A thermodynamically consistent gradient theory for diffusion–reaction–deformation in solids: Application to conversion-type electrodes

Abstract

We develop a thermodynamically consistent phase-field finite strain theory for problems in solids mechanics that couple transport of species into a host material, sharp interface reactions of the species with the host, mechanical deformation and stress. The theory distinguishes between diffusion-limited and reaction-limited kinetics, resolving the manner in which a sharp reaction front can be developed in either case. The phase field formulation has the added benefit of enabling the application of wetting (surface energy) boundary conditions which are critical in reproducing experimentally relevant reaction front morphologies. The theory is fully coupled with diffusion and reaction phenomena impacting mechanical deformation and subsequent stress generation, and conversely these phenomena are coupled to mechanical stress. We derive thermodynamically consistent driving forces for diffusion and reaction through a continuum treatment of these phenomena. Importantly, the resulting formulation makes precise the nature of the material properties driving these thermodynamic forces and in turn makes it amenable to being specialized and calibrated for application.While the framework is quite general, we apply it to modeling conversion electrodes for energy storage using a three-dimensional finite element implementation. We demonstrate the manner in which the theory can be specialized and calibrated in straightforward fashion. Simulations are performed of chemical reactions of FeS2 crystals with lithium and sodium ions, both of which proceed through the formation and propagation of a sharp interface, and are compared to experimental observations of the same system. Our simulations show good qualitative agreement with experimental observations, and elucidate the critical role mechanics plays in determining the morphology of the sharp reaction interface and subsequent stress generation which can lead to mechanical deterioration of these materials. Beyond this application, the theoretical framework should serve useful in a number of engineering problems of relevance in which diffusion and sharp interface reactions occur.