PHYSICAL REVIEW B 77, 174204 共2008兲
Transport exponent in a three-dimensional continuum tunneling-percolation model N. Johner,1,* C. Grimaldi,1,2 I. Balberg,3 and P. Ryser1 1
LPM, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland Max-Planck-Institut für Physik Komplexer Systeme, Nöthnitzer Strasse 38, D-01187 Dresden, Germany 3The Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel 共Received 4 March 2008; published 29 May 2008兲
2
The tunneling-percolation mechanism of conduction in disordered conductor-insulator composites is studied for a realistic continuum model where conducting and impenetrable spherical particles are dispersed in a three-dimensional continuum insulating material. Conduction between particles is via tunneling processes and a maximum tunneling distance d is introduced. We determine the percolation critical concentration for several values of d. By doing so, we relax the restrictions applied in the previous studies of the problem, i.e., the considerations of the underlying lattice and the contribution of only the nearest neighbors. The tunnelingpercolation transport is then analyzed by studying the conductance of the composite at and near the percolation threshold using a decimation procedure and a conjugate gradient algorithm. We show that at the critical concentration, and independently of the tunneling parameters, the critical transport exponent t reduces to the universal value t0 ⯝ 2, while moving away from the percolation threshold, the conductance exponent becomes larger than t0, acquiring a strong concentration dependence. We interpret this feature as arising from the peculiar form of the distribution function for the local tunneling conductances. Consequently, apparent nonuniversality of transport appears when the conductance of the composite is fitted by forcing the exponent to be independent of the concentration. This leads us to believe that our tunneling-percolation theory is sufficient to explain the nonuniversal transport exponents observed in real disordered conductor-insulator compounds. DOI: 10.1103/PhysRevB.77.174204
PACS number共s兲: 64.60.ah, 72.80.Tm, 64.60.F⫺
I. INTRODUCTION
In a disordered conductor-insulator composite material, the electrical conduction is established when at least one macroscopic path 共or region兲 spanning the whole sample allows for the flow of charge carriers.1–3 This can be established either by direct contacts between the conducting particles or by indirect electrical connections, such as when the conductivity between the particles is controlled by tunneling. In the former case, the critical concentration of the conducting phase above which the conductance of the sample is nonzero coincides with the geometrical percolation threshold, while it can be significantly reduced when the interparticle conduction is governed by tunneling. Both theory and experiments have well established that the macroscopic conductance G of a direct-contact composite follows the power-law behavior G = G0共x − xc兲t
共1兲
for concentrations of the conducting phase x sufficiently close to the 共geometrical兲 percolation threshold xc. Furthermore, the transport exponent t is universal, i.e., it is independent of the nature of the metallic phase and of the microstructure, and reduces to the value t = t0 ⯝ 2 for threedimensional systems. The only notable exception for threedimensional systems is found for systems described by the random void model, in which insulating penetrable spheres are randomly immersed in a conducting continuum. In this case, the exponent is found to be t ⯝ 2.38.4 When tunneling is the dominant mechanism of transport, the situation is more intricate. Indeed, Eq. 共1兲 is still found experimentally to describe the behavior of G for small x − xc values; however, besides the difference between the 1098-0121/2008/77共17兲/174204共11兲
measured xc and the expected value for the geometrical percolation threshold, the transport exponent is often reported to exceed t0 ⯝ 2, reaching values even as large as t ⬇ 5 – 10. A recent survey of published experimental data has evidenced that for about 50% of the reports the measured transport exponents were larger than t0 ⯝ 2, with strong spread also within the same class of composites.5 The first theoretical attempt to understand the mechanism of transport nonuniversality in tunneling-percolation systems has identified the distribution function h共兲 of the local interparticle conductances as the key quantity governing the observed behavior in real composites.6 This comes about by observing that the main contribution to the total resistance stems from tunneling processes between nearest-neighboring conducting particles.6 If two such particles are separated by a distance r then, from the corresponding distribution function P共r兲, one can readily obtain h共兲. By assuming that the main contribution of P共r兲 is of the form P共r兲 ⬇ exp共−r / a兲, where a is the mean distance between nearest-neighbor conducting particles 共assumed here to be pointlike for simplicity兲, the resulting distribution function h共兲 develops an “anomalous” 共but normalizable兲 divergence for → 0 of the form h共兲 ⬇ −␣, with ␣ = 1 − 2a ⬍ 1, where is the tunneling decay factor associated with the interparticle conductance ⬇ exp共−2r / 兲 共which is known to yield measurable conductance for distances of the order of a few nanometers7兲. As first shown by Kogut and Straley,8 who applied the effective medium approximation 共EMA兲 to the problem by mapping it onto a regular lattice, such diverging h共兲 would lead, for 0 ⬍ ␣ ⬍ 1, to the breakdown of universality, with the conductance exponent t acquiring an explicit dependence on ␣ of the form t共␣兲 = t0 + 1 / 共1 − ␣兲. Subsequent studies9 have shown that the rigorous expression for the exponent in three dimensions is
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JOHNER et al.
