  Lithium iron phosphate (LiFePO$_4$) is the prototypical two-phase battery
material, whose complex patterns of lithium ion intercalation provide a testing
ground for theories of electrochemical thermodynamics. Using a depth-averaged
(a-b plane) phase-field model of coherent phase separation driven by Faradaic
reactions, we reconcile conflicting experimental observations of diamond-like
phase patterns in micron-sized platelets and surface-controlled patterns in
nanoparticles. Elastic analysis predicts this morphological transition for
particles whose a-axis dimension exceeds the bulk elastic stripe period. We
also simulate a rich variety of non-equilibrium patterns, influenced by
size-dependent spinodal points and electro-autocatalytic control of
thermodynamic stability.
