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Generalized two-body self-consistent theory of random linear dielectric composites:an effective-medium approach to clustering in highly-disordered media

Y.-P. PellegriniCEA, DAM, DIF, F-91297 Arpajon, France

F. WillotMINES Paritech, Centre de Morphologie Mathematique, Mathematiques et Systemes,

35, rue Saint-Honore, F-77305 Fontainebleau Cedex, France(Dated: January 22, 2018)

Effects of two-body dipolar interactions on the effective permittivity/conductivity of a binary, symmetric, ran-dom dielectric composite are investigated in a self-consistent framework. By arbitrarily splitting the singularityof the Green tensor of the electric field, we introduce an additional degree of freedom into the problem, in theform of an unknown “inner” depolarization constant. Two coupled self-consistent equations determine the latterand the permittivity in terms of the dielectric contrast andthe volume fractions. One of them generalizes theusual Coherent Potential condition to many-body interactions between single-phase clusters of polarizable mat-ter elements, while the other one determines the effective medium in which clusters are embedded. The latter isin general different from the overall permittivity. The proposed approach allows for many-body corrections tothe Bruggeman-Landauer (BL) scheme to be handled in a multiple-scattering framework. Four parameters areused to adjust the degree of self-consistency and to characterize clusters in a schematic geometrical way. Giventhese parameters, the resulting theory is “exact” to secondorder in the volume fractions. For suitable parametervalues, reasonable to excellent agreement is found betweentheory and simulations of random-resistor networksand pixelwise-disordered arrays in two and tree dimensions, over the whole range of volume fractions. Com-parisons with simulation data are made using an “effective”scalar depolarization constant that constitutes verysensitive indicator of deviations from the BL theory.

PACS numbers: PACS numbers: 05.60.Cd, 72.80.Tm, 78.20.Bh

I. INTRODUCTION

The problem of the effective transport properties of disor-dered media has a long history [1, 2], and still continues toattract wide attention owing to its intrinsic theoretical interest[3–5], and to its importance to various domains of engineer-ing sciences [6]. Our focus here is on the effective permittivityof binary symmetric random media [7] with quenched disor-der, that undergo percolative behavior [1, 8]. Investigationshave mainly been carried out on prototypical models such asRandom Resistor Networks (RRN) [9–11], random arrays ofpolarizable point elements [12] and checkerboards [13]. Theadvent of full-field numerical methods of computation, suchas Fast Fourier Transform (FFT) calculations on pixel arrays[14, 15] (see [16] for an alternative use of FFTs in this con-text), or finite-elements methods, make it now possible to ad-dress in detail random checkerboards and various other typesof heterostructures [17].

From a theoretical standpoint, accounting for presence ofa percolation threshold in a systematic theory of interac-tions between heterogeneities is extremely difficult, since thehighly-disordered character of the percolation regime involvesmany-body positional correlation functions to all orders [3, 5].There, perturbative methods are inapplicable unless someamount of self-consistency is injected in suitable approxima-tions. The simplest successful [18] self-consistent (s.c.) ap-proach to the effective permittivity problem in percolatingmedia is the well-known Bruggeman-Landauer (BL) theory[1, 19], whose theoretical status is well-established withre-gard to perturbative approaches [20, 21]. It can be applied to

RRNs and to continuum systems. In the BL theory, as wellas in more elaborate s.c. treatments [20, 22], the thresholdisfixed, which is a nuisance since it varies in practice with mi-crostructure [5].

The present work proposes a parametric theory to accountfor corrections to the BL theory in the high-contrast, highly-disordered regime. Simple ways of doing so mostly rely onvarying the shape of the inclusions by modifying their de-polarization coefficients [23]. The related formulations ofMcLachlan and co-workers [24] empirically introduce tunablecritical exponents and threshold in the BL effective-mediumformula. In contrast, our purpose is to incorporate correctionsto the BL theory by considering exact pairwise interactionterms between polarizable pointlike matter elements, whichwill be done in the generic continuum framework of Miller’scell-material model [25, 26]. In this connection, it shouldbe mentioned that Shen and Sheng previously accounted fornearest-neighbor and next-nearest-neighbor interactions – aspecial case of pairwise terms – in a BL-like framework, byusing two-dimensional split inclusions [27], which resultedin a theory with a double percolation threshold [28]. Thisdouble-threshold effect is nowadays actively discussed [29],notably in the context of random checkerboards [30].

Our starting point is the modified BL equation for the ef-fective permittivityεe

⟨ε− εe

ℓeε+ (1 − ℓe)εe

⟩= 0. (1.1)

The brackets denote an average over material cells of vari-able permittivityε, andℓe is an unknown effective depolariza-tion coefficient to be determined. In the original BL theory,

2

ℓe = 1/d whered is the space dimension, which is equal tothe percolation threshold. Eq. (1.1) arises in a variety of con-texts. In particular, the Wu–McLachlan formula reduces tothe above one when its exponents are set to one [24]. Rather,our line of thought will be thatℓe is a dimensionlessfunc-tion of permittivity ratios and of the volume fraction of phases[31, 32].

The phenomenology ofℓe is established in Sec. II by iden-tifying the solution of Eq. (1.1) with results of simulationsof binary RRN and pixelwise-disordered systems in two andthree dimensions. In Sec. III, we develop a formalism rootedin multiple-scattering theory, in which a free, inner, depo-larization constantℓ, analogous to the above effective one,can be introduced without making approximations. Indepen-dently, microstructure-related features are approximately ac-counted for by means of two parameters, which provides arough characterization of single-phase clusters. With this sim-ple device, we make an bypass the need to consider explicitlycorrelation functions. Section IV formulates thetwo s.c. con-ditions required to determineℓ and the effective permittivity.One of these modifies the usualcoherent-potential conditionof vanishing self-energy, into one that distinguishes betweenthe local and non-local parts of the two-body term in the self-energy. It serves to adjust, by means of two supplementaryparameters, the “amount of locality and non-locality” thaten-ter the s.c. condition. This peculiar treatment is justifiedbycomparing the resulting four-parameter theory with the simu-lations of Sec. II. We conclude in Sec. V.

II. PHENOMENOLOGICAL BEHAVIOR OF ℓe

A. Preliminary remarks

We consider ad-dimensional binary medium, with phasesof permittivitiesε1 andε2 in respective proportions1− f andf . For definiteness, we choose phase 2 as that of high permit-tivity. Eq. (1.1) then reduces to the second-order polynomialequation for the effective permittivityεe

(1− f)ε1 − εe

ℓε1 + (1− ℓe)εe+ f

ε2 − εeℓε2 + (1− ℓe)εe

= 0, (2.1)

In the dilute limit, an exact well-known [5, 33] result (due toMaxwell in three dimensions) is that:

εe/ε1 = 1 + fd(ε2 − ε1)

ε2 + (d− 1)ε1+O(f2). (2.2)

On the other hand, expanding (2.1) provides:

εe/ε1 = 1 + fε2 − ε1

ℓε2 + (1− ℓe)ε1+O(f2), (2.3)

whereℓe stands forℓe(f = 0). TheO(f) term arises from theexact polarizability factor that individually characterizes theimpurities (one-body term). Thus,ℓe(f = 0) = 1/d and theO(f) correction to this value is due to theO(f2) term inεe,which accounts for pairwise (two-body) interactions. Similarconsiderations hold nearf = 1 whereℓe(f = 1) = 1/d.

