Nonlinear Phase-field Model Research Articles

A nonlinear phase-field model of corrosion with charging kinetics of electric double layer

Abstract A nonlinear phase-field model is developed to simulate corrosion damage. The motion of the electrode-electrolyte interface follows the usual kinetic rate theory for chemical reactions based on the Butler-Volmer equation. The model links the surface polarization variation associated with the charging kinetics of an electric double layer (EDL) to the mesoscale transport. The effects of the EDL are integrated as a boundary condition on the solution potential equation. The boundary condition controls the magnitude of the solution potential at the electrode$-$electrolyte interface. The ion concentration field outside the EDL is obtained by solving the electro$-$diffusion equation and Ohm's law for the solution potential. The model is validated against the classic benchmark pencil electrode test. The framework developed reproduces experimental measurements of both pit kinetics and transient current density response. The model enables more accurate information on corrosion damage, current density, and environmental response in terms of the distribution of electric potential and charged species. The sensitivity analysis for different properties of the EDL is performed to investigate their role in the electrochemical response of the system. Simulation results show that the properties of the EDL significantly influence the transport of ionic species in the electrolyte.

Analysis of Pressure Characteristics of Ultra-High Specific Energy Lithium Metal Battery for Flying Electric Vehicles

The lithium metal battery is likely to become the main power source for the future development of flying electric vehicles for its ultra-high theoretical specific capacity. In an attempt to study macroscopic battery performance and microscopic lithium deposition under different pressure conditions, we first conduct a pressure cycling test proving that amplifying the initial preload can delay the battery failure stage, and the scanning electron microscope (SEM) shows that the pressure is effective in improving the electrode’s surface structure. Secondly, we analyze how differing pressure conditions affect the topography of lithium deposits by coupling the nonlinear phase-field model with the force model. The results show that the gradual increase in the external pressure is accompanied by a drop in the length of the dendrite and the migration curvature in the diaphragm, and the deposition morphology is gradually geared towards smooth and thick development, which can significantly reduce the specific surface area of lithium dendrite. However, as cyclic charging and discharging continue, the decrease in the electrolyte diffusion coefficient results in higher internal stress inside the battery, and thus the external pressure must be increased so as to achieve marked inhibitory effects on the growth of the lithium dendrite.

Fully decoupled energy-stable numerical schemes for two-phase coupled porous media and free flow with different densities and viscosities

In this article, we consider a phase field model with different densities and viscosities for the coupled two-phase porous media flow and two-phase free flow, as well as the corresponding numerical simulation. This model consists of three parts: a Cahn–Hilliard–Darcy system with different densities/viscosities describing the porous media flow in matrix, a Cahn–Hilliard–Navier–Stokes system with different densities/viscosities describing the free fluid in conduit, and seven interface conditions coupling the flows in the matrix and the conduit. Based on the separate Cahn–Hilliard equations in the porous media region and the free flow region, a weak formulation is proposed to incorporate the two-phase systems of the two regions and the seven interface conditions between them, and the corresponding energy law is proved for the model. A fully decoupled numerical scheme, including the novel decoupling of the Cahn–Hilliard equations through the four phase interface conditions, is developed to solve this coupled nonlinear phase field model. An energy-law preservation is analyzed for the temporal semi-discretization scheme. Furthermore, a fully discretized Galerkin finite element method is proposed. Six numerical examples are provided to demonstrate the accuracy, discrete energy law, and applicability of the proposed fully decoupled scheme.

A modified Crank-Nicolson finite difference method preserving maximum-principle for the phase-field model

In this paper, we mainly study a new Crank-Nicolson finite difference (FD) method with a large time step for solving the nonlinear phase-field model with a small parameter disturbance. To this end, we first introduce an artificial stability term to build a modified Crank-Nicolson FD (MCNFD) scheme, and then prove that the MCNFD scheme satisfies the intrinsic maximum principle of the phase-field model. Secondly, we discuss that the MCNFD scheme satisfies the intrinsic energy stability of the phase-field model, and obtain the optimal error estimation of the MCNFD solution based on the conclusion of the discrete maximum principle. Finally, two numerical examples of one-dimensional and two-dimensional phase-field models are given to verify that the solution of the MCNFD scheme satisfies the above theoretical results.

