Cahn Hilliard Phase Field Equation Research Articles

Dry granular column collapse: Numerical simulations using the partially regularized [formula omitted]-model via stabilized finite elements and phase field formulation

We revisit the gravitational collapse of a 2D column of dry granular material surrounded by air, using a continuum mechanics approximation. By employing the Cahn-Hilliard phase-field equation as an interface capturing technique and by coupling it with the Cauchy equation, we numerically simulate this multiphase system, eliminating the need for any ad-hoc numerical adjustment to prevent the finger formation of light fluid between the material and the solid boundary due to the no-slip boundary condition. We implement the μ(I)-rheology in our stabilized Finite Element method, highlighting the presence of instabilities when using this constitutive law. Our study is characterized by three main goals. First, we address the instability issue by implementing the partially regularized formulation of the μ(I)-rheology proposed by Barker and Gray (2017). An important outcome is that using shock-capturing terms in the momentum equation can significantly smooth these oscillations by adding dissipation in the direction of the gradients. Second, we systematically study the fluid dynamics under realistic conditions. Our results accurately replicate the material dynamics during collapse, confirming three distinct stages: free-fall, spreading, and cessation. We identify two regions in the material during the spreading phase: a quasi-static zone with negligible velocities and deformations, and a flowing layer exhibiting high shear rates. These observations closely align with experimental data. Additionally, we examine the evolution of the yielded/unyielded regions based on the Drucker-Prager criterion, and we also explore an empirical criterion, based on a critical value of the velocity norm, that satisfactorily separates these regions. Finally, we perform an extensive parametric study covering a wide range of rheological parameters.

Multiphysics Electrochemical Impedance Simulations of Complex Multiphase Electrodes

Electrochemical impedance spectroscopy (EIS) is a widely used technique for characterizing materials in electrochemical systems. However, directly connecting the obtained quantities to microstructure-level phenomena is challenging. In this work, we performed detailed electrochemical microstructure simulations to investigate the EIS behavior of phase-separating graphite electrodes. We employed the Cahn-Hilliard phase-field equation to model Li transport and phase transitions in the graphite particles. In single-phase graphite particles, the charge-transfer resistance reflected the total active surface areas. In two-phase coexistence graphite particles with phase boundaries present on the particle surfaces, the simulations exhibited an inductive loop on the EIS curve. In core-shell phase-morphology cases, the EIS measurements reflected only the properties of the shells. The resulting EIS curves were indistinguishable from those in the single-phase cases. While Fick's law of diffusion has been mistakenly employed to model Li transport in phase-separating graphite electrodes, our simulations showed that the EIS curves obtained using the Fickian diffusion model were very similar to those obtained using the Cahn-Hilliard phase-field model. This tool provides unprecedentedly detailed simulations to connect the intrinsic material properties, electrochemical processes in the microstructures, and resulting EIS behavior. Figure 1

Phase-field electrochemical simulations of reconstructed graphite electrodes

Graphite currently serves as the most predominant anode material in lithium-ion batteries due to its favorable balance of energy density, material cost, and relative safety in operations. While graphite anodes are manufactured mainly by randomly packing graphite particles, there is an avenue to improve the anode performance through optimized electrode microstructure. This is especially crucial for fast charging/discharging operations, where lithium plating is exaggerated and poses a critical safety concern. Therefore, the capability to investigate detailed electrochemical processes in electrode microstructures is essential for graphite anode designs. In this work, we incorporate the Cahn–Hilliard phase-field equation into a smoothed boundary method electrochemical simulation framework to account for phase transformations in graphite electrode complex microstructures upon lithiation/delithiation. These simulations quantitatively demonstrate that (i) modeling phase-separating graphite as a Li solid-solution material will overestimate graphite electrodes’ performance, (ii) particle size determines the electrode performance, (iii) pore tortuosity is important to enhance electrodes’ performance noticeably for thick electrodes, and (iv) introducing tunnels can spread insertion reaction more uniformly throughout the electrodes. Moreover, a substantial portion of this work focuses on simulating the behavior of thick electrodes. Thick electrode design holds a promising opportunity to increase overall energy density and our model enables computational predictions for such electrodes in an easy manner without conducting physical experiments. Thus, the presented simulation framework offers a fast and robust tool for designing electrode microstructures and operation protocols on digitally represented complex electrode microstructures. This approach has the potential to advance the electrode design of lithium-ion batteries.

