Detection and Data Analysis in Size Exclusion Chromatography 9780841214293, 9780841211964, 0-8412-1429-8

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Detection and Data Analysis in Siz Chromatography

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

ACS

SYMPOSIUM

SERIES

352

Detection and Data Analysis in Size Exclusion Chromatography Theodor The Glidden Company

Developed from a symposium sponsored by the Division of Polymeric Materials Science and Engineering at the 191st Meeting of the American Chemical Society, New York, New York, April 13-18, 1986

American Chemical Society, Washington, DC 1987

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Library of Congress Cataloging-in-Publication Data Detection and data analysis in size exclusion. chromatography. (ACS symposium series; 352) Includes bibliographies and index. 1. G e l permeation chromatography—Congresses. I. Provder, Theodore, 1939. II. American Chemical Society. Division of Polymeric Materials: Science and Engineering. III. America Society. Meeting (191st: 1986: Ne IV. Series. QD272.C444D47 1987 ISBN 0-8412-1429-8

547.7'046

87-19480

Copyright © 1987 American Chemical Society A l l Rights Reserved. The appearance of the code at the bottom of the first page of each chapter in this volume indicates the copyright owners consent that reprographic copies of the chapter may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc., 27 Congress Street, Salem, M A 01970, for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating a new collective work, for resale, or for information storage and retrieval systems. The copying fee for each chapter is indicated in the code at the bottom of the first page of the chapter. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by A C S of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. Registered names, trademarks, etc., used in this publication, even without specific indication thereof, are not to be considered unprotected by law. P R I N T E D IN T H E U N I T E D S T A T E S O F A M E R I C A

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

ACS Symposium Series M. Joan Comstock, Series Editor 1987 Advisory Board Harvey W. Blanch University of California—Berkele

Alan Elzerman

W. H . Norton

Clemson University

J. T. Baker Chemical Company

John W. Finley

James C. Randall

Nabisco Brands, Inc.

Exxon Chemical Company

Marye Anne Fox

E. Reichmanis

The University of Texas—Austin

A T & T Bell Laboratories

Martin L . Gorbaty

C. M . Roland

Exxon Research and Engineering Co.

U.S. Naval Research Laboratory

Roland F. Hirsch

W. D. Shults

U.S. Department of Energy

Oak Ridge National Laboratory

G. Wayne Ivie

Geoffrey K. Smith

U S D A , Agricultural Research Service

Rohm & Haas Co.

Rudolph J. Marcus

Douglas B. Walters

Consultant, Computers & Chemistry Research

National Institute of Environmental Health

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Foreword The ACS S Y M P O S I U M S E R I E S was founded in 1974 to provide a

medium for publishing symposia quickly in book form. The format of the Series parallels that of the continuing A D V A N C E S IN C H E M I S T R Y SERIES except that, in order to save time, the papers are not typese by the authors in camera-read the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously published papers are not accepted. Both reviews and reports of research are acceptable, because symposia may embrace both types of presentation.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Preface THE

F I E L D O F SIZE E X C L U S I O N C H R O M A T O G R A P H Y ( S E C ) remains a viable

and lively area of polymer characterization. Over the past several years, there has been considerable research activity in the area of S E C detection and data analysis in order to obtain more comprehensive information concerning the composition and molecular architecture of complex polymer systems. In part, this has bee polymeric materials fro y building blocks. These constraints resulted from government legislation in the areas of clean air, toxic substances, hazardous wastes, etc. As a consequence of being restricted to a narrow set of building blocks, it becomes critical to understand how a polymeric material is put together (composition, structure, molecular architecture, morphology) in order to relate fundamental properties to a polymeric material's performance. Therefore, the use of concurrent detectors in S E C along with sophisticated data analysis methods to unravel the nature of complex polymers is growing. Advances in electronics and computer technology are catalyzing the activities of detector development and data analysis. The detection and data analysis activities in the field of S E C applied to polymeric materials is expected to grow in the future. Improved detectors and data analysis systems will become commercially more available as a result of the current research activities in selected industrial and academic labs. I thank the authors for their effective oral and written communications, and the reviewers for their critiques and constructive comments. THEODORE

PROVDER

The Glidden Company Strongsville, O H 44136

ix

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 1

An Overview of Size Exclusion Chromatography for Polymers and Coatings Cheng-Yih Kuo and Theodore Provder The Glidden Company, Research Center, 16651 Sprague Road, Strongsville, OH 44136

Recent technological advances have sparked a new level of activity in the field of Size Exclusion Chromatography (SEC). These include: 1) high performance/high development and increased use of simultaneous multiple in-line detectors such as differential refractometer, ultraviolet and infrared spectrophotometric detectors, viscometers, low angle laser light scattering, and mass detection, and 3) the application of minicomputer and microcomputer technology for instrument control and data analysis. These developments in turn have led to new improved applications of SEC as well as higher quality information. In this paper, the SEC separation mechanism, molecular weight calibration methods including the use of hydrodynamic volume, instrument spreading corrections and polymer chain branching calculations will be discussed. Quantitative and qualitative examples of the application of multiple detectors will be given. Finally, there will be some discusison of the requirements necessary for high resolution SEC analysis of oligomers and examples will be shown. Polymer chemists and c o a t i n g s f o r m u l a t o r s a r e c o n t i n u a l l y b e i n g c a l l e d upon t o t a i l o r - m a k e c o a t i n g s systems which r e q u i r e polymers having s p e c i f i c a l l y designed m o l e c u l a r a r c h i t e c t u r e s and p h y s i c a l p r o p e r t i e s . Knowledge of t h e m o l e c u l a r weight and m o l e c u l a r weight d i s t r i b u t i o n (MWD) o f the polymer components i n a c o a t i n g s system i s e s s e n t i a l f o r the o p t i m i z a t i o n o f polymer d e s i g n f o r s p e c i f i c end-use p r o p e r t i e s . S i n c e i t s i n t r o d u c t i o n over two decades a g o , ( J J g e l permeation chromatography (GPC) o r s i z e e x c l u s i o n chromatography (SEC) has become an important and p r a c t i c a l t o o l f o r the d e t e r m i n a t i o n o f the MWD o f polymers. A l a r g e number of s t u d i e s has been p u b l i s h e d on the use o f SEC i n p l a s t i c s , e L a s t o m e r i c and c o a t i n g s systems. W i t h t h e advent o f h i g h e f f i c i e n c y columns, the r e s o l u t i o n i n t h e lower m o l e c u l a r 0097-6156/87/0352-0002$07.75/0 © 1987 American Chemical Society

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

1.

KUO AND PROVDER

3

SEC for Polymers and Coatings

weight r e g i o n ( m o l e c u l a r w e i g h t s i n the range of 200 t o 10,000) has been g r e a t l y improved and the speed of a n a l y s i s i n c r e a s e d . These f e a t u r e s make h i g h performance SEC (HPSEC) an i n d i s p e n s a b l e c h a r a c t e r i z a t i o n t o o l f o r the a n a l y s i s of o l i g o m e r s and polymers i n e n v i r o n m e n t a l l y a c c e p t a b l e c o a t i n g s systems. SEC. S^J^J^ÀPJL-

e

il Ahani sm

S i z e e x c l u s i o n chromatography i s a l i q u i d chromatography method, whereby, the polymer m o l e c u l e s are s e p a r a t e d by t h e i r m o l e c u l a r s i z e or "hydrodynamic volume" i n s o l u t i o n as s o l v e n t e l u t e s through a column(s) packed w i t h a porous s u p p o r t . The degree of r e t e n t i o n of the polymer m o l e c u l e s i n the pores i s the phenomenon which a f f e c t s the s e p a r a t i o n . S m a l l e r m o l e c u l e s a r e r e t a i n e d i n the pores t o a g r e a t e r degree than the l a r g e r m o l e c u l e s . As a r e s u l t the l a r g e s t s i z e m o l e c u l e ( o r the m o l e c u l e h a v i n g the g r e a t e s t hydrodynamic volume f o l l o w e d by the s m a l l e a s o l u t e e l u t e s from a column or the volume of l i q u i d c o r r e s p o n d i n g t o the r e t e n t i o n of a s o l u t e on a column i s known as the r e t e n t i o n volume ( V ) and can be r e l a t e d to the p h y s i c a l parameters of the column as f o l l o w s : R

v where

= ν

R

ο

+

κν.

(1)

V = r e t e n t i o n volume of the s o l u t e V = i n t e r s t i t i a l volume (dead volume) of the column V\ = i n t e r n a l s o l v e n t volume i n the pores Κ = the d i s t r i b u t i o n c o e f f i c i e n t , based upon the r e l a t i v e c o n c e n t r a t i o n s between phases. R

q

The t o t a l column volume V

=

T

V

o

i s g i v e n by

+ V.

(2)

T h e r e f o r e , the r e t e n t i o n volume i s e x p r e s s i b l e i n terms of the two measurable q u a n t i t i e s V and V as Q

V

K

= V (1-K) + KV^

O

1

T

0 10^ g/mole), flow rates of less than 0.1 mL/min may be required when using a 4mm ID column. Because of the inverse relationship between p a r t i c l e diameter and shear rate, the use of SEC packings much smaller than 5 urn should be avoided for the analysis of >10^ molecular weight polymers. Experimentally i t is d i f f i c u l t to detect the occurrence of polymer shear degradation since concentration e f f e c t s , increased peak dispersion, and u l t r a f i l t r a t i o n of high molecular weight components may also d i s t o r t the peak p r o f i l e or s h i f t the d i s t r i b u t i o n towards the low molecular weight region. Furthermore, 5

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

38

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

because shear degradation may occur anywhere in the chromatographic system where a high force f i e l d exists, shear degradation may not necessarily be accompanied by an increase in elution volume or t a i l i n g , especially i f degradation occurs near the end of or after the SEC column. A q u a l i t a t i v e approach for detecting shear degradation i s to examine the shape of a c a l i b r a t i o n curve generated by the use of a "molecular weight" detector (low-angle laser l i g h t scattering photometer or v i s c o s i t y detector): A downward s h i f t of the log M versus V plot i s indicative of shear degradation. The best approach to determine the extent of shear degradation i s to measure either the i n t r i n s i c v i s c o s i t y or an average molecular weight of the polymer before and after elution through the column. e

Ultrafiltration With decreasing packin physical entrapment of molecular weight l i m i t , must f i r s t calculate an average radius of the i n t e r s t i c e s formed in a packed bed. This i s done by assuming that the packed column consists of a bundle of c a p i l l a r i e s in which the c a p i l l a r y radius can be estimated from the bed hydraulic radius: R

= D e/6(l-e)

h

(7)

p

where Dp i s the average diameter of the packing and e i s the porosity of a packed bed taken as 0.36. Thus, R

h

- 0.094 Dp

(8)

I t should be noted that i f there are any fines present, which would readily f i l l in the i n t e r s t i c e s , in those regions of the packed bed would be considerably smaller. In addition to the packed bed acting as an u l t r a f i l t e r , the porous f r i t s used at both ends of the column may act as very e f f e c t i v e f i l t e r i n g devices. Thus a 2-]im porosity f r i t would have an average pore radius of 1 ]im. Because of the tortuosity and r e l a t i v e l y wide pore-size d i s t r i b u t i o n present in f r i t s , i t would be safe to assume that i t contains much smaller crevices which can entrap macromolecules. The molecular weight of a polymer which begins to approach Rh can be approximated by calculating the radius-of-gyration of a macromolecule l/2 which i s defined as the root-mean-square distance of the elements of the chain from i t s center of gravity, using the Flory-Fox equation (111): 2

f

< 2>3/2 s

β

[ηΐΜ/φ

7

(9)

where^[Ti] i s the i n t r i n s i c v i s c o s i t y , M i s the molecular weight, and Φ i s a constant that is equal to 0.39 χ Ι Ο mol" when [η] i s expressed in terms of cm^/g. A more general equation written in terms of the Mark-Houwink equation i s 2 5

1

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

2. BARTH

Nonsize Exclusion Effects in High-Performance SEC < 2>3/2 S

=

ΚΜ*+1/Φ

39 (10)

where Κ and a are the Mark-Houwink constants. The molecular weight equivalents to for d i f f e r e n t packing sizes for several representative polymers, calculated from eqn. (9), are shown in Table I. As indicated, the use of 5- and 10- μιη packings should pose no serious problem because most polymers commonly encountered have molecular weights below 1 χ 10? g/mol. The use of < 2 urn packings, however, may lead to u l t r a f i l t r a t i o n depending upon the polydispersity of the sample. These conclusions are based on the assumption that there are no fines present in the column. When analyzing ultrahigh molecular weight polymers using high porosity O3000 Â pore diameters) s i l i c a packings, this assumption i s probably not true because of the f r a g i l i t y of these packings (155). Thus, the presence of, l e t us say, l-ym fragments caught in the i n t e r s t i c e s of 10 um packings coul s u f f i c i e n t l y small to trap polyethylene. Furthermore, insoluble material present in the sample may also become trapped either in the f r i t or the column packing and further reduce the i n t e r s t i c e s . Table I.

Molecular Weight Equivalent to the Hydraulic Radius of a Column Packed with 2-, 5-, and 10-μπι P a r t i c l e s

Dpi

Polymer

2

Polystyrene

8

Linear Polyethylene

1.4xl0 15

Poly(styrene s u l f o n a t e ) a. b. c.

0

μ*η

5 7

6.6xl0

10 7

2xl0

8

7.1xl0

6

3.5xl0

7

1.2xl0

8

7.2xl0

6

3.1xl0

7

9.3xl0

7

3

a = 0.766, Κ ^ 6.82 χ 10~ in THF (152). a = 0.7, Κ = 5 . 9 x l 0 in trichlorobenzene (153). a = 0.89, Κ = 2.8xl0- in 0.01M NaCl (154) -2

3

Hydrodynamic Effects Hydrodynamic or flow-rate effects in SEC have been reviewed recently by Aubert and T i r r e l l (156) and Giddings (12). Although Aubert and T i r r e l l propose that nonequilibrium effects are not s i g n i f i c a n t in SEC, they show experimental evidence to support an e f f e c t c a l l e d molecular migration (157) in which K increases with flow rate. One possible mechanism that i s proposed i s that nonhomogeneous and c u r v i l i n e a r flow f i e l d s , which exist in porous media flow, cause macromolecules to migrate to the packing surface (concave side of streamlines). This enhanced concentration forces more solute into the packing, thus increasing KI 500

I

I

I

I

I

I—

600

700

800

900

1000

1100

t

[sec]

F i g u r e 11: P o l y s t y r e n e c a l i b r a t i o n c u r v e s f o r SEC 2: w i t h % η-heptane. (Reproduced from Ref. 6. C o p y r i g h t American C h e m i c a l S o c i e t y . )

variation 1983,

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

4.

BALKE

Orthogonal Chromatography and Related Advances

73

ο C l o u d p o i n t e x p e r i m e n t s showed t h a t some o f the h i g h e r m o l e c u l a r weight p o l y s t y r e n e s t a n d a r d s u s e d i n a n a l y s e s s h o u l d have been p r e c i p i t a t i n g a t η-heptane c o n c e n t r a t i o n s beyond 65%. ο The r e s o l u t i o n of the second SEC was s t r o n g l y a f f e c t e d by t h e o r d e r of columns. F i g u r e 12 shows the r e s u l t o f c h a n g i n g the p o s i t i o n of the s m a l l e s t p o r e s i z e column from the f u r t h e s t from t h e i n j e c t i o n v a l v e , t o c l o s e s t , when a n a l y z i n g a b l e n d o f p o l y s t y r e n e and p o l y ( n - b u t y l m e t h a c r y l a t e ) . Status

of

OC

E f f e c t of I n j e c t e d Solvent. I t was e v e n t u a l l y d e t e r m i n e d t h a t the m o b i l e phase i n j e c t e d from t h e f i r s t SEC ( i . e . , p u r e THF) a f f e c t e d t h e s e p a r a t i o n i n the second SEC. T h i s i s d r a m a t i c a l l y demonstrated i n F i g u r e 13 which shows t h e r e s u l t o f i n j e c t i n g a narrow m o l e c u l a r weight d i s t r i b u t i o n p o l y s t y r e n e sample d i r e c t l y i n t o the second SECM o b i l e phase was of c o n s t a n i n THF)· However, the s o l v e n v a r i e d from 0% η-heptane i n THF t o 50% and p l o t t e d on t h e a b s c i s s a v e r s u s peak r e t e n t i o n time on the o r d i n a t e . Peak r e t e n t i o n time v a r i e d from 915 seconds a t 0% η-heptane t o 960 seconds a t 50%. The consequences of t h i s e f f e c t o f i n j e c t e d s o l v e n t are as follows (6): ο O p e r a t i o n of the second SEC i s e f f e c t i v e l y always i n a g r a d i e n t mode. The " p l u g " of THF i n j e c t e d w i t h t h e polymer sweeps t h r o u g h t h e columns t o form t h i s g r a d i e n t . At h i g h e r η-heptane c o n t e n t m o b i l e p h a s e s , s t y r e n e - r i c h p o l y m e r s would p r e f e r t o remain w i t h t h i s THF plug. However, s m a l l p o r e s i z e p a c k i n g c a u s e s the THF t o be s e p a r a t e d from the s t y r e n e - r i c h polymer because t h e THF e n t e r s more pores. I f t h i s p a c k i n g i s remote from the d e t e c t o r the THF may c a p t u r e t h i s polymer l a t e r i n l a r g e r s i z e p a c k i n g s s i n c e t h e polymer i s a l s o r e t a r d e d by n o n e x c l u s i o n mechanisms ( e . g . , a d s o r p t i o n ) . If t h e p a c k i n g i s near t h e d e t e c t o r t h e n t h e s t y r e n e - r i c h polymer may be s e p a r a t e d from t h e THF p u l s e l o n g enough t o e x i t s e p a r a t e l y , ο P r e c i p i t a t i o n of s t y r e n e - r i c h polymer i s a p o s s i b i l i t y . No column p l u g g i n g o r s i g n i f i c a n t l o s s of p o l y m e r t o the columns (except f o r a few c o n d i t i o n s ) was o b s e r v e d . T h i s c o u l d be due t o t h e sweeping o f the columns by the THF p u l s e . However, s i n c e no p r e c i p i t a t i o n was e v i d e n t i n the d e t e c t o r and s i n c e t h e s t y r e n e - r i c h polymers were a b l e t o be s e p a r a t e d from t h e THF p u l s e (see p r e v i o u s p a r a g r a p h ) i t i s l i k e l y t h a t s o l v a t i o n o f t h e polymer by t h e THF i s a f f e c t i n g the r e s u l t s . ο Sequence l e n g t h can be a f f e c t i n g b o t h f r a c t i o n a t i o n and detection. F r a c t i o n a t i o n i n the f i r s t SEC ( a c c o r d i n g t o m o l e c u l a r s i z e ) i s " u n i v e r s a l " . However, the c o m p o s i t i o n f r a c t i o n a t i o n i n t h e second SEC may be s c r a m b l e d by sequence l e n g t h v a r i a t i o n s . There i s some e v i d e n c e t h a t sequence l e n g t h a f f e c t s UV d e t e c t o r r e s p o n s e . Thus, a diode a r r a y s p e c t r o p h o t o m e t e r c o u l d be u s e d t o o b t a i n s u f f i c i e n t i n f o r m a t i o n t o e l u c i d a t e b o t h c o m p o s i t i o n and sequence length. Recent Advances R e l e v a n t t o OC. HPLC s t u d i e s o f polymers i n c r e a s i n g l y employ columns w i t h v e r y s m a l l p o r e s t o p u r p o s e f u l l y e x c l u d e a l l polymer m o l e c u l e s (8.-13). The r e a s o n s j u s t i f y i n g

