Effective Medium Theory

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I

I

Effective rvlediurn Theory Principles and Applications TUCK C. CHOY National Centrefor Theoretical Sciences Taiwan

CLARENDON PRESS 1999

•

OXFORD

OXFORD UNIVERSITY PRESS

Great Clarendon Street, Oxford ox 6DP Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Athens Auckland Bangkok Bogota Buenos Aires Calcutta Cape Town Chennai Dares Salaam Delhi Florence HongKong Istanbul Karachi KualaLumpur Madrid Melbourne Mexico City Mumbai Nairobi Paris São Paulo Singapore Taipei Tokyo Toronto Warsaw with associated companies in Berlin Ibadan Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries Published in the United States by Oxford University Press Inc., New York © T. C. Choy, 1999 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 1999 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographic rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0 19 851892 7 lpeset by the author Printed in Great Britain by Bookcraft (Bath) Ltd., Midsomer Norton, Avon

FOREWORD

The great classical theories of elasticity and electromagnetism evolved up to the nineteenth century. They were developed largely as continuum theories. As such, these theories remain immensely successful in engineering applications to this day. The early scientists and engineers invoked the idea of atoms only in the most distant way, even though atoms were fundamental to chemistry and electrochemistry, which were emerging at about the same time. The pioneers at that time recognised implicitly that the underlying atomic structure need not get in the way of macroscopic descriptions. The way that the atomic scale linked to the engineering scale was less ob vious. Craftsmen, as well as engineers, recognised that any averaging depended on what process or property was involved. People who worked with wood had no doubt that the fibrous grain structure was important; people who worked with crystals realised that some properties depended on the underlying atomic microstructure arrangements. Those who, like Darcy Thompson, looked at the the properties which on of bone, realised that there was an intermediate scale century, that twentieth of materials depended. It was much later, in the middle scale mesoscopic this to ideas about dislocations helped to rationalise approaches that was clear became for the mechanical properties of metals. What gradually con the appropriate: there were at least three scales at which modelling was intermediate, an and scale, tinuum macroscopic scale of engineering, the atomic mesoscopic, scale at which microstructure is handled systematically. Tuck Choy’s book brings together the major ideas in one of the most impor tant approaches to the mesoscopic scale. Effective medium theory is a systematic approach of very wide application. It draws on a number of linked ideas. One idea is that, within the continuum approaches, there is a systematic way to define an average medium, which replaces all the complexity of a tree, a bone, a polycrys talline diamond film, or a superconducting oxide ceramic. A related idea is that one could define an average medium within which some other action occurs. This has led to some of the density functional analogues of effective medium theory. These powerful ideas address major issues in real materials, which are frequently inhomogeneous and described by a structure which is only statistically defined. The averaging methods and their generalisations can go beyond the predic

tion of properties. They can describe the evolution of this microstructure, such as the way in which dislocation structures develop. Averaging methods are, of course, not the only approaches to the mesoscopic scale. Some properties cannot be represented by an average, but have strong dependences on very local fea tures. Brittle fracture is an example, as is the prediction of currents in ceramic

vi

FOREWORD

superconductors. In these cases, it is necessary to go to many realisations of the microstructure, and to average over the behaviour of this ensemble. The range of mesoscopic methods, and of effective medium theories in particular, has not always been appreciated. Many of the treatments in the literature are special cases, considered only in isolation. What Tuck Choy has done is to draw together the important themes. He has extended the theory, especially to the area of superconducting systems and of time-dependent properties. Further, he has developed some powerful new variational theorems. So this book gives more than a comprehensive and systematic approach to an important class of methods. It looks foward to some of the challenges which these methods face, as new systems and applications emerge. Tuck Choy’s analysis also provides some of the new ideas by which these challenges might be tackled.

I I

PREFACE

I “Each generation must examine and think through again, from its own distinctive vantage point, the ideas that have shaped its understanding of the world.” Richard Tarnas, in The passion of the western mind

Marshall Stoneham October 1998 I

I j I I I

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My main motivation for writing this book is to bring together in a single text an exposition of the basic principles and broad applications of effective medium theory (EMT). I have set myself the task of trying to form a cohesive text, suitable as a reference for experimentalists, as well as a graduate text (with a few problems) for theorists. At appropriate places, I have taken the liberty of including some of my own unpublished results and insights. This avoids excessive pedagogy and provides me with extra impetus during the course of this work. The range of applications of the EMT means that I do have to make selections. However, I hope to set out a clear formulation of EMT with a study of its limitations, and to explore extensions beyond one-body EMT. To provide a wider perspective, I have in Chapter 5 collected together a number of related theories that share the ‘spirit’ of EMT, such as the acclaimed density functional theories, and the final Chapter 6 is devoted to problems in a range of application areas; for example, magnetoresistance, granular superconductors, viscoelastic properties of suspensions, and so on. Unfortunately, much as I would have liked, I had to leave out any discussions of modern computer simulation/numerical techniques. This vast area has important contributions from other disciplines, such as microwave engineering, and would have required an entire manuscript on its own. Finally, owing to the enormity of the literature on the subject, I wish to apologise beforehand if any worker feels that his or her own contributions to EMT have unintentionally been omitted. I would gratefully appreciate any feedback and suggestions, and endeavour to incorporate them when the time comes for a revision. Many people helped with this book. I wish to thank Marshall Stoneham for encouragement over the years, and the late Rudolph Peierls, who unfortunately did not live to see its first draft. The memory of my early discussions with him was a constant source of inspiration. Special thanks also go to the reviewers of my final draft: Marshall Stoneham, Walter Kohn, Roger Elliott, Sam Ed wards, Gaoyuan Wei, Mukunda Das, and others, whose feedback provided many useful improvements. Permission from both the authors and publishers of the original papers/books where some of the figures have been adapted is greatly acknowledged. I wish to thank my wife Debra Ziegeler, graduate student Aris Alexopoulos, and in particular Rob Blundell at OUP, without whose help, with

Viii

PREFACE

the usual chores, the project would still lie dormant. This book is especially ded icated to the memory of my father and, most recently, my mother, who both passed away before its completion.

CONTENTS

T. C. Choy April 1999

1

j 1

j •

1 1

•

1 1

I

1 ESSENTIALS The Lorentz field 1.1 Clausius—Mossotti 1.2 Maxwell—Garnett 1.3 Bruggeman 1.4 1.4.1 Some properties of Bruggeman’s formula Green’s functions formulation 1.5 Summary and equivalence 1.6

1 1 5 7 10 13 16 21

2 2.1 2.2 2.3 2.4 2.5

RIGOROUS RESULTS Introduction Variational bounds The concentric shell model Spectral representation Exactly soluble models 2.5.1 Calculations Reciprocity theorems 2.6

24 24 25 30 33 38 42 44

DYNAMICAL THEORY 3 Introduction 3.1 Review 3.2 Macroscopic electrodynamics 3.3 3.4 The quasi-static regime Displacement current and wave scattering 3.5 Mie scattering 3.6 Dynamical effective medium theory 3.7 Open problems 3.8

47 47 47 49 53 58 61 64 69

LIMITATIONS AND BEYOND 4 Introductory remarks 4.1 Higher-order terms 4.2 Percolation and criticality 4.3 4.4 Mie resonances Multiple scattering 4.5 4.5.1 Single-sphere T matrix 4.5.2 Two-sphere T matrix 4.5.3 n-sphere T matrix Competing interactions 4.6

72 72 73 74 77 80 80 84 85 87

I x

4.7 4.8 5 5.1 5.2

5.3 5.4 5.5

5.6

5.7

5.8 6 6.1 6.2

6.3 6.4

6.5

6.6

6.7 6.8

CONTENTS

CONTENTS Two-body effective medium 4.7.1 Two-body Maxwell—Garnett Non-equilibrium

89 90 92

RELATED THEORIES Comments 5.1.1 General viewpoint Coherent potential approximation 5.2.1 Random phase approximation 5.2.2 ATA 5.2.3 CPA Feynman diagrams Localisation of light Classical theory of liquids 5.5.1 The mean field theory of fluids 5.5.2 Three-particle factorisation Density functional theories 5.6.1 Zero-temperature DFT 5.6.2 Finite-temperature DFT The Hubbard model, CPA, and DFT 5.7.1 LSD/DFT for the Hubbard model 5.7.2 CPA for the Hubbard model Summary

