Equilibrium shapes of semicoherent inclusions
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The equilibrium shapes of stress free inclusions with a small mismatch and possibly a small rotation from the matrix phase are studied. The model used includes both bond-breaking and elastic contributions to the interface energy. The unrotated shape contains only facets. Rotated inclusions sometimes contain smoothly curved parts as well. Interface phase diagrams are used to characterize the stable phase behavior of flat interfaces of fixed orientation.
I. BACKGROUND A. Calculating equilibrium shapes An important component of interface thermodynamics concerns the equilibrium shape of a particle of fixed volume surrounded by a matrix phase.1 By definition, of all shapes of the same (large) volume, this shape has the smallest excess free energy. In nucleation of large particles this shape should be observed more frequently than others. Large precipitates will approach this shape if the kinetics of processes changing the shape is much faster than the growth or shrinkage rate of the inclusion. Equilibrium shapes are also important in determining the orientational stability of (flat) interfaces of specific orientations, since the orientations that are missing from equilibrium shapes are unstable with respect to a macroscopic faceting into two or more interfaces of other orientations determined by the equilibrium shape.1-2 These shapes can be determined theoretically from a microscopic model by finding 7, the interface (free) energy of a flat interface per unit area, as a function of n, the boundary plane orientation (inclination). The Wulff construction2 transforms the Wulff plot 7(11) into the equilibrium shape 9t(r). This procedure, which is actually a manifestation of interface thermodynamics,1 in principle gives the shape of an infinitely large inclusion, since a thermodynamic limit is assumed. However, the lowest energy shape of clusters of fairly small numbers of atoms is often amazingly close to this limiting shape.3 A seemingly similar problem involves the shape of a coherent or partially coherent inclusion in the presence of coherency strains.4 However, in this case the excess (free) energy includes, in addition to a part proportional to the boundary area, an elastic energy, which scales as the volume of the inclusion. Due to kinetic limitations and finite-size effects, second phase particles with volume strain are commonly observed. As the particle size increases, the volume contribution increases relative to that of the interface, resulting in a greater incentive to eliminate volume effects. In those cases in which misfit dislocations are readily nucleated to reduce the contribution of the volume stresses to an J. Mater. Res., Vol. 7, No. 4, Apr 1992
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insignificant amount, the shapes discussed here are relevant. Also, the shapes found here should provide a useful backdrop to future work which, by including volume effects in the shapes, better mimics most situations found experimentally.5 Typically most of the effort in finding equilibrium shapes is devot
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