Wheeler, A.A. and Fadden, G.B.
On the notion of a ?-vector and a stress tensor for a general class of anisotropic diffuse interface models.
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 453, (1963), . (doi:10.1098/rspa.1997.0086).
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We show that, under equilibrium conditions, an anisotropic phase-field model of a diffuse interface leads to the notion of a divergence-free tensor, whose components are negligible away from the interface. Near the interface, it plays the role of a stress tensor, and involves the phase-field generalization of the Cahn-Hoffman ?-vector. We show that this tensor may be used to derive the equilibrium conditions for edges in interfaces. We then extend these ideas from the phase-field model to a general class of multiple-order-parameter models. In this broader context, we demonstrate that it is also possible to construct a ?-vector and we use it to derive the three-dimensional form of the Gibbs-Thomson equation in the sharp-interface limit. We derive the associated stress tensor for the multiple-order-parameter case and show that it also leads to the appropriate equilibrium conditions for edges in interfaces and multiple junctions in the sharp-interface limit. This approach shows that both multiple-order-parameter models and less physically based phase-field models share a common framework, the former providing a generalization of the notion of the ?-vector for phase-field models as well as providing a connection to the classical ?-vector theory for a sharp interface. For the special case of a phase-field model with non-convex surface energy, we show that edges can be represented by weak shocks in which the spatial derivatives of the phase field are not continuous, and we derive the associated jump conditions. In all the situations considered the notion of the ?-vector and the stress tensor play a central role.
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