Amorphous Materials : Research, Technology and Applications [1 ed.] 9781608767144, 9781606922354

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

AMORPHOUS MATERIALS: RESEARCH, TECHNOLOGY AND APPLICATIONS

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

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AMORPHOUS MATERIALS: RESEARCH, TECHNOLOGY AND APPLICATIONS

JASON R. TELLE .

AND

NORMAN A. PEARLSTINE

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EDITORS

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Telle, Jason R. Amorphous materials : research, technology, and applications / Jason R. Telle and Norman A. Pearlstine. p. cm. Includes index. ISBN 978-1-60876-714-4 (E-Book) 1. Amorphous substances--Industrial applications. I. Pearlstine, Norman A. II. Title. TA418.9.A58T45 2009 620.1'1--dc22 2008053071

Published by Nova Science Publishers, Inc.    New York

CONTENTS Preface Chapter 1

Computer Modeling of Amorphous Metals David K. Belashchenko

Chapter 2

Novel Amorphous and Nanocrystalline Soft Magnetic Materials Nuria Iturriza, Juan José del Val, Arkadi P. Zhukov, Ignacio García, José A. Pomposo and Julián González

Chapter 3

Relationship of Fragility and Dilatation with Glass-Forming Ability of Pr-Based Bulk Metallic Glasses Qingge Meng, Shuguang Zhang, Mingxu Xia, Jianguo Li and Xiufang Bian

Chapter 4

Chapter 5

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vii

53

105

Studies of Isolated Pores in Non-Graphitazable Carbon by Solid State NMR Kazuma Gotoh and Aisaku Nagai

133

Ge:Sb:Te Stoichiometric and Eutectic Films for Phase-Change Memory Technology E. Prokhorov and J. González-Hernández

147

Chapter 6

Significance of the Amorphous State – A Pharmaceutical Approach István Antal and Romána Zelkó

Chapter 7

Fundamentals and Technological Applications of Hydrogen Absorbing Mg Amorphous Alloys Sydney F. Santos, Tomaz T. Ishikawa and Edson A. Ticianelli

Chapter 8

Reuse Strategies for Dual-Layer Organic Photoreceptors David M. Goldie and Martin J. Ball

Chapter 9

A New Approach to the Ab Initio Generation of Amorphous Semiconducting Structures: Electronic and Vibrational Studies Ariel A. Valladares

Index

1

185

219 239

255 319

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PREFACE Amorphous materials, also known as soft glass, lack a definite form. Computer modeling of liquid and amorphous metals has a rather long history. Due to computer facilities progress, possibilities of construction of adequate model of liquid or amorphous metal have considerably increased. New results and also some problems that have arisen in the literature are discussed in this book. In particular, problems of ambiguity of correlation structureinteraction potential, topological aspects of structural characteristics of non-crystalline system and feature of diffusion in amorphous systems are discussed. Bulk metallic glasses (BMGs) have enabled the measurements of various physical as well as mechanical properties that were often impossible. Thermal expansion is one of the most important thermodynamic characteristics and is strongly correlated to the amorphous structures of metallic glasses. Thus, the relationship of fragility and dilatation with glassforming ability of Pr-based bulk metallic glasses are explored. Phase-change memory technology is based on the high speed reversible amorphous-tocrystalline transformation of a thin film material and utilizes the difference in the optical or electrical properties between both states. In this book, stoichiometric and eutectic doped with additional elements (oxygen and nitrogen) films, having amorphous and crystalline structures, are reviewed. In addition, the influence of phase-change material composition on amorphous phase stability, crystallization temperature, crystallization rates, optical and electric contrast are discussed. The solid-state characteristics play an important role during the development and manufacture of medicinal products, because they may influence the effectiveness, stability as well as the processibility of pharmaceutical systems. The amorphous state is critical in determining the solid-state physical and chemical properties of many pharmaceutical dosage forms. The main reason of the growing interest toward amorphous materials is the need to improve the bioavailability of compounds with poor aqueous solubility. The amorphous state is also of great importance for excipients used in various dosage forms. These developments are discussed in this book as well as the tracking of the ageing process of amorphous materials that is possible using an array of techniques (calorimetry, scanning electron microscopy, X-ray diffraction). Finally, Hydrogen is emerging as a strong candidate to replace fossil fuels. Hydrogen storage through metal hydrides is the most compact and safe form of storing this gas. Mg alloys, which are known to have large storage capacities and great improvements in activity

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viii

Jason R. Telle and Norman A. Pearlstine

are examined when they are obtained when these alloys are synthesized with amorphous structures. Chapter 1 - Computer modeling of liquid and amorphous metals has a somewhat long history. Development in this area is stipulated by application of interparticle potentials, more suitable for metals, and by an increase in the sizes of models. Earlier principal types of interparticle potentials were pair-wise. Multiparticle potentials that better reflect the specificity of metal bonds are now most popular. The maximum sizes of models have increased by hundreds times thanks to advances in computer facilities. Therefore, the possibilities for construction of adequate models of liquid or amorphous metals have considerably increased. Besides the normal thermodynamic properties, dynamic properties, phase transformations, behavior of substances in extreme conditions of high pressures and temperatures are well studied by computer methods of modeling. The author of this article has been working for a long time in the field of computer modeling of non-crystalline metals, ionic systems and other inorganic objects. Some new results and also some problems that are little reflected in the literature are discussed in this chapter. In particular, problems of ambiguity of correlation structure—interaction potential, topological aspects of structural characteristics of non-crystalline systems and features of diffusion in amorphous systems are discussed below. Chapter 2 - Promising iron-based soft nanocrystalline alloys, which present a softer magnetic behaviour than that of the precursor amorphous materials have been, in the last years, developed. Most investigations were performed on the typical Fe73.6Cu1Nb3Si13.5B9 composition (trademark Finemet) with a nanocrystalline grain structure produced by the partial devitrification of the precursor amorphous ribbons to reach the nanostructure character with optimal soft magnetic properties. This softening effect provides an alternative approach to the development of novel soft magnetic materials. Nanocrystalline soft magnetic alloys prepared by annealing melt-spun amorphous precursors, with which this chapter is concerned, are one of the latest successful outcomes of such a new approach to the development of novel soft magnetic materials. In this chapter we will present the origin of the softness, processing and magnetostrictive properties of the nanocrystalline soft magnetic Fe(Co,Ni)-based Finemet alloys with particular attention paid to: (i) nanostructure-magnetic properties relationships and (ii) the principles underlying material design. The nanocrystallization process will be performed by thermal treatments of current annealing under the action of a mechanical stress and/or magnetic field in order to develop simultaneously a macroscopic uniaxial magnetic anisotropy. The substitutions of Fe by Ni or Co on the mentioned behaviours are also analyzed. Chapter 3 - Bulk metallic glasses (BMGs) have enabled the measurements of various physical as well as mechanical properties that were often impossible. There have been many studies focusing on the Angell’s fragility parameter m [J. Non-Cryst. Solids 131, 13 (1991)] and dilatation of BMGs. However, the fragility parameter m in undercooled melts can not be used to predict the glass-forming ability (GFA) since the fragility parameter m can only be gotten after the glass has been obtained. To predict GFA, in particular, of marginal metallic glasses, a fragility parameter, M, for superheated liquid was empirically proposed by Bian [Phys. Lett. A 335, 61-67(2005)]. On the other hand, thermal expansion is one of the most important thermodynamic characteristics and is strongly correlated to the amorphous structures of metallic glasses. Nonetheless, few research works has paid attention to the relationship between the dilatation characteristics and GFA. Therefore, it is of particular

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Preface

ix

interest to deduce the expression of the parameter M, and to study the correlation of the both fragility parameters, m and M, and dilatation behavior with GFA of BMGs. The results on this topic particularly for Pr-based BMGs will be discussed and summarized in this article. By measuring the viscosity and specific heat capacity of the supercooled liquid, it was found that the Pr55Ni25Al20 alloy behaved much closer to strong glasses than the other reported metallic glasses, and the GFA is correlated more strongly with the liquid/crystal Gibbs free energy difference than with the supercooled melt fragility parameter m. To improve the predictive power for GFA, in particular, of marginal metallic glasses with the parameter M, the refinement to this parameter was made from both the thermodynamic and kinetic perspectives. The refined parameter gives a much better reflection of GFA for metallic glasses. A correlation between αaver and the weighted average of the thermal expansion coefficients αi for the constituent elements was found as αaver=∑fi αi for Pr-based BMGs. By assuming Lennard-Jones (LJ) type potentials, the average nearest-neighbor distances r1 and the depths of the effective pair potentials V0 were calculated. The values of r1 were in good accordance with the experimental results and V0 correlated with the GFA. The above mentioned results will help to gain deep insight into the glass formation of metallic glasses. Chapter 4 - One of the amorphous carbons, non-graphitizable carbon (hard-carbon), after heating to temperatures greater than 1000°C, is expected to be superior as an anode active material in lithium ion secondary batteries for large devices such as electric vehicles. In this chapter, we intend to show examples of solid state NMR analyses of 7Li and 129Xe nuclei for hard-carbon. The Li ion intercalated into graphene sheets, and quasimetallic Li in pores, are observed in the charged hard-carbon by 7Li NMR measurement. The intensity of the NMR signal belonging to quasimetallic Li is generally proportional to 0.0 V constant voltage capacity of charge-discharge curves, and the small difference is explainable by an under potential deposition model. On the other hand, 129Xe NMR can observe the Xe atom adsorbed into pores. Because the chemical shift of 129Xe NMR changes with the pore size, the pore size and carbon structure can be evaluated by NMR method. The growth of the porous structure of hard-carbon samples made from petroleum pitch and phenolic resin were observed by 129Xe NMR, which was difficult to analyze using other porosimetric methods such as nitrogen adsorption. Chapter 5 - Phase-change memory technology is based on the high speed reversible amorphous-to-crystalline transformation of a thin film material and utilizes the difference in the optical or electrical properties between both states. For optical storage and for nonvolatile electrical applications the difference in reflectivity (20-30 %) and in the electrical conductivity (3-4 orders of magnitude), respectively is utilized to write the bits of information. In these technologies the most commonly employed materials have stoichiometric compositions in the Ge:Sb:Te phase diagram mainly along the GeTe–Sb2Te3 pseudobinary line. Some of the frequently used compositions are Ge2Sb2Te5, Ge1Sb2Te4, Ge1Sb4Te7 and Ge4Sb1Te5 as well as the Sb:Te eutectic (Sb69Te31) alloys, modified with Ge. The amorphous-to-crystalline phase transformation in these two groups of alloys is different. The stoichiometric films show a crystallization behavior dominated by nucleation i. e., crystallization proceeds mainly via the generation of nucleus. In contrast, eutectic films have low nucleation probability. As a consequence, crystallization proceeds primarily from the amorphous mark boundaries. However, both types of materials perform well as rewritable phase change media.

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x

Jason R. Telle and Norman A. Pearlstine

In this chapter stoichiometric and eutectic doped with additional elements (oxygen and nitrogen) films, having amorphous and crystalline structures, are reviewed. Films prepared using several techniques such as: thermal evaporation, sputtering, metal organic chemical vapor deposition and pulsed-laser deposition, as well as the influence of the deposition methods on the film properties, such as the optical and electrical properties are analyzed. In addition, the influence of phase-change material composition on amorphous phase stability, crystallization temperature, crystallization rates, optical and electrical contrast (difference in reflection and resistivity between amorphous and crystalline phases) are discussed. Chapter 6 - The solid-state characteristics play an important role during the development and manufacture of medicinal products, because they may influence the effectiveness, stability as well as the processibility of pharmaceutical systems. The amorphous state is critical in determining the solid-state physical and chemical properties of many pharmaceutical dosage forms. The main reason of the growing interest toward amorphous materials is the need to improve the bioavailability of compounds with poor aqueous solubility. However the disordered structure of higher energy state assures increased solubility and faster dissolution rate, the amorphous state is a non-equilibrium state. Materials often go through spontaneous transformations towards lower energy equilibrium states. The amorphous state is of great importance also for excipients used in various dosage forms. Physical ageing of polymers can be initiated by several factors and is manifested in macrostructural (enthalpy and volume) and microstructural (free volume distribution) alterations. These changes also occur during the processing and storage of pharmaceutical dosage forms containing such materials, thus influencing their biopharmaceutical profile. Tracking the possible ageing process of amorphous materials is possible using an array of techniques. Macrostructural changes can be followed by calorimetry and scanning electron microscopy, while microstructural alterations can be sensitively monitored by powder X-ray diffraction and several spectroscopic methods (IR, Raman, NIR, Terahertz Pulsed spectroscopy, solid-state NMR). In addition, positron annihilation lifetime spectroscopy is able to follow the ageing of polymer excipients. The given experimental examples reveal some of these processes and highlight the importance of the amorphous characteristics in the pharmaceutical field. Chapter 7 - Greenhouse gas emissions, increasing prices, and inevitable exhaustion are intrinsic problems associated to the consumption of fossil fuels and are also the main driving forces to the search of sustainable and environmental friendly sources of energy. In this scenario, hydrogen emerges as a strong candidate to replace fossil fuels. The large scale utilization of hydrogen as energy vector has been hindered by a number of technological and economical issues that must be previously solved to allow this change in energy generation. A major technical problem is to find adequate forms of storing hydrogen, especially in transport sector. Hydrogen storage through metal hydrides is the most compact and safe form of storing this gas. Mg alloys are known to have large storage capacities and great improvements in activity and kinetics are obtained when these alloys are synthesized with amorphous structures. In this chapter, fundamentals of gas – solid reaction are introduced and the application of Mg amorphous alloys for hydrogen storage and nickel-metal hydride anodes are overviewed. Chapter 8 - The effect of market use upon the xerographic response of a dual layer organic photoreceptor (DLOP) is evaluated for a commercial photocopier drum product.

Preface

xi

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Continued market use of the drums leads to increased mechanical wear of the DLOP charge transport (CT) layer and an associated reduction in xerographic performance. To identify the principal cause of the xerographic degradation, fundamental operational parameters of the DLOP such as charge carrier mobilities and supply efficiencies are compared between newproduction and market-returned products. It is found that the performance of both the CG and CT regions is diminished in the market-returned products with the CG degradation being more critical. Reuse strategies to remedy these effects include a partial recoat of the CT layer, and complete removal and recoating of the CT layer. The single-step partial recoat strategy is commercially cheaper but the DLOP performance may be prone to hysteresis phenomena (cycle-up) during normal xerographic operation. The complete recoat strategy avoids such hysteresis problems but is found to require careful selection of the stripping solvent and control of the drying temperature used during CT removal to ensure a comparable xerographic performance with new-production DLOP drums. Chapter 9 - We have devised a new method to amorphize semiconducting group IV elements (C, Si, Ge) and their alloys using an ab initio approach based on the Harris functional and computationally thermally-randomized periodically-continued cells with at least 64 atoms. The computational thermal processes consist in linearly heating the samples to just below their melting temperatures, and then linearly cooling them afterwards in what we call the undermelt-quench method. These processes are carried out from initial crystalline conditions using short and long time steps. The different time steps lead to samples having distinctly different topological disorder. We found that a step 4 times the default time step (mmin/5)1/2 was adequate for the simulations. The radial distribution functions (partial and total) of the resulting samples were calculated and compared with measured distributions and the agreement is very good. For some materials we have studied the effect of the topological disorder on their electronic densities of states, their optical properties, and their phonon properties. Here we present our results for the following amorphous systems: a-C, a-Si, a-Ge, a-Si:H, a-Si:N, a-C:N.

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In: Amorphous Materials: Research, Technology… Editors: J. R. Telle, N. A. Pearlstine

ISBN: 978-1-60692-235-4 © 2009 Nova Science Publishers, Inc.

Chapter 1

COMPUTER MODELING OF AMORPHOUS METALS David K. Belashchenko* Moscow Institute of Steel and Alloys, Moscow, Russian Federation

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INTRODUCTION Computer modeling of liquid and amorphous metals has a somewhat long history. Development in this area is stipulated by application of interparticle potentials, more suitable for metals, and by an increase in the sizes of models. Earlier principal types of interparticle potentials were pair-wise. Multiparticle potentials that better reflect the specificity of metal bonds are now most popular. The maximum sizes of models have increased by hundreds times thanks to advances in computer facilities. Therefore, the possibilities for construction of adequate models of liquid or amorphous metals have considerably increased. Besides the normal thermodynamic properties, dynamic properties, phase transformations, behavior of substances in extreme conditions of high pressures and temperatures are well studied by computer methods of modeling. The author of this article has been working for a long time in the field of computer modeling of non-crystalline metals, ionic systems and other inorganic objects. Some new results and also some problems that are little reflected in the literature are discussed in this chapter. In particular, problems of ambiguity of correlation structure—interaction potential, topological aspects of structural characteristics of non-crystalline systems and features of diffusion in amorphous systems are discussed below.

*

[email protected]

2

David K. Belashchenko

THE BASIC METHODS OF MODEL CONSTRUCTION Some known methods of modeling of atomic systems will be considered below: a method of molecular dynamics (MD), allowing building of models at nonzero temperatures in a case when potentials of interparticle interaction are known; a method of a static relaxation (SR), allowing building of models of amorphous substances at a temperature of absolute zero in a case when potentials of interparticle interaction are known; and the methods that allowing building models using known diffraction data about structure. Such powerful methods as the Monte Carlo method and ab initio method of model construction will be not considered here.

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The Method of Molecular Dynamics (MD) The method of molecular dynamics (MD) is described in detail in many works [1,2]. This allows investigating systems at temperatures T > 0. Usually the model of substance (phase) consists of a number of particles placed in the basic cube with periodic boundary conditions (PBC). These PBCs mean that the basic cube together with its content is translated on three co-ordinate axes, forming an infinite simple cubic superlattice (cells of non-cubic form are seldom applied). In the MD method, the forces operating on particles are defined on each calculation step, and then all particles are simultaneously displaced in a new position. The forces operating on particles can be set analytically in the form of formulas where coordinates of particles or interparticle distances enter. Another case occurs when forces are set by the table of values in dependence on distance. Accurate interpolation of the table data on actual interparticle distance for a given pair of particles is required in this case. Interparticle forces can be set in the form of any (but physically reasonable) functions of co-ordinates; they can also be calculated quantum-mechanically or found using known diffraction data about the system structure. These methods are considered below. Calculation of the total force acting on it is required for each particle. Projections of forces on co-ordinate axes are actually calculated. For this purpose, it is necessary to look over all pairs (or triples) of particles, which takes the most time in the calculations. In the case of short-range forces, one can enter the cutoff radius of interaction rc and consider only those pairs with a distance less than rc. In such cases it is useful to prepare periodically the list of nearest neighbors of each atom (L.Verlet list) that are away from the central atom at a distance no greater than rv, which is something greater than cutting radius rc. If such a list is updated, for example, for every k0 = 10 time step, no another particle should enter into the sphere with radius rc for this period. The greater the difference rv - rc, the greater can be k0 and the longer the series of iterations that can be conducted with the same list of nearest neighbors. The speed of the calculations considerably increases at use of the Verlet list, because only those pairs that are contained in the list are searching. Considering the 3rd law of mechanics, it is enough to include in a list of neighbors only those particles whose number is more than the number of the central atom. Usually one chooses such гv that the length of a series k0 is equal to 5, 10, etc. The gain in speed demands expenses of operative memory. If, for example, the model contains 10,000 particles and the Verlet array gets on the average 100 neighbors for an atom (that is, about 200 neighbors at the first atom and any at last atom) the file of neighbors will

Computer Modeling of Amorphous Metals

3

contain 1,000,000 integers. Using ordinary personal computers, it is possible to work with models containing tens of thousand of atoms. Parallel schemes allow building models of millions of particles. The difference in algorithms of the MD method consists of a way to calculate new coordinates and velocities at the end of each time step. It is a standard problem of numerical integration of the movement equations, and for its decision it is possible to involve calculation schemes of a various order—for example, Runge-Kutta formulas, a predictorcorrector method, etc. It is desirable at each step for each pair of particles to calculate force of interaction only once, not to reduce the speed of the calculation. To check the accuracy of the calculations it is possible to determine the degree of conserving the total energy in the absence of external forces, and also to change a sign of particle velocities and inspect how precisely the particles come back to a starting position. Exact enough (to an error of an order τ4 where τ is the length of a time step) is the simple L. Verlet algorithm, allowing calculation of co-ordinates of i-th particles at the end of step via its co-ordinates at the beginning of the given and previous steps: ri (t + τ) = 2 ri (t) - ri (t - τ) + τ2 Fi (t)/mi + O (τ4), Here Fi is total force and mi is mass of a particle. Velocity of particles can be calculated via the formula:

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vi (t) = [гi (t + τ) - ri (t - τ)] / 2 τ + O (τ3) In Verlet algorithm it is enough to remember only co-ordinates of atoms on two subsequent steps, and velocities are necessary only for calculation of kinetic energy. It is necessary to store in memory the initial velocities only for calculation of velocity autocorrelation function. Transition from simple precision of calculations to double precision usually influences results rather little. In practice, more complex algorithms are also applied. However, it makes sense to apply them only in special cases when, for example, it is required to make the time reversal and very high accuracy of calculations is necessary. The internal system of units is used at MD calculations. For example, it is possible to choose length L0 = 1 Å, mass of certain atom (M0) and energy E0 = 1 eV as basic units. Then the time unit t0 will be the derivative unit equal to t0 = L0 (M0/E0)1/2. For example, for an oxygen atom t0 = 4.072.10-14 s. This time order corresponds to the period of vibrations of atoms in a potential well. The time step τ (dimensionless, in units t0) should be optimum. At small τ, roundoff errors may be important, and at large τ a method of finite differences will be incorrect. In both cases, drift of system full energy will be observed. In usual conditions one takes τ ∼0.01. In the run of MD calculations, potential energy of the system U, kinetic energy W and total energy H = U + W are defined. In the absence of external forces, ensemble of N particles in volume V is micro-canonical. It is called the NVE - ensemble. Often in MD modeling the algorithms are applied, allowing to maintain constant temperature or/and pressure. The review of such algorithms is given, for example, in [2]. The model temperature can be changed, multiplying velocities of particles by the fixed number.

4

David K. Belashchenko

This operation can be necessary, if it is required to maintain constant temperature. In this case full energy will change in time. Some variants of MD method are considered, for example, in [3]. Easy is to maintain constant temperature by periodic (to say, on each 10th step) multiplication of particles velocities by [(3/2) NkT0/Ekin]1/2, where N is number of particles in model, k is Boltzmann constant, T0 is the target temperature, Ekin is kinetic energy of particles of model on the given step. In Verlet algorithm a change of velocities is provided with small corrections of co-ordinates on the previous step. Another way is to keep temperature approximately constant via corrections on each step, for example, multiplying all the velocities by factor λ [2]:

λ = [1 +

τ T0 ( − 1)]1/2 tc T

Here τ is a time step, tc is characteristic time in units t0 (which must be chosen), T is actual temperature of model, T0 is the target temperature. In case of pair interaction pressure of system p can be calculated via the virial theorem:

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pV NkT

∑ rij ∂ϕ (rij)/∂rij

= 1−

i< j

3NkT

,

Here V is volume, N is number of particles, k is Boltzmann constant, φ(r) is interaction potential, rij is distance between particles. If there is Coulomb interaction in the system with energy ECoul, the Coulomb contribution to pressure can be calculated via the formula: pCoulV = ECoul/3. In a case when it is required to keep pressure close to given value (for example, to zero), various algorithms also are applied. The simplest solution is to multiply periodically the edge length of the basic cube and co-ordinates of all particles by factor χ = 1 + α (p - p0), where p is actual pressure of model, p0 is the target pressure, and α is the factor, which should be established empirically to avoid sharp changes of length of cube edge in MD run. In [2] factor α is defined as α = βT τ/tc, where βT is isothermal compressibility, and tc is characteristic time in units t0. MD method plays the big role in research of condensed phases, but it has some restrictions concerning amorphous systems. The possibilities to cross potential barriers at low temperatures are rather limited. Therefore it is very difficult to obtain equilibrium condition of amorphous system using MD method. In this case results depend on an initial state and on the way of transition to a final state. The method of static relaxation (SR) discussed below is more productive in this respect. Applications of MD method considered above concerned a case of short-range interparticle potentials which can be cut off on distances lesser than half of basic cube length. In ionic or partially ionic systems Coulomb interaction takes place which drops with distance very slowly. Here the cutting of potential leads to the big errors. Therefore calculation of interparticle forces in the presence of Coulomb interaction should be conducted by a special technique (a method of Evald and its variants).

Computer Modeling of Amorphous Metals

5

Modern computers allow to create rather big models. So, in [4] the model of amorphous silicon was created containing 110592 particles, and in [5] ionic models of liquid and glassy silica were constructed containing 41472 particles. The basic cube of such model has the edge length of ~8.5 nm. Calculations were done on the parallel computer. Thus may be obtained data not only for near, but also for middle order. The correct form of the structural factor (including pre-peak at small values of a scattering vector) is obtained at the number of several thousand particles in a model.

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Method of Static Relaxation (SR) The amorphous structure is most clearly visible on the models constructed at absolute zero when temperature influence is not imposed. In the range from a room temperature to absolute zero the structure characteristics vary usually very little. Therefore the considerable number of works on amorphous substance modeling was executed at Т = 0. For this purpose the method SR may be applied. It differs from MD method only in the way of calculation of particle co-ordinates in the end of given step. Velocities of particles are equal to zero at Т = 0 in the classical mechanics. Therefore displacement of particle can be evaluated using only direction of total force. If the system is classical (in the sense of applicability of classical mechanics) then particles at Т = 0 are in mechanical equilibrium and the total forces acting on each particle are equal to zero. Thus, the structure is defined not by potential ϕ but by force function F(r) = ∂ϕ (r)/∂r. Multiplication of force function by any factor a > 0 should not lead to violation of mechanical equilibrium. Hence, at model construction, the given force function can be normalized from convenience reasons, taking, for example, F (r0) = 1, where r0 is the fixed distance. The structure at Т = 0 will not depend on a choice of r0. On the contrary, if the structure of an amorphous phase at Т = 0 is known, it is possible to define force function (and potential) only within factor a > 0. The particles are displaced at SR modeling sequentially or simultaneously in a direction of total forces. Various algorithms of a static relaxation were applied. For example, it is possible to displace each atom in position of a near local energy minimum. Energy of system thus diminishes. Such algorithms have the shortcoming that the system cannot cross some existing potential barriers. Besides, only one realization of system is created and badly average pair correlation function (PCF) is obtained. The general feature of such algorithms is fast decrease of a displacement step in the course of approach to a final condition. The author [6,7] has applied algorithm of a continuous static relaxation (CSR) at which all particles are simultaneously displaced at each iteration on a constant step in direction of total forces. In the beginning the step is taken big enough (~0.1[V/N] 1/3) to provide crossing of high potential barriers on the way to the equilibrium state. Then the step gradually decreases to the value corresponding to amplitude of zero fluctuations of particles at Т = 0 (∼1 pm). Each decrease of a step is conducted when energy of system in CSR run ceases to diminish. CSR method with variable step is very effective at construction of amorphous structures near absolute zero. It allows to reach the lowest levels of energy. Despite continuous moving of particles, diffusion in a final (relaxed) state is absent.

6

David K. Belashchenko

The central moment in SR method (the same as also in MD method) is calculation of total forces. Here it is also possible to define distance rc after which interaction of particles is negligible and to apply storing of numbers of the nearest neighbors in a special list. The initial state in CSR method can be chosen either in the form of random system of points, or in the form of a crystal, or in the form of already constructed model with other characteristics. Linear transformation of a cube together with model particles allows change the system density easily. CSR method generates long sequence of states so it gives good averaging of PCF. The PCF curve for amorphous iron model (CSR method) shown in Figure 1 is smooth and it has no ripples typical for badly averaged functions.

Ab Initio Method of Construction of Models (“From the First Principles”) The method has been proposed in R. Car and M. Parinello’s work [8] and then it was applied to modeling of various systems: metals and others. Reviews of the works using this method are published in [9,10]. Some features of this method are considered below.

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INTERPARTICLE POTENTIALS

Figure 1. PCF (1) and structure factor (2) of amorphous iron model consisted of 5,000 particles, at T = 0 (author’s data).

Types of interparticle potentials are considered in many works, for example, in [11, 12]. Here, pair-wise, three-particle and multiparticle potentials are used. Pair potentials are characteristic for simple liquids of the type of argon and salt systems and have proven to be rather good for liquid and amorphous metals. In systems with covalent bonds, the interaction potential is a three-particle one because it should lead to the existence of valence angles.

Computer Modeling of Amorphous Metals

7

Recently great attention has been given to multiparticle potentials which allow successful description of properties of solid substances with defects of structure, in the presence of a free surface, etc.

Pair Interparticle Potentials In this case, full potential energy of system U is equal to the sum of interaction energies taken over pairs of particles U (r1, r2... rN) =

∑ ϕ (rij)

(1)

i< j

Here ϕ (rij) is pair interparticle potential. The interaction potentials describing atom collisions are rigid enough usually. They have steep repulsive branch on small distances. A limiting case is hard-sphere potential which is described by the formula: ϕ (r) = ∝ at r < d

and ϕ (r) = 0 at r > d

It contains sole parameter - diameter of sphere d. The system of hard spheres (HSS) is the simple system whose properties have been investigated by methods of MD and Monte Carlo. Important characteristic of HSS is a packing fraction η = (πd3/6) N/V, where N is number of spheres in volume V. The maximum value of η factor for real system of identical metal balls is ηmax = 0.6366 ± 0.0001. At such η value balls cannot move and the system behaves as amorphous at Т = 0. At η < 0.50 structural properties of HSS mimic properties of real liquids. Real potentials smoothly depend on distance and have no discontinuities. Full energy is equal in the pseudo-potential theory of metals to the sum of the pair contribution and the volume part depending only on the volume of system, but not on co-ordinates of particles. Therefore the structure is defined by pair potential ϕ (r). At the construction of non-crystalline models one often applies purely repulsive potential (sometimes called as “potential of soft spheres”) with parameters εdm and m:

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ϕ(r) = ε (d/r) m

(2)

Further, two-parametrical Lennard-Jones potential was proposed for the description of argon:

d d r r ϕLJ (r) = ε [( 0 )12 − 2( 0 )6] or ϕLJ (r) = 4 ε [( )12 − ( )6] , r

r

r

r

(3)

more general “m - n potential”: ϕ (r) =

ε r r [ n ( 0 )m − m ( 0 )n ] , m−n r r

and three-parametrical Morse potential:

(4)

8

David K. Belashchenko ϕМ (r) = ε {exp [ −2 α (

r r − 1)] − 2 exp [ −α ( − 1) ]} r0 r0

(5)

All of these potentials have steep repulsive branch and approach zero at long distances. In formulas (3) - (5) value ε is a depth of a potential minimum, r0 is co-ordinate of minimum, and d is co-ordinate of zero of potential. For reduction of calculation time it is necessary to enter potential cutoff on some distance гc and not to consider interaction with further particles. For example, for potential (3) one takes usually гc = (2 – 2.5)г0. Discontinuities ϕ (r) or ∂ϕ (r)/∂r lead to occurrence of PCF kinks near to them. Therefore it is desirable to smooth potential cutoff. The choice of pair potential usually does not allow to describe simultaneously energy, compressibility and the elastic module. This difficulty is caused by the concept of pair interaction. For example, for body centered cubic lattice (BCC) and also for some loose lattices of diamond type the pair potential of Lennard-Jones (3) does not provide mechanical stability at some deformations [13]. For metals one could include in considerations the volume energy of electrons which reduces compressibility, but does not influence the shear modulus. Pseudo-potential calculations show, that for "good" metals the main contribution to energy is given by pair contributions (Madelung energy and energy of zone structure), and other contributions substantially cancel each other (in case of Na pair contributions give ~90 % of energy). Such pair potentials depend on density and - in multicomponent systems – on concentration.

