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The Gibbs-Thompson relation within the gradient theory of phase transitions. (English) Zbl 0681.49012

Summary: This paper discusses the asymptotic behaviour as \(\epsilon \to 0+\) of the chemical potentials \(\lambda_{\epsilon}\) associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by \(\epsilon^ 2| \nabla \rho |^ 2+W(\rho)\), where \(\rho\) is the density. The main result is that \(\lambda_{\epsilon}\) is asymptotically equal to \(\epsilon E\lambda /d+0(\epsilon)\), with E the interfacial energy, per unit surface area, of the interface between phases, \(\lambda\) the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin [Some results and conjectures in the gradient theory of phase transitions, Inst. Math. Appl., Univ. Minnesota, preprint #156(1985)] and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp [Arch. Ration. Mech. Anal. 92, 205-245 (1986; Zbl 0608.35080)] within the context of the phase field model of free boundaries arising from phase transitions.

MSC:

49J99 Existence theories in calculus of variations and optimal control
82B26 Phase transitions (general) in equilibrium statistical mechanics
49Q05 Minimal surfaces and optimization

Citations:

Zbl 0608.35080
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References:

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