Phase-field Simulations Research Articles

Numerical Analysis to Evaluate the Effect of Cooling Rates on Microstructures in Casted Cu-Ni-Si Alloys by Phase-Field Simulation

Numerical Analysis to Evaluate the Effect of Cooling Rates on Microstructures in Casted Cu-Ni-Si Alloys by Phase-Field Simulation

Chemo-mechanical benchmark for phase-field approaches

Abstract Phase-field approaches have gained increasing popularity as a consequence of their ability to model complex coupled multi-physical problems. The efficient modeling of migrating diffuse phase boundaries is a fundamental characteristic. A notable advantage of phase-field methods is their ability to account for diverse physical driving forces for interfacial motion due to diffusive, mechanical, electro-chemical, and other processes. As a result of this versatility, phase-field methods are frequently employed in the fields of materials science, mechanics, and physics, and are continually undergoing development. To test the accuracy of these developments, it is indispensable to establish standardized benchmark tests, to ensure the thermodynamic consistency of studies carried out. This work presents a series of such tests based on chemo-elastic equilibrium states for Fe-C binary alloys, benchmarking the performance of a phase-field model with chemo-elastic coupling based on the grand potential density. Use of parameters for the Fe-C system from a Calphad database allows for the determination of the Gibbs free energy, thereby enabling the quantification of chemical driving forces. For a circular inclusion, the capillary driving force is derived on a geometrically motivated basis using the lever rule and expressed as a function of the chemical potential. These simulations contribute to the development of standardized benchmark tests that validate chemical, capillary, and mechanical driving forces separately and in combination. The present study compares phase-field simulation results with results from the analytic solution of chemo-elastic boundary value problems and the generalized Gibbs–Thomson equation.

Assessing the Ubiquity of Bloch Domain Walls in Ferroelectric Lead Titanate Superlattices

The observation of unexpected polarization textures such as vortices, skyrmions, and merons in various oxide heterostructures has challenged the widely accepted picture of ferroelectric domain walls as being Ising-like. Bloch components in the 180° domain walls of PbTiO3 have recently been reported in PbTiO3/SrTiO3 superlattices and linked to domain wall chirality. While this opens exciting perspectives, the ubiquity of this Bloch component remains to be further explored. In this work, we present a comprehensive investigation of domain walls in PbTiO3/SrTiO3 superlattices, involving a combination of first- and second-principles calculations, phase-field simulations, diffuse scattering calculations, and synchrotron-based diffuse x-ray scattering. Our theoretical calculations highlight that the previously predicted Bloch polarization in the 180° domain walls in PbTiO3/SrTiO3 superlattices might be more sensitive to the boundary conditions than initially thought and is not always expected to appear. Employing diffuse scattering calculations for larger systems, we develop a method to probe the complex structure of domain walls in these superlattices via diffuse x-ray scattering measurements. Through this approach, we investigate depolarization-driven ferroelectric polarization rotation at the domain walls. Our experimental findings, consistent with our theoretical predictions for realistic domain periods, do not reveal any signatures of a Bloch component in the centers of the 180° domain walls of PbTiO3/SrTiO3 superlattices, suggesting that the precise nature of domain walls in the ultrathin PbTiO3 layers is more intricate than previously thought and deserves further attention. Published by the American Physical Society 2024

Modeling abscission of cacti branches

During evolution, various functional principles have evolved that allow plants to create predetermined breaking points for the spatially defined abscission of organs. In the plant family of cacti, some species, such as Cylindropuntia bigelovii, have fragile branch–branch junctions that serve vegetative reproduction, while in other species, such as Opuntia ficus-indica, they are very stable. The fracture behavior of these junctions has been thoroughly characterized anatomically and mechanically, the data being the prerequisite for the performance of cactus-inspired phase field simulations. We have found that models composed of homogeneous materials or material systems with low elastic modulus contrast (analogous to Cylindropuntia bigelovii) exhibit a fracture mode where cracks initiation occurs at the epidermis of the junction notch. In comparison, heterogeneous material systems with high elastic modulus contrast (similar to Opuntia ficus-indica) show fracture nucleation along the inner vascular bundles, with an increase in the maximum fracture energy by a factor of 2.2. In the high contrast heterogeneous models, the V-notch and stiffening of the dermal tissue (“periderm”) have a negligible effect on their fracture behavior. In addition, the fracture morphologies of these models resemble the rough junction fracture sites found experimentally. The knowledge gained about the geometric influences and the importance of the contrasts in the mechanical properties of the individual materials in the overall system can be transferred as functional principles to bioinspired engineering composites in order to program their fracture behavior.

