Microscopic theory for cross-linked macromolecules. II. Replica theory of the transition to the solid state

We investigate the statistical mechanics of a set of randomly cross-linked macromolecules. Using the replica method, we are led to consider a field theory for an order parameter which describes the breaking of the symmetry of independent translations and rotations between replicas. We construct the Landau free energy in this order parameter by systematically integrating out the polymer degrees of freedom. We find that the system can exhibit two transitions. The first corresponds to the collapse of a swollen rubber, while the second is a continuous transition to an equilibrium amorphous solid once the number of crosslinks exceeds a critical value. Very close to the latter transition, the Landau free energy is of the same form as that for the long-range Ising spin glass, enabling us to use Parisi's replica--symmetry-breaking scheme to describe the ordered state.