Quantitative Phase-field Simulations Research Articles

Thermal-field effects on interface dynamics and microstructure selection during alloy directional solidification

We carry out three-dimensional phase-field simulations to model unique experimental observations of cellular and dendritic solidification structures formed under diffusive growth conditions in the DSI (Directional Solidification Insert) of the DECLIC (DEvice for the study of Critical LIquids and Crystallization) aboard the International Space Station. We had previously shown experimentally that complex thermal conditions affect the stationary position of the solid-liquid interface, as well as its dynamics of relaxation towards this stationary position over a finite time after the onset of sample pulling. Here, we discuss the effects of thermal diffusion within the adiabatic zone of the directional solidification setup and of latent heat release at the solid-liquid interface by means of quantitative phase-field simulations. Simulations and experiments characterize the entire evolution of the primary spacing of cellular/dendritic array structures from the onset of morphological instability to the establishment of the final steady-state spacing, including the transient coarsening regime associated with a sharp increase of spacing. Accounting for these thermal effects leads to a major improvement in the agreement between simulations and microgravity measurements for both the time of occurrence of morphological instability after the start of the experiment and the subsequent spacing evolution, which are not accurately predicted using the standard frozen temperature approximation.

The Diffusion and Solid-Liquid Phase Transformation in Directional Solidification of Alloy: A Quantitative Phase Field Characterization and Real-Time Observation

Directional solidification is a paradigm process to gain the desired microstructure via certain applied solidification parameters. A thorough understanding of the diffusion-limited solid-liquid interface morphology evolution from initial transient to steady state is of uppermost importance to optimize the solidification processes. The rapid development of quantitative phase-field model provides a feasible computational tool to explore the underlying physics of the morphological transition at different stages. On basis of the diffusion-limited quantitative phase-field simulations using adaptive finite element method, the directional solidification of Al-4wt.%Cu alloy is characterized and both the solid interface propagation speed and solute profile are analyzed. The simulations are then compared with the in situ and real-time observation by means of synchrotron radiation x-ray radiography image. Good agreements are obtained between simulations and experimental data. Detailed mechanism that controls the morphological instability and transition are then addressed.

Current development in quantitative phase-field modeling of solidification

The quantitative phase-field simulations were reviewed on the processes of solidification of pure metals and alloys. The quantitative phase-field equations were treated in a diffuse thin-interface limit, which enabled the quantitative links between interface dynamics and model parameters in the quasi-equilibrium simulations. As a result, the quantitative modeling is more effective in dealing with microstructural pattern formation in the large scale simulations without any spurious kinetic effects. The development of the quantitative phase-field models in modeling the formation of microstructures such as dendritic structures, eutectic lamellas, seaweed morphologies, and grain boundaries in different solidified conditions was also reviewed with the purpose of guiding to find the new prospect of applications in the quantitative phase-field simulations.

Anomalous Halo Formation in an Arc-Melted ScNi-Sc2Ni Off-Eutectic Binary Alloy

Diverse non-equilibrium eutectic structures have attracted numerous experimental and theoretical studies. One special type is the formation of a halo of one phase around a primary dendrite of another phase. In our experiments, it was occasionally observed that ScNi halos grow as dendritic morphology around the primary ScNi dendrites in an arc-melted ScNi-ScNi off-eutectic binary alloy. The formation of this anomalous halo structure was then well reproduced by employing quantitative phase-field simulations. Based on the phase-field simulation, It was found that (i) the large undercooling and growth velocity of the ScNi phase during solidification causes the formation of halos; and (ii) the released latent heat induces the recalescence phenomenon, and changes the solidification sequence largely, resulting in the anomalous halo structure in the Sc-34 at % Ni alloy.

Phase-Field Modeling of Facet Hexagonal Spirals with Anisotropy, Deposition, and Kinetic Effects

In this paper, we perform the quantitative phase-field simulations based on the surface morphology and growth regime of the hexagonal GaN spiral structure. We investigate the highly anisotropic energy, the deposition rate and the kinetic attachment and detachment effects. A regularized equation including the modified gradient coefficient is employed to study the anisotropic effect. Results show that the highly anisotropic energy modulates the equilibrium state by changing the local curvature of the tip step and thus leading to the changed spiral spacing. Under the weak anisotropy, the spiral spacing and morphology keep stable with the increase of the anisotropic strength. In the case of facet anisotropy, however, the larger anisotropic strength facilitates the spiral growth due to the local interfacial instability caused by increasing the supersaturation for the tip step. As to the effect of deposition, the deposition rate imposes the reaction on the curvature of interface due to the variations of supersaturation and step velocity. The larger rate of deposition enables the shorter spacing for both anisotropic and isotropic spirals. We carry out a convergence study of spiral spacing with respect to the step width to estimate the precision of the phase-field simulation. Results show that the larger deposition rate and the higher anisotropy give rise to the lower convergence of the spiral model. Moreover, we find that the kinetic attachment affects the instinct regime of spiral growth by changing the step spacing and the scaling exponents of spiral spacing versus deposition rate. The anisotropic spiral exhibits the more significant hexagonal structure and the lower value of step velocity by reducing the value of kinetic coefficient. The scaling exponent decreases with anisotropy increasing, but it increases with kinetic effect strengthening. The highly anisotropic energy contributes to weakening the sensitivity of the spiral spacing to the kinetic effect.