96 96 97 97 101 101 102 104 108 111 113 117 124 124 128 130 131 133 135

EMT APPLICATIONS Introduction Electric and magnetic properties 6.2.1 A polycrystalline metal in an applied magnetic field Optical properties Granular high-Ta superconductors 6.4.1 Susceptibility behaviour 6.4.2 Resistivity behaviour 6.4.3 Microwave properties Hydrodynamics of suspensions 6.5.1 Einstein’s solution 6.5.2 EMT formulae 6.5.3 Brownian motion Mechanical properties 6.6.1 Ultrasonic attenuation 6.6.2 Porous media Non-linear composites Conclusions

136 136 137

1

1

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1

I’

138 140 143 143 145 146 148 150 151 152 153 155 159 161 163

1

I

I

Xi

Appendices A.1 Stationary properties of Up and UR A.2 Evaluation of the last term in eqn (2.18)

164 164 165

Bibliography

168

Index

177

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ESSENTIALS

I I I

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f i

1

I j I

This chapter is devoted to the essentials. It will, in fact, lay down the frame work which forms the theme for the whole book. Although there appears to be some chronological order in the discussions, our purpose is primarily to bring out the main ideas as originally developed by the early pioneers in the field and to re-examine the key assumptions, some of which may in fact be questioned or improved through modern advancements. For this reason we start the initial dis cussions with the Lorentz field in section 1.1, invented by its founder to provide a description of macroscopic fields in a media but later found to be the key idea behind the Clausius—Mossotti relation (section 1.2) This provides the basis for the Maxwell—Garnett (MG) formula (section 1.3) which is the first traditional effective medium theory (EMT). Following on, we shall develop the Bruggeman theory (section 1.4) which has certain advantages, but also disadvantages, over the MG theory. This will be discussed and made explicit through various ex amples. In section 1.5 a modern Green’s functions formulation will be used to redevelop the two theories, which has important advantages for further improve ments, particularly for the extension of the theories to higher orders. This will also lend itself naturally to the ideas in the later chapters where we shall con trast the static and dynamical versions of the EMT, and the problems associated with them. The chapter concludes in section 1.6 with a summary list of tables of equivalent problems and other applications of the theory for later reference. The example problems are especially designed to highlight certain features and limitations of the theory. .

1.1

The Lorentz field

We begin our study by deriving one of the central concepts in this book, the well known Lorentz local field relation, which is a subject treated in many textbooks on solid state physics and electromagnetism; e.g. Kittel (1971), Ashcroft and Mermin (1976), and Reitz and Milford (1970). Originally the idea was invented by Lorentz (1870), as part of his programme to develop macroscopic electro dynamics (see, for example, Jackson 1975). In Lorentz’s treatment, Maxwell’s equations first operate at the microscopic level in terms of the electromagnetic fields E and B, which obviously vary in space and time on microscopic scales. By a suitable averaging process, we arrive at the macroscopic Maxwell equations in volving both E, B and the derived fields D and H which include all polarisation

2

ESSENTIALS

Ch. 1

effects due to the medium; see, for example, Van Vieck (1932).1 It also appears that Lorentz had the idea that there should be considerable flexibility in defining this averaging process. Indeed (see, for example, Ashcroft and Mermin 1976), the form of the averaging function is expected to be irrelevant as long as it is smooth over molecular dimensions and timescales. In the modern context this point has not been sufficiently re-examined. 2 We now know that there are at least three levels in which the physics differ. These are the microscopic, the mesoscopic and the macroscopic. It is forseeable that at the mesoscopic level some essential modi fications to Lorentz’s idea may be necessary. The macroscopic Maxwell equations thus derived, whose forms are independent of the material media, are however incomplete. In addition, Lorentz’s exposition can only be deemed complete if the corresponding constitutive relations: D = D(E, B), H = H(E, B) are both specified. This step depends notably on the material media. The displacement field D = E + 4irP and magnetic field H = B 4ii-M, to linear orders, is valid only when quadrupole and higher order multipole polarisation fields are ignored. They consist of the external field plus a supplement coming from the polarisable entities which constitute the media. These polarisation fields are denoted by P and M respectively. In addition, for conducting media, there is a constitutive relation J = J(E, B), again, Ohm’s law: J = uE being only the linear case. Here o can also depend on the B field if there exists a magnetoresistance. It is for the purpose of establishing these constitutive relations that the concept of the Lorentz field was invented. Without loss of generality, we shall specialise to the case of molecular dipoles arranged on a regular (cubic) lattice. Lorentz’s assumption is that the local field Ei 0 experienced by a molecule is not the macro scopically averaged E field but, instead, . 10 This consists of the electric field E produced by all external sources and by the polarised molecules in the system, except for the one molecule at the point in question. In fact, we must remember that the macroscopic field E, by definition, is the force on an infinitesimal unit test charge in the dielectric that is small and thus unable to disturb the charge distribution in the media, but is large by molecular dimensions. To evaluate the local field, a spherical cavity which is macroscopically small but microscopically large is defined around the given molecule. The argument follows by noting that, from Fig. 1.1, we can replace the dielectric outside the cavity by a system of bound charges. Hence

THE LORENTZ FIELD

§1.1

3

0 E

—

‘Evidently some care has to be exercised here. In general, the spatial average has to be over dimensions of at least several lattice spacings and temporal averages must be longer than all molecular times. It is perhaps noteworthy that, in a vacuum, the derived fields D and H are one and the same with the fields E and B respectively, when using Gaussian units f = = 1, which is slightly advantageous. Henceforth we shall adhere to Gaussian units. A conversion table for electromagnetic units can be found, for example, in the Appendix section of Jackson (1975). 1n fact there are difficulties here, as it can be shown, via a suitable choice of gauge, that the 2 D and H fields are redundant; thereby questioning the physical content of Lorentz’s averaging procedure (Yan 1995).

FIG. 1.1. The Lorentz cavity concept for definition of the local field Ej . 0

E 1 0

=

(1.1)

0 + Ed + E E 8 + Enear,

where E 0 is the external field, Ed the depolarising field due to the bound charges on the outer surface of the dielectric medium, E the field due to bound charges

on the surface S of the cavity, and Enear is due to the configuration of all nearby molecules. Now Ed is given by the charge density on the surface of the system, Up = Pj = +P, and hence (1.2) Ed = —4irP. We can now connect the local field to the macroscopic field E, since the normal

component of the displacement D is continuous across the vacuum-dielectric boundary: D=Eo=E+4irP.

(1.3)

Combining the above eqns (1.1)—(1.3) we have

Eioc = E + E + Enear.

(1.4)

This result is quite general and not specified only to the above geometry. The spherical polarisation field E 8 is now easily evaluated using a continuum ap proximation, since the polarisation P can now be assumed to be uniform on the macroscopic scale. By elementary electrostatics (see, for example, Reitz and Milford 1970), then 8 E

=

P

f f dg5

20 dO sinO cos

=

P.

(1.5)

Now we come to the field Enear that is due to the dipoles inside S. There are a few cases for which this term vanishes, for instance in a gas or a liquid, where

4

ESSENTIALS

Ch. 1

these dipoles are distributed randomly in uncorrelated positions. This is also true for a cubic crystal, since Enear

p.rkrik 3

—rqkp

(1.6)

=

ijk

i,j,k

where r k is the radius vector of a molecule on the lattice point (i, i, k). Scrutiny 3 of the various terms shows that a typical component E, say, has the form Px + .) Py -I- ik p)

Enear =

(z 3 a 2

i,j,k

—

2 + (i

)r 2 + j2 + k

2 i

)p 2 +k

=

E + P,

(1.8)

which is the famous Lorentz local field relation. The reader might note that sometimes, as will the case for the rest of this book, we may assume that the depolarising field Ed is known for a given sample. In any case it is dependent on a given sample surface geometry, which is a peripheral problem, as we are mainly interested in the properties of a bulk material. In this case we shall ignore Ed and define the local field to be

CLAUSIUS-MOSSOTTI

§1.2

5

that the cavity concept must have its limitations. First, the cavity shape is in general ill-defined and it is unclear if it has significance in general. Secondly, the size of the cavity, if it coincides with the length scales of the averaging process must be involved in the very process which defines the macroscopic fields. Some form of self-consistency condition must therefore be required for a satisfactory theory. Finally, the identification of a particular molecule as the centre of the cavity leads to certain diffculties which culminate in the Lorentz catastrophe (see 3 section 1.3), giving the concept a final blow. 1.2

