Abstract
This chapter reviews the present status of the theory of equilibrium crystal shapes, with particular emphasis on the connection between the crystal-shape problem and recent results in two-dimensional statistical mechanics. Interpretation of the Wulff construction as a Legendre transformation allows the crystal shape to be viewed as a free energy. Thus, edges on the crystal shape correspond to thermodynamic phase boundaries, and critical behavior is observed in the neighborhood of certain edges. The thermal evolution of the crystal shape is reinter-preted from this point of view, and results are compared with recent experiments.
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References
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In the sense that the intercept fi(m̂) is not on some fixed ordinate axis but rather on the radial line from the origin in the direction m̂.
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In practice, defect pinning may also play a role here.
Of course, this kind of an unstable region can never come out of a rigorous statistical mechanical calculation.
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Just as in the usual applications, certain hypotheses are needed to prove in a rigorous way the equivalence between ensembles. These hypotheses serve to control the scale of fluctuations about mean values such as (∂z/∂xk). Hypotheses such as short-range pairwise forces normally suffice. We do not know of any work which addresses ensemble equivalence for the interface problem in a rigorous way. There are also important questions involved in the existence of an interface Hamiltonian such as H[C] and in the proof directly from statistical mechanics of the Wulff construction. Some of these have been discussed in the review article of D.B. Abrahams, in: Phase Transitions and Critical Phenomena, ed. C. Domb, J.L. Lebowitz (Academic, New York 1986), Vol.10, p.2. There now exist a few rigorous results in two dimensions but much remains to be done. It is perhaps worth emphasizing, however, that for ordinary force laws everything in the literature suggests that the conventional viewpoint, which we take here, is correct.
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Wortis, M. (1988). Equilibrium Crystal Shapes and Interfacial Phase Transitions. In: Vanselow, R., Howe, R. (eds) Chemistry and Physics of Solid Surfaces VII. Springer Series in Surface Sciences , vol 10. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-73902-6_13
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DOI: https://doi.org/10.1007/978-3-642-73902-6_13
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