Combined Phase Field Research Articles

The phase field method coupled with molecular dynamics parameter for simulating phase separation behavior of SBS modified asphalt

Polymer-modified asphalt is a thermodynamically unstable system, in which the polymer and asphalt are prone to phase separation. Due to the inadequacy of characterization methods for phase separation, this field has not been thoroughly understood or explored. This paper introduces an innovative methodology by employing a simulation approach that couples phase-field theory with molecular dynamics parameters. Using SBS polymer-modified asphalt as an example, validate the method's applicability and establish a more precise parameter library for the phase separation field model of SBS-modified asphalt. The phase separation process of SBS-modified asphalt was simulated. In combination with fluorescence microscopy experiments, the evolution of the micro and meso-phase states of the modified asphalt over time was established and tracked. Results indicate that higher light component content and lower heavy component content in asphalt improve its compatibility. Migration coefficients and interaction parameters for the phase field model were determined, and a more accurate correction factor for the entropy contribution of SBS-modified asphalt was estimated. The phase-field model shows that higher light component content and lower heavy component content result in slower and smaller phase separation, leading to better compatibility. The combined phase field and molecular dynamics simulations accurately reproduce experimental findings from fluorescence microscopy, providing theoretical references for studying phase separation in polymers.

Microstructural and mechanistic insights into the Tension–Compression asymmetry of rapidly solidified Fe–Cr alloys: A phase field and strain gradient plasticity study

Rapid solidification in Additively Manufactured (AM) metallic materials results in the development of significant microscale internal stresses, which are attributed to the printing induced dislocation substructures. The resulting backstress due to the Geometrically Necessary Dislocations (GNDs) is responsible for the observed Tension–Compression (TC) asymmetry. We propose a combined Phase Field (PF)-Strain Gradient J2 Plasticity (SGP) framework to investigate the TC asymmetry in such microstructures. The proposed PF model is an extension of Kobayashi’s dendritic growth framework, modified to account for the orientation-based anisotropy and multi-grain interaction effects. The SGP model has consideration for anisotropic temperature-dependent elasticity, dislocation strengthening, solid solution strengthening, along with GND-induced directional backstress. This model is employed to predict the solute segregation, dislocation substructure and backstress development during solidification and the post-solidification anisotropic mechanical properties in terms of the TC asymmetry of rapidly solidified Fe–Cr alloys. It is observed that higher thermal gradients (and hence, cooling rates) lead to higher magnitudes of solute segregation, GND density, and backstress. This also correlates with a corresponding increase in the predicted TC asymmetry. The results presented in this study point to the microstructural factors, such as dislocation substructure and solute segregation, and mechanistic factors, such as backstress, which may contribute to the development of TC asymmetry in rapidly solidified microstructures.

A combined interface phase field (CIPF) model for interfacial crack propagation

The phase field method has been very popular in the past decade because it can handle complex crack propagation problems without ad hoc criteria. In this paper, a combined interface phase field (CIPF) model is proposed to better investigate the fracture behaviours of composite systems with a finite-thickness interface. To build the CIPF model, a new numerical method named the modified crack surface displacement extrapolation (MCSDE) method is proposed to determine the real interfacial fracture toughness. Then, an effective interfacial fracture toughness is given based on the coupling relationship between the length scale parameter and the interface thickness. We investigate crack propagation behaviours in bi-material structures with varied initial notch positions and sandwich specimens using the CIPF model. The results show that the MCSDE method can effectively and accurately determine the interface fracture toughness, and the difference between the calculated results and the results obtained by the sing-leg bending (SLB) experiment is 6%. The CIPF model can not only simulate the propagation of cracks in bulk, but also simulate the propagation of cracks between interfaces. For different positions of the initial notch, the propagation patterns of cracks and the mechanical response of the structure are very different.

A combined kinetic Monte Carlo and phase field approach to model thermally activated dislocation motion

In this work, a kinetic Monte Carlo (KMC) scheme is integrated into a phase field dislocation dynamics (PFDD) model to account for thermal activation. While the implementation is general and could be applied to many different thermally activated processes, we apply the model to simulate a dislocation transmitting through an interface. Two cases are analyzed: a dislocation transmitting through a strong interface, and a dislocation transmitting through a more realistic grain boundary in tungsten. Results and trends for both cases are discussed in detail, and ultimately show that the combined KMC-PFDD algorithm is effective in capturing thermally activated dislocation processes.