(1.7)

where a is the lattice spacing. Clearly, the cross-terms (ii Py + ik Pz) vanish, while the remaining terms cancel by cubic symmetry. This holds similarly for the other components Ey and E respectively. Finally, we can now write the local field E 10 as Eioc

1

Clausius—Mossotti

The Clausius—Mossotti relation, perhaps one of the earliest formulae advanced, relates a macroscopic property (e.g. the dielectric constant e) to a microscopic property (e.g. the molecular polarisability a). The relation has been used for instance by Einstein in 1910 (see, for example, Jackson 1975), in his treatment of the critical opalescence. This is the phenomenon of enhanced light scattering near the critical point of a fluid due to large density fluctuations. The derivation of the relation is straightforward and it rests solely on the Lorentz local field concept. Its proof consists of first identifying the connection between the dipole moment of the molecule with the local field E 10 via the molecular polarisability a. Thereafter the Lorentz field relation given in eqn (1.9) is invoked to derive the macroscopic polarisation P and hence the susceptibility x in terms of a. Thus we start from the fact that the dipole moment of a molecule p is given by p

= a

(1.10)

10 E ,

where a is the polarisability. Then the polarisation P for the crystal is Ejoc

=

0 + P. E

(1.9) P

There remains the problem of the size of the Lorentz sphere. Obviously, this has to be of the order of the length scales for which Maxwell’s equations are to be averaged, i.e. several lattice spacings or tens of A. While there have been several criticisms of Lorentz’s approach, including Landauer (1978), which caution the assumption of point dipole molecules in the above, the result given in eqn (1.9) is generally held to be true, as long as the polarisable entities are not too patho logical; for example, for fiat discs, see Cohen et al. (1973). Even for such cases, eqn (1.9) can of course be generalised by modifying 4ir/3 to 4ira/3, where u is an appropriate second rank tensor. However, difficulties remain in the case of inclusions whose shapes are not spherical, as we shall see later in deriving the Maxwell—Garnett formula in section 1.3. Before moving on from here, we shall mention a few pitfalls of the Lorentz field concept, some of which will be discussed at length later. Here we should mention Rayleigh’s attempts at finding exactly soluble models, like dipoles on a regular periodic lattice to evaluate the validity of Lorentz’s concept. More recently, computer simulations have allowed more complex systems to be studied in this way. Even without these sophisticated studies, it is perhaps noteworthy

=

=

(j). 1 E 0

(1.11)

Here the sum is over all molecules j, whose polarisabilities are a with the local 10 at that site, and N is the number of molecules per unit volume. E field (j) 4 Assuming that the local field is identical for all sites, we can substitute eqn (1.8) for the local field E 10 in eqn (1.11), which leads to p =

(

Njaj) (E +

(1.12)

We obtain the dielectric susceptibility by solving for P as Although the empirical successes in condensed matter physics attest to Minkowski’s for 3 mulation of the macroscopic Maxwell equations (Jackson 1975, Kong 1990), this failure of Lorentz’s scheme (see also the previous footnote) indicates that an adequate microscopic foun dation for the macroscopic Maxwell’s equations is still presently lacking. 1n a solid we should take c as a ‘renormalised’ polarisability, not necessarily identical with 4 that of a single molecule in free space.

6

ESSENTIALS p X= E

1- 4 j Na

Ch. 1

§1.3

(1.13)

that, at least for the static case, the electrodynamics for the current J and the displacement D are perfectly similar to those given by potential theory; see, for example, Landau, Lifshitz, and Pitaevskil (1984) — see also Table 1.2.

The Clausius—Mossotti relation is usually written in terms of the dielectric con stant = 1 + 4 ’irx, which is easily obtained by rearranging eqn (1.13) thus —1 —-

4ir = 3 --N . a

(1.14)

MAXWELL—GARNETT

7

1.3 Maxwell—Garnett In this section we shall study the extension of the Clausius—Mossotti relation to arbitrary composite systems. The essence of this theory, due to Maxwell— Garnett, is to assume a convenient model for the microscopic polarisability a for the inclusion. If a definite link with microscopic theory is not demanded, an alternative procedure is obviously to evaluate the polarisability a using a simple model, 7 say, a spherical molecule with dielectric constant € and radius a. Elementary electrostatics (see, for example, Reitz and Milford 1970), then gives

In this form the explicit relation between the macroscopic € and the microscopic aj is specially evident. An important point to be emphasised here is the as sumption that the same local field acts at all molecular sites in eqn (1.12). For a crystal this is justified on account of periodicity and the Lorentz averaging process (which is over length scales of a few unit cells), the sum in eqn (1.12) now being over molecules in a unit cell (Kittel 1974, Ashcroft and Mermin 1976). Even so, this is valid only for the same molecular specie, so that the Clausius— Mossotti formula (1.14) should relate € to aj only for that component. We will return to this point shortly in relation to a formula due to Böttcher (1952). For the present we should just note that eqn (1.14) provides the valued link between the macroscopic observable € and the microscopic parameter cxi. A mi croscopic theory is still needed to calculate a. Most textbooks use classical spring models for atomic polarisability or lattice spring models for ionic dis placement polarisability (Ashcroft and Mermin 1976). A full quantum mechani cal calculation is necessary in principle; for example, second-order perturbation theory for the ground state of the hydrogen atom gives aj = where a 0 is the Bohr radius, a result as old as quantum theory itself — see, for example, Pauling and Wilson (1935). It is perhaps clear at this point that a similar analysis would also lead to a Clausius—Mossotti relation for the magnetic permeability Im• The analogue of eqn (1.14) would relate i-tm to the microscopic magnetic polarisability a, which is again to be calculated from a microscopic model in principle also requiring quantum mechanics. Quoting once more the case for the ground state of the hydrogen atom as an example (Landau and Lifshitz 1991), am = as predicted by second-order perturbation theory. Here mc 2 is the rest energy of the electron in eV, i.e. about 0.5MeV; 6 hence am is some five orders of magni tude smaller than its electrical counterpart. Once €, 1 m, and also a are known, we would have completed the programme of macroscopic electrodynamics, since all the necessary constitutive equations are now defined. For example, the mi croscopic quantity analogous to polarisability for the current J is the mobility i, a transport coefficient which again has quantum origins. We emphasise here

This is the Maxwell—Garnett (MG) formula, 8 where is the volume fraction of molecules or inclusions. In view of the assumptions in the model eqn (1.15)

5 U nfortunately there is a slight confusion of terminology here. Lm is sometimes known as the magnetic susceptibility, a term which originated in Langevin (1905). We shall, however, reserve the latter word only for the (macroscopic) quantity Xm. Without further digression, we note here that this is the orbital diamagnetic polarisability, 6 the electron having, of course, an intrinsic magnetic moment due to its spin.

We can of course relate this to a microscopic theory later when a is obtained, as in the 7 case of the hydrogen atom where the equivalent c is then —20/7 with a = a . However, this 0 value of i is unphysical (bc. cit.) and the caveat here is that eqn (1.15) applies only to a classical model. Also sometimes known in the literature as the Clausius—Mossotti formula, but we shall 8 reserve the latter name for eqn (1.14).

a

=

(€1_—_a. €i

(1.15)

+ 2)

The factor of 2 in the denominator on the right-hand side of eqn (1.15) comes as no surprise, for it is similar to the left-hand side of eqn (1.14) and has its origin in spherical geometry. For the convenience of later reference, we shall quote here the polarisabilities for the general ellipsoid with semi-axes a, b, and c: (1.16)

=

where the depolarising factors P are in general given in terms of elliptic integrals and have been tabulated in the literature ( Osborn 1945, Stoner 1945; see also Landau, Lifshitz and Pitaevskii 1984). The polarisabilities here are the principal components of a second-rank tensor. A well known case is the prolate spheroid, a > b = c, in which 1_2 1+ [ln(j-__) 2], (1.17) = 2 —

where is the eccentricity, the two limiting cases being P - for spheres and P —+ 0 for rods. Returning now to the spherical case, upon substituting eqn (1.15) into eqn (1.14) we obtain /€—1\

—-)

(€—l ?li)

(1.18)

