Dynamics in inhomogeneous liquids and glasses via the test particle limit

We show that one may view the self-part and the distinct-part of the van Hove dynamic correlation function of a simple fluid as the one-body density distributions of a binary mixture that evolve in time according to dynamical density functional theory. For a test case of soft-core Brownian particles the theory yields results for the van Hove function that agree quantitatively with those of our Brownian dynamics computer simulations. At sufficiently high densities the free energy landscape underlying the dynamics exhibits a barrier as a function of the mean particle displacement, shedding new light on the nature of glass formation. For hard spheres confined between parallel planar walls the barrier height oscillates in phase with the local density, implying that the mobility is maximal between layers, which should be experimentally observable in confined colloidal dispersions.

Figure 1

The self-part $G_s(r,t)$ (dashed lines, squares) and scaled distinct-part $G_d(r,t)∕\rho$ (solid lines, circles) of the van Hove correlation function for the GCM fluid with bulk density $\rho R^3=0.2263$ and $k_{B}T∕ϵ=0.5$, at times $t^{*}\equiv t∕{\tau }_B=0.01,0.1,0.2,0.4$ [(a),(b),(c),(d), respectively], as a function of the (scaled) distance $r∕R$. Shown are results from DDFT (lines) and BD simulations (symbols).

Figure 2

The scaled free energy $F∕k_{B}T$ as a function of the root-mean-square particle displacement $w$ [see Eq. (3)], calculated with the RY functional for hard spheres at densities $\rho {\sigma }^3=0.6$, 0.7, 0.8, and 0.9 (as indicated by the arrow) and shifted such that $F(w=3)=0$. (a) Self-part of the van Hove function on a logarithmic scale as a function of $r∕\sigma$ for $\rho {\sigma }^3=0.78$ and three typical widths $w$. Note the pronounced non-Gaussian character for $w∕\sigma =0.45$. (b) Comparison of results for $F∕k_{B}T$ as a function of $w∕\sigma$ from free minimization (dot-dashed line) and a Gaussian parametrization for $G_s$ (solid line) for $\rho {\sigma }^3=0.78$. Also shown is the free-energy barrier $\Delta F$ between minimum and maximum (arrow) and the values of $w∕\sigma$ (circles) for which results are shown in (a).

Figure 3

Correlation functions for hard spheres confined between parallel plates of separation distance $H=4\sigma$. (a) Density profile $\rho \left(z\right)$ (solid line) as a function of $z∕\sigma$ and free-energy barrier $\Delta F\left(z_0\right)$ (symbols; the dotted line is to guide the eye). Also shown is the corresponding bulk value of $\Delta F$ (dashed line). (b) The (scaled) self-part ${\rho }_s\left(\mathbf{r}\right){\sigma }^3∕25$ and distinct-part ${\rho }_d\left(\mathbf{r}\right)$ of the van Hove function as a function of the scaled coordinates perpendicular and parallel to the wall, $z∕\sigma$ and $x∕\sigma$, respectively, evaluated at the minimum of the free-energy landscape for bulk density $\rho {\sigma }^3=0.8$.