Dendritic Growth Of Binary Alloy Research Articles

A FFT-based phase-field framework for simulating dendritic growth in binary alloy

In the present study, a Fourier pseudo-spectral-based, phase-field framework is developed to simulate the binary alloy solidification using fixed grids. The motivation behind this proposition of a new model towards overcoming existing limitations is two-fold: firstly, to create a fully validated high-order phase-field model that closely aligns with LKT predictions of tip kinetics across various undercoolings and compositions, and secondly, to achieve accurate simulations using fixed Cartesian meshes with a grid size of order more than unity. In the Fourier pseudo-spectral method, the nonlinear terms of the PF equations are de-aliased using zero padding and high-order Fourier smoothing exponential filters. Accurate growth kinetics during binary alloy solidification are observed despite employing fixed mesh sizes, even when the ratio of grid size to diffuse interface thickness is 1.42. A hybrid, integrating factor (IF)-based, strongly stable third-order Runge-Kutta method (SSPRK3) is implemented to obtain improved temporal stability at high Lewis numbers. A novel scaling relationship between dimensionless tip velocity and undercooling is obtained from the growth of a four-arm equiaxed dendrite at different levels of undercooling. The growth of several randomly oriented dendrites is also accurately simulated without using any mesh refinement schemes. Likewise, the tip velocity closely matched the LKT predictions at dilute concentrations at the boundary. Furthermore, the effects of the coupling parameter and the anti-trapping term on dendritic growth kinetics are explored. Overall, the proposed FFT-based framework is expected to capture the crystals' global chemical wave features precisely with fewer points per wavelength (PPW) and has the potential to be scaled up for large-scale simulations.

Multi-GPU implementation of a cellular automaton model for dendritic growth of binary alloy

A parallel three-dimensional (3D) cellular automaton (CA) model is developed for dendritic growth simulation during solidification of binary alloy. The CA model is implemented using the graphic processing unit (GPU). Compute Unified Device Architecture (CUDA) is combined with Message Passing Interface (MPI) to perform large-scale simulation on the multi-GPU architecture. A CUDA-Aware OpenMPI is used to reduce the communication cost between different GPU cards. The accuracy and the speedup ratio of the CA model are investigated. First, the simulated steady-state tip velocity under different undercooling is compared with the LGK analytical solution. Then, dendritic growth during directional solidification and equiaxed multi-dendrite growth are simulated. Dendritic growth simulation with a domain size up to 453 million grids is performed. Finally, the speedup ratio of the CA model using four GPUs is compared with that using four CPU processes via MPI, which leads to a maximal speedup ratio of 152.19. The developed CA model achieves high parallel performance compared with the MPI-based parallel model. The developed CA model is capable of predicting the columnar to equiaxed transition (CET) event during directional solidification. The results indicate that increasing cooling rate promotes the occurrence of the equiaxed dendrites in the melt ahead of the solidification front, and equiaxed dendrites are prone to nucleate in the columnar dendrite groove zone when a middle cooling rate is applied.

Modelling of dendritic growth during alloy solidification under natural convection

A two-dimensional (2D) lattice Boltzmann method (LBM)-cellular automaton model is presented to investigate the dendritic growth of binary alloys in the presence of natural convection. The kinetic-based LBM is adopted to calculate the transport phenomena by the evolution of distribution functions of moving pseudo-particles. To numerically solve natural convection thermal and solute transport simultaneously, three sets of distribution functions are employed in conjunction with the lattice Bhatnagar–Gross–Krook scheme. Based on the LBM calculated local temperature and concentration at the solid/liquid interface, the kinetics of dendritic growth is determined according to a local solute equilibrium approach. Thus, the physics of a complete time-dependent interaction of natural convection, thermal and solutal transport, and dendritic growth during alloy solidification is embedded in the model. Model validation is performed by comparing the simulated results with literature data and analytical predictions. The model is applied to simulate dendritic growth in binary alloys under the influence of natural convection. The effects of Rayleigh numbers and initial undercooling on dendrite growth are investigated. The results show that natural buoyancy flow, induced by thermal and solutal gradients under gravity, transports the heat and solute from the lower region to the upper region. The dendritic growth is thus accelerated in the downward direction, whereas it is inhibited in the upward direction, yielding asymmetrical dendrite patterns. Increasing the Rayleigh number and undercooling will enhance and reduce, respectively, the influence of natural flow on the dendritic growth.

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.

Quantitative phase-field modeling of nonisothermal solidification in dilute multicomponent alloys with arbitrary diffusivities

A quantitative phase-field model is developed for simulating microstructural pattern formation in nonisothermal solidification in dilute multicomponent alloys with arbitrary thermal and solutal diffusivities. By performing the matched asymptotic analysis, it is shown that the present model with antitrapping current terms reproduces the free-boundary problem of interest in the thin-interface limit. Convergence of the simulation outcome with decreasing the interface thickness is demonstrated for nonisothermal free dendritic growth in binary alloys and isothermal and nonisothermal free dendritic growth in a ternary alloy.

Phase-Field Simulation of Solidification Double Dendritic Growth of Binary Alloy in the Forced Flow

We study the effect of force convection and temperature on the double dendrite growth during the solidification of binary alloy using a phase-field model. The mass and momentum conservation equations are solved using the Simple algorithm, and the thermal governing equation is numerically solved using an alternating implicit finite difference method. The results indicate that dendritic grows unsymmetrically under a forced flow, the growth velocity of the upstream tip is faster than the downstream tip. The downstream tip of the first dendrite and the upstream tip of the second dendrite are influenced each other, the upstream tip of the second dendrite will Coarsen, and the concentration at the boundary between them is the highest. Moreover, the interaction between the two dendrites is more and more obvious with the increasing of the temperature.