________

8

ESSENTIALS

Ch. 1

and the remarks in footnote 7, we should not view this formula as a way to derive the constitutive relations from microscopic theory. Instead, eqn (1.18) should be viewed as a formula relating the bulk dielectric constant € in terms of its constituents € which form inclusions of volume fraction in a host. For reasons spelled out in footnote 7, the value of this theory passes from molecular to granular or composite systems. Hereafter we shall replace molecules with inclusions without further specifying their nature, apart from €. This formula contains much to dwell upon. First, there are two ways to view the system. We could view the system as a host of dielectric constant which contains type 1 inclusions of dielectric constant €i, of volume fraction embedded in it. Then eqn (1.18) is of the form

§1.3

MAXWELL-GARNETT

Another shortcoming of the Maxwell—Garnett formula is found when at tempts were made to generalise it to several components. One of the earliest efforts was the formula due to Böttcher (1952), mentioned in the previous sec tion:

/

,

(6—60)

i(

2€)’

(1.19)

or the alternative form 31]7 6=60+

1

— 11171

(1.20)

where — 71=

fi + 2€

(1.21)

Conversely, we could have viewed the system as a host of type 1 with dielectric constant €, and assumed that inclusions of dielectric constant of volume fraction ij, are now embedded in it. Now we have, for eqn (1.18), /6—fi

€+2€)

‘\

(fo—fi

“°€o+2€

(1.22)

Note that the two formulas do not yield the same results for € at concentrations which are related by 111 and + h = 1—that is, when we expect the two systems to be identical—since the decision of which type of inclusions we wish to consider as the host is a matter of choice. This asymmetry in the results is particularly drastic when the difference in the dielectric constants of the two materials is large; that is, for the case 61/60 >> 1. In fact, the inequality in the magnitude of the slopes 9 d€/di 7 at the limit points 170 —4 1 and ij —* 0 suggest that the theory is only correct if the minority component occurs always as a skin completely surrounding and separating the other material. Such a picture is consistent with the second feature, namely that the Maxwell—Garnett formula does not yield a critical threshold . A conductor remains a conductor in the Maxwell—Garnett theory until every element in it is replaced by an insulator. This is a major shortcoming of the theory, for most composite systems do not behave in this way. 1t follows from eqns (1.19) and (1.22) that, in this limit, the ratio in the magnitude of the 9 slopes —t/I-I is times the ratio of the dielectric constants Co/El

9

6

+ 2€

i — 6

=

+26)

(1.23)

Indeed, this formula appears to follow directly from eqn (1.14) but it has an even greater shortcoming than eqn (1.18). For the case of two materials of i and 2, for example, embedded in a host of €0, the composite € depends on the latter, even when it has been completely replaced, i.e. 11i + 112 = 1! This shortcoming arises from the assumption that the same Lorentz field acts on all elements in the system, as mentioned earlier in the paragraph following eqn (1.14). One other question that follows concerns the shape of the inclusions. Had we started with ellipsoidal inclusions, for example, then eqn (1.18) would be 6—1 €+2

611

111( —

3 1+(€ —1)P

124

This formula fails naturally in the limit 11’ —+ 1 and is unacceptable, in spite of its not so infrequent appearance in the literature. It may, however, be acceptable only in a limiting case 111 —* 0 of the Bruggeman theory in the next section. One way out of the difficulty is to replace the Lorentz sphere by an ellipsoid when calculating the local field (Bragg and Pippard 1953). It is obvious then that eqn (1.24) would have identical depolarisation factors on both sides, thereby avoiding the embarrassing erroneous limit when 111 —+ 1: 6—1

/

1+(€— 1)F

61—1

7’1+(€

—

\ i)P)

(1.25)

Unfortunately, this procedure seems contrived. There appears, a priori, no reason for each constituent to have identical alignment with the Lorentz ellipsoid, and the embarrassment of eqn (1.24) would not go away if the alignment of the inclusions were random, featuring the same problems as the Böttcher formula, eqn (1.23). At this point we shall return to the Clausius—Mossotti formula, eqn (1.14), and indicate that it is the latter which is responsible for the unsatisfactory behaviour mentioned above. The Clausius—Mossotti relation leads to the so called ‘Lorentz catastrophe’, which spells out the underlying physical inadequacy of the theory. For the case under consideration, eqn (1.14) can be easily rearranged in the form 1+ !!na 6=

1

—

--n€

,

(1.26)

where ri is now the number of inclusions per unit volume. This formula suggests a possible divergence of € when o increases. However, the value of € is bounded by the requirement that its maximum value should be that corresponding to the case where all of the material has been replaced by the inclusions, i.e. 6 = 1 + 4irna.

10

ESSENTIALS

Ch. 1

a

FIG. 1.2. The effective medium approximation for a continuous binary compo site. However, eqn (1.26) is unbounded at the critical point n = , whereas the maximum physically acceptable value for € at this critical point can only be 4! This anomalous behaviour for highly polarisable inclusions can be avoided only if the Lorentz local field concept is abandoned altogether. As we shall see later in Chapter 2, an alternative derivation of the Maxwell—Garnett formula, eqn (1.22), can be obtained which bypasses this difficulty of the Lorentz field.

Einside

=

1 cos6 —A

1 sin6 0. +A

3€

L

sion. which are the equivalent dipole moments generating the fields Eoutsjde and Einside of the spheres respectively. With these expressions we can now calcu late the electric flux deviation due to the polarisation by the inclusion. For a spherical inclusion we can calculate the flux by taking a disk whose surface area is lra 2 in a plane normal to the z-axis (the axis of Eo), i.e. 6 = rr/2 (see Fig. 1.3). The flux deviation LS4 is then given by the difference in the following two integrals: =

27r(J dr rD

J

dr r€Eo)

/ f

=

€Eo(\ 2 27ra

—

+

€ \

(1.30)

Example 1.1 Show by taking an arbitrary disk size I?> a (see Fig. 1.3) that eqn (1.30) is now given by =

2€Eo

I

€1

—

€

+2€)

(1.31)

Hence show that the size of the disk R is in fact fairly arbitrary, as long as /R) in eqn (1.31) is 3 I? < oo. The effect of the flux change vanishing as O(a of course due to the static dipole near-field characteristics of eqn (1.27). In the dynamic case, the radiation field goes as 1/R and here R is no longer arbitrary; it must be taken to infinity, as we shall see in later chapters.

Bruggeman’s greatest contribution is the hypothesis that there should be zero average flux deviations, i.e.

(1.28)

and 0 are unit vectors, and C 1 and A 1 are given by 1€)a3Eo,Ai=_( 6 Cl=(

11

FIG. 1.3. The surface integral for the flux deviation through a spherical inclu

Bruggeman

Bruggeman made a significant improvement to the Maxwell—Garnett theory when he discovered an approximation that treats the two composites in a sym metrical fashion, thereby overcoming many of the difficulties discussed in the last section. Here we will discuss the theory in some depth, as it is the most widely known version of the effective medium theories and the ideas associated with it are central to this book. As with all approximate theories, a price will have to be paid for gaining certain grounds while at the same time sacrificing others. We shall first present the theory here and then discuss some of the issues later on in this section. Consider a binary system with fractions hi of dielectric constant €i and h2 of dielectric constant €2 respectively, as before. We first replace this complex system by a simpler model. This model (see Fig. 1.2) consists of a homogeneous system of effective dielectric constant c but with a spherical inclusion of radius a and dielectric constant €i embedded in it. Far from this inclusion the electric field is a constant E , but the field nearby, from elementary electrostatics (see, for 0 example, Reitz and Milford, 1970, Landau, Lifshitz and Pitaevskil 1984), tells us that \ 1 C 0 + 2) cos6 + (_Eo + -m-) sin6 6, Eoutside = (E (1.27)

Here

BRUGGEMAN

0

0

1.4

§1.4

)Eo.

?]ii

(1.29)

Z4 =0. ?] +2

(1.32)

By itself, this hypothesis seems physically reasonable, and one would expect that this statement must be true for the exact flux deviations if they were known,

12

ESSENTIALS

Ch. 1

although to the best of this author’s knowledge, there is no rigorous proof for this theorem. The important point here is that Bruggeman considers that eqn (1.31) would be an excellent approximation if the flux deviations were due to single particle polarisations in an effective medium €. This was a bold suggestion at the time, and for most of this book we shall be revisiting this assumption to contras t with the Maxwell—Garnett formula. Using the single-particle flux deviati ons or polarisations calculated in eqn (1.30), we have f€i—E\

iiL

\€i

+ 2€.’ )

f€2—€\

+7721

\E2

+ 2€, 1

0.

