Framework For Phase Field Modeling Research Articles

Order evolution from a high‐entropy matrix: Understanding and predicting paths to low‐temperature equilibrium

AbstractInterest in high‐entropy inorganic compounds originates from their ability to stabilize cations and anions in local environments that rarely occur at standard temperature and pressure. This leads to new crystalline phases in many‐cation formulations with structures and properties that depart from conventional trends. The highest‐entropy homogeneous and random solid solution is a parent structure from which a continuum of lower‐entropy offspring can originate by adopting chemical and/or structural order. This report demonstrates how synthesis conditions, thermal history, and elastic and chemical boundary conditions conspire to regulate this process in Mg0.2Co0.2Ni0.2Cu0.2Zn0.2O, during which coherent CuO nanotweeds and spinel nanocuboids evolve. We do so by combining structured synthesis routes, atomic‐resolution microscopy and spectroscopy, density functional theory, and a phase field modeling framework that accurately predicts the emergent structure and local chemistry. This establishes a framework to appreciate, understand, and predict the macrostate spectrum available to a high‐entropy system that is critical to rationalizing property engineering opportunities.

Interplay Between Nucleation and Kinetics in Dynamic Twinning

Abstract In this work, we apply a phase-field modeling framework to elucidate the interplay between nucleation and kinetics in the dynamic evolution of twinning interfaces. The key feature of this phase-field approach is the ability to transparently and explicitly specify nucleation and kinetic behavior in the model, in contrast to other regularized interface models. We use this to study two distinct problems where it is essential to explicitly specify the kinetic and nucleation behavior governing twin evolution. First, we study twinning interfaces in 2D. When these interfaces are driven to move, we find that significant levels of twin nucleation occur ahead of the moving interface. Essentially, the finite interface velocity and the relaxation time of the stresses ahead of the interface allow for nucleation to occur before the interface is able to propagate to that point. Second, we study the growth of needle twins in antiplane elasticity. We show that both nucleation and anisotropic kinetics are essential to obtain predictions of needle twins. While standard regularized interface approaches do not permit the transparent specification of anisotropic kinetics, this is readily possible with the phase-field approach that we have used here.

Modeling the electro-chemo-mechanical failure at the lithium-solid electrolyte interface: Void evolution and lithium penetration

The solid-solid contact interface is crucial for the reliability of solid-state energy storage systems. The contact condition becomes more complicated when lithium (Li) metal is used as the anode. The contact between solid electrolyte (SE) and Li metal is inferior compared to the liquid/solid interface in conventional Li-ion batteries. Experimental evidence has shown that improper operating conditions of solid-state batteries can lead to electro-chemo-mechanical failures at the Li/SE interface, including the formation of voids and the penetration of Li. In this study, a unified phase-field model is developed to investigate these two mechanisms. The model considers the coupled electro-chemo-mechanical processes including void diffusion, lattice annihilation, stripping and plating reactions, and plastic deformation of Li metal. The study begins with a revisit of the deformation-mechanism map for Li metal under a wide range of temperatures, stress, and deformation rates. This map serves as the basis for the mechanical characterization in the phase-field model. The large inelastic deformation of Li is considered by introducing an advection term into the Allen-Cahn equation, which is used to describe the dynamic evolution of the Li and void phases. The effects of current density and stack pressure on void evolution and Li penetration are studied based on the model predictions. By combining the simulation results with the experimental data from publications, we obtain the stable operation zone of stack pressure and applied current density. In this zone, the Li/SE interface can enable stable stripping and plating of Li metal. The same phase-field modeling framework is transferred to investigate the Li-Mg alloy/SE interface considering Li-Mg alloy is also used as the anode. The fundamental difference between Li/SE and Li-Mg/SE is analyzed accordingly. This study provides a useful tool for the design, manufacturing, and management of next-generation batteries by providing important scientific insights into the electro-chemo-mechanical processes of different anode materials under various operational conditions.

Phase-field modeling of fracture with physics-informed deep learning

We explore the potential of the deep Ritz method to learn complex fracture processes such as quasistatic crack nucleation, propagation, kinking, branching, and coalescence within the unified variational framework of phase-field modeling of brittle fracture. We elucidate the challenges related to the neural-network-based approximation of the energy landscape, and the ability of an optimization approach to reach the correct energy minimum, and we discuss the choices in the construction and training of the neural network which prove to be critical to accurately and efficiently capture all the relevant fracture phenomena. The developed method is applied to several benchmark problems and the results are shown to be in qualitative and quantitative agreement with the finite element solution. The robustness of the approach is tested by using neural networks with different initializations.

