Our research features diffusion in elastically deformable solids with complex microstructure. A prominent example are electrodes of Li-ion batteries which are typically a porous compound consisting of a binder, conductive particles, and active particles, filled with an electrolyte. The charge and discharge processes are complex multi-physics problems. Based on the used materials, they may include chemical reactions, electrochemically driven diffusion, separation into lithium poor and lithium rich regions in the active particles, or swelling phenomena [1]. Here, we consider a simplified version of this problem restricting ourselves to diffusion in an elastic matrix-particle composite. In the matrix, the transport of the mobile species is described using a Fickian-type flux driven by the gradient of the chemical potential. In the particles, the mobile species tends to accumulate and build separate regions of equal concentration – either high or low. This behavior is modeled using the Cahn-Hilliard equation.