Time scale for the onset of Fickian diffusion in supercooled liquids

We propose a quantitative measure of a time scale on which Fickian diffusion sets in for supercooled liquids, and we use Brownian dynamics computer simulations to determine the temperature dependence of this onset time in a Lennard-Jones binary mixture. The time for the onset of Fickian diffusion ranges between 6.5 and 31 times the $\alpha$ relaxation time (the $\alpha$ relaxation time is the characteristic relaxation time of the incoherent intermediate scattering function). The onset time increases faster with decreasing temperature than the $\alpha$ relaxation time. Mean-squared displacement at the onset time increases with decreasing temperature.

Figure 1

The probability of the logarithm of single-particle displacements $P\mathbf{(}{\log }_{10}\left(\delta r\right);t\mathbf{)}$ at $T=0.45$ for the $A$ particles. The time $t$ is equal to, from left to right, ${\tau }_{ng}=135$, ${\tau }_{\alpha }=729$, ${\tau }_{nng}=1550$, ${\tau }_{0.1}=2841$, and ${\tau }_F=\mathrm{22\hspace{0.17em}391}$ (see text for definition of these times). For a comparison, we also show, as a dotted line, $P\mathbf{(}{\log }_{10}\left(\delta r\right);t\mathbf{)}$ resulting from a Gaussian distribution of displacements.

Figure 2

Left panel: The probability of the logarithm of single-particle displacements $P\mathbf{(}{\log }_{10}\left(\delta r\right);{\tau }_F\mathbf{)}$ for the $A$ particles at $T=1.0$, 0.8, 0.6, 0.55, 0.50, 0.47, and 0.45, listed from left to right. Right panel: the probability distributions from the left panel shifted in such a way that $⟨\delta r^2⟩=1$; for a comparison we also show, as a dotted line, $P\mathbf{(}{\log }_{10}\left(\delta r\right);t\mathbf{)}$ resulting from a Gaussian distribution of displacements with $⟨\delta r^2⟩=1$.

Figure 3

Temperature dependence of characteristic times for the $A$ particles. Diamonds: the $\alpha$ relaxation time, ${\tau }_{\alpha }$. Circles: the onset time for Fickian diffusion, ${\tau }_F$. Inset: temperature dependence of the ratio ${\tau }_F∕{\tau }_{\alpha }$.

Figure 4

The time dependence of the mean-square displacement for the $A$ particles for $T=1.0$, 0.8, 0.6, 0.55, 0.50, 0.47, and 0.45 listed from left to right. The symbols are placed at different characteristic times. Squares: the time at which the standard non-Gaussian parameter reaches the maximum value, ${\tau }_{ng}$. Triangles: the $\alpha$ relaxation time, ${\tau }_{\alpha }$. Circles: the onset time for Fickian diffusion, ${\tau }_F$.

Figure 5

Circles (left vertical axis): the root-mean-squared displacement at the onset time, ${\left[⟨\delta r^2\left({\tau }_F\right)⟩\right]}^{1∕2}$. Triangles (left vertical axis): the square root of the product of the self-diffusion coefficient and the $\alpha$ relaxation time (multiplied by 20 for convenience), $20{\left(D{\tau }_{\alpha }\right)}^{1∕2}$. Squares (right vertical axis): the ratio of the onset time and the $\alpha$ relaxation time, ${\tau }_F∕{\tau }_{\alpha }$. All quantities are plotted vs the $\alpha$ relaxation time; all the data pertain to the $A$ particles. Dashed lines indicate the scaling relationships ${\left[⟨\delta r^2\left({\tau }_F\right)⟩\right]}^{1∕2}\propto {\tau }_{\alpha }^{0.13}$ and ${\tau }_F∕{\tau }_{\alpha }\propto {\tau }_{\alpha }^{0.13}$.