optical and dielectric properties of self-similar structures

12
2s _- ., iii3 j-- .- ELSEVIER Physica A 207 (1994) 185-196 Optical and dielectric properties of self-similar structures R. Fuchsa”, K. Ghoshb “Department of Physics and Astronomy and Ames Laboratory, VSDOE, Iowa State University, Ames, IA, USA bDepartment of Physics and Atmospheric Sciences, Jackson State University, Jackson, MS, USA Abstract The effective dielectric constant and spectral function of a composite system prepared by recursively introducing inclusions of a given component are calculated. Connections are made with differential effective medium theory, and dynamical critical behavior is studied. 1. Introduction The construction of a composite system using a recursive procedure was first proposed by Bruggeman [ 11. In the first stage of this recursive procedure, known as the differential effective medium theory (DEMT), an infinitesimal volume fraction fr of component 1 is added to a second component 2, and the effective dielectric constant of the mixture is found. For the second stage, the same infinitesimal volume fraction of component 1 is added to a host consisting of the effective dielectric constant of the first stage. If this process is continued, the final overall volume fraction 4r of component 1 can reach any desired value. Although the DEMT cannot realistically describe any actual structure, it has been used as a model for brine-filled rocks, in which an infinitesimal volume fraction f, of insulator is recursively added to a conductor (brine) [2,3]. The final volume fraction of conductor, &, can be as small as desired, even though the conducting component is connected at all stages, giving a finite value of the dc conductivity. In this paper we generalize the DEMT by allowing the volume fraction fi of the added component 1 to be finite. The new effective dielectric function at each stage is described using the Bergman representation [4]. As the recursion proceeds, the final volume fraction & of component 2 approaches zero, and there 1 Correspondence to: R. Fuchs, Department of Physics and Astronomy, Iowa State UnivdtY, Ames, IA 50011. USA. 0378-4371194/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI 0378-4371(93)E0528-M

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Page 1: Optical and dielectric properties of self-similar structures

2s _- ., iii3 j-- .- ELSEVIER Physica A 207 (1994) 185-196

Optical and dielectric properties of self-similar structures

R. Fuchsa”, K. Ghoshb

“Department of Physics and Astronomy and Ames Laboratory, VSDOE, Iowa State University, Ames, IA, USA

bDepartment of Physics and Atmospheric Sciences, Jackson State University, Jackson, MS, USA

Abstract

The effective dielectric constant and spectral function of a composite system prepared by recursively introducing inclusions of a given component are calculated. Connections are made with differential effective medium theory, and dynamical critical behavior is studied.

1. Introduction

The construction of a composite system using a recursive procedure was first proposed by Bruggeman [ 11. In the first stage of this recursive procedure, known as the differential effective medium theory (DEMT), an infinitesimal volume fraction fr of component 1 is added to a second component 2, and the effective dielectric constant of the mixture is found. For the second stage, the same infinitesimal volume fraction of component 1 is added to a host consisting of the effective dielectric constant of the first stage. If this process is continued, the final overall volume fraction 4r of component 1 can reach any desired value. Although the DEMT cannot realistically describe any actual structure, it has been used as a model for brine-filled rocks, in which an infinitesimal volume fraction f, of insulator is recursively added to a conductor (brine) [2,3]. The final volume fraction of conductor, &, can be as small as desired, even though the conducting component is connected at all stages, giving a finite value of the dc conductivity.

In this paper we generalize the DEMT by allowing the volume fraction fi of the added component 1 to be finite. The new effective dielectric function at each stage is described using the Bergman representation [4]. As the recursion proceeds, the final volume fraction & of component 2 approaches zero, and there

1 Correspondence to: R. Fuchs, Department of Physics and Astronomy, Iowa State UnivdtY, Ames,

IA 50011. USA.

0378-4371194/$07.00 0 1994 Elsevier Science B.V. All rights reserved

SSDI 0378-4371(93)E0528-M

Page 2: Optical and dielectric properties of self-similar structures

186 R. Fuchs, K. Ghosh I Physica A 207 (1994) 185196

are critical exponents for the dc conductivity and the static dielectric function. The real part of the dielectric function has a frequency dependence described by a critical exponent that does not satisfy the Bergman-Imry relation [5,6].

If the added volume fraction fi is small, the Maxwell Garnett theory (MGT) can be used to find the new effective dielectric function at each stage [7]. We find that the spectral function develops an approximately self-similar structure that approaches the DEMT spectral function in the limit as f, approaches zero.

