Wrinkling instabilities of swelling hydrogels

It has long been known that gels attached to a solid surface and brought into contact with water can form wrinkles, and this has often been explained as resulting from an elastic instability akin to Euler’s analysis of buckling beams. However, experiments also show that the characteristic wavelength of wrinkles increases in time and, in some cases, they are seen to smooth out entirely as the hydrogel approaches its steady state. We use our linear-elastic-nonlinear-swelling model to conduct a linear stability analysis of the interface and calculate growth rates of normal modes. We find that, unlike the Euler beam, there are no incompressible instabilities when the gel is tethered to a rigid base and that elastically-driven instabilities are limited by water transport through the gel. If the base state itself is differentially swollen, such as when a gel is first exposed to water, we find a different, osmotically-driven, instability, and the interplay between osmotically-driven flow and elasticity selects a wavelength that scales with the thickness of the swollen region. As the base state approaches its uniform steady state, the characteristic wavelength increases and the effect of this latter mechanism becomes less important, reducing the growth rate and smoothing the wrinkles.