Theory of correlated two-particle activated glassy dynamics: General formulation and heterogeneous structural relaxation in hard sphere fluids

Abstract

We generalize the nonlinear Langevin equation theory of activated single particle dynamics to describe the correlated motion of two tagged spherical particles in a glass- or gel-forming fluid as a function of their initial separation. The theory is built on the concept of a two-dimensional dynamic free energy surface which quantifies the forces on two particles moving in a cooperative manner. For the hard sphere fluid, above a threshold volume fraction we generically find two relaxation channels corresponding largely, but not exclusively, to a center-of-mass-like displacement and a radial separation of the two tagged particles. The entropic barriers and mean first passage times are computed and found to systematically vary with volume fraction and initial particle separation; both oscillate as a function of the latter in a manner related to the equilibrium pair correlation function. A dynamic correlation length is estimated as the length scale beyond which the two-particle activated dynamics becomes uncorrelated in space and time, and is found to modestly grow with increasing mean relaxation time. The theory is also applied to a simplified model of cage escape, the elementary step of structural relaxation. Predictions for characteristic relaxation times, translation-relaxation decoupling, and stretched-exponential decay of time correlation functions are obtained. A novel mechanism for understanding why strong decoupling emerges in the activated regime, but stretched nonexponential time correlation functions do not change shape as the mean relaxation time grows, is presented and favorably compared with experiment. The theory may serve as a starting point for constructing a predictive model of multiple correlated caging and hopping (forward and backward) events of a pair of tagged particles.