3D Phase-field Model Research Articles

Machine learning surrogate for 3D phase-field modeling of ferroelectric tip-induced electrical switching

Phase-field modeling offers a powerful tool for investigating the electrical control of the domain structure in ferroelectrics. However, its broad application is constrained by demanding computational requirements, limiting its utility in inverse design scenarios. Here, we introduce a machine-learning surrogate to accelerate 3D phase-field modeling of tip-induced electrical switching. By dynamically handling the boundary conditions, the surrogate achieves accurate reproduction of switching trajectories under various tip locations and applied voltages. With stable predictions throughout entire morphological evolution pathways and a relative error inferior to 10% compared to direct solvers, the model efficiently emulates intricate switching sequences. By successfully replicating the boundary conditions, the presented framework strides towards a holistic surrogate for the ferroelectric phase field. With up to 2500-fold speed-ups over classical methods, our approach opens the path for the tractable design of the domain structure and the resolution of realistic inverse problems.

Voronoi-based 3D phase field model and numerical simulations of granite crack propagation

The distribution of mineral grains within underground rocks and materials is crucial for understanding geological processes and the stability of engineering structures. However, numerical simulation methods that consider 3D mineral particles are still immature. This study presents a 3D phase-field fracture model of mineral grains based on Voronoi polygons, making some beneficial attempts in this regard. The phase-field method does not rely on mesh reconstruction or crack path tracking and can simulate complex crack propagation behavior. Meanwhile, the Voronoi polygon can better describe the microscopic structure of rock with grain boundaries. Rooted in Voronoi polygons, this model efficiently captures the complex geometric shapes of mineral grains and accurately characterizes their positions, shapes, and sizes. Therefore, this study combines Voronoi polygons to establish a 3D phase-field model. Through two numerical simulation cases, the model demonstrates its effectiveness in simulating the impact of three-dimensional mineral grains on crack propagation behavior, showing that cracks tend to propagate at the boundaries or inside softer mineral grains. The modeling approach proposed in this paper has good reference value for numerical simulation studies considering the microscopic structure of rocks.

Self-organization of ovarian cancer organoids revealed by 3D image segmentation and phase-field model

Self-organization of ovarian cancer organoids revealed by 3D image segmentation and phase-field model

Multiscale dynamics of charging and plating in graphite electrodes coupling operando microscopy and phase-field modelling

The phase separation dynamics in graphitic anodes significantly affects lithium plating propensity, which is the major degradation mechanism that impairs the safety and fast charge capabilities of automotive lithium-ion batteries. In this study, we present comprehensive investigation employing operando high-resolution optical microscopy combined with non-equilibrium thermodynamics implemented in a multi-dimensional (1D+1D to 3D) phase-field modeling framework to reveal the rate-dependent spatial dynamics of phase separation and plating in graphite electrodes. Here we visualize and provide mechanistic understanding of the multistage phase separation, plating, inter/intra-particle lithium exchange and plated lithium back-intercalation phenomena. A strong dependence of intra-particle lithiation heterogeneity on the particle size, shape, orientation, surface condition and C-rate at the particle level is observed, which leads to early onset of plating spatially resolved by a 3D image-based phase-field model. Moreover, we highlight the distinct relaxation processes at different state-of-charges (SOCs), wherein thermodynamically unstable graphite particles undergo a drastic intra-particle lithium redistribution and inter-particle lithium exchange at intermediate SOCs, whereas the electrode equilibrates much slower at low and high SOCs. These physics-based insights into the distinct SOC-dependent relaxation efficiency provide new perspective towards developing advanced fast charge protocols to suppress plating and shorten the constant voltage regime.

