Surface scaling analysis of a frustrated spring-network model for surfactant-templated hydrogels

We propose and study a statistical-mechanical model, inspired by recent atomic force microscopy studies of the surface structures of crosslinked polymer gels into which voids are introduced through templating by surfactant micelles [M. Chakrapani et al., e-print cond-mat/0112255]. The gel is represented by a frustrated, triangular network of nodes connected by springs of random equilibrium lengths. The nodes represent crosslinkers, and the springs correspond to polymer chains. The boundaries are fixed at the bottom, free at the top, and periodic in the lateral direction. Voids are introduced by deleting a proportion of the nodes and their associated springs. The model is numerically relaxed to a representative local energy minimum, resulting in an inhomogeneous, "clumpy" bulk structure. The free top surface is defined at evenly spaced points in the lateral (x) direction by the height of the topmost spring, measured from the bottom layer, h(x). Its scaling properties are studied by calculating the root-mean-square surface width and the generalized increment correlation functions C(q)(x)=<|h(x(0)+x)-h(x(0))|(q)>. The surface is found to have a nontrivial scaling behavior on small length scales, with a crossover to scale-independent behavior on large scales. As the vacancy concentration approaches the site-percolation limit, both the crossover length and the saturation value of the surface width diverge in a manner that appears to be proportional to the bulk connectivity length. This suggests that a percolation transition in the bulk also drives a similar divergence observed in surfactant templated polyacrylamide gels at high surfactant concentrations.