Phase-field modeling of elastic microphase separation

We propose a novel phase-field model to predict elastic microphase separation in polymer gels. To this end, we extend the Cahn-Hilliard free-energy functional to incorporate an elastic strain energy and a coupling term. These contributions are naturally obtained from a derivation that starts from an entropic elastic energy density combined with the assumption of weak compressibility, upon second-order approximation around the swollen state. The resulting terms correspond to those of a poroelastic formulation where the coupling energetic term can be interpreted as the osmotic work of the solvent within the polymer matrix. Additionally, a convolution term is included in the total energy to model non-local forces responsible for coarsening arrest. With analytical derivations in 1D and finite element computations in 2D we show that the mechanical deformation controls the composition of the stable phases, the initial characteristic length and time, the coarsening rates and the arrested characteristic length. Moreover, we demonstrate that the proposed coupling is able to predict the arrest of coarsening at a length scale controlled by the stiffness of the dry polymer. The numerical results show excellent agreement with the experimental evidence in terms of phase-separated morphology and scaling of the characteristic length with the stiffness of the dry polymer.