Concentration-driving pinning effect in lead-free Mn-substituted BCZT ferroelectric ceramics

Domain wall motion in ferroelectrics, closely related to electric characteristics and amenable to tuning through doping, remains incompletely understood, particularly with regard to the pinning defect in the (Ba0.85Ca0.15)(Zr0.1Ti0.9)O3 (BCZT) system, which has attracted significant interest. Here, we investigate the defect dipole-assisted pinning effect in Mn-substituted BCZT (abbreviated as BCZT-xMn) ceramics with Mn concentrations ranging from x = 0 to 0.04. As x increases, we observe a gradual decrease in the maximum permittivity, accompanied by a leftward shift in temperature, leading to a diffuse phase transition. Analysis of the dielectric spectrum reveals a consistent trend in the activation energy within a specific concentration range, suggesting the incorporation of oxygen vacancies to maintain electric neutrality. The resulting oxygen vacancies, combined with Mn dopants, form defect dipoles Mn4++e′− VO⋅⋅ −Mn4++e′, attaining sufficient concentration at x = 0.02, and enabling precise establishment of the pinning effect. This phenomenon is extensively documented by the observation of pinched polarization-electric field loops and double current peaks in the current-electric field curves, visually corroborated by experimental data and a 3D phase-field simulation model. Consequently, this study not only elucidates the concentration-driven pinning effect in the BCZT system but also provides valuable insights into doping effects in other lead-free ferroelectrics.

Mesoscale simulations of spherulite growth during isothermal crystallization of polymer melts via an enhanced 3D phase-field model

A finite difference-based 3D phase-field model is developed to investigate the spherulite growth at the mesoscopic scale during the isothermal crystallization of polyamide (PA) 12. The model introduces a phase-field variable to distinguish the crystalline and amorphous phases of polymers. The phase-field evolution equation is coupled with the heat conduction equation that considers the latent heat of crystallization. The evolution equations introduce both the dimensionless diffusivity and latent heat that are dependent on the crystallization temperature. A high-order finite difference-based numerical framework is applied to the phase-field model. Both the qualitative simulation results of the phase-field model such as the crystal morphologies and the quantitative results including the radial crystal growth rate, degree of crystallinity, and lamellar thickness are validated against experiments. The simulation for single-crystal growth shows that a high crystallization temperature results in a large crystal with a slow radial growth rate. The simulation for multi-crystal growth shows that the crystals impinge on each other and finally fill the whole domain during crystallization, which further demonstrates the capability of the model in simulating the spherulite growth during isothermal crystallization of polymer melts.

Transverse penny-shaped hydraulic fracture propagation in naturally-layered rocks under stress boundaries: A 3D phase field modeling

The ground (tectonic) stress and layered structures of rocks are the among the main factors that influences the hydraulic fracturing behavior. The purpose of this study is to examine the suitability of the phase field model (PFM) in simulating transverse penny-shaped hydraulic fracture propagation and to investigate the phase field feature for the transverse penny-shaped hydraulic fracture in layered rocks under stress boundaries. A phase field model for 3D transverse penny-shaped hydraulic fracture propagation in naturally-layered rocks considering the effect of initial stress field is established. The mathematical model is based on Biot poroelasticity theory and the phase field fracture theory, while the governing equations are solved by using the finite element method in a staggered manner. The established PFM is validated experimentally and analytically by 2D and 3D examples. At last, the influences of the initial stress field, stiffness contrast and inclination angle of the layer interface on the penny-shaped fracture evolution in naturally-layered rocks are investigated by using the PFM. The study indicates that the phase field model has excellent feasibility and practicability in predicting penny-shaped hydraulic fractures. The layer inclination has nearly no effect on the penny-shaped fracture evolution in naturally-layered rocks. The stress ratio Sv/Sh on the boundaries has a significant effect on propagation of the penny-shaped hydraulic fracture. With the increase in Sv/Sh, the hydraulic fracture deflects and propagates along the direction of the maximum in-situ stress. For a high Sv/Sh, branching scenarios can be observed. The stiffness contrast of the rock layers determines whether the penny-shaped hydraulic fracture can penetrate into the adjacent layer. The predictions on the effects of the initial stress field, stiffness contrast and inclination angle of the layer interface provide new understanding of penny-shaped hydraulic fracture propagation in underground geological environment.