t=
冦
for
+
1 ⱕ t0 1−␣
1 for 1−␣
+
1 ⬎ t0 , 1−␣
t0
+
冧
共2兲
where ⯝ 0.88 is the correlation length exponent.2 Hence, according to Eq. 共2兲 and provided that ␣ ⱖ 0.107, the transport exponent may display values much larger than the universal limit t0 ⯝ 2, yielding then a possible explanation for the experimental results. The original formulation of the tunneling-percolation model, and its prediction of nonuniversality, was based on the assumption that a simple exponential decay of P共r兲 is able to capture the essential physics of the problem.6 Later investigations confirmed that such form of P共r兲 could be rigorously obtained for a particular distribution of the conducting matter on a regular lattice.10 However, it is rather well known that ensembles of spherical particles dispersed in a three-dimensional space display P共r兲 functions which decay much faster than exp共−r / a兲 for large distances 共in comparison to the particle size兲.11 For example, the interparticle distance for a Poisson distribution of pointlike particles follows the Hertz distribution function,6,11 P共r兲 ⬇ 3
r2 exp共− r3/a3兲. a3
共3兲
In such a case, the behavior of the system remains universal at the percolation threshold because P共r兲 decays much faster than ⬇ exp共−2r / 兲 for r → ⬁, so that the resulting h共兲 is no longer divergent for → 0.10 This situation remains unchanged also for the case of nonoverlapping spheres, i.e., when particles with a hard core are considered. Recently, a theoretical analysis has evidenced that, notwithstanding the universality at the percolation threshold, the total conductance G, for both overlapping and nonoverlapping spheres in three dimensions, still follows Eq. 共1兲, but with a strongly x-dependent exponent t共x兲 larger than the universal value of t0 ⯝ 2, and that only very close to xc the exponent reduces to t0.12 This behavior has been shown to stem from the local conductance distribution h共兲 which is characterized, for sufficiently small / a values, by a strong peak at a small 共=ⴱ兲 value, so that h共兲 is not very much different from the diverging Kogut–Straley8 distribution for ⬎ ⴱ. From the practical point of view, this behavior would be manifested by an “apparent” nonuniversal transport. In fact, in real experiments, where the measured conductance is sampled over a limited range of a couple of orders of magnitude of x − xc and only for a few x − xc values, the fit to Eq. 共1兲 would lead to an exponent which is basically an average of t共x兲, say tⴱ, and since t共x兲 ⬎ 2, the average exponent will appear as tⴱ ⬎ 2.13 Despite that, the use of a realistic distribution function P共r兲 has lead to an interpretation of the tunneling-percolation mechanism of nonuniversality. On the other hand, as in the pioneering work,6 the above models were based on a mapping on a regular lattice and on the inclusion of only nearest-neighbor particles.12 In this paper, we relax these restrictions by considering an off-lattice model where the conducting 共impenetrable兲 particles are
φ
(a)
φ+d
(b)
FIG. 1. 共Color online兲 Two-dimensional representation of the composite particles model used in the present work. 共a兲 The filled circle denotes a conducting impenetrable sphere of diameter , while the concentric shell of thickness d / 2 defines the upper cutoff for the conductance. 共b兲 When several particles are arranged together, their microstructure is that of impenetrable spheres with penetrable shells. When the separation between the closest surfaces of two 共black兲 spheres is larger than d, the interparticle conductance is set equal to zero, otherwise it follows Eq. 共4兲.
given by equally sized spheres dispersed in an insulating continuum, and tunneling between particles is allowed to extend also beyond the nearest-neighbor distances. By using exact numerical renormalization techniques and relaxation methods, we demonstrate that, right at the percolation threshold, the system is universal with the electrical transport exponent t0, while away from xc, the conductance follows Eq. 共1兲 with an x-dependent exponent t共x兲. Furthermore, we show that t共x兲 depends on the tunneling decay factor and on the mean distance between nearest-neighboring particles, verifying therefore the effective medium results of Ref. 12 and the on-lattice calculations of Ref. 13. II. MODEL
Let us consider equal size conducting spheres of diameter dispersed in a continuous three-dimensional insulating medium. Furthermore, let us assume that the spheres are impenetrable and that the conductance between two spheres whose centers are separated by r 共with r ⱖ 兲 is given by
冉
共r兲 = 0 exp −
2共r − 兲
冊
共4兲
where is the characteristic tunneling distance and the prefactor 0 can be set equal to unity without loss of generality. In principle, Eq. 共4兲 applies to all pairs of particles regardless of their relative distances. However, in practice, particles which are set apart by a distance much larger than / 2 can actually be considered as electrically disconnected. In order to deal with such situation, we introduce an upper distance cutoff d beyond which the tunneling conductance between two particles is set equal to zero. If the cutoff length is sufficiently large in comparison to , the exponential decay of Eq. 共4兲 should assure results independent of the cutoff. The introduction of a cutoff length d leads to the composite particle model shown in Fig. 1共a兲, where the black circle is the impenetrable metallic sphere while the attached concentric shell has a thickness d / 2. From the structural point of view, our model is a penetrable-concentric-shell model 共known also as cherry-pit model or semipermeable particles model兲. As shown in Fig. 1共b兲, the composite particles define a spa-
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tially correlated continuum percolation system where two hard-core particles are considered linked, and the corresponding interparticle conductance is given by Eq. 共4兲, only when the separation between their closest surfaces is less than the cutoff distance d. Due to the composite nature of the particle system, hardcore spheres of diameter and semipermeable spheres of diameter + d, the definitions of the particle concentration and the percolation threshold depend on which aspect of the composite particles is relevant for the problem. For the electrical transport properties, it is, of course, the whole composite sphere which must be considered, since the electrical connectivity is governed by the overlap of the permeable shells of two or more composite particles. Hence, given N spheres in a volume V, for the determination of the conductorinsulator transition of the system, we shall use as a density variable the following dimensionless quantity:
= 共 + d兲3 , 6
共5兲
where = N / V is the number density. The corresponding critical concentration for the onset of electrical conduction c determines the conductor-insulator percolation threshold of the system. However, it must be kept in mind that such a definition depends on the value d of the cutoff parameter, which is certainly of computational convenience, but does not represent a real physical length. Instead, the volume concentration of hard-core particles x = 3 / 6 has the direct significance of volume percentage of the conducting phase dispersed in the whole conductor-insulator composite. The concentration x is related to Eq. 共5兲 through the relation x = 3 ,
共6兲
FIG. 2. 共Color online兲 Spanning probability as a function of the density for a cube with several edge lengths and for the penetrability coefficient value = 1 / 2. The solid lines are fits to Eq. 共8兲.
