Phase-field Simulations Research Articles

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.

Deciphering the Effect of Defect Dipoles on the Polarization and Electrostrain Behavior in Perovskite Ferroelectrics.

Defect dipoles are crucial for regulating electromechanical properties in piezoelectric ceramics, but their effects on polarization and electrostrain behaviors are still unclear. Here, a reasonable theoretical model is proposed and evidenced by experiments to address a long-standing puzzle of the relationship between the internal bias field and defect dipoles. By incorporating the additional polarization induced by defect dipoles, we refine the classical theory to account for the recently reported asymmetric giant-strain behaviors. Phase-field simulation reveals the electrostrain evolution in response to defect dipole elastic distortion and additional polarization. This work not only elucidates the effect of defect dipoles on polarization and electrostrain but also advances the theoretical understanding of defects in piezoelectrics.

Data-driven bio-mimetic composite design: Direct prediction of stress–strain curves from structures using cGANs

Designing high-performance composites requires integrating tasks, including material selection, structural arrangement, and mechanical property characterization. Accurate prediction of composite mechanical properties requires a comprehensive understanding of their mechanical response, particularly the failure mechanisms under high deformations. As traditional computational methods struggle to exhaustively explore every composite configuration in the vast design space for optimal design search, machine learning offers rapid identification of optimal composite designs. This study presents a cGAN-based deep learning model for predicting stress–strain curves directly from composite structures using an image-to-vector approach. The model incorporates fully connected layers within a U-Net generator for stress–strain curve generation and utilizes a PatchGAN discriminator for realism assessment. This end-to-end mapping from structures to mechanical response effectively eliminates the need for extensive simulations and labor-intensive post-analyses. Phase-field simulations were conducted to model the material failure process, generating stress–strain curves for various composite structures used as ground truth data to train and test the surrogate model. This study incorporates various composite structures in the dataset, including random (RS), layered (LS), chessboard-like (CS), soft-scaffold (SS), and hard-scaffold (HS), enhancing the representation of design diversity. Despite being trained on a limited dataset (approximately 1.5% for each bio-mimetic structure and 10−72% for RS composites), the model achieves highly accurate predictions in stress–strain curves, with MAE loss converging to 0.01 for training and 0.05 for testing after 2 million iterations. High evaluation scores on training data (R2>0.997, MAPE <1.08%) and testing data (R2>0.946, MAPE <5.53%) demonstrate the model’s accuracy in predicting mechanical properties such as Young’s modulus, strength, and toughness across all composite structures. Overall, the study provides a proof of concept for using machine learning to simplify the design process, demonstrating its potential for solving inverse composite design problems.

Quantitative phase-field simulation of Pu oxidation at low temperatures through a pragmatic strategy for coupling with thermodynamic and diffusion descriptions of stoichiometric compounds

In this letter, we developed a pragmatic strategy for coupling the phase-field model with the thermodynamic/kinetic descriptions and applied it to quantitatively describe the complex oxidation process of Pu. The microstructure evolution of Pu during oxidation at 423 K and 473 K was carefully simulated, and the thickness curves of various oxides were successfully predicted, which can provide the theoretical support for the storage and protection of Pu related materials. Furthermore, it is anticipated that the pragmatic strategy is universally applicable for quantitative simulation of the complex oxidation process.

Phase-field formulated meshless simulation of axisymmetric Rayleigh-Taylor instability problem

A formulation of the immiscible Newtonian two-liquid system with different densities and influenced by gravity is based on the Phase-Field Method (PFM) approach. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in all the terms. The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in Cartesian and axisymmetric coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. This solution procedure has improved stability compared to Chorin's pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the authors’ previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris, Open-source Field Operation and Manipulation (OpenFOAM®) code and previously existing results.

Ohta–Kawasaki energy for amphiphiles: Asymptotics and phase-field simulations

We study the minimizers of a degenerate case of the Ohta–Kawasaki energy, defined as the sum of the perimeter and a Coulombic nonlocal term. We start by investigating radially symmetric candidates which give us insights into the asymptotic behaviors of energy minimizers in the large mass limit. In order to numerically study the problems that are analytically challenging, we propose a phase-field reformulation which is shown to Gamma-converge to the original sharp interface model. Our phase-field simulations and asymptotic results suggest that the energy minimizers exhibit behaviors similar to the self-assembly of amphiphiles, including the formation of lipid bilayer membranes.

