Quantitative Phase-field Simulations Research Articles

Toward multiscale simulations for solidification microstructure and microsegregation for selective laser melting of nickel-based superalloys

In this study, we carried out multiscale simulations that integrate a macro-scale finite element method simulation for mass and heat transfer of the transient molten pool and a micro-scale quantitative phase-field model for dendritic growth and solute distribution based on selective laser melting (SLM) experiments. Macroscopic simulations reveal an approximately inverse coupling of solidification velocity Vs and thermal gradient G near the solid–liquid interface at the tail end of the molten pool. It was found that the maximum flow velocity of the Inconel 718 alloy melt in the molten pool exceeds 2×10−2 m/s and the maximum negative pressure reaches −5.28×10−4 Pa. Through quantitative phase-field simulations, the map about G/Vs for each solidification morphology was obtained, and it was found that solidification velocity dependent solute partition coefficients k(Vs) can strongly influence the estimated stability regions. The power law relationship of the primary dendritic arm spacing (PDAS) λ1 versus solidification velocity λ1∝Vsα agrees with the previous experimental results for lower G, and the exponent α decreases with increasing G. The PDAS λ1 versus cooling rate R following λ1∝R−0.512 is consistent with previous investigations. We found that λ1 is a nonlinear function with G−0.5Vs−0.25, i.e., λ1∝(G−0.5Vs−0.25)α, with α ranging from 0.887 to 2.466 for various G. Comparison of the predicted λ1 with Trivedi's model reveals that λ1 is still nonlinear with G−0.25Vs−0.25.

Scaling law for growth of misoriented equiaxed Al-Cu dendrites: A phase-field study with in situ experiment validation

We consider the effect of a mutual misorientation between two interacting equiaxed dendrites in polycrystalline materials on the scaling law of growth from undercooled melt. This effect is investigated by three-dimensional quantitative phase-field simulations of Al-Cu alloy solidification in thin samples. It has been found that the equiaxed dendritic growth kinetics changes due to the reduced solute interaction when considering mutual grain misorientation, and both the characteristic growth rate and primary dendritic arm length can be approximately correlated linearly to the misorientation of two dendrites. The scaling law, which originally uses the solute composition and the distance between two nuclei as physical parameters to describe the dynamics of primary dendritic arms, is then adapted with consideration of the misorientation. Compared with the previous scaling law that only concerns face-to-face growth of two dendrites, the predictions of the newly proposed scaling law taking the grain orientation in account are in better agreement with the experimental data.

Hot cracking susceptibility prediction from quantitative multi-phase field simulations with grain boundary effects

Hot cracks are notorious defects in casting, welding, and additive manufacturing. Grain boundaries play a key role in hot cracking formation. Still, their effects are often neglected in hot cracking susceptibility predictions, and thus a quantitative picture of hot cracking susceptibility and liquid fracture is still lacking. In this work, we construct a multi-order parameter phase-field model for binary alloy solidification to simulate attractive, neutral, and repulsive grain boundaries in the thin-interface limit. Phase-field simulations of different combinations of grain boundary and bi-crystal growth types are used to obtain solidification path data, which is then used as the input into the Rappaz, Drezet and Gremaud (RDG) model (Rappaz et al., 1999). We study the so-called Λ-shape variation of hot cracking susceptibility as a function of solute concentration, showing that the magnitude of the Λ curve peak shifts to higher values and to lower concentrations as grain boundary energy increases. Meanwhile, the peak magnitude becomes higher and shifts to a higher value and higher concentrations when convergent grain growth is applied. Furthermore, in all cases examined, the corresponding pressure drops predicted for hot cracking exceed the theoretical liquid rupture stress, consistent with experimental observations. These results demonstrate that quantitative phase-field simulations, when coupled with the RDG model, are capable of relatively accurately predicting liquid rapture states relevant to hot cracking since the large disparity between the predicted pressure drop and critical stress of liquid film is eliminated through the consideration of grain boundary effects in the solidification path calculation.

