Thermodynamics of multiphase systems A treatment of closed systems with homogeneous free-energy functions

The thermodynamic equilibrium conditions for closed multiphase systems are examined in detail. All systems are constrained in some way, most simply by conservation of atomic species, though more interesting systems can have complicated sets of constraints. For sets of constraints which are linear with respect to the independent variables, necessary conditions for system free-energy minima can be found using the technique of Lagrange multipliers. The equilibrium conditions are represented by the simple, but general, matrix equations 0 = α + Bn, 0=μ+B^Tλ, where the first relationship represents the system constraints on the independent variables n and the second relationship represents the necessary conditions prevailing at all system free-energy minima which are consistent with the system constraints. Free-energ functions are commonly taken to be homogeneous functions of degree one, which supposes that there is an intensive free-energy density. When this is the case, the Lagrange multipliers, λ, can be given the physical interpretation that they represent the thermodynamic (chemical) potentials conjugate to the various external parameters, α, of the aggregate system. This has the useful implication that determination of global properties in a multiphase system can yield specific information about each phase in the aggregate. In the present work the general developments for the determination of thermodynamic equilibria conditions in constrained multiphase systems are applied to the problems of stoichiometric phases, paraequilibrium conditions, and fixed levels of point defects in existing phases. The method is also applied to the determination of equilibrium conditions for systems described by the cluster variation model of free energy.