Phase-field model for binary alloys

Mesomorphology of clathrate hydrates from molecular ordering

Clathrate hydrates are crystals formed by guest molecules that stabilize cages of hydrogen-bonded water molecules. Whereas thermodynamic equilibrium is well described via the van der Waals and Platteeuw approach, the increasing concerns with global warming and energy transition require extending the knowledge to non-equilibrium conditions in multiphase, sheared systems, in a multiscale framework. Potential macro-applications concern the storage of carbon dioxide in the form of clathrates, and the reduction of hydrate inhibition additives currently required in hydrocarbon production. We evidence porous mesomorphologies as key to bridging the molecular scales to macro-applications of low solubility guests. We discuss the coupling of molecular ordering with the mesoscales, including (i) the emergence of porous patterns as a combined factor from the walk over the free energy landscape and 3D competitive nucleation and growth and (ii) the role of molecular attachment rates in crystallization-diffusion models that allow predicting the timescale of pore sealing. This is a perspective study that discusses the use of discrete models (molecular dynamics) to build continuum models (phase field models, crystallization laws, and transport phenomena) to predict multiscale manifestations at a feasible computational cost. Several advances in correlated fields (ice, polymers, alloys, and nanoparticles) are discussed in the scenario of clathrate hydrates, as well as the challenges and necessary developments to push the field forward.

Experimental-numerical Investigation of ⍺/β-phase formation within thin electron beam melted Ti-6Al-4V

Electron beam melting is a powder bed fusion process capable of manufacturing thin structural features. However, as the thickness of these features approaches typical microstructure grain sizes, it becomes vital to understand how the manufacturing process contributes to local crystallographic texture and anisotropy in micromechanical response. Therefore, this article investigates Ti-6Al-4V ⍺/β-phase formation within thin components using a variety of experimental and numerical approaches. Optical and scanning electron microscopy are used to determine through-thickness distributions of prior-β width ([top, middle, bottom]:[81.2 ± 44.2, 76.02 ± 30.4, 75.6 ± 31.2] μm), ⍺-lath thickness ([top, middle, bottom]:[1.0 ± 1.3, 1.3 ± 1.2, 1.4 ± 1.8] μm; average), and ⍺/β-phase fractions ([top, middle, bottom]:[0.87 ± 0.05, 0.82 ± 0.03, 0.88 ± 0.03]; average). Manufacturing process (i.e., "logfile") data is used within a layer-by-layer finite element "birth/death" model. This model is loosely coupled with the Kim-Kim-Suzuki phase field model and a CALPHAD thermodynamic database to predict ⍺-lath growth throughout the process. In general, good correlation is found between the experimental data and the predicted temperature history, ⍺-lath coarsening, and phase fraction. This indicates that these tools would be useful in predicting process-structure-properties-performance relationships for thin features.

Deterministic Formation and Growth of Dendritic Crystals of Attractive Colloids

Dendrites are ubiquitous crystals produced in supersaturated solutions and supercooled melts, but considerably less is known about their formation and growth kinetics. Here, the key factors are explored that dictate dendrite formation and growth, utilizing experimental colloidal models in which the particles act as molecules with Mie potential. Depletion attraction is employed to colloids and manipulate their strength to control supersaturation. Dendrites are predominantly produced under conditions of low supersaturation, where the separation between crystals is large due to slow nucleation. The dendrites do not emerge directly from nuclei. Instead, isotropic grains, initially produced from nuclei, morph into polygons. Arms then sprout from the vertices of these polygons, eventually giving rise to dendrites. Triggering this polygon-to-dendrite transformation requires a high diffusional flux. This necessitates a prolonged diffusion time to maintain a steep concentration gradient in the surrounding environment even after the transformation from circular grains to polygons.

Phase field modelling of hopper crystal growth in alloys

Here we use phase field to model and simulate "hopper" crystals, so named because of their underlying cubic structure but with a hopper-like depression on each of the six faces. Over the past three decades simulations of single phase solidification have successfully explored dendritic structures, in two and three dimensions, formed under high undercooling from a slight perturbation in anisotropy. More recently we see the modelling of faceted structures at near equilibrium, and also, under high undercooling, the formation of dendritic-like structures in two dimensions which retain some faceting in the dendrite arms. A cubic hopper crystal appears to be a hybrid structure, somewhere between a perfect cube and a dendrite, and, to date, has not appeared in the modelling literature. In this paper we describe a model for faceted cubic growth and explore results, necessarily in three dimensions, that include perfect cube, hopper and dendritic. We also touch briefly on one other morphology-octahedral.

