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

Abstract

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 ButlerVolmer 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).

Interplay of phase boundary anisotropy and electro-auto-catalytic surface reactions on the lithium intercalation dynamics in LiXFePO4 plateletlike nanoparticles

“…The strongly anisotropic diffusivity of LiFePO 4 [46] favors diffusion along ionchannels in the direction of the b-axis, which is expressed in the model by diffusivity in the b direction which is six orders of magnitude higher than in the ac plane. Surface energy has a significant effect on the phasemorphology [20] and the performance [68] of Li X FePO4. Comparing ab initio computations of the surface energies of facets of LiFePO 4 and FePO 4 (exposed to vacuum) [63] and the LiFePO 4 / FePO 4 phase-boundary energy reveals that the bc and ab side-facets have a much lower energy for Li-rich phases and tend to be fully "wetted" with a surface layer of intercalated lithium (c ≈ 1).…”

“…It is important to note that the theory of suppressed phase separation at the surface also does not rule out the possibility of phase separation below the surface, where the surface overpotential cannot directly influence thermodynamic driving forces. Indeed, previous phase-field models of bulk phase separation with surface reactions, assuming 1D spherical symmetry [68,61,60,35], 2D spheroidal symmetry [26] or 2D planar symmetry (with concentration variations confined to the ab plane) [24], have predicted subsurface phase separation, albeit without including crystal anisotropy, elastic coherency strain, surface wetting and/or thermodynamically consistent reaction kinetics [7]. Some three-dimensional simulations have also been performed, although the computational expense limits the particle size, time-dependence and model complexity that can be considered [3,26,15].…”

Phase Transformation Dynamics in Porous Battery Electrodes

“…Recent work on LFP has shown that the voltage plateau is an emergent property of the active particles [11,[35][36][37][38][39][40][41] and the porous electrode [30,[42][43][44][45] for any material with multiple stable phases at different compositions. In single particles, phase separation occurs within the miscibility gap (range of unstable homoge-neous compositions), which depends on temperature and particle size [11,37], overpotential [11,[46][47][48], and current density [36,37].…”

Phase Separation Dynamics in Isotropic Ion-Intercalation Particles

“…In recent years, the shrinking-core hypothesis has been called into question because different phase behavior has been observed experimentally [48,17,1,30,19] and predicted theoretically [10]. It has become clear that a more realistic particle model must account for two-phase thermodynamics [39,65,50,49,80], crystal anisotropy [65,3,67], coherency strain [20], surface energy [21], and reaction limitation in nanoparticles [65,3,2], and electrochemical interactions between large numbers of such particles in porous electrodes [36,4,37,60]. In larger, micron-sized particles, the shrinking-core model may still have some relevance due to solid diffusion limitation and defects (such as dislocations and micro cracks) that can reduce coherency strain [65,12,26].…”