Guidelines for optimizing the architecture of battery insertion electrodes based on the concept of wiring lengths

The kinetics of storing mass in a battery electrode are typically limited by slow diffusion in storage particles. The diffusion timescale can be made faster by decreasing the size of the particles, but then it becomes more difficult to efficiently contact each particle with ionic and electronic current collectors, e.g., electrolyte and carbon. To achieve an optimal balance, the dimensions of the various phases in the electrode architecture should be tuned to the transport properties of the storage phase. Here we quantify this strategy by modeling the kinetics of galvanostatic charging for several particle geometries using the Nernst-Planck formalism and assuming mass storage via a solid solution. We show that when ions and electrons are inserted at separate contact surfaces, in general the storage kinetics depend on two length scales - the ionic and electronic wiring lengths - that characterize the transport distances within the storage material to the respective current collectors. Quantitative guidelines for the optimal wiring lengths are derived for two model geometries, and the dependence on transport parameters, particle shape, and contact geometry is discussed. These results can guide the optimization of various aspects of the architecture of a battery electrode, including the size and shape of individual particles and the configuration of the electrolyte and current collector networks.