Multi-phase Field Method Research Articles

Crystallization and crystal morphology of polymers: A multiphase-field study

In this paper, we introduce a coarse-grained model of polymer crystallization using a multiphase-field approach. The model combines a multiphase-field method, Nakamura’s kinetic equation, and the equation of heat conduction for studying microstructural evolution of crystallization under isothermal and non-isothermal conditions. The multiphase-field method provides flexibility in adding any number of phases with different properties making the model effective in studying blends or composite materials. We apply our model to systems of neat PA6 and study the impact of initial distribution of crystalline grains and cooling rate on the morphology of the system. The relative crystallinity (conversion) curves show qualitative agreement with experimental data. We also investigate the impact of including carbon fibers on the crystallization and grain morphology. We observe a more homogeneous crystal morphology around fibers. This is associated with the higher initial volume fraction of crystal grains and higher heat conductivity of the fiber (compared to the polymer matrix). Additionally, we observe that the crystalline grains at the fiber surface grow perpendicular to the surface. This indicates that the vertical growth observed in experiments is merely due to geometrical constraints imposed by the fiber surface and neighbouring crystalline regions.

Unraveling elastochemical effects in microstructural evolution of Al–Cu–Ni system through DFT-informed multi-phase field simulations

Through the multi-phase field method, this research explores microstructure evolution within the Al–Cu–Ni system comprising Al-rich FCC, Al2Cu, and Al3Ni intermetallic compounds (IMCs). Notably, the Al3Ni exhibits substantial growth due to its higher free energy magnitude, while the Al2Cu phase diminishes within the FCC matrix. Additionally, this study investigates the impact of heterogeneous elastic effects on these microstructural changes. Density functional theory calculations were employed to determine temperature-dependent elastic tensor at a simulation temperature of 723 K. Initially both cases, simulations with elastic constants at 0 K and 723 K, start with identical grain areas. But by 21.45 ms, deviations notably intensify to 18.18% for single grains, underlining the importance of temperature-calibrated elastic tensor in the phase evolution driven by elastochemical effects.

Combined study of phase transitions in the P2-type NaXNi1/3Mn2/3O2 cathode material: experimental, ab-initio and multiphase-field results

The research of new electrode materials such as sodium intercalation compounds is key to meet the challenges of future demands of sustainable energy storage. For these batteries, the intercalation behavior on the micro-scale is governed by a complex interplay of chemical, electrical and mechanical forces strongly influencing the overall cell performance. The multiphase-field method is a suitable tool to study these multi-physics and bridge the scale from ab-initio methods to the cell level. In this work, we follow a combined approach of experiments, density functional theory (DFT) calculations and multiphase-field simulations to predict thermodynamic and kinetic properties for the P2-type NaXNi1/3Mn2/3O2 sodium-ion cathode material. Experimentally, we obtain the thermodynamic potential and diffusion coefficients at various sodium contents using electrochemical techniques and discuss limitations of the experimentally applied methods. DFT is used to identify stable phases by calculating an energy hull curve. Then, the influence of long-range dispersion interactions and the exchange-correlation functional on the voltage curve is investigated by comparison with experimental results. Finally, multiphase-field simulations are performed based on inputs from experiments and DFT. The fitting of phase-specific chemical free energies from DFT calculations and experimental data is discussed. Our results highlight the thermodynamic consistency of all three approaches close to thermodynamic equilibrium. Furthermore, the phase-field method accurately describes the kinetics of the system including multiple phase transitions, by which we unravel the mechanism of the P2-O2 phase transition in a single crystal under the influence of intercalation reaction, bulk diffusion and elastic deformation. The model is able to predict the kinetic capacity loss depending on charging rate in agreement with C-rate experiments.

Bubble flow analysis using multi-phase field method

In simulations of gas-liquid two-phase flows using conventional interface capture methods, we observed that when bubbles come close to each other, they tend to merge numerically, despite experimental evidence indicating that they would repel each other. Given the significant impact of sequential numerical coalescence on flow patterns, it is necessary to regulate the merging behavior of close bubbles. To address this issue, we introduced the Multi-Phase Field (MPF) method, which mitigates bubble coalescence by applying an independent fluid fraction function to each bubble. In this study, we employed the MPF based on the N-phase model [7] to minimize numerical errors associated with surface interactions at triple junction points. Additionally, we implemented the Ordered Active Parameter Tracking (OAPT) method [9] to efficiently store several hundreds of fluid fraction functions. To validate the MPF method, we conducted analysis of turbulent bubbly pipe flows and compared the results against experimental data from Colin et al [12]. The validation results showed reasonable agreements with respect to the bubble distribution and the flow velocity profiles.

