Towards a chemo-mechanical coupled theory of physical hydrogel for sol–gel transition identified by crosslink density

This paper presents a chemo-mechanical coupled theory of physical hydrogel for bulk phase behavior during reversible sol–gel transition. A free energy density is proposed to account for the crucial elastic, mixing, bonding, and chemical contributions, based on statistical mechanics with Gaussian chain distribution. In particular, the crosslink density is included as a novel mean-field parameter in the free energy density functional, such that modulating the density changes the state of the gel-sol phase, non-trivially. The governing equations are then formulated for force equilibrium and mass conservation, and the constitutive relations are obtained based on the second law of thermodynamics. Moreover, an Allen-Cahn-type kinetic equation is imposed on the crosslink density capturing its spatial and temporal evolution, in which the deformation heterogeneity prescribes the inhomogeneous distribution of the crosslinks. Equilibrium analysis is first conducted for physical hydrogels, indicating that the formation of physical crosslinks is promoted at (i) higher monomer volume fraction, (ii) more extensive polymer chemical potential, (iii) more repulsive Flory interaction parameter, (iv) higher bond activation energy, and (v) longer bonding length. Next, dynamic analysis is performed on a cylindrical specimen subject to a controlled torsion ramp in the framework of continuum mechanics, revealing the fundamental mechanism of the reversible sol–gel transition for physical hydrogels. To show the robustness of our theory, the stress yield shear-thinning phenomenon is simulated, for the first time, achieving a good agreement between the reported experiment data and the numerical results from our approach. The crosslink density is found to play key role in determining the mechanical behavior of physical hydrogels.