The solid–liquid interfacial free energy of close-packed metals: Hard-spheres and the Turnbull coefficient

Abstract

Largely due to its role in nucleation and crystal-growth, the free energy of the crystal-melt interfacial free energy is an object of considerable interest across a number of scientific disciplines, especially in the materials-, colloid-, and atmospheric sciences. Over 50 years ago, Turnbull observed that the interfacial free energies ͑scaled by the mean interfacial area per particle͒ of a variety of metallic elements exhibit a linear correlation with the enthalpy of fusion. This correlation provides an important empirical ''rule-of-thumb'' for estimating interfacial free energies, but lacks a compelling physical explanation. In this work we show that the interfacial free energies for close-packed metals are linearly correlated with the melting temperature and are therefore primarily entropic in origin. We also show that the slope of this linear relationship can be determined with quantitative accuracy using a hard-sphere model, and that the correlation with the enthalpy of fusion reported by Turnbull follows as a consequence of the fact that the entropy of fusion for close-packed metals is relatively constant. © 2001 American Institute of Physics. ͓DOI: 10.1063/1.1391481͔The crystal-melt interfacial free energy, ␥, defined as the reversible work required to form a unit area of interface between a crystal and its coexisting fluid, plays a central role in determining the kinetics of crystal nucleation and growth. 1,2 Unfortunately, direct experimental determinations of this quantity are difficult and exist for only a handful of materials. For most materials, knowledge of the interfacial free energy is obtained indirectly from measurements of crystal nucleation rates from undercooled fluids from which ␥ is determined within the approximations of classical nucleation theory.In a seminal 1950 paper, 3 Turnbull reported values of the crystal-melt interfacial free energy, ␥, for a variety of materials, mostly metallic elements, which were obtained indirectly from nucleation rate experiments. For systems in which directly determined values exist with which to compare the values so obtained typically are accurate within about 10-20%. ͑For example, the interfacial free energy for bismuth was determined 4 using grain boundary angles to be 61.3ϫ10 Ϫ3 J m Ϫ2 , as compared to the value of 54.4 ϫ10 Ϫ3 J m Ϫ2 obtained from nucleation rate data.͒ In order to compare the results for various systems, Turnbull defined a ''gram-atomic'' ͑or molar͒ interfacial free energy as the free energy of an interface ͑one atom thick͒ containing Avagadro's number, N A , of atoms ͑or molecules͒:The data for ␥ was found to exhibit a strong correlation with the latent heat of fusion. Empirically, Turnbull foundwhere the Turnbull coefficient, C T , was found to be approximately 0.45 for metals ͑especially close-packed metals͒ and 0.32 for many nonmetals.There have been attempts to explain this empirical result through the analysis of simple models for the structure of the interface, 5-7 but the results are quite sensitive to the nature of the assum...

Phase-field crystal model for a diamond-cubic structure

“…We believe the addition of atoms into the system as σ increases is the cause for a decreasing solid-liquid interfacial energy, which is in disagreement with the trend measured experimentally [25,26] and calculated using atomistic simulations [27][28][29] for closed systems. In order to directly compare the dependence of γ p (σ, α 0 , α 0 ) on σ from the PFC model to the dependence of γ p (σ, α 0 , α 0 ) on melting temperature from experiments and atomistic simulations, it is required to keep the number of particles constant as σ is varied, which is similar to what has been implemented for calculating elastic constants [30].…”

Atomistic and continuum modeling of dendritic solidification

“…The positive temperature-dependence of g is consistent with the negative values of the excess entropy (DS ¼ À@g=@T) resulting from a number of phenomenological and structural models of the solid-liquid interface [77][78][79][80][81][82][83][84][85][86][87][88], as well as density-functional theory [33,34] and recent simulation results [28,38,89] for elemental systems. As one specific example, in 1994 Spaepen [77] published a reanalysis of the measured data for the temperature-dependent (homogeneous) nucleation rates in mercury [72] and gallium [90] within the framework of classical nucleation theory, assuming a linear temperature dependence of interfacial free energy.…”

“…From data reported using a variety of techniques-MU, DMP and DA-Eustathopoulos [76] obtained a value of C T ¼ 0:55 AE 0:08 and from a survey of solid-liquid dihedral angle measurements in fcc metals [80], Granasy et al [93] derived a value for C T of approximately 0.6. In an important recent contribution, Laird [89] built upon earlier evidence (see above) suggesting that g for fcc metals is largely entropic in nature, i.e. the excess enthalpy is nearly zero.…”

“…This result lends further evidence to the proposal that the interfacial free energy in simple fccbased metals is dominated by entropic effects. [89] of the hard sphere system and the least squares fit to the EAM data is shown by the solid line. For the MD results, the Turnbull coefficient is found to be 0:521 AE 0:025.…”

Micro-structure evolution of wall based crystals after casting of model suspensions as obtained from Bragg microscopy

“…62 Recent nucleation studies have shown that the meltcrystal surface tension of charged spheres contains an energetic contribution 63 in addition to the well known entropic contribution of hard sphere systems. 64 This energetic contribution increases linearly with increased meta-stability (i.e., interaction strength). If we assume a similar dependence also for the twin-twin surface tension, the observed WilsonFrenkel type dependence of the initial coarsening velocities on meta-stability is a straight forward consequence.…”