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 equations have 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.

Morphological instability of solid tumors in a nutrient-deficient environment

A phenomenological reaction-diffusion model that includes a nutrient-regulated growth rate of tumor cells is proposed to investigate the morphological instability of solid tumors during the avascular growth. We find that the surface instability could be induced more easily when tumor cells are placed in a harsher nutrient-deficient environment, while the instability is suppressed for tumor cells in a nutrient-rich environment due to the nutrient-regulated proliferation. In addition, the surface instability is shown to be influenced by the growth moving speed of tumor rims. Our analysis reveals that a larger growth movement of the tumor front results in a closer proximity of tumor cells to a nutrient-rich region, which tends to inhibit the surface instability. A nourished length that represents the proximity is defined to illustrate its close relation to the surface instability.

Phase-field modeling of pitting and mechanically-assisted corrosion of Mg alloys for biomedical applications

A phase-field model is developed to simulate the corrosion of Mg alloys in body fluids. The model incorporates both Mg dissolution and the transport of Mg ions in solution, naturally predicting the transition from activation-controlled to diffusion-controlled bio-corrosion. In addition to uniform corrosion, the presented framework captures pitting corrosion and accounts for the synergistic effect of aggressive environments and mechanical loading in accelerating corrosion kinetics. The model applies to arbitrary 2D and 3D geometries with no special treatment for the evolution of the corrosion front, which is described using a diffuse interface approach. Experiments are conducted to validate the model and a good agreement is attained against in vitro measurements on Mg wires. The potential of the model to capture mechano-chemical effects during corrosion is demonstrated in case studies considering Mg wires in tension and bioabsorbable coronary Mg stents subjected to mechanical loading. The proposed methodology can be used to assess the in vitro and in vivo service life of Mg-based biomedical devices and optimize the design taking into account the effect of mechanical deformation on the corrosion rate. The model has the potential to advocate further development of Mg alloys as a biodegradable implant material for biomedical applications. STATEMENT OF SIGNIFICANCE: A physically-based model is developed to simulate the corrosion of bioabsorbable metals in environments that resemble biological fluids. The model captures pitting corrosion and incorporates the role of mechanical fields in enhancing the corrosion of bioabsorbable metals. Model predictions are validated against dedicated in vitro corrosion experiments on Mg wires. The potential of the model to capture mechano-chemical effects is demonstrated in representative examples. The simulations show that the presence of mechanical fields leads to the formation of cracks accelerating the failure of Mg wires, whereas pitting severely compromises the structural integrity of coronary Mg stents. This work extends phase-field modeling to bioengineering and provides a mechanistic tool for assessing the service life of bioabsorbable metallic biomedical devices.

Finite element analysis of FGM dental crowns using phase-field approach

Functionally graded materials (FGMs) - categorized in advanced composite materials - are specially designed to reduce the stresses and failure due to material mismatches. Advances in manufacturing techniques have brought FGMs into use in a variety of applications. However, the numerical analysis is still challenging due to the difficulties in simulations of non-homogeneous material domains of complex parts. Presenting a numerical procedure that both facilitates the implementation of material non-homogeneity in geometrically complex mediums, and increases the accuracy of the calculations using a phase-field approach, this study investigates the usage of FGMs in dental prostheses. For this purpose, a porcelain fused to metal (PFM) mandibular first molar FGM crown is simulated and analyzed under the maximum masticatory bite force, and eventually the results are compared to a PFM crown prepared conventionally.

A Phase-Field Model for In-Space Manufacturing of Binary Alloys

The integrity of the final printed components is mostly dictated by the adhesion between the particles and phases that form upon solidification, which is a major problem in printing metallic parts using available In-Space Manufacturing (ISM) technologies based on the Fused Deposition Modeling (FDM) methodology. Understanding the melting/solidification process helps increase particle adherence and allows to produce components with greater mechanical integrity. We developed a phase-field model of solidification for binary alloys. The phase-field approach is unique in capturing the microstructure with computationally tractable costs. The developed phase-field model of solidification of binary alloys satisfies the stability conditions at all temperatures. The suggested model is tuned for Ni-Cu alloy feedstocks. We derived the Ginzburg-Landau equations governing the phase transformation kinetics and solved them analytically for the dilute solution. We calculated the concentration profile as a function of interface velocity for a one-dimensional steady-state diffuse interface neglecting elasticity and obtained the partition coefficient, k, as a function of interface velocity. Numerical simulations for the diluted solution are used to study the interface velocity as a function of undercooling for the classic sharp interface model, partitionless solidification, and thin interface.