Multiphysics Electrochemical Impedance Simulations of Complex Multiphase Graphite Electrodes

Electrochemical impedance spectroscopy (EIS) is a widely used technique to measure the properties of materials in electrochemical systems. However, the obtained quantities are difficult to directly connect to the microstructure-level phenomena. In this work, detailed electrochemical microstructure simulations were performed to investigate the EIS behavior of the phase-separating graphite electrodes. The Cahn–Hilliard phase-field equation was employed to model Li transport in the graphite particles. In graphite electrodes that were in single-phase stages, the obtained charge-transfer resistance reflected the total active surface areas. The effect of pore tortuosity, which dominates an electrode’s high-rate performance, cannot be reflected in the EIS behavior. In two-phase coexistence graphite electrodes, when phase boundaries were present on the particle surfaces, the simulations exhibited an inductive loop on the simulated EIS curve. In the core–shell phase-morphology cases, the EIS measurements reflected only the properties of the shells. The resulting EIS curves are indistinguishable from those in the single-phase cases. While Fick’s law of diffusion has been mistakenly employed to model Li transport in phase-separating graphite electrodes, our simulations showed that the EIS curves obtained using the Fickian diffusion model are very similar to those obtained using the Cahn–Hilliard phase-field model. This presented tool provides unprecedented detailed simulations to connect the intrinsic material properties, the electrochemical processes in the microstructures, and the resulting EIS behavior.

An anisotropic immerse precipitation process for the preparation of polymer membranes.

We study the immerse precipitation process in a ternary polymer/solvent/non-solvent system by numerically solving the two-dimensional Cahn-Hilliard phase field equation. In particular, we introduce anisotropic mobility, namely the mobility of a polymer varies over different spatial directions, and focus on the porosity morphology of the obtained polymer membrane. Simulations reveal that as the anisotropy increases in the polymer mobility, the polymer pattern changes from nearly isotropic and random voids to strips parallel to the direction with smaller mobility. The influence of anisotropy quickly saturates. The anisotropic mobility model is also applied to a ternary system mimicking the preparation of a hollow fiber membrane, and shows strong effects on the membrane porosity pattern.

Machine learning materials physics: Multi-resolution neural networks learn the free energy and nonlinear elastic response of evolving microstructures

Many important multi-component crystalline solids undergo mechanochemical spinodal decomposition: a phase transformation in which the compositional redistribution is coupled with structural changes of the crystal, resulting in dynamically evolving microstructures. The ability to rapidly compute the macroscopic behavior based on these detailed microstructures is of paramount importance for accelerating material discovery and design. Here, our focus is on the macroscopic, nonlinear elastic response of materials harboring microstructure. Because of the diversity of microstructural patterns that can form, there is interest in taking a purely computational approach to predicting their macroscopic response. However, the evaluation of macroscopic, nonlinear elastic properties purely based on direct numerical simulations (DNS) is computationally very expensive, and hence impractical for material design when a large number of microstructures need to be tested. A further complexity of a hierarchical nature arises if the elastic free energy and its variation with strain is a small-scale fluctuation on the dominant trajectory of the total free energy driven by microstructural dynamics. To address these challenges, we present a data-driven approach, which combines advanced neural network (NN) models with DNS to predict the homogenized, macroscopic, mechanical free energy and stress fields arising in a family of multi-component crystalline solids that develop microstructure. The microstructures are numerically generated by solving a coupled, mechanochemical spinodal decomposition problem governed by nonlinear strain gradient elasticity and the Cahn–Hilliard phase field equation. The hierarchical structure of the free energy’s evolution induces a multi-resolution character to the machine learning paradigm: We construct knowledge-based neural networks (KBNNs) with either pre-trained fully connected deep neural networks (DNNs), or pre-trained convolutional neural networks (CNNs) that describe the dominant characteristic of the data to fully represent the hierarchically evolving free energy. We demonstrate multi-resolution learning of the materials physics; specifically of the nonlinear elastic response for both fixed and evolving microstructures.

Convergence to equilibrium of global weak solutions for a Cahn–Hilliard–Navier–Stokes vesicle model

In this paper, we consider a model describing the dynamics of vesicle membranes within an incompressible viscous fluid in 3D domains. The system consists of the Navier–Stokes equations, with an extra stress tensor depending on the membrane, coupled with a Cahn–Hilliard phase-field equation associated with a bending energy plus a penalization related to the area conservation (volume is exactly conserved). This problem has a dissipative in time free energy which leads, in particular, to prove the existence of global in time weak solutions. We analyze the large-time behavior of the weak solutions. By using a modified Lojasiewicz–Simon’s result, we prove the convergence as time goes to infinity of each (whole) trajectory to a single equilibrium. Finally, the convergence of the trajectory of the phase is improved by imposing more regularity on the domain and initial phase.

A fractional step lattice Boltzmann model for two-phase flow with large density differences

In this paper, a fractional step lattice Boltzmann method is proposed to model two-phase flow with large density differences by solving a modified Cahn-Hilliard phase-field equation and the incompressible Navier-Stokes equations. In order to maintain the interface profile and conserve the volume, the normal flux in the original Cahn-Hilliard equation is removed and a profile-corrected term to enforce the interface to be a hyperbolic tangent profile is added. The modified Cahn-Hilliard equation is split to two sub-equations. One is the original CH equation and the other is the correction equation. Both the original CH and the hydrodynamic equations are solved by the lattice Boltzmann method. The correction equation is solved by the finite difference method. With the multi-relaxation-time collision model and a high-order compact selective filter operation employed in the lattice Boltzmann method, numerical stability is much improved. Compared with the previous lattice Boltzmann methods, the proposed method is able to maintain the order parameter within a physically meaningful range, which is capable of tracking the interface accurately. The proposed method can also simulate two-phase fluid flows with the density ratio up to 1000. Several benchmark problems, including single vortex deform of a circle, translation of a drop, stationary droplet, capillary wave and rising bubble with large density ratios, are presented to validate the accuracy and efficiency of the present method.