Nonlinear Phase-Field Modeling of Lithium Dendritic Growth during Electrodeposition

Lithium batteries have been widely used, but the growth of lithium dendrites does lead to some hazards, such as internal short circuit, thermal runaway and even explosion. In paper, a nonlinear phase field model combined with Butler-Volmer expression was developed to study dendrite growth during electrodeposition. Two types of initial electrode-electrolyte interface are applied in numerical simulations. Results show that the cation concentration enrichment occurs near the dendritic tip due to the electromigration, and the curve of the growth velocity versus time follows a simple power law relation, indicating that the stability of the dendritic array originating from initial electrode-electrolyte interface can be broken with the applied electrical potential increasing. Moreover, we found that applied electric potential gradient can strongly influence the dendritic growth velocity and the dendritic growth during electrodeposition is less relative to the surface tension anisotropy.

(Invited) Atomically-Informed Phase-field Modeling of Li and Mg Electrodeposition Morphologies

Exploring the origin of non-dendritic Mg electrodeposition might provide a new prospect of realizing the Li metal anode application in next-generation batteries. To fundamentally understand the morphological difference between Li and Mg plating, we hereby developed a multiscale model for a general metal electrodeposition process based on the phase-field method and transition state theory, connecting the atomic level charge-transfer physics to the mesoscale morphological evolution. At the atomic level, density functional theory (DFT) based half-cell calculations were used to predict the interfacial structures, de-solvation processes and Li+/Mg2+ ion energy landscapes at Li/SEI/ethylene carbonate (EC)-electrolyte and Mg/ tetrahydrofuran(THF)-electrolyte, and the effects of electric field on them. With thermodynamic and kinetic parametric inputs from DFT calculation (i.e. interfacial energy and anisotropy of Li/SEI (Li2CO3), surface energy and anisotropy of Mg, diffusion coefficient of Li+ in SEI (Li2CO3), and relative difference in charge-transfer energy barrier of Li/Mg plating), our non-linear phase-field model captures the morphological evolution difference between dendritic Li plating and faceted Mg plating. Furthermore, through systematical parametric analyses, we conclude that the cation desolvation-induced exchange current difference between Li and Mg is mainly responsible for their deposition morphological differences. This study provides a strategy of connecting the phase-field method and atomistic calculations and sheds lights on the dendrite-free battery anode design.

Prospect of Thermal Shock Induced Healing of Lithium Dendrite

Dendritic growth plagues the development of rechargeable lithium metal anodes. Recently, it has been reported that ( Science 2018, 359, 1513−1516) self-heating of the cell provides a mitigation strategy for suppressing dendrites. In order to study this phenomenon, we extend our recently developed nonlinear phase-field model to incorporate an energy balance equation allowing a full thermally coupled electrodeposition model using the open-source software package MOOSE. In this work, we consider the interplay between ionic transport and electrochemical reaction rate as a function of temperature and explore the possibility of using thermal shock induced dendrite suppression. We discover that, depending on the electrochemical reaction barrier and ionic diffusion barrier, self-heating could accelerate (larger reaction barrier) or decelerate (larger diffusion barrier) dendrite formation. Given that the electrolyte constituents can be used to tune both barriers, this study could provide an important avenue to exp...

Phase-Field Simulations of Lithium Dendrite Growth with Open-Source Software

Dendrite growth is a long-standing challenge that has limited the applications of rechargeable lithium metal electrodes. Here, we developed a grand potential based nonlinear phase-field model to study the electrodeposition of lithium as relevant for a lithium metal anode, using open-source software package MOOSE. The dynamic morphological evolution under large/small overpotential is studied in 2-dimensions, revealing important dendrite growth/stable deposition patterns. The corresponding temporal-spatial distributions of ion concentration, overpotential and driving force are studied, which demonstrate an intimate, dynamic competition between ion transport and electrochemical reactions, resulting in vastly different growth patterns. Given the importance of morphological evolution for lithium metal electrodes, wide-spread applications of phase-field models have been limited in part due to in-house or proprietary software. In order to spur growth of this field, we utilize an open-source software package and make all files available to enable future studies to study the many unsolved aspects related to morphology evolution for lithium metal electrodes.

