Phase-field Model Research Articles

Triple junction benchmark for multiphase-field models combining capillary and bulk driving forces

Abstract A benchmark problem is formulated which is well suited for the validation of mesoscopic phase-field models for grain-boundary migration in polycrystals. First, an analytical steady-state solution of the sharp moving boundary problem is derived for a symmetric lamellar structure, which is valid for arbitrary bulk driving forces and triple junction angles. Characteristic quantities are identified to reduce the parameter space which in turn allows a systematic comparison of simulations and analytical results. Various multiphase-field (MPF) formulations are compared which approximate the sharp interface problem in terms of a diffuse regularization. An interfacial thickness convergence study reveals that the model error is largely dependent on the ratio of bulk to interfacial stabilizing force as well as the underlying model formulation. An additional grid convergence study highlights the efficiency of a more advanced discretization scheme. The results can be used to guide the selection of appropriate models and to estimate the interface thickness and spatial resolution required to achieve a given accuracy target. The post-processing framework consists of a fully automated determination of well-defined metrics from the phase field simulation data, eliminating human bias and facilitating reproducibility. The corresponding code is made openly available to assist the materials science and engineering community in validating MPF, multi-order parameter and similar model developments. We believe that this work provides a reliable benchmark procedure to better understand the potentials and limitations of current multiphase-field models as well as alternative approaches.

A data-driven strategy for phase field nucleation modeling

We propose a data-driven strategy for parameter selection in phase field nucleation models using machine learning and apply it to oxide nucleation in Fe-Cr alloys. A grand potential-based phase field model, incorporating Langevin noise, is employed to simulate oxide nucleation and benchmarked against the Johnson-Mehl-Avrami-Kolmogorov model. Three independent parameters in the phase field simulations (Langevin noise strength, numerical grid discretization and critical nucleation radius) are identified as essential for accurately modeling the nucleation behavior. These parameters serve as input features for machine learning classification and regression models. The classification model categorizes nucleation behavior into three nucleation density regimes, preventing invalid nucleation attempts in simulations, while the regression model estimates the appropriate Langevin noise strength, significantly reducing the need for time-consuming trial-and-error simulations. This data-driven approach improves the efficiency of parameter selection in phase field models and provides a generalizable method for simulating nucleation-driven microstructural evolution processes in various materials.

A critical review on free edge delamination fracture criteria

Laminates experience three-dimensional singular stress near their free edges due to elastic mismatches between layers, which can cause delamination. This paper critically evaluates methods for predicting free edge delamination and highlights the limitations of conventional strength-of-materials and fracture mechanics approaches. The Theory of Critical Distances (TCD) uses a material-dependent critical distance parameter, while Finite Fracture Mechanics (FFM) employs a combined stress-energy criterion without needing a predefined length parameter. This review compares TCD and FFM, also discussing Cohesive Zone Models and Phase-Field Models, and aims to guide the selection of appropriate methods for analyzing free edge delamination.

A Ternary Phase-Field Model for Separation of Water from a Water/Oil Mixed Drop Using Lamb Waves

A Ternary Phase-Field Model for Separation of Water from a Water/Oil Mixed Drop Using Lamb Waves

Order evolution from a high‐entropy matrix: Understanding and predicting paths to low‐temperature equilibrium

AbstractInterest in high‐entropy inorganic compounds originates from their ability to stabilize cations and anions in local environments that rarely occur at standard temperature and pressure. This leads to new crystalline phases in many‐cation formulations with structures and properties that depart from conventional trends. The highest‐entropy homogeneous and random solid solution is a parent structure from which a continuum of lower‐entropy offspring can originate by adopting chemical and/or structural order. This report demonstrates how synthesis conditions, thermal history, and elastic and chemical boundary conditions conspire to regulate this process in Mg0.2Co0.2Ni0.2Cu0.2Zn0.2O, during which coherent CuO nanotweeds and spinel nanocuboids evolve. We do so by combining structured synthesis routes, atomic‐resolution microscopy and spectroscopy, density functional theory, and a phase field modeling framework that accurately predicts the emergent structure and local chemistry. This establishes a framework to appreciate, understand, and predict the macrostate spectrum available to a high‐entropy system that is critical to rationalizing property engineering opportunities.