Oscillatory cellular patterns in three-dimensional directional solidification.

We present a phase-field study of oscillatory breathing modes observed during the solidification of three-dimensional cellular arrays in microgravity. Directional solidification experiments conducted onboard the International Space Station have allowed us to observe spatially extended homogeneous arrays of cells and dendrites while minimizing the amount of gravity-induced convection in the liquid. In situ observations of transparent alloys have revealed the existence, over a narrow range of control parameters, of oscillations in cellular arrays with a period ranging from about 25 to 125 min. Cellular patterns are spatially disordered, and the oscillations of individual cells are spatiotemporally uncorrelated at long distance. However, in regions displaying short-range spatial ordering, groups of cells can synchronize into oscillatory breathing modes. Quantitative phase-field simulations show that the oscillatory behavior of cells in this regime is linked to a stability limit of the spacing in hexagonal cellular array structures. For relatively high cellular front undercooling (i.e., low growth velocity or high thermal gradient), a gap appears in the otherwise continuous range of stable array spacings. Close to this gap, a sustained oscillatory regime appears with a period that compares quantitatively well with experiment. For control parameters where this gap exists, oscillations typically occur for spacings at the edge of the gap. However, after a change of growth conditions, oscillations can also occur for nearby values of control parameters where this gap just closes and a continuous range of spacings exists. In addition, sustained oscillations at to the opening of this stable gap exhibit a slow periodic modulation of the phase-shift among cells with a slower period of several hours. While long-range coherence of breathing modes can be achieved in simulations for a perfect spatial arrangement of cells as initial condition, global disorder is observed in both three-dimensional experiments and simulations from realistic noisy initial conditions. In the latter case, erratic tip-splitting events promoted by large-amplitude oscillations contribute to maintaining the long-range array disorder, unlike in thin-sample experiments where long-range coherence of oscillations is experimentally observable.

Microsegregation in multicomponent alloy analysed by quantitative phase-field model

Microsegregation behaviour in a ternary alloy system has been analysed by means of quantitative phase-field (Q-PF) simulations with a particular attention directed at an influence of tie-line shift stemming from different liquid diffusivities of the solute elements. The Q-PF model developed for non-isothermal solidification in multicomponent alloys with non-zero solid diffusivities was applied to analysis of microsegregation in a ternary alloy consisting of fast and slow diffusing solute elements. The accuracy of the Q-PF simulation was first verified by performing the convergence test of segregation ratio with respect to the interface thickness. From one-dimensional analysis, it was found that the microsegregation of slow diffusing element is reduced due to the tie-line shift. In two-dimensional simulations, the refinement of microstructure, viz., the decrease of secondary arms spacing occurs at low cooling rates due to the formation of diffusion layer of slow diffusing element. It yields the reductions of degrees of microsegregation for both the fast and slow diffusing elements. Importantly, in a wide range of cooling rates, the degree of microsegregation of the slow diffusing element is always lower than that of the fast diffusing element, which is entirely ascribable to the influence of tie-line shift.

Stability range of tilted dendritic arrays during directional solidification

The stability range of primary spacing of the tilted dendritic arrays in directional solidification has been studied by quantitative phase-field simulations. Results show that both the real growth direction and morphological shapes of dendritic arrays change with the primary spacing for different misorientation angles (θ0). It has been found that the lower limit of primary spacing is independent of θ0, but the upper limit of primary spacing is strongly influenced by that. The two kinds of tertiary branching instabilities result in different behaviors of the variation of the upper limit with misorientation angle for different pulling velocities.

Onset of the initial instability during the solidification of welding pool of aluminum alloy under transient conditions

Onset of initial morphological instability is predicted by using a new analytic model and quantitative phase field model during the solidification of the welding pool of Al–Cu alloy under transient conditions. In the linear growth stage of the welding pool, the dynamic evolution of the interface instability is analyzed, and the interface behaviors under infinitesimal fluctuations are also investigated. The results show that the mean wavelength at the crossover time evaluated from this analytic model is in good agreement with those obtained by the quantitative phase field simulations and the experiments. The linear growth stage takes up quite a long time of the whole solidification of welding pool and thus it should be primarily considered in investigating the transient growth of welding pool. This study establishes a valid numerical framework for studying the dendrite growth under transient solidification conditions and provides a new approach for studying the transient solidification of welding pool.

