Scaling behavior of self-diffusion coefficients: Dependence on the mass ratio of a binary mixture

In the present work we demonstrate how the diffusion constants $D_S$ for a tagged particle of species $s\left(=1,2\right)$ of a binary mixture depend on the mass ratio $\kappa =m_2/m_1$ of the two constituents. This dependence has also been reported earlier from simulations of a mixture. We use here a proper formulation of the self-consistent mode-coupling theory (MCT) of a two-component system. The effects of nonlocal and nonlinear coupling of slowly decaying hydrodynamic fluctuations on the long-time dynamics of a dense mixture is calculated here. The MCT memory functions are estimated here in the adiabatic approximation which assumes for the glassy state fast decay of momentum fluctuations compared to that of the number density. Plots $D_1$ and $D_2$ obtained from evaluation of the self-consistent MCT formulas, shows that for nearly equal-sized particles, the two self-diffusion coefficients are related as $D_1\sim D_2^a$ with a nonuniversal exponent $a$ and this relation holds over a wide range of $\kappa$ values. The simple MCT used here predicts an ideal ergodic-nonergodic transition from liquid to glassy state for critical packings $\{{\eta }_2^c,{\eta }_1^c\}$ of the two species. For high packing of one species ${\eta }_2$, at which the mixture is already in a nonergodic state, transition to an ergodic state occurs at a small ${\eta }_1^c$ and a reenetry into the nonergodic phase can occur again at an even larger value of ${\eta }_1^c$, while ${\eta }_2$ remains same as before.

Figure 1

The main (inset) shows tagged particle correlation ${\psi }_2(q,t) \left[{\psi }_1(q,t)\right]$ for particles of species 2 (1) vs time in units of $\sqrt{\beta m_1{\sigma }_2^2}$. $q$ is near the structure factor peak. For the mixture, size ratio $\alpha =1$, mass ratio $\kappa =10$, and total packing fraction $\eta =0.5$. Fraction of species 2 is $x=0.95$.

Figure 2

The main (inset) shows total density correlation ${\varphi }_{\rho \rho }$ (${\varphi }_{cc}$) near the structure factor peak vs time in units of $\sqrt{\beta m_1{\sigma }_2^2}$. Binary parameters are the same as given in Fig. 1.

Figure 3

Same as in Fig. 1 for an identical system, but with size ratio $\alpha =10$.

Figure 4

Same as in Fig. 2 but with a size ratio $\alpha =10$.

Figure 5

A comparison of theory and simulations in a log-log plot of self-diffusion coefficients $D_s$ for species $s$ ($=1,2$) of a binary mixture of hard sphere. Size ratio $\alpha =1.2$, mass ratio $\kappa =1$, and relative abundance $x=0.50$ for the mixture. For each species $s=1,2$ the respective $D_s$ from the MCT model presented here (squares) and from simulations of Ref. [11] (circles) are normalized by the corresponding diffusion constant values at packing fraction $\eta =0.400$. The scaling exponents are $a=0.95$ (theory) and 0.88 (simulations).

Figure 6

Same as in Fig. 5 for a mixture with size ratio $\alpha =1.67$, mass ratios $\kappa =1$, and relative abundance $x=0.46$. For each species $s=1,2$ the respective $D_s$ from the MCT model presented here (squares) and from simulations of Ref. [11] (circles) are normalized by the corresponding diffusion constant values at packing fraction $\eta =0.400$. The scaling exponents are $a=0.80$ (theory) and 0.71 (simulations).

Figure 7

Self-diffusion coefficients (expressed in units of ${\sigma }_2/\sqrt{\beta m_1}$) for a hard-sphere mixture with size ratio $\alpha =1$ and respective mass ratios $\kappa =5$ (circle), 10 (square), and 20 (triangle). Inset shows simulation results [34] of a truncated and shifted Lennard-Jones mixture for $\alpha =1$ and $\kappa =10$. The plots follow a scaling relation $D_1=\vartheta D_2^a$ with $a=1.30$, 1.15, and 1.12, respectively (main) and $a=0.90$ (inset). Using the respective amplitude factors $\vartheta \left(\kappa \right)$, the straight lines for different $\kappa$ values have been shifted along the horizontal axis so as to pass through the common point of the filled circle representing the one-component system.

Figure 8

Bare self-diffusion coefficients obtained in Ref. [35] for the mixture of Fig. 7 having size ratio $\alpha =1$ and respective mass ratios $\kappa =5$ (circle), 10 (square), and 20 (triangle). All the data points follow scaling relation $D_1=\vartheta D_2^a$ with the same $a=0.95$ and $\vartheta \left(\kappa \right)=0.8$.

Figure 9

Self-diffusion coefficients (expressed in units of ${\sigma }_2/\sqrt{\beta m_1}$) for a binary mixture of hard spheres of size ratio $\alpha =1.2$, mass ratio $\kappa =1$, and relative abundance $x=0.8$ (circle), 0.5 (square). The data corresponds to the total packing fraction $\eta$ range $\{0.500–0.520\}$. Inset shows simulation results [36] with circle (square) for a Kob-Anderson (square-well) mixture with size ratio $\alpha =1.25\left(1.20\right)$, and mass ratio $\kappa =1\left(1\right)$. The range of temperatures in Lennard-Jones units is $\{0.46,6.0\}\left(\right\{0.31–10\left\}\right)$. The scaling exponents are $a=0.95$ (main) and 0.94 (inset).

Figure 10

Self-diffusion coefficients of species 1 (filled) and 2 (blank) symbols vs mass ratio $\kappa$ in a mixture of equal-sized hard sphere with packing fraction $\eta =0.5$. The relative proportion of species 2 is varied from $x=0.9,0.5,0.2,$ and 0.1 as shown in the box.

Figure 11

Relaxation time $\tau$ (in units of $\sqrt{\beta m_1}{\sigma }_2$) for the total density correlation ${\varphi }_{\rho \rho }$ near the structure factor peak vs mass ratio $\kappa$ for a mixture of hard spheres, with size ratio $\alpha =10$, relative abundance $x=0.95$, and total packing fraction $\eta =0.5$. Inset: $\tau$ vs $x$, for size ratio $\alpha =10$. The curves are labeled with corresponding mass ratio $\kappa$.

Figure 12

The critical packing fraction ${\eta }_c$ corresponding to the ENE transition of MCT vs the relative abundance $x$ of species 2 in a hard-sphere mixture with $\alpha =1$ and mass ratio $\kappa =1$ (circle), 5 (squares), and 10 (traingles).

Figure 13

The mass ratio ${\kappa }_c$ and corresponding size ratio ${\alpha }_c$ at the ENE transition of MCT for a $50:50$ mixture with packing $\eta =0.50$ (solid) and 0.55 (dashed).

Figure 14

The respective packing fractions of the two species at the ENE transition points of MCT for a hard-sphere mixture of size ratio $\alpha =10$ and mass ratio $\kappa =1$ (main), 0.5 (upper inset), and 5 (lower inset). The $X$ and $Y$ axes respectively show ${\eta }_2^c$ and ${10}^3{\eta }_1^c$. At a fixed ${\eta }_2^c$, reentry into the ENE transition point occurs in the parameter space at a higher value of ${\eta }_1^c$.

Figure 15

Self-diffusion coefficients (expressed in units of ${\sigma }_2/\sqrt{\beta m_1}$) for a hard-sphere mixture with size ratio $\alpha =10$ and mass ratio $\kappa =10$ at packing $\eta =0.5$. The fraction of bigger particles $x$ is changing from 0.58 to 0.95. The scaling relation stated in Fig. 7 does not hold here.