Finally, the spanning probability is obtained by recording the number of times that a percolating cluster appears for several realizations. In Fig. 2, we show the so-obtained spanning probability for L / = 10, 15, 20, and 30 and for d / = 1 共 = 0.5兲, calculated by imposing open boundary conditions. The calculations have been performed for number of particles N ranging from N = 200 共with 2000 realizations兲 up to N = 100 00 共with 250 realizations兲. According to the finitesize scaling method described, for example, in Ref. 14, the critical density c共L兲 for finite L can be extracted by imposing c共L兲 to be the value for which the spanning probability is 1/2. This can be evaluated by fitting the discrete data with some suitable function. Here, we use14
冉
where
= +d
共7兲
is the penetrability coefficient which has a limiting value = 0 共 = 1兲 for completely penetrable 共impenetrable兲 spheres. III. PERCOLATION THRESHOLD
Since the electrical connectivity is established only if the overlaps of the particle shells span the entire sample, it is essential for the evaluation of the composite conductance to estimate the percolation threshold c and its dependence on the penetrable shell thickness 共and hence on 兲. Here, we obtain c as a function of by first computing the spanning probability for finite systems with linear size L, i.e., the probability of having a percolating cluster which spans the system from one side to the opposite one, and then by extrapolating the results to the L → ⬁ limit.14 In practice, we consider a box with edges of length L where N spheres of diameter are placed randomly but with no overlaps. A simple Metropolis algorithm is used to attain equilibrium.15,16 Next, for a given value of d 共hence of 兲, we use a modified Hoshen–Kopelman17 algorithm to extract, if it exists, the percolating cluster for a given realization.18
冋
1 − c共L兲 1 + tanh 2 ⌬共L兲
册冊
,
共8兲
where ⌬共L兲 is the width of the percolation transition. Equation 共8兲 is presented by the solid lines in Fig. 2, and the resulting best fitting values of c共L兲 and ⌬共L兲 are reported in Fig. 3. Using scaling arguments,2 it can be shown that the functions ⌬共L兲 and c共L兲 follow the scaling relations, ⌬共L兲 ⬀ L−1/ ,
共9兲
c共L兲 − c ⬀ L−1/ ,
共10兲
where is the correlation length exponent. Hence, the exponent can be extracted from Eq. 共9兲 and used in Eq. 共10兲 to evaluate the critical density c for an infinite system. The results we obtained by this procedure are shown in Fig. 3 for = 1 / 2. From relation 共9兲, we have obtained 共inset of Fig. 3兲 = 0.87⫾ 0.02, which is in excellent agreement with other estimations yielding ⯝ 0.88,14,19 while from Eq. 共10兲 we have deduced 共main frame of Fig. 3兲 c = 0.3203⫾ 0.0003 by using our value for . By using Eq. 共6兲, we also find the corresponding value in terms of the volume fraction of impenetrable spheres. This value, xc ⬇ 0.04, is of course much lower than the geometrical percolation threshold for direct
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FIG. 3. Evaluation of the critical density as a function of L for = 1 / 2. The solid line follows Eq. 共10兲 and c is given by the intercept at L−1/ = 0. The value of the correlation length exponent is extracted from Eq. 共9兲 by fitting the data of ⌬共L兲 as derived from Fig. 2 and Eq. 共8兲 共inset兲.
contact between hard-core spheres 共xc ⬇ 0.64 for random close packing兲.20 The behavior of c for different values of the penetrability coefficient is plotted in Fig. 4 共crosses兲 together with the results of Ref. 19 共empty circles兲. For = 0, we obtain c = 0.3423⫾ 0.0003 which is in very good accord with most accurate value to date 共c = 0.341 889⫾ 0.000 003兲 reported in Ref. 21 共open square in Fig. 4兲. Such agreement holds true also for ⬎ 0, for which our results are perfectly compatible with the original work of Balberg and Binenbaum in 1987 共Ref. 22兲 and with the more recent results of Lee and Yoon.19 Having established the behavior of c as a function of the penetrability coefficient, let us now consider the problem of choosing some representative values of the cutoff parameter for the evaluation of the system conductance. From the results of Fig. 4, and by using Eq. 共6兲, it is straightforward to
FIG. 4. 共Color online兲 The percolation critical density c as a function of the penetrability coefficient . Crosses: our results. Circles: results from Ref. 19. Open square: 共at = 0兲 from Ref. 21.
FIG. 5. 共Color online兲 共a兲 The percolation critical concentration xc as a function of the cutoff length d / . 共b兲 The corresponding values of d / 共ac − 兲, where ac − is the mean distance between the closest surfaces of the two nearest-neighbor conducting spheres. Crosses: our results. Circles: results from Ref. 19.
obtain the critical concentration xc of the conducting particles as a function of the cutoff parameter d / . This dependence is shown in Fig. 5共a兲, from which one infers that the electrical connectivity of the system is established for concentration values that are rapidly decreasing as d / increases. Hence, for an efficient numerical evaluation of the system conductance, it is certainly preferable to consider large enough values of d / so that not too many particles ought to be considered in the calculation of the conductance. Yet, large values of d / may lead to important finite-size effects and/or poor statistics. Hence, intermediate values of d / should represent the best compromise. We have chosen then for our work the values of d / = 1 and d / = 2 which, according to Fig. 5共a兲, amount to describe a system with a relatively low particle concentration. Such values of the cutoff length, however, do not limit the validity of the subsequent calculations, since the ratio between the cutoff length d and the typical interparticle separation, 共ac − 兲, where ac is the mean nearest-neighbor particle distance at percolation, changes little with d / . This is demonstrated in Fig. 5共b兲, where the plotted d / 共ac − 兲 values have been obtained from the data of Fig. 5共a兲 and by using ac = 兰drrP共xc , r兲, where P共xc , r兲 is the distribution function for the distance between two neighboring impenetrable spheres at concentration xc, as given in Ref. 11. From the results in the figure, it is seen that
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FIG. 6. 共Color online兲 The calculated values of the dc transport exponent from fits to Eq. 共12兲 for different values of the tunneling decay factor and for d / = 1 共filled circles兲 and d / = 2 共open squares兲. The horizontal dashed line is the universal value t / = 2.27, while the solid lines refer to the corresponding Kogut– Straley 共Ref. 8兲 exponent of Eq. 共13兲.