Emergence and transformation of polar skyrmion lattices via flexoelectricity

As analogies to magnetic skyrmions, polar skyrmions in ferroelectric superlattices and multilayers have garnered widespread attention for their non-trivial topology and novel properties like negative capacitance and nonlinear optical effect. So far, they have only been theoretically predicted to be able to assemble ordered hexagonal skyrmion lattices (SkLs) in ferroelectric thin films. Here, based on phase-field simulations, we report the critical roles of flexoelectricity playing in the stabilization and transformation of polar SkLs. Different polar SkL patterns can emerge in the ferroelectric thin films, including tetragonal-SkL, and hexagonal-SkLs with diverse orientations, as summarized by phase diagrams. These emergent SkL states are attributed to the material anisotropy modified by the flexoelectric effect. Interestingly, we further found that the hexagonal-SkLs can be rotated by applying strain gradient or in-plane electric field to the films. Moreover, a nonreciprocal bending response of tetragonal-SkL is also induced by the flexoelectric effect. Our results provide useful guidelines for the implementation of polar skyrmion lattices in experiments.

Quaternary phase-field simulation of the effect of additives on the structure of polyvinylidene fluoride microporous membranes

This study builds upon the existing phase-field model of the non-solvent-induced phase separation (NIPS) method, simulating factors that influence membrane formation in a ternary system. A polyvinylidene fluoride (PVDF) membrane system is utilized, and a fourth substance, an additive, is introduced based on the original mathematical model of the ternary system. The interaction parameters of the additives within the system are calculated using the Krevelen method. The simulation is configured to set the values of additive mobility and loss coefficients of the concentration gradient, thereby simulating the effect of additives in the quaternary system on the evolution of the micropore membrane structure. Finally, simulated images of different concentrations of additives with various types are compared with experimental results of microporous structure and porosity formation. This comparison serves to validate the model.

Stability and deformation of a vacancy defect in skyrmion crystal under external magnetic and temperature fields

Skyrmion, a local bubble-like topological magnetization structure, can collectively emergent in magnets in a lattice form skyrmion crystal (SkX). SkX has great application potential in functional devices because it can manipulate material properties via coupling with atomic lattices. The lattice defects such as vacancy widely exist in the SkX as well, and they have rich dynamic behaviors and have great implications for the host material. However, although the nature of ideal SkX is well studied, the characteristics of SkX defects are relatively underdeveloped. Here, we deeply studied the structural properties of a vacancy defect in the SkX by a thermodynamic phase-field simulation. We found that the higher external magnetic and temperature fields favor rigid skyrmions (crystal), in which the SkX vacancy is less deformed, while the lower fields favor softer skyrmions where the SkX vacancy structure is considerably deformed. Such unique deformation and stability of the SkX vacancy are mainly the results of the competition between free energies in the view of thermodynamics. Our study demonstrated that the external field-controlled static properties of SkX vacancy highly depend on the quasiparticle nature of skyrmions. This indicates the properties of SkX defects can be controlled by the SkX features under different fields, which should open an avenue for the study and design of smart materials and advanced devices by engineering skyrmion crystals.

Chemo-electro-mechanical phase-field simulation of interfacial nanodefects and nanovoids in solid-state batteries

Solid electrolytes encompass various types of nanodefects, including grain boundaries and nanovoids at the Li-metal/solid electrolyte interface, where lithium dendrite penetration has been extensively observed. Despite the importance of ion transport near grain boundaries with different anisotropy and the combinatorial effects with interfacial nanovoids, a comprehensive understanding of these phenomena has remains elusive. Here, we develop a chemo-electro-mechanical phase-field model to elucidate how Li penetrates Li7La3Zr2O12 in the co-presence of grain boundaries and interfacial nanovoids. The investigation unveils a grain-boundary-anisotropy-dependent behavior for Li-ion transport correlated with the presence of interfacial nanovoids. Notably, the Σ1 grain boundary exhibits faster Li dendrite growth, particularly in the co-presence of interfacial nanovoids. The model quantitatively reveals whether interfacial electronic properties dominate Li dendrite morphology and penetration, providing a strategy for designing stable Li/solid electrolyte interfaces. These findings help prioritize approaches for optimally tailoring nanodefects and exploiting synergetic effects at the interface to prevent dendrite formation.

