hys. stat. sol. b 36, 65-633 003 Effective conductivity in a lattice model for binary disordered media with comlex distributions of grain sizes R. PIASECKI Institute of Chemistry, University of Oole, Oleska 48, 45-05 Oole, Poland PACS: 05.90.m, 7.80.Ng, 7.80.Tm Using numerical simulations and analytical aroximations we study a modified version of the twodimensional lattice model [R. Piasecki, hys. stat. sol. b 09, 403 1998] for random :1 systems consisting of grains of high low conductivity for the --hase, resectively. The modification reduces a sectrum of model bond conductivities to the two ure ones and the mixed one. The latter value exlicitly deends on the average concentration γ of the -comonent er model cell. The effective conductivity as a function of content of the -hase in such systems can be modelled making use of three model arameters that are sensitive to both grain size distributions, GSD and GSD. owever, to incororate into the model information directly connected with a given GSD, a comuter simulation of the geometrical arrangement of grains is necessary. By controlling the olydisersity in grain sizes and their relative area frequencies, the effective conductivity could be raised or decreased and correlated with γ. When the hases are interchanged, a hysteresis-loo like behaviour of the effective conductivity, characteristic of dual media, is found. We also show that the toological non-equivalence of system s microstructure accomanies some GSDs, and it can be detected by the entroic measure of satial inhomogeneity of model cells. 1. Introduction The effective roerties of binary disordered media are often redicted by simle models on the basis of the comonent roerties see [1] for a recent review. Both theoretical and exerimental investigations devoted to article size influence on macroscoic roerties attract increasing attention. Incororation of grain size distributions GSDs into the effective medium aroach EA by a two-dimensional lattice model [] for the effective conductivity of binary granular systems was motivated by recent onte Carlo simulations [3, 4 ]. The latter, though erformed for mono-sized grains, revealed oscillations of the critical ercolation coverage as a function of the size of grains. or finite-sized objects with circular symmetry in two dimensions and sherical one in three dimensions, the ercolation threshold is a well-defined quantity as shown by numerical investigations on a lattice [5]. On the other hand, the effect of the geometrical arrangement of equally sized hard sheres on the average electrical roerties of two- and three-comonent comosites was simulated using a modified transfer matrix algorithm for a triangular network [6]. The results comared with some common mixture formulas indicate that the geometrical arrangement effects become more imortant as the samle size decreases and the filling fraction aroaches the effective ercolation threshold. It was assumed there that the sheres fill the sace in a closeacked attern. In this context, it is worth to mention the exerimental studies [7] how several shaes comosed of welded sheres doubles, hexagons, small and large triangles, diamonds, triles, traezoids ack in two dimensions. Ordered domains occurred near a acking density of 0.8 excet traezoids, which all ackings remain disordered and near the transition density, even e-mail: iaser@uni.oole.l
after annealing by shaking. A numerical work on mono-sized square and rod-like articles was done for the descrition of the conductivity of more secified materials, solid electrolytes [8]. A comlementary study within EA in both two and three dimensions for disersed ionic conductors revealed analytical exressions for the article size deendence of the ercolation thresholds of the model [9]. Also the overall deendence of the conductivity of the three-dimensional continuum ercolation model on the concentration of the insulating hase, enhanced interface conductance and article size, was in good qualitative agreement with exeriments. Recently, the same model has been used [10] for studying the ionic transort in nano- and microcrystalline comosites with thermally activated comonent conductivities. It was found that the ionic conductivity deends drastically on exlicitly taking into account the two different grain sizes of both comonents. The above mentioned examles show the real imortance of incororation of the changes in the size of the disersed articles for effective media theories, which describe at least qualitatively the conduction roerties. A novel, so called local linearization method of determination of the effective nonlinear conductivity has been given recently [11]. Also the network extension [1] of EA embodies essential hysics of macroscoic roerties of random heterogeneous materials. Within such an aroach, the macroscoic nonlinear resonse characteristics of metalinsulator mixtures was obtained for a minimal model bond ercolative network [13] with three tyes of bonds: ohmic, tunnelling and urely insulating ones. In Ref.