On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field Equation

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn-Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank-Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $1/\varepsilon$, where $\varepsilon$ is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes.

Numerical analysis of the Rayleigh–Taylor instability in an electric field

A numerical analysis is presented of the Rayleigh–Taylor instability (RTI) in the presence of an external electric field, with an emphasis on nonlinear phenomena associated with the evolution of complex interfacial morphology. The Poisson equation for the electric field and the Navier–Stokes equation for fluid flow field are solved simultaneously along with the Cahn–Hilliard phase field equation for interface deformation and morphology development. Numerical model is validated against the existing data and the results of linear analysis. Extensive numerical simulations are carried out for a wide range of fluid flow and electric field conditions. Computed results show that, in both linear and nonlinear regimes, a horizontal field suppresses the RTI, while a vertical electric field aggravates it. However, the vertical field does not affect the secondary instability; specifically, it does not contribute to the baroclinical generation of vorticity and consequently does not affect the roll-up formation. Linear analysis predicts that the RTI remains the same with the interchange of the dielectric constants of the two fluids, which is also confirmed by the numerical model for small interface deformations. This prediction, however, does not hold true in the nonlinear regimes in that complex interfacial morphology may evolve quite differently if the dielectric constants of two fluids are interchanged.

Homogenization of two fluid flow in porous media.

The macroscopic behaviour of air and water in porous media is often approximated using Richards' equation for the fluid saturation and pressure. This equation is parametrized by the hydraulic conductivity and water release curve. In this paper, we use homogenization to derive a general model for saturation and pressure in porous media based on an underlying periodic porous structure. Under an appropriate set of assumptions, i.e. constant gas pressure, this model is shown to reduce to the simpler form of Richards' equation. The starting point for this derivation is the Cahn–Hilliard phase field equation coupled with Stokes equations for fluid flow. This approach allows us, for the first time, to rigorously derive the water release curve and hydraulic conductivities through a series of cell problems. The method captures the hysteresis in the water release curve and ties the macroscopic properties of the porous media with the underlying geometrical and material properties.

Continuous finite element schemes for a phase field model in two-layer fluid Bénard–Marangoni convection computations

In this article, we study a phase field model for a two-layer fluid where the temperature dependence of both the density (buoyancy forces) and the surface tension (Marangoni effects) is considered. The phase field model consisting of a modified Navier–Stokes equation, a Cahn–Hilliard phase field equation and an energy transport equation is derived through an energetic variational procedure. An appropriate variational form and a continuous finite element method are adopted to maintain the underlying energy law to its greatest extent. A few examples for Bénard–Marangoni convection in an Acetonitrile and n-Hexane two-layer fluid system heated from above will be computed to justify our phase field model and further show the good performance of our methods. In addition, an interesting experiment will be performed to show the competition between the Marangoni effects and the buoyancy forces.

A Variational Model for Two-Phase Immiscible Electroosmotic Flow at Solid Surfaces

AbstractWe develop a continuum hydrodynamic model for two-phase immiscible flows that involve electroosmotic effect in an electrolyte and moving contact line at solid surfaces. The model is derived through a variational approach based on the On-sager principle of minimum energy dissipation. This approach was first presented in the derivation of a continuum hydrodynamic model for moving contact line in neutral two-phase immiscible flows (Qian, Wang, and Sheng, J. Fluid Mech. 564, 333-360 (2006)). Physically, the electroosmotic effect can be formulated by the Onsager principle as well in the linear response regime. Therefore, the same variational approach is applied here to the derivation of the continuum hydrodynamic model for charged two-phase immiscible flows where one fluid component is an electrolyte exhibiting electroosmotic effect on a charged surface. A phase field is employed to model the diffuse interface between two immiscible fluid components, one being the electrolyte and the other a nonconductive fluid, both allowed to slip at solid surfaces. Our model consists of the incompressible Navier-Stokes equation for momentum transport, the Nernst-Planck equation for ion transport, the Cahn-Hilliard phase-field equation for interface motion, and the Poisson equation for electric potential, along with all the necessary boundary conditions. In particular, all the dynamic boundary conditions at solid surfaces, including the generalized Navier boundary condition for slip, are derived together with the equations of motion in the bulk region. Numerical examples in two-dimensional space, which involve overlapped electric double layer fields, have been presented to demonstrate the validity and applicability of the model, and a few salient features of the two-phase immiscible electroosmotic flows at solid surface. The wall slip in the vicinity ofmoving contact line and the Smoluchowski slip in the electric double layer are both investigated.