Quantitative Phase-field Model Research Articles

Quantitative phase-field model to simulate low carbon steel aqueous corrosion phenomena

A quantitative phase-field model has been developed to simulate aqueous corrosion on low carbon steel under chloride-based electrolytes at different pHs. All relevant transport phenomena are included through physically interpretable variables. In particular, both diffusion and migration charge transport are modelled. Separated electrical potentials for the electrolyte and the metal are considered, and physically meaningful boundary conditions are provided. Using this model, different polarization conditions can be predicted and many parameters not observable/challenging during experiments, can be obtained. Finally, since the proposed model includes different time scales, a segregated numerical integration algorithm is proposed to speed up simulations.

Quantitative Phase Field Model for Electrochemical Systems

Modeling microstructure evolution in electrochemical systems is vital for understanding the mechanism of various electrochemical processes. In this work, we propose a general phase field framework that is fully variational and thus guarantees that the energy decreases upon evolution in an isothermal system. The bulk and interface free energies are decoupled using a grand potential formulation to enhance numerical efficiency. The variational definition of the overpotential is used, and the reaction kinetics is incorporated into the evolution equation for the phase field to correctly capture capillary effects and eliminate additional model parameter calibrations. A higher-order kinetic correction is derived to accurately reproduce general reaction models such as the Butler-Volmer, Marcus, and Marcus-Hush-Chidsey models. Electrostatic potentials in the electrode and the electrolyte are considered separately as independent variables, providing additional freedom to capture the interfacial potential jump. To handle realistic materials and processing parameters for practical applications, a driving force extension method is used to enhance the grid size by three orders of magnitude. Finally, we comprehensively verify our phase field model using classical electrochemical theory.

Influence of the temperature gradient and the pulling velocity on solidification cracking susceptibility during welding: A phase field study

The influence of the temperature gradient and the pulling velocity on solidification cracking susceptibility (SCS) during welding is studied by simulating the liquid channel evolution with a quantitative phase field model. Increasing the pulling velocity or decreasing the temperature gradient increases the pressure drop from the dendrite tip to the coalescence point, leading to an increase in SCS. Decreasing the primary dendrite arm spacing (PDAS) decreases the permeability of the liquid channel and the liquid channel length at the same time, resulting in a decrease in the pressure drop and SCS when the PDAS is small. Consideration of the PDAS dependency on the temperature gradient and the pulling velocity influences the value of the pressure drop but does not change the tendency of SCS. The conclusions are also valid for an alloy with strong back diffusion. As the temperature gradient and the pulling velocity within the melt pool are controlled by process parameters, the findings from this work provide a theoretical basis to optimize process parameters to avoid solidification cracking.

Optimizing solidification dendrites and process parameters for laser powder bed fusion additive manufacturing of GH3536 superalloy by finite volume and phase-field method

Investigating the surface morphology and microstructure of laser powder bed fusion (L-PBF) in solidification and optimizing process parameters is crucial for improving the accuracy and performance of cladding parts of GH3536 superalloy. In this study, a computational model incorporating Computational Fluid Dynamics (CFD) and Discrete Element Method (DEM) was utilized to simulate the mesoscale evolution of the molten pool in L-PBF for GH3536 superalloy. Furthermore, the growth of columnar dendrites along the fusion boundary under transient conditions was simulated utilizing a quantitative phase-field model. The impact of scanning speed and laser power on temperature distribution, molten pool size, surface morphology of cladding layer, and columnar dendrite spacing during solidification of single-track L-PBF were examined. The findings suggest that the molten pool undergoes a fast process characterized by significant temperature gradients. Increasing the laser power or reducing the scanning speed results in higher maximum molten pool temperature, as well as increased molten pool depth and width. Overlarge or undersized scanning speed and laser power can result in poor surface quality of the cladding layer. When the laser power was 120 W and the scan speed was 0.4 m/s, the cladding layer exhibited improved surface morphology, with finer columnar dendrites observed at the molten pool's base. This study delved into understanding the evolution characteristics of the molten pool and the growth process of columnar dendrites at the molten pool's base, providing valuable insights for determining appropriate experimental process parameters for L-PBF of GH3536 superalloy.

Variational quantitative phase-field modeling of nonisothermal sintering process.