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

F i g u r e 12: E f f e c t o f column r e o r d e r i n g on p o l y s t y r e n e (AA) and p o l y (η-butyl m e t h a c r y l a t e ) (BB) r e t e n t i o n i n SEC 2. Key: top, s m a l l p o r e s i z e column l a s t ; and bottom, s m a l l p o r e s i z e column first. (Reproduced from R e f . 6. C o p y r i g h t 1983, American C h e m i c a l S o c i e t y . )

1000 990 980 COMBINED WITH

970

IMPURITY PEAK 960 j

950 940 930

-

920

-

910

10

20 30 40 % η HEPTANE INJECTED

50

60

70

F i g u r e 13: E f f e c t o f i n j e c t e d s o l v e n t c o m p o s i t i o n on p o l y s t y r e n e r e t e n t i o n i n SEC 2. (Reproduced from R e f . 6. C o p y r i g h t 1983, American C h e m i c a l S o c i e t y . )

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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75

t h i s approach a r e : The h i g h e x t e r n a l s u r f a c e areas o f h i g h r e s o l u t i o n p a c k i n g s ( e s t i m a t e d a t 20% o f t h e t o t a l s u r f a c e a r e a o f a p a c k i n g m a t e r i a l w i t h p o r e s l a r g e enough t o accommodate a l l o f the polymer m o l e c u l e s ) (TO); t h e p o s s i b l e a b i l i t y o f polymer m o l e c u l e s t o u n c o i l s u f f i c i e n t l y t o be a f f e c t e d by s m a l l p o r e s i n a d s o r p t i o n - t y p e mechanisms (_12). A l t h o u g h t h e r e remains some disagreement (12,13), the p r e s e n c e of a p r e c i p i t a t i o n mechanism i n s e p a r a t i o n s u s i n g g r a d i e n t chromatography o f polymers i n HPLC s e p a r a t i o n s appears v e r y l i k e l y (8,9J. T h i s r e i n f o r c e s t h e h y p o t h e s i s mentioned above t h a t p r e c i p i t a t i o n i s a l s o an important mechanism i n s e p a r a t i o n i n the s e c o n d SEC u s e d i n OC. However, i t i s i m p o r t a n t t o n o t e t h a t c o n s i d e r i n g the c o m p l e x i t y o f the polymers t o be a n a l y z e d and the v a r i e t y o f important p a c k i n g v a r i a b l e s , mixed mechanisms must be anticipated. Attempting t o arrange f r a c t i o n a t i o n t o s y n e r g i s t i c a l l y use t h e s e mechanisms appears as a more r e l i a b l e and more g e n e r a l l y a p p l i c a b l e approach t h e mechanism. However, i p o l y m e r c o m p l e x i t y and c u r r e n t u n c e r t a i n t y a s s o c i a t e d w i t h these mechanisms demands use o f modem d e t e c t o r t e c h n o l o g y t o i d e n t i f y t h e e x i t i n g m o l e c u l e s as t h o r o u g h l y as i s r e q u i r e d by the p u r p o s e o f t h e analysis. G l o c k n e r e t a l . (9) have shown v e r y good s e p a r a t i o n s o f s t y r e n e a e r y I o n i t r i l e copolymers u s i n g an SEC and a HPLC s e t up as a r e t h e two SEC s i n OC. They term the s e p a r a t i o n o c c u r r i n g i n the HPLC " h i g h performance p r e c i p i t a t i o n l i q u i d chromatography". The r e t a r d i n g e f f e c t o f p o r e s on s o l v e n t r e l a t i v e t o the p o l y m e r d e s c r i b e d i n S e c t i o n 6.1 has now been p r o p o s e d as the r e a s o n f o r t h e good s e p a r a t i o n s o b t a i n e d by p r e c i p i t a t i o n mechanisms (8-10, 13). The polymer i s v i s u a l i z e d t o c o n t i n u a l l y r e - p r e c i p i t a t e and r e - d i s s o l v e as the s o l v e n t f r o n t o f a g r a d i e n t r e p e t i t i v e l y o v e r t a k e s and t h e n l o s e s the p o l y m e r . G a r c i a Rubio e t a l . (14,15) have a c c o m p l i s h e d s i g n i f i c a n t development i n t h e UV a n a l y s i s o f c o p o l y m e r s . In e x a m i n i n g t h e d a t a j u s t i f y i n g the use o f UV s p e c t r a f o r d e t e r m i n i n g b o t h c o m p o s i t i o n and sequence l e n g t h o f p o l y m e r s , a p p l i c a t i o n o f e r r o r p r o p a g a t i o n t h e o r y showed t h a t the p u b l i s h e d r e s u l t s on measurement o f sequence l e n g t h by UV can be e x p l a i n e d by c o n s i d e r i n g o n l y c o m p o s i t i o n , a t l e a s t f o r c e r t a i n wavelengths. F u r t h e r m o r e , they showed t h a t UV a b s o r p t i o n s p e c t r a a r e s i g n i f i c a n t l y a f f e c t e d by benzoate groups on t h e polymer. These groups are p r o d u c e d d u r i n g p o l y m e r i z a t i o n when b e n z o y l p e r o x i d e i s u s e d as the i n i t i a t o r . F o r a c c u r a t e q u a n t i t a t i v e work t h e s p e c t r a must be c o r r e c t e d f o r t h i s a b s o r p t i o n . Some o f the i n a c c u r a c y i n F i g u r e 9 c o u l d w e l l be due t o t h i s source o f e r r o r . Much UV development work u t i l i z e s " o f f - l i n e " a n a l y s i s ( d e t e c t o r not c o n n e c t e d t o an SEC) o f p r e c i p i t a t e d p o l y m e r . I t i n v o l v e s very d e t a i l e d i n t e r p r e t a t i o n o f d i f f e r e n c e s i n s p e c t r a . One c a u t i o n which must be o b s e r v e d i s t o e n s u r e t h a t r e s i d u a l monomer o r o t h e r s m a l l m o l e c u l e s are not c a u s i n g s p u r i o u s r e s u l t s . Acquiring spectra " o n - l i n e " u s i n g a d i o d e a r r a y UV/vis s p e c t r o p h o t o m e t e r a t t a c h e d t o an SEC i s one answer t o t h i s p r o b l e m . However, t h e n d e t e r m i n i n g t h e c o n c e n t r a t i o n o f polymer examined can be t r o u b l e s o m e . F u r t h e r m o r e , i t s h o u l d be n o t e d t h a t i n t e r p r e t i n g polymer spectra i n solvent mixtures t y p i c a l of gradient operation i n l i q u i d chromatography may be an a d d i t i o n a l p r o b l e m . A l t h o u g h a l l s o l v e n t s 1

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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may be t r a n s p a r e n t t o UV, t h e c o n f o r m a t i o n o f t h e polymer i s e x p e c t e d t o change w i t h s o l v e n t c o m p o s i t i o n and t h i s i n t u r n can a f f e c t t h e observed s p e c t r a . Conclusions ο C o n v e n t i o n a l SEC a n a l y s i s o f complex polymers t o e l u c i d a t e i n d i v i d u a l p r o p e r t y d i s t r i b u t i o n s ( o t h e r than t h e d i s t r i b u t i o n o f molecular s i z e s ) encounter d i f f i c u l t i e s i n both e f f e c t i v e f r a c t i o n a t i o n and unambiguous d e t e c t i o n . ο The use o f new d e t e c t o r t e c h n o l o g y has been t h e main emphasis i n attempts t o u s e SEC f o r t h e a n a l y s i s o f complex p o l y m e r s . ο HPLC attempts t o a n a l y z e complex polymers have emphasized f r a c t i o n a t i o n employing g r a d i e n t o p e r a t i o n and a d s o r p t i o n o r r e v e r s e d phase p a c k i n g s . Recent HPLC. ο OC i s a m u l t i - d i m e n s i o n a l SEC method which emphasizes f r a c t i o n a t i o n and d e t e c t i o n .

both

ο OC has been s u c c e s s f u l l y a p p l i e d t o a c c o m p l i s h a c o m p o s i t i o n s e p a r a t i o n o f s t y r e n e / m e t h a c r y l a t e homopolymers and copolymers. ο A mechanism has been p o s t u l a t e d t o account f o r t h e o b s e r v e d s e p a r a t i o n s and p r o p o s e s t h a t t h e v a r i o u s mechanisms i n v o l v e d can a c t synergistically. ο The p a r t i c i p a t i o n o f t h e m o b i l e phase from t h e f i r s t SEC i n t h e s e p a r a t i o n o b s e r v e d i n t h e second SEC e f f e c t i v e l y c r e a t e s a g r a d i e n t o p e r a t i o n i n t h e second. ο S u c c e s s i v e p r e c i p i t a t i o n and d i s s o l u t i o n o f t h e polymer i n t h e second SEC i s l i k e l y an a d d i t i o n a l i m p o r t a n t s e p a r a t i o n mechanism. ο Q u a n t i t a t i v e a n a l y s i s has been attempted l e n g t h i s needed. ο of to

b u t d e t e c t i o n o f sequence

Development o f UV d e t e c t o r i n t e r p r e t a t i o n and i n c r e a s e d automation column s w i t c h i n g and d a t a a c q u i s i t i o n / i n t e r p r e t a t i o n a r e important f u t u r e OC development.

Literature Cited 1. 2. 3. 4. 5.

Balke, S.T. "Quantitative Column Liquid Chromatography, A Survey of Chemometric Methods"; Elsevier: Amsterdam, 1984. Balke, S.T. Sep. Purif. Methods 1982, 11, 1. Balke, S.T. In "Modern Methods of Polymer Analysis"; Barth, H.G., Ed.; Wiley: New York, 1987 (in press). Hamielec, A.E.; Styring, M. Pure & Appl. Chem. 1985, 57, 955. Balke, S.T.; Patel, R.D. In "Size Exclusion Chromatography (GPC)"; Provder, T., Ed.; ACS SYMPOSIUM SERIES No. 138, American Chemical Society: Washington, DC, 1980; p. 149.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

4.

BALKE

Orthogonal Chromatography and Related Advances

6.

11

Balke, S.T.; Patel, R.D. In "Polymer Characterization: Spectroscopic, Chromatographic, and Physical Instrumental Methods"; Craver, C.D., Ed.; ADVANCES IN CHEMISTRY SERIES No. 203, American Chemical Society: Washington, DC, 1983; p. 281. 7. Balke, S.T.; Patel, R.D. J . Polym. Sci., Polym. Letters 1980, 18, 453. 8. Glockner, G. TRAC 1985, 4, 214. 9. Glockner, G.; Van Den Berg, J.H.M.; Meijerink, N.L.J.; Scholte, T.G.; Koningsveld, R. J . Chromatogr., 1984, 317, 615. 10. Glockner, G. Pure & Appl. Chem. 1983, 55, 1553. 11. Mourey, T.H.; Noh, I.; Yu, H. J . Chromatogr. 1984, 303, 361. 12. Snyder, L.R.; Stadalius, M.A.; Quarry, M.A. Anal. Chem. 1983, 55, 1412A. 13. Armstrong, D.W.; Boehm, R.E. J . Chromatogr. Sci. 1984, 22, 378. 14. Garcia-Rubio, L.H.; Ro, N.; Patel, R.D. Macromolecules 1984, 17, 1998. 15. Garcia-Rubio, L.H. RECEIVED

March

10,

1987

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 5

A New Stand-Alone Capillary Viscometer Used as a Continuous Size Exclusion Chromatographic Detector W. W. Yau, S. D. Abbott, G. A . Smith, and M. Y. Keating Central Research and Development Department, Ε. I. du Pont de Nemours & Co., Wilmington, DE 19898

This paper describes a new design of a forced-flow-through-type capillary viscometer used for batch sample viscosity measurements as well as continuous viscosity detectio (SEC). In one version of our viscometer design, an analytical capillary is connected in series with a reference capillary in the flow stream and the pressure drop across the capillaries is measured by pressure transducers. A differential log-amplifier is used to convert the two transducer signals into an output signal that is directly proportional to the natural logarithm of the relative viscosity of the sample fluid. The output signal is highly insensitive to flow rate fluctuations and thus gives a very sensitive and accurate means to measure viscosity. The sample fluid could be any neat liquid or a sample of polymer solution. Under favorable conditions, a single viscosity determination on a polymer solution at high dilution can provide a direct measure of the polymer intrinsic viscosity, without the need of polymer concentration extrapolation. With this viscometer used as a continuous viscosity detector for SEC, it is possible to achieve SEC molecluar weight calibration by way of the universal SEC calibration methodology without the need of molecular weight standards for the unknown polymers. Background Accurate measurements of f l u i d v i s c o s i t y are important i n many industries f o r such diverse uses as monitoring syrup manufacture or studying polymer structures such as polymer branching, chain conformation, solvent interactions or polymer molecular weight (MW). H i s t o r i c a l l y , the drop-time type glass c a p i l l a r i e s , such as the Ubbelohde or Cannon and Fenske types, have been widely used to measure f l u i d v i s c o s i t y . However, t h i s t r a d i t i o n a l method i s tedius and labor intensive, and lacks the desired speed and s e n s i t i v i t y to 0097-6156/87/0352-0080$06.75/0 © 1987 American Chemical Society

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

81

meet the needs of v i s c o s i t y measurements, e s p e c i a l l y f o r d i l u t e polymer solution characterizations. The following i s a b r i e f review of the v i s c o s i t y parameters that are commonly used i n polymer analyses. The r e l a t i v e v i s c o s i t y (ri , ) of a polymer sample solution as defined i n Equation 1 can be determined experimentally from the measured v i s c o s i t y value f o r the polymer sample solution (η) and that of the solvent ( f i ) . From the Υ) , value and the polymer sample concentration (c), tne calculations f o r the other v i s c o s i t y parameters are possible i n accordance to Equations 2 through 5:

Relative V i s c o s i t y :

\el

Specific Viscosity:

Y\

Inherent V i s c o s i t y :

n

i n h

=

^^o

= \

e

l

" l

= (In \

(2) e l

)/c

(3)

Reduced V i s c o s i t y Intrinsic Viscosity:

[Y\] = l i m η. , - l i m η c-»o c->o

, (5)

1

where the mathematical symbol In means natural logarithm, and J^g means the l i m i t i n g value f o r the v i s c o s i t y parameter as the sample concentration c approaches zero at i n f i n i t e d i l u t i o n . The experimental determination of polymer i n t r i n s i c v i s c o s i t y i s done through the measurement of polymer solution v i s c o s i t y . The connotation of i n t r i n s i c v i s c o s i t y [ή], however, i s very d i f f e r e n t from the usual sense of f l u i d v i s c o s i t y . I n t r i n s i c v i s c o s i t y , or sometimes c a l l e d the l i m i t i n g v i s c o s i t y number, c a r r i e s a f a r more reaching significance of providing the size and MW information of the polymer molecule. Unlike the f l u i d v i s c o s i t y , which i s commonly reported i n the poise or centipoise units, the value i s reported i n the dimension of inverse concentration units of dl/g, for example. The value of [η] for a l i n e a r polymer i n a s p e c i f i c solvent i s related to the polymer molecular weight (M) through the Mark-Houwink equation: [ri] = KM

a

(6)

where Κ and ce*are Mark-Houwink v i s c o s i t y constants, some of which are available i n polymer handbooks. The usual value f o r α f a l l s between 0.5 and 0.8 for polymers of the random-coil type conformation i n solution. A more sensitive viscometer than the drop-time glass c a p i l l a r y method i s also needed i n size exclusion chromatography (SEC) such as the gel permeation chromatographic (GPC) analysis of polymer molecular weight d i s t r i b u t i o n (MWD). In an SEC system, a concentration detector i s commonly used for providing the weight concentration p r o f i l e of the polymer e l u t i o n curve. The MW and MWD information of the sample i s provided i n d i r e c t l y by the retention time of the d i f f e r e n t polymer components i n the sample. Quite obviously, i t i s highly desirable to have additional detectors available for SEC that are sensitive to the molecular weight of the d i f f e r e n t polymer components e l u t i n g from

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

82

DETECTION

A N D DATA ANALYSIS I N SIZE E X C L U S I O N

CHROMATOGRAPHY

the SEC columns. The on-line v i s c o s i t y and l i g h t scattering detection c a p a b i l i t i e s i n addition to the usual SEC concentration detector are very useful i n achieving absolute SEC-MW c a l i b r a t i o n . The required features of such a MW-specific detector f o r SEC include: continuous mode of monitoring, high s e n s i t i v i t y , low noise, and low dead volume for minimal SEC band broadening. E a r l i e r experiments involved the c o l l e c t i o n of SEC e f f l u e n t aliquots to measure s o l u t i o n v i s c o s i t y i n batches with the very time consuming Ubbelohde drop-time type viscometers. A continuous c a p i l l a r y type viscometer was f i r s t proposed f o r SEC by Ouano . B a s i c a l l y , as shown i n Figure 1, a single c a p i l l a r y tube with a d i f f e r e n t i a l pressure transducer was used to monitor the v i s c o s i t y of SEC e f f l u e n t at the e x i t of the SEC column. As l i q u i d continuously flows through the c a p i l l a r y (but not through the pressure transducer), the detected pressure drop (ΔΡ) across the c a p i l l a r y provides the measure f o r the f l u i d v i s c o s i t y (ή) according to the P o i s e u i l l e ' 4

ΔΡ - k Q η

(7)

where Q = flow rate, k = c a p i l l a r y geometrical constant, 4

k = 8L/JIR

(8)

with L « c a p i l l a r y length, and R = c a p i l l a r y inside radius. I n i t i a l t e s t i n g of such a detector was encouraging mainly because of i t s c a p a b i l i t y to continuously detect and record the SEC-viscosity e l u t i o n p r o f i l e . The detector f e l l short of being e n t i r e l y successful due to unfavorable signal-to-noise problems. There have been other attempts atgthe SEC v i s c o s i t y detector based on single c a p i l l a r y design. ' The performance of these viscometers, however, remains marginal because the pressure drop ΔΡ signal of a single c a p i l l a r y i s highly subjective to the unavoidable flow rate and temperature fluctuations. Attempts to compensate f o r flow rate f l u c t u a t i o n have been made by the use of multiple c a p i l l a r y tubes. Both B l a i r ' s and Haney's viscometers use a p a r a l l e l bridge design of four capillaries. ' The desired f l u i d v i s c o s i t y response i s detected by measuring the d i f f e r e n t i a l pressure across the c a p i l l a r y bridge. Haney's design has l e d to a commercial c a p i l l a r y viscometer (Viscotek Corp., Porter, Texas). The device provides much superior s e n s i t i v i t y over the e a r l i e r single c a p i l l a r y designs. Flow rate and temperature fluctuations are l a r g e l y eliminated to provide a very stable baseline. The size of the sample v i s c o s i t y s i g n a l , as measured by the d i f f e r e n t i a l pressure across the c a p i l l a r y bridge, however, i s s t i l l highly affected by flow rate changes. The measured d i f f e r e n t i a l pressure i s d i r e c t l y proportional to the o v e r a l l flow rate across the c a p i l l a r y bridge. The s e r i e s c a p i l l a r y design of Abbott and Yau overcomes, the problem of flow rate dependency of the viscometer response. This l a t t e r viscometer design i s the subject of t h i s paper. While t h i s viscometer i s described herein, with reference p a r t i c u l a r l y to polymer-solvent solutions, i t should be noted that the viscometer may be used with other type sample l i q u i d s as well.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

