Tagged-particle dynamics in a hard-sphere system: Mode-coupling theory analysis

The predictions of the mode-coupling theory of the glass transition (MCT) for the tagged-particle density-correlation functions and the mean-squared displacement curves are compared quantitatively and in detail to results from Newtonian- and Brownian-dynamics simulations of a polydisperse quasi-hard-sphere system close to the glass transition. After correcting for a 17% error in the dynamical length scale and for a smaller error in the transition density, good agreement is found over a wide range of wave numbers and up to five orders of magnitude in time. Deviations are found at the highest densities studied, and for small wave vectors and the mean-squared displacement. Possible error sources not related to MCT are discussed in detail, thereby identifying more clearly the issues arising from the MCT approximation itself. The range of applicability of MCT for the different types of short-time dynamics is established through asymptotic analyses of the relaxation curves, examining the wave-number and density-dependent characteristic parameters. Approximations made in the description of the equilibrium static structure are shown to have a remarkable effect on the predicted numerical value for the glass-transition density. Effects of small polydispersity are also investigated, and shown to be negligible.

Figure 1

Comparison of the static structure factor $S\left(q\right)$ for the simulated polydisperse soft-sphere system at $\phi =0.58$ (symbols) with the Percus-Yevick approximation to the hard-sphere $S\left(q\right)$ at the same density (solid line). The dashed line shows the Percus-Yevick result for $\phi =0.505$. The region around the first diffraction peak is enlarged in the inset.

Figure 2

MCT results for the critical nonergodicity parameters $f^c\left(q\right)$, using as input the static structure factor $S\left(q\right)$ from Percus-Yevick theory for hard spheres (dashed line). The crosses connected by the solid line show the results using $S\left(q\right)$ as obtained from the polydisperse nearly-hard-sphere simulation.

Figure 3

Simulation results for $\phi =0.50$, 0.53, 0.55, 0.57, 0.58, 0.585, 0.59 (from left to right) at wave vector $\mathit{qd}=7.8$. Heavy solid lines are the results using Brownian dynamics, thin lines the results for Newtonian dynamics. For the latter curves, times $t$ have been multiplied by factors $t_{*}$ given in the inset. The dotted line is the BD result for $\phi =0.01$, indicating the dilute limit of the correlation function. The solid diamonds indicate the points where Newtonian and Brownian dynamics results start to agree at a 2% level.

Figure 4

Comparison of the BD data for $\phi =0.58$ (plus symbols) at wave vectors $\mathit{qd}=4.0$, 7.8, 13.8, and 19.8 (from top to bottom). The long-time part of the data for $\phi =0.55$ (I symbols), 0.57 (crosses), 0.585 (circles), and 0.59 (star symbols) is also shown, scaled in $t$ to match the long-time part of the $\phi =0.58$ data. Solid lines are the MCT master curves for shifted $q_{\mathrm{MCT}}$ (see text for details).

Figure 5

Demonstration of the variability between different simulation runs for the ND and BD simulations at $\phi =0.59$. Data is shown for $\mathit{qd}=7.8$, with open (filled) symbols denoting BD (ND) results. Triangles indicate averages over a small subset of the data (8 out of 30 sets for BD; 25 out of 70 for ND) only; inverted triangles are the averages over the remaining data sets. The solid lines without symbols are the total averages. Dotted lines indicate the time-scaled MCT $\alpha$-master functions.

Figure 6

Example for Kohlrausch fits to the simulation data at $\phi =0.58$ (symbols: Brownian dynamics; dots: Newtonian dynamics), $\mathit{qd}=4.0$, 7.8, 9.0, 13.8, 17.0, and 19.8. The fit range was $t⩾55$.

Figure 7

Critical nonergodicity parameter $f^{s,c}\left(q\right)$ calculated from MCT for the one-component hard-sphere system with PY approximation (solid line), and for a five-component polydisperse system (dashed line). Symbols are the amplitudes $A\left(q\right)$ from Kohlrausch fits to the data, where error bars estimate deviations depending on $\phi$ and BD∕ND. The dash-dotted line indicates $f^{s,c}\left(q\right)$, but transformed with the wave-number shift applied in the discussion of the dynamical data (see text for details).