A Review on Phase Field Modeling for Formation of η-Cu6Sn5 Intermetallic

Formation of intermetallic compounds (IMCs) exhibits remarkable microstructural features and provides opportunities for microstructure control of microelectronic interconnects. Excessive formation of brittle IMCs at the Cu/Sn interface such as η-Cu6Sn5 can deteriorate the reliability and in turn lead to solder joint failure in the Pb-free Sn-based solder joints. Phase field method is a versatile tool for prediction of the mesoscopic structure evolution in solders, which does not require tracking interfaces. The relationships between the microstructures, reliability and wettability were widely investigated, and several formation and growth mechanisms were also proposed for η-Cu6Sn5. In this paper, the current research works are reviewed and the prospective of the application of phase field method in the formation of η-Cu6Sn5 are discussed. Combined phase field simulations hold great promise in modeling the formation kinetics of IMCs with complex microstructural and chemical interactions.

Computational modelling of hydrogen assisted fracture in polycrystalline materials

We present a combined phase field and cohesive zone formulation for hydrogen embrittlement that resolves the polycrystalline microstructure of metals. Unlike previous studies, our deformation-diffusion-fracture modelling framework accounts for hydrogen-microstructure interactions and explicitly captures the interplay between bulk (transgranular) fracture and intergranular fracture, with the latter being facilitated by hydrogen through mechanisms such as grain boundary decohesion. We demonstrate the potential of the theoretical and computational formulation presented by simulating inter- and trans-granular cracking in relevant case studies. Firstly, verification calculations are conducted to show how the framework predicts the expected qualitative trends. Secondly, the model is used to simulate recent experiments on pure Ni and a Ni–Cu superalloy that have attracted particular interest. We show that the model is able to provide a good quantitative agreement with testing data and yields a mechanistic rationale for the experimental observations.

Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach

Adaptive modelling of dynamic brittle fracture - a combined phase field regularized cohesive zone model and scaled boundary finite element approach

Lithium Plating Kinetics Under Fast Charge Conditions

Long range electric vehicular technology requires extreme fast charging under 10-15 minutes. There are numerous existing knowledge gaps inhibiting this feat, including the assessment of lithium plating kinetics on the graphite anode under fast charging conditions. In order to bridge this gap, we performed a combined phase field modeling and experimental analysis to understand the onset of lithium nucleation on graphite particles in fast charging batteries [1,2]. Also, the phase field model is used to understand the effect of lithium nucleation and plating kinetics and their overall impact on lithium intercalation. The study herein presents a model which incorporates solid electrolyte interphase (SEI) effects, interfacial electrochemical reactions such as re-intercalation of plated lithium into graphite, and dendrite growth during fast charge. Finally, the developed framework is a good starting point to spatially predict the amount of plated lithium in the complex graphite anode microstructures, including effects such as surface tension, particle size, porosity, and tortuosity. Ultimately, this work will aid in the design of optimized physical electrolyte properties and charging conditions to delay the onset of lithium nucleation during fast charge.

Dispersion compensation for orbital angular momentum mode based on circular photonic crystal fiber

In this work, a circular dual-core photonic crystal fiber is proposed for dispersion compensation for the controllable orbital angular momentum (OAM) mode by the inner-outer core mode coupling. This design can simultaneously realize a negative dispersion of up to −3039.45 ps nm−1 km−1 for OAM mode HE31 and a high effective index separation of 2.15 × 10−3 within scalar linearly polarized (LP)21 mode group (HE31 and EH11) at 1.55 µm, ensuring HE31 vortex mode robust propagation without channel crosstalk. Evolution of the combined effective index, OAM phase, and electric field is systemically verified functioning as wavelength during the inner-outer mode coupling. Structural parameter dependence of negative dispersion is demonstrated with respect to lattice constant, air hole size and air-filling fraction, remarkably suggesting that negative dispersion dip at phase-matching wavelength exhibits an enhanced blue shift with lattice constant of inner cladding increasing but a red shift with lattice constant of outer cladding. The feasibility of this design is demonstrated with an achievement of negative dispersion of −1773.21 ps nm−1 km−1 for EH11 mode at wavelength 1.55 µm by tuning the structural parameter. This design is expected for fiber applications in OAM-based optical communication systems.

A combined adaptive phase field and discrete cutting method for the prediction of crack paths

Crack propagation is modelled in this contribution by combining different methods. The main idea is to use the newly developed virtual element method in combination with the phase-field methodology and a specific cutting technology. The idea is that the direction of a crack path can be easily predicted using the phase-field method. However this method needs a very fine mesh to resolve a real crack and thus this method has to be coupled with an adaptive approach. Due to the flexible virtual element method that allows to add arbitrary number of nodes to an element a robust cutting technique can be used to replace the refined mesh by a discrete crack. In total this combination of different methods allows a very efficient and robust solution of crack growth in solids.

