Phase-field Simulations Research Articles

Construction of high-performance multi-main-phase Nd–Dy–Fe–B sintered magnets with strengthened chemical heterogeneity

By designing the [(Pr20Nd80)100-xDyx]30.5FebalM1.3B1.0 components (x = 0, 10, 25, 50, 75 wt.%, denoted as Dy0, Dy10, Dy25, Dy50, Dy75), we constructed a series of multi-main-phase (MMP) and single-main-phase (SMP) Nd–Dy–Fe–B bulk magnets with same average composition of (Pr20Nd80)27.5Dy3.0FebalM1.3B1.0 but with different chemical heterogeneities, as evaluated by Dy concentration discrepancy between Dy-rich shell and Dy-lean core (△C). △C increases from 2.8 wt.% for MMP Dy3-I magnet (∼39.4 % Dy25 and ∼60.6 % Dy0), to 3.1 wt.% for MMP Dy3-II magnet (∼19.7 % Dy50 and ∼80.3 % Dy0), further to 3.4 wt.% for MMP Dy3-III magnet (∼13.1 % Dy75 and ∼86.9 % Dy0), compared to 0 wt.% for SMP Dy3 magnet (100 % Dy10). Corresponding magnetic performance at 20∼180 °C is positively related with △C value, as evidenced by the best magnetic properties of MMP Dy3-III magnet with highest △CIII value than others. Further phase-field simulation underlies the governing mechanism for strengthened chemical heterogeneity in MMP Nd–Dy–Fe–B magnets, while also provides design principles for concentration graded magnetic materials.

Mother-leaf-method accelerated parallel-GPU AMR phase-field simulations of dendrite growth

Parallel computing with multiple graphics processing units (GPUs) for adaptive mesh refinement (AMR) is a powerful method that improves the computational efficiency of phase-field (PF) simulations of dendrite growth. Octree-type block-structured AMR is often used in parallel computation because of its good parallelism; however, the process of filling the buffer region with physical data from adjacent blocks constitutes a major overhead cost that reduces computational performance. In this study, the mother-leaf (ML) method, which can reduce this overhead, was applied to accelerate parallel-GPU AMR PF simulations of dendrite growth. To evaluate the computational performance of the ML method, single- and parallel-GPU AMR PF simulations of dendrite growth were performed. The results show that single- and parallel-GPU AMR computations with the ML method accelerate the PF simulations of dendrite growth approximately 1.2–1.6 times and 1.2–1.4 times, respectively, compared to the conventional method without the ML method. In large-scale dendrite-growth PF simulations, such accelerations are significant for investigating various dendrite growth phenomena.

Phase-field simulation study on dendritic growth behavior during bilateral directional solidification

The directional solidification technique can uniform the crystal growth direction within the material, but does not remove the lateral side branches of the axial main arm during the growth process. The influence of such lateral side branches existing in the directionally solidified dendritic organisation on the mechanical properties of the material cannot be neglected. In this paper, based on the Kim-Kim-Suzuki (KKS) phase-field model, the competitive behaviour of lateral secondary dendrites between the two main arms under different boundary conditions and non-synchronous growth conditions is investigated for Fe-0.5 %C alloys. In addition, the diffusion and enrichment rules of solutes are analysed by the distribution of solutes. The results show that the molten pool is spherical without external force, and the size distribution of the molten pool is as follows: dendrite tip collision zone > dendrite root > dendrite wall, and the solute segregation rule is the same; The spacing of the main arms has little effect on the concentration values between the dendrites, and the solutes are very symmetrically over the entire computational region; The dendrite tip interface concentration during the stable phase of dendrite growth is 0.89 %. Without considering the initial liquid-phase concentration, the concentration in the collision zone at the dendrite tip is approximately twice the concentration at the tip of the dendrite in the stable period, which is 1.28 %, when the bilateral dendrites are encountered. Dendrites always collide preferentially on the centreline of the included angle under non-parallel conditions, and solute-enriched zones also occur on the centreline of the included angle; During the curved surface bilaterally directional solidification process, when the left and right sides have the same shape of surfaces, the solute enrichment at the end of solidification is on the centreline. For the case of a single planar boundary, the solute-enriched region is influenced by the curved surface and is similar to the initial curved surface profile.

Dislocation Density-Mediated Functionality in Single-Crystal BaTiO3.