A convergence result for the derivation of front propagation in nonlocal phase field models

We prove that the mean curvature of a smooth surface in \R^{n} , n\geq 2 , arises as the limit of a sequence of functions that are intrinsically related to the difference between an n - and 1 -dimensional fractional Laplacian of a phase transition. Depending on the order of the fractional Laplace operator, we recover the fractional mean curvature or the classical mean curvature of the surface. Moreover, we show that this is an essential ingredient for deriving the evolution of fronts in fractional reaction-diffusion equations such as those for atomic dislocations in crystals.

Simulation of Dendrite Remelting via the Phase-Field Method

The solidification of alloys is a key physical phenomenon in advanced material-processing techniques including, but not limited to, casting and welding. Mastering and controlling the solidification process and the way in which microstructure evolution occurs constitute the key to obtaining excellent material properties. The microstructure of a solidified liquid metal is dominated by dendrites. The growth process of these dendrites is extremely sensitive to temperature changes, and even a small change in temperature can significantly affect the growth rate of the dendrite tip. Dendrite remelting is inevitable when the temperature exceeds the critical threshold. In this study, a temperature-induced-dendrite remelting model was established, which was implemented through the coupling of the phase field method (PFM) and finite difference method (FDM). The transient evolution law of dendrite remelting was revealed by simulating dendritic growth and remelting processes. The phase field model showed that the lateral dendrites melt first, the main dendrites melt later, and the main dendrites only shrink but do not melt when the lateral dendrites have not completely melted or the root is not broken. The long lateral branches break into fragments, while the short lateral branches shrink back into the main dendrites. The main dendrites fracture and melt in multiple stages due to inhomogeneity.

Phase Field Modeling of Hydraulic Fracturing with Length-Scale Insensitive Degradation Functions

A length-scale insensitive degradation function is applied to extend the cracks during hydraulic fracturing under stress boundary conditions in this study. The phase field method is an effective modeling technique that has great potential for use in hydraulic fracturing. Nonetheless, current hydraulic fracturing research is still concentrated on small scales. The phase field model employs a degradation function that is insensitive to length scale, allowing for the decoupling of the phase field length scale from the physical length scale. This facilitates the simulation of hydraulic fracturing crack extensions in larger structures with a consistent mesh density. The correctness of the phase field method is verified firstly by comparing with the experimental results, and the accuracy and efficiency of the proposed method are further verified through a series of numerical calculations.

Phase Field Coupled Finite Deformation Plasticity Formulation of Ductile Fracture With Nonlinear Kinematic Hardening and Modified Energy Release Function

ABSTRACTA ductile damage theory is presented by coupling the covariant formulation of finite deformation plasticity with the phase field modeling of fracture, including kinematic hardening for the ductile response of the materials. A phase field coupled nonlinear kinematic hardening equation is proposed in the reference configuration having equivalent representation through the Lie derivative of the kinematic hardening tensor by push‐forward operation in the spatial configuration, thus ensuring the satisfaction of frame invariance. To capture the correct physical response of the material by the phase field evolution equation, the fracture driving function, that is, the difference between the sum of elastic and plastic energies and threshold energy, after the damage initiation is modified by an energy release controlling function, which is an empirical relation of equivalent plastic strain. In defining the energy release controlling function, well‐defined points with physical significance in the experimental load versus displacement curve are used. To simulate the response of the material in relatively large time steps, a modified staggered scheme is presented, evaluating the fracture driving and energy release controlling functions from the previous converged step and using the updated phase field variable in the weak form of the momentum balance equation. To quantify different material parameters from available experimental results in the literature, the developed phase field coupled elasto‐plastic model uses a neural network optimization procedure consisting of neural network training together with optimization in MATLAB and finite element model evaluation in Abaqus user element subroutine UEL. Model capabilities are demonstrated by simulating the crack propagation in complex 3D geometries such as the second and third Sandia Fracture Challenges.

