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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
I
I
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
I
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
I
ESSENTIALS
I I I
I I
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,
<77i1,
0
(1.40)
The critical value as shown here for spherical inclusions is 1/3 and the behaviour near this value is linear in the concentration , Both of these features are charac teristic of the Bruggeman theory. The theory predicts that for a metal/insulator composite, the material behaves as an insulator for small metal concentrations until a critical value (in this case 7)1 = 1/3) is reached. Conversely, the system behaves as a metal for small concentrations of insulators until a critical con centration of insulating composites (in this case 772 = 2/3) is added, whence it reverts to behaving as an insulator. Since no theory is ever perfect, and without pre-empting the later discussions (see Chapter 4), we shall briefly discuss the inadequacies of this theory as well (Kirkpatrick 1973). First, experiments and computer simulations are in agreement with eqn (1.39) for the case e For the case € >> 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
—
Pi
; 1,
—
(1.43)
The difficulty here is the close coupling between the critical threshold value and the shape of the inclusions. If the inclusions were needle shaped then the crit ical threshold value predicted by this theory would be close to zero. Here we shall have the same difficulty then as with the Maxwell—Garnett formula; see the comments after eqn (1.22). This result is in violation of a well known result from another area known as percolation theory. Percolation studies (Kirkpatrick 1971) show that the percolation probability, defined as the probability of being connected to an infinite cluster, has a finite percolation threshold concentration which is independent of the detail shape of the inclusions—the universality con cept. One would expect that in our case the critical threshold is identical to the percolation threshold (see Shante and Kirkpatrick 1971) and therefore should not vanishJ° Lest the reader might think that the problem with the formula only exists near percolation, we conclude this section with the following exam ples. The first example establishes the general d-dimensional low volume fraction expansions for the Maxwell—Garnett and Bruggeman formulae for comparison. The superconducting limit is the extreme case where the disagreements between both theories are the largest in magnitude.
Example 1.3 In the (superconducting) limit e -4 cc, show that the Maxwell— Garnett formula for such inclusions of volume fraction , can be written as =1+311+3112+...
(1.44)
Show that for the same system, the Bruggeman formula is given by
=
1 (1—3)
(1.45)
Show also that, not surprisingly, the factor 3 in the above equations should be replaced by d for d-dimensional spherical inclusions. We remind the reader once again (see Table 1.2) that the dielectric constant c used here 10 maps directly on to the conductivity u in percolation theory (Kirkpatrick 1973). This is often used interchangeably in the early literature (Landauer 1978), but we avoid doing so here due to the potential for confusion.
The next example is an important one. Here we see immediately the inferior ity of the Bruggeman formula in that it consistently overcorrects the second-order term.
Example 1.4 Why do you think the two formulas disagree at the second order O(u2)? Note that neither formula agrees with the exact second-order solutions. For three dimensions, this is given by (1.46) where the factor 4.51 (accurate to the last decimal place) comes from a slowly convergent series (of a hundred terms; Jeifreys 1973) and is known in principle to any desired accuracy. For two dimensions, the exact result is e
=
1 + 21] + 2.744989676172 +...,
(1.47)
where the second-order coefficient is again known in principle to any desired ac curacy, a result obtained by an infinite sum of dipole images (Djordjevié, Hether ington, and Thorpe 1996). This has now been extended to arbitrary d dimensions by Choy, Alexopoulos and Thorpe (1998a). Writing the second order coefficient as d + Fd(1), Table 1.1 gives the idea. Since the Fd(1) function diverges like d, the error for the Maxwell—Garnett formula is a ratio d+Fd(1) which approaches a constant for d —* cc. The corresponding error ratio for the Bruggeman formula, as shown in Table 1.1, unfortunately diverges like d. It remains an open question if this poor behaviour can be improved.
The above problems bring out the question as to when the Bruggeman theory could be and when it could not be trusted. This naturally leads to the concept of a validity criterion, and flux fluctuations due to correlations between pairs of
I 16
ESSENTIALS
Ch. 1
inclusions etc. Chapter 4 is devoted to examination of these issues. In the next section, we shall discuss an alternative formulation of the Bruggeman theory using Green’s functions. The mathematical advantage gained will be seen to be useful for extension to dynamics and discussion of perturbation methods, as well as various other topics, in later chapters. 1.5
Green’s functions formulation
The method of Green’s functions is an established powerful mathematical tool for dealing with boundary value problems. In this section we shall generalise the effective medium formulae developed in the last two sections to inclusions of arbitrary shape. For this purpose the use of Green’s functions lends itself readily to the task. The Green’s functions for the problems we have discussed turn out to be dyadics (second rank tensors) and the symmetry considerations become transparent through them. In addition, the approximations used by Maxwell—Garnett (section 1.3) and Bruggeman (section 1.4), will transpire in this formulation to be decoupling approximations for the infinite set of coupled integral equations defining the exact Green’s function for our complex boundary value problem (Stroud 1975). This is a powerful result which will facilitate a critical review of these approximations in future chapters. Thus in what follows we shall reformulate the problem more generally. We return to our inhomogeneous material of volume V which is bound by a surface S, that tends to infinity in the thermodynamic limit. The system is characterised by a spatially varying dielectric tensor 6(x) which reflects its inhomogeneity. We assume of course that the inhomogeneity is random and the purpose here is to calculate the ensemble average (6(x)), which has to be inde pendent of x. This is the quantity that can be experimentally assessed in static measurements. It is convenient to further assume that there exists an ‘ergodic hypothesis’ in that (6(x))
=
€(x)dx,
V
oo.
(1.48)
This is not strictly an ergodic hypothesis in the sense of statistical mechanics (Khinchin 1949). Rather, it is an assumption (so far supported empirically) that the random average over large volumes, given in eqn (1.48) is independent of the choice of the statistical distribution of the inhomogeneities. It is again one of those assumptions that will have to be re-examined for mesoscopic systems (cf. Hadjipanayis and Siegel 1994). Our objective is to calculate E, the effective dielectric tensor, if an external electric field E 0 were to be applied to our sample such that the scalar potential is o(x) = —E .x on the surface S. By definition, 0 e is related to E 0 by (D(x)) = (E(x)) = eE , where (D(x)) is the averaged 0 displacement. In order to calculate f we first expand 6(x) as follows: ’ 1 See section 1.1 and note that E is in general not equal to (6(x)). The latter is in fact an 11 upper bound for q see section 2.2 later.
GREEN’S FUNCTIONS FORMULATION
§1.5
+ 6€(x).
e(x) = €
17 (1.49)
Here 60 is an as yet undetermined constant which we shall fix later. It is com pletely arbitrary at this stage and can range from 6 min to 6 max; that is, the minimum to maximum values among the components respectively. This includes and (€(x)), and the choice, as we shall see, will determine the approximation scheme. Using this expansion, we have (D(x)) The calculation depends us first derive the exact electrostatics, div D = 0 relation D(x) = €(x)E(x)
=
(e(x)E(x))
=
E + (6e(x)E(x)). 0 €
(1.50)
on making good approximations for (6€(x)E(x)). Let integral equation obeyed by 66(x). The equations of and curl E = 0, when combined with the constitutive imply that we have (1.51)
div(€(x) grad (x))= 0. Using eqn (1.45), the above equation becomes
div(€o grad (x))= —div(6€(x) grad (x)),
(1.52)
inside the sample volume V with the boundary condition on the surface S given by .x. 0 (1.53) (x) = o(x) = —E As usual (see, for example, Morse and Feshbach 1953), the solution to this bound ary value problem can be written in terms of the Green’s function G(x, x’): (x)
=
o(x) +
f
G(x,x’)div(ö€(x) grad (x))dx’.
(1.54)
Here our Green’s function is defined by
div(€o grad G(x, x’)) G(x,x’)
=
—6(x
=
0
—
x’)
in V,
for x’ on S.
,
(1.55)
The integral in eqn (1.50) can now be performed by parts, and using the above boundary conditions we arrive at the following equation for the electric field, after taking the gradient on both sides: E(x)
=
0+ E
f
(x,x’)6€(x’)E(x’)dx’,
(1.56)
V
where the dyadic G(x, x’) is here defined by ã(x,x’)
=-—-
8Xcr
P_G(x,x’). 3 1 (9X
(1.57)
18
ESSENTIALS
The formal exact solution for tiplying by 66(x); thus
can now be obtained from eqn (1.56) upon mul
&(x)E(x)
=
&(x)Eo + &(x)
f
Ch.
O(x,x’)&(x’)E(x’)dx’,
1
=
&(x)E(x).
(1.58)
(1.59)
Formally, this satisfies the integral equation
x(x)
=
66(x) + 6e(x)
f
(x, x’)(x’)dx’.
(1.60)
Upon comparison with eqn (1.50) and the definition of , we find that =
o + (x(x)).
(1.61)
It is for the formidable problem posed by eqn (1.60) that ingenious approximation methods are needed. The nature of these approximations depends largely on the inhomogeneous medium we are concerned with, for example a composite material as in the previous sections, or others like atmospheric fluctuations, which generally give 6 both spatial and temporal fluctuations (see, for example, Manning 1993). Needless to say, the appropriate approximation procedure must be guided by the length and time scales associated with the randomness. For a granular material consisting of composite crystallites or cells of different size, shape, and orientation, a class of approximations leading to the effective medium approach in the last sections is obtained as follows. We consider a typical x which lies in an ith crystallite which has a volume v; then for eqn (1.60) we may write
x(x)
=
6e
+öEif
O(x,x’)(x’)dx’ +6eif
(x,x’)(x’)dx’,
19
This approximation decouples eqn (1.60), for now (X(x)) is a homogeneous quan tity, but there still remains the non-trivial task of obtaining the Green’s function G(x, x’) in eqn (1.55). The trick is to avoid having to solve eqn (1.55), which is nasty for an arbitrary shape S of the sample surface. Moreover, its solution depends on the choice of 60. Of the many choices, the ‘best’ one seems to be that which ensures that (x(x)) = 0. (1.64)
It is convenient to define the tensor x(x) by (x)Eo
GREEN’S FUNCTIONS FORMULATION
§1.5
I
The sense in which this is the best will be clear in Chapter 4 when we examine the limitations of EMT. We see from eqn (1.61) that this choice implies that = 60. Notice also that this is a self-consistency condition, for x(x) itself is de termined from eqn (1.62) with eqn (1.63), whose solution depends on a Green’s function containing o (eqn (1.55)). This approximation is analogous to the co herent potential approximation (CPA) as used in the study of electronic states for disordered alloys (see section 4.4). Other choices for 60 and hence (X(x)) in eqn (1.63) are not self-consistent and they do not satisfy eqn (1.64). The latter are analogous to the averaged t-matrix approximation (ATA) in disordered alloy theory (section 4.4) and lead to the Maxwell—Garnett theory, as we shall see. In what follows, we shall specialise to ellipsoidal inclusions. It can be shown that x(x) is in fact uniform inside the cavity. This follows from a theorem of Kinoshita and Mura (1971), whose proof was rather cumbersome. As pointed out by Mills (see ref. 12 of Stroud (1975)), the theorem can be established in a few lines. For if we return to eqn (1.58) with our approximation, which amounts to embedding our ellipsoidal cavity with dielectric constant c in an effective medium of dielectric constant 6 in the infinite volume limit, it is well known from electrostatics-’-see for example Landau, Lifshitz and Pitaevskii (1984) or Moon (1986)—that the field E inside is uniform, 12 since & is uniform there. The latter follows from eqn (1.49) and the fact that 6j is uniform inside. Thus from eqn (1.59) it follows that x(x) is uniform there. Upon substituting eqn (1.63) into eqn (1.62), we can partially integrate both terms, noting that Xi is now uniform, to obtain
(1.62)
where now co is the contrast of the ith crystallite. The reader may = query the last statement, because we have taken 6€, to be independent of x. This is unique to the type of inhomogeneous system we are considering here, for we have postulated a granular composite each of whose elements has a homogeneous e. It is, however, erroneous to assume that x(x) is also homogeneous over the size of the grain, because the electric field E(x) is in general not uniform there. The approximation method consists of noting that the last term in eqn (1.62) couples to all the other cells. It is tempting to replace x by its average in the integrand, i.e.
Xr
= 66
—
6e
dS’
8G(x,x’)
n) (X613
-
)). 13 (X
(1.65)
—
G(x, x’)X(x’)dx’
O(x,x’)(X(x’))dx’. fVv,
(1.63)
Here we have Xi as the value of x(x) for x within the ith cell, and n is the 6th component of the unit normal outwards from the surface 5’ of the cavity.’ 3 In eqn (1.65) Greek superscripts denote Cartesian components and the Einstein summation convention (of repeated Greek indices) is used. We have also used the boundary condition eqn (1.55), i.e. G(x, x’) vanishes for x’ on S in its derivation. This convenient property of the solution to the Laplace equation is unfortunately lost when 12 we move to finite frequencies, as it does not hold for solutions of the Helmhotz equation (see Chapter 3). We have used the terms ‘cell’, ‘grain’, ‘cavity’ and ‘inclusion’ interchangeably so far. A 13 distinction between them will unfortunately be necessary at finite frequencies; see Chapter 3.
20
ESSENTIALS
Ch. 1
We can now pass to the infinite volume limit, whereby G(x, x’) goes over to the ‘free space’ Green’s function satisfying eqn (1.55) with permittivity € and as such is a function of x x’l only. Thus the surface integral in eqn (1.65) becomes a constant independent of x and so —
öfi
Xi
—
öfifi (Xi
(1.66)
(Xi)),
where we have, for the depolarisation tensor:’ 4
3 r
=
—
L
dSIôCfl.
(1.67)
The solution of eqn (1.66) is now straightforward: it is given in terms of Xi
(1
.—
&jI’j)— c 1 5fj (1
—
(Xi)
as
(1.68)
j(Xj)).
On averaging eqn (1.68), we can now solve for (Xi), to obtain (Xi)
=
(((1
—
öEF)-’)) ‘ ((1
—
6€F’&).
(1.69)
From eqn (1.64) we see that the self-consistency equation to be solved for is given by ((1 &r) o€) = 0. 1 (1.70) This equation is in fact identical to eqn (1.41), for a binary system of spherical grains. It is also interesting to show that the solution as given by eqn (1.69), namely
1
§1.6
SUMMARY AND EQUIVALENCE
21
In summary, the procedures involved in the effective medium approach for ellipsoids consist of evaluating the integral equation (1.67) and with a suitable choice for e 0 lead to either via eqn (1.70) or eqn (1.71). The former is the Bruggeman theory and the latter is the Maxwell—Garnett theory. At this point we shall leave the application aspects of the theory for later chapters. Needless to say, the formulae to be derived will depend to some extent on the type of system and properties we are modelling. For example, in superconductors, London elec trodynamics will alter the form of the Green’s functions (Choy and Stoneham 1990) and in elasticity applications they are tensors of higher ranks (Wu 1966), so that the integral equation (1.67) is more involved. Nonetheless, the princi remain the same as discussed here. The advantages of the Green’s function pies formulation as we have seen include: (a) its generality for including anisotropic grains; (b) bypassing the need for a Lorentz field; and, last but not least, (c) the approximation is mathematically more transparent as a decoupling scheme for the infinite set of integral equations. The latter lends itself to ‘Feynman dia grammatic’ analysis, as we shall see later. In the next section we shall conclude this chapter with a summary of the principal results and then discuss various problems that can be embodied in the general formalism described above. In certain cases, due to their exact mathematical equivalence our results can be carried over immediately, while in other cases similar results can be derived (see Chapter 6).
—
= o + (((1 o€F)-’))’ ((1 1 6EF & ), )corresponds to the Maxwell—Garnett formula, eqn (1.19), when that of the host medium.
(1.71)
—
—
€
is chosen as
Example 1 .5 For a spherical inclusion of size a, show by evaluating eqn (1.67) that eqn (1.70) recovers eqn (1.41).
Example 1.6 Similarly, show that eqn (1.71) recovers the Maxwell—Garnett formula eqn (1.19).
This tensor, if suitably diagonalised (via a principal 4 ‘ axes transformation), is related to the depolarising factors; cf. eqn (1.17).
1.6 Summary and equivalence To summarise, we shall provide a set of tables identifying the key equations in this chapter for which the two versions of effective medium theory are commonly employed. We shall refer to both spherical and ellipsoidal inclusions as separate cases. However, it should be noted that the Maxwell—Garnett and Bruggeman formulas are by no means the only two effective medium theories. They are the two most common theories, and like all approximation methods (see Examples 1.1—1.4), have their separate virtues and limitations. The choice of which formula to use will depend on the nature of the problem one is studying and the following Table 1.2 hopes to provide a guide to the reader for this purpose. In fact, as we shall see later in section 2.2, Hashin (1968) has shown that there is an infinity of such theories, which can vary from the Maxwell—Garnett formula to the Brugge man formula depending on an arbitrary continuously varying free parameter, the physical significance of which we shall see later. The comments in the last row of Table 1.2 give an idea of what to expect with either approximation. By the nature of their construction, both Maxwell—Garnett and Bruggeman are exact theories to the first order in the volume fraction. The big discrepancies show up at the second order (see Examples 1 .3 and 1 .4) , where the Maxwell—Garnett is bad, up to 50% error, while the Bruggeman is even worse, at up to 100% error. Yet for the purpose of seeking an estimate of the percolation threshold or critical point T lc, Bruggeman is certainly qualitatively correct and hence superior. Next, in view of the mathematical equivalence of a variety of problems in thermal conductivity, electrical conductivity, magnetostatics , diffusivity, etc , we .
h
Ii
22
ESSENTIALS
Ch. 1
1
§1.6
Table 1.2 Summary of effective medium formulas.
Type of inclusion Maxwell—Garnett Spherical Ellipsoidal Comments
Bruggeman
Eqn (1.22) Eqn (1.25)
Eqn (1.32) Eqn (1.36) Exact at first order, Exact at first order, 50% error at second 100% error at second order, order, critical r, = 1 reasonable critical Tic < 1
SUMMARY AND EQUIVALENCE
23
Table 1.4 Mathematically equivalent problems
Problem
Field
Flux Transport Differential Effective medium Coefficient equation formula
Hydro-
Velocity
Stress
Viscosity
Eqn (6.30)
Eqn (6.42)
Stress
Elastic
Eqn(6.52)
Eqn(6.63)
Eqn(6.68)
Eqn(6.71)
dynamics field Elasticity Strain
constants Porous
Pressure Fluid flux
medium
gradient
Darcy constant
Table 1.3 Mathematically equivalent problems.
Problem Electrostatics
Potential,
E
=
—V
Electric
Electric
potential
field
Magnetostatics Magnetic Electricity
c7
e
D
Permittivity
Electric displacement
Magnetic
Permeability
Magnetic
potential
field
Permeability
induction
Electric
Electric
Electrical
Current
potential
Heat conduction Diffusion
field conductivity Temperature Temperature Heat gradient conductivity Density Density Diffusion gradient
constant
density Heat flux Particle current density
provide in Table 1.3 a list of such problems for which our present formulae can be straightforwardly transcribed. This follows from the universality of the Laplace/Poisson equations, eqns (1.51)—(1.54), which are the underlying laws of physics in these problems. This list must be augmented by Table 1.4, which details problems that can be treated by the same basic principles of the effec tive medium theory, but which differ from the above mathematically due to their different underlying differential equations. In most of these cases we can still derive similar formulas, the only difference being mathematical rather than physical. We shall, however, defer discussios of these formulae until Chapter 6, as we would not want to get too bogged down with technical details at this stage. The above list, while not entirely exhaustive, represents a fairly comprehensive selection of the problems that we shall study in this book. Readers who are inter ested only in applications can turn to the formulae given in the list. Applications to other problems, for example sedimentation, radiation damage, fracture, etc., should follow the same methods once the basic principles as described here and
in the following chapters are fully appreciated. Readers may like to consult the bibliography for related topics.
VARIATIONAL BOUNDS
§2.2
2 RIGOROUS RESULTS 2.1
Introduction
In this chapter we shall study some rigorous results concerning the dielectric properties of composite systems which offer some alternative insights into the problems which the effective medium theory attempts to treat. These results take the form of rigorous bounds (section 2.2), derived on the basis of a variational principle, the well known one initially due to Hashin and Shtrikman (1962a) and its other generalisations, e.g. to elastic composites Hashin and Shtrikman (1962b,c). The variational theorems are special cases of the complementary vari ational principles studied by Arthurs (1980) (see also Sewell 1992). Although Arthur’s work is fairly general, it is unfortunately less useful for the derivation of rigorous bounds than that used here. The reason is that Hashin and Shtrikman were the first to recognise the importance of the polarisation fields in formulat ing their theorems. This approach enables realistic approximations to be made in the form of suitable ansatz for these fields in the system. This led to the famous Hashin—Shtrikman bounds, eqn(2.3) below. Thus, as we shall see, while the vari ational principles provide a rigorous formulation for the effective permittivity in terms of stationary properties of an appropriate energy functional upon arbitrary variations of the polarisation fields, the bounds derived are only as good as the ansatz for these fields. Nonetheless, using the concentric shell model in section 2.3, which furnishes an alternative derivation of the above bounds and the effec tive medium theory, Hashin (1968) was able to show that these bounds are the best possible bounds for a two-component system, where the volume fraction of one of the components is the only available information. We shall explore these variational principles at some length, mainly because of their physical appeal. The complementary variational methods have not been very popular over the years, primarily due to the lack of more imaginative ansatz for the polarisation fields. Yet in systems where an inherent coupling between canonical fields is essential to the physics of the problem, as in inhomogeneous superconductors, or in electromagnetic properties of composites, the complementary variational approach is a powerful tool (Choy 1997). Moreover, these principles are inti mately connected with the Bergman spectral representation (section 2.4) for the dielectric constant, but these connections remain insufficiently explored up to the present day. In this representation, Bergman (1978) first showed that the ef fective permittivity can be expressed in terms of various characteristic functions whose analytical properties are in fact quite general. They depend entirely on geometrical information from the microstructure and in fact provide some useful physical information on the excitations of the system at finite frequencies. It is
25
an interesting but ill understood result that the pole structure of these spectral functions, which are in the unphysical region for the ratio of the dielectric con stants, and which should be real and positive in the static case, should depend only on geometrical information, and that they in turn yield information on the low-frequency excitations in the composite. We shall also examine some exactly soluble models in this context and derive their spectral density functions in sec tion 2.5. Some interesting questions and associated problems will be discussed along the way in the accompanying sections. We end the chapter by discussing various reciprocity theorems and their uses in section 2.6. 2.2
Variational bounds
The existence of certain rigorous bounds for the effective dielectric constant of a composite system has been known for a long time, since the early days of the subject. One of the earliest results was due to Wiener (1912) and was later re derived in the context of thermal conductivity by Woodside and Messmer (1961). These bounds can be written as (2.1) where the averages are defined as in
(c(x))
(2.2)
= T1jEj,
for an rn-component composite, where each component i has a dielectric constant j and volume fraction rj. We shall not discuss the derivation of the above bounds, not just because of their limited use, but because they can be derived more easily as a special case of the variational bounds below. The variational bounds of particular interest to us here were first derived by Hashin and Shtrikman (1962a) and then later Bergman (1976) generalised them to other cases, in particular when partial information is also known about an analogous transport coefficient. For a two-component composite, the Hashin—Shtrikman bounds are given by +
1712 (€2—El)
+
< e < 2 + 3
2 (1—E2)
+
(2.3) 3
A few words are in order before we discuss the derivation of these bounds. First the reader should by now recognise that the factor 3 in these formulae comes from the use of spherical inclusions in three dimensions; the appropriate change for spheres in d dimensions merely replaces this factor by d. Here we will show that the assumption of spherical inclusions is not a necessary condition for these bounds, as long as the system has overall isotropic, or at least cubic symmetry. Secondly, both bounds are in fact intimately related to the Maxwell—Garnett formulae in the previous chapter. For by simple manipulations we can easily see
Ch. 2
RIGOROUS RESULTS
26
that the right-hand side of eqn (2.3), i.e. the upper bound, is equivalent to eqn (1.14); while the left-hand side, i.e. the lower bound, is equivalent to eqn (1.15), if Co and Jo were substituted for €2 and i2 respectively. This is a remarkable result, for regardless of how good the Maxwell—Garnett formula is, the true dielectric constant must lie between these two bounds. Moreover, these are the best bounds we can have in the sense that they either maximise or minimise the stationary value of a variational integral to be shown below. For two-component systems they are also the best bounds that can be obtained in terms of volume fractions alone. For, as first shown by Brown (1955), and indeed as we shall see later when we study the concentric shell model in section 2.4, that further knowledge about the effective dielectric constant necessitates information on the statistical distribution of the inhomogeneities in the composite system. In this section we shall sketch the main steps in the proof of the above bounds, highlighting not only the mathematical results but the physical principles as well. We start from the following well known expression for the internal (electrostatic) energy stored by the field in the dielectric system (Landau, Lifshitz and Pitaevskil 1984) U
=
8ir
—--
f
(2.4)
dr E.D.
