Effect of Elastocapillarity on the Swelling Kinetics of Hydrogels

The swelling behavior of hydrogels, involving coupled diffusion and large deformations, makes them ideal for biomedical applications such as micro- and nano-scale drug delivery systems. Understanding the transient swelling or drying behavior of hydrogels at relevant length-scales will provide insight for the development of specialized and controlled drug release.  Understanding how elastocapillary phenomena affect hydrogel behavior can improve the ways hydrogels are utilized in technology today. It is well known that surface energy has an effect on the behavior of liquids, but typically has a negligible effect in solids in the macroscale. However, soft elastomers and hydrogels are known to exhibit elastocapillary effects at length-scales up to the order of millimeters. Surface effects are also critical in soft-tissue morphogenetic processes such as embryogenesis and wound healing, which involves feature formation and growth. Surface effects are prominent when the surface energy contribution is comparable to the bulk energetic contributions. The elastocapillary length-scale dictates when surface effects are a significant driving force of material response.  In general, surface stresses minimize the surface area and round-off sharp features of solid surfaces. Classically, surface stress in hydrogels is treated as a constant isotropic tensor (equivalent with surface tension which is a scalar), but recent experiments have demonstrate that surface stress is strain dependent. This finding has implications for adhesionless and adhesive contact, where classical theories such as the Hertz theory or Johnson-Kendell-Robert theory have to be reformulated. In this work, we present a novel non-linear theory and mixed finite element formulation that takes into account mass transport, large deformations, and elastocapillary effects for hydrogels. Focusing on hydrogel micro-spheres, we provide a comparison of swelling kinetics between the presented non-linear theory and an analytical solution using linear poroelasticity that incorporates surface stresses. Our results demonstrate that when the surface free energy is constant per unit current area (fluid-like) and the elastocapillary length-scale is on the order of the size of the hydrogel micro-sphere, the transient response equilibrates approximately an order of magnitude faster in time compared to the case without surface effects irrespective of swelling or drying. This difference in equilibration time suggests the interplay between competing processes of solvent diffusion, large deformations, and surface effects. Furthermore, we demonstrate that a Neo-Hookean type surface free energy can result in an even faster equilibration as compared to a fluid-like surface free energy. We obtain a map regarding the surface instabilities present in swelling hydrogels spheres as a function of material and surface properties along with initial swelling state. Lastly, our finite element implementation predicts the transient response of complex shapes and constrained structures.