Abstract

Melting is a fundamental process in which a crystal undergoes a phase transition from a solid to a melt. Despite its common occurrence, understanding this process still a challenge.

A number of theories, which consider melting as a process occurring homogeneously throughout the crystal have been proposed during the past century. For example, according to Lindemann, melting is triggered by a mechanical instability of the solid, which caused by enhanced vibration of the atoms. Solids liquefy when the amplitude of atomic thermal vibrations exceeds some fraction of interatomic spacing. According to Born, melting arises from the onset of a mechanical instability of the crystal lattice, which manifests itself in an imaginary phonon frequency and the vanishing shear elastic moduli, accompanied by the collapse of the crystal lattice. Other models are based on spontaneous thermal production of the intrinsic lattice defects (vacancies, interstitials and dislocations) near the melting point and this leads to break-down of the long-range crystalline order and melting transition. However, the extrinsic defects (free surfaces and grain boundaries) were not considered as an important ingredient of a melting scenario. Those models are based on the concept of one-phase melting or continuous melting, i.e. they imply that the phase change might be continuous or nearly so; given sufficient resolution it should be possible to track the breakdown of the solid throughout its transition to the liquid state. These models are capable of calculate the melting temperature $T_m$, but in the most it is overestimated.

The existing theories of melting are still far from being complete and raise new questions. Hence, the purpose of the present research is to gain a better understanding of the mechanism of melting transition, and especially to investigate the rôle of point defects and the surface of the solid in the melting transition. Despite the fact that we learned very much recently about the melting of fcc metals, it is not clear if those results were specific to fcc structure of these metals, so we decided to study melting of a bcc metal, vanadium, by means of computer simulations.

An interatomic potential proposed by Finnis and Sinclair [42], was chosen for our simulations. The potential was tested by calculation of various properties of a perfect crystal of vanadium. The results are in good agreement with available experimental data. Afterwards point defects were introduced into the bulk either by the removal an atom (vacancy) or by the addition one (self-interstitial). The most stable configuration of defects at low temperatures was found to be a dumb-bell, the $< 110 >$ split-interstitial. Point defects change the physical properties of the solid. Interstitials expand the sample, while vacancies decrease its volume. The change of the volume is less noticeable for vanadium than for copper which is attributed to the less close-packed structure of its bcc lattice. We found also that the shear moduli are softened as a result of the volume expansion of the solid which is associated either with an increase in temperature or interstitial concentration. This softening of the moduli is less pronounced for vanadium in comparison with copper.

There is a strong evidence that Born instability is the trigger for bulk melting. The instability is set in by interstitials which expand the solid up to a critical volume, at which the lattice of the crystal becomes mechanically unstable and collapses. This defect-mediated mechanical melting occurs at the temperatures below the melting temperature of the perfect crystal. We verified that the critical volume at which the crystal melts is independent of the path thru the phase space by which it reached, i.e. either by heating the perfect crystal or by adding defects at a constant temperature. We performed simulations with various concentrations of point defects and found that bulk melting temperature $T_b$ is lowered by interstitials, but this effect is less pronounced in comparison with the same effect for copper.

The mechanical melting can not be observed directly in the laboratory, because a real crystal will eventually melt at $T_m$ which is lower than $T_b$ via thermodynamic melting process that nucleates at its surface. The process can be suppressed experimentally if the surface is eliminated, for example, by coating one material with another one, with larger $T_m$. In this way silver coated by gold was superheated by $25$K above $T_m$ [93]. In computer experiments we are able to eliminate the surface using periodic boundary conditions in all directions and thus can investigate bulk melting.

In order to study surface melting we use periodic boundary conditions only in two directions and create a free surface in the third one. The thermodynamic melting temperature was found to be $T_m = 2220 \pm 10 K$ by means of the method proposed by Lutchko et al. [77] (the bulk melting temperature of a perfect sample is $T_b=2550\pm5$). Melting of crystals begins at the surface, because the activation energy for formation of a liquid phase is lower at the surface, than in the bulk. The liquid layer at the surface eliminates the barrier for nucleation of the liquid phase, and thus no metastability effects (superheating) exist.

Most of the theoretical models of surface melting phenomenological in nature, and therefore neglect the atomistic details of the phenomenon. Only recently the first microscopic theory has been proposed [36], which is capable of describing static properties of the rare-gas crystals. The microscopic description of surface melting phenomena emerged mainly from computer simulations.

We studied surface premelting of vanadium using molecular dynamics. The structural, transport and energetic properties of the various low-index surfaces of vanadium, namely Va(001), Va(011) and Va(111), at different temperatures were investigated. We found that upon increasing the temperature the vibrations of atoms at the surface region becomes so large, that they ``disturb'' each other. As a result, point defects are generated which begin to migrate between the surface layers and an adlayer appears on the top of the first surface layer. The disorder begins to spread from the topmost layer to a deeper ones. At higher temperatures a thin quasiliquid film appears in the surface region. The observed premelting phenomena are most pronounced in the surface region of the least packed Va(111) face, and is less noticeable for the closest packed Va(011) face. Similar results were obtained in simulations of fcc metals, where the least packed face (011) exhibits premelting, while the closest packed face (111) remains ordered almost up to the melting point.

In order to understand the relation between the bulk and surface melting, we applied the Born criterion of melting to the surface region and found a linear relation between the activation energy of surface defects and the melting temperature. This relation was confirmed by results of experiments and computer simulations of metals with fcc structure [92]. In order to test the model for metals with bcc structure, we calculated the activation energy of the surface defects for the least packed Va(111) and compared it with theoretical prediction. The agreement between the theory and simulations was found to be reasonable. A general conclusion was made that the Born criterion correctly describes both surface and bulk melting, and may provide the ``missing link'' which will finally tie together these two scenarios for melting transitions.