Glass transition of two-dimensional binary soft-disk mixtures with large size ratios

We simulate binary soft-disk systems in two dimensions and investigate how the dynamics slow as the area fraction is increased toward the glass transition. The “fragility” quantifies how sensitively the relaxation time scale depends on the area fraction, and the fragility strongly depends on the composition of the mixture. We confirm prior results for mixtures of particles with similar sizes, where the ability to form small crystalline regions correlates with fragility. However, for mixtures with particle size ratios above 1.4, we find that the fragility is not correlated with structural ordering, but rather with the spatial distribution of large particles. The large particles have slower motion than the small particles and act as confining “walls” which slow the motion of nearby small particles. The rearrangement of these confining structures governs the lifetime of dynamical heterogeneity, that is, how long local regions exhibit anomalously fast or slow behavior. The strength of the confinement effect is correlated with the fragility and also influences the aging behavior of glassy systems.

Figure 1

(Color online) The orientation pair correlation as a function of distance. The straight line corresponds to $R^{-1/4}$, the behavior expected for hexatic phases. A sample with size ratio ${\sigma }_L/{\sigma }_S=1.2$ and area fraction ratio ${\varphi }_L/{\varphi }_S=0.5$ at $\varphi =0.66$ does not have hexatic order since $g_6\left(R\right)/g\left(R\right)$ decays faster than the power of $-1/4$ (the lower curve, marked “Glass”). In contrast, the upper curve (marked “Hexatic”) shows the correlation for ${\sigma }_L/{\sigma }_S=1.15$, ${\varphi }_L/{\varphi }_S=0.5$, and $\varphi =0.66$, which has long-range orientational ordering.

Figure 2

(Color online) The contour plot of (a) fragility index $D$, (b) the growth rate of hexagonal order $\partial ⟨{\psi }_6⟩/\partial \varphi$, (c) the growth rate of dynamical correlation length $\partial \xi /\partial \varphi$, and (d) the dynamical correlation around a large particle $\Theta$ in a plane of $\left({\sigma }_L/{\sigma }_S,{\varphi }_L/{\varphi }_S\right)$. All those figures except (a) are obtained at $\varphi =0.66$. The shading in each figure is darker when each value is small; thus, the darker regions correspond to (a) fragile liquids, (b) slow growth of hexagonal order, (c) slow growth of dynamical correlation length, and (d) minimal correlations of motion between a large particle and its neighbors. The circles in (a) are the simulated points, and the specific states A–F are labeled.

Figure 3

(Color online) The area fraction $\varphi$ dependence of (a) the relaxation time ${\tau }_{\alpha }$, (b) average hexagonal ordering $⟨{\psi }_6⟩$, and (c) the dynamic correlation length $\xi$ at states A–F indicated in Fig. 2a. The solid lines in (a) are fitting lines with the Vogel-Fulcher function at each state. Filled and open symbols correspond to fragile liquids and less fragile liquids, respectively.

Figure 4

(Color online) The fragility index $D$ as a function of (a) $\partial ⟨{\psi }_6⟩/\partial \varphi$ and (b) $\partial \xi /\partial \varphi$ at $\varphi =0.66$. The triangles correspond to systems where size ratio is close to 1 $\left({\sigma }_L/{\sigma }_S<1.4\right)$, and the circles correspond to large size ratios $\left(>1.4\right)$. The lines are added as guides to the eye.

Figure 5

(Color online) (a) and (b) are snapshots of the system at state B, $\varphi =0.66$; (c) and (d) are at state E, $\varphi =0.66$. The simulation box is four times as large as those snapshots in each side. In (a) and (c), particles are colored based on mobility $\Delta r_j^2$, with darker colors indicating more mobile particles. The darkest color corresponds to $\Delta r_j^2=0.25{\sigma }_s^2$. The displacement time scale $\Delta t$ is chosen to maximize the non-Gaussian parameter, and is $\Delta t=5\times {10}^4$ for (a) and $2\times {10}^4$ for (c). In (b) and (d), particles are colored based on the hexagonal order parameter ${\psi }_6^j$, with the darkest color corresponding to ${\psi }_6^j=0.8$. The outlined regions are guides to the eye.

Figure 6

(Color online) ${\tau }_{\alpha }$ as a function of $\xi$ for our six representative states. We find two trends of behaviors. One is an exponential correlation for systems with size ratio close to 1 (A and B, ${\sigma }_L/{\sigma }_S=1.25$), and the other is a faster divergence of ${\tau }_{\alpha }$ with $\xi$ for the systems with larger size ratios.