Unlike metals where dislocations carry strain singularity but no charge, dislocations in oxide ceramics are characterized by both a strain field and a local charge with a compensating charge envelope. Oxide ceramics with their deliberate engineering and manipulation are pivotal in numerous modern technologies such as semiconductors, superconductors, solar cells, and ferroics. Dislocations facilitate plastic deformation in metals and lead to a monotonous increase in the strength of metallic materials in accordance with the widely recognized Taylor hardening law. However, achieving the objective of tailoring the functionality of oxide ceramics by dislocation density still remains elusive. Here a strategy to imprint dislocations with {100}<100> slip systems and a tenfold change in dislocation density of BaTiO3 single crystals using high-temperature uniaxial compression are reported. Through a dislocation density-based approach, dielectric permittivity, converse piezoelectric coefficient, and alternating current conductivity are tailored, exhibiting a peak at medium dislocation density. Combined with phase-field simulations and domain wall potential energy analyses, the dislocation-density-based design in bulk ferroelectrics is mechanistically rationalized. These findings may provide a new dimension for employing plastic strain engineering to tune the electrical properties of ferroics, potentially paving the way for advancing dislocation technology in functional ceramics.

Understanding the grain size dependence of functionalities in lead-free (Ba,Ca)(Zr,Ti)O3

Grain size effects in ferroelectric ceramics have long been exploited to tailor their functional properties. However, the underlying mechanism for the grain size effects is not yet fully understood. Here, we study the grain size dependence of domain wall activities in a lead-free piezoceramics system, (Ba,Ca)(Zr,Ti)O3, with grain sizes in the range of 4 − 22 μm. The dielectric permittivity is highest at intermediate grain sizes (∼12 μm) where there are moderate lattice distortion and the most active domain wall motion under low voltages. Despite the larger fraction of switched domains in the material with larger distortion (∼22 μm), as revealed by in situ electric-field synchrotron X-ray diffraction, time-resolved characterizations demonstrate an easier and faster domain wall dynamics in the sample with moderate lattice distortion. Our analysis and phase-field simulations show that the grain size dependence of the domain wall dynamics of polycrystalline ferroelectrics is dictated by the interaction between an external electric field and the grain size-related change in intergranular stress, and this interaction is most effective in stimulating the movement of non-180° domain wall at intermediate grain sizes during the initial phase of polarization reversal. Our results provide a new fundamental understanding to guide the future design of materials for improving functionalities.

A real space convolution‐based approximate algorithm for phase field model involving elastic strain energy

AbstractPhase field models have been employed extensively in the study of microstructure evolution in materials. Elasticity plays an important role in solid‐state phase transformation processes, and it is usually introduced into phase field models in terms of the elastic strain energy by applying Khachaturyan–Shatalov microelasticity theory. Conventionally, this energy is derived in the reciprocal space and results in full‐space Fourier transformation in practice, which becomes bottle‐neck in large‐scale and massively‐parallel applications. In this article, we propose an error‐controlled approximation algorithm for scalable and efficient calculation of the elastic strain energy in phase field models. We first derive a real‐space convolutional representation of the elastic strain energy by representing the equilibrium displacements in the Khachaturyan–Shatalov microelasticity theory using Green's function. Then we propose an error‐controlled truncation criterion to approximate the corresponding terms in the phase field model. Finally, a carefully designed parallel algorithm is presented to carry out large‐scale simulations. The accuracy and efficiency of the proposed algorithm are demonstrated by real‐world large‐scale phase field simulations.

A simple displacement perturbation method for phase-field modeling of ferroelectric thin film

A displacement perturbation method is proposed for phase-field simulations of the ferroelectric domain structures. A temperature-misfit strain phase diagram is computed to validate this method's accuracy, encompassing rhombohedral, orthorhombic, tetragonal, cubic, and mixed phases by comparing with previous phase-field simulations. The change in free energy density surface with temperature and strain distribution is computed to clarify the mechanism of the mixed phase. The phase diagram, ferroelectric hysteresis, and domain structure all demonstrate that the numerical method of displacement discretization is reliable and practical in solving the time-dependent Ginzburg–Landau equation, consistent with previous simulated and experimental results. This new method offers the advantages of simple programming and easy parallelism, paving the way for advancements in phase-field modeling.

Wetting Effect Induced Depletion and Adsorption Layers: Diffuse Interface Perspective.

