  We present a multiscale theoretical framework to investigate the interplay
between diffusion and finite lattice deformation in phase transformation
materials. In this framework, we use the Cauchy-Born Rule and the Principle of
Virtual Power to derive a thermodynamically consistent theory coupling the
diffusion of a guest species (Cahn-Hilliard type) with the finite deformation
of host lattices (nonlinear gradient elasticity). We adapt this theory to
intercalation materials--specifically Li$_{1-2}$Mn$_2$O$_4$--to investigate the
delicate interplay between Li-diffusion and the cubic-to-tetragonal deformation
of lattices. Our computations reveal fundamental insights into the
microstructural evolution pathways under dynamic discharge conditions, and
provide quantitative insights into the nucleation and growth of twinned
microstructures during intercalation. Additionally, our results identify
regions of stress concentrations (e.g., at phase boundaries, particle surfaces)
that arise from lattice misfit and accumulate in the electrode with repeated
cycling. These findings suggest a potential mechanism for structural decay in
Li$_2$Mn$_2$O$_4$. More generally, we establish a theoretical framework that
can be used to investigate microstructural evolution pathways, across multiple
length scales, in first-order phase transformation materials.