Figure 8

Stretching exponents $\beta \left(q\right)$ from fits to the BD simulation data at $\phi =0.58$ using Kohlrausch laws, Eq. (9). Error bars indicate deviations estimated from fits to ND data and to $\phi =0.57$. The dashed horizontal line indicates the $b$ value calculated from MCT using the Percus-Yevick approximation for $S\left(q\right)$, $b\approx 0.583$, and the dash-dotted line is $b$ as determined from MCT with simulation-data input for $S\left(q\right)$, $b\approx 0.521$.

Figure 9

$\tau \left(q\right)$ from Kohlrausch fits, Eq. (9), to the BD data at $\phi =0.58$ (diamonds) and at $\phi =0.59$ (squares; scaled by a factor of 0.021). Error bars estimate the uncertainty from the fits; see text for details. The dotted line shows a $1∕q^2$ law.

Figure 10

Plots of ${\tau }_{\alpha }^{-1∕\gamma }$ for wave vectors $\mathit{qd}=4.0$ (diamonds), 7.8 (squares), 9.0 (up triangles), and 13.8 (down triangles); using $\gamma =2.46$. (a) Results for the simulation data using Brownian dynamics (open symbols) and Newtonian dynamics (filled symbols). The values of $\tau$ for the latter have been multiplied by 4.5 for this plot. Dashed lines are linear fits to the Brownian dynamics data in the range $0.54⩽\phi ⩽0.58$.

Figure 11

(a) $\beta$ analysis of the BD simulation data at $\phi =0.58$: the curves marked by symbols show $X(q,t)=[{\varphi }^s(q,t)-{\varphi }^s(q,t^{\prime })]∕[{\varphi }^s(q,t^{\prime })-{\varphi }^s(q,t^{″})]$ with $t^{\prime }=8.234$ and $t^{″}=20.8075$. Wave vectors are $\mathit{qd}=4.0$ (diamonds), $\mathit{qd}=7.8$ (squares), $\mathit{qd}=9.0$ (up triangles), $\mathit{qd}=13.8$ (down triangles), and $\mathit{qd}=17.0$ (circles). The dashed line is the equivalent of the MCT $\beta$ correlator; see text for details. The dash-dotted line indicates the plateau value estimated from the root of the $\beta$ correlator. (b) Same for the ND simulation data, $t^{\prime }=2.31$ and $t^{″}=5.845$.

Figure 12

MCT (solid lines) and simulation results (symbols) for ${\varphi }^s(q,t)$. For the simulation data, packing fractions are $\phi =0.50$, 0.53, 0.55, 0.57, 0.58, 0.585, and 0.59 (from left to right), and $\mathit{qd}=7.8$. For the MCT results, packing fractions have been adjusted to ${\phi }_{\mathrm{MCT}}=0.445$, 0.47, 0.484, 0.499, 0.505, 0.508, and 0.5135, and the wave number has been adjusted to $q_{\mathrm{MCT}}d=9.13$; see text for details.

Figure 13

MCT and simulation data for ${\varphi }^s(q,t)$ with symbols and packing fractions as in Fig. 12, but for $\mathit{qd}=9.0$, adjusted in the MCT calculations to $q_{\mathrm{MCT}}d=10.3$.

Figure 14

MCT and simulation data for ${\varphi }^s(q,t)$ with symbols and packing fractions as in Fig. 12, but for $\mathit{qd}=13.8(q_{\mathrm{MCT}}d=15.1)$.

Figure 15

MCT and simulation data for ${\varphi }^s(q,t)$ with symbols and packing fractions as in Fig. 12, but for $\mathit{qd}=17.0(q_{\mathrm{MCT}}d=18.3)$.

