User:Ldm1954/Sandbox/Duality

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General form

The thermodynamic Wulff describe the relationships between the shape of a single crystal and the surface free energy of different surfaces facets. It has the form that the perpendicular distance from a common center to all the external facets is proportional to the surface free energy of each one. This can be viewed as a relationship between the different surface energies and the distance from a Wulff center $h_{j}=\lambda \gamma _{j}$ , where the vector $h_{j}$ is the "height" of the $j$ th face, drawn from the center to the face with a surface free energy of $\gamma _{j}$ , and $\lambda$ a scale. A common approach is to construct the planes normal to the vectors from the center to the surface free energy curve, with the Wulff shape the inner envelope. This is represented in the figure where the surface free energy is in red, and the single crystal shape would be in blue. In a more mathematical formalism it can be written describing the shape as a set of points $S_{w}$ given by^[1]^[2]

$S_{w}=x:x.{\widehat {n}}\leq \lambda \gamma ({\widehat {n}})$ for all unit vectors ${\widehat {n}}$

For the extended constructions an additional term is included for an interface free energy $\gamma _{i}$ which is marked in purple with dashes in the figure. The dashed interface is included which may be a solid interface for the Winterbottom case, two interfaces for Summertop or one or more twin boundaries for the modified Wulff construction.

^ Cahn, J.l; Hoffman, D.l (1974-10). "A vector thermodlnamics for anisotropic surfaces—II. Curved and faceted surfaces". Acta Metallurgica. 22 (10): 1205–1214. doi:10.1016/0001-6160(74)90134-5. {{cite journal}}: Check date values in: |date= (help)
^ Hoffman, David W.; Cahn, John W. (1972). "A vector thermodynamics for anisotropic surfaces". Surface Science. 31: 368–388. doi:10.1016/0039-6028(72)90268-3.

[1] Cahn, J.l; Hoffman, D.l (1974-10). "A vector thermodlnamics for anisotropic surfaces—II. Curved and faceted surfaces". Acta Metallurgica. 22 (10): 1205–1214. doi:10.1016/0001-6160(74)90134-5. {{cite journal}}: Check date values in: |date= (help)

[2] Hoffman, David W.; Cahn, John W. (1972). "A vector thermodynamics for anisotropic surfaces". Surface Science. 31: 368–388. doi:10.1016/0039-6028(72)90268-3.

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