Phase-field simulations of forced flow effect on dendritic growth perpendicular to flow

The effect of supercooled melt forced laminar flow at low Reynolds Number on dendritic growth perpendicular to melt flow direction was investigated with the phase-field method by incorporating melt convection and thermal noise under non-isothermal condition. By taking the dendritic growth of high pure succinonitrile （SCN） supercooled melt as an example, side-branching shape difference of melts with flow and without flow was analyzed. Relationships among supercooled melt inflow velocity, deflexion angle of dendritic arm and dendritic tip growth velocity were studied. Results show that the melt inflow velocity has few effects on the dendritic tip growth velocity. A formula of relationship between the velocity of the melt in front of primary dendritic tip and the dendritic growth time was deduced, and the calculated result was in quantitative agreement with the simulation result.

Phase-field modeling of dendritic growth under forced flow based on adaptive finite element method

A mathematical model combined projection algorithm with phase-field method was applied. The adaptive finite element method was adopted to solve the model based on the non-uniform grid, and the behavior of dendritic growth was simulated from undercooled nickel melt under the forced flow. The simulation results show that the asymmetry behavior of the dendritic growth is caused by the forced flow. When the flow velocity is less than the critical value, the asymmetry of dendrite is little influenced by the forced flow. Once the flow velocity reaches or exceeds the critical value, the controlling factor of dendrite growth gradually changes from thermal diffusion to convection. With the increase of the flow velocity, the deflection angle towards upstream direction of the primary dendrite stem becomes larger. The effect of the dendrite growth on the flow field of the melt is apparent. With the increase of the dendrite size, the vortex is present in the downstream regions, and the vortex region is gradually enlarged. Dendrite tips appear to remelt. In addition, the adaptive finite element method can reduce CPU running time by one order of magnitude compared with uniform grid method, and the speed-up ratio is proportional to the size of computational domain.

Modeling of equiaxed and columnar dendritic growth of magnesium alloy

Based on the cellular automaton （CA） method, a numerical model was developed to simulate the dendritic growth of magnesium alloy with HCP crystal structure. The growth kinetics was calculated from the complete solution of the transport equations. By defining a special neighborhood configuration with the square CA cell, and using a set of capturing rules which were proposed by BELTRAN-SANCHEZ and STEFANESCU for the dendritic growth of cubic crystal metals during solidification, modeling of dendritic growth of magnesium alloy with different growth orientations was achieved. Simulation of equiaxed dendritic growth and columnar dendritic growth under directional solidification was carried out, and validation was performed by comparing the simulated results with the experimental results and those in the previously published works.

Free dendritic growth model incorporating interfacial nonisosolutal nature due to normal velocity variation

Considering both the effects of the interfacial normal velocity dependence of solute segregation and the local nonequilibrium solute diffusion,an extended free dendritic growth model was analyzed.Compared with the predictions from the dendritic model with isosolutal interface assumption,the transition from solutal dendrite to thermal dendrite moves to higher undercoolings,i.e.,the region of undercoolings with solute controlled growth is extended.At high undercoolings,the transition from the mainly thermal-controlled growth to the purely thermal-controlled growth is not sharp as predicted by the isosolute model,but occurs in a range of undercooling,due to both the effects of the interfacial normal velocity dependence of solute segregation and the local nonequilibrium solute diffusion.Model test indicates that the present model can give a satisfactory agreement with the available experimental data for the Ni-0.7% B（mole fraction） alloy.

Phase-field modeling of dendritic growth of magnesium alloys with a parallel-adaptive mesh refinement algorithm

A two-dimensional phase field(PF)model was developed to simulate the dendritic solidification in magnesium alloy with hcp crystal structure.By applying a parallel-adaptive mesh refinement(Para-AMR)algorithm,the computational efficiency of the numerical model was greatly improved.Based on the PF model,a series of simulation cases were conducted and the results showed that the anisotropy coefficient and coupling coefficient had a great influence on the dendritic morphology of magnesium alloy.The dendritic growth kinetics was determined by the undercooling and equilibrium solute partition coefficient.A significant finding is acquired that with a large undercooling,the maximum solute concentration is located on both sides of the dendrite tip in the liquid,whereas the maximum solute concentration gradient is located right ahead of the dendrite tip in the liquid.The dendrite tip growth velocity decreases with the increase of the equilibrium solute partition coefficient,while the variation trend of the dendrite tip radius is the opposite.Quantitative analysis was carried out relating to the dendritic morphology and growth kinetics,and the simulated results are consistent with the theoretical models proposed in the previously published works.

