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The role of microscopic anisotropy in the macroscopic behavior of a phase boundary. (English) Zbl 0639.58038
A Landau-Ginzburg approach is implemented to study the influence microscopic anisotropy on the interfacial boundary between two phases. The continuum limit for a lattice spin system with anisotropic interactions \(J_ i\) and spacing \(a_ i\) leads to the system of equations \[ \tau (\partial \phi /\partial t)=\sum^{d}_{i=1}\xi^ 2_ i(\partial^ 2\phi /\partial x^ 2_ i)+(\phi -\phi^ 3)+2u, \]
\[ \partial u/\partial t+(\ell /2)(\partial \phi /\partial t)=K\Delta u. \] Here, \(\phi\) is an order parameter, u is the temperature, \(\tau\) is a relaxation time, \(\xi_ i\) are related to \(J_ i\) and \(a_ i\), \(\ell\) is the latent heat of fusion, and K is the diffusivity. Both equilibrium and non-equilibrium conditions are considered. A modified Gibbs-Thompson relation \[ \Delta su(r,\theta)=-[\sigma (\theta)+\sigma ''(\theta)]\kappa -\tau \nu \sigma (\theta)/\xi^ 2_ A(\theta)+{\mathcal O}(\xi^ 2) \] is obtained, where s is entropy density, \(\kappa\) is curvature, \(\nu\) is normal velocity of the interface, and \(\xi_ A(\theta)\) is a measure of the thickness of the interface.

MSC:
58Z05 Applications of global analysis to the sciences
82B99 Equilibrium statistical mechanics
58J99 Partial differential equations on manifolds; differential operators
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