Not so well known, however, is the fact that the above expression can be obtained from a variational principle of Brown (1962) and Hashin and Shtrikman (1962a), whose theorem is stated below. Theorem 2.1 The following functional integral, given by
Up
= -
f
dr
(€0
2 p
E —
0 + P.E’), + 2P.E
(2.5)
—
is stationary for arbitrary variations of the generalised polarisation P when P where E
=
=
(€
—
)E 0 €
=
D
—
(2.6)
E, 0 €
0 + E’, subject to the subsidiary conditions E =
—Vb’, o div E’
=
—div P
,
(2.7)
and the boundary condition on all surfaces S that (2.8) In the above equations, E(r) and D(r) are the local electric and displacement fields and €(r) is the permittivity that varies in space in the composite system: €o is an arbitrary constant and Eo(r) is the field that would result from the same boundary conditions if the system were homogeneous. The stationary value of Up, as easily shown by substitution and use of the subsidiary and boundary
§2.2
i
I
VARIATIONAL BOUNDS
27
conditions, eqn (2.7) and (2.8) (Appendix A.1), is exactly the electrostatic energy stored in the system equation (2.4). In fact, a complementary variational theorem exists, which will be of considerable use later on, that is formulated in terms of the inverse dielectric constant = 1/c and a generalised polarisation function R, given by (2.9) The complementary theorem is stated as follows. Theorem 2.2 The following functional integral, given by UR
dr (o D
8
2 R
—
(
0 + R.D’), + 2R.D
(2.10)
—
is stationary for arbitrary variations of the generalised polarisation R when (2.11) where D
=
0 + D’, subject to the subsidiary conditions D div D’
=
0,
curl D’
=
—curl R,
(2.12)
and the boundary condition on all surfaces that the normal component of D’ is zero: D(S) = 0. (2.13) Here D(r) is the electric displacement produced by an assumed fixed sys tem of charges on the surface and thus Do(r) would be that produced by the same charges if the system were homogeneous. The stationary value of (JR is eas ily shown again by substitution (Appendix A.1) and use of the subsidiary and boundary conditions, eqns (2.11)—(2.13), to be exactly the electrostatic energy equation (2.4). Moreover, for €o < € the stationary value of Up is an absolute maximum, while for €o > € the stationary value of Up is an absolute minimum. This situation is reversed for UR, where for € < € the stationary value of UR is an absolute minimum, while for €o > € the stationary value of UR is an absolute maximum (Appendix A.1). In order to derive the bounds in eqn (2.3) we now have to consider the proper choice for the trial functions for P and R. We start by assuming a constant trial •function for P of the form P E P , E 0 (2.14) and, using this in Theorem 2.1, we find upon remembering €eE 1
€ Ce
—
=
(€
—
)
. 0 + 2P
The stationary value of eqn (2.15) with respect to P 0 is then
=
8’irUp/V that (2.15)
28
RIGOROUS RESULTS 1 \(€—€) /
=
€+(
)
Ch. 2
Equation (2.5) shows that the first three terms are trivial:
—
K
€
_€))
0+ + 2(P)E
f
dr P.E’,
(2.18)
where the averages are now defined in terms of the P ‘s; for example, m
(P)
m ij
=
) 2 (F
=
2 r,?, p
.
(2.19)
Most of the work consists of the evaluation of the last term in eqn (2.18). Hashin and Shtrikman used a technique analogous to the Ewald method of summation, by dividing the system into cubic cells L 3 whose size is small compared to the system but large in comparison to the inhomogeneities, and thus a Fourier series expansion for P is valid within each cell. We shall see in Appendix A.2 that an alternative method as used by Bergman (1976) is more useful, as it relaxes the requirement of full rotational symmetry to cubic symmetry and it further shows that the result given in eqn (2.20) below is strictly dependent on the assumption given in eqn (2.17). The evaluation of the last term in eqn (2.18) (see Appendix A.2), gives the expression of the variational integral, as 8Up =
Eg 0 €
-
K
€
€))
+ 2(P)Eo -
29
(2.16)
.
.
E 0 €
VARIATIONAL BOUNDS
—1
Equation (2.16) shows that 6 e is a monotonically decreasing function of €0. There fore the upper and lower bounds can be obtained by choosing co = oo or 0 respectively. We immediately recover from these the bounds eqn (2.1) of Wiener (1912) and Woodside and Messmer (1961). Obviously, we can now see that for an improvement of these bounds a more sophisticated ansatz for P is necessary. Hashin and Shtrikman’s (1962a) choice of trial function P(r) is a constant for each component i of the m components, 15 and thus in place of eqn (2.14) we have P E PE 0 (2.17)
=
§2.2
± [(p2) - (P) ]. 2
(2.20)
In a similar way as the last example, eqn (2.20) is minimised or maximised with respect to P,, and by a suitable choice of eo in the appropriate cases (Example 2.3), we obtain the Hashin—Shtrikman bounds (eqn (2.3)), in the case of m = 2 components.
This cannot be satisfied in general, except for two components in a concentric shell model, 5 ‘ as we shall see in the next section.
Example 2.3 From eqn (2.20), derive the Hashin—Shtrikman bounds for a twocomponent system by extremising with respect to P and choosing for € the minimum or maximum values, i.e. € or €2 respectively. Show also that if the choice €o = €e is made, we recover the Bruggeman formula, eqn (1.25).
If, instead of eqn (2.17), we use the trial function for R in the form R
0 RD
,
(2.21)
where R is again constant in each component, then exactly similar argumerts as the above lead to the same bounds as given in eqn (2.3). The reader should note the importance of the choice of €o. In fact, according to Example 2.3, the lower and upper bounds in eqn (2.3) follow from the choice of € = €min and €0 = €max respectively which, for a two-component system, depends on whether €1 < 2 or € > 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
=
Ca cos8
U
+ Ca sine ue
(2.22)
.
In the concentric shell a < r
Eb
=
(—Gb +
C’ r
cosO u + (Gb + \
C’
-_-) r
sinO
(2.23)
1
§2.3
THE CONCENTRIC SHELL MODEL
31
Outside the whole composite sphere b < r < oo, if there is to be no change in the energy, the only allowable solution is of the form E
=
0 cosO P2
U
—
0 sin6 u E . 0
(2.24)
Now the boundary conditions, which require the continuity in D and in P2 11 across the interfaces at r = a and r = b, lead to the following set of equations: 2 =0, Eob—Cbb—Cb — fbCb + 2ebCb 3 = 0, 2 Caa = , 0 Cba + Ca —
EbCb
—
— EaCa
3 2ebCba
=
0.
(2.25)
These equations can provide a solution for the C’s iff the determinant vanishes. Indeed, the determinant
o
—1 —o 0
b 6
1
—2e,,O 1
0
b 6
€b 2
1
01 0 =0 —1 Ea
(2.26)
I
leads us to the following illuminating formula: = E,
+
,
ab
+
(2.27)
€b 3
where 6 = (a/b) , which is a free parameter in this model. Let us now con 3 sider this replacement process to be carried for every infinitesimal particle in the system, however, maintaining the same ratio 6 in volumes. Then we have Tia = =
6, 1
— 0.
(2.28)
Here we see that by the choice 61,2 = 6 a,b and q1,2 = T ia,b eqn (2.27) is the lower bound of eqn (2.3), while by the choice 1,2 = 6 b,a and 71,2 = T lb,a, eqn (2.27) is the upper bound of eqn (2.3). Hence it follows that the bounds given in eqn (2.3) are the best possible bounds obtainable in terms of volume fractions and indi vidual component permittivities alone. To improve these bounds would require additional information on the statistical distribution of the inhomogeneities, con sistent with the theorem first proved by Brown (1955). Hashin (1968) later considered the effect of such a composite sphere in the presence of an external field P2 0 without invoking the constant energy condition. Then the field outside the composite sphere, i.e. for b < r < oc, now has the form E=(Eo+2) cos6u+(—Eo+) sin6u , 0
(2.29)
32
RIGOROUS RESULTS
§2.4
Ch. 2
where Co is the important dipole polarisability term. The above set of equations suffices in the determination of all the coefficients, Ca, Cb, C, and Co in terms of E . The coefficient Ca is easily solved to be 0
[
SPECTRAL REPRESENTATION
grounds. The free parameter 6 is an interesting quantity. On account of the arguments leading to eqn (2.27) previously and the monotonic character of ic given by eqn (2.36), this parameter can take on only values between
0 9E Ca =
2(1 —6) + (26 + 1) + 2[2 + 6 + (1
—
6)j
=
where 9 ?a and
rib
+ D)b,
(2.31)
are the respective volume fractions as before, while =
CaE,
=
CbE,
(2.32)
are the appropriate averages in the respective phases, and further by definition D = CeEO, so that C0 = Ce, then we have immediately CeEO = CallaE + CbllbE.
(2.33)
Now for a system that is statistically homogeneous with homogeneous boundary conditions, the field is on average just E ; thus 0 0 = 1laE + ribE. E
(2.34)
Upon eliminating Eb, we see that the effective permittivity is entirely determined by the average in the phase a, i.e. through Ca: Ce
=
Cb
+ Tla(Ca
—
Cb)
(2.35)
—.
0 E
Substituting eqns (2.22) and (2.30) into eqn (2.35), we find the following quadratic equation for Ce: 2[2 +6 + /3(1
—
2 6)]ic
—
[2(1 + 26) + 3(1 —[2(1
—
46) + 9(i3 l)ria]k 6) + /3(1 + 26)] = 0,
—
—
(2.36)
where K(?]a, 6) =
Ce(1)a, ö)/Cb,
r’ <6 < 1,
(2.30)
If we now assume that the same procedure of replacing every particle in the system with the same ratio 6 for every particle is used, then the average electric field of the a phase is determined entirely by Ca. Ifl fact, Ce is also determined by Ca in this case. This is easily seen as follows. Since the average displacement ] is given by
(2.37)
and /3 = Ca/Cb. It is easy to prove that this quadratic always has a pair of real roots of opposite signs, whereby the negative root is discarded on physical
33
(2.38)
where rj’ is the volume fraction of the minority or particulate phase, in this case = 1 )a The reader should have no difficulty in showing that the choice 6 = leads us to our familiar Bruggeman formula, eqn(1.25) Ca Ca
—
Ce
+ 2Ce
+?
Cb Cb
—
+
Ce
=0.
(2.39)
Ce 2
The other choice, 6 = 1, leads us back to the Maxwell—Garnett formula, eqn (2.27). The introduction of the free parameter 6 can be thought of as a means of introducing more information than just the volume fractions and thereby improving on the Hashin—Shtrikman bounds. The fact that 6 = 1 provides the Bruggeman formula seems to show that 6 parametrises some nearest neighbour correlation effect, neglected in the latter theory. The exact relation between this parameter and near neighbour particle distributions remains to be investigated. In summary, we have seen that a self-consistent field theory can be con structed that provides some better understanding of the Hashin—Shtrikman bounds and the earlier effective medium formulas. The free parameter introduced in the concentric shell model provides partial compensation for the lack of statistical information in the derivation of the Hashin—Shtrikman bounds. There are re maining problems that are of special interest. In particular it is not clear how the ansatz equation (2.17) should be modified in order to reproduce the results of the concentric shell model. A further understanding of this will lead to improvements in the effective medium theory in general. In the next section, we shall examine yet another alternative view. Here, the perspective on the statistical information is retained in the form of a spectral density function. It has many interesting features, including generalisation to finite frequencies, as well as an intimate relation to the variational principles discussed earlier. 2.4
Spectral representation
The central problem in the study of the dielectric properties of a composite system is the calculation of the effective permittivity Ce of the system in terms of the permittivities of the individual components, and the volume fractions. As discussed previously, Brown (1955) first proved that information about the statistical distribution of the inhomogeneities, in the form of suitable N-point distribution functions, is necessary for a complete calculation of Ce. We shall not prove this statement here, since it will become obvious when we study the exact perturbation series expansion in Chapter 6. Considerable credit should therefore be given to Bergman (1978) who first found that one can separate out the
RIGOROUS RESULTS
34
Ch. 2
SPECTRAL REPRESENTATION
§2.4
35
dependence of Ce on the microstructure of the system by defining a set of ‘geo metrical’ characteristic functions whose analytical properties are rather general. It is particularly important that the poles and residues of these functions also contain certain experimentally observable information on physical excitations at frequencies low enough for the static approximation to Maxwell’s equations to be valid. These spectral functions are also useful for the derivation of variational bounds in the complex domain, thereby extending in a sense the results of the last section. Here we first derive Bergman’s theorem for a two-component system, leaving the more general case as an exercise for the reader. For a two-component system with permittivities-€ 1 and e 2 respectively, Bergman’s results can be stated in the following theorem.
our system to form the dielectric of a parallel plate capacitor. The curl-free electric field in the system must therefore satisfy
Theorem 2.5 In terms of the variables: s —— and t —f-’— = 1 effective permittivity Ce admits the following spectral representations:
where E 0 is the average field over the total volume V, i.e.
—
s the
V.(e(r)E) = 0.
(2.43)
The definition of the effective permittivity follows from the expression for the electrostatic energy stored in this system: feE
fdr 2 e(r)E ( r),
0 = jfdr E(r). E F(s)
E
1-
=
1 C
(2.44)
(2.45)
The latter obviously depends on the applied potential, which we shall set to be zero on one plate and L on the other. We can represent the spatially varying permittivity e(r) as e(r) = e (r) + e 8 1 (r), 6 2 (2.46)
Aa tt
H(t)E1E(s)E1—=.
(2.40)
where 9(r) = 1 if r lies inside the ith phase and is zero if it lies outside. It is clear now that E(r) is a homogeneous function of order zero while Ce 1S a homogeneous function of order one in e, by definition. Thus this allows us to substitute for e and Ce the following variables:
The poles sa, L, t, = 1 sa, and I,- = 1 — , and the residues Acr, Ba, Ca, and Da in the above equations are all real and are between 0 and 1. Moreover, the residues satisfy the following sum rules related to the volume fractions: —
h=
,
m(h)
= C2
fdr Oh(V) , 2
(2.47)
where Ba=Ca7)i,
a
E=—IEoIV.
a
Aa=Daii 1 2 ii a
.
(2.41)
a
Additional sum rules can also be derived if more information is known about the system. For example, if the composite has a cubic or isotropic rotational symmetry, then
:
BaS
— ‘71112
(2.42)
where d is the spatial dimensionality The original proof of the above Theorem 2.5 follows a series of intricate arguments (Bergman 1978) which we shall not reproduce here. Moreover, that proof lacks mathematical rigour and is now replaced by other works, such as that of Golden and Papanicolaou (1983). For mathematical convenience, we choose
The problem is completely defined formally if boundary value problem:
(2.48)
is obtained from the following
V.(OhV) = 0,
t94’/8n = 0
at the walls.
(2.49)
The latter condition is equivalent to a large parallel conductance and is intro duced to facilitate the discussion of a finite system. By analogy with the the last section, we can also define the inverse quantity h = 1/h by a complementary process. This is of course through the definition of the displacement field D(r) and its average D 0 respectively, as defined by
36
RIGOROUS RESULTS
D(r) The effective permittivity expression
e
€(r)E(r),
0 D
SPECTRAL REPRESENTATION
Ch. 2
§2.4
(2.50)
where we admit complex solutions to eqns (2.49) and (2.54) for mathematical convenience. Equation (2.49) can be easily rewritten as
37
can now be given in terms of D(r) via the energy = uV.(8iVi/’),
(2.56)
D(r)2
D/Ee
=
fdr
(2.51)
where a E 1 (61/62). We can now cast the boundary value problem, eqn (2.49), into an equivalent integral equation by the use of Green’s functions. Define the Green’s function for the Laplace equation as —
We can now define a scalar field ç5, given by
D
—IDol
8hV,
(2.52)
in terms of which we have the expression for ñi as =
Here the field
fdr 8h(V) . 2
(2.53)
V G 2 (r,r’)
—
r’),
=
z
1 G(r,r’) r’)V’(r uf dr’ ( V’.(O ’)),
—
V.(8hV)
=
0,
(r’)V’G(r,r’).V’(r’), 1 = z+uf dr’ 9
q(0) (L)
=
0,
Ez+uGb.
=
constant,
5 q 8
—=0
(2.58)
Here we have integrated by parts and used the boundary conditions on G and atthewalls,
fdSeh = 1.
the last line defines the integral operator G. With respect to the scalar product (2.54)
The last condition applies to any plane parallel to the plates, and it follows from div D = 0 and that D vanishes at the walls. The function ñ(h) 1/m(h) may appear to be just an alternative way to calculate m. However, this is deceptive, since we know from the previous section that this quantity can yield further information through the existence of complementary bounds. In fact, the two functions m and rYi include all geometrical information about the system orig inally introduced via the theta functions in eqn (2.46) and they will of course depend in detail on the statistical distribution of the inhomogeneities. Their usefulness, however, comes from certain general analytical properties which we shall now examine. We shall formulate the spectral representation in terms of Green’s functions, which has more physical appeal. Apart from making contact with the previous chapter and the mathematical elegance of that formulation, this bears a close resemblance to the spectral properties of Green’s functions in many-body theory (Mahan 1993) that needs further illumination. There is potential for further new results here that are of particular importance at finite frequencies, i.e. beyond the static approximation (cf. Chapter 4). We first define the scalar product E
(2.57)
where the boundary condition on G(r, r’) is the same as that for except that it also vanishes for z = L. Then eqn (2.49) is equivalent to the integral equation (r)
satisfies a similar boundary problem to eqn (2.49):
—6 ( 3 r
=
fdr Oi(V*.V),
in eqn (2.55), the operator G is Hermitian and bounded. The former follows from the fact that G(r,r’) = G(r’,r) (2.59) is symmetric, which is a consequence of the reciprocity principle of electrostatics, while the boundedness follows from the spectrum of eigenvalues being between 0 and 1. In fact, we can write the solution of eqn (2.58) formally as (r)
=
s(s
—
O)’z,
(2.60)
where s = 1/u as in eqn (2.40). We can now see that the properties of the spectral function follow from the properties of G, since by considering the total charge on the plates Q and the potential difference L we have, for the capacitance, C
=
=
fdS €(r)()
()€e
.
(2.61)
Here A is the area of the plate and we now note that, as before, the integral can be taken over any plane parallel to the plates because the difference is a volume integral which vanishes by virtue of eqn (2.49). This permits us to replace the surface integral by an integral over volume divided by L thus 62
(2.55) =
1d
(1 -u9i)
(2.62)
RIGOROUS RESULTS
38
Ch. 2
§2.5
(2.63)
2.6). We have three cases where the effective permittivity c can be written down easily. First, we have random platelets perpendicular to two capacitor plates, which is equivalent to a set of parallel capacitors. Thus
Thus, by the definition of f(u), we have
f(u) El—
()
=
1 —m(h)
=
fdr
o()
=
EXACTLY SOLUBLE MODELS
= 17161
Following from eqn (2.60), we arrive at the expression
f(u)
=
(z,(s_G)1z).
+
39
(2.67)
72E2,
and the spectral function F(s) is given by (2.64) F(s)
=
(2.68)
-,
In terms of the eigenvalues s, and eigenvectors A’’ of G, where symbolically G A’’
=
s,LA’,
(2.65)
whose only pole is at infinity and the residue is i. Note, surprisingly, that the other trivial case is the complementary one which is equivalent to a set of capacitors in series. Thus
we now see that the spectral representation has the familiar form f(u)
=
1
I(z4)I
E
F(s).
(2.66)
The reader can now be convinced that the rest of the results in Theorem 2.5 follow by similar arguments. Readers who are interested in a rigorous derivation using more advanced analytical function theory should consult the work of Golden and Papanicolaou (1983). In the next section, we shall employ these formalisms to study the spectral properties of a number of exactly soluble models. The reader should by now see that the spectral functions are analogous to the concept of the spectral representation of Green’s functions and the density of states in electronic structure. The latter concept is of profound significance in disordered systems, since the Bloch theory of solids become inapplicable here. Likewise, we shall see that the few non-trivial systems in which the spectral functions can be exactly solved are periodic systems.
Example 2.6 From eqns(2.36)—(2.38) for the concentric shell model, derive the pole structure for the spectral function, there denoted by k(h), as the Hashin parameter 6 is varied. By examining the pole structure of the spectral functions and a consideration of the variations of these functions upon variations of the
poles and residues, derive once again the Hashin—Shtrikman bounds of eqn (2.3).