A probabilistic computational framework for the prediction of corrosion-induced cracking in large structures

Corrosion can initiate cracking that leads to structural integrity reduction. Quantitative corrosion assessment is challenging, and the modeling of corrosion-induced crack initiation is essential for model-based corrosion reliability analysis of various structures. This paper proposes a probabilistic computational analysis framework for corrosion-to-crack transitions by integrating a phase-field model with machine learning and uncertainty quantification. An electro-chemo-mechanical phase-field model is modified to predict pitting corrosion evolution, in which stress is properly coupled into the electrode chemical potential. A crack initiation criterion based on morphology is proposed to quantify the pit-to-cracking transition. A spatiotemporal surrogate modeling method is developed to facilitate this, consisting of a Convolution Neural Network (CNN) to map corrosion morphology to latent spaces, and a Gaussian Process regression model with a nonlinear autoregressive exogenous model (NARX) architecture for prediction of corrosion dynamics in the latent space over time. It enables the real-time prediction of corrosion morphology and crack initiation behaviors (whether, when, and where the corrosion damage triggers the crack initiation), and thus makes it possible for probabilistic analysis, with uncertainty quantified. Examples at various stress and corrosion conditions are presented to demonstrate the proposed computational framework.

Macro-Micro-Coupling Simulation and Space Experiment Study on Zone Melting Process of Bismuth Telluride-Based Crystal Materials

Zone melting is one of the main techniques for preparing bismuth telluride-based crystal thermoelectric materials. In this research, a macro-micro-coupled simulation model was established to analyze the distribution of temperature and heat flow during the zone melting process. The simulation results show the melting temperature tends to affect the length of the melting zone, while the moving velocity of the melting furnace tends to affect the curvature of the melting and solidification interface. There are two small plateaus observed in the temperature curve of the central axis of bismuth telluride ingot when the moving velocity of the heat source is higher than 20 mm/h. As the moving velocity of the heat source increases, the platform effect is becoming more obvious. Based on the simulation results, the zone melt experiments were carried out both under microgravity condition on the Tiangong II space laboratory and conventional gravity condition on the ground. The experimental results indicate that the bismuth telluride-based crystal prepared in microgravity tends to possess more uniform composition. This uniform composition will lead to more uniform thermoelectric performance for telluride-based crystals. In the space condition, the influence of surface tension is much higher than that of gravity. The bismuth telluride ingot is very vulnerable to the influence of surface tension on the surface morphology during the solidification process. If the solidification process is not well controlled, it will be easier to produce uneven surface morphology.

Phase field modeling for the morphological and microstructural evolution of metallic materials under environmental attack

The complex degradation of metallic materials in aggressive environments can result in morphological and microstructural changes. The phase-field (PF) method is an effective computational approach to understanding and predicting the morphology, phase change and/or transformation of materials. PF models are based on conserved and non-conserved field variables that represent each phase as a function of space and time coupled with time-dependent equations that describe the mechanisms. This report summarizes progress in the PF modeling of degradation of metallic materials in aqueous corrosion, hydrogen-assisted cracking, high-temperature metal oxidation in the gas phase and porous structure evolution with insights to future applications.

Multiphase-field model for surface diffusion and attachment kinetics in the grand-potential framework

Grand-potential based multiphase-field model is extended to include surface diffusion. Diffusion is elevated in the interface through a scalar degenerate term. In contrast to the classical Cahn-Hilliard-based formulations, the present model circumvents the related difficulties in restricting diffusion solely to the interface by combining two second-order equations, an Allen-Cahn-type equation for the phase field supplemented with an obstacle-type potential and a conservative diffusion equation for the chemical potential or composition evolution. The sharp interface limiting behavior of the model is deduced by means of asymptotic analysis. A combination of surface diffusion and finite attachment kinetics is retrieved as the governing law. Infinite attachment kinetics can be achieved through a minor modification of the model, and with a slight change in the interpretation, the same model handles the cases of pure substances and alloys. Relations between model parameters and physical properties are obtained which allow one to quantitatively interpret simulation results. An extensive study of thermal grooving is conducted to validate the model based on existing theories. The results show good agreement with the theoretical sharp-interface solutions. The obviation of fourth-order derivatives and the usage of the obstacle potential make the model computationally cost-effective.