Ni coarsening in Ni-yttria stabilized zirconia electrodes: Three-dimensional quantitative phase-field simulations supported by ex-situ ptychographic nano-tomography

In-depth understanding of nickel (Ni) coarsening is helpful for improving the service life of Ni-yttria stabilized zirconia (YSZ) electrodes in solid oxide cells. Unfortunately, very few quantitative experimental/theoretical descriptions of Ni coarsening in Ni-YSZ electrodes during long-term operation exist. In this paper, quantitative modeling of Ni coarsening in Ni-YSZ electrodes was achieved through three-dimensional (3D) phase-field simulation supported by ex-situ ptychographic nano-tomography and input of reliable thermophysical parameters. A pragmatic procedure was proposed to refine and verify the materials parameters for the simulations. Moreover, the microstructures of the Ni-YSZ electrode in the pristine and annealed states obtained via the ex-situ ptychographic nano-tomography were used as the initial input and experimental validation for the phase-field simulations. After that, comprehensive comparison between the simulated and the experimental 3D microstructures was conducted, indicating the successful quantitative phase-field simulation of Ni coarsening in Ni-YSZ electrodes presented here. The success of this work is expected to pave the way for accurate prediction of the service life and even design of high-performance Ni-YSZ electrodes.

Time evolution of interface shape distribution of equiaxed dendrite: A phase-field study

An understanding of the morphology of growing dendrites in alloys is needed for an analysis of microsegregation, as well as an estimation of the permeability for macroscopic fluid dynamics. Quantitative phase-field simulations were used to study the growth process of three-dimensional (3D) equiaxed dendrites in an Al-1.0 mass%Cu alloy during continuous cooling. The dendrites were analysed using an interface shape distribution (ISD) map, which provides the probability of the local interface having a morphology with a given curvedness (C) and shape factor (S). Morphological changes in the microstructure can be measured sensitively from the change in the average value of the curvedness 〈C〉 relative to the solid volume fraction. The ISD map continued to change over time during continuous cooling, implying that it was not time-invariant. Furthermore, when microstructural changes occurred, similarities between the ISD maps were observed, independent of the cooling rates and system sizes.

High-Throughput Screening of Optimal Process Parameters for PVD TiN Coatings With Best Properties Through a Combination of 3-D Quantitative Phase-Field Simulation and Hierarchical Multi-Objective Optimization Strategy

Physical vapor deposition (PVD) is one of the most important techniques for coating fabrication. With the traditional trial-and-error approach, it is labor-intensive and challenging to determine the optimal process parameters for PVD coatings with best properties. A combination of three-dimensional (3-D) quantitative phase–field simulation and a hierarchical multi-objective optimization strategy was, therefore, developed to perform high-throughput screening of the optimal process parameters for PVD coatings and successfully applied to technically important TiN coatings. Large amounts of 3-D phase-field simulations of TiN coating growth during the PVD process were first carried out to acquire the parametric relation among the model parameters, microstructures, and various coating properties. Experimental data were then used to validate the numerical simulation results and reveal the correlation between model parameters and process parameters. After that, a hierarchical multi-objective method was proposed for the design of multiple coating properties based on the quantitative phase–field simulations and key experimental data. Marginal utility was subsequently examined based on the identification of the Pareto fronts in terms of various combinations of objectives. The windows for the best TiN coating properties were, therefore, filtered with respect to the model/process parameters in a hierarchical manner. Finally, the consistent optimal design result was found against the experimental results.

Thermal diffusion coupled quantitative phase-field simulations with large undercooling

The numerical solution of phase-field model depends on the input interface-width particularly when the undercooling is high. In order to make the computation feasible, the interface is required to be thick, but such choice of interface-width renders the numerical solution inaccurate. In the present work, we investigate the effect of interface-width on phase-field computation with thermal diffusion under high undercooling. We show that the accuracy of phase-field computation is actually related to the interfacial-energy. A numerical strategy based on mobility-correction is formulated and applied to the transformation of γ→α-iron under various conditions. We also numerically solve the moving-boundary model based on the sharp interface and compare it with the phase-field computations to assess its accuracy. It has been observed that convergence of the phase-field computations to sharp-interface based predictions is remarkably good with the proposed computational strategy over a wide ranges of input interface widths and undercoolings.