2.5
Exactly soluble models
We shall first examine a few cases where the boundary value problem is trivially soluble and observe the properties of the spectral functions (see also Example
1
1
e 6
61
1
(2.69)
and the spectral function is now F(s) =
(s
(2.70) —172)’
The third case is the low-density distribution of spherical inclusions of ci in a host consisting of Here we meet, once again, the Maxwell—Garnett formula: + = lji 2
(6e
62)
(——-
+
362
(2.71) (61
62)
and now the spectral function is slightly more interesting: F(s)
= ,
(2.72)
(s—172)
in which the pole is now shifted to ii2. Now we see that for a system of different pure dielectrics at dc, where all the permittivities are real and positive, the sys tem can be modelled as a network of lumped capacitors if each grain is modelled in detail with capacitances. However, if the system is near one of the poles of the spectral function, then we require that at least one of the component c’s be negative. Therefore the only way in which the composite can be modelled as a lumped circuit is to include inductances and to change the situation to ac, i.e. to apply a finite-frequency alternating voltage to the system. In fact, we can imme diately conclude that any pole of the characteristic function rn(h) corresponds to a series resonance of the network, where the total admittance diverges, while every zero of m(h) corresponds to a parallel resonance of the network, where the total admittance vanishes. In reality, this property is inherent to metal- insulator composites, since for a metal for frequencies below the plasma frequency of the electron gas, the dielectric constant is by nature real and negative:
40
RIGOROUS RESULTS meta1 = 1
—
Ch. 2
§2.5
(2.73)
assuming that the eigenfunctions /‘aa are an orthonormal set, we have converted eqn (2.58) into a set of linear algebraic equations in the coefficients Aa. This has the form (2.77) Aa = Zaa + IL Ga,bAb
In other words, the poles provide information on the excitations of the sys tem which, for a metal—insulator system, can be directly probed experimentally. Example 2.6 also discusses the analytical structure of spectral functions in the Bruggeman formula. This case is special because it has a branch cut, which can be viewed as an infinite sequence of poles forming a continuous band. Of particular interest to us is the behaviour of this analytical structure when the percolation threshold is reached, i.e. when i -÷ 7 Since the Bruggeman 1c = formula is only an approximation, an interesting question is whether the be haviour of this singularity holds true for all real systems near criticality. There are heuristic arguments (see, for example, Bergman 1978) suggesting that one of the singularities must approach zero, i.e. h —* 0 at criticality. However, the nature of this singularity is uncertain. Without doubt, it is here where an exactly soluble model would yield useful information. Thus we shall visit an old problem of a simple cubic lattice of identical spheres, first solved by Rayleigh (1892). We shall re-examine this problem using the spectral function representation. Sev eral authors have also studied this problem in greater depth, among which are Meredith and Tobias (1960), Doyle (1978), McPhedran and McKenzie (1978), and Bergman (1979). In his classic paper, Lord Rayleigh (1892) introduced a cubic array of identical spheres of permittivity € and radius a 0 embedded in a medium of permittivity E2. What makes this problem soluble is of course the periodicity of the system. In fact, this problem has been re-invented many times in different guises, a well known example being the muffin-tin potential in the quantum theory of solids. To solve this problem, we note that we can write
EXACTLY SOLUBLE MODELS
b13
where the matrix elements are given by (Obaa,z) = fdr 0 a
Za E
.
01 = 0a,
GGa,
Gaa,b
(2.75)
where the superscript + is to ensure that we always consider a volume that is infinitesimally larger that a , in order to properly define expressions such as 0
0i(r)V(0jF(r)(r)) = 0a(r)V’/(r)
(0
=
Sb
(2.76)
Substituting eqn (2.75) into eqn (2.58) and taking the scalar product with 0’aa,
(2.78)
az
aa,G0tbb) = (0aa,Gbbb)
f
dr 0aV V.
(2.79)
.
Here sb is an eigenvalue of Ob, i.e. Gb’b = sb,3/)b3. The integral occurring in eqn (2.79) is not the usual scalar product and has to be evaluated with some care. It is in fact an overlap type integral involving the product of eigenfunctions of sphere b with sphere a, integrated over the volume of a. The key to the solubility of our problem is of course the translational symmetry, which in view of Bloch’s theorem means that our coefficients Aaa must be of the form Aa = Aa(k)
ei(a
(2.80)
where k is the usual wave vector in the first Brillouin zone of the lattice. Sub stituting this form into eqn (2.77), we have the equation ZSk,O
+UGac,b/3(k) A(k),
(2.81)
where in view of the equivalent spheres, Ga,b is a function only of the transla tion vector a — b, so that
where the sum is over all spheres a. Now any eigenfunction of G can be expressed as a linear combination of the eigenfunctions ‘t ’ac. of each of the Ga, for points r 4 that are inside a sphere. Hence
8(r)(r) = Aaa 0(r)?J’ac(r),
E
A ( 0 k) =
(2.74)
41
Gao,b(k)
Gaa,b
E
eik_l,
(2.82)
a—b
We see that the only non-zero solutions for A (k) when k 0 must be eigenvec tors of Ga,b(k). Infact, for the spectral density function we need only concern ourselves with the k = 0 components, for by inserting the expansion of b in eqn (2.75) into eqn (2.63) we can easily see that f(u)
= VZAacr =
= 0),
where N here is the total number of unit cells in the lattice.
(2.83)
42
RIGOROUS RESULTS
Ch. 2
2.5.1 Calculations The calculations now proceed in three steps: (i) evaluating the matrix elements , (ii) evaluating the overlap matrix elements Ga,b and finally (iii) inverting 3 z, the matrix equation, eqn (2.81). In order to proceed with step (i) we shall need the eigenfunctions for a single sphere. This is a by now familiar problem, for the solution of eqn (2.49) which satisfies the appropriate boundary conditions, i.e. the continuity of and c is, as expected, given by spherical harmonics r < a 0
= rYjm(),
02+1 a
—
§2.5
EXACTLY SOLUBLE MODELS
—
G
— —
0I
—
=
—<m<,
(2.84)
Qim,’m’ = Q’m’,m =
jI
‘
1(2 + 1)(2’ +
( + £‘ ( + £‘
—
rn’
—
mj)!u(f + £‘, rn’ — ml) m)!u( + £‘,m’ + m),
m’ 1J + £‘ m’)!u( + = (+f’)!cr(f+’,0), +
—
—
—
r
£ = 1,m = 0,
£1,m0.
(2.85)
—
m,m’ > 0,
m’),
+ 2 H)[( b b 1)(2’ + (+e’ +m—m’)! —
(2.90)
Further progress can only be made by a numerical evaluation of the latter sum in general. All our results will depend of course on the volume fraction i, which in this case is merely the ratio of the volume of the spheres to the unit cell: This completes our discussion of the exact solution of the Rayleigh = 4 j model, for once all the matrix elements has been completed step (iii) can proceed, for example, by diagonalising G and the normalised eigenvectors A’m will give us the residue of each pole. In view of eqns (2.66) and (2.88), the spectral function has poles at s , with residues as given by 1 I(z,A)I2
m()t
m)!(f’ + m’)!(f’
—
—
m’)!)
em1_m)P(m’_m)(cos8b).
(2.86)
In view of the degeneracy in m, it is appropriate to take odd or even linear combinations of m which form two mutually orthogonal subspaces: bim
seöm,’m’
b0
—
(( + m)!(f
+
m = 0, m’ > 0, m=m’ =0, (2.89)
£‘,
= 3’\2
The evaluation of Ga&,b in step (ii) is the first non-trivial exercise in this prob lem. One approach (Bergman 1979) is to find an expansion of the multipole eigenfunctions ?/bem which are centred around b to the origin using the matrix elements of the tranlation operator. This expansion will, not surprisingly, involve a series of 3 j coefficients. Other alternative methods might involve the use of the Gegenbauer polynomials, since 1/)r bI’ is its generating function. In any case, the matrix elements for a pair of spheres with radii a and a’ at the origin and at b = (b, 0 b, b), in the states £m, £‘m’ are given by
X
I
where b 0 is the cubic lattice constant, s is as given in eqn (2.84), and
dr
=0,
GOm,b’m’ = (_l)
V
[( + m)!( — m)!(’ + m’)!(’ — m’)!jQem,t’m’
where r is measured from the centre of the sphere whose radius is a 0 and Y,m (el) is a spherical harmonic function. Using this result, it is now straightforward to calculate the matrix element Zm, by the use of the properties of the spherical harmonics:
,‘4ir
1 e uia0e+e!+ir
—
(2.88)
£
f
(2.87)
Upon summing eqn (2.86) over b for a cubic lattice, in which case the only nonvanishing elements require £ + £‘ to be even and m’ m to be an odd integral multiple of four, we need only consider the following matrix elements:
x
1 f 2 Se
=
.
E
r > a , 0
I—i-j--jYem(I),
Zm
43
±
-m),
m >0,
3’41L
‘2
t ’ 1 V3 °
=
0 I 1 2 l A
(2.91)
This simple-looking result is in fact deceptive. We shall see, in particular, that more interesting information can be obtained in the low-density limit as well as at the percolation limit i —+ , where T k = r/6 is the critical volume fraction when the spheres begin to touch.
Example 2.7 From the low-density expansion of eqn (2.91) for the Rayleigh model and by employing the results given in eqns (2.88)—(2.90), show that the spectral density function to the lowest order in i is given by
44
RIGOROUS RESULTS
F(s)
2 AI s—s’
+
3 A
Ch. 2
2
3 s—s
— 252i (1 — 1 + 24i 13
s
—
i)
3
13
25224]1 W
+ S
—
(2.92)
10
+
—
24ij
where the lattice structure factors are given by
earlier than that they were also well known in the context of electromagnetism, especially in antenna theory. In Chapter 3 we shall also mention the important modern formulation of scattering theory in which reciprocity relations provide the fundamental framework. Here we shall discuss an exact rigorous result first obtained in a special case by Keller (1964), but later proved more generally by Dykhne (1971). Unfortunately, the result only holds in two dimensions, and states that, for a two-component system, for example with intrinsic permittivities i and 2, then the effective permittivity €‘, which by definition is a function of these intrinsic permittivities, must satisfy the simple relation E:(61,62)6;(62,61) = 6162
ri+1
a
(n+1)
P(cosOb)
fl(--)
.
(2.93)
bO
45
RECIPROCITY THEOREMS
§2.6
(2.95)
,
where the second factor on the left-hand side pertains to a reciprocal medium 2 in terchanged. The proof of this relation is quite straightforward and is based on an elegant reciprocity transformation. We start from the intrinsic constitutive relation D(x) = (x)E(x), (2.96)
that is one with the same microstructure but with the roles of e and
Observe the changes to the spectral function at this order in comparison with eqn (2.72) and discuss their physical origin. The behaviour of the spectral function, eqn (2.93), at the percolation point for the cubic array is an interesting = one. Show that the dominant singularity at the ‘conductivity threshold’ u = —cc is given by f(u)
U\
/
—ir 1n1 —
i.),
and the two static Maxwell equations
(2.94)
a result due to Batchelor and O’Brien (1977). Studies of the behaviour of the spectral function near the threshold (Bergman 1979) show an essential singular
ity, which is thus not very useful for critical properties. It is unclear, however, if this behaviour is general or pathological to the periodic lattice.
curl E(x)
=
div D(x)
0,
=
0
,
(2.97)
together with the definition of the effective permittivity: (D(x))
=
*(E(x))
(2.98)
Following Dykhne (1971), we define new ‘rotated’ vector fields: Example 2.7 contains some important results associated with this model. Even though the random character of our problem has been simplified by the periodic lattice structure, we see here that beyond the lowest-order expansion,
where the Maxwell—Garnett formula holds, higher corrections bring in new poles as well as the requirement of more lattice information than just the volume fraction, in this case In fact, the Maxwell—Garnett pole and its weight are now modified, with an additional pole appearing at a higher order. While the usefulness of such exactly soluble models may seem to be of limited relevance to actual composite materials, it is obvious from the discussions here that they furnish good testing grounds for any approximation methods we might envisage using. In fact, this example shows that with regard to the important percolation limit, even this model is highly non-trivial. In the next section we shall conclude this chapter by summarising various reciprocity theorems which are often useful for the solution of conjugate problems. Reciprocity theorems Reciprocity theorems are an important part of physics and have been exploited for a long time, with famous examples such as that due to Onsager, but even 2.6
D’(x)
=
€R E(x),
E’(x)
=
--ã D(x),
(2.99)
where c, is a constant and H = x is the rotation operator in the (x, y) plane through 90 degrees. This choice of fields is selected to satisfy the static Maxwell equations curl E’(x) = 0, div D’(x) = 0, (2.100)
which follow from eqn (2.97). By comparing eqn (2.96) with eqn (2.99), the following constitutive relation is obtained for the ‘primed’ fields D’(x)
=
‘(x)E(x),
ê’(x)
=
. 1 6’(x)ñZ
(2.101)
For an isotropic binary mixture, the tensor jj (x) = e(x)ö, with €(x) taking on the values € or 2, respectively in phases I and II. One can now choose = det = 6162, so that 6(x) = c’(x)S is also isotropic. Hence 6
(x)
=
6162
€(x)
(2.102)
46
RIGOROUS RESULTS
Ch. 2
which take on the reciprocal values 62 and 61, respectively, in phases I and II. Thus eqns (2.100) and (2.101) are similar to the original eqns (2.97) and (2.96), which describe a new system with €i and 62 interchanged. Thus in the primed system the effective permittivity will be defined as in eqn(2.98) (D’(x)) where
‘
= *(6261)
=
*I(E!(x)),
3
(2.103)
DYNAMICAL THEORY
Now, from eqn (2.99) we have 3.1
(D’)
=
112 ) 2 € 1 (€ (E),
(E’)
=
) (eie ( 2 D). /
(2.104)
By comparison of eqn (2.98) with eqn (2.103), subject to the constraint imposed by eqn (2.104), these equations are compatible if the tensor )_l:ñ,_le*Iãe* = 2 (ei€
I,
(2.105)
where I is the unit tensor. This concludes the proof of the Keller—Dykhne reci procity relation, eqn (2.95). In the case of an isotropic medium where = = 6’, then the relation becomes 6*(61,62)6*(62,61) = 6162.
(2.106)
Some applications of this reciprocity relation can be seen in the following exam ple.
Example 2.8 Do the Maxwell—Garnett and Bruggeman formulas satisfy the Keller—Dykhne reciprocity relation? Show that for the case of statistically equiv alent distribution of the two phases we have the simple formula 6* = (61 62)1/2 and hence deduce that the percolation threshold occurs at the fraction c 1 =c 2 = 1/2.
As mentioned earlier, unfortunately there is no reciprocity relation in dimen sions greater than two. In fact, it has been proved by Schulgasser (1976) that reciprocity relations of the type given above does not exist. In three dimensions, eqn (2.106) is replaced by a weaker inequality: 6*(61,62)6*(62, 61)
6162,
(2.107)
which was derived by Schulgasser (1976) using a variational principle. In this case the relation has much less value and the bounds we have obtained in the earlier part of this chapter are more useful. This concludes our survey of the main rigorous results in the field. In the next chapter we shall commence our study of the dynamical effective medium theory.
Introduction
In this chapter, we shall embark on the study of the dynamical theory. When considering the dynamical theory for random media based on the studies we have made so far in this book, there are several issues that must be raised and clearly spelled out before we can proceed. In sections 3.2 and 3.3 we shall devote ourselves to this task. The first complication we will encounter as we shall see, is the loss of the universality (and simplicity) of Laplace’s equation. In the subse quent section 3.4 we shall explore the theory in the so-called quasi-static regime. This in fact comprises two cases, the static limit, appropriate to dielectrics, and the skin-depth regime, suitable for metals or conductors. The latter occurs when losses due to the skin effect become significant. In these regimes, exten sion of the EMT is in fact straightforward, but non-trivial to justify. We shall discuss the latter problem in section 3.8. In section 3.5 we shall look at the full Maxwell theory, whose single-particle properties are exhibited in the well known Mie scattering formulae (section 3.6). In section 3.7 we shall formulate the dynamical effective medium theory and discuss their simplifications in the limit of the various regimes, recovering the results of the earlier sections 3.4 and 3.5. In section 3.8 we shall end by discussing various problems associated with dynamical theory. 3.2
Review
The purpose of this section is to first review some of the issues that concern dynamical properties in general and to orientate the reader towards the devel opment of the dynamical effective medium for electromagnetic systems. In the context of the latter, on which most of this book is based, the underlying equa tions of motion will now be the full set of Maxwell’s equations of electrodynamics. These consist of the eight linear differential equations, the two curl and the two divergence equations, for the five vector fields E, B, D, H, andJ plus the three constitutive relations (see section 3.3). This set of equations is unique to electro dynamics, and since we have 17 equations for 15 variables, this implies that the equations are not all independent, a point we have to be concerned with. In other contexts, such as in the theory of elasticity or in hydrodynamics, the equations of motion are different, so that no general conclusions can be made on the basis of studies in any one field. Nonetheless, it is our hope that the principles used for the study of an electromagnetic system, particularly in the case of the effective medium theory, can with suitable modifications be adopted to other fields, even though the underlying mathematical structure may differ widely. It is with this
48
DYNAMICAL THEORY
Ch. 3
goal in mind that we shall begin our studies in this chapter. Moreover, there is a certain phenomenon associated with wave propagation in a random medium which is universally described by a Helmholtz type differential equation for the appropriate vector and/or scalar fields. In electrodynamics this wave equation is exact, while in other cases it is approximate, valid only in the linearised regime, as in acoustic waves in solids or in fluids. Thus the theory developed will be useful for a broad range of problems other than for electromagnetic systems. To start with, time-dependent or dynamical phenomena can take on many forms, even if we were to neglect transient phenomena, which we will not deal with in this book. So far we have been treating only ‘frozen’ composites whose un derlying disorder is quenched and thus its probability distribution is fixed during the process of manufacture. Although this class of system is large and significant, nature contains other equally important, if not more interesting, cases. For exam ple, even if the electric field applied to a sample is static, the composite particles can move, as in colloidal suspensions, and time is now involved. In other cases the particle properties may be subjected to random fluctuations due to thermal noise, shot noise, atmospheric fluctuations, or other intrinsic quantum noise ef fects. Next even in the case of ‘frozen’ composites, which is the main focus in this book, when the external fields are time varying, the response of the system will generally be an average over time scales of the order of the period of the applied field. This procedure averages all electronic or molecular processes within each grain, so that the intrinsic frequency and spatial dependence of the dielectric constant e(r, w) will contain all of the information we need to characterise each grain. This description, which in turn implies that all the internal fields now have a related (r, ) dependence, owes its validity to macroscopic electrodynamics. We shall therefore review the latter theory in section 3.3. This picture is of course not specific to electrodynamic properties, since in other areas such as acoustics, the field equations of continuum mechanics are already macroscopic averages in some sense. 16 Just as the equations of hydrodynamics can be derived by a coarse graining of microscopic laws of motion, in this case Newton’s laws, applied to an ensemble of particles, the same is true of macroscopic electrodynamics, but with one exception. The latter is a point of debate at the moment and it pertains to the question as to whether the equations of macroscopic electrodynamics can be derived solely by a statistical averaging process from the microscopic Maxwell equations, as in Lorentz’s approach. We shall not debate the matter in this book (Yan 1995), but we will accept Maxwell’s equations in continua as an empirical truth. The main focus for us will in fact be in Maxwell’s case, the propagation of electromagnetic waves in a random media. We will define the latter along the way when needed, as the range is quite wide, spanning from dilute systems like aerosols, colloids, and fluids to dense systems like amorphous solids, polymers, and biological structures. The macroscopic description fortunately also has a wide regime of validity: it is in fact valid (with suitable modifications) up to The proper treatment of this averaging process is a problem of statistical mechanics, de 16 Groot (1969), de Groot and Suttorp (1972).
1
§3.3
MACROSCOPIC ELECTRODYNAMICS
49
near X-ray frequencies for most laboratory systems (see, for example, Landau, Lifshitz and Palevskii 1984). However, when time, and therefore motion, is involved, one problem we must contend with in electrodynamics is relativity. This is because electrodynamics is fundamentally relativistic, as the speed of light c in Maxwell’s equations is finite and not infinite as in Newton’s theory. While it is not necessary that a covariant formulation of electrodynamics be used, since the motion of the particles that we shall consider is much less than c, it is necessary to note that the phase difference arising from path differences due to scattering on the scale of wavelengths will dominate the physics in nearly all cases. This is where the dynamical problems differ from the static problems we have considered so far. It helps to remem ber that the equations of electrodynamics are ultimately derived on the basis of a variational principle, and the finiteness of c entails a covariant Lagrangian description. It is in fact useful to think in this way and then, thereafter, for mathematical convenience and empirical purposes, transfer to the physical fields E, B, D, H, and J which are the measureable quantities.’ 7 The reason is that for time-varying fields and motion, and therefore electrodynamics, the characters of these measured fields are in no way unique: they can be transformed into one another via Lorentz’s transformations. The fundamental fields in the theory are in fact the four-vector fields A’ 2 and J1i, and the measured fields are admixtures of these obtained via covariant derivatives. Insufficient care in treating these fields can lead to errors and paradoxes. Fortunately, this situation arises only for electrodynamics or magnetohydrodynamics, since under most laboratory condi tions the relativistic nature of hydrodynamics and acoustics can be neglected altogether. The price to pay here, however, is that these theories are inherently non-linear, a complication we will unfortunately not be able to fully address in this book (cf. Chapter 6). Thus a linearised wave equation has only a limited regime of validity, as for low-velocity Reynold’s transport in fluids (Landau and Lifshitz 1989). 3.3
Macroscopic electrodynamics
The macroscopic equations of electrodynamics were written in their original form in 1860 by Maxwell (1873) on the basis of a theoretical analysis of the well known empirical laws due to Faraday, Ampere and Gauss (see also Whittaker 1989). We shall first write them in the form given in most standard texts (Stratton 1941, Jackson 1975, Landau, Lifshitz and Palevskil 1984) and curlH=-J c curl E
=
18D
1013 c
(3.1)
(3.2)
Quantum mechanics does enable us to measure the fundamental electromagnetic fields, 17 which are A(r, w) and (r, w), but only indirectly via the Aharanov—Bohm effect.
50
DYNAMICAL THEORY
Ch. 3
We note here that in vacuum, where B = H and D = E, eqn (3.1) and (3.2) completely determine the two vector fields E and H. The two divergence equations associated with Gauss’ law are in fact not independent of the above equations, but can be derived from them by the requirement of causality.
Example 3.1 By taking the divergence of eqn (3.2) in vacuum, we have
div H
=
0,
(3.3)
which upon integration between the times to and t gives div H(r,t)
—
div H(r,to)
=
0.
