Free Dendritic Growth Research Articles

Modeling for free dendrite growth based on physically-informed machine learning method

In order to overcome limitations of traditional theoretical and numerical models, based on machine learning (ML) method, three models were developed for dendrite growth in undercooled alloy melts to predict solid-liquid interfacial velocity, including purely data-driven model-1, model-2 based on Galenko-Danilov (GD) model and model-3 coupled with an extended GD model newly proposed by introducing thermo-kinetic correlation. A thoroughly comparative analysis was carried out by applying the three ML models to existing data of five alloys. Results verify that model-3 has the relatively better fitting ability to experimental data in case of interpolation, due to the varied effective kinetic coefficient introduced and the thermo-kinetic correlation considered. Both cases of complete and partial extrapolations were discussed. It is concluded that on the whole the two physics-based ML models, especially model-3, are superior to the purely data-driven ML model for extrapolation ability. Thus, ML model-3 is finally proposed in modeling dendrite growth.

Grid anisotropy reduction method for cellular automata based solidification models

The reliability of a cellular automata (CA) simulation for a free dendritic growth problem relies heavily on its ability to reduce the artificial grid anisotropy. Hence, a computationally efficient, accurate and elegant cell capturing methodology is essential to achieve reliable results. Therefore, a novel cell capturing method termed limited circular neighbourhood (LCN) is proposed in the present study for solidification models. The LCN method is applied to the canonical test cases with an isotropic growth rate and is compared with other grid anisotropy reducing methods. It is observed that the LCN method is able to capture the growth orientation accurately. Moreover, the mass loss and shape error in the proposed method is significantly reduced as compared with the other methods. In addition, its performance is also evaluated for a free dendrite growth problem in a pure material in which the growth captured by the LCN method is found to be accurate. Finally, its efficacy is also demonstrated in the results presented for a constrained dendritic growth problem in a binary alloy with multiple growth sites.

Introducing density variation and pressure in thermodynamically self-consistent continuum phase-change models including phase-field

We present a thermodynamically self-consistent method to introduce dilation/density variation in continuum-scale phase-change models. Dilation incurs a pressure response via a hyperelastic contribution to the free energy that generalizes the lattice constraint. The dilation is represented entirely by species concentrations and permits composition, temperature, and phase-dependent specific volumes in a robust form. The impact of this approach is compared with the common assumption of the lattice constraint through demonstrative Stefan and phase-field models applied to the equilibrium of a Ni-Cu nanoparticle in equilibrium with its melt. Furthermore, the effect of phase and composition-dependent specific volume is explored in free dendritic growth behavior.

Simple Method of Including Density Variation in Quantitative Continuum Phase-Change Models.

We present a simple, quantitative, and thermodynamically self-consistent method of capturing density and pressure variation in continuum phase-change models. The formalism shows how the local state of homogenous dilation may be entirely given by species concentration in an Eulerian formulation. Ahyperelastic contribution to the thermodynamic potential generalizes the lattice constraint while permitting composition, temperature, and phase-dependent specific volumes. We compare the results of models implementing this paradigm to those with the lattice constraint by examining the composition and size-dependent equilibrium of a Ni-Cu nanoparticle in its melt and free dendritic growth.

Induced side-branching in smooth and faceted dendrites: theory and phase-field simulations.

The present work is devoted to the phenomenon of induced side branching stemming from the disruption of free dendrite growth. We postulate that the secondary branching instability can be triggered by the departure of the morphology of the dendrite from its steady state shape. Thence, the instability results from the thermodynamic trade-off between non monotonic variations of interface temperature, surface energy, kinetic anisotropy and interface velocity within the Gibbs-Thomson equation. For the purposes of illustration, the toy model of capillary anisotropy modulation is prospected both analytically and numerically by means of phase-field simulations. It is evidenced that side branching can befall both smooth and faceted dendrites, at a normal angle from the front tip which is specific to the nature of the capillary anisotropy shift applied. This article is part of the theme issue 'Transport phenomena in complex systems (part 2)'.