Phase-field modeling of free dendritic growth in binary alloy under forced flow

A phase-field model (PFM) coupling with phase field, flow field and diffuse equation is presented for simulating isothermal dendrite growth of a nickel-copper alloy under a forced flow. Based on the finite difference method with uniform grid, the C programming code is implemented to complete the phase-field simulations. The simulation results indicate that the interfacial morphology, the symmetry of dendrite formation, the tip growth velocity and the concentration distribution are strongly influenced by the fluid flow.

Influence of Anisotropy on Dendritic Growth in Binary Alloy with Phase–Field Simulation

Some computational results on dendritic growth in binary alloy are obtained by using a phase–field model coupled the solute gradient term. The effect of crystalline anisotropy on the morphological formation, tip steady state and the solute partition is investigated for different dendrites. The interface formation and tip steady state are affected evidently with increase in anisotropy for ‹100› dendrite growth, but the solute partition coefficient is not significantly influenced. For ‹110› preferred growth directions, when the anisotropy strength is lower than the critical value, the tip velocity of [110] direction is lower than [101] and [011] directions. As the anisotropy strength crosses the critical value, the tip velocity of [110] direction increases suddenly, larger than the tip velocity of [101] and [011] directions, showing strong solute trapping.

An enthalpy method for modeling dendritic growth in a binary alloy

An enthalpy fixed grid method is developed for modeling dendritic growth in an under-cooled binary alloy. The proposed numerical method couples explicit finite difference solutions of equations expressing the conservation of enthalpy and solute to an iterative procedure that enforces node by node consistency between enthalpy, solute, liquid fraction, and interface under-cooling. Calculations made with the scheme, are consistent with previously reported work, agree well with limit analytical solutions, approach the correct steady-state tip operating conditions, show grid size independence, are relatively free of grid anisotropy, and can be obtained with a low CPU cost.

A pseudo-front tracking technique for the modelling of solidification microstructures in multi-component alloys

A two-dimensional model for the simulation of microstructure formation during solidification in multi-component systems has been developed. The model is based on a new pseudo-front tracking technique for the calculation of the evolution of interfaces that are governed by solute diffusion and the Gibbs–Thomson effect. The diffusion equations are solved in the primary solid phase and in the liquid using an explicit finite volume method formulated for a regular hexagonal grid. Volume elements located in the liquid phase undergo a transition to interfacial (or mushy) cells before being incorporated in the solid phase. This layer of interfacial elements, which always separates the solid from the liquid sub-domains, permits to handle the displacement of the interface in agreement with the flux condition at the interface. The interface curvature is obtained from the field of the signed distance to the interface, as reconstructed with a PLIC (piecewise linear interface calculation) technique. The concentrations at the solid–liquid interface are calculated using thermodynamic data provided by the phase diagram software Thermo-Calc [Sundman et al. CALPHAD 1987;9:153] . Different coupling strategies between the microstructure model and Thermo-Calc have been developed, in particular a computationally-efficient direct coupling using the TQ-interface of Thermo-Calc. After testing the accuracy of the model with respect to curvature calculation, comparisons are made with predictions obtained with the marginal stability theory, a one-dimensional front-tracking method and two-dimensional phase-field simulations of dendritic growth in binary alloys. The model is then used to describe the formation of several grains in an Al–1%Mg–1%Si alloy, as a function of the heat extraction rate and inoculation conditions. It is shown that the model is capable of reproducing the transition between globular and dendritic morphologies.

Time Resolved X-Ray Imaging of Dendritic Growth in Binary Alloys

Direct beam x-ray imaging, with intense coherent monochromatic synchrotron radiation and a fast readout low noise detector has been used for in situ studies of cellular and dendritic solidification of Sn-Pb alloys. Temporal and spatial resolutions down to $\ensuremath{\sim}0.7\mathrm{s}$ and $\ensuremath{\sim}2.5\ensuremath{\mu}\mathrm{m}$, respectively, with a field of view up to $1{\mathrm{mm}}^{2}$ were obtained. The collected images display features from both phase and amplitude contrast which, combined with the attainable temporal resolution, provide a method with the potential to uncover dynamic processes in nonequilibrium solidification.

Phase diagram effects in phase field models of dendritic growth in binary alloys

Phase field models offer the prospect of being able to perform realistic numerical experiments on dendritic growth in binary alloys. Previous work demonstrated this potential for a nickel-copper system. Simulations were carried out to investigate the effects of varying the grid size, the interface thickness, the noise amplitude and the diffusivity in the solid. In this paper, we investigate phase diagram effects in phase field models. We show how changing various features of the phase diagram changes the length scale and velocity dendritic growth, the degree of solute segregation, and the formation of secondary tertiary dendrite arms. Simulations are based on material parameters for four different binary alloy systems—nickel-copper, gold-silver, antimony-bismuth and aluminium-zinc.

Microsegregation in cellular dendritic growth in binary alloys of Al-Cu

Microsegregation in binary alloys of Al-2%wtCu in a large range of cooling rates using the unidirectional growth technique was studied. The spacing of substructures was determined and the value of the minimum composition in the center of the dendritic cells applying metallographic techniques and electronic microprobe was measured. The results obtained confirmed that the tendency of the experimental points can be explained with a good accuracy by the theoretical models of Solari and Biloni and of Trivedi.