Phase-field Framework Research Articles

Apparent microstructurally induced phase separation in porous LiNi1/3Mn1/3Co1/3O2 cathodes

A thermodynamically consistent phase field framework is presented to analyze the combined effects of internal grain microstructure and the particle size polydispersity on the microstructural mechanisms that control lithium transport and intercalation kinetics in porous LiNi1/3Mn1/3Co1/3O2 (NMC111) cathodes. Physically, Fickian, chemomechanical, and interfacial reaction driving forces are identified and summarized into simple dimensionless numbers, amenable to define different regimes of behavior. Microstructurally NMC111 cathodes comprised of a monodispersed distribution of secondary particle sizes display a unimodal lithium population in the absence of transport limitations in the electrolyte. In contrast, cathodes with a bimodal distribution of secondary particle sizes display two clearly distinguishable populations of lithiated particles due local to differences in the chemomechanical and Fickian transport mechanisms in large and small particles. For larger particles, lithium transport is driven by stress-induced chemomechanical driving forces that preferably lithiate the grain boundaries leading to the formation of lithium-rich networks. For smaller particles, lithiation is driven by Fickian diffusion from the surface inwards due to a lack of chemomechanical driving forces. The compound microstructural and particle size contributions to lithium intercalation result in the development of macroscopic populations of lithium distributions that may appear at first glance as two-phase in character, and highlight the possibility of simultaneously engineering the interior of the particle and the particle size distribution.

A dual resolution phase-field solver for wetting of viscoelastic droplets.

We present a new and efficient phase‐field solver for viscoelastic fluids with moving contact line based on a dual‐resolution strategy. The interface between two immiscible fluids is tracked by using the Cahn‐Hilliard phase‐field model, and the viscoelasticity incorporated into the phase‐field framework. The main challenge of this approach is to have enough resolution at the interface to approach the sharp‐interface methods. The method presented here addresses this problem by solving the phase field variable on a mesh twice as fine as that used for the velocities, pressure, and polymer‐stress constitutive equations. The method is based on second‐order finite differences for the discretization of the fully coupled Navier–Stokes, polymeric constitutive, and Cahn–Hilliard equations, and it is implemented in a 2D pencil‐like domain decomposition to benefit from existing highly scalable parallel algorithms. An FFT‐based solver is used for the Helmholtz and Poisson equations with different global sizes. A splitting method is used to impose the dynamic contact angle boundary conditions in the case of large density and viscosity ratios. The implementation is validated against experimental data and previous numerical studies in 2D and 3D. The results indicate that the dual‐resolution approach produces nearly identical results while saving computational time for both Newtonian and viscoelastic flows in 3D.

A phase field model for mass transport with semi-permeable interfaces

In this paper, a thermaldynamical consistent model for mass transfer across permeable moving interfaces is proposed by using the energy variation method. We consider a restricted diffusion problem where the flux across the interface depends on its conductance and the difference of the concentration on each side. The diffusive interface phase-field framework used here has several advantages over the sharp interface method. First of all, explicit tracking of the interface is no longer necessary. Secondly, interfacial conditions can be incorporated with a variable diffusion coefficient. Finally, topological changes of interfaces can be handed easily. A detailed asymptotic analysis confirms the diffusive interface model converges to the existing sharp interface model as the interface thickness goes to zero. An energy stable numerical scheme is developed to solve this highly nonlinear coupled system.Numerical simulations first illustrate the consistency of theoretical results on the sharp interface limit. Then a convergence study and energy decay test are conducted to ensure the efficiency and stability of the numerical scheme. To illustrate the effectiveness of our phase-field approach, several examples are provided, including a study of a two-phase mass transfer problem where droplets with deformable interfaces are suspended in a moving fluid.

The phase-field simulations of blasting failure in granites

In this study, an explicit phase-field model with new mixed-mode fracture driving forces is developed to simulate the blasting fracture phenomena in granites. The new fracture driving forces make our new phase-field model able to capture pure tensile crack, pure shear cracks and other mixed-mode tensile/compressive-shear cracks in rock/rock-like materials. To validate the correctness and accuracy of our new dynamic phase-field framework, two benchmarked examples, i.e., a Kalthoff–Winkler test and a laboratory borehole blasting experiment, are performed by the qualitative and quantitative comparisons. Furthermore, a series of borehole blasting tests of granites are simulated by our new numerical model. The effects of blast loading parameters (i.e., peak pressure, loading rate and blasting attenuation) and mineral components on the dynamic failure characteristics of granites, including ultimate failure patterns, breakage ratio, energy evolution, peak particle velocity, are systematically investigated. Overall, this study not only offers a useful computational tool to the researchers in rock mechanics, but also provides deep insights on the dynamic rock fracture to the engineers in rock engineering.

