The elastic strain energy of growth ledges on coherent and partially coherent precipitates
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INTRODUCTION
PRECIPITATE morphology can influence the properties of many types of alloys. The toughness of Fe-, AI-, or Ti-based alloys, for example, depends to varying degrees upon whether the precipitate phase is plateshaped or equiaxed. It is, however, difficult to predict precipitate shapes a priori. Precipitates do not necessarily adopt an equilibrium shape, and kinetic considerations are often important. Several approaches to predicting growth shapes have been proposed when the precipitate does not assume the equilibrium morphology. Bywater and Dyson suggested that in the case of needle-shaped precipitates, growth occurs in the direction of minimum interatomic mismatch between the precipitate and parent crystal lattices, t~l It was argued that this direction minimizes the strainenergy density during growth. Using elements of the crystallographic theory of the martensite reaction, Dahmen I2,31 and Dahmen and Westmacott I4,5J expanded this concept into the invariant line (IL) hypothesis. According to this theory, precipitates grow along a crystallographic IL a direction of zero misfit determined by a transformation strain and a lattice rotation. The transformation strain and lattice rotation are obtained from a chosen lattice correspondence between the parent and product phases. The IL hypothesis has achieved considerable success predicting both the growth direction of precipitates and the orientation relationships between the matrix and precipitateJ 3'6-9J On a related approach, the shape and habit plane of a coherent precipitate are predicted by minimizing the strain energy of the precipitate, ll~ Khachaturyan suggests a growing precipitate selects a habit plane so as to G. CHEN and J.K. CHEN, Graduate Students, and W.T. REYNOLDS, Jr., Associate Professor, are with the Materials Science and Engineering Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0237. J.K. LEE, Professor, is with the Department of Metallurgical and Materials Engineering, Michigan Technological University, Houghton, MI 49931. Manuscript submitted January 20, 1993. METALLURGICALAND MATERIALSTRANSACTIONS A
minimize strain energy. Although this approach, like the IL theory, employs a transformation strain specified by a lattice correspondence, the habit plane is selected by minimizing an explicit expression for strain energy rather than by finding a good match between the precipitate and matrix lattices. A third approach to predicting precipitate shape emphasizes the mobility of the boundary between the precipitate and matrix phases, t~l,~2s The precipitate shape is taken to depend upon the anisotropy of boundary mobility. Boundary orientations with a higher mobility "grow out," leaving behind a precipitate enclosed by low mobility boundaries.lll.~2] The boundary mobility, in turn, is determined by the migration mechanism of the boundary. When the precipitate and matrix phases differ in crystal structure and composition, the interphase boundary between them is generally coherent or partially coheren
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