[], for certain GSDs the non-monotonous critical behaviour of conductor-insulator and conductorsuerconductor systems was demonstrated. Within this aroach every grain for --hase of high low conductivity is comosed of grains of tye 1 s 1, which can occuy alone the centre of a unit bond on a reference square lattice marking a ure unit --bond. The effective dc conductivity, where is the fraction of unit -bonds, was considered on a coarsened lattice for systems with grains simle in form only. ere we study a modified two-dimensional model that allows for differently shaed grains and comlex GSDs. A main goal is to find how both GSD and GSD and their reverse combination affect. This rovides insight into how emirical size histogram information can be used to redict effective roerties, e.g. for random granular films and their dual counterarts.. odel and results Consider now the coarsened square lattice with model bonds each of length l > 1. To each of the bonds a square model cell is assigned consisting of l ositions, that is, the centres of neighbouring unit bonds. Now, besides square shae, e.g., for 4 s 4 grains, one allows also rod-like if l 4 or T-like grains, letter or zigzag shaed grains and their mirror reflections. In ig.1, exemlary configurations at length scale l=3 are resented for certain GSDs. Given that the area of an s n grain is measured by the number n of the occuied ositions, a simle rule analogous to the shaking rocedure used in numerical simulations reads: the sum of all - and -grain areas inside of every model cell must be equal to l. et N and N K for a given be, resectively, the number of all model bonds and bonds of tye K =,, and. We comment further on mixed case below. The bond fractions defined as K N K N fulfil the obvious condition = 1. 1 Similarly to Ref.[], γ i n i l and 1 γ i is the local fraction of n i and l n i of unit - and - bonds in i-th model cell. Thus, for a ure --bond we have γ i = 1 = 0, while for i-th - bond 0 < γ i < 1. Now, for the fraction of unit -bonds the elementary relation holds
3 { } = i i N N ] [ 1 γ γ, where γ is an average fraction of unit -bonds er mixed cell. ig. 1. Examles of the model bonds each of length l=3 and cells marked by dashed lines, each consisting of l = 9 ositions for chosen distributions {s 1, s, s 4 }:{s 1, s } of - hase filled circles and -hase oen circles, resectively. In contrast to the reviously used model [], differently shaed grains here of s 4 area are also allowed. Note that the resented model bonds and cells belong to the mixed tye,. In order to simlify the roblem the following aroximation is made: each mixed-cells with a certain i conductivity can be relaced by a reresentative mixed cell with γ and 1γ fractions of - and -bonds, and the mixed conductivity be calculated within the EA from a corresonding quadratic exression, 1 γ = 0. Thus, 4 1 1 = γ γ. 3 The general distribution of the corresonding bond conductivities < < is now of a simler form comared to Eq.1 in Ref.[] on age 405, it should read without a misrinted = 0 on r.h.s., P δ δ δ =. 4 ollowing [1] the equation for the overall effective conductivity can now be written as 0 =. 5 The fractions,, and are difficult to calculate analytically due to the large number of ossible local configurations in a model cell. The same refers to the γ function. owever, using Eqs.1 and at least two of the four quantities can be treated as indeendent ones, for instance, and γ. Thus, from Eqs.1 and we have. 1 1, γ γ = = 6 As observed in numerical simulations for various GSDs, the fractions and are concave functions while the mixed fraction is a convex one reaching a maximum, say β, at certain =α. The toological equivalence of equally sized - and -grains imlies α = 0.5 while for comlex GSDs we exect α 0.5. Suorted by reliminary comuter simulations, a trial function aroximating the behaviour of is roosed as l = 3