Late stage coarsening kinetics induced by a phase-dependent mobility: A phase field study of FeCr in the spinodal regime

An efficient algorithm is presented in this work to compute the late stage coarsening kinetics within the phase-field framework. Microstructures resulting from off-critical quenches over large time periods have been simulated in the FeCr system, which exhibits substantial variations in atomic mobilities between the Fe-matrix and Cr-rich precipitates. To understand the impact of such a phase-dependent mobility on the microstructure, 2D simulations have been performed in the spinodal regime using a constant, symmetric, and phase-dependent mobility. Detailed analyses of these simulations highlight the fact that the more realistic situation where atomic mobility is non-uniform induces a competition between coalescence and coarsening of precipitates. We use these simulations to shed light upon previously published numerical and experimental results.

A phase field model for partially saturated geomaterials describing fluid–fluid displacements. Part I: The model and one-dimensional analysis

A model describing immiscible fluid–fluid displacements in partially saturated porous media is presented. This is based on a phase field approach that interprets the mixture of wetting (liquid water) and non-wetting (air) fluids within the pore space as a single saturating non-uniform pore fluid characterized by a phase field parameter, which is considered to be the saturation degree of the wetting fluid. While the standard retention curve provides for the retention properties of the pore walls, a Cahn–Hilliard like double-well energy is employed to describe the possible co-existence of the immiscible fluid phases. An enhanced description of the macroscopic surface tension between the fluid phases is obtained naturally within the phase field framework due to a regularization that depends on the spatial gradient of the water content. A generalized Darcy’s law is used to describe dissipation due to fluid flow driven by the gradient of a generalized chemical potential. Thus, in the context of soil hydrology this model is interpreted as an extension to the classical Richards equation governing the spatio-temporal evolution of the phase field parameter. Employing a convex–concave flux function it is shown, using one-dimensional analysis, that both imbibition and drainage fronts can be modeled in this phase field framework. The non-monotonicities observed in the resolved solutions are explained using a combination of asymptotic matching techniques and dynamical systems analysis.

Phase-field framework with constraints and its applications to ductile fracture in polycrystals and fatigue

Modeling of ductile fracture in polycrystalline structures is challenging, since it requires integrated modeling of cracks, crystal plasticity, and grains. Here we extend the typical phase-field framework to the situations with constraints on the order parameters, and formulate two types of phase-field models on ductile fracture. The Type-I model incorporates three sets of order parameters, which describe the distributions of cracks, plastic strain, and grains, respectively. Crystal plasticity is employed within grain interiors accommodated by J2 plasticity at grain boundaries. The applications of the Type-I model to single crystals and bicrystals demonstrate the influences of grain orientations and grain boundaries on crack growth. In the Type-II model, J2 plasticity is assumed for the whole system and grain structures are neglected. Taking advantage of the efficiency of the fast Fourier transform, our Type-II model is employed to study low cycle fatigue. Crack closure and striation-like patterning of plastic strain are observed in the simulations. Crack growth rate is analyzed as a function of the J-integral, and the simulated fatigue life as a function of plastic strain agrees with the Coffin–Manson relation without a priori assumption.

The concept of Representative Crack Elements (RCE) for phase-field fracture: transient thermo-mechanics

The phase-field formulation for fracture based on the framework of representative crack elements is extended to transient thermo-mechanics. The finite element formulation is derived starting from the variational principle of total virtual power. The intention of this manuscript is to demonstrate the potential of the framework for multi-physical fracture models and complex processes inside the crack. The present model at hand allows to predict realistic deformation kinematics and heat fluxes at cracks. At the application of fully coupled, transient thermo-elasticity to a pre-cracked plate, the opened crack yields thermal isolation between both parts of the plate. Inhomogeneous thermal strains result in a curved crack surface, inhomogeneous recontact and finally heat flow through the crack regions in contact. The novel phase-field framework further allows to study processes inside the crack, which is demonstrated by heat radiation between opened crack surfaces. Finally, numerically calculated crack paths at a disc subjected to thermal shock load are compared to experimental results from literature and a curved crack in a three-dimensional application are presented.