Setting the standard for machine learning in phase field prediction: a benchmark dataset and baseline metrics

Phase field models are an important mesoscale method that serves as a bridge between the atomic scale and the macroscale, used for modeling complex phenomena at the microstructure level. Machine learning can be employed to accelerate these simulations, enabling faster and more efficient analyses. However, the development of new machine learning algorithms depends on access to extensive datasets. This work introduces an accessible and well-documented dataset aimed at benchmarking new machine learning algorithms. We validate the dataset with a benchmark using U-Net regression, a widely used neural network architecture. Although direct comparisons are limited by the lack of existing benchmarks, our model’s error metrics are competitive with previous work and generalize across multiple domain sizes. This contribution provides a valuable resource for future efforts in machine learning model development for phase field simulations and demonstrates the potential of U-Net regression, highlighting the scope for novel method development in this area.

Degradation evaluation on the mechanical properties of composite electrode through an integrated phase field and continuum simulation

The electrochemical degradation phenomena in lithium-ion batteries imply that the mechanical properties of a composite electrode may deteriorate during cycling, however, there are still lack of detail information on their direct correlations. Here, we proposed an integrated phase field-continuum simulation method to track the process of mechanical degradation. Based on scanning electron microscope images of a composite electrode, a representative volume element microstructural model was constructed and the damage evaluations caused by lithiation and delithiation were estimated though the phase-field method. Then a subsequent continuum mechanical model was performed on meso-structures with damage to evaluate their mechanical properties. The results show that damage primarily occurs during the initial cycle, and decreasing particle sizes and the cycling rate can effectively mitigate damage. This work bridges the relationship between the particle damage and mechanical properties of composite electrodes, and thus, provides an important insight on the stress evaluation and optimal design in service.

Splitting behavior of lamella

Lamella is a unique microstructure in matters that possesses special properties. The formation and evolution of lamellar microstructures are crucial for achieving super abilities, while the mechanisms of lamellar formation and evolution at the nanoscale are unclear. Driving by the interesting while uncovered micromechanism in lamellar microstructure evolution, we performed the phase-field simulation and experiment to investigate the multiple splitting behaviors of lamellar Ti-Al alloys. The splitting mode of lamella is discovered, inner splitting (IS) and outside splitting (OS). The originations of splitting are found to come from the high interfacial energy of step-like interface between γ variants for the IS, and the stress concentration at α2/γ interface drives step-like interface for the OS. The lamellar splitting is complied with the energy change in matter, decreasing in total free energy and elastic energy, while increasing in interfacial energy. These findings provide the new mechanisms of internal lamellar interface breaking driven by the interfacial energy. Therefrom, the expected matter features will be achieved by the strategy of microstructure modulation and optimization.

Time series forecasting of multiphase microstructure evolution using deep learning

Microstructure evolution, which plays a critical role in determining materials properties, is commonly simulated by the high-fidelity but computationally expensive phase-field method. To address this, we approximate microstructure evolution as a time series forecasting problem within the domain of deep learning. Our approach involves implementing a cost-effective surrogate model that accurately predicts the spatiotemporal evolution of microstructures, taking an example of spinodal decomposition in binary and ternary mixtures. Our surrogate model combines a convolutional autoencoder to reduce the dimensional representation of these microstructures with convolutional recurrent neural networks to forecast their temporal evolution. We use different variants of recurrent neural networks to compare their efficacy in developing surrogate models for phase-field predictions. On average, our deep learning framework demonstrates excellent accuracy and speedup relative to the “ground truth” phase-field simulations. We use quantitative measures to demonstrate how surrogate model predictions can effectively replace the phase-field timesteps without compromising accuracy in predicting the long-term evolution trajectory. Additionally, by emulating a transfer learning approach, our framework performs satisfactorily in predicting new microstructures resulting from alloy composition and physics unknown to the model. Therefore, our approach offers a useful data-driven alternative and accelerator to the materials microstructure simulation workflow.