Effect of grain structure and strain rate on dynamic recrystallization and deformation behavior: A phase field-crystal plasticity model

In this work, we combined phase field method and crystal plasticity to study the effects of initial grain structures, recrystallized grain orientation, and strain rates on dynamic recrystallization and deformation behavior under stress states typical of the Shear Assisted Processing and Extrusion (ShAPE) process. The geometrically necessary dislocation density was used as the nucleation criterion for recrystallization, and the minimization of the deformation energy drove the growth of the recrystallized grains. With the set of material properties and model parameters, the simulation results demonstrate that (1) a polycrystalline structure with the texture observed in ShAPE process has the lowest yield stress under the ShAPE stress conditions, (2) the recrystallized grains with the texture observed in ShAPE process largely soften the materials, and (3) higher shear strain rate or rotation speed results in larger magnitude of material softening.

Modelling distinct failure mechanisms in composite materials by a combined phase field method

A numerical computational framework that combines a phase field model (PFM) and cohesive element (CE) for modelling progressive failure in composite material is proposed. In this setting, cracks in single material and the interface are captured separately by using the PFM and CE. A unified PFM that incorporates general cohesive softening laws is adopted for quasi-brittle fracture process. An energy split scheme is used to prevent material cracking under compressive state. The irreversibility of the damage evolution is enforced by introducing a history related energy density. The compatibility issue between the CE and PFM is considered for accurately capturing complicated failure mechanisms in composite. The developed approach is implemented into commercial software ABAQUS through user-defined subroutine element (UEL). Validation is made by modelling an experiment on delamination migration of a cross ply composite. Failure mechanisms including delamination, matrix crack, and location of delamination migration that observed in the experiment are all well captured. Moreover, the importance of considering the compatibility between CE and other numerical methods is discussed through a complementary modelling.

Study of the Influence of Elastic Anisotropy of Cu on Thermo-Mechanical Behavior and Cu Protrusion of Through Silicon Vias Using Combined Phase Field and Finite Element Methods

Drastic reduction in the diameter of the Cu-filled through silicon via (TSV) leads to the TSV feature size becoming comparable with the grain size in the Cu filler, and accordingly Cu grain characteristics strongly influence thermo-mechanical behavior and Cu protrusion of the TSV due to the elastic anisotropy of Cu, which poses new emerging issues and challenges for reliability analysis of the TSV. In this paper, a 2-D phase field model is combined with a finite element model to study comprehensively and quantitatively the grain morphology characteristics, thermo-mechanical behavior and Cu protrusion in a typical Cu-filled TSV structure, while considering the mechanical property anisotropy of Cu. Results show that Cu grains with different orientations make distinct contributions to the deformation behavior of the Cu filler which undergoes three stages of deformation, that is, elastic, elasto-plastic, and plastic as temperature increases, consequently resulting in the anisotropic thermo-mechanical behavior of the Cu-filled TSV and the dependence of Cu protrusion on the average grain size of the Cu filler. Moreover, both the average von Mises stress and equivalent plastic strain of the Cu filler decrease with the increase in Cu grain size, and the average von Mises stress in fine grains is higher than that in coarse grains. Furthermore, Cu protrusion decreases with the increase of the average size of Cu grains. However, for limited grains in the Cu-filled TSV, the average von Mises stress, equivalent plastic strain, and Cu protrusion increase with an increase in the average size of Cu grains.

A phase field approach to shape optimization in Navier–Stokes flow with integral state constraints

We consider the shape optimization of an object in Navier--Stokes flow by employing a combined phase field and porous medium approach, along with additional perimeter regularization. By considering integral control and state constraints, we extend the results of earlier works concerning the existence of optimal shapes and the derivation of first order optimality conditions. The control variable is a phase field function that prescribes the shape and topology of the object, while the state variables are the velocity and the pressure of the fluid. In our analysis, we cover a multitude of constraints which include constraints on the center of mass, the volume of the fluid region, and the drag of the object. Finally, we present numerical results of the optimization problem that is solved using the variable metric projection type (VMPT) method proposed by Blank and Rupprecht, where we consider one example of topology optimization without constraints and one example of maximizing the lift of the object with a state constraint, as well as a comparison with earlier results for the drag minimization.

Combined finite element and phase field method for simulation of austenite grain growth in the heat-affected zone of a martensitic steel weld

Engineering components operating at high temperature often fail due to the initiation and growth of cracks in the heat-affected zone adjacent to a weld. Understanding the effects of microstructural evolution in the heat-affected zone is important in order to predict and control the final properties of welded joints. This study presents a combined finite element method and phase field method for simulation of austenite grain growth in the heat-affected zone of a tempered martensite (P91) steel weld. The finite element method is used to determine the thermal history of the heat-affected zone during gas tungsten arc welding of a P91 steel plate. Then, the calculated thermal history is included in a phase field model to simulate grain growth at various positions in the heat-affected zone. The predicted mean grain size and grain distribution match well with experimental data for simulated welds from the literature. The work lays the foundation for optimising the process parameters in welding of P91 and other ferritic/martensitic steels in order to control the final heat-affected zone microstructure.