Next, percolative behavior takes place whenε1 ≪ εe ≪ε2. Letting εe/ε1 → ∞ andε2/ε1 → ∞ at the percolationthresholdf = fc in (2.1) provides the following equation forfc [31]:

ℓe(fc) = fc. (2.4)

In the critical region, differences are expected between two-and three-dimensional cases. For infinite contrast, the mod-ified BL model (2.1) reduces toεe = ε1(1 − f/ℓe)

−1 forf < fc, andεe = ε2(f − ℓe)/(1 − ℓe) for f > fc. On theother hand, the well-known critical behavior of binary mediain the infinite-contrast limit is [5]εe ∝ ε1(1 − f/fc)

−s forf < fc andεe ∝ ε2(f/fc − 1)t for f > fc, with d-dependentcritical exponentss and t. Comparing these equations pro-vides

ℓe(f) ≃ f + a−fc(1− f/fc)s (f <∼ fc), (2.5a)

≃ f − a+(1− fc)(f/fc − 1)t (f >∼ fc).(2.5b)

wherea± > 0 are coefficients of order one. Ford = 2, ex-ponents are suchs = t ≃ 1.3 > 1, so that derivatives atfcare

ℓ′e(f±c ) = 1 (d = 2), (2.6)

andℓe(f) is smooth. Ford = 3 instead, exponents are nows < 1 andt > 1 and

ℓ′e(f−c ) = −∞, ℓ′e(f

+c ) = 1 (d = 3), (2.7)

so thatℓe(f) must display a cusp atfc.These considerations must be modified when applied to

a mean-field theory such as the one developed hereafter, inwhich the critical exponents would have typical effective-medium valuess = t = 1 irrespective ofd. Then,

ℓ′e(f−c ) = 1− a−, (2.8a)

ℓ′e(f+c ) = 1 + a+(1− f−1

c ). (2.8b)

Depending on the values ofa± andfc, ℓ′e(f±c ) can then (a

priori) be of either sign.To close, we point out that the double-threshold effect for

two-dimensional checkerboards (see Introduction), translatesin terms ofℓe into the propertyℓe(f) ≃ f for f ∈ (pc, 1−pc),where pc is the two-dimensional site-percolation threshold[28].

B. Simulation data

The BL theory compares well to binary RRNs, providedthat an identification is made between the phase volume frac-tion and the volume fraction of bonds, because both are sym-metric with respect to phase interchange [21, 34]. In particu-lar, for d = 2 the percolation threshold coincide, which leadsto an overall reasonable match. Moreover, the effective con-ductivity (permittivity) in the BL theory coincides with thatof RRNs up to the third order (included) of perturbations inpowers of the scaled contrastδε/〈ε〉 [21], and it is exact to

3

HaL

RRN d=2512x512

20

50102

108

0.0 0.2 0.4 0.6 0.8 1.00.48

0.49

0.50

0.51

0.52

f

{ eHfL

HbL

RRN d=364x64x64

102

103

108

0.0 0.2 0.4 0.6 0.8 1.00.26

0.28

0.30

0.32

0.34

f

{ eHfL

HcL

Pixel d=2512x512

20

50

102

108

0.0 0.2 0.4 0.6 0.8 1.0

0.48

0.49

0.50

0.51

0.52

f

{ eHfL

HdL

Voxel d=364x64x64

20

50

102

103

104

0.0 0.2 0.4 0.6 0.8 1.00.29

0.30

0.31

0.32

0.33

f

{ eHfL

FIG. 1. (Color online) Permittivity (conductivity) data analyzed in terms ofℓe(f) for two- and three-dimensional models, and contrastsε2/ε1as indicated in the plots: (a) and (b), random-resistor networks of sizeL = 512 (resp.,L = 64) for d = 2 (resp.,d = 3); (c) and (d),pixelwise-disordered media of same sizes (see text). Dashed: 1/d. Quasi-vertical grey line: functionℓe = f . Solid lines between data pointsare guides to the eyes.

one-body order (included) in the sense of Sec. III C. However,strong discrepancies with RRNs arise ford ≥ 3, primarily be-cause the percolation threshold of the BL theory (fc = 1/d)overestimates that of the lattice system.

Discrepancies can be analyzed by expressingℓe(f) as afunction ofεe by means of (2.1), as [35]

ℓe =εe(〈ε〉 − εe)

(εe − ε1)(ε2 − εe), (2.9)

and by using forεe effective-permittivity data obtained frommodels. We consider three numerical models.

To begin with, Fig. 1(a) and (b) show an analysis of the ef-fective conductivity of simulations of bond-disordered RRNs,carried out for the present purpose. The functionℓe(f) is com-puted from (2.9) for various contrasts. Statistical averages ofthe overall conductivity of a sufficiently large number of sam-ples have been carried out to reduce standard deviations to thetypical error bar values represented in Fig. 1(a) forf < 1/2.Although not represented, they are of same order of magni-tude in the domainf > fc, and in Figs. 1(b), (c) and (d).Under-sampling of the dilute configurations makes the statis-tical errors larger in the limitsf → 0, 1. System sizes arelarge enough to make finite-size effects negligible to our pur-pose. According to (2.4),fc is the volume fractionf at thecrossing point between the infinite-contrast plots and the greyline that representsℓe = f . It is seen thatfc(d = 2) = 1/2,andfc(d = 3) ≃ 0.26, close to the expected theoretical bond-percolation value≃ 0.249 . The dilute-limit valueℓe = 1/d isrepresented by the dashed horizontal line, and is approachedwith negativeslopes in all cases. Note that in three dimensionsℓe slightlyexceeds1/3 for f >∼ 0.55. In this high-conductivityregion, the closeness to Bruggeman’s theory is remarkable.

We also carried out noiseless calculations usingBernasconi’s Real-Space Renormalization-Group modelfor RRNs [36]. Up to some limitations of the approach, andto a lower percolation threshold in three dimensions, resultsare good qualitative agreement with Figs. 1(a) and (b) [? ].

Figs. 1(c) and (d) display further results of simulationson pixelwise-disordered arrays (PDA) [15], solved using theFFT method developed for elastic composites by Michel et al.[14]. Its adaptation to linear dielectric media is straightfor-

ward. The formulation employed is the original one, that usesthe continuum Green function (other types of implementationwere considered in [15]). Each sample is a regular squareor cubic array ofLd pixels (voxels, in three dimensions), ofpermittivity chosen at random according to the binary prob-ability density. Conceptually closer to a random array withsubstitutional disorder, than to a bond network or a randomcheckedboard (the fields are not resolved within the pixels orvoxels), this system nonetheless features in two dimensionsthe same percolation thresholdfc = 1/2 as a bond-disorderedRRN; for d = 3, fc ≃ 0.315, a value reminiscent of site per-colation (fc ≃ 0.312), close to the Bruggeman value. Ford = 2 its ℓe(f) function resembles that of RRNs, with asharper variation at threshold. The situation changes in threedimensions, especially in the high-permittivity phase wheredeviations from the dilute limit markedly differ from that inRRNs: the approach off = 1 has apositiveslope; the con-cavity at smallf is opposite; moreover, in the infinite-contrastlimit, ℓe(f) develops a valley just after the percolation thresh-old. The cusp atf ≃ 0.36 marks out a transition from thecritical region where Eq. (2.7) (right) applies, to behavior ofthe effective-medium type (2.8b). The critical regionf <∼ fcwhere Eq. (2.7) (left) would lead to a cusp is not observed.This suggests that the critical behavior of this system, whichhas no contact interactions, might be different from the RRNone, with eithers ≃ 0, or s = 1 with a− ≪ 1. This point–not crucial to our purpose– has not been investigated further,due to some difficulties in achieving high-quality numericalconvergence forf < fc in the infinite-contrast limit (this isthe reason why contrast is limited to104).

The rich typology ofℓe(f) behaviors, even in the dilute re-gion, illustrates the dramatic influence of microstructural fea-tures on the overall response. This makesℓe(f) an interestingmeans of analysis, since plots ofℓe provide nontrivial infor-mation over the whole range of concentrations (trying to useit at low contrast on noisy data may however lead to an incon-clusive outcome [31]).

