Phase Field Research Articles

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.

Creep-thermal fatigue behavior of thin-walled structures with holes and a creep-thermal fatigue-oxidation phase field model

Creep-thermal fatigue behavior of thin-walled structures with holes and a creep-thermal fatigue-oxidation phase field model

Parameterization of a phase field model for ferroelectrics from molecular dynamics data

Parameterization of a phase field model for ferroelectrics from molecular dynamics data

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.

Phase synthesis method in multi-aperture optical systems based on iterative image processing algorithms

Background. Modern technologies in the field of optics and photonics place high demands on the quality and output power of the radiation source. The fulfillment of these requirements can currently be achieved by creating multi-aperture laser systems with coherent beam addition. The main problem of creating such systems is the development of phase synthesis methods in a multichannel optical system. Aim. In this paper, we consider a phase synchronization method without a reference beam for a coherent addition system with active feedback, which is based on the Gerchberg–Saxton algorithm. This algorithm makes it possible to reconstruct complex field amplitudes in the aperture and focal planes from intensity distributions in these fields. The application of this algorithm for multichannel systems is analyzed and its features, such as the occurrence of stagnation and ambiguity of the solution, are revealed. Methods. Solutions are proposed to eliminate the problem of convergence of the algorithm to side solutions and getting the iterative procedure into the local extremum using global optimization algorithms, methods of reducing the dimension of the problem and the introduction of antisymmetric amplitude modulation. Results. The paper demonstrates the results of phase field reconstruction for a large number of optical sources. For a seven-aperture system, a physical simulation was performed to restore phase information, confirming the results of numerical modeling. Conclusion. The proposed approach is a reasonable alternative to currently existing methods and can be used in the problem of coherent addition in multichannel optical systems.

A Simulation Study on the Effect of Supersonic Ultrasonic Acoustic Streaming on Solidification Dendrite Growth Behavior During Laser Cladding Based on Boundary Coupling

Laser cladding has unique technical advantages, such as precise heat input control, excellent coating properties, and local selective cladding for complex shape parts, which is a vital branch of surface engineering. During the laser cladding process, the parts are subjected to extreme thermal gradients, leading to the formation of micro-defects such as cracks, pores, and segregation. These defects compromise the serviceability of the components. Ultrasonic vibration can produce thermal, mechanical, cavitation, and acoustic flow effects in the melt pool, which can comprehensively affect the formation and evolution for the microstructure of the melt pool and reduce the microscopic defects of the cladding layer. In this paper, the coupling model of temperature and flow field for the laser cladding of 45 steel 316L was established. The transient evolution laws of temperature and flow field under ultrasonic vibration were revealed from a macroscopic point of view. Based on the phase field method, a numerical model of dendrite growth during laser cladding solidification under ultrasonic vibration was established. The mechanism of the effect of ultrasonic vibration on the solidification dendrite growth during laser cladding was revealed on a mesoscopic scale. Based on the microstructure evolution model of the paste region in the scanning direction of the cladding pool, the effects of a static flow field and acoustic flow on dendrite growth were investigated. The results show that the melt flow changes the heat and mass transfer behaviors at the solidification interface, concurrently changing the dendrites’ growth morphology. The acoustic streaming effect increases the flow velocity of the melt pool, which increases the tilt angle of the dendrites to the flow-on side and promotes the growth of secondary dendrite arms on the flow-on side. It improves the solute distribution in the melt pool and reduces elemental segregation.

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.

3D model of a stable triangle LiF–NaBr–KBr four-component reciprocal system Li+, Na+, K+ || F-, Вr-

A 3D model of the phase equilibrium states of the quasi-three-component system LiF–NaBr–KBr, which is a stable triangle of the four-component reciprocal system Li+, Na+, K+ || F-, Br-, has been constructed. Based on the 3D-model, polythermal, isothermal sections and the polytherm of phase crystallization were constructed for the first time. Two polythermal sections contain wide areas of boundary solid solutions based on sodium and potassium bromide. In an isothermal section at 650 оC, the fields of the liquid phase and the coexisting two and three phases are delimited. The crystallization polytherm is represented by three fields. In the crystallization field of lithium fluoride, the area of separation of two liquids is limited. The direction of the ion exchange reaction 2LiBr + NaF + KF = 2LiF + NaBr + KBr was confirmed by thermodynamic calculations at temperatures of 400, 600, 800, 1000K. The exothermic nature of the exchange reaction is confirmed by taking a DTA heating curve for a mixture of powders from 50% LiBr + 25% NaF + 25% KF, and the phase composition of the reaction products LiF + NaBr(OTR) + KBr(OTR) is confirmed by X-ray phase analysis data, where OTR is limited solid solution.

