A linear-elastic-nonlinear-swelling theory for hydrogels: displacements and differential swelling

The shapes of hydrogels as they swell or dry in one spatial dimension, for example when a bead of gel is placed in water, can be determined straightforwardly using polymer conservation. For problems in higher dimensions, we derive an expression for the displacement field of individual gel elements, which allows us to describe the shape of arbitrarily complicated gel geometries given the polymer-fraction field. In a result with parallels in classical linear elasticity, we find that the displacement field satisfies a biharmonic equation forced by gradients in the polymer fraction. As a demonstration, we investigate the drying of slender cylinders of hydrogel by evaporation into the air, with their bases submerged in reservoirs of water. At leading order, the gel locally contracts isotropically as it loses water, with the deviatoric shrinkage arising from differential drying remaining small. Experiments show the formation of a concave top surface and convex bottom surface, phenomena that are explained qualitatively by our model. We also show how our results are equivalent to a mathematical description of the cylinders as stacked disks, with each disk satisfying equations of classical plate theory and coupled dynamically to the disks above and below it.