4

III. THEORY

The theoretical framework we adopt to account for theabove observations heavily relies on the well-known multiple-scattering formalism [37, 38], widely used in solid-statephysics [39], and repeatedly employed to study dielectric me-dia [40–42]. Because some additions to the classical setupare needed, the following sections review it briefly with em-phasis on our modifications. A number of the equations alsoarise in the context of dilute alloys, a related problem, whereBruggeman’s EMA is known as theCoherent-Potential Ap-proximation(CPA) [43]; see [44] for a recent review.

A. Split-up of the Green function

We consider ad-dimensional dielectric medium with per-mittivity ε(r) fluctuating from cell to cell. The latter are spher-ical on average, of infinitesimal typical radiusa → 0 and vol-umev = adSd/d, whereSd = 2πd/2/Γ(d/2) is the area ofthed-dimensional sphere of unit radius. We use the notationsI for thed × d identity matrix, andr = r/r for the unit po-sition vector. Following the usual treatment, we introduceanarbitrary background permittivityεb, and re-write the equilib-rium equation∇.(εE) = 0, whereE is the electric field, asεb∇.E = −∇.δεE, whereδε = ε − εb is the dielectric con-trast. The problem can then be cast in the form of an integralequation forE [45]:

E(r) = E0(r) +

∫ddr′ G1/d(r− r′)

δε(r′)

εbE(r′), (3.1)

whereE0 is the applied field, andG1/d is thed-dimensionaldipolar Green function (θ is the Heaviside function, used toimplement a principal value prescription at the origin)[46]

G1/d(r) = − I

dδ(r) − lim

η→0θ(r − η)

1

Sd

1

rd(I− drr). (3.2)

In operator notation where, e.g.,ε(r) is understood as the bi-variate operatorε(r|r′) ≡ ε(r)δ(d)(r− r′) so that(εE)(r) =∫ddr ε(r|r′)E(r′), Eq. (3.1) reads:

E = Eb +G1/d(δε/εb)E. (3.3)

From this point on, we depart from the usual treatment. Amodified Green operatorGℓ is considered, in which the sin-gularity at the origin is “renormalized” by introducing [31] anarbitrary “inner” depolarization parameterℓ, of a more funda-mental nature thanℓe. By definition,

G1/d(r) ≡ −ℓ I δ(r) + Gℓ(r). (3.4)

Writing the prefactor of the local (Dirac) part ofGℓ as

∆ ≡ ℓ− 1/d, (3.5)

one has by Eq. (3.2) and definition (3.4),

Gℓ(r) = ∆Iδ(r) − limη→0

θ(r − η)1

Sd

1

rd(I− drr). (3.6)

Introducing moreover a “screened” electric field

Eℓ ≡[1 + ℓ

δε(r)

εb

]E, (3.7)

and a modified permittivity contrast

uℓ(r) ≡[δε(r)/εb]

1 + ℓ[δε(r)/εb]=

ε(r)− εbℓε(r) + (1− ℓ)εb

, (3.8)

Eq. (3.1) in transformed into the equivalent equation [31]

Eℓ = E0 +Gℓ uℓEℓ. (3.9)

For ℓ = 0, u0(r) is the scaled dielectric contrast. Forℓ =1/d, u1/d is (up to a prefactorvεb) the dielectric polarizabilityof a spherical cell, andE1/d(r) is the local (Lorentz) fieldimpinging on it [7]. Forℓ 6= 1/d, some screening effectsare implemented at the level of the cell. This possibility ofsplitting the Green function has been noticed previously [47]to the purpose of improving convergence in solving Eq. (3.1)by iterations (see also [4] p. 302). Although related, our aimis somewhat different.

B. Multiple-scattering expansions

Both (3.3) and (3.9) are continuum analogues of scatteringequations for finite-size scatterers. It proves convenientto em-phasize the connection by casting the continuum problem intoa multiple-scattering framework. Thescattering potentialuℓyof the material element aty is introduced as

uℓy(r|r′) ≡ v uℓ(y)δ(r − y)δ(r′ − y) I. (3.10)

SettingGℓ(r|r′) ≡ Gℓ(r− r′) Equ. (3.9) is rewritten as

Eℓ(r) = E0(r) +

∫ddy

vddr2d

dr1 Gℓ(r|r2)uℓy(r2|r1)Eℓ(r1).

(3.11)In order to distinguish between integrations over ‘in’ and ‘out’space variablesr1,2 and summations over scattering cells, letuℓyj

≡ uℓj and make the formal replacement∫(ddy/v) →∑

j . Then (3.9) takes the form

Eℓ = E0 +Gℓ

∑

j

uℓjEℓ. (3.12)

Define now the Green functionG associated toEℓ by theequationEℓ = GG−1

ℓ E0, whereG−1ℓ E0 is the source. Then

G = Gℓ +Gℓ

∑

j

uℓjG. (3.13)

The individual transition operator, also calledT -matrix (see[42] for a comprehensive list of references), completely char-acterizes the polarizability properties of theith scatterer. Itreads

ti ≡ uℓi(1−Gℓuℓi)−1. (3.14)

5

Introducing the Green function for thelocal field impingingon the scatterer, namely,Gi ≡ (1 − Gℓuℓi)G; the polar-ization of the scattererPi ≡ tiGi; and finally its “dressed”T -matrix, Ti ≡ PiG

−1ℓ , which accounts for corrections due

to the other scatterers, one obtains from (3.13) the familiarmultiple-scattering equations [37]:

G = Gℓ +GℓT Gℓ, (3.15a)

T ≡∑

i

Ti, Ti = ti + tiGℓ

∑

j 6=i

Tj, (3.15b)

whereT is theT -matrix of the whole system. We observe thatsince scattering events occur in succession between differentscatterers, the Dirac term ofGℓ is irrelevant inT . Thus, wereplaceGℓ byG1/d in (3.15b) to indicate that this propagatordoes not depend onℓ.

Next, a n-body (“virial”) expansion of the formT =∑n≥1 T (n) is written down, where then-body term

T (n) =∑

′ T(n)(i1,i2,··· ,in)

(3.16)

is a sum over all possible scattering sequences onn dis-tinct scatterers. The prime indicates that the sum is carriedout first over labeli1, then overi2 6= i1, etc., then overin 6∈ {i1, i2, · · · , in−1}, so thatT (n)

(i1,i2,...,in)depends on la-

bel ordering. This question was solved long ago by Petersonand Strom [48]. We provide in Appendix A another differentapproach to their result. Specifically, we show that:

T(n)(i1,i2,...,in)

= T(n−1)(i1,i2,...,in−1)

G1/dtin

×(1−G1/dSi1...in−1

G1/dtin)−1

×(1 +G1/dSi1...in−1

), (3.17a)

Si1...in = G−11/d

[(1 +G1/dtin

)

×(1−G1/dSi1...in−1

G1/dtin)−1

×(1 +G1/dSi1...in−1

)− 1

], (3.17b)

T(1)i = Si ≡ ti. (3.17c)

In particular, the two-body term, to be used hereafter, is [48,49]:

T(2)(i1,i2)

=ti1G1/dti2(1−G1/dti1G1/dti2

)−1(1+G1/dti1)

= ti1G1/dti2

+ti1G1/dti2G1/dti1G1/dti2(1−G1/dti1G1/dti2

)−1

+ ti1G1/dti2G1/dti1(1−G1/dti2G1/dti1

)−1. (3.18)

In the last writing, the termti1G1/dti2 has been singled out forconvenience, in the perspective of using Eq. (3.25b) below.Although we disregard it for simplicity, the three-body term[48] has been considered by Cichocki and Felderhof [50].