Which Interfaces Matter Most? Variability in Grain Boundary Defect Chemistry and Conductivity in a Concentrated Solid Electrolyte

Inorganic solid electrolytes with fast ionic conductivity are critical components of fuel/electrolysis cells, all-solid-state batteries, membrane and separation reactors, sensors and other electrochemical devices. For example, solid oxygen ion (O2-) conductors are used in solid oxide fuel/electrolysis cells (SOFCs/SOECs) for high-efficiency conversion between chemical and electrical energy and chemical production from CO2. One major challenge in engineering such materials is the diminished grain boundary (GB) conductivity in polycrystalline oxide ceramics. This phenomenon, which was initially accredited to highly resistive silica-based glassy impurity phases, is attributed to the formation of intrinsic space charge layers (SCLs) directly adjacent to GB cores. The positive potential at the GB core (due to large concentrations of oxygen vacancies) results in a depletion of oxygen vacancies in the space charge layer as well as the segregation of charge-compensating acceptor solutes from grain interiors which ultimately reduces GB conductivity. Several strategies have been explored to improve GB conductivity by understanding and mitigating the SCL effects. While the standard Poisson–Boltzmann approach models the near-grain boundary defect behavior in dilute solid solutions and recent models by Mebane et al [1] and Virkant et al [2] incorporate the effects of defect-defect interactions (in concentrated solid solutions) and chemomechanical stress, respectively, there is yet to be a model developed that can predict individual GB conductivities in concentrated solid solutions based on direct experimental measurements of near-GB defect chemistry.Here, we present an experimental-computational framework that predicts which GBs in a polycrystalline electrolyte likely facilitate ionic conductivity. We developed a thermodynamic phase-field modeling framework and applied it to a model oxygen electrolyte Gd0.25Ce0.75O2-δ, wherein the GB-to-GB variability in defect concentrations (Gd3+ solutes, electrons localized at Ce3+, and oxygen vacancies) was measured microscopically using scanning transmission electron microscopy electron energy-loss spectroscopy (STEM EELS). These data were used to predict the GB-to-GB variability in cross-GB ionic conduction using a unique model, which prioritizes reproducing microscopically observed GB defect concentrations and considers defect-defect interactions in highly concentrated solid solutions, making the framework applicable to other technologically relevant solid electrolytes. Across the GBs studied, we revealed a non-monotonic relationship between depletion and GB conductivity, with the highest conductivity predicted for intermediate depletion amounts. This work lays the foundation for future experimental-computational research and advanced data-driven design of solid electrolytes needed for functional and energy storage/conversion devices.[1] D. S. Mebane, R. A. D. Souza, Energy & Environmental Science, 2015.[2] K. S. N. Vikrant, R. Edwin García, Energy & Environmental Science, 2018.[3] H. Vahidi, W.J. Bowman (Submitted). Acknowledgments HV and WJB acknowledge primary support from the UCI new faculty startup funding. This research was partially supported by the National Science Foundation Materials Research Science and Engineering Center program through the UC Irvine Center for Complex and Active Materials (DMR-2011967). We acknowledge the use of facilities and instrumentation at the UC Irvine Materials Research Institute (IMRI) supported in part by the same NSF MRSEC (DMR-2011967). AM, DM, and AC acknowledge funding from the NSF [WB1] [hv2] and the WVU MAE research department for the use of the Thorny Flat HPC Cluster for the numerical modeling work and computational resources.

Permeability evolution in open fractures during precipitation and dissolution - A phase-field study

In dilated fractures in the Earth’s crust fluid flow in combination with precipitation or dissolution processes can occur, which in turn influences the mechanical and transport properties of the rock system. We use a phase-field modeling framework to investigate these processes of epitaxial crystal growth and dissolution on the fracture walls of crystalline rock systems on microscale. Fluid flow simulations are performed and analyzed during intermediate crystallization and dissolution stages and the obtained hydraulic properties of the partly open fractures are compared to existing literature. The systematic simulation studies show how the rock properties are affected by factors as mineral type with different crystal morphologies, fracture type (inter- vs. transgranular), aperture of the open fracture, and presence of accessory minerals. The results indicate that within the considered parameter space the flow paths remain open until late stages of fracture sealing. Moreover, the long-term permeability and porosity evolution is strongly affected by fracture surface heterogeneities and initial fracture apertures. The simulations enable insights into the evolution of microstructural and fluid flow characteristics and can lay the basis for applications in fractured porous media as groundwater protection, geothermal and hydrocarbon reservoir prediction, water recovery, or storing H2 or CO2 in the subsurface.

