Spatially heterogeneous dynamics investigated via a time-dependent four-point density correlation function

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Abstract
Relaxation in supercooled liquids above their glass transition and below the onset temperature of “slow” dynamics involves the correlated motion of neighboring particles. This correlated motion results in the appearance of spatially heterogeneous dynamics or “dynamical heterogeneity.” Traditional two-point time-dependent density correlation functions, while providing information about the transient “caging” of particles on cooling, are unable to provide sufficiently detailed information about correlated motion and dynamical heterogeneity. Here, we study a four-point, time-dependent density correlation function g 4 (r,t) and corresponding “structure factor” S 4 (q,t) which measure the spatial correlations between the local liquid density at two points in space, each at two different times, and so are sensitive to dynamical heterogeneity. We study g 4 (r,t) and S 4 (q,t) via molecular dynamics simulations of a binary Lennard-Jones mixture approaching the mode coupling temperature from above. We find that the correlations between particles measured by g 4 (r,t) and S 4 (q,t) become increasingly pronounced on cooling. The corresponding dynamical correlation length ξ 4 (t) extracted from the small- q behavior of S 4 (q,t) provides an estimate of the range of correlated particle motion. We find that ξ 4 (t) has a maximum as a function of time t, and that the value of the maximum of ξ 4 (t) increases steadily from less than one particle diameter to a value exceeding nine particle diameters in the temperature range approaching the mode coupling temperature from above. At the maximum, ξ 4 (t) and the α relaxation time τ α are related by a power law. We also examine the individual contributions to g 4 (r,t), S 4 (q,t), and ξ 4 (t), as well as the corresponding order parameter Q(t) and generalized susceptibility χ 4 (t), arising from the self and distinct contributions to Q(t). These contributions elucidate key differences between domains of localized and delocalized particles.