Phase field simulation of dendrite growth with boundary heat flux
© Du and Zhang.; licensee Springer 2014
Received: 7 February 2014
Accepted: 29 April 2014
Published: 28 June 2014
Boundary heat flux has a significant effect on solidification behavior and microstructure formation, for it can directly affect the interfacial heat flux and cooling rate during phase transition. In this study, a phase field model for non-isothermal solidification in binary alloys is employed to simulate the free dendrite growth in undercooled melts with induced boundary heat flux, and an anti-trapping current is introduced to suppress the solute trapping due to the larger interface width used in simulations than a real solidifying material. The effect of heat flux input/extraction from different boundaries was studied first. With heat input from boundaries, the temperature can be raised and the dendritic morphology changed with gradient temperature distribution caused by the heat flux input coupling with latent heat release during the liquid-solid phase transition. Also, the concentration distribution can be also influenced by this irregular temperature distribution. Heat flux extraction from the boundaries can decrease the temperature, which results in rapid solidification with small solute segregation and concentration changes in the dendrite structures. Also, dendrite growth manner changes caused by undercooling variation, the result of competition between heat flux and latent heat release from phase transition, are also studied. Results indicate that heat flux in the simulation zone significantly reduces the undercooling, thus slowing down the dendrite formation and enhancing the solute segregation, while large heat extraction can enlarge the undercooling and lead to rapid solidification with large dendrite tip speed and small secondary dendrite arm spacing, while solute segregation tends to be steady. Therefore, the boundary heat flux coupling with the latent heat release from the solidification has an effective influence on the temperature gradient distribution within the simulation zone, which leads to the morphology and concentration changes in the dendritic structure formation.
KeywordsComputer simulations Metals and alloys Rapid solidification Microstructure Diffusion
The mechanical properties of many materials have a significant relationship with the microstructure formation process [], but in practice, it is difficult to observe the microstructure formation process with experimental methods, and computer simulations can visually show the phase transition process and provide much more information to calculate many other parameters that are related to the mechanical properties with the data achieved from simulations. For decades, the phase field method has become a popular technique to model various types of complex microstructure changes qualitatively, such as solidification, spinodal decomposition, Ostwald ripening, crystal growth and recrystallization, domain micro-structure evolutions in ferroelectric materials, martensitic transformation, dislocation dynamics, and crack propagation [,]. The phase field model has been used for computing solidification morphologies to avoid the explicit boundary tracking needed to solve the classical sharp-interface model. With this advantage, phase field methods have attracted considerable interest in the last decades as a means of simulating the solidification process. In phase field models for solidification process, a variable ϕ(r, t), called the phase field order parameter, is introduced to indicate the physical state of the system in time and space. This order parameter ϕ(r, t) is assigned the value 0 (or −1) in the bulk solid phase and 1 in the bulk liquid phase, and it changes smoothly between these values over a thin transition layer that plays the role of the classical sharp interface. The governing equations for the growth of a solid phase from a liquid need to be derived from the irreversible thermodynamic law and conservation laws for both mass and energy, and the resulting equations need to be applicable to the entire space being modeled without any discontinuities between the various phases present. It also must be possible to determine the parameters in the governing equations from classical thermodynamic and kinetic quantities.
In the past decades, many researches have studied the solidification process using the phase field method [,]. Kobayashi [] developed a simple phase field model for one-component melt growth including anisotropy and used this model to study the formation of various dendritic patterns. He found that the qualitative relations between the shapes of crystals and some physical parameters and noises gave a crucial influence on the side branch structure of dendrites. The first phase field model for alloys was developed by Wheeler et al. [,], called the WBM model. Kim et al. [,] presented another model for alloys by adopting the thin-interface limit, which is known as the KKS model. Karma [] presented a phase field formulation to quantitatively simulate microstructural pattern formation in alloys, and the thin-interface limit of this formulation yielded a much less stringent restriction on the choice of interface thickness than previous formulations and permitted one to eliminate non-equilibrium effects at the interface. Dendrite growth simulations with vanishing solid diffusivity showed that both the interface evolution and the solute profile in the solid were accurately modeled by this approach. Recently, solidifications with forced flow or convection were studied in binary alloys in 2D and 3D [–]; solidifications of multi-component and multiphase were also studied using phase field methods [–].
