  We develop a continuum phase-field model for the simulation of diffusion
limited solid-solid phase transformations during lithium insertion in
LiFePO4-nano-particles. The solid-solid phase boundary between the LiFePO4
(LFP)-phase and the FePO4 (FP)-phase is modeled as a diffuse interface of
finite width. The model-description explicitly resolves a single
LiFePO4-particle, which is embedded in an elastically soft electrolyte-phase.
Furthermore, we explicitly include anisotropic (orthorhombic) and inhomogeneous
elastic effects, resulting from the coherency strain, as well as anisotropic
(1D) Li-diffusion inside the nano-particle. The effect of the nano-particle's
size on the kinetics of FP to LFP phase transformations is investigated by
means of both model. Both models predict a substantial increase in the steady
state transformation velocity as the particle-size decreases down to dimensions
that are comparable with the width of the interface between the FP and the
LFP-phase. However, the extra kinetic parameter of the Allen-Cahn-type
description may be used to reduce the strength of the velocity-increase with
the decreasing particle size. Further, we consider the influence of anisotropic
and inhomogeneous elasticity on the lithiation-kinetics within a rectangularly
shaped LiFePO4-particle embedded in an elastically soft electrolyte. Finally,
the simulation of equilibrium shapes of LiFePO4-particles is discussed. Within
a respective feasibility study, we demonstrate that also the simulation of
strongly anisotropic particles with aspect ratios up to 1/5 is possible.
