Modeling diffusion in colloidal suspensions by dynamical density functional theory using fundamental measure theory of hard spheres

We study the dynamics of colloidal suspensions of hard spheres that are subject to Brownian motion in the overdamped limit. We obtain the time evolution of the self- and distinct parts of the van Hove function by means of dynamical density functional theory. The free-energy model for the hard-sphere fluid that we use is the very accurate White Bear II version of Rosenfeld's fundamental measure theory. However, in order to remove interactions within the self-part of the van Hove function, a nontrivial modification has to be applied to the free-energy functional. We compare our theoretical results with data that we obtain from dynamical Monte Carlo simulations, and we find that the latter are well described by our approach even for colloid packing fractions as large as 40%.

Figure 1

The distinct density profiles ${\rho }_d(r,t)$ for a hard-sphere fluid obtained by means of dynamical test particle theory and a full linearization (FL) vs partial linearization (PL) of the tensorial White Bear II functional with respect to the self-component; see text for details. The dashed-dotted lines show the profiles obtained by dynamical Monte Carlo simulations. Parts (a) and (b) display the results for a bulk density ${\rho }_b{\sigma }^3=0.4$, while (c) and (d) display results for a density ${\rho }_b{\sigma }^3=0.6$. The times are $t/{\tau }_B=0.1$ in (a) and (c) and $t/{\tau }_B=0.2$ in (b) and (d).

Figure 2

The self- and distinct density profiles, ${\rho }_s(r,t)$ and ${\rho }_d(r,t)$, corresponding to the self- and distinct parts of the van Hove function $\mathcal{G}(r,t)$ for a density ${\rho }_b{\sigma }^3=0.2$. The plots on the left-hand side show the results obtained from DDFT, while on the right-hand side we display the DMC data. The lines correspond to times $t/{\tau }_B=0.01$, 0.05, 0.1, 0.2, and 1. Note that the ordinate axes of the self-parts feature a logarithmic scale.

Figure 3

The self- and distinct density profiles, ${\rho }_s(r,t)$ and ${\rho }_d(r,t)$, for a density ${\rho }_b{\sigma }^3=0.4$. The plots on the left-hand side show the results obtained from DDFT, while on the right-hand side we display the DMC data. The lines correspond to the same times as in Fig. 2.

Figure 4

The self- and distinct density profiles, ${\rho }_s(r,t)$ and ${\rho }_d(r,t)$, for a density ${\rho }_b{\sigma }^3=0.6$. The plots on the left-hand side show the results obtained from DDFT, while on the right-hand side we display the DMC data. The lines correspond to the same times as in Fig. 2.

Figure 5

The self- and distinct density profiles, ${\rho }_s(r,t)$ and ${\rho }_d(r,t)$, for a density ${\rho }_b{\sigma }^3=0.8$. The plots on the left-hand side show the results obtained from DDFT, while on the right-hand side we display the DMC data. The lines correspond to the same times as in Fig. 2

Figure 6

Mean-square displacement $\langle r^2\rangle$ of the self-part ${\rho }_s(r,t)$ as a function of $t/{\tau }_B$ for densities (a) ${\rho }_b{\sigma }^3=0.2$, (b) ${\rho }_b{\sigma }^3=0.4$, (c) ${\rho }_b{\sigma }^3=0.6$, and (d) ${\rho }_b{\sigma }^3=0.8$. The solid lines are DDFT results, while the points represent data obtained by dynamic Monte Carlo simulations. The data are normalized with respect to the ideal-gas solution, Eq. (35), thus the latter is a line with slope 1.