2. Spectral representation

The mathematical formalism for the recursive construction has already been presented by Ghosh and Fuchs [8], so we will only present a summary here. In the first stage we randomly place unconnected inclusions of component 1 with volume fraction f, into a host of component 2. The effective dielectric function ~2) of the mixture can be written as

4$ = M(a1, ez, fi) , (1)

where the form of the function M depends on the geometry of the composite. In the second stage we insert inclusions of component 1 into a host consisting of the first-stage composite. Assuming that the second-stage inclusions are large enough that the first-stage composite is effectively a homogeneous medium, we find E:), the effective dielectric function of the mixture at the second stage, by replacing the host dielectric function c2. in eq. (1) by E:). Repeating this procedure recursively, the effective dielectric function after stage J’ is

$1 = M(E1, p, fi) . (2)

Eq. (1) can be written in the form of a spectral representation,

1

E(l) = El + &If2 c, (E2/E* - 1) + ( I G2 (4 m o (E*IE,-l)-l+n dn . 1 (3)

All information about the geometry of the mixture at stage 1 is contained in the value of the percolation strength C, and the spectral function G*(n). These quantities depend on f2 (= 1 -fi), although the dependence is not explicitly shown. Peaks in G,(n) are directly related to frequencies of surface modes that may appear in frequency regions where E~IE~ < 0, and n is the depolarization factor of the mode. For example, if G,(n) has a peak at IZ = IZ~ [G(n) - A6(n -

n,)], there is a surface mode if c2 and F~ have values such that the integral in eq. (3) gives an infinite contribution to E, . (l) This occurs when [(E~/E~) - 11-l + IZ~ = 0, or E~ /Ed = -n,l( 1 - n,,). For dilute spherical inclusions of component 1, there is an infinite number of surface modes with depolarization factors n,(l) = (1+ l)/ (21+ l), where 1 = 1,2,3, . . . is the multipole index. For the dipole (I = 1) mode,

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R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196 187

which is the only mode that has a nonzero weight A, we have n,(l = 1) = f , or E1/EZ = -2.

An equivalent spectral representation that uses the spectral function G,(n) can be obtained by interchanging the indices 1 and 2 on the right-hand side of eq. (3). G,(n) obeys the sum rules C, + 1: G,(n) dn = 1 and J,’ nG,(n) dn = +(l -f2), and there are similar sum rules for G,(n) [4,9]. Furthermore, G, and G, are related

by frnG,(n) =.&(I - n)G,(I -n), and the percolation constants C, and C, obey the relation

’ G(n) flC,+f,C*+f,J*dn=l. 0

For our specific system, the inclusions of component 1 are disconnected, so that c, =o.

3. Recursive procedure

If we introduce the variables x, x1, x2, . . . , xj, defined by E~/E~ - 1 = -l/x

> . . . 2 E:)/E~ = -l/xi, eqs. (1) and (2) can be written

x1 = h(x), . . . ,xj = h(Xjpl) ) (4)

h(x) = [f2(2+ j$$dn)]-’ .

0

(5)

Using the recursive map given by eqs. (4) and (5), one can calculate the variable xi associated with the stage j of the recursive procedure. The spectral function g2( j, n) for the recursive structure at stage j appears in an expression similar to eq. (3); ~2) on the left-hand side of the equation must be replaced by E:), and on the right-hand side, f2, C, and G,(n) are replaced by the quantities &2(j), c2( j) and g2( j, n), respectively. The actual volume fraction of component 2 at the end of stage j is +y) = (f2)j; the percolation constant is

cl” = 4,’ lim (E~)/Q) = (C,)’ , (6) E*-+-=

and the spectral function is given by

g2( j n) = (r+(j))-* lim(E”‘/E ) > 2 s-0 In 1 3 (7)

where, in eq. (7), we have set &Z/&l - 1 = -l/x = -ll(n + is). From the above expressions, the final spectral function at the end of stage j can

be found if it is known for the first stage. The procedure will be illustrated using the Maxwell-Garnett theory (MGT) as the mixture equation for the first stage: G,(n) = (1 - C,)s(n - no) where C, = 2/(2 +fr) and no = (2 +f,)/3.

Page 4: Optical and dielectric properties of self-similar structures

188 R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196

Fig. 1. The map h(n) for the iterated MGT with initial volume fraction f, = 0.1. Depolarization factors

of surface modes at successive stages can be located geometrically using the curve y = h(x) and the line

y = n, as shown for the first three stages. The x values of the points marked 2 and 3 correspond to the

depolarization factors n of the new surface modes that appear in stages 2 and 3.