4 a0 a1 a β b0 b1 b for for for 0 < < α, = α, α < < 1, where constant coefficients a k and b k can be determined from the conditions i ii iii lim lim lim The final form is 0 α α = 0 and = lim α d = lim d lim α 1 = 0, d = 0. d α 1 β [α ] β α α = α 1 β [1 α α 1α 7 = β, 8 ] β 1 α for for 0 < α, α < 1. rom Eqs.6 and 9 the remaining and bond fractions can be also arameterized by α, β, and γ. Note that the final form of the model function ossesses the desirable symmetry roerty i, α, β = 1, 1α, β. As comuter simulations revealed, such a symmetry is connected with the hase interchanging. So, the more detailed form is, ; 1, ; α, β =, ; 1, ; 1α, β, where the bold symbols clearly show that in the simlified notation both the concentrations and 1 refer to the -hase. Remembering this, in the following the simler notation will be used. urther symmetry relations are: ii γ γ1 = 1, iii, α, β, γ = 1,1α, β,1γ and vice versa, see Eqs.6. urther, Eq.3 for the mixed conductivity as well as Eq.5 for the effective conductivity is invariant under the simultaneous interchange γ 1γ,. A general identity linking the twodimensional effective conductivity of a two-hase macroscoically isotroic system to the effective conductivity of the same microstructure but with the hases interchanged, called the hase-interchange relation, has been obtained by Straley [14]. The corresonding exression is α β γ α β γ =, ;1, ;,,, ;1, ;1,,1. 10 This relation was checked making use of comuter simulations. or examle, see ig.a, where the exemlary air of = 0. and 1 = 0.8 = 0.6 and 1 = 0.4 shows that the sum of logarithms of the corresonding values indicated by black white arrows always equals 3, as exected for the given hase conductivities, =1 and =10 3 in a.u. et us check the limiting behaviour of our model. Its examination shows that for l = 1 there are no mixed model cells, γ = 0 or 1, = 0, =, = 1, and the Bruggeman model with a standard binary distribution of ure bonds is recovered. or l >> 1 and medium there is a negligible robability of aearance of ure model cells, so γ, 1, 0, 0, and from Eq.5 results that, i.e., the mixed conductivity becomes the global one, as exected. If both hases consist of equally monosized grains and the system is considered at larger scales, then using an auxiliary arameter, e.g. δ =α, where the derivative is taken over, a linear function for an aroximated 9
5 descrition of γ can be deduced, γ β δ 1 β 1 δ β. In general, however, the γ values for a given concentration are evaluated by erforming comuter simulations. At this stage, we would like to concentrate on the numerical evaluation of model functions γ and to calculate. Also and values are evaluated and then Eq.5 is solved for the most interesting range of length scales, i.e., when l is comarable with the largest grain size of the given GSD and GSD. Comuter simulations on a square lattice with 4900 bonds each of length l = 6 are erformed for = 0.1, 0.,..., 0.9. The robability of selecting a --grain is roortional to the initial area fraction of grains of a given tye. or the considered concentrations this rule is good enough to overass a blockage roblem and it is not so restrictive for randomness as, e.g., the sequential drawing. The effective conductivity is averaged over 1000 run trials. With a high accuracy the same results are obtained by first averaging the bond fractions and then solving Eq.5. To illustrate how the effective conductivity deends on the resective grain sizes we consider first -hase lognormal distributed among the mono-sized grains of the -hase for two different distributions. Then, a slightly erturbed log-normal GSD and mono-sized -grains as well as the arbitrarily chosen GSD are considered. Simultaneously, the corresonding haseinterchanged situations are also illustrated in igs. and 3. or commercially available metal owders used in metal oxide/metal comosites, see for instance Ref.