Microstructure evolution accounting for crystal plasticity in the context of the multiphase-field method

The role of grain boundaries (GBs) and especially the migration of GBs is of utmost importance in regard of the overall mechanical behavior of polycrystals. By implementing a crystal plasticity (CP) theory in a multiphase-field method, where GBs are considered as diffuse interfaces of finite thickness, numerically costly tracking of migrating GBs, present during phase transformation processes, can be avoided. In this work, the implementation of the constitutive material behavior within the diffuse interface region, considers phase-specific plastic fields and the jump condition approach accounting for CP. Moreover, a coupling is considered in which the phase-field evolution and the balance of linear momentum are solved in each time step. The application of the model is extended to evolving phases and moving interfaces and approaches to strain inheritance are proposed. The impact of driving forces on the phase-field evolution arising from plastic deformation is discussed. To this end, the shape evolution of an inclusion is investigated. The resulting equilibrium shapes depend on the anisotropic plastic deformation and are illustrated and examined. Subsequently, evolving phases are studied in the context of static recrystallization (SRX). The GB migration involved in the growth of nuclei, which are placed in a previously deformed grain structure, is investigated. For this purpose, three approaches to strain inheritance are compared and, subsequently, different grain structures and distributions of nuclei are considered. It is shown, how the revisited method contributes to a simulation of SRX.

A multiphase-field approach to small strain crystal plasticity accounting for balance equations on singular surfaces

AbstractAn implementation of the crystal plasticity theory in the context of the multiphase-field method provides a numerically efficient tracking of evolving grain boundaries, modeled as diffuse interfaces. In literature, several approaches exist for the implementation of the plastic material behavior within the diffuse interface, based on interpolation, homogenization, or the mechanical jump conditions. Among these, only the jump condition approach exhibits an intrinsic relationship to the sharp interface (SI) theory. Therefore, in the work at hand, the implementation of the crystal plasticity theory within the jump condition approach, referred to as phase-specific plastic fields approach (PSPFA), is discussed in detail. The PSPFA is compared to the interpolation approach, referred to as common plastic fields approach (CPFA), using three-dimensional benchmark simulations of a bicrystal set-up. The comparison reveals that the PSPFA and SI coincide convincingly regarding the accumulated plastic slip and the Mises stress. In contrast, a significant deviation of CPFA and SI is observed both quantitatively and qualitatively, not only within the diffuse interface region, but throughout the complete simulation domain. A variation of the interface width illustrates that this observation can be transferred to the normal components of the total strain, even for smaller interface widths. Consequently, a quantitative estimate of the plastic material behavior, which is crucial for the prediction of the dynamic behavior of grain boundaries, is only provided by the PSPFA. The application of the crystal plasticity in the context of PSPFA to more complex microstructures is illustrated with respect to a periodic honeycomb-structure and an octotuple.

A phase-field model for ferroelectric materials—Based on the multiphase-field method

A novel phase-field model is developed to investigate the domain structure in ferroelectric materials subjected to an external field. Differing from the time-dependent Ginzburg–Landau (TDGL) theory that describes domain structures based on the spatial and temporal evolution of polarization, our model utilizes the multiphase-field approach to calculate domain structures by minimizing the total energy functional with respect to the multiphase-field order parameter ϕ. The energy functional in the proposed model comprises the general multiphase interfacial energy and the phase-dependent bulk energy, accounting for the contributions from domain walls, and mechanical as well as electric fields. By considering the interplay between the different energy components, the model provides insights into the complex domain structure of ferroelectric materials and their response to external stimuli. This comprehensive approach is validated through simulations of both single-crystal and polycrystalline structures. The approach presented in this study simplifies the computational coefficients required to analyze the domain structure in ferroelectric materials, compared to the TDGL approach, by disregarding the Landau potential. This makes it possible to analyze the domain structure of materials without Landau coefficients. Furthermore, the proposed model serves as a reference for applying the multiphase-field concept to ferroelectric materials.

Investigation of microstructure evolution accounting for crystal plasticity in the multiphase‐field method

AbstractRegarding microstructured materials, a quantitative prediction of phase transformation processes is highly desirable for a wide range of applications. With respect to polycrstalline materials, the plastic material behavior is commonly investigated using a crystal plasticity (CP) theory, since it accounts for the underlying microstructure, that is, slip systems of the crystal lattice. In classical continuum mechanics, grain boundaries (GBs) are commonly modeled as material singular surfaces. However, the tracking of moving GBs, present during phase transformation processes, is numerically challenging and costly. This can be circumvented by the use of a multiphase‐field method (MPFM), which provides a numerically highly efficient method for the treatment of moving interfaces, considered as diffuse interfaces of finite thickness. In this work, the microstructural evolution is investigated within the MPFM accounting for CP. The implementation of the constitutive material behavior within the diffuse interface region accounts for phase‐specific plastic fields and the jump condition approach. To improve the understanding of the impact of plastic deformation on the phase evolution, a single inclusion problem is analyzed. Within a plastically deformed matrix, the shape evolution of a purely elastic inclusion with a different Young's modulus, referred to as inhomogeneity, is investigated. It is shown, how the anisotropic plastic behavior affects the phase evolution. The resulting equilibrium shapes are illustrated and examined.

