Wulff’s theorem

The equilibrium shape of a macroscopic crystal is an old problem first addressed by Wulff [37], who showed the equilibrium shape at OK to be a polyhedron. At the equilibrium, the surface energy is given by the famous Wulff s theorem ... 

Wulff s theorem [25] states that cr/r is invariant for all faces. Therefore, the result obtained from the Kelvin s equation must be independent of the choice of a face. 

If the growth of an ice crystal from the vapour were a simple near-equilibrium process, then the resulting crystal habit could be determined by Wulff s theorem (Wulif, 1901) which states that, in equilibrium, the distance of any face from the centre of the crystal... 

The value of a for a crystal face relative to the other crystal faces can be found using Wulff s theorem, which states that there exists an interior point such that the surface free energy of a crystal face is proportional to the perpendicular distance from that point to the crystal face (Wolff and Gualtieri, 1962). 

The ratio 7a/ha can advantageously be expressed by averaged quantities , as 7/f, where 7 = Sjaj7j/Sjaj and f = Sja hj/Eja with j being a running index of the equilibrium planes. Note that Sjaj7j/a = (7j/hj)Sjhjaj/a on accoimt of Wulff s theorem. For spherical bodies (fluid phases) the well-known Kelvin-equation (7/f = 7/r) follows from Eq. (5.73). 

Making use of Eqs. (2) and (3) the condition for the equilibrium form dO = 0, dV = 0 leads to Wulff s rule (Gibbs-Curie-Wulff theorem [iii-v]) generalized by - Kaischew [i] to account for the crystal-substrate interaction ... 

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