Phase-field-simplified lattice Boltzmann method for modeling solid-liquid phase change

This article proposes a phase-field-simplified lattice Boltzmann method (PF-SLBM) for modeling solid-liquid phase change problems within a pure material. The PF-SLBM consolidates the simplified lattice Boltzmann method (SLBM) as the flow solver and the phase-field method as the interface tracking algorithm. Compared with conventional lattice Boltzmann modelings, the SLBM shows advantages in memory cost, boundary treatment, and numerical stability, and thus is more suitable for the present topic which includes complex flow patterns and fluid-solid boundaries. In contrast to the sharp interface approach, the phase-field method utilized in this work represents a diffuse interface strategy and is more flexible in describing complicated fluid-solid interfaces. Through abundant benchmark tests, comprehensive validations of the accuracy, stability, and boundary treatment of the proposed PF-SLBM are carried out. The method is then applied to the simulations of partially melted or frozen cavities, which sheds light on the potential of the PF-SLBM in resolving practical problems.

Uncertainty Quantification in Atomistic Modeling of Metals and Its Effect on Mesoscale and Continuum Modeling: A Review

The design of next-generation alloys through the integrated computational materials engineering (ICME) approach relies on multiscale computer simulations to provide thermodynamic properties when experiments are difficult to conduct. Atomistic methods such as density functional theory (DFT) and molecular dynamics (MD) have been successful in predicting properties of never before studied compounds or phases. However, uncertainty quantification (UQ) of DFT and MD results is rarely reported due to computational and UQ methodology challenges. Over the past decade, studies that mitigate this gap have emerged. These advances are reviewed in the context of thermodynamic modeling and information exchange with mesoscale methods such as the phase-field method (PFM) and calculation of phase diagrams (CALPHAD). The importance of UQ is illustrated using properties of metals, with aluminum as an example, and highlighting deterministic, frequentist, and Bayesian methodologies. Challenges facing routine uncertainty quantification and an outlook on addressing them are also presented.

Grand-potential based phase-field model for systems with interstitial sites

Existing grand-potential based multicomponent phase-field model is extended to handle systems with interstitial sublattice. This is achieved by treating the concentration of alloying elements in site-fraction. Correspondingly, the chemical species are distinguished based on their lattice positions, and their mode of diffusion, interstitial or substitutional, is appropriately realised. An approach to incorporate quantitative driving-force, through parabolic approximation of CALPHAD data, is introduced. By modelling austenite decomposition in ternary Fe–C–Mn, albeit in a representative microstructure, the ability of the current formalism to handle phases with interstitial components, and to distinguish interstitial diffusion from substitutional in grand-potential framework is elucidated. Furthermore, phase transformation under paraequilibrium is modelled to demonstrate the limitation of adopting mole-fraction based formulation to treat multicomponent systems.

Phase-field simulation of microstructure evolution in electron beam additive manufacturing

Electron beam additive manufacturing (EBAM) is an emerging additive manufacturing technology with extremely high energy beam. The rapid solidification in the molten pool is of interest but not fully understood. In EBAM, with both large thermal gradient and cooling rate, the microstructure evolution during solidification is difficult to be described. The quantitative multi-phase-field model provides an effective way to reveal the dynamic evolution of dendrites in the molten pool of EBAM. In this study, the thermal profile is interpolated from the macroscale simulation at each time-step, to couple the realistic thermal evolution in the molten pool. The microstructure evolution and competitive growth have been investigated in details. Simulations of dendrite arrays with the same orientation showed how the growth velocity and the primary spacing of columnar dendrites depend on thermal gradient and cooling rate. The results are in agreement with theoretical models qualitatively. Moreover, the Gaussian nucleation model was introduced so as to give a better prediction of the microstructure in EBAM.

Phase-field modeling of complex dendritic structures in constrained growth of hexagonal close-packed crystals

We perform the phase-field modeling to investigate the growth pattern selections of the complex dendritic structures in constrained growth with different solidification and orientation conditions. The results show that hexagonal close-packed (hcp) crystals emerge as dendritic and cellular arrays in different planes, originating from the specific hcp anisotropy that allows different growth preferences between the basal and cylindrical planes. A morphological transition of the titled dendrites to tip-splitting dendrites arises reflecting the competition between the preferred orientation induced primary growth and the misorientation induced sidebranching formation. Furthermore, the dendritic patterns exhibit sharper tips and the more significant sidebranches, while the cellular pattern is changed from the symmetric cells to the tip-splitting cells, and to seaweeds with the increase of anisotropy strength, indicating the competitive mechanism of the in-plane anisotropy induced growth promotion and the out-plane anisotropy induced growth restriction. We expect to understand the growth competition, the morphology selection, as well as the orientation dependence of the complex dendritic structures in the three-dimensional (3D) constrained growth.

