Swelling Kinetics of a Microgel Shell

Tanaka's approach to swelling kinetics of a solid gel sphere is extended to a spherical microgel shell. The boundary condition at the inner surface is obtained from the minimization of shear elastic energy. Temporal evolution of a shell is represented in a form of expansion over eigenfunctions of the corresponding diffusion equation. The swelling of Tanaka's solid spherical gel is recovered as a special case of our general Solution if the inner radius approaches zero. In another limiting case of it thin (balloon-like) shell, the set of eigenfunctions is reduced to a single exponential term. In the general case, a solid sphere swells slightly faster than the same sphere with in internal cavity. To test Our theoretical model, we prepared monodisperse poly-N-isopropylacrylamide (PNIPAM) hydrogel shells using a microfluidic device. The temporal dependence of the inner and outer radii of the shell were measured, and the data were fitted to our theoretical model. As a result, we obtained the collective diffusion constants for shrinking and for swelling processes. The obtained values for microgel shells are in excellent agreement with the previous results obtained for submillimeter PNIPAM solid spheres in the same temperature interval. Our model shows that the characteristic swelling time of a gel shell should be proportional to the square of the outer radius not to the thickness of the shell, agreeing with experimental observation.