Mathematical modelling of lithium intercalation dynamics in battery electrodes - Wang, C (2016)
This thesis discusses mathematical models for phase separation pattern in electrode materials for lithium ion batteries. The material is assumed to be composed of discrete "compartments", which might model individual particles in nanoparticle Lithium Iron Phosphate cathode material, for example, or individual layers in graphite anode material. We first present deterministic ODE models to describe quasi-equilibrium and out-of-equilibrium lithiation/delithiation of such systems. In sequence, we examine a single compartment under voltage-control, a multicompartment system under current-control, and a multicompartment system under voltage-control. We show that the dynamics of a single compartment can be reduced to rapid switching between an "empty" state and a "full" state, and we identify the critical voltage at which the switch occurs under both static and dynamic conditions. We also identify the critical voltages for the multicompartment system to change stage. We find that the multicompartment system supports a large number of stable states in quasiequilibrium, and we further reveal that its dynamics may be extremely sensitive to initial conditions and control parameters. We also explore how the dynamics are affected by discreteness and thermal noise by investigating the probability distribution of the lithium concentration. We first study a single-compartment and two-compartment systems, and then extend the analysis to more interacting compartments. We start with the discrete chemical master equations, and then derive a discrete-to-continuum model to explore the probability distribution during dynamic lithiation/delithiation. We identify distinct asymptotic regimes in which either discreteness and thermal noise are important or the dynamics is well captured by the deterministic model. We then compare the different phase-separation patterns predicted by the deterministic model and by the stochastic model, for quasi-equilibrium and out-of-equilibrium evolutions. Finally, we explore how different interaction laws could impact the observed phase-separation pattern.