Temperature dependence of the crystal–melt interfacial energy of metals

Abstract

A model to express the dependence of the crystal–melt interfacial energy on the temperature for metals is proposed. The crystal–melt interfacial energies, the homogeneous nucleation undercoolings and the critical cooling rates to form ideal metallic glasses of silver, copper and nickel have been predicted according to the present model and simulated by the molecular dynamics method. The results show that the crystal–melt interfacial energy of metals increases nonlinearly with temperature. Over a wide temperature range from the melting point to the glass transition temperature the predicted results for the crystal–melt interfacial energy, the homogeneous nucleation undercooling and the critical cooling rate to form ideal metallic glasses from the present crystal–melt interfacial energy model are in good agreement with the experimental results reported, as well as the results of molecular dynamics simulations based on different EAM potentials of the metals.

Introduction

The crystal–melt interfacial energy plays a key role in a wide range of metallurgical and materials phenomena, from wetting and sintering to solidification. Without a clear knowledge of the crystal–melt interfacial energy it is impossible to completely comprehend the solidification behaviour, such as the nucleation rate [1], [2], [3], [4], [5], the growth rate [5], [6], [7], [8] and the growth mode [9], [10] of crystals, and thereby effectively control the structures and properties of materials.

Measurement of the crystal–melt interfacial energy σ^T is generally carried out by the maximum nucleation undercooling (MU) technique [1], [2], [3], [4], [5], based on homogeneous nucleation theory, which is used to measure the crystal–melt interfacial energy at the homogeneous nucleation temperature. In the seminal work of Turnbull [1] in 1950 the first measurements of σ^T were derived from nucleation studies in undercooled melts. Turnbull [1] demonstrated a strong correlation between the crystal–melt interfacial energy and the ratio of the latent heat of melting (L, per atom) to the average interfacial area ($V_{\text{a}}^{2/3}$, per atom): ${\sigma }^T=\mathit{CL}/V_{\text{a}}^{2/3}$, where C is termed the Turnbull coefficient and was originally reported to have a value of approximately 0.45 for metals. As more experimental data for σ^T became available the value of the Turnbull coefficient has been refined. From maximum undercoolings in silver [11], [12], copper [13], [14] and nickel [13], [15] C was determined to be in the region 0.46–0.52 for face-centered cubic (fcc) metals. In addition, in a compilation of 26 maximum undercooling studies, Kelton [3] found C = 0.49 ± 0.08 for metals.

The dihedral angle (DA), the contact angle (CA) and the grain boundary groove (GBG) techniques [16], [17], [18], [19], [20], [21], [22], [23], [24], [25] are used to measure the crystal–melt interfacial energy at the melting point. From a survey of solid–liquid dihedral angle measurements in fcc metals [16] Granasy et al. [17] derived a value for C of approximately 0.6. According to the data for crystal–melt interfacial energies for fcc metals based on the CA technique [19], [20] the derived value of C is found to be in the region 0.63–0.68. In terms of the data for crystal–melt interfacial energies measured by the GBG technique C was determined to be in the region 0.57–0.68 for aluminum alloys (Al–Si [21], Al–Cu [21], Al–Mg [22], Al–Ni [23], Al–Ti [23] and Al–Ag [24]). Also, the crystal–melt interfacial energy at the melting point can also be determined by atomistic simulation. Since the pioneering work of Broughton and Gilmer [25], atomistic simulations have been applied extensively in calculations of σ^T for a variety of systems [5], [26], [27]. The average value of C = 0.55 at the melting point for fcc metals can be derived from the atomistic simulations [2].

Comparing the magnitudes of C obtained via different techniques it is noteworthy that the MU data tend to be lower than those derived by the DA, CA and GBG methods. This trend has been previously noted by a number of authors and it can be rationalized based on the fact that MU data provide values of σ^T at nucleation temperatures that are typically a few hundred degrees below the melting point (T_m), whereas DA, CA and GBG measurements are performed near the T_m. This means that the crystal–melt interfacial energy varies with temperature. The lower values of σ^T derived from MU measurements would thus be consistent with a positive temperature dependence for the crystal–melt interfacial energy. As is well known, a certain amount of undercooling is needed for all solidification processes. So the crystal–melt interfacial energy in the undercooled state is more important in the study of solidification behaviour. Unfortunately, with the exception of the homogeneous nucleation temperature and the melting point, the crystal–melt interfacial energy at temperatures between the melting point and the homogeneous nucleation temperature cannot be measured. Therefore, it is necessary to explore the correlation between the crystal–melt interfacial energy and temperature.

At present the most widely used model for correlating the crystal–melt interfacial energy with temperature is that proposed by Spaepen [28], [29]. In this theory the crystal–melt interfacial energy of metals was proposed to increase linearly with temperature. A value of C = 0.86 is derived for fcc crystals under the assumption that the liquid structure is characterized by tetrahedral packing. This value has been widely applied in modeling experimental nucleation data [2]. However, the crystal–melt interface in Spaepen’s model is assumed to be perfectly smooth, whereas the real interface is rough. Consequently the results for the crystal–melt interfacial energy predicted from Spaepen’s model are much higher than the data obtained by the MU [1], [2], [3], [4], [5], [11], [12], [13], [14], [15], DA [16], [17], [18], CA [19], [20] and GBG [21], [22], [23], [24] techniques and atomistic simulations [5], [26], [27].

Recently Jian et al. [10], [30], [31] proposed a model to express the correlation between the crystal–melt interfacial energy and temperature for faceted materials. In this model the crystal–melt interfacial energy of a faceted material is predicted according to the critical growth transition undercoolings (i.e. the critical undercooling for a faceted material to grow from lateral to intermediary mode and the critical undercooling for a faceted material to grow from intermediary to continuous mode). It is found that the results for the crystal–melt interfacial energies predicted from the critical growth transition undercoolings for silicon [10], [30], [31], germanium [10] and bismuth [32] are consistent not only with the experimental results for the undercooled state according to the MU technique [33], [34] but also with that at the melting point using the GBG technique [21]. However, this model cannot be used to predict the crystal–melt interfacial energy of metals.

The purposes of this paper are, through modeling and molecular dynamics simulation of the crystal–melt interfacial energy of fcc metals, to determine the dependence of the crystal–melt interfacial energy on temperature and introduce a useful method which can be used to predict the crystal–melt interfacial energies of metals at the melting point and the homogeneous nucleation temperature as well as the temperature difference between the melting point and the homogeneous nucleation temperature.