Phase-field modeling has become a powerful tool in describing the complex pore-structure evolution and the intricate multiphysics in nonisothermal sintering processes. However, the quantitative validity of conventional variational phase-field models involving diffusive processes is a challenge. Artificial interface effects, like the trapping effects, may originate at the interface when the kinetic properties of two opposing phases are different. On the other hand, models with prescribed antitrapping terms do not necessarily guarantee the thermodynamics variational nature of the model. This issue has been solved for liquid-solid interfaces via the development of the variational quantitative solidification phase-field model. However, there is no related work addressing the interfaces in nonisothermal sintering, where the free surfaces between the solid phase and surrounding pore regions exhibit strong asymmetry of mass and thermal properties. Also, additional challenges arise due to the conserved order parameter describing the free surfaces. In this work, we present a variational and quantitative phase-field model for nonisothermal sintering processes. The model is derived via an extended nondiagonal phase-field model. The model evolution equationshave naturally cross-coupling terms between the conserved kinetics (i.e., mass and thermal transfer) and the nonconserved one (grain growth). These terms are shown via asymptotic analysis to be instrumental in ensuring the elimination of interface artifacts, while also examined to not modify the thermodynamic equilibrium condition (characterized by a dihedral angle). Moreover, we demonstrate that the trapping effects and the existence of surface diffusion in conservation laws are direction-dependent. An anisotropic interpolation scheme of the kinetic mobilities that differentiates between the normal and tangential directions along the interface is discussed. Numerically, we demonstrate the importance of the cross-couplings and the anisotropic interpolation by presenting thermal-microstructural evolutions.

Quantitative phase-field model for the isothermal solidification of a stoichiometric compound in a ternary liquid

The phase-field (PF) model is one of the most powerful mathematical models to simulate the solidification of multicomponent alloys. A major challenge faced by the PF model is the treatment of stoichiometric compounds (SCs). In the PF model, compositional derivatives of the free energies of phases need to be treated; however, those of SCs are thermodynamically defined only at a certain composition with respect to the stoichiometric components. Therefore, conventional studies used parabolic functions to provide a pseudo-compositional dependence to free energies. In this paper, we propose a methodology to treat SCs without using the parabolic free energy function. Time-evolution equations were rederived assuming that the free energy of the SC is independent of the composition of its stoichiometric component. We performed numerical calculations for a ternary system comprising two phases, olivine (stoichiometric solid phase) in a basaltic melt (liquid phase), whose free energies were gathered from a thermodynamic database, and we confirmed that the model represents the stoichiometry of the solid phase with high accuracy. This model provides a simple and straightforward methodology to handle SCs in the quantitative PF model.

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.

Modeling of solid-air multi-dendrite growth evolution driven by coupled thermal-solute using non-isothermal quantitative phase field method

The safety hazard posed by residual or infiltrated air condensing into external oxygen-rich solid-air (SA) particles in the liquid hydrogen container deserves attention. In this study, the patterns of morphological evolution, oxygen solute distribution, and growth rate of SA single- and multi-dendrite under various thermal environments are investigated using a non-isothermal quantitative phase field model for the first time. The lattice anisotropy problem of six-fold symmetric dendrites is effectively avoided. The results show that the Type II boundary condition is more suitable for the simulation of dendrite growth, which can avoid the influence of boundary conditions on dendrite growth. In addition, the suitability of the isotropic difference method for simulating the growth of SA multi-dendrites with six-fold symmetry is verified in this paper. The maximum values of solid fraction and oxygen solute concentration within the simulated domain are 0.62 and 0.435, respectively, for an initial subcooling degree of 4 K. The temperature distribution is significantly altered by the application of boundary heat flux and affects the growth pattern of SA dendrites. Compared to the scenario with no boundary heat flux, unilateral heat input and heat extraction decreased the solid fraction of SA dendrites by 31% and increased it by 43%, respectively, while the maximum concentration of oxygen solute was reduced to 144.8% and 97.3%, respectively. When boundary heat flux was applied to all quadrilateral boundaries, the solid fraction under heat extraction and heat input conditions is reduced by 82% and enhanced by 138%, respectively, compared to the scenario with no boundary heat flux. Additionally, the maximum concentration of oxygen solute is found to be 197.6% and 69.5% of that without boundary heat flux, respectively. This study improves the understanding of the evolution of solid-air dendrite growth and can provide theoretical guidance for the safe use of liquid hydrogen systems.