[15], the article diameters d can be characterized by a log-normal distribution 1 f d = ex [ ln d µ ], 11 d π where µ is the average of the logarithms of the diameters and is the standard deviation of the distribution. Assuming that the side lengths d of square grains satisfy Eq.11 and using the equality d = s n, this formula can be rewritten to account for the grain area distribution dn s = fsds, A f s = ex [ ln s µ s s ], 1 s π s where µ s is the average of the logarithms of the grain areas and s is the standard deviation of the distribution. Because we consider discrete values of the areas of grains s n, a factor A allows a set of ossible values of corresonding relative area frequencies to be constrained by Σf i = 1. et us consider two {s 4, s 9, s 16, s 5, s 36 } distributions for -grains with the relative area frequencies f i 1 f i 0.007 0.138, 0.04 0.551, 0.417 0.50, 0.7 0.051, 0.100 0.010, deicted in the inset in ig.a, filled oen diamonds, and common {s 1 } uniform distribution for -grains. The log-normal distribution curves have the same s = 0.4, but different µ s 1 = 1.5 and µ s = 1.. The average values are shown in ig.a, filled and oen diamonds. The dashed solid lines 1 are guides to the eye only. or both log-normal distributions considered the figure shows, that -grains olydisersed in size when embedded into a matrix of mono-sized -grains, favour a higher effective conductivity in comarison to the reverse case. Note that for these distributions with a fixed s the higher increase in, referring to the transition 1-r 1, is connected with the higher average of the logarithms of the grain areas. So, the frequency f i 1 with a relatively higher number of larger -grains is more effective in this context than f i. A hysteresis-loo like behaviour of the effective conductivity, see igs. a and 3a, strikingly remains the behaviour of conductivity obtained for the two-dimensional cooer-grahite dual media, see igs.1 and in Ref. [16]. There, the former figure shows examles of so called NEG negative board and POS ositive board atterns. In the latter figure the exerimental
6 oints of conductivity versus grahite area fraction are fitted by a general effective medium GE equation, which combines most asects of ercolation and effective medium theories [17]. og 0.0-0.5-1.0-1.5 -.0 -.5 a 1 -r frequency 1-r 0.6 f i 0.4 0. f i1 0.0 0 18 s n 36-3.0 0.00 0.5 0.50 0.75 1.00 γ 1.00 0.75 0.50 0.5 b -r 1 1-r 0.00 0.00 0.5 0.50 0.75 1.00 1.00 c 1 d 10 0.75 0.50 S 8 6 1 1-r 0.5 -r 4 -r 1 1-r 0.00 0.00 0.5 0.50 0.75 1.00 0 0.00 0.5 0.50 0.75 1.00 ig.. Examles of GSD deendence of the model related quantities for = 0.1, 0.,..., 0.9 and for = 1 and = 10 3 in a.u. at length scale l = 6. a The effective conductivity log for case 1 with {f i 1;}:{s 1 } filled diamonds and to dashed line, for case 1-r with {s 1 }:{f i 1;} filled diamonds and bottom dashed line, for case with {f i ;}:{s 1 } and -r with {s 1 }:{f i ;} oen diamonds and middle solid lines. Black and white arrows illustrate the hase-interchange relation 10. Area frequencies f i 1 filled diamonds and f i oen diamonds are shown in the inset. b The corresonding γ values. c The corresonding fractions of mixed bonds. d The satial inhomogeneity of coarsened lattice quantified by entroic measure S in the thermodynamic limit [18, 0]. In ig.b the corresonding γ functions are ordered from the to to the bottom, in the same sequence as the conductivity curves. The inversion symmetry of the aroriate airs of γ functions with resect to the oint = 0.5, γ = 0.5 aear in agreement with the already mentioned symmetry relation ii. The corresonding mixed-bond fractions with the reflection symmetry under the relacement 1 described in relation i are resented in ig.c. Now, for most values the following tendency aears for the two log-normal distributions: the lower mixed bonds fraction 1 < favours the higher effective conductivity 1 >, while for the reverse cases the oosite connection is observed,
7 namely 1-r < -r and 1-r < -r. Consider now the quantitative evaluation of microstructure attributes for searching their ossible connections with the macroscoic roerties of a disordered system. The quantity of interest is the entroic measure of the length scale deendent satial inhomogeneity for systems of finite-sized objects, which was fully described and alied in recent aers [180]. Using the notation aroriate for the resent work, the measure can be written at fixed length scale l in the thermodynamic limit as a function of concentration, S = l { ln 1 ln 1 [ γ lnγ 1 γ ln 1 γ ] }. 