  In the lithium-ion battery literature, discharges followed by a relaxation to
equilibrium are frequently used to validate models and their parametrizations.
Good agreement with experiment during discharge is easily attained with a
pseudo-two-dimensional model such as the Doyle-Fuller-Newman (DFN) model. The
relaxation portion, however, is typically not well-reproduced, with the
relaxation in experiments occurring much more slowly than in models. In this
study, using a model that includes a size distribution of the active material
particles, we give a physical explanation for the slow relaxation phenomenon.
This model, the Many-Particle-DFN (MP-DFN), is compared against discharge and
relaxation data from the literature, and optimal fits of the size distribution
parameters (mean and variance), as well as solid-state diffusivities, are found
using numerical optimization. The voltage after relaxation is captured by
careful choice of the current cut-off time, allowing a single set of physical
parameters to be used for all C-rates, in contrast to previous studies. We find
that the MP-DFN can accurately reproduce the slow relaxation, across a range of
C-rates, whereas the DFN cannot. Size distributions allow for greater internal
heterogeneities, giving a natural origin of slower relaxation timescales that
may be relevant in other, as yet explained, battery behavior.
