Change in relaxation scenario at the order-disorder transition of a colloidal fluid of hard spheres seen from the Gaussian limit of the self-intermediate scattering function

Self-intermediate scattering functions (ISFs) are measured by dynamic light scattering for the colloidal fluid of hard spheres for both equilibrium and nonequilibrium (undercooled) conditions, i.e., for volume fractions below and above the known freezing transition of the hard-sphere system. The delay time ${\tau }_{\mathrm{m}}$ where the mean-squared displacement, or the low wave-vector limit of the ISF, exhibits its maximum stretching is identified as a characteristic of the non-Markovian process(es) and is used to separate the ISF into fast $(\tau <{\tau }_{\mathrm{m}})$ and slow $(\tau >{\tau }_{\mathrm{m}})$ contributions. Each of these contributions exposes qualitative differences in the dynamics of the particles between the equilibrium and nonequilibrium colloidal fluids. These changes in the relaxation scenario signal the colloidal fluid’s awareness of its traversal of the freezing volume fraction.

Figure 1

Double logarithm plot of the MSD vs delay time at the volume fractions indicated. The arrows indicate the points $({\tau }_{\mathrm{c}},R_{\mathrm{c}}^2)$ and $({\tau }_{\mathrm{m}},R_{\mathrm{m}}^2)$, respectively (see text for more information). The dashed line is the MSD for the ideal dilute suspension.

Figure 2

Characteristic delay times discussed in the text: ${\tau }_{\mathrm{m}}$ is the time of maximum stretching; ${\tau }_{\mathrm{c}}$ is the average interval between particle encounters; ${\tau }_{\mathrm{m}}^{*}$ is the interval where $N(q,\tau )$ [Eq. (7)] has a maximum. The vertical dashed lines, shown in all figures where $\varphi$ is the abscissa, are located at the observed freezing, melting, and glass transition volume fractions, ${\varphi }_{\mathrm{f}}=0.494$, ${\varphi }_{\mathrm{m}}=0.535$, and ${\varphi }_{\mathrm{g}}=0.565$.

Figure 3

Short-time self-diffusion coefficients $D_{\mathrm{s}}$ obtained from present and previous experiments.

Figure 4

Self-ISF, $F_{\mathrm{s}}(q,\tau )$, the thermal mode, $T(q,\tau )=\exp [-D_{\mathrm{s}}{q}^2\tau ]$, and the non-Markovian mode, $N(q,\tau )$ for several volume fractions. The vertical lines mark the delay times ${\tau }_{\mathrm{m}}^{*}$ where $N(q,\tau )$ is a maximum.

Figure 5

Long-time self-diffusion coefficients $D_{\mathrm{l}}$ obtained from present and previous experiments.

Figure 6

The amplitudes $T(q,{\tau }_{\mathrm{m}}^{*})$ and $N(q,{\tau }_{\mathrm{m}}^{*})$ of the thermal and non-Markovian modes at ${\tau }_{\mathrm{m}}^{*}$ [defined by Eqs. (6, 7)], and the amplitude $A\left(q\right)$ defined by Eq. (9). See text for details.

Figure 7

Stretching index $\nu$ defined by Eq. (8), as a function of volume fraction. The horizontal lines represent the value of $\nu$ as it crosses freezing $\left(\nu =\frac{1}{2}\right)$ and melting $\left(\nu =\frac{1}{3}\right)$.

Figure 8

The quantity, $P(q,\tau )$, defined by Eq. (10) for the volume fractions indicated.

Figure 9

The fast decay, $P(q,\tau <{\tau }_{\mathrm{m}})$, of the self-ISF at volume fractions indicated. (a) $\varphi <{\varphi }_{\mathrm{f}}$: The dashed curve is Eq. (12) with $\beta =1$ and $\log {\tau }_{\mathrm{e}}=0.22$. (b) $\varphi >{\varphi }_{\mathrm{f}}$: The curves are Eq. (12) for various values of $\beta$. Note the difference in scale of the abscissas of (a) and (b).

Figure 10

Characteristic times and scaling times: (a) shows ${\tau }_{\mathrm{m}}$, the time of maximum stretching; ${\tau }_{\mathrm{s}}$ [defined by Eq. (14)]; ${\tau }_{\mathrm{e}}$ [defined by Eq. (12)]. (b) shows these quantities on an expanded scale and also shows the ratio ${\tau }_{\mathrm{m}}∕{\tau }_{\mathrm{e}}$.

Figure 11

The exponent $\beta$ defined by Eq. (12) as a function of volume fraction. The line of best fit is also shown.

Figure 12

Slow decay, $P(q,\tau >{\tau }_{\mathrm{m}})$, of the self-ISF at the representative volume fractions indicated. (a) $\varphi <0.35$, (b) $0.35<\varphi <{\varphi }_{\mathrm{f}}$, and (c) $\varphi >{\varphi }_{\mathrm{f}}$. In (a) and (b) the solid curves represent the best fits to the data with Eq. (13) and $\gamma =1$. In (b) the dashed curves represent the best fit to the data with Eq. (13) and $\gamma <1$. The horizontal lines in (c) mark the range over which Eq. (14), shown by the solid lines, has been fitted to the data. For comparison, the dashed curve in (c) represents the best fit with Eq. (13). The fit parameters are shown in Fig. 13.

Figure 13

Shows the parameters $\gamma$ [defined in Eq. (13)], $\lambda$ [defined in Eq. (14)], and the quantity $\log \left[(q^2{D}_{\mathrm{l}}∕6){\tau }_{\mathrm{s}}\right]$ as function of volume fraction.