A Phase Field Model for Binary Alloy Solidification with Boundary Interface Intersection

A phase field model for binary alloy solidification with boundary interface intersection is developed. In the phase field model, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface, and a relaxation boundary condition for the phase field variable is introduced to balance the interface energy and boundary surface energy in the multiphase contact region. The thin interface asymptotic analysis is applied on the phase field model to yield the free interface problem with dynamic contact point condition. Introduction The study of the evolution of the interface structure in alloy solidification near physical boundary is of great technological interest. The solidification near the boundary interface multiphase intersection is governed by heat and solute flux as well as the boundary interface surface energy balance. To simulate the micro-structure evolution during solidification, the phase field approach has been known as a widely accepted technique[2, 8, 10, 11, 12, 16, 17, 18, 20]. This approach has the well known advantage that it avoids explicitly tracking a sharp boundary by introducing an order parameter, i.e., a phase field variable, which takes constant values in the bulk phases and varies smoothly but steeply in a diffuse interface region with small thickness. In order to model the coupled heat and solute diffusion in binary alloy solidification, the heat and solute conservation equations are appropriately modified to account for the presence of heat and solute rejection inside the diffuse interface in the phase field model[17]. However, the above mentioned work didn’t consider the case when the interface intersect with the external physical boundary of the domain, where additional mechanism to balance the interface energy and boundary surface energy in the multiphase contact region should be imposed[19]. To recover the additional mechanism in the multiphase contact region, we consider the solidification of a dilute binary alloy in a channel Ω = {x = (x, y) : x ∈ R, 0 ≤ y ≤ L} with a moving interface Γ(t) separating solid and liquid states, Ω+(t) and Ω−(t), where Ω+(t) denotes the liquid region, Ω−(t) denotes the solid region and Γ(t) is the solid/liquid interface. The evolution of the interface is extensively studied when the interface is a closed curve that never intersect the external boundary. We will concern the case when Γ(t) intersect ∂Ω at {y = 0} and {y = L}, and study the dynamics of the boundary-interface intersection. The motion of the interface is controlled by coupled heat and solute diffusion in the whole domain. We assume the alloy have equal thermal diffusivity α in solid and liquid, the solutal diffusivity is D in liquid and zero in solid. The governing free-interface problem is ∂tU = D∇U (liquid), (1) ∂tθ = α∇θ (liquid and solid), (2) (1 + (1− k)Ui)Vn = −D∂nU |l (mass conservation), (3) Vn = α(∂nθ|s − ∂nθ|l) (heat conservation), (4) International Frontier Science Letters Submitted: 2016-03-15 ISSN: 2349-4484, Vol. 8, pp 9-18 Revised: 2016-04-15 doi:10.18052/www.scipress. com/IFSL.8. 9 Accepted: 2016-04-15 © 2016 SciPress Ltd., Switzerland Online: 2016-06-29 SciPress applies the CC-BY 4.0 license to works we publish: https://creativecommons.org/licenses/by/4.0/ θi +Mc∞Ui = −d0κ− βVn (Gibbs-Thomson), (5) where U is the dimensionless concentration (mole fraction of solute) and θ is the temperature relative to the equilibrium freezing temperature scaled by latent heat of fusion and the heat capacity, k is the equilibrium partition coefficient, Vn is the normal velocity of the interface, n is the direction normal to the interface pointing from solid to liquid, M is the scaled magnitude of the liquidus slope, c∞ is the initial concentration of the alloy, d0 is the thermal capillary length, κ is the local curvature of the interface, β is the kinetic coefficient. Note that the subscript i on θ and U is present when these quantities are being evaluated at the interface. In this setting, we need to solve the coupled heat and solute diffuse equations, and the shape and position of the free boundary Γ(t). The evolution of the interface Γ(t) is governed by the above kinetic Gibbs-Thomson relation (5), this equation can be understood similarly as curvature flow equation, which is a second order parabolic type partial differential equation. When the free boundary Γ(t) intersect the external boundary, to complete the formulation, we need to impose conditions on the boundary-interface contact points to incorporate the boundary-interface interaction in the evolution of the interface, thus additional dynamic conditions on the contact points are required[19]. To get some insight on the dynamic contact point condition, we study this problem by phase field model, in which the sharp interface is replaced by a diffuse interface, thus an auxiliary order parameter φ, the phase field, is introduced to indicate the phase. The quantityφ is a continuous variable that takes constants in the bulk phases, say +1 in the solid and−1 in the liquid, and varies smoothly but steeply from−1 to +1 over a thin layer, the diffuse interface. Then a phase field model will be constructed to govern the time evolution of φ, U and θ, which incorporate the interfacial physics of the problem and the diffuse of heat and solute in the system. The standard phase field model, a version of the Model C of Halperin, Hohenberg and Ma [9] was originally derived on the basis of a gradient functional for the free energy of the system, with dynamics that ensure the free energy decreasing with time. The time rate of change of the order parameter is coupled to the energy equation to incorporate the release of latent heat as solidification occurs. This model was further elaborated in Caginalp [3], Fix [8] and Langer [14] etc. However, the phase field model has the disadvantage that it is hard to use quantitatively because it is often computationally too stringent to choose the interface thickness small enough to resolve the desired sharp interface limit of the phase field model. To be useful for real simulation, phase-field models need to bridge the scale gap between the thickness of the physical solid-liquid interfaces and the typical scale of the microstructures. This is achieved by increasing the orders of magnitude of the interface width such that the width of the diffuse interface is about one order of magnitude smaller than the radius of curvature of the interface but much larger than the real microscopic width of a solid-liquid interface. This procedure magnifies the physical effect that is due to the diffuseness of the interface, therefore, to guarantee precise simulations, all these effects have to be controlled or, if possible, eliminated. The privileged tool to achieve this is the so-called thin-interface limit presented by Karma and Rappel [11, 12], Almgren [1] for pure substances, and by Karma [10] for isothermal solidification of dilute binary alloys: the equations of the phase field model are analysed under the assumption that the interface thickness is much smaller than any other physical length-scale present in the problem, but much larger than the capillary length. The procedure of matched asymptotic expansions then yields the effective interface conditions valid at the macroscale, which contain all effects of the finite interface thickness up to the order to which the expansions are carried out. Note that the isothermal solidification of dilute binary alloys considered in Karma [10], the socalled antitrapping current was introduced to eliminate nonequilibrium effects at the interface, and it was shown that quantitative simulations of alloy solidification are possible with this model [5]. However, the concentrations in dilute binary alloys is small, the thermal diffusivity may be much larger than the solutal diffusivity, the solutal and thermal effect are still comparable and the solute and heat conservation equations need to be solved simultaneously. For this purposes, a phase-field model 10 Volume 8