Figure 16

Plot of ${\phi }_{\mathrm{MCT}}$ vs $\phi$ for the fits shown in Figs. 12, 13, 14, 15, 17, 18 (diamonds). The dashed line is a linear fit, ${\phi }_{\mathrm{MCT}}\approx 0.81\phi +0.037$. The circles are for the independent fit of the MSD, Fig. 19. The dotted horizontal line indicates the calculated critical point, ${\phi }^c\approx 0.516$.

Figure 17

MCT and simulation data for ${\varphi }^s(q,t)$ with symbols and packing fractions as in Fig. 12, but for $\mathit{qd}=4.0(q_{\mathrm{MCT}}d=5.0)$.

Figure 18

Comparison of the mean-squared displacements $\delta r^2\left(t\right)$ from simulation and MCT calculations; all fit parameters have been taken from Fig. 12. In addition, the MCT curves have been multiplied by 1.1 to account for an error in localization length; see text for details.

Figure 19

Comparison of the mean-squared displacements $\delta r^2\left(t\right)$ from simulation and MCT calculations. In this plot, values ${\phi }_{\mathrm{MCT}}$ used for the MCT calculations have been adjusted to fit the long-time diffusion regime of the simulated $\delta r^2\left(t\right)$; the values are ${\phi }_{\mathrm{MCT}}=0.438$, 0.46, 0.478, 0.494, 0.502, 0.503, and 0.511 for $\phi =0.50$, 0.53, 0.55, 0.57, 0.58, 0.585, and 0.59, respectively.

Figure 20

Self-diffusion coefficients $D$ as a function of the packing fraction $\phi$, as obtained from the Brownian dynamics simulation (diamonds, with connecting lines to guide the eye) and from MCT calculations (solid line). The dashed line indicates the MCT asymptote, $D\propto {({\phi }^c-\phi )}^{\gamma }$.

Figure 21

Product $D\bullet \tau \left(q\right)$ at wave vector $\mathit{qd}=7.8$ for the BD simulation (crosses, connected by lines to guide the eye), and for the MCT curves (solid line). The latter curve has been transformed along the horizontal axis according to Fig. 16. The inset shows a magnification of the MCT result versus ${\phi }_{\mathrm{MCT}}$.

Figure 22

Averaged static structure factor $S\left(q\right)$ for a monodisperse system (dashed lines) and the polydisperse system studied in this work (solid lines) at the same state point. The main figure shows the simulation results at $\phi =0.54$. The inset shows the Percus-Yevick $S\left(q\right)$ at the respective MCT-critical packing fractions ${\phi }_{\mathrm{MCT}}^c$, for both the monodisperse (dashed) and a five-component system (solid lines).

Figure 23

(a) Correlation functions ${\varphi }_{\alpha }^s(q,t)$ for the BD simulation, binned into three particle sizes, $\alpha =\text{small}$ (diamonds), medium (plus symbols), and large (circles); see text for details. The data refer to packing fraction $\phi =0.58$ and $\mathit{qd}=7.8$. The averaged quantity ${\varphi }_{\mathrm{pd}}^s(q,t)$ analyzed in the discussion in detail is plotted as a solid line but coincides with the $\alpha =\text{medium}$ curve on the scale of the figure. The solid and dashed lines decaying at shorter times are the results for $\phi =0.54$ from the simulation of the polydisperse and the monodisperse system, respectively. (b) As in (a), but results from the MCT calculations for a three-component mixture at packing fraction $\phi =0.505$ and $\mathit{qd}=7.8$. Again, the averaged ${\varphi }_{\mathrm{pd}}^s(q,t)$ is included as a solid line and is hidden by the $\alpha =\text{medium}$ curve. The solid line decaying at shorter times is the averaged result from the three-component mixture at $\phi =0.4$. For comparison, one-component results at $\phi =0.42$ and $\phi =0.507$ are included as dashed lines, the latter being obscured by the polydisperse-averaged $\phi =0.505$ result.