Quantitative theory of a time-correlation function in a one-component glass-forming liquid with anisotropic potential

The Shintani-Tanaka model is a glass-forming system whose constituents interact via an anisotropic potential depending on the angle of a unit vector carried by each particle. The decay of time-correlation functions of the unit vectors exhibits the characteristics of generic relaxation functions during glass transitions. In particular it exhibits a stretched exponential form, with the stretching index $\beta$ depending strongly on the temperature. We construct a quantitative theory of this correlation function by analyzing all the physical processes that contribute to it, separating a rotational from a translational decay channel. These channels exhibit different relaxation times, each with its own temperature dependence. Interestingly, the separate decay function of each of these processes is a temperature-independent function, and is shown to scale (exhibit data collapse) at different temperatures. Taken together with temperature-dependent weights determined a priori by statistical mechanics this allows one to generate the observed correlation function in quantitative agreement with simulations at different temperatures. This underlines the danger of concluding anything about glassy relaxation functions without detailed physical scrutiny.

Figure 1

(Color online) Relaxation function $C_R\left(t\right)$ as a function of linear time (upper panel), and of logarithmic time in the lower panel. The leftmost curve (in red) pertains to $T=0.4$ and in order to the right the temperatures are $T=0.3$, 0.25, 0.22, 0.20, 018, and 0.17. Note the extreme slowing down in this range of temperatures.

Figure 2

(Color online) Relaxation function $C_R\left(t\right)$ as a function of rescaled time $t∕{\tau }_{\alpha }\left(T\right)$. Note that the functions do not collapse on each other due to the change in functional form which is carried by $\beta \left(T\right)$.

Figure 3

(Color online) Potential curves for particle pairs with two spins, one spin, or no spin in favored position (blue continuous, green dashed, or red dashed-dotted line, respectively). Inset: the measured energies of particle pairs, falling in three distinct ranges with gaps between them.

Figure 4

(Color online) An example of an $n$-star with $n=5$, $i=2$, $j=2$, and $k=1$. The central particle has a spin with favored orientation with respect to edges 1 and 2. Thus these edges can be either blue or green, and this central spin cannot be favored with respect to any other edge. In the interesting range of temperatures we observe 36 $n$-stars with $4⩽i+j+k⩽6$. Colors are as used in Fig. 3.

Figure 5

(Color online) Test of data collapse to validate the assumption (12). All the temperature shown in Fig. 1 are employed in this test. Note that the data for the higher temperatures ($T=0.3$ and 0.25) do not contribute throughout the range of $t∕\tau$ since the raw functions are already extremely small at long times.

Figure 6

(Color online) The logarithm of the relaxation time ${\tau }_{\mathrm{rot}}(k=2)$ as a function of inverse temperature.

Figure 7

(Color online) Comparison of the theoretical relaxation function (7) (circles) with data from the simulations (continuous line). The temperatures shown from left to right are $T=0.4$, 0.3, 0.25, 0.22, 0.20, 018, and 0.17.