Field variable diffusion cellular automaton model for dendritic growth with multifold symmetry for the solidification of alloys

The substantial mesh dependency caused by the discrete properties of cells and limitations of the sharp interface model are the inherent disadvantages of the cellular automaton (CA) model, which leads to difficulties in simulating dendritic growth with multifold symmetry and random preferred orientations for the solidification of alloys. A novel field variable diffusion CA (FCA) model is proposed by referring to the concept of the gradient energy term for diffuse interfaces in the phase field method. The diffusion equation for the field variable based on the constructed gradient functional is derived using the gradient descent method and is introduced into the CA model to reduce the mesh dependency. The FCA model deals with the solidification/remelting growth kinetics of the solid–liquid interface according to the lever rule and by considering the constitutional supercooling and Gibbs–Thomson effect. Free dendritic growth from undercooled melts and competitive dendritic growth during the directional solidification of Al–4 wt% Cu and Mg–5.26 wt% Al alloys were calculated to validate the accuracy of the FCA model for four-fold, six-fold symmetric dendritic morphologies and growth kinetics. The FCA model can efficiently reproduce multi-equiaxed and columnar dendrites with multifold symmetry and random preferred orientations as well as graphically reveal essential dendritic growth information, such as coarsening of secondary arms and side branch fusion.

CA Modeling of Microsegregation and Growth of Equiaxed Dendrites in the Binary Al-Mg Alloy.

A two-dimensional model based on the Cellular Automaton (CA) technique for simulating free dendritic growth in the binary Al + 5 wt.% alloy was presented. In the model, the local increment of the solid fraction was calculated using a methodology that takes into account changes in the concentration of the liquid and solid phase component in the interface cells during the solidification transition. The procedure of discarding the alloy component to the cells in the immediate vicinity was used to describe the initial, unstable dendrite growth phase under transient diffusion conditions. Numerical simulations of solidification were performed for a single dendrite using cooling rates of 5 K/s, 25 K/s and 45 K/s and for many crystals assuming the boundary condition of the third kind (Newton). The formation and growth of primary and secondary branches as well as the development of component microsegregation in the liquid and solid phase during solidification of the investigated alloy were analysed. It was found that with an increase in the cooling rate, the dendrite morphology changes, its cross-section and the distance between the secondary arms decrease, while the degree of component microsegregation and temperature recalescence in the initial stage of solidification increase. In order to determine the potential of the numerical model, the simulation results were compared with the predictions of the Lipton-Glicksman-Kurz (LGK) analytical model and the experimental solidification tests. It was demonstrated that the variability of the dendrite tip diameter and the growth rate determined in the Cellular Automaton (CA) model are similar to the values obtained in the LGK model. As part of the solidification tests carried out using the Derivative Differential Thermal Analysis (DDTA) method, a good fit of the CA model was established in terms of the shape of the solidification curves as well as the location of the characteristic phase transition temperatures and transformation time. Comparative tests of the real structure of the Al + 5 wt.% Mg alloy with the simulated structure were also carried out, and the compliance of the Secondary Dendrite Arm Spacing (SDAS) parameter and magnesium concentration profiles on the cross-section of the secondary dendrites arms was assessed.

Free dendritic growth model considering both interfacial nonisothermal nature and effect of convection for binary alloy

Considering both the effect of the nonisothermal nature of the interface as well as the effect of forced convection, an extended free dendritic growth model for binary alloys was proposed. Comparative analysis indicates that the effect of convection on solute diffusion is more remarkable compared with the ignorable effect of convection on thermal diffusion at low bath undercooling, due to the fact that solute diffusion coefficient is usually three orders of magnitude less than thermal diffusion coefficient. At high bath undercooling, the effect of convection on the dendritic growth is very slight. Furthermore, a satisfying agreement between the model predictions with the available experiment data for the Cu70Ni30 alloy was obtained, especially at low bath undercoolings, profiting from the higher values of interfacial migration velocity predicted by the present model with nonideal fluid case than that predicted by the one ignoring the effect of convection.

