Phase-field Method Research Articles

Microstructure Characterization of Battery Materials based on Voxelated Image Data: Computation of Active Surface Area and Tortuosity

Abstract The reliable computation of microstructure metrics such as specific surface area and tortuosity factors is key to bridge the gap between the battery microscale and fast, homogenized cell models.
In this work, we present an approach to compute the surface area of phases based on pixelated image data which is both easy-to-implement and computationally efficient. The concept is inspired from the diffuse surface representation in phase-field methods. Subsequently, the approach is validated and compared with common python libraries on two benchmark cases and actual battery microstructure data. The results underline the reliability and fast computational performance of the approach.
Furthermore, the concept of through-feature connectivity in pixelated image data is introduced and explored to quantify the reliability of tortuosity factor computations.
Overall, this work enhances the computational tools to bridge the scale from battery microstructures to cell models and gives an overview of state-of-the-art methodology. The developed code is published to further accelerate the scientific progress in this field.

A nonlocal macro-meso-scale damage model based modeling for crack propagation in ferroelectric materials

The accurate simulation of crack growth for ferroelectric materials plays a crucial role in the application of ferroelectric materials in electronic devices. In recent years, a nonlocal macro-meso-scale consistent damage (NMMD) model has been proposed for simulating the crack propagation in brittle materials, which offers the advantage of higher efficiency compared to the phase-field method. Different from the phase field method, this model does not need to solve the phase field equation at the same time. Therefore, the degree of freedom in solving can be reduced, thereby reducing the computational workload. In this paper, the authors proposed a new integration strategy for microscopic damage and devised a simple and efficient implementation of the NMMD model for the modelling of quasi-static fracture in the general purpose commercial software developer, COMSOL Multiphysics based on the finite element method (FEM) and studied the crack propagation for ferroelectric materials under the applied stress and electric fields. Different from the original NMMD model, only the tensile stress induced geometric damage is accounted for crack propagation by using the decomposition of elastic strain energy. The effects of different crack-face conditions, electrical boundary conditions and the electromechanical loading for crack growth of ferroelectric materials have been considered in our FEM simulation are discussed in this work.

Efficiently and consistently energy-stable [formula omitted]-phase-field method for the incompressible ternary fluid problems

Ternary incompressible fluid flows extensively exist in atmospheric science, chemical engineering, and energy and power engineering, etc. The phase-field method is popular in multi-phase fluid modeling thanks to its efficient ability of interface capturing. This work aims to develop an energy dissipation law-preserving and temporally second-order accurate algorithm for a ternary phase-field fluid system. To establish a simple energy estimation, an adapted auxiliary variable approach is used to transform the original model into its equivalent form. Later, a second-order backward difference strategy is used to design the fully decoupled and linear time-marching scheme. To improve the consistency between original and modified discrete energy functionals, a practical energy correction technique is presented. We analytically prove the discrete energy dissipation property and show that the energy estimation can be easily established without considering the complex treatments of nonlinear and coupling terms. To facilitate the interested readers, we briefly describe the numerical implementation in each time step. The numerical tests indicate that the proposed method not only has desired accuracy, but also satisfies the energy stability even if a larger time step is used. Moreover, the proposed method can well simulate various ternary fluid phenomena, such as the liquid lens, phase separation, droplet dynamics, Kelvin–Helmholtz instability, and billowing cloud.

Spatiotemporal prediction of solidified dendrites based on convolutional long-short-term neural network

The formation and growth of dendrites during the solidification of metals are critical factors influencing material properties. Accurate prediction of dendritic growth morphology and evolution is of great significance for materials science research and industrial applications. Traditional simulation methods, such as the phase-field method, provide high-precision results but are computationally expensive and time-consuming. Recently, with the advancement of deep learning technology, data-driven approaches have shown great potential in materials science. This paper proposes the use of Convolutional Long-Short-Term Memory Network (ConvLSTM) to capture spatiotemporal dependencies for predicting the spatiotemporal evolution of dendritic solidification in materials. By training on historical data of dendritic growth, the ConvLSTM neural network model predicts the future growth morphology of the dendritic microstructure during solidification. The model accurately forecasts the future morphology and appearance of the microstructure. Case studies were conducted using a generated dataset of dendritic solidification simulations, and the results demonstrate that the ConvLSTM model effectively captures the complex dynamic characteristics of dendritic growth and provides high-precision prediction results. This offers new insights for material design and optimization.