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Calculation of Coulomb Forces In the course of calculation of total forces summation over other particles of system is conducted. For short-range potentials it is enough to consider rather small number of near neighbors. It is much more difficult to work with ionic systems where the Coulomb forces decrease very slowly with distance. Here potential cutoff is inadmissible. Therefore in systems with Coulomb interaction it is necessary to count all pairs of atoms in a model. Moreover, in the case of periodic boundary conditions it is necessary to consider interaction of i-th particle (together with its images) not only with everyone j-th, but also with all images of j-th particles in the cubes obtained at translations of the basic cube along three co-ordinate axes. These images are located in sites of a simple cubic superlattice. Therefore the interaction potential of pair ij is equal to the lattice sum: ϕ(r) = Zi Zj e2

1 ij + Ln |

∑| r n

(6)

Here Zie and Zje - charges of ions, and vector Ln begins in the origin of co-ordinate system and runs over sites of a simple cubic superlattice with an edge L. Actually for each definite pair of ions separately the sum (6) diverges, though full energy of system with respect to one particle converges. One can enter two fictitious distributions of positive and negative charges. Both backgrounds cancel each other and do not give the contribution to energy. As a result,

Computer Modeling of Amorphous Metals

9

calculated energy of interaction of i-th ions with j-th ion, with its images and with corresponding neutralizing backgrounds has limited value. P. Evald (1921) has developed a special technique in which slowly converging sum is replaced with two quickly converging sums in direct and inverse spaces. Energy of interaction of ion pair (taking into account images of i-th and j-th ions) is equal to the sum of spherically symmetric contribution and the second one having cubic symmetry. For acceleration of calculations J. Hansen [14] had presented formulas for Coulomb energy in the form of exponent-polynomial serials in direct space. The error of calculation of full energy in this method has an order of 0.1 %. At differentiation of Evald potential on the co-ordinates of i-th particle one can obtain expression for Coulomb force in a superlattice of ions. Symmetry of a lattice shows that the forces acting on i-th ion in the centers of edges, in the centre and vertices of cubic cells of a jlattice, are equal to zero. Therefore force decreases with distance faster, than the simple Coulomb one. It is known that the system with purely Coulomb interaction is unstable, and it is necessary to enter repulsion of ionic cores. Rather good results are obtained by approach of rigid ions in which properties of given particle do not depend on a kind of other particles. For alkali halides M. Tosi and F. Fumi [15] have checked up applicability of pair potentials in the form of Born-Mayer: ϕ (r) = ZiZj e2/r + ϕrep (r) - Сij r-6 - Dij r-8,

(7)

Here ϕrep (r) = Bij exp (-r / ρij) (Bij and ρij are empirically chosen parameters of interaction). In the form of Born-Mayer-Huggins (BMH) it will be:

r +r −r ϕrep (r) = bij exp i j ρij

(8)

Here, ri and rj are “ion radiuses,” ρ is screening parameter. Terms with r-6 and r-8 represent contributions from a dipole-dipole and dipole-quadruple interactions. According to BMH, the factor bij is equal:

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bij = b (1 +

Zi + ni

Zj ), nj

(9)

where b = 3.38.10-20 J, and ni is number of electrons in outer shell of an ion. Characteristics of ions in alkali halides are presented in [15, 16]. Contributions from a dipole-dipole and higher interactions are small. So, for the melt KCl at 1045 K they give about 4.4 % of total energy. At high pressures or temperatures when ions approach on small distances, these contributions are not described any more by simple expression (7). In most cases they may be neglected. Expression BMH also is applied to ions with higher charges. The model of rigid ions can be improved considering the polarizability of ions. As a result it is possible to exploit the assumption of central interaction and to receive the good agreement with experiment both for elastic constants and for their derivatives on pressure that is for the values containing third derivative of potential.

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David K. Belashchenko

Quantum-Mechanical Calculation of Pair Potentials Methods of calculation of the effective interparticle potentials have been developed for liquid metals and alloys using a pseudo-potential method. The interparticle potential contains the contribution from direct Coulomb interaction of ions and the contribution from the indirect interaction, caused by interaction of screening electronic charges. The elementary expression for effective ion-ion potential in a one-component case had obtained Harrison [17]: ∞

Z2e2 1 sin (qr) 2 φ (r) = + 2 ∫ F(q) ε(q) q dq r π n0 qr

(10)

0

Here Z is an ion charge, ε(q) is a Fourier transform of dielectric function, and F (q) - the characteristic function: F (q) = −

q2 | ua (q) | 2 [ε (q) − 1] 8 π e2 n0 [ε (q)]2 H

(11)

ua (q) is a Fourier transform of atom non-screened pseudo-potential, εH(q) is a Fourier transform of Hartree dielectric function, n0 is number of atoms in volume unit. Formulas for dielectric functions are cited in textbooks on the theory of metals. In the case of an alloy it is necessary to write ZiZj instead of Z2 in the formula (10), and uiuj instead of ua2 in (11). If internal cores of atoms overlap themselves it is necessary to add the corresponding contribution in expression (10). The effective interparticle potentials calculated by a pseudopotential method, depend on density and concentration of a metal solution.

Ab Initio (“From the First Principles”)

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Calculation of Interparticle Interactions The method is offered in R. Car and M. Parinello's work [8]. In this method the forces acting on atoms (ions) are calculated quantum-mechanically for each definite arrangement of particles in the basic cube. At the heart of a method the density functional of Kohn-Sham lays. Exact wave functions (orthonormal) minimize the value of a functional and they can be found, solving a variational problem. Car and Parinello had suggested to solve this problem by method of simulated annealing when displacements of ions and changes of wave functions are calculated by means of quasi-classic equations of movement obtained via special Lagrangian. This method allows to construct rather small models (usually about hundred atoms) and it is suitable mainly for investigation of electronic density distribution in a space and for calculation of electronic properties.

Computer Modeling of Amorphous Metals

11

Embedded Atom Model (EAM) In recent years, the embedded atom model (EAM) has been used fairly extensively in metal simulations. This model includes collective interactions. The potential energy of a metal is written in the form (e.g., see [18]): U=

∑ Φ(ρi) + ∑ϕ (rij)

(12)

i< j

i

Here Φ(ρi) is the “embedding potential” of the i-th atom, which depends on the effective electron density ρ at the center of the atom. The second term in (12) contains the usual pair potential. The “effective electron density” at the center of the atom is created by the surrounding atoms. It is written as ρi =

∑ ψ(rij) ,

(13)

j

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where ψ(rij) is the contribution of neighbor j to the electron density. Calculations use three adjustable functions, Φ(ρ), φ(r), and ψ(r). There is therefore considerable possibility for fitting the calculated properties to experiment. For crystals where the number of nearest interatomic distances at equilibrium is small, the model allows density, energy, elastic constants, vacancy formation energy, surface properties, etc. to be described correctly. Embedded atom model potentials obtained for crystalline metals are sometimes inapplicable to their melts [19]. It is therefore desirable to develop a special technique for calculating embedded atom potentials of liquid metals. Such potentials should correctly describe both the thermodynamic properties and structure of liquids. This technique should include the possibility of using diffraction structural data. Two different versions of the corresponding procedure were suggested in [20, 21, 22, 23]. In metal simulations by MD method, the total force that acts on atom i is obtained by the differentiation of the total energy with respect to the coordinates of this atom. The effective i-j pair force in a one-component system is given by Fij = - [ (

∂ψ ∂ϕ ( r) ∂Φi ∂Φ ) | + ( j ) | )] | | ∂ρ ρj ∂r rij ∂ρ ρi ∂ r r ij

(14)

The first sum is the result of embedding potential action, and the second is the contribution of the pair potential. The ∂ψ/∂r derivative is negative by the meaning of the ψ(r) function, because the effective electron density created by an atom should decrease as the distance increases. The ∂Φ/∂ρ derivative is negative in low-density and may be positive in compressed states. The embedded atom model was also applied to liquid metals and alloys. With liquid or amorphous metals, an important role is played by the main structural characteristic, the pair correlation functions (PCF). The structural characteristics of real metals and their models are usually considered by comparing their pair correlation functions. The degree to which two

12

David K. Belashchenko

PCF’s g1(r) and g2(r) coincide is determined as the standard deviation (residual, or misfit) calculated by the equation

Rg = {

1 n2 −n1+1

n2

∑

[ g1 (rj) – g2 (rj) ]2 }1/2

(15)

n1

where n1 and n2 are the lower and upper values for the compared histogram points. If the residual is on the order of 0.01, two pair correlation function plots are almost indistinguishable visually. For example, the functions Φ(ρ) and ψ(The selection of the Φρ, ψ(r) and φ(r) functions is not unambiguous. Some transformations of these functions do not change the properties of models. We can therefore use various function forms. The Φ(ρandφ(r) functions were represented by piece-wise continuous combinations of splines in several works. The usual form of Φ(ρ is as follows: it starts from zero into the negative region as ρ grows, passes a minimum, and then again begins to increase. r) were taken in [24] in fairly simple forms for liquid rubidium, ψ(r) = p1 exp(- p2 r) Φ(ρ) = a1 + a2 (ρ - ρ0)2 + a3 (ρ - ρ0)3

(16) at ρ ≥ ρ1 = 0.9ρ0

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Φ(ρ) = b0 + b1(ρ - ρ1) + b2(ρ - ρ1)2 + b3(ρ - ρ1)3

at ρ < ρ1 = 0.9ρ0 ,

(17) (18)

The Φ(ρ) function itself and its first derivative are continuous at ρ = ρ1 = 0.9ρ0. This allows the b0 and b1 coefficients to be written as functions of a1, a2, and a3. As a result, the embedded atom model potential is determined by seven parameters (p1, p2, a1, a2, a3, b2, and b3), which makes it possible in principle to fit to experimental data such properties of the model as density, potential energy (atomization energy), bulk compression modulus, and thermal expansion coefficient. Equations (17) and (18) are the initial terms of the expansions of Φ(ρ). These expansions can be fairly accurate if the deviations of density ρ from ρ0 or ρ1 are not too large. It is very practical to assume that ρ0 = 1. Equations (17) and (18) are used to simulate states with elevated and lowered densities, respectively. The a1 parameter is largely determined by the potential energy value, a2, by the bulk modulus, and a3 can be adjusted using the thermal expansion coefficient or density in some high-temperature state. The accuracy of the embedded atom model potential is determined by the closeness of ρvalues to ρ0. At low densities, the mean value decreases substantially, and the selected Φ(ρ) function (18) can be incorrect. Conversely, if a metal is strongly compressed, ρ>can increase to the extent that expansion (17) will require refinement by the introduction of additional terms. There, however, remains such an important property of liquid or amorphous metal as its structure, which is primarily characterized by the form of PCF. To reproduce the structure of a liquid, one must carefully select the pair interparticle potential φ(r). The scheme of calculations suggested in [19, 22, 23] was as follows. Let the standard state be selected as the

Computer Modeling of Amorphous Metals

13

state of a liquid metal close to the melting temperature, and let denote the mean effective electron density ρiat the atoms and be the mean Φ(ρ) value for the given state of the liquid. At the stage of selecting potentials, the sum of the derivatives s = ∂Φi(ρ)/∂ρ+∂Φj(ρ)/∂ρ can approximately be replaced in (14) by the mean value of this sum over all particles 2/∂ρat ρ= . In reality, the sum s depends on the selected pair of particles and time. However, if the system is fairly dense and density fluctuations are not too strong, the sum s can fluctuate with rather small amplitude and can approximately be considered constant. On this assumption, Eq. (14) for the total effective force can be written as: F(r) = -

dϕ total dΦ ( ρ) dψ dϕ = -2 − dρ dr dr dr

,

(19)

where φtotal is the total effective potential, which determines the structure of the liquid, and the /∂ρ derivative is calculated at ρ = . Potential φtotal(r) may be calculated in different ways on the known PCF (for instance, using Schommers algorithm [25,26]). The selected Φ(ρ) and ψ(r) functions can then be used to calculate the pair potential φ(r), and liquid or amorphous metal simulation by the molecular dynamics method can be performed.

CONSTRUCTION OF MODELS OF ONE- AND TWO-COMPONENT CLASSICAL SYSTEMS

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Non-Crystalline Structures with Reverse Power Potential (“Canonical Structures”) The initial state at modeling of an amorphous phase can be chosen differently. The simplest way is to use already available model of any liquid or an amorphous phase and to relax it at given density and temperature. Results usually do not depend on a choice of an initial state. For example, at construction of amorphous model it is possible to take initially random system of points in the basic cube, or to cool down (quickly or with definite rate) the model of a liquid. Systems with reverse power potential (2) were repeatedly investigated both in liquid and in amorphous state by computer modeling. Their feature is that structure and properties of such systems are defined by dimensionless parameters n = Nd3/V and Т* = кТ/ε. It is useful to enter the reduced density and pressure: n* = (Nd3/V) (ε/kT) 3/m,

р* = (рV/NkТ) n*

(20)

Then the state equation will look like р*(n*) = 0. Internal energy U is connected with pressure by the relation: рV/NkТ = (m/3) (U/NkТ) + (1 - m/2)

(21)

14

David K. Belashchenko

At Т = 0 the simple relation рV = (m/3)U appears. Energy of non-crystalline system with potential (2) is expressed by the formula: ∞ U/NkТ = 3/2 + (2π/kT) (N/V) ε (d/r)m g(r) r2 dr

∫

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0

At m ≤ 3 integral diverges. Hence, at such m potential (2) is physically unacceptable, however is can be applied at potential cutoff on certain distance. Let’s consider non-crystalline system with pair potential (2) near to absolute zero. If the system is mechanically equilibrium, all total forces Fi for each particle are equal to zero. At a homogeneous stretching or compression along three co-ordinate axes all interatomic distances are multiplied by a constant λ, pair forces change in λm+1 time, and the condition of mechanical balance Fi = 0 remains true. Therefore the structure remains equilibrium and after deformation. Then initial PCF gini (r) and stretched one gλ (r) are expressed one from another: gini (r) = g λ (r / λ). In this sense the structure of amorphous system at Т = 0 does not depend on density (and also, of course, on factors ε and d in (2)). Such structure will be mentioned as "canonical". At m → ∝ system will be equivalent to HSS with diameter of sphere d. If to enter the length unit d0 = (V/N) 1/3 and the reduced distance ρ = r/d0 at Т = 0, the PCF of canonical structure should depend only on co-ordinate ρ and on an power m, that is looks like gm (ρ) and should not depend on density [27]. The structural factor of canonical structure also does not depend on density and is defined only by power m. In [27] a series of models of canonical structures with m = 3 - 30 had been constructed by CSR method. PCF for values m = 3, 6, 12 and 20 are represented on Figure 2. The height of first PCF peak increases with the growth of power m (that is with increase of potential rigidity). The co-ordinate of this peak ρ1 ≅ 1.08 varies very little. It is very close to the value 1.0673 found for dense non-crystalline HSS. Second peak of PCF behaves rather special. For all m it is split. With reduction of m splitting gradually smoothes out, however even at m = 3 it does not disappear completely. With increase of m the height of the right subpeak of the second maximum increases. In Figure 3, the structure factors am (κ) are shown for the models investigated (κ = Kd0, where K is scattering vector). They differ rather little. The height of the first peak slightly increases with reduction of m. The form of the second maximum also slightly varies and a "shoulder" is well visible on right slope of it. At m = 3 the second peak is split. Very steady oscillations are characteristic in the region of a minimum to the right of the first peak. There is a monotonous fall of height of the third peak (in area κ ≅ 18.5) and its flattening at the diminishing of m, and also reduction of heights of further peaks. The similar effects are observed for liquid metals [28].

Computer Modeling of Amorphous Metals

15

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Figure 2. PCF of canonical structures at T = 0. Power m = 3 (1), 6 (2), 12 (3), 20 (4). Author’s data.

Figure 3. Structure factors of canonical structures at T = 0. Designations as on Figure 2.

The dynamic behavior of canonical structures has not been investigated yet. For potentials (2) spectrum of frequencies depends in reduced units only on power m, and at

16

David K. Belashchenko

volume change all frequencies change via the formula ω ∼ V-(m+2)/6 [29]. Thus, the similarity theory is easily applicable to the description of temperature dependence of structure and properties of canonical amorphous systems.

Gauss Potential Models of liquid, amorphous and crystal phases with Gauss potential ϕ (r) = ε exp [-(r / r0) 2]

(22)

were constructed by MD method with densities n* = Nr03/V = 1.0 and 0.4 [30, 31]. An amorphous phase was received by cooling of liquid one at constant volume. The minimum reached temperature was equal to T* = kT / ε = 7, that is ∼0.013 from equilibrium temperature of fusion at n* = 1. PCF of amorphous phase had a shape usual for liquids, with height of the first peak ∼3.6. Energy of amorphous systems with potential (22) very little depends on the structure [32]. Reduced potential energies u = U/Nε of canonical structures with potential (22) switched on them are shown below (at Т = 0 and n* = 1):

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Hence, strong changes of a near order change the energy calculated with Gauss potential very little (maximum on 0.11%). Stability of a system is defined at absolute zero by potential energy. If this energy is almost identical for various structures, the stimulus to transition of one structure in another disappears. This may lead to simultaneous coexistence of various types of a near order in different volumes of a real amorphous phase. For studying of such possibility, the models were built in [32] with potential (22) using CSR method. Parameter in (22) was chosen as r0 = d0 = (V/N) 1/3, that leads to n* = 1.0. Energy of initial canonical structure at the switching of potential (22) (without relaxation) was u = 2.29897.

Model C1 At CSR with step 2 pm (with cutting radius of potential ρc = 3.965) energy diminished to u = 2.27771. The PCF of received state C1 is shown in Figure 4; some its characteristics are shown in Table 1. The anomaly of the second peak typical for 3d-metals is obviously visible. The second peak of structure factor is abnormal and has the form, characteristic for 3d metals. Model С2 In the following series of simulations the same initial state was relaxed in some stages: 1) 260 iterations with length of a step 20 pm; 2) 120 iterations with step 5 pm; 3) 420 iterations with step 2 pm; 4) 30 iterations with step 1 pm. At stages 3) and 4) a new steady structure appeared with height of the first peak 2.04 and energy u = 2.27807 (Table 1). PCF of С2 model is shown on Figure 4. It differs strongly from structure С1. Splitting of first PCF peak

Computer Modeling of Amorphous Metals

17

is appreciable. Co-ordinates ρ2 of the second peaks of the states С1 and С2 considerably differ. The first peak of structure factor of model С2 is lowered in comparison with C1.

Figure 4. PCF of the models with Gauss potential. Number of model is shown near the curve. Author’s data.

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Table 1. Characters of amorphous states of a system with potential (22) at T = 0 State

С1

С2

СЗ

С4

С5

n*

1.0

1.0

0.4

1.0

1.0

ρmin

0.870

0.820

0.918

0.890

0.875

ρ1

1.10

1.10

1.075

1.075

1.10

ρ12

1.45

1.68

1.42

1.40

1.50

ρ2

1.95

2.25

2.02

2.10

1.93

ρ3

2.87

3.10

2.87

2.90

2.88

g(ρ1)

3.27

2.04

3.72

3.42

4.13

g(ρ2)

1.41

1.29

1.46

1.42

1.45

g(ρ3)

1.30

1.15

1.26

1.22

1.22

κ1

7.25

6.55

6.92

6.92

7.00

κ2

11.88

13.60

12.25

12.38

12.42

a(κ1)

3.64

2.71

3.74

3.48

3.85

a(κ2)

1.51

1.14

1.39

1.37

1.47

u

2.27771

2.27807

0.632495

2.27762

2.27724

Some other characters of the models are shown in Table 1: the co-ordinates of third peak ρ3, first minimum ρ12, zero point of PCF ρmin, the peak heights g(ρi), co-ordinates of structure factor peaks κ1 and κ2, the respective peak heights a(κ1) and a(κ2). Comparison of PCF and

18

David K. Belashchenko

structure factors of models C1 and C2 shows, that the first one is much more ordered. However, energies of these models are very close (2.27771 and 2.27807). Compressibility of states C1 and С2 coincide to within three significant figures, and pressure of system changes at transition C1 → С2 only in the fourth significant figure.

Model СЗ Model СЗ with density n* = 0.4 has been received by expansion of model С2. Energy of model diminished to 0.632495. PCF of this model is shown on Figure 4. At the reduction of density the height of peak increased and PCF get the form, normal for liquids. The structure factor behaves similarly.

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Model С4 Then model СЗ has been again compressed to density n* = 1.0. After a relaxation the system came to a new state with energy 2.27762. This energy differs from energy of C1 model on 0.004 %. As a result the steady state С4 appeared with the height of first PCF peak ∼3.42, that is more than in model C1 (Figure 4). It was not possible to observe the transition from С4 back to С2. All states C1-C4 are various metastable realizations of system with Gauss potential. Model С5 In [32] the relaxation of system with Gauss potential was further conducted having the model of canonical structure with m = 30 as an initial state. After a relaxation the height of first PCF peak has gone down from 9.42 to 4.12. Energy of this condition (С5) is equal 2.27724. PCF and structure factor of the state С5 are very similar to functions of C1 and С4 (Figure 4). Crystallization in models C1-C5 was absent. Thus, at modeling of amorphous system with Gauss potential it is possible to create the states with strongly differing structure, but with very close values of energy. If such phenomenon can meet in real systems, then will be possible the simultaneous occurrence of areas with various structure, that is stratification on two amorphous phases. The facts of simultaneous presence of various structures in amorphous system were really observed for alloys Pd-Au-Si and Pt-Ni-P [33]. Change of a near order can be initiated by temperature change. Structural transformations of such type are found out in amorphous alloys Fe40Ni40BxP20-x at х = 6 and 20 (Verkin B.I., Sidorov G.V. et al., 1986). So, the same pair potential can generate various non-crystalline structures if their energies are close enough in the same conditions. This possibility is shown now only for Gauss potential but other cases are also possible. Therefore we have the ambiguity of dependence potential - structure. Ambiguity of a reverse problem “dependence structure – potential” will be considered below.

Computer Modeling of Amorphous Metals

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Amorphous Structures with Lennard-Jones Potential If the rate of liquid model cooling is great enough, it is possible to avoid crystallization and to receive an amorphous phase. In [34] liquid model was abruptly cooled to T* = kT/ε = 0.108, and in [35] – by steps from T* = 1.026 to 0.094. Splitting of second PCF peak appears at Т* ≅ 0.25 (at ρ0 = r0/d0 = 1.0837) and disappears at back heating at Т* > 0.25. Sharp reduction of self-diffusion rate (at amorphization) occurs close to Т* = 0.5 at ρ0 = 1.0837, but if to increase ρ0 to 1.1031 then amorphization is observed in the range of 0.8 < Т* < 0.9 and pressure 200-1500 at. [34]. The lower is density, the less is amorphization temperature. Splitting of the second maximum does not allow to distinguish liquid and amorphous states because it appears gradually at cooling or compression of a normal liquid already below temperature where the self-diffusion coefficient becomes immeasurably small in MD run (below 10-7 cm2/s). In connection with the general problems of topology of non-crystalline structures the systems with the lowered density are of interest. Some models with potential (3) have been constructed by CSR method at values ρ0 = 0.8, 0.9 and 1.0. The system with ρ0 = 0.8 appeared to be extremely non-uniform ("loose") and the distribution of volumes of Voronoi polyhedrons (VP) in it is as wide as in random system of points. The low coefficient of sphericity of VP (0.59) also specifies strong variations of VP form. However angular distributions demonstrate accurate peaks at 59 and 108 °, characteristic for an icosahedral arrangement. Value ρ0 = 0.8 respects to packing factor ∼0.27. In this situation the structure loses uniformity, and large cavities - "pores" appear in it. Atoms gather in clusters packed rather densely. High value of g(ρ1) ≅ 14.0 is received here and the co-ordinate of 1st peak is almost equal to ρ0. The second peak is very distinctly split. At ρ0 = 0.9 structure still remains loose, however height g(ρ1) is less than at ρ0 = 0.8. At further increase of ρ0 to 1.08 the structure transforms to dense one. Formation of close-packed clusters in loose structures is well visible on the distribution of VP volumes where group with small volumes is seen. It proves that in loose systems with Lennard-Jones potential (and, obviously, with other potentials having a minimum) the clusterization really takes place. Large cavities (pores) of the irregular form are formed between clusters. The maximum VP volume may exceed the average value in 2 - 3.5 times and more. Maximum VP volume does not decrease with ρ0 growth from 0.8 to 1.0; number of VP with a great volume decreases only. Therefore at increase ρ0 in loose systems the pore sizes can remain almost constant and disappear only at ρ0 ≅ 1.08.

Embedded Atom Model (EAM) Parameters of EAM potentials are evaluated usually, using properties of crystal phases. The potentials obtained can be applied to construction of models of liquid and amorphous substances and calculation of their structural characteristics. For example, in [36] structure factors of liquid Cu, Ag, Au, Ni were calculated using the equations of the theory of liquids. These factors agree very well with experimental ones.

20

David K. Belashchenko

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In [21] parameters of embedded atom potential were calculated for some BCC, FCC (face-centered cubic) metals and liquid iron. Here the bad accordance with experiment on height of the PCF first peak is received; the calculated structure is more ordered, than a real liquid. It is possible to improve an agreement if to include the information on interaction forces on small distances which are absent in a lattice, but appear in a liquid. The model of liquid iron was constructed for this purpose consisted of 100 atoms in the basic cube, and interparticle forces were calculated via quantum-mechanical ab initio method. Inclusion of these data in procedure of EAM potential calculation allowed to improve considerably the agreement of the structure with experimental data (Figure 5). Here the misfit Rg = 0.0387 and agreement of two PCF is good. Potential energy of a model is equal to -353.6 kJ/mol (respective to ideal gas at the same temperature). Experimental value is -339.2 kJ/mol [23]. That means appreciable difference in 14.4 kJ/mol. Rather good agreement with experiment was obtained for calculated structure factors, molar volumes and volume expansion coefficients of liquid Cu, Ag, Au, Ni, Pd, Pt [38] with the use of EAM potentials calculated for crystals. Atomization energy agrees with experiment in limits of 6 %. Volume change at fusion is in worse accordance with real data (the maximum divergence 19.0 % instead of actual 3.8 % is received for silver). MD models of the same liquid metals have been constructed with use of EAM potentials in [39]. The structure factors calculated agree well with diffraction data. Agreement for the energies and entropies of the liquids close to the melting points of the metals was also obtained. EAM potential for Ag was calculated also in [102]. PCF of real and model Ag at 1273 K and real density 9.26 g/cm3 are shown on Figure 6. Here the difference between diffraction data and MD data obtained with EAM potential is clearly visible (Rg = 0.0687). Potential energy of real liquid is equal -251.3 kJ/mol and for the model U = -247.0 kJ/mol. So it is seen that EAM potentials invented for crystals may be not good for liquid and amorphous solids. Embedded atom potentials were used in [41–43] to determine the static structure factors and some thermodynamic properties of liquid transition metals. Embedded atom potentials were also applied to construct the phase diagrams of the Al–Pb [44] and Cu–Pb [45] systems. EAM potentials invented specially for liquids were used to simulate liquid Ga and Bi [19], liquid mercury [22], liquid and amorphous iron at high temperatures and pressures [23,46], liquid Li [47], Pb [48] and Cu [49].

Figure 5. Pair correlation functions of liquid iron at 1820 K. Solid line – experiment [37], circles – model created with EAM potential proposed in [21].

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Figure 6. Pair correlation functions of liquid Ag at 1273 K. Solid line – diffraction data [40,101], circles - MD modeling with EAM potential proposed in [102].

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MOLECULAR DYNAMICS COOLING OF THE LIQUID MODEL It is known, that at cooling of liquid model it is possible to receive as a result either model of an amorphous phase or model of a nano-crystal phase (which contains usually many defects). For creation of amorphous model high speed of cooling is required. The reason is that at strong supercooling of a liquid specific non-classical mechanism of solidification begins to work which doesn’t demand the formation of a crystal seed. This mechanism has been discovered at MD modeling of rubidium [24] and then investigated on nickel [50] and silver [51]. Firstly the phenomenon of homogeneous crystallization of liquid models was detected and investigated by MD method in [52-56]. The models contained up to 500 particles. Lennard-Jones potential was applied, then the pair potential calculated by a pseudo-potential method for rubidium and these potentials in truncated form. The review of early works with other potentials is given in [55]. In the course of long MD runs crystallization of a liquid was observed with formation of BCC or FCC crystals depending on the form of the pair potential used. The structure was characterized via the Voronoi polyhedrons distribution. In the case of BCC crystal VP had the indices 0-6-0-8 and 0-4-4-6. In the classical nucleation theory it is supposed, that in the undercooled liquid the seeds of a new phase (heterophase fluctuations) should arise and disappear fluctuationally. The probability of detection a seed is defined by the change of Gibbs energy ΔG at its formation. Usually one considers a seed of spherical form of radius r. The value ΔG may be expressed in a form valid for macroscopic phase: ΔG = 4πr2σls + (4/3) πr3 (μl - μs)

(23)

Here σls is an interphase tension, and μl and μs are Gibbs energies of liquid and solid phases with respect to unit volume. The probability of formation of a seed of the size r is defined by the formula:

22

David K. Belashchenko W = A exp (-ΔG/kT),

(24)

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where frequency factor A has dimension of cm-3s-1. At increase of radius the value ΔG increases at first from zero to some maximum, and then starts to decrease and passes in negative area. The critical radius is defined by condition of ΔG maximum. After an achievement of the critical size seed must grow constantly because this is thermodynamically favourable. It follows from these assumptions that in the undercooled liquid the seeds of different size must exist with various probability, and the bigger is a seed of pre-critical size the less is probability of its occurrence. These conclusions are correct for a stable or metastable liquid and are not connected with the mechanism of creation and growth of seeds. However till now there are few data on the actual mechanism of crystallization of liquids, especially at its initial stages and in the case of strong supercooling. Computer methods allow to reproduce the processes occurring in the supercooled liquid at atomic level. In particular, it is possible to watch process of preparation of a liquid to crystallization, formation and growth of crystal seeds by means of MD method. Therefore this method is quite useful to verify the adequacy of the classical nucleation theory. The homogeneous crystallization of liquid rubidium models containing N = 500, 998, and 1968 particles in the basic cube was studied by the MD method. The pair potential calculated by a pseudo-potential method had been chosen. One may watch for solidification process, analyzing the atoms of a liquid having Voronoi polyhedrons of type 06-0-8 (0608-atoms), characteristic for BCC crystals, and also calculating the structural factor in various directions. Concentration of 0608-atoms increases at fall of temperature and they are aggregated in clusters of various size. Dependence of the average cluster size along MD run at 185 K is shown on Figure 7 Strong fluctuations of average (and maximal) cluster sizes are clearly seen.

Figure 7. Time dependence of mean size of 0608-clusters in Rb model. N = 1968, T = 185 K.

Below 185 K, the liquid crystallized predominantly with the formation of BCC structure. The mechanism of crystallization was different from that accepted in classic nucleation theory. Crystallization developed as an increase in the number of atoms with Voronoi

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23

polyhedrons of the 0-6-0-8 and 0-4-4-6 types, the formation of bound groups (clusters) from these atoms, and growth of these groups as in the coagulation of an impurity from a supersaturated solution [57]. At the initial stage, bound groups had a very loose structure and included a fairly large number of atoms with Voronoi polyhedrons of other types. The linear dimension of the largest group rapidly approached the basic cube size. The atoms with the 06-0-8 and 0-4-4-6 VP played a leading role in crystallization and activated the transition of bound group atoms with other coordination types into BCC coordination. The probability of formation of a bound group of a given size was found to be independent of the volume of the liquid model. Cluster size fluctuations especially strong over the temperature range 180 - 185 K played an important role in the formation of 0608 clusters of a threshold (“critical”) size. Dependence of the maximum cluster size on time in MD run at 100 K is shown in Figure 8.