Low hysteresis in composites ceramics achieved by building polarization field and restoring force

Large strain hysteresis and remnant strain are one of the vital reasons for the absence of BiFeO3-BaTiO3-based ceramics in commercial actuator fields. Here, we elaborately propose a strategy, preparing 0–3 type composite ceramics, to reduce the hysteresis and remnant strain, and the target is successfully achieved by building restoring force and polarization field. Normal strain constant and electric field-induced strain in 0–3 composites have enhanced by 260% and 196% compared to those of non-composite ceramics, respectively. Also, hysteresis and remnant strain in 0–3 composites have decreased by 35.9% and 50.6% in contrast to those of non-composites. Superior electrostrain properties under the low electric field are attributed to the construction of polarization field, restoring force, and micro-capacitance, coinciding with phase field simulation, and the strategy will pave a useful way to optimize the hysteresis and remnant strain in BiFeO3-BaTiO3-based high-temperature ceramics.

Mechanical Writing of Polar Skyrmionic Topological States via Extrinsic Dzyaloshinskii-Moriya-like Flexoelectricity in Ferroelectric Thin Films.

Exploring complex topological structures in condensed matter has shown promising applications in nanotechnology. Although polar topologies such as chiral vortices and skyrmions have been observed in ferroelectric heterostructures, their existence in simple systems has posed challenges due to the absence of intrinsic noncollinear interaction (like Dzyaloshinskii-Moriya interaction in ferromagnetics). Here, we demonstrate that a nanoindentation mechanically switches local polarizations to stable polar topologies, including skyrmions, within a room-temperature PbTiO3 thin film via the flexoelectric effect as a noncollinear (Dzyaloshinskii-Moriya-like) driving force using phase-field simulations. In addition, by moving the indenter, the continuous polarization switching leads to the "writing" of arbitrary polar patterns (such as donut-like skyrmionium). Furthermore, the written topologies can be "erased" by applying a voltage with the same conducted indenter. Therefore, this study shows the writing and erasing process of room-temperature polar topologies in a ferroelectric thin film, which significantly advances their potential applications.

Advances in machine learning methods in copper alloys: a review.

Advanced copper and copper alloys, as significant engineering structural materials, have recently been extensively used in energy, electron, transportation, and aviation domains. Higher requirements urge the emergence of high-performance copper alloys. However, the traditional trial-and-error experimental observations and computational simulation research used to design and develop novel materials are time-consuming and costly. With the accumulation of material research and rapid development of computational ability, the thorough application of material genome engineering has sped up the development of novel materials and facilitates the process of systematic engineering application. This review summarizes the benefits of data-driven machine learning techniques and the state of the art of machine learning research in the area of copper alloys. It also displays the widely used computational simulation approaches (e.g., the first-principles calculation, molecular dynamics simulation, phase-field simulations, and finite element analysis) and their combined applications in material design and property prediction. Finally, the limitations of machine learning research methods are outlined, and future development directions are proposed.

Quasi-2D and 3D phase-field simulation studies of polar vortex domain structures for high-density domain engineering

Phase-field simulations were performed on the polar vortex array structure appearing in the SrTiO3/PbTiO3/SrTiO3 heterostructure to investigate the Kittel law relationship (w ∝ d1/2) between the vortex array period (w) and the PbTiO3 ferroelectric layer thickness (d). Quasi-two-dimensional simulation results show that the equilibrium period can be determined from the relationship between the size of the simulation box and the total free energy density and that the obtained period obeys the Kittel law. However, three-dimensional simulation results show that the polar vortex arrays are not homogeneous in the direction perpendicular to the vortex plane. In the pure vortex state, no obvious correlation exists between the simulation size and energy because of the dislocations caused by the vortex period mismatch. However, in the mixed state with ferroelectric a1/a2, the vortex domains exhibit equilibrium periods that satisfy the Kittel law relationship because of the confinement effect caused by phase separation. Further study of the confined vortex domains in these mixed states and the vortex domains of finite size due to mismatched dislocations are expected to enable domain engineering of higher domain densities.

Parameterization of a phase field model for ferroelectrics from molecular dynamics data

In this work, molecular dynamics (MD) simulations of ferroelectric BaTiO3 are used to generate parameters for a phase-field (PF) model. The MD simulations provide important input parameters, such as elastic and piezoelectric properties, as well as domain wall (DW) widths and velocities. A parameterization technique is proposed to incorporate the MD results into the PF simulations, allowing for the generation of relevant parameters. Anisotropic interface energy coefficients are used to match the widths of both the 180° and 90° DWs in the PF simulations to the MD data. The velocities of the DWs are calculated and compared to the results obtained from MD simulations, demonstrating excellent agreement. Furthermore, we investigate the disparity between coercive fields obtained from PF simulations and MD simulations. Initially, our PF simulations yield coercive fields approximately twice as high as those observed in MD simulations. To address this discrepancy, we introduce a nucleus into the PF simulation system. The parameters of this nucleus are derived from a statistical analysis of coercive field data from MD simulations. By incorporating the nucleus model, we achieve a coercive field in PF simulations that closely aligns with the MD results. Validation of the parameterized PF model using MD data obtained with different initial conditions and thermodynamic constraints shows good agreement, further confirming the model’s accuracy.

