稀有金属(英文版) 2020,39(02),147-155

Phase-field-lattice Boltzmann simulation of dendrite growth under natural convection in multicomponent superalloy solidification

Cong Yang Qing-Yan Xu Bai-Cheng Liu

Key Laboratory for Advanced Materials Processing Technology (Ministry of Education),School of Materials Science and Engineering,Tsinghua University

作者简介：*Qing-Yan Xu,e-mail:scjxqy@tsinghua.edu.cn;

收稿日期：1 August 2018

基金：financially supported by the National Key Research and Development Program of China (No. 2017YFB0701503);the National Science and Technology Major Project (No.2017ZX04014001);the National Natural Science Foundation of China (No.51374137);

Phase-field-lattice Boltzmann simulation of dendrite growth under natural convection in multicomponent superalloy solidification

Cong Yang Qing-Yan Xu Bai-Cheng Liu

Key Laboratory for Advanced Materials Processing Technology (Ministry of Education),School of Materials Science and Engineering,Tsinghua University

Abstract：

The thermosolutal convection can alter segregation pattern,change dendrite morphology and even cause freckles formation in alloy solidification.In this work,the multiphase-field model was coupled with lattice Boltzmann method to simulate the dendrite growth under melt convection in superalloy solidification.In the isothermal solidification simulations,zero and normal gravitational accelerations were applied to investigate the effects of gravity on the dendrite morphology and the magnitude of melt flow.The solute distribution of each alloy component along with the dendrite tip velocity during solidification was obtained,and the natural convection has been confirmed to affect the microsegregation pattern and the dendrite growth velocity.In the directional solidification simulations,two typical temperature gradients were applied,and the dendrite morphology and fluid velocity in the mushy zone during solidification were analyzed.It is found that the freckles will form when the average fluid velocity in the mushy zone exceeds the withdraw velocity.

Keyword：

Dendrite growth; Natural convection; Phasefield model; Lattice Boltzmann method;

Received： 1 August 2018

1 Introduction

Nickel-based superalloys [ 1, 2, 3] are among the most important high-temperature structure materials in the modern turbine engines.Since the performance of superalloy parts is closely related to the casting microstructure,it is vital to understand the microstructure formation [ 4, 5, 6, 7, 8] during solidification.The thermosolutal convection,which is caused by density inversion along gravity direction,can change the microstructure features,such as dendrite morphology [ 4] ,dendrite spacing [ 5] and microsegregation.Moreover,the freckle defect formation [ 6] in superalloy directional solidification was directly related to the melt convection in the mushy zone.

In recent years,both the experimental and numerical methods have been developed to investigate the dendrite growth under melt convection during alloy solidification.The early studies use the transparent aqueous and organic system to investigate the dendrite growth.With the development of X-ray and synchrotron X-ray techniques,the in situ observation on dendrite growth during solidification becomes possible in lightweight Ga-In [ 9] and AlCu [ 10] alloys.As reported,channel plume flow,dendrite remelting and overgrowth have been discovered and analyzed.However,due to low-attenuation contrast between the solid and melt,the direct observation of dendrite growth and melt flow in superalloy solidification is still impossible.As for the numerical methods,the phase-field method [ 11, 12, 13] has emerged as a powerful tool to simulate dendrite growth in alloy solidification.When coupled with computer coupling of phase diagrams and thermochemistry(CALPHAD)-based thermodynamic database,the multiphase-field model [ 14, 15] is able to simulate dendrite growth in multicomponent super alloy.To model melt convection during solidification,the Navier-Stokes equation must be solved.However,the traditional finite volume method requires huge amount of calculation,which is not suitable for massive parallel implementation.Recently,Sun et al. [ 16, 17, 18] have developed a lattice Boltzmann scheme for simulating melt convection and dendrite growth in alloy solidification.The lattice Boltzmann method is easy to implement and suitable for parallelization.Takaki et al. [ 19, 20] successfully developed the phase-field-lattice Boltzmann method to simulate large-scale dendrite growth with fluid flow in binary alloy solidification.However,the modeling and simulation of complex multicomponent superalloy dendrite growth with melt convection have not been accomplished.