Pinning Effect of Cerium Inclusions during Austenite Grains Growth in SS400 Steel at 1300 °C: A Combined Phase Field and Experimental Study

The pinning effect of cerium inclusions in the austenite grain growth of SS400 steel at 1300 °C is investigated by using a semi-empirical-simulation. Firstly, steel samples containing cerium inclusions are prepared; then the properties of inclusions are determined using SEM. In situ observation of austenite grain growth is performed by LSCM, to determine the fitting parameters of the model such as the grain mobility and the pinning parameter. These parameters are directly inserted into our phase field simulation. The time-dependent Ginzburg-Landau (TDGL) equation is implemented in our phase field model, where the effects of inclusion and grain boundary interaction are inserted as a potential term in the local free energy. The results proved that the optimal size of austenite grains can be achieved by changing the volume fraction of inclusions. In fact, by increasing the volume fraction of inclusions from 0 to 0.1, the austenite grain growth can be decreased where the boundary mobility reduces from 2.3×10−12 m4/Js to 1.0×10−12 m4/Js. The results also demonstrated that increasing the temperature can provide more energy for grain to overcome the inclusions’ pinning force. Moreover, it was shown that the classical Zener model, R c = 0.45 r p f i − 1 , describes the pinning effect of cerium inclusions.

Grain growth in U–7Mo alloy: A combined first-principles and phase field study

Grain size is an important factor in controlling the swelling behavior in irradiated U–Mo dispersion fuels. Increasing the grain size in U–Mo fuel particles by heat treatment is believed to delay the fuel swelling at high fission density. In this work, a multiscale simulation approach combining first-principles calculation and phase field modeling is used to investigate the grain growth behavior in U–7Mo alloy. The density functional theory based first-principles calculations were used to predict the material properties of U–7Mo alloy. The obtained grain boundary energies were then adopted as an input parameter for mesoscale phase field simulations. The effects of annealing temperature, annealing time and initial grain structures of fuel particles on the grain growth in U–7Mo alloy were examined. The predicted grain growth rate compares well with the empirical correlation derived from experiments.

Cross-Permeability of the Semisolid Region in Directional Solidification: A Combined Phase-Field and Lattice-Boltzmann Simulation Approach

Based on the results of microstructure simulations, fluid flow through the semisolid region during directional solidification of the technical Ni-base alloy 718 has been studied. Three-dimensional microstructures at different positions in the semisolid region were obtained by using a multicomponent multiphase-field model that was online coupled to a commercial thermodynamic database. For the range of five different primary dendrite distances λ 1 between 50 µm and 250 µm, the flow velocity and the permeability perpendicular to the dendrite growth direction was evaluated by using a proprietary Lattice-Boltzmann model. The commercial CFD software ANSYS FLUENT was alternatively applied for reference. Consistent values of the average flow velocity along the dendrites were obtained for both methods. From the results of the fluid flow simulations, the cross-permeability was evaluated as a function of temperature and fraction liquid for each of the five different primary dendrite distances λ 1. The obtained permeability values can be approximated by a single analytical function of the fraction liquid and λ 1 and are discussed and compared with known relations from the literature.

Martensitic transformations and damage: A combined phase field approach

AbstractA combined continuum phase field model for martensitic transformations and damage is introduced. The present approach considers the eigenstrain within the martensitic phase which leads in the surrounding material to both tensile and compressive stresses. The damage model needs to account for an appropriate differentiation thereof, since compressive stresses should not promote fracture. Interactions between micro crack propagation and the formation of the martensitic phases are studied in two dimensions. In agreement with experimental observations, martensite forms at the crack tip and influences the crack formation. For the numerical implementation finite elements are used while for the transient terms an implicit time integration scheme is employed. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Hybrid level set phase field method for topology optimization of contact problems

The paper deals with the analysis and the numerical solution of the topology optimization of system governed by variational inequalities using the combined level set and phase field rather than the standard level set approach. The standard level set method allows to evolve a given sharp interface but is not able to generate holes unless the topological derivative is used. The phase field method indicates the position of the interface in a blurry way but is flexible in the holes generation. In the paper a two-phase topology optimization problem is formulated in terms of the modified level set function and regularized using the Cahn-Hilliard based interfacial energy term rather than the standard perimeter term. The derivative formulae of the cost functional with respect to the level set function is calculated. Modified reaction-diffusion equation updating the level set function is derived. The necessary optimality condition for this optimization problem is formulated. The finite element and finite difference methods are used to discretize the state and adjoint systems. Numerical examples are provided and discussed.