C. Ensemble averages, self-energy and effective permittivity

To address substitutional or positional randomness, ensem-ble averages over disorder, denoted by〈·〉, are carried out

[38]. Due to statistical homogeneity, averaged operators aretranslation-invariant. Introducing the complete scattering po-tential Uℓ ≡ ∑

i uℓi associated to the whole set of hetero-geneities, the so-called self-energy (or coherent potential) op-eratorΣℓ of the averaged Green function〈G〉 [44] is definedby 〈UℓG〉 ≡ Σℓ〈G〉, and from (3.13) follows the Dyson equa-tion [51]

〈G〉 = (G−1ℓ − Σℓ)

−1. (3.19)

The kernelΣℓ(r|r′) = Σℓ(r−r′) possesses one local and onenon-local part. It can be written as the Green function (3.2)inthe form

Σℓ(r) = Σlocℓ Iδ(r) + Σ

nlocℓ (r), (3.20)

whereΣlocℓ is a scalar, and where the same principal-value

prescription as in (3.2) applies to the non-local term. The ne-cessity of distinguishing between the local and non-local partwill show up when comparing theory to our reference data.

Similarly, the effective permittivityεe is a non-local opera-tor [52], of the same generic form, and is defined through theequality〈εE〉 = εe〈E〉; that is,

〈ε(r)E(r)〉 =∫

ddr′ εe(r− r′)〈E(r′)〉. (3.21)

Definitions ofEℓ, εe andΣℓ lead to [31]

εe = εb[1 + (1− ℓ)Σℓ][1− ℓΣℓ]−1. (3.22)

This formal equation becomes algebraic when Fourier trans-forms of the kernels are used. In particular, it holds for thevolume integrals of the kernels. We recall that the effectiveℓe(f) is obtained fromεe by means of (2.9).

The configurational average of (3.15a) yields

〈G〉 = Gℓ +Gℓ〈T 〉Gℓ. (3.23)

Comparing this equation with (3.19) provides the relationship:

Σℓ = 〈T 〉[1 +Gℓ〈T 〉]−1. (3.24)

For an infinite system, this equation is merely formal and onlyhas a meaning as a perturbative series, because〈T 〉 involvesconditionally-convergent integrals. Expanding (3.24), and us-ing (3.16) and (3.17), givesΣℓ in the form of an-body expan-sion, where one- and two-body contributions are read from(3.17c) and (3.18):

Σℓ =∑

n≥1

Σ(n)ℓ , where Σ

(1)ℓ =

∑

i1

〈ti1〉; (3.25a)

Σ(2)ℓ =

∑

i1,i2 6=i1

〈T (2)(i1,i2)

〉 −∑

i1,i2

〈ti1〉Gℓ〈ti2〉; etc. (3.25b)

The long-range part of the second term ofΣ(2)ℓ cancels out the

first term of the average of (3.18). The remaining terms in-volve only absolutely-convergent integrals, with an integranddecaying at least asG2

1/d ∼ r−2d as r → ∞. This prop-erty holds at each order (the perturbative expansion ofΣ hasa special – ordered – type of cumulant structure [53, 54]),which implies thatΣ is independent of the sample shape inthe infinite-volume limit [54, 55].

6

D. Parameters s and q, and clustering

Before computing the self-energy, we remove some inde-terminacies of the continuum theory. The latter admits twonatural adjustable parameters. A first parameters > 0 stemsfrom remarking thatη in Eq. (3.2) must be of ordera, whichwe writeη = sa. We interprets/2 as a measure of aneffectiveradiusof neighboring inclusions, to be further constrained inSec. IV A. The values = 2 (η = 2a) corresponds to set-ting a hard-sphere-type exclusion distance between polariz-able point inclusions. However, we allow here for smaller orlarger values to tune the strength of dipolar interactions,so asto compensate for the lack of explicit higher-order multipolesin interactions between finite-size inclusions [56, 57].

To work out the above equations, we need an expressionof ti, defined formally by Eq. (3.14). Dropping the indexifor brevity, and expanding, one hast = uℓ

∑k≥0(Gℓuℓ)

k.Powers are evaluated by means of definitions (3.6) and (3.10),but this involves squares of Dirac distributions: considerforinstanceuℓ(Gℓuℓ)

2, which reads∫ 4∏

k=1

ddrk uℓy(r|r1)Gℓ(r1 − r2)uℓy(r2|r3)×

×Gℓ(r3 − r4)uℓy(r4|r′)= v3uℓ(y)

3Gℓ(0)

2δ(r− y)δ(r′ − y)

= v uℓ(y)3∆2[vδ(0)]2δ(r− y)δ(r′ − y). (3.26)

As discussed in Appendix B, we use the prescription

vδ(r)2 ≡ qδ(r). (3.27)

The numberq > 0, a mathematical and physical necessaryaddition when∆ 6= 0, is the second parameter of the theory.We can then writevδ(0) = q. The “infinitely large” numberv−1 gives the physical “order of magnitude ofδ(0)” in thisproblem. Thus,

t = uℓ

∑

k≥0

(q∆)kukℓ = uℓ(1− q∆uℓ)

−1 = uℓ, (3.28)

where

ℓ ≡ ℓ− q∆ = (1− q)ℓ + q(1/d); (3.29)

that is, witht(y) ≡ uℓ(y)

ty(r|r′) = v t(y)δ(r − y)δ(r′ − y) I. (3.30)

We assume that0 ≤ q ≤ 1, so thatℓ can be interpreted as aweighted average ofℓ and1/d.

We propose the following interpretation ofq, illustrated byFig. 2, where ellipsoidal shapes are meant to indicate that in-clusions have an effective depolarization factor different from1/d (the orientation of the ellipsoids in the drawing is irrele-vant). Each scattererti is viewed as a aggregate of screenedpolarizable elements of polarizability proportional touℓ, ofsame permittivity, embedded in mediumεb. Their degree ofclustering is adjusted throughq. We interpret the latter as acoverage/spreading parameter for aggregated elements. Inthe

FIG. 2. (Color online) Clustering interpretation of thes andq pa-rameters.

figure, the small bar associated to∆ represents some spread-ing of δ(r) (see Appendix B). Whenq = 0, t = uℓ and theelements are considered separately; whenq = 1 they gatheras a compact isotropic (spherical) inclusion of polarizabilityt = u 1

d. Settingℓ ≡ 1/d suppresses the influence ofq:

the aggregate reduces to one single spherical inclusion in thiscase also. The aggregate is represented as a whole in a mean-field way in the sense that electrostatic interactions betweenits components are approximated by the local part∆ of Gℓ.Moreover, aggregates are treated as point-like polarizable ob-jects [Eq. (3.30)] when it comes to considering their mutualdipolar interactions viaGℓ or G1/d (of which only the non-local part is relevant here; see Sec. III B).

To summarize, introducing0 < q < 1 makes the abovetheory a simplified one of interacting aggregates, in which thedifference betweenℓ and ℓ distinguishes between inclusionsand aggregates thereof.

E. Self-energy to two-body order

The self-energy follows from (3.25). Considering onlyvolume-integrated kernels (with a slight abuse of notation),

Σ(1) = Σ(1)loc =

∫ddrΣ(1)(r− r′) =

∫ddr

∑

i

〈ti(r|r′)〉

=

∫ddr ddy δ(r− y)δ(r′ − y)〈t〉 = 〈t〉. (3.31)

Likewise, from (3.18) and (3.25b),Σ(2) = Σ(2)loc +Σ(2)nloc

with

Σ(2)loc = −∆〈t〉2 +∫

r>η

ddr

⟨v t2 tG2

1/d(r)

1− v2t tG21/d(r)

⟩,(3.32a)

Σ(2)nloc =

∫

r>η

ddr

⟨v2 t2 t2 G3

1/d(r)

1− v2t tG21/d(r)

⟩, (3.32b)

whereη = sa, and wherer is the separation vector betweentwo statistically uncorrelated cells of volumev. We denote

7

their polarizabilities byt andt to distinguish them. The con-figurational average over permittivities must be carried out in-dependently on these quantities.

