composition of the solid phase is fixed because it cannot contain any solvent, whereas the composition of the liquid phase is variable. Where the liquid phase is in equilibrium with the solid phase, the composition of the liquid phase is fixed at any given temperature, as shown by the solubility curve. This treatment is also valid if the liquid contains two solvents in a fixed ratio, neither of which can form solvates. It can also be applied to recrystallization of salts and cocrystals provided that only one solid can crystallize. In most cases, additional components must be treated explicitly. However, the most common complication is where a single component can exist in more than one solid phase – as in polymorphism, solvates, and hydrates.

13.3.1 Polymorphs It is possible for a single component to exist in more than one solid phase. This phenomenon, polymorphism, has been known for over 200 years, since the discovery by Tennant (1797) that diamond and graphite were both forms of carbon. Many minerals exist as polymorphs; for example, calcium carbonate exits as aragonite, vaterite, and calcite, and many organic systems exhibit the ability to control this polymorphism to ensure that, for example, eggshells are made of calcite. There are several examples of the industrial significance of polymorphism, for example, the importance of selecting the correct polymorph of titanium dioxide as a whitener in paints (rutile) and fibers (anatase; Clegg et al. 2001). Polymorphism in organic solids is well known in dyes (different colors) and explosives (different detonation properties; Foltz et al. 1994). The significance of polymorphism in the pharmaceutical industry is exemplified by ritonavir (Norvir) and ranitidine HCl (Zantac). In the former case, a formulation of a launched product started to fail stability tests as a result of crystallization of a new polymorph. The full story is told in two excellent articles (Chemburkar et al. 2000; Bauer et al. 2001). In the case of Zantac, the discovery of a second polymorph led to the granting of a second patent (Bernstein 2002). These two cases in particular increased interest within the pharmaceutical industry in finding new polymorphs of APIs that could be

385

Simon N. Black

160 140

Concentration (g/liter)

120 100 80 60 40 20 0 0

10

20

30

40

50

60

70

80

Temperature (°C) Figure 13.5 Solubility curves for a monotropic system

patentable and might affect the processing and properties of the drug. “Same in the liquid and gas phase, different in the solid phase” is a useful definition of polymorphs. Each solid phase will have its own solubility curve. It follows that a crystallization process, or a formulation, designed using solubility data for one polymorph may not work for another one. There are many other differences between polymorphs that can influence the preparation, processing, and performance of APIs. Fuller accounts are given in several textbooks (Byrn et al. 1999; Brittain 2009; Hilfiker 2006; Bernstein 2002). For the development of crystallization processes, the key consequences are 1. The process must make the required polymorph robustly. The simplest way to achieve this is to use seeded cooling crystallizations. 2. If the desired polymorph changes, so must the crystallization process to make it, based on the appropriate solubility data. Figure 13.5 shows solubility curves for two polymorphs that do not cross. The polymorphs are related monotropically. By definition, the more stable polymorph has lower solubility. A characteristic of monotropic systems is that the curves become closer as temperature increases. In the region between the two solubility curves, crystals of the more stable polymorph will grow, and crystals of the less stable polymorph will dissolve. This illustrates a further important use for solubility plots such as Figure 13.5, which is that they describe system compositions that will give only the desired solid. The robustness of this process is linked to the size of the gap between the two polymorphs, which is quantified by the ratio of their solubilities. Figure 13.6 is a representation of the data from a literature compilation (Pudipeddi and Serajuddin 2005) of values of this solubility ratio for different polymorphs.

386

As can be seen from this graph, the ratio of approximately 5 that was observed for Norvir (Chemburkar et al. 2000) is unusually high. In designing crystallization processes, a solubility ratio of 1.5 or more generally corresponds to a temperature gap of greater than 10°C, which is sufficient for a robust seeding strategy. By definition, processes for making metastable polymorphs rely on kinetics – as discussed further later. The solubility ratio (strictly the ratio of activities; Svard et al. 2010) is a function of the relative energies of the two solids, and it is independent of solvent. However, as exemplified in Figure 13.5, the ratio can and usually does vary with temperature. In some cases, the solubility curves cross each other, as shown in Figure 13.7. Such enantiotropic cases are characterized by a transition temperature below which one polymorph is stable and above which the order of stability is reversed. The existence of no more than one transition temperature for any pair of polymorphs is a direct consequence of the phase rule. A review of literature data (Yu 1995) for 96 polymorphic compounds showed that in only 14 cases was there an enantiotropic transition temperature within the range 0–100°C. The definitive test for enantiotropy is reversibility, which may be difficult to establish. As the name implies, enantiotropy describes the relationship between two polymorphs, which can be a clumsy concept for systems with three or more polymorphs. It is usually only necessary to establish the transition temperature if it lies within the operating temperature range, and this is best done by performing slurries at various temperatures. If the transition temperature is within the range of storage and body temperatures, then specific precautions may be necessary to preserve the desired polymorph. If the transition temperature is between ambient conditions and the temperatures used in accelerated storage tests (e.g., 45°C), then these tests may give misleading results.