Iteratively Coupled Flow and Geomechanics in Fractured Poroelastic Reservoirs: A Phase Field Fracture Model

Fluid-solid coupling in fractured reservoirs plays a critical role for optimizing and managing in energy and geophysical engineering. Computational difficulties associated with sharp fracture models motivate phase field fracture modeling. However, for geomechanical problems, the fully coupled hydromechanical modeling with the phase field framework is still under development. In this work, we propose a fluid-solid fully coupled model, in which discrete fractures are regularized by the phase field. Specifically, this model takes into account the complex coupled interaction of Darcy-Biot-type fluid flow in poroelastic media, Reynolds lubrication governing flow inside fractures, mass exchange between fractures and matrix, and the subsequent geomechanical response of the solid. An iterative coupling method is developed to solve this multifield problem efficiently. We present numerical studies that demonstrate the performance of our model.

Permeability Measurements of 3D Microstructures Generated by Phase Field Simulation of the Solidification of an Al-Si Alloy during Chill Casting

The permeability of the semi-solid state is important for the compensation of volume shrinkage during solidification, since insufficient melt feeding can cause casting defects such as hot cracks or pores. Direct measurement of permeability during the dynamical evolution of solidification structures is almost impossible, and numerical simulations are the best way to obtain quantitative values. Equiaxed solidification of the Al-Si-Mg alloy A356 was simulated on the microscopic scale using the phase field method. Simulated 3D solidification structures for different stages along the solidification path were digitally processed and scaled up to generate 3D models by additive manufacturing via fused filament fabrication (FFF). The Darcy permeability of these models was determined by measuring the flow rate and the pressure drop using glycerol as a model fluid. The main focus of this work is a comparison of the measured permeability to results from computational fluid flow simulations in the phase field framework. In particular, the effect of the geometrical constraint due to isolated domain walls in a unit cell with a periodic microstructure is discussed. A novel method to minimize this effect is presented. For permeability values varying by more than two orders of magnitude, the largest deviation between measured and simulated permeabilities is less than a factor of two.

Controlling Equilibrium Morphologies of Bimetallic Nanostructures Using Thermal Dewetting via Phase-Field Modeling.

Herein, we report a computational model for the morphological evolution of bimetallic nanostructures in a thermal dewetting process, with a phase-field framework and superior optical, physical, and chemical properties compared to those of conventional nanostructures. The quantitative analysis of the simulation results revealed nano-cap, nano-ring, and nano-island equilibrium morphologies of the deposited material in thermal dewetting, and the morphologies depended on the gap between the spherical patterns on the substrate, size of the substrate, and deposition thickness. We studied the variations in the equilibrium morphologies of the nanostructures with the changes in the shape of the substrate pattern and the thickness of the deposited material. The method described herein can be used to control the properties of bimetallic nanostructures by altering their equilibrium morphologies using thermal dewetting.

A continuum theory for mineral solid solutions undergoing chemo-mechanical processes

Recent studies on metamorphic petrology as well as microstructural observations suggest the influence of mechanical effects upon chemically active metamorphic minerals. Thus, the understanding of such a coupling is crucial to describe the dynamics of geomaterials. In this effort, we derive a thermodynamically consistent framework to characterize the evolution of chemically active minerals. We model the metamorphic mineral assemblages as a solid-species solution where the species mass transport and chemical reaction drive the stress generation process. The theoretical foundations of the framework rely on modern continuum mechanics, thermodynamics far from equilibrium, and the phase-field model. We treat the mineral solid solution as a continuum body, and following the Larché and Cahn network model, we define displacement and strain fields. Consequently, we obtain a set of coupled chemo-mechanical equations. We use the aforementioned framework to study single minerals as solid solutions during metamorphism. Furthermore, we emphasise the use of the phase-field framework as a promising tool to model complex multi-physics processes in geoscience. Without loss of generality, we use common physical and chemical parameters found in the geoscience literature to portrait a comprehensive view of the underlying physics. Thereby, we carry out 2D and 3D numerical simulations using material parameters for mineral solid solutions to showcase and verify the chemo-mechanical interactions of mineral solid solutions that undergo spinodal decomposition, chemical reactions, and deformation.

A general phase-field framework for predicting the structures and micromechanical properties of crystalline defects

This work provides a phase-field simulation framework that bridges the structure of defects (e.g., dislocations and grain boundaries (GBs)) to their characteristic properties, such as stresses and energies. The validity of the current methodology is examined first by predicting the stress field of a single infinitely long screw dislocation using both the analytical solutions based on anisotropic elasticity and the current phase-field framework. The well-known stress singularity associated with the dislocation core in the former method has been effectively avoided in the latter. The framework is then applied to predicting the dislocation network of {0 0 0 1} twisted GB in Mg, which is found to consist of triangular-shaped regions of stacking faults and perfect crystals separated by partial dislocations. This prediction is consistent with some existing atomistic simulations and the underlying formation mechanism is analyzed rigorously using the displacement field predicted by our model, revealing the energy minimization process via the dissociation of a 〈112¯0〉/3 screw dislocation into a pair of 〈1100〉/3 screw dislocations. Based on the structure prediction, the associated stress field is further simulated, which provides critical information in evaluating the interaction of GBs and other crystalline defects such as impurities and dislocations.