The map h(x) for the MGT, using the initial volume fraction fi = 0.1, is shown

in Fig. 1. The surface mode resonance at n, = 0.7 corresponds to a zero in h(x) at

x = 0.7, and the positive percolation constant C, corresponds to the zero in h(x)

at x = 0. As the recursion proceeds, to higher stages, x2 = h(x,) = h(h(x)), etc.,

the zeros of xi correspond to the n values for surface modes of the stage j. The

location of these zeros can be determined from the geometrical construction

shown in Fig. 1, using the curve y = h(x) and the straight line y =x. At stage 2,

for example, there are two new modes labeled 2, and the original first-stage

modes, labeled 1. At each subsequent stage, each of the newest modes of the

previous stage yields two new modes, and all modes of the previous stages are still

present, with reduced weights.

The evolution of the spectral function for the first three stages is shown in Fig.

2. In the calculations, the parameter s appearing in x = n + is has been made

small but nonzero, giving a finite width to the peaks. In order to understand the

behavior of the spectral function, we assume that the spheres (component 1) are

metallic, with Re(c,) < 0, Im(.s,) = 0, and the surrounding medium (component 2)

is vacuum (Ed = 1). Consider stage 2, for which there are three peaks in the

spectral function. Fig. 3 shows a large stage 2 sphere surrounded by a few typical

smaller stage 1 spheres, with instantaneous patterns of dipole moments on the

spheres.

Fig. 3a is the mode corresponding to the central peak at IZ = 0.7. This is the

same n-value as that of the single stage 1 mode, for which si’ becomes infinite.

Because the effective dielectric constant of the medium around the large sphere is

infinite, the average E field is zero. The field lines shown, corresponding to

average displacement or polarization, do not penetrate the large sphere, and

there is no dipole moment on the large sphere. Fig. 3b is the mode with the

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R. Fuchs, K. Ghosh I Physica A 207 (1994) 18.5-196 189

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.0 0.9 1.0

Fig. 2. Spectral functions for the first three stages of the recursive MGT.

largest IZ value. In this mode E:) > 1, so the dipoles on the small spheres are

parallel to the average E field in the region outside the large sphere, which has a

dipole moment as shown. Fig. 3c is the mode with the smallest n value. In this

mode ~2) < 1, so the dipoles on the small spheres are antiparallel to the average

field in the region outside the large sphere, which again has a dipole moment. The

splitting of these last two modes arises from the coupling between the dipole

moment on the large sphere and the dipole moments on the small spheres, which

can have either of two directions.

A similar analysis applies to the mode behavior in the following stages. Every

mode in a given stage also appears in the following stage when the spheres

introduced in the following stage have zero dipole moment. Also for each mode

in a given stage, a pair of new modes appears in the following stage, because for

these modes, the spheres introduced in the following stage have dipole moments

which interact with the dipole moments on the spheres at the given stage.

The concepts introduced in the preceding paragraph show that a recent

criticism of the Bruggeman DEMT (and by extension, of our recursive theory) by

Fu, et al. [lO,ll] is invalid. These authors have shown that for any system of

unequal spheres with positions described by a two-particle distribution function,

and within the mean-field approximation, the Maxwell-Garnett theory is valid, so

that the spectral function should only have a single peak. The essence of their

proof lies in the facts that (1) in the mean-field approximation, the dipole or

multipole moment on every sphere of a given size is the same, and (2) the

Page 6: Optical and dielectric properties of self-similar structures

190

W

R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196

(a)

Fig. 3. Dipole moments on spheres for the three modes in stage 2. Average E or D fields are shown

only in the region outside the large sphere. (a) Mode corresponding to the central peak. (b) Mode

with the largest n value. (c) Mode with the smallest n value.

configuration average field, at a given sphere, due to a fixed multipole moved to

all possible positions about the given sphere, is zero. It is clear from Fig. 3 that

the dipole moments on the small spheres are not constant, but vary with angle

and distance from the large sphere, so that, for the modes shown in Figs. 3b and

3c, their effect on the large sphere does not cancel. Hence, the mean-field

approximation cannot be considered to apply to the final effective dielectric

function found either by our recursive theory or by Bruggeman’s DEMT theory.

This is true even if the mean-field approximation is used at each stage of the

Page 7: Optical and dielectric properties of self-similar structures

R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196 191

recursive procedure, as in our example, where we have used the Maxwell- Garnett theory.