The first two terms of (3.18) are built on scattering se-quences that start and end on the different scatterersi1 6= i2,and thus contribute toΣ(2)nloc, which stands as a non-localsusceptibility. Instead, the third one is made of closed scatter-ing sequences that start and end on the same scattereri1, andso contributes to the local partΣ(2)loc as a renormalization(“dressing”) of theT -matrix of individual scatterers [49]. In-deed, for∆ = 0, in the diagrammatic representation [21, 40]where a line stands for a “propagator”G1/d and a2n-leggeddot stands for thenth power of aT -matrix t,

Σ(1)loc = (3.33a)

Σ(2)loc = + + + · · · (3.33b)

Σ(2)nloc = + + + · · · (3.33c)

As recalled in Sec. III C, the absolute convergence of the inte-grals inΣ(2) results from the fact that the latter involves onlyterms with at least two propagators. When∆ 6= 0, the addi-tional contribution−∆〈t〉2, which stems from the local partof Gℓ in the last term of (3.25b), is counted withinΣ(2)loc

rather thanΣ(2)nloc, in view of definition (3.20) of the localand nonlocal parts of theΣ operator.

In spite of the shorthand scalar notation employed in (3.32)(all terms commute) matrix inverses are required. They followfrom writing the nonlocal part ofG1/d as a linear combinationof the projectorsrr andI− rr. Carrying out the integrals overr in (3.32) yields [58]

Σ(2)loc = −∆〈t〉2 + d−2(d− 1)⟨t(tt)

12 g(z)

⟩,(3.34a)

Σ(2)nloc = d−2(d− 1)⟨tt h(z)

⟩, (3.34b)

g(z) = tanh−1(z) + tanh−1((d− 1)z

), (3.34c)

h(z) =1

2log

1− z2

1− (d− 1)2z2, (3.34d)

z =√tt/(d sd). (3.34e)

We note for further use that, with principal determinations,

g(z) + h(z) = log1 + z

1− (d− 1)z. (3.35)

Due to scale invariance, sizea is absent from these equations.Gathering one- and two-body terms, the explicit expression

of Σℓ after having taken averages is as follows. We setp1 =(1− f), p2 = f , and introducet1 andt2 such that

t1,2 =ε1,2 − εb

ℓε1,2 + (1 − ℓ)εb. (3.36)

The quantitiest and t in Eq. (3.32) take on valuest1 or t2with respective probabilitiesp1 andp2. Introduce moreoverz11 = t1/(ds

d), z12 =√t1t2/(ds

d), z22 = t2/(dsd). Then,

from Eqs. (3.31) and (3.34), with〈t〉 = p1t1 + p2t2,

Σlocℓ = 〈t〉 −∆〈t〉2 + d−2(d− 1)

[(p1t1)

2g(z11)

+ (p2t2)2g(z22) + p1p2(t1 + t2)(t1t2)

12 g(z12)

],(3.37a)

Σnlocℓ = d−2(d− 1)

[(p1t1)

2h(z11)

+ (p2t2)2h(z22) + 2(p1t1)(p2t2)h(z12)

]. (3.37b)

We remark thath(z) ≡ 0 for d = 2, so that in two dimensionsΣℓ ≡ Σloc

ℓ .Two-body interactions terms in the effective permittivityof

composites have been considered by numerous authors, manyof who focused on interactions between spherical inclusionsof finite size [56, 59]. Here, takingd = 3, ℓ = 1/3, s = 2,εb = ε1, andt1 = α/(ε1v) whereα is a polarizability, andt2 = 0, and expandingεe to orderO(f2) returns a known ex-pression of the two-body correction in the effective permittiv-ity of a suspension of polarizablepoint inclusions distributedaccording to the law of a hard-sphere gas [57, 60]. An often-cited expression for this term [61] is only an approximate one.

IV. EFFECTIVE-MEDIUM CONDITIONS

In this section, the general theory is completed by s.c. con-ditions, and exploited. Several schemes are possible. For clar-ity, a non-self-consistent setting is considered first.

A. Theory of the Clausius-Mossoti type

When used with the volume integrals of the kernels, Eq.(3.22) resembles the Clausius-Mossoti (CM) formula [62],in which the polarizability of inclusions replaces the self-energy, and constitutes a generalization of the Maxwell–Garnett effective-medium formula [63]. The latter holds fordilute systems of spherical inclusions embedded in a matrix.It is retrieved by lettingℓ = 1/d, fixing the background per-mittivity to that of the matrix, and keeping only inΣ theone-body contribution (3.25a). Similar many-body general-izations of the CM formula by means of cluster expansionshave previously been worked out by Felderhof and co-workers[50, 55, 64, 65], among others.

WheneverΣℓ 6= 0 in Eq. (3.22), which produces CM-typeestimates, a distinction must be made between the backgound“inner” permittivity εb and the overall oneεe. It follows thatthe “inner” ℓ of the theory needs to be distinguished from the“effective” one,ℓe(f) obtained fromεe by means of Eq. (2.9).

In the rest of this Section and in the next one, we fixℓ to itsusual value1/d. This eliminates altogether theq parameter,see Sec. III D, and implies that∆ = 0 and

t1,2 =d(ε1,2 − εb)

ε1,2 + (d− 1)εb. (4.1)

Dilute behavior in the CM-type approach is then as follows.Settingεb = ε1 theO(f) term ofℓe is readily obtained. Using

8

identity (3.35), the result reads

ℓCMe (f) =

1

d+

d− 1

d2

[dζ2 − log

1 + ζ2/sd

1− (d− 1)ζ2/sd

]f,

ζ2 ≡ ε2 − ε1ε2 + (d− 1)ε1

. (4.2)

The medium being symmetric, theO(f − 1) behavior stemsfrom interchangingε1 andε2, and from replacingf by 1− f .Letting ℓe(f) = 1/d + σ0f + O(f2) = 1/d + σ1(f − 1) +O((f − 1)2), the dilute-limit slopes of theℓe(f) graph in theinfinite-contrast limitε2 ≫ ε1 follow as

σCM0 =

d− 1

d2

[d− log

sd + 1

sd − (d− 1)

], (4.3a)

σCM1 =

d− 1

d2

[d

d− 1− log

sd + 1

sd − (d− 1)−1

]. (4.3b)

By self-duality [9],σCM0 = σCM

1 for d = 2.The cut of the logarithm in (4.2) materializes the two-body

resonance spectrum [11] of the theory, and such resonancesshould not occur for positiveε1,2; that is,ℓe must be real instatics. Assuming thats does not depend on volume fractions,the requirement that the infinite-contrast limits ofσ0,1 be realimplies the constraint

s > smin ≡ (d− 1)1/d. (4.4)

This is a direct consequence of requiring that(d −1)max[|z11|, |z12|, |z22|] < 1 in Eqs. (3.37) to avoid the cutswhateverd and the contrast.

Fig. 3(a) represents the functionsσCM0,1 (s). Slopeσ0 blows-

up logarithmically nearsmin. Thus, ford = 2, negative slopesσ0,1 such as in Figs. 1(a) and (c) are possible fors closeenough to1. Ford = 3, negative slopesσ0 are obtained forsclose tosmin(d = 3) ≃ 1.25992, and the theory admits pos-itive σ1 slopes: this is evocative of the PDA behavior of Fig.1(d) in the dilute limit, but the correspondingσ1 values, of or-der0.2, outstrip those of simulations. Besides, this approachcannot reproduce the weak negative slope atf = 1 of Fig.1(b).

HaL

CM-type theory

Σ0,1Hd=2L

Σ0Hd=3L

Σ1Hd=3L

1.0 1.5 2.0 2.5 3.0 3.5 4.0

-0.5

0.0

0.5

1.0

s

slop

esΣ

0,1

HbL

Self-consistent

Σ0,1Hd=2L

Σ1Hd=3L

Σ0Hd=3L

1.0 1.5 2.0 2.5 3.0-0.4

-0.3

-0.2

-0.1

0.0

0.1

s

slop

esΣ

0,1

FIG. 3. (Color online) Slopesσ0,1 in (a) the CM-type theory, and (b)generalized self-consistent theory, withℓ = 1/d, vs. parameters.Vertical dashed lines represent thesmin lower bounds.