( P o i s e u e l l ' s Law Δ Ρ « k7/ , F

L

0

W

IN

k = 8QL

CAPILLARY

/qjR«) FLOW OUT

Pi

ΔΡ = Pi - P

u

VARIABLE RELUCTANCE TRANSDUCER

F i g u r e 1. D i f f e r e n t i a l Pressure Detection of F l u i d

Viscosity.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

84

The Key Design Features The new viscometer design u t i l i z e s two sets of the c a p i l l a r y and pressure transducer assemblies l i k e the one shown i n Figure 1. The c a p i l l a r i e s are connected i n series as shown i n Figure 2 f o r the stand-alone viscometer configuration. At the time the sample solution passes through the a n a l y t i c a l c a p i l l a r y , the reference l i q u i d or the c a r r i e r solvent continuously flows through the reference c a p i l l a r y . The d i f f e r e n t i a l pressure signals from the two capillary-transducer systems are fed to a d i f f e r e n t i a l logarithmic a m p l i f i e r . The d i f f e r e n t i a l logarithmic amplifier compares the input signals and gives the r a t i o between the a n a l y t i c a l and the reference pressure drops ΔΡ /ΔΡ . The output of the log-amplifier gives the desired measure of the natural logarithm of the sample r e l a t i v e v i s c o s i t y , that i s In η ,. Real time signal processing of the simultaneous pressure arops across the a n a l y t i c a l and reference c a p i l l a r i e s eliminate the e f f e c t s of flow rate an log-amplifier output s i g n a l noise i s the key to the high s e n s i t i v i t y and the signal-to-noise performance of the present viscometer. The following i s a mathematical analysis of the d i f f e r e n t i a l log-amplifier output signal (s) of the viscometer with the series c a p i l l a r y design: Α

S - In Δ Ρ

- In

Α

AP

R

= In (ΔΡ /ΔΡ ) Α

Κ

- In ( G k Q V G k Q n ) A

A

A

R

R

R

o

where G i s the e l e c t r o n i c gain, k i s the c a p i l l a r y geometrical constant as defined before i n Equation 8, Q i s again the flow rate, ri i s the v i s c o s i t y of the f l u i d i n the a n a l y t i c a l c a p i l l a r y , and n i s the solvent v i s c o s i t y ; the subscripts A and R refer to the a n a l y t i c a l and the reference c a p i l l a r y , respectively. Under the usual solvent flow rate and v i s c o s i t y conditions of SEC and viscometric measurements, the laminar flow requirement of the P o i s e u i l l e ' s equation (Eq. 8) i s e a s i l y s a t i s f i e d . The solution flow through the measuring c a p i l l a r y usually does not exceed a Reynold's number of 100, f a r below the condition for the on-set of turbulence. Since the flow rates i n two c a p i l l a r i e s connected i n series have to be the same, that i s Q. = Q , flow rate e f f e c t s are cancelled out from Equation 9 to give trie flow rate independent s i g n a l : R

s = m

(G k vcyyy

= In n

A

r e l

A

+

In ( G k / G k ) A

A

R

R

(10)

The second term i n Equation 10 i s a zero o f f s e t factor for the desired viscometer signal of In ri of the f i r s t term, where ï) - VU as noted before. 1

r e

rel

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

STAND-ALONE VISCOMETER

Figure 2

e

Viscometer-Configuration 1.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

86

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION

CHROMATOGRAPHY

By adjusting the gains of the ΔΡ signal e l e c t r o n i c a l l y so as to match G.k and G_k , that i s ln(G.k /G_k ) = In 1 = 0 to give the d i r e c t l f i n Mdout: Α Α κ κ A

r e l

S

-

l n

= n

\ e l i n h

χ c

(11)

And, at s u f f i c i e n t sample d i l u t i o n , S = [ή] χ c

(12)

The e l e c t r o n i c matching of the c a p i l l a r y performances can be e a s i l y accomplished by n u l l i n g the log-amplifier output when the viscometer pumps solvent through both the a n a l y t i c a l and the reference c a p i l l a r i e s , that i s when η = ft and l n = l n 1 = 0. The unique advantages of t h i s new viscometer design include: (1) true l n η , readou match the c a p i l l a r y flo factors between the a n a l y t i c a l and the reference c a p i l l a r y , (3) quick and convenient rematching of c a p i l l a r y performance i s e a s i l y done e l e c t r o n i c a l l y to o f f s e t any long term d r i f t of the c a p i l l a r y resistance due to the polymer build-up on c a p i l l a r y walls or any response factor v a r i a t i o n s of the pressure transducers, (4) high accuracy over a wide dynamic range of 0.0001 to about 5 i n the r e l a t i v e v i s c o s i t y u n i t s . On the other hand, the viscometer i s equally matched with the Viscotek viscometer i n enjoying the following a d d i t i o n a l advantages: (1) high s e n s i t i v i t y of better than the 10 relative viscosity units, (2) due to the high s e n s i t i v i t y , single point [η] determination i s possible without the need of sample concentration extrapolation, (3) c a p i l l a r i e s are of high length-to-diameter r a t i o and require no k i n e t i c energy and end-effect corrections, (4) shear-rate can be c o n t r o l l e d and varied to study the non-Newtonian behavior of a polymer s o l u t i o n or other neat l i q u i d samples, (5) easy adaptation to automation, (6) option to add a concentration detector to the batch viscometer to allow i n - s i t u sample concentration monitoring. Configurations As a Stand-Alone Viscometer Figures 2 and 5 i l l u s t r a t e two d i f f e r e n t configurations of the viscometer that can be used i n batch sample v i s c o s i t y determinations. In the viscometer-configuration 1 shown i n Figure 2, the reference c a p i l l a r y i s placed before the sample i n j e c t i o n valve. Carrier solvent i s i n the reference c a p i l l a r y a l l the time. Sample s o l u t i o n i s introduced into the solvent flow stream from a sample loop v i a a sample i n j e c t i o n valve. The solvent flow pushes the sample s o l u t i o n through the a n a l y t i c a l c a p i l l a r y where the v i s c o s i t y of the sample i s detected. The ΔΡ signal from each pressure transducer i s fed to a d i f f e r e n t i a l logarithmic amplifier as shown i n the Figure. The viscometer output i s the l n ft , s i g n a l of the sample s o l u t i o n . A pump i s used to c i r c u l a t e the solvent through the viscometer. The viscometer c a p i l l a r i e s are immersed i n a temperature c o n t r o l l e d l i q u i d bath

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

87

A concentration detector such as the d i f f e r e n t i a l refractometer i s shown here connected i n s e r i e s with the c a p i l l a r i e s . The following components are t y p i c a l l y used i n the viscometer: s t a i n l e s s s t e e l c a p i l l a r i e s of 1/16-in. o . d . and 0.016 i n . i . d . X 8 i n . long, 2 ml. sample loop, Celesco pressure transducers of 1 p s i r a t i n g , Valco 6 port sample valve, Burr Brown Log 100 J P . type d i f f e r e n t i a l l o g amplifier, VWR-1145 c i r c u l a t i o n temperature bath (-15 to 1 5 0 ° C ) . Several l i q u i d chromatographic pumps have been used. A Du Pont 860 pump was used to obtain the data reported i n t h i s work. In operation, the viscometer of Figure 2 w i l l generate two separate signal detector traces for recording. The d i f f e r e n t i a l log-amplifier w i l l generate a v i s c o s i t y (In h J trace while the concentration detector w i l l generate a concentration (c) trace. Both w i l l occur simultaneously and repeatedly from successive sample i n j e c t i o n s as shown i n Figure 3 for a polystyrene sample. From the In η , and the c s i g n a l s , the inherent v i s c o s i t y of the polymer sample can be calculated d i r e c t l y and accurately from the r a t i o of the signal amplitude obvious that the use o has the advantage of reducing operator errors i n preparing sample solutions of desired concentrations. The flow rate independence of t h i s viscometer has been demonstrated by i n t e n t i o n a l l y varying the flow rate during the sample analyses. In Figure 4, the log-amplifier signal (In η , ) i s recorded at the top, the ΔΡ and ΔΡ signals are also recorded as the middle and the bottom traces respectively. The upsets i n the bottom ΔΡ trace r e f l e c t the i n t e n t i o n a l flow rate v a r i a t i o n s manipulated by upsetting the pump flow rate c o n t r o l . Such flow rate upsets have greatly disturbed the ΔΡ. signal as w e l l , e s p e c i a l l y at the top of the ΔΡ. response to an injected sample viscosity. The log-amplifier s i g n a l , however, i s not affected by the flow rate upsets. The i n t e g r i t y of the log-amplifier signal gives credence to the true In η , measurement of the viscometer. An a l t e r n a t i v e configuration for a batch viscometer i s shown i n Figure 5. In t h i s case, the reference c a p i l l a r y i s also placed downstream from the sample i n j e c t i o n valve. A delay volume i s added between the a n a l y t i c a l and the reference c a p i l l a r i e s . The function of the delay volume i s to prevent sample solution reaching the reference c a p i l l a r y during the time that the sample v i s c o s i t y i s being monitored i n the a n a l y t i c a l c a p i l l a r y . With the delay volume, the sample ΔΡ signal i s s t i l l referenced against the solvent ΔΡ signal to give the true In η , measurement for the sample. At the completion of the sample measurement, the viscometer w i l l reset i t s e l f as the sample solution flushes through the reference c a p i l l a r y . This viscometer-configuration 2 i s shown i n Figure 5 with an optional UV concentration detector connected i n the p a r a l l e l arrangement. In operation, the viscometer i n Figure 5 w i l l generate the dual trace v i s c o s i t y and concentration signals shown i n Figure 6 for a polyethylene-terephthalate sample. The flushing of the delay volume can be monitored by the returning of the negative l o g - a m p l i f i e r signal back to baseline. Compared to the e a r l i e r viscometer configuration of no delay volume, t h i s viscometer configuration offers a better signal-to-noise performance, however, at the cost of longer sample analysis time due to the a d d i t i o n a l time required to flush out the delay volume. A c o i l e d large i . d . tubing of about 4 to 6 ml volume i s t y p i c a l l y used as the viscometer delay volume.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

DETECTION

A N D DATA A N A L Y S I S IN SIZE E X C L U S I O N C H R O M A T O G R A P H Y

PRECISION OF S T A N D - A L O N E VISCOMETER

P S - 1 7 . 5 K M W in THF h ] -

0.135 d l / g . 1% Cone.

1.0 ml/min. Flowrate , n 7

?rel

v

J

1

I

I

L

Time (minutes) — -

Figure 3

• .

P r e c i s i o n o f the D i f f e r e n t i a l Pressure Viscometer.

e

LOB-AMP VISCOSITY READOUT INDEPENDENT OF FLOWRATE !.. ! p.,. .............. . .|. j .:. : ! ; : : | ! . SI BNAL UNP FF ICI ED BY FLOW RAI ϋφφϋ ι.

....,....)....

• ·

;

=

In

Λ

i— i

!

;

.

:

ΔΡα' , i "':j"~

.. : . . . .

:..

4

::•·!•

1

ΔΡ„ , ; : ! RFFRFNCF: CAPILLARY :

• .i :

: :

ô

:.N

;

r

t &

;:.·

^

:: Γ :.!::"!" 1 :

. . . · | - · · · | ν |; :•

l'y.

I

:::Ί vt::::!. ' ι.

Γ::

:|:::.r

-:/

s'

ii;:N::-| •

:-:.|:::-

1

—j • • · ·

1

FLOWRATE UPSET BY — j · - • VARYING · THE PUMP SETTING ::: |: : ;

: : :y

ι

'''ï"Œ

::::

; :

_ I , :: ί. :! . j . . - !:. :

JA

!·

Ί

0

U-

;

i ••'

\

/

Figure 4

!

...)....

\ ul ?

! : . ι

1 1

j

1

\ •

:

'•['•'

::::|::::

rΓB

1

: '

i

VA

Si

'

.!

: !

::::!:

-

jj

ι , , 05

ANALYTICAL CAPILLARY Γ

J

\ V

j r ο:

uv 03;

/

!, i

|

. ί

Λ

i

-J M

:

::·:

/ΔΡη)

7]

:

i l

ι

In ( Δ Ρ

... 1 ···•··;·• ;

, ........... .:t." : •*":.:..:'

1 IM ....

:

:

:

-f

.'

,l,

\h

-

: ;

i

T

j

]

L.l.1;... ...i

i... :.. ;

....

Flow r a t e Independent V i s c o m e t e r Response. Sample: p o l y s t y r e n e 17500 MW S o l v e n t : THF, c o n c e n t r a t i o n : 1%.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

STAND-ALONE

89

VISCOMETER

< P o i * « u e l i ' » Law: Δ Ρ - kQ7/ , k - 8L

Figure 5 « Viscometer-Configuration 2.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

)

90

D E T E C T I O N A N D DATA ANALYSIS IN SIZE E X C L U S I O N

CHROMATOGRAPHY

Due to the high s e n s i t i v i t y of the viscometer, accurate readings of v i s c o s i t y responses can be made with sample values much l e s s than 1.1. Therefore, sample solutions of very low concentration can be analyzed i n the viscometer to produce a single point i n t r i n s i c v i s c o s i t y determination without the need of concentration extrapolation. As shown i n Figure 7 for a polystyrene sample, any Y\. , determination at the r i - , value of much less than 1.1 gives jbPactically the i n t r i n s i c v i s c o s i t y of the sample. The approximation can be made even better by c a l c u l a t i n g the sample i n t r i n s i c v i s c o s i t y from the Solomon-Gatesman equation: [Vl] = l i m 7 2 ( n

r e l

-l-lnn

r e l

)/c

(13)

c-X) Configurations as a SEC Detector Figures 8 and 10 i l l u s t r a t viscometer used as an on-lin In the detector-configuration 1 shown i n Figure 8, a SEC column set i s placed between the sample i n j e c t i o n valve and the a n a l y t i c a l c a p i l l a r y of the viscometer. A large depository column has been added between the a n a l y t i c a l and the reference c a p i l l a r y . The SEC concentration detector such as the d i f f e r e n t i a l r e f r a c tometer (R.I.) i s shown here connected i n the series arrangement following d i r e c t l y a f t e r the a n a l y t i c a l c a p i l l a r y . T y p i c a l l y , a large low pressure column of about 300 ml volume f i l l e d with large packing beads can be used as the depository column. In operation, the SEC detector of Figure 8 w i l l generate both the v i s c o s i t y (In η .) and the concentration traces for recording the SEC elutiofi curve p r o f i l e . The polymer bands e l u t i n g from the SEC column and the a n a l y t i c a l c a p i l l a r y w i l l be dras­ t i c a l l y d i l u t e d i n concentration when they f i n a l l y emerge from the depository column and reach the reference c a p i l l a r y . For a l l p r a c t i c a l purposes, the pressure drop across the reference c a p i l l a r y w i l l be responding to the solvent v i s c o s i t y and any flow rate changes, while the pressure drop of the a n a l y t i c a l c a p i l l a r y w i l l respond to the v i s c o s i t y of the e l u t i n g polymer bands as well as any flow rate changes. Some build-up of polymer concentration i n the depository column w i l l occur a f t e r many sample analyses i n close time i n t e r v a l s . Flushing out the depository column may be necessary at times. This can be done for example by the continuous pumping of solvent through the system overnight. The flow rate independence of the new SEC v i s c o s i t y detector design has been demonstrated by i n t e n t i o n a l l y working with a very large pump flow rate noise as shown i n Figure 9. The noisy pump response was created when two of the three pistons of a Du Pont 860 pump were made inoperative, leaving only a single reciprocating piston to do the pumping. The Figure shows the ΔΡ and the ΔΡ signals at two flow rate l e v e l s , while the SEC e l u t i o n peaks are barely v i s i b l e i n the noisy ΔΡ signal at the top trace, they are, however, c l e a r l y detected i n tne log-amplifier signal shown at the bottom of the Figure. This i s the result of the very e f f e c t i v e cancellation of pump noise by the log-amplifier i n the present viscometer design. Another thing to notice i s the size of the e l u t i o n peaks i n the log-amplifier trace. The fact that the size e

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector STAND-ALONE VISCOMETER RESULT (Dacron in HFIP) Sample: 0.1% Concentration

Figure 6. Typical Viscometer Output Signal Traces.

(Differential-pressure

C a p i l l a r y Viscometer)

POLYSTYRENE STANDARD- 17. 500 WW SOLVENT- TETRAHYDROFURAN FLOHRATE- 1.0 ML/MIN. 0.2

/SINGLE POINT INTRINSIC 0.15

o.i

0.05

77—

-

77..

7).„*.

- In 77

C 77 3

-V2

'C-C77 3*Ki.t77 3 * C -

< 7/

/ C - t 7 7 3 - k ^ t 7 7 3*C » - I - In 7 7 - - » ) / C

k„ -» Κ» - 0.313 • 0.184 - 0.499

KREAMER SOLOMON & BATESMAN

EXPECTED O.S

1 CONCENTRATION (g/dl)

2

Figure 7. Single Point I n t r i n s i c C a p a b i l i t y of the Viscometer

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

D E T E C T I O N A N D DATA ANALYSIS IN SIZE E X C L U S I O N C H R O M A T O G R A P H Y

D I F F E R E N T I A L P R E S S U R E C A P I L L A R Y V I S C O M E T E R A S AN I N - L I N E V I S C O S I T Y DETECTOR

WASTE

LOG

Figure

7) r e l

8

C

SEC

Detector-Configuration

1.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

5. YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

Figure 9o Flow Rate-Independent SEC-Viscosity Detection Sample: polystyrene mixture (1.8M + 100K + 9K) Solvent: THF, Concentration: 0.08% 2X Du Pont PSM-Bimodal columns 100 μΐ sample loop.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

93

94

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION

CHROMATOGRAPHY

of the e l u t i o n peaks remain the same at two flow rate l e v e l s i s indicative of the v i s c o s i t y detector providing the true v i s c o s i t y information of the polymer sample. T y p i c a l l y , with an average good working LC pump, the SEC v i s c o s i t y e l u t i o n curves can be obtained with very l i t t l e baseline noise, much better than the noise shown here. The usual SEC Sample loop size of 25 or 50 μΐ i s normally adequate for the SEC v i s c o s i t y analyses as w e l l . An a l t e r n a t i v e configuration for a SEC v i s c o s i t y detector i s shown i n Figure 10. The depository column i n the Figure 8 detector i s replaced here by a considerably smaller delay volume. The delay volume element i s nothing more than a c o i l e d large i . d . tubing connecting the a n a l y t i c a l and the reference c a p i l l a r y . The delay volume should be s u f f i c i e n t l y large so that, when the separated polymer bands are e l u t i n g through the a n a l y t i c a l c a p i l l a r y , only pure solvent i s present i n the reference c a p i l l a r y at the time. T y p i c a l l y , a delay volume of about 10 ml i s s u f f i c i e n t for SEC systems where a set of two high performance SEC columns are used. The v i s c o s i t y detector i t r a t i o n detector connecte operations, t h i s v i s c o s i t y detector w i l l reset i t s e l f as the e l u t i n g polymer bands completely sweeps through the delay volume. Compared to the e a r l i e r detector configuration, t h i s detector configuration o f f e r s a better signal-to-noise performance. The self-cleaning and reset feature i s a considerable advantage. However, a d d i t i o n a l time i s required with t h i s detector configuration to f l u s h out the delay volume a f t e r every sample. Figure 11 i l l u s t r a t e s an SEC separation of a sample of 3-component polystyrene mixture with the dual concentration and v i s c o s i t y detectors of Figure 10. The top trace shows the concentration e l u t i o n p r o f i l e of the SEC separation as detected by a UV-photometer. The bottom trace records the same SEC separation, except with the viscometer s i g n a l from the log-amplifier output. The viscometer response i s highly noise free and i s shown here, as expected, being highly biased i n favoring the detection of the early e l u t i n g high MW component. The l a s t e l u t i o n peak occurring before the f l u s h i n g of the delay volume i s caused by the impurity i n the sample preparation. When the polymer sample i s flushing through the delay volume and the reference c a p i l l a r y , a negative log-amplifier s i g n a l r e s u l t s as shown i n the Figure. The f l u s h i n g of the delay volume can be watched through the log-amplifier signal. The high v i s c o s i t y of some high MW samples i s known to cause flow rate upsets as the sample passes through the SEC column f r i t s . Such flow rate upsets often occur at the time of e l u t i o n of the sample. While the flow rate upsets l i k e t h i s are l i k e l y to cause v i s c o s i t y detection errors i n most other SEC v i s c o s i t y detectors, the signal of the present v i s c o s i t y detector, however, w i l l remain true, and unaffected by the high sample v i s c o s i t y problem. Placement of the reference c a p i l l a r y ahead of the SEC columns and the sample i n j e c t i o n value i s not an acceptable configuration for an SEC v i s c o s i t y detector. Being at the high back pressure location, the flow rate noise sensed by the reference c a p i l l a r y would be out of phase with that sensed by the downstream a n a l y t i c a l c a p i l l a r y . The compressibility of the column l i q u i d volume under high pressure acts as an hydraulic capacitance causing the phase s h i f t s of the flow noise between the two c a p i l l a r i e s . The r e s u l t i s incomplete c a n c e l l a t i o n of flow rate f l u c t u a t i o n s .

