Phase-field model of dendritic growth

doi:10.1016/S0022-0248(01)01891-7

Journal of Crystal Growth

Volumes 237–239, Part 1, April 2002, Pages 125-131

https://doi.org/10.1016/S0022-0248(01)01891-7 Get rights and content

Abstract

The phase-field models for binary and ternary alloys are introduced, and the governing equations and phase-field parameters for dilute alloys are derived at a thin interface limit. The phase-field simulations on isothermal dendrite growth for Fe-C, Fe-P and Fe-C–P alloys are carried out and the effect of the ternary alloying element on dendrite growth is examined. The secondary arm spacing for Fe-C, Fe-P and Al–Cu alloys is numerically predicted using the phase-field model and compared to the experimental data. The change in the arm spacing, and the exponent of local solidification time depending on alloy is systematically examined by imposing artificial set of physical properties. The phase-field simulation for the microstructure evolution during rapid solidification is also successfully carried out. Through the numerical examples, the wide potentiality of the phase-field model to the applications on solidification has been demonstrated.

Introduction

The phase-field model is known to be a powerful tool for describing the complex pattern evolution of the interface between mother and new phases in nonequilibrium state because all the governing equations are written in unified manner in the whole space of system. The model was originally proposed for simulating dendrite growth in undercooled pure melts [1], [2], [3], [4], [5], [6], [7], [8], [9] and has been extended to solidification of alloys [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. The first model for alloy solidification was proposed by Wheeler et al. (WBM model) [10], [13]. The WBM model has been the most widely used [10], [11], [12], [13], [21], [22] and is derived in a thermodynamically consistent way [13]. In the model, any point within the interfacial region is assumed to be a mixture of solid and liquid both with a same composition. The phase-field parameters are determined not only under a sharp interface condition [10], but also under a finite interface thickness condition [23].

Recently, Kim et al. have proposed a new phase-field model for binary alloys (KKS model) [24]. The model is equivalent with the WBM model, but has a different definition of the free energy density for the interfacial region. An extra-potential originated from the free energy density definition in the WBM model disappears and the governing equations completely correspond to those for pure materials [25] as pointed out by Langer [26]. In the present study, the phase-field model for binary alloys by Kim et al. and its extension to ternary alloys are briefly introduced. The examples of numerical analyses for different types of dendrite growth are demonstrated and the validity of the models is discussed.

Section snippets

Free energy density

In the KKS model, the free energy density, f(c,φ), where φ is phase-field, is defined as the sum of the free energies of liquid and solid phases with different compositions of c_L and c_S, respectively, and an imposed double-well potential, Wg(φ). The solute composition in interface region, c, is determined to be the fraction-weighted value of both liquid and solid compositions. The chemical potentials, μ, defined as the difference between the chemical potentials of solute and solvent, are

Calculations

For the simulation, the phase-field equation was modified so as to include the anisotropy of interface. The four-hold anisotropy was introduced by putting the coefficient in the phase-field parameter as follows: $ε(θ)=ε{1+v cos (4θ)},$ where v was the magnitude of anisotropy (v=0.03) and θ is the angle between normal direction of interface and x-axis. Note that the ε² should be located inside the divergence in Eq. (4) and the following terms should be added when the anisotropy is considered [15]: $∂ ∂ x$

Isothermal dendrite growth

The dendrite shapes of Fe–0.5 mol%C binary alloy and Fe–0.5 mol%C–0.001 mol%P ternary alloy at 1780 K are shown in Fig. 1. The secondary arms develop well and the arm spacing becomes narrow with a small additional phosphorous because the phosphorous is likely to enrich at the interface and reduces the interface stability. However, the dendrite shape becomes coarse with the further increase of phosphorus concentration. It is due to the decrease of the phase-field mobility because the value of ζ₂ in

Conclusion

The phase-field model for a dilute binary alloy is applied to the solidification problems of binary and ternary alloys. The governing equations are written into simplified forms with a dilute solution approximation and the relationship between the phase-field mobility and the kinetic coefficient is obtained at a thin interface limit. The model is free from the restrictions on the kinetic coefficient and the mesh size in calculation, and has a great advantage to applications. The numerical

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Phase-field model of dendritic growth

Abstract

Introduction

Section snippets

Free energy density

Calculations

Isothermal dendrite growth

Conclusion

Physica D

Physica D

Physica D

Physica D

Acta Metall. Mater.

Physica D

Phys. Rev. B

Phys. Rev. A

Phys. Rev. E

Phys. Rev. E

Phys. Rev. A

Phys. Rev. E

Phys. Rev. E