UNIVERSALITY AND ITS ORIGINS AT THE AMORPHOUS SOLIDIFICATION TRANSITION

Systems undergoing an equilibrium phase transition from a liquid state to an amorphous solid state exhibit certain universal characteristics . Chief among these are the fraction of particles that are randomly lo calized and the scaling functions that describe the order parameter an d (equivalently) the statistical distribution of localization lengths far these localized particles. The purpose of this paper is to discuss the origins and consequences of this universality, and in doing so, t hree themes are explored. First, a replica-landau-type approach is for mulated for the universality class of systems that are composed of ext ended objects connected by permanent random constraints and undergo am orphous solidification at a critical density of constraints. This form ulation generalizes the cases of randomly cross-linked and end-linked macromolecular systems, discussed previously. The universal replica fr ee energy is constructed, in terms of the replica order parameter appr opriate to amorphous solidification, the value of the order parameter is obtained in the liquid and amorphous solid states, and the chief un iversal characteristics are determined. Second, the theory is reformul ated in terms of the distribution of local static density fluctuations rather than the replica order parameter. It is shown that a suitable free energy can be constructed, depending on the distribution of stati c density fluctuations, and that this formulation yields precisely the same conclusions as the replica approach. Third, the universal predic tions of the theory are compared with the results of extensive numeric al simulations of randomly cross-linked macromolecular systems, due to Barsky and Plischke, and excellent agreement is found. [S0163-1829(98 )04102-2].