PHYSICAL REVIEW B VOLUME 31, NUMBER 3 1 FEBRUARY 1985 Position-space renormalization for elastic percolation networks with bond-bending forces Shechao Feng Schlumberger Doll -Research, P. O. Box 307, Ridgefield, Connecticut 06877 and Physics Department, Harvard University, Cambridge, Massachusetts 02138 Muhammad Sahimi Department of Chemical Engineering, University of Southern Cali'fornia, Los Angeles, California 90089-1211 (Received 15 October 1984) We develop a three-parameter position-space renormalization-group method and study the percolation properties of a two-dimensional elastic network in which both central and rotationally invariant borid- bending forces are present. The critical exponent f, which describes the power-law behavior of the elastic moduli near the percolation threshold, is estimated for a square network and is found to be consistent with recent estimates obtained by other methods. There has recently been considerable interest in the prob- lem of elastic properties of random networks near the per- colation threshold. Until recently this problem was mostly vie~ed as analogous to the problem of electrical conductivi- ty of percolating networks. de Gennes' discussed the rela- tionship between the conductance, conductivity, and bulk modulus of a percolating network which is a simple model for gelation without solvent. He set out the explicit equa- tions governing the behavior of the last two properties and pointed out that the bulk elastic modulus of a gel, modeled by a nonrotationally invariant isotropic force constant, is analo- gous to the electrical conductivity of the system. His intui- tive argument resulted in some controversy, but the con- troversy seems to have been settled by the scaling argu- ments of Yu, Chaikin, and Orbach which is supportive of de Gennes's argument. Feng and Sen4 recently considered a different model, namely, the central force elastic percolating network model, and provided numerical evidence that the critical properties of this model belong to a new universality class than that of percolation conductivity problem. This model, which is ba- sically a network of springs, is rotationally invariant, but suffers from a few peculiarities, as was pointed out by Feng, Sen, Halperin, and Lobb. 5 For simple cubic lattices at all dimensions the elastic threshold is p, = l. Thus a meaning- ful study of this problem is limited to certain lattices, e. g. , triangular and fcc lattices. Another shortcoming of this model is that the significance of the straight-bond chains in transmitting elastic forces is ambiguous, because in a non- linear model, the straight bonds could "buckle" under compression, but not under extension. An effective medi- um approximation has also been developed for this model which correctly predicts that p, = 1 for d-dimensional simple cubic networks of randomly occupied springs and provides very reasonable estimates of p, for many other lattices. In this Rapid Communication we study another rotation- ally invariant model for the elastic moduli of percolating networks. We study a model in which both central and bond-bending forces are included and develop a three- parameter position-space renormalization-group7 (PSRG) method to estimate the critical exponent f which describes the power-law behavior of the elastic moduli of the system near the elastic threshold p, . If E and W are the bulk and shear modulus of the network, respectively, we may write (as p p, ) EC, N — (p — p, )~ . The potential energy E of the network is given by E = TA X [ (U/ ui) ' rttl gtt + Qp X (50 JIJt) gttgtk &Ok& (2) where uI and u& are displacements of sites i and j and r& a unit vector from site i to site j. g~ is a random variable which takes the values 1 and G with probabilities p and q=1 — p, for bonds that are occupied and empty, respec- tively. The bond-bending forces between two occupied bonds ij and ik having site i in common are given in terms of the change in angle 88jq, at site i, which is expressed in turn as a linear function of u&, u&, and uk. The sums are, respectively, over all bonds, and over all pairs of bonds with a site in common. This model is essentially the same as that of Kirkwood who studied vibrational properties of rod- like molecules. Keating9 studied the elastic properties of co- valent crystals with essentially the same model. The only, difference here is that we have included the bending of 180' bonds. Kantor and Webman'0 have recently studied the critical properties of this model by using "the nodes and links" model of percolation networks together with scaling arguments and have proposed that f ~ 3. 6 at d = 2, while Feng et al. and Bergman have employed numerical simula- tions and finite-size scaling technique. An experimental study of hole-punched sheets by Benguigui5 also gave f = 3.5 at d=2. The elastic threshold of this system can be easily shown to be the same as that of the ordinary bond percolation. ' Our PSRG method is similar to that developed for per- colation conductivity" but with a few differences which we note. We use a standard 0 shape RG cell that has been used extensively in the past. 7 The basic idea is to develop three RG transformations to relate n, p, and p to their re- scaled values a', p', and p'. Since the RG cell used here is self-dual, the recursion relation, ' p'= p + 5p q+ Sp q +2p q, for the rescaled probability of occupied bonds reproduces the exact result, p, - Y. For the RG cell used here the displacements u& - (0, 6 ), u2 = (0, 5), us = (0, 0), . and us = (0, 0) of the exterior sites of the cell are held fixed (similar to the PSRG treatment of percolation conductivity 31 1671 1985 The American Physical Society