Phase-field Method Research Articles

Effects of the diffusion path on the effective diffusion coefficient of hydrogen isotope in tungsten with helium bubbles

In irradiated tungsten (W), the formation of numerous helium (He) bubbles has a significant impact on the diffusion behaviors of hydrogen (H) isotopes. To investigate the influence of the microstructure of He bubbles on the diffusion behaviors of deuterium (D), we utilize the reconstructed experimental data, the phase-field method, and the steady-state diffusion equation to calculate the effective diffusion coefficients of D based on four possible diffusion paths, i.e., D atoms diffuse in the bulk W, inside the He bubbles, and along the outer and inner surface of He bubbles. Simulation results based on the reconstructed depth-dependent distribution of He bubbles reveal that the diffusion paths remarkably affect the effective diffusion coefficient of D. By varying the radius and number density of He bubbles, the effective diffusion coefficient of D undergoes significant changes. Subsequently, we fit an empirical formula of the effective diffusion coefficient of D as a function of the radius and number density of He bubbles for different diffusion paths of D based on simulation data. Besides, the growth behaviors of He bubbles are simulated by the phase-field model, and the effective diffusion coefficient of D as a function of the evolutionary time for different diffusion paths is also discussed. At last, we investigate the influence of the D concentration and temperature on the effective diffusion coefficient of D along different diffusion paths. These results indicate that the effective diffusion coefficient of D increases with decreasing the D concentration and increasing the temperature for four diffusion paths. The current study provides a reference for investigating the diffusion behaviors of D in W in the presence of He bubbles, and may benefit the efforts of developing low D retention W-based materials.

Lattice Boltzmann simulation of breaking waves

Water wave breaking at the ocean’s surface is a turbulent phenomenon and encompasses a wide range of physical processes such as generation of bubbles/drops with different spatio-temporal scales. In the present study, breaking of periodic waves in deep water regime has been simulated using phase-field lattice Boltzmann method (LBM). Compared to direct Navier–Stokes equation based numerical models, the present LB model is Poisson-free and explicit in time integration. Both the flow field and interface are solved using LBM with two sets of distribution functions. The normalized pressure-velocity formulation is used for the flow field, whereas the conservative Allen–Cahn equation is used to capture the interface. An impulsively started wave approach with Stokes third order wave profile has been used to initiate the breaking process. Through systematic variation of Weber number and initial steepness, the effect of surface tension on the kinematic, geometric, and energy dissipation characteristics has been studied for three breaker types, viz., spilling, weak plunging, and plunging breakers. The results are in good agreement with the past experimental observations and direct numerical simulation. In particular, the obtained breaking strength for different Weber numbers agrees well with the semi-empirical relation used in a wind-wave forecasting models.

Design of advanced steels by integrated computational materials engineering

AbstractThe integrated computational materials engineering (ICME) has achieved great success in accelerating the rational design and deployment of new materials. It is a new route of designing new materials and processes and highlighted by Materials Genome Initiative/Engineering that stresses the high‐throughput computation in addition to high‐throughput experimentation and materials informatics. This article presents a brief review on the basic theories and multi‐scale computational tools of ICME to design advanced steel grades, including the first‐principles calculations, the CALPHAD method (i.e., computational thermodynamics) fueled by dedicated databases, diffusion and phase‐field simulations, as well as finite analysis methods and machine learning. In the ICME scheme to deal with steels, the CALPHAD method is considered as the core to readily consider multi‐component systems and integrated to link the microscopic simulations (such as diffusion and phase field method to predict microstructure evolutions in response to external conditions) and macroscopic finite analysis method to deal with mechanical properties. Two applications are also presented to address the new routes to carry out materials design, especially for advanced steels.

Multiphase transformation and mechanical analysis of polycrystalline [formula omitted] nanoparticle during lithiation via phase diagram-guided phase-field approach

The fabrication of tin (Sn)-based materials with exceptional electrochemical and mechanical properties requires a thorough understanding of the multiphase evolution that occurs during lithiation. However, predicting the microstructural changes during this process using analytical models is challenging due to the intricate interactions among electrode materials, cell operating parameters, complex geometries, and polycrystalline structures. To tackle this challenge, we propose the application of the phase-field method in a grand-potential formulation, that is combined with the smooth boundary method to apply constant-current (de-)lithiation boundary conditions. This approach effectively describes the migration of lithium and the multiphase transitions in polycrystalline electrodes with arbitrary geometries. To investigate nanoparticles with hundreds of grains in two/three dimensions efficiently, we have incorporated this model into the self-developed open-source phase-field solver package MInDes in a highly optimized manner. Further, comparing the output voltage of the simulated lithiation process of LixSn nanoparticles in three dimensions demonstrates an excellent correlation with experimental results. Based on this, we analyze the mechanical performance of polycrystalline nanoparticles with varying amounts of copper doping and evaluate the maximal von Mises stress during lithiation to predict the onset of crack formation.