B. Generalized self-consistency with ℓ = 1/d.Parameters α and β.

We keep usingℓ = 1/d. The exceeding high slopes (4.3)are a consequence of their going to a finite limit ass → ∞,because of the termdζ2 within the brackets of Eq. (4.2). Thisterm is traced to the one-body contributionΣ(1) to the self-energy, i.e., it already exists in the standard Maxwell-Garnettformula interpreted in terms ofℓe. Powers ofΣ(1) are gener-ated to all orders of perturbations when expanding Eq. (3.22).The self-energy expansion (3.25) is essentially an asymptoticone, and the CM-like expression (3.22) does not perform wellin reorganizing it into a physically-meaningful effectiveper-mittivity for all volume fractions.

A widely-used reshuffling device is to determineεb self-consistently by the CPA conditionΣ ≡ Σ(1) + Σ(2) = 0,which implies thatεe = εb. However this is not the only pos-sibility: we can also consider the restricted one-body versionΣ(1) = 0 [43], i.e., the BL condition; or even an intermediate“local” conditionΣloc ≡ Σ(1) + Σ(2)loc = 0, etc. To handlethem all, and much more, we introduce new parametersα andβ, and put forward thegeneralized s.c. condition

Σ(1) + αΣ(2)loc + βΣ(2)nloc = 0 0 ≤ α, β ≤ 1. (4.5)

Parameterβ is irrelevant ford = 2, according to our remarkfollowing Eq. (3.37b). With the above condition, and settingδΣ ≡ (1−α)Σ(2)loc +(1− β)Σ(2)nloc, Eq. (3.22) reduces to

εe = εb1 + (1− 1/d)δΣ

1− δΣ/d, (4.6)

which, unlessα = β = 1, remains of the CM type in spite ofthe s.c. condition.

As far as theO(f) term in ℓe is concerned, the outcomeof condition (4.5) is independent of(α, β); this property doesnot hold for ℓ. With ζ2 as in (4.2), working out the diluteexpansion indeed gives

ℓsce (f) =1

d+

d− 1

d2

[dζ2sd

− log1 + ζ2/s

d

1− (d− 1)ζ2/sd

]f.

(4.7)It is important to remark that with self-consistency, the slopeatf = 0 becomes a function ofζ2/sd, which turns parameters into a polarizability rescaling factor. Infinite-contrastslopesfollow as:

σsc0 =

d− 1

d2

[d

sd− log

sd + 1

sd − (d− 1)

], (4.8a)

σsc1 =

d− 1

d2

[ds−d

(d− 1)− log

sd + 1

sd − (d− 1)−1

]. (4.8b)

They are drawn in Fig. 3(b). They vanish in the limits → ∞where two-body interactions are suppressed, so that the slopesare now apure two-body effect. However, whereasσ1 takeson more realistic values, it still cannot be made negative ford = 3.

Percolation occurs throughεb. A second-degree polyno-mial equation for the percolation thresholdfc is obtained as

9

HaL

Β= 0.00

d=3 H{=1�3LGeneralized s.c.

0.0 0.2 0.4 0.6 0.8 1.0

0.29

0.30

0.31

0.32

0.33

f

{ eHfL

HbL

Β= 0.33

d=3 H{=1�3LGeneralized s.c.

0.0 0.2 0.4 0.6 0.8 1.0

0.29

0.30

0.31

0.32

0.33

f

{ eHfL

HcL

Β= 0.67

d=3 H{=1�3LGeneralized s.c.

0.0 0.2 0.4 0.6 0.8 1.0

0.29

0.30

0.31

0.32

0.33

f

{ eHfL

HdL

Β= 1.00

d=3 H{=1�3LGeneralized s.c.

0.0 0.2 0.4 0.6 0.8 1.0

0.29

0.30

0.31

0.32

0.33

f

{ eHfL

FIG. 4. (Color online) Dimensiond = 3. Dependence onβ (as indicated) ofℓe(f) in the generalized s.c. approach at infinite contrast, withs = smin + 0.05 andα = 0.00, 0.33, 0.67 and1.00 in each figure (top to bottom).

the infinite-contrast limit of (4.5), lettingε1 = 0 first, thenεb = 0. For brevity, we no not reproduce its lengthy coeffi-cients. The solution ford = 2 is fc = 1/2 irrespective of theparameters. Ford = 3, the model percolates at the BL thresh-old fc = 1/3 if α = β = 0; otherwise, the leading-order termof the equation fors near tosmin provides the asymptotic es-timate

fc ∼1

2

[6 + α log 5− β log(5/4)

(α+ β)| log(s− smin)|

]1/2. (4.9)

The lowest value offc is obtained withα = β = 1, at fixeds,andfc can be made as small as needed by lettings → smin.

Fig. 4 illustrates typical consequences onℓe(f) of the gen-eralized s.c. condition, in the three-dimensional case wherevariations are most conspicuous. As long as self-consistencyinvolves two-body interactions, the threshold is lowered withrespect to the BL value, with an intricate dependence onα andβ. Their influence onℓe is confined to a definite region aroundfc, because the dilute-limit slopes do not depend onα andβ.The effect of removing part ofΣ(2)nloc from the s.c. conditionis interesting: in Fig. 4(a), a graph shape with an upward cuspat fc, akin to that of Fig. 1(d) (highest contrast), is producedwith β = 0 and a small amount ofΣ(2)loc (α = 0.33); onthe other hand, a downward cusp results from injecting a highamount ofΣ(2)nloc [β = 1, Fig. 4(d)].

C. Generalized self-consistency with variable ℓ

We can now discuss the effect of the “inner” depolarizationvariableℓ. Relaxing the conditionℓ = 1/d, the s.c. condition(4.5) for εb still holds, but this time expressed in terms ofΣℓ

with ℓ arbitrary. Since∆ = ℓ − 1/d 6= 0, the self-energyhas one additional term in its “loc” part (3.37a), and theqparameter introduced in Sec. III D becomes operative. Thetheory of Sec. IV B is retrieved ifq ≡ 1. To determineℓ weimpose the supplementary s.c. condition,

〈uℓ〉 = 0, (4.10)

which, in the interpretation of Sec. III D, makesεb the localeffective medium surrounding the elementary screened ele-ments within aggregates. This is Eq. (1.1), withεe and ℓe

replaced byεb andℓ, respectively. Its solutions are

ε±b =ε1

1− ℓ

[τ ±

√ℓ(1− ℓ)τχ+ τ2

],

τ =1

2[(1 − f − ℓ) + (f − ℓ)χ], χ ≡ ε2/ε1. (4.11)

Solutionsℓ follow from using these expressions in the gener-alized CPA condition (4.5). Ifα = 1 (cased = 2 whereβ isirrelevant), orα = β = 1 (cased ≥ 3), thenΣ = 0 so thatεe = εb andℓe = ℓ.

s=3q=0.99 10-3

Α=1.0

HaL

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

{HfL

s=3q=1.02 10-3

Α=1.0

HbL

physical branch

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

f

{HfL

FIG. 5. Dimensiond = 2 and contrast108. Typical conformation ofthe branches ofℓ for d = 2 with (a) a ‘bad’ case; (b) a ‘good’ one(see text).

Depending on parameters{f, s, q, α, β} the theory admitsup to three real solutions. Admissible values of{s, q, α, β}must be such that a continuous real functionℓ(f) existsforall contrastsin the intervalf ∈ (0, 1), with endpoint valuesℓ = 1/d. We call it the “physical branch”. A criterion thatgeneralizes (4.4) to the whole parametric domain seems outof reach. Still, by requiring real solutions for positive permit-tivities and any0 ≤ ℓ ≤ 1, an argument similar to the oneused in Sec. IV A leads to the constraint

qsd > (d− 1), (4.12)

which extends (4.4) to values ofq 6= 1. The actual permitteddomain depends on(α, β) and is certainly wider, since solu-tionsℓ take on values in a much more restricted interval. Thus,the above constraint is too restrictive in practice.