Multi-phase field modeling and simulation of magnetically driven grain boundary migration in SmCo polycrystals

There is a growing demand for magnetic materials straight forward in wind turbines and electric motors. Their functional properties depend critically on their microstructure and thus on the microstructure evolution during sintering or heat treatment. Field-assisted selective grain growth allows to optimize the microstructure. However, the simultaneous modeling of the structural and magnetic degrees of freedom on the micrometer scale is not possible with most simulation packages. Therefore, we extend the open-source software project OpenPhase and implement the micromagnetic equations needed to treat both degrees of freedom in the framework of the multi-phase field method. We apply our model to the field-assisted grain growth in Sm2Co17 polycrystal films.

Grain-refining effects and mechanism of novel Al-Nb-B refiner on Al-Mg-Si alloy: phase-field simulation and experiment study

In this paper, the grain-refining effect of Al-Nb-B refiner in wrought Al alloys and corresponding refining mechanism was investigated using phase-field simulation and experiment methods. Through experimental statistics and data correction, the seed radius introduced by Al-Nb-B refiner into Al alloy melt and the corresponding quantitative density distribution data were obtained, and a seed density model (SDM) model has been established. On this basis, a multi-phase field method (MPFM) combined with the calculation of phase diagram (CALPHAD) has been employed to simulate the α-Al grain evolution of Al-Mg-Si alloy during the solidification process with the help of thermodynamic database. Experimental studies showed that the addition of 0.03 wt% Nb (Adding by Al-1.93Nb-0.22B master alloy) significantly reduced the grain size from 956.4 μm to about 219.4 μm, and the grain size slowly decreased to 192.3 μm by continuing to add refiner to 0.09 wt% Nb. Meanwhile, the simulation results demonstrated that after the addition of refiners, dendrite morphology transformed from the relatively developed dendrites with secondary and tertiary dendrites to a fine and uniform equiaxed shape. Simulation and experimental studies showed quantitative agreement. In addition, the results also prove that the addition of Al-Nb-B refiner can provide high-quality and stable heterogeneous nucleation particles.

Thermomechanically coupled theory in the context of the multiphase-field method

The modeling and simulation of microstructure evolution is usually subject to the multiphase-field method. In this context, thermomechanical coupling is often neglected, even when non-isothermal phase transformations are considered. Using a simplified example, the present work shows that this assumption is not justified for small strains and small strain rates with respect to a non-vanishing coefficient of thermal expansion. To this end, both a thermomechanically coupled and a thermomechanically weakly coupled theory are briefly revisited. The difference between the coupled and the weakly coupled theory regarding the growth of an inclusion is discussed. The considered elastoplastic inclusion, subjected to eigenstrains, is embedded in an elastoplastic matrix under load. It is shown, that the weakly coupled theory overestimates the growth of the inclusion, and, thus, the volume concentration, compared to the coupled theory. Moreover, only the application of the coupled theory reflects a load-induced anisotropic growth of the inclusion, which exhibits an isotropic material behavior, due to non-vanishing uniaxial Neumann boundary conditions. In addition, it is shown that the smaller the heat conduction coefficient, the more pronounced the anisotropic growth of the inclusion.

Multiphase-field modelling of anisotropic elasticity at finite deformation in Eulerian space

This paper presents a novel computational framework for simulating finite deformation elasticity in Eulerian space combined with the multi-phase field method. A key novel aspect of this work is the phase-field mapping method developed to track interfaces (e.g. grain boundaries) in a multi-phase system (e.g. electrode microstructure) undergoing large deformation. The elastic material response is modelled employing a polyconvex, cubic anisotropic material law which has been implemented in the multiphase-field context and validated using benchmark cases. The performance of this framework is demonstrated through representative numerical examples. The model is well-suited for the simulation of multi-crystalline samples where different phases may possess distinct elastic properties and anisotropies which is shown exemplarily for the electrode calendering process at the nano-scale. Additionally, Kelvin–Voigt visco-elasticity is included in the model which allows for simulation of large deformation in an aerogel under compressive load and, furthermore, enables treatment of fluid–structure interaction within a common configuration. Finally, a broad outlook of technical applications as well as model extensions is given.

