Strong-contrast expansions and approximations ... - Semantic Scholar
JOURNAL OF APPLIED PHYSICS
VOLUME 94, NUMBER 10
15 NOVEMBER 2003
Strong-contrast expansions and approximations for the effective conductivity of isotropic multiphase composites D. C. Pham and S. Torquatoa) Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544
共Received 27 May 2003; accepted 27 August 2003兲 We extend the previous approach of one of the authors on exact strong-contrast expansions for the effective conductivity e of d-dimensional two-phase composites to case of macroscopically isotropic composites consisting of N phases. The series consists of a principal reference part and a fluctuation part 共a perturbation about a homogeneous reference or comparison material兲, which contains multipoint correlation functions that characterize the microstructure of the composite. The fluctuation term may be estimated exactly or approximately in particular cases. We show that appropriate choices of the reference phase conductivity, such that the fluctuation term vanishes, results in simple expressions for e that coincide with the well-known effective-medium and Maxwell approximations for two-phase composites. We propose a simple three-point approximation for the fluctuation part, which agrees well with a number of analytical and numerical results, even when the contrast between the phases is infinite near percolation thresholds. Analytical expressions for the relevant three-point microstructural parameters for certain mixed coated and multicoated spheres assemblages 共extensions of the Hashin–Shtrikman coated-spheres assemblage兲 are given. It is shown that the effective conductivity of the multicoated spheres model can be determined exactly. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1619573兴
I. INTRODUCTION
that render the integrals absolutely convergent in the infinitevolume limit. Thus, no renormalization analysis is required because the procedure used to solve the integral equation systematically leads to absolutely convergent integrals. Another useful feature of the expansions is that they can provide accurate estimates for all volume fractions when truncated at finite order, even when the phase conductivities differ significantly. In this paper, we extend the strong-contrast expansion approach of Torquato4 to the case of macroscopically isotropic composites consisting of N isotropic phases. The series expansions, which perturb about a homogeneous reference material, involve a principal reference term and a fluctuation term that perturbs about the reference medium. Based on a specified level of correlation information about the microstructure of the composite, we devise accurate approximations for the effective conductivity by choosing an appropriate equation for the conductivity of the reference 共comparison兲 material, a free parameter in the theory. The approximation is tested against available benchmark analytical and numerical results for various models of dispersions. We find that the approximation generally provides very good agreement with these benchmark results. In Sec. II, we derive strong-contrast expansion for the effective conductivity of macroscopically isotropic multicomponent composites. Approximation schemes based on the expansion are constructed in Sec. III, including a threepoint approximation, i.e., one that contains microstructural parameters that depends on three-point corelation functions. Section IV investigates the restrictions upon the three-point parameters. Section V collects well-known three-point bounds for comparison with our three-point approximation.
The prediction of the effective properties of composite materials has a rich history1–3 and is still an active area of research. In general, the effective properties of a composite depend on an infinite set of correlation functions that statistically characterize the medium.4 In the case of the effective conductivity e of composite materials, our concern in the present work, a number of different approximation schemes have been devised.1,5–7 Upper and lower bounds on the effective conductivity have been derived using variational principles.8 –13 For those composites in which the variations in the phase conductivities are small, formal solutions to the boundary-value problem have been developed in the form of weak contrast perturbation series.14,10 Due to the nature of the integral operator, one must contend with conditionally convergent integrals, which can be made convergent using an ad hoc normalization procedure.15 Alternatively, Brown16 constructed a strong-contrast expansion of the effective conductivity of three-dimensional two-phase isotropic media in powers of rational functions of the phase conductivities. The strong-contrast expansion approach has been further developed for the effective conductivity of d-dimensional macroscopically anisotropic composites consisting of two isotropic phases by introducing an integral equation for the cavity intensity field.4,17,18 The expansions are not formal but rather the nth-order tensor coefficients are given explicitly in terms of integrals over products of certain tensor fields and a determinant involving n-point statistical correlation functions a兲
Author to whom correspondence should be addressed; electronic mail: [email protected]
0021-8979/2003/94(10)/6591/12/$20.00
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© 2003 American Institute of Physics
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
Our three-point approximation is applied to a number of coated spheres and dispersion models and is compared with simulation results and bounds in Sec. VI.
phase having conductivity 0 , which is subjected to an applied electric field E0 (x) at infinity. Introducing the polarization field defined by P共 x兲 ⫽ 关 共 x兲 ⫺ 0 兴 E共 x兲
enables us to reexpress the flux J, defined by Ohm’s law 共4兲, as follows:
II. STRONG-CONTRAST EXPANSIONS
We derive strong-contrast expansions of the scalar effective conductivity of a macroscopically isotropic multiphase composite. The derivation is a direct and straightforward extension of the one given by Torquato4 for a macroscopically anisotropic two-phase composite. The reader is referred to this reference for greater details of the derivation. Consider a large macroscopically isotropic composite specimen in arbitrary space dimension d comprised of N isotropic phases having conductivities ␣ and volume fractions ␣ ( ␣ ⫽1,...,N). The microstructure is perfectly general but possesses a characteristic microscopic length scale that is much smaller than that of the specimen. Thus, the specimen is virtually statistically homogeneous. Ultimately, we shall take the infinite-volume limit and hence consider statistically homogeneous media. The local scalar conductivity at position x is expressible as
J共 x兲 ⫽ 0 E共 x兲 ⫹P共 x兲 .
兺
where 共␣兲
I
␣ I共 ␣ 兲 共 x兲 ,
␣ ⫽1
共 x兲 ⫽
再
1,
x in phase ␣ ,
0,
otherwise
共1兲
共2兲
is the indicator function for phase ␣ ( ␣ ⫽1,...,N). For statistically homogeneous media, the ensemble average of the indicator function is equal to the phase volume fraction ␣ , i.e.,
具 I共 ␣ 兲共 x兲 典 ⫽ ␣ ,
共3兲
where angular brackets denote an ensemble average. The local conductivity 共x兲 is the coefficient of proportionality in the linear constitutive relation J共 x兲 ⫽ 共 x兲 E共 x兲 ,
共4兲
where J共x兲 denotes the local electric 共thermal兲 current or flux at position x, and E共x兲 denotes the local field intensity. Under steady-state conditions with no source terms, conservation of energy requires that J共x兲 be solenoidal “"J共 x兲 ⫽0,
共5兲
while the intensity field E共x兲 is taken to be irrotational “ÃE共 x兲 ⫽0,
共6兲
which implies the existence of a potential field T(x), i.e., E共 x兲 ⫽⫺“T 共 x兲 .
共7兲
Thus E共x兲 and T(x) represent the electric field 共negative of temperature gradient兲 and electric potential 共temperature兲 in the electrical 共thermal兲 problem, respectively. Now, following Torquato,4 let us embed this d-dimensional composite specimen in an infinite reference
共9兲
The vector P共x兲 is the induced flux polarization field relative to the reference medium. Using the infinite-space Green’s function of the Laplace equation for the reference medium corresponding to the problem, Eqs. 共4兲–共7兲, we find that the electric field satisfies the integral relation4 E共 x兲 ⫽E0 共 x兲 ⫹
冕
dx⬘ G共 0 兲 共 r兲 •P共 x⬘ 兲 ,
共10兲
where G共 0 兲 共 r兲 ⫽⫺D共 0 兲 ␦ 共 r兲 ⫹H共 0 兲 共 r兲 , D共 0 兲 ⫽
N
共 x兲 ⫽
共8兲
1 I, d0
H共 0 兲 共 r兲 ⫽
1 dnn⫺I , ⍀0 rd
共11兲 共12兲
r⫽x⫺x⬘, n⫽r/兩r兩, ␦共r兲 is the delta Dirac function, I is the second order identity tensor, ⍀(d) is the total solid angle contained in a d-dimensional sphere given by ⍀共 d 兲⫽
2 d/2 , ⌫ 共 d/2兲
共13兲
and ⌫(x) is the gamma function. In particular, ⍀共2兲⫽2, ⍀共3兲⫽4. In relation 共11兲, the constant second order tensor D(0) arises because of the exclusion of the spherical cavity, and it is understood that integrals involving the second order tensor H(0) are to be carried out by excluding at x⬘⫽x an infinitesimal sphere in the limit that the sphere radius shrinks to zero. Moreover, the integral of H(0) (r) over the surface of a sphere of radius R⬎0 is identically zero, i.e.,
冕
r⫽R
H共 0 兲 共 r兲 d⍀⫽0.
共14兲
Substitution of Eq. 共11兲 into expression 共10兲 yields an integral equation for the cavity intensity field F共x兲 F共 x兲 ⫽E0 共 x兲 ⫹
冕
⑀
dx⬘ H共 0 兲 共 x⫺x⬘ 兲 •P共 x⬘ 兲 ,
共15兲
where we define
冕
⑀
dx⬘ f 共 x,x⬘ 兲 ⫽ lim
⑀ →0
冕
兩 x⫺x⬘ 兩 ⬎ ⑀
dx⬘ f 共 x,x⬘ 兲 .
共16兲
The cavity intensity field F共x兲 is related to E共x兲 through the expression F共 x兲 ⫽ 兵 I⫹D共 0 兲 关 共 x兲 ⫺ 0 兴 其 •E共 x兲 .
共17兲
Combination of the expressions 共8兲 and 共17兲 gives a relation between the polarization and cavity intensity fields P共 x兲 ⫽L 共 x兲 F共 x兲 ,
共18兲
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
Multiplying this relation by the scalar L(x) defined by Eq. 共19兲, yields
where N
L 共 x兲 ⫽ 0 d
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共 x兲 ⫺ 0 ⫽0 d b ␣ 0 I共 ␣ 兲 共 x兲 , 共 x兲 ⫹ 共 d⫺1 兲 0 ␣ ⫽1
兺
P⫽LE0 ⫹LH共 0 兲 P.
共25兲
共19兲
A solution for the polarization P in term of the applied field E0 can be obtained by successive substitutions using Eq. 共25兲, with the result
and we have used Eqs. 共1兲 and 共2兲. The effective conductivity e for the macroscopically isotropic composite can be defined through an expression relating the average polarization to the average Lorentz field, i.e.,
P⫽LE0 ⫹LH共 0 兲 LE0 ⫹LH共 0 兲 LH共 0 兲 LE0 ⫹...⫽SE0 , 共26兲
␣⫺ 0 , b ␣0⫽ ␣ ⫹ 共 d⫺1 兲 0
具 P共 x兲 典 ⫽Le • 具 F共 x兲 典 ,
共20兲
where Le ⫽L e I,
L e⫽ 0d
e⫺ 0 . e ⫹ 共 d⫺1 兲 0
共21兲
The constitutive relation 共20兲 is localized, i.e., it is independent of the shape of the composite specimen in the infinitevolume limit. This relation is completely equivalent to the averaged Ohm’s law that defines the effective conductivity
具 J共 x兲 典 ⫽ e 具 E共 x兲 典 .
共22兲
We want to obtain an expression for the effective conductivity e from relation 共22兲 using the solution of the integral Eq. 共15兲, which is recast as F共 1 兲 ⫽E0 共 1 兲 ⫹
冕
共0兲
d2H 共 1,2兲 •P共 2 兲 ,
where the second-order tensor operator S is given by S⫽L 共 I⫺LH共 0 兲 兲 ⫺1 .
共27兲
Ensemble averaging Eq. 共26兲 gives
具 P典 ⫽ 具 S典 E0 .
共28兲
The operator 具S典 involves products of the tensor H(0) , which decays to zero like r ⫺d for large r, and hence 具S典 at best involves conditionally convergent integrals. In other words, 具S典 is dependent upon the shape of the composite specimen. In order to obtain a local 共shape-independent兲 relation between average polarization 具P典 and average Lorentz field 具F典 as prescribed by Eq. 共22兲, we must eliminate the applied field E0 in favor of the appropriate average field. Inverting Eq. 共28兲 yields E0 ⫽ 具 S典 ⫺1 具 P典 . Averaging Eq. 共24兲 and eliminating the applied field E0 yields
具 F典 ⫽Q具 P典 ,
共29兲
where Q⫽ 具 S典 ⫺1 ⫹H共 0 兲 .
共30兲
共23兲
Comparing expressions 共20兲 and 共29兲 yields the desired result for the effective tensor Le :
where we have adopted the shorthand notation of representing x and x⬘ by 1 and 2, respectively. In schematic operator form, this integral equation can be tersely rewritten as
⫺1 共0兲 ⫽H共 0 兲 ⫹ 具 L 共 I⫺LH共 0 兲 兲 ⫺1 典 ⫺1 . L⫺1 e ⫽Q⫽H ⫹ 具 S典 共31兲 The first few terms of the expansion Eq. 共31兲 are explicitly given by
⑀
F⫽E0 ⫹H共 0 兲 P.
L⫺1 e 共 1 兲⫽
冕
共24兲
d2Q共 1,2兲 ⫽
I共 1 兲 ⫺ 具L共 1 兲典 ⫺
冕冕
冕 冉具 d2
d2d3
冉
冊
L 共 1 兲 L 共 2 兲 典 ⫺ 具 L 共 1 兲 典具 L 共 2 兲 典 共 0 兲 H 共 1,2兲 具 L 共 1 兲 典具 L 共 2 兲 典
The general term contains the n-point correlation functions
具 L(1)...L(n) 典 . The explicit expression for any term in the series can be given as in Ref. 4. Let us introduce the property-independent dipole tensor
t共 r兲 ⫽ 0 H共 0 兲 共 r兲 ⫽
dnn⫺I ⍀r d
.
冊
具 L 共 1 兲 L 共 2 兲 L 共 3 兲 典 具 L 共 1 兲 L 共 2 兲 典具 L 共 2 兲 L 共 3 兲 典 共 0 兲 ⫺ H 共 1,2兲 •H共 0 兲 共 2,3兲 ⫺... 具 L 共 1 兲 典具 L 共 2 兲 典 具 L 共 1 兲 典具 L 共 2 兲 典具 L 共 3 兲 典
共33兲
Substituting Eqs. 共19兲 and 共33兲 into Eq. 共32兲 and taking into account that t is traceless, we take the trace of both sides of Eq. 共32兲. We then multiply them with 0 ( 兺 ␣ ␣ b ␣ 0 ) 2 to obtain the relation for our macroscopically isotropic media
e ⫹ 共 d⫺1 兲 0 e⫺ 0
冉兺 ␣
冊
共32兲
2
␣b ␣0 ⫺ 兺 ␣b ␣0 ␣
⫽A共 0 , 1 ,..., N ,microstructure兲 .