(1.33)

This is the most widely known form of the effective medium formulas. Unlike the Maxwell—Garnett formula, it can give rise to a critical threshold, as we shall see, and it can be generalised to include any number of components withou t difficulty: 77(€i€)

=0.

(1.34)

§1.4

BRUGGEMAN

Some properties of Bruggeman’s formula We shall first summarise the main property of eqn (1.33), which the reader can readily verify. In the small 771 limit it agrees with the same limit derived from the Maxwell—Garnett formula of eqn(1.21) to 0(77,). Similarly, in the small 7)2 limit it agrees with the same limit derived from the formula of equ (1.22). We now write down the solution to the quadratic eqn (1.33): 1.4.1

=

(+ 1 /T8e ) 2 e ,

(1.39)

where j3 = (37)1 — l)ei + (3772 — 1)e2. The symmetry of the result is now obvious from this expression, in that for 771 —+ 1, e — e and for €2 —* 1, e -4 €2, the negative square root solution is discarded by considering these limits. Unlike the Maxwell—Garnett formula the value of e for the same material composition is now unique. In the case when both species differ greatly, i.e. €, >> €2, we now find a critical threshold. For from eqn (1.39) we have in this case:

Example 1.2 An alternative way of calculating the flux deviation is to compute the dipole polarisation for each sphere in the infinite volume limit V —* oc. This follows from the definition of the average displacement field: D = lim -- [dr e(r)E, v- V iv

13

--=(377i—1), =0,

62 discussed above, it seems that there is agreement with eqn (1.39) in two dimensions but not three. In general, the Bruggeman theory fails at the critical point on two counts: (a) it predicts too large a critical value and (b) the power law behaviour at the critical point is incorrect. Both features are now well known as typical of effective medium theories. One of the clear signs of trouble was pointed out by Davidson and Tinkham (1976). Consider the general ellipsoidal case of eqn (1.33) (see also eqn (1.25)) namely .

= eE 0 + lim --(€ V—*ooV

—

e)

f

1 v

dr E.

(1.35)

In the last step, we have converted the integral over V into an integral over the volume occupied by the inclusion spheres V,. The explicit reference to each sphere is now contained in defining the dipole polarisation for each sphere, as 1 S= _ e)f (e

drE,

Vspher

(1.36)

where the integral is only over the volume of one sphere . S is the strength of the dipole component of the disturbance, which we shall loosely call the dipole polarisation. If S is the average over all spheres in the system , then we have 0 D= + nS eE ,

(1.37) where n is the number of spheres per unit volume. Show that, for a single sphere, S = a33e(’)Eo. S can be substituted for M in the Bruggeman equatio n (1.32).

(1.38)

‘

(€

+

-‘_

1)€)

+772 (€2

The solution for this quadratic is given by

+

1)€)

= 0.

(1.41)

14

ESSENTIALS 1 2(P’

—

1)

(

Ch. 1

+ 4(P’

+

), 2 1)eie

—

(1.42)

where 3 = (Pr 11 — 1)ei + (P 112 1)e . From this, once again we can deduce 2 that the critical threshold is given by

§1.4

BRUGGEMAN

15

Table 1.1 Second-order coefficient function Fd(1). Dimension d Fd(1)

Error factor,

0 1

0 0

0 1

2

0.744989676 1.45720038

3 4

1.51

1.995

1.758244

2.778

d+(1)

—

e —

ei

1 111 1), 1 (P (P 1) =0, 0 2 accordingly. Notice also that if the choice of = €e 5 taken (Example 2.3), then we recover none other than the Bruggeman formula, eqn (1.25). While the freedom of choice for o was already alluded to in eqn (1.33), here we gain a different insight in the sense that €ç is now seen to be an arbitrary variational parameter pertaining to a homogeneous reference system. It represents our lack of information or, more specifically, consideration for the statistical distribution of the inhomogeneities in the formulation of the theory. Some authors, Hashin (1968) for example, used this as an argument that the effective medium theory should be avoided in preference to rigorous bounding methods, as the former yields no further information than is available by the latter. Others like Berryman (1980) take the view that effective medium theory does contain many attractive features, like the ability to predict a percolation threshold, albeit only approximately, and that no other contemporary theory when considered in toto has yet been conclusively demonstrated to be better on either theoretical or experimental grounds.’ The present author agrees with these sentiments and, moreover, there are inter esting questions that remain (see Example 2.4) which invite further explanation, so that in spite of their beautiful results, there are still fundamental physical insights lacking in using the rigorous bounding methods alone.

Example 2.4 Show by expanding Hashin’s formula, eqn (2.36), for the super conducting case (Example 1.3) that the value of ó has to be about 0.25 for agreement with the exact second-order expansion of Jeffrey (Example 1.4). In fact, for all ratio’s of €1/€2, the value of ö is remarkably constant (Jeffrey 1973). What possible explanation can one give for this result and, indeed, would this value of 6 give consistent corrections at higher orders (cf. Chapter 6)?

Ch. 2

RIGOROUS RESULTS

30

In the next section we shall yet again obtain similar results by exploiting a concentric shell model first introduced by Hashin (1968), who argued that the ö < 1, which when self-consistent scheme really contains a free parameter i varied obtains a value of e between the Hashin—Shtrikman bounds. Remarkably, Jeffrey (1973) (Example 2.4) noted that this value for 6 0.22 is approximately constant, up to second order in j. Finally, the reader who is interested in a fuller mathematical development of the variational theory should consult the excellent monograph by Arthurs (1980) and more modern developments; for ex ample, in Sewell (1992) and also Atkinson and Appleby (1994). A more complete variational theorem for a system containing coupled canonical fields, e.g. an inhomogeneous superconducting system, embodying these principles is given by Choy (1997). 2.3

The concentric shell model

The concentric shell model was introduced by Hashin and Shtrikman (1962a) when they attempt to answer the question concerning the latitude introduced by the above variational bounds (eqn (2.3)). Are these bounds an inherent prop erty of the model or are they due to the limitations of the method based on the ansatz eqn (2.17)? The answer to this question is in general difficult, but the case of a two-component composite with the use of a concentric shell model provides some additional insight, both on the self-consistent effective medium theory, and for the origin of the bounds, eqn (2.3). A modification of this model (Hashin 1968) furnishes yet another alternative derivation of the Maxwell—Garnett and Bruggeman formulae which are the central themes in this book. More than any thing else, it reveals the limitations resulting from the lack of detailed statistical information on the system and could, if studied in greater depth, yield higherorder improvements (Jeffrey 1973), as discussed in the last section. , where the surface poten 0 Consider a homogeneous body with permittivity e 0 within the body, i.e. tial is prescribed to be /o so as to create a uniform field E the reference of the last section. Suppose a sphere of radius b in this system is now replaced by a composite sphere whose inner radius is of radius a and permit Then the question tivity a, while the outer concentric shell is of radius b and one may ask is: Under what conditions is there no change in the energy stored E, then it 0 in our system? Since the original energy of our system is U = e 0 outside our composite sphere re is obvious that this requires that the field E mains unchanged by the replacement. From our previous results (section 1.4), we can write down the solution for the fields immediately. In the innermost sphere 0 < r > a, temperature gradients are induced in addition to mechanical deformations which are anomalously large. Absorption is now dominated by thermal conduction, and not by the viscosity. Two regimes were identified by Zener in which w > X/a , where 2 = ii/C the is thermometric conductivity which , is the ratio of the thermal x conductivity to the specific heat. In the former case we have the sound absorp tion , while in the latter case we have y T/ (Landau and Lifshitz 2 Tw 1989b). As such, a proper account will require incorporation of the effect due to the induced temperature gradients. Fortunately, for solids the only means of heat transfer is thermal conduction, since convection plays no role. This means that eqn (6.47) can be supplemented by 82u

p—-

=

u + (K + !)grad div 2 guV

u

—

oc7T,

(6.67)

where is the thermal expansion coefficient. In most solids the specific heats C and C do not differ significantly and the equation for the temperature field is

159

MECHANICAL PROPERTIES

=

(6.68)

VT.