Comparison of the Interfacial Reaction Kinetics and Plating Morphology of Lithium and Magnesium Anodes

Li dendrite growth is a critical safety issue for batteries with lithium metal anodes. Mg metal electrodes offer a promising way to overcome this safety limitation. Yao et.al. has observed that Li and Mg plating show different morphology, where Li plating is more dendritic but Mg plating is more faceted [1]. The difference of the morphology evolution of the plated Li and Mg is related to the interfacial reactions kinetics at the electrode and electrolyte interface. One major difference is that an electrical insulating but ionic conducting solid electrolyte interphase (SEI) always covers Li anode, but Mg electrode surface is believed to be SEI free[2]. Here, atomistic simulations are combined with phase-field model to evaluate the effect of interfacial reaction kinetics on the plated Li/Mg structural morphology. Reactivity of Li with liquid electrolytes, typically ethylene carbonate (EC), forms SEI layer on the Li metal. Magnesium aluminum chloride complex in tetrahydrofuran (THF) is one of the few electrolytes showing good reversible Mg deposition/stripping with a relatively large anodic stability (~3.1 V)[3]. Thus we considered Li/SEI/EC-electrolyte and Mg/THF-electrolyte interface models. Using both static first-principles calculations and ab initio molecular dynamics, the interfacial structures, de-solvation processes and Li+/Mg2+ ion energy landscapes at Li/SEI/EC-electrolyte and Mg/THF-electrolyte, and the effects of electric field on them were investigated. The preliminary results showed that the Li and Mg metal are all negatively charged at zero voltage, but the negative charge density of Li metal is more than that of Mg metal, which is due to SEI can block the electrons into electrolyte from anode. While the Mg plating free energy change for the electrode reaction is higher than Li due to its larger solvation energy (2.5 eV for Mg vs 1.2 eV for Li from thermodynamic cycle). Coupling with atomistic calculations, phase-field simulations were performed to investigate the Li/Mg plating morphological evolution from mesoscale. With thermodynamic and kinetic parametric inputs from first-principles calculation (i.e. interfacial energy and anisotropy of Li/SEI (Li2CO3), surface energy and anisotropy of Mg, diffusion coefficient of Li+ in SEI (Li2CO3), and relative difference in charge-transfer energy barrier of Li/Mg plating), our non-linear phase-field model captures the notable morphological evolution difference between dendritic Li plating and faceted Mg plating. Furthermore, the simulation results indicate that, among all possible parameters for Li/Mg electrodeposition processes, the plating charge-transfer energy barrier difference is the major contributing factor that accounts for the morphological discrepancy. Specifically, in our systematical tests on the impact of charge-transfer energy barrier on plating morphology (a range of 0.5-10 times of the Li plating reaction energy barrier height were tested), dendrites were prominently suppressed from around 2 times and faceted morphology (hexagonal for Mg) were produced at about 4 times. This study result fundamentally explains the origin of the Li/Mg plating morphological difference, and reveals that the proper elevation of the charge-transfer barrier height (e.g. by anodic surface treatment, protective coating, upgraded electrolyte, etc.) is a promising approach to dendrite suppression in Li metal batteries.

A Non-Linear Multi-Phase Field Model for Li Plating and Dendrite Growth at Li Metal Anode Incorporating Solid Electrolyte Interphase (SEI) Layer

Despite the highest specific capacity and lowest voltage of all secondary Li battery anode candidates, the application of Li metal anode has been a challenge due to dendritic Li electrodeposition in liquid electrolytes, leading to battery degradation and even safety issues. We previously generated a dendritic pattern map using a non-linear phase field model for a Li metal/liquid electrolyte binary system. However, the solid electrolyte interphase (SEI), the key to understand the mechanism of Li dendrite formation, is not explicitly included in the model. In this study, we hereby further advance our phase field model by incorporating the effect of SEI layer (or any artificial electrode protective coating layers (PCLs)) as an additional phase. A new set of phase-fields are re-defined to distinguish the spatial distribution of each individual phase in the electrode/SEI/electrolyte microstructure. Butler-Volmer type non-linear electrochemical kinetics and the SEI phase rigid body translation are tested to validate the new model. The impacts of different SEI properties (e.g. conductivity, Li diffusion coefficient, electron tunneling range, etc.) on Li plating and dendrite formation will be investigated. The possibility of incorporating the SEI mechanical effects on Li plating will also be discussed.