13 or a given this measure quantifies the average er model cell deviation of a general configurational macrostate related to the actual system s configuration from the reference macrostate related to the most satially uniform distribution. ormula 13 includes combinations of both γ and functions. owever, taking into account relations i and ii, the measure shows a symmetry of the latter one, S, α, β = S 1, 1α, β. Now, the higher satial inhomogeneity S 1 > S, see ig.d, favours the higher effective conductivity 1 >, while for the reverse cases we have the oosite relation, namely S 1-r > S -r and 1-r < -r. Note the asymmetry of the S curves regarding = 0.5, which confirms the toological non-equivalence of - and -hases [0]. et us now consider a erturbed log-normal {s 4, s 9, s 16, s 5, s 36 } distribution denoted as I, with the slightly deviated in comarison to the revious case 1 frequencies f i 1; 0.0463, 0.1531, 0.4535, 0.68, 0.104, deicted in the inset in ig.3a filled triangles, and {s 1 } uniform distribution of -grains. Next, we relace {s 1 } by {s 1, s, s 4, s 1, s 36 } distribution with arbitrarily selected relative area frequencies g i 0.653, 0.1361, 0.0680, 0.004, 0.510 deicted in the inset in ig.3a oen triangles. This way we obtain a modification of case I denoted here as mi. The two reverse cases with the hases mutually interchanged are marked as I-r and mi-r. Now, the clear increase of with regard to case I aears in ig.3a filled circles. Quite surrisingly, the resence of the largest s 36 - grains seem to be necessary for that. or the reverse case mi-r the largest reduction of can be also seen in ig.3a oen circles. The deendence of on GSDs can be roughly understood on the basis of igs.3b and c. et the arrows and symbolise increasing and decreasing values of the quantities followed by them and, for a given, consider the exemlary transition : I mi. This imlies γ see ig.3b, not shown here, see ig.3c, not shown here, not shown here, and finally see ig.3a. The decrease of induces the favourable higher increase of that is more significant than the unfavourable increase of. On the other hand, the increase of the γ function is also essential for the increase of, articularly for the middle concentration values. All these effects lead finally to a higher. This kind of sensitivity to various GSDs is a eculiar feature of our model in which satially distributed mixed cells with various conductivities i are relaced by the identically distributed average mixed cells, each of conductivity. It is worth to mention that a quite intriguing similarity aears between the γ behaviour shown in ig.3b, e.g. for the mono-sized cases with {s 9 }:{s 9 }, symbols, and with {s 6 }:{s 6 }, oen squares, and the so called microstructure arameter ζ 1 see Table 1 and ig.1 in Ref. [1]. A similar observation refers also to ig.b. Continuing the comarison of cases I with mi and I-r with mi-r we must remember that both modifications were done by relacing {s 1 } with {s 1, s, s 4, s 1, s 36 } distribution. Due to the reduced for both cases and the roer behaviour of the γ function we observe
8 for the two modifications a higher satial inhomogeneity than for the initial situations, see ig.3d. og 0.0-0.5-1.0-1.5 -.0 -.5 a I mi I-r frequency mi-r 0.0 0 18 s n 36-3.0 0.00 0.5 0.50 0.75 1.00 0.6 0.4 0. f i ' g i γ 1.00 0.75 0.50 0.5 b I-r mi mi-r 0.00 0.00 0.5 0.50 0.75 1.00 I 1.00 c 1 d 0.75 I-r I 10 0.50 mi-r mi S 8 6 mi-r mi 4 0.5 I-r I 0.00 0.00 0.5 0.50 0.75 1.00 0 0.00 0.5 0.50 0.75 1.00 ig. 3. urther examles of GSD deendence for the model related quantities with the same concentrations as in ig., ure hase conductivities and length scale. a The effective conductivity log for case I with {f i }:{s 1 } to dashed line, for case I-r with {s 1 }:{f i } bottom dashed line, for the modified case mi with {f i }:{g i } filled circles, for the reverse modified case mi-r with {g i }:{f i } oen circles, for mono-sized cases with {s 9 }:{s 9 } symbols and with {s 6 }:{s 6 } oen squares. In the inset the area frequencies are shown, f i 1 dashed line, its erturbed version f i black triangles, and secifically chosen g i oen triangles. b The corresonding γ values. c The corresonding fractions of mixed bonds with an analytical result Eq.9 for both modified cases with estimated α 0.77 for mi and α 0.3 for mi-r and common β 0.71, thick solid lines. d The corresonding satial inhomogeneity of coarsened lattice. The figure also shows the asymmetry of the S curves regarding = 0.5, which again confirms the toological non-equivalence of olydisersed - and -hases in contrast to the two symmetrical mono-sized cases. Toologically non-equivalent microstructure of a system means that the two hases are not ideally randomly disersed and deviations from randomness exist because of local ordering correlations. We can see again that the higher satial inhomogeneity S mi > S I is connected with the higher effective conductivity mi > I, and for the reverse cases the oosite relation S mi-r > S I-r and mi-r < I-r aears. Such a corresondence is found in both investigated cases.