Application of a meshless space-time adaptive approach to phase-field modelling of polycrystalline solidification

We have developed a 2-D numerical meshless adaptive approach for phase-field modelling of dendritic solidification. The quadtree-based approach decomposes the computational domain into quadtree sub-domains of different sizes. The algorithm generates uniformly-distributed computational nodes in each quadtree sub-domain. We apply the meshless radial basis function generated finite difference method and the forward Euler scheme to discretise governing equations in each computational node. The fixed ratio between the characteristic size and the node spacing of a quadtree sub-domain ensures space adaptivity. The adaptive time-stepping accelerates the calculations further. In the framework of previous research studies, we used the approach to solve quantitative phase-field models for single dendrite growth in pure melts and dilute binary alloys. In the present study, we upgrade the solution procedure for the modelling growth of multiple differently oriented dendrites. Along with the space-time adaptive approach, we apply non-linear preconditioning of the phase-field equation to increase computational efficiency. We investigate a novel numerical approach’s accuracy and computational efficiency by simulating the equiaxed dendrite growth from a dilute binary alloy.

Phase-field study of polycrystalline growth and texture selection during melt pool solidification

Grain growth competition during solidification determines microstructural features, such as dendritic arm spacings, segregation pattern, and grain texture, which have a key impact on the final mechanical properties. During metal additive manufacturing (AM), these features are highly sensitive to manufacturing conditions, such as laser power and scanning speed. The melt pool (MP) geometry is also expected to have a strong influence on microstructure selection. Here, taking advantage of a computationally efficient multi-GPU implementation of a quantitative phase-field model, we use two-dimensional cross-section simulations of a shrinking MP during metal AM, at the scale of the full MP, in order to explore the resulting mechanisms of grain growth competition and texture selection. We explore MPs of different aspect ratios, different initial (substrate) grain densities, and repeat each simulation several times with different random grain distributions and orientations along the fusion line in order to obtain a statistically relevant picture of grain texture selection mechanisms. Our results show a transition from a weak to a strong ⟨10⟩ texture when the aspect ratio of the melt pool deviates from unity. This is attributed to the shape and directions of thermal gradients during solidification, and seems more pronounced in the case of wide melt pools than in the case of a deeper one. The texture transition was not found to notably depend upon the initial grain density along the fusion line from which the melt pool solidifies epitaxially.

Prediction of Primary Dendrite Arm Spacing of the Inconel 718 Deposition Layer by Laser Cladding Based on a Multi-Scale Simulation.

Primary dendrite arm spacing (PDAS) is a crucial microstructural feature in nickel-based superalloys produced by laser cladding. In order to investigate the effects of process parameters on PDAS, a multi-scale model that integrates a 3D transient heat and mass transfer model with a quantitative phase-field model was proposed to simulate the dendritic growth behavior in the molten pool for laser cladding Inconel 718. The values of temperature gradient (G) and solidification rate (R) at the S/L interface of the molten pool under different process conditions were obtained by multi-scale simulation and used as input for the quantitative phase field model. The influence of process parameters on microstructure morphology in the deposition layer was analyzed. The result shows that the dendrite morphology is in good agreement with the experimental result under varying laser power (P) and scanning velocity (V). PDAS was found to be more sensitive to changes in laser scanning velocity, and as the scanning velocity decreased from 12 mm/s to 4 mm/s, the PDAS increased by 197% when the laser power was 1500 W. Furthermore, smaller PDAS can be achieved by combining higher scanning velocity with lower laser power.

Study of the peritectic phase transformation kinetics with elastic effect in the Fe–C system by quantitative phase-field modeling

A quantitative phase-field model developed for two-phase growth process (Folch and Plapp, 2005) is applied to the simulation of peritectic phase transformation in a Fe–C binary alloy system with elastic effects due to the misfit on the solid/solid interface. The model includes the anti-trapping current, elastic effect, and the quantitative model parameters estimated based on the thin-interface limit. The diffusion of carbon in the solid phases (i.e. δ and γ phases) has not been neglected and an appropriate free-energy functional has been developed such that there is no presence of any extra phase in the interface. The model’s behavior over a wide range of interface width has been studied and the effect of anti-trapping current and elastic effect on the phase transformation has been closely analyzed.