13.3.2 Hydrates Unlike enantiotropy, hydrates are very common. In the presence of only water, the solubilities of hydrates and neat (nonhydrated or solvated) forms can be plotted on a simple solubility curve. Higher hydrates contain more water than lower hydrates or neat forms and have steeper solubility curves. This phenomenon is well known for inorganic salts (Mullin 2001) and is related to the greater enthalpies of dissolution of higher hydrates. Thus the solubility curves for a hydrate and the corresponding anhydrous form in water are similar to those of enantiotropic systems (see Figure 13.7), and the transition temperature is a useful parameter in process development (Black et al. 2009). Similar considerations apply to the stability of solvates in their pure solvent. It is possible to exclude water from a process. It is much harder to exclude water during process development because the necessary experimental procedures such as dry nitrogen blanketing become more irksome at smaller scales. However, the API will meet water again, as water vapor, during processes such as wet granulation and some tablet coating methods, in some excipients, and in the body. If one or more hydrates exist, the risks associated with hydrate formation need to be evaluated.

Crystallization in the Pharmaceutical Industry Figure 13.6 Solubility ratios for polymorphs – cumulative frequency plot (line is a guide to the eye)

Table 13.2 Relative Stability and Solubility of Two Forms That Are Not Polymorphs

160 140

Solvent (v/v)

T(°C)

Stable form

Solubility ratio

Percent water

Water

25

B

10

100

Water + 15 % DMSO

25

B

10

85

Water + 50 % acetone

25

B

9

50

40

Water + 50 % acetonitrile

25

B

6

50

20

Water + 50 % acetonitrile

75

B

4

50

Water + 85 % DMSO

25

A

1

15

Anisole

75

A

1

0

Concentration (g/liter)

120 100 80 60

0 0

10

20

30

40

50

60

70

80

Temperature (°C) Figure 13.7 Solubility curves for an enantiotropic system

Where a hydrate and a neat form exist, at a given temperature there will be a critical relative humidity (RH) above which the hydrate is stable and below which the anhydrous form is stable (Zhu and Grant 1996; Zhu et al. 1996). This critical RH will increase with temperature, reaching 100 percent at the transition temperature in pure water. Knowledge of this critical RH is helpful for the design of drying processes, as well as for crystallizations from mixed solvents, because the critical RH is directly related to the critical water activity in such systems, as discussed in more detail later. It is common practice to use gravimetric vapor sorption (GVS, sometimes dynamic vapor sorption [DVS]) to evaluate this RH. As in true enantiotropy, reversibility is the key to establishing that the critical RH has been identified. Table 13.2 shows the solubility assessment of two forms of a pharmaceutical intermediate. It is immediately clear that this behavior is not consistent with the assertion that the relative

solubility of true polymorphs is independent of solvents. However, analysis of the two forms showed that both contained less than 1 percent water. The samples had not (as is sometimes the case) been dried in an oven prior to analysis. If the data are ranked in terms of water content in the solvent, as in the table, it immediately becomes clear that form B must be some sort of hydrate because it becomes less soluble (i.e., more stable) as the water content increases. A more detailed investigation revealed that this was a channel hydrate, in which the water was loosely bound and evaporated from the solid during drying. Thus far the discussion has considered the thermodynamics of two-component systems – a single solute and a single solvent, where polymorphism, hydrates, and solvates may introduce additional solid phases. The next complication to

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Simon N. Black

consider is the effect of adding an additional component. This may be another solvent or another solute.