Figure 7

(Color online) (a) Schematic showing a group of small particles temporarily confined within a region bounded by large particles. Each small particle is assigned a value $R_w$, which is its closest distance to any large particle (the large particles act as walls). Within each region, the local maximum of $R_w$ identifies the confinement pore size $R_c$ for that region. Thus, $R_c$ is the maximum radius of a circle that fits within the bounded region. (b) A contour plot for the mean-square displacement $⟨\Delta r^2⟩$ (at fixed $\Delta t=\Delta t^{\ast }$) as a function of the effective pore size $R_c$ where the given particle is located, and the distance of that particle from the nearest wall $R_w$, for state E with $\varphi =0.66$. Darker shading corresponds to high mobility. The motion of particles is faster in larger pores (large $R_c$) and when further from the walls (larger $R_w$ for a constant $R_c$). The length scales are in terms of the small particle diameter. For these data, $R_c^{\ast }=3.6$; only 10% of regions have $R_c>R_c^{\ast }$.

Figure 8

(Color online) (a) The time dependence of $F\left(k,\Delta t\right)$ for small particles [thick solid (black) line; $k=2\pi /{\sigma }_S$] and for large particles [thick dashed (blue) line; $k=2\pi /{\sigma }_L$] at state E and $\varphi =0.66$. The thin solid (red) line indicates the breakup of the largest confined regions, and the thin dashed (green) line indicates when the most mobile particles become less mobile. (b) The typical structural relaxation time ${\tau }_S$ for small particles (filled circles), the structural relaxation time ${\tau }_L$ for large particles (open circles), the confinement lifetime ${\tau }_c$ (diamonds), the dynamical heterogeneity lifetime ${\tau }_{DH}$ (triangles), and the peak time for the non-Gaussian parameter $\Delta t^{\ast }$ (squares). The data are for state E. (c) Five time scales (${\tau }_L$, ${\tau }_c$, ${\tau }_{DH}$, $\Delta t^{\ast }$, and ${\tau }_{hex}$) normalized by ${\tau }_S$ as functions of ${\sigma }_L/{\sigma }_S$ at ${\varphi }_L/{\varphi }_S=2.0$ at $\varphi =0.66$. ${\tau }_{hex}$ is the lifetime of hexagonal order and corresponds to the filled hexagon symbols. (These data go through state points A, C, and E; see Fig. 2.) (d) Five time scales (${\tau }_L$, ${\tau }_c$, ${\tau }_{DH}$, $\Delta t^{\ast }$, and ${\tau }_{hex}$) normalized by ${\tau }_S$ as functions of ${\sigma }_L/{\sigma }_S$ at ${\varphi }_L/{\varphi }_S=0.5$ at $\varphi =0.66$. (These data go through state points B, D, and F.) There are only a few confinement regions when ${\sigma }_L/{\sigma }_S<1.5$, preventing a clear determination of ${\tau }_c$, so those curves are clipped in (c) and (d).

Figure 9

(a) $⟨\Delta r^2⟩$ as a function of lag time $\Delta t$ for small particles (solid lines) and for large particles (dashed lines) at $t_w={10}^3$, ${10}^4$, ${10}^5$, and ${10}^6$ at state E and $\varphi =0.72$. The mobility of both particle species decreases with respect to aging time. (b) The dark black lines correspond to $⟨\Delta r^2⟩$ at $\Delta t={10}^5$ as a function of aging time $t_w$ for small particles (solid) and large particles (dashed). The smooth curves are fits with a stretched exponential, and we find that the decay time of large particles is slightly shorter than that of small particles. The gray line shows the waiting time dependence of $\Theta$, the correlation between the displacements of a large particle and its surrounding small particles. The choice of $\Delta t={10}^5$ is arbitrary although (a) shows that this is a reasonable choice to capture the slowing dynamics.

Figure 10

(a) The probability distribution $P\left(R_c\right)$ of confinement pore sizes at $t_w={10}^3$ (black line) and $t_w={10}^6$ (superimposed white line), showing that the distributions are essentially identical. This is for state E with $\varphi =0.72$. (b) Mean-squared displacements $⟨\Delta r^2⟩$ for fixed $\Delta t={10}^5$ as a function of the confinement pore size $R_c$ at $t_w={10}^3$ (circles) and $t_w={10}^6$ (squares), for the same state as (a). The upward trend (faster motion for larger pores) is similar to what is seen in equilibrated liquids [Fig. 7b]. For these data $\Delta t={10}^5$ to capture time scales over which the dynamics age [see Fig. 9b].

Figure 11

(Color online) Scatter plot of the mean-square displacement $⟨\Delta r^2⟩$ as a function of $\Theta$ at state E and $\varphi =0.72$, using $\Delta t={10}^5$. Each point corresponds to a different $t_w$, the time since the start of the simulation. The solid line is a least-squares fit. The data indicate that mobility is linked to the correlation of the motion between large particles and their nearest neighbors.