Effect of Elastic Stress on Phase Selection in a Binary System
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EFFECT OF ELASTIC STRESS ON PHASE SELECTION IN A BINARY SYSTEM JOO-YOUL HUH AND WILLIAM C. JOHNSON Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890 ABSTRACT Elastic stress arising from differences in lattice parameters between phases is known to alter both qualitatively and quantitatively the characteristics of phase equilibria in coherent systems. One important consequence of misfit or epitaxial strain is the possible existence of several linearly stable equilibrium states: For a given composition, temperature and applied stress, different combinations of volume fraction and corresponding phase compositions render the free energy of the system a minimum. Here, we examine how epitaxial stresses influence phase equilibria in a binary alloy when the system can select from three different phases. In particular, we show the existence of several equilibrium states with different combinations of phases that minimize the system free energy.
INTRODUCTION Elastic strains occur naturally in solid-state reactions. Epitaxial or misfit strains are
present in heterostructures when the stress-free lattice parameters of the phases are different. Coherency strains are induced through compositional inhomogeneity when the shape and dimensions of the unit cell depend upon composition. Likewise, thermal- and piezoelectric-induced strains are present when the unit-cell dimensions depend on temperature and electric field and the temperature and electric fields within the system are spatially non-uniform [1]. The elastic stresses engendered by the above sources are long-range in nature. Compositional changes at one point in the crystal induce stresses at other points in the crystal which, in turn, can lead to composition changes at those other positions. This long-range coupling is known to render the diffusion equation a functional of the composition field [2]. Similarly, the long-range elastic fields produced by misfitting second-phase particles renders the free energy of the two-phase system a function of the spatial distribution of the particles. The long-range, elastically-induced coupling between phases is known to affect the characteristics of phase equilibria in two-phase coherent systems [3, 4, 5, 6, 7, 8]. The specific characteristics depend upon the system geometry and mechanical loading conditions. However, it is known that the ends of tie lines, which give the equilibrium compositions of the phases, need not coincide with the phase boundaries as in unstressed systems. In addition, an elastically stressed, two-phase system can possess several thermodynamically stable equilibrium states [6]. This means that, for a given temperature, bulk composition and imposed state of stress, more than one combination of volume fraction and equilibrium phase compositions will give a minimum in the free energy extremized at equilibrium. The existence of the (meta)stable equilibrium states is not a consequence of nucleation barriers but results from the long-range coupling between phases t
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