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

YAU ET A L .

Capillary Viscometer as a Continuous SEC Detector

DIFFERENTIAL PRESSURE CAPILLARY VISCOMETER AS AN IN-LINE

yiSCTSITY DETECTOR.

LOG

7) REL

Figure 10. SEC Detector-Configuration

2.

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

96

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

DUAL GPC CONCENTRATION & VISCOSITY TRACES

DuPont Bimodal PSM Columns Polystyrene Mixture in THF

Impurity

P S - M W : 1.8M + 100K + 9K Flowrate«1.5ml/min.

UV CONC.

Inject } I < r >f o

g

br

g

-P >Tj

·_ _G

;~

.,-· ·

*

r—' rn {J

μ-,

r-1 fO

C •

2

0 > b

/

0 ( J l

)

(10)

M

2 2 where , and ^ a r e the unperturbed r a d i i of g y r a t i o n f o r a branched and a l i n e a r polymer, r e s p e c t i v e l y , of t h e same m o l e c u l a r w e i g h t . The v a l u e of ε l i e s between 1/2 and 3/2 f o r a branched polymer d i s s o l v e d i n good s o l v e n t . F o r a s t a r - s h a p e d polymer, g can be e s t i m a t e d by u s i n g t h e random walk model (45) q

*R.W.

"

( 3 f

"

2 ) / f 2

0

1

)

where f i s t h e number of branches t h a t r a d i a t e from a branch p o i n t . F e t t e r s and co-workers i n a r e c e n t s t u d y (49) on 12- and 18-arm s t a r polymers found t h a t i n t h e t a s o l v e n t , g i s always g r e a t e r than g ^ s u g g e s t i n g t h a t these s t a r polymers a r e expanded a t t h e t h e t a temperature. They a l s o found t h a t i n t h e t a s o l v e n t , t h e v a l u e of ε as d e s c r i b e d i n E q u a t i o n 9 i s around 0.62 and seems t o be a lower l i m i t f o r branched polymers. I n good s o l v e n t s , t h e v a l u e s of ε a r e h i g h e r than t h a t i n t h e t h e t a s o l v e n t . T h i s i s c l e a r l y seen from Table 6 which l i s t s some of t h e hydrodynamic d a t a t a k e n from t h a t paper a l o n g w i t h our r e s u l t s . I n f a c t , both v a l u e s of g and ε i n THF a r e i n e x c e l l e n t agreement w i t h the v a l u e s o b t a i n e d i n t o l u e n e . R

T

American Chemical Society Library 1155 16th St., N.W. In Detection and Data Washington, Analysis in Size D.C. Exclusion Chromatography; Provder, T.; 20036 ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

F i g u r e 13. P l o t of l o g [η] v s . l o g M f o r a Randomly Branched Polystyrene.

F i g u r e 14. P l o t s of l o g [η] v s . l o g M f o r a L i n e a r and a Branched P o l y s t y r e n e .

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

8.

Viscometric Detector for SEC Characterization of MWD

KUO ET AL. TABLE 6.

HYDRODYNAMIC DATA FOR

149

12-ARM STAR POLYSTYRENE*

2

exptl =

= 0.276

star

=

g R.W.

0.236 (Random Walk)

(3f-2)/f

S ' c y d o h e x a n e " ^ 1 star/[η ]

£

g ,

Toluene

=

= 0.41

= g ^ ;

^Istar/[η 1^ = 0.35

ε =

; ε =

0.62 0.72

THIS WORK g' * See Reference

= 0.35

T H p

49

F i g u r e 16 shows the v i s c o m e t e r and DRI t r a c e s of another s t a r - b r a n c h e d p o l y s t y r e n e . T h i s sample c o n t a i n e d about 12% of the s t a r t i n g l i n e a r arm p r e c u r s o r which e l u t e d at r e t e n t i o n volume c a . 52 ml. The k i n e t i c m o l e c u l a r weight of the l i n e a r p r e c u r s o r was 260,000. The r e s u l t s o b t a i n e d f o r the i n d i v i d u a l peak through the S E C / V i s c o s i t y methodology are summarized i n Table 7. I t i s seen t h a t the measured M^ of the l i n e a r arm i s v e r y c l o s e d to the k i n e t i c v a l u e . The average f u n c t i o n a l i t y of t h i s s t a r polymer i s c a l c u l a t e d to be f = 10.

TABLE 7.

FRACTION

SEC/VISCOMETRY RESULTS OF STAR PS-W

Μ

n

S t a r polymer L i n e a r Arm

g

f

χ

10

1,920 241

= 0.325

-3

Μ

w

χ

10

h](dl/g)

1.72

2,530 254 (260) ε =

K i n e t i c v a l u e of m o l e c u l a r weight

(WHOLE POLYMER)

0.914

f

10.0 1.0

3

0.88

of l i n e a r p r e c u r s o r

P o l y v i n y l A c e t a t e . Two p o l y v i n y l a c e t a t e samples (PVAc #1 and PVAc #3) a l s o were a n a l y z e d . Both samples have been shown to be branched by Hamielec (50) by a SEC/LALLS s t u d y . Table 8 shows the r e s u l t s o b t a i n e d from SEC/Viscometry a l o n g w i t h some of the a v a i l a b l e d a t a . I t i s seen t h a t the i n t r i n s i c v i s c o s i t y v a l u e s f o r

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

F i g u r e 15. P l o t of B r a n c h i n g Index as a F u n c t i o n of M o l e c u l a r Weight f o r a Randomly Branched P o l y s t y r e n e .

2*

32

40

48 RETENTION

56 VOLUME

6ft (ML)

72

80

88

F i g u r e 16. DRI and V i s c o m e t e r Traces of an U n f r a c t i o n a t e d Star-shaped P o l y s t y r e n e .

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

8. KUO ET A L .

151

Viscometric Detector for SEC Characterization of MWD

both samples a r e i n good agreement w i t h the ASTM Round Robin r e s u l t s o b t a i n e d w i t h o f f - l i n e measurements. The m o l e c u l a r weight averages a r e comparable w i t h the r e s u l t s o b t a i n e d from v a r i o u s s o u r c e s . The Mark-Houwink parameters, Κ and a, o b t a i n e d from l i n e a r e x t r a p o l a t i o n of l o g [η] _vs. l o g M, f e l l between the two v a l u e s o f t e n c i t e d f o r l i n e a r p o l y v i n y l a c e t a t e . ( 8 , 5 1 ) Dawkins and co-worker (52) i n t h e i r most r e c e n t p u b l i c a t i o n a l s o found t h a t i s the c a s e . F i g u r e 17 shows the d e v i a t i o n of the p l o t l o g [η] v s . l o g M from l i n e a r i t y . A l s o shown i n the p l o t i s the m o l e c u l a r weight d i s t r i b u t i o n c u r v e . The b r a n c h i n g i n d e x , g , as a f u n c t i o n of m o l e c u l a r weight i s shown i n F i g u r e 18 a l o n g w i t h the m o l e c u l a r weight d i s t r i b u t i o n c u r v e . T

TABLE 8. SEC/VISCOMETRY RESULTS OF TWO POLYVINYL ACETATE SAMPLES

SOURCE

Mn χ 10

S E C / V i s c o m e t r y ( T h i s work)101 Aldrich 83.4 Hamielec, e t . a l . 90.2 ASTM Round Robin 8_3_._4_

Mw χ 10

[n](dl/g)

287 331 300.2 2_63_

0.79

Κ , χ 10

0.89

0.757

a

0.81

PVAC #3

SOURCE

Mn χ 10

S E C / V i s c o m e t r y ( t h i s work)109 Aldrich 103 Hamielec, e t . a l . 146 ASTM Round Robin 102 Graessley, e t . a l . D i e t z , e t . al.° Dawkins, e t . a l . ^ Extrapolated See Reference ^ See Reference See Reference

b

Mw χ 10

695 840 626 587_

[n](dl/g)

Κ , χ 10

1.48

0.86

K51

0.51 1.56 0.942

0.761

0.791 0.708 0.737

from the l i n e a r p o r t i o n of l o g [η] v s . l o g M c u r v e . 8. 51. 52.

Summary I n t h i s paper, the enhancement of the s i g n a l - t o - n o i s e r a t i o of the v i s c o m e t e r s i g n a l through the i m p l e m e n t a t i o n of a n u m e r i c a l FFT t e c h n i q u e i s d i s c u s s e d and the c o m p u t a t i o n a l procedures a r e d e s c r i b e d . A number of examples of q u a n t i t a t i v e a p p l i c a t i o n s t o a

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

r r n iiiii—ι ι ι i i n i | — ι

ι ι mill—ι

ι ι iniij—ι ι ι iiim|—r—H

F i g u r e 17. P l o t of l o g [η] v s . l o g M f o r a Branched A c e t a t e (PVAc) ( A l d r i c h 18250-8 L o t #3).

Polyvinyl

DC Û.

Ο

MOLECULAR WEIGHT

F i g u r e 18. P l o t of B r a n c h i n g Index as a F u n c t i o n of M o l e c u l a r Weight f o r a Branched PVAc ( A l d r i c h 18250-8 L o t #3).

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

8.

KUO ET AL.

Viscometric Detector for SEC Characterization of MWD

153

wide v a r i e t y of l i n e a r and branched polymers are shown. Overall, the SEC/Viscometry d e t e c t o r methodology d e s c r i b e d i n t h i s paper p r o v i d e s v e r y a c c u r a t e a b s o l u t e m o l e c u l a r weight d i s t r i b u t i o n s , m o l e c u l a r weight a v e r a g e s , b u l k i n t r i n s i c v i s c o s i t y v a l u e s , and Mark-Houwink Κ and α parameters from a s i n g l e SEC e x p e r i m e n t . I n a d d i t i o n , a c c u r a t e i n t r i n s i c v i s c o s i t y - m o l e c u l a r weight r e l a t i o n s can be o b t a i n e d f o r l i n e a r and branched polymers as w e l l as a b r a n c h i n g i n d e x as a f u n c t i o n o f m o l e c u l a r w e i g h t .

Literature Cited 1. Moore, J. C., J. Polym. Sci., A,2, 835(1964). 2. Meyerhoff, G., Makromol. Chem., 118, 265(1968). 3. Goedhart, D., and Opschoor, Α., J. Polym. Sci., A2,8, 1227(1970). 4. Meyerhoff, G., Separ. Sci., 6, 239(1971). 5. Grubisic-Gallot, Z., Picot, M., Gramain, P., and Benoit, H., J. Appl. Polym. Sci. 6. Gallot, Z., Marais, L., and Benoit, H., J. Chromatogr., 83, 363(1973). 7. Servotte, Α., and DeBruille, R., Makromol. Chem., 176, 203(1975). 8. Park, W. S., and Grassley, W. W., J. Polm. Sci., Polm. Phys. Ed., 15, 71(1977). 9. Constantin, D., Eur. Polym. J., 13, 907(1977). 10. Janca, J., and Kolinsky, S. J., J. Chromatogr., 132, 187(1977). 11. Janca, J., and Pokorny, S., J. Chromatogr., 134, 273(1977). 12. Ouano, A. C., J. Polym. Sci., A-1, 10, 2169(1972). 13. Lesec, J., and Quivoron, C., Analusis, 4, 399(1976). 14. Letot, L., Lesec, J., and Quivoron, C., J. Liq. Chromatogr., 3, 427(1980). 15. Malihi, F. Β., Koehler, Μ. Ε., Kuo, C., and Provder, T., Pittsburgh Conference on Analytical Chemistry and Applied Spectroscopy, Paper No. 806(1982). 16. Malihi, F. B., Kuo, C., Koehler, Μ. Ε., Provder, T., and Kah, A. F., "Size Exclusion Chromatography Methodology and Characterization of Polymers and Related Materials", T. Provder, Ed., ACS SYMPOSIUM SERIES NO. 245, 281(1984). 17. Lecacheux, D., Lesec, J., and Quivoron, C., J. Appl. Polym. Sci., 27, 4867(1982). 18. Viscotek Corporation, Porter, Texas. 19. Haney, Μ. Α., J. Appl Polym Sci., 30, 3037(1985). 20. Abbott, S. D. and Yau, W. W. U. S. Patent 4 578 990, April 1, 1986. 21. Miller, Τ. Ε . , and Small, Η., Anal. Chem., 54, 907(1982). 22. Miller, Τ. Ε., Chamberlin, Τ. Α., and Tuinstra, Η. Ε., Am. Lab, January, 1983. 23. Koehler, Μ. Ε., Kah, A. F., Niemann, T. F., Kuo, C., and Provder, T., "Size Exclusion Chromatography Methodology and Characterization of Polymers and Related Materials", T. Provder, Ed., ACS SYMPOSIUM SERIES, NO. 245, 57(1984). 24. Hamielec, A. E. and Ouano, A. C., J. Liq. Chromatogr., 1(1), 111()1978).

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

25. Foster, G. Ν., MacRury, Τ. Β., and Hamielec, Α. Ε., in "Liquid Chromatography of Polymers and Related Materials II", J. Cazes, Ed., Dekker, N.Y., 1979, pp.143. 26. Zimm, Β. Η., and Stockmayer, W. H., J. Chem., Phys., 17, 1301 (1949). 27. Digital Equipment Corporation (Maynard, MA), Laboratory Subroutine Manual #AA-C984A-7. 28. Provder, T. and Rosen, Ε. Μ., Separ. Sci., 5(4), 437(1970). 29. Busnel, J. P., Polymer, 23, 137(1982). 30. Yau, W. W., and Malone, C. P., J. Polym. Sci., Polym. Letters Ed., 5, 663(1967). 31. Olsson, D. M., J. Quality Technology, 6, 53(1974). 32. Olsson, D. Μ., and Nelson, L. S., Technometrics, 17, 45(1975). 33. Smith, J. Μ., "Mathematical Modeling and Digital Simulation for Engineers and Scientists", John Wiley and Sons, Ν. Y., 1977. 34. Stickler, Μ., and Eisenbeiss, F., Eur. Polym. J., 20, 849(1984). 35. Hamielec, Α. Ε., an Polymeric Materials: Science and Engineering, 51, 541(1984). 36. Rosen, Ε. M. and Provder, T., Separ. Sci., 5(4), 485(1970). 37. Rudin, Α., and Wagner, R. Α., J. Appl. Polym. Sci., 20, 1483(1976). 38. Lecacheux, D., and Lesec, J., J. Liq. Chromatogr., 12, 2227 (1982). 39. Hellman, Μ. Y., in "Liquid Chromatography of Polymers and Related Materials"; J. Cazes, Ed., Dekker, NY, 1977, p. 29. 40. Provder, T., Woodbrey, J. C., and Clark, J. Η., Separ. Sci., 6, 101(1971). 41. Samay, G., et al, Makromolecular Chemistry, 72, 185(1978). 42. Takahashi, Α., Ohara, Μ., and Kagawa, I., Kogyu Kagaku Zasshi, 66, 960(1963). 43. Cane, F., and Capaccioli, T., Eur. Polym. J., 14, 185(1978). 44. Freeman, Μ., and Manning, P. Β., J. Polym. Sci., A, 2, 2017(1964). 45. Lyngaae-Jorgensen, J., 7th International GPC Seminar, 1969, p. 188. 46. Bohdanecky, Μ., Sole, K. Kratochvil, P., Kolinsky, Μ., Ryska, Μ., and Lim, D., J. Polym. Sci., A-2, 5, 343(1967). 47. Rudin, Α., private communication. 48. Zimm, Β. Η., and Stockmayer, W. Η., J. Chem. Phys., 17, 1301(1949). 49. Roovers, J., Hadjichristidis, Ν., and Fetters, L. J., Macromolecules, 16, 214(1983). 50. Foster, G. Ν., MacRury, T. B., and Hamielec, Α. Ε., in "Liquid Chromatography of Polymers and Related Materials II"; J. Cazes and X. Delamare, Eds., Marcell Dekker, Inc., NY, 1980, pp. 143. 51. Atkinson, C. Μ., and Dietz, R., Eur. Polym. J. 15, 21(1979). 52. Coleman, Τ. Α., and Dawkins, J. V., J. Liq. Chromatogr., 9, 1191(1986). R E C E I V E D April 21, 1987

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 9

A New Detector for Determining Polymer Size and Shape in Size Exclusion Chromatography L. Brower, D. Trowbridge, D. Kim, P. Mukherjee, R. Seeger, and D. McIntyre The Institute of Polymer Science, The University of Akron, Akron, OH 44325 A viscometer has been constructed using membrane pores as capillary viscometers and can be readily adapted to conventional SE shown to give solvent flow and intrinsic viscosities for moderate molecular weight polymers of M W approximately 100,000. The viscometer has been used to analyze different types of polymer structures including microgel. The microgel pressure difference measurements correlate with conventional measurements of gel content. The chromatograms of microgels in both natural rubbers and commercial acrylic polymers can be obtained and give a rapid method of detecting microgel. Also the detailed chromatographic patterns show the possibility of differentiating between types of microgel. The observed gel viscometry results are discussed in terms of polymer entanglements. Traditional SEC has had difficulties handling extremely large molecules in which linear polymers have molecular weights exceeding 10 g/mol.(1,2), Also the different shapes of large molecules are difficult to pin point unless there is a marked difference from the universal ca1ibration.(3) Finally very large molecules commonly known as microgel (consisting of internally cross!inked and branched chains of colloidal dimensions) present a scientific challenge to characterize adequately by any technique. In addition microgel is also a laboratory hazard to be avoided in a routine SEC measurements because of its inadvertent plugging of costly columns. In response to the above characterization problems and an interest in understanding the topology of intramolecular entanglement a membrane viscometer was developed.(4 ; In the membrane viscometer a solution is passed through a thin CM0 μrn) membrane with well-defined pores of fixed diameter that are nearly perpendicular to the membrane surface. The important feature is 0097-6156/87/0352-0155$06.00/0 © 1987 American Chemical Society

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

156

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

t h a t the pore (or h o l e ) d i a m e t e r s , D , be a v a i l a b l e i n a range of s i z e s such t h a t the polymer (or molecule) of d i a m e t e r , D , can pass through the large holes f r e e l y or the s m a l l h o l e s w i t h somedifficulty. Both the p r e s s u r e drop a c r o s s the membrane and the c o n c e n t r a t i o n of e f f l u e n t are measured c o n t i n u o u s l y w h i l e the flow r a t e s of e f f l u e n t are i n c r e a s e d i n s t e p - w i s e or c o n t i n u o u s fashion. At t h i s stage of development o n l y the unconfined time-average r a d i u s of g y r a t i o n i s c o n s i d e r e d as ^ D . However other v i s c o m e t r i c or p r o j e c t i o n r a d i i may u l t i m a t e l y be more u s e f u l . E a r l y measurements i n a s t e a d y - s t a t e f l o w a p p a r a t u s showed that the membrane viscometer a l l o w s the d i r e c t c a l c u l a t i o n of kinematic viscosities t h a t are i n good agreement w i t h independent capillary viscometer measurements under l i m i t e d c o n d i t i o n s . Agreement i s e x c e l l e n t when (1) the average polymer diameter i s s m a l l e r than the membrane h o l e , that i s , D < D , and (2) the e f f l u e n t f l o w r a t e or i t | r e l a t e d maximum shear r a t e , dv/dx, i s not too l a r g e , (dv/dx > < 10 s *.(4) However, i t a l s becam c l e a i th e a r l measurement that to have a g e n e r a l l c h a r a c t e r i z a t i o n of polymer c o n c e n t r a t i o n p o l a r i z a t i o n and d e l i v e r a p u l s e of polymer through the membrane. To that end an i n j e c t i o n loop was used to introduce the polymer s o l u t i o n to the membrane.(5) The performance of the p u l s e d f l o w ( i n j e c t i o n loop) i s s i m i l a r to that of the unpulsed u n i t when l i n e a r polymer m o l e c u l e ^ are not too c o n f i n e d by the h o l e , the shear r a t e s are l e s s than 10 s , and c o n c e n t r a t i o n s are l e s s than 100 pom. The p u l s e d f l o w apparatus has s i n c e been used to e x p l o r e i n a p r e l i m i n a r y way the a n a l y s i s of extremely h i g h molecular weight polymers, e n t a n g l e d polymers, h i g h l y branched polymers, and m i c r o g e l s . In t h i s paper a b r i e f d e s c r i p t i o n of the a p p a r a t u s i s p r e s e n t e d and discussed, and then some i n t e r e s t i n g p r e l i m i n a r y r e s u l t s on m i c r o g e l s are g i v e n . F i n a l l y a s p e c u l a t i v e d e s c r i p t i o n of the m o l e c u l a r rearrangements that occur during these membrane measurements i s f o l l o w e d by a few remarks on the a n a l y t i c a l p o t e n t i a l of the membrane v i s c o m e t e r detector. Special c o n s i d e r a t i o n i s g i v e n to i t s use as a g e n e r a l purpose addition to SEC equipment i n l a b o r a t o r i e s a l r e a d y analyzing polymers. h

m

n

m

h

Membrane Viscometer The membrane viscometer must use a membrane w i t h a sufficiently w e l l - d e f i n e d pore so that the f l o w of i s o l a t e d polymer m o l e c u l e s i n s o l u t i o n can be analyzed as P o i s e u i l l e f l o w i n a long capillary, whose length/diameter i s > 10. As such the v i s c o s i t y , η,, of a Newtonian f l u i d can be determined by measuring the p r e s s u r e drop across a s i n g l e pore of the membrane, knowing i n advance: the t h i c k n e s s , L, and c r o s s s e c t i o n . A, of the membrane, the r a d i u s of the p o r e , R | _ , the f l o w r a t e per pore, Q; , and the number of pores per u n i t a r e a , N. The v i s c o s i t y , the maximum shear s t r e s s , σ, and the v e l o c i t y g r a d i e n t , r , can be c a l c u l a t e d from laboratory measurements of the above i n s t r u m e n t a l parameters where = Q^^/N. η, =

TTR^

8L

ΔΡ Q t

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

9.