13.3.3 Two Solvents The use of solvent mixtures to obtain a desired solubility– temperature profile has already been discussed. If the solvent ratio is fixed throughout the process, the system can be treated in the same way as a single solvent. Specifically, solubility– temperature curves such as those shown in Figure 13.1 describe the thermodynamics of the system. It is common practice to add a second solvent (an antisolvent) in which the product has poor solubility in order to isolate it as a solid. Generally, this procedure works quickly and gives high yield – two attributes of high importance to the bench synthetic chemist. Moreover, identifying a suitable antisolvent is not difficult. For example, the data in Figure 13.3 suggest either water or heptanes as antisolvents, not unusually. There are several disadvantages to antisolvent crystallizations, and these grow in importance as processes move from the bench to larger scales. The most significant difference is the increased sensitivity to mixing. At the point at which the antisolvent first makes contact with the solution, the local supersaturation may be extremely high. In the example given in Figure 13.3, the solubilities in water and heptanes are approximately 0.1 mg/ml. At the addition of antisolvent, supersaturations of up to 100–1000 times are accessible – compared with less than 10 for cooling crystallizations. Furthermore, as the solvents mix, the supersaturation falls at a rate determined by the mixing. This is the underlying cause of numerous scale-up problems. Differences in chemical purity, polymorph, filtration, drying, and downstream processing often can be traced back to differences in mixing during antisolvent additions. The following general advice is offered when one is confronted with antisolvent crystallizations: •

• • •

Avoid. Is it possible to achieve the desired result by cooling or by a combination of adding an antisolvent to reach saturation and then cooling? If the yield is still too low, add more antisolvent after cooling when the available crystal surface area is largest. Test the mixing sensitivity by performing rapid forward and reverse additions. If the outcomes are significantly different, then there is a scale-up problem. Use baffles to improve the mixing. The often-quoted advice to add the antisolvent into the vortex does not alter the fact that a vortex is evidence of poor mixing. Seed (see below). However, finding a seed point is more difficult than for cooling crystallizations.

The second disadvantage of antisolvent crystallizations is their inherent complexity. This becomes apparent when plotting solubility data for such systems. Figure 13.4 indicates some of the issues that arise. There are no set rules for how solubility changes with solvent compositions. In some cases, solubility may be extremely sensitive to small amounts of antisolvent or cosolvent, particularly if this is water. Care is also needed with the units used for solvent ratios. Mole fractions, weight ratios, and volume ratios are all used commonly, with the later being particularly popular.

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Still greater care is necessary in using solubility data to design crystallization processes and in setting operating ranges. For example, consider the design of an antisolvent crystallization for a compound with a solubility of 100 g/liter at a solvent ratio of 9:1 by volume and 10 g/liter at a solvent ratio of 1:1. Adding sufficient antisolvent to reach a solvent ratio of 1:1 does not give a yield of 90 percent. In fact, 800 ml of antisolvent must be added to 1 liter to achieve a 1:1 mixture. The total volume is now 1.8 liters. This will dissolve 18 g, so the yield is 82 percent, not 90 percent, and the productivity also will be reduced. The underlying problem here is that solubility plots such as Figure 13.4 are not capable of representing compositions of a system containing three components.

13.3.4 Oiling One further problem with antisolvent addition is the possibility of oiling. This confusing term covers many different phenomena, one of which is the thermodynamic separation of one liquid phase into two liquid phases. For binary solvent mixtures, the phase behavior is well known and available in lookup tables or with physical properties software. As a rough guide, this type of phase separation in the absence of either water or an alkane is rare. This phase behavior will be influenced by the solute. While such phase separation is common, it may disappear when seed is added. In one case study, this effect was found to depend on temperature. The picture in Figure 13.8 was taken at room temperature. Seed crystals are growing in the continuous phase, from the left of the image, by transfer of solute from the drops into the continuous phase. The two solvents are miscible when not supersaturated with solute, giving rise to the halo effect, as reported previously by Bonnett et al. (2003). This slide was prepared by sampling the two-phase mixture at an elevated temperature and allowing it to cool on a microscope slide. When the oil was cooled and seeded, crystallization ensued. Heating the slurry up again resulted in rapid dissolution of the solids and a reappearance of the second liquid phase. At elevated

Figure 13.8 Crystallization after oiling. Note the halo effect.

Crystallization in the Pharmaceutical Industry

temperatures, this was stable to seeding. It should be noted that nucleation occurred in the continuous phase in this case. Had nucleation occurred in the oil phase, the solid product would have been spheres. Although supersaturations are the same in both phases, the concentrations of impurities may not be.

