Kinetic Coefficient for Hard-Sphere Crystal Growth from the Melt

Using molecular-dynamics simulation, we determine the magnitude and anisotropy of the kinetic coefficient ($\mu$) for the crystal growth from the melt for the hard-sphere system through an analysis of equilibrium capillary fluctuations in interfacial height. We find ${\mu }_{100}=1.44(7)$, ${\mu }_{110}=1.10(5)$, and ${\mu }_{111}=0.64(3)$ in units of $\sqrt{k_B/(m{T}_m)}$, where $k_B$ is Boltzmann’s constant, $m$ is the particle mass, and $T_m$ is the melting temperature. These values are shown to be consistent, with some exceptions, with those obtained in recent simulation results a variety of fcc metals, when expressed in hard-sphere units. This suggests that the kinetic coefficient for fcc metals can be roughly estimated from $C\sqrt{R/(M{T}_m)}$, where $R$ is the gas constant, $M$ is the molar mass, and $C$ is a constant that varies with interfacial orientation.

Figure 1

Plot of $\ln ⟨\tilde{h}(q,t){\tilde{h}}^{*}(q,0)⟩$ vs $t[m{\sigma }^2/(k_{B}T){]}^{1/2}$ at a variety of $q$ values for (a) (100), (b) (110), and (c) (111). The lines are obtained from weighted linear regression fit. (The dotted lines represent $q$ values not used in the final determination of $\mu$—see text).

Figure 2

Plot of $\tau$ versus $q^{-2}$ for (a) (100), (b) (110), and (c) (111) orientations. The lines are obtained from a weighted linear regression to determine the value of $\mu$ using Eq. (4, 5). The circled points represent data that was excluded from the calculation of $\mu$.