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Figure 8. Time dependence of 0608-clusters sizes. N = 998, T = 100 K. Circles – maximal cluster size, dashed line – mean size of clusters.

In Figure 9, configurations of 0608-atoms are shown in the course of solidification at temperature 150 K. It is seen that clusters gradually increase in sizes, smaller clusters aggregate with larger, but there are no the crystal seeds in system, considered in the classical nucleation theory. The visual analysis shows that at initial stages (Figure 9a) solid-like clusters have loose and non-regular structure and thread over all volume of the model. The distinct boundary between liquid and solid phases is not seen here. Further (Figures 4b, c), aggregation of atoms with large solid-like clusters begins that is accompanied with the increase of global order parameter. At the final stages of growth the cluster looks like usual crystal. Further, the relaxation of crystal takes place.

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David K. Belashchenko

Figure 9. Location of 0608-atoms in Rb model with N = 998 at T = 100 K in the course of crystallization. a – step No 10000, clusters distribution 12023 5182341, mean CN of the system of chosen atoms 2.22, mean CN of maximal cluster 3.47; b – step No 20000, clusters distribution 1102131711211, mean CN of the system of chosen atoms 4.61, mean CN of maximal cluster 5.36; c - step No 30000, clusters distribution 19211881, mean CN of the system of chosen atoms 4.96, mean CN of maximal cluster 5.24. Time step = 1.88 fm. Designation 11021…means 10 monomers, 1 dimer, etc.

The important feature of cluster mechanism is an existence of the bottom limit of supercooling above which cluster mechanism of solidification cannot work. In a case of Rubidium this temperature is equal Tb = 185 K. Temperature of model fusion was defined by a method of the back heating of preliminary crystallized model. It is equal Tm = 312 ± to 1 K. Real melting temperature of Rb is 312.4. So we have a relation Tb/Tm = 0.59.

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The reason of existence of the bottom limit is shown in [58] as follows. While the liquid is metastable and crystallization doesn’t begin, there is a series of reactions providing the formation of clusters with various sizes: n A1 ↔ An

(25)

Let's designate constants of equilibrium of these reactions as Kn: Kn =

c (An, T) cn (A1, T)

(26)

Here c (A1, T) and c (An, T) - concentration the monomers and clusters at the temperature T. Dependence of equilibrium constant Kn on T is described by the equation of an “reaction isochore” dlnKn/dT = ΔUn/RT2, where ΔUn - change of internal energy at reaction (11). Accepting that the value ΔUn poorly depends on temperature, we obtain:

ln

ΔUn 1 Kn (T) 1 = ( − ) Kn (Tm) R Tm T

(27)

Here Tm - fusion temperature. From (26) and (27) it is found: ln c (An, T) = ln c (An, Tm) + n ln

c(A1, T) ΔUn 1 1 + ( − ) c(A1, Tm) R Tm T

(28)

If equilibrium concentration c (An, T) monotonously and quickly enough decreases with n, the crystallization via cluster mechanism will not go. If after any value n, concentration c (An,) increases with n, spontaneous growth of cluster number in a liquid will be observed. Concentration of monomers should not vary. We may calculate a derivative:

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d ln c(An, Tm) d ln c(An, T) d ΔUn 1 1 c(A1, T) 1 = + ln + ( − ) dn dn c(A1, Tm) dn R Tm T

(29)

The value δUn = dΔUn/dn is a change of internal energy at reaction An + A1 = An+1. In the case of big cluster sizes the value δUn should be close to heat of crystallization. The concentration c(A1, T) increases at cooling and δUn 0, and the second generates the same structure at absolute zero (with the same density and functions g(r) and R3(r1,r2,r3)). Then the first one satisfies BGB equation (39) at T > 0, and the second one satisfies the equation (40) with a zero left part. It is obvious, that the total potential u(r) = u1(r) + λu2(r) will also satisfy the equation (39) at any factor λ. It is required only for stability of system the value λ to be positive. Thus, there should be a family of hybrid pair potentials

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u (r) = u1 (r) + λu2 (r),

(41)

which can generate the same structure of non-crystalline (liquid or amorphous) systems at temperature Т. The first check of this theorem was conducted on an example of one-component system with power potential (2). The system consisted of 3998 particles in the basic cube with edge length L = 40 Å with periodic boundary conditions (d = 2.27 Å and ε = 0.1 eV). The cutting radius of interaction was equal 9.0 Å. Initial model of amorphous system has been constructed by CSR method at T = 0. The final relaxation step was equal 0.005 Å. Some properties of this model (model М0) are shown in Table 8. PCF of this model is shown on Figure 13. As model М0 is constructed at Т = 0 its interparticle potential (2) is potential u2 (r) in expression (41). For determination of u1 (r) it is required to construct model with the same PCF ("target"), but at the given temperature T > 0. Required model (М1) was built at 300 K using MD method, Schommers algorithm and PCF of model М0 as target one. Model M1 had the same size as model М0. The cutting radius of interaction was 9.0 Å and time step Δt = 0.03t0, where t0 is an internal time unit. Standard deviation Rg between PCF’s of M0 and M1 models was rather small (0.051) so these PCF’s are visually indistinguishable in Figure 13, except for very a small difference in first peak heights.

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David K. Belashchenko

Figure 13. PCF’s of one-component system. Dashed line 1 – PCF of model М0 with interparticle potential (2) at T = 0, CSR method. Markers 2 – PCF of model М6 with hybrid potential (41) at T = 300 K and λ = 500, MD method.

Table 8. Properties of one-component systems with potentials (2) and (41). N = 3998 Модель M0 M1 M2 M3 M4 M5 M6

Т, К 0 300 300 300 300 300 300

λ 1 10 50 100 500

Rg 0.051 0.046 0.030 0.048 0.063 0.084

P, GPa 5.501 16.85 23.01 72.11 291.4 565.6 2762

U, eV 1648.1 1255.5 2902.0 17722 83530 165830 824090

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Comment: P – pressure, U – total potential energy of a model.

Some properties of models are shown in Table 8. The temperature of М1 model is higher and its interparticle potential differs from initial potential (2) (see Figure 14) (it is necessary to remember, that at Т = 0 structure of an amorphous phase does not vary at multiplication of potential (2) on any positive number!). Unlike purely repulsive potential (2), the restored potential has the oscillating form. Pressure and energy of M1 model differs also from М0 model. Further a series of models with hybrid potentials (41) has been constructed by MD method with values of λ = 1 - 500. Results are shown in Table 8. In all cases, except the last, the misfits (in relation to PCF of М0 model) almost don’t vary. Hence the structure of models with hybrid potentials (41) practically coincides with the structure of М0 and М1 models. PCF of M6 model is shown on Figure 13 in comparison with PCF of М0 model. Even at very great value of parameter λ = 500 agreement between these PCF’s is quite good. Model М6 behaves as an amorphous substance with very small diffusion coefficient. Obviously, at additive potentials the energy and pressure also should be additive.

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41

These results show that the theorem of hybrid potentials family stated above is correct within calculation errors. It means also that reasons about coincidence of three-particle functions at identical PCF seem to be true even near to absolute zero. This theorem was checked successively also on an example of binary system with Lennard-Jones potentials. Therefore it looks working well for liquids. However its correctness isn’t proved for amorphous structures at absolute zero. It would be interesting to find out, whether there are families of the potentials generating the same PCF’s at Т = 0.

Figure 14. Interparticle potentials of one-component system. 1 – potential (2). 2 – potential restored via Schommers algorithm on the model М1 at 300 K.

DIFFUSION MECHANISMS IN AMORPHOUS METALS

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The Basic Characteristics Diffusion in solids goes via the mechanism of activated jumps (hopping mechanism). For this mechanism temperature dependence D = D0exp (-Eact/kT) is characteristic, where Eact is activation energy. Direct application of molecular dynamics method for studying of hopping mechanism is difficult enough because Boltzmann factor exp (-Eact/kT), describing probability of a particle jump, is usually very low. Besides, thermal activation in small model is extremely improbable. Therefore the method of Monte-Carlo appears the most suitable. For study of the basic features of hopping diffusion in non-crystalline systems it is convenient to investigate so-called disordered lattices in which the ordered arrangement of sites remains, but properties of particles in different sites are various. Let's consider regular lattice on which the particle can migrate. Presence of distributions of the ground and transition energies in disordered system even in the presence of a regular lattice leads to two specific effects [82, 83, 84]. The first one is that the particle prefers to leave the site via lower barriers. Therefore the real trajectory of migrating particle is enriched by lower barriers in comparison with their general set in hole system. This effect lowers average time of being a particle in the site and raises diffusion coefficient. However the number of reversal jumps of a particle from new site in previous one is increased because the

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potential barrier between them has the lower height. This "correlation effect” leads to fall of diffusion rate. Usual expression for an average square of particle displacement for n jumps is written as = n, where is expressed in terms of one jump length (we omit here the correlation effects inherent to diffusion in crystals). In the case of disordered crystal it is necessary to write [82]: = Fn, where F is the correlation factor. Presence of these two effects specific to diffusion in disordered system was mentioned in [85]. Taking into account these reasons the self-diffusion coefficient can be written down in the form of D = γ (d2 /τ) F, where γ is usual geometric factor, d is mean length of a jump, τ is mean time between jumps. The terms «correlation factor» that used in the literature often differ in sense because they may be connected with the various physical origins. Here the correlation factor is used as a measure of fall of an average square of displacement of particles in relation to this value in a regular ordered lattice for the same number of particle jumps. Let’s designate the energies of the ground states as εi and transition states as εij (between sites i and j). They are described by corresponding distributions. In [83] the average time of a settle life of a particle in the given site is defined as: τсрi =

z

zτ0exp(−εi)

∑ exp(−εij /kT) j=1

Here τ0 is period of vibration and z is coordination number. This time is different for various sites. For calculation of diffusion coefficient it is required to find average time τср for the hole system of sites. Simple averaging along sites is not correct because the attendance of sites is different. Generally this time depends on the size of area which the particle has visited in the course of diffusion. In work [84] calculation τср is conducted in the assumption that the particle wanders long time along the sites of system and many times visits each site. Then time which it spends in i-th site, is proportional to the Boltzmann factor exp(-εi/kT). "Equilibrium" average time defined by this method is: N

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∑ exp(−εi/kT)

τр = zτ0

i =1 N z

(42)

∑∑ exp(−εij)/kT)

i =1 j=1

At summation here each transition site is considered twice. The formula (42) is correct when the diffusion covers sufficiently large volume. The similar formula for average time is obtained in [86]. It is possible to apply the formula (42) for concrete distributions of ground and transition energies. For example, for the normal distributions which have been cut off at deviations ±2σ, it is obtained in [84]:

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⎡ (σ )2 − (σtr)2 ⎤ ξ0 τp = exp ⎢ 0 2 ⎥ξ τ* ⎦ tr ⎣ 2(kT) Here σ0 and σtr - standard deviations of ground and transition energies, and τ* - settle time in the case of the ordered system when all energies of ground and transition positions are equal to corresponding average values for disordered system. We will name such system as dual in relation to our disordered system. Values ξs are defined by formula: ⎞ σs ⎛ σ ⎞ + 2 ⎟⎟ − erf ⎜ s − 2 ⎟ ⎝ 2kT ⎠ ⎝ 2 kT ⎠ ⎛

ξ s = erf ⎜⎜

(43)

Here the index s means either “0” or “tr”. It is visible from (43) that growth of distribution width of ground energies increases average time between jumps, and growth of distribution width of transition energies reduces it. If σ0 ≅ σtr then average settle time of disordered system will be equal to average time of dual ordered system [84, 85]. The estimation of correlation factor has been fulfilled in [83]. Diffusing particle prefers to leave sites via the lowest barriers. At low temperatures the exit will practically occur always via the lowest barrier. It is useful to enter the concept of 1N-deadlock as such site which the particle (vacancy) can leave for N repeated jumps always via the same barrier. The probability that given i-th site is 1N-deadlock, is equal [82]: z

∑ exp(− NEs/kT)

fN = s =1 ⎡ z

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⎤N ⎢ ∑ exp(− Es/kT) ⎥ ⎣⎢s =1 ⎦⎥

(44)

Here Es is the height of s-th barrier. Further, the N-deadlock is such site which the particle leaves during N jumps always via one barrier, and on N+1-th jump via any another one. The probability that given i-th site is N-deadlock is equal wN = fN - fN+1. Two neighbors Ndeadlock and (N+1)-deadlock are called N - trap if the particle makes reversal jumps from one deadlock in another. It will make 2N or 2N-1 jumps in a trap and then will leave a trap via any other exit. The probability that any site of amorphous metal is a component of trap, is equal wtrap = f 22 where f 2 - average value f2 on all ground sites. The correlation factor defines intensity of capture of migrating particles in the traps. It is connected directly with probability fN from the equation (44). Having in mind that Es = εij - εi it is visible from (44) that the value fN and accordingly the correlation factor F do not depend on distribution of ground energies εi. Results for factor F were obtained in [83] with the use of various approximations. More exact values of the correlation factor can be found via direct self-diffusion modeling of disordered systems.

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The scheme of hopping diffusions in disordered system looks as follows [82]. The atom makes random wanderings, overcoming activation barriers of various heights. From time to time he meets a trap, makes certain number of jumps in it, then leaves a trap and again migrates from site to site before will be captured by new trap. For calculation of self-diffusion coefficient it is necessary to define an average number of atom jumps on a way from one trap to another and an average number of jumps in a trap. The average number of jumps between two traps depends on f 2 . In [83] these average numbers are calculated analytically. The average number of particle jumps between two traps is equal: ν = [1 − (1 − f 22)1 − 1 / z ] − 1 The correlation factor is equal:

⎡

F = ⎢1 +

⎢⎣

∞ ⎤ −1 1 (2.5 + 2 ∑ fN2/f22)⎥ ν ⎥⎦ N=2

At sufficiently low temperatures the particle is so strongly grasped by traps that the correlation factor practically drops to zero. Analytical calculations were conducted for various distributions of barrier heights normal, triangular, etc. If the set of barriers in the given site is random then effective activation energy of self-diffusion usually increases with temperature. At not random sets of barriers in the sites effective energy of activation can decrease with temperature. The correlation factor is less for wider distribution of barrier heights. Factor D0 can differ from usual value in every side [82]. The expression for self-diffusion coefficient in disordered system was analytically obtained in [85]:

⎡ f(σint)2 − (σo)2 ⎤ ⎥ 2 ⎣ 2(kT) ⎦

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D/D* = exp⎢

Here D* - self-diffusion coefficient in the dual ordered system. The correlation factor is presented implicitly as a factor f < 1, lowering the diffusion mobility. In our designation, factors F and f are connected as: F = exp [(f-1)(σtr)2/2 (kT) 2]. The f factors were calculated by Monte-Carlo method and at σtr/kT = 3.6 have the values from 0.15 to 0.64 for different lattices. This leads to F values from 0.004 to 0.1, and they decrease with reduction of coordination number from 12 to 4. As a result temperature dependence of self-diffusion coefficient remained to have Arrhenius form within 8 orders of D value.

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Diffusion via Interstitials of an Amorphous System

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The review of works on self-diffusion in models of amorphous canonical one-component systems with pair potentials (2) and also in models of amorphous iron with Pak-Doyama potential (described in [92]) and in some crystal structures is presented in [82]. The interaction potential of an interstitial atom with other ones was the same, as between atoms of an amorphous model. At the first stage the distributions of energies were calculated for ground and transition states. Further the self-diffusion coefficients of interstitial atoms were calculated using a method of atom drift under an external force. On Figure 15 the distributions of energies of the ground and transition states in amorphous canonical structures with m = 6, 18 and 30 are shown. Self-diffusion coefficients obey an Arrhenius-like temperature dependence in all cases. Parameters of these dependences for BCC, FCC lattices, amorphous iron and canonical structures are close enough, and activation energy of self-diffusion lays between values for BCC and FCC. This corresponds to the fact that average coordination numbers in an amorphous phase are intermediate between coordination numbers of BCC and FCC phases. Thus, the self-diffusion coefficient behaves as monotonous function of average coordination number.

Figure 15. Distributions of energy in ground (1) and transition (2) states in canonical structures with m = 6 (a), 18 (b) and 30 (c). Author’s data.

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David K. Belashchenko

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Hydrogen Diffusion in Amorphous Alloys Solubility of hydrogen in amorphous alloys on a basis of hydride-forming metals (Pd, Zr, Nb, etc.) may consist of tens percent and the enthalpy of hydrogen dissolution in them is negative (heat is evolved) [87]. In these systems strong dependence of some properties of the dissolved hydrogen (partial volume, heat of dissolution, activation energy of diffusion and D0 factors) on its concentration is found out. Deviations from Siewerts law (when the solubility must be proportional to a square root of hydrogen pressure) are noted even at rather small concentration [88]. In amorphous metallic alloys (“AMA”) of type 3d - metal (Fe, Co, Ni) metalloid (B, P, Si), solubility of hydrogen СH is much lower (of an order of 10-4 atom Н/атom Fe at usual pressure [89]). For example, in AMA Fe83B17 the dependence of lnCH on 1/T is non-linear, that means strong temperature dependence of entropy and heat of dissolution of hydrogen in AMA (heat of dissolution increases from -5.66 kJ/g-atom at 393 to -0.55 at 573). Siewerts law is not fulfilled also. These effects are caused by presence of positions for hydrogen atoms in an alloy structure with various energies. The most part of hydrogen atoms is grasped by the low-energy centers and these atoms remain in the bound state at heating in vacuum up to 523 K. The pores and micro-cracks also can play a role of the capture centers in which molecular hydrogen accumulates. The energy distributions of the ground and transition states for hydrogen in amorphous iron were analyzed in comparison with solutions in crystal iron at a choice of interaction potential (5) for an impurity–solvent pair [82]. The good accordance was obtained with experiment for activation energies of hydrogen in BCC and FCC iron. The diffusion coefficients were equal to 6.1⋅10-4, 1.0⋅10-7 и 1.6⋅10-5 cm2/s at 700 K for BCC, FCC and amorphous iron respectively. Statistical calculation of properties of the diluted hydrogen solutions in amorphous iron was carried out in [90]. Potential of interaction H-Fe was chosen in the form (5). Entropy of the hydrogen atoms was evaluated using the vibration frequencies of Н atoms in ground sites via formulas for three-dimensional oscillator. Wide distributions of energies and vibration frequencies of H atoms in the stable states were found in models of amorphous and liquid iron. At transition from liquid to amorphous state the distributions are narrowed. At low temperatures Н atoms locate in the lowest energy positions, namely the positions close in coordination to octahedric positions in FCC iron. Lower vibration frequencies of H atoms are realized also in these positions. Existence of energy and frequency distributions leads to strong temperature dependence of entropy and heat of dissolution of hydrogen in amorphous iron. At ∼500 K there is a change of a sign of dissolution heat. The calculated values of hydrogen solubility in amorphous iron are higher than in BCC iron in all interval of BCC phase existence [82, 90]. Behavior of hydrogen in amorphous iron was considered also in [91]. Models of liquid and amorphous iron were constructed by MD and CSR methods with the use of Pak-Doyama pair potential [92]. Potential in the form of a 4th power polynomial was applied to pairs H-Fe. Concentration of equilibrium hydrogen positions (∼4.3 on 1 iron atom) is intermediate between values for BCC and FCC iron. One equilibrium octahedric position with respect to 1 Fe atom is found in FCC iron with energy -2.17 eV. Energy of the lowest state of hydrogen in amorphous iron (-2.55 эВ) is lower than in crystal lattice.

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Diffusion coefficients of hydrogen in models of amorphous, BCC and FCC iron were calculated at a choice of potential (5) for Fe-H pairs [82]. Energy distributions of the ground and transition states do not depend almost on model density. For BCC and FCC-iron the calculated activation energies agree well with the experimental ones. Calculated diffusion coefficient of hydrogen in amorphous iron lies between values for BCC and FCC iron.

Carbon Interstitial Diffusion in Amorphous Iron Calculation of diffusion coefficient of carbon migrating via interstitials of amorphous iron models was carried out in [93]. The heights of activation barriers were evaluated and Monte Carlo method was applied to define carbon mobility. In this approach Arrhenius law was shown to fulfill. Calculated diffusion coefficients of carbon were higher than in FCC iron (approximately in 60 times at 500 K).

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Vacancies (“Pores”) in Amorphous Systems: Vacancy Mechanism of Diffusion in Amorphous Metals It is known that in real AMA, the micro-pores exist similar to vacancies in crystals. At deformation of AMA the number of micro-pores (“quasi-vacancies”) increases and at heating their number can decrease via processes of defect annihilation. Micro-pores can generate micro-cracks in an amorphous alloy that essentially influences its properties [82]. Annihilation of micro-pores in the course of structural relaxation leads usually to small compression of a substance, named «an exit of free volume». The analysis of pores in an amorphous structure is conducted usually on models, calculating radiuses of rigid spheres which can be enclosed in a model without crossing with host atomic spheres. The HSS models have been investigated in [94, 95]. The maximal pore has a diameter near 0.66 of host sphere radius. The behavior of pores in amorphous canonical structure was investigated. The maximum pore radiuses in amorphous iron models with PakDoyama potential were about 65-70 pm [82]. They decreased a little in the structural relaxation of model. A pore with radius above 80 pm disappears in the course of CSR and consequently can be considered as eliminated defects of structure of amorphous iron. Defects of pore type can be found also in models of binary amorphous alloys, especially such where the sizes of different atoms strongly differ. For example, in models of amorphous Fe2Tb, constructed via CSR method, the maximum pore radius in model was 134 pm. The largest pore could accommodate without distortion not only such solute atoms as H, C, O, N, B, P, Si but even atoms Fe, Co and Ni. The concept of vacancy in amorphous phase was defined in [96] as such a pore that is capable of several place exchange with the neighbor atoms. The potential profile at atom transition into the next pore of amorphous iron model looks like activation barrier only in the case of a pore with a radius more than 80 pm. If conducting a static relaxation after an exchange between an atom and such a pore, then atom and pore remain in their new places. In other cases (at R < 80 pm) the moved atom returns back to a former place. Hence, a pore with R < 80 pm cannot play a role of vacancies in amorphous iron.

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In further research, the artificial vacancies in models of amorphous iron were created by removal of certain atoms. Then CRS was conducted. The pores with a radius from 60 to 110 pm were thus formed. The analysis of potential energy profiles has shown that normal activation barriers are observed only for pores with radius > 80 pm. Activation barrier heights for the atoms surrounding a large pore lay within 0.4 - 2.7 eV. The minimal barrier does not exceed 1.4 эВ [96]. If the pore having a radius R > 80 pm will have again R > 80 pm after an exchange with a neighbor atom, then it does not disappear and will be capable of making a new exchange of places. Hence it is vacancy. However vacancy finally disappears after some exchange of places, that is gets to “drain.” For models investigated, the atomic ratio of drains decreased with a fall of model energy (that is, with growth of their stability) from 0.53 to 0.125. Concentration of drains defines an average number of jumps which vacancy can make before the disappearance—that is, before transformation into a usual small pore. If concentration of drains is equal to α, the probability to make exactly n jumps and to disappear after that is equal to wn = (1 - α)n α. The average number of vacancy jumps before its disappearance is equal to nav = 1/α. Maximal nav was equal to 8 jumps. As a result, in [96] expression for vacancy self-diffusion coefficient in amorphous iron was obtained: D = 5.8⋅10-3 exp (-177.4 kJ/RT) cm2/s. At 500 K D(Fe) = 8.1⋅10-20 cm2/s (computer calculation). Experiment data for alloy Fe40Ni40P14B6 at temperature 500 K are: D(Fe) = 9.5⋅10-18 ([97]) and 4.5⋅10-18 ([98]). Excess of experiment data is caused, possibly, by presence of non-equilibrium vacancies or participation of other diffusion mechanisms. Therefore, the following scheme of diffusion via vacancy mechanism in an amorphous substance is established. Firstly, a large pore arises as a result of thermal fluctuation. It possesses properties of vacancy and can migrate via interchanging the position with neighbor atoms and so doing on average 1/α jumps. After that, vacancy drains and transforms to a rather small pore which does not participate in the diffusion process [82]. In the case of crystal substances quickly cooled from high temperature, actual concentration of vacancies can be ready above the equilibrium. In amorphous metals, nonequilibrium vacancies (large pores) perish on drains already via some jumps. Therefore, it is rather probable that they can work only at the initial stages of structural relaxation. The behavior of vacancy-like defects in metal glasses with Lennnard-Jones potential was investigated by MD method (up to 10976 atoms in the basic cube, temperature 0 – 10 K) in [99,100]. The atom was taken out from amorphous system and then the relaxation of a system was conducted. Three possible ways of events were possible: the relaxation of surrounding area with the vacancy preservation, a jump of one of the neighbor atoms in the vacancy, accompanied by reorganization of the first coordination sphere round the former vacancy, and disappearance of a vacancy with an exit of free volume on a surface. The result has not been connected definitely either with the level of local shear stress, or with the symmetry of nearest environment of vacancy. However, low local pressure and high degree of environment sphericity probably stabilize the vacancy. Irrespective of the size of a system, the disappearance of vacancy was observed in approximately 25% of cases, in 10-30 % of cases vacancy remained steady, and in another cases there was a jump of neighbor atom in vacancy. According to [100], vacancies can arise in those sites of an amorphous phase which behave as drains, migrate to other drains and disappear there, etc. This process was modeled via time reversal of MD experiment after disappearance of the vacancy in a drain. Thus it was

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possible to restore in reverse sequence the process of migration of vacancy from initial site to a drain. Therefore, the quasi-vacancy mechanism of diffusion is possible in amorphous metals. It consists of the activated formation of a pore (quasi-vacancy) and further exchange of it with the neighboring atom. In comparison with the vacancy mechanism in crystals, a large pore is created in special, structurally-defective positions (“springs”) by thermal activation. Quasivacancies have different sizes. A pore with a size less than the critical one loses the ability to migrate. Quasi-vacancies disappear not on macroscopic defects as in a crystal, but on defect sites of amorphous structure—drains of vacancies. Finally, there exists “an average length of free run” of vacancy up to its disappearance, defined by concentration of drains. Springs and drains can arise in various places of the system as a result of thermal fluctuations.

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Kuksin, A.Y.; Morozov, I.V.; Norman, G.E.; Stegailov, V.V.; Valuev, I.A. Mol. Simulation. 2005, vol. 31, 1005-1017. Berendsen, H.J.C.; van Gunsteren W.F. In Molecular liquids - dynamics and interactions. Ed. Barnes A.J. et al. D. Reidel Publ.; 1984; pp 475-500. Nose, Sh. J. Chem. Phys. 1984, vol. 81, No 1, 511-519. Holender, J.M.; Morgan, G.J. J. Phys.: Condens. Matter. 1991, vol. 3, No 38, 72417254. Nakano, A.; Kalia, R.K.; Vashishta, P. J. Non-Cryst. Solids. 1994, vol. 171 (2), 157163. Belashchenko, D.K. Structure of liquid and amorphous metals; Ed.: Metallurgy, Moscow, 1985, pp 1-192. Belashchenko, D.K.; Tomashpolsky, M. Ю. Metals. 1967, No 6, 137-144 (in Russ.). Car, R.; Parinello, M. Phys. Rev. Letters. 1985, vol.55, No 22, 2471-2474. Tuckerman, M.E. J. Phys. Condens. Matter. 2002, vol.14, R1297-R1355. Marx, D.; Hutter, J. In Modern methods and algorithms of quantum chemistry. Ed. J. Grotendorst. NIC series. Vol.1. Forschungszentrum Uelich, 2000, pp. 301-449. Kaplan, I.G. Introduction in the theory of molecular interactions. Science Ed., Мoscow, 1982, pp. 1-312 (in Russian). Vaks, V.G. Interparticle interactions and bond in solids. Atomic Science and Technics Ed., Moscow, 2002, pp. 1-256 (in Russian). Takai, Т.; Halicoglu,Т.; Tiller, W.А. Scr. Met., 1985, vol. 19, 715. Hansen, J.P. Phys. Rev. 1973, vol. A8, No 6, 3096-3109. Tosi, M.P.; Fumi, F.G. J. Phys. Chem. Liquids. 1964, vol. 25, 45-52. Adams, D.J.; McDonald, I.K. J. Phys. C. 1974, vol.7, No 16, 2761-2775. Harrison, W. Pseudopotentials in the Theory of Metals. Ed. W.A. Benjamin. N.Y., Amsterdam, 1966, pp. 1-366. Daw, M.S.; Baskes, M.I. Phys. Rev. B. 1984, vol. 29, No 12, 6443-6453. Belashchenko, D.K.; Ostrovski, O.I. Russ. J. Phys. Chem. 2006, vol. 80, No 4, 509-522. Mendelev, M. I.; Srolovitz, D. J. Phys. Rev. B. 2002, vol. 66, 014205 (1-9).

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[21] Mendelev, M.I.; Han, S.; Srolovitz, D.J.; Ackland, G.J.; Sun, D.Y.; Asta, M. Phil. Mag. A. 2003, vol. 83, 3977-3994. [22] Belashchenko, D.K. High Temperature. 2006, vol. 44, No 5, 675-686. [23] Belashchenko, D.K. RJPC. 2006, vol. 80, No 5, 758-768. [24] Belashchenko, D. K. RJPC. 2006, vol. 80, No. 10, 1567–1577. [25] Schommers, W. Phys. Rev. A. 1983, vol.28, 3599-3605. [26] Belashchenko, D. K. Computer modeling of liquid and amorphous substances. Moscow Institute of Steel and Alloys Ed., Moscow, 2005, pp.1 – 408 (in Russ.). [27] Belashchenko, D. K. Metal Physics and Metallography. 1987. V. 63. No 4. P. 665-671 (in Russ.). [28] Belashchenko, D. K. Structure of Liquid and Amorphous Metals. Metallurgy Ed. Moscow, 1985, pp. 1- 192 (in Russ.). [29] Hoover, W.G.; Gray, S.G.; Johnson, K.W. J. Chem. Phys. 1971, vol. 55, No 3, 11281136. [30] Stillinger, F.Н.; Weber T.A. J. Chem.Phys. 1978, vol.68, No 8, 3837-3844. [31] Stillinger, F.H.; Weber T.A. J. Chem. Phys. 1979, vol. 70, No 11, 4879-4883. [32] Belashchenko, D. K.; Belashchenko K. D. Melts. 1989, No 2, 32-39 (in Russ.). [33] Susuki K., Fudzimori H., Hasimoto K. In Amorphous Metals. Ed. Masumoto C. (In Jap.) Russ. Transl., Metallurgy Ed. Moscow, pp. 1 - 328. [34] Rahman, A.; Mandell, M.J.; McTague, J.P. J. Chem. Phys. 1976, vol. 64, No 4, 15641568. [35] Kristensen, D.W. J. Non-Cryst. Solids. 1976, vol. 21, 303-318. [36] Bhuiyan, G.M.; Rahman, A.; Khaleque, M.A.; Rashid, R.I.M.A.; Mujibur Rahman, S.M. J. Non-Cryst. Solids. 1999, vol. 250-252, 45-47. [37] Il’inskii, A.; Slyusarenko, O.; Slukhovskii, O.; Kaban, I.; Hoyer, W. Materials Science and Engineering. A. 2002, vol. 325, 98-102. [38] Holzman, L.M.; Adams, J.B.; Foiles, S.M.; Hitchon, W.N.G. J. Mater. Res. 1991, vol. 6, No 2, 298-302. [39] Alemany, M.M.G.; Calleja, M.; Rey, C.; Gallego, L.J.; Casas, J.; Gonzalez, L.E. J. Non-Cryst. Solids. 1999, vol. 250-252, 53-58. [40] Waseda, Y. The Structure of Non-Crystalline Materials. McGraw-Hill, N.Y., 1980, pp. 1-315. [41] Foiles, S.M. Phys. Rev. B. 1985, vol. 32, No 6, 3409-3415. [42] Foiles, S.M.; Adams, J.B. Phys. Rev. B. 1989, vol. 40, No 9, 5909-5915. [43] Sadigh, B.; Grimvall, G. Phys. Rev. B. 1996, vol. 54, No 22, 15742-15746. [44] Landa, A.; Wynblatt, P.; Siegel, D. J.; Adams, J. B.; Mryasov, O. N.; Liu, X.-Y. Acta Mater. 2000, vol. 48, 1753-1761. [45] Hoyt, J.J.; Garvin, J.W.; Webb, E.B. III; Asta, M. Modelling Simul. Mater. Sci. Eng. 2003, vol. 11, 287-299. [46] Belashchenko, D.K.; Kravchunovskaya, N.E.; Ostrovski, O.I. Inorganic Materials. 2008, vol. 44, No 3, 248-257. [47] Belashchenko, D.K.; Ostrovski O.I. High Temperatures, 2009, vol. 47, in press. [48] Belashchenko, D.K. RJPC. 2008, vol. 82, No 7, 1138-1144. [49] Belashchenko, D.K.; Zhuravlev, Yu.V. Inorganic Materials, 2008, vol. 44, No 9, 939945. [50] Belashchenko, D.K.; Ostrovski, O.I. RJPC. 2008, vol. 82, No 3, 364-375.

