Dynamics of premelted films: Frost heave in a capillary

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Abstract
Due either to van der Waals or to short range interactions, some materials will interfacially premelt against a foreign substrate. We present a theoretical study of this phenomenon for a situation in which the material is confined within a deformable capillary tube. At a temperature below the bulk melting transition an annulus of premelted liquid separates the solid from the capillary walls. For an isothermal capillary, at finite reduced temperature, the film is of uniform thickness and is static. On imposition of an axial temperature gradient the thickness of the film varies with position along the axis. A thermomolecular pressure gradient transports fluid towards regions of colder temperatures, where it solidifies and deforms the confining capillary. For the case of van der Waals interactions we formulate a mathematical model and solve it numerically and by matched asymptotic expansions. The main result is the temporal and spatial deformation of the capillary tube; a measurable quantity. In the case of a transient thermal field, we find that the deformation of the capillary is small and that it is uniform over most of its length. For a steady thermal field, large deformation occurs in a region of small reduced temperature and grows towards the cold end of the capillary. We focus on ice monocrystals, and offer our theory as a model for frost-heave phenomena, with the advantage of having exposed the essential physics of the problem in the absence of impurity and curvature effects. The experiments conducted by Wilen and Dash [Bull. Am. Phys. Soc. 38, 747 (1993); Phys. Rev. Lett. (to be published)] provide information that is unavailable using equilibrium techniques, and form the relevant test of this theoretical approach.

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