Hybrid lattice-Boltzmann finite-difference simulation of ternary fluids near immersed solid objects of general shapes

In this paper, a hybrid lattice-Boltzmann finite-difference method is developed for the simulation of ternary fluids near immersed solid objects of general shapes. The flow equations are solved by the lattice-Boltzmann method and the coupled Cahn–Hilliard equations for interface evolutions are solved by the finite-difference method. A special implementation of the wetting boundary condition on a surface of general shapes immersed inside the domain was extended for ternary fluids within the phase-field framework with no need to use complicated interpolations. Several two and three dimensional problems with three immiscible fluids were studied by using the proposed method and the results agree well with analytical predictions and/or previous numerical and experimental studies. In particular, the inclusion of properly chosen free energy to handle total spreading enabled us to numerically reproduce the encapsulation of a small droplet by another bigger one of different component on a round fiber. The proposed method is expected to be useful to investigate a variety of multiphase problems involving ternary fluids and surfaces with different configurations, including the challenging total spreading regime.

A rate-dependent phase-field framework for the dynamic failure of quasi-brittle materials

In recent years, phase-field theory has become an efficient approach for predicting damage and fracture of engineering materials. However, the rate-dependence of quasi-brittle materials is not considered in most exiting models. To address this issue, we develop a new phase-field damage model for the dynamic failure of quasi-brittle solids based on the microforce balance law, within which a linear viscoelastic constitutive relation in effective stress space and a hyperbolic phase-field evolution equation are incorporated. Then several representative numerical examples are presented to demonstrate the ability of the proposed model in characterizing the dynamic failure process of quasi-brittle solids. Good agreements are achieved between numerical predictions and theoretical and experimental results. In particular, the increase of tensile strength and the transition of failure modes of concrete-like materials with the increase of loading rates can be well reproduced.

Unified phase field model to simulate both intergranular and transgranular failure in polycrystalline aggregates

In this work we develop a new phase field method for modeling fracture in polycrystalline materials. Many studies addressed this problem by combining cohesive models at the grain boundaries and damage models for the bulk. In this paper, a unified formulation is proposed in the phase field framework. Anisotropic failure is considered within the grain. Each of the preferential failure direction is associated with a damage variable. An additional damage variable is dedicated to the grain boundaries which are considered of finite thickness. Furthermore the value of the fracture energy at the grain boundaries is set depending on the local misorientation. The obtained model allows modeling competition and interactions between intergranular and transgranular fracture. The proposed model is illustrated through several numerical examples including crack initiation and propagation in polycrystalline aggregates. • Modeling both intergranular and transgranular fractures in polycrystalline materials. • Implementation of a damage variable dedicated to grain boundaries. • Implementation of effects of misorientation on grain boundaries fracture energy. • Numerical correction of the phase field method directionality formalism.

CALPHAD-informed phase-field modeling of grain boundary microchemistry and precipitation in Al-Zn-Mg-Cu alloys

The grain boundary (GB) microchemistry and precipitation behaviour in high-strength Al-Zn-Mg-Cu alloys has an important influence on their mechanical and electrochemical properties. Simulation of the GB segregation, precipitation, and solute distribution in these alloys requires an accurate description of the thermodynamics and kinetics of this multi-component system. CALPHAD databases have been successfully developed for equilibrium thermodynamic calculations in complex multi-component systems, and in recent years have been combined with diffusion simulations. In this work, we have directly incorporated a CALPHAD database into a phase-field framework, to simulate, with high fidelity, the complex kinetics of the non-equilibrium GB microstructures that develop in these important commercial alloys during heat treatment. In particular, the influence of GB solute segregation, GB diffusion, precipitate number density, and far-field matrix composition, on the growth of a population of GB η-precipitates, was systematically investigated in a model Al-Zn-Mg-Cu alloy of near AA7050 composition. It is shown that the GB solute distribution in the early stages of ageing was highly heterogeneous and strongly affected by the distribution of GB η-precipitates. Significant Mg and Cu GB segregation was predicted to remain during overageing, while Zn was rapidly depleted. This non-trivial GB segregation behaviour markedly influenced the resulting precipitate morphologies, but the overall precipitate transformation kinetics on a GB were relatively unaffected. Furthermore, solute depletion adjacent to the GB was largely determined by Zn and Mg diffusion, which will affect the development of precipitate free zones during the early stages of ageing. The simulation results were compared with scanning transmission electron microscopy and atom probe tomography characterisation of alloys of the similar composition, with good agreement.