In this study,the multiphase-field model was for the first time coupled with the lattice Boltzmann method.The developed phase-field-lattice Boltzmann model was used to simulate dendrite growth with natural convection during multicomponent superalloy solidification.The realistic superalloy thermodynamic and kinetic data were directly coupled into the multiphase-field model,and a previously developed graphics processing unit (GPU)-based parallel computing scheme [ 21] was employed to accelerate the computation.In the isothermal solidification simulations,zero and normal gravitational accelerations were applied to investigate the effects of gravity on the magnitude of melt convection and the resulting dendrite morphology.In the directional solidification simulations,the effects of temperature gradient on melt convection and dendrite morphology were investigated.The magnitude of melt convection in the mushy zone and microsegregation was obtained and fully analyzed.

2 Methods

2.1 Multiphase-field model

The multiphase-field model [ 11, 12] has been used to simulate the dendrite growth in multicomponent superalloy solidification during solidification.In the multiphase-field model,the scalar fields and were used to represent inpidual phases and concentrations in each phase,respectively.The free energy functional (F) is the integration of local free energy density (f) over the domainΩ,which is given as:

And the local free energy density is the combination of interface free energy density f^intf and chemical free energy density f^chem

whereσ_αβis the interface energy betweenαandβphases in a v phase junction,andη_αβ=ηis the phase interface.The chemical free energy ofαphase can be furtherly rewritten as ,where V_m is molar volume,gαis the molar Gibbs free energy ofαphase,and the latter can be obtained from commercial CALPHAD-type superalloy thermodynamic database.The multiphase-field kinetic equation is developed with the aim of minimizing the free energy function as follows:

where M_αβis the interface mobility.

To reduce the amount of computation,only solid (fcc)and liquid phases are considered in this study.And the solute kinetic equation considering the anti-trapping current [ 13] and the melt convection can be written as:

where u^*is the physical fluid velocity and is the diffusion coefficient of ith component inαphase.

2.2 Lattice Boltzmann method

The lattice Boltzmann method [ 22, 23] has been widely used to calculate the fluid flow for its simplicity and amenability to parallelization.The Navier-Stokes equation was solved using the discrete lattice Boltzmann method with Bhatnagar-Gross-Krook (BGK) scheme [ 22] .The general form of the lattice Boltzmann evolution equation considering external force can be written as:

where fk(x,t) and are the local and equilibrium distribution function,F_k(x,t) is the external force,τis the LBM (Lattice Boltzmann Method) relaxation time,δt is the time step and e_k is the lattice velocity components.The equilibrium distribution function for solving Navier-Stokes equation can be express as:

where w_k is the weight factor,ρ(x,t) is the lattice density and e_s represents the lattice speed of sound.The external force F_k(x,t) contains two parts:dissipative drag force(G_D) and the buoyancy force (G_B)

The drag force G_D(x,t)=-(2ρvh/η²)φ²u [ 20, 24] is imposed to maintain the no-slip boundary condition near the liquid and solid interface,where v is the liquid kinetic viscosity and h is a constant which equals to 2.757 [ 24] .The buoyancy force is related to local temperature and solute concentration,where g is the gravity,β_T andβ_c are the thermal and solutal expansion coefficients,and T₀and are the reference temperature and reference solute concentrations,respectively.

3 Simulations

3.1 Alloy and simulation parameters

In this work,a second-generation nickel-based superalloy CMSX-4 was investigated.The superalloy chemical compositions and the solutal expansion coefficient of each alloy component are listed in Table 1 [ 25, 26] .To reduce the amount of computation,the alloy components Hf and Mo are removed from the computation due to the low concentration,and the model alloy was used in the following simulations.

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Table 1 Chemical compositions of CMSX-4 and model alloys and solutal expansion coefficient

The extrapolation method [ 11] was used to solve the quasi-equilibrium condition and the chemical driving force in multiphase-field model.The alloy thermodynamic and kinetic data were obtained from PANDAT software.The thermodynamic data contain all the thermodynamic factors and phase concentrations under equilibrium state.The kinetic data include the solid and liquid diffusion coefficients of each alloy component.The equilibrium concentrations of the model alloy component during solidification range from 1656 to 1573 K are shown in Fig.1.The other thermodynamic data such as molar Gibbs free energy and its partial derivatives are stored in the database with a temperature interval of 0.2 K.In phase-field and lattice Boltzmann simulations,the lattice size was set to 1μm,and the relaxation time was set toτ=1.0 for accuracy.The scaling constant (S_f) was used to expand the time step by scaling both kinematic viscosity and gravitational acceleration while keeping the Rayleigh number constant.The other parameters used in simulations are listed in Table 2 [ 27, 28] .

