Effect of mixing and spatial dimension on the glass transition

We study the influence of composition changes on the glass transition of binary hard disk and hard sphere mixtures in the framework of mode coupling theory. We derive a general expression for the slope of a glass transition line. Applied to the binary mixture in the low concentration limits, this method allows a fast prediction of some properties of the glass transition lines. The glass transition diagram we find for binary hard disks strongly resembles the random close packing diagram. Compared to three dimensions from previous studies, the extension of the glass regime due to mixing is much more pronounced in two dimensions where plasticization only sets in at larger size disparities. For small size disparities we find a stabilization of the glass phase quadratic in the deviation of the size ratio from unity.

Figure 1

(Color online) Normalized slopes of the GTLs at $x_s=0$ for the binary HSM models in two dimensions and three dimensions. It is ${\phi }_0^c\cong 0.6914$ for $d=2$ and 0.5159 for $d=3$. For $1/\delta >3.5$ in the 3D model, the tagged-particle NEPs indicate a delocalization transition of the smaller spheres. This regime is indicated by open symbols (see text).

Figure 2

(Color online) Slopes of the GTLs at $x_b=0$.

Figure 3

(Color online) (a) Relative variation of the glass transition lines for the binary HSM model in two dimensions. The open squares calculated with $K=400$ grid points (instead of $K=250$) give an estimate for the error due to the high wave number cutoff. (b) Relative variation of the glass transition lines for the binary HSM model in three dimensions using the numerical data of Götze and Voigtmann [7] calculated with $\Delta k=0.4$ and $K=200$.

Figure 4

(Color online) Relative variation in the random close packing fraction using the numerical data of Okubo and Odagaki [30]. ${\tilde{\phi }}_0^{rcp}$ is determined such that the relative variation vanishes below but close to $x_s=1$.

Figure 5

(Color online) (a) Glass transition lines for the binary HSM model in two dimensions plotted as functions of the packing contribution of the smaller particles ${\widehat{x}}_s={\phi }_s/\phi$. (b) and (c) $\alpha$-relaxation times defined by ${\Phi }_k^{bb}\left[{\left({\mathit{\tau }}^{rel}\right)}_k^{bb}\right]=0.1{\left({\mathit{F}}^c\right)}_k^{bb}$ for the correlators of the big particles at $k=5.1909$ for fixed $\delta$ and $\phi$ close below the corresponding GTLs for the binary HSM model in two dimensions. In both (a) and (c), the open triangles calculated with $K=400$ grid points (instead of $K=250$) give an estimate for the error due to the high wave number cutoff.

Figure 6

(Color online) Normalized correlators of the big particles for the binary HSM model in two dimensions at $\phi =0.686$, $\delta =5/7$, and $k=5.1909$ for different packing contributions ${\widehat{x}}_s={\phi }_s/\phi$. Filled diamonds mark the crossings of the normalized critical plateau values ${\left({\mathit{F}}^c\right)}_k^{bb}/{\left({\mathit{S}}^c\right)}_k^{bb}$. Open diamonds mark the crossings of the values $0.1{\left({\mathit{F}}^c\right)}_k^{bb}/{\left({\mathit{S}}^c\right)}_k^{bb}$.

Figure 7

(Color online) The same as Fig. 6 but for $\delta =1/3$ and $\phi =0.691$. The thin lines show the short-time asymptotes given by Eq. (2) for ${\widehat{x}}_s=0.01$ and ${\widehat{x}}_s=0.3$ (from left to right).

Figure 8

(Color online) Normalized critical NEPs for the big particles at $\delta =1/2$ for the binary HSM model in two dimensions.