A comprehensive characterization of fracture in unit cell open foams generated from Triply Periodic Minimal Surfaces

The main scope of the proposed study is to assess the occurrence of fracture in Triply Periodic Minimal Surfaces (TPMS) foams subjected to compressive loading. TPMS, developed by the mathematics community, may be exploited as a backbone for developing a new class of foams with open porosity for a wide range of engineering and biomedical applications. Therefore, a comprehensive analysis of their fracture response is fundamental and is herein attempted. To this aim, a 3D phase field model is herein proposed and applied to TPMS foam structures under compression, with the goal to predict critical points for crack nucleation, potential crack paths, and the stiffness and maximum force of the unit cell, which can be related to the apparent Young’s modulus and apparent strength of a macro-scale composite made of such TPMS unit cells. A careful mesh sensitivity analysis was conducted on the specimens, to provide guidelines on how to identify the optimal finite element discretization consistent with the internal length scale parameter of the phase field approach to fracture. The major predicted mechanical properties for five different TPMS open foams, and for different levels of porosity, are summarized in Ashby plots. The predicted trends are in agreement with previous results on TPMS taken from the literature and show that TPMS can outperform standard Aluminium open foams.

Phase-Field Modeling of Fused Silica Cone-Crack Vickers Indentation.

In this paper, a 3D phase-field model for brittle fracture is applied for analyzing the complex fracture patterns appearing during the Vickers indentation of fused silica. Although recent phase-field models for the fracture caused by the indentation loading have been verified by some simpler academic axis-symmetric examples, a proper validation of such models is still missing. In addition, heavy computational costs, and a complicated compression stress field under the indenter, which demands different energy decompositions, have been identified as the most important impediments for the successful application of the phase-field method for such problems. An adaptive strategy is utilized for reducing the computational costs, and some modifications are introduced, which enable an accurate simulation of the Vickers indentation fracture. Here, the fracture initiation ring outside the contact zone is detected by using different energy decompositions, and the dominant cone-crack formation under the Vickers indenter is observed. Different contact conditions are investigated. The proposed model is validated by experimental measurements, and a quantitative and qualitative comparison between experimental and numerical results is conducted.

Adaptive finite element method for hybrid phase-field modeling of three-dimensional cracks

Phase-field modeling of cracks in three dimensions (3D) is challenging due to expensive computational efforts caused by inefficient solution of non-linear governing equations as well as the requirement of extremely fine mesh for crack resolution. In this paper, a combination technique aiming at these two issues is presented. Firstly, a hybrid phase-field model is implemented in 3D with a simple and robust one-pass staggered solution scheme, yielding a linear system of governing equations which can be solved efficiently. Secondly, adaptive mesh refinement using the predictor-corrector algorithm is developed such that this hybrid model can be solved with minimal number of nodes. Numerical results show that computational cost of 3D phase-field modeling is dramatically reduced and complex crack events including initiation, mixed mode propagation, non-planar extension and twisting can be efficiently modeled with preferable robustness. Perfect agreement between numerical and experimental results on crack paths and application of the proposed method to the fracture of complicated engineering structures are demonstrated.

Three-dimensional phase-field modeling of temperature-dependent thermal shock-induced fracture in ceramic materials

Current theoretical and experimental methods cannot fully reveal the mechanisms of the rapid and complex thermal shock-induced crack initiation and propagation processes in ceramic materials. Herein, a three-dimensional (3D) coupled thermo-mechanical phase-field model (PFM) is developed for thermal shock-induced fracture with the consideration of the temperature dependence of material properties. Compared with other PFMs, the present model can eliminate the unexpected damage evolution at the initially intact area of materials by introducing a temperature-dependent fracture energy threshold. Both the two-dimensional (2D) and 3D phase-field modeling results of thermal shock-induced fracture show strong agreement with the experimental results. The net-like topologies of thermal shock-induced cracks on the specimen surfaces are captured. Specifically, the crack topologies on the bottom surface (i.e., the first part submerged in water) are significantly different from those on the top surface in 3D cases. These essential findings reveal the mechanism that the tensile part of the strain energy mainly dominates the thermal shock-induced cracking in ceramics.