共34兲
Here A( 0 , 1 ,..., N ,microstructure) is a scalar quantity that depends on the reference conductivity 0 , phase conductivities 1 ,..., N , and the microstructure via n-point correlation functions. The left side of Eq. 共34兲 is referred to as the principal reference term of the strong-contrast expansion, while the right side, given by A is the fluctuation term relative to the reference medium of conductivity 0 . Through the lowest three-point term in the series, A is explicitly given by
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
A共 0 , 1 ,..., N ,microstructure兲 ⫽⫺d ⫺
冋兺
␣,,␥
兺 ␣,,␥,␦
D. C. Pham and S. Torquato
b ␣0b 0b ␥0
冕冕 冕冕
b ␣0b 0b ␥0b ␦0 兺 b 0
d2d3 具 I共 ␣ 兲 共 1 兲 t i j 共 1,2兲 I共  兲 共 2 兲 t i j 共 2,3兲 I共 ␥ 兲 共 3 兲 典 d2d3 具 I共 ␣ 兲 共 1 兲 t i j 共 1,2兲 I共  兲 共 2 兲 典具 I共 ␥ 兲 共 2 兲 t i j 共 2,3兲 I共 ␦ 兲 共 3 兲 典
⫺... .
共35兲
Here conventional summation on repeating Latin indices 共from 1 to d兲 is assumed, and the Greek indices under the sum run from 1 to N.
A. Connections to previous estimates
The effective medium approximation 共EMA兲 EMA 共also known as the self-consistent scheme兲5,7 is the solution of the equation
⫺
␣ EMA ␣ ⫽0. 兺 ⫹ d⫺1 兲 EMA 共 ␣ ⫽1 ␣
共36兲
Equivalently, we can rewrite this via the property function P
e ⫽ EMA⫽ P 共共 d⫺1 兲 EMA兲 , where
冋兺 N
P 共 兲 ⫽
␣ ␣⫹
册
共37兲
⫺1
⫺ . 共38兲 * * This approximation is exact for a certain hierarchical composite consisting of spherical grains.19 As 0 → EMA 共while e ⫽ EMA), the left-hand side of Eq. 共34兲 approaches 0, and hence for the EMA microstructure we have
*
␣ ⫽1
A共 e , 1 ,..., N ,EMA microstructure兲 ⫽0.
共39兲
For two-phase microstructures that are optimal when the volume fraction is specified, such as the Hashin–Shtrikman coated-spheres assemblage,8 we have
e ⫽ P 关共 d⫺1 兲 M 兴 ,
referred to as TPA1兲 for the effective conductivity of twophase composites that can be regarded to perturb about the Hashin–Shtrikman structures4 2 e 1⫹ 共 d⫺1 兲 2 b 21⫺ 共 d⫺1 兲 1 2 b 21 ⫽ 2 1 1⫺ 2 b 21⫺ 共 d⫺1 兲 1 2 b 21
III. APPROXIMATION SCHEMES
N
册
共40兲
where M is the property of the connected matrix phase in the optimal geometry. Thus Eq. 共34兲 also yields A共 M , 1 , 2 ,optimal two-phase microstructure兲 ⫽0. 共41兲 Formula 共40兲 is sometimes called the Maxwell approximation 共MA兲 共also known as the Maxwell–Garnett or Clausius–Mossotti approximation兲.
The fluctuation term A depends on the microstructure of the composite and generally can only be evaluated by some numerical scheme. Torquato17 used the strong-contrast expansion to develop a three-point approximation 共henceforth
共42兲
where ␣ is a three-point microstructural parameter defined by Eqs. 共48兲 and 共49兲. Here the reference phase is taken to be phase 1. We will now derive a different three-point approximation for multiphase composites based on the results Eqs. 共36兲–共41兲. Let us assume that we have an explicit approximation Aapprox for the fluctuation term A for a specific composite. Guided by the discussion of Sec. III A, our strategy is to choose 0 to be the solution of the equation Aapprox⫽0, and from Eq. 共34兲 deduce the respective approximation e ⫽ P 关 (d⫺1) 0 兴 . Let us assume that Aapprox⫽An , where An is the series A truncated after the n-point correlation term. From Eq. 共34兲, we deduce the respective n-point approximation for the effective conductivity
e ⫽ P 共共 d⫺1 兲 0 兲 ,
An 共 0 兲 ⫽0.
共43兲
Clearly, at n→⬁, we should get the exact value of the effective conductivity for an arbitrary microstructure. For any microstructure, we expect that 0 should lie within the interval 关 min ,max兴 for e in order to satisfy the Hashin– Shtrikman bounds,8 which can be expressed as P 共共 d⫺1 兲 max兲 ⭓ e ⭓ P 共共 d⫺1 兲 min兲 ,
共44兲
where
max⫽max兵 1 ,..., N 其 ,
min⫽min兵 1 ,..., N 其 . 共45兲
For example, let us take Aapprox⫽A3 . Since three-point microstructural information is now available for a variety of different microstructures, we are especially interested in the three-point approximation 共TPA2兲:
e ⫽ P 共共 d⫺1 兲 0 兲
共 TPA2 兲 ,
共46兲
where 0 is the solution of equation A3 共 0 兲 ⫽0.
B. Multipoint approximations
共 TPA1 兲 ,
共47兲
We call this TPA2 to distinguish it from TPA1 given by Eq. 共42兲. Both three-point approximations are exact through third order in the difference in the phase conductivities. In the special two-phase case, three-point correlation function information arises via the microstructural parameters ␣ ( ␣ ⫽1,2), 4,20,21 which for d⫽2
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
4 ␣⫽ 1 2
冕 冕 冕 ⬁
⬁
dr r
0
0
S 共2␣ 兲 共 r 兲 S 共2␣ 兲 共 s 兲
⫺
ds s
S 共1␣ 兲
册
0
d cos共 2 兲
D. C. Pham and S. Torquato
冋
S 共3␣ 兲 共 r,s,t 兲
C. Alternative approaches
We note that Le of Eq. 共31兲 can also be expanded as 共 0 兲 ⫺1 ⫽ 具 S典 ⫽ 具 L 共 I⫺LH共 0 兲 兲 ⫺1 典 共 L⫺1 e ⫺H 兲
共48兲
,
⬁
⫽ 具 L 典 I⫹
and for d⫽3 9 ␣⫽ 2 1 2
冕 冕 冕 ⬁
0
dr r
⫻ P 2 共 cos 兲
冋
⬁
0
ds s
1
⫺1
S 共2␣ 兲 共 r 兲 S 共2␣ 兲 共 s 兲 S 共1␣ 兲
册
,
共49兲
where P 2 is the Legendre polynomial of order 2 and is the angle opposite the side of the triangle of length t, and S 共n␣ 兲 共 x1 ,¯,xn 兲 ⫽ 具 I共 ␣ 兲 共 x1 兲 ¯I共 ␣ 兲 共 xn 兲 典 .
共50兲
It is also known4,20,21 that 1 ⭓0, 2 ⭓0, and 1 ⫹ 2 ⫽1. Through the microstructural parameters 1 and 2 , relation 共47兲 can be given simply as
1
1⫺ 0 2⫺ 0 ⫹2 ⫽0. 1 ⫹ 共 d⫺1 兲 0 2 ⫹ 共 d⫺1 兲 0
冕冕 V
V
I共 ␣ 兲 共 x兲 t i j 共 x,y兲 I共  兲 共 y兲 dx dy⫽0.
兺
b ␣ 0 b  0 b ␥ 0 A ␣␥ ⫽0,
共52兲
共53兲
冕冕冕 V
V
V
d1d2d3I共 ␣ 兲 共 1 兲 t i j 共 1,2兲 I共  兲 共 2 兲
⫻t i j 共 2,3兲 I共 ␥ 兲 共 3 兲 .
共57兲
共58兲
where 0 is the solution of the equation ˆ 共 , ,..., ,microstructure兲 ⫽0, A 0 1 N ⬁
ˆ⫽ A
兺
dn
n⫽2
兺
␣ 1 ,..., ␣ n
b ␣ 1 0 b ␣ 2 0 ...b ␣ n 0
共59兲
冕 冕 ¯
d2...dn
⫻ 具 I共 ␣ 1 兲 共 1 兲 t i j 共 1,2兲 I共 ␣ 2 兲 共 2 兲 ...t li 共 n⫺1,n 兲 I共 ␣ n 兲 共 n 兲 典 . 共60兲 At the three-point approximation level, Eqs. 共58兲–共60兲 leads to the same result Eqs. 共46兲 and 共53兲 of the previous subsection. Moreover we see that Eq. 共31兲 can also be given as Le 共 I⫺Le H共 0 兲 兲 ⫺1 ⫽ 具 L 共 I⫺LH共 0 兲 兲 ⫺1 典 ,
冋
⬁
兺
k⫽1
册
共61兲
⬁
具 共 L e H共 0 兲 兲 k 典 ⫽ 具 L 典 I⫹ 兺 具 L 共 LH共 0 兲 兲 k 典 . 共62兲 k⫽1
Taking nth approximation of Eq. 共62兲 gives
冋
Le I⫹
where A ␣␥ ⫽
e ⫽ P 共共 d⫺1 兲 0 兲 ,
Le I⫹
Then Eq. 共47兲 can be written as ␣,,␥
共56兲
In the same manner as that of the previous section, we obtain the expression
which can be expanded as
I共 ␣ 兲 共 x兲 t i j 共 x,y兲 I共  兲 共 y兲 典 dy
⫽
具 L 共 LH共 0 兲 兲 n 典 ,
⫺1 共 0 兲 ⫺1 共0兲 ⫽ 具 L 典 I→L⫺1 I. 共 L⫺1 e ⫺H 兲 e ⫽H ⫹ 具 L 典
共51兲
We can further simplify Eq. 共47兲 for N⭓2. According to the ergodic hypothesis, we could substitute the ensemble averages in Eq. 共35兲 by volume averages over the volume of the macroscopic sample V, which we can take for convenience to be spherical. Thus, for a statistically homogeneous and isotropic medium of spherical volume V, we have
冕具
兺
n⫽1
where we have used an obvious short-hand notation in the second line of Eq. 共56兲. If we take 0 such that the last sum in Eq. 共56兲 vanishes, then we deduce
d 共 cos 兲
S 共3␣ 兲 共 r,s,t 兲 ⫺
6595
n⫺1
兺
k⫽1
册
n⫺1
具 共 L e H共 0 兲 兲 k 典 ⫽ 具 L 典 I⫹
兺
k⫽1
具 L 共 LH共 0 兲 兲 k 典 . 共63兲
If we take 0 as to make the last sum of the right-hand of Eq. 共63兲 vanish 关c.f. Eqs. 共59兲 and 共60兲兴, then at the three-point approximation, Eq. 共63兲 leads to the same result Eqs. 共46兲 and 共53兲.
共54兲
IV. THREE-POINT MICROSTRUCTURAL PARAMETERS
Note the symmetry of A ␣␥ in the indices ␣ and ␥, but not in . The sum on the left side of Eq. 共53兲 is of quadratic form in the variables b ␣ 0 , b ␥ 0 ( ␣ , ␥ ⫽1,...,N), the sign of which depends on those of b  0 (  ⫽1,...,N). Hence, at 0 ⫽ min , the sum is positive, while at 0 ⫽ max , it is negative, with the solution 0 of Eq. 共53兲 lying between these extreme values. Consequently, the TPA2 from Eqs. 共46兲 and 共53兲 should always fall inside the Hashin–Shtrikman bounds Eq. 共44兲. For N⫽2, the microstructural parameters ␣ is related to A ␣␣␣ via
As in the two-phase case, there exist some relations between the microstructural parameters A ␣␥ . From Eq. 共54兲, we see that they are symmetric in the indices ␣ and ␥. Consider a spherical representative element V of the composite with phase ␣ occupying regions V ␣ ( ␣ ⫽1,...,N). For simplicity and without loss of generality we take the volume of V to be unity, and hence the volume of V ␣ is equal to the volume fraction ␣ . Let us introduce the harmonic potentials
A ␣␣␣ ⫽
d⫺1 1 2 ␣ , d
␣ ⫽1,2.
共55兲
共 x兲 ⫽
冕
V
G 共 x⫺y兲 dy,
“ 2 共 x兲 ⫽1,
x苸V;
共64兲
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
␣ 共 x兲 ⫽
冕
␣
“ 2 ␣ 共 x兲 ⫽ ␦ ␣ ,
G 共 x⫺y兲 dy,
D. C. Pham and S. Torquato
x苸  ; 共65兲
where G(r) (r⫽ 兩 x⫺y兩 ) is the respective Green function
G共 r 兲⫽
冦
⫺
冉冊
1 1 ln , 2 r
d⫽2 ,
1 1 ⫺ , 共 d⫺2 兲 ⍀ r d⫺2
ⵜ 2 G⫽ ␦ 共 r 兲 .
d⭓3 共66兲
We have G ,i j ⫽ ti j⫽
␦ij
dn i n j ⫺ ␦ i j ⍀r
d
共67兲
x i ⫺y i n i⫽ , r
,
where the Latin indices after a comma designate differentiation with respective to position. If we denote 1 ␣i j 共 x兲 ⫽ ␣,i j ⫺ ␦ i j ␦ ␣ , d
x苸  ,
共68兲
d⫺1 1 2 . d
共75兲
Generally, a N-phase material with microstructural parameters A ␣␥ ( ␣ , ␥ ,  ⫽1,...,N) can also be considered to be a two-phase material occupying regions ¯V I⫽艛 ␣ 苸⌫ IV ␣ , ¯V II ⫽艛 ␣ 苸⌫ IIV ␣ with respective microstructural parameters ¯A ␣␥ ( ␣ , ␥ ,  ⫽I,II). Here ⌫ I⫽ 兵 ␣ 1 ,..., ␣ k 其 , ⌫ II  ⫽ 兵 ␣ k⫹1 ,..., ␣ N 其 , and 兵 ␣ 1 ,..., ␣ k , ␣ k⫹1 ,..., ␣ N 其 is any permutation of 兵 1,...,N 其 , and k is any number between 1 and N⫺1. By combining the relations Eqs. 共65兲, 共68兲, 共69兲, and 共75兲, one can verify that d⫺1 d
␦ 共 x⫺y兲 ⫺t i j ,
d
22 A 11 1 ⫹A 2 ⫽
冉 兺 冊冉 兺 冊 ␣ 苸⌫ I
⫽ ⫽
␣
␣ 苸⌫ II
␣
d⫺1 ¯ ¯ ¯ I I⫹A ¯ II I II⫽A I II d
兺
A ␣␥ ⫹
 , ␥ , ␣ 苸⌫ I
兺
II
 , ␥ , ␣ 苸⌫ II
A ␣␥ ,
共76兲
¯ and ¯ are the volume fractions of regions I and II, where I II respectively.
then we have A ␣␥ ⫽
冕
V
␣i j ␥i j dx⫽ 共兲
⫻t i j 共 x,y兲 I
冕冕冕 V
V
V
V. THREE-POINT BOUNDS
dx dy dzI共 ␣ 兲 共 x兲 共␥兲
共 y兲 t i j 共 y,z兲 I
共69兲
共 z兲 .
