Microscopic Theory of Softness in Supercooled Liquids

We introduce a new measure of the structure of a liquid which is the softness of the mean-field potential developed by us earlier. We find that this softness is sensitive to small changes in the structure. Then, we study its correlation with the supercooled liquid dynamics. The study involves a wide range of liquids (fragile, strong, attractive, repulsive, and active) and predicts some universal behaviors such as the softness being linearly proportional to the temperature and inversely proportional to the activation barrier of the dynamics with system dependent proportionality constants. We establish a master equation between the dynamics and the softness parameter and show that, indeed, the dynamics, when scaled by the temperature and system dependent parameters, show a data collapse when plotted against softness. The dynamics of fragile liquids show a strong softness dependence, whereas that of strong liquids show a much weaker softness dependence. We also connect the present study with the earlier studies of softness involving machine learning (ML), thus, providing a theoretical framework for understanding the ML results.

Figure 1

Softness, $S$, against temperature for all the systems studied here [30]. Softness is calculated by fitting the mfpt potential, (Eq. (1), to a harmonic form near the minimum. The density of the WCA and the LJ systems are mentioned next to them and WAHN stands for Wahnström model [31]. $f_0=0.5$, 1.00, 1.75, are LJ 1.2 systems with activity (self-propulsion force) $f_0$ (See reference [30] for details). Inset: We have zoomed in near the origin and plotted the $y$ axis in logarithmic scale to show the error bars.

Figure 2

(a) The potential barrier $\beta \mathrm{\Delta }\mathrm{\Phi }(r)=-\beta \mathrm{\Phi }(r=0)$ [Eq. (1)] against inverse of softness. The plot shows linear behavior. Inset: $[\ln ({\tau }_{\mathrm{mfpt}})-C]/B$ shows a master curve for all the systems when plotted against softness, $S$. Here, $C$ and $B$ are the system dependent parameters obtained from the linear fit of $\ln ({\tau }_{\mathrm{mfpt}})$ against $1/S$. (b) Simulated relaxation time ${\tau }_{\alpha }$ as a function of inverse temperature [30]. The systems cover a wide range of fragility [36]. (c) $\mathrm{\Delta }E=T*\ln ({\tau }_{\alpha }/{\tau }_0)$ plotted against the inverse of softness. For each system, the data can be described by a linear fit with system dependent parameters. The inset enlarges some of the systems. (d) $[T*\ln ({\tau }_{\alpha }/{\tau }_0)-C^{\prime }]/B^{\prime }$ vs softness, $S$, shows a master curve for all the systems which is a proof of validity of Eq. (3). Here, $C^{\prime }$ and $B^{\prime }$ are obtained from Fig. 2.

Figure 3

The softness parameter obtained in the ML study,$S_{\mathrm{ML}}$ (Ref. [18]) vs the present softness parameter $S$. A linear correlation exists between them with system independent slope.