On the inapplicability of gibbs phase rule to coherent solids
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I.
INTRODUCTION
IT has been recognized recently that phase equilibria in nonhydrostatically stressed coherent solids may differ markedly from phase equilibria in fluids or incoherent solids. ~-6 Work on coherent phase diagrams has demonstrated the existence of a number of equilibrium features that are clearly prohibited in multiphase fluid systems. These features are definite violations of the Gibbs phase rule yet are true equilibrium phenomena in coherent solids. In this paper, a relationship is derived between the number of phases, independent chemical components, and degrees of freedom in a specially constructed multiphase coherent solid experiencing nonhydrostatic stresses. This "phase rule" applies to a system that is in thermodynamic equilibrium, i.e., thermal, chemical, and mechanical equilibrium. It is not necessarily a general phase rule for coherent solids. However, the derivation does demonstrate unequivocally two important points. First, is that if a general phase rule exists for coherent solids, it is not equivalent to the Gibbs phase rule for fluids. Second, Gibbs' contention that "we may call such bodies as differ in composition or state, different phases of matter''~ may lose its meaning, or at least its usefulness, in nonhydrostatically stressed solids. Specifically, it is shown that for the coherent system studied here, the number of degrees of freedom depends only on the number of components and is independent of the number of phases in the system. Although this coherent system may appear highly artificial, the thermodynamic description must apply to it as well as to any other coherent system. The behavior of other, more complicated systems, will display a richness at least equivalent to that discussed here. These results are in complete agreement with recent work on coherent phase diagrams. 3-6
II.
MODEL SYSTEM
The phase rule is a simple relationship that exists between the number of degrees of freedom, the number of phases,
WILLIAM C. JOHNSON is Associate Professor with the Department of Metallurgical Engineering and Materials Science, Carnegie Mellon University, Pittsburgh, PA 15213. Manuscript submitted August 21, 1986.
METALLURGICALTRANSACTIONS A
and the number of independent components in a system. It is derived by equating the number of degrees of freedom, F, to the difference between the number of variables needed to describe the system, V, and the number of system constraints, C. That is F = V-
C
[1]
In what follows, a system of ~ phases and K independently variable components is considered. All of the K components are found in each of the phases. The system is assumed to be coherent in the sense that there exists a lattice to which all of the phases can be unambiguously referred. The reference lattice is assumed to be homogeneous and to possess a density of lattice points, per unit reference volume, of por. For reasons to be discussed, the system is assumed to possess radial symmetry. Since equilibrium cannot depend on the quantity of the material present, a unit reference vo
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