Note that
,i j ⫽
␦ij d
N
⫽
兺 ␣,i j , ␣ ⫽1
共70兲
V␥
,i j ,i j dx⫽
冕
1  ␦ ␥ ,ii dx⫽ , d ␥ V␥ d
共71兲
V␥
冕 兺 冕 冋 兺
␣ ⫽1
V␥
␣ ,i j
 ,i j dx
N
⫽
␣ ⫽1 N
⫽
V␥
兺 A ␥␣ ⫹ ␣ ⫽1
␣ ij
 i j⫹
␦ ␥ d
⫺
册
1 ␦ ␦ dx d ␣␥ ␥
A ␥␣ ⫽0,
␥ .
᭙  , ␥ ⫽1,...,N.
共72兲
共73兲
22 12 A 11 2 ⫽A 2 ⫽⫺A 2 .
Keeping in mind Eq. 共55兲 and 1 ⫹ 2 ⫽1, we have
1 2共 1⫺ 2 兲 2 1 2 ⫹ 2 1 ⫹ 共 d⫺1 兲共 1 / 1 ⫹ 2 / 2 兲 ⫺1
.
Milton obtained a sharper lower bound for the case d⫽3 and 2 ⭓ 1
共 3L 兲
/ 1⫽
2 1⫹ 共 1⫹2 2 兲 b 21⫺2 共 1 2 ⫺ 2 兲 b 21 2 1⫹ 1 b 21⫺ 共 2 1 2 ⫹ 2 兲 b 21
2⫺ 1 . b 21⫽ 2 ⫹2 1
, 共80兲
For three-dimensional N-phase composites, Phan-Thien and Milton10 derive the following bounds:
In the two-component case Eq. 共73兲 yields 22 12 A 11 1 ⫽A 1 ⫽⫺A 1 ,
共78兲
共79兲
N
兺
1 2共 1⫺ 2 兲 2 , 1 2 ⫹ 2 1 ⫹ 共 d⫺1 兲共 1 1 ⫹ 2 2 兲
22
Comparing Eqs. 共71兲 and 共72兲, we obtain relations between the microstructural parameters
␣ ⫽1
⫺
共 3L 兲 ⫽ 1 1 ⫹ 2 2
N
,i j ,i j dx⫽
共77兲
共 3U 兲 ⫽ 1 1 ⫹ 2 2
and
冕
共 3U 兲 ⭓ e ⭓ 共 3L 兲 , where
and hence we find that
冕
Here we summarize previous three-point bounds that we will subsequently apply. The d-dimensional three-point Beran bounds14 for two-phase composites derived by Torquato4 are given by
共74兲
共 3U 兲 ⭓ e ⭓ 共 3L 兲 ,
共81兲
where
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
共 3U 兲 ⫽ 具 典 ⫺ ␦ •⌫• 共 ⌫⫹ ␦ •A/ 具 典 兲 ⫺1 •⌫• ␦ /3具 典 , 共82兲
⌬ ␣␥ ⫽
再
共 1⫺2  兲 ⌫ ␥ ,
␣⫽
⫺ ␣ ⌫ ␥ ⫺  ⌫ ␣␥ ,
␣⫽
6597
.
1/ 共 3L 兲 ⫽ 具 ⫺1 典 ⫺2 ␦ ⫺1 •⌫• 共 ⌫⫹ ␦ ⫺1 • 共 ⌬ ⫹A兲 /2具 ⫺1 典 兲 ⫺1 •⌫• ␦ ⫺1 /3具 ⫺1 典 ,
共83兲
Here 具 f ( ) 典 for any function f of is given by
with (N⫺1)-rank vectors and matrices
⌫⫽ 兵 ⌫ ␣ 其 ␣N⫺1 ,  ⫽1 ,
冕冕 冕冕冕 1
16
2
V
V
⫽
V
具 f 共 兲典⫽ 兺 ␣ f 共 ␣ 兲.
A⫽ 兵 A ␣␥ 其 ␣N⫺1 ,  , ␥ ⫽1 ,
⌬⫽ 兵 ⌬ ␣␥ 其 ␣N⫺1 ,  , ␥ ⫽1 ,
A ␣␥ ⫽
N
␦ ⫺1 ⫽ 兵 ␣⫺1 ⫺ N⫺1 其 ␣N⫺1 ⫽1 ,
␦ ⫽ 兵 ␣ ⫺ N 其 ␣N⫺1 ⫽1 ,
⌫ ␣ ⫽
再
共84兲
␣ 共 1⫺ ␣ 兲 ,
␣⫽
⫺ ␣  ,
␣⫽
; 共85兲
冉 冊 1
兺 A ␦␥ ⫹ 3 共 ␣ ␦ ␣ ␦ ␣␥ ⫺ ␣  ␦ ␥ ⫺  ␥ ␦ ␣␥ ⫺ ␣ ␥ ␦ ␣ ⫹2 ␣  ␥ 兲 . ␦ ⫽1
Three-point bounds for d-dimensional N-phase composites derived by Pham13,23 are given by P 共共 d⫺1 兲 共03U 兲 兲 ⭓ e ⭓ P 共共 d⫺1 兲 共03L 兲 兲 , and
(3L) 0
共88兲
are the solutions of the equations
Q 共3U 兲 共共 d⫺1 兲 共03U 兲 兲 ⫽0,
共89兲
Q 共3L 兲 共共 d⫺1 兲 共03L 兲 兲 ⫽0,
兺
共 ␣ ⫺ 0 兲 A ␣␥ X  X ␥ ,
共90兲
兺
␥ 共 ␣⫺1 ⫺ ⫺1 0 兲A ␣ X X ␥ ,
共91兲
X  ⫽ 具 关 ⫹ 共 d⫺1 兲 0 兴 ⫺1 典 ⫺ 关  ⫹ 共 d⫺1 兲 0 兴 ⫺1 .
共92兲
Q 共3U 兲 共共 d⫺1 兲 0 兲 ⫽ Q 共3L 兲 共共 d⫺1 兲 0 兲 ⫽
␣,,␥
␣,,␥
In the two-phase case, relations 共89兲 are solved explicitly as
共03U 兲 ⫽
The PhanThien–Milton microstructural parameters A ␣␥ can be related to our A ␣␥ via the expression
dr ds 2 2 具 ⍀ ␣⬘ 共 0兲 ⍀ ⬘ 共 r兲 ⍀ ␥⬘ 共 s兲 典 rs r s
N
where
共86兲
dx dy dz共 I共  兲 共 x兲 ⫺  兲 G ,i j 共 x,y兲共 I共 ␣ 兲 共 y兲 ⫺ ␣ 兲 G ,i j 共 y,z兲共 I共 ␥ 兲 共 z兲 ⫺ ␥ 兲
⫽A ␣␥ ⫺ ␣
(3U) 0
␣ ⫽1
22 1 A 11 1 ⫹ 2A 2 22 A 11 1 ⫹A 2
⫽ 1 1⫹ 2 2 ,
22 A 11 1 ⫹A 2 共03L 兲 ⫽ 11 ⫽ 共 1 / 1 ⫹ 2 / 2 兲 ⫺1 . A 1 / 1 ⫹A 22 / 2 2
共93兲
共87兲
proximation P ( 0 ) from Eqs. 共46兲 and 共51兲 should fall inside the three-point bounds Eqs. 共88兲, 共93兲, and 共94兲 for our real composite. In the two-phase case, the bounds Eqs. 共88兲, 共93兲, and 共94兲 as well as the bounds Eqs. 共81兲–共83兲 coincide with the bounds Eqs. 共77兲–共79兲. The bounds Eqs. 共81兲–共83兲, and 共88兲 and 共89兲 have been compared in Ref. 24 for the class of N-phase (N⭓3) quasisymmetric 共symmetric cell兲 materials using a symbolic algebra program and numerical simulation. It appears that the bounds yield the same results for N-phase spherical cell materials. For N-phase platelet cell materials, the bounds Eqs. 共81兲–共83兲 appear tighter. However, the bounds Eqs. 共88兲 and 共89兲 are simpler in functional form as well as computational aspects. In the case of N-phase spherical cell composites, Eq. ⫽ 具 典 , (3L) 共89兲 are also solved explicitly and yield (3U) 0 0 ⫺1 ⫺1 ⫽ 具 典 . Unsuccessful attempts have been made to transform Eqs. 共81兲–共83兲, which involve multiplications and inversions of (N⫺1)-rank matrices and vectors, into some simple form similar to that of Eqs. 共88兲 and 共89兲. VI. APPLICATIONS OF THE THREE-POINT APPROXIMATION
共94兲
If we imagine a fictitious composite with 1 and 2 beand (3L) ing the volume fractions of the phases, then (3U) 0 0 in Eqs. 共93兲 and 共94兲 are, respectively, the ‘‘arithmetic’’ and ‘‘harmonic’’ averages. Moreover, the solution 0 of Eq. 共51兲 is the ‘‘effective medium approximation’’ value, hence (3U) ⭓ 0 ⭓ (3L) , and consequently the three-point ap0 0
In this section, we apply our three-point approximation 共TPA2兲 to certain multicoated spheres assemblages and dispersions of identical spheres. In each of these instances, the relevant three-point microstructural parameters are known. A. Analytical two-phase models
There are only very few nontrivial models, in which the three-point microstructural parameters A ␣␥ have been deter-
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
mined analytically. Here we discuss three such instances. For the two-phase EMA microstructure,19,22 we have
1⫽ 1 ,
2⫽ 2 .
共95兲
With Eq. 共95兲, we see that relation 共51兲 for 0 has the identical form as Eq. 共36兲 for EMA in the two-phase case. Hence our TPA2 Eqs. 共46兲 and 共51兲 coincides with the exact value Eq. 共37兲 for two-phase EMA microstructure, i.e.,
TPA⫽ P 共共 d⫺1 兲 0 兲 ⫽ EMA⫽ P 共共 d⫺1 兲 EMA兲 . 共96兲 Thus, in the two-phase case, our TPA2 Eqs. 共46兲 and 共51兲 may be interpreted as a three-point generalization of EMA, while the classical EMA expression Eq. 共37兲 is the corresponding two-point version. Next, consider the Hashin–Shtrikman two-phase coated spheres model, which consists of composite spheres that are composed of a spherical core, of conductivity 2 , and radius a, surrounded by a concentric shell of conductivity 1 and outer radius b. The ratio (a/b) 3 is fixed and equal to the inclusion volume fraction 2 . The composite spheres fill all space, implying that there is a distribution in their sizes ranging to the infinitely small 关see Fig. 1共a兲兴. For this coated spheres model, A ␣␥ have been determined explicitly,25 which for general d-dimensional composites can be given as 22 12 A 11 1 ⫽A 1 ⫽⫺A 1 ⫽
d⫺1 1 2 , d
22 12 A 11 2 ⫽A 2 ⫽⫺A 2 ⫽0.
共97兲
Hence, from Eq. 共55兲 one finds
1 ⫽1,
2 ⫽0.
共98兲
Consequently, from Eqs. 共93兲 and 共94兲, one obtains (3U) 0 ⫽ (3L) ⫽ 1 , and the bounds Eq. 共88兲 coincide to yield the 0 exact effective conductivity
e ⫽ P 共共 d⫺1 兲 1 兲 .
共99兲
This relation coincides with the Maxwell approximation and Hashin–Shtrikman upper bound 共when 2 ⬍ 1 ) or lower bound 共when 2 ⬎ 1 ). The result Eq. 共99兲 was also obtained by direct solution of the respective conductivity problem,8 which in turn leads to Eq. 共98兲. Relation 共99兲 is also realizable by certain laminates4 and thus are also optimal. With Eq. 共98兲, the solution of Eq. 共51兲 should be 0 ⫽ 1 , and from Eq. 共46兲 we get the same formula Eq. 共99兲. Thus, for two-phase coated spheres 共as well as other optimal models兲, our three-point approximation also coincides with the exact result. A generalization of the Hashin–Shtrikman coatedspheres model is the mixed-coated-spheres model.26 This microstructure consists of a mixture of the two types of coated spheres corresponding to the Hasin–Shtrikman upper and lower bounds at a fixed volume fraction. Thus, an additional parameter is the proportion of coated spheres in which phase ␣ is the included phase and phase 共⫽␣兲 is the matrix, which we denotes by ␣ . Clearly, 12⫹ 21⫽ 1 ⫹ 2 ⫽1. For this geometry 关Fig. 1共b兲兴, the microstructural parameters have been determined analytically.26 The d-dimensional generalizations are given by
FIG. 1. Schematic illustrations of three different coated-spheres models: 共a兲 Hashin–Shtrikman two-phase coated-spheres assemblage; 共b兲 two-phase mixed-coated-spheres assemblage; and 共c兲 multiphase doubly coated spheres assemblage.
22 12 A 11 1 ⫽A 1 ⫽⫺A 1 ⫽
d⫺1 1 2 21 , d
22 12 A 11 2 ⫽A 2 ⫽⫺A 2 ⫽
d⫺1 1 2 12 , d
共100兲
and from Eq. 共55兲
1 ⫽ 21 ,
2 ⫽ 12 .
共101兲
As an example, consider the case 2 / 1 ⫽20. The Hashin–Shtrikman 共HS兲 bounds Eq. 共44兲, three-point 共TP兲 bounds Eqs. 共78兲 and 共80兲, and the three-point approximation 共TPA2兲 Eqs. 共46兲 and 共51兲 are compared in the plane e / 1 versus 2 关see Fig. 2共a兲兴. In Fig. 2共b兲, the three-point approximation is plotted versus 21 in the range 0⭐ 21⭐1, the extreme cases corresponding to the Hashin–Shtrikman’s upper and lower bounds, respectively.
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
FIG. 2. 共a兲 Comparison of the three-point approximation 共TPA2兲 共solid curve兲 for a two-phase mixed-coated-spheres model for 2 / 1 ⫽20, 21 ⫽0.6 to the HS bounds 共dotted curves兲 and the TP bounds 共dashed curves兲. 共b兲 The TPA2 for the mixed-coated-spheres model for a range of 21 with 2 / 1 ⫽20.
B. Periodic and random dispersions of spheres
Here we apply our approximation to various periodic and random dispersions of spheres. The three-point microstructural parameters ␣ for these models are available in the literature.4,27–37 We consider the infinite-phase contrast cases: superconducting inclusions in a normal conductor ( 2 / 1 ⫽⬁) and perfectly insulating inclusions in a normal conductor ( 2 / 1 ⫽0). These are the most stringent test of our approximation. Relation 共51兲 in these instances yields 1 1 / 共 1⫺d 2 兲 if 2 ⭐ d 0⫽ , 1 ⬁ if 2 ⭓ d 共102兲 共 2 / 1 ⫽⬁ 兲 ,
0⫽
再 冉 再
0
1⫺
冊
d d⫺1 2 1 if 2 ⭓
共 2 / 1 ⫽0 兲 .
d⫺1 d
if 2 ⭐
d⫺1 d ,
共103兲
6599
FIG. 3. Comparison of simulation data 共Refs. 32 and 33兲 for the effective conductivity of equisized ordered superconducting particles to relations 共37兲, 共40兲, 共42兲, and 共46兲 for the EMA, MA, TPA1, and TPA2, respectively: 共a兲 face-centered-cubic spheres 共Ref. 32兲 and 共b兲 hexagonal array of aligned circular cylinders 共see Ref. 33兲.
