Phase-field Simulations Research Articles

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.

A brief review on strain engineering of ferroelectric KxNa1−xNbO3 epitaxial thin films: Insights from phase-field simulations

A brief review on strain engineering of ferroelectric KxNa1−xNbO3 epitaxial thin films: Insights from phase-field simulations

Impact of Secondary‐Phase Particle Morphology on Grain Growth in Pure Copper Using a Phase Field Simulation Approach

Impact of Secondary‐Phase Particle Morphology on Grain Growth in Pure Copper Using a Phase Field Simulation Approach

Phase-Field Simulation of Counter-Current Imbibition and Factors Influencing Recovery Efficiency

Phase-Field Simulation of Counter-Current Imbibition and Factors Influencing Recovery Efficiency

Strain-controlled switching of magnetic skyrmioniums in ultrathin nanodisks

Magnetic skyrmions and skyrmioniums have garnered significant attention due to their distinctive topologically nontrivial spin structures. Gaining a deep understanding of the magnetization dynamics of these structures and their interconversion processes is essential for fully leveraging their potential in magnetic storage technology. Here, the dynamics of strain-controlled generation and annihilation of skyrmions and skyrmioniums are investigated using phase field simulation methods. It is discovered that tensile strain can induce the transformation of a single domain into skyrmions and skyrmioniums, which can still exist stably after the strain is released. Notably, skyrmioniums demonstrate robust stability within a specific strain window of −0.2% to 0.5%. Beyond this, escalating the compressive strain magnitude induces a phase transition from skyrmioniums to skyrmions, culminating in a direct collapse to a single-domain state at a critical compressive strain of −0.8%. This study reveals that strain can effectively control a variety of topological magnetic domain structures and achieve their interconversion, providing guidance for the design of low-power, nonvolatile, multi-state spin storage devices.

Breakdown of Volterra's Elasticity Theory of Dislocations in Polar Skyrmion Lattices.

Emerging polar skyrmion crystals (SkX) have raised much interest for technological applications owing to their nontrivial topologies of electric dipoles, quasiparticle-like behaviors, and unique electrical responses. Understanding SkX defects, especially dislocations, is crucial for their unique lattice dynamics and responses; however, it still remains elusive. Here, we have not only demonstrated that a SkX dislocation exhibits an anomalously deformed core structure with over 50% elongation of skyrmions but also discovered that Volterra's elasticity theory of dislocation is broken down in SkX. Our phase-field simulation reveals that these distinct features of SkX dislocation arise from a rigid to soft quasiparticle transition of skyrmions depending on the electric field and temperature. In SkX, there exist inherent mechanics that mitigate the mismatch by both migration and deformation of skyrmions. This work provides novel insights into a new class of lattice mechanics and related functionality arising from the unique properties of quasi-particle SkX.

Topological data analysis assisted machine learning for polar topological structures in oxide superlattices

Ferroelectric topological phases and phase transitions have been extensively investigated recently due to the rich physical insights and potential applications in next-generation electronic devices. However, precisely predicting the topological phase transitions under different internal and external conditions in polar oxide superlattice systems is challenging due to the complex energy competitions and highly nonlinear kinetics involved. Herein, we adopted a state-of-the-art mathematical concept called “persistent homology” from topological data analysis to extract the essential topological features for the polarization data in various topological structures. By implementing the persistent image as the descriptor, support vector regression (SVR) based convolutional neural network (CNN) models are developed for the automated and high precision classification and regression of topological states based on high-dimensional phase-field simulation data of the PTO/STO superlattice. Using this method, we can automatically construct the strain and electric field phase diagrams in seconds with high throughput phase-field data. We hope to spur further interest in the integration of state-of-the-art mathematical tools, machine learning algorithms, and condensed matter physics for predictions of topological phase transitions.

Realizing Ultrahigh Energy Storage Density in (Bi0.5Na0.5)0.94Ba0.06TiO3-Based Ceramics via Manipulating the Domain Configuration and Grain Boundary Density.