Multi-Phase Field Method for Solidification Microstructure Evolution for a Ni-Based Alloy in Wire Arc Additive Manufacturing

Wire arc additive manufacturing achieves high efficiency and low costs by using a melting wire for directional depositions. Thermal analyses and the finite element method have been applied to predict residual stress and the deformation of fabricated parts. For Ni-based alloy production, a method for predicting solidification microstructure evolution with segregation is needed in order to design precise heat treatment procedures. In this study, a multi-phase field method coupled with a CALPHAD database is developed to simulate the solidification microstructure evolution of a practical Ni-based alloy. Thermal analyses of a wire arc additive manufacturing model were performed by the process modeling of multi-pass depositions with a running cyclic arc. Solidification microstructure evolution was obtained using the temperature profile in each deposited layer by the multi-phase field method. These predicted microstructures are compared with experimental measurements. It is confirmed that the multi-phase field method coupled with the CALPHAD database is effective for predicting solidification microstructure and segregation in the engineering of Ni-based alloys.

An ordered active parameter tracking method for efficient multiphase field simulations

The multiphase field (MPF) method has been widely used for microstructural evolution and immiscible multiphase flow simulations. In this study, we have developed an ordered active parameter tracking (OAPT) method for efficient simulations of the multiple phases using the MPF method. The algorithm and performance of the OAPT method are presented and compared to those of the conventional active parameter tracking (CAPT) method. The grain growth problems were simulated for demonstration, and the results show good agreement with the conventional one and about four to seven-fold performance improvement. Moreover, we demonstrate the application of the proposed method for the multiphase flow problem, namely the turbulent bubbly downflow. The OAPT method can speed up the calculation by four folds while maintaining the original accuracy.

Comparison of evolving interfaces, triple points, and quadruple points for discrete and diffuse interface methods

The evolution of interfaces is intrinsic to many physical processes ranging from cavitation in fluids to recrystallization in solids. Computational modeling of interface motion entails a number of challenges, many of which are related to the range of topological transitions that can occur over the course of the simulation. Microstructure evolution in a polycrystalline material that involves grain boundary motion is a particularly complex example due to the extreme variety, heterogeneity, and anisotropy of grain boundary properties. Accurately modeling this process is essential to determining processing-structure–property relationships in polycrystalline materials though. Simulations of microstructure evolution in such materials often use diffuse interface methods like the phase field method that are advantageous for their versatility and ease of handling complex geometries but can be prohibitively expensive due to the need for high interface resolution. Discrete interface methods require fewer grid points and can consequently exhibit better performance but have received comparatively little attention, perhaps due to the difficulties of maintaining the mesh and consistently implementing topological transitions on the grain boundary network. This work explicitly compares a recently-developed discrete interface method to a multiphase field method on several classical problems relating to microstructure evolution in polycrystalline materials: a shrinking spherical grain, the steady-state triple junction dihedral angle, and the steady-state quadruple point dihedral angle. In each case, the discrete method is found to meet or outperform the multiphase field method with respect to accuracy for comparable levels of refinement, demonstrating its potential efficacy as a numerical approach for microstructure evolution in polycrystalline materials.

Modeling intercalation in cathode materials with phase-field methods: Assumptions and implications using the example of [formula omitted

The multiphase-field method has great potential to advance future research on battery materials. In this work, we discuss the modeling of phase-separating intercalation compounds based on the Cahn-Hilliard equation as well as a multi-phase, multi-component model operating in an Allen-Cahn framework. Modeling assumptions are introduced step-by-step to facilitate future developments and bridging the gap across scales. Dimensionality reduction reflecting the material anisotropies can be critical for combining the phase-field approach with Newman-type models. Furthermore, we discuss a nano-particle battery model starting from intercalation in a defect-free, single-crystalline platelet. The underlying modelling assumptions are reduced step-by-step, first, including in-plane diffusion as a result of crystal defects and finally accounting for a polycrystalline material section. We show how faster in-plane diffusion promotes phase separation while higher C-rates and coherency strain lead to the opposite effect. This work highlights the importance to consider pre-existing grain boundaries for nucleation at higher-order junctions, heterogeneity of the intercalation fluxes and grain-by-grain filling behaviour. Anisotropic elastic deformation leads to high stresses at the evolving phase boundaries, especially at high misorientations between neighbouring grains. The maximum principle stress is used as an estimate for degradation by fracture in polycrystals.

