Pores and Hausdorff dimension in fractal site-percolation systems

Physica A 185 (1992) 222-227 North-Holland

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Pores and Hausdorff dimension in fractal site-percolation systems E.P. Stoll and M. Kolb 1 IBM Research Division, Zurich Research Laboratory, 8803 Riischlikon, Switzerland

Site-percolation systems have been generated to simulate fractal structures. In d = 2, after removal of all finite clusters, the voids between the percolating clusters are considered to represent pores of various areas. We show that the pore size can be expressed in terms of the Hausdorff dimension D of the percolating clusters. The scaling of the pore size distribution is shown to lead to an excellent determination of D, even when the fractal persistence length ~ is rather short. This determination of D is compared with those obtained by box counting, by finite size scaling, or via the pair correlation function g(r). The crossover to the homogeneous regime for systems of finite Ornstein-Zernike ~ is sharp and occurs at a pore size of order 4~ 2. Granularity effects at large q are more important for q-space methods than in real space. Comparing systems of different sizes clearly separates the three regimes where granularity, scaling, or homogeneity dominate.

1. Introduction

Recently, due to the great interest in fractal materials, electron microscopy images of silica aerogel surfaces were published [1, 2]. To better grasp the properties of pores seen in such micrographs we have simulated two-dimensional percolation systems close to the percolation limit. To characterize the fractal, the correlation length ~ and the effective Hausdorff dimensionality D are determined via the average density p(L) in boxes of linear size L, the Fourier transform S(q) of the two-point pair correlation function g(r), and by finite size scaling. While the asymptotic properties of the percolation model are well understood and have universal features that are not influenced by material details, the applicability of such a simple theoretical approach crucially depends on the size of the fractal range. In practice [2] one observes scaling over about one order of magnitude, and the influence of the finite range effects cannot be ignored completely. Furthermore, different ways to determine the fractal t Permanent address: Laboratoire de Chemie Th4orique, l~cole Normale Sup4rieure, 46 all6e d'Italie, 69364 Lyon Cedex 07, France.

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properties sometimes yield different results, which can also be attributed to a limited scaling range. In order to estimate these effects, we study static (geometrical) properties for site percolation in two dimensions. In contrast to the usual approach of considering very large systems, we also consider small systems in order to determine the smallest size that still provides meaningful results.

2.

Theory

Site percolation in two dimensions [3] generates fractal structures near the percolation threshold p = Pc ~ 0.59275 where p is the fraction of sites occupied. The mass M of isolated large clusters is related to their size R by M ~ R D with D = 91/48. For p > Pc, the infinite cluster should scale at lengths less that the correlation length ~ = ~ 0 ( P - P c ) -~, and it is homogeneous above ~. These properties are universal (independent of the lattice, microscopic features) provided that a ~ R ~ ~, where a is a typical microscopic scale (grain size). No general theoretical prediction exists for how fast the granularity affects the scaling properties when R---> a, or how fast the system becomes homogeneous when R---> ~. In a real system (e.g. aerogels) the maximum value for ~ is determined by its mechanical stability. In numerical calculations it can be chosen at will by selecting p close enough to Pc. Nevertheless, the maximum linear size L used in the computation plays a role analogous to ~:. Several different ways have been used to test finite range effects. In one m e t h o d a large L = 6800 system is considered at or above Pc. By randomly placing boxes of linear size b the exponent D and the crossover towards homogeneity for p > p c can be obtained from the number of boxes which overlap the cluster, N b ~ b -D. Box counting is a very general and unbiased way to determine fractal exponents. A closely related quantity is the fraction of pore space in a box of size b, P ( b ) ~ 1 - cb D-d, c = constant. For b >> ~, it becomes constant. Furthermore, the derivative O P ( b ) / O b ~ b D-d-~ can be used to determine D. A third method consists of calculating the Fourier transform S(q) of the two-point correlation function g(r). S ( q ) scales as q - D in the fractal domain with a change at high q due to the granularity and a fiat portion at small q when the structure is homogeneous. A n o t h e r m e t h o d to estimate D, introduced in ref. [4], is to determine the average density p ( L ) in a box of size L centered on an occupied site. This average density is simply the sum of the two-point correlation g ( l r - r c e n t l ) between the sites denoted by r and those of the central site denoted by rcent divided by the system size L 2.