3.2 Solidification conditions

The dendrite growth and microsegregation of superalloy in isothermal and directional solidification conditions were investigated.Table 3 shows the solidification conditions used in simulations.In isothermal solidification simulations,to investigate the effect of gravity on the natural convection and dendrite morphology,the gravitational acceleration was set to 0 and-9.8 m·s^-2 in Cases 1 and 2,respectively.A seed was initially placed at the center of the computational domain.Zero-flux boundary condition was applied in isothermal solidification.In the directional solidification simulations,two typical temperature gradients were settled.The former solidification condition of G=2.0 K·mm^-1 was reported to have freckles formation [ 6] ,while the latter one of G=10.0 K·mm^-1 was frecklefree.The domain size was set to 3072×3072 grids in isothermal solidification simulations and 4096×4096grids in directional solidification simulations.At the initial time,the bottom of the computational domain was set with a slice of solid phase,and the dendrites were developed from the planar interface by Mullins-Sekerka instability.As for the boundary conditions,zero-flux boundary condition and periodic boundary condition were applied to xdirection and y-direction,respectively.

Fig.1 Equilibrium concentrations of model alloy during solidifica-tion ranging from 1656 to 1573 K [15] (noting that components Ti and W are not shown here)

4 Results and discussion

4.1 Isothermal solidification

Figure 2 shows the Al concentration distributions and fluid velocity profile under the isothermal solidification condition with gravitational acceleration of g=0 m·s^-2 and g=-9.8 m·s^-2.Under zero gravity,the dendrite develops symmetry primary and secondary arms.The Al concentration also exhibits symmetry distribution,and no fluid flow is observed because the buoyancy which drives the natural convection equals to zero.When the gravity is applied,the solute-rich low-density liquid near the dendrite becomes the driving force of the melt flow.A plume forms above the growing dendrite,and two vortexes can be observed in the upper left and upper right sides of the dendrite.Owing to the solute segregation in front of the dendrite,the growth of the upper,left and right sides of the dendrite arms is inhibited,while the growth of dendrite arm in the lower side is enhanced due to low concentration of the solute.The magnitude of the fluid velocity is about50μm·s^-1.The limited computational domain may restrict the development of the natural convection,as reported by Takaki et al. [ 20] .

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Table 2 Parameters in phase-field-lattice Boltzmann simulations

The concentration distributions of the alloy components Al,Co,Cr,Re,Ta and W are shown in Fig.3.The components Al,Ta and Ti segregate to the liquid,while components Co,Cr,Re and W segregate to the solid dendrite.The simulated segregation pattern is generally consistent with previous experimental findings [ 29] .In Eq.(7),the buoyancy force is related to the segregation behavior and solutal expansion coefficient of each alloy component.According to Table 1 and the simulation results,the components Al,Re,Ti and W can promote the natural convection,while Co,Cr and Ta will inhibit the natural convection.

The dendrite tip velocities along y-direction were recorded during simulations,and the results are shown in Fig.4a.Under normal gravity ofg=-9.8 m·s^-2,the dendrite tip exhibits the lowest growing velocity due to the high-concentration solute brought by the natural convection.Under zero gravity and inverse gravity,the dendrite tip velocity becomes much larger.The simulated Al concentration profile along y-direction is shown in Fig.4b.The Al concentration in the melt at the solidification front is the highest when g=-9.8 m·s^-2 and the lowest when g=9.8 m·s^-2.These results are consistent with the previous analysis.

4.2 Directional solidification

Most of the modern turbine blades are produced by the directional solidification techniques.The processing conditions,especially the temperature gradient,can strongly affect the final casting microstructure [ 8] in superalloy directional solidification.The coarse dendrite and even freckle defects are observed under low-temperature gradients,and higher-temperature gradient is favored for producing superalloy parts with reduced primary dendrite armspacing.To investigate the effects of temperature gradient on dendrite growth and natural convection,the directional solidification cases in Table 3 were simulated,where V_p is the pulling velocity.