Dielectric capacitors with a high power density are widely used in various pulsed power electronic systems. However, their low comprehensive energy storage performance severely limits the development of these systems in terms of miniaturization and lightweight design. Herein, we achieved decent energy storage performance in a class of (Bi0.5Na0.5)0.94Ba0.06TiO3 (BNTBT)-based ceramics by synergistically manipulating domain configurations and grain boundary densities. High-resolution transmission electron microscopy and piezoresponse force microscopy confirm that composition-driven refined domain configurations with weak polarity effectively improve the polarization response of BNTBT-based ceramics. The results of experimental and phase-field simulation analysis indicate that the refined grain size contributes to its high breakdown electric strength (Eb). Benefiting from the high polarization difference (ΔP) of 32.62 μC/cm2, delayed saturation polarization behavior, and an ultrahigh Eb of 815.00 kV/cm, BNT-based ceramics simultaneously achieve a high energy storage density (Wrec) of ∼12.25 J/cm3 and an efficiency (η) of ∼86.90%. Because of these structure-induced advantages, the ceramics also exhibit good energy storage temperature stability in the range of 30-150 °C. We believe that the findings of this work may provide practical guidance for the development of high-performance energy storage ceramics.

Deep operator network surrogate for phase-field modeling of metal grain growth during solidification

A deep operator network (DeepONet) has been constructed that generates accurate representations of phase-field model simulations for evolving two dimensional metal grain morphology growing from melt. These representations serve as lower resolution, computationally efficient stand-ins for quick parameter space exploration of solutions to the Allen–Cahn equations that dictate the phase-field model simulations. The experimental target for the phase-field model is a uranium casting system cooling a 434 g uranium charge from a maximum temperature of 1400 °C at an average rate of 30 °Cmin, traversing the crystallographic phases of the pure metal. Experimental parameters inform the phase-field model, whose higher resolution computational model solutions are used to train the DeepONet in a given parameter space with the aim of developing a faster, more efficient method for predicting the solidifying metal’s microstructure at different potential experimental values. The final DeepONet generates high accuracy, lower resolution predictions with cumulative relative approximation error over all timesteps of less than 0.5%, while ensuring solutions remain within physically feasible ranges. These relative error values are comparable with other state-of-the-art DeepONet models for microstructure evolution, while significantly reducing the amount of training data required. Training a convolutional neural network simultaneously with the DeepONet, enforcing realistic values at the complex metal grain boundaries, and mathematically encoding boundary conditions into the structure of the DeepONet improved prediction accuracy and computational efficiency over a standard DeepONet model.

Investigating the influence mechanism of aluminum doping amount on the density and grain size of AZO target through phase-field simulation

This study proposes a three-phase model (primary, secondary, and pore phases) utilizing the phase-field method to simulate sintering densification in AZO targets. The impact of secondary phase content and dimensions on the densification of AZO targets was investigated. The findings indicated that the density of the AZO target exhibited an initial increase followed by a subsequent decline with the augmentation of the secondary phase content and size. Conversely, the grain size demonstrated a continuous reduction. The Brook kinetic equation yields activation energies for grain growth of 607.40kJ/mol, 999.32kJ/mol, and 1093.20kJ/mol for 1wt%, 2wt%, and 3wt% Al2O3, respectively. Furthermore, the viability of the model was corroborated through experimentation. Based on the simulation outcomes, AZO targets doped with 2wt% Al₂O₃ achieved 99.78% densification, an average grain size of 3.92 μm, and a resistivity of 1.38 mΩ·cm.

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

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

Insight of the microstructure evolution and performance enhancement of spinodal decomposition in (Ti, Zr)C composite carbide ceramics: Multiscale simulation and experimental investigation