Quantitative Phase Field Modeling of Li Dendrite Growth

Modeling and understanding dendrite growth on lithium metal anodes during charging is vital for improving battery safety. The phase field method has become an increasingly popular tool to achieve this due to its flexibility in handling complex morphologies and coupled physical processes. However, quantitative phase field models under realistic electrochemical conditions are rare. In this work, we derive a thermodynamically consistent phase field model in the grand potential framework. The design of this model emphasizes quantitative modeling of electrodeposition with experimental loading conditions and realistic length and time scales. First, electrostatic potentials in both the electrode and the electrolyte are considered to account for the interfacial potential difference. Second, the electrochemical driving force can be large even with an applied voltage of a few hundred millivolts; therefore, the grid size usually must be as small as a few nanometers to maintain a stable phase field profile. This hinders the modeling of system sizes of practical interest. To solve this problem, a mapping approach is introduced to maintain stability with moderate grid sizes and realistic applied voltages. Third, we thoroughly compare the phase field model with results from a sharp interface model for various cases that permit analytical solutions. Numerical studies demonstrate that the proposed model can quantitatively reproduce the sharp interface results. Moreover, lithium has highly anisotropic interfacial energy, which leads to corners on the equilibrium electrode-electrolyte interface shape. A convexification method is used to remove the ill-posedness due to this strong anisotropy. Finally, a parallel solver is developed to handle large systems. The overall framework is demonstrated by studying lithium dendrite morphologies under various conditions. In future work, the model can be used as a computational tool for the high-throughput screening process in battery design.

An efficient and quantitative phase-field model for elastically heterogeneous two-phase solids based on a partial rank-one homogenization scheme

This paper presents an efficient and quantitative phase-field model for elastically heterogeneous alloys that ensures the two mechanical compatibilities—static and kinematic, in conjunction with chemical equilibrium within the interfacial region. Our model contrasts with existing phase-field models that either violate static compatibility or interfacial chemical equilibrium or are computationally costly. For computational efficiency, the partial rank-one homogenization (PRH) scheme is employed to enforce both static and kinematic compatibilities at the interface. Moreover, interfacial chemical equilibrium is ensured by replacing the composition field with diffusion potential field as the independent variable of the model. Its performance is demonstrated by simulating four single-particle and one multi-particle cases for two binary two-phase alloys: Ni–Al γ′/γ and UO2/void. Its accuracy is then investigated against analytical solutions. For the single-particle γ′/γ alloy, we find that the accuracy of the phase-field results remain unaffected for both planar and non-planar geometries, when the PRH scheme is employed. Fortuitously, in the UO2/void simulations, despite a strong elastic heterogeneity – the ratio of Young’s modulus of the void phase to that of the UO2 phase is 10−4 – we find that the PRH scheme shows significantly better convergence compared to the Voigt–Taylor scheme (VTS) for both planar and non-planar geometries. Nevertheless, for the same interface width range as in the γ′/γ case, the interface migration in these simulations shows dependence on interface width. Contrary to the γ′/γ simulations, we also find that the simulated elastic fields show deviations from the analytical solution in the non-planar UO2/void case using the PRH scheme.

Isotropic finite-difference approximations for phase-field simulations of polycrystalline alloy solidification

Phase-field models of microstructural pattern formation during alloy solidification are commonly solved numerically using the finite-difference method, which is ideally suited to carry out computationally efficient simulations on massively parallel computer architectures such as Graphic Processing Units. However, one known drawback of this method is that the discretization of differential terms involving spatial derivatives introduces a spurious lattice anisotropy that can influence the solid-liquid interface dynamics. We find that this influence is significant for the case of polycrystalline dendritic solidification, where the crystal axes of different grains do not generally coincide with the reference axes of the finite-difference lattice. In particular, we find that with the commonly used finite-difference implementation of the quantitative phase-field model of binary alloy solidification, both the operating state of the dendrite tip and the dendrite growth orientation are strongly affected by the lattice anisotropy. To circumvent this problem, we use known methods in both real and Fourier space to derive finite-difference approximations of leading differential terms in 2D and 3D that are isotropic at order h2 of the lattice spacing h. Importantly, those terms include the divergence of the anti-trapping current that is found to have a critical influence on pattern selection. The 2D and 3D discretizations use an approximated form of the anti-trapping current that facilitates the Fourier-space derivation of the associated isotropic differential operator at O(h2), but we also derive a 2D discretization of the standard form of this current. Finally, we present 2D and 3D phase-field simulations of alloy solidification, showing that the isotropic finite-difference implementations dramatically reduce spurious lattice anisotropy effects, yielding both the tip operating state and growth direction of the dendrite that are nearly independent of the angle between the crystal and finite-difference lattice axes.

