A linear-elastic-nonlinear-swelling theory for hydrogels. Part 1. Modelling of super-absorbent gels

Abstract

We introduce a new approach for modelling the swelling, drying and elastic behaviour of hydrogels, which leverages the tractability of classical linear-elastic theory whilst incorporating nonlinearities arising from large swelling strains. Relative to a reference state of a fully swollen gel, in which the polymer scaffold may only comprise less than $1\,\%$ of the total volume, a constitutive model for the Cauchy stress tensor is presented, which linearises around small deviatoric strains corresponding to shearing deformations of the material whilst allowing for a nonlinear relation between stress and isotropic strains. Such isotropic strains are considered only to be a consequence of losses and gains of water, while the hydrogel is taken to be instantaneously incompressible. The dynamics governing swelling and drying are described by coupling the interstitial flow of the water through the porous gel with the elastic response of the gel. This approach allows for a complete description of gel behaviour using only three macroscopic polymer-fraction-dependent parameters: an osmotic modulus, a shear modulus and a permeability. It is shown how these three material parameters can, in principle, be determined experimentally using a simple rheometry experiment in which a gel is compressed between two plates surrounded by water and the total force on the top plate is measured. To illustrate our approach, we solve for the swelling of a gel under horizontal confinement and for a partially dried hydrogel bead placed in water.

A theory of hydrogel mechanics that couples swelling and external flow

“…ref. 19, 31 and 64. Note also that σ s contributes to the overall force balance of the gel via the Terzaghi stress ϕ s σ s , whereas prior models often adopted an elastic stress tensor for the entire gel as a continuum.…”

“…19,24 The alternative description uses Darcy's law, making the flux proportional to the spatial gradient of an osmotic pressure, with a permeability coefficient. 20,25,30–32 Insofar as the chemical potential and the osmotic pressure are essentially synonymous, and the diffusivity and permeability are both phenomenological material properties on the continuum level, these two descriptions are equivalent. 10 Note, however, that such a Darcian description does not represent the Darcy flow in the classical sense, driven by an external pressure gradient.…”

A linear-elastic–nonlinear-swelling theory for hydrogels. Part 2. Displacement formulation

“…The dynamic viscosity of the interstitial water combines with these parameters to determine a basic slow diffusive poroelastic time scale (where is a characteristic length scale) over which water moves through the gel to cause swelling or drying. However, the diffusivity and the associated time scale are modified by deviatoric stresses and depend additionally on the dimensionless parameter representing the relative contribution of deviatoric stresses to the total stress compared with isotropic osmotic pressure (Webber & Worster 2023).…”

“…The development in recent years of super-absorbent polymers (Mignon et al 2019) has brought to the fore the importance of modelling hydrogels that can swell or dry to a much greater extent than seen in the earliest synthetic gels, taking on several hundred times their dry weight in water (Zohuriaan-Mehr et al 2010), involving swelling strains that are much larger than the approximately 10 % beyond which linear elastic theory is expected to be invalid (Landau & Lifshitz 1986). In Part 1 (Webber & Worster 2023), we introduced a new continuum-mechanical approach to the modelling of swelling and drying in super-absorbent hydrogels, allowing for large isotropic strains associated with extreme swelling but linearising with respect to small deviatoric strains. In effect, this treats a hydrogel swollen to any degree as an incompressible linear-elastic material, encapsulating the (potentially highly nonlinear) swelling behaviour of such materials while retaining the analytic tractability of linear poroelasticity (Doi 2009).…”

“…Combining this constitutive relation for the gel with polymer and water conservation, Cauchy's momentum equation and expressions for the interstitial flux of water, allows for the derivation of a polymer fraction evolution equation, dependent only on φ and the volume-averaged material flux vector q, equal to the sum of the interstitial fluid flux and polymer velocity. This strong formulation in terms of differential transport equations, derived from macroscopic conservation laws and expressed in terms of macroscopically measurable material properties, contrasts with energy minimisation approaches (Doi 2009;Cai & Suo 2012;Bertrand et al 2016) and nonlinear poroelasticity invoking large-deformation stress-strain relationships (MacMinn, Dufresne & Wettlaufer 2016), and is used in Webber & Worster (2023) to solve some uniaxial swelling problems. The importance of the deviatoric strain in determining effective diffusivities for swelling is underlined by the examples in Part 1, where post hoc justification is given for the assumption of small deviatoric strain, as it is shown that -given suitable choices of material properties -it is possible to swell to a large extent, introducing extreme isotropic strains, without the presence at any time of large deviatoric strains anywhere in the hydrogel.…”