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An orientation-field model for polycristalline solidification with a singular coupling between order and orientation

The solidification of polycrystalline materials can be modelled by orientation-field models, which are formulated in terms of two continuous fields: a phase field that describes the thermodynamic state and an orientation field that indicates the local direction of the crystallographic axes. The free-energy functionals of existing models generally contain a term proportional to the modulus of the orientation gradient, which complicates their mathematical analysis and induces artificial long-range interactions between grain boundaries. We present an alternative model, in which only the square of the orientation gradient appears, but in which the phase and orientation fields are coupled by a singular function that diverges in the solid phase. We show that this model exhibits stable grain boundaries whose interactions decay exponentially with their distance. Furthermore, we demonstrate that the anisotropy of the surface energy can be included while preserving the variational structure of the model. Illustrative numerical simulations of two-dimensional examples are also presented.