Abstract
Super-absorbent polymers can form hydrogels with polymer fractions of under 1% by volume when placed in water, with the water molecules being adsorbed by the hydrophilic polymer to form an elastic material. This water is not fixed in place: the polymer scaffold creates a porous structure through which the water can flow to drive swelling and drying processes. Existing studies of the behaviour of these two-phase materials tend to either rely on poroelasticity, coupling the interstitial flow with a constitutive relation to describe the deformation stresses on the gel, or use a microscopic chemical understanding of the water-polymer interactions, minimising the energy density to find both the steady and transient swelling states. In our work, we have derived a constitutive and dynamic model which allows for nonlinearity in the swelling strains but linearises around the macroscopic elastic behaviour of the gel, in effect treating such hydrogels as instantaneously linear-elastic. For one-dimensional problems, such as a gel sphere swelling in water, the swelling state can be described fully by the polymer fraction, which also gives the radial extent of the sphere through a polymer conservation constraint. However, in higher-dimensional problems, the polymer fraction alone cannot describe the gel, since there may be differential swelling in different directions. In such cases, we illustrate how an equation can be derived to describe the displacement field for the gel, which is seen to satisfy a modified biharmonic equation forced by the polymer fraction field, a direct analogue of the biharmonic equation for the displacement field as seen in linear elastostatics. Relying solely on the founding assumption of small deviatoric strains, we can determine the shape of a gel as it is allowed to swell and dry, as well as the transient state as water flows throughout. As an illustration of the utility of this approach, the problem of a cylinder, with its base immersed in water, drying in the air, is considered. In this situation, there is both radial shrinkage and shrinkage along the axis of the cylinder, with the top of the cylinder drying to a greater extent than the base as water is drawn up and evaporated away, with such a problem requiring more than just a polymer fraction for a full description. These modelling assumptions are shown to result in a description of the gel which both agrees with experiments and with a Lagrangian description of the differential drying of the cylinder as a series of stacked elastic plates. Experiments have shown that the cylinder becomes concave on its top surface and convex at its base before reaching a steady state, and both of these phenomena can be described using this displacement formulation, providing further evidence of its ability to solve these more complicated problems.