(Invited) Phase-Field Method of Li Dendrite Formation during Electrodeposition

The recent attempts to model the Li-dendrite formation using the phase-field method will be briefly reviewed. In particular, a nonlinear phase-field model, accounting for the Butler-Volmer electrochemical reaction kinetics, will be presented. The dendritic patterns are examined as a function of applied voltage and initial electrode surface morphology. A design map is proposed to tailor the electrode surface morphology and the applied voltage to avoid undesired dendritic patterns. The possibility of incorporating the effect of SEI on the Li-dendrite formation in a phase-field model will be discussed.

Modulation of dendritic patterns during electrodeposition: A nonlinear phase-field model

A nonlinear phase-field model, accounting for the Butler–Volmer electrochemical reaction kinetics, is developed to investigate the dendritic patterns during an electrodeposition process. Using lithium electrodeposition as an example, the proposed model is first verified by comparison with the Nernst equation in a 1D equilibrium system. The nonlinear electrochemical kinetics is also confirmed at non-equilibrium condition. The dendritic patterns are examined as a function of applied voltage and initial electrode surface morphology. A design map is proposed to tailor the electrode surface morphology and the applied voltage to avoid undesired dendritic patterns.

Nonlinear phase field model for electrodeposition in electrochemical systems

A nonlinear phase-field model has been developed for describing the electrodeposition process in electrochemical systems that are highly out of equilibrium. Main thermodynamic driving forces for the electrode-electrolyte interface (EEI) evolution are limited to local variations of overpotential and ion concentration. Application of the model to Li-ion batteries describes the electrode interface motion and morphology change caused by charge mass transfer in the electrolyte, an electrochemical reaction at the EEI and cation deposition on the electrode surface during the charging operation. The Li electrodeposition rate follows the classical Butler-Volmer kinetics with exponentially and linearly depending on local overpotential and cation concentration at the electrode surface, respectively. Simulation results show that the Li deposit forms a fiber-like shape and grows parallel to the electric field direction. The longer and thicker deposits are observed both for higher current density and larger rate constant where the surface reaction rate is expected to be high. The proposed diffuse interface model well captures the metal electrodeposition phenomena in plenty of non-equilibrium electrochemical systems.

Understanding and Predicting the Lithium Dendrite Formation in Li-Ion Batteries: Phase Field Model

Lithium (Li) dendrite formation compromises the reliability of Li-metal batteries, either because dendrite pieces lose electrical contract or growing dendrite penetrates the separator and leads to internal short-circuiting. In this paper, a nonlinear phase-field model is formulated to predict Li dendrite formation at the electrode/electrolyte interface. The phase field evolves by electrochemical reaction of which the rate depends on nonlinearly the thermodynamics driving force involving overpotential and ion concentration. A revised Poisson-Nesters-Planck Equation is further solved for ionic transport and local overpotential variation. The model is validated by 1-D fields distribution involving phase field, Lithium ion concentration and electrostatic potential. The 2-D tree-type lithium dendrite during Li deposition was produced if anisotropic surface energy is assumed. Finally, the 2D morphological evolution under different electrochemical conditions specified by the charging current density, and the anisotropy of surface energy was discussed.

Decoupled Energy Stable Schemes for a Phase-Field Model of Two-Phase Incompressible Flows with Variable Density

We consider in this paper numerical approximations of two-phase incompressible flows with different densities and viscosities. We present a variational derivation for a thermodynamically consistent phase-field model that admits an energy law. Two decoupled time discretization schemes for the coupled nonlinear phase-field model are constructed and shown to be energy stable. Numerical experiments are carried out to validate the model and the schemes for problems with large density and viscosity ratios.

