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. At sub-millimeter length-scales, surface stresses have been shown to significantly influence material behavior for soft polymers and hydrogels, but very little is known about the influence of surface stresses on the swelling kinetics of hydrogels. In this paper, we present a 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. Lastly, our finite element implementation predicts the transient response of complex shapes and constrained structures.