Phase-field modeling of the particle size and average concentration dependent miscibility gap in nanoparticles of LixMn2O4, LixFePO4, and NaxFePO4 during insertion

Abstract The particle size and average concentration dependent miscibility gap of the nanoscale insertion materials Li x Mn 2 O 4 , Li x FePO 4 , and Na x FePO 4 are investigated during insertion, using a coupled phase-field model based on the Cahn-Hilliard theory and finite deformation elasticity. For the three cathode materials Li x Mn 2 O 4 , Li x FePO 4 , and Na x FePO 4 , a reduced miscibility gap in the center of the spinodal region of average concentration is found for radii below 55.5 nm, 20.5 nm, and 17.5 nm, respectively. For each material, there is a critical particle size below which phase segregation is even inhibited completely. The dependence of the miscibility gap on the particle size is explained by the ratio of the thickness of the interface between phases and the particle size. Namely, the miscibility gap shrinks to accommodate in particles of reduced size the concentration gradient in the interface, the thickness of which is basically determined by a material constant. Concerning the evolution of the miscibility gap during the process of insertion, it is found that for smaller particles it increases with increasing average concentration. However, for each of the three mentioned materials, a threshold value of the particle radius is determined, above which the miscibility gap is constant during the whole process of insertion. The average concentration dependent miscibility gap is physically explained by the related evolution of the system gradient energy. Among the three investigated cathode materials, although the tensile stresses are the smallest in Li x Mn 2 O 4 , the mechanics has the strongest influence on the miscibility gap of this material. The elastic strain energy leads to a reduction of the miscibility gap, a larger threshold value of the particle radius for the size independent miscibility gap, and a larger critical particle size below which phase segregation is completely inhibited. Thus, in view of future design of nanoscale storage particles, of the three investigated cathode materials, Li x Mn 2 O 4 possesses the largest particle size for expanding the solid solution range in which particle stresses are eliminated.