A Detector for Polymer Size and Shape in

BROWER ET AL. y

=

H

SEC

157

Q,

ÏÏFT

σ = R _ ΔΡ 2L Design Concept The o b j e c t i v e of t h i s r e s e a r c h was to b u i l d an a p p a r a t u s which c o u l d measure the f l o w behavior of polymer s o l u t i o n s flowing through porous media. The concept of the d e s i g n was to p r o v i d e c o n t r o l l e d f l o w to a Nuclepore membrane by u s i n g a p r e c i s i o n pump and to measure the p r e s s u r e drop a c r o s s a s i n g l e membrane w i t h a s e n s i t i v e pressure transducer. The dimensions of the flow c h a n n e l s were c o n t r o l l e d by s e l e c t i o n of the proper pore-size membrane. The c o n c e n t r a t i o n of polymer i n the s o l u t i o n downstream from the membrane was measured c o n t i n u o u s l y w i t h a d i f f e r e n t i a l UV absorbance d e t e c t o r . A l l s o l v e n t s p a s s i n g to the a p p a r a t u s were prefiltered d i r e c t l y i n t o the primary d e l i v e r y pump to reduce the p o s s i b i l i t y of b l o c k i n g and dust. The main advantag continuously measuring membrane p r e s s u r e drop and solution concentration. Thus the a p p a r a t u s c o u l d be used to conduct t r a n s i e n t as w e l l as steady s t a t e experiments. Figure 1 presents a diagram of the major s e c t i o n s of the a p p a r a t u s and t h e i r i n t e r c o n n e c t i o n s . P r e f i 1 t e r Sect i o n The p r e f i l t e r system f o r the s o l v e n t was a pump and filter h o l d e r arrangement s i m i l a r to t h a t of the measurement system o n l y u s i n g a c o a r s e f i l t e r of 0.5 μπι pores. Pump S e c t i o n P u l s e l e s s f l o w at c o n t r o l l e d f l o w r a t e s was provided by an Isco model 314 pump. T h i s pump i s b a s i c a l l y a 350 ml motorized s y r i n g e w i t h a c a p a b i l i t y of g e n e r a t i n g constant flow rates. Membrane Se^tj_gjn The N u c l e p o r e membranes p r o v i d e d controlled geometry f l o w c h a n n e l s of polymer molecular d i m e n s i o n s . The pores in a N u c l e p o r e membrane are made by c h e m i c a l l y e t c h i n g the damaged membrane m a t e r i a l c r e a t e d when an atomic n u c l e u s passes through a polycarbonate or p o l y e s t e r f i l m . This patented p r o c e s s allows the manufacturer to c o n t r o l the dimensions of the pores s i m p l y by controlling the l e n g t h of time that the f i l m i s subjected to etching. The number of pores per u n i t area of membrane s u r f a c e i s c o n t r o l l e d by c o n t r o l l i n g the f l u x of n u c l e i p a s s i n g through the f i l m during the i r r a d i a t i o n p r o c e s s . The pores produced by this p r o c e s s are r o u g h l y c y l i n d r i c a l i n shape and o r i e n t e d normal to the plane of the membrane s u r f a c e ( o ) . The pores are arranged on the s u r f a c e i n a random manner and thus i t i s p o s s i b l e to have two pores q u i t e c l o s e t o g e t h e r , however the f r a c t i o n of pores which e x i s t as doublets or h i g h e r m u l t i p l e pores i s l e s s than 10% of the total number of pores. Other workers have measured the pore r a d i i of Nuclepore membranes and found t h a t the reduced s t a n d a r d d e v i a t i o n of the pore r a d i a l dimensions i s 0.05*/.. (7) The Nuclepore membranes used f o r t h i s work were standard p o l y e s t e r membranes 25 mm i n diameter. The pore number, N, is calculated from the pore d e n s i t y and the e f f e c t i v e f l o w i n g area of

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

D E T E C T I O N A N D DATA ANALYSIS I N SIZE E X C L U S I O N

CHROMATOGRAPHY

P*£F ITER

MAIN PUMP

Ρ high

F i g u r e 1.

Ρ cd higftp

Diagram o f major a p p a r a t u s s e c t i o n s

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

9.

A Detector for Polymer Size and Shape in SEC

BROWER ET A L .

159

the 25 mm membrane i n the f i l t e r h o l d e r which i s the area (3.9 cm ) i n s i d e the Q-ring s e a l that h o l d s the membrane. The membranes were mounted i n a standard f i l t e r h o l d e r purchased from Nuclepore. This filter h o l d e r was connected to the pump by s t a i n l e s s s t e e l tubing and the f i l t e r assembly and i n l e t t u b i n g were c o n t a i n e d in a temperature c o n t r o l l e d water b a t h . Metal t u b i n g and a high p r e s s u r e pump were used to g i v e the apparatus the c a p a b i l i t y o f running e x p e r i m e n t s at l i n e p r e s s u r e s above atmospheric p r e s s u r e downstream of the membrane. However, a l l of the work r e p o r t e d here was done w i t h t h e membrane o u t l e t a t atmospheric p r e s s u r e . D i f f e r e n t i a l P r e s s u r e Measurement S e c t i o n The p r e s s u r e drop across the membrane i s the dependent v a r i a b l e i n the experiment and t h e r e f o r e the a c c u r a t e measurement of s m a l l d i f f e r e n t i a l p r e s s u r e i s the p r i m a r y f u n c t i o n o f the a p p a r a t u s . The apparatus uses a p r e s s u r e t r a n s d u c e r to mak drop a c r o s s the membrane p r e s s u r e t r a n s d u c e r was s e l e c t e p r e s s u r e drops. Mod i f ιed Membrane Viscometer For the p u l s e d system a c o i l of t u b i n g (the i n j e c t i o n loop) was p l a c e d a f t e r the p r e f i l t e r and b e f o r e the membrane h o l d e r as shown i n F i g u r e 2. D i r e c t i o n a l v a l v e s a t each end o f the loop c o n t r o l l e d t h e f l o w p a t h . S o l v e n t or s o l u t i o n c o u l d be pumped d i r e c t l y to the UV to e s t a b l i s h b a s e l i n e absorbance or f o r calibration. To make Ρ measurements the f l o w was d i r e c t e d through the membrane and then i n t o the d i f f e r e n t i a l UV spectrophotometer. The f l o w c o u l d a l s o be brought to the upstream p o r t i o n o f the membrane h o l d e r and then to the UV d e t e c t o r i n an e f f o r t to measure the c o n c e n t r a t i o n a t the membrane s u r f a c e . Cai i b r a t i o n of Instrument When 10 μm and 0.6 μm membranes were used to determine the v i s c o s i t y o f THF u s i n g the manufacturer's determination o f Ν from the f l o w o f water, the v i s c o s i t i e s o f THF were measured to be an average o f 85'/. of the t r u e v a l u e . The d i r e c t experimental Ρ vs Q c u r v e s a r e shown i n F i g u r e 3. (There i s , however, a s y s t e m a t i c t r e n d below 65'/. as membranes o f even lower pore s i z e s a r e used. Although t h i s t r e n d i s puzzling i t i s unimportant f o r polymer r e s e a r c h s i n c e most polymer s o l u t i o n s t u d i e s need relative viscosity, specific viscosity, Π..-, measurements.) The Vel ^ f u n c t i o n of f l o w r a t e , Q, a r e shown f o r a 10 g/mol m o l e c u l a r weight p o l y s t y r e n e i n F i g u r e 4. Both the Ubbelohde v i s c o m e t r i c d a t a and the membrane v i s c o m e t e r data a r e plotted on the same graph f o r a 0.6 urn pore membrane at a low c o n c e n t r a t i o n o f 100 ppm. The f l o w i s Newtonian. The a c t u a l agreement of the c a p i l l a r y and membrane v i s c o s i t i e s at low f l o w rates i s always e x c e l l e n t when D , 0.1 μπι (α)

σ CL Ε

ο

-I

1

1

1

·

r

0

4000

8000

12000

16000

20000

Shgar RatG (1/sec) 5

F i g u r e ^. V i s c o s i t y v s . shear r a t e f o r 1 0 MW p o l y s t y r e n e i n membrane v i s c o m e t e r (o) and i n g l a s s c a p i l l a r y v i s c o m e t e r (·)

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

161

162

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

ml/hr, w h i l e the p r e s s u r e d i f f e r e n c e s are s t i l l measurable but s m a l l (10-10 P a ) , and the maximum shear s t r e s s e s are e x t r e m e l y low. The Reynolds numbers are an o r d e r of magnitude below t u r b u l e n c e . The D/l r a t i o s are l e s s than 1/10 except f o r the 10 pm pore membrane i n which P o i s e u i l l e f l o w f o r m u l a s need to be c o r r e c t e d f o r end effects to determine t r u e v i s c o s i t i e s or shear r a t e s . Measurements o f Ge_l_ Çpjntent and Type Convent iona 1 Ge_l_ C h a r a c t e r i z a t ι on The g e l s chosen f o r a p r e l i m i n a r y study were two n a t u r a l rubber g e l s (from g u a y u l e , parthemum argentatum, (GNR) and hevea b r a z i l i e n s i s (HNR) p l a n t s ) and three commercial a c r y l i c g e l s of unknown c o m p o s i t i o n from the G l i d d e n Co. Both types of g e l are t y p i c a l of those encountered i n polymer l a b o r a t o r i e s and r e q u i r e d f o r product s p e c i f i c a t i o n s . A l l of the g e l s were f i r s t c h a r a c t e r i z e d both by s t a n d a r d f i l t r a t i o n t e c h n i q u e s ( u s i n g 5 μ 3

ΪΥ The r e f r a c t i v e i n d e x , t h e r e f r a c t i v e index increment, 5

1

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

242

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

the density, and the v i s c o s i t y of methyl ethyl ketone (MEK) at 25°C were 1.3760 at λ = 514.5 nm, 0.081 ml/g at λ = 514.5 nm,(9) 0.80 g/ml and 0.401 cP, respectively. ° 4.

Methods of Data Analysis

4.1

Light scattering data.

(a) Intensity of scattered

light.

From measurements of scattered l i g h t i n t e n s i t y as a function of concentration and scattering angle, we can determine the weight average molecular weight, Μ , the z-average radius of gyration, R and the second v i r i a l c o e f f i c i e n t , A^. According to the c l a s s i c a l Rayleigh-Debye theory, we have 2 up

1

-ι

ÎTTë) - ΤΓw w where H = ( 4 π

2

2 η

(1

+

2

6

- Γ Τ 3

o

A

/Ν X )(B /3C) n

Ο

cl

2

with λ_ ,

η , Ν

Ο

and

(3n/3C) being,

α

Ο

respectively, the wavelength of l i g h t i n vacuum, the solvent r e f r a c t i v e index, Avogadro's number and the r e f r a c t i v e index increment. Κ[Ξ (4?τ/λ )sin(6/2) with λ = λ /η ] i s the magnitude of the momentum transfer vector. R ( θ) i s ?he°excess Rayleigh r a t i o of the solution at concentration C and scattering angle θ. The intercept at θ = 0 and C=0 y i e l d s the weight average molecular weight, M^. The slope i n a ( H C / R ^ ) ^ ^ vs concentration plot and the angular dependence of ( H C / R ^ ) ^ ^ y i e l d the second v i r i a l c o e f f i c i e n t A^ and the z-average radius of gyration, R , respectively. ^ (b) Spectrum of scattered l i g h t . The spectrum of scattered light contains dynamical information related to t r a n s l a t i o n a l and i n t e r n a l motions of polymer chains. In the self-beating mode, the i n t e n s i t y - i n t e n s i t y time c o r r e l a t i o n function can be expressed (10) as

G

( 2 )

( t ) = = A ( l + b | g

( 1 )

2

(t)| )

(2)

where A i s t h e b a s e l i n e , b i s a coherence function of the spectrometer, g ( t ) i s the normalized time c o r r e l a t i o n function of^the scattered e l e c t r i c f i e l d and t i s the delay time. Although g ( t ) i s a single exponential decay curve for monodisperse, noni n t e r a c t i n g and structureless macromolecules, g ( t ) i s related to the normalized c h a r a c t e r i s t i c linewidth d i s t r i b u t i o n G(T) through a Laplace i n t e g r a l equation for polydisperse systems by eq. (3). ίτ\

i >(t) = _ | L_i r G(De- dr (3)

#

r t

r t

1

g

=

0

s

s

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

14.

WANG ET A L .

243

Light Scattering of Polyvinyl Acetate

where Τ i s t h e c h a r a c t e r i s t i c linewidth. F o r narrow size d i s t r i b u t i o n s , the second-order cumulant e x p a n s i o n can be used to d e t e r m i n e t h e average c h a r a c t e r i s t i c l i n e w i d t h Γ

ln|g

( 1 )

(t)|

« -rt

+ 1/2

p t

2

(A)

2

—2 — ? where t h e v a r i a n c e i s ρ /Γ with = f(T-Γ) G(F)dr. Eq. (3) i s v a l i d f o r p o l y d i s p e r s e p a r t i c l e s w i t h i n t e r n a l m o t i o n s . Then G ( r ) becomes a complex f u n c t i o n o f b o t h s i z e and internal motions w h i c h , under f a v o r a b l e c o n d i t i o n s , may be s e p a r a t e d by e x a m i n i n g t h e a n g u l a r dependence o f G ( K , r ) . I n our c a s e , we know t h a t as K-*0, i n t e r n a l motions become l e s s i m p o r t a n t and measurements o f G(T) at low s c a t t e r i n g angles r e v e a l e s s e n t i a l l y only ^ d i s t r i b u t i o n of t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t ( D E Γ/Κ ). The presence of i n t e r n a a l s o be r e p r e s e n t e d a s c a t t e r i n g a n g l e θ and a f i n i t e c o n c e n t r a t i o n C, we have 2

a

Γ/Κ

2

2

2

= D ° ( 1 + f R K ) ( 1 + k_C) z g d

(5)

where D i s t h e z-average t r a n s l a t i o n a l d i f f u s i o n c o e f f i c i e n t at infinite dilution, f i s a c o n s t a n t depending upon _the c h a i n structure, polydispersity and solvent quality; and k^ i s an a v e r a g e d i f f u s i o n second v i r i a l c o e f f i c i e n t . In order to estimate t h e n o r m a l i z e d l i n e w i d t h d i s t r i b u t i o n G ( T ) , we chose an a l g o r i t h m d e v e l o p e d by Provencher(12)and commonly known as CONTIN. At t=0, /Âb g ^ ( t = 0 ) was made e q u i v a l e n t w i t h t h e i n t e g r a t e d excess scattered intensity ± ;

/Âb

g

( 1 )

( t = 0 ) = Skb f°° G(r)dT = /Âb =

(6)

0

If t h e CONTIN a l g o r i t h m was r u n n i n g i n e q u a l s p a c i n g on t h e l o g a r i t h m i c s c a l e and t h e l i n e w i d t h d i s t r i b u t i o n was n o r m a l i z e d by t h e a r e a , t h e n t h e i n t e n s i t y o f s c a t t e r e d l i g h t f o r each f r a c t i o n Γ. can be e x p r e s s e d as G ( l n r . ) w h i c h i s r e l a t e d t o G ( l \ ) by the relation

Γ /

G(r)dT

( i n e q u a l s p a c i n g on Γ

scale)

^min ΙηΓ = / ΙηΓ . mm or

G(r)Γ

dlnT

GUnT.) = r.G(r.)

( i n e q u a l s p a c i n g on ΙηΓ

scale)

(7)

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

244

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

The next step i s to transform G(r .)[or G(lnT.)] to the weight d i s t r i b u t i o n (MWD). 1

molecular

1

(c) Molecular weight d i s t r i b u t i o n . According to eq. (1), we have Μ Ρ (θ)

w

"

V

(1 + 2A M C) 2 w

J

0

In eq. (8), the term M / ( l + 2A^M C) can be considered to be a bulk property, i . e . , i s independent o î each f r a c t i o n . I n o t h e r words, the second v i r i a l c o e f f i c i e n t i s assumed t o be a constant, independent of molecular weight i n the polydiserse system under consideration. However, the form factor, P(Θ) can play a very important role i n the transformatio

= / f(M)M P(K,M)dM

/f(M)M

*

2

(equal spacing on M scale)

P(K,M) dlnM (equal spacing on InM scale)

ΔInM Σ f(M.)M i

2

P(K,M.)

(9)

Here f(M.) i s the weight f r a c t i o n of molecular weight M. using equal spacing i n M scale and P(K,M.) i s the form factor ior the molecular weight M. at the scattering vector K, and the f i n a l expression i n eq. (9) approximates the MWD as a discrete d i s t r i b u t i o n . I f Γ. i s related to M. by an empirical power law (Γ ~ M D ) ΔΐηΓ can èe linked with ΔInM through a proportionality constant. In our case, the weight f r a c t i o n of each M. i n lnMspace, F^(lnM^) has the form a

9

F j l n M . ) = f(M.)M. -

^

„ 1

(10) 1

as C-K). In computing the molecular weight d i s t r i b u t i o n , we also need the molecular weight dependence of D(= _^ Γ/Κ ) and of R . From laser l i g h t scattering characterization of unfractionatld polymer samples of d i f f e r e n t molecular weight, we obtained K

7> _ ,. w -«D Ζ R g

=LM °R R w

(11)

(12)

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

14.

