Slow relaxations and stringlike jump motions in fragile glass-forming liquids: Breakdown of the Stokes-Einstein relation

We perform molecular dynamics simulation on a glass-forming liquid binary mixture with the soft-core potential in three dimensions. We investigate crossover of the configuration changes caused by stringlike jump motions. With lowering the temperature $T$, the motions of the particles composing strings become larger in sizes and displacements, while those of the particles surrounding strings become smaller. Then the contribution of the latter to time-correlation functions tends to be long-lived as $T$ is lowered. As a result, the relaxation time ${\tau }_{\alpha }$ and the viscosity $\eta$ grow more steeply than the inverse diffusion constant $D^{-1}$ at low $T$, leading to breakdown of the Stokes-Einstein relation. At low $T$, the diffusion occurs as activation processes and may well be described by short-time analysis of rare jump motions with broken bonds and large displacements. Some characteristic features of the Van Hove self-correlation function arise from escape jumps over high potential barriers. We also visualize the particle motions at string formation taking place in a very short time.

Figure 1

(a) B particles [those with ${\mathcal{B}}_i={\mathcal{B}}_i(t_0,t_1)>0$] composed of their strings and their neighbors. (b) BL particles [those with ${\mathcal{B}}_i^{>}(t_0,t_1)>0$] composed of strings, which have large ${\mathcal{B}}_i\ge 3$ due to large displacements. These are snapshots at $t=t_1-t_0={10}^4\cong {10}^{-1}{\tau }_{\alpha }$ or at ${\varphi }_b^{>}\left(t\right)=0.1$ for $T=0.24$ (see Table ). The particle colors represent ${\mathcal{B}}_i$ according to the color bar below. Arrows represent $\Delta {\mathit{r}}_i(t_0,t_1)$.

Figure 2

Formation of a string in a small region in short time intervals $[t_0,t_1]$ for $T=0.24$, where $t_1-t_0=4k$ with $1\le k\le 8$. BL particles with broken bonds and with relatively large displacements $\Delta r_i(t_0,t_1)>0.8$ are written as spheres (small ones in blue and large ones in red). Also shown by arrows are non-B particles with $\Delta r_i(t_0,t_1)>0.2$. Arrows are from the initial positions at time $t_0$ to those at time $t_1$.

Figure 3

BL particles [those with ${\mathcal{B}}_i^{>}(t_0,t_1)>0$] for (a) $T=0.28$ and $t=t_1-t_0=450$ and for (b) $T=0.32$ and $t=t_1-t_0=50$. The corresponding snapshot for $T=0.24$ is given in Fig. 1b. Here, $t=t_1-t_0$ is determined from ${\varphi }_b^{>}\left(t\right)=0.1$. The particle colors represent ${\mathcal{B}}_i$ according to the color bar. The strings become ill defined with increasing $T$. Arrows represents $\Delta {\mathit{r}}_i(t_0,t_1)$.

Figure 4

Fractions of the B and BL particles, ${\varphi }_b\left(t\right)$ and ${\varphi }_b^{>}\left(t\right)$, respectively, for $T=0.24$. They are shown for $t<800$ (left) and on long times (right). Also displayed is the fraction of surviving bonds $1-F_b\left(t\right)$ nearly coinciding with ${\varphi }_b^{>}\left(t\right)$ on long time scales (right).

Figure 5

(a) Averages $\langle {\mathcal{B}}^{>}\rangle$ and $\langle {\mathcal{B}}^{<}\rangle$ in Eqs. (3.13) and (3.14) vs $T$. (b) Averages $\langle {\Delta r}_{\mathrm{BL}}\rangle$ and $\langle {\Delta r}_{\mathrm{BS}}\rangle$ in Eqs. (3.15) and (3.16) vs $T$. Here, ${\varphi }_b^{>}\left(t\right)=0.1$, from which $t=t_1-t_0$ is determined for each $T$ as in Table .

Figure 6

$F_s(q,t)$ in Eq. (2.4) and $F_s^b(q,t)$ in Eq. (3.17) for (a) $T=0.24$, (b) $T=0.28$, and (c) $T=0.32$. Shown in (d) is $F_s^b(q,{\tau }_{\alpha })$ multiplied by $e$ vs $T$, which increases at low $T$.