d / 共ac − 兲 is comprised between 1.7 and 2.1 for 2 orders of magnitude change in d / . In the limit of pointlike particles 11 共d / = ⬁兲, it is found that d / ac = 21/3 c / ⌫共4 / 3兲, where ⌫ is the gamma function, so that for c ⬇ 0.342 one finds d / ac ⬇ 1.6. The weak dependence of d / 共ac − 兲 on the cutoff length points out therefore that the above values of d / , which we have chosen for the conductance simulations, below, not only have a computational convenience but are actually representative of quite a general situation.
IV. TRANSPORT EXPONENT FROM FINITE-SIZE SCALING
We are now in the position of evaluating the behavior of the conductance at the percolation threshold c and the value that the transport exponent t acquires at c. To this end, we have used, as outlined below, a finite scaling analysis of systems of linear size L. In general, the conductance G is a function which depends on both the concentration and the size L. For sufficiently close to the percolation threshold c共L兲 for a finite system of size L, one expects, in view of the above, G共,L兲 = G0共L兲关 − c共L兲兴t ,
共11兲
where G0共L兲 is a size-dependent prefactor. By setting = c, where c is the percolation threshold for L → ⬁, and by using Eqs. 共10兲 and 共11兲, one sees that the main dependence of G共c , L兲 upon L is of the form L−t/, so that the transport exponent t 共as the exponents of other properties; see Ref. 2兲 can be extracted from the linear dependence of ln关G共c , L兲兴 on ln共L兲. However, rather generally, fluctuations at finite L may result to be important and, furthermore, the prefactor G0共L兲 provides some, although weaker, dependence on L. These finite-size contributions can be taken into account by considering the following, more general, scaling relation:
G共c,L兲 = 1L−t/共1 + 2L−兲,
共12兲
where the quantity within parentheses is an ansatz aimed to capture the main L dependence of the finite-size corrections. The evaluation of t then goes as follows. The conductance at c is calculated for several values of L, and the resulting values of G共c , L兲 are fitted by varying t / and the parameters 1, 2, and of Eq. 共12兲. Since is known, the best fit provides then the resulting value for t. In our work, we have extracted the conductance of a system of size L by isolating first the percolating cluster as described in Sec. III. Then, we implemented a numerical decimation procedure which replaces, by applying iterative exact transformations, the initial set of conductances 共belonging to the percolating cluster兲 by a single conductance.23,24 Since the iterative transformation is exact, the final single conductance value coincides with G共c , L兲. The transformation simply proceeds by successively eliminating each node in the cluster and replacing the adjacent conductances by following Kirchoff’s laws. A collection of t / exponents obtained for different / values is plotted in Fig. 6 for d / = 1 共full circles兲 and d / = 2 共open squares兲. Within the error bars, determined from the statistics and the error fits to Eq. 共12兲, the calculated t / exponents for d / = 1 practically coincide with the universal value t0 / ⯝ 2.0/ 0.88⯝ 2.27 共horizontal dashed line兲 independently of the tunneling factor value. For d / = 2, the mean value of t / lies slightly above 2.27 for / larger than about 6, but universality is still confirmed within the error bars in the whole range of / . The results reported in Fig. 6 confirm therefore our previous suggestion12 that, at the percolation threshold c of the composite system, the transport exponent is universal, independently of the tunneling characteristic distance and of the value of the cutoff parameter d. The significance of this result with respect to the classical tunneling-percolation 共one-dimensional-like6兲 theory of
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FIG. 7. 共Color online兲 The calculated conductance as a function of the proximity 共x − xc兲 / xc to the percolation threshold for different values of / for L / = 40. This is for 共a兲 d / = 1 and 共b兲 d / = 2. The solid lines represent the universal dependence 共x − xc兲2.
