Preface

Of all the phenomena exhibited by condensed matter, changes of state are among the most dramatic, and of these melting and freezing are specially striking.

A melting transition occurs when a particular phase becomes unstable under a given set of thermodynamic conditions. Classical thermodynamics offers a sound framework for understanding of phase transition in term of a free energy. Usually the Gibbs free energy $G$ is used. We relate it to a particular phase state of matter by a subscript, e.g. for a solid $G_s$ and for a liquid $G_l$. When two states of matter are in thermodynamic equilibrium the Gibbs free energies are equal $G_s(P,T)=G_l(P,T)$. The free energy of the system is a continuous function of $P$ and $T$ during the phase transitions (See Fig. 1.1), but other thermodynamic quantities such as internal energy $U$ entropy $S$ volume $V$ and head capacity $C$ undergo discontinuous changes.

Figure 1.1: Variation of the Gibbs free energy of a simple atomic substance near the melting point as a function of temperature.

The first derivative of the free energy are then given by: Almost all substances expand on melting, i.e. $\Delta V_{m}>0$, ice being one of very few exceptions [19] (among them Sb, Bi, Ga). The behavior of ice is generally attributed to its 'open' network structure (it has a very small coordination number of the nearest neighbors), which collapses at the melting point, allowing atoms to adopt smaller average separation between them.

In all known materials (except He) the entropy of the liquid phase is higher than the entropy of the solid phase at the melting point:

$${\Delta S_m }=Rln\left( \frac{W_l }{W_s } \right)$$ (2.1)

where $\Delta S_m$ is the entropy difference between the two phases, $R$ is the gas constant and $W_l$ is the number of independent ways of realizing the molten state and the $W_s$ is the same quantity for the solid state. Therefore, the melting transition is a transition from an ordered state to a less ordered state, which increases the 'randomness' of the material structure.

However, the thermodynamic equations can not explain the mechanism of melting. The melting transition is determined by the detailed microscopic structure of the crystalline and melting state. Therefore, the process of melting can not be explained without knowledge of the structure of the material.

All theoretical models of melting can be divided into two groups. The first group is the group of models of "two phases" (solid and melt). These models always involve reference to both phases, expressing the equilibrium between a solid and its melt in terms of the Gibbs free energy and calculating $T_m$. The second group is the group of "one-phase" models, and consider melting as a homogeneous process occurring in the bulk of the solid. Some of those models will be considered in detail below. These theories focus on a certain type of lattice instabilities (anharmonic vibrations, anharmonic crystal elongation, vanishing of resistance to the shear stress, etc) and structural defects (vacancies, interstitials, dislocations, disclinations) which arise at particular range of temperatures and cause the solid to become unstable or ``unrealizable'' above the melting temperature. The bulk melting models usually overestimate the melting temperature. Real crystals, which are finite and always have boundaries, start to melt from the surface at a temperature which is lower than the temperature predicted by the theories of mechanical instability of the crystal lattice. Nevertheless, these theories play a very important rôle in our understanding of the mechanism of melting, and especially in emphasizing of the possible scenarios of melting which include point defects, dislocations, etc.