A consistent Lagrangian theory for phase separation in chemoelastic polymeric media at large deformations

Abstract

Large volume and shape changes may occur in polymer solutions and gels subject to mechanical loads or solvent loss. These deformations can lead to local alterations in composition, affecting or even triggering the phase separation into polymer-rich and solvent-rich phases. Studying microstructure evolution arising under these conditions thus necessitates a large strain formulation of the classical phase-field model for phase separation. To this end, a recent chemomechanical theory for biphasic media at large strains (Stracuzzi et al. ZAMM 8, 2018:2135–2154) was amended to incorporate the energetic contribution of interfaces between phase domains. The resulting model employs the well-established Cahn-Hilliard equation for diffusion and the Flory–Huggins mixing energy for polymer-solvent systems, expressed in a Lagrangian framework. The analysis of previous work on such referential reformulations of the original equations revealed different treatments, particularly with regard to the dependence of interfacial energy on deformation. In the present contribution, a consistent Lagrangian representation of the original model is rigorously derived, which entails a highly non-linear dependence of the interfacial energy on volume changes. The governing equations of this and a common alternative model were solved by a finite element approach, and were compared with regard to their predictions of microstructure formation in boundary value problems that lead to very large deformations, either induced by solvent evaporation or mechanical loads. In addition to the fully coupled chemomechanical theory for phase separation under large strains provided herein, the present work thus specifically highlights the importance of the constitutive model that specifies the free energy density associated with the formation of interfaces in the geometrically non-linear regime.