13.3.5 Solvates and Hydrates Again As noted earlier, conventional concentration versus temperature plots may be difficult to interpret for systems in which two solvents are present in varying ratios. The underlying reason for this is the phase rule (Ricci 1966). When the number of components in the system is increased to three, it is no longer possible to express the complete variation of solubility with temperature and composition on a flat sheet of paper. Instead, isothermal ternary-phase diagrams are the tool of choice for understanding such systems. Although it is rare to determine such diagrams accurately in the pharmaceutical industry, schematic diagrams are extremely helpful in understanding how such systems behave and in guiding process development toward conditions that will yield pure solid products. There are two nonintuitive hurdles to be overcome before phase diagrams become familiar. The first is the distinction between a component and a phase. The solubility curves in Figures 13.1, 13.5, and 13.7 are extracts from the phase diagrams for two-component systems and delineate how a system will respond at equilibrium to changes in composition and temperature. Note that systems that contain two components (solute and solvent) may contain many more phases (polymorphs and solvates as well as solutions). The second hurdle is the use of triangular coordinates. These will be familiar to chemical engineers from their understanding of distillations but can cause confusion and even fear among some scientists and should be introduced with care. Figure 13.9 shows a system containing one solid and two liquid components, with the apex representing 100 percent solid. One of the liquids is water, and the system forms a

Figure 13.9 Schematic phase diagram for a system containing water, an organic solvent, and a solid that forms a hydrate Source: Reproduced with permission of the publisher from Black et al. 2009. Copyright © 2009, American Chemical Society.

hydrate. The diagram displays a number of important features of such systems: • • •

Both hydrate and anhydrate can coexist in equilibrium over a range of system compositions. Where hydrate and anhydrate coexist, the composition of the solution is constant. In concentrated slurries, significant amounts of water may be present in the solid phase.

This approach has been used to understand the phase behavior of formoterol fumarate (Jarring et al. 2006). If the diagram is plotted using mole fractions, a 1:1 solvate–hydrate will appear halfway down one of the sides. A quantitative phase diagram such as that showing carbamazepine and its dehydrate in ethanol–water, plotted in grams per liter, may be more helpful for process understanding (Li et al. 2008). In another example (Black et al. 2009), the critical water activity was determined and corresponded to a water level of less than1 percent in the solvent system for the crystallization. This created a demanding technical challenge to reduce the water activity from unity following the separation of an aqueous layer, which was overcome by understanding and controlling the distillation that preceded the crystallization.

13.3.6 Impurities When a solution is spray dried or evaporated to dryness, all the nonvolatile impurities are isolated with the product in the solid phase. In crystallization, soluble impurities are removed at filtration with the liquid phase. This combination of phase separation and purification in a single technology is responsible for much of crystallization’s popularity. Yet the fundamental reasons for this purification are often overlooked. Consider the example shown in Figure 13.1, and imagine that the solid contains an additional 5 percent of an impurity. If the impurity has a similar solubility to the product and does not affect the solubility of the product, then how much impurity will crystallize with the product if the yield is 90 percent? Process-development scientists typically will give answers in the region of 1 percent. However, the answer to the question, as posed, is definitively 0 percent. If 10 percent of the product remains in solution, then so should a similar mass of the impurity. In practice, the removal of impurities may be complicated, as shown in Figure 13.10. Very insoluble impurities may crystallize separately (Wood 2001), as indicated by the black rectangle (3). Impurities also can be adsorbed onto surfaces of aggregates (1) or crystals (2) or be present in inclusions within crystals (5) or occlusions between crystals in aggregates. The least favorable possibility is that the impurity forms a partial or complete solid solution with the product (4), possibly restricted to certain growth sectors (6). Judicious washing or slurry experiments followed by analysis will distinguish among these options. If the level of impurity is reduced by washing, then it was adsorbed on the surface. If the level of impurity is reduced by grinding the sample and then washing it, then the impurity is included or occluded. In these cases, seeding and control of supersaturation often result in significant improvements to purity. If the

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Figure 13.10 The fate of impurities: (1) adsorbed on aggregate surface, (2) adsorbed in internal surfaces, (3) separate crystals, (4) distributed evenly through host crystals, (5) occluded in host crystals, (6) included in selected growth sectors