Numerical experimentations show that ford = 3, percola-tion thresholds markedly lower than1/3 are obtained for pa-rameter values close to the boundary of the domain, much as

10

in the simpler case of the previous Section, illustrated by ex-pression (4.9) of the threshold. As a rule, close to boundary,the smallerq, the largers, which is consistent with Eq. (4.12)and with the interpretation of Fig. 2: the more dispersed theclusters, the larger their effective radius. But for lack ofanysimple analytical expression of the boundary of the alloweddomain, no systematic study of the threshold ind = 3 wasmade.

Fig. 5 displays some typical conformation patterns of so-lution branches ofℓ for d = 2. Parameters, as indicated (re-call thatβ is irrelevant ford = 2), were chosen such thatℓ′(f = 1/2) ≃ 1. The expression ofℓ′(f = 1/2) is easilydeduced by series expansions. The solid (resp. dashed) linesrepresent solutions arising from using the ‘plus’ (resp. ‘mi-nus’) branch in Eq. (4.11). Lines interrupt themselves wheresolutions become imaginary. Bad parameter values, such asin Fig. 5(a), lead to discontinuous real solutionsℓ(f). Fig.5(b) illustrates a “good’ case, with the physical branch indi-cated. Although lying close tof in a region aroundfc = 1/2,the functionℓ(f) could not be made close enough to repro-duce the double-threshold effect of checkerboards mentionedin Sec. II A. The physical branch was always found to be gen-erated from the ‘plus’ solution of (4.11) in all cases exam-ined where it exists. Whereas in the infinite-contrast limitandfor some special parameter sets, joining ‘plus’ and ‘minus’branches can produce a composite continuousℓ(f) graph withendpoints1/d, such special solutions do not survive at lowercontrasts, and cannot be considered physical in the presentcontext (although the situation might change upon includingthree-body interactions).

With ζ2 as in (4.2), and introducing

αq = [q + (1− q)α]/q, βq = [q + (1 − q)β]/q, (4.13)

and the functiong2(z) ≡ dz−g(z), the dilute expansion reads

ℓscℓe (f) =1

d+

d− 1

d2

[αqg2

(ζ2sd

)− βqh

(ζ2sd

)]f. (4.14)

Whenq = 1, it reduces to (4.7) thanks to identity (3.35). Theinfinite-contrast slopes atf = 0, 1 become:

σscℓ0 =

d− 1

d2[αqg2

(s−d

)− βqh

(s−d

)], (4.15a)

σscℓ1 =

d− 1

d2

[αqg2

(s−d

d− 1

)+ βqh

(s−d

d− 1

)].(4.15b)

With q 6= 1, the slopes now depend onα andβ, with an over-all scaling by1/q. The effect of varyingαq andβq can bestudied by factoring outγq ≡ (α2

q + β2q )

1/2, and by introduc-ing the angle0 ≤ φ ≤ π/2 such thatφ = arctan(βq/αq), forvarious values ofφ.

In two dimensions the common slope isαq times that ofFig. 3(b). Fig. 6 displays graphs of the scaled slopesσscℓ

0,1/γqvs.s in three dimensions for some values ofφ. The importantpoint is that values ofσ1 with either sign are now available.

d=3SC-{ theory

Σ0

Σ1

0.0 0.2 0.4 0.6 0.8 1.0-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

s-smin

scal

edsl

opesΣ

0,1�Γ

q

FIG. 6. (Color online) Dimensiond = 3. Scaled slopes vs. parameters, in s.c. theory with variableℓ, for 2φ/π = 0.0, 1/3, 1/2, 2/3, and1. Dashed:σ0; solid: σ1. Arrows indicate the direction of variationasφ increases (color online).

Whensd ≫ 1 andq ≪ 1 the slopes behave asymptotically as

σscℓ0 = σscℓ

1 ∼ −α/(6qs6) if d = 2, (4.16a)

σscℓ0 ∼

{−β/(3qs6) if d = 3, β 6= 0

−2α/(3qs9) if d = 3, β = 0,(4.16b)

σscℓ1 ∼

{β/(12qs6) if d = 3, β 6= 0

−α/(12qs9) if d = 3, β = 0.(4.16c)

Figure 7 compares the simulation data of Fig. 1 with theo-retical plots drawn using suitable parameter values in the s.c.equations. Parameter values were constrained by imposing thetheoretical slopes atf = 0, 1 to match on average those readfrom the data, and by requiring a high slope atf = 1/2 in thetwo-dimensional case, which is achieved by takings arbitrary,but large. This does not uniquely determine the parameters,sosome variations are admissible.

In both two-dimensional cases of Figs. 7(a) and (c), goodagreement with the data is obtained for low to moderate con-trasts, but the high-contrast data values forf close tofc can-not be matched. Actually, the peak values ofℓe(f) at highcontrast could have been reached by using other parameters,but at wrong volume fractions and at the expense of the low-contrast fits. It can be shown that ford = 2 the theoreticalslope atf = 1/2 is at mostσ(f = 1/2) = −4σ(f = 0, 1),which is obtained fors large. This explains why the high-contrast data cannot be matched. Indeed,σ(f = 1/2) shouldideally be equal to1, whereasσ(f = 0, 1) ≃ −0.05 in bothsets of two-dimensional data.

The best overall fit is obtained in thed = 3 RRN case,Fig. 7(b). Good match is obtained in the three-dimensionalPDA case of Fig. 7(d) also, but in the low-permittivity region(f < fc) only. In the high-permittivity region (f > fc), thedilute slope is well reproduced, and a cusp nearf+

c at highcontrast is retrieved; however,ℓe(f) is too low, and cannotbe turned into matching the data by varying parameters. Thiscase requires aq value much larger than for the other threeand, correspondingly, a relatively smalls value.

11

HaL

RRN d=2

s = 5.00q = 8.9 10-6

Α = 0.05

20

50102

108

0.0 0.2 0.4 0.6 0.8 1.00.48

0.49

0.50

0.51

0.52

f

{ eHfL

RRN d=3HbL

s = 2.71q = 5.9 10-4

Α = 1.00Β = 9.10-3

102

103

108

0.0 0.2 0.4 0.6 0.8 1.00.26

0.28

0.30

0.32

0.34

f

{ eHfL

HcL

Pixel d=2

s = 5.00q = 9.37 10-6

Α = 0.05

20

50

102

108

0.0 0.2 0.4 0.6 0.8 1.0

0.48

0.49

0.50

0.51

0.52

f

{ eHfL

HdL

Voxel d=3

s = 1.297q = 0.34Α = 0Β = 0.1

20

50

102

103

104

0.0 0.2 0.4 0.6 0.8 1.00.29

0.30

0.31

0.32

0.33

f

{ eHfL

FIG. 7. (Color online) Theoretical plots ofℓe(f) obtained from the s.c. theory with variableℓ (solid) fitted on the data of Fig. 1 (dots).Contrasts and parameters values are as indicated.

V. CONCLUDING DISCUSSION

We summarize our findings, and discuss possible improve-ments and directions for further work. First, we proposedto interpret effective-permittivity data in terms of an overalldepolarization-coefficient functionℓe, which proved a sensi-tive probe to highlight differences between models in a neatway, for all concentrations and contrasts.

Our theory rests on the introduction of an inner free depo-larization coefficientℓ. Together with the usual free back-ground permittivity,εb, these quantities are determined bycoupled s.c. equations: a Bruggeman-like equation on the po-larizatibility of clusters, and a CPA-like equation on the over-all self-energy. They reduces to Bruggeman’s withℓ = 1/d inthe absence of many-body corrections. The possibility of in-troducingℓ andεb stems from an overall invariance propertyof multiple-scattering theory under such parametrizations. Al-though we assumedεb andℓ to be scalars, it is clear that themost general reparametrization should involve tensors. Thispossibility was not considered, but would be necessary, e.g.,to compute field fluctuations, since this involves consider-ing anisotropic perturbations to the permittivities [66].As itstands, the theory only applies to cases where the one-bodyterm in the effective permittivity is isotropic in the dilute limit,given by Eq. (2.2).

