  Rechargeable batteries that incorporate shaped three-dimensional electrodes
have been shown to have increased power and energy densities for a given
footprint area when compared to a conventional geometry, i.e., a planar cathode
and anode that sandwich an electrolyte. Electrodes can be shaped to enable a
higher loading of active material, while keeping the ion transport distance
small, however, the relationship between electrical and mechanical performance
remains poorly understood. A variety of electrode shapes have been explored,
where the electrodes are individually shaped or intertwined with one another.
Advances in manufacturing and shape and topology optimization have made such
designs a reality. In this paper, we explore sinusoidal half cells and
interdigitated full cells. First, we use a simple electrostatics model to
understand the cell resistance as a function of shape. We focus on
low-temperature conditions, where the electrolyte conductivity decreases and
the governing dimensionless parameters change. Next, we use a chemo-mechanics
model to examine the stress concentrations that arise due to
intercalation-driven volume expansion. We show that shaped electrodes provide a
significant reduction in resistance, however, they result in unfavorable stress
concentrations. Overall, we find that the fully interdigitated electrodes may
provide the best balance with respect to this trade-off.
