Microscopic theory for cross-linked macromolecules. I. Broken symmetry, rigidity, and topology

This paper is concerned with the onset of rigidity in randomly cross-linked macromolecules. We discuss in detail the possible partitionings of configuration space, which may accompany the spontaneous breaking of translational invariance, treating separately the cases of crystals, amorphous solids, and cross-linked macromolecules. We describe the order parameters for these systems, drawing the distinction between solids with discrete translational symmetry and solids with macroscopic translational invariance, such as randomly cross-linked macromolecular solids. We show that the latter may be described by a sequence of probability distributions for the overlaps of equilibrium states. In a cross-linked system of impenetrable linear chains, the configuration space of the solid state is partitioned into two categories of equilibrium states: those related by translational and rotational symmetry, and those unrelated by these symmetries. The latter are a consequence of the distinct topologies of the network, which are consistent with a given set of cross links. We show how the overlap-probability distributions may be calculated.