Evolutionary Mechanism of Solidification Behavior in the Melt Pool during Disk Laser Cladding with 316L Alloy

Laser cladding is an emerging environmentally friendly surface-strengthening technology. During the cladding process, the changes in molten pool temperature and velocity directly affect the solidification process and element distribution. The quantitative revelation of the directional solidification mechanism in the molten pool during the cladding process is crucial for enhancing the quality of the cladding layer. In this study, a multi-field coupling numerical model was developed to simulate the coating process of 316L powder on 45 steel matrices using a disk laser. The instantaneous evolution law of the temperature and flow fields was derived, providing input conditions for simulating microstructure evolution in the molten pool’s paste zone. The behavior characteristics of the molten pool were predicted through numerical simulation, and the microstructure evolution was simulated using the phase field method. The phase field model reveals that dendrite formation in the molten pool follows a sequence of plane crystal growth, cell crystal growth, and columnar crystal growth. The dendrites can undergo splitting to form algal structures under conditions of higher cooling rates and lower temperature gradients. The scanning speed of laser cladding (6 mm/s) has minimal impact on dendrite growth; instead, convection within the molten pool primarily influences dendrite growth and tilt and solute distribution.

Effect of molybdenum addition on precipitate coarsening kinetics in Inconel 740H: A phase-field study

Inconel 740H (IN740H) has emerged as an important candidate for the advanced ultra-supercritical (AUSC) steam turbines due to its superior microstructural stability and creep resistance under service conditions. In the present work, we have reparameterized a multicomponent Ni-based superalloy (IN740H) into an equivalent ternary Ni-Al-Mo superalloy, based on the partitioning coefficients of the elements, using the thermodynamic database (CALPHAD). We use this thermodynamic description to employ a quantitative phase-field model to assess the long-term stability of γ′ precipitates in IN740H utilizing GPU-based supercomputing architecture. The assessment helps us to enhance our understanding of the effect of the atomic diffusivity of Mo on coarsening kinetics of the γ′-precipitates in equivalent ternary Ni-Al-Mo superalloy. Investigation reveals that our phase-field model can accurately predict the experimentally observed coarsening kinetics in IN740H.

An adaptive acceleration scheme for phase-field fatigue computations

Abstract Phase-field models of fatigue are capable of reproducing the main phenomenology of fatigue behavior. However, phase-field computations in the high-cycle fatigue regime are prohibitively expensive due to the need to resolve spatially the small length scale inherent to phase-field models and temporally the loading history for several millions of cycles. As a remedy, we propose a fully adaptive acceleration scheme based on the cycle jump technique, where the cycle-by-cycle resolution of an appropriately determined number of cycles is skipped while predicting the local system evolution during the jump. The novelty of our approach is a cycle-jump criterion to determine the appropriate cycle-jump size based on a target increment of a global variable which monitors the advancement of fatigue. We propose the definition and meaning of this variable for three general stages of the fatigue life. In comparison to existing acceleration techniques, our approach needs no parameters and bounds for the cycle-jump size, and it works independently of the material, specimen or loading conditions. Since one of the monitoring variables is the fatigue crack length, we introduce an accurate, flexible and efficient method for its computation, which overcomes the issues of conventional crack tip tracking algorithms and enables the consideration of several cracks evolving at the same time. The performance of the proposed acceleration scheme is demonstrated with representative numerical examples, which show a speedup reaching up to four orders of magnitude in the high-cycle fatigue regime with consistently high accuracy. Graphical abstract

Dendrite Growth and Dead Lithium Formation in Lithium Metal Batteries and Mitigation Using a Protective Layer: A Phase-Field Study.