Investigation of Al-Cu-Ni alloy solidification: thermodynamics, experiments and phase-field modeling

The investigation of solidification in ternary Al-Cu-Ni alloys is carried out by means of experiments and phase-field modeling. For three alloys in the Al-rich corner of the phase diagram differential thermal analysis (DTA) is performed. Then the alloys were analyzed using scanning electron microscopy with energy dispersive X-ray microanalysis. For the understanding of the general features of the alloy solidification quantitative phase-field simulations are carried out additionally to the theoretical Scheil calculation. It is found that many experimental DTA signals and the microstructure parts cannot be explained by simple Scheil calculation. We apply the multi-phase-field model previously developed for the simulation of peritectic reaction and extended it to three components and four-phases reactions. The main advantage of this model is the application of the equilibrium parameters evaluated from the refined free energies of the phases. It is shown that the simulated microstructure is comparable to the experimental one for two investigated alloys. The final phase fractions in the modeling correspond to the theoretical predictions of Scheil calculation but the time evolution of fractions is more complicated. In particular the kinetics (relation between tangential and normal growth velocities) of the peritectic-like reaction in ternary Al-Cu-Ni alloys shows differences compared to the peritectic reaction in binary Al-Ni alloys.

Phase field modelling of stressed grain growth: Analytical study and the effect of microstructural length scale

We establish the correlation between the diffuse interface and sharp interface descriptions for stressed grain boundary migration by presenting analytical solutions for stressed migration of a circular grain boundary in a bicrystalline phase field domain. The validity and accuracy of the phase field model is investigated by comparing the phase field simulation results against analytical solutions. The phase field model can reproduce precise boundary kinetics and stress evolution provided that a thermodynamically consistent theory and proper expressions for model parameters in terms of physical material properties are employed. Quantitative phase field simulations are then employed to investigate the effect of microstructural length scale on microstructure and texture evolution by stressed grain growth in an elastically deformed polycrystalline aggregate. The simulation results reveal a transitional behaviour from normal to abnormal grain growth by increasing the microstructural length scale.

Phase-Field Modeling of Polycrystalline Solidification: From Needle Crystals to Spherulites—A Review

Advances in the orientation-field-based phase-field (PF) models made in the past are reviewed. The models applied incorporate homogeneous and heterogeneous nucleation of growth centers and several mechanisms to form new grains at the perimeter of growing crystals, a phenomenon termed growth front nucleation. Examples for PF modeling of such complex polycrystalline structures are shown as impinging symmetric dendrites, polycrystalline growth forms (ranging from disordered dendrites to spherulitic patterns), and various eutectic structures, including spiraling two-phase dendrites. Simulations exploring possible control of solidification patterns in thin films via external fields, confined geometry, particle additives, scratching/piercing the films, etc. are also displayed. Advantages, problems, and possible solutions associated with quantitative PF simulations are discussed briefly.

Phase-field simulation of solidification morphology in laser powder deposition of Ti–Nb alloys

A phase-field model of alloy solidification is coupled to a new heat transfer finite element model of the laser powder deposition process. The robustness and accuracy of the coupled model is validated by studying spacing evolution under the directional solidification conditions in laser powder deposition of Ti–Nb alloys. Experimental Ti–Nb samples reveal the microstructure on a longitudinal section with significant change in the size of the dendrites across the sample. Quantitative phase-field simulations of directional solidification under local steady-state conditions extracted from the results of the finite element thermal model confirmed this behavior. Specifically, the phase-field simulations agree with the results of the analytical model of Hunt in predicting a minimum spacing value, which is due to the mutual effects of the increasing temperature gradient and decreasing solidification velocity towards the bottom of the microstructure. This work demonstrates the potential of coupling the phase-field method to complex heat transfer conditions necessary to simulate topologically complex microstructure morphologies present in laser powder deposition and other industrially relevant casting conditions.

Phase-field-crystal calculation of crystal-melt surface tension in binary alloys

A phase field crystal (PFC) density functional for binary mixtures is coarse grained and a formalism for calculating the simultaneous concentration, temperature, and density dependence of the surface energy anisotropy of a solid-liquid interface is developed. The methodology systematically relates bulk free energy coefficients arising from coarse graining to thermodynamic data, while gradient energy coefficients are related to molecular properties. Our coarse-grained formalism is applied to the determination of surface energy anisotropy in two-dimensional Zn-Al films, a situation relevant for quantitative phase field simulations of dendritic solidification in zinc coatings.

