Dynamical heterogeneity and the freezing transition in hard-sphere suspensions: Further analysis of the mean square displacement and the velocity autocorrelation function

The velocity autocorrelation function is derived from the mean-squared displacement measured on a colloidal suspension of particles with hard-sphere-like interactions. It decays to zero from below and follows a stretched exponential function of delay time for the thermodynamically stable suspension. For the metastable suspension a power-law decay emerges. The results are discussed in terms of the classical Lorentz gas and the model that describes diffusion confined to one dimension. With the aid of these models, the experimental results provide a characterization of the dynamical heterogeneities which are observed microscopically, and an explanation for the enhanced resistance to flow and diffusion usually found in undercooled fluids upon approaching the glass transition.

Figure 1

(Color online) Double logarithm plot of the MSD vs delay time for volume fractions and stretching indices indicated. The lower and upper bold triangles denote the points $(R_c^2,{\tau }_c)$ and $(R_m^2,{\tau }_m)$, respectively. The heavy dashed line is the MSD for the ideal dilute suspension. The fits are made using Eq. (12). See text for further details.

Figure 2

Stretching index, $\nu$, defined in Eq. (4), vs volume fraction. The dashed vertical lines are located at the freezing and glass transition volume fractions ${\phi }_f=0.493$ and ${\phi }_g=0.565$. These lines are shown in all figures where $\phi$ is the abscissa.

Figure 3

Logarithm of characteristic delay times vs volume fraction. See text for further details.

Figure 4

Characteristic displacements vs volume fraction. See text for further details

Figure 5

(Color online) Logarithm of the modulus of the VAF vs logarithm of delay time: (a) For values of $\phi (<{\phi }_f)$ and corresponding values of $\nu (>1∕2)$ indicated; (b) values of $\phi (>{\phi }_f)$ and corresponding values of $\nu (<1∕2)$ indicated; (c) for $\phi =0.533$. The dashed line in all cases is the stretched exponential [Eq. (7)] that best describes the VAFs for $\phi <{\phi }_f$. The dashed-dotted line in (c) is the stretched exponential that passes through $Z\left(\tau \right)$ for $\tau <1$. The solid line in (c) is the power law [Eq. (8)] that passes through $Z\left(\tau \right)$ for $\tau >1$.

Figure 6

Exponent $\mu$ of the power law, Eq. (8), and $2-\nu$, where $\nu$ is the stretching index defined in Eq. (4), vs volume fraction. The upper and lower lines are lines of best fit to $\mu$ and $2-\nu .$

Figure 7

Logarithm of characteristic delay times vs volume fraction. The open diamonds represent values of ${\tau }_f$ obtained by fitting power laws directly to the VAFs, whereas the closed diamonds are obtained by the analysis with Eq. (12). See text for further details.

Figure 8

Coefficient of the first term in Eq. (12) vs volume fraction. The solid line is the line of best fit. Typical error bars are shown.

Figure 9

Long-time self-diffusion coefficients from previous measurements compared with those calculated from Eq. (15).