9 In general however, the relations between S and are more comlicated. Qualitatively different behaviour can be seen when the two different mono-sized cases and are comared. Now, γ; = γ; = 0.5 for = α = 0.5 and the relation between satial inhomogeneity and effective conductivity changes for the reverse one when the concentration asses this α value, although the satial inhomogeneity S > S for every. The γ function rovides the simlest way to redict what the sequence of curves is. or a given and -hase log-normal distributed among mono-sized or olydisersed grains of -hase, simulation results show that the larger γ values correlate with higher effective conductivity, comare igs.a and b and igs.3a and b. Another common consequence of olydisersity in grain sizes is a blockage roblem for low and high concentrations, usually for < 0.1 and > 0.9, linked also with a tye of GSD. owever, the model can be readily extended to one containing ores. Then, a conductance equal to zero is attributed to each unoccuied unit bond. Such urely insulating unit bonds can be resent only in mixed cells of our model. Denoting by γ o its average fraction er mixed cell, the corresonding quadratic exression for the mixed conductivity is now [ 1 γ γ o ] 1 γ o = 0. Some symmetry roerties of the model functions are lost but the blocking is ractically eliminated. One more remark is in order. Concerning the question of extending the model ability to three dimensions, the answer is ositive for all odd lengths of model bonds, l=k 1, and for those even lengths that are of l=4k form, where k is a natural number. or examle, for l=3 the model cell of the coarsened regular lattice consists of 19 unit bonds. All we need is relacing the basic conductivity formula within EA by its well-known three-dimensional counterart. 3. Conclusions In this aer, the two-dimensional lattice model describing, within EA, the concentration deendence of the effective conductivity for binary disordered systems has been examined. Aart from the simlicity of the model, a wide variety of grain areas and shaes is included. Such a model can be also alied to artificially synthesised comosite systems with different shaes of grains of both hases. Its analytical form allows for consideration of the whole three-arameter sace, 0 < α, β, γ < 1. The model is highly sensitive to a tye of GSDs as well as their mutual interchanging. In such cases it behaves in a way tyical for dual media. We show that there exists a simle correlation between the model cell average concentration of the -comonent and the effective conductivity. The toological non-equivalence of the system s hases can be easily detected by the entroic measure of satial inhomogeneity of the model cells. Incororation of the orosity of disordered media is straightforward in this model. inally, it should be stressed that the model ossesses a characteristic feature. Namely, it allows for a descrition of the effective conductivity from different size scales oint of view. Thus, it may exhibit information not necessarily revealed by other models of disordered media. References [1] S. TORQUATO, Random eterogeneous aterials, Sringer-Verlag New York 00. [] R. PIASECKI, hys. stat. sol. b 09, 403 1998. [3] S. TOYOUKU, T. ODAGAKI, at. Sci. Eng. A 17/18, 381 1996. [4] S. TOYOUKU, T. ODAGAKI, J. Phys. Soc. Ja. 66, 351 1997. [5] R.E. ARITKAR, ANOJIT ROY, Phys. Rev. E 57, 169 1998. [6] DAE GON AN, GYEONG AN COI, Solid State Ionics 106, 71 1998. [7] I.C. RAKENBURG, R.J. ZIEVE, Phys. Rev. E 63, 061303 001.
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