Two empirical parameterss andq were introduced to de-scribe clusters. While valuesℓ 6= 1/d suggests that the rel-evant clusters of the theory are not spherical, computationswith two-body terms were carried out with a spherical exclu-sion volume (parametrized bys), for simplicity; we note thata parametern, similar to ours, was previously introduced byCichocki and Felderhof in two-body integrals to the purposeof studying scaling relationships in corrections to the CM for-mula [67].

Allowing for ℓ 6= 1/d involves products of Dirac func-tions. The latter arise from a rough treatment of clusters aspoint-like polarizable entities, within which dipolar interac-tions occur only through the local part of the Green function.The resulting mathematical ambiguity was handled by intro-ducingq, interpreted as a covering parameter between innerinclusions. The main operational role of parameterss andq isto modify the three-dimensional percolation threshold of thetheory. Inasmuch as they represent microstructutral features

of the medium, these parameters are akin to the geometricalones introduced by Miller [25], although we cannot claim anyprecise connection at this point.

We moreover needed to introduce two supplementary pa-rametersα andβ to handle independently the local and non-local two-body parts in the CPA self-consistency conditiononthe self-energy. When only part of the latter is canceled out,the theory stands as intermediate between CM-like and BL-like approaches. This additional flexibility was required toproduce realisticℓe(f) functions.

Although our emphasis was on versatility, the present the-ory could be rigorously reformulated in the discrete frame-work adapted to RRNs, for which the two-body interactionterm is known exactly (without needing to introduce param-eterss andq) [11, 49, 58], in order to investigate preciselythe role of parametersα andβ. We also remark that whilethe percolation threshold can be adjusted ford = 3, whereit depends on all four parameters, it remains stuck to its ex-act bond valuefc = 1/2 in d = 2. This does dot allow oneto handle site percolation [5]. Improving the behavior nearfc, as well as attempting to reproduce site-percolation effectswithin the present theory, would presumably require includingthree-body interactions [50]. It would be possible, if needed,to introduce additional parameters similar toα andβ whenincluding their contribution in the generalized CPA condition.

Finally, the multiple-scattering formulation in the contin-uum allows in principle for extensions to elasticity [68], or tothe frequency domain, including wave propagation dynamics[69].

Appendix A: N -body expansion in multiple-scatteringframework

To prove (3.17), we first demonstrate that

T(n)(i1,i2,...,in)

= T(n−1)(i1,i2,...,in−1)

Gℓuℓin

(1−Gℓ

n∑

p=1

uℓip

)−1

,

(A1)for n ≥ 2, the recursion being initiated withT (1)

(i) ≡ ti. We

start from the expressionT =∑

i uℓi(1−Gℓ

∑j uℓj)

−1, ob-tained from the equivalence between (3.13) and (3.15a), and

12

to which we apply systematic transformations. Introduce

ai ≡ (1−Gℓuℓi)−1, Ai ≡

(1−Gℓ

∑

k 6=i

uℓkai

)−1

. (A2)

Then, withT (1)(i) ≡ ti = uℓiai,

T =∑

i

uℓi

(1−Gℓuℓi −Gℓ

∑

j 6=i

uℓj

)−1

=∑

i

tiAi

=∑

i

ti

(1 +Gℓ

∑

j 6=i

uℓjaiAi

)

=∑

i

T(1)(i) +

∑

i

j 6=i

T(1)(i) GℓuℓjaiAi, (A3)

Let nowaij ≡ [1−Gℓ(uℓi + uℓj)]−1. One has

A−1i = 1−Gℓuℓj ai −Gℓ

∑

k 6=i,j

uℓk ai

=[1−Gℓ

∑

k 6=i,j

uℓkai (1−Gℓuℓjai)−1

](1−Gℓuℓjai)

=(1−Gℓ

∑

k 6=i,j

uℓkaij

)a−1ij ai. (A4)

Whence, withAij ≡(1−Gℓ

∑k 6=i,j uℓk aij

)−1

, and

T(2)(i,j) ≡ T

(1)(i) Gℓuℓj aij , (A5)

it follows that

T =∑

i

T(1)(i) +

∑

i

j 6=i

T(2)(i,j)Aij

=∑

i

T(1)(i) +

∑

i

j 6=i

T(2)(i,j) +

∑

ij 6=i

k 6=i,j

T(2)(i,j)Gℓuℓkaij Aij ,

(A6)

The progression from (A3) to (A6) represents one transfor-mation step. Going on by applying to the last term of (A6) atransformation similar to (A4), namely,

A−1ij = 1−Gℓuk aij −Gℓ

∑

l 6=i,j,k

uℓl aij , (A7)

and so on, proves (A1). Proving (3.17) is now simple. Assume(3.17a) to hold at rankn, and evaluateT (n+1)

(i1i2...in+1). Let

Bn =(1−Gℓ

n∑

p=1

uℓip

)−1

.

Because of (A1), it suffices to show that

uℓin+1Bn+1 (A8)

= tin+1

(1−GℓSi1i2...inGℓtin+1

)−1(1 +GℓSi1i2...in)

with Si1i2...in expressed in terms ofSi1i2...in−1as in (3.17b),

knowing that

uℓinBn = tin(1−GℓSi1i2...in−1

Gℓtin)−1

×(1 +GℓSi1i2...in−1

). (A9)

Since by definition [cf. (3.14)]uℓin = tin(1+Gℓtin)−1, (A9)

implies

B−1n =

(1 +GℓSi1i2...in−1

)−1

×(1−GℓSi1i2...in−1

Gℓtin)(1 +Gℓtin)

−1. (A10)

Therefore

B−1n+1 = B−1

n −Gℓtin+1(1 +Gℓtin+1

)−1

= (1 +GℓSi1i2...in)−1

×[1− (1 +GℓSi1i2...in)Gℓtin+1

(1 +Gℓtin+1)−1

]

= (1 +GℓSi1i2...in)−1

×(1−GℓSi1i2...inGℓtin+1

)(1 +Gℓtin+1

)−1, (A11)

where use has been made of (3.17b) in the second line. Theresult follows.

Appendix B: Square of the Dirac distribution

One possible definition of distribution products, due toColombeau [70, 71], is as equivalence classes, whose repre-sentative chosen for calculations must be inferred from thecontext. This concept allows one to work properly with ob-jects such asδ2, which is proportional toδ, but with non-standard (infinite) proportionality constant [71].

In the one-dimensional case for instance, takeδ(x) =

limσ→0 δσ(x) whereδσ(x) = e−(x/σ)2/2/(√2πσ) is a Gaus-

sian delta-sequence. Then∫dx δ2σ(x) = 1/(σ

√4π). This al-

lows one to setσδ2(x) ≡ qδ(x) with q = 1/√4π, asσ → 0.

A different choice of delta-sequence would lead to some otherq. This quantity therefore depends on the choice of representa-tion of δ(x), motivated by the physical nature of the problemconsidered. Thed-dimensional generalization of this argu-ment leads to (3.27).

We note that, in principle, a more fundamental treatmentof thed-dimensional case would require acknowledging thatthe Dirac contribution to the Green function (3.2) stems froma representation (using a superscript to emphasize the dimen-sion) [71]

δ(d)(r) = limη→0

δ(d)η (r), δ(d)η (r) =δ(r − η)

Sd rd−1, (B1)

where δ(r − η), with η identical to that in the principalvalue prescription in Eq. (3.2), materializes the surface of theLorentz cavity associated to the exclusion volume between in-teracting polarizable elements [46]. Doing so would providetheq of Eq. (3.27) as a function ofη, but also inevitably intro-duce other arbitrary constants, leading to unnecessary compli-cations.

13

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