Electric field control of piezoelectricity in textured PMN-PZT ceramics

Ferroelectric ceramics generally develop macroscopic piezoelectric effects only after undergoing suitable electric field treatment, referred to as poling. In this study, we examine the influence of various poling conditions, including both direct current (DC) and alternative current (AC) electric fields, as well as temperature, on the piezoelectric properties of [001]-textured PMN-PZT ceramics. The results show that the piezoelectric properties under alternative current poling (ACP) condition include d33 = 1330 pC/N, ɛr = 3280, and k31 = 0.646. ACP shows a 9% increase in d33 compared to direct current poling (DCP). Increasing the temperature during DCP can raise d33 to 1350 pC/N and k31 to 0.66, with a high TR−T of 121°C. Due to the influence of grain boundaries, there is a significant difference between [001] textured ceramics and single crystals, as the domain structure switching and growth are constrained by orientation differences between grains. Phase-field simulations further reveal that the growth of domains is impeded by grain boundaries in polycrystals, which hinders the formation of larger-size domains. The results suggest that ACP may be more effective in larger grains in textured ceramics.

Phase Field Simulation Study of the β′ Precipitation Phase of Mg-Gd-Y Alloys

Abstract The key reinforcing precipitation phase in the aging precipitation sequence of Mg-Gd-Y alloys is the bamboo-leaf-like substable β′ phase, which can effectively inhibit the crystal base slip to enhance the age-hardening response of magnesium alloys. In this work, a phase field model of the aging precipitation process of Mg-Gd-Y ternary alloys was created by combining the phase field method and thermodynamic database. The microstructure evolution of the β′ precipitated phase in the isothermal aging process of Mg-Gd-Y alloys was simulated, and the impact of anisotropic parameters on the precipitated phase’s morphology was examined. Finally, the effect of Gd content on the β′ phase in the Mg-Gd-Y alloys was disclosed from the standpoint of the kinetics. β′ precipitation phase precipitation mechanism of Mg-Gd-Y alloys from the kinetic point of view. The results show that the anisotropic elastic energy and the interfacial energy work together to influence the shape of the β′ precipitation phase. Gd content increases lead to refined grains, an increase in the volume fraction of the Mg-Gd-Y alloy β′ precipitated phase, and a greater aspect ratio of the precipitated phase.

Unlocking Electrostrain in Plastically Deformed Barium Titanate.

Achieving substantial electrostrain alongside a large effective piezoelectric strain coefficient (d33*) in piezoelectric materials remains a formidable challenge for advanced actuator applications. Here, a straightforward approach to enhance these properties by strategically designing the domain structure and controlling the domain switching through the introduction of arrays of ordered {100}<100> dislocations is proposed. This dislocation engineering yields an intrinsic lock-in steady-state electrostrain of 0.69% at a low field of 10kVcm-1 without external stress and an output strain energy density of 5.24Jcm-3 in single-crystal BaTiO3, outperforming the benchmark piezoceramics and relaxor ferroelectric single-crystals. Additionally, applying a compression stress of 6MPa fully unlocks electrostrains exceeding 1%, yielding a remarkable d33* value over 10000pmV-1 and achieving a record-high strain energy density of 11.67Jcm-3. Optical and transmission electron microscopy, paired with laboratory and synchrotron X-ray diffraction, is employed to rationalize the observed electrostrain. Phase-field simulations further elucidate the impact of charged dislocations on domain nucleation and domain switching. These findings present an effective and sustainable strategy for developing high-performance, lead-free piezoelectric materials without the need for additional chemical elements, offering immense potential for actuator technologies.