impurity level increases after washing, then the impurity is insoluble in the wash solvent, and washing with an alternative solvent may remove it. If the impurity level remains unchanged after washing or slurrying, this indicates a partial/complete solid solution. In this case, the only crystallization option remaining is to isolate a different crystalline form of the product – e.g., a salt or a hydrate (Black et al. 2004). If an impurity is present in large amounts, it can be helpful to treat the impurity as a third component in the system and sketch that ternary phase diagram (Figure 13.11). A common example is diastereosiomers, typically present in solution in equal amounts. Figure 13.11 clarifies that the ability of a crystallization process to separate diastereoisomers depends on their relative solubilities and works best when the desired diastereoisomer A is much less soluble than the undesired diastereoisomer B. The key parameter is the eutectic – the solution composition at which the solubility is at maximum. If the solubilities of the two diastereoisomers are independent (as shown in the figure), then the eutectic can be estimated directly from the solubilities of A and B. A counterintuitive property of system compositions in which both A and B are present as solids is that the composition of the solution (the eutectic, or maximum solubility) does not change. If A and B are diastereoisomeric salts, then the common ion effect needs to be considered. The figure also indicates how very insoluble impurities can affect certain solubility measurements. In an experiment designed to measure the solubility of B, the presence of a less soluble material A can lead to misleading results because a suspension can exist in which the only solid present is A (Black et al. 2013). This can be particularly troublesome if A has not been identified – a common enough occurrence with insoluble impurities in the early stages of development. Impurities can have other effects on solubility measurement. The presence of a different polymorph, with a different solubility, could be related to impurity. Impurities that affect the pH could have dramatic effects on the solubility of ionizable species. Small amounts of water can have dramatic effects

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Figure 13.11 Schematic phase diagram for two compounds with different solubilities

on solubility, particularly in solvents such as ethyl acetate, in which water is very active. However, careful solubility measurements (see Section 13.2.2) should give solubility data that do not vary with purity. The author is not aware of any counterexample in the literature.

13.3.7 Racemates, Cocrystals, and Salts One assumption made in Figure 13.11 is that A and B do not combine to form a solid compound. In many cases they do, and often this compound is the desired product. Here phase diagrams are a useful way of defining the system compositions that will yield only the desired solid product, avoiding the other solids that pose risks to the process. Racemates, cocrystals, and salts are common examples, for which the corresponding phase diagrams all have the same basic form as that shown in Figure 13.12. The phase diagram approach has been applied extensively to racemates (Jacques et al. 1994). A particularly lucid account of the use of such diagrams for mandelic acid is given by Lorenz et al. (2001). A distinctive feature of these diagrams is that the solubilities of the two components (enantiomers) must be equal, so the diagrams are symmetrical. This makes them easier to construct and interpret than those for salts and cocrystals. It follows that the eutectic positions (solubility maxima) are relatively insensitive to choice of solvent and can be estimated reliably from the solubilities of the enantiomer and the racemate (Klussmann et al. 2006). The key knowledge, which is counterintuitive, is that the enantiomeric excess (e.e) in solution is generally different from the e.e in the solid and that the position of the eutectic imposes fundamental constraints on e.e. and yield (Wang et al. 2005). Pharmaceutical cocrystals contain at least two components and may offer advantages in terms of physical properties and exposure (Remenar et al. 2003; McNamara et al. 2006). Figure 13.12 also applies, assuming that the only cocrystal has a 1:1 stoichiometry. However, A and B are no longer constrained to have the same solubilities. If the solubilities are similar, then process design is straightforward. For example, the phase diagram for nicotinamide–cinnamic acid–methanol resembles Figure 13.12 (left), whereas that for nicotinamide–cinnamic

Crystallization in the Pharmaceutical Industry

Figure 13.12 (Left) Schematic phase diagram for salts, cocrystals, and racemates; (right) experimental phase diagram for a cocrystal Source: Reproduced with permission of the publisher from Chiarella et al. 2007. Copyright © 2007, American Chemical Society.

acid–water is reproduced in Figure 13.12 (right). The very different solubilities of the two cocrystal formers in water skew the phase diagram so that a considerable excess of the more soluble component is required to ensure that only the cocrystal crystallizes (Chiarella et al. 2007). Moreover, the presence of one component can have a dramatic effect on the solubility of the other via complex formation in solution (Nehm et al. 2006). The strength of these interactions in solution may correlate with the ease of cocrystal formation (He et al. 2008). Washing the cocrystal may cause partial disproportionation, as may exposure to water/high humidity. By far the most common pharmaceutical compounds (solids containing more than one component) are salts. Like cocrystals, salts are often prepared to improve exposure or avoid solids with undesirable physical properties (Stahl and Wermuth 2002). Surprisingly, the use of phase diagrams to describe such systems is comparatively rare but has been described schematically for a series of ephedrine salts (Black et al. 2007). The eutectic is sometimes referred to as pHmax and the property that the solution composition stays constant as the system composition changes is familiar as a buffer. A recent quantitative study of ephedrine pimelate (Cooke et al. 2010) shows how dramatically solubility can vary in the presence of a counterion and demonstrates the complications arising from multiple salt stoichiometries. This is discussed in more detail in Section 13.8.3.