Multiscale dynamics of charging and plating in graphite electrodes coupling operando microscopy and phase-field modelling

The phase separation dynamics in graphitic anodes significantly affects lithium plating propensity, which is the major degradation mechanism that impairs the safety and fast charge capabilities of automotive lithium-ion batteries. In this study, we present comprehensive investigation employing operando high-resolution optical microscopy combined with non-equilibrium thermodynamics implemented in a multi-dimensional (1D+1D to 3D) phase-field modeling framework to reveal the rate-dependent spatial dynamics of phase separation and plating in graphite electrodes. Here we visualize and provide mechanistic understanding of the multistage phase separation, plating, inter/intra-particle lithium exchange and plated lithium back-intercalation phenomena. A strong dependence of intra-particle lithiation heterogeneity on the particle size, shape, orientation, surface condition and C-rate at the particle level is observed, which leads to early onset of plating spatially resolved by a 3D image-based phase-field model. Moreover, we highlight the distinct relaxation processes at different state-of-charges (SOCs), wherein thermodynamically unstable graphite particles undergo a drastic intra-particle lithium redistribution and inter-particle lithium exchange at intermediate SOCs, whereas the electrode equilibrates much slower at low and high SOCs. These physics-based insights into the distinct SOC-dependent relaxation efficiency provide new perspective towards developing advanced fast charge protocols to suppress plating and shorten the constant voltage regime.

An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events

An SPH-based FSI framework for phase-field modeling of brittle fracture under extreme hydrodynamic events

A phase-field framework for brittle fracture in quasi-crystals

Most of quasi-crystals are brittle at room temperature due to their specific cluster structures. Phase-field fracture models have demonstrated a powerful ability to predict brittle crack evolution. This paper develops a phase-field framework for modeling macroscopic brittle fracture in quasi-crystal solids. The kind of quasi-crystals is not preset in the phase-field model; therefore, this model is valid for almost all quasi-crystal solids. The phase-field model for the first time introduces a volume fraction parameter into the fracture toughness to reflect the effect of phason wall. Furthermore, a new numerical implementation approach of phase-field fracture models in Comsol Multiphysics is presented. Several examples of 1D hexagonal and 2D decagonal quasi-crystals are performed. The developed model is validated against the reported results of benchmark problems. Further investigations indicate that the phason field peculiar to quasi-crystals has a certain influence on the crack paths in asymmetric fracture problems and on the force–displacement curves in both symmetric and asymmetric fracture problems. When the fracture toughness is fixed, the increase of the volume fraction parameter results in the decrease of the peak force and the increase of the failure displacement. When the influence of the phason field on the fracture toughness is considered, both the peak force and failure displacement would sharply decrease, which indicates that the degradation of fracture toughness due to the phason field plays a more important role than the phason elastic energy in the fracture of quasi-crystals. Furthermore, the relation between the phonon-phason energy transformation and the elastic constants of QCs has been qualitatively analyzed. The developed phase-field modeling framework provides a guidance for understanding the fracture mechanism of quasi-crystals and assessing the safety of quasi-crystal structures in engineering practice.

Phase-field modeling of coupled spall and adiabatic shear banding and simulation of complex cracks in ductile metals

In this work, a unified phase field theory for two coupled fracture types is proposed. Compared with the existing unified phase field theory for single fracture type, it considers the coupling behavior between different fracture types and is consistent with the single fracture type theory when one type of fracture is suppressed. Based on a new form of energy decomposition, that is, the strain energy is decomposed into deviatoric, tensile volumetric and compressive volumetric parts, and specifying the driving energy for each fracture type, a double phase-field model for coupled spall and adiabatic shear banding, which are two kinds of typical dynamic ductile fracture, is realized with the proposed unified phase field theory for two coupled fracture types. This coupled model can be used to predict the potential spall and adiabatic shear banding failures in ductile metals under dynamic loading, revealing the corresponding fracture types under different loadings without any additional criteria. The entire damage and fracture evolution (that is, from damage evolution, through crack expansion, and to fragmentation) are captured with this unified phase-field modeling framework. In addition, as an advantage of the double phase-field model, the complex multicrack distributions of spall and adiabatic shear bands in ductile metals can be presented with the proposed model. Based on these features, the expanding characteristics of complex cracks in metallic expanding shells are numerically studied. It is found that, if multilayer spall occurs, the specimen tends to fracture into three layers, that is, a complete spall layer, fragmentations in multilayer areas, and the main body, which is consistent with the experimental observations under certain conditions and is not the regular multiple layers predicted by the multilayer spall theory with ideal material assumption. For multiple adiabatic shear bands (ASBs) in the shell, the spiral patterns of ASBs’ distributions, the initial positions, and the propagating path are clarified for the collapsed and expanding shells. The collapsed shell tends to form a single-direction spiral pattern whereas the expanding shell tends to form a double-direction spiral pattern. In addition to the inner surface, the intersection points of two ASBs can initiate cracks, and, as the intersection points have the chance of being distributed on the outer surface, the expanding shell has a greater potential to initiate a crack on the outer surface than the collapsed shell. The propagating path will be deflected after interactions of two ASBs, and the quantitative relationship between the deflection angle and the stress state before the interactions is given, which explains why the deflection angle is significantly larger in the collapsed shell compared with that in the expanding shell. Based on the above results and analysis, four typical stages of fractures in the expanding shell under internal explosive loading are identified.

