Phase Separation Dynamics in Isotropic Ion-Intercalation Particles

Lithium-ion batteries exhibit complex nonlinear dynamics, resulting from diffusion and phase transformations coupled to ion intercalation reactions. Using the recently developed Cahn-Hilliard reaction (CHR) theory, we investigate a simple mathematical model of ion intercalation in a spherical solid nanoparticle, which predicts transitions from solid-solution radial diffusion to two-phase shrinking-core dynamics. This general approach extends previous Li-ion battery models, which either neglect phase separation or postulate a spherical shrinking-core phase boundary, by predicting phase separation only under appropriate circumstances. The effect of the applied current is captured by generalized Butler-Volmer kinetics, formulated in terms of diffusional chemical potentials, and the model consistently links the evolving concentration profile to the battery voltage. We examine sources of charge/discharge asymmetry, such as asymmetric charge transfer and surface "wetting" by ions within the solid, which can lead to three distinct phase regions. In order to solve the fourth-order nonlinear CHR initial-boundary-value problem, a control-volume discretization is developed in spherical coordinates. The basic physics are illustrated by simulating many representative cases, including a simple model of the popular cathode material, lithium iron phosphate (neglecting crystal anisotropy and coherency strain). Analytical approximations are also derived for the voltage plateau as a function of the applied current

Interplay of phase boundary anisotropy and electro-autocatalytic surface reactions on the lithium intercalation dynamics in LiX_XFePO4_4 platelet-like nanoparticles

Experiments on single crystal Li$_X$FePO$_4$ (LFP) nanoparticles indicate rich nonequilibrium phase behavior, such as suppression of phase separation at high lithiation rates, striped patterns of coherent phase boundaries, nucleation by binarysolid surface wetting and intercalation waves. These observations have been successfully predicted (prior to the experiments) by 1D depth-averaged phase-field models, which neglect any subsurface phase separation. In this paper, using an electro-chemo-mechanical phase-field model, we investigate the coherent non-equilibrium subsurface phase morphologies that develop in the $ab$- plane of platelet-like single-crystal platelet-like Li$_X$FePO$_4$ nanoparticles. Finite element simulations are performed for 2D plane-stress conditions in the $ab$- plane, and validated by 3D simulations, showing similar results. We show that the anisotropy of the interfacial tension tensor, coupled with electroautocatalytic surface intercalation reactions, plays a crucial role in determining the subsurface phase morphology. With isotropic interfacial tension, subsurface phase separation is observed, independent of the reaction kinetics, but for strong anisotropy, phase separation is controlled by surface reactions, as assumed in 1D models. Moreover, the driven intercalation reaction suppresses phase separation during lithiation, while enhancing it during delithiation, by electro-autocatalysis, in quantitative agreement with {\it in operando} imaging experiments in single-crystalline nanoparticles, given measured reaction rate constants

Nonequilibrium Thermodynamics of Porous Electrodes

We reformulate and extend porous electrode theory for non-ideal active materials, including those capable of phase transformations. Using principles of non-equilibrium thermodynamics, we relate the cell voltage, ionic fluxes, and Faradaic charge-transfer kinetics to the variational electrochemical potentials of ions and electrons. The Butler-Volmer exchange current is consistently expressed in terms of the activities of the reduced, oxidized and transition states, and the activation overpotential is defined relative to the local Nernst potential. We also apply mathematical bounds on effective diffusivity to estimate porosity and tortuosity corrections. The theory is illustrated for a Li-ion battery with active solid particles described by a Cahn-Hilliard phase-field model. Depending on the applied current and porous electrode properties, the dynamics can be limited by electrolyte transport, solid diffusion and phase separation, or intercalation kinetics. In phase-separating porous electrodes, the model predicts narrow reaction fronts, mosaic instabilities and voltage fluctuations at low current, consistent with recent experiments, which could not be described by existing porous electrode models

Suppression of Phase Separation in LiFePO4 Nanoparticles During Battery Discharge

Using a novel electrochemical phase-field model, we question the common belief that LixFePO4 nanoparticles separate into Li-rich and Li-poor phases during battery discharge. For small currents, spinodal decomposition or nucleation leads to moving phase boundaries. Above a critical current density (in the Tafel regime), the spinodal disappears, and particles fill homogeneously, which may explain the superior rate capability and long cycle life of nano-LiFePO4 cathodes.Comment: 27 pages, 8 figure

Li-diffusion accelerates grain growth in intercalation electrodes: a phase-field study