Figure 7

Left: Times ${\tau }_{\alpha }$ in Eq. (2.5), ${\tau }_{\mathrm{bp}}$ in Eq. (3.11), ${\tau }_{\mathrm{st}}$ in Eq. (3.12), and ${\tau }_b$ in Eq. (3.4) vs $T$. Right: ${\tau }_{\alpha }$, ${\tau }_{\alpha }^1$, ${\tau }_b$ vs $1/T$, where ${\tau }_{\alpha }^1$ is the $\alpha$ relaxation time for the small species.

Figure 8

Contributions to the mean-square displacement for $T=0.24$. (a) $M\left(t\right)$ from all the particles (top curve), $M_B\left(t\right)$ from B particles, $M_B^{>}\left(t\right)$ from BL particles, $M^{>}\left(t\right)$ from those with $\Delta r_i>0.8$, and $\overline{M}\left(t\right)$ for time-smoothed positions on a logarithmic scale. [See Eqs. (4.4, 4.5, 4.6, 4.7).] (b) Top five curves represent contributions divided by $6Dt$ on a semilogarithmic scale, tending to unity with increasing $t$. The approach is very rapid for $M^{>}\left(t\right)/6Dt$ and $M_B^{>}\left(t\right)/6Dt$. Shown also is the contribution divided by $6Dt$ from BL particles belonging to the second species (bottom line), which is very close to $D_2/(D_1+D_2)\cong 0.2$ for $t\gtrsim 10$.

Figure 9

(a) $M_B^{>}\left(t\right)$ vs ${\varphi }_b^{>}\left(t\right)$ for various $T$, where both grow linearly in time in the early stage. (b) Ratio ${\ell }_{\mathrm{jp}}^2=M_B^{>}\left(t\right)/{\varphi }_b^{>}\left(t\right)$ vs $T$, where ${\ell }_{\mathrm{jp}}$ is a characteristic jump length of a particle.

Figure 10

(a) Diffusion constants $D_1$ for small particles, $D_2$ for large particles, and $D=(D_1+D_2)/2$ as functions of $T$. (b) $D_1{\tau }_{\alpha }$, $D_2{\tau }_{\alpha }$, $D{\tau }_{\alpha }$, and $D{\tau }_b$ as functions of $T$. The curves of $D_1{\tau }_{\alpha }$ and $D_2{\tau }_{\alpha }$ indicate the SE violation from $\eta /T\propto {\tau }_{\alpha }$, whereas $D{\tau }_b$ only weakly depends on $T$.

Figure 11

$4\pi r^4{G}_s(r,t)$ vs $r$ at various $t$. (a) For $T=0.24$, there is a deep minimum at $r=r_{\mathrm{m}}\sim 0.7$, which separates a nearly stationary part for $r<r_{\mathrm{m}}$ and a growing part with multiple maxima for $r>r_{\mathrm{m}}$. (b) For $T=0.28$, a minimum at $r=r_{\mathrm{m}}\sim 0.7$ increases in time with one maximum for $r>r_{\mathrm{m}}$. A secondary peak in the outer region $r>r_{\mathrm{m}}$ starts to appear as a shoulder.

Figure 12

$4\pi r^4{G}_s(r,t)$ is decomposed into the small-particle part $2\pi r^4{G}_{s1}(r,t)$ and the large-particle part $2\pi r^4{G}_{s2}(r,t)$ from Eq. (4.12) at $t={10}^4$ for $T=0.24$. The minimum at $r=r_{\mathrm{m}}$ is common for the small and large particles.

Figure 13

$4\pi r^4{G}_s(r,t)$ vs $r$ at (a) $t=500$, (b) 10$$000, and (c) 50$$000 for $T=0.24$. The contributions from the particles with ${\mathcal{B}}_i(t_0,t_0+t)=k$ are shown, which have $k$ broken bonds in the time interval $[t_0,t_0+t]$.

Figure 14

$M_B^{>}\left(t\right)/6Dt$ from the BL particles with $r>{\ell }_{\mathrm{m}}=0,7,0.8$, and $0.9$ for $T=0.24$, where ${\ell }_{\mathrm{m}}=0.8$ in the other figures in this paper. They are close to the minimum distance $r_{\mathrm{m}}\sim 0.7$ of $4\pi r^{4}G(r,s)$ in Fig. 11a. These three curves converge to unity for $t\gtrsim 1000$.