Effect of solid–liquid density change on dendrite tip velocity and shape selection

doi:10.1016/j.jcrysgro.2009.07.031

Journal of Crystal Growth

Volume 311, Issue 19, 15 September 2009, Pages 4447-4453

https://doi.org/10.1016/j.jcrysgro.2009.07.031 Get rights and content

Abstract

Phase-field simulations are used to examine tip velocity and shape selection in free dendritic growth of a pure substance into an undercooled melt in the presence of a density change between the solid and liquid. The dendrite is assumed to grow two-dimensionally inside a Hele-Shaw cell. The phase-field model is coupled with a previously developed two-phase diffuse interface model to simulate the flow in the liquid that is induced by the density change. The predicted dependence of the dendrite tip growth Péclet number on the relative density change is compared with an available analytical solution and good agreement is obtained. The simulations verify that the dendrite tip selection parameter, modified to account for the different densities of the solid and liquid phases, is independent of the relative density change.

Introduction

Dendritic growth is of fundamental importance in the solidification of many materials. Usually, such crystal growth involves a density change between the solid and liquid phases. This density change induces a contraction or expansion flow in the undercooled melt ahead of the dendrite. McFadden and Coriell [1], for an axisymmetric paraboloidal dendrite, and Emmerich [2], for a two-dimensional parabolic plate dendrite, extended the Ivantsov solution for free dendritic growth of a pure substance into an undercooled melt to include the effects of a density change upon solidification. In the extended Ivantsov solution, the tip growth Péclet number, Pe, is a function not only of the imposed dimensionless melt undercooling, Δ, but also of the relative density change, β. The relative density change is defined as β=ρ_s/ρ_l−1, where ρ_s and ρ_l are the densities of the solid and liquid, respectively. From the solution of the Navier–Stokes equation, the magnitude of the flow in the melt was found to be of the order of β, with the velocity decaying to zero with increasing distance from the dendrite. Despite the fact that the density change induced flow plays an important role, the extended Ivantsov solution does not contain the Prandtl number or, more directly, the liquid viscosity. This result is different from the case of dendritic growth with forced convection in the melt, where the growth Péclet number depends also on the Prandtl number [3].

Emmerich [2] also modified the microscopic solvability theory of Kessler et al. [4] for dendrite tip velocity and shape selection to include the effects of a density change. According to this theory, a unique dendrite tip velocity and shape is selected by the action of anisotropic surface tension via a selection parameter σ*, and at low undercoolings, σ* is a constant that only depends on the crystalline anisotropy strength ε. In the presence of a density change, Emmerich [2] found that the relation for the selection parameter must be modified to include the relative density change. With this modification, σ* should take the same value as for diffusion limited dendritic growth without a density change. To date, this theory has not been validated by either experiments or direct numerical simulations.

In recent years, phase-field methods and other diffuse interface approaches have been popular tools in the direct numerical simulation of dendritic growth from the melt [5], [6]. By introducing an order parameter, φ, to represent the transition between the solid and liquid, a unique set of evolution equations is solved over the entire domain without explicitly tracking the solid–liquid interface. Due to the strong influence of melt motion on the evolution of microstructures during solidification, efforts have been made to include convection within phase-field models [7], [8], [9], [10]. For the sake of simplicity, most phase-field simulations of solidification assume equal densities in the solid and liquid. Recently, several phase-field simulations [2], [11], [12], [13], [14], [15], [16] have been carried out that account for the density change induced flow during free dendritic growth. Although these studies have revealed much interesting physics, it is not always clear if the results correspond to the same sharp interface model for which analytical solutions were obtained previously and/or if the results are fully converged with respect to the width of the diffuse interface. None of the previous studies have examined the effect of the density change on the dendrite tip selection parameter, σ*.

In this paper, the phase-field model of Karma and Rappel [17] is used in junction with the two-phase diffuse interface approach of Sun and Beckermann [18], [19] to examine tip velocity and shape selection in two-dimensional free dendritic growth of a pure substance into an undercooled melt in the presence of a density change. The remainder of this paper is organized as follows. In Section 2, an available theory for two-dimensional (2D) dendritic growth with density change is reviewed. A sharp interface description of the problem being solved is provided in Section 3. The corresponding phase-field model is summarized in Section 4. Section 5 describes the numerical procedure used to solve the model equations. The results and conclusions are presented in 6 Results and discussion, 7 Conclusion, respectively.

