Kinetics of swelling of polyelectrolyte gels: Fixed degree of ionization

The swelling kinetics of uncharged and charged polymer (polyelectrolyte) gels in salt-free conditions is studied in one dimension by solving the constitutive equation of motion (Newton's law for the elementary gel volume) of the displacement variable by two theoretical methods: one in which the classical definition of stress is used with the bulk modulus taken as a parameter, and the other in which a phenomenological expression of the osmotic stress as a function of polymer density and degree of ionization is taken as an input to the dynamics. The time-evolution profiles for spatially varying polymer density and stress, along with the location of the gel-solvent interface, are obtained from the two methods. We show that both the polymer density (volume fraction) and stress inside the gel follow expected behaviours of being maximum for the uniformly shrunken gel, and relaxing slowly to the lowest values as the gel approaches equilibrium. We further show that, by comparing the temporal profiles of the gel-solvent interface and other variables between the two methods, one may attempt to assign an effective bulk modulus to the polyelectrolyte gel as a function of the degree of ionization and other parameters of the gel such as hydrophobicity, cross-link density, and the temperature. The major result we get is that the effective bulk modulus of a polyelectrolyte gel increases monotonically with its degree of ionization. In the process of identifying the parameters for a monotonic swelling, we calculated using a well-known expression of the free energy the equilibrium results of two-phase co-existence and the critical point of a polyelectrolyte gel with a fixed degree of ionization.