Inverse analysis of anisotropy of solid-liquid interfacial free energy based on machine learning

A machine leaning-based approach is proposed for the inverse analysis of the anisotropy parameters of solid–liquid interfacial free energy. The interface shape distribution (ISD) map, which characterizes the details of the dendrite morphology, was selected as the input of a convolutional neural network (CNN). The ISD maps for a free-growing dendrite during the isothermal solidification of a model alloy system were obtained by quantitative phase-field simulations and used as the training and test data for the CNN. Two anisotropy parameters were estimated with errors of less than 5%, which can be further improved by increasing the size of the training dataset.

Cellular growth during rapid directional solidification: Insights from quantitative phase field simulations

In this paper, columnar cellular growth with kinetic effects including kinetic undercooling and solute trapping in rapid directional solidification of alloys was investigated by using a recent quantitative phase-field model for rapid solidification. Morphological transition and primary spacing selection with and without kinetic effects were numerically investigated. Numerical results show that doublon structure is an intermediate state in the primary spacing adjustment of cellular arrays. It was found that the inclusions of kinetic effects result in the increase of the solute in the solid phase and the solute enrichment in the interdendritic liquid channel. Moreover, predicted results indicate that the growth directions of the cellular arrays in rapid directional solidification with and without kinetic effects are independent of the Péclet number. Therefore, the kinetic effects play important roles in numerical simulations of the growth pattern selection and solute distribution during rapid solidification. Neglecting them will result in the inaccurately predicted results.

Effects of dislocation boundary spacings and stored energy on boundary migration during recrystallization: A phase-field analysis

The local migration of recrystallization boundaries into spatially varying deformation fields which are typical for metals deformed to low and high strains is investigated using quantitative phase-field simulations. It is found that the average migration velocities as well as the local velocities of the recrystallization boundaries critically depend on the spacing between planar geometrically necessary boundaries (GNBs) which are present in the deformation microstructure. Also the morphology of the deformation field reflecting low and high strain deformation cases strongly affect the migrating boundary during recrystallization. Additional simulations were performed as a function of the two critical parameters- stored energy and GNB spacing - independently. Two regimes are found suggesting that these two parameters are closely linked. Previous experimental results are interpreted based on these novel simulation results.

Quantitative phase-field simulations of globulitic solidification from the critical nucleus with the effects of interface curvature and attachment kinetics

A one-dimensional quantitative phase-field model (thin-interface approximation) considering the effects of interface curvature and attachment kinetics was implemented to simulate the solidification of a solid sphere (globulitic solidification) of a pure substance in an undercooled infinite melt, beginning from the critical nucleus. The classical sharp interface model for the same problem was also solved and the growth velocity and interface temperature were calculated and compared with the results of the phase-field model. Numerical conditions of the phase-field model, namely interface thickness, numerical mesh spacing, and critical nucleus model, that give close agreement with the sharp interface model (quantitative modeling) were determined. The possibility of a unique controlling step of the overall solidification kinetics by heat transfer or interface attachment kinetics or even of a mixed control was analyzed by defining relative undercoolings at the solid–liquid interface. For both metallic and non-metallic materials, growth is initially controlled by interface attachment kinetics, but, as the solid sphere grows and the melt heats up, it changes to the mixed control regime with heat transfer before spherical growth becomes unstable. The spherical radius for the change to mixed control increases when the kinetic coefficient or the magnitude of melt undercooling increases, because the heat transfer step is facilitated relative to that of attachment kinetics.

Quantitative nondiagonal phase field modeling of eutectic and eutectoid transformations

We develop a three-phase field model for the simulation of eutectic and eutectoid transformations on the basis of a nondiagonal model obeying Onsager relations for a kinetic cross coupling between diffusion and the phase fields. This model overcomes the limitations of existing phase field models concerning the fulfillment of local equilibrium boundary conditions at the transformation fronts in the case of a finite diffusional contrast between the phases. We benchmark our model in the well understood one-sided case with diffusion only in the parent phase against results from the literature. In addition to this solidification scenario, the case of solid-state transformations with diffusion in the growing phases is investigated. Our simulations validate the relevance of the theory developed by Ankit et al. [Acta Mater. 61, 4245 (2013)], that describes in a single frame the two limiting regimes where diffusion mainly takes place whether in the mother phase or in the growing phases. In both the one-sided and two-sided cases, we verify the necessity of the kinetic cross coupling for quantitative phase field simulations.