=
0,
(3.5)
in all regions of space—time. In the same way, we can derive the second divergence equation div E(r,t) = 0. (3.6)
Equations (3.5) and (3.6) are thus not independent Maxwell equations: they are sometimes known as the compatibility relations, which must hold for all causal fields. This is also our first example where the special relativity principle mani fests itself, since in the relativistic formulation, the above equations are derived from a single variational principle, whose Lagrangian density is
L= -F’”F +
_i) —JA,
(3.7)
—
where the T are the Lagrange multipliers, introduced to allow arbitrary varia tions of both fields F 1 and A (see, for example, Lanczos 1986). This variational treatment leads to the well known field equations (3.2) and (3.6), which in rela tivistic form are part of a single expression: DI 0F 8F DxT + Dx1L + Ox’
—
—
0
§3.3
MACROSCOPIC ELECTRODYNAMICS DFIIV_j ‘ Dxv’
51 (3.9)
see, for example, Lanczos (1986), Landau and Lifshitz (1987), or Eddington (1975). Thus the causality conditions are built into the relativistic treatment as required by the finite speed of light. In a material medium the causality conditions applied in the same way as above to eqn (3.1) lead to the solenoidal equation for B: div B(r,t) = 0. (3.10) This equation is now fundamental, which incidentally implies the popular state ment for the non-existence of magnetic monopoles, and as we have seen here is a consequence of causality. 18 The second compatibility relation, which is obtained by taking the divergence of eqn (3.1), leads to a continuity equation:
(3.4)
We now assume that the field H(r, t) was switched on at time to, so that H(r, t) = 0 and therefore div H(r, t) = 0 for all t
1
38 (.)
The other two equations, eqns (3.1) and (3.5), also arise from this variational procedure as part of a single relativistic formula:
div J + div D
=
0,
(3.11)
where we have assumed the absence of all external charge or current sources for convenience. It is through this continuity equation that the electric charge is historically defined as div D = 4irp. (3.12) Here we should note that the quantity p in eqn (3.12) contains only free charges and that we have taken some special care with the above exposition to avoid mistakes which can easily occur, even for static fields.’ 9 We now come to the constitutive relations. Since we have, as noted in section 3.2, an underdetermined set, eqns (3.1) and (3.2), for five vector fields, we need to have three vector relations among the five fields to completely determine all the fields in the system. Conventionally, these are the constitutive relations, the first of which is given by D = eE. (3.13) This is an innocent-looking expression, as we often suppress all the arguments for convenience, and it hides a range of possibilities which we must elucidate. We should first note that the dielectric properties can depend not just on the E field but also on the B or even J fields, depending on the material; for example, one that is magneto-optically active. Secondly, for most materials that we shall be concerned with, the constitutive relations are assumed to be local. Thus the fields D and E are functions of the same space point. The obvious exception to this is a BCS superconductor, in which case an integral over non-local contributions must he formed, but we shall only treat special cases like this as they arise. 1t is for this reason that conventionally we associate (E,B) with the electromagnetic field 18 while (D,H) are constitutive fields due to material polarisation. This is especially true if the constitutive equations are non-linear, as for example the energy 19 density of the electromagnetic field is no longer simply E.D, Panofsky and Phiilips (1964), see also problems in Chapter 6.
52
DYNAMICAL THEORY
Ch. 3
Thirdly, and of far more importance, is the time dependence, which for linear systems takes the form: ° 2 D(r, t) =
f
e(r, t
—
t’)E(r, t’)dt’,
leading to the familiar local, instantaneous relation =
e(r)E(r,t).
and the statement of Ohm’s law given by =
uE.
Q—
2 are the inner , 1 where B 0 is the axial field, w is the rotational frequency, and r and outer radii of the cylinder respectively. A modern account is given by Pel legrini and Swift (1995) and Ridgely (1998) and is, in brief, a consequence of the Poincaré—Lorentz group. The reader is thus warned that macroscopic elec trodynamics involving rotations is particularly troublesome, and we shall avoid ’ We shall conclude this section by merely mentioning 2 it altogether in this book. that, from our macroscopic Maxwell equations, we can derive the fundamental energy conservation equation or Poynting’s theorem as follows.
Example 3.2 By taking the dot product of eqn (3.2) with H and eqn (3.1) with E, show using the vector identity for the divergence of a curl, and an integration over an arbitrarily large volume V enclosing the sample, that
(3.16)
This is actually connected with the quasi-static limit, which we shall look at more closely in the next section. Equation (3.14), however, generally implies that the dielectric constant is frequency dependent, as in the case of a metal or semiconductor above microwave or optical frequencies respectively. With these provisos in mind, we shall write down the remaining two constitutive relations as B = huH, (3.17)
J
53
surfaces of a rotating material cylinder in an axial magnetic field. This is a non-trivial result: , 0 (3.19) V= )(r —rB
(3.14)
where the range of integration is dictated by causality. This means that Only events from the infinite past up to the present time t are able to determine the system’s response. Here we also note that the convolution structure of the integrand is dictated by the consequence of time-invariance, as clearly there is to be no preferred time origin for the system. In some instances, and especially at frequencies much lower than all microscopic relaxation processes, the system’s response can be assumed to be instantaneous, in which case we shall have a delta function for e; thus e(r,t—t’) =e(r)6(t—t’), (3.15)
D(r,t)
THE QUASI-STATIC REGIME
§3.4
(3.18)
These constitutive relations, eqns (3.13),(3.17), and (3.18), and the two Maxwell equations, eqns (3.1) and (3.2), are the starting point for the study of dynamical phenomena as focused upon in this chapter. We now come to the second example of the significance of special relativity in electrodynamics: the above constitutive equations are in general dependent on the frame of reference. Minkowski (1908) was known to have written them down in covariant form for uniform linear motion. We will not worry about this here, since the effects are of order O(v/c), where v is the frame’s velocity. What does concern us here is the case of rotating frames whose effects are not O(v/c) even for small velocities. This is the famous case of Thomas’ precession or, in the macroscopic case, the experiment of M. Wilson and H. A. Wilson (1913), who measured a voltage zV, induced on both Non-linear systems need more care and we shall not have the occasion to deal with them: 20 cf. the previous footnote; see also Landau, Lifshitz and Palevskii (1984).
+ .E)dV =
-
f
E.J dV + f(E x H).dS,
(3.20)
where the last integral over the surface S of the volume contains the Poynting vector, S = E x H, associated only with radiation energy.
3.4
The quasi-static regime
The quasi-static approximation, (see, for example, Landau, Lifshitz and Palevskil 1984), pertains to low-frequency situations, in which the applied magnetic field varies slowly with time. The characteristic feature here is that the wavelength of any electromagnetic field arising from the applied field A = 2irc/w must be large compared with the dimensions £ of the sample. In this case the external field can be treated as static and is completely determined by
div B curl H
=
0,
=
0
(3.21)
.
Inside the conductor the approximation consists of retaining only the equations
div B
=
curl H
=
0 4irJ ,
(3.22)
Rotating frames may be unavoidable, however, if we have to consider general elliptical 21 polarisation of light.
54
DYNAMICAL THEORY
Cli. 3
1 OD (3.27) c--’ (cf. eqn (3.23)) is now of a higher order of smallness, in fact O(/A). Thus the quasi-static limit for dielectrics corresponds to the static limit, where we have the convenience of the Laplace equation as before, although the c’s may of course now be frequency dependent(cf. eqn (3.14)), while for conductors we have a more difficult diffusion type Helmholtz equation, eqn (3.25), to determine the fields. With these preliminaries, we shall use the results of the previous chapters an effective medium theory for both cases. Needless to say, for the formulate to the Maxwell—Garnett and Bruggeman formulae remain unaltered case, dielectric for the frequency dependence in the c’s. Thus we need not display except form, in them here; instead, we shall concentrate on the conductor case. The key to this is the solution of the Helmholtz equation, eqn (3.25), for a single sphere, and by considering a harmonic external field Hoe_t, we have
where J uE by Ohm’s law, and the electric field E is determined solely via the time variation of the magnetic field through curl E
=
curl H
(3.23)
c at
From this point of view 22 the electric field in the conductor plays a secondary role; it is induced by the varying magnetic field through the Maxwell equation (3.23. In fact, the role of E can be eliminated as such and we can easily derive 4ir OB
—i-
= --
—curl (u’curl H),
(3.24)
which for uniform electrical conductivity u and magnetic permeability p gives a diffusion type equation for H as 23 curl curl H
OH =
—
2 c
(3.25)
This equation, together with the divergence equation for H, suffices to com pletely determine the problem when appropriate boundary conditions are specified. They take the form of the requirement for the continuity of the normal component B± and the tangential component H 11 respectively, since these are required by the static field outside the sample. 24 It is clear from the discussions so far that the validity of the quasi-static approximation in this case is the neglect of the displacement current term, OD/Ot, and this requires a low frequency, <
Thus the electric field E is of a higher order in smallness, i.e. O(/), than the B 22 field. This estimate follows from the fact that the curl on the LHS of eqn (3.23) varies as 1/i, while the RHS varies as w/c. i cartesian co-ordinates, curl curl H = grad div H V 23 H. 2 As in most textbook treatments, this can be shown by integrating eqn 24 (3.21) over an infinitesimal region near the surface.
=
curl curl H
=
(3.28)
—H,
where 6 is the electromagnetic skin depth (see, for example, Landau, Lifshitz and Palevskil 1984), given by
=
c V2iiuw
(3.29)
Fortunately, for spheres, eqn (3.28) is in fact separable. The solution follows the same dipole field ansatz as in section 1.4. Here we shall follow the treatment of London (1950) (see, for example, Choy and Stoneham 1990).
Example 3.3 We now define a complex penetration depth, ,\ which eqn (3.28) now has the form
=
(1
—
i)6/2, by
I
I
(3.26)
These are of course the same equations as the static limit and the magnetic field, which is now given by
55
THE QUASI-STATIC REGIME
§3.4
I I
curl curl H
I
(3.30)
—H.
As in section 1.4, we propose that, outside the sphere of radius a, we have a dipole field: H
=
(Ho + 2Smr ) cos6 3
?
) sin& 3 + (—Ho + Smr
§,
(3.31)
25 while inside we now have a 0 is the magnitude of the applied field, where H more complicated form: H
—
=
=
u(r) cosO i + v(r) sin0
We assume harmonic time dependence, so that H€ 25
=
O.
Hoe_it.
(3.32)
56
DYNAMICAL THEORY
These functions
u(r)
u(r)
and
=
Ch. 3
are given by: 26
v(r)
(c 0 — \rJ
[sinh() —
experiments, the complex surface resistance and Palevskii 1984) :28
()
coshQ)]
,
0 C A = --Bigl(_) {(i
2 r
+
)
sinh() -
()
()
—
cosh()].
(3.34)
i],
(3.35)
where i is now associated with the medium and we have redefined in the definition of A as that of the inclusion. Here GL (x) is the complex Langevin function well known in the theory of magnetism (see Kittel 1974): GL(x) = coth(x)
1/x.
(3.36)
From the above results we can now write down the Maxwell—Garnett formula for conductors in the quasi-static limit, as (IL ho 11+2110 .
)=
—_ZL(_), 2 1 A
(3.37)
where ZL(X) = 1 (3/x)GL(x), ,ii is the effective permeability for the medium, 110 that of the host material, and A 1 is the complex penetration length of the inclusions with skin depth 5j, and is defined with a permeability ,u 1 by eqn (3.29). For non-magnetic inclusions we can take = 1, as is usually the case. Following section 1.4, we can also write down the Bruggeman formula for conductors in the quasi-static limit: —
111+211
2
2 A
(3.39) and it is the real part of ( which determines the dissipation, since the timeaveraged Poynting vector through the conductor surface is given by
By applying the usual boundary conditions at the interface, we can determine the constants Co and Sm. From this we can show (cf. Example 1.1) that the complex flux deviation due to the inclusion is given by
= ra 0 [3GL H 2
57
(see, for example, Landau, Lifshitz
(3.33)
and v(r)
THE QUASI-STATIC REGIME
§3.4
(3.38)
where A 2 is that of the second component or inclusions. Equations (3.37) and (3.38) are the obvious extensions of section 1.4. We note that they are complex equations whose solution yield the effective permeability i which has real and imaginary parts. 27 Here it is to be related to the quantity often measured in 26 continuation of the hyperbolic functions for complex ) is implied. Analytical We have assumed that io in eqn (3.37) and jzj in eqn (3.38) are real, since the losses are 27 associated with the inclusions, as is often the case.
S
=
i—Re (E x Fit)
=
i—Re 2 clHt( n ,
(3.40)
where n is a inward unit vector normal to the surface of the sample and the subscript t denotes the tangential fields. The alert reader will note that there are certain physical assumptions in writing down the above equations for the EMT. First, we have assumed that the sample is sufficiently transparent so that there exists a quasi-static field B which penetrates the sample almost completely. Secondly, the inclusions which are conductors form a minority component which is characterised by the penetration length A. In particular, the symmetric nature of the Bruggeman formula is lost in eqn (3.38). A symmetric Bruggeman formula can of course be written down, but that is for an entirely different problem where both components are lossy and each have their own different skin depths. There is in fact a third scenario, whereby the sample of concerned is a thin surface layer whose majority component is lossy. Here we can assume an induced surface eddy current, and that the minority phase is non-conducting. The appropriate equations now are (3.41) curl curl J = —J outside the inclusion, while inside we have, for a non-conductor div J
=
0,
curl E
=
0.
(3.42)
The latter follows since the field E is of a higher order in smallness in the in clusions. This is an interesting case and has a well known example. Readers will recall that in the days when toroidal transformer constructions were less com mon, the design of conventional transformers was based on the use of E and I laminations forming the iron core, in order to prevent eddy current losses and thus overheating of the core. We shall leave the details of this version of the EMT as an exercise for the reader. We conclude this section by making two more remarks. One concerns eqn (3.39), which has considerable significance in transmission line applications (Dworsky 1979). A complex 11 adds an imaginary part to the distributed line inductance L, and the skin resistance loss is now as where C is the sociated with a complex part in the line impedance Z = is irrelevant to our discussions here for conductors and can be taken as unity Note that 28 without loss of generality.
58
DYNAMICAL THEORY
Ch. 3
distributed capacitance of the line. The other remark concerns the justification of our EMT theory as written. This can be done presumably using a Green’s function formulation, cf. section 1.5 (see also Webman et al. 1977), but the more convenient approach is probably via the use of the technique of complementary variational theorems (Choy 1997). Both of these approaches are fascinating but not without difficulties, and thus the issues are not completely resolved (see sec tion 3.8). We shall not attempt to justify eqns (3.37) and (3.38) here, but later we shall show that these theories are appropriate limiting cases of the full dy namical EMT based on wave scattering theory (see section 3.7). The ultimate justification of these theories will of course rest with their empirical value. 3.5 Displacement current and wave scattering In this section we shall treat the full Maxwell theory for electromagnetic waves (eqns (3.1), (3.2), (3.13), (3.17), and (3.18)) without incurring any approxima tions on the frequency, in particular when the displacement current term in eqn (3.1) is no longer negligible. As is easily shown, as a consequence of this term, we can, by taking the curl of both eqn (3.1) and eqn (3.2), derive the following wave equation for either E or H, as written down below for E: — V E 2
—
4iri8E =
2 8t c 2
0,
(3.43)
where we have used Ohm’s law as given in eqn (3.18) to eliminate J and we have assumed cartesian co-ordinates without loss of generality. We now note that the skin depth term, as discussed in the last section, constitutes a damping term in this second-order differential equation of Helmholtz type. Again, by considering harmonic waves, 29 with a time dependence of the form we can easily rewrite eqn (3.43) as the familiar Helmholtz equation E 2 V
-
E =0, 2 k
(3.44)
where the wave vector k has its magnitude given by 2 k
=
2+ pEk
2i
(3.45)
in which k = 2 /c is the square magnitude in vacuum. The quantity that will w concern us most in this case is the complex refractive index: 2
47ripu
w
(3.46)
which governs the speed of propagation of the electromagnetic wave and its damping in the medium. Physically, the displacement current regime charac tenses the case when the electromagnetic nature of light manifests itself fully, as Non-harmonjc waves can also be treated by Laplace transform techniques, so that there is 29 no loss in generality in what follows.
1
§3.5
DISPLACEMENT CURRENT AND WAVE SCATTERING
59
both E and H are of equal magnitudes. In this region the wavelength is small and for our purpose usually smaller than typical grain sizes. Here we shall need to consider various length scales that are important. There are essentially three such lengths for our purpose: (a) the skin depth ö, (b) the wavelength A, and (c) the grain size a. Thus, the ratio 6/A determines the significance of the quasistatic eddy current regime versus the displacement current regime. For a metal like copper with a resistivity of 0.2 p cm, the eddy current effects become insignificant above about 3 x i0 GHz, whereas for a ceramic material with a GHz. Elementary typical resistivity of 30 ,u cm, this frequency becomes or the form radiator a dipole of field E the examining by as such s, consideration and (Panofsky (3.44) eqn of function Green’s dyadic tic of the electromagne These consider. to zones three are there that show 1975), Jackson Phillips 1964, are the well known near-field or static zone, where the typical distance we shall consider r to satisfy is a << r << A, the intermediate wave zone, a <
=
c’(r) + if”(r),
(3.48)
with real and imaginary parts. For the present, we shall ignore the fact that c may also fluctuate with time, which is of importance, for example, in remote sensing, We shall find it more convenient later to consider circular polarisation. 30
DYNAMICAL THEORY
60
Ch. 3 , 0 E
particle
r) E ( 0
!o
61
MIE SCATTERING
§3.6
by scattering of the incident field at the particle and is hence proportional to the incident field times the volume; thus
k
IEscattl
Escatt(r) FIG. 3.1. Scattering geometry for an arbitrary object sample with an incident plane wave E . 0 or in cases where relaxation processes are important; for example, in a metal. Near the object, i.e. when r << a /A, where a is the typical size of the object and 2
A the wavelength, we have complicated amplitude and phase variations and thus interference of the various components of the scattered wave. As stated above, we shall not be interested in these near-field effects but focus on the far-field region, where r >> a /A. In this region, the scattered field is that of a spherical wave, 2 whose polarisation and magnitude is determined by the scattering amplitude vector f(, k): =
f(, k)
—,
(3.49)
where A is the unit vector of the incident field and i that of the scattered field. In general, the polarisation of the scattered field as given by f is elliptical even though the incident field may be linearly polarised. For our purpose, the differ ential scattering cross-section contains all the information we can obtain about the scatterer: f(i, )I2, ad(r, ) = lim [(r (3.50) Sscatt)/So] 2 where So and 5 scatt are the magnitudes of the Poynting vectors for the incident and scattered power flux densities. The quantity given by eqn (3.50) has the dimensions of area per unit solid angle, and when integrated over all solid angles gives the total cross-section:
=
f
d
d =
f
f(, )I2 dft
(3.52)
constant x EoI—.
As a consequence dimensionally, the constant of proportionality must vary as ; thus also o must be 2 k) 2 as a function of wavelength and hence 1/A
AN
Escatt(r)
=
(3.51)
The above definitions contain the main quantities that we shall be using in the following sections. We shall conclude by making two more comments. First, it is easy to show, by an elegant dimensional argument due to Rayleigh (1881), that for small particles, i.e. a << A, the total scattering cross-section is inversely proportional to the fourth power of the wavelength and directly proportional to the square of the volume, i.e. to a . This is because the scattered field is caused 6
. The second remark concerns the formal solution of the 4 proportional to a°/A scattering problem. Here the technique we shall discuss and use is the familiar one of seeking solutions for the wave fields both inside and outside the scattering body, with appropriate boundary conditions at the surface of the body. Since the total wave field in all space is a sum of both an incident and a scattered part, by the principle of superposition, it can be shown that there are two equivalent approaches to this problem for the scattered fields (de Hoop 1995). In one ap proach, as discussed above in the case of Rayleigh scattering, the scattered field Escatt and Hscatt can be viewed as being excited through both the presence of
the contrast in the medium properties e, p and the presence of the incident field. In this case the perturbations can be treated as due to source currents, both electric and magnetic, which are proportional to the contrast & = € €o and Eo; 0 respectively. This approach is convenient for numeri and St = o and H 1 i cal computations, especially those using finite element methods. The alternative picture is to view the above contrast source terms as radiators into the rest of the medium. These sources are then proportional to the contrast and to the total fields. This method constitutes the integral equation and Green’s function ap proach, which is analytically tractable only for simple geometries, like spheres or cylinders, but nevertheless is useful for developing approximations such as EMT. Readers should note that the modern treatment of wave scattering, and the re lated forward source, inverse scattering, and inverse source problems, actually belong to a whole family of remote sensing type problems that can be cast in an elegant framework. This framework employs the power of reciprocity relations (see de Hoop 1995), whose usefulness remains to be fully explored. —
—
3.6
Mie scattering
The purpose of this section is to summarise the key results from the complete electromagnetic scattering solution of plane waves by a sphere due to Mie (1908) and Debye (1909) (see, for example, Van de Huist 1981). As mentioned at the end of the last section, this is one rare case where a complete analytical solution of the scattering problem is available, and it will form the basis of our discussions in the next two sections. Various limiting approximations will also be discussed at the end of this section, which can be shown to follow from the exact solutions for the sphere. The geometry, like that of Fig. 3.1, is shown in Fig. 3.2. The symmetry of the sphere enables us to write the electromagnetic fields in terms
_________
62
________
DYNAMICAL THEORY
Ch. 3
i(2n + 1 r11= n(n+l)
Z
63
MIE SCATTERING
§36
(kr) P(cos) cos,
-E 0 E
r112
2n + i ( 1 (kr) P(cosO) sin. n(n +1)
=
(3.57)
7
J+(x) are the Riccati—Bessel functions and is the complex impedance of the medium, cf. eqn (3.39). Similarly, = the scattered fields outside the sphere (r > a) are given by the same form but with Mie scattering coefficients a and b containing the information on the scattering:
Here ‘i/i(x) 1
0 H
= xjn(X) =
rflrtt
x
—1
=
lfll(2fl+1)an(n(kr) P(cos8)cos, n(n + 1)
n=1
3.2. Scattering geometry for a sphere with an incident plane wave E 0 polarised in the x direction.
FIG.
rllrtt
=
JZ[mag =
11i7, r, 2 11
1 nil
1
H
=
curl curl 1 (rile + ) curl curl (rHmag)
iW[t —
0,
curl (rllmag), curl (rHei),
i’/-t
(3.56)
where c, ,u are the permittivity and permeability inside the sphere, while 6 = = I’o are those outside, respectively. The incident field E 0 is readily decomposed into incident Hertz fields, as Readers will recall that the spherical electromagnetic dyadic Green’s function are given in 31 terms of the derivatives of the scalar Green’s function satisfying the scalar Helmholtz equation with a point source (see, for example, Jackson 1975, Papas 1988).
2n + 1) 4 ( dj(kr) P(cos 0) sin n(n +1)
.