nonuniversality is best appreciated by looking at the corresponding values of the Kogut–Straley8 exponent which, by taking into account the fact that the particles are impenetrable, reduces to
␣=1−
/2 . a−
共13兲
For d / = 1, the percolation threshold value used in Fig. 6 is c ⬇ 0.32, which corresponds to a hard-core sphere density
of xc ⬇ 0.04. The corresponding value of the mean nearestneighbor distance a can be obtained either from Fig. 5 or, more transparently, by identifying a with the radius of a sphere containing, on average, only one particle at its center.6,12 Since for x ⬇ 0.04 the system is well within the low concentration regime, a simple estimate of a is obtained by assuming that the particles are implanted randomly with a concentration , so that a can be extracted from 共4 / 3兲a3 = 1, leading to a / ⯝ 共1 / x兲1/3 / 2 ⬇ 1.46. By using Eq. 共13兲, the resulting Kogut–Straley8 exponent reduces therefore to ␣ ⯝ 1 − 1.087 / which, according to Eq. 共2兲, would predict nonuniversality as soon as / ⬎ 1.217. The resulting / dependence of t / as obtained by using Eq. 共2兲 is plotted in Fig. 6 by the solid black curve. According to the original tunneling-percolation theory, therefore, t / would exceed 10 already at / ⯝ 8.6, while our continuum model gives t / ⯝ 2.27 for even larger values of / . By repeating the same analysis for d / = 2 共for which c ⯝ 0.332 and xc ⯝ 0.012兲, one would expect nonuniversality for / ⬎ 0.473, with an even steeper enhancement of t / 共red or gray curve in Fig. 6兲 compared to the d / = 1 case. V. TRANSPORT EXPONENT FROM THE CONCENTRATION DEPENDENCE
The universality of the transport exponent for the tunneling-percolation continuum model established in Sec. IV poses the problem of how to account for the observed nonuniversality in real systems within the same theoretical
framework. As pointed out in Sec. I, deviations of the transport exponent t from the universal value t0, when observed, are obtained by measuring the conductance as a function of the 共estimated兲 distance from the percolation threshold. To examine whether our model also displays similar features, we have calculated the conductance G as a function of the concentration x of the conducting particles. For values of x in the vicinity of xc, we have used the same numerical decimation algorithm that was described in Sec. IV, while for larger values of x, we have implemented a preconditioned conjugate gradient algorithm,25,26 which provides a faster computational technique when x is far above xc. The results are plotted in Fig. 7共a兲 for d / = 1 and in Fig. 7共b兲 for d / = 2. To mitigate the effects of the finite size 共L / = 40 in this ˜ 共x兲 = 关G共x兲 case兲 in Fig. 7, we have plotted the quantity G − G共xc兲兴 / G共xc兲, where G共xc兲 is the conductance at the critical concentration xc for L / = ⬁. Note, in particular, that this ˜ 共x兲 ⲏ 10 and that it was normalization is unimportant for G introduced to emphasize the similarity of behavior for all curves close to the percolation threshold xc. As is apparent from the figure, independently of / and of the cutoff parameter, all the calculated conductances follow the same power-law behavior as x − xc → 0, not deviating much from 共x − xc兲2 共solid line兲. We have therefore re-obtained the result of Sec. IV: transport is universal at, or very close to xc, the percolation threshold. However, away from xc, the results of Fig. 7 clearly indicate also that different behaviors arise depending on the value of / . Indeed, for / = 0 共solid circles兲, the conductance matches approximately the 共x − xc兲2 behavior in the whole concentration range. This is, of course, due to the fact that for / = 0, the interparticle conductance 关Eq. 共4兲兴 becomes a constant regardless of the tunneling distance, giving rise to universal transport. On the contrary, for larger values of / , G displays a steeper increase as one moves away from xc and the deviation from the 共x − xc兲2 behavior is stronger for larger / values and for the d / = 2 case compared to d / = 1.
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TRANSPORT EXPONENT IN A THREE-DIMENSIONAL… VI. COMPARISON OF EFFECTIVE-MEDIUMAPPROXIMATION-LATTICE RESULTS WITH OFFLATTICE MODEL RESULTS
The behavior shown in Fig. 7 is qualitatively similar to that obtained in our previous calculations based on a lattice tunneling-percolation model,12 where it was shown that the transport exponent acquires a concentration dependence, recovering the universal value t0 only for x → xc. It is useful then, concerning the present off-lattice model, to discuss the origin of this behavior by considering an EMA of the lattice model. To this end, let us consider a bond percolation model of a cubic lattice with a bond conductance distribution function 共兲. The corresponding EMA equation is1,3
冕
1
d共兲
0
−G = 0, + 2G
共14兲
where G is the effective conductance of the system. If p is the fraction of bonds with nonzero tunneling conductances distributed according to some function h共兲, Eq. 共14兲 can be rewritten more conveniently as 2G
冕
1
d
0
p − pc h共兲 = , + 2G p
共15兲
where pc = 1 / 3 is the EMA bond percolation threshold. Following Ref. 12, we assume that h共兲 is given by tunneling processes between nearest-neighbor particles, so that Eq. 共15兲 reduces to 2G
冕
⬁
0
p − pc P共r兲 = , dr 共r兲 + 2G p
共16兲
where 共r兲 is the tunneling conductance given in Eq. 共4兲 and P共r兲 is the distribution function of the distance r between two nearest neighbors. For simplicity, we consider the limiting case of pointlike particles 共 = 0兲, for which P共r兲 is given by the Hertz distribution 关Eq. 共3兲兴. By introducing the dimensionless variable z = r / a and the function W共z兲 = 2G / 关2G + exp共−2az / 兲兴 and by integrating Eq. 共16兲 by parts, one obtains 2G + 1 + 2G
冕
⬁
0
p − pc dzW⬘共z兲exp共− z3兲 = . p
再 冋 冉 冊册 冎 1 ln 2a 2G
3
=
p − pc . p
ⲏ exp共−2a / 兲, the second term in the left-hand side of Eq. 共18兲 dominates over the first one, and the resulting conductance is given by
冋 冉
2a p ln G ⬇ exp − p − pc
共18兲
From the above result, we obtain that the conductance reduces to the universal EMA power-law G ⬇ 共p − pc兲 only if p − pc is so small that G Ⰶ exp共−2a / 兲. Conversely, when G
冊册冉 冊 1/3
p − pc = p
2a/兵ln关p/共p − pc兲兴其−2/3
, 共19兲
where in the second equality, we have made explicit that in this regime the conductance is governed by a p-dependent exponent. In the whole p − pc region, the solution to Eq. 共18兲 may be expressed conveniently as a conductance of the form G⬇
冉 冊 p − pc p
with p-dependent exponent,
共17兲
For 2a / Ⰶ 1, the function W⬘共z兲 = dW共z兲 / dz depends only weakly on z and it is proportional to 共2a / 兲G, so that the solution of Eq. 共17兲 is G ⬇ 共p − pc兲, i.e., the conductance is universal with EMA exponent t0 = 1. Instead, for 2a / Ⰷ 1, W⬘共z兲 is a strongly peaked function at z = zⴱ ⬅ 共 / 2a兲ln关1 / 共2G兲兴, and its integral between 0 and ⬁ is W共⬁兲 − W共0兲 = 1 / 共1 + 2G兲. Therefore, for small G values, W⬘共z兲 can be replaced by the Dirac-delta function ␦共z − zⴱ兲, thus reducing Eq. 共17兲, for G Ⰶ 1, to 2G + exp −
FIG. 8. 共Color online兲 Bond-percolation dependence of the local exponent as calculated from a numerical solution of Eq. 共17兲 for different values of ␣ = 1 − / 2 / 共a − 兲 for pointlike particles 共filled symbols兲 and for hard-core particles 共open symbols兲. The solid line describes the 关ln p / 共p − pc兲兴−2/3 dependence.