Multiscale prediction of microstructure length scales in metallic alloy casting

Microstructural length scales, such as dendritic spacings in cast metallic alloys, play an essential role in the properties of structural components. Therefore, quantitative prediction of such length scales through simulations is important to design novel alloys and optimize processing conditions through integrated computational materials engineering (ICME). Thus far, quantitative comparisons between experiments and simulations of primary dendrite arms spacings (PDAS) selection in metallic alloys have been mainly limited to directional solidification of thin samples and quantitative phase-field simulations of dilute alloys. In this article, we combine casting experiments and quantitative simulations to present a novel multiscale modeling approach to predict local primary dendritic spacings in metallic alloys solidified in conditions relevant to industrial casting processes. To this end, primary dendritic spacings were measured in instrumented casting experiments in Al–Cu alloys containing 1 wt.% and 4 wt.% of Cu, and they were compared to spacing stability ranges and average spacings in dendritic arrays simulated using phase-field (PF) and dendritic needle network (DNN) models. It is first shown that PF and DNN lead to similar results for the Al-1 wt.%Cu alloy, using a dendrite tip selection constant calculated with PF in the DNN simulations. PF simulations cannot achieve quantitative predictions for the Al-4 wt.%Cu alloy because they are too computationally demanding due to the large separation of scale between tip radius and diffusion length, a characteristic feature of non-dilute alloys. Nevertheless, the results of DNN simulations for non-dilute Al–Cu alloys are in overall good agreement with our experimental results as well as with those of an extensive literature review. Simulations consistently suggest a widening of the PDAS stability range with a decrease of the temperature gradient as the microstructure goes from cellular-dendrites to well-developed hierarchical dendrites.

Uniquely selected primary dendrite arm spacing during competitive growth of columnar grains in Al–Cu alloy

The steady-state value of primary dendrite arm spacing (PDAS) in the columnar dendrites growing between the converging and diverging grain boundaries is investigated by means of quantitative phase-field simulations. The simulations show that there is a unique value of PDAS under a given solidification condition in the system with grain boundaries. This is in contrast to existence of allowable range of PDAS under a given solidification in a system without the grain boundaries, i.e., an infinitely large columnar grain investigated in many early works. Such a unique value of PDAS depends on the pulling speed and inclination angle of the crystal, but not on the initial condition; that is, it is independent of the history of solidification condition. The dependences of the unique value on the pulling speed and inclination angle qualitatively agree with the theoretical models.

Quantitative Phase-field Modeling and Simulations of Solidification Microstructures

Quantitative Phase-field Modeling and Simulations of Solidification Microstructures

Revisiting dynamics and models of microsegregation during polycrystalline solidification of binary alloy

Microsegregation formed during solidification is of great importance to material properties. The conventional Lever rule and Scheil equation are widely used to predict solute segregation. However, these models always fail to predict the exact solute concentration at a high solid fraction because of theoretical assumptions. Here, the dynamics of microsegregation during polycrystalline solidification of refined Al-Cu alloy is studied via two- and three-dimensional quantitative phase-field simulations. Simulations with different grain refinement level, cooling rate, and solid diffusion coefficient demonstrate that solute segregation at the end of solidification (i.e. when the solid fraction is close to unit) is not strongly correlated to the grain morphology and back diffusion. These independences are in accordance with the Scheil equation which only relates to the solid fraction, but the model predicts a much higher liquid concentration than simulations. Accordingly, based on the quantitative phase-field simulations, a new analytical microsegregation model is derived. Unlike the Scheil equation or the Lever rule that respectively overestimates or underestimates the liquid concentration, the present model predicts the liquid concentration in a pretty good agreement with phase-field simulations, particularly at the late solidification stage.