Grain boundary migration is driven by the boundary's curvature and external loads such as temperature and stress. In intercalation electrodes an additional driving force results from Li-diffusion. That is, Li-intercalation induces volume expansion of the host-electrode, which is stored as elastic energy in the system. This stored energy is hypothesized as an additional driving force for grain boundaries and edge dislocations. Here, we apply the 2D Cahn-Hilliard$-$phase-field-crystal (CH-PFC) model to investigate the coupled interactions between highly mobile Li-ions and host-electrode lattice structure, during an electrochemical cycle. We use a polycrystalline FePO$_{4}$/ LiFePO$_{4}$ electrode particle as a model system. We compute grain growth in the FePO$_{4}$ electrode in two parallel studies: In the first study, we electrochemically cycle the electrode and calculate Li-diffusion assisted grain growth. In the second study, we do not cycle the electrode and calculate the curvature-driven grain growth. External loads, such as temperature and stress, did not differ across studies. We find the mean grain-size increases by $\sim11\%$ in the electrochemically cycled electrode particle. By contrast, in the absence of electrochemical cycling, we find the mean grain-size increases by $\sim2\%$ in the electrode particle. These CH-PFC computations suggest that Li-intercalation accelerates grain-boundary migration in the host-electrode particle. The CH-PFC simulations provide atomistic insights on diffusion-induced grain-boundary migration, edge dislocation movement and triple-junction drag-effect in the host-electrode microstructure.Comment: 11 pages, 9 figure

Cahn-Hilliard Reaction Model for Isotropic Li-ion Battery Particles

Using the recently developed Cahn-Hilliard reaction (CHR) theory, we present a simple mathematical model of the transition from solid-solution radial diffusion to two-phase shrinking-core dynamics during ion intercalation in a spherical solid particle. This general approach extends previous Li-ion battery models, which either neglect phase separation or postulate a spherical shrinking-core phase boundary under all conditions, by predicting phase separation only under appropriate circumstances. The effect of the applied current is captured by generalized Butler-Volmer kinetics, formulated in terms of the diffusional chemical potential in the CHR theory. We also consider the effect of surface wetting or de-wetting by intercalated ions, which can lead to shrinking core phenomena with three distinct phase regions. The basic physics are illustrated by different cases, including a simple model of lithium iron phosphate (neglecting crystal anisotropy and coherency strain).National Science Foundation (U.S.) (Graduate Research Fellowship Program under Grant No. 1122374)Samsung (Firm) (Samsung-MIT Alliance

Effects of Nanoparticle Geometry and Size Distribution on Diffusion Impedance of Battery Electrodes

The short diffusion lengths in insertion battery nanoparticles render the capacitive behavior of bounded diffusion, which is rarely observable with conventional larger particles, now accessible to impedance measurements. Coupled with improved geometrical characterization, this presents an opportunity to measure solid diffusion more accurately than the traditional approach of fitting Warburg circuit elements, by properly taking into account the particle geometry and size distribution. We revisit bounded diffusion impedance models and incorporate them into an overall impedance model for different electrode configurations. The theoretical models are then applied to experimental data of a silicon nanowire electrode to show the effects of including the actual nanowire geometry and radius distribution in interpreting the impedance data. From these results, we show that it is essential to account for the particle shape and size distribution to correctly interpret impedance data for battery electrodes. Conversely, it is also possible to solve the inverse problem and use the theoretical "impedance image" to infer the nanoparticle shape and/or size distribution, in some cases, more accurately than by direct image analysis. This capability could be useful, for example, in detecting battery degradation in situ by simple electrical measurements, without the need for any imaging.Comment: 30 page

Anisotropic surface reaction limited phase transformation dynamics in LiFePO4

A general continuum theory is developed for ion intercalation dynamics in a single crystal of a rechargeable battery cathode. It is based on an existing phase-field formulation of the bulk free energy and incorporates two crucial effects: (i) anisotropic ionic mobility in the crystal and (ii) surface reactions governing the flux of ions across the electrode/electrolyte interface, depending on the local free energy difference. Although the phase boundary can form a classical diffusive "shrinking core" when the dynamics is bulk-transport-limited, the theory also predicts a new regime of surface-reaction-limited (SRL) dynamics, where the phase boundary extends from surface to surface along planes of fast ionic diffusion, consistent with recent experiments on LiFePO4. In the SRL regime, the theory produces a fundamentally new equation for phase transformation dynamics, which admits traveling-wave solutions. Rather than forming a shrinking core of untransformed material, the phase boundary advances by filling (or emptying) successive channels of fast diffusion in the crystal. By considering the random nucleation of SRL phase-transformation waves, the theory predicts a very different picture of charge/discharge dynamics from the classical diffusion-limited model, which could affect the interpretation of experimental data for LiFePO4.Comment: 15 pages, 10 figure

Phase Transformation Dynamics in Porous Battery Electrodes

Porous electrodes composed of multiphase active materials are widely used in Li-ion batteries, but their dynamics are poorly understood. Two-phase models are largely empirical, and no models exist for three or more phases. Using a modified porous electrode theory based on non-equilibrium thermodynamics, we show that experimental phase behavior can be accurately predicted from free energy models, without artificially placing phase boundaries or fitting the open circuit voltage. First, we simulate lithium intercalation in porous iron phosphate, a popular two-phase cathode, and show that the zero-current voltage gap, sloping voltage plateau and under-estimated exchange currents all result from size-dependent nucleation and mosaic instability. Next, we simulate porous graphite, the standard anode with three stable phases, and reproduce experimentally observed fronts of color-changing phase transformations. These results provide a framework for physics-based design and control for electrochemical systems with complex thermodynamics