Phase-field fracture modeling for large structures

The phase-field modeling framework for linear elastic fracture mechanics problems is modified for the purpose of analyzing crack growth in large structures. The phase-field approach to fracture replaces sharp crack surfaces with a diffuse fracture zone that represent the traction-free crack faces. The diffuse fracture zone is characterized by a length scale that appears prominently in the governing partial differential equation that governs the evolution of the phase-field variable. Within numerical calculations the diffuse fracture zone must be resolved with a sufficient number of degrees of freedom in order to obtain accurate solutions. Additionally, all material fracture problems possess a physical process zone length scale that scales with the ratio of the fracture energy to the square of the material strength. For the vast majority of problems governed by linear elastic fracture mechanics, this physical process zone length scale is small, for example it is on the order of microns for aluminum alloys. If the material strength is to be captured, then the most commonly used formulation of the phase-field approach to fracture ties the diffuse phase-field crack length scale to the physical process zone length scale. This presents a challenge for extending such models to large structures that may be on the order of meters in size due to the prohibitive meshing requirements for the phase-field crack length scale. This is especially the case for three-dimensional problems. To address this problem a new form of the phase-field degradation function is introduced that allows for a decoupling of the phase-field length scale from the physical process zone length scale. The behavior and limitations of this new formulation are discussed and illustrated with a series of numerical test cases.

Phase-field modeling of magnetic field-induced grain growth in polycrystalline metals

A constitutive model for diffuse interface description of magnetic field-induced grain boundary migration in polycrystalline metals is developed based on the laws of thermodynamics. Within this phase field modeling framework, simultaneous minimization of the total grain boundary energy and the stored magnetic energy within grains provides the driving force for grain growth in a polycrystal material exposure to a magnetic field. Using the available experimental data, phase field simulations of magnetic field induced grain growth in bicrystalline zinc and polycrystalline titanium with two-dimensional columnar microstructure are performed and the model is validated by demonstrating a qualitative and quantitative match with experimental data. Using the validated constitutive model, the effects of magnetic field intensity and direction on evolution of microstructure and polycrystalline texture in titanium are investigated. Quantitative simulation results show that under certain magnetic field directions and sufficient magnetic field intensity, the magnetic energy minimizing driving force can overcome the curvature driving force and cause texture evolution towards less magnetic energy grain orientations during annealing. However, magnetic fields with insufficient intensity or improper direction have negligible effect on grain coarsening by nearly normal grain growth. The desired magnetic field direction and intensity are quantitatively presented for cold-rolled and annealed sheet titanium samples.

Comparative mapping of crawling-cell morphodynamics in deep learning-based feature space

Navigation of fast migrating cells such as amoeba Dictyostelium and immune cells are tightly associated with their morphologies that range from steady polarized forms that support high directionality to those more complex and variable when making frequent turns. Model simulations are essential for quantitative understanding of these features and their origins, however systematic comparisons with real data are underdeveloped. Here, by employing deep-learning-based feature extraction combined with phase-field modeling framework, we show that a low dimensional feature space for 2D migrating cell morphologies obtained from the shape stereotype of keratocytes, Dictyostelium and neutrophils can be fully mapped by an interlinked signaling network of cell-polarization and protrusion dynamics. Our analysis links the data-driven shape analysis to the underlying causalities by identifying key parameters critical for migratory morphologies both normal and aberrant under genetic and pharmacological perturbations. The results underscore the importance of deciphering self-organizing states and their interplay when characterizing morphological phenotypes.

