Nonlinear dynamic response of glass-forming liquids to random pinning

We use large scale computer simulations of a glass-forming liquid in which a fraction $c$ of the particles has been permanently pinned. We find that the relaxation dynamics shows an exponential dependence on $c$. This result can be rationalized by assuming that the configurational entropy of the pinned liquid decreases linearly upon increasing of $c$. This behavior is discussed in the context of thermodynamic theories for the glass transition, notably the Adam-Gibbs picture and the random first order transition theory. For intermediate and low temperatures we find that the slowing down of the dynamics due to the pinning saturates and that the cooperativity decreases with increasing $c$, results which indicate that in glass-forming liquids there is a dynamic crossover at which the shape of the relaxing entities changes.

Figure 1

(a) Arrhenius plot of the $\alpha \text{−}$relaxation time $\tau (c,T)$, normalized by the relaxation time of the bulk, for different concentrations of pinned particles. The data for $c\le 0.031$ are for $N=20000$ particles, the other are for $N=1024$. Note that the error bars in the curve for $c=0$ represent the errors of $\tau (c=0,T)$ and they have been used to obtain the error bars for the curve for $c>0$. (b) $T$ dependence of the activation energy $E\left(T\right)$ for different values of $c$ for $N=20000$.

Figure 2

Time dependence of ${\chi }_4$ for different values of $c$ and $T$. (a) $T=8.0$, (b) $T=5.5$. Note the different scales of the ordinate in the two panels.

Figure 3

Height of peak in ${\chi }_4(t,c,T)$ as a function of $c$ for different temperatures. Also shown is the function $1/c$ (dashed-dotted line), which clearly overestimates the envelope to the data. The latter is described well by the effective power law $c^{-0.5}$ (dashed line).

Figure 4

Value of $Q_c^{\infty }$, the collective overlap at long times, as a function of $c$ for different temperatures.

Figure 5

$\alpha \text{−}$relaxation time $\tau (c,T)$, normalized by the relaxation time of the bulk, as a function of concentration of pinned particles. (a) $N=20000$, (b) $N=1024$. Note that the two panels show a different range in $c$.

Figure 6

(a) $T$ dependence of the slope of the curves in Fig. 5. The $T$ dependence of this slope changes significantly at a temperature that is close to the mode-coupling temperature of the system. (b) $T$ dependence of the ratios $Y_{\mathrm{AG}}\left(T\right)/A$ (circles) and $Y_{\mathrm{RFOT}}\left(T\right)$ (triangles) from the Adam-Gibbs theory and the RFOT theory, respectively.