A microscopic model of a lithium battery is developed, which accounts for lithium diffusion within particles, transfer of lithium from particles to the electrolyte and transport within the electrolyte assuming a dilute electrolyte and Butler–Volmer reaction kinetics. Exploiting the small size of the particles relative to the electrode dimensions, a homogenised model (in agreement with existing theories) is systematically derived and studied. Details of how the various averaged quantities relate to the underlying geometry and assumptions are given. The novel feature of the homogenisation process is that it allows the coefficients in the electrode-scale model to be derived in terms of the microscopic features of the electrode (e.g. particle size and shape) and can thus be used as a systematic way of investigating the effects of changes in particle design. Asymptotic methods are utilised to further simplify the model so that one-dimensional behaviour can be described with relatively simpler expressions. It is found that for low discharge currents, the battery acts almost uniformly while above a critical current, regions of the battery become depleted of lithium ions and have greatly reduced reaction rates leading to spatially nonuniform use of the electrode. The asymptotic approximations are valid for electrode materials where the OCV is a strong function of intercalated lithium concentration, such as LixC6, but not for materials with a flat discharge curve, such as LiFePO4.