Mesoscale simulation on bimodal distribution of nano-sized intragranular fission bubbles in UO2

UO2 is widely used as a nuclear fuel in the light water reactors. Irradiated UO2 nuclear fuel exhibits an intriguing bimodal distribution of bubble number and size at the nanoscale. To uncover the underlying mechanisms, we investigate bubble characteristics under varying damage events using the continuum phase-field method. Our findings suggest that the occurrence of bimodal pattern is primarily due to the biased trapping of supersaturated vacancies and gas atoms by bubbles, rather than gas resolution effect.

Phase-field method of fracture for experimental validation to simulate damage caused by reinforcement corrosion in clam-shell concrete beams

Phase-field method of fracture for experimental validation to simulate damage caused by reinforcement corrosion in clam-shell concrete beams

Modeling the uncoupled damage-healing behavior of self-healing cementitious material with phase-field method

Microbially induced calcium carbonate precipitation (MICCP), which is a biochemical process that utilizes microbial metabolic activities to precipitate calcium carbonate (CaCO3), can be used for damage-healing by crack filling. In this study, we develop a computational uncoupled damage-healing framework to predict the fracture-healing responses of a multifunctional vascular cementitious composite. The framework integrates a numerical model for self-healing cementitious materials into a damage model developed using the phase-field method. The numerical self-healing model is utilized to compute the MICCP healing based on the chemical and enzyme kinetics of microbial activities, whereas the phase-field method is employed to capture the fracture response of the material. This framework comprises three stages: Stage 1 — damage, Stage 2 — healing, and Stage 3 — re-damage of the healed structure. Using the load–displacement curve of the healed cementitious material, the proposed uncoupled damage-healing models can predict the recovery in material stiffness based on the MICCP healing ratio. The proposed framework can be applied to a wide range of fracture-healing problems in self-healing cementitious materials utilizing MICCP as the healing methodology.

Investigation of oxygen transport in porous transport layer with different porosity gradient configurations using phase field method

Minimizing oxygen accumulation in the porous transport layer (PTL) is crucial for reducing mass transfer losses in proton exchange membrane (PEM) electrolyzer. This study develops a two-dimensional transient model of gas-liquid two-phase flow at the anode of PEM electrolyzer using the phase field method. The model investigates the mechanisms of oxygen transport and the interactions among various oxygen paths in PEM electrolyzer. We explore the impact of porosity gradient configurations in the PTL and the presence of a surface microporous layer (MPL) on oxygen transport. The findings indicate that for PTL with an average porosity of 60%, forward gradient configuration—where porosity increases from the catalyst layer (CL) towards the channel (CH)—promotes the merging of bubble sites and path contraction, thereby reducing oxygen saturation. The optimal gradient configuration, with porosities of 50% at the CL and 70% at the CH, achieves a 29.5% reduction in oxygen saturation. Conversely, reverse gradient configuration, with decreasing porosity from CL to CH, results in increased oxygen saturation. The addition of surface MPL further lowers oxygen saturation and shortens oxygen breakthrough time; smaller MPL particle sizes correspond to lower oxygen saturation and shorter breakthrough times. This study provides valuable insights for the optimal design of PTL structures in PEM electrolyzers.

Analysis of CO2 bubble growth detachment kinetics in direct methanol fuel cell flow channels

CO2 bubbles flow in the anode flow channel is an important issue in the commercialization process of direct methanol fuel cells (DMFC). A T-channel model is built in COMSOL using the phase field method to investigate the CO2 bubbles flow at the anode side of the DMFC. The factors and mechanisms of single bubble detachment are discussed by analyzing the methanol inlet velocity (Ul), CO2 inlet velocity (Ug), contact angle of the diffusion layer, and the Weber number during bubble detachment. The research findings indicate that increasing the liquid flow rate leads to the generation of smaller bubbles that detach more rapidly due to the increases of drag force (FD) and the shear-life force (FSL) to overcome the surface tension on the bubble. The CO2 inlet velocity can promote the bubble detachment due to the increase in FSL, but also leads to a larger detachment diameter. Compared to hydrophobic surfaces, hydrophilic surfaces are more conducive to bubble detachment and removal. In all case We (Weber number) is significantly less than 0.6, indicating that liquid momentum dominated the bubble detachment process. Once the ratio of the gas momentum to the liquid one is greater than 1, the bubble is hard to detach. The contour map of bubble flow patterns and the bubble detachment diameters distribute with the ratio of Ug/Ul can further indicate that the bubble detachment is connected with the ratio of Ug/Ul closely, which will have guiding significance for the selection of inlet velocity of DMFC.