Light Scattering of Polyvinyl Acetate

WANG ETAL.

245

By combining eq. (5) with eq. (11), we can imply a r e l a t i o n between the molecular weight, Μ , and the c h a r a c t e r i s t i c linewidth, Γ, at a f i n i t e scattering angle and a f i n i t e concentration

M. = ι

Γ [-_ 1 2 2 ? K k_(l + f R V ) ( l g

1

J

1

1 / α

(13)

+ k.C) α

υ

= (const. Γ . )

ι /^

ϋ

2 2 In eq. (13), we have included the (1 + fR Κ ) term without e x p l i c i t l y inserting eg.^ (12). I t should be 8oted that at small scattering angles, fR Κ =

we h a v e ]

M- MV

c i s a proportionality a

and

V

Eg. 3, 4 and 5 i n t o C

in

M

+A(V

EX

BV

9

which r e l a t e s the spreading f a c t o r t o the retention volume V w i t h f o u r p h y s i c a l l y meaningful parameters a a n a b . T h e f o r m e r t w o p a r a m e t e r s / (16) T h e a v e r a g e s p r e a d i n g f a c t o r < or^> o f n a r r o w MWD poly­ s t y r e n e s t a n d a r d s may b e r e g a r d e d a s t h e s p r e a d i n g factor O^Q o f a m o n o d i s p e r s e p o l y m e r f o r w h i c h V = V. A l l the r e s u l t s thus obtained are l i s t e d i n Taole I I . The v a r i a t i o n o f t h e s p r e a d i n g f a c t o r w i t h r e t e n t i o n v o l u m e i s s h o w n i n F i g . 1. The e x i s t e n c e o f a maximum

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

283

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

284

Table

I . T h e M o l e c u l a r W e i g h t a n d SEC D a t a of P o l y s t y r e n e Standards

Polymer

*1ΰ w

TSK-2 TSK-10 TSK-20 TSK-40 TSK-80 TSK-128 NPS-2 NPS-3 NPS-4 NPS-5 NPS-6 NPS-7

1 .73 9.89 18.4 42.7 79. 1 130 0.563 1 .20 2.78 5.00 12.0 15.4

/ w η 1 .02 .02 1 .07 1 .05 I .01 1 .05 1 .05 1 .04 1 .03 1 .02

1

V

37. 80 32. 96 31 .44 28..49 26..50 25..50 42..05 39..52 36..89 35 .05

ςτ£ Τ 0..67 0..80 1 .51 . 1 .43 . 0..70 0..95 0,.91 0 .86 0..80 0 .77

Table I I . The C o e f f i c i e n t s o f t h e E f f e c t i v e r e l a t i o n s and t h e S p r e a d i n g F a c t o r s o f P o l y s t y r e n e Standards

Polymer

A *

Β *

ξ

TSK-2 TSK-10 TSK-20 TSK-40 TSK-80 TSK-128 NPS-2 NPS-3 NPS-4 NPS-5 NPS-6 NPS-7

16.24 16.68 18.73 18. 17 16.60 19.81 19.15 21 .00 19.27 21 .91 20.50 19.38

0., 172 0., 157 0., 21 1 0.. 184 0.,114 0. 226 0..253 0..299 υ..247 0..316 0..270 û..233

0. 500 0..462 0..623 0..535 0..329 0..663 0 .746 . 0,.873 0..723 0,.928 0,.794 0..685

V V = i " a - b V

R

(17)

the l o g a r i t h m i c term s h o u l d d e c r e a s e s l i n e a r l y w i t h . S u c h a p l o t i s shown i n F i g . 2 i n w h i c h t h e v a l u e o f and V a r e e s t i m a t e d f r o m t h e chromatograms o f t o t a l l y e x c l u d e d s a m p l e s a s 0.37 a n d 25.3 r e s p e c t i v e l y . From t h e i n t e r c e p t a n d s l o p e o f t h e l i n e i n F i g . 2, t h e p a r a m e t e r a a n d b a r e e v a l u a t e d a n d e q u a l t o 1190 a n d 0.309 r e s p e c V tively. The c a l c u l a t e d c u r v e o f ° "O^R ^ with these evalua t e d p a r a m e t e r s i s drawn i n F i g . 1 t o o . T h e c o i n c i d e n c e with the experimental data i s q u i t e well. Q

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

C H E N G ET A L .

1

.6

1

.2

Spreading Factor Dependence on Retention Volume of SEC

o.8h-

0.4h-

0

Figure

. Ol

1.

Dependence o f t h e s p r e a d i n g the r e t e n t i o n

volume.

2 F i g u r e 2.

P l o t o f l n { CT

f a c t o r on t h e

2. - ° * _ , ) / ( j

versus

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

V

D

285

286

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

I t s h o u l d be n o t i c e d t h a t t h e v a l u e o f p a r a m e t e r b ( 0 . 3 0 9 ) i s s l i g h t l y s m a l l e r than the s l o p e o f the c a l i ­ bration function Β ( 0 . 3 4 2 ) a s e x p e c t e d by t h e t h e o r y . From t h e r e l a t i o n s h i p between b and Β„ ( E g . 8 ) , we g e t Λ

jyj

S - 0 . 9 0 f o r t h e p r e s e n t c a s e . The c a l c u l a t e d t h e o r e t i c a l O ^ ( V ^ ) curves a r e very sensitive t o the value of the p a r a m e t e r £ as shown i n F i g . 1 . I t i n d i c a t e s t h a t t h e d i f f u s i o n b e h a v i o r o f m a c r o m o l e c u l e s i n t h e p o r e o f SEC P a c k i n g s c a n be s t u d i e d i n a q u a n t i t a t i v e way by s y s t e m a ­ t i c investigation of instrumental spreading e f f e c t .

Literature Cited 1. Rong-shi Cheng and Shu-qin Bo, ACS Symposium Series 245, "SEC, Methodology and Characterization of Polymers and Related Materials", 1984; p. 125. 2. Rong-shi Cheng and Shu-qi Bo Gaofenzi Tongxu (Polymer Communication 3. M. Kubin, J. Chromatogr., ; , 4. W. W. Yau, J. J. Kirkland and D. D. Bly, "Modern Size Exclusion Liquid Chromatography", 1979; p. 8 2 . 5. R. Groh and I. Halasz, Anal. Chem., 1981; 53, 1325. RECEIVED May 15, 1987

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Chapter 17 Correction for Instrumental Broadening in Size Exclusion Chromatography Using a Stochastic Matrix Approach B a s e d o n Wiener Filtering T h e o r y L. M. Gugliotta, D. Alba, and G. R. Meira

1

Intec (Conicet and Universidad Nacional del Litoral), (3000) Santa Fe, Argentina The correction for non-uniform instrumental broadening in SEC is solve stochastic technique (1) must be reformulated in matrix form, and the meas­ urements assumed contaminated with zero-mean noise. The proposed technique is based on an extension to time-varying systems of Wiener's optimal filtering method (1-3). The estimation of the corrected chromato gram is optimal in the sense of minimizing the estima­ tion error variance. A test for verifying the results is proposed, which is based on a comparison between the "innovations" sequence and its corresponding expected standard deviation. The technique is tested on both synthetic and experimental examples, and com­ pared with an available recursive algorithm based on the Kalman filter (4). Most methods of correction for instrumental broadening in SEC (or hydrodynamic chromatography) are based on the deterministic integral equation due to Tung (_5) : oo z(t) = j g(t,x) U(T) di (1) —OO where t,x: both represent elution time or elution volume; z(t): is the base-line corrected chromatogram; g(t,x): is the time-varying or non-uniform spreading function, which is built up by the set of unit mass impulse responses g(t) of truly monodisperse polymers with dif­ ferent elution times τ ; and u(t): is the corrected chromatogram. When g(t,x) is considered time-invariant, then Equation 1 re­ duces to a convolution integral. There are two basic problems associated to Equation 1: i) the determination of the spreading g(t,x); and Correspondence should be addressed to this author. 1

0097-6156/87/0352-0287$06.00/0 © 1987 American Chemical Society

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

288

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

i i ) t h e e s t i m a t i o n of u ( t ) , based on the knowledge o f z ( t ) and g(t,x). W i t h r e s p e c t t o t h e s p r e a d i n g c a l i b r a t i o n , s e v e r a l methods have been suggested e.g. (6-lk). Numerous t e c h n i q u e s have been proposed f o r s o l v i n g the i n v e r s e f i l t e r i n g problem r e p r e s e n t e d by E q u a t i o n 1, w i t h d i f f e r e n t degrees o f success e.g. (U,15-19)» Only r e f e r e n c e s (]0 , ( l 8 ) and (19) make no assumptions on t h e shape o f g ( t , x ) . I n t h i s work, an i n v e r s e f i l t e r i n g t e c h n i q u e based on Wiener's o p t i m a l t h e o r y ( l - 3 ) i s p r e s e n t e d . T h i s approach i s v a l i d f o r t i m e v a r y i n g systems, and i s s o l v e d i n t h e time domain i n m a t r i x form. A l s o , i t i s i n many r e s p e c t s e q u i v a l e n t t o the n u m e r i c a l l y " e f f i ­ c i e n t " Kalman f i l t e r i n g approach d e s c r i b e d i n (k) . For t h i s r e a s o n , a comparison between the two t e c h n i q u e s w i l l be made. Theory The S p r e a d i n g Model. C o n s i d e r i n and b e a r i n g i n mind t h a l e n g t h , then one may w r i t e k =s I g(k,k ) k =-r 0

z(k)

=

0

(k = 0 , 1 , 2 , . . . ,

u(k ) 0

n)

(2)

0

where k,kQ*. are t h e d i s c r e t e e q u i v a l e n t s o f t and τ , r e s p e c t i v e l y ; -r,s: a r e t h e lower and upper l i m i t s o f t h e sum i n E q u a t i o n 2, w i t h non-zero v a l u e s o f the i n d i c a t e d p r o d u c t . L e t ζ and u denote column v e c t o r s such t h a t : [z(0), z ( l ) ,

., z(n)

[u(0), u ( l ) ,

u(n)]-

(3a) (3b)

_z has n o r m a l l y more non-zero elements than u. Even though t h e t h e o r y can be m o d i f i e d t o a l l o w f o r t h i s f a c t , we assume f o r s i m p l i c i t y t h a t u has the same number o f components as z_. T h e r e f o r e , one can w r i t e E q u a t i o n 2 i n m a t r i x form as f o l l o w s : ζ =

G u

(Ua)

with g(0,0) g(0,l)

. . . g(0,n)"

g(l,0)

. . . g(l,n)

g(l,l)

(kh)

,g(n,0) g ( n , l ) . . . g(n,n) Note the f o l l o w i n g : In g e n e r a l , the f i r s t and t h e l a s t elements of u w i l l be z e r o . Each column o f G c o n t a i n s an impulse response, w i t h t h e impulse a p p l i e d at t h e element i n the d i a g o n a l o f G. (See F i g u r e 1 f o r a 3-D r e p r e s e n t a t i o n o f a t y p i c a l G m a t r i x . )

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

17.

GUGLIOTTA ET A L .

Instrumental Broadening in

SEC

g(k,k ) 0

F i g u r e 1: T r i d i m e n s i o n a l r e p r e s e n t a t i o n o f a t y p i c a l G m a t r i x .

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

289

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

290

Hess and K r a t z as f o l l o w s :

(6) t r i e d t o e s t i m a t e u d i r e c t l y

Û=

from E q u a t i o n s

k

(5)

G-lz

In general, t h i s operation i s numerically i l l - c o n d i t i o n e d , l e a d i n g to i n c o r r e c t r e s u l t s . (The degree o f i l l - c o n d i t i o n i n g may be measured by t h e c o n d i t i o n number, d e f i n e d by t h e r a t i o o f the modulus o f the l a r g e s t t o the s m a l l e s t e i g e n v a l u e o f G ) . A s t o c h a s t i c v e r s i o n o f E q u a t i o n ka. may be w r i t t e n :

where

y_ =

ζ =

G u + v

(6a)

ζ_ =

y + y_

(6b)

[ν(θ),..., v(n y:

i s the n o i s e - f r e e measured chromato­ gram; and _z,u,y_: w i l l be assumed zero-mean s t o c h a s t i c variables. In what f o l l o w s , we s h a l l seek a r e s t o r i n g m a t r i x H such t h a t t h e e s t i m a t e u_ i s c a l c u l a t e d t h r o u g h : u =

Η ζ

(Τ)

Λ.

Let e^ = (u-u) be the e s t i m a t i o n e r r o r a s s o c i a t e d w i t h u . The e s t i ­ mate u i s chosen i n such a way t h a t t h e c o r r e s p o n d i n g mean square e r r o r Ε[(u-u) (u-u)] i s m i n i m i z e d . The Input E s t i m a t i o n Through a Wiener F i l t e r i n g Approach. Equation 6a r e p r e s e n t s a t i m e - v a r y i n g l i n e a r f i l t e r w i t h a measurement no i s e, and t h e s t a t i s t i c s o f such n o i s e may be c o n s i d e r e d n o n - s t a t i o n a r y . Simply s t a t e d , the o p t i m a l i n v e r s e f i l t e r i n g problem i s t h i s : assum­ i n g t h a t a s i g n a l i s f i r s t d i s t o r t e d t h r o u g h a l i n e a r f i l t e r o f known c h a r a c t e r i s t i c s and then contaminated w i t h an a d d i t i v e n o i s e , what l i n e a r o p e r a t i o n on the r e s u l t i n g measurement w i l l y i e l d the b e s t e s t i m a t i o n o f t h e o r i g i n a l s i g n a l ? . " B e s t " i n t h i s case means minimum mean-square e r r o r . T h i s branch o f f i l t e r i n g began w i t h N. Wiener's work i n the 19U0's ( 1_) . R.E. Kalman t h e n made an important c o n t r i ­ b u t i o n i n the e a r l y 1960's; by p r o v i d i n g an a l t e r n a t i v e approach t o t h e same problem u s i n g s t a t e - s p a c e methods ( 2 0 - 2 1 ) . N. Wiener's s o l u t i o n was o r i g i n a l l y d e r i v e d i n the frequency domain f o r t i m e - i n v a r i a n t systems w i t h s t a t i o n a r y s t a t i s t i c s . In what f o l l o w s , a m a t r i x s o l u t i o n d e r i v e d from such approach but developed i n the time domain f o r t i m e - v a r y i n g systems and n o n - s t a t i o n a r y s t a ­ t i s t i c s w i l l be p r e s e n t e d ( 2 2 - 2 3 ) · An e x p r e s s i o n f o r the r e q u i r e d t r a n s f o r m a t i o n H i n E q u a t i o n 7 w i l l be o b t a i n e d . In a l l t h a t f o l l o w s , we s h a l l denote w i t h u the b e s t e s t i m a t e o f u, i . e . an e s t i m a t e such that :

T

T

E[(u-Û) (u-Û)] < E [ ( u - u ) ( u - u ) ]

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

(8)

17.

Instrumental Broadening in

GUGLIOTTA ET AL.

291

SEC

where i s any s u b o p t i m a l e s t i m a t e of u_. The p r i n c i p l e o f o r t h o g o n a ­ l i t y ( 2 4 - 2 6 ) , s t a t e s t h a t E q u a t i o n 8 w i l l be v e r i f i e d i f the e s t i m a ­ t i o n e r r o r v e c t o r i s o r t h o g o n a l t o t h e measurements. I n o t h e r words, t h e f o l l o w i n g must be t r u e : E[(u-u)z ] =

T

0

and

Equation

Τ

6a

S u b s t i t u t i n g Equation a t i n g , one o b t a i n s :

T

E[uu ] G We

T

T

+ E[uv ] =

(9) into

Equation

9 and

oper­

Η E[zz_T]

(lO)

s h a l l assume t h e i n p u t u u n c o r r e l a t e d w i t h v, i . e . : E[uvT] =

E[vuT]

ο

=

( ) U

Let Z and Σ be t h e c o v a r i a n c r e s p e c t i v e l y . (Such m a t r i c e E q u a t i o n 10 may be w r i t t e n u

ζ

Σ

G

u

T

=

Η Σ

(12)

ζ

From: Σ

T

=

ζ

E[(Gu+v)(Gu+v) ]

and b e a r i n g i n mind E q u a t i o n 11, one Σ

=

ζ

G Σ

finds: G

u

(13)

T

+ Σ

(1*0

ν

S u b s t i t u t i n g E q u a t i o n lk i n t o E q u a t i o n 12 and arrives at: Η = In

Σ

G

η

T

[G Σ

G

α

T

+ Σ^'

o t h e r words, t h e o p t i m a l e s t i m a t e may

Û = —

Σ

u

GT

[G

Σ

G

u

T

+

o p e r a t i n g , one

1

(15)

be c a l c u l a t e d

Σ

ν

l"

finally

1

through:

(16)

ζ —

Note the f o l l o w i n g : ^ - F o r any a r b i t r a r y G , t h e e x i s t e n c e o f [ G Σ G + Σ ] ~ i s ensured by t h e i n v e r t i b i l i t y o f Σ . - A d o p t i n g Σ = q I and Σ = 0 , t h e n E q u a t i o n l 6 reduces t o E q u a t i o n 5. - W i t h E = q l and Σ = Γ Ι , E q u a t i o n 16 has a format which i s i d e n t i c a l t o t h e s o l u t i o n d e r i v e d i n (27) t h r o u g h a d e t e r m i n i s t i c minimum l e a s t squares approach f o r t i m e - i n v a r i a n t systems. T h i s i s t o be e x p e c t e d , because t h e Wiener f i l t e r i n g t e c h n i q u e may be i n f a c t i n c l u d e d as p a r t o f t h e g e n e r a l t h e o r y o f l e a s t s q u a r e s . T

Υ

Ν

Ν

ν

u

u

Υ

The F i l t e r Adjustment. The computation o f u t h r o u g h E q u a t i o n 16 i n ­ v o l v e s the p r e s p e c i f i c a t i o n of Σ and Σ · These m a t r i c e s a r e i n g e n e r a l symmetric; and t h e s i m p l i f i c a t i o n o f c o n s i d e r i n g b o t h v. and u w h i t e n o i s e s has been found t o p r o v i d e s a t i s f a c t o r y r e s u l t s . Thus, Σ and Σ w i l l be assumed d i a g o n a l . Η

Α

Ν

Γ

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

292

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

The s t a t i s t i c s of ν may be considered stationary with p h y s i c a l b a s i s . T h e r e f o r e , we s h a l l s i m p l y adopt: Σ

=

γ

sound

r I

(IT)

where t h e s c a l a r r may be o b t a i n e d from the sample v a r i a n c e of the chromatogram b a s e l i n e n o i s e . Note t h a t f o r any p o s i t i v e r , the i n v e r t i b i l i t y o f [G E G + Σ ] i n E q u a t i o n l 6 i s t h e o r e t i c a l l y ensured. C o n s i d e r now the e s t i m a t i o n o f the d i a g o n a l elements o f E . The f o l l o w i n g assumptions can be made: a) Take the v a r i a n c e of u(k) t o be c o n s t a n t . I n t h i s c a s e , and remem­ b e r i n g t h a t u(k) i s assumed o f zero mean, one may w r i t e : t

U

ν

u

Σ where t h e v a l u e o f z ( k ) as f o l l o w s :

q may

=

u

be

simply

J

q =

I k=0

=

u

estimated

[z(k)]

b) A l l o w now t h e v a r i a n c e of u(k) r e a l l i s t i c than b e f o r e ) . C a l l : Σ

(18a)

q I

from the measurement

(18b)

2

t o be t i m e - v a r y i n g . ( T h i s i s more

d i a g . [ q ( 0 ) , q ( l ) , ... q(n)]

(19)

Here, we can e s t i m a t e q(k) i n s e v e r a l ways, f o r example: q(k) =

Ci [z(k)]

2

(20)

q(k) =

C

2

(21)

or 2

[u(k)]

where C^, C a r e p o s i t i v e c o n s t a n t s , and u(k) i s any o t h e r e s t i m a t i o n of u. 2

suboptimal

The S o l u t i o n V a l i d a t i o n . Obvious c o n d i t i o n s t h a t the r e s u l t a n t s o l u ­ t i o n u must s a t i s f y a r e : a) u must be n o n - n e g a t i v e ; b) the o p e r a t i o n Gu s h o u l d p r o v i d e a n o i s e - f r e e measured f u n c t i o n ; and c) t h e areas under t h e measured and the c o r r e c t e d chromatograms must be e q u a l . I t s h o u l d be emphasized t h a t c o n d i t i o n b) i s a n e c e s s a r y but not s u f ­ f i c i e n t f o r good r e s u l t s . Apart from the mentioned c h e c k s , a v a l i d a ­ t i o n procedure based on the a n a l y s i s o f t h e i n n o v a t i o n s w i l l now be presented. C o n s i d e r f i r s t the c o v a r i a n c e m a t r i x Σ~ c o r r e s p o n d i n g t o the .... . u e s t i m a t i o n e r r o r e^, i . e . : e

Σ

(22)

T

= e

E[(u-Û)(u-Û) ]

u

S u b s t i t u t i n g E q u a t i o n 6a i n t o E q u a t i o n 7 and t h e l a t t e r i n t u r n i n t o E q u a t i o n 22, one o b t a i n s : Σ

e

= u

Σ

u

- 2 H G E

u

+ H G Σ

u

G

T

H

T

+ H ϋ

v

H

T

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

(23)

The

293

Instrumental Broadening in SEC

GUGLIOTTA ET A L .