A framework for phase-field modeling of interfacial debonding and frictional slipping in heterogeneous composites

Modeling and simulating interfacial shear load transfer in fiber-reinforced composites is crucial for characterizing fiber pullout behaviors. Addressing this complicated topic, this work presents a novel phase-field framework to simulate interfacial failure behaviors in fiber-reinforced cementitious composites. Specifically, a comprehensive constitutive model is proposed to describe both interfacial debonding and frictional slipping, which accounts for snubbing and bonding effects in fiber-reinforced composites. The developed model has the following novel features: (1) The model can simultaneously characterize interfacial debonding and frictional slipping, and thus thoroughly describe the fiber-reinforcing mechanisms in quasi-brittle solids; (2) An interfacial slipping degradation theory is implemented in the model to describe interfacial slippage and stress degradation during slipping; and (3) The model effectively and precisely incorporates the friction law with interface normal vectors. The developed model is successfully validated against experimental and simulated results. Finally, a series of parameter studies are undertaken to provide strategic insights to improve the mechanical properties of the composites. The proposed method adds efficiency and robustness to the study of features and fracture mechanisms of fiber reinforcements in complex fiber–cement systems.

The development of grain structure during additive manufacturing

Additive manufacturing of structural alloys results in the formation of complex microstructures, often with long, columnar grains that grow epitaxially from existing grains. A phase-field modeling framework is presented that considers solidification along a single track of 316L stainless steel in a regime where the solid-liquid interface is moving sufficiently fast that there is absolute interfacial stability with negligible composition variations. By coupling to a thermal field of a moving laser source, the model captures the trajectory of grains solidifying around a melt pool. The effect of interfacial kinetic anisotropy on the predicted as-solidified microstructures is examined through three-dimensional simulations. Through a combination of qualitative and quantitative analyses, we find that kinetic anisotropy has the largest impact along the center of the laser track and that the grain morphology in the early stages of solidification is predominantly due to the shape of the melt pool.

Phase field simulation of fatigue crack propagation under complex load situations

Within this work, we utilize the framework of phase field modeling for fracture in order to handle a very crucial issue in terms of designing technical structures, namely the phenomenon of fatigue crack growth. So far, phase field fracture models were applied to a number of problems in the field of fracture mechanics and were proven to yield reliable results even for complex crack problems. For crack growth due to cyclic fatigue, our basic approach considers an additional energy contribution entering the regularized energy density function accounting for crack driving forces associated with fatigue damage. With other words, the crack surface energy is not solely in competition with the time-dependent elastic strain energy but also with a contribution consisting of accumulated energies, which enables crack extension even for small maximum loads. The load time function applied to a certain structure has an essential effect on its fatigue life. Besides the pure magnitude of a certain load cycle, it is highly decisive at which point of the fatigue life a certain load cycle is applied. Furthermore, the level of the mean load has a significant effect. We show that the model developed within this study is able to predict realistic fatigue crack growth behavior in terms of accurate growth rates and also to account for mean stress effects and different stress ratios. These are important properties that must be treated accurately in order to yield an accurate model for arbitrary load sequences, where various amplitude loading occurs.

Interphase boundary segregation and precipitate coarsening resistance in ternary alloys: An analytic phase-field model describing chemical effects

Many experimental and first principles studies on precipitation hardening alloys show that segregation of elemental species to the matrix-precipitate interphase boundary (IB) reduces the boundary’s energy. This segregation mechanism can thermally stabilize the microstructure against precipitate coarsening processes and allow for higher operating temperatures in structural applications. In this paper, we develop a phase-field modeling framework to describe IB solute segregation in ternary alloys. The interfacial thermodynamics is effectively described by defining an IB phase with a characteristic free energy-concentration dependence. Equilibrium for the IB phase is established via the parallel tangent plane construction, analogous to classical treatments for segregation to free surfaces and grain boundaries. Analytic steady-steady solutions elucidating the dependence of IB properties on bulk phase composition, temperature and model parameters are derived for a one-dimensional system. Analytic relations for the classical thermodynamic quantities–IB energy and relative solute excess–are derived and the Gibbs adsorption equation is shown to hold; therefore, predictions of the model can be compared with experiments and atomistic simulations. An application of the model is demonstrated for Zn segregation to Mg/Mg2Sn using representative IB parameters. A two-particle coarsening simulation of IB segregation is performed: the result demonstrates enhanced coarsening resistance of the ternary alloy relative to the binary alloy.