Triple junction benchmark for multiphase-field models combining capillary and bulk driving forces

Abstract A benchmark problem is formulated which is well suited for the validation of mesoscopic phase-field models for grain-boundary migration in polycrystals. First, an analytical steady-state solution of the sharp moving boundary problem is derived for a symmetric lamellar structure, which is valid for arbitrary bulk driving forces and triple junction angles. Characteristic quantities are identified to reduce the parameter space which in turn allows a systematic comparison of simulations and analytical results. Various multiphase-field (MPF) formulations are compared which approximate the sharp interface problem in terms of a diffuse regularization. An interfacial thickness convergence study reveals that the model error is largely dependent on the ratio of bulk to interfacial stabilizing force as well as the underlying model formulation. An additional grid convergence study highlights the efficiency of a more advanced discretization scheme. The results can be used to guide the selection of appropriate models and to estimate the interface thickness and spatial resolution required to achieve a given accuracy target. The post-processing framework consists of a fully automated determination of well-defined metrics from the phase field simulation data, eliminating human bias and facilitating reproducibility. The corresponding code is made openly available to assist the materials science and engineering community in validating MPF, multi-order parameter and similar model developments. We believe that this work provides a reliable benchmark procedure to better understand the potentials and limitations of current multiphase-field models as well as alternative approaches.

A data-driven strategy for phase field nucleation modeling

We propose a data-driven strategy for parameter selection in phase field nucleation models using machine learning and apply it to oxide nucleation in Fe-Cr alloys. A grand potential-based phase field model, incorporating Langevin noise, is employed to simulate oxide nucleation and benchmarked against the Johnson-Mehl-Avrami-Kolmogorov model. Three independent parameters in the phase field simulations (Langevin noise strength, numerical grid discretization and critical nucleation radius) are identified as essential for accurately modeling the nucleation behavior. These parameters serve as input features for machine learning classification and regression models. The classification model categorizes nucleation behavior into three nucleation density regimes, preventing invalid nucleation attempts in simulations, while the regression model estimates the appropriate Langevin noise strength, significantly reducing the need for time-consuming trial-and-error simulations. This data-driven approach improves the efficiency of parameter selection in phase field models and provides a generalizable method for simulating nucleation-driven microstructural evolution processes in various materials.

Exploring the effects of solution temperature and cooling rate on α phase in a multicomponent titanium alloy: Insights from phase-field simulations

The solution temperature and cooling rate during heat treatment of titanium alloys will affect their microstructure and thus mechanical properties. In this study, a phase-field model of the β →α solid state phase transformation in multicomponent titanium alloys is developed to investigate the effects of solution temperature and cooling rate on the microstructural evolution of Ti60 alloy. The thermodynamic calculations, the evolution of equiaxed α phase and solute distribution during solid solution process, the orientation of α lamellae during continuous cooling process, the morphology of α lamellae structure under different cooling rates, and the changing trend of α lamellae spacing, all correspond well with the experimental results. Our work not only gives a clear outlook for the influence of solution temperature and cooling rate on α phase of Ti60 alloy, but also provides a feasible approach for developing heat treatment protocols in actual multicomponent titanium alloys.

An active and durable perovskite electrode with reversibly phase transition-induced exsolution for protonic ceramic cells

Protonic ceramic cells (PCCs) possess great application potential, attributed to their cleanliness and high energy conversion efficiency. However, designing active air electrodes with both high stability is challenging. Herein, we report a nanospinel-modified perovskite oxide, Nd0.5Ba0.5Mn0.7Co0.15Ni0.15O3−δ-(CoxNiy)3O4 (CNO@NBMCN), obtained by a reversibly phase transition-induced exsolution process, as a highly active and durable air electrode for PCCs. Phase-field simulations reveal the intrinsic mechanism of nanoparticles strongly pinning to the substrate in a high-temperature oxidizing environment. The CNO@NBMCN electrode exhibits remarkable low area specific resistance (0.38 Ω cm2 at 600 °C) and exceptional stability (degradation rate 0.01 % h−1). It is confirmed by soft X-ray absorption spectroscopy that the construction of the heterostructure leads to more distortion in Mn–O octahedra and weaker metal–oxygen bonds, thereby contributing to the formation of oxygen vacancies. And the exceptionally high durability supported by both fast ion surface exchange and charge transfer at triple-phase boundaries. A single cell with the CNO@NBMCN air electrode achieves outstanding performances in the fuel cell (peak power density of 1.28 W cm−2) and electrolysis modes (current density of −2.14 A cm−2 at 1.3 V), recorded at 700 °C. This work inspires innovative design concepts for robust heterogeneous electrocatalysts.