Slow dynamics in supercooled liquids.

Abstract

Slow dynamics in supercooled liquids is investigated on the basis of the trapping diffusion model which takes account of two types of diffusive dynamics, jump motion and stray motion. Parameters of the model are determined in such a way that the waiting-time distribution of the model agrees with those found for a binary soft-sphere system through molecular-dynamics simulation. With the use of the coherent-medium approximation, the frequency dependence of the self-part of the dynamical structure factor ${\mathrm{S}}_{\mathrm{s}}$(q,\ensuremath{\omega}) and the generalized susceptibility \ensuremath{\chi}${\mathrm{\ensuremath{''}}}_{\mathrm{s}}$(q,\ensuremath{\omega}) is obtained. Above the glass-transition point, there exist frequency regions where ${\mathrm{S}}_{\mathrm{s}}$(q,\ensuremath{\omega}) shows a power-law decay, corresponding to \ensuremath{\alpha} relaxation and \ensuremath{\beta} relaxation, which are shown to be caused by the subanomalous diffusion due to the jump motion and by the stray motion, respectively. This indicates that above the glass-transition point there exists a certain time window where the intermediate scattering function ${\mathrm{F}}_{\mathrm{s}}$(q,t) shows a stretched exponential decay. Below the glass-transition point, ${\mathrm{F}}_{\mathrm{s}}$(q,t) decays in a stretched exponential form in the long-time limit, which is caused by the anomalous diffusion due to the jump motion. Accordingly, ${\mathrm{S}}_{\mathrm{s}}$(q,\ensuremath{\omega}) in the static limit is show to be a cusp between the glass transition and a certain temperature below the freezing point, and to diverge below the glass-transition point. The \ensuremath{\alpha}-relaxation time determined from the position of the \ensuremath{\alpha} peak of \ensuremath{\chi}${\mathrm{\ensuremath{''}}}_{\mathrm{s}}$(q,\ensuremath{\omega}) is shown to diverge at a certain temperature below the glass-transition point, in line with the Vogel-Fulcher equation. The exponent representing the long-time decay of the non-Gaussian parameter is also obtained, which agrees quantitatively with the result obtained for the soft-sphere system by molecular-dynamics simulation.