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[51] Belashchenko, D.K.; Lobanov E.S. Cluster Mechanism of Liquid Metals Crystallization at Great Supercooling: Liquid Silver. Int. Conf. EMMM-2007. Moscow 3-7.09.2007. Institute of Crystallography RAS. Book of abstracts. P. 15. [52] Rahman, A.; Mandell, M.J.; McTague, J.P. J. Chem. Phys. 1976, vol. 64, No 4, 15641568. [53] Mandell, M.J.; McTague, J.P.; Rahman, A. J. Chem. Phys. 1976, vol. 64, No 9, 36993702. [54] Hsu, C.S.; Rahman, A. J. Chem. Phys. 1979, vol.70, No 11, 5234-5238. [55] Hsu, C.S.; Rahman, A. J. Chem. Phys. 1979, vol.71, No 12, 4974-4986. [56] Mandell, M.J.; McTague, J.P.; Rahman, A. J. Chem. Phys. 1977, vol. 66, No 7, 30703075. [57] Belashchenko, D.K.; Lobanov, E.S.; Syrykh, G.F. J. of Alloys and Compounds. 2007, vol. 434-435C, 577-580. [58] Belashchenko, D.K. RJPC. 2006, vol. 80, No 12, 1968-1979. [59] Steinhardt, P.J.; Nelson, D.R.; Ronchetti, M. Phys. Rev. B. 1983, vol. 28, 784-805. [60] van Duijneveldt, J.S.; Frenkel, D. J. Chem. Phys. 1992, vol. 96, No 6, 4655-4668. [61] Cherne, F.J.; Baskes, M.I.; Schwarz, R.B.; Srinivasan, S.G.; Klein, W. Modelling Simul. Mater. Sci. Eng. 2004, vol.12, 1063–1068. [62] Tatarinova, L. I. Structure of solid amorphous and liquid substances. Science Ed., Moscow, 1983. pp. 1-151. [63] Belashchenko, D.K. Russ. Metallurgy (Metally). 1989, No 3, 136-142 (in Russ.). [64] Belashchenko, D.K. Russ. Metallurgy (Metally). 1990, No 1, 166-172 (in Russ.). [65] Belashchenko, D.K. Crystallography Reports. 1998, vol. 5, 733-737. [66] Belashchenko, D.K. Inorganic Materials. 2001, vol.37, No 4, 416-423. [67] Pusztai, L.; Svab, E. J. Non-Cryst. Solids. 1993, vol. 156-158, 973-977. [68] McGreevy, R.L.; Pusztai, L. Proc. Roy. Soc. London. 1990, vol.430, 241-261. [69] Pusztai, L.; Svab, E. J. Phys.: Condens. Matter. 1993, vol. 5, 8815-8828. [70] Svab, E.; Meszaros, Gy.; Konczos, G.; Ishmaev, S.N.; Isakov, S.L.; Sadikov, I.P.; Chernyshov, A.A. J. Non-Cryst. Solids. 1988, vol. 104, 291-299. [71] Schommers, W. Phys. Lett. 1973, vol. 43A, 157-158. [72] Belashchenko, D.K. RJPC. 2001, vol.75, No 3, 387-394. [73] Mendelev, M.I.; Belashchenko, D.K.; Ishmaev, S.N. Inorganic Materials. 1993, vol. 29, No 11, 1483-1489 (in Russ.). [74] Belashchenko, D.K.; Mendelev, M.I.; Ishmaev, S.N. J. Non-Cryst.Solids. 1995, vol. 192-193, 623-626. [75] Ishmaev, S.N.; Isakov, S.L.; Sadikov, I.P.; Svab, E.; Koszegi, L.; Lovas, A.; Meszaros, Gy. J. Non-Cryst. Solids. 1987, vol. 94, No 1, 11-21. [76] Belashchenko, D.K.; Mendelev, M.I. Metal Physics and Metallography. 1991, No 8, 28-38 (in Russ.). [77] Belashchenko, D.K.; Mendelev, M.I. Melts. 1993, No 1, 46-51 (in Russ.). [78] Mendelev, M.I.; Srolovitz, D.J. Phys. Rev. B. 2002, vol. 66, 014205.1-9. [79] Canales, M.; Gonzalez, L.E.; Padro, J.A. Phys. Rev. E.: Stat. Phys., Plasmas, Fluids. 1994, vol. 50, No 5, 3656-3669. [80] Belashchenko, D.K.; Mendelev, M.I. RJPC. 1995, vol. 69, No 3, 543-549. [81] Johnson, M.D.; Hutchinson, P.; March, N.H. Proc. Roy. Soc. Lond. A, 1964, vol.282, 283-302.

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[82] Belashchenko, D.K. Diffusion mechanisms in disordered systems: computer simulation. Physics - Uspekhi. 1999, vol. 42, No 4, 297-319. [83] Belashchenko, D.K. Metal Physics and Metallography. 1982, vol.53, No 6, 1076-1084. [84] Belashchenko, D.K.; Pham Khac Hung; Rosenthal, A.L.; Phan Suan Khien. Russian Metallurgy (Metally). 1986, No 2, 57-63. [85] Limoge, Y.; Bocquet, J.L. Phys. Rev. Lett. 1990, vol.65, No 1, 60-63. [86] Gorbunov, D.A.; Klinger, L.M. Metal Physics and Metallography. 1986, vol.61, No 6, 1084-1088. [87] Stolz, U.; Kirchheim, R.; Sadoc, J.E.; Laridjani, M. J. Less-Common Metals. 1984, vol.103, No 1, 81-90. [88] Nakamura, K. Ber. Bunsenges. Phys. Chem. 1985, Bd. 89, No 1, 191. [89] Gritzenko, A.B.; Andreev, L.A.; Belashchenko, D.K. Metal Physics and Metallography. 1989, vol.67, No 5, 972-978. [90] Gritzenko, A.B.; Belashchenko, D.K. Metal Physics and Metallography. 1988, vol.65, No 6, 1045-1053. [91] Gritzenko, A.B.; Belashchenko D.K.; Kosov I.N. Metal Physics and Metallography. 1991, No 2, 57-63. [92] Yamamoto, R.; Haga, K.; Sibuta, H.; Doyama, M. J. Phys. F: Metal Physics. 1978, vol. 8, No 8, L179-L182. [93] Lançon, F.; Billard, L.; Chambron, W.; Chamberod, W. J. Phys. F: Metal Phys. 1985, vol.15, No 7, 1485-1496. [94] Ahmadzadeh, M.; Cantor, B. J. Non-crystalline Solids. 1981, vol.43, No 2, 189-219. [95] Finney, J.L. Proc. Roy. Soc. A. 1970, vol.319, 479-493. [96] Pham Khac Hung; Belashchenko, D.K. Izv. Vuzov. Ferrous metallurgy. 1987, No 5, 9195. [97] Schuehmacher, J.J.; Guiraldenq, P. Acta met. 1983, vol.31, No 12, 2043-2049. [98] Valenta, P.; Meier, K.; Kronmüller, H.; Freitag, K. Phys. stat. sol. 1981, vol.106, No 1, 129-133. [99] Delaye, J.M.; Limoge, Y. J. Phys. I France. 1993, vol. 3, 2063-2077. [100] Delaye, J.M.; Limoge Y. J. Phys. I France. 1993, vol. 3, 2079-2097. [101] Site: \\www.tagen.tohoku.ac.jp/general/building/iamp/database/scm/LIQ/gr.html [102] Doyama, M.; Kogure, Y. Computational Materials Science. 1999, vol. 14, 80-83. [103] Landa A., Wynblatt P., Girshick A., Vitek V., Ruban A., Skriver H. // Acta mater. 1998. V. 46. No. 9. P. 3027.

In: Amorphous Materials: Research, Technology… Editors: J. R. Telle, N. A. Pearlstine

ISBN: 978-1-60692-235-4 © 2009 Nova Science Publishers, Inc.

Chapter 2

NOVEL AMORPHOUS AND NANOCRYSTALLINE SOFT MAGNETIC MATERIALS Nuria Iturriza1, Juan José del Val1, Arkadi P. Zhukov1, Ignacio García2,3, José A. Pomposo2 and Julián González1 1

Department of Materials Physics, Faculty of Chemistry, University of the Basque Country, Pº Manuel de Lardizabal 3, 20018 San Sebastian, Spain 2 New Materials Department, CIDETEC – Centre for Electrochemical Technologies, Pº Miramón, 20009 San Sebastián, Spain 3 CIC nanoGUNE Consolider, Paseo de Mikelete 56 E-20.009 - San Sebastián, Spain

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1. ABSTRACT Promising iron-based soft nanocrystalline alloys, which present a softer magnetic behaviour than that of the precursor amorphous materials have been, in the last years, developed. Most investigations were performed on the typical Fe73.6Cu1Nb3Si13.5B9 composition (trademark Finemet) with a nanocrystalline grain structure produced by the partial devitrification of the precursor amorphous ribbons to reach the nanostructure character with optimal soft magnetic properties. This softening effect provides an alternative approach to the development of novel soft magnetic materials. Nanocrystalline soft magnetic alloys prepared by annealing melt-spun amorphous precursors, with which this chapter is concerned, are one of the latest successful outcomes of such a new approach to the development of novel soft magnetic materials. In this chapter we will present the origin of the softness, processing and magnetostrictive properties of the nanocrystalline soft magnetic Fe(Co,Ni)-based Finemet alloys with particular attention paid to: (i) nanostructure-magnetic properties relationships and (ii) the principles underlying material design. The nanocrystallization process will be performed by thermal treatments of current annealing under the action of a mechanical stress and/or magnetic field in order to develop simultaneously a macroscopic uniaxial magnetic anisotropy. The substitutions of Fe by Ni or Co on the mentioned behaviours are also analyzed.

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Nuria Iturriza, Juan José del Val, Arkadi P. Zhukov et al.

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2. INTRODUCTION Magnetism at nanoscale presents surprising experimental features that makes to the socalled Nanomagnetism as an exciting branch of the Magnetism and Magnetic Materials. This chapter is aimed on the analysis of the role of microstructure in the technological magnetic properties (for example, coercive and saturation magnetostriction constant) of new advanced magnetic materials exhibiting amorphous, nanocrystalline and nanogranular character. Therefore, the magnetic principles of modern and soft magnetic materials will be presented comprehensively. Magnetization processes in such materials will be considered in detail taking into account that surface effects coming from the macroscopic area of the material as well as from the frontier between nanograins and the matrix where are embedded such nanograins ... As is well known, initially was believed that the ferromagnetism in amorphous solids could not exit because of lack of atomic ordering. Gubanov [1] theoretically predicted in 1960 that amorphous solids would be ferromagnetic. Later it was found, that the 3d-metal based amorphous alloys obtained by rapid-quenching of the melt are excellent soft magnetic materials, i.e. they exhibit very low value of the coercive field and relatively high saturation magnetization [2]. Such magnetic softness is originates from the absence of magnetocrystalline anisotropy in these alloys [2]. The amorphous ribbons obtained by the melt-spinning technique have widely been introduced as the soft magnetic materials in 70-th years. Their excellent magnetic softness and high wear and corrosion resistance made them very attractive in recording head and microtransformer industries. The magnetic behavior of these amorphous alloys is strongly connected to the internal stresses because they result to be as the main source of magnetic anisotropy in amorphous and nanocrystalline materials due to the magnetoelastic coupling between magnetization and internal stresses through magnetostriction. Consequently, these materials result to be very interesting for field and stress-sensing elements because the Fe-rich amorphous alloys exhibiting high magnetostriction values (λS = 10-5) and therefore many of magnetic parameters (i.e.: magnetic susceptibility, coercive field,...) are extremely sensitive to the applied stresses. Recently, promising iron-based soft nanocrystalline alloys have been developed which present a softer magnetic behaviour than that of the precursor amorphous material. Most investigations were performed on the typical Fe73.6Cu1Nb3Si13.5B9 composition (trademark Finemet) with a nanocrystalline grain structure produced by the partial devitrification of the precursor amorphous ribbons to reach the nanostructure character with optimal soft magnetic properties [3-11]. The magnetic softening is brought about by the reduced effect of the intrinsic magnetocrystalline anisotropy due to the mutual transmission of the anisotropy energy among a great number of nanocrystallites. This softening effect provides an alternative approach to the development of novel soft magnetic materials. It is remarkable that these nanocrystalline soft magnetic alloys are one of the latest successful outcomes of such a new approach to the development of novel soft magnetic materials. In this chapter we will discuss the history, origin of the softness, processing and properties of the nanocrystalline soft magnetic alloys with particular attention paid to: (i) nanostructure-magnetic properties relationships and (ii) the principles underlying material design.

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Historically, the discovery of the nanocrystalline soft magnetic alloys seems to have originated from research aimed at the improvement of soft magnetic properties in amorphous alloys. Although optimum soft magnetic properties of melt-spun amorphous alloys were usually obtained after stress-relief annealing [2], it was widely believed that the magnetic softness of any amorphous alloy deteriorates by increasing the annealing temperature far beyond the crystallization temperature. This common belief was wrong. In 1988, Yoshizawa et al. [3] of Hitachi Metals discovered that the magnetic softness of melt-spun amorphous Fe73.5Si13.5B9Nb3Cu1 is improved significantly by crystallization. This alloy (Finemet) exhibited an exceptionally high effective permeability (μe) of ~ 105 with a saturation magnetization (Ms) of 1.25 T. This improved magnetic softness originated from the formation of a unique microstructure composed of extremely fine grains with a size of about 10 nm. In the following year, Herzer [12] analyzed the magnetocrystalline anisotropy energy of nanocrystalline materials based on the concept of the so-called random anisotropy model originally proposed by Alben, Becker and Chi [13] for amorphous systems. Herzer demonstrated that this model is applicable to nanocrystalline systems and pointed out that the coercivity in nanocrystalline materials scales as the 6th exponential power of the grain size. The discovery of the excellent magnetic core properties in nanocrystalline Fe73.5Si13.5B9Nb3Cu1 initiated an era of intense research on the development of new Fe-based nanocrystalline soft magnetic alloys in various alloy systems. As a result, (Cu, Si)-free Femetal based nanocrystalline alloys, known as Nanoperm, whose Fe content is much higher than the Fe-metalloid based alloys such as Fe91Zr7B2 were developed by Suzuki et al. [14,15]. The saturation magnetization of nanocrystalline Fe91Zr7B2 with μe of 3 x 104 was as high as 1.7 T [16-18], the highest value among the nanocrystalline soft magnetic alloys reported to date, while the permeability of the Fe73.5Si13.5B9Nb3Cu1 alloy with Ms of 1.25 T is still the highest in the nanocrystalline soft magnetic alloy family. Besides these bulk-form alloys, thin-film nanocrystalline soft magnetic alloys based on similar principles were also reported for Fe-EM-C [19] and Fe-EM-N (EM = IVa to VIa metals) systems [20-22]. An excellent review on the nanocrystalline Fe-EM-C thin-films can be found elsewhere [23]. The unique microstructure in the nanocrystalline soft magnetic alloys has stimulated intensive investigations of the decomposition behavior in the amorphous precursors. Advanced experimental techniques including in-situ time resolved x-ray diffractometry [24], atom-probe field ion microscopy (APFIM) [25-27] and x-ray absorption fine structure (XAFS) [28] were employed in these investigations. These studies showed that the nanostructural evolution in the Fe-Si-B-Nb-Cu and Fe-Zr-B systems is due to primary crystallization (cf. Köster and Herold [29]) of the precursor amorphous phase and the resultant nanoscale structure consists of the primary bcc-Fe precipitates and the residual amorphous matrix. This peculiar microstructure as well as the mechanism behind the magnetic softness is also attracting growing interest from the viewpoint of the fundamental magnetism problems in heterogeneous nanostructural systems. As reviewed by Gleite [30], the concept of nanocrystalline materials can be found as early as in 1981. The basic idea of these nanocrystalline materials was to generate a new class of disordered solids by introducing a high density of grain boundary defects. However, the idea of recent nanocrystalline soft magnetic alloys is to reduce the effects of the magnetocrystalline anisotropy by simply reducing the grain size below the exchange correlation length, different from the principal idea of the previous nanocrystalline materials.

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Hence, some authors in the early stages of research on nanocrystalline soft magnetic alloys preferred to refer to their materials as ultrafine grain or microcrystalline, rather than nanocrystalline. Another important study on nanostructured materials prior to the discovery of the Fe-Si-B-Nb-Cu alloys can be found in the 1960's. Hoffmann [31] analyzed the effect of grain size on soft magnetic behavior in Ni-Fe (Permalloy) films in the nanocrystalline regime. However, the intrinsic magnetocrystalline anisotropy in his Ni-Fe films was small enough to realize good magnetic softness. On the other hand, recent studies on nanocrystalline soft magnetic alloys have involved attempting to reduce the effect of large magnetocrystalline anisotropy in Fe-rich alloys by means of grain refinement. Consequently, the rationale behind recent work on nanocrystalline soft magnetic alloys appears to be distinct from that of the early studies.

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2.1. Nanoscale of Novel Magnetic Materials Nanostructured materials are of growing importance regarding the macroscopic magnetic properties for industrial applications. In fact, it is well known that the magnetic applications can be classified into two large groups: (a) magnetic flux multiplication which requires soft magnetic materials with high magnetization and narrow hysteresis loop, i.e. low coercivity and (b) magnetic storage of either energy (magnets) or information (magnetic recording), both cases require hard magnetic materials with high magnetization and wide hysteresis loop, i.e. high coercivity and remanence. Since the possibility of increasing the magnetization, always required, is restricted by serious limitation [32] (It is not possible to increase substantially either the number of atoms per unit volume or the magnetic moment), the main task of the research in magnetic materials has consisted in spreading over a wider range the available coercivities. If the saturation magnetization, μ0MS, of the ferromagnetic alloys is typically of the order of 1 T, the coercivity, μ0HC, ranges between 10-7 T for the softer magnetic materials and 5 T for the harder magnets. Therefore, the seven orders of magnitude which separate in coercivity, the softer from the harder alloys should be considered as a suitable index of the success of the science of magnetic materials. From this point of view it is really remarkable the softest material Fe79Zr7B9 (μ0HC ≈ 10-7 T) as well as the hardest material Fe79Nd7B9 (μ0HC ≈ 1 T) which are obtained from amorphous alloys with closely related compositions. Notice that only a difference of 7 at % in content gives rise to a difference of seven orders of magnitude in coercivity. The reason for such an enormous difference is the nanostructure obtained by partial devitrification of the initial amorphous state. Both types of sample consist of a soup of nanograins embedded in a softer residual matrix. In partially crystallized Fe79Zr7B9 Fe nanocrystals with anisotropy constant, K1, of the order of 104 J/m3 are embedded in the magnetically softer remaining amorphous matrix, whose composition depends on the crystallized fraction, whereas in the devitrified Fe79Zr7B9 hard nanocrystals of the Nd2Fe14B phase with anisotropy constant K1 = 107 J/m3 are dispersed in a softer matrix formed by nanocrystalline Fe. It is assumed that the coercivity term depending on K1 can be expressed as: HC = (α2K1)/(μ0MS)

(1)

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The expected difference in coercivity between FeZr and FeNd nanostructures should be that the anisotropy constant three order of magnitude. However, the nanostructure of the FeZr sample reduces by itself the effective macroscopic anisotropy by four orders of magnitude, whereas the nanostructure of FeNd alloy enhances the magnetization and the energy product a factor, which describes the energy stored by a magnet.

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3. SOFT MAGNETIC MATERIALS. ON THE ORIGIN OF SOFT MAGNETIC BEHAVIOR Some fundamental questions related with the magnetic order in amorphous and nanocrystalline materials having structural disorder appear considering the existence of such well-defined magnetic order. In fact, considering the ferromagnetic interactions of the magnetic materials and neglecting the magnetic anisotropy, the magnetic moments tend to arrange their orientations parallel to each other via exchange interactions; this they do when lying along a magnetic easy axis which is in the same direction at every point in the material. However, if the easy axis orientation fluctuates from site to site, a conflict between ferromagnetic coupling and anisotropy arises. As long as we imagine lattice periodicity, a ferromagnetic structure is a consequence of ferromagnetic exchange interactions, the strength of the anisotropy being irrelevant. In this situation we are assuming a major simplification, namely: the direction of the easy axis is uniform throughout the sample. With this simple picture, crucial questions related to the influence of an amorphous structure on magnetic order are open. Regarding the magnetic order in amorphous and nanocrystalline materials, we know that it originates from two contributions: exchange and local anisotropy. The exchange arises from the electron-electron correlations. The mechanism of the electrostatic interactions between electrons has no relation to structural order and is sensitive only to overlapping of the electron wave functions. With respect to magnetic anisotropy is also originated by the interaction of the local electrical field with spin orientation, through the spin-orbit coupling. Therefore, magnetic anisotropy is also a local concept. Nevertheless, the structural configuration of magnetic solids exerts an important influence on the macroscopic manifestation of the local anisotropy. As a consequence, when the local axes fluctuate in orientation owing to the structural fluctuation (amorphous and nanocrystalline materials as examples), calculations of the resultant macroscopic anisotropy becomes quite difficult. In the case of amorphous ferromagnetic alloys the usual approach to the atomic structure of a magnetic order connected to a lattice periodicity is not applicable. These materials can be defined as solids in which the orientation of local symmetry axes fluctuate with a typical correlation length l = 10 A. The local structure can be characterized by a few local configurations with icosahedral, octahedral, and trigonal symmetry. These structure units have randomly distributed orientation. The local magnetic anisotropy would be larger in these units with lower symmetry. In general, are characterized by fluctuations of the orientation local axis. It is remarkable that these types of structures the correlation length, l, of such fluctuation is typically the correlation length of the structure and ranges from 10 A (amorphous) to 10 nm (nanocrystals) and 1 mm (polycrystals).

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Fluctuations in the interatomic distances associated with the amorphous structure should also contribute to some degree of randomness in the magnetic interactions of the magnetic moments. Nevertheless, such randomness is expected not to affect the magnetic behavior qualitatively [33,34]. Moreover, random distribution of the orientation of the easy axis drastically affects to the magnetic properties. Random anisotropy model developed by R. Alben et al. [35] provides a successfully explanation that how the correlation length, l, exerts a relevant influence on magnetic structure. The important question is: What is the range of orientational correlation of the spins? Let is L the correlation length of the magnetic structure. If we assume L >> l, the number of oriented easy axes in a volume L3 should be N = (L/l)3. The effective anisotropy can be written as: Keff = K/N1/2

(2)

where K is the local anisotropy which strength is assumed to be everywhere uniform. By minimizing the total energy with respect to L, the following expression can be deduced: L = (16A2) / (9K2l3)

(3)

where A is the exchange stiffness parameter.

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3.1. Amorphous Alloys There are few reviews dealing with fundamental magnetic properties of amorphous materials [36-40]. In this sense, it must be mentioned that the variations of the average magnetic moment μ and Curie temperature TC for amorphous alloys as a function of transition metal content have been studied by Mizoguchi et al [40]. They studied the dependence of μ and TC on the number of valence electrons for amorphous T80B10P10 thin films. Difference observed for the amorphous and crystalline alloys for the case of strong ferromagnetism were attributed to a charge transfer from the metalloid atoms to the 3D transition metal. In the case of T80B20 and T80P20 amorphous alloys such charge transfer of 1.6 and 2.4 electrons per boron and phosphorus atoms is suggested to explain the observed dependence of μ on average transition metal valence electron concentration for several amorphous ribbons [41]. Significantly various authors have explained the difference in the temperature dependence of the magnetization in crystalline and amorphous alloy [42]. Thus, the fitting at low temperature T3/2 behavior according to the σ(T)/σ(0) = (1-BT3/2+... ) suggested the existence of spin wave excitations in amorphous alloys. Larger value of the B coefficient observed for the amorphous alloys has been attributed to the chemical disorder. Inelastic neutron scattering confirmed the existence of well-defined spin-wave excitations [43]. On the other hand, generally flatter magnetization curves obtained by plotting σ(T)/σ(0) versus T/TC of amorphous metallic alloys have been attributed to exchange fluctuations [34]. Such exchange fluctuations were attributed mainly to the local disorder [43] As has been mentioned, amorphous alloys are characterized by the absence of dislocations and grain boundaries, which result to be the most important pinning centers of the domain wall displacements governing the coercive field, HC, in crystalline materials.

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Therefore, in amorphous alloys only structural defects are linked to fluctuations of mass density, known as free volume [44,45], short-range ordering or agglomeration of free volumes. In addition to the lack of conventional defect structures, long-range magnetocrystalline anisotropy is absent. Nevertheless, the initial susceptibility of amorphous alloys is of the same order of magnitude as those of crystalline alloys, for example Permalloy. Obviously, inhomogeneities exit in amorphous alloys leading to a moderate intrinsic pinning of the domain wall displacement. Five types of intrinsic magnetic inhomogeneities may be of importance in amorphous alloys. Five types of intrinsic inhomogeneities have been proposed by Kronmüller and Fähnle to be of importance in soft magnetic amorphous alloys [46-48]: 1. 2. 3. 4. 5.

Fluctuations of the exchange energy Fluctuations of the local magnetic anisotropy Elastic coupling energy of elastic dipoles Clusters of atomic short-range order Surface irregularities

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All these different types of inhomogeneities should be considered to explain the coercivity behaviour of soft magnetic amorphous alloys. Nevertheless, a critical study of the fluctuations of the exchange energy and local magnetic anisotropy gives a value of μ0HC = 3 x 10-8 T [48], where the long-range anisotropy is assumed to be due to long-range internal stresses being much smaller than the anisotropy constant of the corresponding crystalline material. On the other hand, if we assume that the long-range anisotropy is quite small, the coercive field could be determined by the fluctuations of the exchange energy only leading to a values of μ0HC = 3 x 10-9 T., i.e., the minimum intrinsic coercive field achievable in amorphous alloys is of the order of nanoclusters. Unfortunately, these low values of coercive fields have not been observed. (the lowest values measured are of the order of 10-7 T.). Consequently, other microstructural aspects should be considered to explain such coercivity behaviour of the amorphous alloys.

3.1.1. Effect of Internal Stresses on the Coercivity Because of the coercive field in amorphous alloys due to fluctuations of exchange energy and local magnetic anisotropy results to be a factor 10 – 100 lower than that experimentally obtained, the magnetostrictive interactions should play an important role on such coercivity behaviour. In fact, the compositional dependence of the coercivity, HC(x) of the Fe100-xNixB20 amorphous alloys, which are characterized by a variation of the saturation magnetostriction constant, λS, from 5x10-6 for x = 60 up to λS = 45x10-6 for x = 0 showed in the Figure 1 can be considered as an indication of the influence of the magnetoelastic energy on the coercive field. As can be seen, HC shows the expected monotonous increase as a function of λS. Similarly, the dependence of the anisotropy constant, Keff, which follows that of HC can be understood assuming that it results mainly from elastic long-range stresses, i.e., from the magnetoelastic coupling energy. Two types of stress sources have been assumed to exit in amorphous alloys, namely: long-range quenching stresses and short-range stresses due to socalled “quasidislocation dipoles” [40]. The long-range stresses are the source for magnetic anisotropy energy whereas the elastic dipoles act as pinning centers for domain wall displacements. Such elastic dipoles of extension 1-10 nm are present in amorphous alloys

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with a high density of 1018 m-3 [49], resulting from mass density fluctuation quenched-in during the rapid cooling process. These mass density defects, so-called “free volume” [50], are agglomerated during the quenching process, thus leading to a local collapse of the amorphous structure inducing, at the same time, elastic deformations. Such elastic deformations could give a contribution to HC of the order of some A/m, which is of the some order of magnitude of HC measured in Fe-rich amorphous alloys (with large and positive λS). Nevertheless, that magnetoelastic contribution to HC in the case of non-magnetostrictive (Co, Ni, Fe)-based alloys do not give a relevant contribution, since the expected HC values are lower than those experimental measured in these alloys. Consequently, additional contribution to HC in amorphous alloys should be considered.

Reprinted from Ref. [48].

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Figure 1. Effective anisotropy constant, Keff, coercive field, HC, and χ0/μ02 MS2 as a function of λΣ forFe80-xNixB20 amorphous alloy for nearly nonmagnetostrictive Co alloys the results of χ0/μ02HCMS are shown in the lower left side.

3.1.2. Surface Contribution to the Coercivity in Amorphous Alloys The natural surface roughness in amorphous alloys plays a very dramatically role in the coercivity in the case of soft magnetic amorphous alloys produced by the melt-spinning technique in the ribbon shape. In fact, these ribbons have a natural surface roughness due to the uneven roller as well as to the inhomogeneous solidification process. In general, the contact surface reveals a larger surface roughness than the free surface [51]. In the Figure 2 is schematically shown that, in addition to the different rough nesses, the wavelength of the surface irregularities are different from each other [51]. On the free surface wavelengths of λ ≈ 10 μm are observed. Due to the varying ribbon thickness T(y, z) the domain wall moving along the ribbon changes its area as a function of z.

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Reprinted from Ref. [48]. Figure 2. Domain walls in a in a ribbon with fluctuating thickness, T(y,z), showing a smooth free surface and a rough contact surface.

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An additional probe of the surface contribution to HC was obtained by changing the topological parameters by special treatments. In fact, the Figure 3 shows the influence of thinning a ribbon by special preparations [48]. It was found that HCsurf depends linearly on 1/. The increase of HC with decreasing is largest after grinding by glass-paper. Polishing by diamond paste leading to an intermediate increase of HC, whereas electropolishing leads only to a small increase of HC. This behaviour of HC with respect to different surface treatments is compatible with the different values of ΔT and λ expected for the above mentioned treatments.

