Mechanics of disordered solids. II. Percolation on elastic networks with bond-bending forces.

Abstract

Bond and site percolation on two- and three-dimensional (3D) elastic and superelastic percolation networks with central and bond-bending (BB) forces are studied. We calculate the force distribution and show that, depending on the relative contributions of the central and BB forces, its shape can be unimodal or bimodal, both near and away from the percolation threshold ${\mathit{p}}_{\mathit{c}}$. The Poisson ratios of various 2D and 3D, isotropic and anisotropic BB models are calculated and are shown to take on negative values near ${\mathit{p}}_{\mathit{c}}$. Several experimental realizations of this peculiar property are given. We then analyze various experimental data on elastic and rheological properties of gel polymers near ${\mathit{p}}_{\mathit{c}}$. The scaling laws for elastic properties of gel polmers near ${\mathit{p}}_{\mathit{c}}$ and their associated critical exponents f are divided into two groups. In one group are physical gels in which the contribution of BB forces to the elastic energy dominates that of central forces (CF's), and their scaling properties are described by the BB model, with f\ensuremath{\simeq}3.75. In the second group are chemical gels in most of which CF's are dominant, with f\ensuremath{\simeq}2.1. The scaling laws for the viscosity of a gelling solution near the gel point can also be divided into two groups. In one group are gelling solutions that are close to the Zimm regime. We propose that the scaling properties of the viscosity of such gels is analogous to the shear modulus of a static superelastic percolation network that diverges, as ${\mathit{p}}_{\mathit{c}}$ is approached from below, with an exponent \ensuremath{\tau}=\ensuremath{\nu}-${\mathrm{\ensuremath{\beta}}}_{\mathit{p}}$/2\ensuremath{\simeq}0.68 in 3D, where \ensuremath{\nu} and ${\mathrm{\ensuremath{\beta}}}_{\mathit{p}}$ are the critical exponents of the correlation length and the strength of percolation networks, respectively. In the second group are gelling solutions that are close to the Rouse limit. We propose that the scaling law for the divergence of the viscosity of such gels is the same as that of the shear modulus of a dynamic superelastic percolation network, with \ensuremath{\tau}'=2\ensuremath{\tau}\ensuremath{\simeq}1.35 in 3D.