Phase-field model of oxidation: Kinetics

The kinetics of oxidation is examined using a phase-field model of electrochemistry when the oxide film is smaller than the Debye length. As a test of the model, the phase-field approach recovers the results of classical Wagner diffusion-controlled oxide growth when the interfacial mobility of the oxide-metal interface is large and the films are much thicker than the Debye length. However, for small interfacial mobilities, where the growth is reaction controlled, we find that the film increases in thickness linearly in time, and that the phase-field model naturally leads to an electrostatic overpotential at the interface that affects the prefactor of the linear growth law. Since the interface velocity decreases with the distance from the oxide vapor, for a fixed interfacial mobility, the film will transition from reaction- to diffusion-controlled growth at a characteristic thickness. For thin films, we find that in the limit of high interfacial mobility we recover a Wagner-type parabolic growth law in the limit of a composition-independent mobility. A composition-dependent mobility leads to a nonparabolic kinetics at small thickness, but for the materials parameters chosen, the deviation from parabolic kinetics is small. Unlike classical oxidation models, we show that the phase-field model can be used to examine the dynamics of nonplanar oxide interfaces that are routinely observed in experiment. As an illustration, we examine the evolution of nonplanar interfaces when the oxide is growing only by anion diffusion and find that it is morphologically stable.

Phase-Field Simulation of Grain Boundary Evolution In Microstructures Containing Second-Phase Particles with Heterogeneous Thermal Properties

Understanding the interaction between complex thermal fields and metallic structures at the meso-scale is crucial for the prediction of microstructural evolution during thermomechanical processing. The competitive growth of crystal grains, driven by thermodynamic forces at the grain boundaries, is one of the most fundamental phenomena in metallurgy and solid state physics. The presence of second phase particles, which act as pinning sites for boundaries, drastically alters the coarsening behaviour of the system; particularly when considering that these particles have different thermal properties to the primary phase. In this work a multi-phase field model, incorporating thermal gradient and curvature driving forces, is used to predict grain growth in a Ti6Al4V alloy system with second phase particle inclusions representative of oxide and carbide precipitates. The multi-phase field framework is fully coupled to the heat equation. The incorporation of the thermal gradient driving force enables the detailed behaviour of the grain boundaries around the particles to be predicted. It is shown that the inclusion of particles with a lower thermal conductivity has a significant influence on the coarsening behaviour of various systems of grains, due to the combined effects of thermal shielding and the generation of thermal gradient driving forces between the boundaries and pinning particles.

Phase field model of uranium carbide solidification through a combined KKS and orientation field approach

A Phase Field model is developed combining the Orientation Field approach to modeling solidification with the Kim, Kim, Suzuki method of modeling binary alloys. These combined methods produce a model capable of simulating randomly oriented second phase dendrites with discrete control of the solid-liquid interface energy and thickness. The example system of carbon in a liquid uranium (U) melt is used as a test for the model. The formation of uranium carbide within a liquid U melt is simulated for isothermal conditions and compares well with experiments.

The origin and stability of nanostructural hierarchy in crystalline solids

The structural hierarchy exhibited by materials on more than one length scale can play a major part in determining bulk material properties. Understanding the hierarchical structure can lead to new materials with physical properties tailored for specific applications. We have used a combined experimental and phase-field modeling approach to explore such a hierarchical structure at nanoscale for enhanced coarsening resistance of ordered γ' precipitates in an experimental, multicomponent, high-refractory nickel-base superalloy. The hierarchical microstructure formed experimentally in this alloy is composed of a γ matrix with γ' precipitates that contain embedded, spherical γ precipitates, which do not directionally coarsen during high-temperature annealing but do delay coarsening of the larger γ' precipitates. Chemical mapping via atom probe tomography suggests that the supersaturation of Co, Ru, and Re in the γ' phase is the driving force for the phase separation, leading to the formation of this hierarchical microstructure. Representative phase-field modeling highlights the importance of larger γ' precipitates to promote stability of the embedded γ phase and to delay coarsening of the encompassing γ' precipitates. Our results suggest that the hierarchical material design has the potential to influence the high-temperature stability of precipitate strengthened metallic materials.

