the coexistence of phases in thermodynamic equilibrium with one another in a system consisting of two or more phases. The simplest examples of phase equilibrium are the equilibrium of a liquid and its saturated vapor, the equilibrium of water and ice at the melting point of ice, and the separation of a mixture of water and triethylamine into two immiscible layers, or phases, that differ in density. Two phases of a ferromagnet that have the same axis of magnetization but different directions of magnetization can be in equilibrium in the absence of an applied magnetic field; the normal and superconducting phases of a metal can be in equilibrium in an applied magnetic field.
The energy of a system does not change when a particle undergoes a transition from one phase to another at equilibrium. In other words, the chemical potentials of each component in the different phases are equal at equilibrium. The Gibbs phase rule follows from this situation. The rule states that in a substance consisting of k components, not more than k + 2 phases can coexist in equilibrium. For example, the number of coexisting phases does not exceed three in a substance consisting of one component (seeTRIPLE POINT). The number of degrees of freedom—that is, the number of variables, or physical parameters, that can be changed without violating the conditions for phase equilibrium—is equal to k + 2- φ, where φ is the number of phases in equilibrium. For example, in a binary system, which contains two components, three phases can be in equilibrium at various temperatures, but the pressure and the component concentrations are completely specified by the temperature.
The temperature at which a phase transition occurs—for example, a boiling point or a melting point—changes if the pressure changes. The temperature change that results from an infinitesimal change in pressure is given by the Clapeyron equation. Graphs that represent the interrelation of the various thermodynamic variables at phase equilibrium are called phase transition curves or surfaces; a set of such curves or surfaces is known as a phase diagram, or a structural diagram. A phase transition curve may either intersect two other phase transition curves at a triple point or terminate at a critical point.
Phases not in equilibrium, which can coexist with phases in equilibrium, occur in solids as a result of the slowness of the diffusion processes that lead to thermodynamic equilibrium. In this case, the phase rule may not be satisfied. The phase rule is also not satisfied in the case where the phases on a phase-transition curve do not differ (see).
In bulk specimens, the number of equilibrium phase boundaries is the least when no long-range forces act between the particles. For example, in the case of a two-phase equilibrium, only one phase boundary exists. If a long-range field—such as an electric or magnetic field—were to exist within and outside the substance even in one of the phases, then equilibrium states with a large number of periodically arranged phase boundaries—such as ferromagnetic or ferroelectric domains or the mixed state in superconductors—would be more favorable from the standpoint of energy, as would be an arrangement of the phases such that the long-range field would be confined to the specimen.
The shape of a phase boundary is determined by the condition of minimum surface energy. Thus, the phase boundary is spherical in a mixture of two components if the densities of the phases are equal. The faceting of a crystal is governed by the planes that have the lowest surface energy.
REFERENCESLandau, L. D., A. I. Akhiezer, and E. M. Lifshits. Kurs obshchei fiziki: Mekhanika i molekuliarnaia fizika, 2nd ed. Moscow, 1969.
Frenkel’, Ia. I. Statisticheskaia fizika. Moscow-Leningrad, 1948.
V. L. POKROVSKII