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Investigating a large system using probes of size b ~ L has the advantage that no truncation due to the boundary will affect the results. On the other hand the finiteness of the system is felt when b approaches L. In other words, two lengths L and ~ compete to determine the crossover to homogeneity, which limits the effectively unbiased range. To avoid this effect another approach has been tested. Instead of varying b, the system size L itself is varied, which leads to larger boundary effects (small L) but has no second competing length scale.

3. Results

In fig. 1 we present the structure factor S(q) for several realizations of a L = 6800a percolation lattice. The data for p = pc (solid curve) clearly follows the asymptotic q-D law for small q. For large q there are systematic deviations from the simple power law. The most reliable scaling range is for intermediate q-values (--(1.5 x 10-2-1.5 x 10-1)a -1) where low-q fluctuations and high q granularity are not important. Note that at very small q there is a systematic trend away from scaling even at pc - presumably this is an effect of the periodic boundary. Several values of p >pc show very nicely the crossover from a fractal to a homogeneous state. At large q the curves are indistinguishable,

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indicating that the structure is unaffected by p at scales ~ : . One precise definition of sc (Ornstein-Zernike) can be given by the intercept of the plateau with the q-O law (~:= 1/q.... ). One concludes from fig. 1 that in order to identify a scaling regime one needs a q .... ~<(3-4) x 10-ea -1 (dashed curve) which corresponds to ~: ~ 30a. The slow onset of scaling on the high-q side is due to granularity. To show this, a small L = 12a system is shown (open squares), with a remarkably similar S(q) shape for the possible q-values. Note that for this case the smallest q-value has not even entered the scaling regime. In fig. 2, the change of the pore fraction as a function of the linear pore size b is shown. For b <~5a the granularity significantly affects the curves. For small q and p = Pc (solid curve), and for linear pore sizes of the order of 100a, the finite system size influences the distribution such that it seems to decay with = b -1, which corresponds to a D of =2. For ~: ~ L the initial decay of OP/Ob for b > 5 a is consistent with D =91/48. For b >2~: there is a crossover to a much faster decay, which demonstrates that the occurrence of larger pores is relatively rare, but that such large pores can nevertheless appear [2]. In fig. 3 the scaling of the average density as a function of size is plotted for (1) boxes of size b, whose center sites are occupied in the large L = 6800a system [4], and (2) the finite size approach. In the finite size method one determines the largest cluster in a periodically bounded box of size L. This cluster is the finite size equivalent of the percolation cluster. To obtain the average density one divides the number of particles in it by L:. The figure clearly shows how the two methods probe the system. At Pc (solid curve) the data from the boxes with an occupied central site approach the asymptotic

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slope from above for small b. At large b it starts to deviate from b D - 2 law because of the finite system size. The finite L extrapolation approaches asymptotic scaling from below and converges quite rapidly. The absence of an upper cutoff does not limit the scaling range on the upper end. For p > Pc, the two approaches describe very closely the same crossover at £ ~ 45a, in which case the finite system size is irrelevant. As in fig. 1 the crossover length £ is determined unambiguously by the crossing of the plateau and the scaling range.

4. Summary and conclusions The investigation of the limitations of the scaling range for standard 2D-site percolation shows a considerable dependence on the method. (1) The effect of the granularity extends to larger sizes in reciprocal space than in real space; (2) direct pore-fraction counting yields excellent results even for small sizes; (3) average density of boxes with occupied centered sites is limited at Pc by the finite system size; and (4) finite size scaling works well for small systems and is indistinguishable from the box density approach for large systems, for finite ~. A further unexpected result is that the value of G0 in the relation for the correlation length is much smaller than a. We estimated a value of ~:0 = (0.05 -+ 0.02)a.

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Acknowledgements Many thanks are expressed to S. Alexander, E. Courtens, R. Orbach, L. Pietronero, H.E. Stanley, M. Shalaev and R. Vacher for stimulating discussions.

References [1] J.L. Rousset, A. Boukenter, B. Champagnon, J. Dumas, E. Duval, J.F. Quinson and J. Serughetti, J. Phys.: Condens. Matter 2 (1990) 8445. [2] R. Vacher, E. Courtens, E. Stoll, M. B6ffgen and H. Rothuizen, J. Phys.: Condens. Matter 3 (1991) 6531. [3] D. Stauffer, Introduction to Percolation Theory (Francis and Taylor, London, 1985). [4] A. Kapituinik, A. Aharony, G. Deutscher and D. Stauffer, J. Phys. A 16 (1983) L269.