A multiphase theory for transient osmotic swelling of chemically responsive hydrogels

The osmotic pressure due to a concentration difference of identical solute molecules across a semi-permeable membrane can be determined by van't Hoff's formula for chemical potential equilibration when the number of solute molecules remain constant on both sides of the membrane. This condition can be inherently relaxed at the interface between an aqueous supernatant domain and a chemically responsive hydrogel, when a chemical stimulus freed from inside the gel slowly diffuses into the supernatant while creating a dynamic osmotic pressure balanced by the poroelastic diffusion of water into the gel. We introduce a continuum poroelastic theory for the dynamic build-up and relaxation of osmotic pressure due to an interplay between copper cations as osmosis-driving agents and acid in a polyacrylic acid hydrogel thin film. Our theory relates the non-equilibrium osmotic pressure to the vertical gradients of the solute concentration across the interface. The theory quantitatively captures the osmosis induced swelling and contraction of the gel film, in agreement with experiments.