(3.59)
—
(3.55) inside the sphere, where j = is the refractive index of the sphere. These solutions, once obtained, allow us to determine the fields: =
2 (o(k)
(3.54)
outside the sphere and
E
i(2n + (kr) P(cos8) cos, 0 c n(n + 1) °°
=
k il=o, + 2 (V )
°°
= ()2
r112
=
(3.58)
(3.53)
where all TE (transverse electric) and TM (transverse magnetic) modes are given in terms of 112 and il respectively. These scalar functions are solutions of the following scalar Helmholtz equations:
2 +2 (V 11 ui ) k
i(2n+ ‘bnCn(kr) P(cos8)sin, n(n + 1)
2 k 0 (
where the choice of the Riccati—Bessel function is dictated by ( (x) = xhP (x) = (x) is dictated by the asymptotic spherical wave behaviour of the 1 /jH 2 fl+ required solutions. In the same way, inside the sphere we have the fields
of two scalar functions that define the electric and magnetic Hertz vectors: 31 Hei
=
I I
Again, the choice of the Riccati—Bessel functions is dictated by the requirement of regularity at the origin, and the Mie coefficients c and d contain all the information on the propagation within the sphere. The boundary conditions on , and H at the interface 9 , E, H 9 the continuity of the four tangential fields E r = a suffice to determine all four Mie coefficients. The two quantities of main interest to us are the a and the b coefficients: a
=
‘çb(a)(3)
—
—
—
b
—
—
(3.60)
—
where a = ka and /3 = kra. Note that the above are exact solutions and that the E and H fields obtained upon taking the various curls in eqn (3.56) are unduly complicated, containing near, intermediate and far-field components. The latter is all that interests us here, and this can be easily shown to be (Van de Huist
64
DYNAMICAL THEORY
Ch. 3
=
—-—Si(8)sin,
0 E
ie
=
DYNAMICAL EFFECTIVE MEDIUM THEORY
(6)coscb. 2 ——S
(3.61) =
Here, the S (9) are the scattering amplitude functions and are given in terms 2 , 1 of the Mie a, b coefficients by 00
(6) 1 S (O) 2 S
+
=
+
n=1
where ir and r, are angular distribution functions, as given by
ir(cos&) =
P(cosO) sine
r(cos8)
=
d 1 P(cos8).
=
-
(2n + 1)(IanI2 +
). 2 IbI
The above equations can be modified appropriately for incident waves that are not linearly but elliptically and/or partially polarised, by the use of Stoke’s pa rameters. The details are a matter of book-keeping and we will not concern ourselves with them here; see, for example, Van de Huist (1981) and Ishimura (1978). Appropriate limiting forms of the Mie formulae can be obtained through asymptotic expansion formulae for the Bessel functions, which can now be con veniently treated using symbolic software, like MathematicaTM. For small parti cles the Rayleigh scattering formula is once again obtained. The weak scattering limit where the dielectric of the sphere is close to one is also another limit; the approximation here bears the names of Born and Rayleigh—Debye. Other ap proximations, like the Wentzel, Kramers, and Brillouin (WKB) method, well known to students of quantum mechanics, are also used and can be tested by the Mie formulae above. We note that the above single-particle formulae are the analogues of the single-sphere solutions of section 1.4. The task for us now is to employ them with suitable physical assumptions to formulate a dynamical effective medium theory (DEMT). 3.7
Dynamical effective medium theory The development of the dynamical version of the effective medium theory may at first sight seem to be a relatively straightforward extension of the static theory,
f
... ..)dei,de , .2 P(ei,e , 2
°
+ 2(
60 2 €j+
(3.63)
(3.64)
=
(3.66)
Indeed for spherical grains, in the single-particle approximation for computing the fields, our familiar results of Chapter 1 show that eqn (3.65) takes on the well known expression of Maxwell—Garnett: 6 =
From this, we can obtain the total scattering cross-section by integration, and it is the quantity we will be using frequently:
(3.65)
as an integral over the joint probability distribution of each component: (f(ej))
(3.62)
)(E(e,€o)), 0 €(€
where e pertains to each grain, and in the static case the average is to be defined
[a(cos8) + br(cosO)],
(2n + 1) ){aflrfl(cos6) + bir(cos9)], 1
=
65
since in principle all one needs is to define a medium 60 such that the effective dielectric constant e of our sample is given in terms of suitably defined averages:
1981) ikr
§3.7
))
(i —
(
—60)),
(3.67)
60 2 6j+
by which, the assumption € = e, yields the Bruggeman formula, cf. eqn (1.71). Extending eqn (3.65) to finite frequencies is in general problematic for the fol lowing reasons.
propagating through the composite Consider a plane wave of frequency material. This wave is attenuated and retarded through multiple scattering and absorption as it propagates through the system. In general, the total wave in the system is a sum of the incident wave and scattered waves. These scattered fields
can, however, propagate with a whole range of wave vectors. The question is: How do we consistently define an average that takes into account all these scattered fields in defining the effective dielectric constant, eqn (3.65)? On account of superposition, one could in principle define the averages on a mode-by-mode
basis via (D(e,eo,q,w))
=
e(eo,q,w)(E(e,eo,q,)),
(3.68)
and then average over all modes q. The problem is that while the distribution over grains is known, or assumed to be known through the process of manufac ture, the distribution over modes is far from trivial, and would have required a complete solution of the multiple scattering problem. In addition, since there is absorption and scattering, these modes are complex, so do we average over the real part or the imaginary part? More importantly, the above definition does not automatically satisfy the consistency condition, which follows from the fact that the average D field so obtained must propagate through the system with the required propagating wave vector. This issue is a central problem in dynamical extensions of the effective medium theory. One approach to dealing with this was taken by Stroud and Pan (1978), who used a simple approximation to bypass the need to define the averages in eqn (3.65). In fact, they achieved this by imposing the consistency condition through the Fourier component D(q, w) and relating
DYNAMICAL THEORY
66
Ch. 3
it via Ceff (w) to the electric field E(q, ) at one magnitude of q, that of the 32 propagating mode itself, keff = /777w/c; thus
f
D(r,w)ef1z dr
= Ceff
f
E(r,w)e_iffz dr,
=
CeffI1eff. c
B(r, w)e_iffz dr
=
ILeff
f
H(r, w)e_ikeffz dr,
L.
Ej(r,w)effz dr.
=
(3.73)
Cef f(CO)
This is the dynamical Maxwell—Garnett theory, while for the Bruggeman theory, as usual, we set Cef f = CO by the self-consistency hypothesis, and thus we have
(3.71)
The choice of the phase convention is in fact arbitrary as long as it is consistent, as is the 32 case for all Fourier transform pairs. Here it is dictated by the form of the incident wave given in eqn (3.47), whose exponential dependence is of the form Magnetic properties are generally insignificant in the displacement current regime (Landau, 33 Lifshitz, and Palevskii 1984), but there are exceptions, as for ferromagnetic systems. The requirement of spheres as in the static case can be relaxed, but the calculations are 34 many times more complex than the static case.
(3.72)
The reader should note that our formulation here is a refinement of Stroud and Pan’s (1978) arguments, as they cannot justify replacing their integrals over internal fields within each grain of volume v by an integral over a large volume (cf. eqn(1.35)), without employing the superposition principle for far 5 in eqn (3.72) must be at least of the order of fields. The integral volume V several wavelengths to satisfy the far-field criterion. With these definitions of the averages, we can now return to eqn (3.69) and, by a slight modification of the , we arrive 0 above arguments, in which we consider the embedding host to be e at the equivalent expression
(3.70)
but this introduces a source of ambiguity that has not really been resolved in the literature (Mahan 1988; see also Lamb et al. 1980), and that is to what do we assign keff? Mahan (1988) favours the form given in eqn (3.70) and assumes that both of the averages given in eqns (3.69) and (3.71) are independent, but this raises certain difficulties, associated with double counting (see section 3.8). For the moment, we shall stick with this assumption, to avoid further digression. Now, eqn (3.69) requires further approximations before further progress can be made, since the fields D and E can only be obtained if we have solved the full multiple scattering problem. In the spirit of the static EMT, the approximation here is to consider an isolated spherical 34 grain embedded in a medium whose dielectric constant is some Ce! f (w). In this case, we are back to the single-particle Mie scattering problem treated in section 3.6, with appropriate medium proper ties. This embedding assumption, together with the superposition principle for each of grains—which are now assumed to be independent—make the problem of
67
the integrations in eqn (3.69) mathematically tractable. For in this case the inte gral of the field E over the sample volume V breaks up into a sum over individual grains of the form =
In this case we see that this entails a similar hypothesis for the B field as in eqn (3.69); thus
f
DYNAMICAL EFFECTIVE MEDIUM THEORY
(3.69)
where the average is in fact automatically taken through the process of Fourier transformation. We will not discuss the physics underlying this approximation and the issue of other alternatives, which we shall return to in section 3.8. We shall accept this as a working hypothesis, whose correctness will be decided by its agreement with the various limiting cases, as well as with experiment. One immediate feature of this hypothesis is that it seems to be biased towards the displacement current regime only. The reason is that in the general case, where magnetic properties are also important, 33 then the effective wave vector must be defined via keff
§3.7
6ej
=
0.
(3.74)
However, the Maxwell—Garnett formula given by eqn (3.73) is flawed, since it violates the important consistency condition mentioned earlier (cf. remarks be fore eqn (3.69)), in that the propagating mode defined by ko(eo) through the analogue of eqn (3.70) is inconsistent with Cef f. Thus, while the lack of selfconsistency can be tolerated as in the static case MG theory, this is not the case in the dynamical theory. We shall have more to say about this in section 3.8: here we shall proceed with developing the Bruggeman theory, which at this stage reduces to the evaluation of the integral equation (3.72). We shall now show that (0) given in 2 this integral is related to the forward Mie scattering amplitude Si, 3.6. section
Example 3.4 The integral eqn (3.72) can be performed, assuming circularly polarised incident light:
( + i)e’
Eo(z)
=
url Eo(z), k c 1
(3.75)
so that 1j =
k L.
E(z).& E(r) dr,
(3.76)
—
our exposition of the dynamical effective medium theory (DEMT). We shall conclude this section with the following example.
5 will where ö€ = €o is the contrast as usual, and the integration volume V twice Integrating eqn by the calculation. (3.76) be taken as infinity at the end of parts, we have
=
1 2k 6 2 €
J, E(z).curl curl (S€ E(r))dr.
Example 3.5 We shall now consider the long-wavelength limit wa/c —* 0 for a dielectric. Using the asymptotic properties of the Bessel functions that define the Mie coefficients (eqn (3.60)), show that eqn (3.83) reduces to the standard Bruggeman expression, eqn (1.33). This is also the limit of Rayleigh—Gans scat tering and is in agreement with the quasi-static limit for dielectrics in section 3.4. Show that a second-order term, due to magnetic dipoles, can be included, thus modifying the Bruggeman formula to
(3.77)
We shall now use Maxwell’s equations for a source-free region, which can be combined (see Jackson 1975), into the form )D = —curl curl 2 2 +k (V
(&
H). 1 E) + ik€ocurl (ö
(3.78)
At this point we shall ignore magnetic properties and thus assume that the magnetic contrast 5 t is zero. Note that we have defined k = /w/c in this case. By putting eqn (3.78) into eqn (3.77) with the use of Green’s theorem, we now have 8E* E scatt )ds, (3.79) r = —
(
ikr
(3.80)
we have, finally, —
—€0
ö€ 2 2k
i(1 + cosO)(
—
).S
R
dS.
=
1r1€eff 4
6€ 3 k
+
wa
(—)
2
(€j
—
€eff)) = 0,
(3.84)
(3.81) 3.8
(3.82)
and S(O) is the magnitude of S(0). Thus the Bruggeman theory now takes on the simple form (S(O)) = 0. (3.83) Note that for Mie scattering of spherical particles (see section 3.6), the forward scattering amplitudes S (0) given in eqn (3.62) are identical. This completes 2 , 1 Note that for Mie scattering from a spherical particle, there is no azimuthal angular 35 dependence.
Open problems
Readers will note from the discussions in the previous sections that there are numerous issues and problems that remain unresolved in DEMT. The purpose of this section is to discuss some of these issues and to suggest possible improvements to the current theory. Let us examine these issues in turn. Now, the magnetic counterpart consisting of B(r), tt(r), and therefore n’ was mentioned in Example 3.5. The question is: Do €eff and /Leff each have a separate meaning in that they determine the dispersion relation through
For large R, the integral equation (3.81) is dominated by the leading term, and so I = ( — where 1
+
€eff
where a is the radius of spherical grain i. Estimate the significance of this , where w is the plasma frequency. We 2 term for a metal where € = 1 — (w/w) still consider the case of a dielectric but we shall now assign the electromagnetic properties of the medium to i. Show that a formula similar to eqn (3.84) can also be derived. This example illustrates the ambiguity in whether one should assign the absorption to € or to i. For dielectrics, this does not matter, since the measured quantity is the scattering cross-section, which is determined by 81,2(0) through keff. If one component is a metal with a skin depth ö, and by considering again the long-wavelength limit with the use of the asymptotic properties of the Bessel functions, show that we can now recover eqn (3.38).
where S is the surface of the integration volume and use has been made of the 0 together with D = €o(Eo + Escatt) outside the particle, wave equation for E 35 Using the asymptotic form of the latter, whereby
r
€j — €,
-
Escatt(8,) = —s(0,g),
69
OPEN PROBLEMS
§3.8
Ch. 3
DYNAMICAL THEORY
68
t
kff(w) =
/c €effteffw ? 2
(3.85)
The answer is in general ‘no’, since this demands a further consistency requirement that is not built into the present DEMT. In an exact theory, as in the case of plane waves propagating through a homogeneous media (de Hoop 1995), the determination of the propagation vector is invariant to the choice of the de scription, which is either via €, E or via i, B. Both descriptions must lead to the same propagation modes. However, in an approximate theory like our DEMT
70
DYNAMICAL THEORY
Ch. 3
there is no such guarantee. To construct a theory that manifestly exhibits this consistency is thus a hitherto unmet challenge. Some preliminary results in the long-wavelength limit have been proved, showing that eqn (3.85) can indeed be satisfied for periodic or aperiodic arrays with low-order multiple scattering and for weak scattering (Lamb et al. 1980). There are in fact problems associated with double counting that are inadequately addressed (Choy and Stoneham 1992), in particular the assumed independence of both averages in ceff and /Leff (Mahan 1988). The author feels that this issue can be addressed if we focus not on either the E or B fields separately, but rather on the vector potential A. In section 3.5, we mentioned the difficulty of formulating a DEMT at the intermediate zone. Now we are in a better position to discuss the problems. The chief difficulty here, in spite of the fact that the Mie formulae, while cumbersome, are correct in this zone, is no doubt eqn (3.69). Almost certainly, the assumption that the physics in this zone can be entirely attributed to a keff will be flawed in view of the complicated phase relationships here. Indeed, eqn (3.83), which results from this assumption, can be shown to be equivalent to a zero average scattering field ansatz (Stroud and Pan 1978): E(i,t)
=
0,
(3.86)
a result of relevance only in the radiation zone. The problem of a consistent mode theory for a dynamical Mawell—Garnett theory is a subtle one, and has to date not been resolved. In fact, this problem may be the cause of discrepancies between MG theory and numerical studies (Chew 1990) and is, surprisingly, widely ignored in the engineering literature. This issue might well be related to a more subtle one with which we shall con clude this section, and that is the problem of causality. In the exact solution one presumes that, as a consequence of causality, the complete solution of the mul tiple scattering problem should imply that the wave fields on each sphere must be suitably correlated. As such, the real and imaginary parts of the dielectric function in the complex frequency space must satisfy analytical properties, first derived by Kramers and Kronig; see, for example, Landau, Lifshitz, and Palevs ki (1984) or de Hoop (1995). This is the well known Hubert transform pair for the real part €‘ and the imaginary part c” of the dielectric constant: — 1
1
=
—P ir
=
rc
fl(
‘
I J_X—W
_f°°
-co
‘(w)
dx, —
1
dx,
(3.87)
§3.8
OPEN PROBLEMS
71
where P denotes the principal part integral. While the above causality require ments are desirable, the construction of a theory that respects them is highly non-trivial. Even a proof that our DEMT, eqn (3.83), does or does not satisfy eqn (3.87) is rather non-trivial. Physically, one expects that the independent singleparticle approximations invoked might have unintentionally eliminated the nec essary correlations among fields imposed by causality. This issue alone highlights the richness of the territory in dynamical generalisations of EMT.
4 LIMITATIONS AND BEYOND 4.1
Introductory remarks
Throughout this book, we have developed the effective medium theory for static and dynamical systems, and it is now time to look more closely into the limita tions imposed by the approximations in these theories. Specifically, the purpose here is to have a clearer picture of the shortcomings with a view to identifying pathways for future improvements of the theory: hence the title of this chapter, where we shall study the limitations with a focus on beyond. The reader will find that in what we are about to study, the topics of Chapter 2 are most appro priate, since we shall have occasion to use what we have learned there to further illuminate the topics to be discussed here. In section 4.2, we shall examine the higher-order terms that are neglected in the EMT and hence derive a criterion for its validity. Such a question was first addressed by Kirkpatrick (1973) for the related problem of resistor networks, but we shall look at it here in the context of the theory as developed in this book. In particular, we shall use this insight in section 4.3 to discuss the percolation point which, we recall, is the critical volume fraction when the dielectric constant diverges or vanishes for superconducting or hole inclusions respectively (cf. section 1.4). The properties of the system in this region are dominated by critical fluctuations, and the modern theory is that based on the renormalisation group (RG) (Ma 1976a) where the concept of scaling is the key starting point. However, the analogy is well worth exploring further, since the technique of Wilson’s renormalisation group, as expounded by Ma (1976a), Amit (1984), and Domb and Green (1972—1988) for example, relies heavily on an initial Ginzburg—Landau type free energy expression. Thus the techniques we have studied in Chapter 2, based on variational principles, can be readily extended by adapting the tools of modern critical phenomena. Unfortu nately, as we shall see, technical problems prevent this programme from being carried through in our case. In section 4.4 we look again at the spectral represen tation, but now for the dynamical situation, and we note that the theory here is still relatively underdeveloped as compared to the Bergman representation the ory for the static case (cf. section 2.4). The limitations here are due to the lack of rigorous mathematical theorems that can be applied to a Helmholtz operator as opposed to the Laplacian operator. We take this opportunity, then, to look at multiple scattering in section 4.5, which, for the first time in this book, will be in the perspective of numerical computations, as no better method is known to treat this problem to date. This is followed by competing interactions in section 4.6. For the nature of such interactions can have an effect similar to multiple scattering, and that in general will move the universality class of the system,
73
HIGHER-ORDER TERMS
§4.2
resulting in significant modifications to the theory even at the level of EMT if appropriately considered. Multiple scattering is indeed the culprit that hinders EMT and it needs to be better understood in the context of section 4.2, as well as looking beyond, to the incorporation of two-body effects (section 4.7). We shall conclude this chapter with some discussions about non-equilibrium (section 4.8) as no doubt any study of dynamics and transport must entail. For most of this chapter, our emphasis will shift away from details and be placed more on prin ciples, so that in general we shall use symbolic mathematical notation, avoiding details unless necessary. 4.2
Higher-order terms
We recall that in Chapter 1 we have used a powerful Green’s function formulation of the EMT, (cf. section 1.5). We shall now rewrite eqn (1.60) in the following symbolic way: (4.1) = & + &,
where we recall that the dyadic Green’s function operator is a predominantly non-local operator which couples all the grains, i.e. to all the other components of x using a matrix notation. However, we shall define a local operator F (cf. eqn (1.66)) and the difference Q so that eqn (4.1) is now rewritten as
x=&+&Fx+&Qx,
(4.2)
F contains only off-diagonal, i.e. non-local, bits in where the operator Q = We can now iterate eqn (4.2) and see that —
c.
X
Sf + &F& + &F&F& +... + &F&Q& + &Q& + &Q&F&
(4.3)
This equation can now be re-summed. We do this by extracting and then re summing all the local terms involving F alone, and thus
x
=
(1
=
(1
—
SF
—
—
(4.4)
)‘(l—&F)’&.
We recall once again that the EMT approximation consists of the ansatz (x) = 0, cf. eqn (1.70). In the present symbolic notation, we note that the criterion for the latter equation to be valid requires that -
1
F)’
-
&F)’&)
((i
-
1
-
&F)’&). (4.5)
Equation (4.5) is the criterion for the EMT to be a good approximation. It requires that non-local correlations involving the operator Q be small. The power
74
LIMITATIONS AND BEYOND
Ch. 4
of the Green’s function formulation is that this approach can in principle be extended to the dynamical case, since the operator will now be the appropriate dynamical dyadic (Papas 1988); see also section 4.4. The neglected terms in this case are associated with multiple scattering, which we shall return to in section 4.5. Unfortunately, the neglect of the correlations given in eqn (4.5) fails near the critical point, for in this region the correlations between grains are large on all length scales. In the next section, we shall examine percolation theory and criticality in the context of the systems we have studied and the failure of our EMT. We shall in particular start to address the issue as to how one might improve this aspect of the EMT. 4.3
Percolation and criticality
Percolation theory has a long history and is tied up with the development of critical phenomena during the 1960s, the 1970s, and later in the 1980s. Many reviews of percolation theory were in fact written during this period; see, for example, Deutscher et al. (1983), Essam (1972), Kirkpatrick (1973), Shante and Kirkpatrick (1971), Sokolov (1986) and Zallen (1983). The specific question is a very simple one. On a lattice, we start removing bonds (bond percolation) or sites (site percolation) with a certain probability 1 — p, and hence p is the probability for a bond or a site to be occupied. It is found, both empirically and via computer experiments, that there exists a threshold Pc at which an infinite cluster spans the lattice. Extensive numerical evidence and some exact results limited to two dimensions are now available for the characteristic exponent for the infinite cluster probability near the percolation point, i.e. when p —* Pc: P(p)
9(p—pc), 3 (p—pc
(p_pc)_V,
Table 4.1 Critical percolation points for some common lattices (Isichenko 1992). Bond percolation
Dimension, d Lattice type 2
3
Site percolation
0.5927 ± 0.00003 1/2 = 0.347296 1 — 2 sin = 0.6527014 0.698 ± 0.003 Honeycomb 0.3117 ± 0.0003 0.2488 ± 0.0002 Simple cubic 0.2464 ± 0.0007 Body-centred cubic 0.18025 ± 0.00015 0.1998 ± 0.0006 Face-centred cubic 0.1198 ± 0.0003 ± 0.4299 ± 0.0008 Diamond 0.3886 0.0005 1/2 2 sin
Square Triangular
Table 4.2 Critical exponents for percolation (Isichenko 1992). Exponents a= 2
/3
—
vd
d
2
=
d= 3
-0.64 ± 0.05 5/36 = 0.13888 0.405 ± 0.025 43/18=2.3888 1.77± 0.02
-2/3
y=vd—2/3 ö = vd//3 — 1 91/5 = 18.2 4/3 = 1.3333 u 1.3 II
4.81 ± 0.14 0.905 ± 0.023 1.9
(4.6)
where P(p) is the probability for a given site to belong to the infinite cluster and 9(x) is the Heaviside step function. The exponent /3 is only one of many other exponents, a, /3, -y,... (Domb et al. 1972—1988) and is related to those that are well known in the context of critical phenomena, associated with the specific heat, order parameter, and susceptibility exponents, respectively, etc. In particular, the correlation length exponent v specifies the average size of clusters, which diverges near the critical point as (p)
75
PERCOLATION AND CRITICALITY
§4.3
(4.7)
and, as will be seen (cf. remarks at the end of the last section), is responsible for the breakdown of the EMT. We need not delve too much into the details of percolation theory here, except to note that the current state of knowledge is that some exact results are known in two dimensions, while numerical and only approximate results are available for three dimensions. Exact results for the bond percolation threshold Pc for the three common two-dimensional lattices (square, triangular, and honeycomb) have been known since the early days (Sykes and Essam 1963, 1964) from the use of the familiar duality and ‘star-triangle’ type
transformations, whereas for site percolation and in three dimensions the results are from a combination of series expansions and Monte-Carlo simulations (see Table 4.1, adapted from Isichenko 1992). Through the development of a variety of tools in two dimensions, notably mappings to exactly soluble solid-on-solid models (Nienhuis 1982), exactly soluble models (Baxter 1982), and conformal invariance (Polyakov 1970, Saleur 1987), some of the exponents are in fact known exactly, while in other cases and in three dimensions, Monte-Carlo simulations and series expansions have yielded quite good estimates for the exponents (see Table 4.2, adapted from Isichenko 1992). We note that in Table 4.2, some of the exponents are known to obey scaling relationships like ‘y and ó. In general, the type of lattice is irrelevant and is a consequence of the validity of the universality concept, so far verified for two and three dimensions. However, this is known to be partially violated in continuum percolation for certain ‘dynamical’ exponents, which we shall call ‘transport’ exponents; to avoid confusion, in particular that of ,a. The latter is of special interest to us, as it relates to the behaviour of dc conductivity: 7dc(P)
(PPcY9(Ppc).