A Phase-Field Model Study of Microstructural Effects on the Corrosion of Mg Alloys

Understanding how alloy composition and thermal history determine the corrosion behavior is crucial to designing alloys with lower corrosion rates. We employ a model that combines the phase-field and smoothed boundary methods [1] to investigate the effect of second phases and particles on the corrosion behavior of Mg alloys. In this model, the dissolution rate at the metal/electrolyte interface is determined by the coupling of mass transport within the electrolyte and the Butler-Volmer electrochemical kinetics at the interface. The model is implemented in the PRISMS-PF framework [2,3], an open-source framework for phase-field modeling that employs a matrix-free finite element method. Simulations of corrosion are conducted with varying microstructure and compositions with associated corrosion parameters representative of Mg alloys. Specifically, we show how our approach can describe the experimentally observed microgalvanic effects due to the presence of impurities and second phases. Finally, we show the results of a parametric study of the dependence of the rate for various combination of the corrosion parameters for each phase.This work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering as part of the Center for PRedictive Integrated Structural Materials Science (PRISMS Center) at University of Michigan.[1] A.F. Chadwick, J.A. Stewart, R.A. Enrique, S. Du, K. Thornton, Numerical Modeling of Localized Corrosion Using Phase-Field and Smoothed Boundary Methods, J. Electrochem. Soc. 165 (2018) C633–C646.[2] S. DeWitt, S. Rudraraju, D. Montiel, W. B. Andrews, K. Thornton, PRISMS-PF: A General Framework for Phase-Field Modeling with a Matrix-Free Finite Element Method. Submitted to npj Comput. Mater.[3] http://prisms-center.github.io/phaseField/

PRISMS-PF: A general framework for phase-field modeling with a matrix-free finite element method

A new phase-field modeling framework with an emphasis on performance, flexibility, and ease of use is presented. Foremost among the strategies employed to fulfill these objectives are the use of a matrix-free finite element method and a modular, application-centric code structure. This approach is implemented in the new open-source PRISMS-PF framework. Its performance is enabled by the combination of a matrix-free variant of the finite element method with adaptive mesh refinement, explicit time integration, and multilevel parallelism. Benchmark testing with a particle growth problem shows PRISMS-PF with adaptive mesh refinement and higher-order elements to be up to 12 times faster than a finite difference code employing a second-order-accurate spatial discretization and first-order-accurate explicit time integration. Furthermore, for a two-dimensional solidification benchmark problem, the performance of PRISMS-PF meets or exceeds that of phase-field frameworks that focus on implicit/semi-implicit time stepping, even though the benchmark problem’s small computational size reduces the scalability advantage of explicit time-integration schemes. PRISMS-PF supports an arbitrary number of coupled governing equations. The code structure simplifies the modification of these governing equations by separating their definition from the implementation of the numerical methods used to solve them. As part of its modular design, the framework includes functionality for nucleation and polycrystalline systems available in any application to further broaden the phenomena that can be used to study. The versatility of this approach is demonstrated with examples from several common types of phase-field simulations, including coarsening subsequent to spinodal decomposition, solidification, precipitation, grain growth, and corrosion.

Interphase Boundary Segregation and Precipitate Coarsening Resistance in Ternary Alloys: An Analytic Phase-Field Model Describing Chemical Effects

Many experimental and first principles studies on precipitation hardening alloys show that segregation of elemental species to the matrix-precipitate interphase boundary (IB) reduces the boundary's energy. This segregation mechanism can thermally stabilize the microstructure against precipitate coarsening processes and allow for higher operating temperatures in structural applications. In this paper, we develop a phase-field modeling framework to describe IB solute segregation in ternary alloys. The interfacial thermodynamics is effectively described by defining an IB phase with a characteristic free energy-concentration dependence. Equilibrium for the IB phase is established via the parallel tangent plane construction, analogous to classical treatments for segregation to free surfaces and grain boundaries. Analytic steady-steady solutions elucidating the dependence of IB properties on bulk phase composition, temperature and model parameters are derived for a one-dimensional system. Analytic relations for the classical thermodynamic quantities--IB energy and relative solute excess--are derived and the Gibbs adsorption equation is shown to hold; therefore, predictions of the model can be compared with experiments and atomistic simulations. An application of the model is demonstrated for Zn segregation to Mg/Mg2Sn using representative IB parameters. A two-particle coarsening simulation of IB segregation is performed; the result demonstrates enhanced coarsening resistance of the ternary alloy relative to the binary alloy.