Phase field model for cell spreading dynamics.

We suggest a 3D phase field model to describe 3D cell spreading on a flat substrate. The model is a simplified version of a minimal model that was developed in Winkler (Commun Phys 2:82, 2019). Our model couples the order parameter u with 3D polarization (orientation) vector field [Formula: see text] of the actin network. We derive a closed integro-differential equation governing the 3D cell spreading dynamics on a flat substrate, which includes the normal velocity of the membrane, curvature, volume relaxation rate, a function determined by the molecular effects of the subcell level, and the adhesion effect. This equation is easily solved numerically. The results are in agreement with the early fast phase observed experimentally in Dobereiner (Phys Rev Lett 93:108105, 2004). Also we find agreement with the universal power law (Cuvelier in Curr Biol 17:694-699, 2007) which suggest that cell adhesion or contact area versus time behave as [Formula: see text] in the early stage of cell spreading dynamics, and slow down at the next stages.

Phase-field model of ion transport and intercalation in lithium-ion battery

The unified 3D phase-field model for description of the lithium-ion cell as a whole is developed. The model takes into account the realistic distribution of particles in electrodes, percolative transport of ions, and difference in size of solute and solvent molecules. The model is based on the Cahn–Hilliard equation with spatially dependent interaction and dynamic parameters. The spatial dependences are determined through the order parameter field describing distribution of matter in nanostructured electrodes. The process of charging/discharging of a battery is simulated for constant chemical potential difference across the electrode/electrolyte interface, which naturally determines the electrochemical reaction. The proposed model is applied to study charging and discharging process in 3D lithium-ion cell with nanostructured electrodes. The current density over the cell depends linearly on the chemical potential difference. The obtained results indicate non-diffusive propagation of the concentration front during charging/discharging, nonuniform intercalation flux over the surface of electrode particles and violation of the equipartition of electric current density over the electrode surface. All these facts are not taken into account by most commonly used models.

The creation of raspberry-like droplets and their coalescence dynamics: An ideal model for certain biological processes

HypothesisAlthough a raspberry-like configuration has been long observed in biological processes (e.g., the intimate association between Cajal bodies and B-snurposomes), studies on this morphology are very limited. Raspberry-like droplets created with multiple immiscible liquids are expected to provides an ideal model for such structures in biological systems, including their possible formation mechanism, phase behaviors, and coalescence dynamics. Experiments & SimulationsUsing three liquid phases, one surfactant and some colloidal particles, raspberry-like droplets containing one large central droplet and multiple protrusions embedded on its surface were successfully created. Confocal microscopy studies were carried out to track their formation and coalescence dynamics. A 2D phase-field model was applied to test the influence of the protrusions in the system. FindingsThe formation of this raspberry-like morphology involves a partial inversion process, which was predicted by Friberg et al. with numerical simulations but has never been demonstrated experimentally. A two-step coalescence was revealed, where the protrusions merge first and create a capillary bridge, which drives the droplets to coalesce. Increasing the viscosity of the continuous phase can help to prevent the destabilization. These fundamental features of raspberry-like droplets represent an important step toward producing multi-liquid materials with unique functionality, and can potentially illuminate some biological systems and processes.

Phase Field Modeling of Crystallographic Corrosion Pits

The modeling of corrosion to understand and predict corrosion behavior is a topical issue. Here, a 3D phase field model is developed to simulate the pit morphology, primarily focusing on crystallographic pits. A crystallographic function is employed to incorporate different corrosion rates for different crystallographic planes. The model is benchmarked and validated against an analytical solution for a simple case. 3D crystallographic pits in a single crystal are simulated and the effect of substrate orientation on the pit morphology is studied. The crystallographic pit morphology changes significantly with the substrate orientation and these morphologies have a symmetry consistent with the substrate orientation. This first 3D phase field model of crystallographic pits will help in predicting the intricate shapes of pits, thereby, pushing the frontiers of pitting corrosion modeling.