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.

Time invariance of three-dimensional morphology of equiaxed dendrite: A phase-field study

Dendrite morphology has a significant effect on solute segregation and fluid flow in bulk metallic materials. Therefore, the detailed morphological evolution of dendrites is important to better understand these processes. Recently, three-dimensional (3D) dendrite morphology has been analyzed using interface shape distribution (ISD) maps that are spanned by the curvedness and shape factor of the local solid−liquid interface in an Al−Cu alloy. Data were collected through in-situ observations using synchrotron radiation imaging techniques [J.W. Gibbs et al.: Sci. Rep., 5 (2015) 11824]. This methodology is quite effective for describing 3D dendrites. In this study, we thoroughly investigated the morphological evolution and the related ISD map of free-growing equiaxed dendrites in an Al−3mass%Cu alloy using a quantitative phase-field model. The ISD was found to differ depending on the degree of undercooling. Importantly, the results indicate the presence of a time-invariant feature after sufficient branching and growth of secondary arms, when the degree of undercooling is substantial enough to produce a bunch of branching. The time invariance is considered a universal feature of equiaxed dendrite growth.

Quantitative phase-field modeling of directional growth with kinetic effects in quasi-rapid solidification

Directional growth in quasi-rapid solidification was simulated by using a quantitative phase-field model to investigate the roles of the kinetic effects on the growth pattern selection and solute segregation. A convergence study was carried out to choose proper the interface thickness in numerical simulations. Results show that the stable range of primary spacing with kinetic effects is larger than that without them. The inclusion of the kinetic effects leads to the solute enrichment in the interdendritic region, indicating that kinetic effects can be involved in phase-field simulations for accurate predicted results.

Investigation of δ zirconium hydride morphology in a single crystal using quantitative phase field simulations supported by experiments

• A quantitative phase field model of the α -Zr/ δ -ZrH system was developed. • Elastic anisotropy influences hydride shape, hydrogen distribution, and hydride interaction. • Circumferential stacking of nanoscale hydrides is predicted. • The model’s predictions successfully compare with experimental observations. In light water nuclear reactors , waterside corrosion of the cladding material leads to the production of hydrogen, a fraction of which is picked up by the zirconium cladding and precipitates into brittle hydride particles. These nanoscale hydride particles aggregate into mesoscale hydride clusters. The principal stacking direction of the nanoscale hydrides precipitated in the cladding tube changes from circumferential in the absence of applied stress to radial under circumferential applied stress. A quantitative phase field model has been developed to predict the hydride morphology observed experimentally and identify the mechanisms responsible for nanoscale hydride stacking. The model focuses on nanoscale hydride precipitation in a single zirconium grain with a detailed description of the anisotropic elastic contribution. The model predictions concerning the shape, orientation, and stacking behavior of nanoscale hydride are analyzed and compared with experimental observations. The model accurately accounts for the experimentally observed elongated nanoscale hydride shape and the stacking of hydrides along the basal plane of the hexagonal zirconium matrix. When investigating the role of applied stress in hydride morphology, the model challenges some of the mechanisms previously proposed to explain hydride reorientation. Although hydride reorientation has been hypothesized to be caused by a change in nanoscale hydride shape, the current study shows that these mechanisms are unlikely to occur.

The Morphology and Solute Segregation of Dendrite Growth in Ti-4.5% Al Alloy: A Phase-Field Study

Ti-Al alloys have excellent high-temperature performance and are often used in the manufacture of high-pressure compressors and low-pressure turbine blades for military aircraft engines. However, solute segregation is easy to occur in the solidification process of Ti-Al alloys, which will affect their properties. In this study, we used the quantitative phase-field model developed by Karma to study the equiaxed dendrite growth of Ti-4.5% Al alloy. The effects of supersaturation, undercooling and thermal disturbance on the dendrite morphology and solute segregation were studied. The results showed that the increase of supersaturation and undercooling will promote the growth of secondary dendrite arms and aggravate the solute segregation. When the undercooling is large, the solute in the root of the primary dendrite arms is seriously enriched, and when the supersaturation is large, the time for the dendrite tips to reach a steady-state will be shortened. The thermal disturbance mainly affects the morphology and distribution of the secondary dendrite arms but has almost no effect on the steady-state of the primary dendrite tips. This is helpful to understand the cause of solute segregation in Ti-Al alloy theoretically.