Phase-field model for growth and dissolution of a stoichiometric compound in a binary liquid

We propose a simple formulation of the phase-field model for a stoichiometric compound growing in a binary liquid. In previous models, chemical free energies of stoichiometric compounds have been approximated by parabolic functions of composition; however, the curvature has been determined arbitrarily in spite of the fact that the stoichiometric composition was undesirably modified depending on the curvature. To avoid this uncertainty, we supposed that the chemical free energy of the stoichiometric compound is represented by a single value at a given temperature and derived the phase-field equations without the parabolic free-energy approximation. The phase mobility was derived both for diffusion- and interface-controlled solidification based on a thin interface limit analysis. We carried out numerical simulations of one-dimensional calculations both in diffusion-controlled and interface-controlled cases and found that the growth velocities agreed with the analytic predictions. We also examined two-dimensional solidification of a circular crystal and confirmed that the equilibrium state was shifted, as suggested by the Gibbs-Thomson effect. This study is an important step for phase-field modeling that includes stoichiometric compounds with their accurate thermodynamic properties.

Multiscale Modeling and Simulation of Directional Solidification Process of Ni-Based Superalloy Turbine Blade Casting

Ni-based superalloy turbine blades have become indispensable structural parts in modern gas engines. An understanding of the solidification behavior and microstructure formation in directional solidified turbine blades is necessary for improving their high-temperature performance. The multiscale simulation model was developed to simulate the directional solidification process of superalloy turbine blades. The 3D cellular automaton-finite difference (CA-FD) method was used to calculate heat transfer and grain growth on the macroscopic scale, while the phase-field method was developed to simulate dendrite growth on the microscopic scale. Firstly, the evolution of temperature field of an aero-engine blade and a large industrial gas turbine blade was studied under high-rate solidification (HRS) and liquid-metal cooling (LMC) solidification processes. The varying withdrawal velocity was applied to change the curved mushy zone to a flat shape. Secondly, the grain growth in the aero-engine blade was simulated, and the grain structures in the starter block part and the spiral selector part in the HRS process were compared with those in the LMC process. The simulated grain structures were generally in agreement with experimental results. Finally, the dendrite growth in the typical HRS and LMC solidification process was investigated and the simulation results were compared with the experimental results in terms of dendrite morphology and primary dendritic spacing.

Grand-potential-based phase-field model for multiple phases, grains, and chemical components

Grand-potential-based phase-field model for multiple phases, grains, and chemical components is derived from a grand-potential functional. Due to the grand-potential formulation, the chemical energy does not contribute to the interfacial energy between phases, simplifying parametrization and decoupling interface thickness from interfacial energy, which can potentially allow increased interface thicknesses and therefore improved computational efficiency. Two-phase interfaces are stable with respect to the formation of additional phases, simplifying implementation and allowing the variational form of the evolution equations to be used. Additionally, we show that grand-potential-based phase-field models are capable of simulating phase separation, and we derive conditions under which this is possible.

Phase-field model of pitting corrosion kinetics in metallic materials

Pitting corrosion is one of the most destructive forms of corrosion that can lead to catastrophic failure of structures. This study presents a thermodynamically consistent phase field model for the quantitative prediction of the pitting corrosion kinetics in metallic materials. An order parameter is introduced to represent the local physical state of the metal within a metal-electrolyte system. The free energy of the system is described in terms of its metal ion concentration and the order parameter. Both the ion transport in the electrolyte and the electrochemical reactions at the electrolyte/metal interface are explicitly taken into consideration. The temporal evolution of ion concentration profile and the order parameter field is driven by the reduction in the total free energy of the system and is obtained by numerically solving the governing equations. A calibration study is performed to couple the kinetic interface parameter with the corrosion current density to obtain a direct relationship between overpotential and the kinetic interface parameter. The phase field model is validated against the experimental results, and several examples are presented for applications of the phase-field model to understand the corrosion behavior of closely located pits, stressed material, ceramic particles-reinforced steel, and their crystallographic orientation dependence.