Reprinted from Ref. [48]. Figure 3. The coercive field as a function of the inverse ribbon thickness after different polishing treatments.

For this, two limiting cases have been proposed according to the wavelength of the surface roughness (λ) and the average wavelength of the volume pinning which affects to the

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domain wall displacement. In the case that the average wavelength of the domain wall pinning is of the order of 2δB (δB: width of the domain wall), HC = [(HCsurf)2 + (HCVol)2]1/2

(4)

Finally, if λ >> 2δB the two contributions to HC have to be added because the volume and the surface pinning take place simultaneously and, consequently, both terms contribute similarly to the coercivity, being in this case HC = HCsurf + HCVol

(5)

In general, the condition λ >> 2δB is more reasonable for the amorphous alloys, that is, the total coercive field is just the addition of both individual coercive fields.

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3.2. Nanocrystalline Alloys In the last decade the study of magnetic behaviour of nanocrystalline phases has attracted special attention. These materials are obtained, usually, by suitable annealing of amorphous metallic ribbons owing to their attractive properties as soft magnetic materials. Such soft magnetic character is though to be originated because the magnetocrystalline anisotropy vanishes and the very small magnetostriction value when the grain size approaches 10 nm, average anisotropy for randomly oriented α-Fe(Si) grains is negligibly small when grain diameter does not exceed about 10 nm. Thus, the resulting magnetic behavior can be well described with the random anisotropy model [8,11,52,53]. According to this model, the very low values of coercivity in the nanocrystalline state are ascribed to small effective magnetic anisotropy (Keff around 10 J/m3). However, previous results [8,53] as well as of that recently published by Varga et al. [54] on the reduction of the magnetic anisotropy from the values in the amorphous precursors (∼1000 J/m3) down to that obtained in the nanocrystalline alloys (around 300 – 500 J/m3) is not sufficient to account for the reduction of the coercive field accompanying the nanocrystallization process. The enhancement of the soft magnetic properties should therefore be linked to the decrease of the microstructure-magnetization interactions. These interactions originating in large units of coupled magnetic moments suggest a relevant role of the magnetostatic interactions as well in the formation of these coupled units [8,53]. In addition to the suppressed magnetocrystalline anisotropy, low magnetostriction values provide the basis for the superior soft magnetic properties observed in particular compositions. Low values of saturation magnetostriction results to be essential to avoid magnetoelastic anisotropies arising from internal or external mechanical stresses. The increase of initial permeability with the formation of the nanocrystalline state is closely related to a simultaneous decrease of the saturation magnetostriction. Partial crystallization of amorphous alloys leads to an increase of the frequency range where the permeability presents high values [55]. These high values in the highest possible frequency range are desirable in many technological applications involving the use of ac fields. It is remarkable that a number of workers have investigated the effects on the magnetic properties of the substitution of additional alloying elements for Fe in the

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Fe73.5Cu1Nb3Si13.5B9 alloy composition so-called Finemet, to further improve the properties, e.g. Co [56], Al [57-59] varying the degree of success. Moreover, it was shown in [60] that substitution of Fe by Al in the classical Finemet composition led to a significant decrease in the minimum of coercivity, Hcmin ≈ 0.5 A/m, achieved after partial devitrification, although the effective magnetic anisotropy field was quite large (around 7 Oe) [60]. The coercivity behavior was correlated with the mean grain size and a theoretical low effective magnetic anisotropy field of the nanocrystalline samples was assumed in contradiction with those experimentally found in [53,54,59].

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3.2.1. Coercivity Behaviour Such as has been mentioned, the discovery of Fe-rich nanocrystalline alloys carried out by Yoshizawa et al. [3] was really important, owing to the outstanding soft magnetic character of such materials. Typical composition of the precursor amorphous alloys, which after partial devitrification reach the optimum nanostructure character are FeSi- or FeZr-based with small amount of B to allow the amorphization process; small amounts of Cu, which act as nucleation centers, and Nb which prevents grain growth. This effect is provided by the Zr in FeZr alloys. After the first step of crystallization, FeSi or Fe crystallites are respectively finely dispersed in the residual amorphous matrix. In a wide range of crystallized volume fraction, the exchange correlation length is larger than the average intergranular distance, d, and the exchange correlation length of the grains is larger than the grain size, D. Magnetic softness of Fe-rich nanocrystalline is due to a second complementary reason: the opposite sign of the magnetostriction constant of crystallites and the residual amorphous matrix, which allows the reduction and compensation of the average magnetostriction. The thermal variation of the coercive field (HC) in a Finemet-type (Ta content) amorphous alloy is presented in the Figure 4. This behaviour is quite similar to that reported in the case of Nb-containing ones and, particularly, evidences the occurrence of a maximum in the coercivity linked to the onset of the nanocrystallization process [5,61].

Reprinted from Ref. [11]. Figure 4. Coercivity evolution with the annealing current density in the alloys with nominal composition Fe73.5Si13.5B9Cu1Nb3 (O) and Fe73.5Si13.5B9Cu1Ta3 (P) annealed during 1 minute using the Joule effect heating.

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Considering the grain size, D, to be smaller than the exchange length, Lex, and the nanocrystals are fully coupled between them, the random anisotropy model implies a dependence of the effective magnetic anisotropy, , with the sixth power of average grain size, D. The coercivity is understood as a coherent rotation of the magnetic moments of each grain towards the effective axis leading to the same dependence of the coercivity with the grain size [4]: HC = pc [/JS ] = pc {K14D6/JSA3} [ = (K14 D6/A3)]

(6)

where K1 = 8 KJ/m3 is the magnetocrystalline anisotropy of the grains, A = 10-11 J/m is the exchange ferromagnetic constant, JS = 1.2 T. is the saturation magnetic polarization and pc is dimensionless pre-factor close to the unity. The predicted D6 dependence of the coercivity has been widely accepted to be followed in a D range below Lex (around to 35 to 40 nm).for nanocrystalline Fe-Si-B-M-Cu (M = IV A and VI A metal) alloys [6, 55]. A clear deviation from the predicted D6 law in the range of HC ≈ 1 A/m was reported by Hernando et al. [8] and Murillo and Gonzalez [11]. Such deviation was ascribed to the effect of the induced anisotropy (i.e.: shape, magnetoelastic and field induced anisotropies) on the coercivity, which could be significant with respect those of the random magnetocrystalline anisotropy. We have proposed in [11] that:

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= Ku + (1/2) {[(Ku) 1/2K1D3]/A3}

(7)

where Ku is the induced magnetic anisotropy. Therefore, in the case of an appreciable induced anisotropy, HC according to Equation (7) scales with a D3 law. Experiments reported in [11,60] in fact point to the validity of Equation (7) as demonstrated by Figure 5 (a) and (b) where HC vs. D leads to a D3.35 law for different compositions [11,62-67]. Deviations from D6 law were also reported by Suzuki et al. [55] and ascribed to the role of an induced anisotropy. To remark that the second part of the Equation (7) corresponds to and if Ku is in space or if it’s spatial fluctuations are negligible to , this second part ultimately determines the grain size influence on the coercivity. Such influences changes from the D6 law to a D3-4 one when the coherent uniaxial anisotropies dominate over the random magnetocrystalline anisotropy. Additional point in order to justify the Equation (7) is connected with the fact that the results of Figure 5 (a) and (b) were obtained in samples treated by the Joule effect heating (with and without stress applied during the current annealing). These kinds of annealings could induce some magnetic anisotropy, which could be responsible for this significant change of the grain size dependence of coercive field. As a consequence, it can be assumed that the presence of more long-range uniaxial anisotropies larger than the averaged magnetocrystalline anisotropies . It should also be noted that Equation (7) can, interestingly, account for the occurrence of dipolar and deteriorated exchange intergrain interaction, and thus can be more realistic than the simple anisotropy averaging since those features are involved in the accomplishment of a nanocrystallization process.

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(A)

(B)

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Figure 5. Dependencies of the coercivity with the grain size for several nanocrystalline alloys (a) samples nanocrystallized by conventional furnace and CA treatment; (b) conventional annealing and Ni-Finemet submitted to SA treatment. (■) Ni5-Finemet [67]; (●) Ni10-Finemet [67]; (○) FeCuNbSiB alloy after Ref.[4]; (x) FeNbSiB alloy after Ref.[4]; (∆) FeCuTaSiB alloy after Ref.[62]; (▲) FeCuSiB alloy after Ref.[63]; (+) FeCuVSiB alloy after Ref.[64]; (□) FeZrB alloy after Ref.[65]; and (♦) FeZrCo after Ref.[66].

Nevertheless, both kinds of anisotropies behave similarly regarding the HC(D) dependencies. Therefore, these kinds of annealings could induce inhomogeneous magnetic anisotropy, which could be responsible for this significant change of the grain size dependence of coercive field. As a consequence, it can be assumed that the presence of more long-range uniaxial anisotropies larger than the averaged magnetocrystalline anisotropies and, consequently, the reduction of HC is not detected as in the case of the Finemet alloy. Additionally, it should also be noted that the occurrence of dipolar and deteriorated exchange intergrain interaction can, interestingly, account for the substitution of Fe by Ni atoms. To remark that the analysis of the magnetization processes in this kind of materials can be inferred from the plots μ0HC/MS as a function of the annealing temperature (Ta). The

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μ0HC/MS quantity changes from 4x10-3 below 840 K to a value of 0.6 for 870 K> Lex the magnetization process is best described by the bowing of domain walls at grain boundaries.

4. PROCESSING. INDUCED MAGNETIC ANISOTROPY

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As-prepared amorphous alloys have a non-equilibrium or metastable microstructure, attributed to the ability of the amorphous state as well as to the high quenching stresses. In the case of glass coated microwires with composite (glass-metal) structure such stresses have crucial role. Therefore various technological procedures might significantly change their microstructure and consequently the magnetic properties. The most common technological procedure used for the tailoring of magnetic properties of amorphous magnetic materials is the thermal treatment. Application of magnetic field and/or applied stress can strongly affect atomic order and consequently magnetic properties of amorphous and nanocrystalline materials. Thermal treatment under applied stress and/or stress can also induce macroscopic magnetic anisotropies, described below. Stress relaxation and local environment changes during thermal treatments generally result in evolution of the local atomic structure towards more stable atomic configuration. It is well established that generally the relaxation of internal stress in amorphous materials (before crystallization) produced by the thermal treatments is connected with the magnetic softening [68]. On the other hand there are various factors affecting soft magnetic behavior of amorphous materials. At least five pinning effects have been identified and discussed by H. Kronmüller [47] as contributing to the total coercivity: 1. 2. 3. 4. 5.

Intrinsic fluctuations of exchange energies and local anisotropies (10-3–1 me), HC(i) Clusters and chemical short ordered regions (< 1 me), HC(SO) Surface irregularities (< 5 Me), HC(surf) Relaxation effects due to local structural rearrangements (0.1-10 me), Volume pinning of domain walls by defect structures in magnetostrictive HC(rel) alloys (10-100 Me), Hc( )

Within the framework of the statistical potential theory, the resultant total coercivity was expressed as following (Kronmuller 1979): Hc(total)= [Hc(σ)2 +Hc(surf)2+ Hc(SO) 2+Hc(i)2]1/2 + Hc(rel)

(8)

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In the case when the surface irregularities give largest contribution, the various terms add linearly, i.e.: Hc(total)= Hc(σ) +Hc(surf)+ Hc(SO) +Hc(i) + Hc(rel)

(9)

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a detailed analysis of each term is described in [47,69]. On the other hand it was found that the shape of the hysteresis loop is determined by the magnitude and easy axis of the various magnetic anisotropies presented in the material [7072]. Thermal treatments affect this magnetoelastic anisotropy. After annealing the magnetoelastic anisotropy drastically decreases and the remaining anisotropy should be ascribed to the directional ordering, magnetic field or stress induced anisotropy. The shape of the hysteresis loop is then determined by the resultant direction and magnitude of this directional order anisotropy. It was found that the magnetostriction constant of the amorphous ribbons of systems (Co1-xFex)75Si 10B15 and (Co1-xFex)80P13C7 changes from positive to negative values passing through zero in the vicinity of x = 0.094 [73]. Consequently, amorphous ribbons with x=0.094 exhibit excellent magnetic softness [68]. On the other hand it is well known that so-called "nanocrystalline materials”, that is, systems consisting two-phase (nanocrystalline grains randomly distributed in a soft magnetic amorphous phase) can be obtained in certain amorphous FeSiB-based alloys with small additions of Cu and Nb by the devitrification of the conventional amorphous FeSiB-based alloys after annealing in the range of 500-600 oC, 1 hour (i.e., at temperatures between the first and second crystallization peaks) [3]. After crystallization, such material consists of small (around 10 nm grain size) nanocrystals embedded in the residual amorphous matrix. In addition, the devitrification process of these amorphous alloys (Finemet) leads to the possibility of obtaining rather different microstructures depending on the annealing parameters as well as on the chemical composition. These materials exhibit extremely high magnetic softness at certain conditions of thermal treatment. In fact, the macroscopic magnetic anisotropy averages out (when L>>D) and the domain wall can move without pinning. For some particular compositions and volume crystallized fraction the average magnetostriction constant become vanishing and the magnetoelastic contributions to the macroscopic anisotropy also become negligible.

4.1. Amorphous Metallic Alloys The magnetization characteristics of amorphous magnets (in particular, those of metallic glasses produced in a form of thin ribbons) can be modelled by annealing under the influence of externally applied magnetic field or mechanical stresses, in a similar way as the case of crystalline soft magnetic materials. This modelling is of technological interest since it allows tailoring the magnetization curves of a given material to the specific requirements, thus, broadening a range of its application. In fact, the amorphous alloys, although macroscopically isotropic in the as-quenched state, can exhibit macroscopic magnetic anisotropy if, such as has been mentioned, they are subjected to suitable annealing treatments at the presence of either a magnetic field (field annealing, FA) or a mechanical stress (stress annealing, SA). More recently stress+field annealing has been performed in several amorphous alloys and it

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has been shown, first the stress+field induced magnetic anisotropy can not be supposed as a simple sum of the stress-induced anisotropy and the field induced anisotropy. Second, it seemed, from earlier works on stress+field annealing [74,75], that the direction of the stress+field induced anisotropy was fixed by the direction of the magnetic field applied during the induction process and the applied tensile stress may enhance this anisotropy. But this hypothesis was found to be untrue with posterior investigations whose are presented in the following. Figure 6 (a) and (b) shows the experimental data reported by González and Kulakowski [76] on the annealing temperature dependencies of different kinds of induced magnetic anisotropies (HT: transverse field annealing; HP: longitudinal field annealing; σ: stress annealing; HT + σ: transverse field + stress annealing and HP+σ: longitudinal field + stress annealing) in two CoFe-based (Co-rich) amorphous alloy ribbons. These results on induced anisotropies point out that the field-induced anisotropy has the same direction as the magnetic field, which was applied during the annealing.

(A) Figure 6. (Continued).

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(B) Reprinted from Ref. [76].

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Figure 6. Magnetic anisotropies induced in two nearly-zero magnetostrictive Co-rich amorphous alloys.

The behaviours of the stress and stress + transverse field induced anisotropies are similar. The evolution of these two kinds of anisotropy as a function of the annealing temperature is quite different with respect to field induced anisotropy and they increases with the annealing temperature reaching a maximum value, Kmax, at the temperature, Tmax, and then decreases. This decrease of stress and stress + field induced anisotropies after maximum value allows to assume that their origin could be ascribed to the magnetic ordering of pair atoms at high annealing temperatures probably acting on the metallic glass at equilibrium, while at low temperatures other processes and mechanisms are occurring during the induction process, such as the development of a longitudinal plastic component [77,78] and the change of the local short range order which is introduced by the stress during the thermal treatment. If the stress is absent, the former mechanism seems to be very weak and, in this case, the fieldinduced anisotropy is almost symmetric with respect to the change of the direction of the annealing field. The direction of the stress + longitudinal induced anisotropy strongly depends on the composition. This dependence is presented in the Figure 7, where can be seen that the increase of the relative concentration of Fe and Si gives the decrease of the longitudinal anisotropy.

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Reprinted from Ref.[76]. Figure 7. Influence of the composition on the direction of the stress + longitudinal field induced magnetic anisotropy in Co-Si-B and Co-Fe-Si-B (Co-rich) amorphous alloys.

As can be seen from Figure 6 (a) and (b) field annealing at elevated temperature but below the Curie temperature induces a macroscopic magnetic anisotropy with the preferred axis determined by the direction of the magnetization during the annealing. Moreover, field induced anisotropy increases as the annealing temperature increases, similarly as the magnetization with the temperature, vanishing at Curie temperature. This behaviour has been observed in metallic glasses with different composition (Fe-rich [79,80]; CoFe-based (Corich) [77,78]; CoNi-based [81] and Co-rich [82]). The microscopic origin of this field-induced anisotropy has been successfully explained according to directional ordering of atomic pairs mechanism developed by Néel [79,80,83]. This model predicts a dependence of the fieldinduced anisotropy with the annealing temperature as:

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Kind(T) = k Msn(T)

(10)

where n is a constant, the value of which can be assumed to be equal 2 if the microscopic origin is the directional ordering of atomic pairs. Theoretical predicted value of the index n was experimental found in FeNi-based metallic glasses [83]. Nevertheless, deviations of such theoretical value have been obtained in metallic glasses of composition (Co1-xFex)78Si10B12 [84,85] and (Co1-xFex)75Si15B10 [77]. In this case, an additional contribution coming from the single-ion (initially n = 3) can be accounted, which allows to conclude that depending of the annealing temperature the weight of each two contributions could be different according to the content of magnetic elements. The stress and stress + field induced anisotropies have been a topic of intensive research in the last three decades in amorphous Co, CoFe, CoFeNi and CoNi-based alloys [7477,81,85-90]. It is remarkable the fact that as an important difference with respect to the field annealing, the stress and stress + field annealing in metallic glasses could induce magnetic anisotropy even at annealing temperatures above the Curie temperature. Moreover, these

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kinds of anisotropy can be induced in zero-magnetostrictive amorphous alloys and its origin is, therefore, different to that of field induced anisotropy. Hernando et al. [91]; Blanco et al. [81] and González and Blanco [92] have proposed several contributions to explain the microscopic origin of the stress and stress + field induced anisotropies, namely: (i) atomic pair ordering and (ii) tetrahedral Bernal holes (BTH) surrounded by (3Co-1Fe, 1Co-3Fe), (3Fe-1Ni, 1Fe-3Ni) and (3Co-1Ni, Co-3Ni). These BTHs differ from each other in the local anisotropy according to particular interactions. In fact, the compositional dependence of the stress and stress + field induced anisotropies of several amorphous alloy systems (Figure 8) shows two maximum values for the (Co1-xFex)-based system for around x = 0.75 (Co-rich) and x = 0.25 (Fe-rich) respectively, being the Co-rich maximum larger than that of the Fe-rich maximum. This implies that the 3Co-1Fe BTHs could give a more efficient contribution to the induced anisotropy. Moreover, the absence of these maximums for Ni-containing (Co and Fe) systems or the lower intensity of those maximums in CoFeNi-based system suggest that the Ni atoms seems to "dilute" magnetically the amorphous matrix.

Reprinted from Ref. [81]. Figure 8. Compositional dependence of the stress + field induced magnetic anisotropy (•)(Co1xFex)75Si15B10; (o) [Co1 - x(Fe0.5Ni0.5)x]75Si15B10; (x) (Co1 - xNix)75Si15B10. The values of (⊗) correspond to the stress-induced anisotropy in (Co1-xNix)75Si15B10 amorphous alloys.

As has been mentioned, it is interesting to remark that these stress and stress + field induced anisotropies can be induced by annealing even above the Curie temperature. This characteristic suggests that the source of these induced anisotropies are not related to magnetoelastic effects but rather to transformations of the microstructure probably connected to "viscoelatic" flow. By analogy with the creep strain, these induced anisotropies have been

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divided in two contributions: the anelastic, Kan, of recoverable character and the plastic, Kpl, (permanent) one being: Kind = Kan + Kpl

(11)

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Where Kind is the stress or stress + field induced magnetic anisotropy. The anelastic component, Kan, results in a transverse easy axis and is reversible while the plastic anisotropy, Kpl, results in an axial easy axis and is irreversible in the sense that it is mostly retained even if further treatments without applied stress are performed. When the first stress or stress + filed annealing is carried out, Kan predominates as its kinetics permits rapidly induced anisotropy, Kpl is induced more slowly as viscoelastic flow is being generated. As further treatment is performed at the same annealing conditions but without applied stress, the resultant induced anisotropy changes the preferred direction toward that of the ribbon axis. After the second (stress free) annealing, only plastic deformation and thereby plastic anisotropy remains. The strong influence of preannealing on the stress induced anisotropy is notable (see Figure 9). In general, preannealing favours the anelastic component to be outstanding.

Reprinted from Ref. [81]. Figure 9. Influence of the pre-annealing on the stress + field induced magnetic anisotropy as a function of the annealing temperature in (Co0.50Fe0.25Ni0.25)75Si15B10 amorphous alloy. (x) stress + longitudinal field annealing (as-quenched, AQ); (•) stress annealing (AQ); (o) stress + longitudinal field annealing (pre-annealed, PA); (+) stress + transverse field annealing (AQ); (⊕) stress + transverse field annealing (PA).

Moreover, experiments reported in [81,93] have shown that the plastic component is identical for stress and stress + longitudinal field and stress + transverse field induced

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anisotropies in CoFe-based amorphous alloys. Consequently, the stress and stress + field induced anisotropies can be considerably increased by preannealing treatments. This increase could be related with relaxation processes, reversible in nature. This conclusion seems to rule out the free volume elements origin of the anelastic component of these anisotropies proposed by Argon and Kuo [94].

4.2. Nanocrystalline Alloys

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The development of soft magnetic materials for applications requires the study of a variety of intrinsic magnetic properties as well as development of extrinsic magnetic properties through an appropriate optimization of the microstructure. As intrinsic properties we take to mean microstructure insensitive properties. Among the fundamental intrinsic properties (which depend on alloy composition and crystal structure) the saturation magnetization, Curie temperature, magnetic anisotropy and magnetostriction coefficient are all important. In a broader sense, magnetic anisotropy and magnetostriction can be considered as extrinsic in that for a two-phase material (in aggregate) they depend on the microstructure. A vast literature exists on the variation of intrinsic magnetic properties with alloy composition. Although new discoveries continue to be made on this area, it can be safely stated that a more wide open area in the development of magnetic materials for applications is the fundamental understanding and exploitation of the influence of the microstructure on the extrinsic magnetic properties. Extrinsic magnetic properties important in soft magnetic materials include the magnetic permeability and the coercivity, which typically have an inverse relationship. Thorough discussions of soft magnetic materials are developed in text such as in Refs. [95-98], and review articles as in Ref. [99]. In this section, we explore issues, which are pertinent to the general understanding of the magnetic properties of nanocrystalline materials. As the state of the art for amorphous magnetic materials is well developed and much of which has been thoroughly reviewed (see Ref. [2,7,100,101]), we will concentrate on highlights and recent studies on induced magnetic anisotropy in different families of soft nanocrystalline alloys. The development of nanocrystalline materials for soft magnetic applications is an emerging field for which we will try to offer a current perspective than may well evolve further with time.

4.2.1. Fe-Based Nanocrystalline Alloys The magnetization characteristics of Finemet-type nanocrystalline magnets (FeCuNbSiBalloy) similarly those of metallic glasses, can also be well controlled by the magnetic anisotropy induced by field annealing (FA), stress annealing (SA) and stress + field annealing (SFA). Magnetic field annealing induces uniaxial anisotropy with the easy axis parallel to the direction of the magnetic field applied during the heat treatment. The magnitude of the fieldinduced anisotropy in soft nanocrystalline alloys depends upon the annealing conditions (that is, if the magnetic field is applied during the nanocrystallization process or firstly the sample is nanocrystallized and then submitted to field annealing) [7,102] and on the alloy composition (relative percentage content of Fe and metalloids) [103]. Nevertheless, this field induced anisotropy is induced at temperature range 300-600 ºC (above the Curie temperature of the residual amorphous matrix and below of the Curie temperature of the α-Fe(Si) grains. Thus, the anisotropy induced during nanocrystallization should primary originate from the

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bcc grains. The amorphous matrix has a rather inactive part, since its Curie temperature is far below the typical field annealing temperature. The evolutions of the different types of anisotropy induced in a typical alloy susceptible of nanocrystallized, as a function of current density (thermal treatment carried out by current annealing technique under action of stress and/or field) are shown in Figure 10. As can be seen, stress and stress + field induced anisotropies increase with the current density (temperature) up to a maximum value at 45 A/mm2 which may be related to a maximum of the coercive field. The increase of induced magnetic anisotropy up to 45 A/mm2 could be ascribed to an increase of the intensity of the interactions between the metallic atoms and, consequently, an increase of the induced anisotropy could be expected. This argument is linked to the internal stress relaxation produced by thermal treatment in the metallic glasses. Similarly, field-induced anisotropy decreases with the current density down to minimum value. Such minimum is observed again at around 45 A/mm2. For current densities higher than that of the maximum, induced anisotropy monotonically decreases with current annealing density.

Reprinted from Ref. [105].

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Figure 10. Maximum magnetic anisotropies versus current annealing intensity induced in Fe73.5Cu1Nb3Si15.5B7 samples.

Studies on the stress-induced anisotropy [104-110] indicate that resembling behaviours to those in metallic glasses can also be found in nanocrystalline magnets Finemet-type. Although the occurrence of this effect has been well confirmed, nevertheless, its origin seems to be not entirely interpreted up to the present. Herzer proposed [107] an explanation, claiming that this anisotropy is of magnetoelastic nature and is created in the nanocrystallites α-Fe(Si) grains due to tensile back stresses exerted by the anelastically deformed residual amorphous matrix. The above conclusion seems to be highly probable because of a strong correlation between the stress-induced anisotropy and the magnetostriction of the nanocrystallites found by Herzer [107]. However, Hofmann and Kronmüller [108] and Lachowicz et al. [110] suggested an alternative explanation of the origin of the considered anisotropy. They adapted the Néel´s calculations of atomic pair directional ordering [83] to the conditions of the investigated material obtaining theoretical value of the energy density of the stress-induced anisotropy of the same order of magnitude as that observed experimentally.

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Consequently, besides the magnetoelastic interactions within the nanocrystallites suggested by Herzer, the directional pair ordering mechanism in α-Fe(Si) grains is also a very probable origin of the stress-induced anisotropy in Finemet-type material. The occurrence of dipolar and deteriorated exchange intergrain interaction should be also considered to explain the origin of the stress-induced anisotropy in the nanocrystalline alloys [9,53]. This leads to a more realistic situation than the simple anisotropy averaging, since those features are involved in the accomplishment of a nanocrystallization process. In this way, the procedure to obtain the weighted average anisotropy nicely proposed by Alben et al [13] strongly depends on the degree of magnetic coupling. This stress anisotropy is induced as has been noted previously inside the grains. The maximum value (around 1000 J/m3) is clearly lower than 8000 J/m3 corresponding to the magnetocrystalline anisotropy of the αFe(Si) grains and, therefore, the origin of the stress anisotropy should be strongly connected to the internal stresses in the FeSi nanocrystals. An interesting question should be that related to the coupling between these two phases with large interface area such as is the case of Ferich nanocrystalline alloys. For this, a deep knowledge about the nature of the interface results to be determinant. Unavoidable mixing of atoms of the interface gives rise to the formation of thin layers of alloys of unknown composition, which makes this study to be very complicated.

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4.2.2. Effect of the Substitution of Fe by Co or Ni Regarding the future technological applications (specially those at high frequency range) a significant efforts have been recently made on the study of the induced magnetic anisotropies in Finemet alloys by substituting Fe by Co (Co-Finemet) and/or Ni (Ni-Finemet and CoNi-Finemet). 4.2.2.1. Co-Finemet Samples XRD patterns of the as-cast and thermal treated (by different way) of a Co-Finemet sample as an example of the structural variations induced by such treatments are shown in the Figure 11. It is obvious that the treated samples, in comparison with the as-cast one, show a scattering excess which could be attributed to a crystalline peak overlapping on the amorphous halo, corresponding to the crystallization of bcc-Co phase [111]. The subtraction of the amorphous halo of the pattern of the as-cast sample allows the evaluation of the crystalline phase formed with the treatment. It was reported in [112] that sample treated by CA at 409 ºC for 120 s presented at crystallinity ratio of 17% and at 445 ºC for 450 s was 24%. For Sa treatment at 409 ºC for 120 s was 13% and SFA treated at 445 ºC for 300 s leads to a crystalline ratio of 15%. Figure 12 shows TEM bright field micrographs of the samples annealed which develop the maximum of induced magnetic anisotropy as is will presented below. It can be detected bcc-Fe(Co,Si) and bcc-Co phases and a residual amorphous phase in the SA and SFA samples, while the FA sample presented an amorphous state with some clusters dispersed randomly in the amorphous matrix. This microstructure of the FA sample is typically observed in the Finemet alloy annealed around 450 ºC (pre-primary crystallization) because of the formation of a high number of Cu-enriched clusters that serve as heterogeneous nucleation sites for bcc-Fe particles [61]. Figure 13 shows the grain size distribution obtained for the SA and SFA samples corresponding to the microstructures shown in the Figure 12. A rough fitting allows to estimate that the mean grain size of the SA sample (≈14.2±0.4 nm) is significantly larger than that of the SFA sample (≈10.1±0.2 nm). This fact could indicate that

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the primary crystallization is hindered by the nanocrystallization occurring due to the heterogeneous nucleation mechanism in the SFA sample owing to the presence of the magnetic field during the thermal treatment.

Reprinted from Ref. [113].

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Figure 11. XRD patterns in the Co-Finemet sample submitted to different treatments. AC: as-cast; CA: current annealing; SA: stress annealing; FA: field annealing and SFA: stress and field annealing.

Reprinted from Ref. [113]. Figure 12. TEM patterns and images obtained for the Co-Finemet sample annealed at 409 ºC for 120 s of SA, FA and SFA treated.

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Reprinted from Ref. [113]. Figure 13. Grain size distribution obtained in Co-Finemet samples from TEM data in SA and SFA (409 ºC, 120 s).

It must be noted that the TEM investigations, which can be considered as confirmed with XRD data, have revealed that the dimensions of the stable grains are in the nanometric scale like similar soft magnetic alloys such as Finemet or Nanoperm, widely studied, where spherical bcc-Fe nanoparticles (with dimensions between 5 and 20 nm) randomly distributed are crystallized. It seems that the diffusion field impingement limiting the bcc crystal growth

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[113,114] (effect of Nb and B atoms at the interface nanoparticle-amorphous matrix and Cu clustering) does not occur in this alloy. Figure 14 shows the hysteresis loops of the as-cast and thermal treated sample. As can be observed, the typical soft magnetic character of the as-cast sample is evidenced from its hysteresis loop (large reduced remanence and very small coercive field). The inclination of the hysteresis loop shape of the SA sample indicates that a transverse anisotropy was homogeneously developed by this kind of thermal treatment.

Reprinted from Ref. [113].

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Figure 14. Hysteresis loops of Co-Finemet sample in as-cast and treated by different treatments.

The evolution of the induced magnetic anisotropy as a function of the annealing time for respectively current annealing (CA) and field annealing (FA) Co-Finemet samples with the annealing temperature, Tann, as parameter is shown in the Figure 15. FA treatment induces longitudinal anisotropy which could simultaneously produce a magnetic softening, being the magnetization process mainly caused by a displacement of domain walls. In this case, a mechanism of atomic pair ordering could be ascribed for the origin of this field annealing anisotropy.