(4.8)
76
LIMITATIONS AND BEYOND
Ch. 4
This conductivity pertains to the ‘holes’ limit in our model, which is the com plementary case to Example 1.3, where €j. —+ 0 (cf. eqn(1 .43)), and the predicted EMT behaviour is given by f
—=l— i 2 ,
d=2, d=3.
(4.9)
Thus the EMT gives the ‘mean field’ theory exponent of unity in all dimensions. Note that a special feature of the two-dimensional case is that of duality, so that both the supeconducting and the holes limit are related due to reciprocity (cf. section 2.4). Unfortunately, for transport exponents like i, it is now known that they can violate universality and thus vary from one continuum model to another, even in the same dimension d (Feng et al. 1987).36 This imposes enor mous challenges when one has to consider improvements on EMT with regard to critical properties, for it is known that any form of local cluster approxima tions, e.g. the Bethe-Peierls approximation (see, for example, Huang 1987), will not result in a non-mean field exponent, without recourse to renormalisation. Formally, it is possible to proceed by analogy with critical phenomena, but the details present numerous stumbling blocks. On the lattice, one could formulate our problem for transport in terms of random walks on a percolating network. Then the full machinery of scaling, Monte-Carlo simulation with renormalisation, could be employed to study the conductivity using the equilibrium correlation functions and a Green—Kubo formula. Whether or not such a generalisation will be useful in the continuum case is unclear, in view of the lack of universality. Alternatively, assuming that one could start with a variational procedure based on the theorems developed in Chapter 2, so that the free energy functional could be expanded about an EMT solution, then there are at least two main issues to be resolved: (a) the choice of the EMT solution and (b) the technique for doing the trace. Both of these are formidable problems. Issue (a) is important, as one must first find an EMT type solution that is sufficiently close to the true minimum to give the correct features near criticality and then treat fluctuations around this minimum via (b), which hopefully will allow us to renormalise the fluctuations. Clearly, the single-particle EMT developed so far is lacking in both features. Issue (b) is particularly troublesome in the continuum, since even the solution of two bodies is a highly non-trivial affair (see, for example, Choy et al. 1998a,b), and thus a proper treatment of the trace is confronted with the difficulties of many-body electrodynamics. Yet in the RG theory, it is this trace procedure that iteratively smooths out the fluctuations on all length scales (Ma 1976a; for a modern review, see Fisher 1998). In the Green’s function formal ism of the last section, which is also to be discussed later, in section 5.5, in the context of the Feynman diagrams approach, this trace procedure corresponds to a re-summation of an appropriate infinite set of diagrams. It is fair to say For an early discussion of when universality could fail in critical phenomena, see Kadanoff 36 and Wegner (1971).
1
MIE RESONANCES
§4.4
77
that at the present moment, both procedures are faced with an impasse, so that the best estimates of exponents like u remain those obtained numerically, which unfortunately lack insight into the role of critical fluctuations in this problem. In fact, the age-old method of series expansions is also blocked by the same difficulties, for which reason only the second-order coefficients have so far been calculated (Choy et al. 1998a,b). Fortunately, for non-transport exponents the percolation properties have been found to satisfy universality and estimates for the critical volume fractions are now reasonably well known, albeit only via nu merical methods (Isichenko 1992). While the discussions in this section are by no means exhaustive, they nevertheless summarise the present state of the field, which is badly in need of new ideas and tools for continual advancement. In the next section we shall revisit the spectral representation, but not in the context of the criticality. In fact, earlier studies of the Bergman representation for criti cal properties were rather disappointing (cf. section 2.5; Bergman 1979) in that the analytical structure of the spectral functions acquires essential singularities near the critical point, making the procedure of doubtful value for extracting exponents. We shall instead examine the likelihood of a finite frequency form of spectral representation, which may have greater empirical value. 4.4
Mie resonances
In this section we return once again to our binary composite material, but we shall consider finite frequencies in the displacement current regime (cf. section 3.5). Using notations which must by now by familiar, we incorporate the con ductivity or its related skin depth into a complex dielectric constant ic,j = 1, 2, so that 4iriu ic = e + (4.10) w Thus the spatially varying complex dielectric constant has the form k(r)
=
(r) + K 1 Ki8 (r) 8 2
[1 — u8 2 ic (r)J, 1
(4.11)
where u = 1 i /‘ is the complex spectral parameter, and the 8 functions are unity inside the appropriate inclusions. Equation (3.44) now takes on the form —
E+k 2 V E 2
=
8iE, 2 uk
(4.12)
where the wavenumber is given by k = wJ/c. By analogy with Chapter 2 and 3, this wave equation must be solved with appropriate boundary conditions for the scattering fields, so that for large distances (radiation zone) the solution is a superposition of an incident field Eo(r) and a scattering solution e”/r. The analogous integral equation, cf. eqns (4.1) and (1.56) now looks like: E(r) where
=
Eo(r) + u
f
dr’ 8 (r’)g(r 1
—
r’, k).E(r’),
(4.13)
is now the full electrodynamic dyadic Green’s function (Papas 1988):
Ch. 4
LIMITATIONS AND BEYOND
78
ikR =
, —(k o 2
+vava)
(4.14)
MIE RESONANCES
§4.4
E = (1
—
0 E 1 uA)
+ Escatt,
79 (4.22)
where Escatt is given by where R
=
r
—
r’, and which satifies the wave equation with a unit source:
Escatt =
c 2 v
g 2 +k
=
(r k ö 2
—
(4.15)
r’)I,
where I is the unit dyadic. Once again, we shall write eqn (4.13) in symbolic form as (4.16) +uAE, 0 E=E where the operator A operates on a Hubert space whose scalar product can be defined by analogy with eqn (2.65): E r). r)E(r). 2 ( 1 8 E = fdr ( 2 , 1 (E )
(4.17)
A
=
A*,
(4.18)
it follows that the left eigenfunctions of A are equal to the complex conjugate of the right eigenfunctions. Thus we have, for the eigenvalue A
AlA) = ALA)
(A*IA = (A*IA*))t
=
(A*IA*))t
=
(A*IA
(4.19)
where we shall use a Dirac notation for convenience. These eigenfunctions are bi-orthogonal in that (4.20) (A*Ia)=0 Aa, but they are not in general normalisable, so that it is possible for eqn (4.20) to hold also for A = a. We will have to assume that the eigenfunctions are normal isable in order to proceed: there is thus a need to verify this in every particular case. We do not yet know the conditions for this to be so for a given A operator, and as a consequence there is a lack of rigour in subsequent developments of the theory. Nevertheless, assuming that all eigenstates can be normalised to unity, then we have an eigenfunction expansion of the form
A = IA)A(A*I, from which the formal solution of eqn (4.16) is
(4.21)
(4.23)
Using the eigenfunctions of A, we can now obtain a spectral representation for Escatt; namely,
IEscatt)
=
A
s—A
(4.24)
where the spectral parameter s is now in general complex:
s=—= U
However, this operator is somewhat unusual in that it is not Hermitian but nevertheless symmetric. Thus the eigenfunctions in general form a bi-orthogonal set (see, for example, Morse and Feshbach 1953) and in view of this symmetry, whereby the Hermitian conjugate is equivalent to the complex conjugate,
A . 0 E 1/u—A
(4.25) k—K
From the structure of the dyadic eqn (4.14) we note that the eigenfunctions IA) are all spherical outgoing waves e/r for large r, and the spectral representa tion equation (4.24) is the analogue of eqn (2.76). Just as in the static case, the microscopic geometry of the grains and the incident wavenumber k completely determine the eigenvalues and the eigenfunctions. The poles of the spectral func tion depend on the material paramater s, whose residue is the overlap matrix element (A* lEo) which also depend on the incident wave vector. We note that these poles are frequency dependent (cf. eqn (4.10)) and they occur in the com plex frequency plane so that for real frequencies, the physical case, they do not represent true infinite divergences. These are related to the Mie resonances, first discovered by Mie (1908) (see also Stratton 1941), which are for the case of a single sphere. In this case they occur at frequencies given by the vanishing of the denominators of the a and b Mie coefficients; cf. eqn (3.60). They are also related to the transverse electric (TE) and transverse magnetic (TM) natural oscillation modes of a conducting sphere (Debye 1909). In order for this spectral representation to be of use, we will need to find a convenient scheme to compute ). The obvious 0 the eigenfunctions A) and then the overlap matrix elements (A*IE scheme is to use a vector spherical harmonics representation which is orthogo nal but whose radial part involving spherical Bessel functions is bi-orthogonal, as we shall see. This is a multipole expansion which is tedious, and nowadays amenable to symbolic programming, but like its static counterpart will suffer from convergence problems when spheres are not far apart. Long-wavelength limit expansions have been studied (Bergman and Stroud 1980), which are very instructive, especially as the Clesch—Gordon architecture required by such cal culations are now part of packages like MathematicaTM. This approach is yet to be fully explored, and it offers significant advantages for numerical multipole expansions of the multiple scattering terms, which is the subject of the next section.
80
4.5
LIMITATIONS AND BEYOND
Ch. 4
(TM) : E(r)
Multiple scattering
The problems associated with multiple scattering are central to all discussions for going beyond EMT. There are several issues that require clarification at this point in order to be clear as to what we are trying to achieve. Hitherto, our development of EMT in this book, and indeed in the whole field, has only focused on the one-body problem. Multiple scattering has not been incorporated, because the minimum non-trivial configuration for this is that of two bodies, and even for the static two-body case, an analytical solution (Choy et al. 1998a) is formidable. In other words, the EMT theory we have been studying all along approximates all multiple scattering effects via an effective medium based on the one-body solution. This is an apparent contradiction, but may be successful, as long as we can effectively assume a factorisation of multiple scattering effects into one-body terms (cf. eqn (4.5)). This state of affairs is likely to change in the near future when we gain a better handle on a two-body effective medium theory. Early attempts have been made (Bedeaux 1987, Choy 1995) in the context of hydrodynamics, but these efforts were ad hoc and rather unsatisfactory. We will return to a discussion of the two-body EMT in section 4.7. Here, we shall focus on the numerical treatment of the multiple scattering problem and, in particular, on an efficient algorithm whereby these terms may be computed for small finite systems. While it is beyond the scope of this book to delve into numerical methods, the techniques discussed here are in principle exact. They are important from the viewpoint of being able to furnish results that will test higherorder approximations than the one-body EMT, as well as sharing a structure with Feynman diagrams, which we will explore in section 5.5. Specifically, we shall discuss an iterative T matrix algorithm that is generalisable from one body to n bodies and, above all, is computionally efficient and relies on methods discussed in the last section; namely, specific eigenfunctions of the operator A (see, for example, Bergman and Stroud 1980, Chew et al. 1990). Two main goals for these studies will be (a) to devise schemes to handle multiple scattering and (b) to look for methods to incorporate this knowledge for going beyond one-body EMT. 4.5.1
Single-sphere T matrix
We have begun to look at the eigenfunctions of the A operator in the last section and noted that in general the eigenvalues and eigenfunctions are complex. There are in fact two kinds of eigenfunctions, the longitudinal fields E = —V asso ciated with the trivial eigenvalue u = 1 (Bergman and Stroud 1980), which are of no interest to us here, and the important transverse eigenfunctions for which 1 and div E = 0. These are the well known transverse electric (TE) and u transverse magnetic (TM) eigenfunctions (cf. section 3.6), which can be written in terms of vector spherical harmonics (VSH) as (TE) : E’(r)
=
MULTIPLE SCATTERING
§4.5
(r)Xe,m(1l);
=
k[1
81
{curl (r)X,m()] -
uOi(r)]
},
(4.26)
where the VSH are given by Xi,m()
(4.27)
\/ + 1)
=
Here L is the angular momentum operator acting on the scalar spherical har monic functions Ye,m; see, for example, Schiff (1949) and Jackson (1975). The important functions for us here are the scalar wavefunctions, which can be writ ten in terms of appropriate spherical Bessel functions within (r a) the sphere:
(r)
=
Aj (kr[1
=
Bh’(kr), for r > a.
-
In this problem, the eigenvalues of
for r
u]),
for F
=
(4.28) E, M and the coefficients of
A, B are to be determined by the boundary conditions at r = a. These are the continuity of the tangential components of the E and H fields respectively, 37 noting that hereafter we have 01(r) being non-zero only in a sphere centred at the origin. The important point is that in terms of the radial functions, the boundary conditions take the form of the continuity and appropriate derivatives of the scalar functions: 1 (M)
Yj’ (E)
kr)
dq Y’ dr 1
(r)
(E)
(4.29)
dr
These conditions lead to the following set of equations for determining the eigen values: xj(x) j(x)
(1)i
x=xM) —
x
(i
+ x(x)) 3j(x)
X2
xh (x)j (£ x) x=ka 1 2 X
(i
+ x14’)’(x)) .
h’(x)
x=ka
(4.30)
In the above equations we have denoted the eigenvalues as We have assumed that we are only considering non-magnetic, i.e. 37 out this chapter, without loss of generality.
i
= 1, systems through
82
LIMITATIONS AND BEYOND ka/Z].
E
Ch. 4
—
=
j(kr)X,m(l)ee,m,
(4.32)
where e,m are the well known coefficients for the plane waves (cf. eqn (3.56) and (3.57)). Following section 3.6, the external field outside the sphere is a sum of the above incident field and a scattered field: Eext(r)
=
Eo(r) + Escatt(r),
kr)X,m(fl)a,m. (h 1
=
=
jj(kir)X,m(l)b,m, e,m
where we shall remind ourselves that k
(4.36)
=
0 E
=
Escatt 1 E
=
X(T)(I)j(kr)e
(4.37)
=
where (fl) is a column vector, with the superscript (T) meaning its transpose, which contains the VSH as elements, j and h are diagonal matrices containing the spherical Bessel functions, and e, a, b are column vectors containing the ee,m, ae,m, bt,m coefficients as elements. The unknowns are found by the boundary conditions given in eqn (4.29), which can now be written in our matrix notation as (T)
(1) [J(ka) .e + fi(ka) .aJ
= j(T)
.
k(T) (1) .[1’(ka) .e + fi’(ka) .aj
=
(1) .J(k 1 a) .b,
T) (O k ( 1
.j’ (k 1 a) .b,
(4.38)
where the primes here denote matrices that contain the derivatives of the spher ical Bessel functions. Using the orthogonality of the VSH, the above eqn (4.38) simplifies to
where kr
=
3(kia).b
=
{j(ka).e + fi(ka).aJ,
krJ’(kia).b
=
‘(ka).e + ’1 (ka).a},
a
=
b
=
(4.39)
k. These equations are easily inverted, so that k / 1
[J(kia)ii’ (ka) [J(kia)i’(ka)
-
-
kr3’(ka)i(ka)]’ .[krl’(kia)J(ka) kr3’(ka).(ka)]’.[’(kia)j(ka)
-
-
3(kia)J’(ka)].e,
fi(kia).J’(ka)].e,
(4.40)
(4.34)
the inversion being relatively straightforward since the matrices involved are diagonal. We can now define the T matrix using the above solutions:
Finally, we also expand the field internal to the sphere in the same way: Et(r)
83
For this one-sphere problem, we can of course easily fill in the coefficients using our results of section 3.6.38 We shall, however, reformulate all these results in a convenient compact matrix notation, which we will require later on. This consists of rewriting the above equations as matrix products:
(4.33)
where the scattered field has an expansion similar to that of eqn (4.32): Escatt(r)
1 k
(4.31)
Equations (4.30) yield an infinite sequence of eigenvalues for a given £ and F 0 limit, are related to the zeros of the spher which, in the long-wavelength ka these by the radial quantum number n enumerate ical Bessel functions. We shall lity applies only to these radial wavebi-orthogona and we should note that the normal orthogonality relations in the satisfies functions, as the angular part operator, which commutes with momentum view of the fact that the angular discuss the numerical scheme, to ready are we A, is Hermitian. At this point, also Mackowski 1991). We (see (1990), al. et which is based on that of Chew the eigenvalues, which are furnish to solved is first note that after eqn (4.30) of our single sphere solution scattering the find to generally complex, the task is scalar function whose case, TE the on focus shall and thereby the T matrix. We be easily gen can case TM the while (1990), al. et in fact coincides with Chew problem scattering the in interested are we If eralised using the same approach. which real, is u eigenvalue particular the then and not the spectral problem, can field incident The generality. of loss without we shall assume to be the case point this from shall we and s, eigenfunction above the be expanded in terms of onwards pick out a particular radial wavefunction with quantum number, say, n = 0, as corresponding to the real eigenvalue value of u appropriate to the (cf. eqn (4.25)). Thus we shall first system, i.e. a i sphere in a medium of as expanded is which field, write down the incident Eo(r)
MULTIPLE SCATTERING
§4.5
(4.35)
a
=
te,
(4.41)
so that the full external field can now be written as Note that here we are only considering the TE mode for convenience. Comparison with 38 section 3.6 will require carrying out the same exercise for the TM mode; cf. eqn (3.53) and comments following it.
=
C
Ch. 4
LIMITATIONS AND BEYOND
84 Eext
= :x(T)
( .[/3 20 , T ) 2 2 = .e], a
(4.42)
(O j(kr)e +
Before proceeding, it is worth mentioning that T matrices for non-spherical scat terers are also known, at least for regular bodies like ellipsoids (Bowman et al. 1987). With the results for the one sphere on hand, we can now look at the twosphere solution, by which an algorithm emerges which will provide a recursive solution for the n sphere T matrix, as we shall see. 4.5.2
MULTIPLE SCATTERING
§4.5
Two-sphere T matrix
We now look at the two-sphere problem, which is a dynamical generalisation of the static case (Choy et al. 1998a) and similarly to the latter is presum ably extendable to arbitrary d dimensions, although we shall deal here with the usual three-dimensional case only. Equation (4.42) now contains terms from both
85 (4.48)
where these two-sphere T matrices are formally given by ./3 = [I 10 ) 2 Tl,( .e ( = [I .e ./3 , T ) 2 o
—
—
T .c2lj .(l) .cl , Tl,(l) 2
].e, T (l)./3 . l , 2 10 + O .Ti,(i).[/3
( .al ,(i).a 2 T , .T 21 ) 1 ]
l.Tl,(l)./3lo].e, 2 (i).[/3 + c 20 , 2 .T
(4.49) and where the factor / i0 ensures that the incident field is in the coordinates 3 of the ith sphere. We can see the power of this method in that the two-sphere T matrix is now given entirely in terms of the one-sphere T matrix, and the procedure is in fact generalisable so that the (n + 1)-sphere T matrix can be written in terms of the n-sphere T matrix.
spheres: = x(T)
.a )J(kr T)(I ) 2 (f).j(kro).e + j(Tli)Si(kri).ai + ,
where the first term is the usual incident field and the next two terms are the scattered fields from the two spheres, written in their self-coordinates, respec tively. Translation formulae are known for shifting these coordinates from one sphere to the other (Danos and Maximon 1965; see also Choy et al. 1998a). We shall write these shift formulae in the form r(T) (fl)ii(kr).a = (T)
(fl).J(kr).a
5(T)
= 5(T)
n-sphere T matrix Following the results of the two-sphere solution in the last subsection, we can write down the n-sphere field as
4.5.3 (4.43)
(4.44)
Eext
=
j(T)
()J(kro).e +
j(T)
(4.50)
Here we have written the scattered field in the self-coordinates of each sphere A similar result in terms of the incident field via the n-sphere T matrix: holds for the (n + 1)-sphere solution in terms of its T matrix: n
Eext
Using these shift formulae, we can rewrite eqn (4.43) as Et =
k(T)
(Z)j(kri).io.e +
5T)
(1i)it(kri).ai +
k(T)
. 2 )J(kri).&i .a 2 (12
=
(T)
5(T)
()J(kro).e +
.e 0 (1l).f1(kr).i’,(+l) ./3
+ Tn+l).(krn+l).Ifl+l,(fl+l).&+l,o.e,
(4.51)
(4.45) We can now see that the first and the last terms of eqn (4.45) can be combined to act as an incident field on the first sphere. Thus we can write the T matrix in the form of a one-sphere solution, following the previous subsection 4.5.1, as follows: (4.46) . 12 & ] 2 .e + .a 10 1 = Tl,(l).[/3 a
where we have separated out the (n + 1)th term which, together with the first , n term, can be viewed as an incident field interacting with each of the i = 1,2, have now we following eqn Then, (4.48), spheres. .
T,(+l).o.e =
+
.Tfl+l,(fl+l).fl+1,o.e],
.
.
(4.52)
We can derive a similar formula for the second sphere, which by symmetry (for equal spheres) is easily shown to be
where we have made judicious use of the shift formulae given in eqn (4.44) to
(4.47)
formulae lies in an expression for Tfl+l(+l) in terms of the (presumed) known terms. This can be achieved via the use of eqns (4.47) and (4.52), details of which will be left as an exercise for the reader.
i.ai], 2 ,(l).[/3 + & 2 T .e 2 = 20 a
where we have used a notation in which ‘tj,(i) is the one-sphere T matrix for the ith sphere. Now, eqns (4.46) and (4.47) can be inverted to give the two-sphere T matrices as
1 = a
write all coordinates in terms of theith sphere. The key to obtaining the recursion
Example 4.1 Show that the term Tn+l,(+l) is given by an expression similar to eqn (4.49):
Ch. 4
LIMITATIONS AND BEYOND
86
T+l(+l)fl+1o.e =
[i
1
§4.6
COMPETING INTERACTIONS
The potential of one sphere is given by — Tn+l(l).
n+l,i.Ti,(n)i,n+l].Tn+l,(l)
= y(T)çj
n +
[n+1,O
.e.
