Deterministic and stochastic analysis of the lithiation/delithiation dynamics of a cathode nanoparticle

We use asymptotic methods to analyse the effects of discreteness and random noise on a mathematical model of the lithiation/delithiation dynamics of a single electrode nanoparticle. We begin by presenting a simple ordinary differential equation model for the lithiation dynamics of a nanoparticle under voltage control, assuming that the nanoparticle is sufficiently small that internal diffusion and phase-separation can be neglected. We use the method of matched asymptotic expansions to show that the dynamics of this single particle system can be reduced to rapid switching between an ‘empty’ state and a ‘full’ state, and we identify the critical voltage at which the switch between states occurs under both quasi-equilibrium and out-of-equilibrium conditions. Next we present an alternative model of nanoparticle lithiation, assuming that a large but finite number of ‘lithiation states’ are available and that transitions between states are governed by the discrete chemical master equations. This leads to a discrete, stochastic model of lithiation that has the capacity to exhibit very different behavior from the original differential equation model. We use discrete-to-continuum asymptotic methods to analyse the discrete, stochastic model, and we identify the crucial dimensionless parameter that controls whether or not the discreteness and thermal noise play an important role in the dynamic behavior of the system.