Interestingly, our approximation predicts a nontrivial microstructure-dependent percolation threshold. For the superconducting and perfectly insulating cases, it predicts a percolation threshold at 2c ⫽1/d and 2c ⫽(d⫺1)/d, respectively. The corresponding threshold in terms of volume fraction 2c is easily found from the function 2 ( 2 ) tabulated in the aforementioned literature. One cannot expect a three-point approximation to yield accurate estimates of the percolation threshold, which requires higher-order microstructural 共clustering兲 information. It is interesting to note, however, that for some two-dimensional 共rather than threedimensional cases兲, Eqs. 共102兲 and 共103兲 yield reasonable estimates of 2c . For example, for a two-dimensional square array, Eq. 共102兲 predicts 2c ⬇0.775, which is to be compared to the exact result 2c ⫽ /4⬇0.785. The predictions of our TPA2 for the effective conductivity are compared to simulation data for certain periodic dispersions32,33 关see Figs. 3共a兲 and 3共b兲兴 and random dispersions34 –37 关see Figs. 4共a兲, 4共b兲, and 5兴 over large ranges of volume fractions 2 . We include in the figures relations 共37兲, 共40兲, and 共42兲 for the EMA, MA, and TPA1, respectively. We see that both the TPA1 and TPA2 are rela-
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6600
J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
FIG. 5. Comparison of simulation data 共see Ref. 37兲 for the effective conductivity of equisized random dispersions of overlapping insulating spheres to relations 共37兲, 共40兲, 共42兲, and 共46兲 for the EMA, MA, TPA1, and TPA2, respectively.
volume proportions and coating orders of the phases in all N-compound spheres are the same. The microstructural parameters of this N-phase model have also been determined exactly,26 which in d-dimensional space can be expressed as ( ␣ ,  , ␥ ⫽1,...,N) A ␣␥ ⫽
d⫺1 ␣  ␥ d
,␥⬍␣; FIG. 4. Comparison of simulation data 共see Refs. 34 –36兲 for the effective conductivity of equisized random superconducting particles to relations 共37兲, 共40兲, 共42兲, and 共46兲 for the EMA, MA, TPA1, and TPA2, respectively: 共a兲 random dispersions of hard spheres 共see Ref. 35兲, and 共b兲 random dispersions of hard circular cylinders 共see Refs. 34 and 36兲.
A ␣␣ ⫽⫺ A ␣␣␣ ⫽
␦⬍␣
兺
␦• 兺 ⭐␣
冉兺 冊 冉兺 冊 ␦⭐␣
d⫺1 ␣ ␦• d ␦⬍␣
A ␣␥ ⫽0 tively good predictors of e up to high volume fractions 2 of the included phase close to the percolation threshold, where higher-order information is clearly required to be more accurate. For superconducting cases in three dimensions, the TPA1 is slightly more accurate than the TPA2 for the ordered array but the reverse is true for the disordered array. For superconducting cases in two dimensions, the TPA1 and TPA2 are comparable, except at high volume fractions where the TPA2 is more accurate. In the case of perfectly insulating overlapping spheres, the TPA1 and TPA2 are again comparable, but the TPA2 is superior at high sphere volume fractions. The TPA2 is most accurate for simple periodic systems, followed by random dispersions of hard cylinders and spheres, and least accurate for overlapping cylinders and spheres.
d⫺1 ␣  d
冉兺
␦
⭐␣
⫺1
,
冊
⫺1
,
⬍␣; 共104兲
⫺1
,
␣ ⭓2;
if  ⬎ ␣ or ␥ ⬎ ␣ or ␣ ⫽  ⫽ ␥ ⫽1.
For example, for the three-phase doubly-coated spheres in three dimensions, Eq. 共104兲 becomes 2␣ 3␣ 3␣ A 11 1 ⫽A 1 ⫽A 1 ⫽A 2 ⫽0,
␣ ⫽1,2,3
11 12 2 ⫺1 , A 22 2 ⫽A 2 ⫽⫺A 2 ⫽ 3 1 2 共 1 ⫹ 2 兲 2 A 33 3 ⫽ 3 3共 1⫹ 2 兲 , 2 A 23 3 ⫽⫺ 3 2 3 ,
2 A 13 3 ⫽⫺ 3 1 3 ,
共105兲
2 ⫺1 A 12 , 3 ⫽ 3 1 2 3共 1⫹ 2 兲
2 2 ⫺1 , A 11 3 ⫽ 3 1 3共 1⫹ 2 兲
2 2 ⫺1 A 22 , 3 ⫽ 3 2 3共 1⫹ 2 兲
while Eq. 共87兲 yields 11 A 111⫽ 13 共 1 ⫺3 21 ⫹2 31 兲 ⫺ 1 共 A 11 2 ⫹A 3 兲 , 22 22 A 222⫽ 13 共 2 ⫺3 22 ⫹2 32 兲 ⫹A 22 2 ⫺ 2 共 A 2 ⫹A 3 兲 ,
C. Multicoated spheres
12 A 112⫽ 13 共 2 21 2 ⫺ 1 2 兲 ⫺ 1 共 A 12 2 ⫹A 3 兲 ,
Another generalization of the Hashin–Shtrikman twophase coated-spheres assemblage is the N-phase multicoated spheres model.26 Here spheres of phase 1 are coated with spherical shells of phase 2, which in turn are coated with spherical shells of phase 3,... 关see Fig. 1共c兲兴. The relative
11 11 A 211⫽ 13 共 2 21 2 ⫺ 1 2 兲 ⫹A 11 2 ⫺ 2 共 A 2 ⫹A 3 兲 ,
共106兲
12 12 A 212⫽ 13 共 2 22 1 ⫺ 1 2 兲 ⫹A 12 2 ⫺ 2 共 A 2 ⫹A 3 兲 , 22 A 122⫽ 13 共 2 22 1 ⫺ 1 2 兲 ⫺ 1 共 A 22 2 ⫹A 3 兲 .
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
FIG. 6. Comparison of exact effective conductivity values to the HS bounds Eq. 共44兲 and the TPA2 Eq. 共46兲 for a three-phase doubly coated spheres 4 model: 共a兲 1 ⫽10 3 , 2 ⫽20 3 , 1 ⫽0.1→0.9, 2 ⫽ 5 (1⫺ 1 ), 3 1 ⫽ 5 (1⫺ 1 ), conventional coating order 1-2-3 共phase 1 in phase 2 in phase 1 3兲 and 共b兲 2 ⫽10 1 , 3 ⫽20 1 , 2 ⫽0.1→0.9, 1 ⫽ 5 (1⫺ 2 ), 3 4 ⫽ 5 (1⫺ 2 ), for coating orders: 1-3-2 共phase 1 embedded in phase 3, and then in phase 2兲, 2-1-3, and 3-1-2.
For numerical illustrations, we take 1 ⫽10 3 , 2 ⫽20 3 , 1 ⫽0.1→0.9, 2 ⫽ 54 (1⫺ 1 ), 3 ⫽ 51 (1⫺ 1 ), conventional coating order 1-2-3 共phase 1, then phase 2, then phase 3兲. The HS bounds Eq. 共44兲, TP bounds Eqs. 共81兲–
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共83兲, and TPA2 Eqs. 共46兲 and 共53兲 are plotted in Fig. 6共a兲. It is interesting to find that the TP bounds Eqs. 共81兲–共83兲 with relations 共105兲 and 共106兲 concide to yield the exact effective conductivity of the doubly coated spheres model. In Table I, we take 2 ⫽10 1 , 3 ⫽20 1 ( 1 ⬍ 2 ⬍ 3 ), 2 ⫽0.1→0.9, 1 ⫽ 51 (1⫺ 2 ), 3 ⫽ 54 (1⫺ 2 ), and collect the exact effective conductivity values of the doubly coated spheres model at different coating orders: 1-3-2, 1-2-3, 2-1-3, 2-3-1, 3-2-1, and 3-1-2. The HS upper 共HSU兲 and lower HS 共HSL兲 bounds are also given for comparison. Some of these results are plotted in Fig. 6共b兲 together with the respective TPA2 results 共46兲 and 共53兲. It is interesting to observe that the model with highest conductivity is 1-3-2 共but not 1-2-3兲, and the model with lowest conductivity is 3-1-2 共not 3-2-1兲, in which the matrix phases are not the ones with extremal conductivities. Note also that most of the exact and approximation values are close to the Hashin–Shtrikman upper or lower bounds, except those for the model 2-1-3, which falls between the bounds. The fact that TP bounds Eqs. 共81兲–共83兲 with relations 共105兲 and 共106兲 yield exact effective conductivities for doubly coated spheres that do not coincide with the TPA2 value indicates that our TPA2 Eqs. 共46兲 and 共53兲 is not as accurate for general N-phase composites as for two-phase ones. The TPA2 does not always yield the best possible approximation and it may even violate certain three-point bounds using the same available geometric information 共although it always falls within the two-point Hashin–Shtrikman bounds as confirmed兲. Therefore, in the general N-phase case, this approximation should be used in conjunction with bounds. One can use computer simulation to verify that TP bounds Eqs. 共81兲– 共83兲 with relations 共87兲 and 共104兲 converge to yield the exact effective conductivity of general N-phase multicoated spheres. We can also generalize the model further: consider a random mixture of multicoated spheres of different kinds, each of which has different coating order. The only restriction is that the volume proportions of the constituent materials in all of the compound spheres are the same. For such generalized models, the three-point microstructural parameters A ␣␥ can also be determined explicitly, however the TP bounds Eqs. 共81兲–共83兲 generally should not converge, as evidenced by the two-phase mixed-coated-spheres model considered.
TABLE I. Exact effective conductivities for doubly coated spheres models at various embedding orders, and 1 HSU and HSL bounds. Here we take 2 ⫽10 1 , 3 ⫽20 1 ( 1 ⬍ 2 ⬍ 3 ), 2 ⫽0.1→0.9, 1 ⫽ 5 (1⫺ 2 ), 4 3 ⫽ 5 (1⫺ 2 ).
2
HSU
1-3-2
1-2-3
2-1-3
2-3-1
3-2-1
3-1-2
HSL
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
14.377 13.853 13.339 12.835 12.340 11.855 11.378 10.911 10.451
14.348 13.804 13.277 12.765 12.270 11.789 11.322 10.869 10.428
14.248 13.673 13.146 12.649 12.174 11.715 11.270 10.837 10.414
13.616 12.601 11.809 11.197 10.735 10.399 10.170 10.034 9.981
8.014 8.318 8.614 8.897 9.163 9.404 9.615 9.789 9.920
7.988 8.266 8.538 8.802 9.053 9.288 9.504 9.697 9.864
7.919 8.136 8.356 8.580 8.807 9.038 9.273 9.511 9.754
7.895 8.092 8.296 8.510 8.732 8.963 9.205 9.458 9.723
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VII. CONCLUSIONS
D. C. Pham and S. Torquato Lord Rayleigh, Philos. Mag. 34, 481 共1892兲. A. Einstein, Ann. Phys. 共Leipzig兲 19, 289 共1906兲. 4 S. Torquato, Random Heterogeneous Materials 共Springer, New York, 2002兲. 5 D. A. G. Bruggeman, Ann. Phys. 共Leipzig兲 24, 636 共1935兲. 6 R. Landauer, J. Appl. Phys. 23, 779 共1952兲. 7 R. Landauer, in Electrical Transport and Optical Properties of Inhomogeneous Media, edited by J. C. Garland and D. B. Tanner 共AIP, New York, 1978兲. 8 Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 共1962兲. 9 M. J. Beran, Statistical Continuum Theories 共Wiley, New York, 1968兲. 10 N. Phan-Thien and G. W. Milton, Proc. R. Soc. London, Ser. A 380, 333 共1982兲. 11 S. Torquato, J. Chem. Phys. 84, 6345 共1986兲. 12 S. Torquato and M. D. Rintoul, Phys. Rev. Lett. 75, 4067 共1995兲; 76, 3241 共1996兲; R. Lipton and B. Vernescu, Proc. R. Soc. London, Ser. A 452, 329 共1996兲. 13 D. C. Pham, Arch. Ration. Mech. Anal. 127, 191 共1994兲. 14 M. J. Beran, Nuovo Cimento 38, 771 共1965兲. 15 J. J. McCoy, Q. Appl. Math. 36, 137 共1979兲. 16 W. F. Brown, J. Chem. Phys. 23, 1514 共1955兲. 17 S. Torquato, J. Appl. Phys. 58, 3790 共1985兲. 18 A. K. Sen and S. Torquato, Phys. Rev. B 39, 4504 共1989兲. 19 G. W. Milton, Commun. Math. Phys. 99, 463 共1985兲. 20 S. Torquato, Ph.D. Dissertation, State University of New York, Stony Brook, New York, 1980. 21 G. W. Milton, Phys. Rev. Lett. 46, 542 共1981兲. 22 G. W. Milton, in Physics and Chemistry of Porous Media, edited by D. L. Johnson and P. N. Sen 共AIP, New York, 1984兲. 23 D. C. Pham, Phys. Rev. E 56, 652 共1997兲. 24 D. C. Pham and N. Phan-Thien, Z. Angew. Math 48, 744 共1997兲. 25 D. C. Pham, Mech. Mater. 27, 249 共1998兲. 26 D. C. Pham, Acta Mech. 121, 177 共1997兲. 27 R. C. McPhedran and G. W. Milton, Appl. Phys. A: Solids Surf. 26, 207 共1981兲. 28 S. Torquato, G. Stell, and J. Beasley, Int. J. Eng. Sci. 23, 385 共1985兲. 29 S. Torquato and J. D. Beasley, Int. J. Eng. Sci. 24, 415 共1986兲. 30 S. Torquato and F. Lado, Proc. R. Soc. London, Ser. A 417, 59 共1988兲. 31 C. A. Miller and S. Torquato, J. Appl. Phys. 68, 5486 共1990兲. 32 D. R. McKenzie, R. C. McPhedran, and G. H. Derrick, Proc. R. Soc. London, Ser. A 362, 211 共1978兲. 33 W. T. Perrins, D. R. McKenzie, and R. C. McPhedran, Proc. R. Soc. London, Ser. A 369, 207 共1979兲. 34 I. C. Kim and S. Torquato, J. Appl. Phys. 68, 3892 共1990兲. 35 I. C. Kim and S. Torquato, J. Appl. Phys. 69, 2280 共1991兲. 36 H. W. Cheng and L. Greengard, J. Comput. Phys. 136, 629 共1997兲. 37 I. C. Kim and S. Torquato, J. Appl. Phys. 71, 2727 共1992兲. 2 3
In this article, exact strong-contrast expansions for the effective conductivity e of d-dimensional macroscopically isotropic composites consisting of N phases are presented. The series consists of a principal reference part and a fluctuation part, which contains multipoint correlation functions that characterize the microstructure of the composite. The fluctuation term may be estimated exactly or approximately in particular cases using available information about the given microstructure. We demonstrate that appropriate choices of the reference phase conductivity, such that this fluctuation term vanishes, results in simple expressions for e that agree with the well-known two-phase estimates. We propose a simple three-point approximation for the fluctuation part, which agrees well with a number of analytical and numerical results, including those for the EMA and HS coated-spheres microstructures, and various periodic and random dispersions of spheres and aligned cylinders. Even when the contrast between the phases is infinite, the approximation can yield accurate predictions, sometimes up to the percolation thresholds. In cases where clustering effects are significant, higher-order percolation information may be needed for the effective conductivity to be described accurately. We have also given the analytical expressions of the three-point correlation parameters for certain mixed-coated and multicoated spheres assemblages. It is shown that the effective conductivity of the multicoated spheres model can be determined explicitly from known three-point bounds and exact values of the respective three-point parameters. ACKNOWLEDGMENTS
The work was completed during D. C. P.’s visit to the Materials Institute, Princeton University as a Fulbright Senior Scholar. S. T. was supported by a MRSEC Grant at Princeton University, Grant No. NSF DMR-0213706 and by the Air Force Office of Scientific Research under Grant No. F49620-03-1-0406. 1