Lithium metal batteries (LMBs) are considered one of the most promising next-generation rechargeable batteries due to their high specific capacity. However, severe dendrite growth and subsequent formation of dead lithium (Li) during the battery cycling process impede its practical application. Although extensive experimental studies have been conducted to investigate the cycling process, and several theoretical models were developed to simulate the Li dendrite growth, there are limited theoretical studies on the dead Li formation, as well as the entire cycling process. Herein, we developed a phase-field model to simulate both electroplating and stripping process in a bare Li anode and Li anode covered with a protective layer. A step function is introduced in the stripping model to capture the dynamics of dead Li. Our simulation clearly shows the growth of dendrites from a bare Li anode during charging. These dendrites detach from the bulk anode during discharging, forming dead Li. Dendrite growth becomes more severe in subsequent cycles due to enhanced surface roughness of the Li anode, resulting in an increasing amount of dead Li. In addition, it is revealed that dendrites with smaller base diameters detach faster at the base and produce more dead lithium. Meanwhile, the Li anode covered with a protective layer cycles smoothly without forming Li dendrite and dead Li. However, if the protective layer is fractured, Li metal preferentially grows into the crack due to enhanced Li-ion (Li+) flux and forms a dendrite structure after penetration through the protective layer, which accelerates the dead Li formation in the subsequent stripping process. Our work thus provides a fundamental understanding of the mechanism of dead Li formation during the charging/discharging process and sheds light on the importance of the protective layer in the prevention of dead Li in LMBs.

Parametric sensitivity analysis of the modified phase-field model in polymer membrane formation by immersion precipitation method

Parametric sensitivity analysis of the modified phase-field model in polymer membrane formation by immersion precipitation method

Phase-field approach for precise fracture tracking in anisotropic rocks: Integrating orthotropy-based energy decomposition and two-fold symmetric fracture toughness

Rock formations are known to exhibit material anisotropy, both in terms of elastic and fracture properties. This means that the fracture path in such formations is not a priori known but rather a complex unknown that requires robust numerical techniques to predict accurately. In this context, the phase-field model is considered particularly effective, provided that certain physical considerations are carefully adjusted to align with the physics of the problem. While addressing elastic anisotropy is well-established, the tension–compression asymmetry necessary to inhibit crack interpenetration in phase-field fracture models needs to account for the specific material anisotropy. Additionally, to accurately capture crack propagation, it is critical to simultaneously account for orientation-dependent fracture toughness in such materials. To address this, the present study employs an anisotropic phase-field model that integrates the generalized spectral decomposition proposed in the literature for orthotropic materials with a two-fold symmetric fracture toughness, to predict the fracture trajectories in rock-type samples under fixed mixed-mode loading ratios. While each of the two aspects has primarily been applied to model orthotropic plates under simple tensile and shearing loading conditions in the literature, here we study their applicability in complex loading scenarios. To this end, the experimental data from notched semi-circular specimens of Grimsel Granite undergoing complex mixed-mode loading obtained in our previous work is considered. We focus on two given mode-mixity ratios and perform numerical studies. Our results emphasize the importance of considering this generalized decomposition for phase-field modeling of fracturing in rock-type materials, particularly under loading conditions where the crack might otherwise be unrealistically driven into the compressive region. Although certain features are well captured by considering anisotropy in elasticity alone, our findings demonstrate that incorporating a two-fold symmetric fracture toughness proves to be advantageous for more precise tracking of the fracture path.

An improved phase-field model for fatigue crack growth considering constraint effects

The phase field model is a promising method for simulating fatigue crack growth (FCG) behavior. However, the conventional phase field (PF) model may not adequately account for constraint effects, where fracture toughness is affected by geometries. Therefore, stress triaxiality is incorporated into the PF model by modifying the fracture energy release rates to consider constraint effects. The model successfully simulates the FCG behavior of different geometries, such as CT, SENB, and MT specimens, as well as the mixed-mode FCG behavior of CTS specimens and other complex geometries. All simulations agree well with experiments, proving that our model is capable to capture the constraint effects in FCG behavior. These findings indicate that stress triaxiality is important to capture the constraint effects.

