A Phase-Field Model Coupled with Large Elasto-Plastic Deformation: Application to Lithiated Silicon Electrodes

Abstract

A phase-field model, accounting for large elasto-plastic deformation, is developed to study the evolution of phase, morphology and stress in crystalline silicon (Si) electrodes upon lithium (Li) insertion. The Li concentration profiles and deformation geometries are co-evolved by solving a set of coupled phase-field and mechanics equations using the finite element method. The present phase-field model is validated in comparison with a non-linear concentration-dependent diffusion model of lithiation in Si electrodes. It is shown that as the lithiation proceeds, the hoop stress changes from the initial compression to tension in the surface layer of the Si electrode, which may explain the surface cracking observed in experiments. The present phase-field model is generally applicable to high-capacity electrode systems undergoing both phase change and large elasto-plastic deformation. As a promising anode material for lithium (Li)-ion batteries, 1-3 the theoretical Li capacity of Silicon (Si) is 4200 mAh/g (corresponding to the lithiated phase of Li 4.4 Si), which is one order of magnitude larger than the commercialized graphite anode.1,2 Recent experiments revealed that the lithiation of crystalline Si (c-Si) occurs through a two-phase mechanism, i.e., growth of lithiated amorphous Li x Si (aLi x Si, x ∼ 3.75) phase separated from the unlithiated c-Si phase by a sharp phase boundary of about 1 nm thick.4-8 An abrupt change of Li concentration across the amorphous-crystalline interface (ACI) gives rise to drastic volume strain inhomogeneity. The resulting high stresses induce plastic flow, fracture, and pulverization of Si electrodes, thereby leading to the loss of electrical contact and limiting the cycle life of Li-ion batteries. [9][10][11] Electrochemically driven mechanical degradation in high-capacity electrodes has stimulated enormous efforts on the development of chemo-mechanical models to understand how the stress arises and evolves in lithiated Si electrodes.12-15 These chemo-mechanical models often treated the lithiation-induced stress as the diffusion-induced stress by considering Li diffusion in a solid-state electrode that results in the change of composition from its stoichiometric state. Deviation from stoichiometry usually results in a volume change that generates stress if the Li distribution is non-uniform. Early chemo-mechanical models only involved a unidirectional coupling. Namely, the diffusioninduced mechanical stress was considered, whereas the effect of mechanical stress on diffusion was ignored. Both experimental and computational studies, however, have shown that the mechanical stresses play an important role in the lithiation kinetics of Si electrodes. 4,6,16,17 Recently, fully coupled chemo-mechanical models were developed to incorporate the mechanical stress into the chemical potential. [18][19][20][21][22] In these models, the local stress modulates lithiation kinetics (reaction rate and diffusivity), 4,6,23 and in turn, lithiation kinetics regulates the stress generation in lithiated ...

Chemomechanical modeling of lithiation-induced failure in high-volume-change electrode materials for lithium ion batteries

“…In these regards, a phase field model may be advantageous over the discrete tracking method in finite element framework since phase field models naturally track the phase boundary with a controllable width. A recent phase field model 54 for simulating lithiation of c-Si demonstrated that the phase field model provides nearly equivalent results as the fronttracking finite element method.…”

“…In parallel to these experiments, models of different length scales have been established, ranging from first-principles simulations, [25][26][27][28][29][30][31][32][33][34][35][36] molecular dynamics with empirical force fields on the atomic scale, [37][38][39][40][41][42] continuum-level simulations that couple field equations dictating Li transport and mechanical equilibrium. 9,12,[43][44][45][46][47][48][49][50][51][52][53][54][55] Together, this has opened a bottom-up avenue for developing high-performance LIBs, in contrast to the top-down approach adopted by the conventional battery development.…”

Mitigating mechanical failure of crystalline silicon electrodes for lithium batteries by morphological design

“…The concurrent processes of Li transport, phase transformation and elasto-plastic deformation in Si anodes have inspired numerous modeling works [19][20][21][22][23][24][25] . Several groups developed models to explain the anisometric volume expansion of c-Si nanowires upon lithiation 14,18,24 .…”

Mechanistic Analysis of Mechano-Electrochemical Interaction in Silicon Electrodes with Surface Film

“…31,33 The Cahn-Hilliard formulation based two phase transport model has been utilized by Chen et al to model combined lithium diffusion and motion of the two-phase interface. 34 A similar formulation is adopted to capture two-phase diffusion in silicon in the current study.…”

“…39 Large elastic-plastic deformation of silicon has been incorporated in a few computational studies. 29,34 Lattice spring model based high volume expansion modeling has been shown to replicate the deformation of silicon and tin electrodes. 40,41 However, there are no detailed numerical analyses investigating the effect of the diffusion mechanism on fracture tendencies inside silicon particle with surface film mimicking solid electrolyte interphase or secondary phase layer.…”