These equations allow us to modify the Maxwell—Garnett and Bruggman formu lae accordingly, where in addition to elastic constants each component may or 1 and cr2, k2 respectively. An may not have different thermal coefficients al, K temperature wave, comes into play. We propagation, i.e. a other new mode of who will find the frequency reader, the exercise for leave these results as an interest and impor immense of subject to absorption be a dependence of sound exotic applications for recent and 1991), et Harker al. tance. See, for example, therein. references and Mans (1998) see industry, to the semiconductor 6.6.2

Porous media

In this final subsection we shall combine our knowledge of fluid and mechanical properties to study sound propagation in a porous medium. Imagine the case of a fluid suspension (cf. section 6.5) in which the inclusion volume fraction is large. This is the case of a porous medium: the typical example is geological, such as earth rock, but in condensed matter physics we are now also very interested in porous materials such as porous silicon. The constants characterising such a suspension are the viscosity of the fluid, r, the permeability of the media, k, and, for an incompressible fluid, its density p. The average velocity of the fluid has been known since Darcy’s work in 1856 to obey the following law: (v)

=

—(VP

—

(6.69)

pg),

where P is the fluid pressure and g is the gravity vector. There have been several theoretical derivations of Darcy’s law: the determination of the effective ratio k/ij is itself a classic problem in EMT (see the summary in Chapter 1). Here we shall not concern ourselves with the problem of flow, which has many fascinating aspects on its own (see, for example, Sahimi 1993). Rather, we shall concentrate on sound attenuation. This is extraordinarily more interesting, for we need to combine both the mechanical and hydrodynamic properties of the composite, comprising a porous elastic medium filled with a compressible fluid. The theory for sound propagation in such a system was first worked out by Biot in a series of papers (Biot, 1956a,b, 1962a,b). Biot assumed that the wavelength of sound is much larger than the largest dimension of the grains and that the relative movement between the fluid and the solid frame obeys Darcy’s law, eqn (6.69). Then the macroscopic displacement fields of the fluid U(r, t) and of the solid u(r, t) satisfy equations which are modifications of the acoustic wave equation, eqn (6.47): 82

V [ 2 (P + S)u + QU]

=

u + U) 11 (p 12 + -—(u p

—

U),

160

EMT APPLICATIONS [Qu+RU] 2 V

U) 22 u+p 12 (p

=

§6.7

Ch. 6

—

-—(u—U).

P22

+ +

P12 =

(1

—

6.7

ç&)ps

In particular, P12 for example gives the inertial drag, as opposed to the viscous drag that the fluid exerts on the solid as the latter is accelerated relative to the former, and vice versa. Notice that for the first time in our studies the volume fraction or the porosity q comes directly into play in the equations of motion. Experimentally, the measured quantities are the various compressibilities: cb, that of the bulk composite material, and its shear modulus Gb—which are the effective medium quantities—and that of the fluid Cf and of the solid matrix c respectively. The empirical determination of these quantities uses an apparatus that includes a jacketed and a drained measurement (Geertsma 1957). Once again, EMT is useful for relating these measured quantities to the component elastic constants. Thermodynamic arguments alone (Geertsma and Smit 1961) give an incomplete set of relations: H

(1

=

—

(1——)C + 5 Cf

(1

K=

—

+

5 \c

+

(6.72)

where H = P+2Q+S+R, K = (Q+R)/g, and L = R/q . Biot’s theory in the 2 limit when the fluid compressibility is much larger than the solid leads to two propagating modes with a fast and a slow component of sound. In the limit of a suspension whereby Gb -+ oc and in the absence of inertial drag P12 = 0, these merge into a single mode at low frequencies, with a velocity formula first given by Wood (1941): H

=: — =

p

1 [(1

—

)c +

I

Cf][(l —

)Ps + ffj

(6.74)

where the quantities Xi, X2 etc. have been dubbed non-linear susceptibilities. It is often assumed that the host with fraction 1 p would have an ordinary ohmic I-V characteristic: I=2 V. (6.75) —

Similarly, we can translate this problem to dielectrics which are of interest to non-linear optics: the inclusions are known to have a non-linear constitutive relation of the form (Landau, Lifshitz, and Pitaevskii 1984)

i3)

1

L=

I =u V + iV 1 2 + X2V 3 +...,

I

3)2

Non-linear composites

The problem of non-linear composites has received much attention in recent years due to its importance in physics and engineering; see, for example, Bardhan et al. (1994), Yu (1996), and Castañeda and Kailasam (1997) and references quoted therein. The classic example of such a system would be our old friend the random resistor network. Imagine replacing at random some of the resistors in such a network by a non-linear element, such as a semiconductor diode. The problem at hand is to calculate the effective transport coefficients. For low bias voltages, the non-linear element with fraction p can be approximated by the response I — V characteristics, of the form

(6.71)

P12 = fl9f.

161

Once again, thermal effects will also have to be considered and this involves a higher level of sophistication. In the next section we shall once more switch back to electromagnetic problems. Here we shall discuss, for the first time in this book, the subject of non-linear composites. This is a relatively new and important field, a full exploration of which requires a separate monograph. However, we shall make use of the methods in this book and encourage the reader to address the main problems that non-linearity poses.

(6.70)

In this case, F, Q, R, and S are elastic constants, is the porosity of the medium, and the second term on the RHS follows from Darcy’s law. The important new physics in Biot’s theory lies in the first term on the RHS, which includes mass coupling terms. These densities are related to the densities for the solid, p, and of the fluid, p, via P11

NON-LINEAR COMPOSITES

(6.73)

Unfortunately, none of these theories—or others (see, for example, Harker and Temple 1988)—are adequate due to the lack of self-consistency, thus leaving room for improvements using the EMT techniques as discussed in the previous sections.

D = eE + 2 xlEi E ,

(6.76)

where the term non-linear susceptibility originated, and is of importance to second-harmonic laser light generation, for example. The problem at hand is of course similar to those treated throughout this book, and that is to find the effective coefficients e, Xe, e etc., in terms of the fraction p, and the various intrinsic coefficients. One would expect that the Maxwell—Garnett and Brugge man formulae will also have their counterparts. Unfortunately, the one-body problem is now highly non-trivial, since the governing equations of Maxwell, namely div D = 0 and curl E = 0, lead us to a non-linear partial differential equation, cf. eqn (1.51): V.{e(x)Vq(x) + 2 X(xIV(x)I V (x)] = 0,

(6.77)

for the potential (x), in which the field E = —V(x). This equation can nomi nally be obtained by invoking the variational principle on the energy functional

162

EMT APPLICATIONS

W[çb]

Ch. 6

W { 2 ]+ 4 W [ j,

=

(6.78)

where the two contributions are the linear part,

f

W [ 2 ] =

2 dV, e(x)IV(x)l

(6.79)

dv.

(6.80)

and the non-linear part,

W [ 4 j =

For weak non-linearities, i.e. when xIEI /e +SP.SE’ ( fo) -

1

dr SC.SE’

0,

=

=

8ir

----

/ [ {( dr

.

—

2 (SP 6

—

‘ 60)

(A.6)

—

Thus, for 60 , eqn (A.7) is positive and thus the stationary Up is an absolute minimum. The proof follows the same lines for the stationary properties of UR and need not be repeated here (Hashin and Shtrikman 1962a).

APPENDICES

SUp

App. A.2

together with the boundary condition given in eqn (2.7). Now a well known solution to eqn (A.i0) is given by the Coulomb Green’s function:

=

[dr’ j

V(r’)

(A.ii)

—

which, though it does not satisfy the boundary condition given in eqn (2.8), is satisfactory inside the grain i and its immediate vicinity, since correction terms to correct for this fault are of order O(i/V) if both r and r’ are well away from the surface. This can be seen by considering a parallel plate capacitor sample, as in Fig. 2.1. At large distances from the grain, eqn (A.1i) can be expanded in a series of cylindrical harmonics, which are singular at the origin, whose leading term is a dipole field: (A.12) where p is the radius of the cylinder whose axis is the z-axis. In order to cancel this term at the plates and the walls, we must add another series of cylindrical harmonics which are regular at the origin, whose first term is of the form

166

App. A.2

EVALUATION OF THE LAST TERM IN EQN (2.18) z 2 C

I

1 0 V 4ir€ 1

-

—

—

!