Understanding and Predicting Li Dendrite Formation in Li-Ion Batteries: Phase Field Model

Lithium dendrite formation can lead the degradation and failure of Lithium ion battery, either because dendrite pieces lose electrical contract with the rest of the Li electrode (typically during Li stripping on discharge) or because growing dendrite can penetrate the separator and lead to internal short-circuiting. In this presentation, we will discuss a nonlinear phase-field model for the Li electrochemical deposition process to predict the conditions for dendrite formation along with their morphology. The model incorporates microstructure and evolution of the anode, the SEI (solid-electrolyte interphase layer) and the Li metal phase as functions of temperature, current density and history. It also describes the effect of electrode geometry on the nucleation and growth of Li-deposits. The simulation will suggest the optimum electrode geometry to prevent the Lithium dendrite formation. The 2D morphological evolution during Li deposition under different electrochemical conditions specified by charging current density, the ionic concentrations, reaction rate and applied potentials at the electrodes will be discussed.

Nonlinear phase-field model for electrode-electrolyte interface evolution

A nonlinear phase-field model is proposed for modeling microstructure evolution during highly nonequilibrium processes. We consider electrochemical reactions at electrode-electrolyte interfaces leading to electroplating and electrode-electrolyte interface evolution. In contrast to all existing phase-field models, the rate of temporal phase-field evolution and thus the interface motion in the current model is considered nonlinear with respect to the thermodynamic driving force. It produces Butler-Volmer-type electrochemical kinetics for the dependence of interfacial velocity on the overpotential at the sharp-interface limit. At the low overpotential it recovers the conventional Allen-Cahn phase-field equation. This model is generally applicable to many other highly nonequilibrium processes where linear kinetics breaks down.

Discontinuous Galerkin solution of a phase-field model in isothermal chemical vapor infiltration of SiC

The microstructure evolution of the isothermal chemical vapor infiltration (ICVI) of silicon carbide (SiC) from methyltrichlorosilane (MTS) is simulated by a phase-field model which is implemented numerically using the discontinuous Galerkin method. The model consists of a phase-field equation describing the diffusive interface kinetics and a species transport equation describing the mass balance. For the diffusive interface kinetics, a new form of the free energy density related to the ICVI process is adopted to ensure two stable states of gas phase and solid phase, and the Gibbs free energy changing between these two phases is taken as driving force. For the mass balance, two process intensities depending on the phase-field parameter are introduced to account for the homogeneous gas reaction and the heterogeneous surface reaction including the influence of the by-product hydrogen chloride (HCl). Finally, a discontinuous Galerkin formulation is derived for the nonlinear phase-field model and a priori error analysis is used for the semi-discontinuous formulations based on the proof of the uniqueness of the solution of the discontinuous approximation. Two numerical examples are presented to demonstrate the validity of the discontinuous Galerkin method and to investigate the effect of geometric parameters on microstructure evolution.

A Phase-Field Model and Its Numerical Approximation for Two-Phase Incompressible Flows with Different Densities and Viscosities

Modeling and numerical approximation of two-phase incompressible flows with different densities and viscosities are considered. A physically consistent phase-field model that admits an energy law is proposed, and several energy stable, efficient, and accurate time discretization schemes for the coupled nonlinear phase-field model are constructed and analyzed. Ample numerical experiments are carried out to validate the correctness of these schemes and their accuracy for problems with large density and viscosity ratios.

Phase Field Model of Thermo-Induced Marangoni Effects in the Mixtures and its Numerical Simulations with Mixed Finite Element Method

In this paper, we study the Marangoni effects in the mixture of two Newtonian fluids due to the thermo-induced surface tension heterogeneity on the interface. We employ an energetic variational phase field model to describe its physical phenomena, and obtain the corresponding governing equations defined by a modified Navier-Stokes equations coupled with phase field and energy transport. A mixed Taylor-Hood finite element discretization together with full Newton’s method are applied to this strongly nonlinear phase fieldmodel on a specific domain. Under different boundary conditions of temperature, the resulting numerical solutions illustrate that the thermal energy plays a fundamental role in the interfacial dynamics of two-phase flows. In particular, it gives rise to a dynamic interfacial tension that depends on the direction of temperature gradient, determining the movement of the interface along a sine/cosine-like curve. AMS subject classifications: 65B99, 65K05, 65K10, 65N12, 65N22, 65N30, 65N55, 65Z05