17.

i n n o v a t i o n s sequence e^ i s d e f i n e d by:

Ëz

=

L

"1

(2*0

and t h e r e f o r e ,

£z =

(25)

ζ - Gu

because t h e best e s t i m a t e f o r ζ i s y = Gu, s i n c e Y_ i s zero mean. Sub­ s t i t u t i n g E q u a t i o n 6a i n t o E q u a t i o n 25 y i e l d s : e_2 =

(26)

G e^ + ν

The c o r r e s p o n d i n g c o v a r i a n c e m a t r i x i s found s u b s t i t u t i n g E q u a t i o n 23 into: Σ

= e

E[(Ge +v) u

(G e

(27)

+ v) ]

z

and t h e f i n a l r e s u l t i s : Σ

e

= G Σ G u

z

+ Σ

T

- 2 G H G E

u

G

T

H

H G E G u

T

Η

Τ

G

T

+ G Η Ε }F G? + ν (28)

γ

The proposed check c o n s i s t s i n matching t h e i n n o v a t i o n s sequence o b t a i n e d from E q u a t i o n 25 w i t h t h e c o r r e s p o n d i n g expected t i m e v a r y i n g v a r i a n c e p r o v i d e d by E q u a t i o n 28. I f t h e i n n o v a t i o n s sequence i s assumed zero-mean Gaussian w h i t e , t h e n e ( k ) s h o u l d be w i t h i n t h e + σ (k) bounds f o r a p p r o x i m a t e l y two t h i r d s o f t h e t i m e . ( a (k) r e ­ presents t h e s t a n d a r d d e v i a t i o n o f e ( k ) , found by square rooming t h e d i a g o n a l elements o f Z )· Note t h a t t h e proposed check must be perfomed a f t e r h a v i n g ob­ t a i n e d t h e e s t i m a t i o n o f u . I n c o n t r a s t , i n t h e Kalman f i l t e r t e c h n i q u e (k) , t h e c o r r e s p o n d i n g v a l u e s o f e^ and Σ may be r e c u r s i v e l y ^z c a l c u l a t e d along w i t h the input estimate. z

θ

e

z

e

e

Examples o f A p p l i c a t i o n In o r d e r t o compare t h e p r e s e n t t e c h n i q u e w i t h t h e method based on t h e Kalman f i l t e r (k), t h e same examples p r e s e n t e d i n t h a t p u b l i c a ­ t i o n w i l l be attempted. The f i r s t two examples a r e s y n t h e t i c , w h i l e t h e t h i r d i s based on r e a l e x p e r i m e n t a l d a t a . A l l examples were s o l v e d by means o f a VAX 11/780 computer programmed i n FORTRAN 77. R o u t i n e s f o r m a t r i x o p e r a t i o n from t h e IMSL package (28) were u t i ­ lized. Example 1. By p r o c e s s i n g t h e curve u(k) shown i n F i g u r e 2a t h r o u g h a t i m e - v a r y i n g f i l t e r d e f i n e d by t h e s e t o f impulse responses o f F i g u r e 1, a n o i s e - f r e e chromatogram y ( k ) i s o b t a i n e d . T h i s curve was then contaminated w i t h Gaussian w h i t e n o i s e o f a r e l a t i v e l y low v a r i a n c e (10~5) t o p r o v i d e z(k) . C l e a r l y , t h e b e s t e s t i m a t e f o r r i s 10~5 and i n t h i s case a c o n s t a n t v a l u e f o r q=5xlO*"3 v a s adopted by t r i a l and e r r o r , p r o v i d i n g an a c c e p t a b l e compromise between t h e d i f f e r e n t checks. 5

5

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

294

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

L e t ûj^(k) be t h e o p t i m a l e s t i m a t e o b t a i n e d through the Kalman a p p r o a c h , u(k) and u^(k) a r e a l s o shown i n F i g u r e 2a. The i n n o v a t i o n s c o r r e s p o n d i n g t o û(k) are r e p r e s e n t e d i n F i g u r e 2b. T h i s example was s o l v e d assuming n o i s y b a s e l i n e s e c t i o n s b e f o r e and a f t e r the peak as p a r t of the chromatogram. For t h i s r e a s o n , and because q was assumed c o n s t a n t , o s c i l l a t i o n s are observed i n u(k) and uf[(k) i n t h o s e s e c t i o n s of the c u r v e . In both t e c h n i q u e s , b e t t e r e s t i m a t i o n s a r e obt a i n e d i f q i s adopted t i m e - v a r y i n g t h r o u g h E q u a t i o n 20. In t h i s c a s e , the mentioned o s c i l l a t i o n s around t h e b a s e l i n e s e c t i o n s b e f o r e and a f t e r the peak d i s a p p e a r . Example 2. T h i s example was f i r s t suggested by Chang and Huang (29), and attemped l a t e r on by Hamielec and co-workers (19)· The problem i s i l l u s t r a t e d by F i g u r e 3, which r e p r e s e n t s the f o l l o w i n g : u ( k ) , t h e u n i f o r m s p r e a d i n g f u n c t i o n g ( k ) , t h e broadened curve z ( k ) , and t h e r e c u p e r a t e d u ( k ) by method 2 proposed i n (19)» The s o l u t i o n shown i n Figure 3 i s p r a c t i c a l l (29) o f method 1 i n (19). Clearly p r i a t e l y r e c o v e r the double-peake i n p u t T h i s problem was s o l v e d a d o p t i n g t h e same v a l u e s f o r r and q as i n (k_) , i . e . : r=0.1 and q c a l c u l a t e d t h r o u g h E q u a t i o n 20 w i t h C]_=l. The r e s u l t s are shown i n F i g u r e Ha, where the o r i g i n a l u(k) i s comp a r e d t o the e s t i m a t e s o b t a i n e d through t h e proposed t e c h n i q u e and t h r o u g h the Kalman approach ( F i g u r e 10a of 00). Figure illust r a t e s the i n n o v a t i o n s t e s t . 2

Example 3· Curve z ( k ) i n F i g u r e 5 r e p r e s e n t s the chromatogram o f a PS s t a n d a r d o f MW=525, when f r a c t i o n a t e d through an A-802 Shodex column mounted on a S e r i e s 3-B P e r k i n Elmer l i q u i d chromatograph. The chromatogram o f pure benzene g(k) i s adopted as t h e u n i f o r m s p r e a d i n g f u n c t i o n . The polymer sample i s expected t o be i n t e g r a t e d by the f i r s t PS o l i g o m e r s , w i t h preponderance o f the pentamer. I d e a l l y , d e l t a f u n c t i o n s ought t o be r e c u p e r a t e d , w i t h the h i g h e s t peak at a m o l e c u l a r weight o f 520. Here, a v a l u e o f r=5xlO~5 was a d o p t e d , and q was c a l c u l a t e d t h r o u g h E q u a t i o n 20 w i t h C]_=0.75. In F i g u r e 5, the r e s u l t o f the p r e s e n t t e c h n i q u e i s compared t o the r e s u l t i n F i g u r e 12a o f 00. As w i t h a l l p r e v i o u s examples, t h e e s t i m a t e d n o i s e - f r e e chromatogram y ( k ) i s p r a c t i c a l l y c o i n c i d e n t w i t h the measured z ( k ) . Conclusions The proposed t e c h n i q u e i s n u m e r i c a l l y " r o b u s t " , and i t s r e s u l t s are comparable t o t h o s e o b t a i n e d t h r o u g h a r e c u r s i v e method based on the Kalman f i l t e r {k) . I t s h o u l d be noted t h a t because the p r e s e n t t e c h n i q u e u t i l i z e s a l l o f t h e i n f o r m a t i o n s i m u l t a n e o u s l y , the r e s u l t s have been compared t o those of the o p t i m a l smoother e s t i m a t e s i n 00 , w h i c h a r e " b e t t e r " than the t r u e f i l t e r e d e s t i m a t e s . The main advantage o f t h e s t o c h a s t i c m a t r i x approach i s the s i m p l i c i t y f o r i t s computer i m p l e m e n t a t i o n . E q u a t i o n 17 d i r e c t l y p r o v i d e s the d e s i r e d r e s u l t , and E q u a t i o n 28 i s the b a s i s of a v a l i d a t i o n t e s t which may or may not be performed a c c o r d i n g t o p r e v i o u s e x p e r i e n c e . In o t h e r words, the proposed method i s c o n c e p t u a l l y and p r a c t i c a l l y e a s i e r t o implement than t h e Kalman c o u n t e r p a r t . The

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GUGLIOTTA ET A L .

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b

0

50

100

F i g u r e 2: Example 1: a) Comparison between t h e " t r u e " i n p u t u ( k ) , t h e e s t i m a t i o n o f t h a t i n p u t t h r o u g h t h e present t e c h n i q u e u ( k ) and t h e same e s t i m a t i o n t h r o u g h t h e method d e s c r i b e d i n (h) u^-(k) ; b) I n n o v a t i o n s sequence and +o (k) bounds c o r r e s p o n d i n g t o u ( k ) . e

F i g u r e 3: Example 2:

( a f t e r Hamielec and co-workers

(19))·

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DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

F i g u r e 5: Example 3: a) E x p e r i m e n t a l chromatogram, s p r e a d i n g f u n c t i o n and comparison o f p r e s e n t r e s u l t s w i t h those i n {k); b) V a l i d a t i o n t e s t f o r û(k).

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

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p r i n c i p a l drawback o f t h e p r e s e n t t e c h n i q u e i s i t s r e l a t i v e l y h i g h c o m p u t a t i o n a l c o s t , both i n memory and computation t i m e . T y p i c a l l y , i n o r d e r t o s o l v e a chromatogram o f 128 p o i n t s w i t h a t i m e - v a r y i n g q(k) , a computation time o f 5 mins. was r e q u i r e d t o e s t i m a t e u(k) , and 4.5 more mins. were n e c e s s a r y f o r t h e v a l i d a t i o n t e s t . A p o i n t t h a t has not been i n v e s t i g a t e d i s t h e p o s s i b i l i t y o f c o n s i d e r i n g u ( k ) a c o l o u r e d n o i s e i n s t e a d o f white n o i s e , and t h e r e ­ f o r e a non d i a g o n a l E . F o r example, t h e c h o i c e o f a t r i d i a g o n a l E would imply t h e assumption o f u ( k ) a random walk p r o c e s s . On t h e one hand, by imposing a c o r r e l a t i o n among s u c c e s s i v e v a l u e s o f u ( k ) , t h e f l e x i b i l i t y o f t h e output i s reduced, and f o r example a d e l t a f u n c t i o n c o u l d not be r e c u p e r a t e d . On t h e o t h e r hand, smoother o u t ­ p u t s and b e t t e r s o l u t i o n s c o u l d be o b t a i n e d i f good "a p r i o r i " e s t i ­ m a t i o n s o f t h e r e a l a u t o c o r r e l a t i o n s o f u ( k ) c o u l d be p r o v i d e d . F i n a l l y , i t s h o u l d be noted t h a t a p a r t from i t s use i n chromato­ g r a p h i c data t r e a t m e n t , i n v e r s e f i l t e r i n g t e c h n i q u e s such as t h a t de­ s c r i b e d i n t h i s work hav of p o l y m e r i z a t i o n engineering u

u

Acknowledgments We would l i k e t o thank Mr. M. B r a n d o l i n i f o r h i s h e l p w i t h t h e e x p e r i m e n t a l work, CONICET and U.N.L. f o r t h e i r f i n a n c i a l support and Dr. J . F . Weisz f o r r e v i s i n g t h e m a n u s c r i p t .

Literature Cited 1. Wiener, Ν., "Extrapolation, Interpolation and Smoothing of Stationary time Series"; J. Wiley and Sons, Inc.: New York, 1949, p. 163. 2. Helstrom, C.W., J . Opt. Soc. Am., 1967, 57, 297. 3. Sondhi, M.M., Proc. IEEE, 1972, 60, 842. 4. Alba, D. and Meira, G.R., J . Liq. Chromatogr., 1984, 7(l4), 2833. 5. Tung, L.H., J . Appl. Polym. Sci., 1966, 10, 375. 6. Hess, M. and Kratz, R.F., J . Polym. Sci., Part A-2, 1966, 4, 731. 7. Husain, Α., Hamielec, A.E. and Vlachopoulos, J., J . Liq. Chroma­ togr., 1981, 4, 459. 8. Tung, L.H., Moore, J.C. and Knight, G.W., J . Appl. Polym. Sci., Part A-2, 1966, 10, 126l. 9. Tung, L.H. and Runyon, J.R., J . Appl. Polym. Sci., 1969, 13, 2397. 10. Waters, J . L . , J. Polym. Sci., Part A-2, 1970, 8, 411. 11. Grubisic-Gallot, Ζ., Marais, L. and Benoit, Η., J . Polym. Sci., Polym. Physics Edition, 1976, l4, 959. 12. Gruneberg, J. and Klein, J., J . Liq. Chromatogr., 1980, 3, 1593. 13. McCrakin, F.L. and Wagner, H.L., Macromolecules, 1980, 13, 685. 14. Alba, D. and Meira, G.R., J . Liq. Chromatogr., (in press). 15. Vozka, S. and Kubin, Μ., J . Chromatogr., 1977, 139, 225. 16. Hamielec, A.E., J . Liq. Chromatogr., 1980, 3(3), 38l. 17. Hamielec, A.E., Ederer, H.J. and Ebert, K.H., J . Liq. Chroma­ togr., 1981, 4(10), 1697. 18. Chang, K.S. and Huang, R.Y.M., J . Appl. Polym. Sci., 1972, l6, 329. 19. Ishige, T., Lee, S.I. and Hamielec, A.E., J . Appl. Polym. Sci., 1971, 15, 1607.

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20. Kalman, R.E., Trans. ASME, Series D, J . Basic Eng., 1960, 82, 35. 21. Kalman, R.E. and Bucy, R.S., Trans. ASME, Series D, J. Basic Eng., 1961, 83, 95. 22. Booton, R.C., Proc. IRE, 1952, 40, 977. 23. Davis, M.C., IEEE Trans, on Automatic Control, 1963, AC-8, 196. 24. Papoulis, Α., "Probability, Random Variables and Stochastic Proc­ esses", McGraw-Hill 1965. 25. Srinath, M.D., Rajasekaran, P.K.; "An Introduction to Statistical Signal Processing with Applications", Wiley 1979. 26. Anderson, B.D.O. and Moore, J.B., "Optimal Filtering", Prentice Hall 1979. 27. Rosen, E.M. and Provder, T., J . Appl. Polym. Sci., 1971, 15, 1687. 28. The International Mathematical Statistical Libraries, Inc. 1980. 29. Chang, K.S. and Huang, R.Y.M., J . Appl. Polym. Sci., 1969, 13, 1459. 30. Couso, D., Alassia L d Meira G.R. J Appl Polym Sci. 1985, 30(8), 3249. 31. Gugliotta, L.M. and Meira, G.R., Die Makromoleculare Chemie, (in press). RECEIVED

February

26, 1987

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Author Index Abbott, S. D., 80 Alba, D., 287 Anthony, R. G., 183 Armonas, J. E., 104,119 Balke, Stephen T., 59,202 Barth, Howard G., 29 Brower, L., 155 Cheng, Rong-Shi, 281 Chisum, Matthe Chu, Β., 24 Ekmanis, Juri , Garcia-Rubio, L. H., 220 Gugliotta, L. M , 287 Hamielec, A. E., 104 Haney, Μ Α., 119 Kah, A. F., 130 Keating, M Y., 80 Kim, D., 155 Koehler, M E . , 130

Kohn, Erwin, 169 Kuo, Cheng-Yih, 2,130 Mclntyre, D., 155 Meira, G. R , 287 Mendelson, Robert Α., 263 Moore, P. K., 183 Mukherjee, P., 155 Park, I. H., 240

, , Seeger, R., 155 Smith, G. Α., 80 Styring, Mark G., 104 Trowbridge, D., 155 Wang, Q. W., 240 Wang, Zhi-Liu, 281 Yau, W. W., 80 Zhao, Yang, 281

Affiliation Index Ε. I. Du Pont De Neumours & Co., 80 The Glidden Company, 2,130 Hercules Inc., 29 Intec (Conicet and Universidad Nacional del Litoral, 287 Mason & Hanger-Silas Mason Company, Inc., 169 McMaster University, 104 Millipore Corporation, 47

ModChrom, Inc., 104,119 Monsanto Company, 263 Nanjing University, 281 Pressure Chemical Company, 119 State University of New York, 240 Texas A & M University, 183 University of Akron, 155 University of South Florida, 220 University of Toronto, 59,202 Viscotek Corporation, 119

Subject Index A

Adsorption elimination of hydrogen bonding, 33 elimination of hydrophobic interactions, 33 Alkanes, distribution in coal liquids, 195

Absolute flow rate, calibration, 132 Acrylic polymers, membrane viscometer characterization, 163,165/

300

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

Author Index Abbott, S. D., 80 Alba, D., 287 Anthony, R. G., 183 Armonas, J. E., 104,119 Balke, Stephen T., 59,202 Barth, Howard G., 29 Brower, L., 155 Cheng, Rong-Shi, 281 Chisum, Matthe Chu, Β., 24 Ekmanis, Juri , Garcia-Rubio, L. H., 220 Gugliotta, L. M , 287 Hamielec, A. E., 104 Haney, Μ Α., 119 Kah, A. F., 130 Keating, M Y., 80 Kim, D., 155 Koehler, M E . , 130

Kohn, Erwin, 169 Kuo, Cheng-Yih, 2,130 Mclntyre, D., 155 Meira, G. R , 287 Mendelson, Robert Α., 263 Moore, P. K., 183 Mukherjee, P., 155 Park, I. H., 240

, , Seeger, R., 155 Smith, G. Α., 80 Styring, Mark G., 104 Trowbridge, D., 155 Wang, Q. W., 240 Wang, Zhi-Liu, 281 Yau, W. W., 80 Zhao, Yang, 281

Affiliation Index Ε. I. Du Pont De Neumours & Co., 80 The Glidden Company, 2,130 Hercules Inc., 29 Intec (Conicet and Universidad Nacional del Litoral, 287 Mason & Hanger-Silas Mason Company, Inc., 169 McMaster University, 104 Millipore Corporation, 47