13.4 Crystallization Kinetics The vast literature on crystallization kinetics is rarely consulted within the pharmaceutical industry. For process and product development, there are three main points of interest. First, the relative rates of crystal growth and of other processes (such as secondary nucleation) determine how long it takes to grow crystals large enough to filter. Second, the relative rates of crystal growth in different directions determine the crystal shape (morphology). Third, impurities have significant effects

on crystal growth rates. Because impurity profiles usually change during process development, as a result of either route changes or process improvements upstream, crystal growth kinetics usually change also. This contrasts with thermodynamic data discussed in Section 13.3, which only need to be measured once for each solvent–solute combination. Crystallization kinetics include the kinetics of crystal nucleation and the kinetics of crystal growth. The latter are reasonably well understood. However, nucleation is inherently random, poorly understood, extremely difficult to study, and even more difficult to control. This is a topic of active academic research. Meanwhile, in process development, it is possible to avoid many of the difficulties associated with nucleation by seeding.

13.4.1 Seeding Seeding is the first and essential step for control of a crystallization process. The two key questions are: when to add the seed and how much seed to add. Figure 13.13 illustrates where to add seed during a cooling crystallization. If the seed is added in the “solution” region, it will dissolve. If the seed is added in the “suspension” region, spontaneous nucleation will already have occurred. Seed needs to be added in the metastable zone (MZ). Once seed is added, it is important to maintain the solution concentration within the MZ as the crystallization progresses. The cooling rate must be matched to the surface area of crystals available for growth. Knowledge of the width of the MZ (see Figure 13.13) is essential to determine the seed point. Table 13.3 gives MZ widths for potassium nitrate and four development compounds measured at the 200-ml scale and expressed as degrees of undercooling. In general, MZs are small for inorganic systems, unless certain additives are present. In contrast, MZs in the pharmaceutical industry are typically greater than 10°C and do not vary much at scales larger than a few milliliters. Whatever the fundamental reasons for these behaviors (Black 2007), they

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Compound

Mzwidth (°C)

KNO3

2

AZ5

8

AZ6

19

AZ7

39

AZ8

>50

Source: From Parsons et al. 2003.

Figure 13.13 The metastable zone (MZ)

form the basis for the widespread application of seeding in the pharmaceutical industry. A key consideration when determining the seed point is accuracy. The largest errors arise from temperature measurement. In the laboratory, it is common to measure and control the temperature to within ±1°C. Typically this corresponds to errors in solution concentration of ±3.5 percent (see Section 13.2.4). At pilot-plant and production scales of hundreds of liters, errors in temperature, including calibration errors, are typically ±4°C. Seed points must be defined accordingly to minimize the risk that the process will stray outside the MZ before the seed is added. For full control, variations in the weight and purity of the starting material and solvent amounts must also be considered. Using solvent mixtures introduces addition variability. In seeding polymorphic systems (Beckmann 2002), there is an additional consideration. Figure 13.5 shows schematic solubility curves of two monotropic polymorphs. If the stable polymorph is desired, seeding between the two solubility curves, as shown in Figure 13.14, minimizes the risk of generating the metastable polymorph and has the additional benefit that any undesired polymorph present in the seed will dissolve. Seeding alone does not guarantee control of crystal form and particle size – three different outcomes are possible: 1. A mixture of crystal structures 2. The desired crystal structure but uncontrolled particle size 3. The desired crystal structure and controlled particle size Outcome 2 may be sufficient in some cases, particularly in early stages of process development, and 0.1 percent seed (based on yield) is a typical level for this purpose. For outcome 3, seed levels of 0.5–2.0 percent are typical. It is common practice to specify the weight of seed because this is of crucial importance in ensuring adequate supply of seed, as well as in agreeing to the level of good manufacturing practice (GMP) clearance that is required. However, seed efficacy is determined by surface area and hence particle size. For manufacturing campaigns, the preparation and supply of seed are significant activities that must be planned well in advance. Seed behavior can be monitored using in-line technology, for example, Lasentec FBRM (Barrett and Glennon 2002).

