Anisotropy and crystal morphology

For a reversible change at constant temperature and volume of both phases and for a constant number of moles of the components, the equilibrium shape can be 

Here /, is the surface energy of the crystal surface i. The equilibrium shape of a crystal is thus a polyhedron where the area of the crystal facets is inversely proportional to their surface energy. Hence the largest facets are those with the lowest surface energy. 

The Laplace equation (eq. 6.27) was derived for the interface between two isotropic phases. A corresponding Laplace equation for a solid-liquid or solid-gas interface can also be derived [3], Here the pressure difference over the interface is given in terms of the factor that determines the equilibrium shape of the crystal  

Comparing this expression with eq. (6.27), we see that yv/hv for each crystal face represents cr divided by the radius of curvature for an isotropic spherical phase. As a first approximation we may replace yv/hv with y/r for near-spherical crystals. In this case /represents an average surface energy of all possible crystal faces. 

In the remaining part of the chapter we will use the term y for interfaces that involve solids. It should then implicitly be understood that we are here considering bulk solids that are treated as isotropic systems and that the surface energy thus defined is the average value of the surface energies for different crystal surfaces. Furthermore, we will consistently use superscripts to denote the phases adjacent to the interface in the rest of Section 6.1 and in Section 6.2. 

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