Random walk model of mobility in glass formers

A mechanism responsible for the generic features of the mean squared displacement and the decay of the orientational autocorrelator of a molecule in a glass forming liquid is poorly understood, where such a mechanism would be critical for creating the theory of glass transition. A discrete random walk model is proposed where, instead of a straight line, a walk is along a tortuous path consisting of blocks of switchback ramps. Subdiffusive regime, short-term dynamic heterogeneity, and existence of the α- and β-relaxation processes emerge naturally from the model. The model suggests that slowing of the rate of relaxation may be due to an increase in the number of the switchback ramps per block rather than due to growth of an energy barrier as typically assumed.

Figure 1

Results of MD simulation of a three-dimensional dumbbell system. (Data courtesy of J. Yungbluth.) Details of the molecular potential are in Ref. [1]. (a) MSD of the center of mass of a dumbbell, (b) rotational autocorrelators: $C_1\left(t\right)=\langle cos\theta \rangle$, blue (upper curve at ${log}_{10}t=0$), and $C_2\left(t\right)=3\langle {cos}^2\theta \rangle 2-1/2$, red. Dashed line in (a) is the asymptote with the slope of unity.

Figure 2

Schematic of a switchback structure. The case of $N_r=7,L=5$ is shown. Site A is between adjacent blocks; site B is at the center of a block.

Figure 3

MSD vs time for the switchback model. (a) Effect of varying the number of ramps at fixed $L=4$ in the order indicated by the arrow: orange, $N_r=7$; green, $N_r=9$; blue, $N_r=11$; magenta, $N_r=13$. (b) Effect of varying the ramp length at fixed $N_r=7$ in the order indicated by the arrow: orange, $L=4$; cyan, $L=6$; red, $L=8$; black, $L=10$. Dashed line is the standard case of a discrete random walk on a straight line $\langle x^2\rangle \left(t\right)=t$.

Figure 4

Autocorrelator vs time for the switchback model for $\mathrm{\Omega }=15$ (number of step lengths in $\pi$). (a) Effect of varying the number of ramps at fixed $L=12$ in the order indicated by the arrow: red, $N_r=5$; orange, $N_r=7$; green, $N_r=9$; cyan, $N_r=11$; blue, $N_r=13$; magenta, $N_r=15$. (b) Effect of varying the ramp length at fixed $N_r=13$ in the order indicated by the arrow: red, $L=5$; orange, $L=6$; green, $L=8$; cyan, $L=10$; blue, $L=12$. Dashed line is for the case of a discrete random walk on a circle for $\mathrm{\Omega }=15$.

Figure 5

(a) Orientational autocorrelators for $\mathrm{\Omega }=15,L=6,N_r=13$ case, dashed lines, and $\mathrm{\Omega }=15,L=12,N_r=13$ case, solid lines; $C_1\left(t\right)$, blue (upper), and $C_2\left(t\right)$, red (lower) lines. (b) Spectral strength vs relaxation time for the Prony series expansion of $C_1\left(t\right)$, blue, and $C_2\left(t\right)$, red (filled), for the $\mathrm{\Omega }=15,L=12,N_r=13$ case. Main α and β peaks are indicated.

Figure 6

Effect of initial conditions on the autocorrelator functions $C_1\left(t\right)$ and $C_2\left(t\right)$ for the $\mathrm{\Omega }=15,L=12,N_r=13$ case. Initial position: uniformly distributed within a block, dashed; located between adjacent blocks, solid blue (lower); located in the center of a block, solid red (upper).