t共p兲 = t0 +
冉
t共p兲
共20兲
,
2a p ln p − pc
冊
−2/3
,
共21兲
which contains the limiting behaviors discussed above. The validity of Eqs. 共20兲 and 共21兲 is tested in Fig. 8 where the quantity 共 / 2a兲兵ln共G兲 / ln关共p − pc兲 / p兴 − t0其, obtained by solving Eq. 共17兲 by numerical integration and iteration, is plotted as a function of 共p − pc兲 / p for different values of ␣ = 1 − / 2a 共filled symbols兲. In agreement with the above results, the curves are nearly independent of ␣ and approximately follow 兵ln关p / 共p − pc兲兴其−2/3 共solid black line兲 in the whole range of 共p − pc兲 / p, verifying therefore the validity of Eq. 共21兲. An extension of the EMA analysis presented above is briefly described in the Appendix for the case of impenetrable particles with a hard-core diameter ⫽ 0. The interesting feature is that both results show that the ratio of the particle-surface distance to the tunneling decay length is the basic parameter that governs the nonuniversal behavior, as is
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FIG. 9. 共Color online兲 Concentration dependence of the “local” exponent of the continuum tunneling-percolation model. The symbols refer to the numerical calculations of the conductance replotted as explained in the text. The solid line is Eq. 共23兲.
the case in one-dimensional systems, but in the threedimensional case, this dependence is weakened, reaching universality, as the percolation threshold is approached. The off-lattice results of Fig. 7 can now be interpreted in the light of the EMA approach. We can assume now that also for the off-lattice case, the main contribution to G comes from the tunneling processes between nearest-neighboring particles. Furthermore, since now the relevant quantity is the volume fraction x of the conducting phase, we make use of the site percolation formulation of the EMA equations as described in Ref. 3. This amounts to replace in Eq. 共17兲 共p − pc兲 / p by 共x2 − x2c 兲 / x2 where, in order to approximately map the off-lattice case to the EMA formulation, xc is now regarded as the percolation threshold determined by d. Finally, since the tunneling length is limited by the cutoff d, the upper limit of integration of Eq. 共17兲 is set equal to d / a. As noticed in Sec. III, d / a is approximately 1.6ac / a, where ac is the nearest-neighbor mean distance at percolation, so that for large 2a / the upper limit of integration can be actually be safely set to infinity because the peak of W⬘共z兲 is at zⴱ Ⰶ 1.6ac / a ⬍ 1.6. Therefore, by following the same steps which led to Eqs. 共20兲 and 共21兲 and for finite conducting particle diameters 共see the Appendix兲, the conductance of the tunneling-percolation model in the continuum can be argued to be well described by a generalized power-law behavior with a concentration dependent exponent of the form t共x兲 = t0 +
冉 冊
a − x2 − x2c f , /2 x2
共22兲
where the mean distance a depends implicitly on x, t0 ⯝ 2 is the universal transport exponent for three dimensions, and f共y兲 is a generic function which goes to zero when y → 0. In Fig. 9, we have replotted the results of Fig. 7, together with few more cases for different values of / , in terms of ˜ 兲 / ln关共x2 − x2兲 / x2兴 − t 其 which, according to 关 / 2共a − 兲兴兵ln共G 0 c
Eq. 共22兲, should all follow a single master curve f共y兲, with y = 共x2 − x2c 兲 / x2.27 Such a collapse of data for different / values, and independently of d / , is indeed verified in Fig. 9, where all data approximately follow a master curve of the form
冉 冊
f共y兲 = 1.5 ln
1 y
−4/3
,
共23兲
which is plotted by the solid line. Note that the exponent −2 / 3 appearing in EMA formula 共21兲 is replaced by −4 / 3 in Eq. 共23兲 and that the range of variation of f共y兲 is much broader compared to the EMA results of Fig. 8. The outcome of the off-lattice model of tunneling-percolation summarized in Fig. 9 basically confirms therefore that the main contribution to the conductance stems from the first nearest-neighbor particles, which is reflected in the explicit dependence upon ␣ of the local transport exponent t共x兲.
VII. DISCUSSION
Following the above results, it is interesting at this point to discuss what a strongly varying local exponent t共x兲 would imply when the x dependence of the conductance is forced to be of the form of Eq. 共1兲, with an x-independent exponent, although not necessarily equal to t0. This issue is fundamental in establishing a connection with experiments in real composites, where indeed Eq. 共1兲, with t independent of x, is used to fit the concentration dependence of the measured conductance, without a prior knowledge of the percolation threshold xc. We have fitted then the data of Fig. 7, and few more cases for different values of / , by using Eq. 共1兲 with the prefactor G0, the exponent t, the critical concentration xc, and an additive constant 共to take into account the finite size of the system兲 as fitting parameters. The results are plotted by filled squares in Figs. 10共a兲 and 10共b兲 for d / = 1 and
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FIG. 10. 共Color online兲 Comparison between the Kogut– Straley 共Ref. 8兲 exponent 共solid lines兲, and the exponents obtained from the finite-size analysis 共filled circles兲 and the apparent exponent tⴱ 共filled squares兲 obtained by using Eq. 共1兲. The horizontal dashed line is the universal value t0 = 2. 共a兲 d / = 1 and 共b兲 d / = 2.