J. C. Maxwell, A Treatise on Electricity and Magnetism 共Clarendon, Oxford, 1892兲.
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VOLUME 94, NUMBER 10
15 NOVEMBER 2003
Strong-contrast expansions and approximations for the effective conductivity of isotropic multiphase composites D. C. Pham and S. Torquatoa) Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, New Jersey 08544
共Received 27 May 2003; accepted 27 August 2003兲 We extend the previous approach of one of the authors on exact strong-contrast expansions for the effective conductivity e of d-dimensional two-phase composites to case of macroscopically isotropic composites consisting of N phases. The series consists of a principal reference part and a fluctuation part 共a perturbation about a homogeneous reference or comparison material兲, which contains multipoint correlation functions that characterize the microstructure of the composite. The fluctuation term may be estimated exactly or approximately in particular cases. We show that appropriate choices of the reference phase conductivity, such that the fluctuation term vanishes, results in simple expressions for e that coincide with the well-known effective-medium and Maxwell approximations for two-phase composites. We propose a simple three-point approximation for the fluctuation part, which agrees well with a number of analytical and numerical results, even when the contrast between the phases is infinite near percolation thresholds. Analytical expressions for the relevant three-point microstructural parameters for certain mixed coated and multicoated spheres assemblages 共extensions of the Hashin–Shtrikman coated-spheres assemblage兲 are given. It is shown that the effective conductivity of the multicoated spheres model can be determined exactly. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1619573兴
I. INTRODUCTION
that render the integrals absolutely convergent in the infinitevolume limit. Thus, no renormalization analysis is required because the procedure used to solve the integral equation systematically leads to absolutely convergent integrals. Another useful feature of the expansions is that they can provide accurate estimates for all volume fractions when truncated at finite order, even when the phase conductivities differ significantly. In this paper, we extend the strong-contrast expansion approach of Torquato4 to the case of macroscopically isotropic composites consisting of N isotropic phases. The series expansions, which perturb about a homogeneous reference material, involve a principal reference term and a fluctuation term that perturbs about the reference medium. Based on a specified level of correlation information about the microstructure of the composite, we devise accurate approximations for the effective conductivity by choosing an appropriate equation for the conductivity of the reference 共comparison兲 material, a free parameter in the theory. The approximation is tested against available benchmark analytical and numerical results for various models of dispersions. We find that the approximation generally provides very good agreement with these benchmark results. In Sec. II, we derive strong-contrast expansion for the effective conductivity of macroscopically isotropic multicomponent composites. Approximation schemes based on the expansion are constructed in Sec. III, including a threepoint approximation, i.e., one that contains microstructural parameters that depends on three-point corelation functions. Section IV investigates the restrictions upon the three-point parameters. Section V collects well-known three-point bounds for comparison with our three-point approximation.
The prediction of the effective properties of composite materials has a rich history1–3 and is still an active area of research. In general, the effective properties of a composite depend on an infinite set of correlation functions that statistically characterize the medium.4 In the case of the effective conductivity e of composite materials, our concern in the present work, a number of different approximation schemes have been devised.1,5–7 Upper and lower bounds on the effective conductivity have been derived using variational principles.8 –13 For those composites in which the variations in the phase conductivities are small, formal solutions to the boundary-value problem have been developed in the form of weak contrast perturbation series.14,10 Due to the nature of the integral operator, one must contend with conditionally convergent integrals, which can be made convergent using an ad hoc normalization procedure.15 Alternatively, Brown16 constructed a strong-contrast expansion of the effective conductivity of three-dimensional two-phase isotropic media in powers of rational functions of the phase conductivities. The strong-contrast expansion approach has been further developed for the effective conductivity of d-dimensional macroscopically anisotropic composites consisting of two isotropic phases by introducing an integral equation for the cavity intensity field.4,17,18 The expansions are not formal but rather the nth-order tensor coefficients are given explicitly in terms of integrals over products of certain tensor fields and a determinant involving n-point statistical correlation functions a兲
Author to whom correspondence should be addressed; electronic mail: [email protected]
0021-8979/2003/94(10)/6591/12/$20.00
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© 2003 American Institute of Physics
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
Our three-point approximation is applied to a number of coated spheres and dispersion models and is compared with simulation results and bounds in Sec. VI.
phase having conductivity 0 , which is subjected to an applied electric field E0 (x) at infinity. Introducing the polarization field defined by P共 x兲 ⫽ 关 共 x兲 ⫺ 0 兴 E共 x兲
enables us to reexpress the flux J, defined by Ohm’s law 共4兲, as follows:
II. STRONG-CONTRAST EXPANSIONS
We derive strong-contrast expansions of the scalar effective conductivity of a macroscopically isotropic multiphase composite. The derivation is a direct and straightforward extension of the one given by Torquato4 for a macroscopically anisotropic two-phase composite. The reader is referred to this reference for greater details of the derivation. Consider a large macroscopically isotropic composite specimen in arbitrary space dimension d comprised of N isotropic phases having conductivities ␣ and volume fractions ␣ ( ␣ ⫽1,...,N). The microstructure is perfectly general but possesses a characteristic microscopic length scale that is much smaller than that of the specimen. Thus, the specimen is virtually statistically homogeneous. Ultimately, we shall take the infinite-volume limit and hence consider statistically homogeneous media. The local scalar conductivity at position x is expressible as
J共 x兲 ⫽ 0 E共 x兲 ⫹P共 x兲 .
兺
where 共␣兲
I
␣ I共 ␣ 兲 共 x兲 ,
␣ ⫽1
共 x兲 ⫽
再
1,
x in phase ␣ ,
0,
otherwise
共1兲
共2兲
is the indicator function for phase ␣ ( ␣ ⫽1,...,N). For statistically homogeneous media, the ensemble average of the indicator function is equal to the phase volume fraction ␣ , i.e.,
具 I共 ␣ 兲共 x兲 典 ⫽ ␣ ,
共3兲
where angular brackets denote an ensemble average. The local conductivity 共x兲 is the coefficient of proportionality in the linear constitutive relation J共 x兲 ⫽ 共 x兲 E共 x兲 ,
共4兲
where J共x兲 denotes the local electric 共thermal兲 current or flux at position x, and E共x兲 denotes the local field intensity. Under steady-state conditions with no source terms, conservation of energy requires that J共x兲 be solenoidal “"J共 x兲 ⫽0,
共5兲
while the intensity field E共x兲 is taken to be irrotational “ÃE共 x兲 ⫽0,
共6兲
which implies the existence of a potential field T(x), i.e., E共 x兲 ⫽⫺“T 共 x兲 .
共7兲
Thus E共x兲 and T(x) represent the electric field 共negative of temperature gradient兲 and electric potential 共temperature兲 in the electrical 共thermal兲 problem, respectively. Now, following Torquato,4 let us embed this d-dimensional composite specimen in an infinite reference
共9兲
The vector P共x兲 is the induced flux polarization field relative to the reference medium. Using the infinite-space Green’s function of the Laplace equation for the reference medium corresponding to the problem, Eqs. 共4兲–共7兲, we find that the electric field satisfies the integral relation4 E共 x兲 ⫽E0 共 x兲 ⫹
冕
dx⬘ G共 0 兲 共 r兲 •P共 x⬘ 兲 ,
共10兲
where G共 0 兲 共 r兲 ⫽⫺D共 0 兲 ␦ 共 r兲 ⫹H共 0 兲 共 r兲 , D共 0 兲 ⫽
N
共 x兲 ⫽
共8兲
1 I, d0
H共 0 兲 共 r兲 ⫽
1 dnn⫺I , ⍀0 rd
共11兲 共12兲
r⫽x⫺x⬘, n⫽r/兩r兩, ␦共r兲 is the delta Dirac function, I is the second order identity tensor, ⍀(d) is the total solid angle contained in a d-dimensional sphere given by ⍀共 d 兲⫽
2 d/2 , ⌫ 共 d/2兲
共13兲
and ⌫(x) is the gamma function. In particular, ⍀共2兲⫽2, ⍀共3兲⫽4. In relation 共11兲, the constant second order tensor D(0) arises because of the exclusion of the spherical cavity, and it is understood that integrals involving the second order tensor H(0) are to be carried out by excluding at x⬘⫽x an infinitesimal sphere in the limit that the sphere radius shrinks to zero. Moreover, the integral of H(0) (r) over the surface of a sphere of radius R⬎0 is identically zero, i.e.,
冕
r⫽R
H共 0 兲 共 r兲 d⍀⫽0.
共14兲
Substitution of Eq. 共11兲 into expression 共10兲 yields an integral equation for the cavity intensity field F共x兲 F共 x兲 ⫽E0 共 x兲 ⫹
冕
⑀
dx⬘ H共 0 兲 共 x⫺x⬘ 兲 •P共 x⬘ 兲 ,
共15兲
where we define
冕
⑀
dx⬘ f 共 x,x⬘ 兲 ⫽ lim
⑀ →0
冕
兩 x⫺x⬘ 兩 ⬎ ⑀
dx⬘ f 共 x,x⬘ 兲 .
共16兲
The cavity intensity field F共x兲 is related to E共x兲 through the expression F共 x兲 ⫽ 兵 I⫹D共 0 兲 关 共 x兲 ⫺ 0 兴 其 •E共 x兲 .
共17兲
Combination of the expressions 共8兲 and 共17兲 gives a relation between the polarization and cavity intensity fields P共 x兲 ⫽L 共 x兲 F共 x兲 ,
共18兲
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
Multiplying this relation by the scalar L(x) defined by Eq. 共19兲, yields
where N
L 共 x兲 ⫽ 0 d
6593
共 x兲 ⫺ 0 ⫽0 d b ␣ 0 I共 ␣ 兲 共 x兲 , 共 x兲 ⫹ 共 d⫺1 兲 0 ␣ ⫽1
兺
P⫽LE0 ⫹LH共 0 兲 P.
共25兲
共19兲
A solution for the polarization P in term of the applied field E0 can be obtained by successive substitutions using Eq. 共25兲, with the result
and we have used Eqs. 共1兲 and 共2兲. The effective conductivity e for the macroscopically isotropic composite can be defined through an expression relating the average polarization to the average Lorentz field, i.e.,
P⫽LE0 ⫹LH共 0 兲 LE0 ⫹LH共 0 兲 LH共 0 兲 LE0 ⫹...⫽SE0 , 共26兲
␣⫺ 0 , b ␣0⫽ ␣ ⫹ 共 d⫺1 兲 0
具 P共 x兲 典 ⫽Le • 具 F共 x兲 典 ,
共20兲
where Le ⫽L e I,
L e⫽ 0d
e⫺ 0 . e ⫹ 共 d⫺1 兲 0
共21兲
The constitutive relation 共20兲 is localized, i.e., it is independent of the shape of the composite specimen in the infinitevolume limit. This relation is completely equivalent to the averaged Ohm’s law that defines the effective conductivity
具 J共 x兲 典 ⫽ e 具 E共 x兲 典 .
共22兲
We want to obtain an expression for the effective conductivity e from relation 共22兲 using the solution of the integral Eq. 共15兲, which is recast as F共 1 兲 ⫽E0 共 1 兲 ⫹
冕
共0兲
d2H 共 1,2兲 •P共 2 兲 ,
where the second-order tensor operator S is given by S⫽L 共 I⫺LH共 0 兲 兲 ⫺1 .
共27兲
Ensemble averaging Eq. 共26兲 gives
具 P典 ⫽ 具 S典 E0 .
共28兲
The operator 具S典 involves products of the tensor H(0) , which decays to zero like r ⫺d for large r, and hence 具S典 at best involves conditionally convergent integrals. In other words, 具S典 is dependent upon the shape of the composite specimen. In order to obtain a local 共shape-independent兲 relation between average polarization 具P典 and average Lorentz field 具F典 as prescribed by Eq. 共22兲, we must eliminate the applied field E0 in favor of the appropriate average field. Inverting Eq. 共28兲 yields E0 ⫽ 具 S典 ⫺1 具 P典 . Averaging Eq. 共24兲 and eliminating the applied field E0 yields
具 F典 ⫽Q具 P典 ,
共29兲
where Q⫽ 具 S典 ⫺1 ⫹H共 0 兲 .
共30兲
共23兲
Comparing expressions 共20兲 and 共29兲 yields the desired result for the effective tensor Le :
where we have adopted the shorthand notation of representing x and x⬘ by 1 and 2, respectively. In schematic operator form, this integral equation can be tersely rewritten as
⫺1 共0兲 ⫽H共 0 兲 ⫹ 具 L 共 I⫺LH共 0 兲 兲 ⫺1 典 ⫺1 . L⫺1 e ⫽Q⫽H ⫹ 具 S典 共31兲 The first few terms of the expansion Eq. 共31兲 are explicitly given by
⑀
F⫽E0 ⫹H共 0 兲 P.
