Phase equilibrium in two-phase coherent solids
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I.
INTRODUCTION
P H A S E diagrams are graphical constructions used to depict regions of phase equilibrium. The axes of the phase diagram represent the independent thermodynamic variables appropriate to the system of interest. One usually thinks of these axes as temperature, pressure, and alloy composition but volume, entropy, and chemical potentials are all equally permissible. ~,z Phase diagrams are determined by specifying the independent thermodynamic variables, i.e., imposing certain constraints on the system, and then searching for the combination of phases and their respective compositions, subject to these constraints, that minimize the appropriate thermodynamic potential. For example, when temperature, pressure, and alloy composition are specified, equilibrium is determined when the Gibbs free energy is minimized. If the system is subject to the constraints of constant temperature and volume, then the Helmholtz free energy must be minimized. Extremization of a thermodynamic potential or free energy, subject to the specified system constraints, results in the conditions necessary for thermodynamic equilibrium. These are the conditions, sometimes expressed in terms of differential equations, that must be satisfied for the system to be in equilibrium. If capillarity is neglected, the extremization process in fluids always results in a unique minimum and necessitates that the pressure and chemical potentials of each component in each phase be uniform and equal throughout the system. Under these conditions all of the phases are homogeneous in structure, temperature, and composition. The requirement that the phases be homogeneous at equilibrium in fluids allows the total thermodynamic potential of the system to be equal to the sum of the thermodynamic potentials for each of the phases of the system determined as WILLIAM C. JOHNSON is Associate Professor, Department of Metallurglcal Engineering and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213 P.W. VOORHEES is with the Metallurgy Division, National Bureau of Standards, GaLthersburg, MD 20899. Manuscript submitted November 7. 1986. METALLURGICAL TRANSACTIONS A
if the other phases are not present. These conditions allow for lines drawn tangent to the free energy curves to be related to the chemical potentials of the individual components, and elegant graphical techniques can be employed to determine equilibrium phase compositions and volume fractions. When the phases are not uniform in composition or strain at equilibrium, these powerful graphical techniques for determining phase equilibrium and composition fields are in most cases invalid and cannot be used. Phase equilibrium in coherent solids is subject to what is known as the network constraint. 3'4 This constraint requires that a unique reference lattice exist to which each of the phases in the system may be referred. The reference state is assigned a certain crystal structure and the atoms of each of the phases are associated with a specific lattice point. The crystal structure of the
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