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80 60

273 C 378 C 409 C 445 C

40 20 0 -20 -40 -60 0

20

40

60

80

100 120

(A)

(B)

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Reprinted from Ref. [113]. Figure 15 (a) and (b). Annealing time variation of the induced magnetic anisotropy at different current densities corresponding to the expressed temperatures for the different CA (a) and FA (b) annealed CoFinemet samples.

The stress annealing treatment develops a magnetic anisotropy with the easy axis transverse to the ribbon axis; this fact implies that the magnetization process is mainly due to the magnetization rotation at large applied magnetic fields. Figure 16 presents the evolution of Kind with the annealing time for the SA samples, tann (with the annealing temperature as a parameter). The stress-induced anisotropy increases monotonically with the annealing time reaching a maximum, Kmax, after an annealing time tmax. This tmax value decreases with the annealing temperature, Tann. Such decreasing should be ascribed to the activation of mechanisms, which are in competition with the induction of magnetic anisotropy. Therefore, the maximum of the stress-induced anisotropy for each isothermal treatment can be

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considered as a “maximum value, Kmax”. It must be noted that the small values of coercivities observed for the treatment at 273 ºC, which should be assigned to the structural relaxation of the amorphous phase predominant to the nearly zero transverse-induced anisotropy at this low annealing temperature. It must also note that values of HC ≈ 4-6 A/m (annealing temperature of 445 ºC) are quite similar to those reported for Co-rich amorphous alloys relaxed by the current-annealing technique [115]. Similar remarks can be inferred from the experimental results originated by the SFA treatment (Figure 17). 400 300 200

273 C 378 C 409 C 445 C

100 0 -100 0

20

40

60

80

100 120

Reprinted from Ref. [113].

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Figure 16. Variation of the induced magnetic anisotropy with the annealing time at different current densities corresponding to the expressed temperatures for the different SA annealed Co-Finemet samples.

Reprinted from Ref. [113]. Figure 17. Variation of the induced magnetic anisotropy with the annealing time at different current densities corresponding to the expressed temperatures for the different SFA annealed Co-Finemet samples.

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The evolution of Kmax as a function of the annealing temperature for the three kind of annealing treatments is shown in the Figure 18. It can be observed that the Kmax value corresponding to SA and SFA increases with the annealing temperature up to a maximum value, which may be roughly related to a maximum of the coercive field. The increase of K at higher temperatures (up to 409 ºC) could indicate a rise in the value of the intensity of the interactions between the metallic atoms and, consequently, an increase of the induced anisotropy could be expected. This argument is linked to the internal stresses relaxation produced by thermal treatment in the metallic glasses. Similarly, Kmax, for the FA treatment decreases down to a minimum value. Such minimum is observed again at around 400 ºC. For annealing temperature higher than that of the maximum, Kmax monotonically decreases with Tann.

Reprinted from Ref. [113].

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Figure 18. Variation of the maximum induced anisotropy constant (Kmax) versus the annealing temperatures of the CA, FA, SA and SFA s Co-Finemet amples during 2 min.

On the other hand, the importance of the two-phase system in the nanocrystalline state has been already discussed in detail for the Finemet alloy [76,115,116]. As has been mentioned in two-phase system there is a ferromagnetic coupling, which is responsible for the soft magnetic properties, between the amorphous and nanocrystalline phases. This coupling depends on the size of the crystallites and specially on the induced magnetic anisotropy. Finally, the stress annealing in Finemet alloy causes the distance of the Fe atoms to be elongated along the direction parallel to the tensile stress [117]. Consequently, the magnetic anisotropy induced by stress annealing will appear and the domain wall will array perpendicular to this elongated direction [107,109,117-119]. In addition, the structural origin of this anisotropy is mainly in the crystalline phase [α-Fe(Co,Si) and α-Co phases] and not in the amorphous phase [1191 In fact, Ohnuma et al. [118] obtained values of the elongation parallel to the tensile stress which agree with expected values based on the magnetoelastic and pair ordering models [107,118].

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4.2.2.2. Ni-Finemet Samples From the XRD measurements in the CA and SA Fe73.5-xNixCu1Nb3Si15.5B7 (Ni-Finemet with x = 5, 10 and 20) samples, the fractional volume of crystalline phase and the average grain size of the crystals. Figure 19 (a) and (b) shows the fractional crystalline volume of the CA and SA samples, respectively, as a function of the annealing time in the different NiFinemet samples. For short time (2 minutes) it was not observed crystallites in the CA samples while the crystallization seems to starts at very beginning time for SA samples indicating that the presence of tensile stress promotes the nanocrystallization process in similar way to that observed in Co-Finemet samples [112]. As it is expected, the fraction crystalline volume increases with the annealing time being this increase more significant as the Ni content increases. It must be noted that the crystallization volume fraction developed with these treatments are clearly lower than that induced by conventional annealing (using a furnace) of around 70-80% with annealing parameters of 550 ºC, 1 hour [3]. Therefore, it seems that the current annealing technique can be evaluated not allow, probably by kinetic considerations associated to the proper current annealing, the induction of a large nanocrystalline volume fraction as comparing with the nanocrystalline fraction of the classical Finemet alloy treated by conventional annealing at 550 ºC, 1 hour. This is certainly a disadvantage, however, it is well known that the current annealing procedure results to be more quite convenient by its high heating rate and the less complexity to avoid the use of a furnace. Moreover, the degree of magnetic softness which can be achieved in nanocrystalline alloys treated by current annealing is quite similar to that exhibited by the same nanocrystalline alloy treated by conventional annealing to produce the nanocrystallization process [11].

Figure 19. (Continued) .

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Figure 19. Variation of the crystallinity amount (χ) with the treatment time for the different Ni-Finemet samples treated without (a) and with (b) stress.

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Figure 20 (a) and (b) shows the evolution of the average grain size of the CA and SA samples, respectively. The average grain size, D, increases with the annealing time up to saturate (Dsat) around 10, 15 and 20 nm for x = 5, 10 and 20 respectively. This Dsat is the same value for the CA and SA samples, although the kinetic of segregation of such brains is different as can be observed in Figures 21 (a)-(c) where the variation of the average grain size with the two treatments in each composition are shown. From this Figure 4 is observed that in the three compositions the saturation value of D with the annealing time is reached before in SA than in CA samples. In addition, according our work and that previous of X-ray and transmission electron microscopy (TEM) analysis [120,121], these nanocrystalline NiFinemet alloys contain supersaturated phases α–Fe(Si), Fe3Si; Ni-rich and Fe3NiSi1.5 which play a very important in role in the magnetostrictive behaviour. Obviously, the segregation of Ni-rich and Fe3NiSi1.5 is more significant as increasing the Ni content of the composition.

Figure 20. (Continued)

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Figure 20. Variation of the grain size (D) with the treatment time for the different Ni-Finemet samples treated without (a) and with (b) stress. In the case (a), the short time treatments are shown in the inset.

Figure 21. Variation of the coercive field (Hc) with the treatment time for the different Ni-Finemet samples treated without stress. The short time treatments are shown in the inset.

TEM measurements were carried out on the samples in order to check the later XRD conclusions (see some examples in the Figure 22). The results were consistent with the XRD ones. The first image corresponding to the as-cast (x = 5) sample clearly denotes the non-

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crystalline structure of this sample as XRD results showed. The next images corresponding to CA treatments at different times show a grain structure with sizes increasing with the time of treatment: 8 nm, 10 nm and 11 nm for, respectively, 5 min., 30 min. and 60 min. treatment. These results are in agreement with those obtained ones by XRD measurements and reinforce the above pointed conclusions.

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Figure 22.- TEM images obtained in Fe68.5Ni5Cu1Nb3Si15.5B7 alloy for as-cast (a) and CA samples with 5 min. (b), 30 min. (c) and 60 min. (d) treatment.

AFM images of the x = 5 sample treated by CA and SA at 40 A/mm2 are presented in the Figure 23 (a) and (b) respectively. It can be inferred that the SA treatment induces like a texture with a inclination of around 20 degree with respect to the ribbon axis, which could be connected with the uniaxial macroscopic magnetic anisotropy induces by means of this kind of thermal treatment [116]. To notice the structure of grains (like-flower) observed in both CA and SA samples with a large nucleus, probably of ferromagnetic character (α-Fe,Ni), surrounding by several small grains likely of Nb or Cu.

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0

Data type Z range

Weight 46.05 nm

2.00 mm 0

Data type Z range

Phase 133.7

2.00 mm

n151-7a2min.m02

(A)

0

Data type Z range

Weight 51.15 nm

1.00 mm 0

Data type Z range

Phase 125.5

1.00 mm

n151-7a0-5kg2min.m03

(B)

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Figure 23.- AFM images obtained in Fe68.5Ni5Cu1Nb3Si15.5B7 alloy for (a) CA (x = 5) sample (2 x 2 μm) and (b) SA sample (1 x 1 μm) treated during 2 minutes.

5. MAGNETOSTRICTION 5.1. Amorphous Materials As has been mentioned, magnetic anisotropy is macroscopically negligible, its stress derivative defines the magnetostriction coefficient, which depends on alloy composition. Saturation magnetostriction constant of Fe-base alloys exhibit large and positive values, while Co-rich alloys have negative magnetostriction Vanishing magnetostriction is found for those alloys containing a ratio of Fe to Co content of about 5%. The microscopic origin of magnetostriction is the same as that of magnetic anisotropy and it can be understood by studying its temperature dependence. From such a study, it can be determined that single-ion

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anisotropy is the responsible mechanism for the magnetostriction of Fe-rich alloys. Deviations from this mechanism are found in Co-rich alloys in which a significant component corresponding to two-ion mechanism is presumed. In fact, nearly-zero magnetostriction Corich alloys can exhibit a change of sign of magnetostriction well below of Curie temperature [122,123].

5.1.1. Thermal Dependence of Saturation Magnetostriction Magnetostriction constant depends first of composition such as has been mentioned. Magnetostriction constant also depends on temperature, although experimental study of such a dependence is strongly affected by structural relaxation contribution in the case of nearlyzero magnetostrictive compounds (Co-rich containing small amount of Fe) [124-126]. Figure 24 (a-c) presents the temperature dependence of a Fe-rich (a), CoFeSiB (b) and Co-rich (c) alloys exhibiting positive, nearly-zero and negative magnetostriction, λs, (at room temperature) respectively.

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(A)

(B) Figure 24. (Continued)

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100

200

300

-2.5

-5

-7.5

-10

(C) Figure 24. Thermal dependence of the saturation magnetostriction, λs, of (a) Fe73.5Cu1Nb3Si13.5B9 (Ferich) positive magnetostriction; (b) Co71.1Fe4.9Si12B12 nearly-zeo magnetostriction and (c) Co75Si15B10 (o) and Co45Ni30Si15B10 (•) negative magnetostriction samples.

These thermal dependencies of λs has been successfully analysed in terms of single-ion and two-ion contributions according to the expression [127,128]: λs (T) = λ1 Î5/2 [L-1 (m)] + λ2 m2

(12)

where Î5/2 is modified hyperbolic Bessel function of the inverse Langevin function, L-1(m) [129], and m is the reduced magnetization M(T)/M(0). The function Î5/2 ≈ m3 at lower temperatures and ≈ (3/5)m2 near Curie temperature. For checking the equation (12) it can plotted y = λs/Î5/2[L-1 (m)] against x = m2/Î5/2[L1 (m)] where x will vary from unity at 0 K up to 5/3 at Curie temperature [127]. As a consequence, a linear dependence can be assumed, namely:

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y = λ1 + λ2 x

(13)

where λ1 and λ2 are, respectively, the one-ion and two-ion contributions to the magnetostriction λs at 0 K. The data taken for Figure 24 a for the Fe-rich alloy [104] are analysed in Figure 25 in terms of y(x). A single-ion contribution is inferred from the horizontal part which corresponds to the range of low temperature with x < 1.3, i.e., T < 350 K. But for x > 1.3 a strong deviation from the single-ion behaviour can be observed. Therefore, the single-ion model proposed for iron-rich amorphous alloys seems to be valid at low temperatures in agreement with earlier studies reported on these kinds of alloys. The disagreement of this model above 350 K could be related to a two-ion contribution.

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Reported from Ref. [104]. Figure 25. Analysis of the temperature dependence of the saturation magnetostriction according to the equation (13) in the positive magnetostrictive amorphous sample.

According the experimental data of λs(T) in the CoFeSiB amorphous alloy [126], the dependence of y is parabolic (see Figure 26). Within the experimental uncertainties, dy/dx is null at 0 K, this means that below 200 K the single-ion contribution dominates the temperature dependence of λs. Above this temperature, the curvature can not arise from a two-ion coupling which can not be deduced below 200 K. This curvature could be assigned to a temperature dependence of λ1 as a results of a competition between the negative one-ion contribution of Co-atoms (λ1Co = -4x10-6) and the positive of one-ion contribution of Featoms (λ1Fe = + 41x10-6). When the sample is heated, some reversible structural changes occur inducing a very small positive change for λ1Fe and/or λ1Co thus giving a much more important relative change for:

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λ1 = λ1Co + λ1Fe

(14)

therefore, these zero magnetostriction alloys are in fact very sensitive probe for any slight structural changes. Data of λs(T) for Co-rich negative magnetostriction of Figure 27 [130] are analysed also according to the equation (12). For the sample CoSiB such data are perfectly consistent at low temperature range reported in [130] with those at high temperature data (T > 300 K) from [131]. Since λ2 = 0, they prove that the single-ion model is relevant for describing the magnetostriction of a pure Co-metalloid amorphous alloy, as was the case of Fe-metalloid alloy above discussed. It must be mentioned that the idea of a noticeable two-ion contribution to λs in Co-metalloid alloys was arising from some papers [104,129-132].

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Reported from Ref. [127].

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Figure 26. Analysis of the temperature dependence of the saturation magnetostriction according to the equation (13) in the nearly-zero magnetostrictive amorphous sample.

Reported from Ref. [131]. Figure 27. Analysis of the temperature dependence of the saturation magnetostriction according to the equation (13) in the negative magnetostrictive amorphous samples.

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5.1.2. Stress Dependence of Magnetostriction in Nearly-Zero Magnetostrictive Alloys Since 1986 it was well established that saturation magnetostriction in metallic glasses depends on the applied tensile stress, σ. This phenomenon was observed in very-low magnetostrictive metallic glasses. This dependence, λs(σ), was experimentally found by different authors [133-136] to be of the form: λs(σ) = λ(0) + Aσ

(15)

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where λ(0) is the saturation magnetostriction constant when the applied stress σ is zero, and A is negative coefficient which ranges from -6 to -1 x 10-10 MPa-1. Figure 28a and b presents results given by Blanco et al. [135], as an example of the λs(σ) of nearly-zero magnetostrictive amorphous alloys, on the variations of λ(0) and A coefficient, respectively, of the (Co0.95Fe0.05)72.5Si12.5B15 amorphous wire (thermally treated by current annealing) as a function of the current density. Regarding the evolution of λs(0) (Figure 22a) results to be similar to that reported on thermal dependence of nearly-zero magnetostriction amorphous alloys [136]. This λs(0) behaviour could accounted by considering the coexistence of two magnetostrictive phases of different sign and strength [137,138].

(A) Figure 28. (Continued)

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(B) Data reported from Ref. [139].

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Figure 28. Evolutions of the coefficients λs(0) (a) and A (b) as a function of the annealing parameters of the Co0.94Fe0.06)72.5B15Si12.5 amorphous wire treated by current annealing.

Different microscopic mechanism have been invoked to explain the origin of the parameter A. Fähnle et al. [136] correlates A with the second strain derivative of the local anisotropy coefficients which are determined by the local symmetry and chemistry. Therefore, the thermal treatments giving rise irreversible phase transformations in the local symmetry of the amorphous structure are expected to affect the A value. It is to be noted that different experimental hints of such type of transformations have been observed for Co-rich metallic glasses [139]. The model developed by Szymczak [133] describes the stress dependence λs of as a consequence of the bond orientational anisotropy induced by the stress. Therefore, in the framework of Szymczak´s model, thermally activated processes must be invoked in the mechanism giving rise to the stress dependence of λs. Moreover, it is expected that the action of the tensile stress should be drastically affected by the strength and orientation of any bond anisotropy induced previously at higher temperatures. Hernando et al. [135,137] have analysed the λs(σ) behaviour in amorphous ribbons taking into account the fluctuations of the local anisotropy and, therefore, the local magnetostriction. When the local magnetostriction fluctuates with a correlation length larger than the exchange correlation length, λs varies with the applied stress. Moreover, in this last model, the influence of the thermal treatments on the local magnetostriction fluctuations should be reflected in a similar influence on the coefficient A.

5.2. Nanocrystalline Materials The evolution of the saturation magnetostriction constant, λS, as a function of the thermal treatments in Co-Finemet is presented in Figure 29. Small values of λS (around 2x10-6 – 6x10-

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6

) are expected for this amorphous alloys and such evolution should be explained taking into account the coexistence of two magnetostrictive phases of different signs with a relative maximum around 420 ºC (except for the CA sample). The changes of λS below such temperature could be ascribed to the relaxation process, while around 420 ºC it seems to indicate the first steps of the beginning of the crystallization process.

Reprinted from Ref. [113].

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Figure 29. Evolution of the saturation magnetostriction constant (λ) versus the annealing temperatures of the CA, FA, SA and SFA in Co-Finemet samples during 2 min.

For an annealing temperature above 420 ºC, the presence of a small amount of crystal grains could be considered to explain the magnetostrictive behaviour. The low effective magnetostriction in the nanocrystalline sample (treated at a high temperature of 445 ºC) could be interpreted by considering contributions coming from the nanograins, residual amorphous matrix and a surface (nanograin-amorphous matrix) such as will be discussed below. Experimental results of λS in the CA samples (Ni-Finemet) are presented as a function of the annealing time in the Figure 30. As can be seen λS decreases with the Ni-content in the asquenched and treated samples with the annealing time. This should be attributed to transformation of amorphous into nanocrystalline phase and the resulting λS is affected by the magnetostriction of forming nanocrystalline grains with different chemical composition. Thus, λS increases slightly for short annealing time decreases monotonously with the annealing time for the x = 5 sample while for the x = 10 and 20 samples λS increases for short annealing time (connected to a structural relaxation [140]) and then decreases for long annealing time when the nanocrystallization occurs such as has been observed in the classical Finemet alloy. Therefore, in the course of relaxation λS increases moderately and then decreases in the initial stages of crystallization where transformations quenched-in cluster structures into phases with lower Si content (α-Fe and α–Fe(Si)) takes place [11,61]. As a consequence, the initial increases of λS observed in the x=5 sample (similarly to that previously observed in x=0) could indicate that theactivation of clusters with short-range ordering and composition probably rich in metalloid atoms. For long annealing time it is

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reasonable to assume a high concentration of Si in the bcc-Fe or even the saturated Fe3Si phase in both cases with negative saturation magnetostriction, which leads to decrease of effective magnetostriction (λSeff).

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Figure 30. Evolution of the effective saturation magnetostriction constant (λSeff) with the annealing time in the CA Ni-Finemet alloys.

As has been mentioned, the primary crystallization in these Fe-rich samples gives rise a single phase α-Fe(Si), Fe3Si Ni-rich and α-Fe3NiSi1.5 crystallites embedded in the amorphous residual matrix depleted in Fe content. According to ref. [6] the experimental data of effective saturation magnetostriction (λS) of the nanocrystalline alloy, of the α-Fe(Si) (λScr) [141] is negative depending of the Si atoms. Negative character of the saturation magnetostriction is reported in ref. [121,142] for Fe3Si and Ni-rich grains of -9x10-6 and -30x10-6 respectively, while Fe3NiSi1.5 is non-magnetostrictive. It must be noted that the residual amorphous alloy (λSam) can be assumed of positive sign. Therefore, the crystalline grains and the residual amorphous matrix present saturation magnetostriction with different sign for each one of these contributions negative for the nanocrystals and positive for the amorphous matrix. In this way, we have estimated the term corresponding to the residual amorphous matrix in the equation (1) as a function of the partial crystalline fraction (Figure 31). The decrease of λSam in the x = 5 sample with partial crystalline volume seems to be in contradiction with the expected increase due probably to the residual amorphous matrix in unpoored in Si. In the case of the x = 10 and 20 samples the initial increase of λSam could be connected with possible invar effect [143],.because the segregation of Ni-rich grains with quite negative large value of magnetostriction (-30x10-6) [142] implies large positive magnetostriction of the residual amorphous matrix in accordance with the effective magnetostriction values measured in these nanocrystalline alloys.

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Figure 31. Evolution of the saturation magnetostriction constant of the residual amorphous matrix (λSam) with the crystalline volume fraction in the CA Ni-Finemet alloys according to the equation (16).

The effective magnetostriction, λseff, observed in nanocrystalline alloys at different stages of crystallization has been interpreted as a volumetrically weighted balance among two contributions with opposite sign originating from the bcc-FeSi grains (λscr) and residual amorphous matrix (λsam) according to [144]:

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λseff = λscr + (1 - p)λsam

(16)

where p is the volumetric fraction of the crystalline phase. Therefore, assuming negative and positive sign for the nanocrystalline and amorphous phase respectively, the variations of λseff (including the change of sign observed in some nanocrystalline composition) can be interpreted as a consequence of the variations of p parameter. Although this simple approximation gives the qualitatively explanation of the effective magnetostriction in Febased nanocrystalline alloys [144], nevertheless, it does not consider many effects occurring in the real materials. More exact calculations take into account that the magnetostriction of the residual amorphous phase is not constant but depends on the crystalline fraction due to the enrichment with B and Nb [5,145]. Consequently, new efforts have been done, recently, to explain these discrepancies. For this, an additional contribution arising from the boundary grains has been taking into account for the effective magnetostriction an additional contribution similarly to that reported in [146,147]. In this way, the equation (16) can be modified in the form [146]: λseff = λscr + (1 - p) (λsam + kp)

(17)

where k is a parameter which expresses changes of the magnetostriction in the residual amorphous phase with evolution of the crystallization. In many cases even this model does not fit the experimental results demonstrating that the effective magnetostriction in nanocrystalline material can not be described by a simple superposition of the crystalline and

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amorphous components [5]. In the case of FeZrBCu nanocrystalline system in which the bccFe phase is formed, the model described does not fit the experimental data even through the λsam(p) dependence as was shown by Slawska-Waniewska and Zuberek in [146,147]. They considered an additional contribution to the effective magnetostriction, which arises from the enhanced surface to volume ratio describing interfacial effects [146,147]. Therefore, the equation (17) of the effective magnetostriction could be approximated by: λseff = p λscr + (1 - p) (λsam + kp) + pλss S/V

(18)

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where the last term describes the effects at the interfaces and depends on the surface to volume ratio S/V for the individual grain, as well as on the magnetostriction constant λss which characterizes the crystal-amorphous interface. Equation (18) is the basic dependence which can be used to interpret the experimental data on the effective magnetostriction in Fe-based nanocrystalline alloys at different stage of crystallization. The composition of the Fe(Si) grains (depending on the annealing temperature) should be considered giving rise to different values of the magnetostriction constant for the crystalline phase. The appropriate values of λscr can be obtained from the compositional dependence of the saturation magnetostriction in polycrystalline α-Fe100-xSix shown in Figure 32 [144,148]. Thus, the first term in the equation (18) can be treated as the well defined one.

Figure 32. Saturation magnetostriction of the polycrystalline α−Fe100-xSix.

Figure 33 presents the crystallization behavior and accompanying changes in the magnetostriction of classical Finemet (Fe73.5Cu1Nb3Si13.5B9) alloy published by Gutierrez et al. [149]. The analysis of these data according the equation (17) allows (i) estimation of the contribution from the crystalline phase (see Figure 33a where the values of λscr were found from the combined Figures 28 and 29a), and then (ii) fitting of the experimental (λseff - pλscr) on p dependence to the equation (18). The results, both experimental points and the fitted curve (solid line) are shown in Figure 33b.

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Figure 33. Si content in α−Fe(Si) grains (a) and magnetostriction of the Fe73.5Cu1Nb3Si13.5B9 alloy (b) versus crystalline fraction.

Assuming spherical α-Fe(Si) grains, with radius R, the last term of the equation (16) can be expressed as 3λss/R, and the magnetostriction constant which describes the interface λss can be estimated. For the soft nanocrystalline alloys (Finemet and FeZrBCu alloys) [5,144,146], R = 5 nm and thus λss has been found to vary in the range 4.5 - 7.1 x 10-6 nm. These values result to be one order of magnitude smaller than values of the surface magnetostriction obtained in multilayer systems. However, investigations of Fe/C multilayers have shown that not only the value but also the sign of the surface magnetostriction constant depends on the structure of the iron layer and it has been found that for crystalline iron λss (bcc-Fe) = 45.7 x 10-6 nm, whereas for the amorphous iron λss (am-Fe) = -31 x 10-6 nm [150]. Thus, the value of the interface magnetostriction obtained in the nanocrystalline systems is within the range of the surface magnetostriction constant estimated for thin iron layers. It should be noted that, contrary to Fe/c multilayers, in the nanocrystalline materials, both the crystalline and amorphous phases are magnetic and they are coupled through exchange and dipolar interactions. It must thus be expected that the magnetic interactions in the system can affect the magnetoelastic coupling constant at the grain-matrix interfaces. In addition, the structure and properties of the particular surfaces being in contact as well as local strains at the grain boundaries should also be considered. The problem of the surface/interface magnetostriction requires however further studies which in particular should include measurements at temperatures above Curie point of the amorphous matrix where only ferromagnetic grains should contribute to the effective magnetostriction simplifying a separation between bulk and surface contributions.

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6. CONCLUSION Nanocrystalline materials, which have clearly found numerous applications, include soft magnetic alloys and recording media. The other classes of materials apparently still need further investigation in order to improve their processability, performance and/or cost. The typical demands of the market for a new product are defined by the customer and comprise superior performance, smaller size, lower weight and/or lower price than existing solutions. The larger the number of these criteria that are fulfilled simultaneously, the better the chances for success. It is remarkable the fact that for grain sizes D < 15 nm, the average random anisotropy in nanocrystalline Fe-based alloys is largely averaged out and the soft magnetic properties are controlled by uniaxial anisotropies which are uniform on a scale much larger than the exchange length. The most relevant contributions are magneto-elastic anisotropies and most noticeable, field and/or stress induced anisotropies with a still competition between the random and uniform anisotropy contributions. On the other hand, it was outlined in ref. [151] that the economic demands do not necessarily preclude good science and indeed can provide opportunities for new and interesting scientific challenges. Vice versa, history shows that scientific curiosity can equally lead to new products, which enrich and render more comfortable the human condition. Finally, the introduction of new nanocrystalline materials may also be driven by better cooperation between industry and university laboratories. However, a standardized link for improved cooperation does not yet exist in most countries.

ACKNOWLEDGMENTS The authors are deeply acknowledged to the Industry Department of the Basque Government and Diputación Foral de Guipúzcoa for the financial support (Project NANOTRON) and the Spanish Ministry of Science and Technology (grant MAT2007-66897CO3).

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[149] Gutierrez, G.; Gorria, P.; Barandiarán, J.M.; and García-Arribas, A. in "Nanostructured and Non-Crystalline Materials", Eds. Hernando, A.; Vázquez, M. World Sci.Publ., Singapore, 1995. [150] Zuberek, R.; Szymczak, H.; Krishnan, R.; Sela, C.; and Kaabouchi, M. J. Magn. Magn. Mat. 1993, 121, 510-512. [151] Herzer, G.; Vázquez, M.; Knobel, M.; Zhukov, A.; Reininger, T.; Davies, H.A.; Grössinger, R.; and Sánchez-Llamazares, J.L. J. Magn. Magn. Mat. 2005, 294, 252266.

In: Amorphous Materials: Research, Technology… Editors: J. R. Telle, N. A. Pearlstine

ISBN: 978-1-60692-235-4 © 2009 Nova Science Publishers, Inc.

Chapter 3

RELATIONSHIP OF FRAGILITY AND DILATATION WITH GLASS-FORMING ABILITY OF PR-BASED BULK METALLIC GLASSES Qingge Meng,1,2∗ Shuguang Zhang1, Mingxu Xia1, Jianguo Li1 and Xiufang Bian3 1 2

Shanghai Jiao Tong University, Shanghai 200240, China Baoshan Iron and Steel Co., Ltd, Shanghai 201900, China 3 Shandong University, Jinan 250061, China

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ABSTRACT Bulk metallic glasses (BMGs) have enabled the measurements of various physical as well as mechanical properties that were often impossible. There have been many studies focusing on the Angell’s fragility parameter m [J. Non-Cryst. Solids 131, 13 (1991)] and dilatation of BMGs. However, the fragility parameter m in undercooled melts can not be used to predict the glass-forming ability (GFA) since the fragility parameter m can only be gotten after the glass has been obtained. To predict GFA, in particular, of marginal metallic glasses, a fragility parameter, M, for superheated liquid was empirically proposed by Bian [Phys. Lett. A 335, 61-67(2005)]. On the other hand, thermal expansion is one of the most important thermodynamic characteristics and is strongly correlated to the amorphous structures of metallic glasses. Nonetheless, few research works has paid attention to the relationship between the dilatation characteristics and GFA. Therefore, it is of particular interest to deduce the expression of the parameter M, and to study the correlation of the both fragility parameters, m and M, and dilatation behavior with GFA of BMGs. The results on this topic particularly for Pr-based BMGs will be discussed and summarized in this article. By measuring the viscosity and specific heat capacity of the supercooled liquid, it was found that the Pr55Ni25Al20 alloy behaved much closer to strong glasses than the other reported metallic glasses, and the GFA is correlated more strongly with the liquid/crystal Gibbs free energy difference than with the supercooled melt fragility parameter m. To ∗

Corresponding author. Tel.: +86 21 2664 9170; e-mail: [email protected], [email protected]

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Qingge Meng, Shuguang Zhang, Mingxu Xia et al. improve the predictive power for GFA, in particular, of marginal metallic glasses with the parameter M, the refinement to this parameter was made from both the thermodynamic and kinetic perspectives. The refined parameter gives a much better reflection of GFA for metallic glasses. A correlation between αaver and the weighted average of the thermal expansion coefficients αi for the constituent elements was found as αaver=∑fi αi for Pr-based BMGs. By assuming Lennard-Jones (LJ) type potentials, the average nearest-neighbor distances r1 and the depths of the effective pair potentials V0 were calculated. The values of r1 were in good accordance with the experimental results and V0 correlated with the GFA. The above mentioned results will help to gain deep insight into the glass formation of metallic glasses.