Spacing characterization in Al–Cu alloys directionally solidified under transient growth conditions

We study spacing selection in directional solidification of Al–Cu alloys under transient growth conditions. New experimental results are presented which reveal that the mean dendritic spacing vs. solidification front speed exhibits plateau-like regions separated by regions of rapid change, consistent with previous experiments of Losert and co-workers. Quantitative phase-field simulations of directional solidification with dynamical growth conditions approximating those in the experiments confirm this behavior. The mechanism of this type of change in mean dendrite arm spacing is consistent with the notion that a driven periodically modulated interface must overcome an energy barrier before becoming unstable, in accord with a previous theory of Langer and co-workers.

Fourier synthesis predicting onset of the initial instability during directional solidification

With the combination of the Fourier synthesis method and the time-dependent linear stability analysis, a simple model was developed to describe the dynamic evolution of the initial interface instability. By using this simple model, the early instability stage of a planar interface during directional solidification of binary alloys is predicted and how the interface microstructure generates from the background infinitesimal fluctuations is also revealed. The results from this model are in good agreement with those from quantitative phase field simulations and previous experiments.

Quantitative phase-field modeling of solidification in binary alloys with nonlinear phase coexistence curves

A formalism is presented for performing quantitative phase-field simulations of single-phase solidification in binary alloys with nonlinear solidus and liquidus curves. It is shown that, close to equilibrium, Gibbs free energy of an alloy phase can be approximated by the free energy function of a dilute ideal binary alloy, modified by effective temperature-dependent coefficients. This makes it possible to exploit a recent phase-field technique [A. Karma, Phys. Rev. Lett. 87, 115701 (2001)] to model the free-boundary kinetics of single-phase solidification in binary alloys having nonlinear phase coexistence curves. Simulations of isothermal and nonisothermal dendritic solidification in an isomorphous binary alloy are used to demonstrate convergence of tip speed and radius for different values of the phase-field interface thickness. The effect linear versus nonlinear phase boundaries on dendritic tip speed is examined.

Sharp-interface projection of a fluctuating phase-field model

We present a derivation of the sharp-interface limit of a generic fluctuating phase-field model for solidification. As a main result, we obtain a sharp-interface projection which presents noise terms in both the diffusion equation and in the moving boundary conditions. The presented procedure does not rely on the fluctuation-dissipation theorem, and can therefore be applied to account for both internal and external fluctuations in either variational or nonvariational phase-field formulations. In particular, it can be used to introduce thermodynamical fluctuations in nonvariational formulations of the phase-field model, which permit to reach better computational efficiency and provide more flexibility for describing some features of specific physical situations. This opens the possibility of performing quantitative phase-field simulations in crystal growth while accounting for the proper fluctuations of the system.

Quantitative phase-field modeling of dendritic growth in two and three dimensions

We report the results of quantitative phase-field simulations of the dendritic crystallization of a pure melt in two and three dimensions. These simulations exploit a recently developed thin-interface limit of the phase-field model [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)], which is given here a detailed exposition. This limit makes it possible to perform efficient computations with a smaller ratio of capillary length to interface thickness and with an arbitrary interface kinetic coefficient. Simulations in one and two dimensions are first carried out to test the accuracy of phase-field computations performed within this limit. Dendrite tip velocities and tip shapes are found to be in excellent quantitative agreement with exact numerical benchmarks of solvability theory obtained by a boundary integral method, both with and without interface kinetics. Simulations in three dimensions exploit, in addition to the asymptotics, a methodology to calculate grid corrections due to the surface tension and kinetic anisotropies. They are used to test basic aspects of dendritic growth theory that pertain to the selection of the operating state of the tip and to the three-dimensional morphology of needle crystals without sidebranches. For small crystalline anisotropy, simulated values of ${\ensuremath{\sigma}}^{*}$ are slightly larger than solvability theory predictions computed by the boundary integral method assuming an axisymmetric shape, and agree relatively well with experiments for succinonitrile given the uncertainty in the measured anisotropy. In contrast, for large anisotropy, simulated ${\ensuremath{\sigma}}^{*}$ values are significantly larger than the predicted values. This disagreement, however, does not signal a breakdown of solvability theory. It is consistent with the finding that the amplitude of the $\mathrm{cos}4\ensuremath{\varphi}$ mode, which measures the departure of the tip morphology from a shape of revolution, increases with anisotropy. This departure can therefore influence the tip selection in a way that is not accurately captured by the axisymmetric approximation for large anisotropy. Finally, the tail shape at a distance behind the tip that is large compared to the diffusion length is described by a linear law $r\ensuremath{\sim}z$ with a slope $dr/dz$ that is nearly equal to the ratio of the two-dimensional and three-dimensional steady-state tip velocities. Furthermore, the evolution of the cross section of a three-dimensional needle crystal with increasing distance behind the tip is nearly identical to the evolution of a two-dimensional growth shape in time, in accord with the current theory of the three-dimensional needle crystal shape.