Chemo-mechanical benchmark for phase-field approaches

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

Research on Ni-WC Coating and a Carbide Solidification Simulation Mechanism of PTAW on the Descaling Roll Surface

The descaling roll is a critical component in a hot-rolling production line. The operating conditions are significantly impacted by water with high-pressure and dynamic shocks caused by high-temperature steel slab descaling. Roll surfaces often experience wear and corrosion failures. This is attributed to a combination of high temperatures, intense wear, and repeated thermal, mechanical, and fluid stresses. Production costs and efficiency are significantly affected by the replacement of descaling rolls. Practice shows that the use of plasma cladding technology forms high-performance coatings. Conventional metal surface properties can be significantly improved. In this study, a Ni-WC composite coating was prepared on the descaling roll surface by plasma-transferred arc welding (PTAW) technology. The microstructure and phase composition of the welding overlay were analyzed by scanning electron microscopy (SEM) and X-ray diffraction (XRD). Results show that the WC hard phase added to the molten pool dissolves, and subsequently M7C3 and W2C phases are formed. To further explore the morphological evolution mechanism of the hard phase, numerical simulations were performed using a phase-field method to model M7C3 phase precipitation. The evolution from nucleation, rod-like growth, to eutectic structure formation was revealed. Experimental and simulation results show high consistency, validating the established phase-field model. In this study, a theoretical foundation for designing and preparing high-performance coatings is provided.

An adaptive mesh refinement algorithm for crack propagation with an enhanced thermal–mechanical local damage model

This paper presents a computationally effective approach for crack propagation under mechanical and thermal loads based on an adaptive mesh refinement (AMR) approach tailored for our recently developed enhanced local damage model. The mesh-dependent issue encountered in the classical local theories is effectively mitigated by incorporation of fracture energy and element characteristic length into the damage evolution function. Our previous research has demonstrated that being equipped by a novel equivalent strain derived from the bi-energy norm concept and a new damage criterion recently proposed by Mazars et al., the model provides results comparable to the reference experimental data as well as other numerical models based on non-local/gradient damage and phase field method. In the framework of computational efficiency using finite elements, we significantly enhance the performance of our enhanced local model by considering adaptive mesh refinement (AMR). The finite element mesh is locally refined in the damaged zone, and the mesh refinement is conducted on-the-fly during the analysis. For that purpose, the damage parameter whose information is stored at integration points is selected as an indicator to mark whether an element should be refined or not after every loading step. For quadrilateral element mesh, a quad-tree technique is utilized, meaning that each marked element is further divided into four smaller quadrilateral elements. The so-called hanging nodes appear during the process, and the elements are thus treated as n-gons and are constructed by the Laplace shape functions, instead of the usual Lagranges shape functions. To show the accuracy and effectiveness of the proposed scheme, several numerical examples involving homogeneous and heterogeneous materials are studied. In these examples, the damage is induced either by only mechanical loads or by both mechanical and thermal loads.

Stochastic analysis of dynamic fracture of concrete using CT-image based mesoscale models with a rate-dependent phase field method

Concrete structures are commonly exposed to dynamic loads spanning a wide range of strain rates, and the inherent mesoscale heterogeneities complicate stochastic dynamic fracture mechanisms even more. This work develops a numerical framework using mesoscale concrete models based on micro computed tomography (CT) images to investigate such mechanisms with meaningful stochastic analyses. A rate-dependent phase field model is proposed to characterise the dynamic initiation and propagation of cracks by incorporating both micro-viscosity and macroscopic viscoelasticity, which is described by two standard Maxwell elements with different relaxation times to consider a wide range of strain rates. Moreover, the viscoelastic constitutive relation is formulated in the full strain space, which allows for a spectral decomposition of the strain tensor to determine the effective damage driving force, thus effectively addressing the issue of compressive fracture. A numerical implementation scheme is developed by combining user-defined element and material subroutines in ABAQUS/Explicit solver. Extensive Monte Carlo simulations of dynamic tension up to a strain rate of 200 s−1 are performed with statistical analyses. This work reveals the intricate dynamics associated with mesoscale heterogeneities and identifies the critical transition state at 20 s−1. The transition is characterised by changing modes of fracture patterns, stress wave propagation, and load-carrying capacities. A new TDIF–strain rate–standard deviation relation is also proposed and aligns well with the increasing dispersion of experimental data. The relationship between void content and tensile strength reflects the formation characteristics of crack networks, with the void content exhibiting a positive correlation with the TDIF from 20 s−1 to 100 s−1.