d / = 2, respectively. Also shown are the corresponding finite-size results of Fig. 6 共filled circles兲 and the Kogut–Straley8 exponent 关Eq. 共13兲兴 共solid lines兲. From the plots in Fig. 10, it is clear that forcing the exponent t0 to be independent of the concentration leads to a sort of average, say tⴱ, of the local one, t共x兲. Consequently, the fitted exponent results are confined between the universal value obtained from the finite-size analysis and the nonuniversal Kogut–Straley8 exponent. Hence, by reducing / , the mean value tⴱ of the fitted, or apparent, exponent decreases from a large value at large / toward the universal limit t0 ⯝ 2 as / → 0. Furthermore, since the local exponent t共x兲 has a stronger x dependence for larger / values, the error bars resulting from the fitting procedure get reduced as well when / → 0. Finally, for fixed / , the error bars increase by going from d / = 1 to d / = 2 partially because of the stronger t共x兲 variation and partially because of the increased finite-size effects. A last interesting feature is given by the / dependence of the apparent critical concentration xⴱc resulting from the fits to Eq. 共1兲. In Fig. 11, the apparent exponent tⴱ of Fig. 10 is plotted as a function of the corresponding xⴱc values. This result shows a clear correlation between the increase in tⴱ, resulting from the decrease in / , and the 共apparently counterintuitive兲 reduction in xⴱc . This feature can be understood ⴱ by noticing that in order to maintain 共x − xⴱc 兲t approximately ⴱ constant, as long as x − xc Ⰶ 1, an increase in tⴱ would require an increase in x − xⴱc and so lead to a decrease in xⴱc for fixed x. What makes this feature interesting is that, when collecting the measured exponents and the corresponding critical concentrations in real disordered composites, a trend similar to the one of Fig. 11 is observed: namely, for the same type of particles 共say, spheres in cellular composites10兲, lower percolation thresholds are accompanied by larger and more dispersed values of the measured exponent. This is, of course, not the case when other reasons 共such as when the conductance distribution changes with the particle shape6,28兲 modify the percolation threshold.
VIII. SUMMARY AND CONCLUSION
In the present study, we have considered an off-lattice model of the tunneling-percolation mechanism of electrical conduction in disordered composites. We have shown that, by using a numerically exact decimation procedure and a finite-size analysis, at the percolation threshold, the conductance critical exponent is universal and close to the value t0 ⯝ 2. Conversely, by moving away from the percolation threshold, we have demonstrated that, depending on the value of the characteristic tunneling distance, the exponent acquires a strong concentration dependence, attaining values larger, or much larger, than t0. We have interpreted this feature in terms of the strongly varying distribution of the local tunneling conductances, which leads to a shrinking of the 共universal兲 critical region to concentrations very close to the percolation threshold. In particular, we have shown the simi-
FIG. 11. 共Color online兲 The apparent conductivity exponent tⴱ as a function of the corresponding apparent critical concentrations xⴱc for d / = 1 and d / = 2.
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FIG. 12. 共Color online兲 Bondpercolation dependence of the local exponent as calculated from a numerical solution of Eq. 共A2兲 for different values of 共a兲 a / and of 共b兲 / . In parenthesis are reported the corresponding values of ␣ = 1 − 2共a− 兲 . The solid line dep scribes the 关ln共 p−pc 兲兴−2/3 dependence.
larity of the results obtained from the lattice and off-lattice models, indicating that the first nearest neighbors dominate the values derived for the conductivity exponent in the tunneling-percolation problem. This is not a trivial conclusion since there is a clear competition between the decrease in the particle-to-particle conductance and the increase in the possible number of routes with the increase in the particleto-particle distance. These results therefore confirm similar findings for a realistic continuum model and the lattice models of tunneling percolation. In fact, for the lattice, we were able to derive 共in an asymptotic case兲 an analytic expression for the conductivity exponent in the three-dimensional system, noting that such an expression was given previously only for the much simpler one-dimensional system. Furthermore, we have evaluated the apparent exponent arising from a fit of our numerical results by forcing the conductance exponent to be independent of the concentration. This enables us to show that such apparent exponent has a nonuniversal behavior, despite the fact that the system is strictly universal, leading to a possible explanation of many experimental results reported in the literature.
⫽ 0. We shall use a simplified version of the function reported in Ref. 11. This version still describes rather accurately the distribution of nearest-neighboring particles in the low density regime, P共r兲 ⬇ 3
冋
册
r3 − 3 r2 , 3 exp − 共a − 兲 共a − 兲3
共A1兲
where r ⱖ and a is approximately the distance between the centers of two nearest-neighboring spheres. In this way, Eq. 共17兲 becomes 2G + 1 + 2G
冕
⬁
/共a−兲
冋 冉 冊册 a−
dzW⬘共z兲exp − z3 +
3
=
p − pc , p 共A2兲
where z = r / 共a − 兲 and W⬘共z兲 = dW共z兲 / dz, and 2Gⴱ , 2G + exp关− 2共a − 兲z/兴
W共z兲 =
ⴱ
共A3兲
where Gⴱ = exp共−2 / 兲G. For small Gⴱ, the function W⬘共z兲 is well approximated by the Dirac-delta function ␦共z − zⴱ兲, where zⴱ is given now by
ACKNOWLEDGMENTS
This study was supported in part by the Israel Science Foundation 共ISF兲 and in part by the Swiss National Science Foundation 共Grant No. 200020-116638兲. I.B. acknowledges the support of the Enrique Berman chair in Solar Energy Research at the HU.
