Elastically Collective Nonlinear Langevin Equation Theory of Glass-Forming Liquids: Transient Localization, Thermodynamic Mapping, and Cooperativity.

We analyze multiple new issues concerning activated relaxation in glassy hard sphere fluids and molecular and polymer liquids based on the elastically collective nonlinear Langevin equation (ECNLE) theory. By invoking a high-temperature reference state, a near universality of the apparent dynamic localization length scale is predicted for liquids of widely varying fragility, a result that is relevant to recent simulation studies and quasi-elastic neutron-scattering measurements. In contrast, in the same format, a strongly nonuniversal behavior is found for the activation barrier that controls long-time relaxation. Two measures of cooperativity in the ECNLE theory are analyzed. A particle-level total displacement associated with the alpha relaxation event is found to be only of order 1-2 particle diameters and weakly increases with cooling. In contrast, an alternative cooperativity length is defined as the spatial scale required to essentially recover the full barrier and bulk alpha time. This length scale grows strongly with cooling because of the emergence in the deeply supercooled regime of collective long-range elastic fluctuations required to allow local hopping. It becomes very large as the laboratory Tg is approached, though it is relatively modest at degrees of supercooling accessible with molecular dynamics simulation. The alpha time is found to be exponentially related to this cooperativity length over an enormous number of decades of relaxation time that span the lightly to deeply supercooled regimes. Moreover, the effective barrier height increases almost linearly with the growing cooperativity length scale. An alternative calculation of the collective elastic barrier based on a literal continuum mechanics approach is shown to result in very little change of the theoretical results for bulk properties but leads to a much smaller and less temperature-sensitive cooperativity length scale.