When a multi-component fluid contacts arigid solid substrate, the van der Waals interaction between fluids and substrate induces a depletion/adsorption layer depending on the intrinsic wettability of the system. In this study, we investigate the depletion/adsorption behaviors of A-B fluid system. We derive analytical expressions for the equilibrium layer thickness and the equilibrium composition distribution near the solid wall, based on the theories of de Gennes and Cahn. Our derivation is verified through phase-field simulations, wherein the substrate wettability, A-B interfacial tension, and temperature are systematically varied. Our findings underscore two pivotal mechanisms governing the equilibrium layer thickness. With an increase in the wall free energy, the substrate wettability dominates the layer formation, aligning with de Gennes' theory. When the interfacial tension increases, or temperature rises, the layer formation is determined by the A-B interactions, obeying Cahn's theory. Additionally, we extend our study to non-equilibrium systems where the initial composition deviates from the binodal line. Notably, macroscopic depletion/adsorption layers form on the substrate, which are significantly thicker than the equilibrium microscopic layers. This macroscopic layer formation is attributed to the interplay of phase separation and Ostwald ripening. We anticipate that the present finding could deepen our knowledge on the depletion/adsorption behaviors of immiscible fluids.

Intrinsic-strain-induced ferroelectric order and ultrafine nanodomains in SrTiO3

Nano ferroelectrics holds the potential application promise in information storage, electro-mechanical transformation, and novel catalysts but encounters a huge challenge of size limitation and manufacture complexity on the creation of long-range ferroelectric ordering. Herein, as an incipient ferroelectric, nanosized SrTiO3 was indued with polarized ordering at room temperature from the nonpolar cubic structure, driven by the intrinsic three-dimensional (3D) tensile strain. The ferroelectric behavior can be confirmed by piezoelectric force microscopy and the ferroelectric TO1 soft mode was verified with the temperature stability to 500 K. Its structural origin comes from the off-center shift of Ti atom to oxygen octahedron and forms the ultrafine head-to-tail connected 90° nanodomains about 2 to 3 nm, resulting in an overall spontaneous polarization toward the short edges of nanoparticles. According to the density functional theory calculations and phase-field simulations, the 3D strain-related dipole displacement transformed from [001] to [111] and segmentation effect on the ferroelectric domain were further proved. The topological ferroelectric order induced by intrinsic 3D tensile strain shows a unique approach to get over the nanosized limitation in nanodevices and construct the strong strain-polarization coupling, paving the way for the design of high-performance and free-assembled ferroelectric devices.

Self-organized composite morphologies and interface instability of immiscible alloy thin films during physical vapor deposition: Insights from phase-field simulations

In this paper, we carried out phase-field simulations for investigating self-organized composite morphologies formation and interface instability of two phase immiscible binary alloy thin-film growth during physical vapor deposition. Predicted results show that for lower the deposition rate the solid-gas interface keeps planar and the lateral composition modulation (LCM) configuration occurs. As the deposition rate increases, the self-organized composite morphology gradually transfers to vertical composition modulations (VCM) configuration and the random composition modulations (RCM) configuration with the decrease of the boundary layer of the incident vapor and the unsteady solid-gas interface. Results show that the nanoprecipitate concentration modulation (NCPM) depends strongly on the nominally phase fraction. Moreover, we found that the historical dependence of self-organized composite morphologies and solid-gas interface stability, i.e., the initial condition can strongly influence the microstructure of the thin film during physical vapor deposition. Finally, we provide an analysis method for interface instability based on fast Fourier transform (FFT), and the frequency characteristics of cellular and bump structures are identified.

Generative learning facilitated discovery of high-entropy ceramic dielectrics for capacitive energy storage

Dielectric capacitors offer great potential for advanced electronics due to their high power densities, but their energy density still needs to be further improved. High-entropy strategy has emerged as an effective method for improving energy storage performance, however, discovering new high-entropy systems within a high-dimensional composition space is a daunting challenge for traditional trial-and-error experiments. Here, based on phase-field simulations and limited experimental data, we propose a generative learning approach to accelerate the discovery of high-entropy dielectrics in a practically infinite exploration space of over 1011 combinations. By encoding-decoding latent space regularities to facilitate data sampling and forward inference, we employ inverse design to screen out the most promising combinations via a ranking strategy. Through only 5 sets of targeted experiments, we successfully obtain a Bi(Mg0.5Ti0.5)O3-based high-entropy dielectric film with a significantly improved energy density of 156 J cm−3 at an electric field of 5104 kV cm−1, surpassing the pristine film by more than eight-fold. This work introduces an effective and innovative avenue for designing high-entropy dielectrics with drastically reduced experimental cycles, which could be also extended to expedite the design of other multicomponent material systems with desired properties.