Phase-field simulation of dendritic growth in a binary alloy with thermodynamics data

This paper simulates the dendrite growth process during non-isothermal solidification in the Al-Cu binary alloy by using the phase-field model. The heat transfer equation is solved simultaneously. The thermodynamic and kinetic parameters are directly obtained from existing database by using the Calculation of Phase Diagram （CALPHAD） method. The effects of the latent heat and undercooling on the dendrite growth, solute and temperature profile during the solidification of binary alloy are investigated. The results indicate that the dendrite growing morphologies could be simulated realistically by linking the phase-field method to CALPHAD. The secondary arms of solidification dendritic are better developed with the increase of undercooling. Correspondingly, the tip speed and the solute segregation in solid-liquid interface increase, but the tip radius decreases.

Numerical simulation on dendritic growth of Al-Cu alloy under convection based on the cellular automaton lattice Boltzmann method

A numerical model is developed by coupling the cellular automaton(CA)method and the lattice Boltzmann method(LBM)to simulate the dendritic growth of Al-Cu alloy in both two and three dimensions.An improved decentered square algorithm is proposed to overcome the artificial anisotropy induced by the CA cells and to realize simulation of dendritic growth with arbitrary orientations.Based on the established CA-LBM model,effects of forced convection and gravity-driven natural convection on dendritic growth are studied.The simulation results show that the blocking effect of dendrites on melt flow is advanced with a larger number of seeds.The competitive growth of the converging columnar dendrites is determined by the interaction between heat flow and forced convection.Gravity-driven natural convection leads to highly asymmetric growth of equiaxed dendrites.With sinking downwards of the heavy solute,chimney-like or mushroom-like solute plumes are formed in the melt in front of the columnar dendrites when they grow along the gravitational direction.More details on dendritic growth of Al-Cu alloy under convection are revealed by 3D simulations.

Simulation of Dendritic Growth with Melt Convection in Solidification of Ternary Alloys

A cellular automaton-lattice Boltzmann coupled model is extended to study the dendritic growth with melt convection in the solidification of ternary alloys. With a CALPHAD-based phase equilibrium engine, the effects of melt convection, solutal diffusion, interface curvature and preferred growth orientation are incorporated into the coupled model. After model validation, the multi dendritic growth of the Al-4.0 wt%Cu-1.0 wt%Mg alloy is simulated under the conditions of pure diffusion and melt convection. The result shows that the dendritic growth behavior, the final microstructure and microsegregation are significantly influenced by melt convection in the solidification.

Three-dimensional interfacial wave theory of dendritic growth: (Ⅰ). multiple variables expansion solutions

Dendritic pattern formation at the interface between liquid and solid is a commonly observed phenomenon in crystal growth and solidification process. The theoretical investigation of dendritic growth is one of the most profound and highly challenging subjects in the broad areas of interfacial pattern formation, condensed matter physics and materials science, preoccupying many researchers from various areas. Some longstanding key issues on this subject finally gained a breakthrough in the late of last century, via the ＇Interracial Wave （IFW） Theory＇ on the ground of systematical global stability analysis of the basic state of dendritic growth. The original form of the IFW theory mainly focus on the investigation of various axi-symmetric unsteady perturbed modes solutions around the axi-symmetrie basic state of system of dendritic growth. In reality, the system may allow various non-axi-symmetric, unsteady perturbed states. Whether or not the system of dendritic growth allows some growing non-axi-symmetric modes？ Will the stationary dendritic pattern be destroyed by some of such non-axi- symmetric modes？ Or, in one word, what is the stability property of the system, once the non-axi-symmetric modes can be evoked？ The answers for these questions are important for the solid foundation of IFW theory. The present work attempts to settle down these issues and develop a three-dimensional （3D） interfacial wave theory of dendritic growth. Our investigations verify that dendritic growth indeed allows a discrete set of non-axi-symmetric unstable global wave modes, which gives rise to a set of multiple arms spiral waves propagating along the Ivantsov＇s paraboloid.

Three-dimensional interfacial wave theory of dendritic growth: (Ⅱ). non-axi-symmetric global wave modes and selection criterion of pattern formation

This paper is ttie continuation of part （Ⅰ）, which completes the derivations of the 3D global wave modes solutions, yields the stability criterion and, on the basis of the results obtained, demonstrates the selection criterion of pattern formation.

Simulation of dendritic growth of magnesium al oys with fluid flow

Fluid flow has a significant impact on the microstructure evolution of alloys during solidification. Based on the previous work relating simulation of the dendritic growth of magnesium alloys with hcp(hexagonal closepacked) structure, an extension was made to the formerly established CA(cellular automaton) model with the purpose of studying the effect of fluid flow on the dendritic growth of magnesium alloys. The modified projection method was used to solve the transport equations of flow field. By coupling the flow field with the solute field, simulation results of equiaxed and columnar dendritic growth of magnesium alloys with fluid flow were achieved. The simulated results were quantitatively compared with those without fluid flow. Moreover, a comparison was also made between the present work and previous works conducted by others. It can be concluded that a deep understanding of the dendritic growth of magnesium alloys with fluid flow can be obtained by applying the present numerical model.