Numerical investigation of droplet impact and heat transfer on hot substrates under an electric field

In this study, the effects of the electric field on the impact dynamics and heat transfer of a single droplet impinging on hot substrates in the film evaporation regime were numerically investigated with phase-field method. According to the simulation results, many fancy impact behaviors of the droplets under a vertical electric field were observed. A strong electric field was found to induce upward pinch-off or even tiny jet phenomena on the hydrophilic substrates. In contrast, the bouncing droplet on hydrophobic substrate would be vertically stretched before leaving the substrates. The maximum spreading ratio of impacting droplets was always inversely proportional to the applied electric field intensity. Correlations of the maximum spreading for the charged condition were proposed by considering the effects of the electric force. In addition, the total heat transfer from hot substrates to the impacting droplet was verified as closely related to the maximum spreading area, spreading time, and instantaneous local heat flux. However, the broken droplets from the induced tiny jet would eventually fall back to disturb the sessile droplet, enhancing the droplet cooling efficiency. The modified heat transfer model of impacting droplets under the effect of an electric field on hydrophilic substrates was established. Finally, the surface charge distribution and electric force loading of the impacting droplets were also discussed. This work delved deep into the underlings of droplet spreading heat transfer under an electric field, and the main findings promised to advance high-efficiency cooling technology.

Computationally Efficient and Interface Accurate Dual-Grid Phase-Field Simulation of Turbulent Drop-Laden Flows

Abstract In this work, we develop a dual-grid approach for the direct numerical simulations (DNS) of tur- bulent multiphase flows in the framework of the phase-field method (PFM). With the dual-grid approach, the solution of the Navier-Stokes equations (flow-field) and of the Cahn-Hilliard equa- tion (phase-field) are performed on two different computational grids. In particular, a base grid - fine enough to resolve the flow down to the Kolmogorov scale - is used for the solution of the Navier-Stokes equations, while a refined grid - required to improve the description of small interfacial structures - is used for the solution of the Cahn-Hilliard equation (phase-field method). The proposed approach is validated, and its computational efficiency is evaluated considering the deformation of a drop in a two-dimensional shear flow. Analyzing the computational time and memory usage, we observe a reduction between ≃30% and ≃40% (with respect to the single-grid approach), depending on the grid refinement factor employed for the phase-field variable. The applicability of the approach to a realistic three-dimensional case is also discussed, by focusing on the breakage of a thin liquid sheet inside a turbulent channel flow. Indications on the grid resolution representing a good compromise between accuracy and computational efficiency in drop-laden turbulence are also provided.

Numerical simulation of electric field-induced phase transition evolution and boiling characteristics in the evaporative cooling medium C6F12O

Phase change cooling technology offers high cooling efficiency, safety, and reliability, representing a novel approach to achieving efficient heat dissipation for high-power and large-capacity electrical equipment. The formulation of the cooling medium is pivotal to phase change cooling technology. However, current media exhibit compatibility, stability, economy, and environmental friendliness deficiencies. Consideration could be given to implementing the C6F12O medium due to its superior overall performance and ability to meet the latent heat requirements in phase change cooling equipment. This paper employs a numerical simulation approach that combines the phase field method based on the Cahn-Hilliard equation with the theory of electrohydrodynamics. It investigates the impact of temperature, electric field intensity, and electric field direction on the evolution of bubble motion and the boiling state of the C6F12O medium, considering the interaction of electric-fluid-heat-phase fields. Numerical results indicate that the system undergoes initial nucleate boiling, nucleate boiling, and film boiling stages at T = 330–335 K, T = 335–350 K, and T ⩾ 355 K, respectively. The introduction of an appropriate electric field can enhance the motion evolution of C6F12O bubbles. However, attention must be paid to the formation of bubble channels under high field strength to prevent potential decreases in insulation performance. An inhomogeneous electric field in the vertical direction proves more effective in improving the bubble release rate compared to a uniform electric field. To some extent, an inhomogeneous electric field in the horizontal direction can prevent the mass accumulation of bubbles in regions of high field intensity. This research has the potential to offer theoretical guidance for the engineering application of the C6F12O phase change cooling medium.