As known, the undercooling in the liquid melt has a significant effect on the microstructure forming process and thus affects solute diffusion to change the concentration distributions within the structure. In industry, a cooling/heating system can be applied on the chilling wall during the casting process to control the solidification. So, in this study, we introduce complex boundary conditions in order to present the different cooling/heating conditions to find the relation between the structure forming and these solidification conditions. Experimental and numerical investigations have been presented in the past decades to study the heat transfer problems during solidification [–] and find that heat flow can significantly affect the temperature distribution and thus determine the quality of the solidification process. But few studies have been carried out to study the heat flux at the boundaries and its influence on phase transition using phase field methods []. Especially, the effect of heat flux on microstructures and distributions of temperature and concentration still need to be investigated in detail, for temperature field has a directional influence on the solidification process with undercooling. So, in this study, the boundary heat coupling with latent heat release from solidification is the main factor affecting dendritic structure forming in undercooling melts. Different dendrite patterns are obtained by changing the boundary heat flux at different boundaries and directions. Distributions of the concentration change as the result of temperature change caused by different boundary heat flux couplings with latent heat release are also given and analyzed in detail.
Model description and simulation method
In this model, the phase field variable ϕ varies smoothly between 0 in the solid phase and 1 in the liquid phase as we assumed, xB is the mole fraction of solute B in solvent A, and T is the temperature. ϵ is the coefficient of gradient energy, which is determined by the interfacial energy. The anisotropy is included in the system because the phase change kinetics depends upon the orientation of the interface. Here, we introduce the anisotropy ε , where is related to the surface energy σ and interface thickness λ, γ is the magnitude of anisotropy in the surface energy, κ specifies the mode number, and the expression β = arctan((∂ϕ/∂y)/(∂ϕ/∂x)) gives an approximation of the angle between the interface normal and the orientation of the crystal lattice. is the anti-trapping current introduced by Karma [] to suppress the solute trapping due to the larger interface width used in simulations in order to get a more quantitative prediction.
Melting temperature of Ni
Melting temperature of Cu
Latent heat of Ni
2.35 × 109 J/m3
Latent heat of Cu
1.728 × 109 J/m3
Heat conductivity of Ni
Heat conductivity of Cu
Specific heat of Ni
5.42 × 106 J/m3K
Specific heat of Cu
3.96 × 106 J/m3K
Diffusion coefficient of the liquid phase
1.0 × 10−9 m2/s
Diffusion coefficient of the solid phase
1.0 × 10−13 m2/s
Mole volume of alloy
7.42 × 10−6 m3/mol
Surface energy of Ni
Surface energy of Cu
Interfacial kinetic coefficient of Ni
3.3 × 10−3 m/sK
Interfacial kinetic coefficient of Cu
3.9 × 10−3 m/sK
4.9 × 10−8 m
Amplitude of the noise fluctuations
Model of anisotropy
Magnitude of anisotropy
Results and discussion
Free dendrite growth in undercooling melt
Dendrite growth with boundary heat flux
Different heating/cooling conditions during the solidification can significantly change the temperature distributions, thus affecting the solute diffusion and dendrite structure formation. In the following simulations, boundary heat flux is introduced to present different heating/cooling modes during rapid dendrite growth in undercooling melts. Latent heat release during phase transition is also considered in these non-isothermal simulations.
Effect of different magnitudes of heat flux
A phase field model for simulating solidification in binary alloys under non-isothermal conditions is implemented to study the effect of heat flux input/extraction from boundaries on the dendrite structure forming process in a Ni-Cu alloy. In order to suppress the solute trapping due to the larger interface width used in simulations, the anti-trapping current is introduced into the diffusion equation. Simulations were carried out to study the dendrite structure formation, concentration, and temperature changes with different boundary heat fluxes. It is obvious that heat flux input can raise the temperature near the boundary and results in the prevention of dendrite arm growth and side branching, while heat flux extraction from the boundaries can enhance the dendrite arm growth and secondary arm formations. The heat flux can also lead to the concentration distribution changes by changing the temperature distribution. High temperature can enlarge the solute segregation and decrease the solute level inside the dendrite structure decrease. The effect of different heat fluxes on solute segregation and concentration distributions inside dendrite structures is studied. With heat flux changing from large extraction to large input, the dendrite tip speed of the primary arm shows different changes, and the average secondary dendrite arm spacing decreases. Heat extraction from the boundaries could partly counterbalance latent heat release from solidification and lead to uniform concentration distributions in solid, while heat input, coupling with latent heat, would significantly raise temperatures at interfaces and decreases concentration with time evolution. These effects could be enlarged with increasing heat flux. Therefore, the heat flux input/extraction from the boundary can significantly change the morphology, concentration, and temperature distribution of the dendrite growth in undercooled melts, and these morphology and concentration changes will affect the properties of the alloys.
Availaiblity of supporting data
The computer code to reproduce all of the case studies presented in this manuscript is available upon request.
LD is a Ph.D. Candidate in the Department of Applied Physics in Northwestern Polytechnical University, whose research interest is phase-field simulation of solidification in metal alloys under different conditions. RZ is a Professor in the Department of Applied Physics in Northwestern Polytechnical University, whose research interest is solidifying process under mutiphysics fields.
The authors would like to thank the financial support from the NPU Foundation of Fundamental Research, China (No. JC201272).
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