Dynamical signatures of freezing: Stable fluids, metastable fluids, and crystals

Mean squared displacements and velocity auto correlation functions are calculated using molecular dynamics for hard spheres under a range of conditions (i) for the equilibrium fluid below freezing; (ii) for the metastable fluid above freezing; and (iii) for the hard sphere crystal, both in the metastable region between freezing and melting, and in the stable region above melting. In addition, simulations are carried out for a metastable Lennard-Jones system. The results confirm recent studies that indicated the disappearance of the classical Alder long-time tail, and show that they apply to systems other than the metastable hard sphere fluid. The implications of these results for our understanding of crystallization and the glass transition are discussed.

Figure 1

(Color online) The log-log plot of the absolute values of the partial VAFs for the binary mixture at the volume fraction of $\varphi =0.52$, showing the VAFs calculated from the small species particles $Z_1$, the large species particles $Z_2$, and the average or indiscriminate VAF $Z_{av}$.

Figure 2

(Color online) Shown is the time it takes a low frequency sound wave to traverse the simulation cell. The number of particles is kept fixed (at $N=10976$ or $N=5\times {10}^5$) and at the highest volume fraction the box length is smallest and the speed of sound is fastest, therefore the traversal time is smallest. For dilute fluids reducing the volume fraction results in the VAF decaying more slowly and this effect dominates at low enough volume fraction. Thus, the speed of sound crossing the box before the VAF decays, becomes a significant problem at volume fractions below $\varphi =0.15$.

Figure 3

(Color online) The log-log plot of the absolute values of the VAFs $\mid Z\left(\tau \right)\mid$ calculated from hard sphere fluid simulations with $N=10976$ and $N=5\times {10}^5$ at volume fractions of $\varphi =0.15$ and $\varphi =0.58$. The numbers in the legend are the volume fraction, $s$ denotes the $N=10976$ system, and $b$ denotes the $N=5\times {10}^5$ system.

Figure 4

(Color online) This is a summary of typical results as reported in [10] for hard sphere fluids. The one component fluid is shown for the volume fractions of $\varphi =0.15$, 0.41, and 0.45 which are all below the freezing volume fraction ${\varphi }_f=0.494$. The volume fraction of $\varphi =0.52$ is also shown for the binary fluid which is above the freezing volume fraction ${\varphi }_f=0.506$. In (a) the VAF $Z\left(t\right)$ is plotted against a logarithmic time axis. In (b) the logarithm of the absolute value of the VAF $\ln (\mid Z\left(t\right)\mid )$ is plotted against a logarithmic time axis.

Figure 5

(Color online) A log-log plot displaying the error in the long-time tail of the hard sphere fluid for selected volume fractions. From left to right, the volume fractions are 0.48, 0.513, and 0.56, respectively. The error bars (one standard error) are shown for every third data point at long times. The straight lines are power law fits with nominal exponents of $-3∕2$ and $-5∕2$. For clarity the 0.513 data has been shifted along the time axis by a factor of 4 and the 0.56 data by a factor of 25.

Figure 6

(Color online) The log-log plot of the absolute value of the VAF for the crystal phase at various volume fractions above freezing. As for the metastable fluid, the VAFs decay to zero from below, although here the exponent is of the order of $3.5\sim 4$.

Figure 7

(Color online) Characteristic times for the VAF as functions of volume fraction: (i) the time when the first minimum in the VAF occurs (minimum in legend); (ii) the time when the VAF crosses from positive to negative (goes negative in legend); (iii) the time when the VAF crosses from negative to positive (goes positive in legend). The one component hard sphere fluid was used for volume fractions below freezing ${\varphi }_f=0.494$ and the binary hard sphere fluid was used for volume fractions above freezing ${\varphi }_f=0.506$. Above freezing the VAF is never observed to go positive again, once it has gone negative.

Figure 8

(Color online) The log-log plot of the absolute value of the VAF for a truncated Lennard-Jones fluid with a fixed number density $\rho =N{\sigma }^3∕V=0.936$. The temperatures (as given in the legend) are either side of the freezing temperature $T_f=1.06$. The undercooled liquid never becomes positive, after going negative. The equilibrium liquid has a second sharp minimum as Z(t) crosses zero again and becomes positive.

Figure 9

(Color online) (a) Log-log plot of the absolute value of the VAF at volume fractions of $\varphi =0.513$ and $\varphi =0.58$ for the undercooled binary hard sphere fluid. The sharp minimum occurs where the VAF crosses zero and becomes negative. (b) The log-log plot of the MSDs at the volume fractions of $\varphi =0.513$ and $\varphi =0.58$. The initial short-time ballistic motion has a slope of two while the long-time diffusive motion has a slope of one. The stretching or plateau in the MSD may be seen at the higher volume fraction $\varphi =0.58$.