Traditional carbide ceramics face a sharp trade-off between hardness and toughness, leading to a significantly reduced lifespan under harsh service conditions like wear, impact, corrosion, and high temperatures. The spinodal decomposition leads to the formation of nano-lamellar structures, serving to refine grain sizes, thus paving the way to simultaneously improve the hardness and toughness of composite carbide ceramics. A wide range of composite carbides can be prepared, but their complex microstructure evolution during aging makes performance optimization extremely challenging. In this work, a strategy by combining the multiscale simulations and experiments is proposed to systematically study the influence of the composition and process on the spinodal decomposition structure and performance. Taking (Ti, Zr)C carbide ceramics as a representative example, three distinct compositions of (Ti, Zr)C solid solutions were successfully synthesized via a sol-gel combined with spark plasma sintering method at 1800°C, guided by thermodynamic calculations. The influence of aging temperature and duration on spinodal decomposition microstructure evolution in (Ti, Zr)C carbide ceramics was studied by integrating phase-field simulations and first-principles calculations with key experimental observations. After spinodal decomposition, numerous nanoscale nodular structures form, accompanied by the generation of dislocations, leading to a significant improvement in both hardness and toughness of the composite carbides. After aging at 1300°C for 3 h, the composite carbides achieved peak hardness at 2436 HV, accompanied by a fracture toughness of 3.24 MPa·m1/2. This research provides a scientific approach to improving the hardness and toughness of carbide ceramics through spinodal decomposition, offering essential theoretical foundations for microstructural control and synergistic optimization of performance in innovative carbide ceramics.

Rate theory coupling multi-phase-field simulation for neutron irradiation nanophase and vacancy evolution

Radiation damage is a critical problem for the alloys serving under irradiation circumstances where the numerous defects is induced by the cascade collision of radiation particles, the microstructure and mechanical properties will be damaged by the irradiation. By combination of rate theory (RT) and multi-phase-field (MPF) model, the influences of neutron irradiation dose rate (DR) and irradiation temperature are studied on the radiation-enhanced precipitation, vacancies and interstitial atoms in Fe-35Cr-10Al (at.%) alloys. The RT coupled with phase-field simulations capture the radiation induced nanoscale precipitates, clustering of vacancies and interstitial atoms. With the increase of DR, the diffusion of solute atoms is accelerated for the generation of vacancies and interstitial atoms, and the growth and coarsening rate of α′ phase is quickened. The DR affects the distribution of elements between α/α′ phases, high DR promotes the Fe and Al to partition into α phase, Cr into α′ phase in the formation of α′ phase. Therefore, in the steady-state coarsening stage, the high DR leads to the α′ phase to reach the equilibrium composition earlier. As the increase of neutron irradiation temperature, the separation and coarsening rate of α′ phase are accelerated. The simulation morphology and quantitative characteristics are consistent with the experimental results. The coupling of RT and MPF simulation in alloys with point defects and nanophase under neutron irradiation put forward the study tactics and theory understanding for radiation-enhanced multiple microstructure evolutions.

Phase-field modeling of peritectic coupled growth in the TRIS–NPG model system

Microstructures forming during the directional solidification of the hypo-peritectic TRIS–NPG model alloy were studied by phase-field simulations. Steady state growth forms could be obtained under conditions similar to those in recent space experiments. In two dimensions, the wavelength range of stable lamellar structures has been determined as function of the temperature gradient and the melt composition. Front temperatures extracted from simulations were compared to the predictions of the modified Jackson–Hunt theory. In three dimensions, structures and phenomena known from eutectic systems were reproduced. Transitions between lamellar and rod structures were induced by slowly changing the melt composition. The regularizing effect of tilting the temperature gradient, transforming a random labyrinth pattern to an ordered one, has also been demonstrated. These simulations show the universality of the basic governing mechanism — that is, the competition of solute diffusion with capillary effects — in the pattern formation of these multi-phase systems.

A microstructural thermo‐elastoplastic analysis of the mushy zone during laser beam welding