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Table 3 Solidification conditions in simulations

Fig.2 Al concentration distributions and fluid velocity profile under isothermal solidification condition with cooling rate of-1 K·s^-1 and gravitational acceleration of a-c g=0 m·s^-2,d-f g=-9.8 m-s^-2 at different solidification time:a,dt=12.4 s;b,et=16.1 s;c,ft=19.8 s

Fig.3 Concentration distributions of components a Al,b Co,c Cr,d Re,e Ta and f W under isothermal solidification condition with g=-9.8 m·s^-2 at solidification time of 19.78 s (noting that concentration distribution of Ti is not shown here)

Fig.4 a Dendrite tip velocity and b A1 concentration profile under solidification condition of-1 K·s^-1 and gravitational acceleration of 0,-9.8,9.8 m·s^-2

Fig.5 Al concentration distribution and fluid velocity profile under directional solidification with withdraw velocity of V_p=50μm·s^-1 and temperature gradients of a-c G=2 K·mm^-1 and d-f G=10 K·mm^-1 at different solidification time:a t=65.9 s,bt=123.6 s,ct=250.0 s,d t=24.7 s,et=49.4 s and f t=250.0 s

Figure 5 shows the simulated Al concentration distributions and fluid velocity profile at different solidification times under the two directional solidification cases.At the initial growth time,as the undercooling increases,the planar interface becomes instable and small perturbations emerge.The solute-rich liquid gradually accumulates ahead of the solid phase,and natural convection starts to develop under lower-temperature gradient,as shown in Fig.5a,d.After that,a large number of dendrite arms are quickly eliminated through competitive growth.The dendrite morphology is coarser,and the primary dendrite spacing is lager when the temperature gradient is lower.The coarse dendrites lead to a higher permeability in the mushy zone,which is favorable for the rising of the lowdensity liquid,and the magnitude of the fluid flow is much larger in Fig.5b compared with that in Fig.5e.The dendrite morphologies under steady state in the two solidification conditions are shown in Fig.5c,f.Irregular secondary and high-order dendrite arms can be found at lower-temperature gradients,and the solidification front exhibits erratic shape.These irregular high-order dendrite arms may act as potential nucleus of the freckles if they are melted down by the rising solute-rich liquid.

Fig.6 a Average mushy zone fluid velocity and b dendrite tip velocity during solidification

Fig.7 a Concentration profile of components Al,Co,Cr,Re at root of dendrite arm and b average partition coefficients of each alloy component under two solidification conditions of 10 and 2 K·mm^-1

Because the fluid flow in the mushy zone directly interacts with the dendrite,the average mushy zone fluid velocity and dendrite tip velocity during solidification are recorded and shown in Fig.6.Two distinct growth stages can be observed:the initial growth stage and the steady growth stage.Both the average fluid velocity and dendrite tip velocity increase at the initial growth stage and then stabilize in the steady growth stage.The fluid velocity and the dendrite tip velocity fluctuate under temperature gradient of 2 K·mm^-1,and the maximum mushy zone fluid velocity exceeds 50μm·s^-1.The experimental results reveal the existence of freckles under temperature gradient of 2 K·mm^-1,and the current study indicates that the freckles will form when the average fluid velocity exceeds the withdraw velocity,which is also the nominal dendrite growth velocity.The Flemings'criterion [ 30] also supports our findings in predicting freckle formation.

The concentration profile and the average partition coefficients of components Al,Co,Cr and Re at the root of the dendrite arm in the two directional solidification conditions are shown in Fig.7.The average partition coefficients are obtained by piding the average concentration in the solid and the average concentration in the liquid at the dendrite root.According to the simulation results,the segregation is severer under high-temperature gradient,which is favorable for the development of natural convection.However,due to the low permeability in the mushy zone,the intensity of natural convection in hightemperature gradient is still lower than that in low-temperature gradient.

5 Conclusion

The multiphase-field model coupled with lattice Boltzmann method was developed to simulate the dendrite growth under natural convection in superalloy solidification.Large-scale simulations were performed on the GPU platform.The main conclusions are listed as follows.

In isothermal solidification,the natural convection can break the symmetry of the solute distribution and thus change the dendrite morphology.The rising solute-rich plume is found to inhibit the dendrite growth.The components Al,Re,Ti and W can promote the natural convection,while Co,Cr and Ta will inhibit the natural convection.In directional solidification,the magnitude of melt convection is much larger and fluctuates under lowertemperature gradient.The dendrite morphology and the dendrite tip velocity are significantly influenced by the melt convection under temperature gradient of 2 K·mm^-1.According to the simulation results,the freckles are expected to form when the mushy zone average fluid velocity exceeds the withdraw velocity.

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