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Essentially, this technique provides a “fingerprint” of the solids in a suspension, and is particularly sensitive at low solids loading such as following seed addition. This is illustrated in Figure 13.15, which shows a Lasentec FBRM “total counts” trace of the addition of seed to a pilot-plant reactor with a volume of 600 liters. The total counts trace shows a very clear pattern. For this highly active compound, the seed was added from jars via a flexible glove box. Two jars were placed inside the glove box and then emptied into the reactor. This process was repeated three times so that all eight jars of seed were added. This 4 × 2 pattern is immediately visible from the total counts trace. As part of the installation of this technology in the pilot plant, these FBRM data were displayed in real time to the pilot-plant staff in their control room, providing them with instant feedback on seed addition. This was a significant factor in gaining acceptance of this new technology in the pilot plant. The technique also spots unexpected events during crystallizations, as shown in Figure 13.16. The signal indicates an unexpected event at time 15 hours. This triggered a sampling of the batch and analysis of the solid, which was found to be a hydrate. The final product was the expected anhydrous form. Further investigation revealed that the hydrate was formed at the end of the process following a water addition and transformed back to the anhydrous form during washing and drying. Were it not for the in-line technology and an observant interpreter who acted quickly, this knowledge would have been missed (Bohlin et al. 2009). The Lasentec FBRM technique also detects secondary nucleation, as shown in Figure 13.17. In this case, 0.2 percent of seed was added, and the supersaturation was also monitored by sampling and off-line analysis. The seed level was clearly insufficient to control the supersaturation, and nucleation occurred once the supersaturation had reached 1.7 times. The solid was the same polymorph as the seed, so the nucleation was probably homogeneous secondary nucleation. The addition of 2.0 percent seed gave a smoother crystallization profile. This clarifies that the selection of seed amount and surface area is intimately linked to crystal growth rates – seeding alone does not guarantee control.

13.4.2 How Fast Is Crystallization? For successful filtration, avoiding the loss of solid product, crystals must of a similar size to or larger than the pore size of the

Crystallization in the Pharmaceutical Industry Figure 13.14 Seeding for the stable polymorph. The seed point X lies between the solubility curves of the two polymorphs.

100.00 Seed Point

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Time (minutes) Figure 13.15 Monitoring the addition of seed in a pilot plant

filtration medium – typically approximately 20 µm. So how long does it take to grow a crystal of approximately 20 µm in size? If a saturated solution is placed on a microscope slide and the solvent is allowed to evaporate, crystallization kinetics can be observed directly. Usually seeds will form at the edges of the drop and then grow into the remaining solution. For pharmaceutical systems, a wide range of behaviors can be observed. In some systems, crystals nucleate and grow to several millimeters in size within seconds. In other systems, crystal growth halts once crystals reach a certain size, and then new crystals grow from or near the surface. In other cases, the systems gels or sets in an amorphous or semicrystalline state. Similar behaviors are seen in well plates that are frequently used in crystallization screens, polymorph screens, and salt screens and in many unstirred crystallizations reported in the literature. Such observations are difficult to interpret. As observed earlier, nucleation is a random process, so variability is to be

expected. Furthermore, the supersaturation increases in an uncontrolled manner in such experiments. The supersaturation at the point of nucleation is not usually measured or recorded. This makes it very difficult to conclude anything meaningful about the kinetics of nucleation and crystal growth from such experiments. The following two examples illustrate two extremes of crystal growth. In both cases, crystal growth was monitored using the Lasentec FBRRM probe. Figure 13.18 shows complete dissolution (1) during heating, followed by nucleation (2) soon after the onset of cooling from 77°C, with the crystallization finishing (3) at the end of cooling to 0°C (Black et al. 2006). The relevant solubility curve is given above in Figure 13.1. The cooling time was 1 hour, which was chosen deliberately to match the fastest cooling rate achievable in the pilot plant. The material nucleates spontaneously at a supersaturation of 0.4–0.6. There is no evidence for secondary nucleation, and in this example, the rate-determining step is heat transfer. Figure 13.19 shows a 24-hour cooling profile for a different compound that was seeded at a supersaturation of 0.4. Despite the much longer cooling profile in this system, secondary nucleation still occurs, as is clear from both the Lasentec FBRM data and the in-process image, which shows large crystals (presumably grown from the seed) but also many much smaller crystals This material grows as needles, and if the aspect ratio of the needles is too high, as in the image shown here, the crystals form a network in the vessel and “set” and cannot be processed further (Muller 2009). The evidence from the total counts trace suggests that this secondary nucleation started when the cooling rate was increased. It is clear that at equivalent supersaturations this material grows much more slowly than the material in the Figure 13.18. The rate-determining step in this case is likely to be surface integration rather than heat or mass transfer. It takes 24 hours to grow crystals of this material that can be processed downstream.

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Simon N. Black Figure 13.16 An unexpected event during crystallization (at time 15 hours)

growth rates – the thermodynamically stable gamma form of glycine grows 500 times more slowly than the kinetically favored alpha form at a supersaturation of 0.1 (Chew et al. 2007). There is no way of deducing from crystal structures, even if these are known, how fast crystal growth will be as a function of supersaturation. This information has to be derived from experiments. The molecular structure can only provide a crude guide of what to expect. However, experience indicates that 24 hours is usually long enough to grow crystals of APIs that are large enough for filtration.