4. Properties of the spectral function

In this section we examine some properties of the spectral function found by using the Maxwell-Garnett mixture equation at each stage of recursion. Since the number of new modes that appears is twice the number of new modes in the previous stage, and all modes in the previous stage still appear in the next stage, the total number of modes at stage j (not including the percolation mode at IZ = 0) is 2’ - 1. From Fig. 1 it is clear that these modes will have n-values which extend over the entire two branches of h(x): the left branch, from n = 0 to IZ - 6, and the right branch from n = 0.7 to II = 1. There is a gap from 12 - 0.6 to n = 0.7 from which all modes at subsequent stages are excluded. The second recursion of the function h, x2 = h(x,) = h(h(x)), h as f our branches, the three modes at stage j = 2

correspond to the zeros of h(h(x)), and there is a gap immediately to the left of

each of the three modes from which all modes at subsequent stages are excluded. Continuing this reasoning, at stage j, each of the 2’ - 1 modes has a gap immediately to its left, from which all subsequent modes are excluded. However, to the right of each mode there is a branch on which subsequent modes can come arbitrarily close to the given mode.

Expressions from which the positions and strengths of the modes can be calculated have been given by Fuchs and Claro [12]. As one continues the recursive procedure, the spectral function develops an approximately self-similar structure, with a distribution of scaling indices that can be analyzed by the procedure of Halsey, et al. [13]. Fig. 4 shows g2( 10, n), the spectral function for stage 10. As one goes to higher stages, the central part of the spectrum appears unchanged at a fixed resolution, and the left-hand and right-hand edges approach 12 = 0 and II = 1, respectively.

As the initial volume fraction fi approaches zero, our theory should become equivalent to the DEMT. To examine this limit, we shall find the spectral functions g,(n) and g,(n) for the DEMT. The equation that gives the effective dielectric function E, for the DEMT, in which an infinitesimally small volume fraction fi is added at each stage, is [1,2]

[(&In - &I)/(% - 41” = (dd3d% * (8)

Here, & is the final volume fraction of component 2. The stage index j which appeared previously is absent, since an infinite number of stages is needed to reach a given value of &. The percolation strength cZ can be found immediately from eq. (6):

c2=V&. (9)

Page 8: Optical and dielectric properties of self-similar structures

192 R. Fuchs, K. Ghosh I Physicn A 207 (1994) 185-196

Fig. 4.

0 .2 .4 .6 .8 1

n

Spectral function for stage 10, with initial volume fraction f, = 0.1

To find g,(n), we use eq. (7), omitting the index j. Letting c1 = 1 for simplicity

and taking x = -1 /(Ed - 1) = -(n + is), we find that in the limit as s approaches

zero,

(E,,, - 1)3/~, = (~&)~/n’(l -n) = K2, (10)

which is a cubic equation for E,. Using standard methods for finding the roots of

a cubic equation, one finds that there are two complex roots and one real root if

K2 <F, where K2 has been defined in eq. (10). We keep the complex root in

which E, has a positive imaginary part, and eq. (7) gives

g,(n) = (ti/2rr)P3(1 - n))““[(l + B,)“” - (1 - B,)“3] ) (11)

B,=1/1-$K,.

Using the relation &ng,(n) = +,(l - n)g,(l - n), we find

(12)

gl(n) = (ti/2rr)[4,/(1 - $,)]nm4’“(l - n)““[(l + B,)“” - (1 - B,)1’3] ,

(13)

B,=j/w, K, = (&)3/(1 - n)2n . (14)

The function g,(n) is different from zero only in the region n2= <n < nTu, in

which B, > 0 or K, > 4. The actual limits n2L and n2u are solutions of the cubic

equation K, = 4 or n”( 1 - n) = &($,)“. F or small values of cP~,, the two solutions

are n2L - G(&)3’2 and n2” - 1 - $(+,)‘.

Figure 5 shows g,(n) calculated from eq. (ll), for a few selected values of I$~.

In Fig. 6, the spectral functions g,(n) for the DEMT and g,(j, n) for the recursive

Page 9: Optical and dielectric properties of self-similar structures

R. Fuchs, K. Ghosh I Physica A 207 (1994) 18.5-196 193

0 0 .2 .4 .6 .8 1

n

Fig. 5. Spectral function for the differential effective medium theory (DEMT). The final values of the

volume fraction of component 2 are 4, = 0.4, 0.6, 0.8 and 0.9.

MGT and compared for the same final volume fraction & = 0.4. In the recursive MGT we have chosen to use j = 2.5 stages, so we need the intial volume fraction fi = 1 - (o.4)“25 = 0.036. In the regions where the slightly broadened peaks in the recursive MGT overlap, making the plotted g2(j, n) a slowly varying function, there is good agreement between the two spectral functions.

Cp2 = 0.4

0’ I-J’ ” ” ” ” ” ” ” “11 0 .2 .4 .6 .8 1

n

Fig. 6. Comparison of the spectral functions g,(n) for the DEMT and g,( j, n) for the recursive MGT

at stage j = 25. The final volume fraction is 4, = 0.4.