Simulation of nucleation and growth of austenite in duplex stainless steels – metallurgical study on weld metal of duplex stainless steel SUS329J1

The microstructure of duplex stainless steels is formed by dendrite growth in the primary ferrite-phase and by subsequent solid-state transformation to the secondary austenite-phase. In this study, formation of austenite by solid-state transformation in duplex stainless steels was evaluated. The nucleation and initial growth of austenite was simulated by the software TC-PRISMA. The formation temperatures of austenite in previous works were corresponding to the calculation obtained by adjusting the interfacial energy. Moreover, the microstructure formation process both in grain boundary and intragranular was simulated using a multi-phase field method (MICRESS software). The growth of austenite was assumed to be under paraequilibrium considering the diffusion control of nitrogen and its morphology was assumed to be an acicular growth. The microstructure morphologies were relatively similar to those obtained by the experiments. The simulation method in this study can analyse complex microstructural morphology as well as predict the composition distributions. This method is considered to be highly effective in designing duplex stainless steels and in evaluating the microstructure morphology in welding process.

Numerical study of growth competition between twin grains during directional solidification by using multi-phase field method

A multi-phase field model is established to simulate the growth competition and evolution behavior between seaweed and columnar dendrites during directional solidification. According to the effects of surface tension and interfacial energy, we quantitatively analyze the influences of factors such as inclination angles, pulling velocity, and anisotropic strength on twin growth. The results demonstrate that the pulling velocity and anisotropic strength have an important influence on the morphology and evolution of the seaweed and dendritic growth. The low pulling velocity and anisotropic strength are both key parameters for maintaining the stable morphology of seaweed during competitive growth in a bicrystal, showing that the lateral branching behavior is the root of the dendrites that can ultimately dominate the growth. And it is clarified that the lateral branching behavior and lateral blocking are the root causes of the final dominant growth of dendrites. With the increase of anisotropy strength, the seaweed is eliminated fastest in case 1, the seaweed is transformed into degenerate dendritic morphology, and eliminates the seaweed by promoting the generation and lateral growth of the lateral branches of the dendrites. The increase of pulling velocity is to increase the undercooling of favorable oriented grain and accelerate the growth rate of dendrites, thus producing more new primary dendrites for lateral expansion and accelerating the elimination rate of unfavorable oriented grain.

Non- and Quasi-Equilibrium Multi-Phase Field Methods Coupled with CALPHAD Database for Rapid-Solidification Microstructural Evolution in Laser Powder Bed Additive Manufacturing Condition

A solidification microstructure is formed under high cooling rates and temperature gradients in powder-based additive manufacturing. In this study, a non-equilibrium multi-phase field method (MPFM), based on a finite interface dissipation model, coupled with the Calculation of Phase Diagram (CALPHAD) database, was developed for a multicomponent Ni alloy. A quasi-equilibrium MPFM was also developed for comparison. Two-dimensional equiaxed microstructural evolution for the Ni (Bal.)-Al-Co-Cr-Mo-Ta-Ti-W-C alloy was performed at various cooling rates. The temperature-γ fraction profiles obtained under 105 K/s using non- and quasi-equilibrium MPFMs were in good agreement with each other. Over 106 K/s, the differences between the non- and quasi-equilibrium methods grew as the cooling rate increased. The non-equilibrium solidification was strengthened over a cooling rate of 106 K/s. Columnar-solidification microstructural evolution was performed at cooling rates of 5 × 105 K/s to 1 × 107 K/s at various temperature gradient values under a constant interface velocity (0.1 m/s). The results show that, as the cooling rate increased, the cell space decreased in both methods, and the non-equilibrium MPFM was verified by comparing with the quasi-equilibrium MPFM. Our results show that the non-equilibrium MPFM showed the ability to simulate the solidification microstructure in powder bed fusion additive manufacturing.

Phase-inherent linear visco-elasticity model for infinitesimal deformations in the multiphase-field context

A linear visco-elasticity ansatz for the multiphase-field method is introduced in the form of a Maxwell-Wiechert model. The implementation follows the idea of solving the mechanical jump conditions in the diffuse interface regions, hence the continuous traction condition and Hadamard’s compatibility condition, respectively. This makes strains and stresses available in their phase-inherent form (e.g. varepsilon ^{alpha }_{ij}, varepsilon ^{beta }_{ij}), which conveniently allows to model material behaviour for each phase separately on the basis of these quantities. In the case of the Maxwell-Wiechert model this means the introduction of phase-inherent viscous strains. After giving details about the implementation, the results of the model presented are compared to a conventional Voigt/Taylor approach for the linear visco-elasticity model and both are evaluated against analytical and sharp-interface solutions in different simulation setups.