Multigrain phase-field simulation in ferroelectrics with phase coexistences: An improved phase-field model

A new 3D phase-field model is developed to investigate the evolution of the domain structure and phase transformation in a ferroelectric material near the morphotropic phase boundary (MPB) under the influence of external stimuli. The model combines the Landau–Ginzburg–Devonshire (LGD) theory with a general multiphase approach and enables to analytically compute the microstructure inside the grain structures with coexisting phases. Together with experimental analysis, a numerical energy overlap was employed to find the phase-dependent phenomenological coefficients. Domain structures and phase transformations in single crystals, as well as in the polycrystalline systems with or without an external electric field have been investigated in a MPB ferroelectric material, by employing phase-field simulation. The simulated results reveal the relation between the polarization switching and the phase transformation under an applied electric field, and the role of the external electric field orientation and amplitude in the domain structure and the phase transformation fraction.

Computational engineering of the oxygen electrode-electrolyte interface in solid oxide fuel cells

The Ce0.8Gd0.2O2−δ (CGO) interlayer is commonly applied in solid oxide fuel cells (SOFCs) to prevent chemical reactions between the (La1−xSrx)(Co1−yFey)O3−δ (LSCF) oxygen electrode and the Y2O3-stabilized ZrO2 (YSZ) electrolyte. However, formation of the YSZ–CGO solid solution with low ionic conductivity and the SrZrO3 (SZO) insulating phase still happens during cell production and long-term operation, causing poor performance and degradation. Unlike many experimental investigations exploring these phenomena, consistent and quantitative computational modeling of the microstructure evolution at the oxygen electrode–electrolyte interface is scarce. We combine thermodynamic, 1D kinetic, and 3D phase-field modeling to computationally reproduce the element redistribution, microstructure evolution, and corresponding ohmic loss of this interface. The influences of different ceramic processing techniques for the CGO interlayer, i.e., screen printing and physical laser deposition (PLD), and of different processing and long-term operating parameters are explored, representing a successful case of quantitative computational engineering of the oxygen electrode–electrolyte interface in SOFCs.

3D phase field modeling of the morphology of WC grains in WC–Co alloys: The role of interface anisotropy

Understanding the morphological evolution of tungsten carbides (WC) grains is one of a few key factors to control the mechanical properties of the cemented carbides such as wear resistance and toughness. In this work, the three-dimensional (3D) equilibrium morphology of faceted WC grains in WC-Co system was prospected by means of phase-field simulations with parameters adjusted on ab-initio calculations. To model faceted grain surfaces, a specific class of high anisotropy functions harnessing the vertex vectors of the corresponding Wulff shape construction was implemented. As a result, both truncated trigonal prism in C-poor alloys, and trigonal prism in C-rich alloys in WC-25%Co system morphologies could be reproduced, depending on the magnitude of the anisotropy. Phase-field simulations show an excellent quantitative agreement with the present scanning electron microscopy (SEM) observations to representatively sintered WC-Co alloys. This approach paves a new way to model the morphology of high anisotropy grains.

3D Phase Field Modeling of Multi-Dendrites Evolution in Solidification and Validation by Synchrotron X-ray Tomography.

In this paper, the dynamics of multi-dendrite concurrent growth and coarsening of an Al-15 wt.% Cu alloy was studied using a highly computationally efficient 3D phase field model and real-time synchrotron X-ray micro-tomography. High fidelity multi-dendrite simulations were achieved and the results were compared directly with the time-evolved tomography datasets to quantify the relative importance of multi-dendritic growth and coarsening. Coarsening mechanisms under different solidification conditions were further elucidated. The dominant coarsening mechanisms change from small arm melting and interdendritic groove advancement to coalescence when the solid volume fraction approaches ~0.70. Both tomography experiments and phase field simulations indicated that multi-dendrite coarsening obeys the classical Lifshitz–Slyozov–Wagner theory , but with a higher constant of n = 4.3.

Modelling of grain growth and its effect on mechanical properties of stainless steel components

A 2D phase-field model is developed to study the grain growth behaviour and the effect of simulation parameters on grain growth. The effect of nitrogen on grain growth in 316LN stainless steels is ...