Variational formulation of a quantitative phase-field model for nonisothermal solidification in a multicomponent alloy

A variational formulation of a quantitative phase-field model is presented for nonisothermal solidification in a multicomponent alloy with two-sided asymmetric diffusion. The essential ingredient of this formulation is that the diffusion fluxes for conserved variables in both the liquid and solid are separately derived from functional derivatives of the total entropy and then these fluxes are related to each other on the basis of the local equilibrium conditions. In the present formulation, the cross-coupling terms between the phase-field and conserved variables naturally arise in the phase-field equation and diffusion equations, one of which corresponds to the antitrapping current, the phenomenological correction term in early nonvariational models. In addition, this formulation results in diffusivities of tensor form inside the interface. Asymptotic analysis demonstrates that this model can exactly reproduce the free-boundary problem in the thin-interface limit. The present model is widely applicable because approximations and simplifications are not formally introduced into the bulk's free energy densities and because off-diagonal elements of the diffusivity matrix are explicitly taken into account. Furthermore, we propose a nonvariational form of the present model to achieve high numerical performance. A numerical test of the nonvariational model is carried out for nonisothermal solidification in a binary alloy. It shows fast convergence of the results with decreasing interface thickness.

Linearization-based method for solving a multicomponent diffusion phase-field model with arbitrary solution thermodynamics

This article describes a phase-field model for an isothermal multicomponent, multiphase system which avoids implicit interfacial energy contributions by starting from a grand potential formulation. A method is developed for incorporating arbitrary forms of the equilibrium thermodynamic potentials in all phases to determine an explicit relationship between chemical potentials and species concentrations. The model incorporates variable densities between adjacent phases, defect migration, and dependence of internal pressure on object dimensions ranging from the macro- to nanoscale. A demonstrative simulation of an overpressurized nanoscopic intragranular bubble in nuclear fuel migrating to a grain boundary under kinetically limited vacancy diffusion is shown.

Phase-field model of oxidation: Equilibrium

A phase-field model of an oxide relevant to corrosion resistant alloys for film thicknesses below the Debye length L_{D}, where charge neutrality in the oxide does not occur, is formulated. The phase-field model is validated in the Wagner limit using a sharp interface Gouy-Chapman model for the electrostatic double layer. The phase-field simulations show that equilibrium oxide films below the Wagner limit are charged throughout due to their inability to electrostatically screen charge over the length of the film, L. The character of the defect and charge distribution profiles in the oxide vary depending on whether reduced oxygen adatoms are present on the gas-oxide interface. The Fermi level in the oxide increases for thinner films, approaching the Fermi level of the metal in the limit L/L_{D}→0, which increases the driving force for adsorbed oxygen reduction at the gas-oxide interface.

The OpenCalphad thermodynamic software interface

Thermodynamic data are needed for all kinds of simulations of materials processes. Thermodynamics determines the set of stable phases and also provides chemical potentials, compositions and driving forces for nucleation of new phases and phase transformations. Software to simulate materials properties needs accurate and consistent thermodynamic data to predict metastable states that occur during phase transformations. Due to long calculation times thermodynamic data are frequently pre-calculated into "lookup tables" to speed up calculations. This creates additional uncertainties as data must be interpolated or extrapolated and conditions may differ from those assumed for creating the lookup table. Speed and accuracy requires that thermodynamic software is fully parallelized and the Open-Calphad (OC) software is the first thermodynamic software supporting this feature. This paper gives a brief introduction to computational thermodynamics and introduces the basic features of the OC software and presents four different application examples to demonstrate its versatility.

A diffuse-interface modeling for liquid solution-solid gel phase transition of physical hydrogel with nonlinear deformation

A diffuse-interface model is presented in this paper for simulation of the evolution of phase transition between the liquid solution and solid gel states for physical hydrogel with nonlinear deformation. The present domain covers the gel and solution states as well as a diffuse interface between them. They are indicated by the crosslink density in such a way that the solution phase is identified as the state when the crosslink density is small, while the gel as the state if the crosslink density becomes large. In this work, a novel order parameter is thus defined as the crosslink density, which is homogeneous in each distinct phase and smoothly varies over the interface from one phase to another. In this model, the constitutive equations, imposed on the two distinct phases and the interface, are formulated by the second law of thermodynamics, which are in the same form as those derived by a different approach. The present constitutive equations include a novel Ginzburg-Landau type of free energy with a double-well profile, which accounts for the effect of crosslink density. The present governing equations include the equilibrium of forces, the conservations of mass and energy, and an additional kinetic equation imposed for phase transition, in which nonlinear deformation is considered. The equilibrium state is investigated numerically, where two stable phases are observed in the free energy profile. As case studies, a spherically symmetrical solution-gel phase transition is simulated numerically for analysis of the phase transition of physical hydrogel.