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

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

Revealing the Effects of Pressure on the SDAS and Columnar Dendrite Growth Rate in 30Cr15Mo1N Ingot by Phase Field Method

Revealing the Effects of Pressure on the SDAS and Columnar Dendrite Growth Rate in 30Cr15Mo1N Ingot by Phase Field Method

Time series forecasting of multiphase microstructure evolution using deep learning

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

Study on Mechanism of Microstructure Refinement by Ultrasonic Cavitation Effect

During the solidification process of the alloy, the temperature lies in the range between the solid-phase line and the liquidus. Dendrite growth exhibits high sensitivity to even slight fluctuations in temperature, thereby significantly influencing the tip growth rate. The increase in temperature can result in a reduction in the rate of tip growth, whereas a decrease in temperature can lead to an augmentation of the tip growth rate. In cases where there is a significant rise in temperature, dendrites may undergo fracture and subsequent remelting. Within the phenomenon of ultrasonic cavitation, the release of internal energy caused by the rupture of cavitation bubbles induces a substantial elevation in temperature, thereby causing both dendrite remelting and fracture phenomena. This serves as the main mechanism behind microstructure refinement induced by ultrasonic cavitation. Although dendrite remelting and fracture exert significant influences on the solidification process of alloys, most studies primarily focus on microscopic characterization experiments, which fail to unveil the transient evolution law governing dendrite remelting and fracture processes. Numerical simulation offers an effective approach to address this gap. The existing numerical models primarily focus on predicting the dendrite growth process, while research on remelting and fracture phenomena remains relatively limited. Therefore, a dendrite remelting model was established by incorporating the phase field method (PFM) and finite element difference method (FDM) into the temperature-induced modeling, enabling a comprehensive investigation of the entire process evolution encompassing dendrite growth and subsequent remelting.

A numerical study of rupture of a ferrofluid interlayer in a sandwiched fluid system

Surface rupture in ferrofluid layers is a special case of the well-known Rosensweig instability, which can be triggered by applying a strong magnetic field. This study investigates the rupture dynamics in a ferrofluid interlayer sandwiched between two non-magnetic fluids, influenced by a non-homogenous vertical magnetic field. Simulations are performed using a generalized conservative phase-field lattice Boltzmann method for the flow field and interface with a coupled solution of Maxwell's equations for the evolution of magnetic field. The numerical results demonstrate the complete rupture process of ferrofluid layers. In most cases, the ferrofluid layer ruptures into two parts, while under certain conditions, such as a thinner interlayer or high magnetic field intensity, daughter droplets appear at the meniscus. A parametric analysis involving Weber number (We) and dimensionless magnetic parameter (Nm) elucidates the connection between different rupture conditions, such as a deformed interlayer without rupture, rupture with two semi-spindle shaped domains, and rupture with droplets. Additionally, a phase diagram illustrating the various rupture regions is also provided.

Phase-field based shape optimization of uni- and multiaxially loaded nature-inspired porous structures while maintaining characteristic properties

Triply periodic minimal surfaces (TPMS) are highly versatile porous formations that can be defined by formulas. Computationally based, load-specific shape optimization enables tailoring these structures for their respective application areas and thereby enhance their potential. In this investigation, individual sheet-based gyroid structures with varying porosities are specifically optimized with respect to their stiffness. A modified phase-field method is employed to establish a simulation framework for the shape optimization process. Despite constant volume and the preservation of the periodicity of the unit cells, volume redistribution occurs through displacement of the interfaces. The phase-field-based optimization process is detailed using unidirectional loading on three gyroidal unit cells with porosities of 75 %, 80 %, and 85 %. Subsequently, the gyroidal unit cell with a porosity of 85 % is shape-optimized under multidirectional loading. A subsequent experimental validation of the unidirectionally loaded cells confirms that the shape-optimized structures exhibit, on average, higher stiffness than the non-optimized structures. The highest increase of 40 % in effective modulus is achieved with the gyroid structure having a porosity of 75 %, while maintaining minimal alteration to the surface-to-volume ratio and preserving periodicity. Additionally, the experimental data show that the optimization process resulted in a shift in the linear elasticity and plasticity range. In summary, the phase-field method proves to be a valid optimization technique for complex porous structures, allowing the preservation of characteristic properties.

Comparison of Contact Angle Models in Two‐Phase Flow Simulations Using a Conservative Phase Field Equation

ABSTRACTIn phase field methods based on a second‐order Allen‐Cahn (AC) equation, contact angles are prescribed mostly via a geometric formulation. However, it is of great interest to utilize the surface‐energy formulation, which is often employed in the Cahn‐Hilliard (CH) phase field method, in the AC phase field method. This article thus put forward a surface‐energy formulation of contact angles. The model was compared with the geometric one in a number of impact problems, including both normal and oblique impacts. The governing equations were discretized using a finite difference method on a half‐staggered grid. The Navier–Stokes equation was tackled using an explicit projection method. The major findings are as follows. First, the geometric model can maintain a fixed contact angle throughout contact line motion, while the surface‐energy one predicts a changeable contact angle, with a fluctuation of about 5°. In the oblique drop impact, contact angle hysteresis was captured even if a static contact angle was applied in the surface‐energy formulation.