Oscillatory instability and rupture in a thin melt film on its crystal subject to freezing and melting

• Published in: Journal of fluid mechanics Vol. 586; pp. 423 - 448
• Main Authors: BEERMAN, M., BRUSH, L. N.
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 10.09.2007
• Subjects: Condensed matter: structure, mechanical and thermal properties; Cross-disciplinary physics: materials science; rheology; Crystals; Differential equations; Exact sciences and technology; Fluid dynamics; Fluid mechanics; Freezing; Heat transfer; Interfaces; Latent heat; Materials science; Mechanical and acoustical properties; Mechanical engineering; Melting; Methods of crystal growth; physics of crystal growth; Methods of deposition of films and coatings; film growth and epitaxy; Nonlinear equations; Phase diagrams and microstructures developed by solidification and solid-solid phase transformations; Physical properties of thin films, nonelectronic; Physics; Solidification; Stability analysis; Surfaces and interfaces; thin films and whiskers (structure and nonelectronic properties); Temperature gradients; Theory; Theory and models of crystal growth; physics of crystal growth, crystal morphology and orientation; Theory and models of film growth; Thin films; Fluid mechanics; Crystalline phase; Temperature gradients; Growth interface; Solidification front; Heat of fusion; Solidification; Thin films; Standing waves; Temperature effects; Oscillatory instability; Differential equations; Growth mechanism; Capillary pressure; Stress effects; Lubrication; Heat transfer; Tribology
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/S002211200700729X

Lubrication theory is used to derive a coupled pair of strongly nonlinear partial differential equations governing the evolution of interfaces separating a thin film of a pure melt from its crystalline phase and from a gas. The free melt–gas (MG) interface deforms in response to the local state of stress and the crystal–melt (CM) interface can deform by freezing and melting only. A linear stability analysis of a static, uniform film subject to the effects of MG interface capillary forces, thermocapillary forces, the latent heat of fusion, van der Waals attraction, heat transfer and solidification volume change effects, reveals stationary and oscillatory instabilities. The effect of a temperature gradient (by increasing the gas phase temperature) is to stabilize a film. As the temperature gradient is reduced, the onset of instability is oscillatory and is at a unique, finite wavenumber. Instability is oscillatory for all marginally stable, non-isothermal cases. Crystals with higher density than the melt are more stable, whereas crystals with lower density are less stable in the presence of an applied temperature gradient. Fully nonlinear numerical solutions show that oscillatory instabilities lead to rupture by growth of standing or travelling waves. Rupture times and the number of oscillations to rupture increase as the temperature gradient is increased. For stationary linearly unstable initial conditions, the CM interface retreats by melting away from the tip region of the encroaching MG interface due to a rise in the heat flux there as the film thins and nears rupture. Larger amplitude disturbances increase the maximum allowable temperature for instability, at a given wavenumber, and decrease the time to rupture at fixed temperature and wavenumber.