I

di’ P(r)

j

—

1(P(r)) 4iro 3 o

f f dr

fdrfdrI

Hence this part of the integral in eqn (2.18) is easily evaluated: (A.15)

dr’ gjj(r, r’)VV 1

Ii

1 gj(r,rI)2 1 l 1

=

Now the near-field contributions (a) and (b) have to be evaluated grain by grain, and thus (A.16) P 8(r). [Vear]. di’ dr P.Eear =

=

—ijiij

(A.21)

when use is made of eqn (A.19). Inserting this in eqn(A.17), and summing now over all pair components I and J, then the near-field contributions are given by

o 3

‘(P2(r)) 3 o

(A.22)

Combining both near- and far-field contributions, we arrive at the final result as quoted in eqn (2.20), i.e.

f

f

fdr P.E’ =

Using eqn (A.11), this becomes

f

111,

must be proportional to the unit second-rank tensor 6. This implies that the evaluation of eqn (A.20) is trivial, and it yields

(A.14)

2 (P(r)) (P(r)) 1 = fdr P.Ear = fdr P. o 3 3o

167

—

Since z is of order L, at one of the top plates, where L is the system size, then the . We now 1 ), i.e. O(1/V) smaller than the constant C 3 constant C 2 is of order O(L— (a) three contributions: contains grain each ‘ inside field observe that the total in grains an other all by field produced grain; the (b) that an internal self-field of contribution. field and (c) a far grain; the given surrounding intermediate zone S The latter, for Sj sufficiently large, can be taken as the field due to a uniform polarisation given by P

EVALUATION OF THE LAST TERM IN EQN (2.18)

For arbitrary separations, Ir TI I, gjj should exhibit the rotational symmetry pertinent to the sample, and thus in the case when the system has either isotropic or at least cubic symmetry, 63 the double integral in eqn(A.17)

(A.13)

.

App. A.2

dr P.Eear

=

i

jES

f f

x

dr

-

. (P(r)) ] 2

(A.23)

The evaluation of the corresponding integral for R follows a similar route and will not be repeated here. We merely quote here the result (Bergman 1976)

o/3

1 O(r) 8j(r’)VaV dr

r’’

(A.17) —

where the near-field restriction j E S ensures that the surface term vanishes 1 has also been replaced by on partial integration over r’ and the gradient in r one in r. In evaluating the sum we can restrict ourselves either to one particular phase or component or, alternatively, to pairs of phase I and J, under which P 3 are constants. The sum over such a pair product of 8 functions yields a and P truncated correlation function, as (r)8(r’) = 2 8

l[(p2(r))

, gjj(r,r ) 1

(A.18)

iEI jEI jesi

.

JdrRD’

I j

=

2

(r)) 2 [(R

—

, (R(r)) ] 2

(A.24)

-

where the only difference in form to eqn (A.23) is in the factor of 2. Finally, by employing arguments similar to that used when partial information is known, in the form of an exact measurement of a related dielectric coefficient Ee, Bergman (1976) was able to show that the results given by eqns (A.22) and (A.23) are strictly valid only under the assumption given in eqn (2.17). This unfortunately severely limits the Hashin—Shtrikman bounds given in eqn (2.3), for it is in general far easier to partition a composite system such that P(r) is effectively constant over each grain, but it is in general impossible to ensure that it remains constant over an entire component or phase.

which vanishes for large separations Ir r’ For separations smaller than S, this correlation function tends to the probability of finding r e I and r’ E I, and for r = r’ it is in fact the local volume fraction j: —

gjJ(r,r) =

1J 6 r11

(A.19)

This restriction can be further relaxed for systems with ellipsoidal symmetry, if a principal 63 axes transformation is first performed.

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INDEX

I absorption, 59, 65, 69 acoustic, 48, 49 admittance, 40 algorithms, 80, 84, 86 analytic continuation, 56 function, 38 properties, 24, 34, 36, 70 structure, 40 ansatz, 24, 28, 30, 33, 55, 70 applications, 1, 21, 22, 57 approximation, 12, 16, 24, 40, 53, 58, 64—66 continuum, 3 decoupling, 16, 19 effective medium, 10, 61, 66 limiting, 61 method, 18, 44 quasi-static, 53, 54 scheme, 17 single particle, 65, 71 static, 34, 36 WKB, 64 asymmetry, 8 asymptotic, 63, 68, 69 ATA, 19 autocorrelation, 92, 93 averaging, 20 function, 2 procedure, 2 process, 1, 2, 6, 48 BBGKY, 115—117, 135 Bergman, 25, 28, 33, 35, 40, 42, 44 representation, 24 theorem, 34 bi-orthogonality, 82 binary alloys, 96, 98, 104, 108, 133, 135 composite, 10, 96 mixture, 45 system, 10, 20 Biot’s theory, 159, 160 boundary conditions, 17, 26, 27, 31, 32, 37, 42, 54, 56, 61, 63 Brownian motion, 148—150, 152, 153, 158 Bruggeman

formula, 13—15, 21, 29, 30, 33, 40, 46, 55—57, 65, 69 theory, 1, 9, 10, 13, 15, 67, 68 canonical fields, 24, 30 capacitor, 35, 39 cavity, 2, 5, 19 Clausius—Mossotti formula, 6, 7, 9 relation, 1, 5—7 cluster extensions of CPA, 92 infinite, 14, 74 local, 76 properties, 89 size, 74 Cole—Cole plot, 140—143 colloids, 140 communal entropy, 115, 116 compatibility relations, 50, 51 competing energetically, 88 interactions, 72, 87—89 complementary, 39, 54 bounds, 36 process, 35 theorem, 27 variational, 24, 27, 58 compressibilities, 160 compressibility, 120 equation of state, 120, 122 compressional mode, 154, 156, 158 computations, 87 numerical, 72, 87 conductivity, 14, 97, 109—111 dynamical, 108 electrical, 22, 54 static, 108 thermal, 21, 25 threshold, 44 constitutive relations, 2, 47, 51, 52 constrained search, 125, 128, 130, 131 continuum, 3, 48 correlation direct and indirect, 119 energy, 128, 132

178

exchange, 127 ftmctions, 112—114, 116, 117, 119—121, 123 higher order, 117 length, 108 magnetic, 112 short-ranged, 112 total, 119, 122 covariant, 49, 52 cubic, 2, 4 Darcy’s law, 159, 160 decoupling, 16, 21 density, 22 charge, 3 energy, 51 fluctuations, 5 Lagrangian, 50 depolarisation factors, 9 tensor, 20 depolarising factors, 7 field, 3, 4 dielectric, 2, 35, 39, 47, 54, 55, 64, 69 boundary, 3 constant, 5, 7, 8, 10, 19, 24—27, 40, 48, 52, 59, 65, 66, 70 function, 70 medium, 3 properties, 24, 33, 51 susceptibility, 6 tensor, 16 diffusion constant, 152 correction, 152 equation, 159 dipole field, 55, 59 images, 15 moment, 5, 11 polarisability, 32 polarisation, 12 radiator, 59 static, 11 dipoles, 2—4, 12, 69 discs, 4, 11 disorder alloy, 133 alloys, 98 average, 111 bond, 98 chains, 104

INDEX

degree of, 108 parameter, 111 site, 108 topological, 98 displacement averaged, 16 current, 54, 58, 59, 66 electric, 3, 7, 27 field, 2, 12, 26, 36 ionic, 6 dissipation, 57 distribution angular, 64 charge, 2 density, 39 grains, 65 modes, 65 particle, 33 probability, 48, 65 statistical, 16, 26, 29, 31, 33, 36, 46 disturbance, 12 Drude model, 54 dyadic, 16, 17, 59, 62 eddy current, 57, 59 Einstein hydrodynamic solution, 150 viscosity formula, 149 elasticity, 21, 23, 47 electric charge, 51 field, 2, 10, 17, 18, 26, 32, 48, 54, 59, 61, 66 flux deviation, 11 electromagnetic dyadic, 59, 62 fields, 49, 51, 53, 61 nature of light, 58 properties, 69 scattering, 61 skin depth, 55 systems, 47, 48 waves, 48, 58 electromagnetism, 1, 45 electron gas, 40, 54 electrostatics, 3, 7, 10, 17, 19, 22, 37 ellipsoid, 7, 9, 13, 19, 21 embedded, 8—10, 40, 66 Ewald method, 28 expansion, 14, 17, 41, 42 asymptotic, 64 Fourier, 28 low density, 43

INDEX

179

lowest order, 44 perturbation, 34 second order, 29 exponents characteristic, 74 correlation length, 74 critical, 74, 75, 89—91 dynamical, 75, 94 estimates, 75, 77 exact, 75 mean-field, 76 scaling, 75 susceptibility, 74 transport, 75, 76

integral equation, 16—18, 21, 37, 61, 67, 68 functional, 26, 27 operator, 37 variational, 26, 28 intermediate zone, 59, 70

far field, 60, 63, 67 Foldy-Twersky, 107 fractal, 94 frame of reference, 52 frequencies, 24, 33, 34, 36, 40, 49, 52, 65