The phase field method coupled with molecular dynamics parameter for simulating phase separation behavior of SBS modified asphalt

Polymer-modified asphalt is a thermodynamically unstable system, in which the polymer and asphalt are prone to phase separation. Due to the inadequacy of characterization methods for phase separation, this field has not been thoroughly understood or explored. This paper introduces an innovative methodology by employing a simulation approach that couples phase-field theory with molecular dynamics parameters. Using SBS polymer-modified asphalt as an example, validate the method's applicability and establish a more precise parameter library for the phase separation field model of SBS-modified asphalt. The phase separation process of SBS-modified asphalt was simulated. In combination with fluorescence microscopy experiments, the evolution of the micro and meso-phase states of the modified asphalt over time was established and tracked. Results indicate that higher light component content and lower heavy component content in asphalt improve its compatibility. Migration coefficients and interaction parameters for the phase field model were determined, and a more accurate correction factor for the entropy contribution of SBS-modified asphalt was estimated. The phase-field model shows that higher light component content and lower heavy component content result in slower and smaller phase separation, leading to better compatibility. The combined phase field and molecular dynamics simulations accurately reproduce experimental findings from fluorescence microscopy, providing theoretical references for studying phase separation in polymers.

High-piezoelectric lead-free BiFeO3–BaTiO3 ceramics with enhanced temperature stability and mechanical properties

BiFeO3–BaTiO3 (BF–BT) ceramics exhibit higher piezoelectric coefficients (d33), Curie temperatures (TC), and temperature stability than other high-temperature lead-free piezoelectric materials. However, despite their crucial role in piezoelectric devices, the mechanical properties of BF–BT ceramics have been underexplored. A thorough evaluation of the mechanical properties of BF–BT is crucial for developing cost-effective and durable lead-free piezoelectric ceramics. Moreover, the specific causes of the high piezoelectric response and excellent temperature stability of BF–BT ceramics remain unclear owing to the instrumental detection threshold, which limits experimental studies to temperatures above 140 °C and below the degradation temperature of d33. To investigate the intrinsic origins of the high piezoelectricity and temperature stability of BF–xBT ceramics and to enhance their mechanical properties, a two-step sintering process is used to fabricate these ceramics (0.25 ≤ x ≤ 0.40). Owing to improvements in grain refinement and reduced Bi3+ volatilization, the BF–0.33BT ceramic exhibits enhanced overall performance, with a modified small punch strength of 155 MPa, Vickers hardness of 5.2 GPa, a d33 of 220 pC/N at room temperature, TC of 466 °C, and d33 values exceeding 400 pC/N at 260 °C. Phase-field simulations, which address the limitations of device detection thresholds, reveal that with increasing temperature, the domain structure relaxes, and polarization intensity decreases. This indicates that changes in the high-temperature piezoelectric properties can be attributed to domain structure relaxation and the increase in dielectric constant. Overall, BF–BT ceramics exhibit superior piezoelectric performance, mechanical properties, and temperature stability, making them highly suitable for use in high-temperature and demanding environments.

Twin experiments and detailed investigation of data assimilation system for columnar dendrite growth in thin film

Accurately predicting dendrite growth during alloy solidification is crucial for enhancing the quality of metallic products. Recently, data assimilation has emerged as a promising tool for integrating two cutting-edge methods for studying dendritic growth, viz. in situ X-ray observation experiments and phase-field (PF) simulations, to elucidate important parameter(s) of the simulation and produce a “digital twin” of a growing dendrite. This study was conducted to evaluate the performance of a data assimilation system, employing an ensemble Kalman filter, through systematic twin experiments focused on columnar dendrite growth in a thin film of a binary alloy. The results show that the data assimilation system effectively estimates multiple parameters and dendrite morphology simultaneously. Furthermore, two approaches—domain decomposition method and parameter estimation method from a part of the domain—to reduce computational costs are investigated. Both methods can effectively reduce the computational cost of data assimilation. Additionally, data assimilation using voxel observation data of varying resolutions, mimicking actual X-ray images, is performed. The study findings discourage the direct use of voxel data owing to incompatible PF simulations. PF relaxation computation is explored as a solution for generating observation data with smooth PF profiles. The results show that while relaxation computation enhances the estimation accuracy for high-resolution voxel data, its efficacy diminishes for low-resolution data. These detailed investigations of data assimilation provide valuable insights for utilizing actual in situ X-ray observation data in dendrite growth studies.

