Nearly logarithmic decay in the colloidal hard-sphere system

Nearly logarithmic decay is identified in the data for the mean-squared displacement of the colloidal hard-sphere system at the liquid-glass transition [W. van Megen et al., Phys. Rev. E 58, 6073 (1998)]. The solutions of the mode-coupling theory for the microscopic equations of motion fit the experimental data well. Based on these equations, the nearly logarithmic decay is explained as the equivalent of a $\beta$-peak phenomenon, a manifestation of the critical relaxation when the coupling between of the probe variable and the density fluctuations is strong. In an asymptotic expansion, a Cole-Cole formula including corrections is derived from the microscopic equations of motion, which describes the experimental data for three decades in time.

Figure 1

(Color online.) Time derivative of the mean-squared displacement $D\left(t\right)$ in the hard-sphere system for packing fractions $\phi ={\phi }^c$ and $\phi =0.5145$ (full lines from left to right). The upper dashed line is proportional to $t^{-1}$. The lower dashed curve represents the approximation by Eq. (4). The dotted line displays the critical law $t^{-a-1}$. Here and in Fig. 2 the unit of time is fixed by the short-time diffusion coefficient $D_0^s∕d^2=1∕160$. The inset shows $D\left(t\right)$ for $\phi ={\phi }^c$ on a linear scale for comparison.

Figure 2

(Color online.) Mean-squared displacement at the critical point in the hard-sphere system for Newtonian ($N$, chain curve) and Brownian ($B$, full curve) dynamics. The short horizontal line displays the plateau $6r_s^{c2}$. The dashed lines show the asymptotic approximations by Eqs. (2) (labeled $t^{-2a}$) and (4). The leading-order approximations are shown dotted for Eq. (2) $\left(t^{-a}\right)$ and Eq. (4) $\left(cc\right)$. The nearly logarithmic relaxation is shown by the straight dashed line labeled $\ln t$. The diamond and the square indicate 10% deviations of $B$ from the approximation by Eq. (4) and the solution for $\phi =0.5145$, respectively.

Figure 3

(Color online.) Fit of the mean-squared-displacement data from [11] (full circles and curves) by the solutions of mode-coupling theory (dashed). The dotted curve displays the theoretical result at the critical point and is equivalent to curve $B$ in Fig. 2. The square indicates a 10% deviation between the critical relaxation and the fit for ${\phi }^{\exp }=0.583$. The open diamond is the same as in Fig. 2. The inset shows the mapping of the experimental packing fractions ${\phi }^{\exp }$ onto theoretical values ${\phi }^{\mathrm{theo}}$ on the left axis as closed and open circles for curves shown or left out, respectively. The dotted line shows the function ${\phi }^{\mathrm{theo}}=0.87{\phi }^{\exp }+0.023$. The right axis displays the vertical scaling factor by crosses.