Grain boundary diffusion mechanism in Dy-diffused Nd–Fe–B sintered magnets

Grain boundary diffusion (GBD) is an effective method to enhance the thermal stability of Nd–Fe–B based permanent magnets. When developing a high-performance magnet, it is essential to carry out a study on its mechanisms, in order to reveal the distribution regulation of diffusion solutes and microstructural evolution. In the present work, the phase-field method is applied to investigate the thermodynamic features and the heavy rare-earth Dy migration in a Dy-diffused Nd–Fe–B magnet during the GBD process. In the simulation process, the grain phase transformation and volume diffusion were taken into consideration and the effects of the diffusion mode, initial diffusion source concentration, grain size, and grain boundary (GB) width were explored in a set of magnet models with various grain sizes. An optimized fitting function was introduced to evaluate the solute distribution in grain boundaries and the effective diffusion coefficient. It is shown that the diffusion mode and the GB width have significant impacts on the effective diffusion coefficient. The results provide a theoretical scheme concerning the quantitative evaluation of GBD efficiency based on thermodynamic analysis.

Accelerate microstructure evolution simulation using graph neural networks with adaptive spatiotemporal resolution

Surrogate models driven by sizeable datasets and scientific machine-learning methods have emerged as an attractive microstructure simulation tool with the potential to deliver predictive microstructure evolution dynamics with huge savings in computational costs. Taking 2D and 3D grain growth simulations as an example, we present a completely overhauled computational framework based on graph neural networks with not only excellent agreement to both the ground truth phase-field methods and theoretical predictions, but enhanced accuracy and efficiency compared to previous works based on convolutional neural networks. These improvements can be attributed to the graph representation, both improved predictive power and a more flexible data structure amenable to adaptive mesh refinement. As the simulated microstructures coarsen, our method can adaptively adopt remeshed grids and larger timesteps to achieve further speedup. The data-to-model pipeline with training procedures together with the source codes are provided.

In-situ deformation inhomogeneity and damage evolution of mixed-grain structure with tempered sorbite/bainite in Fe–Cr–Mo–Mn steel

The inhomogeneous tempered sorbite/bainite (TS/B) with various morphologies and orientations play a significant role in deformation coordination and damage evolution. In this paper, the influence of distinct morphologies and orientations of the mixed-grain structure with TS/B on the deformation heterogeneity were extensively investigated by in-situ tensile testing. Furthermore, a multi-scale model of dislocation density-based crystal plasticity (CP) coupled with the phase field method (PFM) was developed to illustrate the damage nucleation and its evolution via the representative volume element (RVE) modelling. The results shows that mixed-grain structure with TS/B exhibits non-homogeneous deformation during the tensile process, and the coarse grains bear the main plastic deformation, resulting in cross-slip traces and the wrinkling phenomenon at the grain boundary and the interior of coarse grains. Furthermore, the activation of slip systems was determined through slip trace analysis, and the slip transfer behavior between adjacent microstructure was analyzed. According to the critical resolved shear stress (CRSS), the {123}<111> slip system was determined to be the most activated slip system during the in-situ deformation. Hard-orientated lath-shaped B (M1) is more prone to accumulate low angle grain boundaries (LAGBs) than hard-orientated TS (M2) and soft-orientated granular B (M3), resulting in higher orientation rotation degree and geometrically necessary dislocations (GNDs) density of M1 than M2 and M3. The M1 exhibits the nucleation of damage during the initial deformation stage. In addition, compared with the M2 and M3, the Schmid factor (SF) of M1 dominates the transformation from hard orientation to soft orientation and the higher value of damage factor (φ) of M1 results in the more activated slip systems and local damage at high strain levels. This study provides new insights into elucidate the effects of TS/B morphologies and orientations on deformation heterogeneity and damage evolution in large-scale forged spindles for offshore wind power.

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.

