Velocity Autocorrelation Functions of Hard-Sphere Fluids: Long-Time Tails upon Undercooling

Molecular dynamics simulations are employed to obtain the velocity autocorrelation function (VAF) for hard spheres, spanning a wide range of volume fractions from dilute to high-density metastable fluids. For all volume fractions below freezing, Alder’s classical positive $3/2$ long-time tail is observed. For volume fractions from 0.45 to 0.48 the VAF becomes negative, before becoming positive and decaying with the positive $3/2$ long-time tail. At the freezing volume fraction (0.494) the Alder $3/2$ tail is not observed. At higher volume fractions a negative tail with an exponent of $5/2$ emerges, which coincides with the long-time tail of a Lorentz gas.

Figure 1

A plot of the velocity autocorrelation function $Z(\tau )$ versus $\log \tau$ (symbols are defined in the legend), calculated from one component hard-sphere molecular dynamics simulations of fluids at various volume fractions $\varphi =$(volume of all the spheres divided by the total system volume). For $\varphi \ge 0.45$ the VAF becomes negative, so in order to expose the long-time behavior, double logarithmic plots of $|Z(\tau )|$ are needed (see Fig. 2).

Figure 2

A double logarithmic plot of $|Z(\tau )|$ for the data shown in Fig. 1. For all curves below ${\varphi }_f=0.494$ a long-time $3/2$ power law (indicated by the dashed line) is clearly observed. As the volume fraction increases a nonmonotonic decay emerges as may be seen distinctly from the curve. We interpret this as indicative of the fluids strengthening visco-elastic behavior. For $\varphi \ge 0.45$, the points where $Z(\tau )$ cross zero may be seen as a sharp minimum in this graph. For $\varphi =0.45$ and $\varphi =0.48$ a second sharp minimum is observed [$Z(\tau )$ has crossed zero again and now becomes positive], followed by a long-time $3/2$ tail. Upon increasing the volume fraction to $\varphi =0.494$ and $\varphi =0.505$ this reentrant positive behavior is no longer observed.

Figure 3

Log-log plot of the amplitude $A$, as defined in the legend, as a function of $|{\varphi }_f-\varphi |$. The one component data is below freezing ${\varphi }_f=0.494$ and the binary data is above freezing ${\varphi }_f=0.506$. The solid line is a fit to the one component data $A=0.19({\varphi }_f-\varphi {)}^{1.5}$, which goes to $A=0$ at freezing.

Figure 4

A semilog plot of the absolute value of the VAF near the freezing volume fraction (symbols are defined in the legend) for both single component (open symbols) and binary systems (filled symbols). The squares are below ${\varphi }_f$, the circles are nominally at ${\varphi }_f$, and the triangles are just above ${\varphi }_f$. The binary data is shifted to the right by 1.5 on the time scale for clarity.

Figure 5

A double logarithmic plot of the absolute value of the velocity autocorrelation function for a range of hard-sphere binary mixtures. For clarity, successive volume fractions have been shifted along the time axis by half a decade. A power law with exponent $5/2$ is also shown for the highest volume fraction.