Super-Arrhenius behavior of molecular glass formers

A theory is developed to calculate values of the potential-energy barriers to structural relaxation in molecular glass formers from the data of static pair-correlation function. The barrier height is shown to increase due to an increase in the number of “stable bonds” a particle forms with its neighbors and the energy of each bond as liquids move deeper into the supercooled (supercompressed) region. We present results for a system of hard spheres and compare calculated values of the structural relaxation time with experimental and simulation results.

Figure 1

(a) The reduced effective potential $\beta W\left(r\right)$ between a pair of particles separated by distance $r$ (expressed in units of hard-sphere diameter $\sigma )$ in a system of hard spheres at a packing fraction $\eta =0.59$. $\beta W_{ui},r_{hi}$ are, respectively, value and location of the $i\mathrm{th}$ maximum and $r_{li}$ is the location on the left-hand side of the shell where $\beta W_i\left(r\right)=\beta W_{ui}$ (shown by a dashed line). The locations $r_{li}^{\prime }$ and $r_{hi}^{\prime }$ are values of $r$ on the left- and the right-hand sides of the shell where $\beta W_i\left(r\right)=\beta W_{ui}-1$ (shown by a solid line). $\beta W_{di}$ is the depth of the $i\mathrm{th}$ shell. (b) Radial distribution functions $g\left(r\right)$ (dash-dotted line), $g_b\left(r\right)$ (solid line) and $g_s\left(r\right)$ (dashed line) vs $r$ at $\eta =0.59$. The various peaks correspond to various shells around a central particle. While $g\left(r\right)$ oscillates around one, $g_b\left(r\right)$ and $g_s\left(r\right)$ become zero at boundaries defined by $(r_{li},r_{hi})$ for $g_b\left(r\right)$ and $(r_{li}^{\prime },r_{hi}^{\prime })$ for $g_s\left(r\right)$. In the inset we show how values in the first shell differ from each other.

Figure 2

(a) Number of total bonds $N_b$, metastable bonds $N_m$, and stable bonds $N_s$ formed by a particle in a system of hard spheres vs packing fraction $\eta$. (b) Number of total particles $n_{t1}$, metastably bonded particles $(m$ particles) $n_{m1}$, and stably bonded particles $(s$ particles) $n_{s1}$ in the first shell. The number $n_{s1}$ increases rapidly and crosses $n_{m1}$ at $\eta \simeq 0.524$. At $\eta =0.524$ the crossover from nonactivated to activated dynamics takes place, due to the formation of cage by $s$ particles.

Figure 3

(a) The potential-energy barrier (activation energy) $\beta E_s$ vs $\eta$. Values found from the expression $\beta E_s\left(\eta \right)=A+\frac{B}{{({\eta }_0-\eta )}^{\delta }}$ with (i) $\delta =1.2,{\eta }_0=0.632$ (dashed line) and (ii) $\delta =1.6,{\eta }_0=0.649$ (dash-dotted line) are compared with the calculated values. (b) Calculated values (solid line) of $ln\left[\frac{{\tau }_{\alpha }}{{\tau }_0}\right]$ are compared with experimental values (filled square [26] and open circles [27]) and simulation values (open triangles [27] and stars [28]). The values of Refs. [27, 28] are shifted to lower density by an amount $\mathrm{\Delta }\eta =0.03$.