A framework for phase-field modeling of interfacial debonding and frictional slipping in heterogeneous composites

Modeling and simulating interfacial shear load transfer in fiber-reinforced composites is crucial for characterizing fiber pullout behaviors. Addressing this complicated topic, this work presents a novel phase-field framework to simulate interfacial failure behaviors in fiber-reinforced cementitious composites. Specifically, a comprehensive constitutive model is proposed to describe both interfacial debonding and frictional slipping, which accounts for snubbing and bonding effects in fiber-reinforced composites. The developed model has the following novel features: (1) The model can simultaneously characterize interfacial debonding and frictional slipping, and thus thoroughly describe the fiber-reinforcing mechanisms in quasi-brittle solids; (2) An interfacial slipping degradation theory is implemented in the model to describe interfacial slippage and stress degradation during slipping; and (3) The model effectively and precisely incorporates the friction law with interface normal vectors. The developed model is successfully validated against experimental and simulated results. Finally, a series of parameter studies are undertaken to provide strategic insights to improve the mechanical properties of the composites. The proposed method adds efficiency and robustness to the study of features and fracture mechanisms of fiber reinforcements in complex fiber–cement systems.

Oscillatory and tip-splitting instabilities in 2D dynamic fracture: The roles of intrinsic material length and time scales

Recent theoretical and computational progress has led to unprecedented understanding of symmetry-breaking instabilities in 2D dynamic fracture. At the heart of this progress resides the identification of two intrinsic, near crack tip length scales — a nonlinear elastic length scale ℓ and a dissipation length scale ξ — that do not exist in Linear Elastic Fracture Mechanics (LEFM), the classical theory of cracks. In particular, it has been shown that at a propagation velocity v of about 90% of the shear wave-speed, cracks in 2D brittle materials undergo an oscillatory instability whose wavelength varies linearly with ℓ, and at larger loading levels (corresponding to yet higher propagation velocities), a tip-splitting instability emerges, both in agreements with experiments. In this paper, using phase-field models of brittle fracture, we demonstrate the following properties of the oscillatory instability: (i) It exists also in the absence of near-tip elastic nonlinearity, i.e. in the limit ℓ→0, with a wavelength determined by the dissipation length scale ξ. This result shows that the instability crucially depends on the existence of an intrinsic length scale associated with the breakdown of linear elasticity near crack tips, independently of whether the latter is related to nonlinear elasticity or to dissipation. (ii) It is a supercritical Hopf bifurcation, featuring a vanishing oscillations amplitude at onset. (iii) It is largely independent of the phenomenological forms of the degradation functions assumed in the phase-field framework to describe the cohesive zone, and of the velocity-dependence of the fracture energy Γ(v) that is controlled by the dissipation time scale in the Ginzburg–Landau-type evolution equation for the phase-field. These results substantiate the universal nature of the oscillatory instability in 2D. In addition, we provide evidence indicating that the tip-splitting instability is controlled by the limiting rate of elastic energy transport inside the crack tip region. The latter is sensitive to the wave-speed inside the dissipation zone, which can be systematically varied within the phase-field approach. Finally, we describe in detail the numerical implementation scheme of the employed phase-field fracture approach, allowing its application in a broad range of materials failure problems.

Phase field model for multiphase alloys under arbitrary thermal history: An application to IN718 super-alloy

Currently, it is still a significant challenge to predict the microstructural evolution of multiphase alloys in complex manufacturing processes such as joining, welding and additive manufacturing due to their complex thermodynamics and thermal history. In this work, we present a phase-field framework, aiming to predict the evolution of precipitate microstructure under arbitrary thermal history. Our model incorporates nucleation, growth and transformation by taking into account interfacial, chemical and elastic energies. In order to quantitatively describe the nucleation of phases, we leverage on experimental time-temperature-transformation data to parameterize the classical nucleation model. For demonstration, we employ super-alloy IN718, where thermodynamic and kinetic data is obtained using CALPHAD method and major precipitate phases, i.e. γ′,γ″ and δ and their variants are considered. The predicted continuous-cooling-transformation diagram is consistent with data existing in literature, thus validating the approach. We further predict the precipitate evolution under rapid thermal cycling to mimic a typical additive manufacturing process.