So far we have studied polymers with a linear structure. This is a consequence of the fact that the monomers have a functionality of two, which means that each monomer can bind to two other monomers. Branched polymers occur when the functionality of the monomers is higher. Branched polymers (BP) can have a fixed topology, meaning that they consist of a fixed number of branches and nodes. We will see that the properties of such polymers are still closely linked to those of linear polymers (section 9.1). In some cases it is more appropriate to consider the functionality as random and to describe the polymers as lattice animals. These lattice animals can again be described by the Potts model, but a description using field theory will turn out to be more instructive. We will consider first a branched polymer in a good solvent and later the phenomena of adsorption and collapse for these polymers. Branched polymers also turn up in the study of vesicles which are simple models for cell membranes.

Branched polymers of fixed topology

As a first simple model for a branched polymer we can consider a star polymer, which consists of Na arms which each have the same number N of monomers. These arms are modelled by SAWs (figure 9.1). In the same figure we also show a more arbitrary polymer which can be described as a graph with specified vertices and edges (more precisely, one can say that the polymer is an embedding of the graph). We will denote in general by Na the number of branches of the polymer.