Kinetic Coefficient for Hard-Sphere Crystal Growth from the Melt Majeed Amini Department of Physics, University of Kansas, Lawrence, Kansas 66045, USA Brian B. Laird Department of Chemistry, University of Kansas, Lawrence, Kansas 66045, USA (Received 12 June 2006; published 20 November 2006) Using molecular-dynamics simulation, we determine the magnitude and anisotropy of the kinetic coefficient () for the crystal growth from the melt for the hard-sphere system through an analysis of equilibrium capillary fluctuations in interfacial height. We find  100  1:447,  110  1:105, and  111  0:643 in units of  k B =mT m  p , 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  R=MT m  p , where R is the gas constant, M is the molar mass, and C is a constant that varies with interfacial orientation. DOI: 10.1103/PhysRevLett.97.216102 PACS numbers: 68.08.De, 02.70.Ns, 81.10.Aj The kinetic coefficient, , of a crystal-melt interface is the constant of proportionality between the growth velocity (v) and undercooling (T  T m  T) v  T; (1) where T m is the melting temperature. Both the magnitude and orientation dependence (anisotropy) of  are crucial factors in determining the crystallization rates and growth morphologies of metals [1]—especially for dendritic growth [2]. Because the crystal-melt interface lies between two condensed phases, experimental measurements of  are difficult [3] (especially for the anisotropy) and exist only for a few materials, such as P 4 [4], and Pb [5]. The lack of experimental data enhances the role of atomistic simulation, which has been used recently to determine  for a variety of systems ranging from Lennard-Jones (LJ) [3,6,7] to close-packed metals [8 –10]. Unlike molecular and covalent network materials, the growth kinetics of simple monatomic, close-packed crys- tals (e.g., simple metals) are not believed to be thermally activated [11]. This view is supported by the results of Broughton, Gilmer, and Jackson (BGJ) [6], who, in a molecular-dynamics (MD) simulation of the fcc (100) interface of a LJ system, found significant crystallization rates even at very low temperatures where the liquid dif- fusivity is negligible, contrary to the predictions of the thermal activation model of Wilson and Frenkel (WF) [12,13]. Based on their MD results, BGJ modified the diffusion-limited WF model to create a collision-limited model in which temperature, not the diffusion constant, plays a central role in the crystal growth of metals. The BGJ model has been interpreted to predict a proportion- ality between  and the interplanar spacing, d lmn for a given interfacial orientation (l, m, n)[14]. Thus, based on relative interplanar spacings, the BGJ model would predict that  111 > 100 > 110 for fcc-forming materials. This prediction works well for (100) and (110) interfaces, where the ratio  100 = 110 is often quite close to the BGJ pre- diction of  2 p , but fails for (111), which is seen in simula- tions [14] to have the smallest  of the three interfaces, not the largest, as predicted by BGJ. The low value of  for the (111) interface in fcc systems has been attributed to the formation of transient hcp stacking faults [7,14]. In this work, we determine, via MD simulation, the kinetic coefficient for the hard-sphere system—a standard reference model for close-packed materials. The potential energy of any realizable hard-sphere configuration is zero; therefore, the thermodynamics and kinetics of phase tran- sitions in the hard-sphere system are entirely entropic. Thus, the hard-sphere model can be used to understand the role of entropic driving forces in solidification. It has been shown that the hard-sphere model gives a quantitative description of the crystal-melt interfacial free energy  for simple close-packed systems [15]. Here, we examine the degree to which this is true for the kinetic coefficient. Recently, two MD simulation methods have been devel- oped to determine  and its anisotropy for pure materials: the free solidification method (FSM) and the capillary fluctuation method (CFM). In the FSM [6,9,16], the solid-liquid interface is first equilibrated at the melting point. Next, the system is simulated at a variety of different undercoolings and the interfacial position is monitored. From this data the interfacial velocity, v, is determined as a function of the undercooling (T), from which  is calculated using Eq. (1). In the CFM [17],  is determined from equilibrium fluctuations of interfacial position. The CFM was chosen for this study because the event-driven nature of hard-sphere MD simulations makes difficult the use of isothermal-isobaric simulation techniques, required for the FSM. A detailed review of these two methods can be found in Ref. [17]. To implement the CFM, separate equilibrium samples of crystal and melt are constructed at the equilibrium coex- istence densities. These two samples are then conjoined in PRL 97, 216102 (2006) PHYSICAL REVIEW LETTERS week ending 24 NOVEMBER 2006 0031-9007= 06=97(21)=216102(4) 216102-1  2006 The American Physical Society