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1. INTRODUCTION The discovery of the non-precious metal-based alloy systems for bulk metallic glasses (BMGs) dates back to the end 1980’s. Since then, numerous efforts have been devoted to the development of more metallic glass formers and making the maximum critical size as big as possible for practical applications [1-4]. The first crucial problem that a researcher often has to face is the glass-forming ability (GFA) of an alloy. In the past, a few empirical parameters has been found to have some correlations with the GFA, such as the quantity of negative heat of mixing ΔHmix, the degree of atomic size mismatch Sσ [5], a reduced glass transition temperature Trg (= Tg/Tl, where Tg is the glass transition temperature, Tl is the liquidus temperature) [6], a supercooled liquid range ΔTxg (= Tx -Tg, where Tx is the onset crystallization temperature) [7], and γ =Tx/(Tg + Tl) [8]. Among them, γ has a much better interrelationship with GFA than Trg and ΔTxg [8]. However, one of the biggest obstacles of making the best use of these metallic glasses is their low GFA, and the nature of the GFA still remains unclear. It is thus of great importance to understand the nature of GFA so as to explore the applicable BMGs. Generally, viscosity can be considered as a key parameter to describe the kinetic slowdown when a melt is undercooled below its liquidus temperature, and it is also an important property in practical and theoretical terms with regards to materials analysis and behavior [9-12]. The concept of the fragility of a liquid proposed by Angell [13] has been widely studied and used to characterize and classify the dynamic behavior of undercooled liquids and glasses. The fragility parameter, m, is defined as the ratio of the dependence of viscosity (logη) on temperature (Tg/T) at the glass transition point Tg. Interestingly, it was found that there are close correlations between m and the glass properties such as the vibration [14], Poisson’s ratio [15] etc., and even there might be a roughly negative correlation between the GFA and the fragility parameter m on the basis of experimental studies on some metallic glass formers [16-18]. Busch et al. investigated the thermodynamics and kinetics of the supercooled liquids of the Zr41.2Ti13.8Cu12.5Ni10.0Be22.5, Zr46.75Ti8.25Cu7.5Ni10.0Be27.5 [19, 20] and Mg65Cu25Y10 [21] BMGs, and revealed the strong liquid behaviors of the BMGs. Nevertheless, it was also reported that the kinetic parameter m cannot reflect the GFA effectively in some other metallic glass systems, e.g., Inoue et al. found that the supercooled liquid of the metal-metalloid type glass Pd40Ni40P20 [22] appears to be more fragile than that of the metal–metal type glass La55Ni25Al20 [23] by using the continuous-strain-rate tensile tests, and moreover it is not useful for predicting the GFA since the m can be obtained only after the amorphous sample has been successfully prepared. It is

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not difficult to get m for oxide glasses and polymers, but for metallic system it is often quite difficult due to their comparatively low GFA. Therefore, Bian et al.[24] studied the viscosity of superheated melts and empirically proposed a parameter M, the fragility of superheated melt, and found that M is closely related to the GFA in Al-Co-Ce amorphous alloys. But the physical meaning and mathematic expression of the M and whether or not the negative relationship between M and γ can be applied to more metallic glass systems still needs to be studied. Further, thermal expansion is another important thermodynamic characteristic and is strongly correlated to the amorphous structures of metallic glasses [25-27]. Upon solidification of a supercooled liquid, some excess free volume ΔVf is trapped in the glassy state, the quantity of which will be greatly influenced by the cooling rate. With the annihilation of the free volume during subsequent annealing, the structural relaxation and crystallization of metallic glasses affects the thermal expansion coefficient α and many other related physical quantities such as volume density, elastic modulus, electric resistivity, special heat and magnetic properties [28]. In the classical solid theory, the Young’s modulus is related to the characteristics of the potential curve close to the bottom, while the dilatation property of solid is attributed to the asymmetry of the potential curve. Both properties are closely associated with the binding strength and the stability of solids [29]. Porshcha and Neuhauser [30] have studied the length and modulus changes with temperature for a Cu64Ti36 metallic glass ribbon and calculated the parameters of the effective pair potential for the glass. Chua [31] et al. measured the specific volumes of Pd-based glass-forming alloys as a function of temperature and found a rough correlation between the GFA and the scaled parameter α/Tg. Therefore, it is particularly interesting to investigate the relationship between the thermal expansion coefficient and the thermal stability as well as GFA. In this article, we briefly summarize our recent investigation in Pr-based BMGs regarding to the physical properties of viscosity, dilatation and GFA. The relationship of fragility and dilatation with GFA is established.

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2. STRONG LIQUID BEHAVIOR OF PR55NI25AL20 BULK METALLIC GLASS A series of Pr-Ni-Al BMGs were prepared by a copper mold casting method. The effect of the alloy composition on the GFA was studied. The simple ternary alloy Pr55Ni25Al20, which shows a large supercooled region and GFA, is chosen as a representative to study the dynamic and thermodynamic characteristics of Pr-based alloys. The viscosities of the supercooled and superheated liquids are measured by the threepoint beam bending and oscillating-vessel methods, respectively. Differential scanning calorimetry (DSC) is used to measure the specific heat capacities and the heats of glass transition and melting so as to determine the thermodynamic properties of the supercooled liquid and the crystal.

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Figure 1. DSC traces of Pr55Ni25Al20 bulk metallic glass at a heating rate of 0.167 K/s indicating the crystallization with the heat release, ΔHx, and subsequent melting with the heat of fusion, ΔHf.

Figure 2. Measured viscosity and DSC traces of Pr55Ni25Al20 bulk metallic glass through the glass transition and the supercooled liquid regions at a heating rate of 0.167 K/s.

2.1. The Viscosity Behavior of the Pr55Ni25Al20 Bulk Metallic Glass Figure 1 shows the DSC trace of Pr55Ni25Al20 bulk metallic glass at a heating rate of 0.167 K/s, exhibiting the crystallization with the heat release, ΔHx =7.52 kJ/g-atom, and subsequent melting with the heat of fusion, ΔHf =9.24 kJ/g-atom. The glass transition

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temperature Tg, the onset temperature of crystallization Tx, the onset temperature of melting Tm and the liquidus temperature Tl are determined to be 480.4, 541.5, 751.0 and 817.4 K, respectively. Thus the GFA indicator γ is calculated to be 0.417, indicating a high GFA. The viscosity measurements for the supercooled liquid state of Pr55Ni25Al20 alloy were performed with constant heating rate as the relaxation time in the supercooled liquid region is much shorter than the time scale allowed by the heating rate for most BMGs [32]. The typical viscosity data measured through the supercooled liquid region together with the relative DSC trace is shown in figure 2. This particular measurement was done at a heating rate of 0.167 K/s under a load of 0.98 N. The viscosity data in the amorphous zone appears somewhat fluctuated. It may be attributed to the difficulty in obtaining the continuous equilibrium viscosity in this zone at the heating rate of 0.167K/s, as the relaxation time scale for the solid state of the glass is much longer than that for the supercooled liquid. From the temperature 480.4K (Tg) to 530 K (near Tx), the measured viscosity decreases from 2.260×1012 Pa⋅s to 1.747×1010 Pa⋅s. At the temperature Tg*, 494K, which is near the starting point of the supercooled liquid region or the offset temperature of the glass transition, the measured viscosity decreases to be about 1.0×1012 Pa.s, below which, the supercooled liquid may be considered to be quenched into a glass. Figure 3 summarizes the viscosity results obtained from the high temperature viscometer as well as from the beam bending rheometry, showing comparative values with the viscosity of other reported Pr-based superheated liquid [12] and La-based supercooled liquid [22], respectively. The viscosity for the superheated melt with temperature changes much slower than that for the supercooled liquid. It is well known that the equilibrium viscosity can be well reproduced with the V-F-T relation [33-35]:

η = η 0 exp[ DT0 /(T − T0 )] ,

(1)

where T0 is the V-F-T temperature that is usually below Tg and D is called the fragility parameter, In real liquids D ranges from 3-5 (fragile liquid such as a pure metal) to 30-40 (strong liquid such as Silica). BMG-forming liquids tend to exhibit “strong” liquid behavior with D typically of 15-25[36]. The pre-factor, η 0 = h /ν m in Eq. (1), with h the Planck’s

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constant, is calculated according to Eyring’s theory [37] to be 3× 10-5 Pa.s for the Pr55Ni25Al20 alloy. Based on the free volume model and the percolation theory, Cohen and Grest [38] derived a fitting equation for viscosity over wide temperature ranges with a form:

{

logη = A + 2 B / T − T0 + [(T − T0 ) 2 + ET ]1 / 2

}

(2)

where A, B and E are fitting parameters, T0 is a characteristic temperature usually higher than Tg. We made a try to fit the viscosity data of the supercooled liquid and the melt by these two relations. The best fits yielded η0 =3.0× 10-5 Pa.s, D = 25.5, T0 = 296.3 K for the V-F-T relation, and A= -2.93, B = 69.50 K, E = 13.4 K and T0=845.8 K for the C-G relation. The fitting results are represented in figure 3. It is clear that the V-F-T equation was incompetent for describing the viscosity data over the whole temperature interval ranging from the supercooled liquid regime to the superheated melt regime. On the contrary, the C-G equation does work well, which was similar to the result of La55Ni25Al20 BMG [39].

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Figure 3. Viscosities of Pr55Ni25Al20 bulk metallic glass as measured by beam bending (○) and oscillating vessel (◆) methods. The best fits to the experimental data by the V-F-T (Eq. (1)) and C-G (Eq.(2)) relations are given.

Figure 4. Angell plot of the viscosities of Pr55Ni25Al20 and Zr41.2Ti13.8Cu10.0Ni12.5Be22.5[13], Zr46.75Ti8.25 Cu7.5Ni10Be27.5 [14], Mg65Cu25Y10[15], La55Ni25Al20[16], Pd40Ni40P20 [17] bulk metallic glasses in comparison with the strong extreme liquid like SiO2 and fragile extreme liquid like O-terphenyl. The temperatures are normalized by the reference glass transition temperature Tg, * showing a viscosity of 1×1012 Pa s.

The fragility parameter of Pr55Ni25Al20 BMG was calculated based on the measured viscosity data shown in figure 3. It is a classification method according to the different descriptions of the temperature dependence of the viscosity: a strong liquid is characterized by an Arrhenius-type behavior (e.g. SiO2), while the viscosity for a fragile one is highly nonArrhenius (e.g. O-terphenyl). In order to compare the measured viscosities of different glass forming systems the viscosity is normalized to the temperature (Tg*) where the viscosity of

Relationship of Fragility and Dilatation with Glass-Forming Ability…

111

the respective alloy is 1012 Pa⋅s. According to Angell et al. [41], a fragility parameter m based on viscosity is defined as:

m=

d log η d (T g* / T )

T =Tg*

.

(3)

Table 1. The fitting equations and related parameters for Pr55Ni25Al20 and Zr46.75Ti8.25Cu7.5Ni10Be27.5 [40], Mg65Cu25Y10 [21], Zr41.2Ti13.8Cu10.0Ni12.5Be22.5 [20], La55Ni25Al20 [39], Pd40Ni40P20 [8] bulk metallic glasses, the critical cooling rate Rc and GFA indicator γ except for Pr55Ni25Al20 BMG are taken from Ref. [5]

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Alloy composition (at. %) Pr55Ni25Al20

Eq. type C-G

Zr46.75Ti8.25Cu7.5Ni10Be27.5

V-F-T

Mg65Cu25Y10

V-F-T

Zr41.2Ti13.8Cu10.0Ni12.5Be22.5

V-F-T

La55Ni25Al20

C-G

Pd40Ni40P20

C-G

Eq. fitting parameters

Tg*(K)

m

Rc(K/s)

γ

A= -2.93, B = 69.50 K, E = 13.4 K, T0=845.8 K η0 =3.0× 10-5 Pa s, D=22.7,T0 =372 K η0 =3.0× 10-4 Pa s, D=22.1,T0 =260 K η0 =4.0× 10-5 Pa s, D=16.5,T0 =426.3 K A= -2.85, B = 49.09 K, E = 9.82 K, T0=777.0 K A= -2.33, B = 202.9 K, E = 22.9 K, T0=774.7 K

494

30.8

----

0.417

595

43

28.0

0.402

404

45

50.0

0.401

602

41.5

1.4

0.415

447

35

67.5

0.388

561

51

0.167

0.409

Figure 4 shows the Angell plot of the Pr55Ni25Al20 liquid together with Zr46.75Ti8.25Cu7.5Ni10Be27.5 [40], Mg65Cu25Y10 [21], Zr41.2Ti13.8Cu10.0Ni12.5Be22.5 [20], La55Ni25Al20 [39], Pd40Ni40P20 [22] BMGs as well as two extremes of SiO2 and O-terphenyl [39] for comparison. The fitting equations and related parameters are listed in table 1. The CG equation usually has wider applications for BMG viscosity fitting. The fragility parameter m for Pr55Ni25Al20 BMG is calculated to be 30.8, which shows a comparative value with the m (35) of La55Ni25Al20. They behave much closer to strong glasses than the other reported Mg65Cu25Y10 and metallic glasses such as Zr46.75Ti8.25Cu7.5Ni10Be27.5, Zr41.2Ti13.8Cu10.0Ni12.5Be22, the supercooled liquids of which also fall into relatively strong liquids. Furthermore, it can be seen from table 1 that there are not necessary correlations between GFA and fragility in the whole BMGs. Notwithstanding this, the strong liquid behavior of BMG generally leads to the sluggish kinetics in the supercooled liquid region, i.e., a low nucleation and growth rate of crystals. Thus, for the metal-metal glasses which have similar liquid structures, it seems reasonable to support the statement that the stronger the liquids the larger the GFA. For example, the m rank is just in reverse order of the γ for the Pr55Ni25Al20 and La55Ni25Al20 supercooled liquids.

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2.2. The Thermodynamic Behavior of the Pr55Ni25Al20 Bulk Metallic Glass

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Figure 5. Measured specific heat capacity of the supercooled liquid (●) and the crystal (○) of Pr55Ni25Al20 alloy. The best fits to the experimental data are given.

Figure 6. Entropy of the Pr55Ni25Al20 supercooled liquid with respect to the crystal, including the entropy of fusion, ΔSf, glass transition temperature Tg (onset at a heating rate of 0.167 K/s) and the Kauzmann temperature, Tk.

The specific heat capacities were measured and the relative thermodynamic parameters were calculated as well to further investigate the properties of the Pr55Ni25Al20 liquid. Figure 5 shows the specific heat capacities of Pr55Ni25Al20 alloy that were measured in the liquid and crystalline state in reference to sapphire. The characteristic temperatures Tg (480.4 K), Tm (751 K) and the Kauzmann temperature Tk (448 K) deduced from figure 6 are marked in the

Relationship of Fragility and Dilatation with Glass-Forming Ability…

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figure. Tk is the temperature at which the entropy of the supercooled liquid reaches that of the crystal and is usually considered to be the lower bound for the glass transition from a thermodynamic point. According to Kubaschewski et al.[42], the temperature dependence of specific heat capacities for the supercooled liquid (cpl) and the crystalline (cpx) state are fitted well by the following Eq. (4) and Eq.(5), respectively.

c lp (T ) = 3R + 1.447 × 10 −3 T + 5.949 × 10 6 T −2 ,

(4)

c px (T ) = 3R − 2.681 × 10 −2 T + 5.0 × 10 −4 T 2 ,

(5)

where R =8.314 J/g-atom⋅K. Furthermore, taken into account the specific heat capacity data and the heat of fusion, the thermodynamic functions of entropy, enthalpy and Gibbs free energy differences between the supercooled liquid and the crystal can be calculated by Eqs.(6)-(8), respectively. Here Tm was used instead of Tf for non-eutectic Pr55Ni25Al20 alloy.

ΔS l − x = ΔS f − ∫

c lp (T ' ) − c px (T ' )

Tf

T

T

ΔH l − x = ΔH f − ∫

Tf

T

[c

l p

dT ' ,

(6)

]

(T ' ) − c px (T ) dT ' ,

Δ G l − x = Δ H f − ΔS f T − ∫

Tf

T

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'

[c

l p

]

(7)

(T ) − c (T ) dT + T ∫ '

x p

'

Tf

T

c lp (T ' ) − c px (T ' ) T

'

dT ' .(8)

Figure 7. Enthalpy of the Pr55Ni25Al20 supercooled liquid with respect to the crystal, including the enthalpy of fusion, ΔHf, glass transition temperature Tg (onset at a heating rate of 0.167 K/s) and the Kauzmann temperature, Tk.

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Figure 8. Gibbs free energy difference between the supercooled liquid and the relative crystal for Pr55Ni25Al20 BMG and Zr41.2Ti13.8Cu10.0Ni12.5Be22, Zr46.75Ti8.25Cu7.5Ni10Be27.5, Mg65Cu25Y10 BMGs [20] from the glass transition temperature Tg to the onset temperature of melting Tm.

Figure 6 and figure 7 show the calculated functions of the entropy and enthalpy differences between the supercooled liquid and the crystal, both of which decrease with increasing undercooling. It is believed that this residual entropy and enthalpy are caused by freezing when cooling the alloy from its superheated melt through the glass transition point. For Pr55Ni25Al20 BMG, the Kauzmann temperature Tk of 448 K determined from figure 6 is about 32.4 K lower than the Tg kinetically observed. The large difference between the thermodynamic Tk and kinetic Tg indicates the easy vitrification of the alloy. Figure 8 shows the calculated Gibbs free energy function with reference to the crystalline state for Pr55Ni25Al20 BMG. Other reported BMGs such as Zr41.2Ti13.8Cu10.0Ni12.5Be22, Zr46.75Ti8.25Cu7.5Ni10Be27.5 and Mg65Cu25Y10 taken from Ref. [20] are also plotted in figure 8 for comparison. It can be seen that the slopes of the Gibbs free energy difference curves for all the BMGs just below the melting point are small and have close values. It means a low driving force for crystallization and thus the Gibbs free energy difference is one key reason for the high GFA of these alloys. With decreasing temperature close to the respective glass transition points, the Pr55Ni25Al20 alloy shows the smallest Gibbs free energy difference with respect to the crystal, and then Zr41.2Ti13.8Cu10.0Ni12.5Be22, Zr46.75Ti8.25Cu7.5Ni10Be27.5 and Mg65Cu25Y10 alloys. This trend agrees well with the GFA indicator γ and the reported critical cooling rate Rc listed in table 1. The smaller the Gibbs free energy difference the higher the GFA. Therefore, Compared with the fragility parameter m, the Gibbs free energy difference is a more universal parameter to rank GFA.

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3. FRAGILITY OF SUPERHEATED MELTS AND GLASS-FORMING ABILITY IN PR-BASED ALLOYS 3.1. Deduction of the Mathematic Expression of the Superheated Liquid Fragility Parameter Generally, the temperature dependence of the viscosity of alloys at superheated temperatures obeys the Arrhenius equation well [43],

η = η 0 exp( E ∞ / k B T ) , η 0 = h / v m

(9)

where η is the kinetic viscosity, η0 is a pre-exponent constant which is determined by the alloy species, kB the Boltzmann constant, h the Planck’s constant, vm the flow unit volume, T the absolute temperature, and E∞ the activation energy at superheated temperature. However, when Arrhenius equation is extrapolated to the supercooled liquid, it is no more valid for most of the metallic glasses. In order to give a universal equation for liquids from superheated state to supercooled state, Kivelson and Tarjus [44] proposed a superArrhenius model based on activation energy refinement, which is given by:

Ε(Τ) = Ε∞ + Τ * Β[(Τ * − Τ)/Τ * ] y Η(Τ * − Τ) .

(10)

where H(T * - T) is added as a ladder function, when T ≤ T* , H = 1; T > T* , H = 0 . E∞ , B, T * are constants which are constants related only to alloy compositions, and T* is the crossover temperature, the viscosity shows Arrenhius temperature dependence above it and superArrhenius behavior below it, which is generally lying near the liquidus temperature. y is an exponent with the value of 8/3. Substituted Eq. (10) into Eq. (9), then the superArrhenius equation is obtained by:

η = η 0 exp( E (T ) / k B T ) .

(11)

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According to Angell’s definition of the fragility parameter and the superArrhenius Eq. (11), we can get: m=

d( log η) d(Tg /T)

= T =Tg

d{ log [η0 exp (E(T)/k BT)]} d(Tg /T)

= T =Tg

E∞ + B(T * − Tg ) ln 10k BTg

=

[

]

1 d η(T)/η(Tg ) d (T/Tg ) ln 10

. (12) T =Tg

Eq. (12) indicates that m is indeed the relative variation rate of the temperature dependence of viscosity normalized to glass transition point. Angell used Tg to normalize temperature because it is one of the most important characteristic temperatures for supercooled liquid which relates to glass transition. Likewise, the liquidus temperature Tl is a more important characteristic temperature that relates to the solidification process for both glass-forming and non-glass-forming alloys. If Tl is used to normalize T, the fragility parameter expressed in Eq. (12) can be rewritten as:

116

Qingge Meng, Shuguang Zhang, Mingxu Xia et al. ml =

d{ log [η0 exp (E(T)/k BT)]} 1 d [η(T)/η(Tl )] d( log η) E∞ = = = ln 10 ln 10 d(Tl /T) T =T d(Tl /T) k T d (T/Tl ) B l T =T l

T =Tl

. (13)

l

As ln10 is a constant, the expression of the fragility parameter can be further simplified as: M=

d [η (T ) / η (Tl )] d (T / Tl )

T =Tl

=

E∞ . k BTl

(14)

Here M is just the fragility parameter of the superheated liquids or superheated liquid fragility empirically defined by Bian et al [24]. When ηl is used to normalize Eq. (9), we will get:

η r = η r 0 exp( E∞ / k BTlTr ) = η r 0 exp( Es / k BTr ) where

(15)

η r = η / η l , η r 0 = η0 /ηl , Tr = T / Tl , E s = E ∞ / Tl . Viscosity reflects the bonding

nature of atoms in liquids, implying a close correlation with the liquid phase stability. A liquid with high value of M experiences rapid structure rearrangement towards the melting point, whereas a liquid with low value of M would hold more liquid structures upon solidification. In this sense, M could manifest the relative stability of liquid phase to some extent. The smaller the M, the stronger the liquid is, and the larger the GFA would be. Since the superheated liquids are easier to be obtained than those of the relative supercooled state, M allows evaluating the dynamic behavior of all kinds of alloys even for those difficultly mdescribed alloys such as Al-based alloys. The negative correlations between M and GFA have been confirmed in Al-Co-Ce amorphous alloys by Bian et al, Thereafter, we will discuss the correlation between M and GFA in Pr-based BMGs.

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3.2. Correlation between Superheated Liquid Fragility and Glass-Forming Ability in Pr-Based BMGs

Figure 9. DSC curves of Pr-based bulk metallic glasses at a heating rate of 10 K/min.

Relationship of Fragility and Dilatation with Glass-Forming Ability…

117

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Figure 10. The experimental viscosity data (as the single points) for Pr-based alloys and the fitting curves (as the continuous curves according to Eq. (9).

The DSC curves of the Pr- based BMGs at a heating rate of 10K/min are shown in figure 9. All of the glass transition temperature Tg can’t be obtained clearly from the DSC curves, in the absence of data for reliable Tg, it is proposed to approximate Tg by the crystallization temperature Tx. If the parameters of reduced glass transition temperature γ (=Tx/(Tx+Tl) are adopted to evaluate the GFA. The GFA of the Pr-based alloys from high to low in the order: Pr60Cu30Al10> Pr60Fe10Cu20Al10> Pr60Fe5Cu25Al10> Pr60Co30Al10> Pr60Fe30Al10> Pr60Fe15Cu15Al10. Figure 10 shows the viscosity data for Pr-based alloys at superheated temperatures and the curves based on non-linear least squares (NLLS) fitting according to Eq.(9). Fitting error analysis confirms that these viscosity data of superheated melts is consistent with the Arrhenius equation. Thus the viscosity at liquidus temperature (ηl) can be calculated by the relevant Arrhenius equation. The values of the fitting parameters E∞i and νmi together with Trx, ηl are shown in table 2. There is no direct relationship between the viscosity at the liquidus temperature (ηl) and the GFA indicator γ. For example, the Pr60Fe30Al10 alloy has the largest γ, but its viscosity at the liquidus temperature (ηl) is not the largest. It is smaller than the Pr60Fe5Cu25Al10, Pr60Co30Al10Pr60 and Fe15Cu15Al10 alloys. Similar observations have also been reported by Bian et al [24] in Al-Co-Ce alloys. Figure 11 shows the experimental viscosity of superheated melts scaled with ηl and the fitting curves based on Eq. (15) for Pr-based alloys. The corresponding parameters derived from figure 3 are listed in Tale 2. From the correlations of the superheated liquid fragility M, and the GFA indicator γ, it is obvious to see that the less the value of superheated liquid fragility, the larger the GFA. This is just what Eq. (14) indicates and figure 11 reflects. In figure 11 the steepness of the slopes of the fitting curves is the inverse of GFA as γ approaches 1. For the Pr-based BMGs in this work, the negative correlation between the GFA indicator γ and superheated liquid fragility M still lies and M can be used as indictor of GFA based on the experimental data.

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Table 3. The fitting parameters of Eq.(9) and Eq.(15), ηl is the viscosity at the liquidus temperature calculated from the fitting Eq. (9), M the fragility of superheated melts, γ is the GFA indicator Alloys

Pr60Fe30Al10 Pr60Cu30Al10 Pr60Fe10Cu20Al10 Pr60Fe5Cu25Al10 Pr60Co30Al10 Pr60Fe15Cu15Al10

E∞i (ev)

0.141 0.128 0.122 0.127 0.153 0.143

ν mi (×10 4.324 4.824 4.460 5.979 6.407 5.096

-25

cm3)

ηl (×10-3Pa.s)

ηrl (×10-3Pa.s)

Es(×10-4 ev)

⎛ ES ⎜⎜ ⎝ kB 9.860 9.002 9.976 8.510 10.115 13.492

0.1554 0.1526 0.1496 0.1302 0.1023 0.0964

1.604 1.620 1.634 1.727 1.964 2.016

γ

M=

1.862 1.880 1.899 2.004 2.279 2.340

⎞ ⎟⎟ ⎠ 0.460 0.436 0.413 0.389 0.381 0.376

Relationship of Fragility and Dilatation with Glass-Forming Ability…

119

Figure 11. The scaled experimental viscosity data (as single points) and fitting curves (as continuous curves) for Pr-based system alloys. The inset picture is the amplified zone of continuous curves as η/ηl approaches 1.

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3.3. Thermodynamic Refinement of Superheated Melt Fragility for Evaluation of Metallic Glass-Forming Ability Upon cooling an alloy liquid below the liquidus temperature, it enters a metastable state against crystallization. That is to say, amorphization is indeed a competing process between liquid phase and the resulting crystalline phases, which is controlled by both kinetic parameters such as viscosity and thermodynamic parameters such as entropy and enthalpy of the liquid [45]. In the above sections, it is found theoretically and experimentally that the superheated melt fragility parameter M has a negative correlation with the GFA in the Pr-FeCu-Al alloy system. Now we would study whether or not the negative relationship between M and γ can be applied to more metallic glass systems. The viscosity data for the superheated melts of Al-Ni-Pr(Si)[46], Al–Co–Ce[24], Pr–Fe–Cu–Al [47], and Al–RE (rare earth)[48] metallic glasses have been summarized. The correlation between the fragility parameter M and the GFA indicator γ of these alloys is shown in figure 12. Strikingly, a negative interrelationship between the γ and M as demonstrated by the dash line is observed for each alloy system. However, the negative correlations are not satisfied among the different alloy systems. According to γ, the Pr-based alloys have the largest GFA, and then followed by the Al–Co–Ce, Al–Ni–Pr(Si), and Al–RE alloy systems. But the sequence of the corresponding fragility parameter is just contrary to this, suggesting that the discrepancy between the GFA and M among different systems may be caused by the difference of the thermodynamic factors. Hence, M would be preferred to act as a kinetic parameter that controls the GFA of an individual alloy system.

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Figure 12. The correlation between the fragility parameter M and the GFA indicator γ of Al-Ni-Pr (Si) [46], Al-Co-Ce [24], Pr-Fe-Cu-Al [47], and Al-RE [48] metallic glasses.

Figure 13. The correlation between the parameter ε and the GFA indicator γ of Al-Ni-Pr (Si) [46], AlCo-Ce [24], Pr-Fe-Cu-Al [47], and Al-Re [48] metallic glasses.

In the previous study [49], we have proposed a calculable parameter ε to evaluate the GFA of metallic glasses. A roughly positive correlation between the parameter ε and the GFA are found in the developed BMG systems. The parameter ε is defined as a negative ratio of mixing entropy (ΔSmix) to mixing enthalpy (ΔHmix) according to the regular melt model, n

ε =−

ΔS mix = ΔH mix

n

R ∑ ci (ln ci ri3 − ln ∑ ci ri3 ) i =1

i =1

n

∑ 4ΔH

i =1, i ≠ j

mix AB i

c cj

,

(16)

Relationship of Fragility and Dilatation with Glass-Forming Ability…

121

where R is the gas constant, ci and cj is the atomic percentage of the ith and jth component, respectively; ri is the radius of the ith element; ΔH AB is the mixing enthalpy of mix

binary

liquid

alloys.

The

calibration

value

of

mix ΔH AB

is

used

as

mix ( cali ) trans ΔH AB = ΔH AB − 1 / 2ΔH trans for containing one non-transition metal (NTM) and mix ( cali ) ΔH AB = (ΔH itrans − ΔH trans ) / 2 for containing two NTMs. ΔH trans are 100, 30, 180, j

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310, 17, 34, and 25 kJ/mol for containing H, B, C, N, Si, P and Ge, respectively. Figure 13 shows the correlation between the ε and γ of Al-Ni-Pr (Si), Al-Co-Ce, Pr-FeCu-Al, and Al-RE metallic glasses. A positive correlation can be seen in the different alloy systems, i.e., Pr-based alloys with the largest ε have the highest γ, and then followed by the Al–Co–Ce, Al-Ni-Pr (Si) and Al–RE systems. This agrees well with the ε criterion mentioned above. However, the positive correlation cannot be seen in some single alloy system, such as Al–Co–Ce, Al-Ni-Pr (Si) and Al–RE systems. In these alloy systems, it seems that the ε and γ even have a negative relationship. This disagreement perhaps results from the loss of kinetic factors in ε. To solve this paradox, it is easy to think that the combination of the both parameters of ε and M would be a good way. Now we use the thermodynamic parameter ε to refine the kinetic parameter M, and a new GFA evaluation parameter M∗ with the simplest combination of the both parameters can be obtained as: M∗=M/ε, which integrates the kinetic factor with the thermodynamic factor. Theoretically, it will have a negative correlation with the GFA of all kinds of alloys. This agrees well with the experimental results shown in figure 14. A negative interrelationship is observed between M/ε and γ in the metallic glasses, and a nonlinear relation, M/ε=3.359+0.042/(γ-0.250), was fitted to show this negative trend, indicating that M∗ overcomes the shortcomings of both M and ε.

Figure 14. The correlation between the parameter M∗(=M/ε) and the GFA indicator γ of Al-Ni-Pr (Si) [46], Al-Co-Ce [24], Pr-Fe-Cu-Al [47], and Al-RE (rare earth) [48] metallic glasses.