Interface kinetics modeling of binary alloy solidification by considering correlation between thermodynamics and kinetics

Abstract By considering collision-limited growth mode and short-range diffusion-limited growth mode simultaneously, an extended kinetic model for solid−liquid interface with varied kinetic prefactor was developed for binary alloys. Four potential correlations arising from effective kinetics coupling the two growth modes were proposed and studied by application to planar interface migration and dendritic solidification, where the linear correlation between the effective thermodynamic driving force and the effective kinetic energy barrier seems physically realistic. A better agreement between the results of free dendritic growth model and the available experiment data for Ni−0.7at.%B alloy was obtained based on correlation between the thermodynamics and kinetics. As compared to previous models assuming constant kinetic prefactor, a common phenomenon occurring at relatively low undercoolings, i.e. the interface migration slowdown, can be ascribed to both the thermodynamic and the kinetic factors. By considering universality of the correlation between the thermodynamics and kinetics, it is concluded that the correlation should be considered to model the interface kinetics in alloy solidification.

A comparative in situ X-radiography and DNN model study of solidification characteristics of an equiaxed dendritic Al-Ge alloy sample

In situ X-radiography imaging during the isothermal solidification of a 200 µm thin Al-24 at.% Ge sample was performed to determine dendritic growth rates, liquid concentrations and projected solid fractions. The experimental data allowed to follow dendrite tip growth velocities with high accuracy all the way from the initial transient growth related decrease of velocity via the expected increase of velocity during free dendrite growth to its final decrease during neighbor interacted growth. Concentration profiles in front of individual dendrite tips were measured to analyze the increase in far-field solute concentration due to solute accumulation. By time-resolved imaging of almost the entire 12 mm diameter sample, the global melt concentration could be derived. This enabled to estimate the initial nucleation undercooling of this un-refined alloy, which was so far unresolved. The experimental results are compared to a three-dimensional dendrite needle network model that simulates the experimental conditions, such as hexagonal dendrite structures and confined sample geometry. Although the simulation results are very sensitive to variations of the parameters undercooling and dendrite selection constant, one model parameter set is found that reproduces all experimentally measured solidification characteristics. The thorough experiment-model comparison indicates that the dendritic growth rates in this experimental configuration can be lower by a factor of 7 depending on undercooling than in the non-confined case.

Modeling of free dendritic growth in a gravity environment by lattice Boltzmann method.

A two-dimensional model is developed to simulate dendrite growth and movement in a gravity environment. The model combines the features of cellular automaton and lattice Boltzmann methods. Two sets of distribution functions are adopted to calculate the melt flow and solute transport simultaneously. The fluid force acting on the dendrite is calculated by extending the basic flow simulation at the solid-liquid interface. Incorporating the force interaction between melt flow and solidified dendrite into the algorithm for dendritic growth, the movement of a growing dendrite in the flowing melt can be simulated. After model validation, the coupled model has been applied to simulate the evolution and motion of an individual nucleus that grows into a dendrite in the presence of gravitational force. It is found that the dendrite growth is strongly influenced by the fluid flow, producing an asymmetrical morphology that the dendrite grows faster in the upstream direction, whereas largely slower in the downstream direction. The growth process of dendritic side-branches is modeled in a high settling velocity without any artificial noise introduced. The melt flow triggered by the dendrite motion enhances the growth of the dendrite in the downward direction, which in turn influences the subsequent dendritic translation.

Free dendritic growth model for binary alloy based on microscopic solvability theory and nonisothermal nature caused by anisotropy and curved interface

Abstract Considering both the effect of interfacial anisotropy and the effect of the non-planar interface, an extended free dendritic growth model for binary alloys was proposed. It can be achieved by introducing the microscopic solvability theory to replace the marginal stability theory and by adopting a more improved interface response function. The thermal diffusion equation is also re-solved strictly with the new interfacial response function as boundary condition. Both the morphological stability model and the interface response function take into account the interfacial nonisothermal nature caused by interfacial anisotropy and non-planar interface. Therefore, the present model is truly self-consistent. Model comparisons with the related models and the available experiment data for the Cu70Ni30 alloy were also made. An agreement between the model predictions with the experiment data was obtained.