Multiscale study on the properties and crack propagation of 2D woven SiCf/SiC with pores using elastoplastic phase field

AbstractTo understand the effect of processing and prefabricated pores on 2D woven SiCf/SiC composites (2DW‐SiCf/SiC), their crack propagation and mechanical response under tension load at meso‐ and macro‐ scale are investigated by a promising elastoplastic phase field method (EPPFM). Of much interest, the computational results are well consistent with the available experimental results, verifying the reliability of EPPFM. A significant dependence on pores is observed in the relationship between the strength, Young's modulus, work of fracture of 2DW‐SiCf/SiC and porosity when pores exist solely in the matrix out of yarns, and both in the matrix and yarns, respectively. Moreover, the more dangerous processing pores at mesoscale near yarn are attributed to the inconsistent deformation of the yarn and matrix. Finally, the EPPFM is used for the effect of geometry size on the joint failure mode of 2DW‐SiCf/SiC plate with prefabricated open holes under pin‐loading. Of importance, the more safe bearing failure mode occurs in 2DW‐SiCf/SiC plate when edge distance‐to‐hole diameter ratio and width‐to‐hole diameter ratio are greater than 1.5 and 3, respectively.

Isogeometric proportional topology optimization (IGA-PTO) for multi-material problems

This study addresses the multi-material optimization problem by the effective and robust isogeometric proportional topology optimization (IGA-PTO) approach. Non-uniform rational B-spline basis functions are used for descriptions of geometry, displacement field, and density fields of material phases. The alternating active-phase algorithm is employed to convert the optimization problem with multiple constraints into a series of sub-problems of single constraint. Several examples are exhibited, and intensive comparisons with the gradient-based optimality criteria (OC) algorithm are presented to highlight the superior performance of the proposed PTO algorithm. Finally, the optimized topologies are prototyped using 3D printing technology to evidence the manufacturing feasibility.

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.

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.

Investigating the influence of geometric configurations and loading modes on mixed mode I/II fracture characteristics of rocks: Part I-Numerical simulation

The influence of geometric configurations and loading modes is a bottleneck that restricts the accurate determination of rock fracture parameters and the precise understanding of fracture mechanisms. Therefore, the phase field method is employed to analyze the influences of specimen geometry and loading modes on the dimensionless stress intensity factors, peak load, fracture toughness, and crack initiation angle during the rock mixed mode I/II fracture process. Three new configurations of specimens are proposed to test rock mixed mode I/II fracture. The research results indicate that as the prefabricated crack inclination angle increases, the peak load of three-point bending type specimen increases, while the peak load of diametric-compression type specimen decreases. Moreover, the influence of geometric configurations on the fracture parameters of three-point bending specimens is greater than that of diametric-compression disk specimens. The research findings of this work can provide basic supporting data for the establishment of mixed mode I/II fracture testing standards.

The localized radial basis function collocation method for dendritic solidification, solid phase sintering and wetting phenomenon based on phase field

Phase-field has been effectively applied to many complex problems according to the mesh based method. However, the computational speed of the numerical method based on phase-field still needs improved. In this paper, an improved localized radial basis function collocation method (LRBFCM) based on the adaptive support domain is employed to the phase-field methods. The proposed adaptive support domain can increase the stability of the LRBFCM, and the improved LRBFCM is much more efficient than the traditional finite element method (FEM) in coupling with phase-field methods. The proposed approach is further applied to the single-phase dendrite solidification, two-phase sintering, and three-phase wetting phenomena. We compare the efficiency of the proposed LRBFCM with different numerical methods, which show that the LRBFCM combined with the Fourier spectral method can deal with the three-phase model with more than ten million nodes easily.

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.

Impact of Secondary‐Phase Particle Morphology on Grain Growth in Pure Copper Using a Phase Field Simulation Approach

Impact of Secondary‐Phase Particle Morphology on Grain Growth in Pure Copper Using a Phase Field Simulation Approach

Strain-controlled switching of magnetic skyrmioniums in ultrathin nanodisks

Magnetic skyrmions and skyrmioniums have garnered significant attention due to their distinctive topologically nontrivial spin structures. Gaining a deep understanding of the magnetization dynamics of these structures and their interconversion processes is essential for fully leveraging their potential in magnetic storage technology. Here, the dynamics of strain-controlled generation and annihilation of skyrmions and skyrmioniums are investigated using phase field simulation methods. It is discovered that tensile strain can induce the transformation of a single domain into skyrmions and skyrmioniums, which can still exist stably after the strain is released. Notably, skyrmioniums demonstrate robust stability within a specific strain window of −0.2% to 0.5%. Beyond this, escalating the compressive strain magnitude induces a phase transition from skyrmioniums to skyrmions, culminating in a direct collapse to a single-domain state at a critical compressive strain of −0.8%. This study reveals that strain can effectively control a variety of topological magnetic domain structures and achieve their interconversion, providing guidance for the design of low-power, nonvolatile, multi-state spin storage devices.