Effect of buoyancy-driven convection on steady state dendritic growth in a binary alloy

The effect of buoyancy-driven convection on the steady state dendritic growth in an undercooled binary alloy is studied. For the case of the moderate modified Grashof number, the uniformly valid asymptotic solution in the entire region of space is obtained by means of the matched asymptotic expansion method. The analytical results show that the buoyancy- driven convection has a significant effect on the needle-like interface of dendritic growth. Due to the buoyancy-driven convection, the needle-like interface shape of the crystal is changed. When the Peclet number that is not affected by the buoyant flow is less than a certain critical value, the interface shape of the dendrite becomes thinner as the Grashof number increases; when it is larger than the critical value, the interface shape becomes fatter as the Grashof number increases. In the undercooled binary alloy the morphology number plays an active role in the interface shape and leads to the buoyancy effect that is different from the situation for the pure melt. The smaller the morphology number is, the more significant change the interface shape has. As the Peclet number further increases, the effect of buoyancy on the interface diminishes eventually.

A Diffuse-Interface Lattice Boltzmann Method for the Dendritic Growth with Thermosolutal Convection

In this work,we proposed a diffuse-interface model for the dendritic growth with thermosolutal convection.In this model,the sharp boundary between the fluid and solid dendrite is firstly replaced by a thin but nonzero thickness diffuse interface,which is described by the order parameter,and the diffuse-interface based governing equations for the dendritic growth are presented.To solve the model for the dendritic growth with thermosolutal convection,we also developed a diffuse-interface multirelaxation-time lattice Boltzmann(LB)method.In this method,the order parameter in the phase-field equation is combined into the force caused by the fluid-solid interaction,and the treatment on the complex fluid-solid interface can be avoided.In addition,four LB models are designed for the phase-field equation,concentration equation,temperature equation and the Navier-Stokes equations in a unified framework.Finally,we performed some simulations of the dendritic growth to test the present diffuse-interface LB method,and found that the numerical results are in good agreements with some previous works.

Growth and inhibition of zinc anode dendrites in Zn-air batteries:Model and experiment

Zinc(Zn)-air batteries are widely used in secondary battery research owing to their high theoretical energy density,good electrochemical reversibility,stable discharge performance,and low cost of the anode active material Zn.However,the Zn anode also leads to many challenges,including dendrite growth,deformation,and hydrogen precipitation self-corrosion.In this context,Zn dendrite growth has a greater impact on the cycle lives.In this dissertation,a dendrite growth model for a Zn-air battery was established based on electrochemical phase field theory,and the effects of the charging time,anisotropy strength,and electrolyte temperature on the morphology and growth height of Zn dendrites were studied.A series of experiments was designed with different gradient influencing factors in subsequent experiments to verify the theoretical simulations,including elevated electrolyte temperatures,flowing electrolytes,and pulsed charging.The simulation results show that the growth of Zn dendrites is controlled mainly by diffusion and mass transfer processes,whereas the electrolyte temperature,flow rate,and interfacial energy anisotropy intensity are the main factors.The experimental results show that an optimal electrolyte temperature of 343.15 K,an optimal electrolyte flow rate of 40 ml·min^(-1),and an effective pulse charging mode.

Phase-field lattice-Boltzmann study on fully coupled thermal-solute-convection dendrite growth of Al-Cu alloy

Dendrite growth is a complex liquid-solid phase transition process involving multiple physical factors.A phase-field lattice-Boltzmann method was developed to simulate the two-and three-dimension dendrite growth of Al-Cu alloy.The effect of fully coupled thermal-solute-convection interaction on the dendrite growth was investigated by incorporating a parallel-adaptive mesh refinement algorithm into the numerical model.By accurately reproducing the latent heat release,solute diffusion and convective transport behaviors at the liquidsolid interface,the interaction mechanism among thermal-solute-convection transport as well as their coupling effects on the dendrite growth dynamics were discussed.The simulation results show that the release of latent heat slows down the dendrite growth rate,and both natural and forced convection disrupt the symmetrical growth of dendrites.Their combination makes the growth of dendrites more complex,capturing important physical aspects such as recalescence,dendrite tip splitting,dendrite tilting,dendrite remelting,and solute plume in the simulation case.Based on the robustness and powerful ability of the numerical model,the formation mechanisms of these physical aspects were revealed.