zⴱ =
共A4兲
and, since zⴱ is always larger than / 共a − 兲, for small Gⴱ, Eq. 共A2兲 becomes
冋冉 冊
APPENDIX
In this Appendix, we extend the EMA analysis of Sec. VI to the case where the conducting particles are impenetrable with a hard-core diameter . To this end, it suffices to replace in Eq. 共16兲 the Hertz distribution of Eq. 共3兲 that is valid for = 0 by the corresponding distribution function for
冉 冊
1 ln , 2共a − 兲 2Gⴱ
2G + exp
a−
3
册
− 共zⴱ兲3 =
p − pc . p
共A5兲
Consider now the case for which the second term in the left-hand side is larger than the first one. By using Eq. 共A4兲 and the definition of Gⴱ, Eq. 共A5兲 reduces to
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冋
冉 冊册 冉 冊 冉 冊
1 + ln a − 2共a − 兲 2G
3
−
a−
3
= ln
p , p − pc 共A6兲
which can be recast in the following form: G ⬇ e−2/
冉 冊 p − pc ␥p
关2共a−兲/兴兵ln关␥ p/共p − pc兲兴其−2/3
,
共A7兲
where ␥ = exp关 / 共a − 兲兴3. We have thus shown that also for the more general case of impenetrable particles, the conductance is governed by a p-dependent exponent, reducing to Eq. 共19兲 of Sec. VI in the very dilute limit, for which / a is
13 N.
*
[email protected] Kirkpatrick, Rev. Mod. Phys. 45, 574 共1973兲. 2 D. Stauffer and A. Aharony, Introduction to Percolation Theory 共Taylor & Francis, London, 1992兲. 3 M. Sahimi, Heterogeneous Materials I 共Springer, New York, 2003兲. 4 B. I. Halperin, S. Feng, and P. N. Sen, Phys. Rev. Lett. 54, 2391 共1985兲. 5 S. Vionnet-Menot, C. Grimaldi, T. Maeder, S. Strässler, and P. Ryser, Phys. Rev. B 71, 064201 共2005兲. 6 I. Balberg Phys. Rev. Lett. 59, 1305 共1987兲; For a recent review, see I. Balberg, D. Azolay, D. Toker, and O. Millo, Int. J. Mod. Phys. B 18, 2091 共2004兲. 7 E. K. Sichel, P. Sheng, J. I. Gittleman, and S. Bozowski, Phys. Rev. B 24, 6131 共1981兲. 8 P. M. Kogut and J. P. Straley, J. Phys. C 12, 2151 共1979兲. 9 O. Stenull and H.-K. Janssen, Phys. Rev. E 64, 056105 共2001兲. 10 C. Grimaldi, T. Maeder, P. Ryser, and S. Strässler, Phys. Rev. B 68, 024207 共2003兲. 11 S. Torquato, B. Lu, and J. Rubinstein, Phys. Rev. A 41, 2059 共1990兲. 12 C. Grimaldi and I. Balberg, Phys. Rev. Lett. 96, 066602 共2006兲. 1 S.
small 共␥ ⬇ 1兲. Despite the differences between the above equations 关Eqs. 共A7兲 and 共19兲兴, the dominant contribution to the conductance in the whole p − pc region is still of the form 共p − pc兲t共p兲, with t共p兲 given by Eq. 共21兲 with a replaced by a − . This behavior is demonstrated in Fig. 12, where 共1 − ␣兲兵ln共G兲 / ln关共p − pc兲 / pc兴 − t0其 关with ␣ given by Eq. 共13兲 and G obtained by a numerical solution of Eq. 共A2兲 is plotted for a / = 2兴. This is for several values of / , Fig. 12共a兲, and for fixed / but different a / values, Fig. 12共b兲. Despite of nearly 1 order of magnitude change in / and a / , the different curves are only weakly dependent on ␣ and do not deviate much from the 兵ln关p / 共p − pc兲兴其−2/3 dependence which is presented by the solid lines.
Johner, C. Grimaldi, and P. Ryser, Physica A 374, 646 共2007兲. 14 M. D. Rintoul and S. Torquato, J. Phys. A 30, L585 共1997兲. 15 J. E. Gubernatis, Phys. Plasmas 12, 057303 共2005兲. 16 N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, J. Chem. Phys. 21, 1087 共1953兲. 17 J. Hoshen and R. Kopelman, Phys. Rev. B 14, 3438 共1976兲. 18 A. Al-Futaisi and T. W. Patzek, Physica A 321, 665 共2003兲. 19 S. B. Lee and T. J. Yoon, J. Korean Phys. Soc. 33, 612 共1998兲. 20 G. D. Scott and D. M. Kilgour, Br. J. Appl. Phys. 2, 863 共1969兲. 21 C. D. Lorenz and R. M. Ziff, J. Chem. Phys. 114, 3659 共2001兲. 22 I. Balberg and N. Binenbaum, Phys. Rev. A 35, 5174 共1987兲. 23 R. Fogelholm, J. Phys. C 13, L571 共1980兲. 24 R. Fogelholm, Proceedings of the Fourth ACM Symposium on Symbolic and Algebraic Computation 共ACM, Snowbird, Utah, United States, 1981兲, p. 94. 25 G. G. Batrouni and A. Hansen, J. Stat. Phys. 52, 747 共1988兲. 26 M. Benzi, J. Comput. Phys. 182, 418 共2002兲. 27 Note that G ˜ = 关G共x兲 − G共x 兲兴 / G共x 兲 is again used here to minic c mize the finite-size effects. 28 Z. Rubin, S. A. Sunshine, M. B. Heaney, I. Bloom, and I. Balberg, Phys. Rev. B 59, 12196 共1999兲.
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