L⫺1 e 共 1 兲⫽
冕
共24兲
d2Q共 1,2兲 ⫽
I共 1 兲 ⫺ 具L共 1 兲典 ⫺
冕冕
冕 冉具 d2
d2d3
冉
冊
L 共 1 兲 L 共 2 兲 典 ⫺ 具 L 共 1 兲 典具 L 共 2 兲 典 共 0 兲 H 共 1,2兲 具 L 共 1 兲 典具 L 共 2 兲 典
The general term contains the n-point correlation functions
具 L(1)...L(n) 典 . The explicit expression for any term in the series can be given as in Ref. 4. Let us introduce the property-independent dipole tensor
t共 r兲 ⫽ 0 H共 0 兲 共 r兲 ⫽
dnn⫺I ⍀r d
.
冊
具 L 共 1 兲 L 共 2 兲 L 共 3 兲 典 具 L 共 1 兲 L 共 2 兲 典具 L 共 2 兲 L 共 3 兲 典 共 0 兲 ⫺ H 共 1,2兲 •H共 0 兲 共 2,3兲 ⫺... 具 L 共 1 兲 典具 L 共 2 兲 典 具 L 共 1 兲 典具 L 共 2 兲 典具 L 共 3 兲 典
共33兲
Substituting Eqs. 共19兲 and 共33兲 into Eq. 共32兲 and taking into account that t is traceless, we take the trace of both sides of Eq. 共32兲. We then multiply them with 0 ( 兺 ␣ ␣ b ␣ 0 ) 2 to obtain the relation for our macroscopically isotropic media
e ⫹ 共 d⫺1 兲 0 e⫺ 0
冉兺 ␣
冊
共32兲
2
␣b ␣0 ⫺ 兺 ␣b ␣0 ␣
⫽A共 0 , 1 ,..., N ,microstructure兲 .
共34兲
Here A( 0 , 1 ,..., N ,microstructure) is a scalar quantity that depends on the reference conductivity 0 , phase conductivities 1 ,..., N , and the microstructure via n-point correlation functions. The left side of Eq. 共34兲 is referred to as the principal reference term of the strong-contrast expansion, while the right side, given by A is the fluctuation term relative to the reference medium of conductivity 0 . Through the lowest three-point term in the series, A is explicitly given by
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6594
J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
A共 0 , 1 ,..., N ,microstructure兲 ⫽⫺d ⫺
冋兺
␣,,␥
兺 ␣,,␥,␦
D. C. Pham and S. Torquato
b ␣0b 0b ␥0
冕冕 冕冕
b ␣0b 0b ␥0b ␦0 兺 b 0
d2d3 具 I共 ␣ 兲 共 1 兲 t i j 共 1,2兲 I共  兲 共 2 兲 t i j 共 2,3兲 I共 ␥ 兲 共 3 兲 典 d2d3 具 I共 ␣ 兲 共 1 兲 t i j 共 1,2兲 I共  兲 共 2 兲 典具 I共 ␥ 兲 共 2 兲 t i j 共 2,3兲 I共 ␦ 兲 共 3 兲 典
⫺... .
共35兲
Here conventional summation on repeating Latin indices 共from 1 to d兲 is assumed, and the Greek indices under the sum run from 1 to N.
A. Connections to previous estimates
The effective medium approximation 共EMA兲 EMA 共also known as the self-consistent scheme兲5,7 is the solution of the equation
⫺
␣ EMA ␣ ⫽0. 兺 ⫹ d⫺1 兲 EMA 共 ␣ ⫽1 ␣
共36兲
Equivalently, we can rewrite this via the property function P
e ⫽ EMA⫽ P 共共 d⫺1 兲 EMA兲 , where
冋兺 N
P 共 兲 ⫽
␣ ␣⫹
册
共37兲
⫺1
⫺ . 共38兲 * * This approximation is exact for a certain hierarchical composite consisting of spherical grains.19 As 0 → EMA 共while e ⫽ EMA), the left-hand side of Eq. 共34兲 approaches 0, and hence for the EMA microstructure we have
*
␣ ⫽1
A共 e , 1 ,..., N ,EMA microstructure兲 ⫽0.
共39兲
For two-phase microstructures that are optimal when the volume fraction is specified, such as the Hashin–Shtrikman coated-spheres assemblage,8 we have
e ⫽ P 关共 d⫺1 兲 M 兴 ,
referred to as TPA1兲 for the effective conductivity of twophase composites that can be regarded to perturb about the Hashin–Shtrikman structures4 2 e 1⫹ 共 d⫺1 兲 2 b 21⫺ 共 d⫺1 兲 1 2 b 21 ⫽ 2 1 1⫺ 2 b 21⫺ 共 d⫺1 兲 1 2 b 21
III. APPROXIMATION SCHEMES
N
册
共40兲
where M is the property of the connected matrix phase in the optimal geometry. Thus Eq. 共34兲 also yields A共 M , 1 , 2 ,optimal two-phase microstructure兲 ⫽0. 共41兲 Formula 共40兲 is sometimes called the Maxwell approximation 共MA兲 共also known as the Maxwell–Garnett or Clausius–Mossotti approximation兲.
The fluctuation term A depends on the microstructure of the composite and generally can only be evaluated by some numerical scheme. Torquato17 used the strong-contrast expansion to develop a three-point approximation 共henceforth
共42兲
where ␣ is a three-point microstructural parameter defined by Eqs. 共48兲 and 共49兲. Here the reference phase is taken to be phase 1. We will now derive a different three-point approximation for multiphase composites based on the results Eqs. 共36兲–共41兲. Let us assume that we have an explicit approximation Aapprox for the fluctuation term A for a specific composite. Guided by the discussion of Sec. III A, our strategy is to choose 0 to be the solution of the equation Aapprox⫽0, and from Eq. 共34兲 deduce the respective approximation e ⫽ P 关 (d⫺1) 0 兴 . Let us assume that Aapprox⫽An , where An is the series A truncated after the n-point correlation term. From Eq. 共34兲, we deduce the respective n-point approximation for the effective conductivity
e ⫽ P 共共 d⫺1 兲 0 兲 ,
An 共 0 兲 ⫽0.
共43兲
Clearly, at n→⬁, we should get the exact value of the effective conductivity for an arbitrary microstructure. For any microstructure, we expect that 0 should lie within the interval 关 min ,max兴 for e in order to satisfy the Hashin– Shtrikman bounds,8 which can be expressed as P 共共 d⫺1 兲 max兲 ⭓ e ⭓ P 共共 d⫺1 兲 min兲 ,
共44兲
where
max⫽max兵 1 ,..., N 其 ,
min⫽min兵 1 ,..., N 其 . 共45兲
For example, let us take Aapprox⫽A3 . Since three-point microstructural information is now available for a variety of different microstructures, we are especially interested in the three-point approximation 共TPA2兲:
e ⫽ P 共共 d⫺1 兲 0 兲
共 TPA2 兲 ,
共46兲
where 0 is the solution of equation A3 共 0 兲 ⫽0.
B. Multipoint approximations
共 TPA1 兲 ,
共47兲
We call this TPA2 to distinguish it from TPA1 given by Eq. 共42兲. Both three-point approximations are exact through third order in the difference in the phase conductivities. In the special two-phase case, three-point correlation function information arises via the microstructural parameters ␣ ( ␣ ⫽1,2), 4,20,21 which for d⫽2
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
4 ␣⫽ 1 2
冕 冕 冕 ⬁
⬁
dr r
0
0
S 共2␣ 兲 共 r 兲 S 共2␣ 兲 共 s 兲
⫺
ds s
S 共1␣ 兲
册
0
d cos共 2 兲
D. C. Pham and S. Torquato
冋
S 共3␣ 兲 共 r,s,t 兲
C. Alternative approaches
We note that Le of Eq. 共31兲 can also be expanded as 共 0 兲 ⫺1 ⫽ 具 S典 ⫽ 具 L 共 I⫺LH共 0 兲 兲 ⫺1 典 共 L⫺1 e ⫺H 兲
共48兲
,
⬁
⫽ 具 L 典 I⫹
and for d⫽3 9 ␣⫽ 2 1 2
冕 冕 冕 ⬁
0
dr r
⫻ P 2 共 cos 兲
冋
⬁
0
ds s
1
⫺1
S 共2␣ 兲 共 r 兲 S 共2␣ 兲 共 s 兲 S 共1␣ 兲
册
,
共49兲
where P 2 is the Legendre polynomial of order 2 and is the angle opposite the side of the triangle of length t, and S 共n␣ 兲 共 x1 ,¯,xn 兲 ⫽ 具 I共 ␣ 兲 共 x1 兲 ¯I共 ␣ 兲 共 xn 兲 典 .
共50兲
It is also known4,20,21 that 1 ⭓0, 2 ⭓0, and 1 ⫹ 2 ⫽1. Through the microstructural parameters 1 and 2 , relation 共47兲 can be given simply as
1
1⫺ 0 2⫺ 0 ⫹2 ⫽0. 1 ⫹ 共 d⫺1 兲 0 2 ⫹ 共 d⫺1 兲 0
冕冕 V
V
I共 ␣ 兲 共 x兲 t i j 共 x,y兲 I共  兲 共 y兲 dx dy⫽0.
兺
b ␣ 0 b  0 b ␥ 0 A ␣␥ ⫽0,
共52兲
共53兲
冕冕冕 V
V
V
d1d2d3I共 ␣ 兲 共 1 兲 t i j 共 1,2兲 I共  兲 共 2 兲
⫻t i j 共 2,3兲 I共 ␥ 兲 共 3 兲 .
共57兲
共58兲
where 0 is the solution of the equation ˆ 共 , ,..., ,microstructure兲 ⫽0, A 0 1 N ⬁
ˆ⫽ A
兺
dn
n⫽2
兺
␣ 1 ,..., ␣ n
b ␣ 1 0 b ␣ 2 0 ...b ␣ n 0
共59兲
冕 冕 ¯
d2...dn
⫻ 具 I共 ␣ 1 兲 共 1 兲 t i j 共 1,2兲 I共 ␣ 2 兲 共 2 兲 ...t li 共 n⫺1,n 兲 I共 ␣ n 兲 共 n 兲 典 . 共60兲 At the three-point approximation level, Eqs. 共58兲–共60兲 leads to the same result Eqs. 共46兲 and 共53兲 of the previous subsection. Moreover we see that Eq. 共31兲 can also be given as Le 共 I⫺Le H共 0 兲 兲 ⫺1 ⫽ 具 L 共 I⫺LH共 0 兲 兲 ⫺1 典 ,
冋
⬁
兺
k⫽1
册
共61兲
⬁
具 共 L e H共 0 兲 兲 k 典 ⫽ 具 L 典 I⫹ 兺 具 L 共 LH共 0 兲 兲 k 典 . 共62兲 k⫽1
Taking nth approximation of Eq. 共62兲 gives
冋
Le I⫹
where A ␣␥ ⫽
e ⫽ P 共共 d⫺1 兲 0 兲 ,
Le I⫹
Then Eq. 共47兲 can be written as ␣,,␥
共56兲
In the same manner as that of the previous section, we obtain the expression
which can be expanded as
I共 ␣ 兲 共 x兲 t i j 共 x,y兲 I共  兲 共 y兲 典 dy
⫽
具 L 共 LH共 0 兲 兲 n 典 ,
⫺1 共 0 兲 ⫺1 共0兲 ⫽ 具 L 典 I→L⫺1 I. 共 L⫺1 e ⫺H 兲 e ⫽H ⫹ 具 L 典
共51兲
We can further simplify Eq. 共47兲 for N⭓2. According to the ergodic hypothesis, we could substitute the ensemble averages in Eq. 共35兲 by volume averages over the volume of the macroscopic sample V, which we can take for convenience to be spherical. Thus, for a statistically homogeneous and isotropic medium of spherical volume V, we have
冕具
兺
n⫽1
where we have used an obvious short-hand notation in the second line of Eq. 共56兲. If we take 0 such that the last sum in Eq. 共56兲 vanishes, then we deduce
d 共 cos 兲
S 共3␣ 兲 共 r,s,t 兲 ⫺
6595
n⫺1
兺
k⫽1
册
n⫺1
具 共 L e H共 0 兲 兲 k 典 ⫽ 具 L 典 I⫹
兺
k⫽1
具 L 共 LH共 0 兲 兲 k 典 . 共63兲
If we take 0 as to make the last sum of the right-hand of Eq. 共63兲 vanish 关c.f. Eqs. 共59兲 and 共60兲兴, then at the three-point approximation, Eq. 共63兲 leads to the same result Eqs. 共46兲 and 共53兲.
共54兲
IV. THREE-POINT MICROSTRUCTURAL PARAMETERS
Note the symmetry of A ␣␥ in the indices ␣ and ␥, but not in . The sum on the left side of Eq. 共53兲 is of quadratic form in the variables b ␣ 0 , b ␥ 0 ( ␣ , ␥ ⫽1,...,N), the sign of which depends on those of b  0 (  ⫽1,...,N). Hence, at 0 ⫽ min , the sum is positive, while at 0 ⫽ max , it is negative, with the solution 0 of Eq. 共53兲 lying between these extreme values. Consequently, the TPA2 from Eqs. 共46兲 and 共53兲 should always fall inside the Hashin–Shtrikman bounds Eq. 共44兲. For N⫽2, the microstructural parameters ␣ is related to A ␣␣␣ via
As in the two-phase case, there exist some relations between the microstructural parameters A ␣␥ . From Eq. 共54兲, we see that they are symmetric in the indices ␣ and ␥. Consider a spherical representative element V of the composite with phase ␣ occupying regions V ␣ ( ␣ ⫽1,...,N). For simplicity and without loss of generality we take the volume of V to be unity, and hence the volume of V ␣ is equal to the volume fraction ␣ . Let us introduce the harmonic potentials
A ␣␣␣ ⫽
d⫺1 1 2 ␣ , d
␣ ⫽1,2.