Simulation method based on phase-field lattice Boltzmann model for long-distance sedimentation of single equiaxed dendrite

In solidification of metals and alloys under earth’s gravity, the growing free equiaxed dendrites either settle or float because of the difference between the solid and liquid densities. In this study, we developed a two-dimensional computational method that can efficiently simulate the growth of free dendrite settling over a long distance. In the developed method, the growth of the dendrite is expressed by a phase-field method, the liquid flow is computed by a lattice Boltzmann method, and the settling of dendrite is expressed by equations of motion. A moving-frame algorithm is employed to track the dendrite settling over a long distance. The computation is accelerated using a multi-level mesh method and GPU computing. The validity of the developed method was evaluated through simulation of the settling of a single circular particle. Specifically, using the developed method, we simulated the growth of a single dendrite in an isothermal undercooled melt of a binary alloy under gravity, and evaluated the time evolutions of the growth velocity of the primary arms, settling velocity, and crystal orientation of the dendrite. The results confirmed that the developed computational method can efficiently express the dendrite growth and settling over a long distance at a reasonable computational cost.

Free dendritic growth model based on nonisothermal interface and microscopic solvability theory

Abstract Considering both the effect of nonisothermal nature of the solid/liquid interface and the microscopic solvability theory (MicST), a further improved version of free dendritic growth model for pure materials was proposed. Model comparison indicates that there is a higher temperature at the tip of dendrite predicted by the present model compared with the corresponding model with the isothermal solid/liquid interface assumption. This is attributed to the sidewise thermal diffusion, i.e. the gradient of temperature along the nonisothermal interface. Furthermore, it is indicated that the distinction between the stability criteria from MicST and marginal stability theory (MarST) is more significant with the increase of bath undercoolings. Model test also indicates that the present model can give an agreement with the available experimental data. It is finally concluded that the nonisothermal nature of the solid/liquid interface and the stability criterion from MicST should be taken into account in modeling free dendritic growth.

Quantitative comparison of dendritic growth under forced flow between 2D and 3D phase-field simulation

Due to distinctions of the heat and mass transport in two (2D) and three (3D) dimensions, the morphology and growth of numerically simulated dendrites are usually different. In order to quantitatively address such differences, the free dendritic growth of a binary alloy from undercooled melt under forced flow is simulated using the phase-field (PF) method. The vector-valued approach has been employed to solve the equations of fluid dynamics to expedite simulations. The classical Ananth-Gill solutions are used to validate the results of PF simulations, and good agreements between the simulated growth Péclet numbers of the upstream tip and analytical predictions are achieved. The tip velocity and radius of the steady-state upstream tip, and the solute profile ahead of the tip solid-liquid (S-L) interface, are compared quantitatively between 2D and 3D simulations. In 3D, since the rejected solute can be rapidly transported away from the S-L interface by liquid flow, the solute concentration at the liquid side of the S-L interface is lower, but its gradient is higher, and the thickness of the solute boundary layer is smaller, thus leading to a higher tip velocity and smaller tip radius. However, the ratios of the tip velocity and tip radius between 2D and 3D are not confined in a certain range but varies with the supersaturation and liquid inflow velocity. Nevertheless, the differences are relieved with increasing the supersaturation or the inflow strength. Quantitative distinctions can be found from the ratios of these characteristics and the tip stability parameter in 2D versus 3D simulations, which follow a power law of supersaturation or growth Péclet number.

Analysis of free dendritic growth considering both relaxation effect and effect of nonisothermal and nonisosolutal interface

Through considering both the relaxation effect of local nonequilibrium solute diffusion in bulk liquid and the effect of nonisothermal and nonisosolutal nature of the solid–liquid interface, a further improved version of free dendritic growth model was proposed, for binary alloys. Comparative analysis indicates that the effect of the nonisothermal nature of the interface is significant and this effect should be taken into account in modeling the free dendritic growth. It is also concluded that the effect of the nonisosolutal nature of interface is very slight and really negligible. Thus, the simplified version of free dendritic growth model taking into account the nonisothermal nature of the interface and with an isosolutal interface assumption is reasonable. Furthermore, experimental comparison demonstrates that it is necessary to introduce the diffusion relaxation effect for achieving a good agreement between the model predictions with the available experiment data, especially at high undercoolings, for the Cu70Ni30 alloy.

