Session G11: Porous Media: Hydrogels

A linear-elastic-nonlinear-swelling theory for hydrogels: constitutive relations and dynamics

Hydrophilic polymers can form hydrogels with polymer volume fractions of less than 1% when brought into contact with water, and the resulting gels find numerous uses in healthcare, agriculture and industry. To describe the behaviour of such super-absorbent gels, we allow for arbitrarily large isotropic strain whilst linearising around deviatoric strains that are assumed to be small, treating the hydrogel as an instantaneously incompressible linear-elastic material. Introducing nonlinearities only in the swelling strain allows us to formulate an analytically-tractable model built only on three macroscopically-measurable material parameters, each dependent on the polymer fraction: an osmotic pressure, a shear modulus and a permeability. We detail a conceptual rheometer to measure the quantitative relationships for all of these properties based on mechanical responses of a hydrogel under compression. Given these constitutive relations, we describe various swelling and drying processes – driven by the flow of water through the polymer matrix – to illustrate the utility of the modelling approach and the role of elastic stresses in determining boundary conditions and the effective diffusivity of a gel.   

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.   

Modeling osmotic swelling of chemically responsive hydrogels induced by non-uniform chemical signals

Experiments have shown that hydrogels that couple external chemical stimuli to their constituent polymers offer great potential for controlled shape transformations. For example, rapid release of a chemical stimulus (e.g., copper) trapped within a polyacrylic acid (PAA) hydrogel thin film upon arrival of a second stimulus (acid) can give rise to transient swelling. In contrast with swelling of most gels caused by the osmotic imbalance due to their polymer composition, we hypothesize that the transient swelling of the PAA gels must be driven by the temporary osmotic pressure accumulation upon rapid release of the trapped copper into the gel's fluid phase. Through an augmented 2-dimensional poroelastic theory, we simulate the mechanical response of the copper-complexed gel film to a non-uniform acid front, the subsequent dynamics of the released copper, and the resulting interstitial fluid flow that drives swelling. Our simulations confirm the emergence of traveling swelling fronts at the gel surface, in agreement with experiments. Overall, our theory elucidates the deformation dynamics of the PAA gel films, paving the way for their rational control by using chemical signals.   

A comparison of four boundary conditions for the fluid-hydrogel interface

In adopting a poroelastic model for a hydrogel, one views its constituent fluid and solid phases as interpenetrating continua, thereby erasing the ``pore-scale" geometry. This gives rise to the need for additional boundary conditions (BCs) at the interface between a hydrogel and a clear fluid to supplement the momentum equations for the fluid and solid phases in the hydrogel. Using a thermodynamic argument on energy dissipation, we propose three sets of boundary conditions for the gel-fluid interface that link the normal and tangential velocity jumps across the interface to the normal and tangential stresses on either side of the interface. Using several flow problems---one-dimensional compression, two-layer Couette and Poiseuille shear flows, and deformation of a gel particle by a planar extension flow---as tests, we compare the predictions of these three BCs with that of a previously proposed BC. Some differences are stark and reveal flaws in certain BCs. Others are subtler and will require quantitative experimental data for validation. Based on these results, we recommend one set of BCs over the other three for computing the flow and deformation of hydrogels in contact with a clear fluid. In addition, we suggest benchmark experiments to validate the BCs and our recommendation.