Comparison of dynamic light scattering measurements and mode-coupling theory for the tagged particle dynamics of a hard-sphere suspension

The mean-squared displacement, velocity autocorrelation function, and the non-Gaussian parameter, obtained by dynamic light scattering on suspensions of particles with hard-sphere interactions, are compared with the results of the idealized version of mode-coupling theory. Both leading order asymptotic and full numerical solutions of the mode-coupling equations are considered. Experiment and the full numerical results of the theory expose similar qualitative changes at the volume fraction of the first order freezing transition. In particular, the emergence of negative algebraic decays in the velocity autocorrelation function of the undercooled suspension suggest the emergence of clusters in which particles are trapped. Consistency of experiment, computer simulation, and theory in this regard suggests that, at particular strengths of the delayed, nonlinear feedback, contained in mode coupling theory, the latter predicts not only structural arrest which, as already established, is symptomatic of a glass transition, but also a more subtle change in dynamics that signals the onset of the first order transition.

Figure 1

Stretching index, $\nu$, versus volume fraction, $\phi$. The two dashed vertical lines are drawn at the first order freezing $({\phi }_f=0.493)$ and glass transition $({\phi }_g=0.565)$ volume fractions. These lines are also drawn in the remaining figures where $\phi$ is the abscissa.

Figure 2

MCT volume fraction versus the experimental volume fraction for the same stretching index. The solid line is the line of best fit to these data. The dashed lines correspond to the freezing and GT volume fractions, ${\phi }_f$ and ${\phi }_g$. See text for explanation.

Figure 3

(Color online) Mean-squared displacements, experiment compared with numerical MCT for volume fractions and stretching indices indicated as $(\phi ,\nu )$; (a) $\phi <{\phi }_f$. (b) $\phi <{\phi }_f$.

Figure 4

Reduced diffusion coefficient versus volume fraction. See text for further details.

Figure 5

(Color online) Reduced diffusion coefficient versus the stretching index. The dashed and solid lines, of slopes 4.9 and 5.2, are the best fitting straight lines through the experimental data and the MCT results, respectively.

Figure 6

Delay time, ${\tau }_m$, at the point of maximum stretching of the MSD, versus volume fraction.

Figure 7

Root-mean-squared displacement, $R_m$, at the point of maximum stretching of the MSD.

Figure 8

(Color online) Velocity autocorrelation functions; numerical MCT indicated by circles; the stretched exponential, Eq. (10), shown by solid lines; the power law, Eq. (11), shown by dashed lines. Stray points result from amplification of numerical noise on taking the second derivative of the MSD [Eq. (7)]. (a) For values $(\phi ,\nu )$ from left to right, (0.438,0.65), (0.478,0.55), $({\phi }_f=0.495,0.50)$, (0.510,0.44), (0.532,0.35), with each successive result displaced along the abscissa by one; the dashed/dotted line represents the stretched exponential through the experimental data of Ref. 12. (b) For values $(\phi ,\nu )$ from left to right, (0.510,0.44), (0.532, 0.35), (0.548,0.24), (0.558,0.11). (c) For (0.535,0.33). The crosses are experimental data. Take note of the changes to the scales of the axes when going from (a) to (c).

Figure 9

Parameters of the fit of Eq. (10) to VAFs of MCT (closed symbols) and experiment (open symbols, dashed lines). See text for further details.

Figure 10

Non-Gaussian parameter from MCT, for values $(\phi ,\nu )$ indicated.

Figure 11

Logarithm of MSD (closed symbols) and absolute value of the non-Gaussian parameter (open symbols) from MCT for pairs $(\phi ,\nu )$ indicated in Fig. 10.

Figure 12

(Color online) Minimum value of the non-Gaussian parameter from MCT. The line of best fit through the points between $\phi =0.49$ and $\phi =0.55$ extrapolates to $\min \left(\alpha \right)=0$ at $\phi =0.491$.

Figure 13

Experimental non-Gaussian parameter for volume fractions, left to right, 0.29, 0.500, 0.517, 0.534, 0.560. Each successive data set is displaced along the abscissa by one.

Figure 14

Logarithm of the delay time, ${\tau }_g$, at the minimum in the non-Gaussian parameter.

Figure 15

Double logarithm plots of delay times, ${\tau }_m$, extracted from the MCT and experimental MSDs, and the time scale, ${\tau }_{\beta }$, of the $\beta$ process from Ref. 8, versus the separation parameter. The solid and dashed lines represent the power laws to the MCT and experimental results. Note that $\log \left[\epsilon \left({\phi }_f\right)\right]\approx -0.91$. See text for additional explanation.

Figure 16

Double logarithm plots of the reduced diffusion coefficient versus the separation parameter. The solid and dashed lines represent the power laws to the MCT and experimental results. See text for explanation.