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.

中文翻译：

---

聚电解质凝胶的电荷调节溶胀动力学：弹性和扩散

我们应用了最近开发的方法[S. Sen和A.Kundagrami，化学杂志。物理 143，224904（2015）]，使用渗透压的现象学表达作为聚合物和电荷密度，疏水性和网络弹性的函数，以在无盐溶剂中溶胀具有固定电荷和可变电荷的球形聚电解质（PE）凝胶。球形PE凝胶的溶胀动力学运动方程式中使用了这种应力表达，以数值计算聚合物在不同时间步长和自由离子密度的空间分布以及凝胶尺寸的时间演变。我们比较了从经典线性弹性理论获得的相同变量的分布，并定量估计了PE凝胶的体积模量。此外，我们从应力的线性化表达式（在较小的变形极限内）获得了弹性模量的解析表达式。我们发现，PE凝胶的估计体积模量随溶胀过程中固定变形程度的有效电荷的增加而降低。最后，我们将凝胶前端位置与实验数据相匹配，该数据来自带电可逆加成-断裂链转移凝胶的测量结果，显示带电时凝胶大小会增加，并且与PNIPAM（不带电）和咪唑鎓相同。基（带电）微凝胶，具体证实了凝胶模量值随电荷的增加而降低。实验结果与理论结果之间的一致性证实，随着电离度（电荷）的增加，体积分数降低的PE凝胶的溶胀具有普遍的扩散行为。新的形式主义也捕获了大的变形，并且凝胶的电荷含量发生了显着变化。发现具有较大变形但相同初始尺寸的PE凝胶膨胀速度更快，电荷更高。