Page 10: Optical and dielectric properties of self-similar structures

194 R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196

5. Critical behavior

As the number of recursive stages j increases, the volume fraction 41” approaches zero, and the lower and upper limits of g2(j, n), which we denote nlj,’ and n:G), approach 0 and 1, respectively. We can introduce critical exponents which describe the behavior of various quantities as functions of the volume fraction &.

If component 1 is a conductor and component 2 is an insulator, the critical exponents are determined by the behavior of g, ( j, n) near n = 0 [or g2( j, n) near II = 11. Letting nlL = 1 - nZU denote the lower limit of g, ( j, n), critical exponents E,, sl, and t, are defined by

nlL - 42’l 9 cr, - 4Zf’ , Re E, - KS1 > (15) where a, is the effective dc conductivity. The stage index j has been omitted in eq. (15).

Similarly, if component 2 is a conductor and component 1 is an insulator, the critical exponents are determined by the behavior of g2( j, n) near n = 0. Critical exponents I,, s2, and t, are defined by

n2L - 42’, > urn - 42 > Re E, - 42-s2 . (16) General expressions for these critical exponents have been derived previously [S], and the results will not be repeated here. We only note that if component 1 is a conductor and component 2 is an insulator, the dc conductivity of the mixture is always zero, so t, = 0. We also find that if component 2 is a conductor and component 1 is an insulator, there is a modest enhancement of the static dielectric constant Re E,, but is value approaches a constant, independent of 42r giving s2 = 0. We find that these exponents obey the relations

1, = s1 + t, ) 1, = s* + t, . (17)

We shall present results for the dynamic behavior, taking component 2 to be an insulator (Ed = 1) and component 1 to be a conductor, with a frequency-depen- dent dielectric function &1 = 1 + 4ria,lw, where m1 is the dc conductivity of component 1. Figure 7 shows Re E, as a function of frequency, on a log-log plot, for selected values of the final volume fraction of component 2: 4* = 0.1, 0.01 and 0.001. We have chosen the number of stages to be fixed at j = 30, and varied the initial volume fraction fi in order to reach these values of 4*. (One could also fix

f, and vary j, with essentially the same result.) At low frequency, Re E, approaches a constant value, which diverges as 42

decreases to zero, in accordance with the critical exponent sl. Above a crossover frequency, which increases as 42 increases, Re E, decreases with frequency according to the power law

ReE,--6“) a-4 3 . (18)

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R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196 195

Fig. 7. The recursive MGT, with 25 stages, is used to calculate log Re Ed as a function of log R, where

R = o/4ruI is a dimensionless frequency (logarithms are in base 10). Final volume fractions are

chosen to be 4, = 0.1, 0.01 and 0.001.

This result does not agree with the standard scaling theory near a percolation threshold [5,6], a = sl/(sl + tl) = 1. Standard scaling theory can be derived from a spectral function of the form,

g1(n) - (n - n,,)‘-“ln 2 n>n,, , n-0, (19)

with nlL - C#J> and (Y = si/(si + ti). In our recursive theory, gi(n) (for small fi) is given by eq. (13), which is clearly different from eq. (19). It can be shown that the value (Y - 4 comes from the n-4’3 factor in g*(n). It is possible that this breakdown of the usual scaling behavior arises from the fact that the volume fraction at the percolation threshold is C& = 0, so & only approaches & from one side, instead of passing through &.

6. Concluding remarks

If the starting volume fraction fi in our recursive construction is not in- finitesimally small, the initial spectrum function G,(n) will have a finite width; therefore most, or perhaps all, of the fine structure that we find in the g2(i, n) functions will be missing. However, the dominant behavior, that the overall width of the spectral function increases as the recursion proceeds, must always occur. A detailed examination of the surface modes in our recursive MGT has provided a physical picture of the reasons for the width increase. We have also given physical arguments to justify the requirement that ratio of sizes of the inclusions for successive stages must be large. However, we have not addressed the issue of

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196 R. Fuchs, K. Ghosh I Physica A 207 (1994) 185-196

exactly how large this ratio of sizes must be, or just how the theory breaks down if the size ratio is too small. We have shown that the DEMT, with an infinitesimally small starting fi, can serve as a good approximation to the recursive MGT with a finite starting fi.

Acknowledgements

We thank Francisco Claro for supplying valuable insight about the recursive procedure. The Ames Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. W-7405Eng-82. Part of this work was performed at Jackson State University within the Lawrence Berkeley Laboratory/Jackson State University/Ana G. Mendez Universities System Sci- ence Consortium under U.S. Department of Energy, Office of Energy Research, Contract No. DE-FG 0586ER75274.

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