Phase-field formulated meshless simulation of Rayleigh-Taylor instability problem

The interface between two immiscible Newtonian liquids with different densities and the same viscosity, influenced by gravity, is based on the Phase-Field Method (PFM) formulation. The solution of the related governing coupled Navier-Stokes (NS) and Cahn-Hillard (CH) equations is structured by the meshless Diffuse Approximate Method (DAM) and Pressure Implicit with Splitting of Operators (PISO). The variable density is involved in the inertial and buoyancy terms (non-Boussinesq formulation). The related moving boundary problem is handled through single-domain, irregular, fixed node arrangement in two-dimensional Cartesian coordinates. The meshless DAM uses weighted least squares approximation on overlapping subdomains, polynomial shape functions of second-order and Gaussian weights. Implicit time discretisation is performed for the NS and CH equations in the momentum predictor and Phase-Field (PF) variable corrector steps of PISO, while the momentum corrector steps solve the NS equation explicitly. This solution procedure has improved stability compared to Chorin’s pressure-velocity coupling, previously used in meshless solutions of related problems. The Rayleigh-Taylor instability problem simulations are performed for an Atwood number of 0.76. The DAM parameters (shape parameter of the Gaussian weight function and number of nodes in a local subdomain) are the same as in the author’s previous studies on single-phase flows. The simulations did not need any upwinding in the range of the simulations. The results compare well with the mesh-based finite volume method studies performed with the open-source code Gerris.

Quantitative characterization of imbibition in fractured porous media based on fractal theory

In low-permeability reservoirs, such as shale and tight sandstone, imbibition is an important mechanism for enhancing oil recovery. After hydraulic fracturing treatment, these reservoirs create a network of fracture pathways for fluid flow. Therefore, understanding the imbibition mechanisms in fractured porous media and quantitatively characterizing oil–water distribution are crucial for the development of low-permeability reservoirs. In this study, a mathematical model of two-phase flow in porous media with branching fractures was established. The phase-field method was employed to track the oil–water interface, and quantitative characterization of imbibition was conducted based on fractal theory, and the effects of wetting phase injection rate, the number of disconnected fractures, fracture spacing, and fracture morphology on imbibition in branched fracture porous media were discussed. The research findings indicate that in branched fracture porous media, both co-current and countercurrent imbibition processes occur simultaneously, and there exists a diffusion interface layer with a certain thickness at the oil–water interface. The hydraulic pressure generated by the wetting phase injection rate provides the driving force for imbibition oil recovery, but it also affects the contact time between the wetting and non-wetting phases. The presence of disconnected fractures hinders the propagation of hydraulic pressure, reducing the effectiveness of imbibition. The imbibition displacement zone is limited and occurs only within a certain range near the fractures. As the number of branching fractures increases, the channels for the wetting phase to enter matrix pores are enhanced, resulting in higher efficiency of imbibition displacement of the oil phase. The results of this research can provide guidance for the design of fracturing programs and recovery prediction in low-permeability reservoirs.

Weak Solutions to an Initial-Boundary Value Problem of a Consistent Three-phase-field Model

Global warming has caused the melting of glaciers, especially the Greenland sea ice which is a great threat to human beings. The phase-field method is used to describe the evolution of multi-phase microstructure of Greenland sea ice. This paper shall investigate the global existence, uniqueness and regularity of weak solutions to an initial-boundary value problem for a three-phase-field model based on criteria that lead to both physical and mathematical consistency. For global solutions, it is calculated by applying the method of continuation of local solutions. This model allows for the formulation of spurious phases, which is completely consistent with physical reality.

Droplet dynamics passing through the flexible constriction in the channel

We conducted a numerical investigation into droplet dynamics within a flexible constriction using the phase-field lattice Boltzmann method. Our study focused on constriction bending stiffness, Weber number, and constriction-to-droplet diameter ratio. Flexibility impedes droplet passage at low Weber numbers but significantly facilitates it at moderate and high Weber numbers. Passage times decrease with increasing Weber numbers and are proportional to the droplet's maximum deformed length. An anomalous phenomenon is observed: “more haste, less speed.” The underlying mechanics arising from the interaction between the flexible constriction and the droplet are elucidated. The findings enhanced our understanding of droplet behavior in constrained environments.

