Random-walk analysis of displacement statistics of particles in concentrated suspensions of hard spheres

Mean-squared displacements (MSDs) of colloidal fluids of hard spheres are analyzed in terms of a random walk, an analysis which assumes that the process of structural relaxation among the particles can be described in terms of thermally driven memoryless encounters. For the colloidal fluid in thermodynamic equilibrium the magnitude of the stretching of the MSD is able to be reconciled by a bias in the walk. This description fails for the under-cooled colloidal fluid.

Figure 1

Mean-squared displacement shown for $\varphi =0.48$. The dashed line shows the result, $⟨\Delta r^2\left(\tau \right)⟩=\tau$, for an ideal dilute suspension. The left and right solid lines are $⟨\Delta r^2\left(\tau \right)⟩=D_{\mathrm{s}}\tau$ and $⟨\Delta r^2\left(\tau \right)⟩=D_{\mathrm{l}}\tau$. Left and right arrows indicate the points $({\tau }_{\mathrm{c}},R_{\mathrm{c}}^2)$ and $({\tau }_{\mathrm{m}},R_{\mathrm{m}}^2)$, respectively.

Figure 2

Various characteristic times versus volume fraction, read directly from the MSD (“exp”) and those calculated from expressions indicated. See text for explanations and definitions. The vertical dashed lines indicate the freezing and glass transition volume fractions, ${\varphi }_{\mathrm{f}}=0.494$ and ${\varphi }_{\mathrm{g}}=0.565$.

Figure 3

Characteristic rms displacements versus volume fraction. See text for further details.

Figure 4

Short-time diffusion coefficient vs volume fraction.

Figure 5

The asymmetry, $H$, of the random walk defined by Eq. (8) and the ratio, $D_{\mathrm{l}}∕D_{\mathrm{s}}$, of long- and short-time diffusion coefficients vs volume fraction. The linear least-squares fit to $H$ vs $\varphi$, shown in (a) and (c), is $H=1-1.6\varphi$.

Figure 6

Ratio of the measured long-time diffusion coefficient, $D_{\mathrm{l}}\left(\exp \right)$ and the product, $D_{\mathrm{l}}\left(\mathrm{calc}\right)=H{D}_{\mathrm{s}}$, vs volume fraction. The symbols are Ref. 12 (diamonds), Ref. 8 (circles), and Ref. 9 (triangles).

Figure 7

Ratio of net displacement, $X_{\mathrm{m}}$ [Eq. (10)], and rms Gaussian displacement, $R_{\mathrm{m}}$, vs volume fraction.