(4.53)
Equations (4.52) and (4.53) now furnish a recursion algorithm, whereby , n + 1, can all be calculated in terms of the previously for i = 1,2,.. , n (Chew et al. 1990, Mackowski 1991). values known will show that this scheme does not require all the details of the Careful study simple counting shows that if there are m coefficients in A spheres to be identical. ) floating point operations, 3 m 2 requires O(n algorithm fields, this the multipole (see, for example, Peterson ) m 3 O(n are of which algorithms other as opposed to are less, since requirements memory the In addition, 1974). and Strom 1973, x that the Note nm. then rim rather x m m of sizes carried are the matrices mode, TM for the repeated also be must section in this given entire exercise that so principles, no new but book-keeping more little require a whose details further leave We shall is involved. power in computing of 2 factor a further discussions about computing algorithms at this point, as this will carry us too far afield from the theme of this book. Nevertheless, we shall mention two useful comments to conclude this section. First, numerically exact results for the two spheres are now known, in fact for any dimensions (Choy et al. 1998a), which can serve as useful test beds for such algorithms. The latter, though at this stage only applying to the static case, can be obtained as the long-wavelength limit of 40 the above equations by replacing as follows: ..
.
j(kr)
—*
he(kr) kj(kr)
—*
kh(kr)
—k
r
—
. 2 —( + 1)r’
(4.54)
The a and b column vectors have elements which are now the familiar one-sphere values: 1 + 1) + €r 2+1 ] + 1) + —
am =
bm
=
(e
(4.55)
39 implementations may well reduce this to an 0(n) algorithm. Supercomputer A reinsertion of t which is in fact € is needed, as we are dealing with the scalar Laplace 40 equation.
+ Y(T)(c).(jr)_l.T.e,
(4.56)
where Y is now the column vector containing the scalar spherical harmonics, j is the vector containing the elements rt, and T is diagonal here, with elements Tem,’m’ =
Ti,(+l), for i = 1, 2,.
87
[)]
2 La
6im,t’m’.
(4.57)
Numerical computations for both the static case (Chew et al. 1990) and the dynamical case (Rouleau 1996) have now been reported with up to 30 spheres, which required only fairly modest computations using SUN workstations. The second and final comment for us here is that the scheme presented can be recast in terms of Feynman diagrams (see section 5.5). Apart from its elegance, this will have considerable usefulness for exploring extensions of EMT. For this purpose, we leave the reader with the exercise of once again rederiving the Bruggeman and Maxwell—Garnet approximations using the T matrix formulation for the static case, as given in this section. An insight into the dynamical case will require the tool of Feynman diagrams, as we shall see later in section 5.5.
4.6
Competing interactions
Throughout this chapter, we have studied the limitations and beyond of EMT, and up to now we have placed our emphasis on multiple scattering as the way forward to improvements of EMT. In this section, we would like to discuss an al ternative viewpoint that has so far remained underexploited. This is the concept of competing interactions. In the electronic problem and CPA theory (cf. section 5.4), one does encounter real competing interactions, in the sense that the Hamil tonian contains terms involving electron scattering by random impurities as well as via phonons, magnons, excitons, and/or other collective excitations. In the problems of electromagnetic properties which have been the focus of this book, one may encounter competing interactions in the same sense. An example would be the case when the inclusions are not fixed but are free to move. There will then be, in the simplest case, hard sphere interactions among the inclusions and also with the host, which may be hydrodynamic in nature if the latter is a fluid. These are additional complications to the theory so far developed, but we would not like to delve into them, as each such system has complexities of its own. Nevertheless, the principle of EMT may be generalised to include these forces (see, for example, Chapter 6). What we wish to emphasise here is the concept of fictitious competing interactions. We have an idea of this concept previously, when we recall that the electromagnetic energy stored in a medium, as given by u=f{E.D+B.H] dV
(4.58)
Ch. 4
LIMITATIONS AND BEYOND
88
appears to consist of mutually non-interacting parts, i.e. the E and B are inde pendent and are decoupled, but only in the static case. In the dynamical case, we note from our studies in Chapter 3 that these fields are coupled via Maxwell’s equations. It is in this sense that we have a new perspective, i.e. we shall view the electric and magnetic parts of the field as competing energetically. This sit uation is not restricted to ac, for in the case of superconductors, there can be ’ Therefore the subject of compet 4 competing interactions even for the dc case. ing interactions in this context and its connection with multiple scattering is well worth considering. In the dynamical situation we can consider the time-averaged energy U and rewrite eqn (4.58), assuming harmonic waves only of frequency w and wavelength A, and neglecting dispersion (Landau and Lifschitz 1984) as U
=
2 1 IrA I —Icurl I l6irj L
—
2 HI
1
+
2 JI dV, iiIHI
(4.59)
whereby the competing interactions is in this case manifested as of magnetic type. This formulation is not unique, however, for the same competing interactions can also be described as of electric type:
U
1 =
Ir
2 I I€1E1
l6irJL
+
2 A
1
I.1 dV. 2 —Icurl E
(4.60)
Thus we now have a new viewpoint whereby the effects of multiple scattering are seen to be competing interaction terms. This viewpoint is, however, not restricted to the dynamical case. Let us return to the static case to look more closely at this. In this case, the significant energy terms are in fact those arising from the near fields (cf. section 2.2 and Appendix A.2). The results of the previous section 4.5 can be easily generalised for the internal fields int, and in the same way the relevant T matrix, whose one-body matrix is again diagonal (cf. eqn (4.55)). However, we shall, for convenience, limit our discussions here to the external fields only, leaving the details for the internal fields to the reader. We shall now write the external potential as n =
y(T)(ç)je
+ Y(T)(cli).ji.!3i,o.Ti,(fl).e.
(4.61)
i= 1
it is now easy to show using this potential that the energy stored in the fields, or the ‘Hamiltonian’, in the language of statistical mechanics, is given by Uext
=
f
j.Ti,(fl).Tj,(fl)] dV,
e(r)[i.Ti,(n) + i= 1
(4.62)
i,j=1
up to unimportant constants, where K and J, 3 are appropriate interaction ma trices. We emphasise once again, the T,() ‘s are generated from a set of recursion The competition there is between the London diamagnetic and the magnetic energies. 41
[
§4.7
TWO-BODY EFFECTIVE MEDIUM
89
formulae (cf. section 4.5.3). The significance here is, of course, in the second term in eqn (4.62), which contains competing interactions. The reader will see the rel evance of the above results to our studies in earlier chapters. Once again, the EMT theory we have studied so far seeks to replace the T matrices by appro 42 This ‘Hamiltonian’ is particularly important when we priate one-body terms. come to study the system in the vicinity of the critical point (cf. section 4.3), where the techniques of the RG theory, which has been demonstrated to have wide applications (Benfatto and Gallavotti 1995) is the appropriate tool. We have already remarked previously that the EMT theory hitherto developed, where the T matrix is an effective one-body object, fails to capture the essential physics at criticality. The correction terms, as can seen from a closer scrutiny of eqn (4.62), arise from a sequence of competing interaction terms. 43 Qualitatively, we can see that they may conspire with the percolation properties of the clusters to change the universality class of the system away from mean field behaviour. In the language of RG theory, the renormalised Hamiltonians ‘flow’ to a new fixed point. We will not pursue this subject further, as it will only carry us too far afield and will require a book on its own, except to conjecture that eqn (4.62), when combined with the Monte-Carlo RG methods (Ma 1976b, Burkhart and van Leeuwen 1982), is likely to yield better estimates for the critical exponent i for the conductivity or dielectric constant. Instead, we shall contend ourselves in the next section with looking at an attempt to develop a two-body EMT which, even though it is unlikely to yield better critical exponents, will hopefully be a better theory in the sense of being able to overcome some of the shortfalls in one-body EMT. 4.7
Two-body effective medium
Our studies of the last few sections permit us to now consider the question of going beyond the one-body EMT. The subject is an important one for a variety of reasons. These are of course connected with the perceived objectives that such an improved theory may provide. The first reason that one has to develop such a theory is the hope that it may improve the low-density (or fractional volume) behaviour of the theories so far presented. In recent years, the derivation of exact second-order virial type coefficients based on a multipole series solution of the two-body electrostatic problem (Choy et al. 1998a), as well as the more efficient multiple image series method (Choy et al. 1998b), have renewed the search for such a theory. Presumably, one would like to have a theory that, with a proper incorporation of two-body effects, would be able to reproduce the above exact results. The second reason for developing such a theory is the hope that it might improve the behaviour of EMT at criticality. This hope must, however, be assessed on the basis of what we now know from critical phenomena theory. If two-body effects were to be incorporated in the same way as in the well known Bethe—Peierls approximation, which is now known to be exact on 42 the next section, we will look at a two-body EMT. 1n As generated by the recursion algorithm. 43
______
90
LIMITATIONS AND BEYOND
Ch. 4
field type a Bethe lattice, then one would not expect anything beyond a mean behaviour. This in itself is a highly non-trivial exercise for the continuum models which we have been studying, but less so for resistor networks (Day et al. 1996), improved and is well worth pursuing. However, the essence of the search for an seek out to be must extensions higher-order or two-body using EMT theory behaviour of a simplified theory to understand and then to model the critical been the system. To the best of this author’s knowledge, this aim has not yet or Ising) achieved, even for the well known two-dimensional Ising model (2-D two in others, and this is especially intriguing, given that we now know that dimensions all critical exponents are fractions. In this case, one would expect for that a self-consistent theory would be constructible, whose order parameter Unfortunatel y, the 2-D Ising should satisfy an eighth-order polynomial equation. or theory there do not seem to have been any efforts devoted to developing such a theories, a state of myopia created by the success of RG theory. For our problem improved at hand, this must be the more important reason for the search for an EMT. Efforts to develop an improved EMT have been made, but only in the context of the CPA theory of alloys (section 5.4) and these have not met with a lot of success (see the review by Elliott and Krumhasl 1974). Nickel and Butler (1974) earlier identified that the primary reason for the failures of CPA extensions were associated with the failure to preserve the Herglotz property of the Green’s functions. Attempts at improving this shortcoming have been proposed (Mills and Ratanavararaksa 1978), but successful implementations of these schemes have so far been restricted to one-dimensional linear chains (Kaplan et al. 1980). The extent to which extensions of EMT for the continuum electrostatic problems at hand will face similar problems and can be circumvented by the proposals of Mills and Ratanavararaksa (1978) remain to date unexplored questions. In this section, we shall merely discuss two obvious extensions based on what we have studied so far: many details remain to be worked out for the future. 4.7.1 Two-body Maxwell—Garnett Extensions of the Maxwell—Garnett theory, or the ATA approximation (cf. sec tion 1.5), appear straightforward, and we shall discuss this first. For the twocomponent composite we can propose the following extension, where the dipole polarisability of the Lorentz sphere constituents now includes all n-body terms using the T matrices of the last two sections: =
(4.63)
following the same notation as section 1.3 (cf. eqn (1.19)). For the two-body case, we shall write this as (4.64) ± ) is rather hard to calculate and will in general 2 Unfortunately, the quantity (S, require knowledge of the two-body radial distribution functions. This quantity
§4.7
TWO-BODY EFFECTIVE MEDIUM
91
has so far been calculated only for the low-density limit, up to O(ji) (Choy et al. 1998a,b), which yields in general very slowly converging series. In most cases, except for the perfect conducting limit, it can be approximated by S i,2
—
1 \2S+3
2
/S
q\
(
s
—
)
-(s+1)+s+2
(4.65)
It is easy to show that this theory, by its very construction, gives an exact value for the second-order coefficient ii for the low-density expansion (Choy et al. 1998a): /
2 i
+e 2 o
(4.66)
if the exact (S ) known to O(rii) is used. The theory predicts a finite percolation 2 , 1 threshold for perfect conducting inclusions of iii, = 0.7313... and for holes of 11,c = 0.9103... with a critical exponent of one, as expected. Neither the critical exponents nor the percolation thresholds are particularly good values here, but they nevertheless represent an improvement over the traditional Maxwell— Garnett theory, which predicts 1i,c = 1.00, which is unphysical (cf. section 1.3). The extension of the Bruggeman theory appears to be relatively straightfor ward as well, and we may assume that this consists of the self-consistent equation, which requires zero polarisation by both components of eqn (1.41) to include all n-body terms using the T matrix, embedded in an effective medium e, thus: n
0,
=
(4.67)
i= 1
where the T matrices are all defined in terms of an effective medium e. This is a natural generalisation and is attractive, but unfortunately the calculations are formidable even at the two-body level, where we assume that the above is to hold for n = 2 (cf. eqn (4.49)). One of the central questions to be asked is: Will this theory improve upon the the one-body theory at low densities, or will it encounter unphysical solutions as a consequence of analyticity difficulties, as faced by its CPA counterparts? This point is particularly worth emphasising, since while we expect that the exact T matrix for the medium should indeed satisfy eqn (4.67), we have no reason to expect a priori that this equation should be satisfied for a truncated T matrix. Thus the fact that we have a reasonable approximation for a truncated one-body T matrix is no guarantee that the same will hold for a two-body one: in fact, a naive truncation as suggested here may even degrade the quality of the theory achieved by the one-body EMT. If that is the case, a systematic study is required using the method of variational bounds (cf. section 2.2; for some preliminary work, see Torquato 1985), or the Green’s function method (cf. sections 1.5 and 4.2). Suffice it to say that the whole subject of n-body extensions of the EMT is still full of open questions, and one expects that progress could be made in the near future, given that some exact results
LIMITATIONS AND BEYOND
92
Ch. 4
1998a,b). We shall for the two-body T matrices are now available (Choy et al. in section 5.2, comment further on the analogous situation in the CPA theory promising results. To where some cluster extensions of CPA theory have yielded examine some conclude this chapter of our book, we shall in the next section issues in dynamical theory, relating to non-equilibrium. 4.8
[
§4.8
NON-EQUILIBRIUM
and at equilibrium this is assumed to be replaced by an ensemble average. The importance here is that for stationary random processes, where all correlation functions like eqn (4.68) are unaffected by time shifts, i.e. are only a function of r in accord with the requirement of time-invariance (cf. eqn (3.14) and the comments following it), we have the well known Wiener—Khinchin theorem (Van Kampen 1981):
C(r)
Non-equilibrium
the macroscopic In Chapter 3, we have developed DEMT solely on the basis of fields are aver equations of electrodynamics. As discussed there, the macroscopic and while mechanics, ages or coarse-grained quantities in the sense of statistical media, random we have looked at the propagation of electromagnetic waves in the fluctuations these these waves will undergo fluctuations in time. Microscopically, will ignore, as are either of thermal or of quantum origin: the latter however, we are, however, it is beyond our scope to consider them here. Thermal fluctuations leads to which an important part of the whole process of statistical averaging,
C(T) = (E(r,t).E*(r,t
—
r)),
(4.68)
where (...) denotes a time average, over a sufficiently long time T that loses memory of to, (E(r, t) ) = 2
1
to+T
L
2 dt, E(r,t)
(4.69)
=
f
G()
cos(wT)
dw.
(4.70)
This relates the autocorrelation function to the Fourier cosine transform of the
spectral fluctuation density G(w). As is now well known (see, for example, Wan nier 1966), the demand that our system must reach equilibrium after a sufficiently long time imposes conditions on the structure of the spectral distribution at equi librium. These are given by the Nyquist relation, originally derived via a circuit analysis (Nyquist 1928), which implies a flat ‘white noise’ frequency spectrum whose weight is proportional to the temperature and the transport coefficient:
temperatures. the macroscopic irreversibility of the equations of motion at finite
limit In this sense, all the results of Chapter 3 pertain to the zero temperature non-equilibrium of only. More importantly, we will have to address the question and fluctuations, relaxation time, the approach towards the equilibrium state, w) €(r, permittivity the relationship of the transport coefficients, in our case the embraces adequately or the conductivity a(r, ). Unfortunately, a DEMT that appropriate these issues is still to be developed. It is clear at the outset that the which all by viewpoint is to employ a mesoscopic description (Van Kampen 1981), stochastic the above non-equilibrium features are to be prescribed by appropriate An processes. What is not so clear here is conveyed by the word ‘appropriate’. they will when example would be the use of Markov processes, and the issue is any of be useful and when will they break down? We will not be able to address to do try will the above questions in detail; that is an unrealistic goal. What we identify and is to define the problems, discuss some of the modern perspectives, of the issues with empirical situations that are of current interest. The subject their manifest dynamical scaling and renormalisation group theory once again usefulness. The empirical situation that concerns us mostly will be the case of an in terference spectroscopy of some kind, where our sample may interject the light beam of one arm of a Michelson interferometer, say, as illuminated by a coher an ent laser light source. In this case, the central quantity of interest will be autocorrelation function defined by
93
G(w) = 4 Re(Z)kT,
(4.71)
where Re(Z) denotes the real part of some circuit equivalent impedance. This is
actually connected with the famous Einstein—Nernst relation (see, for example, Wannier 1966, Aschroft and Mermin 1976), for the mobility ILe and the diffusion constant De: Ite
where
e
=
(4.72)
is the electronic charge. Nowadays, these formulae are perceived as
specific cases of the fluctuation—dissipation theorem. This relates the equilibrium
fluctuations to the dissipation, i.e. the imaginary part of the permittivity whose classical formula is ) 2 (E(r,t) =
-f
e”(w)
C”,
(4.73)
which can be inverted, or alternatively derived through a linear response theory as the Kubo formula:
e(w) = feT(E(r,_r)E(0,0))dr.
(4.74)
It is to be noted that the above description entails a linear response assumption, i.e. small perturbations (in some sense), and therefore the system is gently kicked out of and then allowed to relax back into equilibrium. For the cases that we are concerned with, the more general description of non-equilibrium must still require a study of the appropriate correlation functions as defined by an extension of eqn (4.68): C(R, T) = (E(r + R, t).E* (r, t r)). (4.75) —
94
Ch. 4
LIMITATIONS AND BEYOND
One approach—by no means the only one—for calculating these correlation func tions will be to treat the appropriate equations of motion by a Langevin theory. To see this, we shall rewrite the wave equation, eqn (3.43),44 as O E 2 2 at
2 C
&8 E 2 2 8t C 2
(4.76)
where we have separated the grain permittivity in terms of a mean ë and a fluctuating part Se: (4.77) e(r,t) = (r) +Se(r,t), 45 is usually assumed to be The fluctuating part, being of microscopic origin, Gaussian noise: (Se(r, t)Se(r’, t’))
=
kTS(r 7
—
r’)S(t
—
t’).
(4.78)
Thus the effect on the RHS of eqn (4.76) may be assumed to be a Langevin noise term, while the LHS has been studied as through the scattering theories developed in Chapter 3. There are many interesting effects that such fluctuations may induce, but the one of considerable interest is the theory of dynamical scaling which suggests that the time correlation functions like eqn (4.75) should scale universally in the form: (4.79)
C(R,r)
where (t) is some characteristic scattering length which varies with time in tL), so that the plot of C(R, r)/RT accordance with some power law (t) is independent of time. The underlying reason for the time dependence of (t) could be a model of domain coarsening (see, for example, Bray 1994), but this is only one of several possible models. Fractal networks too have a characteristic length scale (w) in frequency space which leads to a dynamical structure factor, i.e. the Fourier transform of eqn (4.79), which has the form
S(q,w)
=
qYF(q(w)),
(4.80)
whose long- and short-wavelength limits are predicted to behave as
S(q,w)
=
, 2 q’w
(4.81)
with appropriate exponents y and z (Nakayama et al. 1994). So far, dynamical scaling has not been proved, but the theory is justified using the modern tool of RG theory. However, for the analogous problem in polymer dynamics (Doi and Edwards 1986), an effective medium type theory appears to be in agreement 44 magnetism, i.e. ji = 1 as before, and also the skin effect, i.e. o = 0. Neglecting 1n atmospheric situations as in pollution monitoring, these fluctuations are caused by 45 metereological or chemical reasons.
§4.8
NON-EQUILIBRIUM
95
with dynamical scaling in three dimensions, but there are anomalies in two di mensions (Shannon and Choy 1997). The application of dynamical scaling theory to electromagnetic systems is likely to be of significance for an understanding of other proposed exotic forms of behaviour, such as the localisation of light, claimed to have been observed recently (Wiersma et al. 1997).
This is an appropriate point to conclude our chapter on the discussion of limitations and beyond of EMT. In the next chapter, we shall now come to the task of relating EMT as developed in this book to other areas of physics, where the fundamental approach is identical or nearly identical.
§5.2
5 RELATED THEORIES 5.1
Comments
At this point, we have completed the development of the effective medium the ory (EMT) for the static (time independent) case and the dynamical effective medium theory (DEMT) for the dynamical (time dependent) case based on the electromagnetic properties of a binary composite. Along the way, we have oc casionally made references to related subjects like the Coherent Potential Ap proximation (CPA), Feynman diagrams, and others, which belong to other areas of condensed matter physics, and which are historically later developments to the traditional Bruggeman and Maxwell—Garnett theories. Some of these related subjects are merely modern viewpoints which developed independently of EMT. They were invented, or reinvented if one adopts a more general picture, to ad dress specific questions relating to microscopic systems—in some cases to deal with quantum properties. The purpose of this chapter is twofold. First, we shall review and then estab lish the analogy of EMT with related theories that employ the same operational principles as discussed above. In particular, we shall examine CPA theory and, with it, the related Random Phase Approximation (RPA), for the electronic properties of binary alloys, in section 5.2. We shall then discuss CPA and, by analogy, the scattering theories for electromagnetic composites (cf. Chapter 3), using a Feynman diagrams approach in section 5.3. The interesting phenomenon of Anderson localisation and its more modern manifestation, the localisation of light (cf. section 4.8), will be the subject of section 5.4. We shall show that these phenomena require extensions beyond CPA theories. The second purpose of this chapter is to stretch the outlook of EMT as pro pounded in this book by extending, in particular, the effective field concept. We shall achieve this by drawing the analogy with mean field type theories of sta tistical mechanics, but we shall also emphasise that more subtle variations are sometimes needed, as in the Kirkwood and Percus—Yerwick theories of liquids (section 5.5). In the quantum case, these mean field theories have now come a long way since the early days of Hartree—Fock, (Fischer 1977), through the de velopment of density functional and the lesser known density matrix functional theories (Levy 1979, Parr and Yang 1989). We shall find it worthwhile to dis cuss these theories in section 5.6. They have some philosophical similarities with EMT concepts, especially through their variational treatments, but however, practical implementations, as through the Kohn—Sham local density functional approximations (LDA) (Kohn and Sham 1965), contain deep physics which led to, and is obviously connected with, their success in many situations. We shall
COHERENT POTENTIAL APPROXIMATION
97
end this chapter by discussing the Hubbard model (section 5.7) which, inter estingly enough, can be reformulated through the viewpoint of a CPA theory or, alternatively, via a density functional theory (DFT), thus demonstrating an important link between CPA and DFT. We shall conclude this chapter with a summary.
