A phase-field model of solidification with convection

Abstract

We develop a phase-field model for the solidification of a pure material that includes convection in the liquid phase. The model permits the interface to have an anisotropic surface energy, and allows a quasi-incompressible thermodynamic description in which the densities in the solid and liquid phases may each be uniform. The solid phase is modeled as an extremely viscous liquid, and the formalism of irreversible thermodynamics is employed to derive the governing equations. We investigate the behavior of our model in two important simple situations corresponding to the solidification of a planar interface at constant velocity: density change flow and a shear flow. In the former case we obtain a non-equilibrium form of the Clausius-Clapeyron equation and investigate its behavior by both a direct numerical integration of the governing equations, and an asymptotic analysis corresponding to a small density difference between the two phases. In the case of a parallel shear flow we are able to obtain an exact solution which allows us to investigate its behavior in the sharp interface limit, and for large values of the viscosity ratio.

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.…”

“…Using an irreversible thermodynamic procedure based on the application of the first and second laws of thermodynamics to the system with energy depending on the gradient of the order parameter (see ref. 12 and references in it) and assuming linear relation between thermodynamic force and flux, one obtains expression for the stress tensor σ and the Ginzburg-Landau (GL) equation in which ρ 0 and ρ are the mass densities in the non-deformed and deformed states, χ is the kinetic coefficient, and ε = const while evaluating ∂/∂η. The kinematics relationship between displacement u and strain ε = 1/3ε 0 I + e, decomposition of ε and the equilibrium equation are given by in which e is the deviatoric strain.…”

“…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.…”

“…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.…”