Phase-field model of dendritic growth
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 cL and cS, 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: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]:
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
References (29)
Physica D
(1993)- et al.
Physica D
(1993) - et al.
Physica D
(1990) - et al.
Physica D
(1993) - et al.
Acta Metall. Mater.
(1995) - et al.
Physica D
(1998) - et al.
Phys. Rev. B
(1985) Phys. Rev. A
(1989)- et al.
Phys. Rev. E
(1993)
Phys. Rev. E
Phys. Rev. A
Phys. Rev. E
Phys. Rev. E
Cited by (86)
Simulation of dendritic grain structures with Cellular Automaton–Parabolic Thick Needle model
2023, Computational Materials ScienceEffect of surface topography on dendritic growth in lithium metal batteries
2022, Journal of Power SourcesA phase-field method for three-phase flows with icing
2022, Journal of Computational PhysicsA review on the application of lattice Boltzmann method for melting and solidification problems
2022, Computational Materials ScienceA GPU-accelerated 3D PF-LBM modelling of multi-dendritic growth in an undercooled melt of Fe–C binary alloy
2022, Journal of Materials Research and TechnologyCitation Excerpt :With the fast development of computational material science, numerical simulation becomes an alternative high-efficient method to investigate the solidification phenomena in metallic alloys. At present, there are many powerful computation methods for predicting the micro/macro solidification phenomena in metallic alloys, such as the traditional Enthalpy method [4–6], Cellular Automaton method (CA) [7–12], Monte Carlo method (MC) [13–15], Front tracking method [16] and Phase Field method (PF) [17–22]. Among them, the Phase Field method (PF) has proven to be a powerful tool to simulate complex phase interface changes in phase transitions, especially in alloy solidification [18–20].
A temperature-dependent atomistic-informed phase-field model to study dendritic growth
2022, Journal of Crystal Growth