AbstractAs a modern method of contact free fusion, laser beam welding has gained importance in recent years. This is due to both the possibiliy of a high advance speed of the laser and low thermal distortion of the components compared to other welding processes, see Dal and Fabbro. These beneficial process properties follow from a focused laser, which thus results in a precise energy input. In addition, laser beam welding processes can be automated which makes the process attractive for a wide range of applications, cf. Dilthey. Nevertheless, various influences, like the temperature gradient, the chemical composition or process parameters, may have an effect on the occurrence of so‐called solidification cracks. Solidification cracks arise during solidification and occur in the area of the mushy zone, which is located behind the melt pool. The material behavior in the mushy zone is governed by a dendritic morphology. Due to this, liquid phase inclusions may occur in some parts of the mushy zone, see Hills et al.. When these trapped areas solidify, no further liquid phase can follow and a negative pressure is created. The resulting stresses can then lead to solidification cracks. In further studies, the geometry of the dendritic morphology will be generated on the basis of previous phase field simulations. These describe the dendrite growth controlled by the chemical concentration. The resulting geometry will then be simulated using the possibilities of a thermo‐elastoplastic material model, which will enable temperature‐driven imitation of growth. This contribution addresses the evolving state of the dendritic microstructure in a simplified manner. Therefore, for the first study, a simplified morphology is used, that is, no secondary arms and an elliptical geometry of the dendrites is assumed. Thermal as well as elastoplastic effects are already taken into account, in order to represent the inherent stress and strain states in the microstructure. These may be related to critical states leading to failure in the future.

A novel semi-explicit numerical algorithm for efficient 3D phase field modelling of quasi-brittle fracture

Phase field models have become an effective tool for predicting complex crack configurations including initiation, propagation, branching, intersecting and merging. However, several computational issues have hindered their utilisation in engineering practice, such as the convergence challenge in implicit algorithms, numerical stability issues in explicit methods and significant computational costs. Aiming to providing a more efficient numerical algorithm, this work integrates the explicit integral operator with the recently developed neighbored element method, for the first time, to solve the coupled governing equations in phase field models. In addition, the damage irreversibility can be ensured automatically, avoiding the need to introduce extra history variable for the maximum driving force in traditional algorithms. Six representative fracture benchmarks with different failure modes are simulated to verify the effectiveness of the proposed method, including the multiple cracks in heterogeneous concrete at mesoscale. It is found that this semi-explicit numerical algorithm yields consistent crack profiles and load capacities for all examples to the available experimental data and literature. In particular, the computational cost is significantly reduced when compared to the traditional explicit modelling. Therefore, the presented numerical algorithm is highly attractive and promising for phase-field simulations of complicated 3D solid fractures in structural-level engineering practices.

Three-Dimensional Phase-Field Simulation of Stress-Assisted Two-Way Shape Memory Effect and Its Cyclic Degradation of Single-Crystal NiTi Shape Memory Alloy

Three-Dimensional Phase-Field Simulation of Stress-Assisted Two-Way Shape Memory Effect and Its Cyclic Degradation of Single-Crystal NiTi Shape Memory Alloy

Remarkable energy storage performance of BiFeO3-based high-entropy lead-free ceramics and multilayers

Electrostatic energy storage capacitors featuring fast charge–discharge capability play an indispensable role in pulsed power capacitors. However, the inverse correlation between polarization and dielectric breakdown strength (Eb) is the main obstacle limiting the access to high recoverable energy storage density (Wrec) and high efficiency (η). In this work, we designed a series of novel (1-x)BiFeO3-x(Ba0.2Sr0.2Ca0.2Bi0.2Na0.2)TiO3 (BF-xBSCBNT, x = 0.4–1.0) high-entropy lead-free relaxor ferroelectric ceramics. The synergy of refined grain size, broadened band gap, and core–shell microstructure is well-investigated by experimental results and phase-field simulations. High polarization difference is achieved by the strengthened relaxation behavior and the dominated polar nanoregions (PNRs). Both high Wrec of 6.0 J/cm3 and η of 81.1 % are achieved in the BF–0.8BSCBNT ceramics. In particular, a remarkable Wrec of 12.1 J/cm3 with a η of 86.1 % are obtained in the multilayer ceramic capacitors (MLCCs) fabricated from the x = 0.8 composition. Excellent temperature stability (<4.0 % in the range of 25–140 °C) and frequency stability (<6.2 % in the range of 1–500 Hz) are also achieved in MLCCs. The excellent energy storage performance demonstrates that high-entropy strategy is effective to develop novel lead-fee ceramics and devices for energy storage applications.