ModChrom, Inc., 104,119 Monsanto Company, 263 Nanjing University, 281 Pressure Chemical Company, 119 State University of New York, 240 Texas A & M University, 183 University of Akron, 155 University of South Florida, 220 University of Toronto, 59,202 Viscotek Corporation, 119

Subject Index A

Adsorption elimination of hydrogen bonding, 33 elimination of hydrophobic interactions, 33 Alkanes, distribution in coal liquids, 195

Absolute flow rate, calibration, 132 Acrylic polymers, membrane viscometer characterization, 163,165/

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301

INDEX Applications, SEC to oligomers, 18-19 Aromatics, distribution in coal liquids, 196

Batch viscometer configuration, 87,89/ output signal traces, 87,91/ single-point intrinsic capabilities, 90,91/ Bed hydraulic radius calculation, 38 molecular weight equivalent, 39/ Branched poly(vinyl acetate) degree of branching, 247-251 hydrodynamic radius, 258,260-262/ intrinsic viscosity vs. molecula weight, 247,250/ molecular parameters, 251,253 radius of gyration vs. molecular weight, 247,248/ radius ratio vs. molecular weight, 251,252/ synthesis, 241 Zimm plot, 247,248/ Branched poly(vinyl acetates) degree of branching, 114 determination of branching, 106 molecular weight data, 112, 114/ molecular weight measurement, 106,108 synthesis, 106 Branched polymers randomly branched polystyrene, 145-148,1 star-branched polystyrene, 147,149-151 Branching data analysis methods analysis of size exclusion chromatogram, 245-247 light-scattering data, 242-245 Branching index, calculation, 135 Bulk intrinsic viscosity, determination, 134 C Calibration curve description, 3,5 error, 219 molecular weight related to retention volume, 5 Chemometrics, definition, 202 Coal, analysis by SEC, 184 Coal liquefaction process, data analysis, 184 Coal liquid samples, description, 185 Coal liquids alkane distribution, 195 aromatic distribution, 196 overlapping of species in SEC fractions, 196-197

Coal liquids—Continued phenol distribution, 195-196 probable molecular structure based on SEC, 190/, 193 quantitative analysis by S E C - G C - M S , 194-195 SEC output, 197,198/ SEC separation of major chemical species, 188,189/ Coefficients of effective relations, polystyrene standards, 283,284/ Complex polymers ambiguous detector response, 60 analysis by H P L C , 62,64 analysis by SEC detector technology, 62,63/ characterization, 221 -222 definition 60,62,221

Compositional distribution, vs. molecular weight, 12-13,15/ Concentration effects, elimination, 36-37 Concentration polarization, definition, 40 Continuous capillary type viscometer multiple-tube system, 82 single-tube system, 82,83/ Contraction of polyelectrolytes, 35-36 Contribution of longitudinal dispersion and extracolumn effect, 282 Contribution of the SEC process, calculation, 282 Cross-fractionation, description, 64

D Data reduction procedures for SEC-viscometer system, scheme, 134-136/ Dead volume determination, 139 effect between detectors, 141/ effect on viscosity-molecular weight plot, 139,140/,141 Deformation, cause, 40 Degree of branching, estimation, 247-251 Deterministic integral equation expression, 287 problems, 287-288 Differential log-amplifier output signal, calculation, 84 Differential pressure, measurement, 159 Differential pressure transducer capillary viscometer description, 131 design, 16 Differential pressure viscometer flow rate independence, 87,88/ precision, 87,88/

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D E T E C T I O N A N D DATA ANALYSIS IN SIZE E X C L U S I O N C H R O M A T O G R A P H Y

Differential refractometer description, 5 disadvantages, 9 Discrete batch type operation, description, 131 Distillation vs. SEC, 197 Distributed properties, definition, 60 Distribution coefficient calculation, 282 definition, 29 for nonideal SEC behavior, 30-31 physical significance of values, 29-30 vs. conformational entropy of solute, 30

Gel permeation chromatography—Continued preparative, 48 system, 47-48 See also Size exclusion chromatography Gel-solute interaction, effect on SEC, 193 Gels, conventional characterization, 162/ GPC, See Gel permeation chromatography, Size exclusion chromatography Graphics definition, 208 misleading plots, 213 moment analysis plots, 210,213,215/ plots of residuals, 210,212/ plotting of digitized chromatogram heights, 210,211/

Ε Electrostatic interactions, 33 Enthalpic interactions adsorption, 33 electrostatic interactions, 33,34/,35 elimination, 32 types, 32 Error propagation analysis definition, 213-214 error in data fit by regression, 214 estimation of error in results, 214,215-216/ Error propagation equations choice of objective functions, 236-237 sources of experimental error, 235 theory, 234-235 Error propagation in static measurements, sources, 235

H

preparative GPC, 50-55 High-performance liquid chromatography, complex polymers, 62,64 High-performance SEC, oligomer applications, 18-22 Hydrodynamic chromatograph, definition, 40 Hydrodynamic effects, description, 39-40 Hydrodynamic volume calibration curve generation, 6-7 refinements, 7 schematic, 7,8/ Hydrogen bonding, elimination, 33 Hydrophobic interactions, elimination, 33

I Filter adjustment, calculation, 291-292 Flame-ionization detector, identification of coal liquids, 185 Flory-Fox equation, 38 Fluid viscosity differential pressure detection, 83/,84 measurements, 80-81 Forced flow-through type capillary viscometer configuration 1, 84,85/,87 configuration 2, 86,87, 88/ differential log-amplifier output signal, 84,86 zero offset factor, 84 Fractionation, poly(styrene-co-«-butyl methacrylate), 68 G Gel permeation chromatography calibration curve, 120,124/ molecular weight distribution, 254,256,257/,258

Inherent viscosity, definition, 81 Injected solvent, effect on orthogonal chromatography, 73,74/ Input estimation through a Wiener filtering approach optimal estimate, 290-291 problem, 290 Instrument spreading correction, calculation, 7,9 Instrumental broadening, correction methods, 287 Instrumentation SEC, essential components, 5 SEC-viscometer system, 132 Intermolecular electrostatic effects, elimination, 35-36 Intrinsic viscosity calibration, 97,100-101/ definition, 81 dependence on number of backbone carbon atoms, 265 determination, 97,134 effect of solvent, 273,274/

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303

Intrinsic viscosity—Continued experimental determination, 81 logarithmic plot vs. molecular weight, 120,123/ measurement, 267 poly(styrene-acrylonitrile), 269-279 relationship to molecular weight, 81 Inverse filtering technique, examples of applications, 293-294,295-296/ Ion exclusion of water-soluble polymer, example, 33,34/,35 Ion inclusion, description, 35

L LALIS, See Low-angle laser light scattering, 13,14/, 16 Laser light scattering, molecula distribution, 254,255/,257/,25 Light scattering intensity of scattered light, 242 molecular weight distribution, 244-245 spectrum of scattered light, 242-244 Linear calibration function, calculation, 282 Linear copolymer, property distributions, 59,61/ Linear homopolymer, property distribution, 59,61/ Linear poly(vinyl acetate) hydrodynamic radius, 258,260-262/ molecular parameters, 251,253/ Linear polymers cumulative and differential distribution curves, 141,144/ differential refractometer trace, 141,142/ molecular weight data, 110,112/ poly(methyl methacrylate), 145,146/ poly(vinyl chloride), 145,146/ polystyrene, 145/ properties, 106,107/ secondary molecular weight curve, 141,143/ viscometer trace, 141,142/ viscosity vs. molecular weight, 141,144/ viscosity vs. retention volume, 141,143/ Long-chain branching, characterization, 240-262 Low molecular weight epoxy resin, preparative GPC, 55-58 Low-angle laser light scattering (LALIS) detector chromatogram, 13,14/, 16 features, 13

M Macromolecular crowding, 36 Mark-Houwink equation, 38-39

Mark-Houwink-Sakurada equation application, 263 derivation for multispecies polymers, 264-267 parameters at fixed acrylonitrile concentration, 275,276/ Mass spectrometers, identification of coal liquids, 185 Membrane viscometer applications, 167 calculation of kinematic viscosities, 156 characterization of acrylic polymers, 163,165/, 166/, 167 characterization of natural rubber, 163,164/ description, 155,156 design, 157,158/ diagra with loo system 159,160/

maximum shear stress calculation, 156-157 membranes, 157-159 prefilter system, 157 pressure/flow rate vs. flow rate, 159,161/ pump, 157 velocity gradient calculation, 156-157 viscosity calculation, 156 viscosity vs. shear rate, 159,161/ Microgel, definition, 155 Microparticulate packings, list, 5,6/ Molecular migration, definition, 39 Molecular size, dependence on retention volume, 3,4/ Molecular weight calculation, 246-247 effect of U V detector noise vs. elution volume, 225,226/ error propagation techniques, 225,227 poly(styrene-acrylonitrile), 269,270/ polystyrene standards, 283,284/ universal calibration curve, 257/,258 variance, 227 Molecular weight calibration curve generation, 13 secondary, generation, 134-135 Molecular weight distribution determination for poly(dimethylsiloxanes), 171,173/ gel permeation chromatography, 254,256,257/,258 laser light scattering, 254,255/,257/,258 Molecular weight distribution curve, generation, 5-7,8/ Molecular weight specific detector, features, 82 Molecular weights branched poly(vinyl acetates), 112 determination for poly(styrene-acrylonitrile), 267 linear polymers, 110,112/

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Multicomponent system, quantitative Orthogonal chromatography—Continued analysis, 13,15/ multidimensional chromatography, 64 Multidetector SEC poly(ethyl methacrylate), 68,71 comparison between static measurements and poly(lauryl methacrylate), 68,71 poly(styrene-co-H-butyl estimates, 228,232/ methacrylate), 68,69/ interpretation of signals, 228,229-231/ polystyrene, 68,71 Multidimensional chromatography, recent advances, 73,75-76 description, 64 separation mechanisms, 65,67/ Multiple capillary tube viscometer, size fractionation, 65,66/ advantages over single-tube design, 82 Multispecies polymers, derivation of Mark-Houwink-Sakurada equation, 264-267 Ν Narrow polystyrene standards, viscosities, 120,121-122/ Natural rubber, membrane viscomete characterization, 163,164/ Non-size-exclusion effects, overview Nonlinear hydrodynamic volume calibration curve effect of polymer concentration, 138-139 example, 138,140/ generation, 138 Nonlinear regression definition, 203 detector nonlinearity assessment, 207 determining calibration curves from polydisperse samples, 205 fitting of calibration curves determined using monodisperse samples, 205,207 fitting of shape functions, 208,209/ parts of method, 203-204 reasons for liquid chromatographic application, 204 resolution correction, 208 schematic of calibration curve, 205,206/

Ο Oligomers acrylic resins, 19,20/ applications, resolution, 18-19 epoxy resins, 19,21/,22 melamine cross-linkers, 19,21/ polyester resin screening, 19,20/ Orientation, elimination of effects, 40-41 Orthogonal chromatography arrangement of SEC instruments, 64,66/ axial dispersion characterization, 68 complications, 68,72/,73,74/ copolymer composition distribution, 68,70/ cross-fractionation, 64 description, 64 detector technology, 65 effect of injected solvent, 73,74/ initial studies, 65

Phenols, distribution in coal liquids, 195-196 Phenylmethylsiloxane

Poly(dimethylsiloxanes) molecular weight distribution, 171,173/ phenyl analysis, 177 quantitation of phenyl content, 179 Poly(ethyl methacrylate), fractionation, 68,71 Poly(lauryl methacrylate), fractionation, 68,71 / Poly(methyl methacrylate), SEC-viscometry results, 145,146/ Poly(styrene-acrylonitrile) intrinsic viscosity, 269-272,275-279 molecular structure data, 275,277/ molecular weights, 269,270/ sample preparation, 267 Poly(vinyl acetate) branched, See Branched poly(vinyl acetate) branching index vs. molecular weight, 151,152/ SEC-viscometry results, 149,151/ viscosity vs. molecular weight, 151,152/ Poly(vinyl chloride), SEC-viscometry results, 145,146/ Polyelectrolytes chain expansion, 36 contraction, 35-36 Polymer complexity, effects on conventional SEC analysis, 60 Polymer concentration, effect on hydrodynamic volume, 138-139 Polymer molecular weight characterization, Mark-Houwink-Sakurada equation, 263 Polymer properties determination, 224 effect of functional group position, 170 errors in estimation, 234-237 variance, 225 Polymer shear degradation, See Shear degradation

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INDEX

305

Polymeric silicones, applications, 169 Polystyrene fractionation, 68,71 / SEC-viscometry results, 145/ Polystyrene broad standards, universal calibration data, 125,127/ Polystyrene narrow standards, effect of sample concentration, 125,128/ Polystyrene standards coefficients of the effective relations, 283,284/ molecular weight, 283,284/ spreading factors, 283,284/ Poly(vinyl acetates), branched, See Branched polyvinyl acetates), 106 Preparative GPC applications, 48 flow rate, 49 high molecular weight polystyren analysis, 53,56/ fractions, 50 maximum loading capacity, 50,51 / molecular weight averages, 50,53/,55/ molecular weight distributions, 50,52/ separations, 53,54/,55/ low molecular weight epoxy resin analysis, 55,56/ fractionation, 57-58/ loading study, 55,57/ preparative separation, 49 procedure, 48/,49 Styragel column, 49

Q Quantitative analysis multicomponent system, mathematical formulation, 13,15/ two-component system, mathematical formulation, 13,15/

R Randomly branched polystyrene branching index vs. molecular weight, 147,150/ size, 145,147 viscosity vs. molecular weight, 147,148/ Ratio of intrinsic viscosities, calculation, 114 Ratios of experimental quantities, error, 218 Raw viscometer data smoothing example from polystyrene standard, 135,137/,138 power spectrum of signal, 135,137/ with dampers removed, 135

Relative viscosity, definition, 81 Resolution, influencing factors, 18 Retention volume calculation, 282-283 definition, 3 dependence of molecular size in solution, 3,4/

SEC, See Size exclusion chromatography Shape functions, definition, 208 Shear degradation detection, 37-38 influencing parameters, 37 Shear stress, maximum, calculation, 156-157 Silanol groups

IR spectra, 174,175/ Silicon-phenyl groups, determination, 174,177-181 Silicone hydride groups determination, 171 percent content, 172/ stretching frequencies, 172/, 173/ Simple polymer, definition, 60 Size exclusion chromatogram, analysis, 245-247 Size exclusion chromatographic analyses, experimental setup, 170-171 Size exclusion chromatographic detection system, calibration, 237-239 Size exclusion chromatographic detector alternative configuration, 94,95/ configuration, 90,92/ flow-rate independence, 90,93/,94 flow-rate upsets, 94 preferred configuration, 102 viscosity analysis, 94,96/ Size exclusion chromatographic measurements, experimental procedure, 241 Size exclusion chromatography application of computers, 202 application to coal liquids, 184 calibration curve, 5-7,8/ complex polymers, 60-63 correction for instrumental broadening, 287-296 description, 3 dual detector, 62,63/ effect of column overloading, 191 effect of difference in detector sensitivity, 222,223/ effect of gel-solute interaction, 193 effect of solvent-solute interaction, 192-193 error propagation analysis, 213-214

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

306

D E T E C T I O N A N D DATA ANALYSIS IN SIZE E X C L U S I O N C H R O M A T O G R A P H Y

Size exclusion chromatography-viscometer Size exclusion chromatography—Continued system—Continued fractionation, 60,63/ analysis of linear polymers, 141-146 future trends and needs, 21 data reduction procedures, 134-135,136/ graphics, 208,210-213 dead volume, 139-141 instrument spreading correction, 7,9 differential pressure transducer capillary instrumentation, 5,6/ microparticulate packings, 5,6/ viscometer, 131 molecular weight calibration, 97,98/ hardware design, 131-132 molecular weight calibration vs. universal instrumentation, 132 calibration, 97,99/ materials, 133 molecular weight distribution nonlinear hydrodynamic volume calibration determination, 171 curve, 138-139 multiple detectors, 9-18 polystyrene standards used for nonlinear regression, 203-208 calibration, 133/ probable molecular structure of coal raw viscometer data smoothing, 135-138 liquids, 190/, 193 Viscotek detector, 131 relationship to other chromatographic Size exclusion data, polystyrene techniques, 31,34/ standards 283,284/ sample spreading, 190/, 193-19 separation of major chemical coal liquids, 188,189/ Solubility parameter model, treatment of separation of Wyodak recycle adsorption effects, 32 solvent, 186,188,189/ Solvent-solute interaction, effect on system setup, 119-120 SEC, 192-193 types of detectors, 131 Specific viscosity, definition, 81 universal calibration, 97,99/ Spreading factor universal calibration curve, 125,126/ applications, 281 viscosity calibration, 97,98/ polystyrene standards, 283,284/ vs. distillation, 197 variation with retention See also Gel permeation chromatography volume, 283-284,285/ Size exclusion chromatography-differential Spreading function variance, definition, 281 refractometer-UV spectrometer-IR Spreading model spectrometer, chromatogram, 10,11/ representation of G matrix, 288,289/ Size exclusion chromatography-GC theory, 288,290 alkane distribution, 195 Star-branched polystyrene analysis of coal liquids, 197,198/ branching index, 147 aromatic distribution, 196 differential refractometer trace, 149,150/ quantitative analysis, 194-195 g value, 147 Size exclusion chromatography-GC interface, hydrodynamic data, 147,148/ system, 186,187/ molecular weight, 147 Size exclusion chromatography-GC-mass SEC-viscometry results, 149/ spectrometry viscometer traces, 149,150/ instrumentation, S E C - G C Styrene-acrylonitrile copolymers, interface, 186,187/ synthesis, 227 phenol distribution, 195-196 Sylgard addition system Size exclusion chromatography-IR curing reaction, 170,173/ spectrometer, chromatogram, 10,12,14/ description, 170 Size exclusion chromatography-IR system, Τ quantitation of silanol groups, 174,176/ Size exclusion chromatography-laser light Tetrahydrofuran, use in SEC scattering system, design, 267-268 analyses, 179,181/ Size exclusion chromatography-UV Truncated power series model, equation, 234 spectrometer, chromatogram, 10,12,14/ Two-component system, quantitative Size exclusion chromatography-UV analysis, 13,15/ spectrometer-IR spectrometer, chromatogram, 10,11/ U Size exclusion chromatography-viscometer system Ubbelohde-type capillary viscometer, experimental setup, 131 analysis of branched polymers, 146-151

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.

307

INDEX Ultrafiltration, molecular weight limit, 38 Universal calibration, description, 264 Universal calibration curve effect of sample concentration, 125,127/ plot, 125,126/ polystyrene broad standards, 125,127/

V Viscometer-differential refractometer detector system calculation of branching, 114-115 comparison of molecular weight data, 108,110/ description, 108 detection of high MW impurities, 115,116/,117 errors, 110,111/ experimental plan, 108 universal calibration curves, 108,109/ Viscosity, calculation, 156 Viscosity detector advantages and disadvantages, 18 design, 16 intrinsic viscosity vs. molecular weight, 16,17/, 18 sensitivity, 16,17/ Viscosity gradient, calculation, 156-157

Viscotek detector, description, 131 Viscous fingering, 36-37 Void volume, definition, 3

W Wiener filtering approach filter adjustment, 291-292 input estimation, 290-291 solution validation, 292-293 Wyodak recycle solvent effect of fraction collectors, 188 G C analysis of SEC fractions, 188,190/ identification of species in SEC fractions, 188,190/,191-192 SEC separation, 186,188,189/

Yau-Malone equation, expression, 138 Ζ Z-average translational diffusion coefficient, determination, 249,250/,252/ Zimm plot, for polyvinyl acetate), 247,248/

In Detection and Data Analysis in Size Exclusion Chromatography; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1987.