共55兲
共 x兲 ⫽
冕
V
G 共 x⫺y兲 dy,
“ 2 共 x兲 ⫽1,
x苸V;
共64兲
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6596
J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
␣ 共 x兲 ⫽
冕
␣
“ 2 ␣ 共 x兲 ⫽ ␦ ␣ ,
G 共 x⫺y兲 dy,
D. C. Pham and S. Torquato
x苸  ; 共65兲
where G(r) (r⫽ 兩 x⫺y兩 ) is the respective Green function
G共 r 兲⫽
冦
⫺
冉冊
1 1 ln , 2 r
d⫽2 ,
1 1 ⫺ , 共 d⫺2 兲 ⍀ r d⫺2
ⵜ 2 G⫽ ␦ 共 r 兲 .
d⭓3 共66兲
We have G ,i j ⫽ ti j⫽
␦ij
dn i n j ⫺ ␦ i j ⍀r
d
共67兲
x i ⫺y i n i⫽ , r
,
where the Latin indices after a comma designate differentiation with respective to position. If we denote 1 ␣i j 共 x兲 ⫽ ␣,i j ⫺ ␦ i j ␦ ␣ , d
x苸  ,
共68兲
d⫺1 1 2 . d
共75兲
Generally, a N-phase material with microstructural parameters A ␣␥ ( ␣ , ␥ ,  ⫽1,...,N) can also be considered to be a two-phase material occupying regions ¯V I⫽艛 ␣ 苸⌫ IV ␣ , ¯V II ⫽艛 ␣ 苸⌫ IIV ␣ with respective microstructural parameters ¯A ␣␥ ( ␣ , ␥ ,  ⫽I,II). Here ⌫ I⫽ 兵 ␣ 1 ,..., ␣ k 其 , ⌫ II  ⫽ 兵 ␣ k⫹1 ,..., ␣ N 其 , and 兵 ␣ 1 ,..., ␣ k , ␣ k⫹1 ,..., ␣ N 其 is any permutation of 兵 1,...,N 其 , and k is any number between 1 and N⫺1. By combining the relations Eqs. 共65兲, 共68兲, 共69兲, and 共75兲, one can verify that d⫺1 d
␦ 共 x⫺y兲 ⫺t i j ,
d
22 A 11 1 ⫹A 2 ⫽
冉 兺 冊冉 兺 冊 ␣ 苸⌫ I
⫽ ⫽
␣
␣ 苸⌫ II
␣
d⫺1 ¯ ¯ ¯ I I⫹A ¯ II I II⫽A I II d
兺
A ␣␥ ⫹
 , ␥ , ␣ 苸⌫ I
兺
II
 , ␥ , ␣ 苸⌫ II
A ␣␥ ,
共76兲
¯ and ¯ are the volume fractions of regions I and II, where I II respectively.
then we have A ␣␥ ⫽
冕
V
␣i j ␥i j dx⫽ 共兲
⫻t i j 共 x,y兲 I
冕冕冕 V
V
V
V. THREE-POINT BOUNDS
dx dy dzI共 ␣ 兲 共 x兲 共␥兲
共 y兲 t i j 共 y,z兲 I
共69兲
共 z兲 .
Note that
,i j ⫽
␦ij d
N
⫽
兺 ␣,i j , ␣ ⫽1
共70兲
V␥
,i j ,i j dx⫽
冕
1  ␦ ␥ ,ii dx⫽ , d ␥ V␥ d
共71兲
V␥
冕 兺 冕 冋 兺
␣ ⫽1
V␥
␣ ,i j
 ,i j dx
N
⫽
␣ ⫽1 N
⫽
V␥
兺 A ␥␣ ⫹ ␣ ⫽1
␣ ij
 i j⫹
␦ ␥ d
⫺
册
1 ␦ ␦ dx d ␣␥ ␥
A ␥␣ ⫽0,
␥ .
᭙  , ␥ ⫽1,...,N.
共72兲
共73兲
22 12 A 11 2 ⫽A 2 ⫽⫺A 2 .
Keeping in mind Eq. 共55兲 and 1 ⫹ 2 ⫽1, we have
1 2共 1⫺ 2 兲 2 1 2 ⫹ 2 1 ⫹ 共 d⫺1 兲共 1 / 1 ⫹ 2 / 2 兲 ⫺1
.
Milton obtained a sharper lower bound for the case d⫽3 and 2 ⭓ 1
共 3L 兲
/ 1⫽
2 1⫹ 共 1⫹2 2 兲 b 21⫺2 共 1 2 ⫺ 2 兲 b 21 2 1⫹ 1 b 21⫺ 共 2 1 2 ⫹ 2 兲 b 21
2⫺ 1 . b 21⫽ 2 ⫹2 1
, 共80兲
For three-dimensional N-phase composites, Phan-Thien and Milton10 derive the following bounds:
In the two-component case Eq. 共73兲 yields 22 12 A 11 1 ⫽A 1 ⫽⫺A 1 ,
共78兲
共79兲
N
兺
1 2共 1⫺ 2 兲 2 , 1 2 ⫹ 2 1 ⫹ 共 d⫺1 兲共 1 1 ⫹ 2 2 兲
22
Comparing Eqs. 共71兲 and 共72兲, we obtain relations between the microstructural parameters
␣ ⫽1
⫺
共 3L 兲 ⫽ 1 1 ⫹ 2 2
N
,i j ,i j dx⫽
共77兲
共 3U 兲 ⫽ 1 1 ⫹ 2 2
and
冕
共 3U 兲 ⭓ e ⭓ 共 3L 兲 , where
and hence we find that
冕
Here we summarize previous three-point bounds that we will subsequently apply. The d-dimensional three-point Beran bounds14 for two-phase composites derived by Torquato4 are given by
共74兲
共 3U 兲 ⭓ e ⭓ 共 3L 兲 ,
共81兲
where
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
共 3U 兲 ⫽ 具 典 ⫺ ␦ •⌫• 共 ⌫⫹ ␦ •A/ 具 典 兲 ⫺1 •⌫• ␦ /3具 典 , 共82兲
⌬ ␣␥ ⫽
再
共 1⫺2  兲 ⌫ ␥ ,
␣⫽
⫺ ␣ ⌫ ␥ ⫺  ⌫ ␣␥ ,
␣⫽
6597
.
1/ 共 3L 兲 ⫽ 具 ⫺1 典 ⫺2 ␦ ⫺1 •⌫• 共 ⌫⫹ ␦ ⫺1 • 共 ⌬ ⫹A兲 /2具 ⫺1 典 兲 ⫺1 •⌫• ␦ ⫺1 /3具 ⫺1 典 ,
共83兲
Here 具 f ( ) 典 for any function f of is given by
with (N⫺1)-rank vectors and matrices
⌫⫽ 兵 ⌫ ␣ 其 ␣N⫺1 ,  ⫽1 ,
冕冕 冕冕冕 1
16
2
V
V
⫽
V
具 f 共 兲典⫽ 兺 ␣ f 共 ␣ 兲.
A⫽ 兵 A ␣␥ 其 ␣N⫺1 ,  , ␥ ⫽1 ,
⌬⫽ 兵 ⌬ ␣␥ 其 ␣N⫺1 ,  , ␥ ⫽1 ,
A ␣␥ ⫽
N
␦ ⫺1 ⫽ 兵 ␣⫺1 ⫺ N⫺1 其 ␣N⫺1 ⫽1 ,
␦ ⫽ 兵 ␣ ⫺ N 其 ␣N⫺1 ⫽1 ,
⌫ ␣ ⫽
再
共84兲
␣ 共 1⫺ ␣ 兲 ,
␣⫽
⫺ ␣  ,
␣⫽
; 共85兲
冉 冊 1
兺 A ␦␥ ⫹ 3 共 ␣ ␦ ␣ ␦ ␣␥ ⫺ ␣  ␦ ␥ ⫺  ␥ ␦ ␣␥ ⫺ ␣ ␥ ␦ ␣ ⫹2 ␣  ␥ 兲 . ␦ ⫽1
Three-point bounds for d-dimensional N-phase composites derived by Pham13,23 are given by P 共共 d⫺1 兲 共03U 兲 兲 ⭓ e ⭓ P 共共 d⫺1 兲 共03L 兲 兲 , and
(3L) 0
共88兲
are the solutions of the equations
Q 共3U 兲 共共 d⫺1 兲 共03U 兲 兲 ⫽0,
共89兲
Q 共3L 兲 共共 d⫺1 兲 共03L 兲 兲 ⫽0,
兺
共 ␣ ⫺ 0 兲 A ␣␥ X  X ␥ ,
共90兲
兺
␥ 共 ␣⫺1 ⫺ ⫺1 0 兲A ␣ X X ␥ ,
共91兲
X  ⫽ 具 关 ⫹ 共 d⫺1 兲 0 兴 ⫺1 典 ⫺ 关  ⫹ 共 d⫺1 兲 0 兴 ⫺1 .
共92兲
Q 共3U 兲 共共 d⫺1 兲 0 兲 ⫽ Q 共3L 兲 共共 d⫺1 兲 0 兲 ⫽
␣,,␥
␣,,␥
In the two-phase case, relations 共89兲 are solved explicitly as
共03U 兲 ⫽
The PhanThien–Milton microstructural parameters A ␣␥ can be related to our A ␣␥ via the expression
dr ds 2 2 具 ⍀ ␣⬘ 共 0兲 ⍀ ⬘ 共 r兲 ⍀ ␥⬘ 共 s兲 典 rs r s
N
where
共86兲
dx dy dz共 I共  兲 共 x兲 ⫺  兲 G ,i j 共 x,y兲共 I共 ␣ 兲 共 y兲 ⫺ ␣ 兲 G ,i j 共 y,z兲共 I共 ␥ 兲 共 z兲 ⫺ ␥ 兲
⫽A ␣␥ ⫺ ␣
(3U) 0
␣ ⫽1
22 1 A 11 1 ⫹ 2A 2 22 A 11 1 ⫹A 2
⫽ 1 1⫹ 2 2 ,
22 A 11 1 ⫹A 2 共03L 兲 ⫽ 11 ⫽ 共 1 / 1 ⫹ 2 / 2 兲 ⫺1 . A 1 / 1 ⫹A 22 / 2 2
共93兲
共87兲
proximation P ( 0 ) from Eqs. 共46兲 and 共51兲 should fall inside the three-point bounds Eqs. 共88兲, 共93兲, and 共94兲 for our real composite. In the two-phase case, the bounds Eqs. 共88兲, 共93兲, and 共94兲 as well as the bounds Eqs. 共81兲–共83兲 coincide with the bounds Eqs. 共77兲–共79兲. The bounds Eqs. 共81兲–共83兲, and 共88兲 and 共89兲 have been compared in Ref. 24 for the class of N-phase (N⭓3) quasisymmetric 共symmetric cell兲 materials using a symbolic algebra program and numerical simulation. It appears that the bounds yield the same results for N-phase spherical cell materials. For N-phase platelet cell materials, the bounds Eqs. 共81兲–共83兲 appear tighter. However, the bounds Eqs. 共88兲 and 共89兲 are simpler in functional form as well as computational aspects. In the case of N-phase spherical cell composites, Eq. ⫽ 具 典 , (3L) 共89兲 are also solved explicitly and yield (3U) 0 0 ⫺1 ⫺1 ⫽ 具 典 . Unsuccessful attempts have been made to transform Eqs. 共81兲–共83兲, which involve multiplications and inversions of (N⫺1)-rank matrices and vectors, into some simple form similar to that of Eqs. 共88兲 and 共89兲. VI. APPLICATIONS OF THE THREE-POINT APPROXIMATION
共94兲
If we imagine a fictitious composite with 1 and 2 beand (3L) ing the volume fractions of the phases, then (3U) 0 0 in Eqs. 共93兲 and 共94兲 are, respectively, the ‘‘arithmetic’’ and ‘‘harmonic’’ averages. Moreover, the solution 0 of Eq. 共51兲 is the ‘‘effective medium approximation’’ value, hence (3U) ⭓ 0 ⭓ (3L) , and consequently the three-point ap0 0
In this section, we apply our three-point approximation 共TPA2兲 to certain multicoated spheres assemblages and dispersions of identical spheres. In each of these instances, the relevant three-point microstructural parameters are known. A. Analytical two-phase models
There are only very few nontrivial models, in which the three-point microstructural parameters A ␣␥ have been deter-
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D. C. Pham and S. Torquato
mined analytically. Here we discuss three such instances. For the two-phase EMA microstructure,19,22 we have
1⫽ 1 ,
2⫽ 2 .
共95兲
With Eq. 共95兲, we see that relation 共51兲 for 0 has the identical form as Eq. 共36兲 for EMA in the two-phase case. Hence our TPA2 Eqs. 共46兲 and 共51兲 coincides with the exact value Eq. 共37兲 for two-phase EMA microstructure, i.e.,
TPA⫽ P 共共 d⫺1 兲 0 兲 ⫽ EMA⫽ P 共共 d⫺1 兲 EMA兲 . 共96兲 Thus, in the two-phase case, our TPA2 Eqs. 共46兲 and 共51兲 may be interpreted as a three-point generalization of EMA, while the classical EMA expression Eq. 共37兲 is the corresponding two-point version. Next, consider the Hashin–Shtrikman two-phase coated spheres model, which consists of composite spheres that are composed of a spherical core, of conductivity 2 , and radius a, surrounded by a concentric shell of conductivity 1 and outer radius b. The ratio (a/b) 3 is fixed and equal to the inclusion volume fraction 2 . The composite spheres fill all space, implying that there is a distribution in their sizes ranging to the infinitely small 关see Fig. 1共a兲兴. For this coated spheres model, A ␣␥ have been determined explicitly,25 which for general d-dimensional composites can be given as 22 12 A 11 1 ⫽A 1 ⫽⫺A 1 ⫽
d⫺1 1 2 , d
22 12 A 11 2 ⫽A 2 ⫽⫺A 2 ⫽0.
共97兲
Hence, from Eq. 共55兲 one finds
1 ⫽1,
2 ⫽0.
共98兲
Consequently, from Eqs. 共93兲 and 共94兲, one obtains (3U) 0 ⫽ (3L) ⫽ 1 , and the bounds Eq. 共88兲 coincide to yield the 0 exact effective conductivity
e ⫽ P 共共 d⫺1 兲 1 兲 .
共99兲
This relation coincides with the Maxwell approximation and Hashin–Shtrikman upper bound 共when 2 ⬍ 1 ) or lower bound 共when 2 ⬎ 1 ). The result Eq. 共99兲 was also obtained by direct solution of the respective conductivity problem,8 which in turn leads to Eq. 共98兲. Relation 共99兲 is also realizable by certain laminates4 and thus are also optimal. With Eq. 共98兲, the solution of Eq. 共51兲 should be 0 ⫽ 1 , and from Eq. 共46兲 we get the same formula Eq. 共99兲. Thus, for two-phase coated spheres 共as well as other optimal models兲, our three-point approximation also coincides with the exact result. A generalization of the Hashin–Shtrikman coatedspheres model is the mixed-coated-spheres model.26 This microstructure consists of a mixture of the two types of coated spheres corresponding to the Hasin–Shtrikman upper and lower bounds at a fixed volume fraction. Thus, an additional parameter is the proportion of coated spheres in which phase ␣ is the included phase and phase 共⫽␣兲 is the matrix, which we denotes by ␣ . Clearly, 12⫹ 21⫽ 1 ⫹ 2 ⫽1. For this geometry 关Fig. 1共b兲兴, the microstructural parameters have been determined analytically.26 The d-dimensional generalizations are given by
FIG. 1. Schematic illustrations of three different coated-spheres models: 共a兲 Hashin–Shtrikman two-phase coated-spheres assemblage; 共b兲 two-phase mixed-coated-spheres assemblage; and 共c兲 multiphase doubly coated spheres assemblage.
22 12 A 11 1 ⫽A 1 ⫽⫺A 1 ⫽
d⫺1 1 2 21 , d
22 12 A 11 2 ⫽A 2 ⫽⫺A 2 ⫽
d⫺1 1 2 12 , d
共100兲
and from Eq. 共55兲
1 ⫽ 21 ,
2 ⫽ 12 .
