Universality and its origins at the amorphous solidification transition
During the last decade there has been an ongoing effort to obtain an ever more detailed understanding of the behavior of randomly cross-linked macromolecular systems near the vulcanization transition.1-4 This effort has been built from two ingredients: (i) the Deam-Edwards formulation of the statistical mechanics of polymer networks;5 and (ii) concepts and techniques employed in the study of spin glasses.6 As a result, a detailed mean-field theory for the vulcanization transition-an example of an amorphous solidification transition-has emerged, which makes the following predic tions: (i) For densities of cross links smaller than a certain critical value (on the order of one cross-link per macromol ecule) the system exhibits a liquid state in which all particles (in the context of macromolecules, monomers) are delocal ized. (ii) At the critical cross-link density there is a continu ous thermodynamic phase transition to an amorphous solid state, this state being characterized by the emergence of ran dom static density fluctuations, (iii) In this state, a nonzero fraction of the particles have become localized around ran dom positions and with random localization lengths (i.e., rms, displacements), (iv) The fraction of localized particles grows linearly with the excess cross-link density, as does the characteristic inverse square localization length, (v) When scaled by their mean value, the statistical distribution of lo calization lengths is universal for all near-critical cross-link densities, the form of this scaled distribution being uniquely determined by a certain integrodifferential equation. For a detailed review of these results, see Ref. 4; for an informal discussion, see Ref. 7.