Figure 1

Time-dependent diffusion coefficients $D_{\alpha }\left(t\right)$ as a function of time [upper curves corresponding to $D_1\left(t\right)$] and partial intermediate scattering functions $F_{11}\left(k,t\right)$ and $F_{22}\left(k,t\right)$ as a function of the wave vector of a repulsive Yukawa mixture with $z=0.15$, $A_1=10$, and $A_2=10\sqrt{5}$ for $t=t_0$ and $t=10t_0$. The volume fraction of the more interacting species is kept fixed at ${\varphi }_2=2.2\times {10}^{-4}$, and ${\varphi }_1$ takes the values ${\varphi }_1=0.725\times {10}^{-4}$ (left column), ${\varphi }_1=2.2\times {10}^{-4}$ (middle column), and ${\varphi }_1=6.6\times {10}^{-4}$ (right column). Solid lines are the SCGLE theoretical results, and circles are Brownian dynamics data. In the middle column we also include the partial static structure factors $S_{\alpha \alpha }\left(k\right)\equiv F_{\alpha \alpha }$ $\left(k,t=0\right)$, with the dotted lines being a smooth HNC-fit of the simulation data.Reuse & Permissions

Figure 2

SCGLE results for the collective diffusion propagators ${\Psi }_{\alpha \alpha }\left(k_{\text{max}},t\right)$ at fixed wave vector $k_{\text{max}}$ [the position of the main peak of $S_{\alpha \alpha }\left(k\right)$] for the Yukawa mixture with $z=0.15$, ${\varphi }_1={\varphi }_2=2.2\times {10}^{-4}$ calculated using HNC partial static structure factors evaluated at effective coupling parameters given by $A_{\alpha }=A_{\alpha }^0\chi$, with $A_1^0=10$ and $A_2^0=10\sqrt{5}$, for values of $\chi$ corresponding to the reference state in Fig. 1 (${\chi }^{\left(1\right)}=1.225$, curves labeled “1”) and to states slightly below (${\chi }^{\left(2\right)}=4.255$, labeled “2”) and above (${\chi }^{\left(3\right)}=4.330$, labeled “3”) the dynamic arrest transition. The dashed plateaus indicate the value of the nonergodicity parameters right at the transition $\left(\chi =4.257\right)$.Reuse & Permissions

Figure 3

(a) Collective diffusion propagators ${\Psi }_{\alpha \alpha }\left(k_{\text{max}},t\right)$ and (b) self-diffusion propagators ${\Psi }_{\alpha \alpha }^{\left(s\right)}\left(k_{\text{max}},t\right)$ at fixed wave vector $k_{\text{max}}$ [the position of the main peak of $S_{\alpha \alpha }\left(k\right)$] for the hard-sphere mixture with $\delta =0.3$ and $x_1=0.2$ for different total packing fraction $\varphi$. The results for $\varphi =0.53$ (dotted lines) illustrate the fully ergodic states, whereas those for $\varphi =57$ (dashed lines) illustrate the fully arrested states, in which both species are arrested. The results for $\varphi =0.55$ (solid lines) illustrate the hybrid states in which the large particles are arrested while the smaller particles continue to diffuse.Reuse & Permissions

Figure 4

Collective diffusion propagators ${\Psi }_{\alpha \beta }\left(k,t\right)$ and diagonal collective diffusion correlators ${\Phi }_{\alpha \alpha }\left(k,t\right)$ at a common wave vector corresponding to the position of the main peak of $S_{22}\left(k\right)$ (see the inset, which contains static structure factors at this state) for the hard-sphere mixture with $\delta =0.3$ and $x_1=0.2$ for the hybrid state with $\varphi =0.55$.Reuse & Permissions

Figure 5

Phase diagram of dynamic arrest states of the binary hard-sphere mixture. The three-dimensional state space $\left(\delta ,\varphi ,x_1\right)$ is spanned by the size disparity parameter $\delta \equiv {\sigma }_1/{\sigma }_2\le 1$, the total volume fraction $\varphi \equiv {\varphi }_1+{\varphi }_2$ with ${\varphi }_{\alpha }\equiv \pi n_{\alpha }{\sigma }_{\alpha }^3/6$, and the molar fraction of the smaller particles, $x_1\equiv n_1/\left(n_1+n_2\right)$, where ${\sigma }_{\alpha }$ and $n_{\alpha }$ are the hard-core diameter and the number concentration of particles of species $\alpha$. This state space is partitioned in three regions corresponding to fully ergodic, mixed, and fully nonergodic states. In this figure we show the intersection lines of this transition surface with the planes of constant $\delta$. The total volume fraction $\varphi$ has been scaled with the volume fraction ${\varphi }_g^{\left(m\right)}$ of the ideal glass transition of the monodisperse hard-sphere system. In (a) we illustrate the regime of moderate-size disparities with the values $\delta =1.0$ (horizontal line, corresponding to $\varphi ={\varphi }_g^{\left(m\right)}$), 0.8, 0.6 (thicker line), and 0.4, where the respective line divides the subspace $\left(\varphi ,x_1\right)$ in the regions of fully ergodic states (below the line) and fully nonergodic states. The four dots linked by the arrow at fixed total volume fraction $\varphi /{\varphi }_g^{\left(m\right)}=1.0175$ correspond to the states prepared in the experiment of Williams and van Megen [29] for $\delta =0.6$. In (b) we illustrate the regime of severe size disparities with the values $\delta =0.3$ and 0.2. In this regime the region of fully ergodic states also lies below the solid line but a region of mixed states appear (shaded region for the case $\delta =0.3$). This region is bounded from above by the localization transition of the small particles (dashed lines).Reuse & Permissions

Figure 6

Nonergodicity parameters $f_{\alpha \alpha }\left(k\right)$ associated with the collective correlators ${\Phi }_{\alpha \alpha }\left(k,t\right)$ of a hard-sphere mixture with size-asymmetry $\delta =0.8$ at three states along the glass transition curve corresponding to molar fractions $x_1=0$ (solid line), $x_1=0.4$ (dotted line), and $x_1=1.0$ (dashed line).Reuse & Permissions

Figure 7

Evolution of the squared localization length ${\gamma }_1$ and ${\gamma }_2$ of a hard-sphere mixture of size asymmetry $\delta =0.2$ when we increase the total volume fraction $\varphi$ at fixed molar fraction $x_1=0.2$. The solid curve represents $\left[{\sigma }_2^2/{\gamma }_2\right]{\delta }^4$, and the dashed curve represents $\left[{\sigma }_1^2/{\gamma }_1\right]$. The insets display the evolution of the nonergodicity parameters ${\psi }_{\alpha \alpha }\left(k\right)$ as the total volume fraction is increased at fixed $x_1=0.2$. The results for ${\psi }_{22}\left(k\right)$, from bottom to top, correspond to $\varphi =0.5402$ $\left(={\varphi }_g^{\left(1\right)}\right)$, 0.56, and 0.60, and the results for ${\psi }_{11}\left(k\right)$ correspond to $\varphi =0.586$ $\left(={\varphi }_g^{\left(2\right)}\right)$, 0.59, and 0.60.Reuse & Permissions