Insight into element segregation mechanisms during creep in γʹ-strengthened Co-based superalloy by elastoplastic phase-field simulation

The element segregation accompanying the creep process has been shown to significantly affect the deformation resistance of the superalloys. However, the processing and mechanism of element segregation are still unclear. This paper investigated the concentration evolution of a model Co–9Al–9W (at. %) alloy during 900 ​°C/275 ​MPa using developed ternary elastoplastic phase-field model coupled with CALPHAD method and crystal plasticity model. The results of simulation show that co-depletion of Al and W element occurs in γʹ precipitate and in γ side at γ/γʹ interface, and this depletion is gradually increasing with the accumulation of plastic strain. From the perspective of changes of driving force of element diffusion, it is found that these segregation phenomena are attributed to the high elastic potential caused by the large local plastic strain. In addition, the effects of these segregations on creep property are also predicted. The current research provides a new method for exploring the mechanism of element diffusion.

Balancing Polarization and Breakdown for High Capacitive Energy Storage by Microstructure Design.

The compromise of contradictive parameters, polarization, and breakdown strength, is necessary to achieve a high energy storage performance. The two can be tuned, regardless of material types, by controlling microstructures: amorphous states possess higher breakdown strength, while crystalline states have larger polarization. However, how to achieve a balance of amorphous and crystalline phases requires systematic and quantitative investigations. Herein, the trade-off between polarization and breakdown field is comprehensively evaluated with the evolution of microstructure, i.e., grain size and crystallinity, by phase-field simulations. The results indicate small grain size (≈10-35nm) with moderate crystallinity (≈60-80%) is more beneficial to maintain relatively high polarization and breakdown field simultaneously, consequently contributing to a high overall energy storage performance. Experimentally, therefore an ultrahigh energy density of 131 J cm-3 is achieved with a high efficiency of 81.6% in the microcrystal-amorphous dual-phase Bi3NdTi4O12 films. This work provides a guidance to substantially enhance dielectric energy storage by a simple and effective microstructure design.

Antiferroelectric domain modulation enhancing energy storage performance by phase-field simulations

Antiferroelectric materials represented by PbZrO3(PZO) have excellent energy storage performance and are expected to be candidates for dielectric capacitors. It remains a challenge to further enhance the effective energy storage density and efficiency of PZO-based antiferroelectric films through domain engineering. In this work, the effects of three variables, misfit strain between the thin film and substrate, defect dipoles doping, and film thickness, on the domain structure and energy storage performance of PZO-based antiferroelectric materials are comprehensively investigated via phase-field simulations. The results show that applying tensile strain to the films can effectively increase the transition electric field from antiferroelectric to ferroelectric. In addition, the introduction of defect dipoles while applying tensile strain can significantly reduce the hysteresis and improve energy storage efficiency. Ultimately, a recoverable energy density of 38.3 J/cm3 and an energy storage efficiency of about 89.4% can be realized at 1.5% tensile strain and 2% defect dipole concentration. Our work provides a new idea for the preparation of antiferroelectric thin films with high energy storage density and efficiency by domain engineering modulation.

Deep artificial neural network-powered phase field model for predicting damage characteristic in brittle composite under varying configurations

This work introduces a novel artificial neural network (ANN)-powered phase field model, offering rapid and precise predictions of fracture propagation in brittle materials. To improve the capabilities of the ANN model, we incorporate a loop of conditions into its core to regulate the absolute percentage error for each observation point, that filters and consistently selects the most accurate outcome. This algorithm enables our model to better adapt to the highly sensitive validation data arising from varying configurations. The effectiveness of the approach is illustrated through three examples involving changes in the microgeometry and material properties of steel fiber-reinforced high-strength concrete structures. Indeed, the predicted outcomes from the improved ANN phase field model in terms of stress–strain relationship, and crack propagation path demonstrates an outperformance compared with that based on the extreme gradient boosting method, a leading regression machine learning technique for tabular data. Additionally, the introduced model exhibits a remarkable speed advantage, being 180 times faster than traditional phase field simulations, and provides results at nearly any fiber location, demonstrating superiority over the phase field model. This study marks a significant advancement in the application of artificial intelligence for accurately predicting crack propagation paths in composite materials, particularly in cases involving the relative positioning of the fiber and initial crack location.

