Swelling and growth of polymers, membranes, and sponges.

Abstract

Polymers can be formed into a wide range of structures depending on the monomer chemistry and the kinetic conditions of growth. A general model of polymers having higher-order connectivity is introduced that reduces to flexible linear polymers, membranes, and sponges as special cases. This ``Wiener sheet'' model, which extends the conventional Wiener path model of linear polymers, is argued to describe various classes of branched polymers, as well as different types of interacting random surfaces. For example, lattice animals and percolation clusters are considered to be perforated sheets whose large-scale dimensions are described by the Wiener sheet model with excluded volume interactions. To within the approximations of the model calculations, the properties of the Wiener sheet ``membrane'' are consistent with this correspondence. The influence of the excluded volume and the kinetics of growth of membrane and sponge structures are treated at a Flory-level approximation, although the Wiener sheet model should admit to a renormalization-group treatment as in the case of linear polymers. Predictions of the self-interacting Wiener sheet model are contrasted with an alternative and complementary random surface model introduced by Nelson and co-workers and are compared with recent simulations and experiment.