Section snippets

Theory of 2D dendritic growth with density change

The basic theory of free dendritic growth predicts the steady-state tip velocity, V, and tip radius of curvature, R, of a dendritic needle crystal of a pure substance growing into an infinite undercooled melt. For a two-dimensional parabolic plate dendrite, and equal densities of the solid and liquid, the analytical solution of the heat equation for diffusion limited growth is given by $Δ = I v_{2} (Pe)$ where Δ=(T_m−T_∞)/(L/c_p) is the dimensionless melt undercooling, T_m and T_∞ are the equilibrium melting

Sharp interface description

Before presenting the present phase-field model, it is important to state the conventional sharp interface equations for the dendritic growth problem being solved. In order to reduce computational effort, the growth is assumed to take place inside of a horizontal Hele-Shaw cell with a narrow gap of width b that is much smaller than the cell dimension L (i.e., b⪡L). This assumption renders the problem two-dimensional. Due to the small gap width, inertia effects can be neglected so that the

Phase-field model

Based on the two-phase diffuse interface model for Hele-Shaw flow recently developed by Sun and Beckermann [19], the continuity equation can be written in a dimensionless form as $\nabla^{'} \cdot [(1 - φ) {u^{'}}_{l}] = - β \frac{\partial φ}{\partial t^{'}}$ where the phase-field φ varies from unity in the solid phase to zero in the liquid over a distance of approximately $3 \sqrt{2} W_{0}$ , where W₀ is a measure of the diffuse interface width. The other dimensionless variables are defined as u′_l=u_lτ₀/W₀, ∇′=∇W₀, and t′=t/τ₀, where τ₀ is the relaxation time. Based on

Numerical procedures

The model described in the preceding section is used to numerically simulate the growth of a two-dimensional dendrite with flow induced by a density difference between the solid and liquid. The simulation domain is schematically illustrated in Fig. 1. Taking advantage of symmetry, only one quadrant of a dendrite is computed; thus the initial seed is actually only a quarter of a circle. The crystal axes are aligned with the coordinate axes. Symmetry boundary conditions are applied for all fields

Results and discussion

All dendritic growth simulations in the present study were performed for a melt undercooling of Δ=0.55 and an anisotropy strength of ε=0.05. Fig. 2 shows the predicted evolution of the phase-field contours (ψ=0), every 100,000 time steps, for three different values of the relative density change: β=0.1, β=0, and β=−0.1. It can be seen that the growth of the dendrite is strongly affected by the density change. Compared to the equal density case (Fig. 2b), the dendrite grows slower when shrinkage

Conclusion

The effect of a density change between the solid and liquid phases on free dendritic growth of a pure material into an undercooled melt is examined using phase-field simulations. The dendritic growth is assumed to be two-dimensional and inside of a Hele-Shaw cell. Good agreement is obtained with the analytical solution of Emmerich [2] for the dependence of the dendrite tip growth Péclet number on the relative density change, β. The modified dendrite tip selection parameter σ*, as defined by

Acknowledgement

This work was supported in part by the US National Science Foundation under Grant no. DMR-0132225.

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Effect of solid–liquid density change on dendrite tip velocity and shape selection

Abstract

Introduction

Section snippets

Theory of 2D dendritic growth with density change

Sharp interface description

Phase-field model

Numerical procedures

Results and discussion

Conclusion

Acknowledgement

J. Cryst. Growth

J. Comput. Phys.

Physica D

Physica D

Physica D

Physica D

Acta Mater.

J. Cryst. Growth

The Diffuse Interface Approach in Materials Science: Thermodynamics Concepts and Applications of Phase-Field Models, Lecture Notes in Physics M73

Forced-convection heat and momentum-transfer to dendritic structures (parabolic cylinders and paraboloids of revolution)

J. Heat Mass Transfer

Pattern selection in fingered growth phenomena

Adv. Phys.