Phase-Field Simulation for Non-isothermal Solidification of Al-Cu Binary Alloy

Based on the principle of dilute solution approximation, a phase field model for dendritic growth during the non-isothermal solidification process was proposed by coupling the phase field, concentration field and temperature field, and was adopted to investigate the dendritic growth of Al-Cu binary alloy during the non-isothermal solidification process. Also, the simulations of free dendritic growth in an undercooled melt of Al-4.5%Cu binary alloy with different perturbation intensity and anisotropy were carried out by the present phase field model. The results show that during the non-isothermal solidification process of Al-4.5%Cu binary alloy, the dendrite grow into undercooled melt with the solute rejection and latent heat release at the front of solid/liquid interface, and the solute enriches at the dendrite root and high temperature appears at the dendritic growth front. With the increase of perturbation intensity, the dendritic growth becomes more developed and more branches appear. Moreover, the anisotropy coefficient also has a great effect on the dendritic growth, and the growth speed of dendrite increases with the increase of anisotropy coefficient.

Free dendritic tip growth velocities measured in Al-Ge

We determine the tip growth velocities of equiaxed dendrites in an Al-24 at. % Ge alloy using in situ x-ray radiography. The tip growth velocity is a main characteristic of dendritic growth and allows for an accurate comparison against theories and simulations describing the growth of a single dendrite tip or an array of dendrites. The disk-shaped samples are 200 or 350 $\ensuremath{\mu}\mathrm{m}$ thin and are positioned horizontally in a near-isothermal furnace to suppress buoyancy and gravity-driven melt flows. We obtain a large data set of free dendritic tip growth velocities of Al dendrites over one order of magnitude (4--140 $\ensuremath{\mu}\mathrm{m}\phantom{\rule{0.16em}{0ex}}{\mathrm{s}}^{\ensuremath{-}1}$) and over a decade of global undercooling (1--10 K). No indications for a significant deviation from three-dimensional growth kinetics due to the sample confinement are found.

Orientation Dependence of Columnar Dendritic Growth with Sidebranching Behaviors in Directional Solidification: Insights from Phase-Field Simulations

In this study, a thin-interface phase-field model was employed to study the orientation dependence of the columnar dendritic growth with sidebranching behaviors in directional solidification. It was found that the dimensionless tip undercooling increases with the increase of misorientation angle for three pulling velocities. The primary spacing is found to be a function of misorientation angle, and the dimensionless primary spacing with respect to the misorientation angle follows the orientation correction given by Gandin and Rappaz (Acta. Metall. 42:2233–2246, 1994). For the analysis of the dendritic tip, the two-dimensional (2-D) form of the nonaxisymmetric needle crystal was used to determine the radius of the tilted columnar dendrite. Based on the definitions of open side and constrained side of the dendrite, the analysis of the width active sidebranches and the dendritic area in 2-D with respect to the distance from the dendritic tip was carried out to investigate the asymmetrical dendrite envelop and sidebranching behaviors on the two sides in directional solidification. The obtained prefactor and exponent with respect to misorientation angle are discussed, showing that the sidebranching behaviors of a tilted columnar dendritic array obey a similar power-law relationship with that of a free dendritic growth.

Cellular automaton modeling of dendritic growth of Fe-C binary alloy with thermosolutal convection

Embedded the thermal and solutal buoyancy into the momentum conservation equation as an additional force term using the Boussinesq approximation, a 2D CA-FVM model is extended to simulate the dendritic growth with thermosolutal convection. The model is firstly validated by comparison of numerical predictions with the benchmark test of Rayleigh – Bénard convection and the analytical solutions of the stagnant film model for the free dendritic growth with thermosolutal convection, and good agreements between the numerical results with analytical solutions are obtained. Later, numerical simulations for both the equiaxed and columnar dendritic growth of Fe-0.82wt%C binary alloy with thermosolutal convection are performed. The results show that, for the equiaxed dendritic growth in an undercooled melt, the dendrite tip growth rapidly decreases from the high velocity to a relative low steady-state value. With the further growth of dendrite, the thermosolutal convection induced by the solute rejection and latent heat release is enhanced and four vortexes are developed between the dendrite arms. Thus, the asymmetries of the dendrite morphology, temperature and solute profiles are intensified. For the columnar dendritic growth with thermosolutal convection under the unidirectional solidification process, the thermosolutal convection transports the rejected solute downward and makes the solute enrich at the interdendritic region. The thermosolutal convection facilitates the upstream dendritic growth, but inhibits the downstream dendritic growth. Moreover, with the increase of deflection angle of gravity, the advection on the top region and the clock-wise vortex flow at the interdendritic region intensified, and finally the columnar dendrite morphology becomes more asymmetrical.