Programming the equilibrium swelling response of heterogeneous polymeric gels

Heterogeneous gels are formed when heterogeneous polymer networks are submerged in a solvent. The heterogeneities stem from spatial variations in local properties such as the chain-density and the chain-length. In turn, these directly influence the local stiffness and the local solvent-mixture interaction parameter. Practically, heterogeneities in gels can arise spontaneously or be programmed into the network through polymerization mechanism and reaction conditions, thus allowing to control the macroscopic response. The aim of this work is to understand the influence of the heterogeneities on the equilibrium swelling response. To account for the response of heterogeneous gels, we employ a free energy that includes four contributions - the entropic energy of the deformed polymer network, the energy of mixing, the hydrostatic work of the external pressure exerted by the solvent, and a total pressure term stemming from the solvent molecules and the network in the gel. The equilibrium equations, the boundary conditions, and the continuity conditions across an interface in the heterogeneous gel are presented. To illustrate the merit of the proposed work, we study the influence of the stiffness and the polymer-mixture interaction parameter on the equilibrium swelling response of a two-phase polymeric gel block and a two-phase polymeric gel tube under different boundary conditions. Next, we examine the dependence of the local properties on the response of heterogeneous polymeric gel blocks and heterogeneous polymeric gel tubes that are characterized by two micro-structures: (1) a spatially varying stiffness and a constant interaction parameter and (2) a spatially varying interaction parameter and a constant stiffness. The proposed framework sheds light on the influence of local properties on the macroscopic behavior and provides a micro-structural design guide that can be used to program the equilibrium swelling response by appropriately distributing the heterogeneities in the polymer network.