An atomistically informed mesoscale model for growth and coarsening during discharge in lithium-oxygen batteries

An atomistically informed mesoscale model is developed for the deposition of a discharge product in a Li-O2 battery. This mescocale model includes particle growth and coarsening as well as a simplified nucleation model. The model involves LiO2 formation through reaction of O2(-) and Li(+) in the electrolyte, which deposits on the cathode surface when the LiO2 concentration reaches supersaturation in the electrolyte. A reaction-diffusion (rate-equation) model is used to describe the processes occurring in the electrolyte and a phase-field model is used to capture microstructural evolution. This model predicts that coarsening, in which large particles grow and small ones disappear, has a substantial effect on the size distribution of the LiO2 particles during the discharge process. The size evolution during discharge is the result of the interplay between this coarsening process and particle growth. The growth through continued deposition of LiO2 has the effect of causing large particles to grow ever faster while delaying the dissolution of small particles. The predicted size evolution is consistent with experimental results for a previously reported cathode material based on activated carbon during discharge and when it is at rest, although kinetic factors need to be included. The approach described in this paper synergistically combines models on different length scales with experimental observations and should have applications in studying other related discharge processes, such as Li2O2 deposition, in Li-O2 batteries and nucleation and growth in Li-S batteries.

Microscopic treatment of solute trapping and drag

The long wavelength limit of a recent microscopic phase-field crystal (PFC) theory of a binary alloy mixture is used to derive an analytical approximation for the segregation coefficient as a function of the interface velocity, and relate it to the two-point correlation function of the liquid and the thermodynamic properties of solid and liquid phases. Our results offer the first analytical derivation of solute segregation from a microscopic model, and support recent molecular dynamics and numerical PFC simulations. Our results also provide an independent framework, motivated from classical density functional theory, from which to elucidate the fundamental nature of solute drag, which is still highly contested in the literature.

Variational formulation and numerical accuracy of a quantitative phase-field model for binary alloy solidification with two-sided diffusion

We present the variational formulation of a quantitative phase-field model for isothermal low-speed solidification in a binary dilute alloy with diffusion in the solid. In the present formulation, cross-coupling terms between the phase field and composition field, including the so-called antitrapping current, naturally arise in the time evolution equations. One of the essential ingredients in the present formulation is the utilization of tensor diffusivity instead of scalar diffusivity. In an asymptotic analysis, it is shown that the correct mapping between the present variational model and a free-boundary problem for alloy solidification with an arbitrary value of solid diffusivity is successfully achieved in the thin-interface limit due to the cross-coupling terms and tensor diffusivity. Furthermore, we investigate the numerical performance of the variational model and also its nonvariational versions by carrying out two-dimensional simulations of free dendritic growth. The nonvariational model with tensor diffusivity shows excellent convergence of results with respect to the interface thickness.

Homogeneous nucleation and microstructure evolution in million-atom molecular dynamics simulation

Homogeneous nucleation from an undercooled iron melt is investigated by the statistical sampling of million-atom molecular dynamics (MD) simulations performed on a graphics processing unit (GPU). Fifty independent instances of isothermal MD calculations with one million atoms in a quasi-two-dimensional cell over a nanosecond reveal that the nucleation rate and the incubation time of nucleation as functions of temperature have characteristic shapes with a nose at the critical temperature. This indicates that thermally activated homogeneous nucleation occurs spontaneously in MD simulations without any inducing factor, whereas most previous studies have employed factors such as pressure, surface effect, and continuous cooling to induce nucleation. Moreover, further calculations over ten nanoseconds capture the microstructure evolution on the order of tens of nanometers from the atomistic viewpoint and the grain growth exponent is directly estimated. Our novel approach based on the concept of “melting pots in a supercomputer” is opening a new phase in computational metallurgy with the aid of rapid advances in computational environments.

Quantitative phase-field modeling of dendritic electrodeposition

A thin-interface phase-field model of electrochemical interfaces is developed based on Marcus kinetics for concentrated solutions, and used to simulate dendrite growth during electrodeposition of metals. The model is derived in the grand electrochemical potential to permit the interface to be widened to reach experimental length and time scales, and electroneutrality is formulated to eliminate the Debye length. Quantitative agreement is achieved with zinc Faradaic reaction kinetics, fractal growth dimension, tip velocity, and radius of curvature. Reducing the exchange current density is found to suppress the growth of dendrites, and screening electrolytes by their exchange currents is suggested as a strategy for controlling dendrite growth in batteries.