A crystal plasticity phase-field model for microstructure sensitive fatigue crack growth in a superalloy

In this work, a dislocation density-based crystal plasticity phase-field model (CP-PFM) is developed to simulate fatigue crack growth in nickel-based superalloys. Through normalization validation, the plastic dissipation work and crystallographic work are shown to be consistent with the fatigue indicator factors (FIPs), cumulative equivalent plastic strain and cumulative shear strain, and the two energies are computed as the main driving forces of the phase field. Both driving force models are able to obtain fatigue crack growth in close approximation to the experimental rate. However, the model with crystallographic work as the main driving force obtains crack growth paths that are in better agreement with electron backscattering pattern (EBSD) observations, which is attributed to its greater ability to characterize the microstructural susceptibility of fatigue crack growth. Specifically, the model is able to capture the tendency of cracks to crack along the close-packed planes and the hindering effect of grains with large misorientation angles on fatigue crack growth, which together contribute to the curved morphology of fatigue cracks. The combination of large grains or grains with small misorientation angles favors persistent slip band (PSB) formation and leads to softening of the crack tip, which results in lower fatigue crack growth rates.

Cracking Mechanism and Inhibition Strategies of Polycrystalline NCM Electrode Particles.

Developing a high-energy-density cathode material (LiNi1-x-yCoxMnyO2, NCM) for lithium-ion batteries is crucial to the electric vehicle and energy storage industries. However, the continuous insertion/extraction of Li+ generates diffusion-induced stress, causing NCM particles to crack or even pulverize, leading to battery capacity loss and limiting its wider commercial application. Current experimental studies are primarily postmortem examinations, and it is difficult to capture the particle cracking evolution. Simulation studies frequently ignore or simplify anisotropic volume contraction, demonstrating an insufficient understanding of the cracking mechanism of NCM polycrystalline particles, and cracking prevention strategies still need improvement. Therefore, we develop an anisotropic polycrystalline fracture phase-field model (AP-FPFM) that focuses on the anisotropic volume contraction of primary particles and precisely generates grain boundary distribution, coupling with Li+ diffusion, mechanical stress, and particle cracking. We employ AP-FPFM to demonstrate the behavior and mechanism of NCM polycrystalline particle cracking and illustrate the necessity and importance of anisotropic volume contraction to understand particle cracking. Furthermore, we explore the effects of average primary particle size, secondary particle size, and core-shell structure modulation on crack initiation and propagation and propose strategies to inhibit or migrate NCM polycrystalline particle cracking. This work provides theoretical support for revealing the cracking mechanism of anisotropic polycrystalline NCM particles and supplying optimization strategies to suppress particle cracking and improve the mechanical stability.

Effect of grain size and grain boundary on the energy storage performance of polycrystalline ferroelectrics

Dielectric capacitors based on polycrystalline ferroelectrics have attracted much attention due to their significant power density and fast charge–discharge speed. The energy storage performance of polycrystalline ferroelectrics is highly dependent on the grain size and grain boundary. Here, the effect of grain size and grain boundary on the domain structures and polarization–electric field (P–E) hysteresis loops of polycrystalline ferroelectrics are investigated by using a phase-field model based on the time dependent Ginzburg–Landau (TDGL) equation. It is found that the depolarization field in the grain boundary induces the vortex domain when the grain size is reduced or the grain boundary thickness increases in certain extent, resulting in slender P–E loops, which contributes to an improvement in the energy storage efficiency and density. However, as the grain size further decreases or the grain boundary thickness further increases, the energy storage density decreases, which is attributed to the concurrent reduction in both the remnant and saturation polarizations. This study provides a considerable insight for optimizing the energy storage performance by carefully adjusting the grain size and grain boundary thickness in polycrystalline ferroelectrics.

Fracture simulation of fiber reinforced composite panels with holes

Fiber reinforced composite (FRC) with holes have broad applications in various fields. In this study, the influence of fiber orientation and hole distribution on the fracture behavior of FRC was investigated. A phase-field modeling was established to simulate the fracture process of the composite, and the mechanical performance of unidirectional fiber reinforced composite and woven fiber reinforced composite were analyzed, respectively. Our numerical results showed that fiber orientation and hole distribution have a significant impact on the fracture behavior of FRC. We observed that aligning the fibers parallel to the loading direction led to an increase in the maximum load bearing capacity of the composite. A more uniform hole distribution can enhance the overall mechanical performance of FRC. Furthermore, in the presence of thermal shock, crack propagation tends to grow towards the hole. These findings are of great significance for understanding the fracture behavior of FRC, and for optimizing material design and fabrication processes.