Charge-regularized swelling kinetics of polyelectrolyte gels: Elasticity and diffusion.

We apply a recently developed method [S. Sen and A. Kundagrami, J. Chem. Phys. 143, 224904 (2015)], using a phenomenological expression of osmotic stress, as a function of polymer and charge densities, hydrophobicity, and network elasticity for the swelling of spherical polyelectrolyte (PE) gels with fixed and variable charges in a salt-free solvent. This expression of stress is used in the equation of motion of swelling kinetics of spherical PE gels to numerically calculate the spatial profiles for the polymer and free ion densities at different time steps and the time evolution of the size of the gel. We compare the profiles of the same variables obtained from the classical linear theory of elasticity and quantitatively estimate the bulk modulus of the PE gel. Further, we obtain an analytical expression of the elastic modulus from the linearized expression of stress (in the small deformation limit). We find that the estimated bulk modulus of the PE gel decreases with the increase of its effective charge for a fixed degree of deformation during swelling. Finally, we match the gel-front locations with the experimental data, taken from the measurements of charged reversible addition-fragmentation chain transfer gels to show an increase in gel-size with charge and also match the same for PNIPAM (uncharged) and imidazolium-based (charged) minigels, which specifically confirms the decrease of the gel modulus value with the increase of the charge. The agreement between experimental and theoretical results confirms general diffusive behaviour for swelling of PE gels with a decreasing bulk modulus with increasing degree of ionization (charge). The new formalism captures large deformations as well with a significant variation of charge content of the gel. It is found that PE gels with large deformation but same initial size swell faster with a higher charge.