Phase field study on the grain size dependent electrical resistivity-strain response in superelastic nanocrystalline NiTi alloys

The superelastic NiTi alloy's electrical resistivity exhibits significant variations in response to deformation, making it a highly attractive material for sensing applications. However, there exists a lack of comprehensive knowledge of the effects of grain size (GS) on the electrical properties of NiTi alloy. To address this gap, a novel electro-mechanical coupled phase field simulation model is developed, with crucial parameters determined through experimental investigations. The results demonstrate a notable change in the relationship between electrical resistivity and strain, transitioning from a quasi-linear pattern to a piecewise nonlinear one as the GS increases. The observed modification is found to be primarily influenced by the martensitic transformation, characterized by a transition from a uniform mode to a localized mode. As the GS increases, there is an initial drop in the total change of electrical resistivity at peak strain, followed by a subsequent stabilization. This is because the GS, through the stress and the degree of phase transformation, exerts two opposing effects on the variation of electrical resistivity. Furthermore, the combined influence of hysteresis and hysteresis in the martensite fraction contributes to a consistent decrease in resistivity hysteresis as the GS diminishes. This study improves the understanding of how GS affects NiTi alloy's electrical resistivity-strain response and lays the foundation for future sensing performance regulation, expanding its multi-functionality as an intelligent material.

Stoichiometric and non-stoichiometric Mn modification on high-power properties in PYN-PZT piezoelectric ceramics

The types of dopants lead to distinctive microstructural evolution behavior and physical properties in materials. In this study, the effect of stoichiometric and non-stoichiometric Mn modification, namely Pb(Mn1/3Nb2/3)O3 (PMnN) and MnO2, on the microstructure and properties of Pb(Yb1/2Nb1/2)O3-PbZrO3-PbTiO3 (PYN-PZT) piezoelectric ceramics are systematically investigated. It was found that stoichiometric PMnN modification inhibits the grain growth while non-stoichiometric MnO2 modification promotes it, and thus the former yields stronger high-power characteristics (higher internal bias field Ei and larger mechanical quality factor Qm) than the latter. Specifically, with an equivalent amount of Mn modification (2 mol%), PMnN and MnO2 modification PYN-PZT ceramics exhibit significantly different values for average grain size (1.21 μm vs. 14.12 μm), Ei (8.5 kV/cm vs. 5 kV/cm), and Qm (2376 vs.1134). To further evaluate high-power performance, the vibration velocity v of these two modified PYN-PZT under high driving conditions was measured. Under an AC electric field of 3.5 V/mm, the PYN-PZT+6PMnN ceramics exhibit a v of up to 0.95 m s−1, larger than both MnO2-doped PYN-PZT (0.72 m s−1) and unmodified PYN-PZT ceramics (0.1 m s−1), and far outperformance than both PZT-4 and PZT-8 ceramics. Furthermore, to elucidate the origin of the exceptional high-power performance of PMnN-modified PYN-PZT, we performed phase-field simulations revealing a pinning effect of the grain boundary on domain wall motion. Consequently, the small grain size (high grain boundary density) in PMnN-modified PYN-PZT exhibits a strong pinning effect, resulting in a large Qm and outstanding high-power performance.

Nucleation-dependent early growth of dendritic grains in Al-Cu alloys: The real-time observations and large-scale phase-field simulations

Formation of a dendritic grain starts from nucleation and then growth propagation occurs. Through the in situ and real-time solidification experiments of Al-Cu alloys observed by synchrotron X-ray imaging, it is found that the early-stage free growth rate of dendritic grains goes far beyond the concept described by the classical crystal growth theory. The rate gradually drops down even though the liquid is continuously cooled down. Quantitative 3D phase-field simulations demonstrate that this abnormal early-stage growth behavior depends strongly on the nucleation. A critical nucleus can grow rapidly to a peak rate driven by the initial nucleation undercooling, and then its rate gradually drops down to a minimum before it approaches the steady-state free growth regime. This strong dependence of the early-stage growth on the nucleation is then supported by an analytical model, and this correlation enables the accurate identification of the nucleation undercooling for each grain in the experiment and thus allows for a large-scale quantitative simulation of the real-time observed polycrystalline growth. This progress provides a comprehensive understanding on the crystal growth kinetics from a critical nucleus to the growth end, and thus will provide a new theoretical framework to design novel technology to control the solidification microstructures.

Grain boundaries are Brownian ratchets.

We demonstrate that grain boundaries (GBs) behave as Brownian ratchets, exhibiting direction-dependent mobilities and unidirectional motion under oscillatory driving forces or cyclic thermal annealing. We observed these phenomena for nearly all nonsymmetric GBs but not for symmetric ones. Our observations build on molecular dynamics and phase-field crystal simulations for a wide range of GB types and driving forces in both bicrystal and polycrystalline microstructures. We corroborate these simulation results through in situ experimental observations. We analyze these results with a Markov chain model and explore the implications of GB ratchet behavior for materials processing and microstructure tailoring.