13.4.3 Supersaturation Figure 13.17 The effect of seed loading

These two materials illustrate the extremes of crystallization kinetics – the behavior of most materials lies somewhere in between these two. The fundamental reasons for the difference between these two materials must be related to the molecular and crystal structures of these materials. It is tempting to seek correlations between molecular structure and crystal growth kinetics. The material in Figure 13.18 has only one rotatable bond, which makes it unusually rigid for an API. The other molecule is more typical, with several rotatable bonds. However, the crystal structure must also be important. Figure 13.19 shows needles that arise because crystal growth is faster along the needle axis than parallel to it. The very existence of anisotropic shapes illustrates the influence of crystal as well as molecular structure on kinetics. There are examples of crystals that grow in one direction but not in the opposite direction – so-called polar morphologies. These phenomena are discussed more fully in Section 13.5.2. There are also many examples of polymorphs that have very different

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For a given material, the most important factor controlling the rate of crystallization is the thermodynamic driving force – the supersaturation. However, the rates of other competing processes also depend on supersaturation and may dominate at higher supersaturations. The most significant but not the only such process is secondary nucleation. Hence supersaturation has to be optimized to maximize crystal growth rates while avoiding unwanted competing processes. This supersaturation has to be determined experimentally for each system. In the example shown in Figure 13.18, this optimum is in the region 0.2–0.5. Linear cooling profiles are unlikely to be optimal. Toward the end of crystallization, the surface area available for growth increases. Assuming a linear variation of solubility with temperature and constant growth rates, the cube-law cooling profile will be optimal (Mullin 2001). This can be approximated by two or three linear profiles with different rates. However, as discussed earlier, the variation in solubility with temperature is generally exponential not linear, as in Figure 13.1. Moreover, this approach assumes that equilibrium has been reached before cooling starts, which is often not the case. Hence the optimal cooling profile may be even further from linear. By this point it should be clear why the common

Crystallization in the Pharmaceutical Industry

100 temperature

90 80

Temperature (ºC)

70 60 50 total counts 40 30 20 10 1 0

0

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3 120

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Time (minutes)

Figure 13.18 Cooling crystallization for a material that grows quickly. Total counts and temperature (scale refers) on left. Optical micrograph (image width = 1mm) on right. Source: Adapted with permission of the publisher from Black et al. 2006. Copyright © 2006, American Chemical Society.

27

mean CL

Chord Length (CL; μm)

22 temperature 17

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Figure 13.19 Cooling crystallization for a material that grows slowly: (left) temperature, total counts, and mean chord length (microns, scale refers); (right) image from in-process camera (image width = 2 mm)

practice of leaving hot solutions to cool naturally is not recommended (Figure 13.20). Using in-line technology and feedback control, it is possible to derive an optimal cooling profile experimentally (Liotta and Sabesan 2004). The solution concentration and temperature are monitored in line and used to calculate the supersaturation in real time. The cooling rate is then adjusted

continuously via closed-loop feedback control to maintain the supersaturation at a preset value. Once this cooling profile is determined, it should be valid at all scales, provided that the seed surface area is constant and surface integration is the rate-determining step. Hence it is not necessary to implement in-line measurement and feedback control at larger scales. An example of such a cooling profile is shown in Figure 13.20

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Figure 13.20 Cooling profiles: (left) cube law + linear; (right) constant supersaturation Source: Reproduced with permission of the publisher from Liotta and Sabesan 2004. Copyright © 2004, American Chemical Society.

80 water–acetonitrile

70

50 40 30 20 10 0

13.4.4 Solvents If mass transfer is rate limiting, then higher solubilities would be expected to correlate with faster growth. Where surface integration is rate limiting, the effect of solvent is more subtle – higher surface concentrations will help, but the molecular conformations may change with solvent, and the activation energy for “disengaging” from the solvent may vary. These last two effects are extremely difficult to analyze and model. In a notable study of the effect of solvent on rates of solvent-mediated transformations of sulfamerazine polymorphs at a supersaturation of 0.24 or less (Gu et al. 2001), a qualitative correlation between transformation time and solubility was observed. For mixtures of water and methanol or water and acetonitrile, the relationship is quantitative, as shown in Figure 13.21. This is consistent with surface integration as the rate-limiting step. In unseeded experiments, no transformation at all was observed after 2 weeks in eight of the 18 solvents studied. This correlated with low solubility (