A phase-field model with convection: sharp-interface asymptotics

Abstract

We have previously developed a phase-field model of solidification that includes convection in the melt [Physica D 135 (2000) 175]. This model represents the two phases as viscous liquids, where the putative solid phase has a viscosity much larger than the liquid phase. The object of this paper is to examine in detail a simplified version of the governing equations for this phase-field model in the sharp-interface limit to derive the interfacial conditions of the associated free-boundary problem. The importance of this analysis is that it reveals the underlying physical mechanisms built into the phase-field model in the context of a free-boundary problem and, in turn, provides a further validation of the model. In equilibrium, we recover the standard interfacial conditions including the Young-Laplace and Clausius-Clapeyron equations that relate the temperature to the pressures in the two bulk phases, the interface curvature and material parameters. In nonequilibrium, we identify boundary conditions associated with classical hydrodynamics, such as the normal mass flux condition, the no-slip condition and stress balances. We also identify the heat flux balance condition which is modified to account for the flow, interface curvature and density difference between the bulk phases. The interface temperature satisfies a nonequilibrium version of the Clausius-Clapeyron relation which includes the effects of curvature, attachment kinetics and viscous dissipation.

Size and mechanics effects in surface-induced melting of nanoparticles

“…20), and we have σ st = β| η| 2 (I − kk), that is, it represents two equal normal stresses along the interface. Thus, σ st is consistent with the SIA, unlike previous approaches [11][12][13][14] , which resolves a long-standing problem in PFA for melting.…”

“…1), in which shear modulus µ varies from that for solid to zero, and this results in generation of internal elastic stresses at the interface. Thermodynamic potential in this model results in surface-tension stresses consistent with the SIA, in contrast to known models with incoherent interface (for which µ = 0) [11][12][13] and coherent interface 14 . Also, the necessity of introducing a non-spherical transformation strain, which, in addition to volumetric expansion during melting, describes change in shape, is demonstrated.…”

“…While for martensitic phase transformations t is a tensor connecting two crystal lattices, for melting it is always pure volumetric strain, 9,[11][12][13][14] . We will show that the usual assumption causes very high internal stresses and elastic energy within the interface, which suppresses melt nucleation and contradicts with the experiment.…”

“…Traditionally, solids and liquids are described in completely different continuum mechanical frameworks (for example, solids are described in undeformed states, whereas liquids are described in a deformed state), which sophisticates the description of intermediate state. Some works consider solid as a very viscous liquid 13 , that is, neglect elastic shear modulus µ. Such a liquid-liquid interface is incoherent and does not generate internal elastic stresses, similar to sharp interface (Fig.…”

“…Although this model is supposed to be more precise, the surface tension in it is not consistent with sharp-interface limit, even if µ = 0. Also, even for liquid-liquid interface, expression for surface tension in PFA differs by a hydrostatic pressure from that in SIA [11][12][13] . This is a fundamental thermodynamic inconsistency, which is especially important for nanoscale interface radii.…”