Modeling of dendritic growth using a quantitative nondiagonal phase field model

The phase field method has emerged as the tool of choice to simulate complex pattern formation processes in various domains of materials sciences. For the phase field model to faithfully reproduce the dynamics of a prescribed free-boundary problem with transport equations in the bulk and boundary conditions at the interfaces, the so-called thin-interface limit should be performed. For a phase transformation driven by diffusion, the kinetic cross-coupling between the phase field and the diffusion field has recently been introduced, allowing a control on interface boundary conditions in the general case where the diffusivity in the growing phase ${D}_{S}$ neither vanishes (one-sided model) nor equals the one of the disappearing phase ${D}_{L}$ (symmetric model). Here, we investigate the capabilities of this nondiagonal phase field model in the case of two-dimensional dendritic growth. We benchmark our model with Green's function calculations (sharp-interface model) for the symmetric and one-sided cases, and our results for arbitrary ${D}_{S}/{D}_{L}$ allow us to propose a generalization of the theory by Barbieri and Langer [Phys. Rev. A 39, 5314 (1989)] for finite anisotropy of interface energy. We also perform simulations that evidence the necessity of introducing the kinetic cross-coupling and of eliminating surface diffusion. Our work opens up the way for quantitative phase field simulations of phase transformations with diffusion in the growing phases playing an important role in the pattern and velocity selections.

A Simplified Scaling Law of Cell-Dendrite Transition in Directional Solidification

To describe the cell-dendrite transition (CDT) during directional solidification, a new simplified scaling law is proposed and verified by quantitative phase field simulations. This scaling law bears clear physical foundation with consideration of the overall effects of primary spacing, pulling velocity, and thermal gradient on the onset of sidebranches. The analysis results show that the exponent parameters in this simplified scaling law vary within different systems, which mediates the discrepancy of exponent parameters in previous experiments. The scaling law also presents an explanation for the destabilizing mechanism of thermal gradient in sidebranching dynamics.

A parametric study of morphology selection in equiaxed dendritic solidification

Morphological change of equiaxed dendritic structure in fcc-based alloys associated with transition in preferred growth direction was investigated by means of three-dimensional quantitative phase-field simulations. The anisotropy parameters of interfacial energy ε1 and ε2 were systematically changed to investigate the growth direction and growth morphology. The growth morphologies can be classified into four types; 〈1 0 0〉 growth, 〈1 0 0〉-like hyperbranched growth, 〈1 1 0〉-like hyperbranched growth and 〈1 1 0〉 growth morphologies. An orientation selection map for these morphologies was constructed in a space spanned by ε1 and ε2. Furthermore, dependence of the orientation selection map on solidification condition and alloy system was investigated. When initial supersaturation is large and/or partition coefficient of alloy is small, the region of 〈1 0 0〉 growth, typical growth direction of fcc alloy, diminishes in the orientation selection map.

Investigation of the Orientation of δ Zirconium Hydrides Using Quantitative Phase Field Simulations Supported by Experiments

Light water reactor fuel cladding integrity is sensitive to the morphology of zirconium hydride precipitates,which can vary depending on the thermomechanical history of the cladding. This study provides insights on the orientation and stacking of nanoscale δ hydrides in a single grain of α zirconium using results from a newly developed quantitative phase field model and experimental observations. The model is informed by CALPHAD free energies and incorporates misfit strains due to the volume difference between the two phases, hydrogen in solid solution in the matrix, and thermal expansion. Simulation results on the growth, orientation, and stacking of nanoscale hydrides are supported by SEM observations of recrystallized zirconium.The model accurately predicts the elongated growth of δ hydrides along the basal plane of the α zirconium. Moreover, nanoscale hydrides are observed to stack circumferentially into long mesoscale hydrides. The effect of applied stress on the microstructure is investigated to probe the experimentally observed hydride reorientation phenomenon. This model predicts that hydride reorientation cannot be explained by radial growth of nanoscale hydrides, nor by a change of the most favorable stacking configuration in a single grain, since the applied stress needed to observe these phenomena is significantly greater than experimentally observed.