Quantitative Phase-Field Modelling of Solidification at High Lewis Number

A phase-field model of non-isothermal solidification in dilute binary alloys is used to study the variation of growth velocity, dendrite tip radius and radius selection parameter as a function of Lewis number at fixed undercooling. By the application of advanced numerical techniques we have been able, for the first time, to extend the analysis to Lewis numbers of order 10000, which are realistic for metals. A large variation in the radius selection parameter is found as the Lewis number is increased from 1 to 10000. PACS: 81.30.Fb, 64.70.dm, 02.60.Cb Introduction The growth of dendritic structures during solidification has been a subject of enduring interest within the scientific community, both because it is a prime example of spontaneous pattern formation and due to the pervasive influence of dendrites on the engineering properties of metals. As dendrites are self-similar when scaled against the tip radius, ρ, the ability to accurately predict ρ is a problem of central importance to the theory of dendritic growth. However, the prediction of ρ has proved exceptionally challenging. Early analytical solutions ] 1 predicted that it was the dimensionless Peclet number, Pt = Vρ/2D, (V = growth velocity, D = diffusivity in the liquid), that was related to undercooling, ∆T, during growth, leading to a degeneracy in the product Vρ not observed in nature. Various models based on the stability of planar solidification fronts were proposed , ] 2 3 to break this degeneracy, although ultimately the application of boundary integral methods established that it is crystalline anisotropy ] 4 rather than stability per se that is responsible for breaking the degeneracy. The analysis reveals that in the limit of vanishing Pt an equation similar to that arising from stability arguments is recovered, but with a radius selection parameter, σ*, that varies as ε, where ε is the anisotropy strength. In recent years significant progress towards understanding solidification processes has also been afforded by the advent of phase-field modelling. However, the application of phasefield modelling has largely been restricted to two limiting cases; namely the thermally controlled growth of pure substances and the solidification of relatively concentrated alloys [e.g. 5] where growth is sufficiently slow that the problem may be considered isothermal. However, this omits alloy solidification problems where the isothermal approximation is not valid, specifically the solidification of very dilute alloys and rapid solidification processes. To date, relatively few attempts have been made to use phase-field techniques to simulate coupled thermo-solutal solidification due to the severe multi-scale nature of the problem (typically Lewis number, Le = α/D, is 10 – 10, where α is the thermal diffusivity). Loginova et al. ] 6 have developed a coupled model using a derivation based on the solutal model of Warren & Boettinger ] 7 , although there are doubts about the quantitative validity of this model ] 8 as the numerical results appear to suggest excess solute trapping and have an unresolved interface width dependence. This methodology has been extended by Lan et al. ] 9 , who introduced an adaptive finite volume solver, which allowed them to use realistic values of Le, although this did not overcome either the excess solute trapping or the interface-width dependence of the solution. An alternative formulation of the coupled phase-field problem has been presented by Ramirez & Beckermann ] 10 , based on the Karma ] 11 . thin interface model. As the thin interface model has been shown to be independent of the length scale chosen for the mesoscopic diffuse interface width, it is capable of giving quantitatively correct predictions for dendritic growth, although Ramirez & Beckermann only used the model at relatively modest Lewis numbers (typically 40). In a previous paper ] 12 we used a fully implicit, adaptive finite difference implementation of the model due to [8] to investigate the dependence of ρ upon ∆T at Le = 200, demonstrating for the first time that ρ pass through a minimum with increasing ∆T, as predicted by stability models such as that due to Lipton, Trivedi & Kurz (LKT). We also showed that the radius selection parameter, σ*, not only varies with ∆T, but that the variation is non-monotonic. In this paper we now consider the extent to which ρ, V and σ* vary as Le is increased at fixed ∆T. This quantitative analysis of the Lewis number dependency has previously been considered in [8], albeit in the restricted range 1 ≤ Le ≤ 200, wherein it was found that the predictions of the LKT theory were valid for Le ≤ 5, with significant deviations thereafter. Here we extended the analysis to higher values of Le, including for the first time values up to 10, which are of appropriate order for metals, in which dendritic growth is most common. Description of the Model The model adopted here is based upon that of [8] in which, following non-dimensionalization against characteristic length and time scales, W0 and τ0, the evolution of the phase-field, φ, and the dimensionless concentration and temperature fields U and θ are given by