Lagrangian, 49, 50 Langevin force, 152 function, 56 LDA, 96, 124, 126, 131 limitations, 72, 87, 95 liquid, 116, 118, 123 metal, 98, 123 molecular, 123 phase, 114 state, 112, 115, 116 theories, 96, 111, 116, 128, 130, 135 local field, 1—5, 9, 10 relations, 51, 52, 54 localisation Anderson, 96, 108, 109 criteria, 110 length, 111 of light, 96, 108, 109, 111 London boundary condition, 144 electrodynamics, 144—146, 148 equation, 145 penetration length, 144, 146 Lorentz approach, 48 averaging, 1, 2, 6 catastrophe, 5, 9 cavity, 3 ellipsoid, 9 field, 1, 2, 4, 5, 9, 10, 21 group, 53 sphere, 4, 9 transformations, 49

Gegenbauer polynomials, 42 generating function, 42 geometrical information, 24, 36 geometry, 3, 4, 7 grain anisotropic, 21 averages, 48 cell, 19 characteristics, 48 distribution, 65 size, 18, 59 spherical, 20, 65, 66, 69 granular, 8, 18 Green’s functions, 1, 16, 21, 36 Hartree—Fock, 96, 124, 126, 133, 134 Hashin—Shtrikman bounds, 24, 25, 28—30, 33, 38 Helmholtz equation, 48, 55, 58, 59, 62 Herglotz property, 90 Hertz fields, 62 vectors, 62 Hilbert space, 78 host embedding, 67 material, 56 hypothesis, 11, 12, 16, 66, 67 inclusions, 4, 7—10, 14, 15, 22, 56, 57 insulator, 8

Kirkwood, 115 superposition approximation, 117, 118 theories, 96 Kohn—Sham finite temperature LDA, 130 LDA, 96, 124 orbitals, 131

macroscopic, 2, 6, 52 averages, 48 description, 48

180

electrodynamics, 1, 6, 48, 49, 53 fields, 1—3, 5 Maxwell equations, 1, 2, 5, 53 observable, 6 polarisation, 5 property, 5 scale, 3 magnetic contrast, 68 dipoles, 69 field, 53, 54 monopoles, 51 permeability, 54 properties, 66, 68 magnetohydrodynamics, 49 Markov process, 92 maximum, 29 absolute, 27 Maxwell—Garnett formula, 1, 4, 7, 9, 10, 12—15, 20, 21, 25, 26, 30, 33, 39, 44, 46 formulae, 55, 56, 65, 67 theory, 8, 10, 19, 21, 67 mean field, 112, 114 metal, 40, 47, 52, 54, 59, 60, 69 metal-insulator, 40 microscopic, 1, 2 laws, 48 Maxwell equations, 48 relaxation, 52 requirements, 54 microstructure, 24, 34, 45 Mie coefficients, 63, 64, 69 formulae, 64, 70 scattering, 47, 61, 63, 66—68 minimum, 29 absolute, 27 Monte-Carlo, 75, 76, 89 multiple scattering, 65, 66, 70 Navier—Stokes equations, 148—152 noise Gaussian, 94 Langevin, 94 white, 93 non-linear, 117, 118 characteristics, 137 composites, 137, 161, 162 dielectrics, 162 differential equation, 127, 161, 162 element, 161 equation, 147

INDEX

integral equation, 121, 122 integro-differential equation, 117 optics, 161 problem, 162 sigma model, 109 susceptibilities, 161, 162 systems, 162 non-linearities, 158, 161, 162 Ohm’s law, 52, 54, 58 orthonormal, 41 Oseen tensor, 149, 152 pair interactions, 113 correlation, 113, 117, 119 distribution, 114, 118 interactions, 113 particles, 119 potential, 118, 121, 123 penetration depth, 55 length, 56, 57 percolation bond, 74, 75 continuum, 75 exponents, 77 limit, 43, 44 point, 44, 72, 74, 75 site, 75 studies, 14 theory, 74 threshold, 14, 21, 29, 40, 46, 74, 91 Percus—Yerwick approximation, 122 theories, 96 permeability, 6, 22, 54, 56, 59, 62 effective, 56 permittivity, 20, 22, 26, 30, 40, 59, 62 effective, 24, 32—36, 39, 45, 46 perturbation, 6, 16, 34, 61 plasma frequency, 39, 69 Poisson ratio, 154, 155 polarisability, 5—7 polarisable, 2 polarisation, 2, 3, 5, 11, 12, 51 circular, 59 elliptical, 53 fields, 2, 24 generalised, 26, 27 scattered field, 60 spherical wave, 60

INDEX vector, 59 polarised circularly, 67 linearly, 59, 60 partially, 64 plane, 62 pole structure, 25, 38 polymer, 94 porous media, 137, 159, 163 silicon, 159 potential, 22 scalar, 16 theory, 7 vector, 70 Poynting vector, 53, 57, 60 Poynting’s theorem, 53 quadrupole, 2 quantum mechanics, 6, 49, 64 noise, 48 origins, 6 theory, 6 quasi-static approximation, 53, 54 fields, 57 limit, 52, 55, 56, 59, 69 regime, 47, 53, 59 radial functions, 81, 90 part, 79 quantum number, 82 wavefunctions, 82 radiation energy, 53 loss, 59 zone, 59, 70 random media, 47, 48 Rayleigh scattering, 60, 61, 64 Rayleigh—Debye approximation, 64 Rayleigh—Gans scattering, 69 reciprocity principle, 37 relations, 45, 46, 61 theorems, 25, 44 transformation, 45 recursion algorithms, 86, 89 formulas, 85, 89 reference system, 29 refractive index, 58, 62

181 regime, 47—49 displacement current, 58, 59, 66 linearised, 48 quasi-static, 47, 53, 59 skin-depth, 47 relativity, 49, 50, 52 relaxation processes, 52, 60 remote sensing, 60, 61 renormalisation, 76 group, 72, 92 rotating frames, 52, 53 RPA, 96, 101 scalar fields, 48 scaling concept, 72, 76 dynamical, 92, 94, 95 exponents, 75 relations, 75 scattering, 49, 59, 61 amplitude, 60 body, 61 cross section, 60 electromagnetic, 61 geometry, 60, 62 inverse, 61 Mie, 47, 61 problem, 61 Rayleigh, 61 theory, 45, 58, 59 wave, 58, 61 self-consistency, 5, 19, 20, 67 self-consistent, 30 field, 33 scheme, 30 self-energy, 99, 102, 104—106, 133 semiconductors, 140 shear mode, 154, 158 modulus, 153—156, 160 skin depth, 47, 55—59, 69 effect, 47 resistance, 57 soluble models, 25, 38, 40, 44 spectral functions, 25, 34, 37—41, 43, 44 representation, 24, 33, 34, 36, 38, 40 spherical, 21, 22 cavity, 1, 2 dyadic, 62 geometry, 7

182

grains, 20, 65, 66, 69 harmonics, 42 inclusions, 10, 11, 13, 14, 20, 25, 39 molecule, 7 particles, 68 polarisation, 3 shapes, 4 wave, 60, 63 static, 1, 7, 11, 16, 25, 48, 49, 53, 65—67 approximation, 34, 36 fields, 51, 54 limit, 47, 54, 55 Maxwell equations, 45 theory, 65 zone, 59 superconductors, 143—145, 148 composites, 143 copper oxide, 143 granular, 143—145, 163 high T, 137, 143, 146 type II, 144 superposition, 61, 65—67 surface conductor, 57 current, 57 layer, 57 resistance, 57 symmetry, 4, 13, 16, 61 thermal conduction, 158, 159 conductivity, 158 effects, 161 expansion, 158 translation formulae, 84 trial functions, 27—29 truncation, 91 ultrasonic attenuation, 137, 153, 155, 157 uniqueness theorem, 162 universality, 47 variational approach, 24 bounds, 25, 30, 34 integral, 26, 28 methods, 24 parameter, 29 principle, 24, 26, 33, 46 theorems, 24, 27, 30 theory, 30 VCA, 101, 103, 107, 133 vector fields, 47—51

INDEX viscosity, 148, 150, 152, 158, 159 dynamic, 148 effective, 151, 152 volume fraction, 7, 8, 14, 21, 24—26, 31—34, 43, 44 Young’s modulus, 154