Summary and conclusions

The existing theories aimed to explain melting transition are still far from being complete and raise futher questions. Hence the main purpose of the present project was 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 this phenomenon.

An interatomic potential proposed by Finnis and Sinclair [42] was chosen to investigate the bulk and the surface of vanadium at different temperatures. In order to validate the computer program and to check the potential we calculated thermal expansion coefficient and the shear moduli of a perfect sample of vanadium at various temperatures. The results were compared with available experimental data, and the agreement between was found to be reasonable.

After validation of the pure system, point defects were introduced into the lattice by either removal an atom (vacancy) or by addition (self-interstitial). The most stable, energetically favorable configuration of atoms around an interstitial at low temperatures was found by using simulated tempering method. The configuration of point defects in the bulk of vanadium (bcc lattice) was found to be a dumb-bell $< 110 >$ split-interstitial with the formation energy $E_f=4.18\pm 0.02$ eV, in contrast to $< 100 >$ split-interstitial of copper (fcc lattice) with the formation energy $E_f=3.28$ eV.

Point defects change the volume of the solid, while the shape of the sample, below the melting point, is almost unchanged, provided that the concentration of point defects is small enough. Self-interstitials expand the sample, while the introduction of vacancies leads to decrease in the volume of the sample. These changes of the volume due to point defects are less noticeable for vanadium in comparison with copper. This can be ascribed to the more loose-packed structure of a bcc lattice, which is less distorted by the presence of defects than a fcc lattice. In addition, variation of the shear elastic moduli with the concentration of defects and temperature was investigated. We found that the shear moduli are softened as a result of the volume expansion of the solid, which is associated with increase in temperature and/or interstitial concentration. Comparison of our results with the corresponding data for copper shows that softening of the shear moduli is less pronounced for vanadium. This again is attributed to the more compact structure of the fcc lattice of copper.

There is a strong evidence that the Born instability is the trigger for melting. This instability could be set in by self-interstitials which expand the solid up to a critical volume, where the lattice of crystal becomes mechanically unstable and collapses. This defect mediated mechanical melting occurs at temperatures below the melting temperature of the perfect crystal. We found that the critical volume at which crystal melts is independent of the path throw the phase space by which it reached, either by heating the crystal without defects, or by adding defects at a constant temperature. We performed computer simulations with various concentrations of point defects using the Parinello-Rahman method. Starting with a given concentration of point defects we gradually heat the solid up to the melting transition at $T_b$. In addition, $T_b$ is determined relying on the dependence of the shear elastic moduli on the temperature at a given concentration of point defects. The melting point is extracted by a continuous extrapolation of the shear modulus $C'$ to zero (at the zero external stress). We found that interstitials lower $T_b$ by approximately 50 K for the defect concentration $0.5\%$, however for copper the effect is more pronounced. Vacancies almost do not affect $T_b$(at least at small concentrations).

The temperature at which bulk melting transition occurs is found to be $T_b=2500 \pm 8 K$. This temperature is larger than the thermodynamic melting temperature, $T_m = 2220 \pm 10 K$ at which the vanadium crystal with the free surface melts (the same effect of was observed for copper [67] which has $T_b=1580 K$ and $T_m=1323 K$). The thermodynamic melting temperature $T_m$ of vanadium was determined using the method proposed by Lutchko et al.[77]. The simulations were carried out for various low-index surfaces of vanadium, and it was found that a sample with the least packed surface melts at lower temperature, than a sample with the close packed surface structure (the difference is about $10$ K).

Surface premelting, i.e. formation of the quasiliquid melt in the surface region, was studied in detail. The structural, transport and energetic properties of low-index faces Va(001), Va(011) and Va(111) were examined at various temperatures. We found that upon increasing the temperature, the vibrations of atoms in the surface region become larger than in the bulk, they start to disturb each other. As a result a formation of point defects begins in the surface region. These defects migrate between the surface layers, opening the way to a ``disordered'' surface regime (premelting). The regime of premelting is characterized by enhanced diffusion of atoms in the surface region, and by an increased formation of the surface defects. Finally, an adlayer emerges on the top of the surface layer and a quasiliquid phase appears as an intermediate layer between the solid and gas phases. The transition region between the solid and quasiliquid is the sharpest one for the close packed Va(011) surface. The surface premelting of the Va(011) is observed only in the close proximity of $T_m$, reflecting a large activation energy need for formation of surface defects. The surface premelting is more noticeable for the loose packed surface, and especially for the least packed face Va(111).

The results of our simulations of surface premelting of the bcc metal vanadium are similar to the result obtained for various fcc metals. The least packed surface (011) of those metals (with fcc lattice) begins to disorder first at low temperatures, while the close packed (111) surface preserves its crystalline structure almost up to $T_m$ (in some cases could be superheated).

Why do crystals start to melt at their surface? To answer this question we applied the Born criterion of melting transition to the surface region. The generation of point defects is enhanced with increase of the temperature in the surface region. Therefore, these defects expand the solid up to the point where the mechanical instability of the lattice (in the surface region)sets in. According to E. Polturak [92], the Born criterion is being applied to the surface melting leads to a simple linear relation between the activation energy of surface defects and $T_m$. This theoretical prediction was tested by results of experiments and computer simulations of fcc metals. In order to test the validity of the theory for vanadium (bcc lattice), we evaluated the activation energy of the surface defects for the least packed Va(111) face and compared it with the value predicted by the theory. The available experimental data and results of bulk simulations were used as input parameters in the theoretical calculations of surface defects formation energy. The activation energies of surface defects estimated from the Born criterion and obtained in our computer simulations are in reasonable agreement. Therefore, a general conclusion is made that the Born criterion correctly describes both surface and bulk melting, and may well provide the ``missing link'' which will finally tie together these two scenarios of the melting transition.

Many aspects of melting transition were investigated in our computer simulations, nevertheless many other questions remain to be answered. First, one could investigate the mechanism of bulk and surface melting using alternative versions of the many-body potential for vanadium (or any other bcc metal) and compare with the present results. In particular, it would be very interesting to apply the latest version of an interaction potential for bcc metals, the second nearest-neighbor modified embedded atom potential ( MEAM) developed by B. Lee et al. [100]. In order to considerably improve the accuracy of calculations we one could to increase the number of atoms (and number of the free layers in case of surface melting). This would be achieved by substantional improvement the parallel version of our MD code and more intensive use of supercomputers.

Another interesting problem, which was outside of the scope of the present research, is the nature of the phonon spectrum in the vicinity of point defects. It is expected that point defects create additional local and resonance modes in the phonon spectrum. These modes play an essential role in the bulk and surface melting transition. In addition, it would be very interesting to verify whether the inhomogeneous bulk melting transition (i.e. formation of the ``clusters'' of the liquid phase inside the crystalline phase) observed in computer simulation with the simple $LJ$ potential [6] could be observed if more a complicated $EAM$ potential is used. Besides that, it would be instructive to investigate whether the bulk melting transition could be initiated around complexes of point defects (e.g. polyvacancies or polyinterstitials). It would be very important to investigate the relation between surface premelting and surface roughening as well as the surface reconstruction at low temperatures.

Finally, it is a fascinating problem to investigate bulk melting as well as surface premelting of metals with hcp lattice structure. Furthermore, next in order of complexity to crystals containing point defects are crystals with more complex defects (dislocations, disclinations, etc). The challenge is how to study the mechanism of the melting transition of those crystals by means of computer simulations?