A three-dimensional phase-field model for studying the orientation-dependent interface evolution in stress-induced martensitic phase transformation

In this study, we address the challenging task of predicting the motion of interfaces in stress-driven martensitic phase transformations within shape memory alloys. The novelty of our approach lies in the introduction of a monolithically solved thermodynamic-based phase-field method, specifically tailored for large strain conditions at the nanoscale for three-dimensional problems. To achieve this, we have developed a custom finite element software seamlessly integrated into the FEniCS open-source framework, providing a sophisticated and efficient tool for the examination of nanostructure evolution. Our investigation delves into five distinct and complex scenarios under diverse loading conditions for two– and three-dimensional problems, each presenting unique challenges: (i) a straightforward uniaxial tension scenario featuring a square domain with a pre-existing martensitic nucleus; (ii) the evolution of interfaces in a square sample incorporating a circular central nanovoid under biaxial tensile stress; (iii) the analysis of a rectangular beam subjected to horizontal and vertical compressive loads; (iv) the assessment of an initially voided rectangular beam experiencing mixed loading conditions; and (v) the three-dimensional simulations of cubic-to-tetragonal phase transformation using various orientation of the habit plane. In contrast to prior studies, our analysis not only explores the standard factors of habit plane reorientation and strain transformation values but also delves into the impact of nanovoids on austenite–martensite interface evolution.

On different implementations of boundary conditions in the meshless RBF-FD method for phase-field modelling of dendritic solidification

Dendritic morphology is one of the most common microstructures in solidifying metallic materials. The phase-field method represents one of the most suitable approaches for modelling the evolution of dendritic morphology. In this paper, the meshless RBF-FD method and forward Euler scheme are used to solve the partial differential equations arising from the phase-field model for dendritic growth. We consider the growth of a single dendrite into a supercooled pure melt. On the computational domain’s surfaces, we apply symmetric boundary conditions. Special care has to be taken in the RBF-FD method to satisfy boundary conditions accurately. In this paper, we test four different implementations of boundary conditions. In the first one, the boundary conditions are incorporated when constructing an interpolation problem in local support domains containing boundary nodes. In the second one, the interpolation problem in the local support domains does not consider boundary conditions, which are satisfied by solving a system of linear equations for values in all boundary nodes at each time step. The third one complements the second one with the use of ghost nodes. The fourth implementation is an alternative one, where the values in ghost nodes are determined by direct mirroring. The accuracy and computational efficiency of all four implementations are compared. We discuss the advantages and disadvantages of each implementation. We show that using ghost nodes is recommended for implementing Neumann boundary conditions in the RBF-FD method.

Evaluation and associated mechanism of mitigating Cr depletion by vanadium in modified 9Cr-1Mo steel: Experiment and phase-field modelling

The 9Cr-1Mo steel is extensively utilized in nuclear heat exchangers due to its exceptional corrosion resistance properties. The Chromium (Cr) element plays a pivotal role in enhancing the corrosion resistance. The Cr accumulates at grain boundaries (GBs) where defects and mismatches are located during long-term usage, leading to the depletion of Cr within the grain interior. This results in the Cr-rich carbide precipitates form along GBs, with a corresponding the depletion zone residing in the adjacent regions. In this study, we employed aberration-corrected high-resolution electron microscopy in high-angle annular dark field microscopy (HAADF) mode, equipped with energy dispersive spectroscopy (EDS), to analyse carbides formation along GBs and obtain elemental distribution information. Our investigation reveals that the Cr-rich carbide promotes the formation of chromium-depletion zone (CDZ) by consuming the surrounding Cr in the matrix, and the presence of vanadium-rich carbide inhibits the formation of (CDZ) from atomic scale.A phase-field modelling approach is employed to investigate the diffusive formation process in a polycrystalline system. Our modelling results are consistent with experimental observations. Cr preferentially diffuses towards grain boundaries (GBs) and accumulates, facilitating the formation of Cr-rich phases by consuming adjacent Cr in the surrounding region. The bilateral region of GBs serves as a source for supplying Cr to these newly formed phases while depleting Cr content, ultimately leading to the development of CDZ.By employing experimental techniques and phase-field method, this research provides valuable insights into the role of Cr, carbide precipitates, influence of vanadium, and CDZ formation.

Topology Optimization of Brittle Composites for Optimizing Fracture Resistance Incorporating Phase Field Method with Strain Orthogonal Decompositions

Topology Optimization of Brittle Composites for Optimizing Fracture Resistance Incorporating Phase Field Method with Strain Orthogonal Decompositions

Modeling the Propagation of a Prefabricated Brittle Crack Using Phase-field Method within the Framework of ABAQUS

Modeling the Propagation of a Prefabricated Brittle Crack Using Phase-field Method within the Framework of ABAQUS