Tests of mode-coupling theory in two dimensions

We analyze the glassy dynamics of binary mixtures of hard disks in two dimensions. Predictions of the mode-coupling theory (MCT) are tested with extensive Brownian dynamics simulations. Measuring the collective particle density correlation functions in the vicinity of the glass transition, we verify four predicted mixing effects. For instance, for large size disparities, adding a small amount of small particles at a fixed packing fraction leads to a speedup in the long-time dynamics, while for small size disparities it leads to a slowing-down. Qualitative features of the nonergodicity parameters and the $\beta$ relaxation, which both depend in a nontrivial way on the mixing ratio, are found in the simulated correlators. Studying one system in detail, we are able to determine its ideal MCT glass transition point as $\phi {}^c=0.7948$ and test MCT predictions quantitatively.

Figure 1

Normalized critical nonergodicity parameters of the simulated collective density correlators ${\Phi }_k^{\mathit{bb}}(t)$ of big particles, extracted from Kohlrausch fits for size ratio $\delta =5/7$ and packing fraction $\phi =0.79$. Number concentrations of small particles $x_s$ vary as labeled in the key. The solid (red) line gives the MCT results calculated with a simulated structure factor input at $\phi {}_{\mathrm{MCT}}^c=0.6920$ and $x_s=0.5$.

Figure 2

Normalized critical nonergodicity parameters of the simulated collective density correlators ${\Phi }_k^{\mathit{ss}}(t)$ of the small particles, extracted from Kohlrausch fits for the size ratio $\delta =5/7$ and the packing fraction $\phi =0.79$. Number concentrations of small particles $x_s$ are the same as in Fig. 1. The solid (red) line shows the MCT results calculated with simulated structure factors as in Fig. 1.

Figure 3

Normalized critical nonergodicity parameters of simulated collective density correlators ${\Phi }_k^{\mathit{bb}}(t)$ of big particles, extracted from Kohlrausch fits for size ratio $\delta =1/3$ and packing fraction $\phi =0.81$. The various number concentrations $x_s$ are as shown in the key. MCT results using simulated structure factors as input with ${\phi }_{\mathrm{MCT}}^c=0.6991$ are shown as the solid (black) line for $x_s=0.5$.

Figure 4

Normalized critical nonergodicity parameters of the simulated collective density correlators ${\Phi }_k^{\mathit{ss}}(t)$ of small particles, extracted from Kohlrausch fits for size ratio $\delta =1/3$ and packing fraction $\phi =0.81$. The color and symbol coding for the different $x_s$ is the same as in Fig. 3. The solid (black) line shows MCT results calculated with simulated structure factors as used in Fig. 3.

Figure 5

Critical amplitudes obtained with Eq. (35) for big and small particles with $k_0{d}_b=2.5$. Data were extracted from the collective correlators at $\phi =0.79$, $\delta =5/7$, and $x_s=0.5$. Solid (black) and dashed (red) lines depict MCT results obtained with the same simulated structure factors as in Figs. 1 and 2.

Figure 6

Critical amplitudes obtained by Eq. (35) for $\delta =1/3$ and $x_s=0.5$ with $k_0{d}_b=5.37$ from the simulated correlators at $\phi =0.81$, $\delta =1/3$, and $x_s=0.5$. Solid (black) and dashed (red) lines depict MCT results obtained with the same simulated structure factors as used in Figs. 3 and 4.

Figure 7

Simulated normalized collective correlation functions of big particles for size ratio $\delta =5/7$ at $\phi =0.79$ and ${\mathit{kd}}_b=8.5$ for varying $x_s\in \{0.4,0.5,0.6,0.7\}$ as labeled in the key. Solid lines show examples of Kohlrausch fits to the $x_s=0.4$ correlator (black) and the $x_s=0.7$ correlator (blue).

Figure 8

Simulated normalized collective correlation functions of small particles for size ratio $\delta =5/7$ at $\phi =0.79$ and ${\mathit{kd}}_b=8.5$ for varying $x_s$. Color and symbol coding is the same as in Fig. 7. Solid lines show examples of Kohlrausch fits to the $x_s=0.4$ correlator (black) and the $x_s=0.7$ correlator (blue).

Figure 9

Simulated normalized collective correlation functions of big particles for size ratio $\delta =1/3$ at $\phi =0.81$ and ${\mathit{kd}}_b=9.0$ for varying $x_s\in \{0.5,0.6,0.7,0.8\}$ as labeled in the key. Solid lines show exemplary Kohlrausch fits to the $x_s=0.5$ correlator (black) and the $x_s=0.8$ correlator (blue).

Figure 10

Simulated normalized collective correlation functions of small particles for size ratio $\delta =1/3$ at $\phi =0.81$ and ${\mathit{kd}}_b=9.0$ for varying $x_s$. Color and symbol coding is the same as in Fig. 9. Solid lines show exemplary Kohlrausch fits to the $x_s=0.5$ correlator (black) and the $x_s=0.8$ correlator (blue).

Figure 11

Functions $X_k^{\alpha \alpha }(t)$ calculated from Eq. (36) with the simulated correlators at $\phi =0.79$, $\delta =5/7$, and $x_s=0.5$ by fixing $t^{\prime }{D}_0/d_b^2=0.7648$ and $t^{″}{D}_0/d_b^2=9.117$ for big and small particles.

Figure 12

Rescaled correlators for $\phi ⩽0.79$, $\delta =5/7$, and $x_s=0.5$ to collapse on one $\alpha$ master function at long times. Rescale times are independent of ${\mathit{kd}}_b$.

Figure 13

Normalized total collective correlators from BD simulations for $\delta =5/7$, $x_s=0.5$, and ${\mathit{kd}}_b=8.5$. Solid lines show corresponding MCT results, where the packing fractions are mapped according to Eq. (38). Inset: $\alpha$-relaxation time scales and the fitted power law $\tau /\tau {}^{*}=A|\phi -\phi {}_{\mathrm{sim}}^c|{}^{-\gamma }$, with $\gamma =2.4969$ (fixed from MCT), $A=1.633\times {10}^{-6}$, and $\phi {}_{\mathrm{sim}}^c=0.7948$. See text for details.

Figure 14

Normalized collective correlators from BD simulations for $\delta =5/7$, $x_s=0.5$, and ${\mathit{kd}}_b=8.5$. Solid lines show corresponding MCT results where the packing fractions have been mapped according to Eq. (38).

Figure 15

Normalized collective correlators from BD simulations for $\delta =5/7$, $x_s=0.5$, and ${\mathit{kd}}_b=6.4$. Solid lines show corresponding MCT results where the packing fractions have been mapped according to Eq. (38).

Figure 16

Normalized collective correlators from BD simulations for $\delta =5/7$, $x_s=0.5$, and ${\mathit{kd}}_b=13.0$. Solid lines show corresponding MCT results where the packing fractions have been mapped according to Eq. (38).

Figure 17

Kohlrausch stretching exponent ${\beta }_k^{\alpha \alpha }$ for big [(black) circles] and small [(red) squares] particles at $\delta =5/7$, $x_s=0.5$, and $\phi =0.79$. The MCT von Schweidler exponent $b=0.5571$ is indicated by the horizontal dashed (black) line.

Figure 18

Three correlators above the MCT glass transition point at $\delta =5/7$, $x_s=0.5$, $\phi =0.81$, $\phi =0.80$, and $\phi =0.795$, together with the highest liquid density at $\phi =0.79$. The correlators violate the scaling property shown in Fig. 12. The plateau values increase upon going more deeply into the glass.