5.1.1 General viewpoint Before we begin, it is time to recast the formulation of EMT in this book into a more general viewpoint than the case of electromagnetic composites, that will facilitate the pursuit of the analogy with related theories. This general viewpoint can be elaborated symbolically as follows (Elliott et al. 1974). We start with a field theory which is either classical or quantum of the problem at hand. The field q(r) obeys a field equation that can be the equation of Maxwell, Schrödinger, Navier—Stokes, Cauchy—Poisson, or others, which is denoted by a linear, or ap propriately linearised, operator £(e), so that £(e) q
=
0.
(5.1)
The operator £(€) depends on one or more parameters €, which may be the con ductivity, permittivity, porosity, elastic constants, etc. In a homogeneous medium these parameters are single-valued, while in a random medium they may vary from place to place, in accordance with a probability distribution. There may also be other parameters which are not random, like the frequency w which we have encountered earlier or the energy E as in Schrödinger’s equation. The essence of CPA or EMT is to break up £ into £ = (L) + £, where the averaged (L) is to be defined intelligently. The purpose for doing this is to employ various approxi mations, self-consistent or otherwise, in order to calculate average quantities like () and (0) from which one can obtain effective parameters via e = (Ø)/(q). In CPA theory this is achieved via an approximation to the Green’s function (cf. section 1.5) which is the inverse operator to £, i.e. £ G = 6(r r’), whence, by Dyson’s equation, +G G=G G 1 L 0 , (5.2) where G 0 is the inverse operator of £o = (L) and by which there is a scattering field representation (cf. eqn (4.16)) —
=
—
G£ o
=
co +
scatt,
(5.3)
where = 0. In practice, eqn (5.2) is used to obtain a self-consistent local field approximation by employing the best low-order local approximation to a re-summed perturbation series (cf. section 4.2). Using this framework, the reader may now translate the electromagnetic problems discussed earlier in this book into other contexts. We shall begin this task by looking at electronic properties. 5.2 Coherent potential approximation Condensed matter physics has to contend with a random ‘world’ at the mi croscopic level, since crystalline (ordered) systems form merely a small subset
Ch. 5
RELATED THEORIES
98
1
§5.2
COHERENT POTENTIAL APPROXIMATION
99
and (jj) = c 2 for i j. In the above Hamiltonian, u(z) remains an as yet undefined ‘self-energy’ which will be adjusted later on, analogous to our € in section 1.5. The unperturbed Hamiltonian K is that given by a regular lattice:
I I
I I
I
I I
I
I
k=
I I
I
FIG. 5.1. A model substitutionally disordered alloy on a regular square lattice. of the structure of most materials. These materials include disordered alloys, impurity semiconductors, amorphous metals, amorphous semiconductors, amor phous magnets, liquid metals, etc., all of which can be classified as random systems. The Coherent Potential Approximation (CPA) was originally devel oped to study one-particle properties of various elementary excitations: elec trons, phonons, magnons, excitons, and others, for substitutionally disordered alloys (see Fig. 5.1). Numerous empirical systems have been studied using CPA theory (Yonezawa 1982): examples include optical absorption spectra of mixed ionic crystals, like KC1—KI and KBr—RbBr, photoemission of transition metal and noble metal alloys, in particular CuNi, and the magnetic properties of NiFe alloys, while of more recent interest are the shape memory CuAlNi alloys. The original idea of CPA (Soven 1967, Taylor 1967), has since been extended to eval uate other physical properties in addition to one-particle properties, as well as to systems with other types of disorder, like bond disorder and topological dis order. Our purpose here is not to review the entire field of CPA-related subjects: excellent reviews already exist in the literature; see Yonezawa (1982), Elliott and Krumhansl (1973), and others. Our main aim is to illustrate CPA via an example system, i.e. the binary substitutional alloy, in order to highlight the equivalence in operational principles between CPA and EMT. The quantum Hamiltonian of the model binary AB alloy of Fig. 5.1 is given 46 by: fI=I?+Ij)[(EA -u(z))+[(1-)(EB
-u(z))](iI,
(5.4)
Ii)u(z)(il +
li)”(iI,
(5.5)
where I’j is the hopping integral, assumed to be a constant Vjj W, for all nearest neighbour pairs i, j, and whose Green’s function and density of states are assumed to be well known and calculated (see, for example, Morita and Horiguchi 1971, Horiguchi 1972). For the purpose of studying single-particle properties, the main quantity of interest is the one-particle Green’s function for the Hamiltonian, eqn (5.4): G(z) = (zI H) , 1 (5.6) —
but we are mostly interested in its configuration average Oav(Z) =
(O(z))
E
(zI
-
Heff)’,
(5.7)
by which we can calculate in particular the density of states: p(E)
=
Tr[(ö(EI
-
1))] = -j-Tm[Tr Oav(E + iO+)].
(5.8)
Here Tr implies the quantum mechanical trace, while Tm denotes the imaginary part taken in the limit 0+, i.e. to the real energy axis from the upper half complex plane of z. Also, Heff is an effective Hamiltonian operator and is defined by Oav(Z). The task here is to calculate the latter, for which we shall find it convenient to introduce the function 1 zf(k)
k
—
po(E) dE, z-E
(5.9)
where po(E) is the density of states for the standard unperturbed Hamiltonian , i.e. when u(z) = 0, and is given in terms of elliptic integrals in two dimensions 0 K or integrals over elliptic integrals in three (Morita and Horiguchi 1971, Horiguchi 1972). Now the Hamiltonian K has its own Green’s function P(z), whose trace defines an F(z) function which is a trivial transformation of Fo(z):
3
where the A sites have site energy EA and the B sites have site energy EB, and the random variables j = 1 or 0, depending on the site being A or B respectively. These variables are used to affect the configuration average of all calculated quantities, so that (j) = () = c, where c is the fraction of A atoms 1n this section, all hats 46
A
denote quantum operators.
F(z)
=
Tr[P(z)]
=
(OIP(z)10)
=
Fo(z
-
u(z)).
(5.10)
Let us now rewrite the Hamiltonian, eqn (5.4), in the following form:
Hk+j)vj(j .2
EK+i,
(5.11)
RELATED THEORIES
100
Ch. 5
by which, from eqn (5.2), a perturbation series expansion of the Green’s function G(z) is easily written down:
O
P+
§5.2
COHERENT POTENTIAL APPROXIMATION
5.2.1
Random phase approximation A naive decoupling scheme employs a random phase approximation in eqn (5.12):
(5.12)
j
j
j
£
£
1 G=P+PVP+P P VPVP+...=(1 , -PVY
(5.13)
which can also be rearranged in various ways; for example, in terms of a T matrix, as (5.14)
(5.22)
(j)(j)(ik)...,
m
This we shall write symbolically as
101
so that the averaged Green’s function looks like the perfect crystal with a simple energy shift: Oav(z) = (zI K (V))’. (5.23) -
The self-energy
-
operator is now rather simple: E(z)
Ii)[u(z) + 1(z)}(jI,
=
(5.24)
where the T matrix is defined by (5.15)
where (VCA):
(z) here has the form known as the Virtual Crystal Approximation =
or in terms of a Dyson equation (eqn (5.2)):
O=P+PO.
(5.16)
=
P + PDGav,
(5.17)
and hence the self-energy is given by =
(I’)[I + P(i’)]’.
(5.18)
We emphasise here that the difficulty in all these computations is not so much in taking the averages, since this is in principle conveniently effected through the random variables j as i>j j
(5.19) j
£j
j
£j
m
where the j’s are easily evaluated as
£ =I[I_P&]—1
j)t(jl,
(5.20)
in which t, is the Slater—Koster t matrix element—in our case easily written down as 3 = v[1 vFo(z u(z))j’. t (5.21) The main difficulty is that after taking the averages, the terms are progressively more complicated functions and the sums in eqn (5.19), with their restrictions, are quite formidable. —
—
+ (1
—
C)EB
=
t,
(5.25)
which is merely an arithmetic mean, as denoted by ti. The density of states (DOS) is also now rather trivial, being
The trick now is to find an approximate Green’s function decoupling scheme (cf. section 1.5) so that the averaged Green’s function Gay can be computed Gay
CEA
FVCA(Z)
=
Fo(z
—
o).
(5.26)
Needless to say, this approximation has numerous shortcomings, among which the most serious is that the shape of the DOS remains unchanged as c goes from 0 to 1, which is unphysical. A realistic theory must be able to model the transition of the DOS from one pure crystal structure to the other. To improve the approximation requires a more sophisticated application of the RPA, now applied to the T matrix instead, leading to the ATA approximation, as discussed in section 1.5. 5.2.2
ATA
However, in applying the RPA to the T matrix we must first note that the sums in eqn (5.19) are restricted in order to prevent further successive scatterings from the same site. In the ATA, these restrictions are included partially by projecting out the diagonal terms in F, which are the dominant terms excluded in the sums; symbolically,
(0)
P + P()P + P()P’(iP + PP’()P’(P + = P + PQ(I P’(’)P, =
(5.27)
-
where (ilP’Ij) = (IIj) 5,(0IPI0). From eqns (5.27) and (5.14) that our approximate T matrix is given by the expression: —
=
()(I
—
we
conclude
(5.28)
102
RELATED THEORIES
where the averaged
1
—
§5.2
(5.29)
This, however, is impossible to achieve in practice, since the exact T matrix, eqn (5.14), contains infinitely many multiple scattering terms. CPA theory consists of taking only the single-particle scattering term
can be easily evaluated, using eqn (5.21), to be
C[EA —
—
—
(j)
Ch. 5
u(z)j
[EA — u(z)]Fo(z — u(z))
+
1
—
(1 C){EB — u(z)j [EB — u(z)]Fo(z —
COHERENT POTENTIAL APPROXIMATION
()
Using eqn (5.17), we can now obtain the self-energy 1 (z), as ) 3 (t —
cZ
—
1+(t)(oo)
—
l—(l—c)Fo(z—EB)’
(5.30) CtA
—
Example 5.1 Show that if we were to ignore the restrictions in the sums in eqn (5.19) altogether, the resultant self-energy would merely remove the factor (1 c) in the denominator in eqn (5.30), thus:
(ti)
cZ
—
—
)(OIPIO) 3 (t
I
—
(5.31)
+ (1
— C)tB
= 0,
(5.34)
which can now be viewed as a self-consistent equation for the determination of u(z). Note that the latter is in general complex.
Example 5.2 Show that there are various equivalent expressions for the above formulae. By writing F(z) = TrOav(z) = Fo(z — u(z)),
—
1+
(5.33)
This equation can be cast in the familiar form of the Bruggeman EMT formula,
where / = EB, and in the latter equation we have invoked the unperturbed host system as the pure B crystal. Equations (5.29) and (5.30) constitute the ATA and are analogous to eqn (1.71), the Maxwell—Garnett formula.
—
= 0.
103
(5.35)
show that eqn (5.34) can be expressed as
—
u(z) — EB =
The latter is the dilute limit, since by ignoring the restrictions on the sums, the effects of the impurities have been regarded as being independent. However, the ATA is an improvement for when c —+ 1 we recover the pure A crystal. On the other hand, the ATA is not symmetric; the formulation is not invariant to the interchange of EA —+ EB and c —* (1 — c). There are also problems with the failure to reproduce the correct band edges. For these, one has to look towards CPA theory. 5.2.3 CPA The reader will by now see the analogy with EMT, that a self-consistent CPA theory can be constructed through a clever choice of the Hamiltonian K; for, in fact, if K = Heff, then the effects of the perturbation terms are eliminated. Unfortunately, we do not know what ‘ eff is, and even if we do, the expression for 1 K becomes rather complicated, as it can no longer be expressed as an equation like eqn (5.11), since Heff would in general contain numerous off-diagonal terms. Nevertheless, if one has chosen such a K, then of course = P, which implies via eqn (5.14) that (T) =0. (5.32) This is a self-consistency equation, since the T matrix is now a complicated function of K, and eqn (5.32) is to be seen as an equation to determine K.
cZ
1 —F(z)[— (u(z) —EB)]’
(5.36)
or alternatively as F(z)
c
= F(z)’ + u(z)
—
EA
+
1—c F(z)’ + u(z) — EB’
(5.37)
where the symmetric nature of the theory is manifest.
The remarkable success of CPA is that it is a unified theory in the sense that it can recover the limits of all the previous approximations. When the parameter = I/2w << 1, where w is the bandwidth, we obtain the VCA. Similar to the Bruggeman formula, the dilute c — 0 limit coincides with the ATA. Other features, such as the correct sum rules and the Herglotz analyticity property 47 of the Green’s function, can also be proved. In addition, numerous computer calculations on finite but large systems have clearly demonstrated the superiority of CPA theory compared to the previous approximations (Elliott et al. 1974). Unfortunately, extensions of CPA to include higher-order scattering terms in eqn (5.14) have not met with much success. Early attempts (Nickel and The analyticity properties of a non-interacting single particle Green’s function require that 47 it must be analytical throughout the complex plane, with poles or branch cuts confined to the real axis (Economou 1979).
104
RELATED THEORIES
Ch. 5
Krumhansl 1971) to straightforwardly include two-particle scattering have led to analyticity problems, first identified by Nickel and Butler (1973). Attempts to improve these schemes have been proposed (Mills and Ratanavararaksa 1978). Butler (1973) had earlier introduced an ad hoc scheme that only includes bound ary sites of a cluster for the effective medium equation and sets to zero only the diagonal elements of the averaged T matrix. While little understood, this method has to date produced the most satisfactory results, by comparison with computer calculations dating back to Dean (1961) and Agacy and Borland (1964). A red erivation of the scheme of Mills and Ratanavararaksa (1978) using an augmented space representation with applications to linear disordered chains has been rather successful (Kaplan et al. 1980). However, no extensions to higher dimensions have yet been made, due to various complexities. We shall not pursue further details of the CPA theory of alloys, as we have fulfilled our purpose, which is to demon strate the equivalent operational principles of EMT and CPA. In a certain sense, this equivalence is not unexpected. In CPA theory we have electrons being scat tered by random potentials at different sites, while in EMT we have light being scattered randomly by different grains (cf. section 5.4). The only difference is the underlying equations of motion. In the next section, we shall explore the Feyn man diagrams reformulation, which offers additional insight into the operational principles of CPA.
§5.3
FEYNMAN DIAGRAMS ,
I
¶Ki>
j_L
+
105
__/_;_
+
I
+
+
+
+
+
1*\
FIG. 5.2. A diagrammatic expansion for
LL
+
Gay. (Reprinted from Elliott et al.
(1973))
J
2
3
4
1 -U
L_
AiA i!
+
--/\
d -
j’.”
L
+
5.3
Feynman diagrams
Without doubt, the use of Green’s functions and the representation of their per turbation series in terms of Feynman diagrams have revolutionalised the way we think of numerous problems in physics and even engineering. In the previous section, we have derived CPA on the basis of an intuitive decoupling scheme. In this section, we shall rederive the scheme using a diagrammatic expansion. The purpose is to illuminate the basis for CPA. We shall do this using the electronic structure problem of binary alloys in the last section. Later, we shall also use the same diagrammatic approach to look at the electromagnetic scattering prob lem studied in the earlier section 4.2. There are, however, some differences in the re-summation of diagrams in both theories that are not exactly analogous. They are a reflection of the difference in their underlying configuration averaging procedures and not in their operational principles. To begin with, one must first establish the rules for the diagrams in accor dance with eqn (5.12) as shown in Fig. 5.2. Here we have taken u(z) = EB: the single lines denote the Green’s function or propagator F, and the dots denote scattering off a site or sequence of sites whose averages are affected through the random variables a’s. If we ignore the fact that there are restrictions on the sums due to these averages when i = j for example, the single-site scattering series can be easily summed. This is given by the first column in Fig. 5.3. The result is the self-energy expression of eqn (5.31). The correction terms due to the restrictions on the sums, or multiple occupancy as this is called by its initiators, can also be worked out. These are shown in the subsequent columns and the
=
[
-ZP(O)’P(O)E’
-
-P(O)Z’P(O)Z’ P(O) E’
FIG. 5.3. Re-summation of self-energy diagrams for E within ATA and correc
tion terms. (Reprinted from Elliott et al. (1973)) summations given by the factors in the last row, using the same rules for sum ming the diagrams as given above (Elliott et al. 1973). These correction terms can be easily summed, since they each contain n irreducible parts of the form ]’, where F(0) = (OIPIO), so that 1 —Ei[P(0)D
= et
—
F(0) E 1
—
E F 1 (0)E (0)>
—...,
(5.38)
where we have abbreviated ct = (ti) for convenience here. The above selfconsistent relation for E 1 is easily solved to yield ct = 1 + ctP(0) = 1
—
(1
ciX c)F(0)’
(5.39)
—
which is none other than the ATA equation (5.30). We note in passing that the correction terms discussed here are peculiar to the electronic structure problem. In electromagnetic scattering problems there are no restrictions on the sums imposed by the configuration averages. However, these averages are not easily effected, since in general additional information on the microstructure, for exam-
RELATED THEORIES
106
J
2
3
Ch. 5 4
§5.3
FEYNMAN DIAGRAMS
whose self-consistent solution for E
5
=
107
(G(0)) is easily obtained:
0
-U
+
=
III
-
+
-
4
-,I,I\ ,
,‘\
*
¶
-A
A -“‘ -¶ =
-
+
FIG. 5.4. Re-summation of self-energy diagrams for tion terms. (Reprinted from Elliott et al. (1973))
within CPA and correc
G=A+A’uG, by which we can identify symbolically Go
ple, N-particle distribution functions, are required to obtain them. . Let us now 48 consider the diagrams for CPA. Here the internal propagators P(0) in the above ATA diagrams are replaced by the full propagator G(0), inserted self-consistently (see Fig. 5.4). The single-site self-energy is again easily summed, giving rise to the selfenergy first obtained by Davies and Langer (1963): cZ CT=
1—G(0)’
—
where the self-energy E is treated as a functional of its internal propagator and we have defined G(0) (5.42) 1—y)G(0) The other columns are similarly summed to obtain the final result, =
cr + (G(0))
- 1- ( )G(0)’ 7
=
A and L
(5.45) =
—u.
Example 5.3 Using the same diagrammatic approach as in the electronic prob lem, show that in the long-wavelength limit one can again rederive the ATA (Maxwell—Garnett) and CPA (Bruggeman) formulae of section 1.5. In particu lar, show that the single-sphere ATA T matrix is proportional to eqn (5.31), where now P(0) = —l/ €o and L = 3 — eo.
(5.40)
which, however, must be corrected for multiple occupancy (Leath 1968). This entails summing the diagrams in the other columns of Fig. 5.4 which are then subtracted from the first. Though complicated-looking, these series are eventually summable as geometrical series; as, for example, is the second column, which sums to (5.41)
E(G(0))
(5.44)
which, upon simplification and identification of G(0) = F(z) and E = u(z) — EB, is exactly the CPA equation in the form of eqn (5.36). This concludes our rederivation of the ATA and CPA using the diagrammatic approach. In a certain sense, the electronic problem contains the more complicated multiple occupancy correction terms that make the above derivations more involved than necessary. Let us now return to the electromagnetic scattering problem. We shall recall eqn (5.3), which is to be compared with the symbolic form given in eqn (4.16). Then the appropriate Dyson equation (5.2) now takes on the form
,,
“I
c/ 1 — (1— c)G(0)/{1 + EG(0)]’
(5.43)
1n section 5.4, we shall look at a simplified Gaussian model which still leads to a complicated 48 cumulant series that is not easily summed.
The common property of all the above Feynman diagram approaches is that a partial re-summation of an infinite set of diagrams is achievable in the form of a geometric series. Attempts to obtain the same for the finite frequency case will encounter the difficulty that the series are not easily summable even within the single-site approximation. The problems are twofold. First, the tensorial struc ture of the field equations generates complicated terms at each order in the perturbation series that nevertheless have a recursion relation (cf. section 4.5). However, these terms are not, easily re-summable. An exception is the case of scalar waves with discrete scatterers (Gloaguen and Lavergnat 1988). Even so, these are no better than the Foldy—Twersky approximation, an electromagnetic counterpart of VCA 49 (see, for example, Ishimaru 1978), and a planar ATA whose quality is dubious since it fails to agree with the Twersky formula for low concentrations, where the latter is known to be good. In conclusion, we note that the Feynman diagram approach is powerful from the point of view of illu minating the processes involved in making the ATA and CPA approximations. However, the difficulties in the earlier chapters in the treatment of DEMT at Note that unlike the electronic VCA the Twersky integral equation is somewhat non-trivial, 49 even for the simplest geometries like a slab (Ishimaru 1978).
RELATED THEORIES
108
Ch. 5
finite frequencies are not resolved. They are now transferred to the problem of an infinite series of complicated tensorial terms that are not re-summable easily, unlike the static limit, which yields geometrical series similar in structure to the electronic binary alloy problem. In spite of these difficulties, we shall discuss in the next section the problem of the localisation of light, which in recent years has motivated much experimental interest (Wiersma 1997), but whose theory draws on a perceived analogy with Anderson localisation for electrons, which we shall first review. 5.4
Localisation of light
The localisation of electronic states for the Hamiltonian equation (5.4) was first noted by Anderson in his classic paper (Anderson 1958). There he predicted that for random site disorder energies whose distribution function as characterised by a width (5 >> w, where w is the bandwidth, the system becomes an insulator. It is now known through arguments of Mott and Twose (1961) and Borland (1963) that in one dimension, all states are localised for (5> 0. There now exist scaling theory predictions for two and three dimensions (see, for example, Be litz and Kirkpatrick 1994), but the consensus is that the Anderson transition is analogous in many ways to classical percolation theory (cf. section 4.3). It is indeed quite similar to a diffusionless transition, as found in many classical models dating back to Lorentz; see, for example, Hauge (1974). This was further demonstrated by Wegner (1979), who succeeded in transforming the Anderson localisation problem defined by the vanishing of the dynamical conductivity into an equivalent field theory of critical phenomena. The latter could be treated by the renormalisation group (RG) method, much in the same way as percolation theory, which can be mapped on to critical phenomena via a q state Potts model in the limit q —p 0 (Wu 1982). The essence of these developments was the argu ment based on RG theory (Wegner 1976), of a scaling formula for the dynamical electrical conductivity u(t, l): (t, 1)
b_(d_ ) 2 f(tbl/v, 1b”).
(5.46)
In eqn (5.46), t is some dimensionless �