Quantitative theory of a relaxation function in a glass-forming system

We present a quantitative theory for a relaxation function in a simple glass-forming model (binary mixture of particles with different interaction parameters). It is shown that the slowing down is caused by the competition between locally favored regions (clusters) that are long-lived but each of which relaxes as a simple function of time. Without the clusters, the relaxation of the background is simply determined by one typical length, which we deduce from an elementary statistical mechanical argument. The total relaxation function (which depends on time in a nontrivial manner) is quantitatively determined as a weighted sum over the clusters and the background. The “fragility” in this system can be understood quantitatively since it is determined by the temperature dependence of the number fractions of the locally favored regions.

Figure 1

(Color online) The relaxation function $C\left(t\right)$ as a function of logarithmic time (upper panel) and of linear time in the lower panel. The leftmost curve (in red) pertains to $T=1.0$, and in order to the right, the temperatures are $T=0.80$, 0.7, 0.65, 0.6, 0.56, 054, 0.52, 0.50, 048, 0.46, 0.44, and 0.42. Note the extreme slowing down in the range $0.56<T<0.42$. The black (dashed) line at $T=0.6$ separates “simple” time dependence at higher temperature, which can be well fitted to a stretched exponential function from “nontrivial” time dependence at lower temperatures, which is ill-fitted by any stretched exponential form. Our aim is to predict quantitatively the form and value of the relaxation function for all temperatures and times.

Figure 2

(Color online) A snapshot of the system at $T=0.44$. In colors we highlight the clusters of large particles in local hexagonal order. The colors have no meaning.

Figure 3

(Color online) A test of Eq. (3) with $\Delta =1.31$. In the inset we demonstrate Eq. (2) with $\mu =4.212$. Here ${\tau }_w$ was measured directly as the decay time of the relaxation function of the whey Eq. (5), and $\xi \left(T\right)$ was computed from Eq. (4).

Figure 4

(Color online) Demonstration that the relaxation of individual clusters of size $s$ (circles) follows the same relaxation function as the whey (continuous line), but with a different relaxation time ${\tau }_s$. In blue (left) we show $s=7$ at $T=0.5$, and in red (right) $s=10$ at $T=0.46$. Note that we do not have similar fits for large clusters, and Eq. (6) is proposed as a model for all $s$.

Figure 5

(Color online) Comparison of the measured relaxation functions with the theoretical prediction on the basis of the cluster decomposition formula (7). From left to right, the temperatures are $T=$1, 0.8, 0.65, 0.56, 0.50, 0.46, and 0.44. Note the quantitative agreement at all times and temperatures. The only parameter fitted here is $\sigma$ of Eq. (6).