A Modified Cellular Automaton Model for the Quantitative Prediction of Equiaxed and Columnar Dendritic Growth

Since the characteristic of dendrite is an important factor determining the performance of castings, a twodimensional cellular automaton model with decentered square algorithm is developed for quantitatively predicting the dendritic growth during solidification process. The growth kinetics of solid/liquid interface are determined by the local equilibrium composition and local actual liquid composition, and the calculation of the solid fraction increment is based on these two compositions to avoid the solution of growth velocity. In order to validate the developed model, quantitative simulations of steady-state dendritic features over a range of undercooling was performed and the results exhibited good agreement with the predictions of LGK（Liptone Glicksman-Kurz） model. Meanwhile, it is demonstrated that the proposed model can be applied to simulate multiple equiaxed dendritic growth, as well as columnar dendritic growth with or without equiaxed grain formation in directional solidification of AleC u alloys. It has been shown that the model is able to simulate the growth process of multi-dendrites with various preferential orientations and can reproduce a wide range of complex dendritic growth phenomena such as nucleation, coarsening of dendrite arms, side branching in dendritic morphologies, competitive growth as well as the interaction among surrounding dendrites.

Study on Dendritic Growth in Pressurized Solidification of Mg–Al Alloy Using Phase Field Simulation

A multi-grain phase field model coupled with thermodynamic calculation was adopted to describe the dendritic growth in pressurized solidification of Mg-A1 alloy during squeeze casting, in which the effects of the pressure on the Gibbs free energy and chemical potential of solid and liquid phases, the solute diffusion coefficient, and the solute partition coefficient were considered. The individual effect of solute diffusion coefficient, and the Gibbs free energy on the dendritic growth was studied. With the compar- ison of the dendritic growth under atmospheric and elevated pressures, the effect of pressure on the microstructure evolution was discussed. The results showed that the grains are refined, the dendritic growth rate tends to increase and the secondary dendrite arms are more developed as the pressure is increased from 0.1 to 100 MPa, which showed a good agreement with the experimental results of direct squeeze casting of Mg-AI alloy. As the pressure increases, the largest dendritic growth rate can be obtained under the pressure between 200 and 250 MPa, while the growth rate decreases with a further increase of pressure.

A three-dimensional cellular automaton model for simulation of dendritic growth of magnesium alloy

A numerical model based on the cellular automaton method for the three-dimensional simulation of dendritic growth of magnesium alloy was developed. The growth ki- netics was calculated from the complete solution of the transport equations. By con- structing a three-dimensional anisotropy model with the cubic CA cells, simulation of dendritic growth of magnesium alloy with six-fold symmetry in the basal plane was achieved. The model was applied to simulate the equiaxed dendritic growth and columnar dendritic growth under directional solidification, and its capability was addressed by comparing the simulated results to experimental results and those in the previously published works. Meanwhile, the three-dimensional simulated results were also compared with that of in two dimensions, offering a deep insight into the microstructure formation of magnesium alloy during solidification.

Dendritic Growth Pattern and Dendritic Network Distortion in the Platform of a Ni-based Superalloy

The pattern of dendritic growth and distortion of dendritic network in the platform have been investigated by one mold casting with different platform length during directional solidification. As the platform length elongates, the symmetry of dendritic growth along left and right edges gradually worsens in platform base. While the platform length reaches 14 mm, the distortion of dendritic network is first observed in outward platform. It is found that the distortion of dendritic network along platform inside is more serious than that along platform edges. Both [001] deviation and accumulated misorientation along platform inside, up to 9~ and 16.3~, respectively, are far greeter than those along left-outward-right edges. The deformation of dendritic network in a platform may be caused by the asymmetry of the solidification front at the mush zone.

Dendritic growth and solute trapping in rapidly solidified Cu-based alloys

Rapid solidification of binary Cu-22%Sn peritectic alloys and Cu-5%Sn-5%Ni-5%Ag quaternary alloys was accomplished by glass fluxing, drop tube and melt spinning methods. The undercooled, by glass fluxing method, Cu-22%Sn peritectic alloy was composed of a（Cu） and δ（Cu41Snll） phases. If rapidly solidified in a drop tube, the alloy phase constitution changed from α（Cu） and δ（Cu41Sn11） phases into a single supersaturated （Cu） phase with the reducing of droplet diameter, and the maximum solubility of Sn in （Cu） phase extended to 22%. The Cu-5%Sn-5%Ni-5%Ag quaternary alloy was composed of （Cu） and （Ag） phases under the containerless processing condition in a drop tube, and the solute microsegregation of （Cu） phase was obvious. When the Cu-5%Sn-5%Ni-5%Ag quaternary alloy was solidified by melt spinning method, microsegregation was suppressed and solute trapping occurred. The experimental results show that the microstructures of primary （Cu） phase in the two alloys transfer from coarse dendrites into equiaxed grains with the increase of cooling rate and undercooling, which is accompanied by the grain refinement effect.