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4. RELATIONSHIP OF DILATATION WITH GLASS-FORMING ABILITY OF PR-BASED BMGS In this article, the Pr-based BMG rods of 2 mm in diameter were prepared by copper mould suction casting in an arc furnace. The dilatation measurements were conducted with a conventional dilatometer (Netzsch DIL 402C). The initial length of the sample was 20 mm and the compression load during measurement was 25N. Calibration was performed using a pure iron sample before the experiments. Figure 15 shows the DSC and the dilatometer (DIL) traces for Pr55Ni25Al20, Pr60Ni25Al15 and Pr60Ni30Al10 BMGs at a heating rate of 5 K/min. The values of ΔTx and γ together with Tg, Tx and Tl for the three BMGs are summarized in table 4. It can be seen from figure 15 that Pr55Ni25Al20 and Pr60Ni25Al15 BMGs have distinct glass transition points, while Pr60Ni30Al10 BMG shows no transition point. The variation of Tg (except for Pr60Ni30Al10 BMG) and Tx for the three BMGs with composition, exhibit the same trend as that of the GFA in terms of ΔTx and γ. The absence of Tg for Pr60Ni30Al10 BMG is attributed to the low thermal stability of its amorphous structure [50]. The dilatation curves are basically linear from about 20 K below the Tg or Tx (for Pr60Ni30Al10 BMG) until they approach room temperature. This is attributed to the normal thermal expansion of materials. The average thermal expansion coefficients αaver in the linear expansion zone for the three BMGs are also listed in table 4, together with the dilatation transformation temperatures Tr1 and Tr2, which correspond to the Tg and the second DSC crystallization peak, respectively. Tr1 is about 6.5±0.5 K higher than Tg and is located in the supercooled liquid region for both of the Pr55Ni25Al20 and Pr60Ni25Al15 BMGs. For Pr60Ni30Al10, Tr1 is 451.8 K, and thus Tg can be estimated to be 445.3±0.5 K using a similar differential of about 6.5±0.5 K. Then the estimated ΔTx is 19.9±0.5 K, which is still the smallest for the three BMGs and follows the same GFA trend, indicated by γ. Based on the onset temperature of crystallization, the samples at Tr1 are still in the amorphous state. The non-linear feature about 20 K below Tg and in the SLR results from the structural relaxation and viscous flow under the action of the compression load for the dilatometric measurement, respectively. Tr2 has almost the same value as Tx2 for the three BMGs with a maximum difference of 2 K, indicating that Tx2 and Tr2 are related to the same crystallization process. The maximum length changes (dL/L0)max for Pr60Ni30Al10 and Pr60Ni25Al15 BMGs through the SLRs are about -1.8% and -49.6%, respectively. For Pr55Ni25Al10 BMG, however, the amount of contraction far exceeds the measuring range of the dilatometer because it has the largest ΔTx value. Thus, it seems that the values of (dL/L0)max correlate with the GFA.

Relationship of Fragility and Dilatation with Glass-Forming Ability…

123

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Figure 15. DSC and DIL traces for Pr55Ni25Al10, Pr60Ni25Al15 and Pr60Ni30Al10 BMGs at the heating rate of 5K/min.

Figure 16 shows the DIL traces of the Pr60Ni30Al10 BMG at different heating rates of 5, 7.5, 10 and 20 K/min, respectively. It can be seen that the length changes dL/L0 and the deduced thermal expansion coefficients αaver become lower with increasing the heating rates. The values of αaver for Pr60Ni30Al10 BMG are determined to be 10.57, 9.36, 9.25 and 9.14 ×106 /K at the heating rates of 5, 7.5, 10 and 20 K/min, respectively. According to the free volume model [51-53], the decrease in the length changes and the average thermal expansion coefficients are directly related to the annihilation of the free volume, which is captured into the solid during cooling of the glass-forming supercooled liquids. When the BMGs are heated up to the same temperature with different heating rates, the total relaxation time for the BMG at a fast heating rate will be shorter than that at a slower heating rate. Thus, less free volume is annealed out of the BMG at the fast heating rate. The volume thermal expansion with temperature will be partially counteracted by the free volume. The more free volume left in the BMG (i.e., under a fast heating rate), the larger the fraction of the volume thermal that expansion will be reduced. Therefore, the BMG shows a smaller αaver at a higher heating rate. Similar results were also reported in the typical non-metallic glass SiO2 [54] and the metalbased glass Pd40Ni40P20 [26], the glass state of which with greater free volume resulted in a smaller αaver than that of the respective crystal.

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Figure 16. DIL traces for Pr60Ni30Al10 BMG at the different heating rates and the inset is the Kissinger plot of Tr2.

The temperature dependence of the length changes and the deduced thermal expansion coefficients (heating rate Φ = 5 K/min) of Pr55Ni25Al20 alloy in the as-quenched and fully crystallized state are shown in figure 17(a). The value of αaver for the fully crystallized sample is 17.12×10-6 /K in the temperature range of the linear thermal expansion zone, which is much larger than that (αaver = 12.19×10-6 /K) of the as-quenched sample. Three repeated heating and cooling cycles (Φ = 5 K/min) well into the glass transition regime of this as-quenched specimen are shown in figure 17(b), where the irreversible plastic flow around Tg appeared due to the compression stresses in the sample grip region. The amount of plastic flow becomes smaller with successive heating and cooling cycles due to structural relaxation and partial crystallization. In the 3rd cycle, the amount of the plastic flow and length change at the same temperature is so small (dL/L0 < 1×10-5) that the cycle can be considered as nearly reversible. The calculated αaver for the three heating processes are 12.57×10-6, 14.38 ×10-6 and 14.53×10-6 /K, respectively. This trend agrees with the results shown in figure 17(a). For the Pr-based BMGs in the relaxed and partially crystallized state, some excess free volume has disappeared, and the shrinkage effect of the amorphous structure has been reduced. This is likely to be the main reason for the increase in the αaver after heat treatment.

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Table 4. Thermal and dilatation properties of Pr55Ni25Al20, Pr60Ni25Al15 and Pr60Ni30Al10 BMGs deduced from the DSC and DIL measurements at a heating rate of 5 K/min, Tx is used instead of Tg for the γ calculation for Pr60Ni30Al10 BMG, + represents the estimated value obtained from the DIL data BMGs

Tg

Tx1

(K) Pr55Ni25Al20 Pr60Ni25Al15 Pr60Ni30Al10

481.9 460.1 +

445.3

Tx2

Tl

(K)

(K)

ΔT x (K)

γ

(K) 533.5 484.5

-----

804.9 802.0

51.6 24.4

0.415 0.384

774.7

19.9+

0.375, 0.381+

465.2

535.0 536.1

αaver

(×106 /K) 12.19 10.79 10.57

(dL/L0)max (%)

Tr1

Tr2

(K)

(K)

too large -49.6

488.3 467.0

---536.0

451.8

534.5

-1.8

126

Qingge Meng, Shuguang Zhang, Mingxu Xia et al.

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Figure 17. The temperature dependence curves of the length change and the deduced thermal expansion coefficients (Φ= 5K/min) of Pr55Ni25Al20 alloy in the as-quenched and fully crystallized states (a); Repeated heating and cooling cycles (Φ= 5K/min) of as-quench Pr55Ni25Al20 alloy well into the glass transition regimes (b).

The XRD diffraction patterns of the as-quenched Pr-based alloy rods of 2 mm in diameter are shown in figure 18. The broad diffraction peaks indicate the full vitrification of the samples. A broad main peak is observed around 2θ from 25° to 40° in every diffraction curve. The peak positions k1 with the maximum diffraction intensity can be obtained exactly by smoothing and interpolating the diffraction curves. A simple expression, r1=7.7/k1, can be used to estimate the nearest-neighbor atomic distances r1 of the Pr-based BMGs, where k1=4πsinθ/λ, and λ is the diffraction wave length [55]. The deduced values of r1, the thermal properties and the αaver for the Pr-based BMGs are summarized in table 5.

Relationship of Fragility and Dilatation with Glass-Forming Ability…

127

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Figure 18. Results of X-ray diffraction patterns of as-quenched Pr-based alloy rods with the diameter of 2mm.

Figure 19. Correlations between the effective depth of pair potentials V0 and GFA indicator .γ.

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Table 5. Thermal parameters were deduced from DSC traces at a heating rate of 10K/min; The average thermal expansion coefficients (αaver) were obtained by DIL experiments at a heating rate of 5 K/min and calculated by Eq. (17); The nearest-neighbor atom distances r1 were obtained by XRD experiments and calculated by Eq. (19); The depths of the effective pair potentials V0 were calculated by Eq. (20) Tg

Tx

Tm

Tl

(K)

(K)

(K)

(K)

Pr60Cu30Al10

------

451.6

695

794

Pr60Cu25Ni5Al10

413.0

462.7

706.7

820.5

Pr60Cu20Ni10Al10

411.5

449.1

696.6

784.0

Pr54Cu15Ni15Al10

459.0

504.5

698.3

755.7

Pr60Cu10Ni20Al10

435.6

454.8

709.1

744.0

Pr60Ni30Al10

------

470.5

752.1

784.7

Pr60Ni25Al15

465.1

490.5

752.5

810.0

Pr55Ni25Al20

485.3

543.5

751.3

816.5

Pr60Fe30Al10

-------

744.7

862.1

901.0

BMGs

γ

0.36 3 0.37 5 0.37 6 0.41 5 0.38 6 0.37 4 0.38 5 0.41 7 0.45 3

αaver (×10-6 /K) Exp. (±0.05) 11.98

r1 (Å) Cal.

V0 (eV)

11.28

Exp. (±0.07) 3.32

Cal. 3.39

0.125

11.59

11.13

3.35

3.40

0.126

10.51

10.97

3.46

3.41

0.128

11.12

11.80

3.28

3.13

0.142

10.69

10.66

3.34

3.42

0.131

10.57

10.35

3.42

3.44

0.136

10.79

10.84

3.42

3.42

0.130

12.19

11.66

3.30

3.16

0.144

9.78

9.87

3.51

3.49

0.142

Relationship of Fragility and Dilatation with Glass-Forming Ability…

129

The average thermal expansion coefficients αaver can be calculated using the following expression:

α αver = ∑ f i .α i ,

(17)

where αi denotes the thermal expansion coefficient and fi is the atomic percentage of the ith constituent element. The data of αi are cited from Ref. [56]. The calculated αaver are also listed in table 5. The largest deviation between the calculated and experimental αaver values is less than 6 %, indicating good agreement. Therefore, we can use the calculated results as an approximation if the experimental data for a BMG is unknown. According to the classical solid theory and the computer simulation work, Lennard-Jones (LJ) and Morse type soft potentials compare with the experimental results so well as to be used to describe the average atomic interactions in amorphous alloys. The pair potential V(r) of two atoms can be represented by a LJ type potential in the form:

V ( r ) = Ar -m − Br -n ,

(18)

where r is the inter-atomic distance, while A, B, m, n are constants. Based on Eq. (18), the average nearest-neighbor distance r1 and the depth of effective potential V0 in the amorphous alloys can be deduced as [30]: 1

⎡ (m + n + 3 )k ⎤ 3 r1 = ⎢ ⎥ , ⎣ 12Eα aver ⎦ V0 =

Er13 , mn

(19)

(20)

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where k is the Boltzmann constant and E is the Young’s modulus. For metallic BMGs, E can be calculated by [57]:

E −1 = ∑ f i .Ei−1 ,

(21)

where Ei denotes the Young’s modulus of the constituent elements with the data obtained from Ref.[56], and fi is the atomic percentage of the constituent elements. Generally, a higher glass-forming alloy has a complicated structure with an atomic configuration corresponding to a higher degree of dense randomly packed structure and high viscosity of the supercooled liquid. Hence, for Pr55Ni25Al20 and Pr54Cu15Ni15Al10 BMGs with a relatively large GFA, a 126 LJ type potential (m=12, n=6) is used, while a 14-7 LJ type potential [30] is adopted for other BMGs, to calculate the r1 and Vo. The calculated results are also listed in table 5. It can be seen from table 5 that the differences between the calculated and experimental r1 are less than 5%, indicating that the adopted LJ type potentials have reasonably described the atomic interactions of the Pr-based BMGs. However, it should be noted that the results of the

130

Qingge Meng, Shuguang Zhang, Mingxu Xia et al.

r1 and Vo here are only the mean values averaged over all possible combinations of atom pairs within the macroscopic dimensions of the BMGs. Nevertheless, these averaged values provide information on the bonding strength and stability of the alloys and are of clear physical significance. The correlation between the V0 and the γ is shown in figure 19. The GFA indicators of Prbased alloys are approximately proportional to their depths of the effective pair potentials, i.e., γ ∝V0. A roughly linear relation exists between γ and V0, with the correlation γ = 3.335V0-0.052, fitted to show this trend in figure 19 with the solid line, and the dashed lines show the 80% confidence levels. The larger deviations of the Pr60Fe30Al10 and Pr60Ni30Al10 points from the solid line are attributed to the substitution of Tg with Tx due to the absence of Tg for the both BMGs. By combining Eq. (18), Eq. (19) and Eq. (20), we can get Eq. (22) and the expression (23) as follows:

V0 =

( m + n + 3 )k , 12mnα aver

γ∝

( m + n + 3 )k . 12mnα aver

(22)

(23)

Thus, the thermal expansion coefficients αaver of the Pr-based BMGs are directly correlated with the GFA. In particular, the GFA rank can be solely determined by αaver when m and n are fixed constants. We consider Eq. (23) is a convenient method for preliminary GFA forecasting as the αaver can be approximately calculated without experiments. However, more work needs to be done to determine whether the correlations between the GFA and the αaver apply to other amorphous alloy systems.

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CONCLUSION The Angell’s fragility parameter m of Pr55Ni25Al20 supercooled liquid is calculated to be 30.8. Compared with those of the other BMGs, the supercooled liquid of Pr55Ni25Al20 BMG showed much closer to the strong liquid side. There does not necessarily exist a correlation between GFA and fragility parameter m and superheated liquid fragility parameter M in the whole BMGs. It is regarded that the correlation between fragility parameter m, M and GFA exists in the same alloy system with the similar liquid structure. Compared with the fragility parameter m, the Gibbs free energy difference is a more universal parameter to rank GFA. The mathematic expression of the fragility parameter of superheated liquids M has been deduced. It is found theoretically and experimentally that the superheated liquid fragility parameter M has a negative correlation with the GFA in each Pr-based or Al-based alloy system. However, the negative correlations are not satisfied among the different alloy systems. A parameter for GFA of metallic glasses, M∗(=M/ε), has been proposed by using the thermodynamic parameter ε to refine the kinetic parameter M. It was showed that M∗ could reflect the nature of the GFA more effectively than both M and ε.

Relationship of Fragility and Dilatation with Glass-Forming Ability…

131

The characteristic temperatures obtained from DSC and DIL measurements agreed closely with each other for the Pr-based BMGs. The thermal expansion coefficients αaver obtained increased with decreasing heating rates due to the annihilation of the free volume during the structural relaxation and crystallization process. A correlation between αaver and the weighted average of the thermal expansion coefficients αi for the constituent elements was found as αaver=∑fi αi. By assuming Lennard-Jones type potentials, the average nearestneighbor distances r1 and the depths of the effective pair potentials V0 for the Pr-based BMGs were calculated. The values of r1 were in good accordance with the experimental results and V0 correlated with the glass-forming ability.

ACKNOWLEDGEMENTS This work is supported by the National Science Foundation for Outstanding Young Scientists of China (Grant No.50125101) and the Natural Science Foundation of China (Grant No. 50231040, No. 50274051) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20050422024).

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Qingge Meng, Shuguang Zhang, Mingxu Xia et al. Bian, X. F.; Sun, B. A.; Hu, L. N.; Jia, Y. B. Phys. Lett. A. 2005, 335, 61–67. Gonchukova, N.O.; Drugov, A.N. Glass Phys. Chem. 2003, 29, 184. Schermeyer,D.; Neuhäuser,H. Mater. Sci. Eng. A 1997, 226-228, 846. Ota, K.; Botta, W.J.; Vaughan, G.; Yavari, A.R. J. Alloys Compd. 388 (2005) L1-3. Varshneya, A.K. Fundamentals of Inorganic Glasses, Academic Press, San Diego, 1994. Shek, C.H.; Lin, G.M. Mater. Lett. 2003, 57, 1229. Porscha, B.; Neuhäuser, H. Phys. Status Solidi. (b) 1994, 186, 119. Chua, L.F.; Yuen, C.W.; Kui, H.W. Appl. Phys. Lett. 1995, 67, 615. Fan, G. J.; Fecht, H. J.; Lavernia, E. J. Appl. Phys. Lett. 2004, 84, 487. Vogel, H. Phys. Z. 1921, 22, 645. Fulcher, G.S. J.Am. Ceram. Soc. 1925, 8, 339. Tammann, G.; Hesse, W.Z. Anorg. Allgem. Chem. 1926, 156, 245. Johnson, W. L.; Lu J.; Demetriou, M. D. Intermetallics. 2002, 10, 1040. Glasstone, S.; Laidler, K. J.; Eyring, H. The theory of rate processes, McGraw-Hill Press, New York, 1941. Cohen, M. H.; Grest, G. S. Phys. Rev. B. 1979, 20, 1077. Kawamura, Y.; Nakamura, T.; Kato, H.; Mano, H.; Inoue, A. Mater. Sci Eng. A. 2001, 304-306, 677. Busch, R.; Bakke, E.; Johnson, W. L. Acta Mater. 1998, 46, 4729. Böhmer, R.; Ngai, K. L.; Angell, C. A.; Plazek, D. J. J. Chem. Phys. 1993, 99, 4201. Kubaschewski, O.; Alcock, C. B.; Spencer, P. J. Materials Thermochemistry, 6th ed., Pergamon Press, Nek, 1993. Suzuka, N.; Satoru, M.; Kazutaka, T. J. Crystal Growth. 2002, 237, 1667 . Kivelson, D.; Tarjus, G. J. Non-Cryst. Solids. 1998, 235, 86-100. Fecht H.J., Mater. Trans., JIM. 1995, 36, 777. Meng, Q. G.; Zhang, S. G.; Xia, M. X.; Li, J. G.; Zhou, J. K. Appl. Phys. Lett. 2007, 90, 031910. Meng, Q. G.; Zhou, J. K.; Zheng, H. X.; Li, J. G. Scr. Mater. 2006, 54, 779. Sun, B. A.; Bian, X. F.; Hu, J.; Mao, T.; Zhang, Y. N. Mater. Chara. 2008, 59, 820– 823. Xia, M.X.; Zhang, S.G.; Ma, C.L.; Li, J.G. Appl. Phys. Lett. 2006, 89, 091917. Fan, G.J.; Löser, W.; Roth, S. Eckert, J. Acta Mater. 2000, 48, 3283. Cohen, M. H.; Turnbull, D. J. Chem. Phys. 1959, 31, 164. Duine, P.A.; Sietsma, J. A. van den Beukel, Acta Metall. 1992, 40, 743. Russew,K.; Zappel,B.; Sommer,F. Scr. Metall. 1995, 32, 271. Chen, S.C.; Chen, L.B. Materials Physical Properties, Press of Shanghai Jiao Tong University, Shanghai, 1999, pp. 21. Birac, C. Phys. Stat. Sol. A. 1975, 36, 247. http://www.webelements.com. Wei,Y. X.; Zhang ,B.; Wang, R.J.; Pan, M.X.; Zhao, D.Q.; Wang, W.H. Scripta Mater. 2006, 54, 600.

In: Amorphous Materials: Research, Technology… Editors: J. R. Telle, N. A. Pearlstine

ISBN: 978-1-60692-235-4 © 2009 Nova Science Publishers, Inc.

Chapter 4

STUDIES OF ISOLATED PORES IN NON-GRAPHITAZABLE CARBON BY SOLID STATE NMR Kazuma Gotoh1 and Aisaku Nagai2 1

Graduate School of Natural Science and Technology, Okayama University, 3-1-1, Tsushima-naka, Okayama 700-8530, Japan 2 Head Office, Kureha Corporation, 3-3-2, Nihonbashi-Hamacho, Chuo-ku, Tokyo 103-8552, Japan

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ABSTRACT One of the amorphous carbons, non-graphitizable carbon (hard-carbon), after heating to temperatures greater than 1000°C, is expected to be superior as an anode active material in lithium ion secondary batteries for large devices such as electric vehicles. In this chapter, we intend to show examples of solid state NMR analyses of 7Li and 129Xe nuclei for hard-carbon. The Li ion intercalated into graphene sheets, and quasimetallic Li in pores, are observed in the charged hard-carbon by 7Li NMR measurement. The intensity of the NMR signal belonging to quasimetallic Li is generally proportional to 0.0 V constant voltage capacity of charge-discharge curves, and the small difference is explainable by an under potential deposition model. On the other hand, 129Xe NMR can observe the Xe atom adsorbed into pores. Because the chemical shift of 129Xe NMR changes with the pore size, the pore size and carbon structure can be evaluated by NMR method. The growth of the porous structure of hard-carbon samples made from petroleum pitch and phenolic resin were observed by 129 Xe NMR, which was difficult to analyze using other porosimetric methods such as nitrogen adsorption.

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Kazuma Gotoh and Aisaku Nagai

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1. INTRODUCTION Amorphous carbon has contributed to many industrial uses as carbon black, carbon fiber, activated carbon, and so on. The application is developing and expanding to higher technology year after year. One of the uses involves a type of amorphous carbon material— non-graphitizable carbon (hard-carbon). After a heating treatment above 1000°C, hard-carbon has been used as an anode active material in lithium ion secondary batteries (LIBs) from the dawn of LIBs’ history [1]. In the present day, graphite is mostly used in anodes because LIBs with graphite have a higher energy density per cell volume—which is preferable for smallsize instruments such as portable PCs and cellular phones—than LIBs of amorphous carbon. However, hard-carbon is reconsidered as a prominent anode active material of LIBs for larger-size instruments. Its properties of high power and long durability are expected to be superior as a power source for large-size batteries—for example, in batteries used for hybrid electric vehicles (HEVs) or plug-in HEVs. It is suggested that hard-carbon has a porous structure because of the lower density (≈1.5 g cm-3) than that of graphite (2.26 g cm-3). The pores are, however, hardly observed by gas adsorption isotherm, probably due to the isolated pores from the outer surface (closed pores). The framework of hard-carbon has attracted interest since a structural report in the early twentieth century by Franklin [2]. Some structural models including micropores have been proposed by others (Conard et al. [3], Jenkins et al. [4], Azuma [5]) after the Franklin model. Structural descriptions of micropores in hard-carbon are roughly classifiable into two: the space surrounded by edges of graphene sheets based on the “card house” (Franklin’s) model, and the void between graphene sheets [6] based on the “wave” (Conard’s) model. However, previously employed methods including microscopic observations (TEM and SEM), XRD, and N2 gas isotherm measurements have not yielded direct evidence of a porous structure. Nevertheless, results of neutron scattering and X-ray small angle scattering have suggested that a nano-scale void exists between graphene sheets [6]. Solid-state Nuclear Magnetic Resonance (NMR) is one of the excellent methods to investigate the environment of adsorbates in carbon materials. In the development of LIB anodes, it is essential to understand the state of lithium intercalated in carbon materials as well as the amorphous structure of carbon materials. Li NMR have been applied to execute the former purpose. (The detailed theory of Li NMR is summarized by Böhmer et al. in Ref. [7].) We show mainly our recent 7Li NMR study of lithium intercalated in hard-carbon electrochemically in the following section. It is important to investigate the structure of pores in hard-carbon not only for development of LIB but also for various uses of hard-carbon, such as carbon black and fiber. 129 Xe NMR spectroscopy is known as a useful medium to study the structure and porosity of porous materials [8, 9]. Since the report of Xenon-129 NMR as a probe for the inner space of materials by Ito and Fraissard [10] in 1980, the porosimetry has been developed and the application is expanding to a lot of porous materials. We show 129Xe NMR study of xenon adsorbed in hard-carbon with mentioning our novel NMR results, following to the Li NMR section.

Studies of Isolated Pores in Non-Graphitazable Carbon by Solid State NMR

135

2. LI-7 NMR STUDIES OF HARD-CARBON PORES

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It is known that lithium metal and first-stage Li-GIC (graphite intercalation compound) shows each peak of 7Li NMR at 263 and 45 ppm, respectively, because of the night shift of lithium, when lithium chloride saturated aqueous solution is adopted as 0 ppm standard. The shift value changes sensitively with reflecting the conductive electronic density around the observed nucleus. Generally, a large shift to lower magnetic field in NMR spectrum can be interpreted as a stronger metallic property of lithium. Many studies of the states of intercalated lithium in hard-carbon by Li NMR have been reported [11-30] and summarized in Refs. [31-33]. The shift values spread in a range between 0 and ca. 200 ppm. In particular, the lithium fully-doped into some hard-carbon samples showed a higher night shift at about 50∼120 ppm than first stage Li-GIC (LiC6) with broad signal at room temperature [14, 16, 21, 22, 28, 29, 34-37]. In the case of our sample [37], a pitch based hardcarbon, the peak at 9.4 ppm attributable to lithium intercalated between graphene sheets at low lithiation shifted to 92 ppm at high loading level (Figure 1). Tatsumi et al. have shown that the signal for Li-doped non-graphitizable carbon at 111 ppm splits to two peaks at 192 ppm and 18 ppm at 143 K [14]. The 18 ppm peak was attributed to the Li ion intercalated into graphene layers and the Li ion existing on the edge of carbon. The 192 ppm signal was explained by quasimetallic lithium forming a lithium cluster in the pore of hard-carbon structure. The model of the growing lithium cluster has been accepted as the ground of the property for the higher capacity of hard-carbon anode (400∼700 mAh g-1) than graphite anode (372 mAh g-1: LiC6 theoretical capacity).

Figure 1. Room-temperature 7Li MAS NMR spectra of a hard-carbon sample lithiated at 0.1 V (155 mAh g-1), 0.0 V (300 mAh g-1) and 0.0 V (500 mAh g-1). The peaks at about 0 and 263 ppm in the 500 mAh g-1 spectrum are originated from small particles’ contribution in the sample and lithium metal, respectively. Peaks marked by symbol (∗) are spinning side bands [37].

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A speculation can be suggested: the amount of lithium constructing the cluster depends on the pore volume of hard-carbon, namely, a sample showing higher NMR shift has higher charge-discharge capacity. Generally, the charge-discharge curve of LIB using a hard-carbon anode shows two stages, viz. the stage proportional to charge voltage and the stage of constant voltage (CV). The cell can be charged (lithiated) quickly by constant current (CC) in the former stage, then charged slowly with keeping the potential at 0 V in the CV stage. The deposition quasimetallic lithium in NMR spectra is observed at a high loading level, which is on the CV stage of charge curve. Although the CV stage extends the charge capacity, it is not preferred for large-size batteries because of following reasons: i) the power sources of large instruments such as HEV and plug-in HEV require a high charge-discharge rate (10～40 C), which cannot be achieved at the CV stage of hard-carbon, while the rate property of CC stage is generally superior to graphite; ii) the state of charge (SOC) of a battery can be easily observed and controlled in the CC stage since it is proportional to the voltage of cell; iii) strong pulse-like input and output can be received because of the great enough margin of electronic potential. We investigated the properties of a hard-carbon optimized for use in large instruments (specimen A) and compared the results with a present hard-carbon (specimen B) [37]. The former has a little CV capacity and higher CC capacity than the latter (Table 1 and Figure 2). The 7Li static NMR spectrum at room temperature had a strong main signals at 85∼106 ppm, with a weak peak at about 10 ppm originating from small particles included in samples to for each sample (Figures 3 and 4). With decreasing temperature, the former signals shifted gradually with broadening and split to 180∼210 ppm and about 10 ppm at 180 K. The former signal broadened further and disappeared below 120 K because of slowing exchange rate of lithium nuclei in cluster. These samples showed similar behavior in Li NMR, however, the ratio of the intensities of two peaks for each sample was different. We deconvoluted the spectra at 180 K to two components, i.e., cluster lithium and intercalated lithium. Comparing the ratio of components’ intensities to CV/CC ratio, the fact that Li cluster was formed before CV processes was implied. We could roughly estimate the starting voltage of quasimetallic lithium deposition to 0.004∼0.010 V by attributing a model of under potential deposition suggested by Zheng et al. [38]. The deposition and extraction of quasimetallic lithium over 0.0 V had also been suggested by Letellier et al. [24]. They constructed LIB cells in a probe of NMR and analyzed the result of in situ 7Li NMR measurement. The method has a merit on knowing a correlation between SOC and the shift of NMR signals precisely. Table 1. The results of electrochemical evaluation of cells using samples A and B Sample (A) (B)

Whole charge capacity / mAh g-1 352 530

CC capacity / mAh g-1 305 264

CV capacity / mAh g-1 47 266

Initial irreversible capacity / mAh g-1 53 77

Studies of Isolated Pores in Non-Graphitazable Carbon by Solid State NMR

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Figure 2. The results of electrochemical evaluation of two hard-carbon samples. Charge (lithiation process) is shown as solid line and discharge (lithium extraction process) is shown as broken line. [37]

Figure 3. Temperature dependence of static 7Li NMR spectrum on fully charged hard-carbon A. [37]

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Kazuma Gotoh and Aisaku Nagai

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Figure 4. Temperature dependence of static 7Li NMR spectrum on fully charged hard-carbon B. [37]

It is conjectured that the origin of the electrochemical difference (the capacity of CV stage) of hard-carbon specimens A and B is accessibility to pores by lithium. Pores of hardcarbon having CV capacity (specimen B) are always “closed” from outer surface for gas molecules such as nitrogen, however, they are accessible by lithium ion through between the graphene sheets. On the other hand, pores of specimen A are probably “isolated” not only for gas molecules but also lithium ions. Although the hard-carbon samples are expected to have both of pore types which are pores of 'card house' model and the void between graphene sheets based on the “wave” model, the isolated pores of specimen A are predicted to the type of the former because the latter is easily accessible by Li ion two-dimensionally. Moreover, the diffusion of Li ion into ink-bottle type of the former can be easily blocked by transformation of the entrance structure. Our NMR result suggests that a little structural difference on the edge concerning to pores changed the properties of samples as anode of LIB because no difference on frameworks of hard-carbon structures between samples was observed by powder X-ray diffraction (XRD). The application of Li NMR analysis is expanding to various new carbon anodes consisting of not only amorphous carbon but also graphite, nanotubes, hybrid carbons and so on. Li NMR will be having an impact for developing of carbon materials with using varied measurements, for instance, in situ measurement, pulsed field gradient (PFG) NMR, etc.

Studies of Isolated Pores in Non-Graphitazable Carbon by Solid State NMR

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3. APPLICATION OF 129-XE NMR POROSIMETRY TO HARD-CARBON PORES The 129Xe NMR chemical shift value (δ) for xenon gas adsorbed in porous materials can be expressed as [9]

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δ = δ0 + δS + δXe-Xe･ρXe + δSAS +δE + δM

,

(1)

where δ0 is the reference (xenon gas at zero density). δS arises from collisions between xenon and the channels’ surfaces, δXe-Xe·ρXe arises from Xe-Xe collisions and is expected to be linear with Xe density under low-loading conditions. δSAS contributes at a very low level of xenon loading by the effect of strong adsorption sites (SAS), δE and δM account respectively for the presence of electric fields and paramagnetic species. The shift value at zero density is related to the pore size through the Xe-wall interaction term (δS), in other words, Xe adsorbed in smaller pore has higher NMR shift at low xenon density. For example, zeolites in 0.1 MPa xenon atmosphere show peaks at 50 ∼ 200 ppm depending on the mean pore size of samples when a signal of xenon gas at 0.1 MPa is used as 0 ppm standard. Meanwhile, the shift value in the high-loading region relies on the local density of adsorbed xenon. The applications of 129Xe NMR porosimetry has been spreading to varied porous materials such as zeolites [8, 39-46], mesoporous materials [47-49], glasses [50], polymers [51-54], ices [55], fullerenes [56, 57], nanotubes [58-60], and porous (activated) carbons [6165]. The intensity of 129Xe gas is so weak that hyper-polarized xenon is often used as an enriching method [8]. 129Xe NMR signal can have 4 orders of magnitude by the Laser irradiation to Xe gas before introducing the gas to sample tube. However, this method has little advantage in the case of hard-carbon because the hyper-polarized xenon does not have a lifetime enough for adsorption into hard-carbon. We intend to find the micropore of hard-carbon and investigate the inner structure and surface in xenon atmosphere using 129Xe NMR [66]. The 129Xe NMR spectrum of pitch based hard-carbon sample with 4.0 MPa of xenon gas pressure is shown in Figure 5. Two strong peaks were apparent at 27 and 34 ppm, with a weak peak at 210 ppm. Although the intensity of the latter peak was less than 1/10 that of the 34 ppm peak, it was observed surely so that it could not be attributed to noise of the spectrum. The peak at 210 ppm in a sample at 4.0 MPa is assignable to xenon in micropores (