Semidilute Principle for Gels

Polymer gels such as jellies and soft contact lenses are soft solids consisting of three-dimensional polymer networks swollen with a large amount of solvent. For approximately 80 years, the swelling of polymer gels has been described using the Flory--Huggins mean-field theory. However, this theory is problematic when applied to polymer gels with large solvent contents owing to the significant fluctuations in polymer concentration. In this study, we experimentally demonstrate the superiority of the semidilute scaling law over the mean-field theory for predicting the swelling of polymer gels. Using the semidilute scaling law, we experimentally determine the universal critical exponent $\nu$ of the self-avoiding walk via swelling experiments on polymer gels. The experimentally obtained value $\nu\simeq 0.589$ is consistent with the previously reported value of $\nu\simeq 0.588$, which was obtained by precise numerical calculations. Furthermore, we theoretically derive and experimentally demonstrate a scaling law that governs the equilibrium concentrations. This scaling law contradicts the predictions made by de Gennes' $c^{*}$ theorem. A major deficiency of the $c^*$ theorem is that the network elasticity, which depends on the as-prepared state, is neglected. These findings reveal that the semidilute scaling law is a fundamental principle for accurately predicting and controlling the equilibrium swelling of polymer gels.