Phase-field simulation of crack propagation in particulate nanocomposite materials considering surface stresses

Phase-field simulation of crack propagation in particulate nanocomposite materials considering surface stresses

Elasto-plastic residual stress analysis of selective laser sintered porous materials based on 3D-multilayer thermo-structural phase-field simulations

Residual stress and plastic strain in additive manufactured materials can exhibit significant microscopic variation at the powder scale, profoundly influencing the overall properties of printed components. This variation depends on processing parameters and stems from multiple factors, including differences in powder bed morphology, non-uniform thermo-structural profiles, and inter-layer fusion. In this research, we propose a powder-resolved multilayer multiphysics simulation scheme tailored for porous materials through the process of selective laser sintering. This approach seamlessly integrates finite element method (FEM) based non-isothermal phase-field simulation with thermo-elasto-plastic simulation, incorporating temperature- and phase-dependent material properties. The outcome of this investigation includes a detailed depiction of the mesoscopic evolution of stress and plastic strain within a transient thermo-structure, evaluated across a spectrum of beam power and scan speed parameters. Simulation results further reveal the underlying mechanisms. For instance, stress concentration primarily occurs at the necking region of partially melted particles and the junctions between different layers, resulting in the accumulation of plastic strain and residual stress, ultimately leading to structural distortion in the materials. Based on the simulation data, phenomenological relation regarding porosity/densification control by the beam energy input was examined along with the comparison to experimental results. Regression models were also proposed to describe the dependency of the residual stress and the plastic strain on the beam energy input.

Atomic-Scale Tracking Topological Phase Transition Dynamics of Polar Vortex-Antivortex Pairs.

Non-trivial topological structures, such as vortex-antivortex (V-AV) pairs, have garnered significant attention in the field of condensed matter physics. However, the detailed topological phase transition dynamics of V-AV pairs, encompassing behaviors like self-annihilation, motion, and dissociation, have remained elusive in real space. Here, polar V-AV pairs are employed as a model system, and their transition pathways are tracked with atomic-scale resolution, facilitated by in situ (scanning) transmission electron microscopy and phase field simulations. This investigation reveals that polar vortices and antivortices can stably coexist as bound pairs at room temperature, and their polarization decreases with heating. No dissociation behavior is observed between the V-AV phase at room temperature and the paraelectric phase at high temperature. However, the application of electric fields can promote the approach of vortex and antivortex cores, ultimately leading to their annihilation near the interface. Revealing the transition process mediated by polar V-AV pairs at the atomic scale, particularly the role of polar antivortex, provides new insights into understanding the topological phases of matter and their topological phase transitions. Moreover, the detailed exploration of the dynamics of polar V-AV pairs under thermal and electrical fields lays a solid foundation for their potential applications in electronic devices.

Electrochemical grand potential-based phase-field simulation of electric field-assisted sintering

An electrochemical grand potential functional was proposed to describe the sintering of an ionic ceramic green body. The resultant phase-field description enables simulation of the consolidation of an arbitrary number of granular particles and their interactions with the surrounding void phase. The model includes the effects of charged vacancies and the associated interactions between internal and applied electric fields. Defect segregation to grain boundaries is also accounted for, as well as enhanced interfacial defect mobilities. The model was parameterized for Y2O3. Simulations of two-particle systems showed that the applied electric field had an increasingly important impact on neck growth as particle size increased. A sudden rapid increase in temperature occurred for larger field strengths, which has been reported to be correlated to the onset of a flash event in flash sintering. Simulations of many particles showed that internal heat generation by Joule heating was localized at particle–particle contacts (grain boundaries), even though their conductivities were lower than nearby internal particle-void interfaces. A percolative path for ionic charge across the green body and the ceramic sintered solid was thus defined, accelerating the Joule heating process as the porosity of the green body is removed.

Design of compositionally modulated materials for controlled strain release during deformation through phase-field simulations

Design of compositionally modulated materials for controlled strain release during deformation through phase-field simulations