共101兲
As an example, consider the case 2 / 1 ⫽20. The Hashin–Shtrikman 共HS兲 bounds Eq. 共44兲, three-point 共TP兲 bounds Eqs. 共78兲 and 共80兲, and the three-point approximation 共TPA2兲 Eqs. 共46兲 and 共51兲 are compared in the plane e / 1 versus 2 关see Fig. 2共a兲兴. In Fig. 2共b兲, the three-point approximation is plotted versus 21 in the range 0⭐ 21⭐1, the extreme cases corresponding to the Hashin–Shtrikman’s upper and lower bounds, respectively.
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
FIG. 2. 共a兲 Comparison of the three-point approximation 共TPA2兲 共solid curve兲 for a two-phase mixed-coated-spheres model for 2 / 1 ⫽20, 21 ⫽0.6 to the HS bounds 共dotted curves兲 and the TP bounds 共dashed curves兲. 共b兲 The TPA2 for the mixed-coated-spheres model for a range of 21 with 2 / 1 ⫽20.
B. Periodic and random dispersions of spheres
Here we apply our approximation to various periodic and random dispersions of spheres. The three-point microstructural parameters ␣ for these models are available in the literature.4,27–37 We consider the infinite-phase contrast cases: superconducting inclusions in a normal conductor ( 2 / 1 ⫽⬁) and perfectly insulating inclusions in a normal conductor ( 2 / 1 ⫽0). These are the most stringent test of our approximation. Relation 共51兲 in these instances yields 1 1 / 共 1⫺d 2 兲 if 2 ⭐ d 0⫽ , 1 ⬁ if 2 ⭓ d 共102兲 共 2 / 1 ⫽⬁ 兲 ,
0⫽
再 冉 再
0
1⫺
冊
d d⫺1 2 1 if 2 ⭓
共 2 / 1 ⫽0 兲 .
d⫺1 d
if 2 ⭐
d⫺1 d ,
共103兲
6599
FIG. 3. Comparison of simulation data 共Refs. 32 and 33兲 for the effective conductivity of equisized ordered superconducting particles to relations 共37兲, 共40兲, 共42兲, and 共46兲 for the EMA, MA, TPA1, and TPA2, respectively: 共a兲 face-centered-cubic spheres 共Ref. 32兲 and 共b兲 hexagonal array of aligned circular cylinders 共see Ref. 33兲.
Interestingly, our approximation predicts a nontrivial microstructure-dependent percolation threshold. For the superconducting and perfectly insulating cases, it predicts a percolation threshold at 2c ⫽1/d and 2c ⫽(d⫺1)/d, respectively. The corresponding threshold in terms of volume fraction 2c is easily found from the function 2 ( 2 ) tabulated in the aforementioned literature. One cannot expect a three-point approximation to yield accurate estimates of the percolation threshold, which requires higher-order microstructural 共clustering兲 information. It is interesting to note, however, that for some two-dimensional 共rather than threedimensional cases兲, Eqs. 共102兲 and 共103兲 yield reasonable estimates of 2c . For example, for a two-dimensional square array, Eq. 共102兲 predicts 2c ⬇0.775, which is to be compared to the exact result 2c ⫽ /4⬇0.785. The predictions of our TPA2 for the effective conductivity are compared to simulation data for certain periodic dispersions32,33 关see Figs. 3共a兲 and 3共b兲兴 and random dispersions34 –37 关see Figs. 4共a兲, 4共b兲, and 5兴 over large ranges of volume fractions 2 . We include in the figures relations 共37兲, 共40兲, and 共42兲 for the EMA, MA, and TPA1, respectively. We see that both the TPA1 and TPA2 are rela-
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D. C. Pham and S. Torquato
FIG. 5. Comparison of simulation data 共see Ref. 37兲 for the effective conductivity of equisized random dispersions of overlapping insulating spheres to relations 共37兲, 共40兲, 共42兲, and 共46兲 for the EMA, MA, TPA1, and TPA2, respectively.
volume proportions and coating orders of the phases in all N-compound spheres are the same. The microstructural parameters of this N-phase model have also been determined exactly,26 which in d-dimensional space can be expressed as ( ␣ ,  , ␥ ⫽1,...,N) A ␣␥ ⫽
d⫺1 ␣  ␥ d
,␥⬍␣; FIG. 4. Comparison of simulation data 共see Refs. 34 –36兲 for the effective conductivity of equisized random superconducting particles to relations 共37兲, 共40兲, 共42兲, and 共46兲 for the EMA, MA, TPA1, and TPA2, respectively: 共a兲 random dispersions of hard spheres 共see Ref. 35兲, and 共b兲 random dispersions of hard circular cylinders 共see Refs. 34 and 36兲.
A ␣␣ ⫽⫺ A ␣␣␣ ⫽
␦⬍␣
兺
␦• 兺 ⭐␣
冉兺 冊 冉兺 冊 ␦⭐␣
d⫺1 ␣ ␦• d ␦⬍␣
A ␣␥ ⫽0 tively good predictors of e up to high volume fractions 2 of the included phase close to the percolation threshold, where higher-order information is clearly required to be more accurate. For superconducting cases in three dimensions, the TPA1 is slightly more accurate than the TPA2 for the ordered array but the reverse is true for the disordered array. For superconducting cases in two dimensions, the TPA1 and TPA2 are comparable, except at high volume fractions where the TPA2 is more accurate. In the case of perfectly insulating overlapping spheres, the TPA1 and TPA2 are again comparable, but the TPA2 is superior at high sphere volume fractions. The TPA2 is most accurate for simple periodic systems, followed by random dispersions of hard cylinders and spheres, and least accurate for overlapping cylinders and spheres.
d⫺1 ␣  d
冉兺
␦
⭐␣
⫺1
,
冊
⫺1
,
⬍␣; 共104兲
⫺1
,
␣ ⭓2;
if  ⬎ ␣ or ␥ ⬎ ␣ or ␣ ⫽  ⫽ ␥ ⫽1.
For example, for the three-phase doubly-coated spheres in three dimensions, Eq. 共104兲 becomes 2␣ 3␣ 3␣ A 11 1 ⫽A 1 ⫽A 1 ⫽A 2 ⫽0,
␣ ⫽1,2,3
11 12 2 ⫺1 , A 22 2 ⫽A 2 ⫽⫺A 2 ⫽ 3 1 2 共 1 ⫹ 2 兲 2 A 33 3 ⫽ 3 3共 1⫹ 2 兲 , 2 A 23 3 ⫽⫺ 3 2 3 ,
2 A 13 3 ⫽⫺ 3 1 3 ,
共105兲
2 ⫺1 A 12 , 3 ⫽ 3 1 2 3共 1⫹ 2 兲
2 2 ⫺1 , A 11 3 ⫽ 3 1 3共 1⫹ 2 兲
2 2 ⫺1 A 22 , 3 ⫽ 3 2 3共 1⫹ 2 兲
while Eq. 共87兲 yields 11 A 111⫽ 13 共 1 ⫺3 21 ⫹2 31 兲 ⫺ 1 共 A 11 2 ⫹A 3 兲 , 22 22 A 222⫽ 13 共 2 ⫺3 22 ⫹2 32 兲 ⫹A 22 2 ⫺ 2 共 A 2 ⫹A 3 兲 ,
C. Multicoated spheres
12 A 112⫽ 13 共 2 21 2 ⫺ 1 2 兲 ⫺ 1 共 A 12 2 ⫹A 3 兲 ,
Another generalization of the Hashin–Shtrikman twophase coated-spheres assemblage is the N-phase multicoated spheres model.26 Here spheres of phase 1 are coated with spherical shells of phase 2, which in turn are coated with spherical shells of phase 3,... 关see Fig. 1共c兲兴. The relative
11 11 A 211⫽ 13 共 2 21 2 ⫺ 1 2 兲 ⫹A 11 2 ⫺ 2 共 A 2 ⫹A 3 兲 ,
共106兲
12 12 A 212⫽ 13 共 2 22 1 ⫺ 1 2 兲 ⫹A 12 2 ⫺ 2 共 A 2 ⫹A 3 兲 , 22 A 122⫽ 13 共 2 22 1 ⫺ 1 2 兲 ⫺ 1 共 A 22 2 ⫹A 3 兲 .
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J. Appl. Phys., Vol. 94, No. 10, 15 November 2003
D. C. Pham and S. Torquato
FIG. 6. Comparison of exact effective conductivity values to the HS bounds Eq. 共44兲 and the TPA2 Eq. 共46兲 for a three-phase doubly coated spheres 4 model: 共a兲 1 ⫽10 3 , 2 ⫽20 3 , 1 ⫽0.1→0.9, 2 ⫽ 5 (1⫺ 1 ), 3 1 ⫽ 5 (1⫺ 1 ), conventional coating order 1-2-3 共phase 1 in phase 2 in phase 1 3兲 and 共b兲 2 ⫽10 1 , 3 ⫽20 1 , 2 ⫽0.1→0.9, 1 ⫽ 5 (1⫺ 2 ), 3 4 ⫽ 5 (1⫺ 2 ), for coating orders: 1-3-2 共phase 1 embedded in phase 3, and then in phase 2兲, 2-1-3, and 3-1-2.
For numerical illustrations, we take 1 ⫽10 3 , 2 ⫽20 3 , 1 ⫽0.1→0.9, 2 ⫽ 54 (1⫺ 1 ), 3 ⫽ 51 (1⫺ 1 ), conventional coating order 1-2-3 共phase 1, then phase 2, then phase 3兲. The HS bounds Eq. 共44兲, TP bounds Eqs. 共81兲–
6601
共83兲, and TPA2 Eqs. 共46兲 and 共53兲 are plotted in Fig. 6共a兲. It is interesting to find that the TP bounds Eqs. 共81兲–共83兲 with relations 共105兲 and 共106兲 concide to yield the exact effective conductivity of the doubly coated spheres model. In Table I, we take 2 ⫽10 1 , 3 ⫽20 1 ( 1 ⬍ 2 ⬍ 3 ), 2 ⫽0.1→0.9, 1 ⫽ 51 (1⫺ 2 ), 3 ⫽ 54 (1⫺ 2 ), and collect the exact effective conductivity values of the doubly coated spheres model at different coating orders: 1-3-2, 1-2-3, 2-1-3, 2-3-1, 3-2-1, and 3-1-2. The HS upper 共HSU兲 and lower HS 共HSL兲 bounds are also given for comparison. Some of these results are plotted in Fig. 6共b兲 together with the respective TPA2 results 共46兲 and 共53兲. It is interesting to observe that the model with highest conductivity is 1-3-2 共but not 1-2-3兲, and the model with lowest conductivity is 3-1-2 共not 3-2-1兲, in which the matrix phases are not the ones with extremal conductivities. Note also that most of the exact and approximation values are close to the Hashin–Shtrikman upper or lower bounds, except those for the model 2-1-3, which falls between the bounds. The fact that TP bounds Eqs. 共81兲–共83兲 with relations 共105兲 and 共106兲 yield exact effective conductivities for doubly coated spheres that do not coincide with the TPA2 value indicates that our TPA2 Eqs. 共46兲 and 共53兲 is not as accurate for general N-phase composites as for two-phase ones. The TPA2 does not always yield the best possible approximation and it may even violate certain three-point bounds using the same available geometric information 共although it always falls within the two-point Hashin–Shtrikman bounds as confirmed兲. Therefore, in the general N-phase case, this approximation should be used in conjunction with bounds. One can use computer simulation to verify that TP bounds Eqs. 共81兲– 共83兲 with relations 共87兲 and 共104兲 converge to yield the exact effective conductivity of general N-phase multicoated spheres. We can also generalize the model further: consider a random mixture of multicoated spheres of different kinds, each of which has different coating order. The only restriction is that the volume proportions of the constituent materials in all of the compound spheres are the same. For such generalized models, the three-point microstructural parameters A ␣␥ can also be determined explicitly, however the TP bounds Eqs. 共81兲–共83兲 generally should not converge, as evidenced by the two-phase mixed-coated-spheres model considered.
TABLE I. Exact effective conductivities for doubly coated spheres models at various embedding orders, and 1 HSU and HSL bounds. Here we take 2 ⫽10 1 , 3 ⫽20 1 ( 1 ⬍ 2 ⬍ 3 ), 2 ⫽0.1→0.9, 1 ⫽ 5 (1⫺ 2 ), 4 3 ⫽ 5 (1⫺ 2 ).
2
HSU
1-3-2
1-2-3
2-1-3
2-3-1
3-2-1
3-1-2
HSL
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
14.377 13.853 13.339 12.835 12.340 11.855 11.378 10.911 10.451
14.348 13.804 13.277 12.765 12.270 11.789 11.322 10.869 10.428
14.248 13.673 13.146 12.649 12.174 11.715 11.270 10.837 10.414
13.616 12.601 11.809 11.197 10.735 10.399 10.170 10.034 9.981
8.014 8.318 8.614 8.897 9.163 9.404 9.615 9.789 9.920
7.988 8.266 8.538 8.802 9.053 9.288 9.504 9.697 9.864
7.919 8.136 8.356 8.580 8.807 9.038 9.273 9.511 9.754
7.895 8.092 8.296 8.510 8.732 8.963 9.205 9.458 9.723
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VII. CONCLUSIONS
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In this article, exact strong-contrast expansions for the effective conductivity e of d-dimensional macroscopically isotropic composites consisting of N phases are presented. The series consists of a principal reference part and a fluctuation part, which contains multipoint correlation functions that characterize the microstructure of the composite. The fluctuation term may be estimated exactly or approximately in particular cases using available information about the given microstructure. We demonstrate that appropriate choices of the reference phase conductivity, such that this fluctuation term vanishes, results in simple expressions for e that agree with the well-known two-phase estimates. We propose a simple three-point approximation for the fluctuation part, which agrees well with a number of analytical and numerical results, including those for the EMA and HS coated-spheres microstructures, and various periodic and random dispersions of spheres and aligned cylinders. Even when the contrast between the phases is infinite, the approximation can yield accurate predictions, sometimes up to the percolation thresholds. In cases where clustering effects are significant, higher-order percolation information may be needed for the effective conductivity to be described accurately. We have also given the analytical expressions of the three-point correlation parameters for certain mixed-coated and multicoated spheres assemblages. It is shown that the effective conductivity of the multicoated spheres model can be determined explicitly from known three-point bounds and exact values of the respective three-point parameters. ACKNOWLEDGMENTS
The work was completed during D. C. P.’s visit to the Materials Institute, Princeton University as a Fulbright Senior Scholar. S. T. was supported by a MRSEC Grant at Princeton University, Grant No. NSF DMR-0213706 and by the Air Force Office of Scientific Research under Grant No. F49620-03-1-0406. 1
J. C. Maxwell, A Treatise on Electricity and Magnetism 共Clarendon, Oxford, 1892兲.
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