Physical and Thermodynamic Origins of Dynamic tetraPEG Hydrogel Shear Elasticity and Phase Behavior

Star-like PEGs with dynamic covalent bonds (DCBs), short-lived covalent interactions, self-assemble into thermo-responsive hydrogels with self-healing and viscoelastic properties. The shear modulus, G'_o, of dynamic hydrogels made from tetra-coordinated PEGs (tetraPEGs) follow a dynamic rubber elasticity model developed by Parada and Zhao (2018). Within the model, elastic strand density is governed by a gel association constant, K_gel. The DCB chemistry is known to affect K_gel; however, a physical model for K_gel has not been established in order to predict G'_o a priori. Here, through probing the bulk rheology of dynamic tetraPEG hydrogels as function of tetraPEG arm length, we find K_gel is directly related to the critical DCB bond concentration, c_gel, along the sol-gel temperature-concentration boundary. To explain this coupling, we develop a simple mean-field model showing that Flory-Stockymayer network theory and equilibrium thermodynamics dictate that (i) K_gelc_gel = 3/2 and (ii) K_gel ~ e^-ΔH/RTe^ΔS/R,  where ΔH and ΔS are the enthalpy and entropy of DCB formation. Collapse of sol-gel phase boundaries at tetraPEG concentrations of c < 0.5c* across a variety of arm lengths show c_gel is related to c*, the tetraPEG overlap concentration, and ΔH, implying c_gel ~ c*e^-ΔH/RT. Our work here thus provides both a thermodynamic and physical origin for K_gel, namely ΔH, the tetraPEG size (c*), and Flory-Stockmayer network theory (e.g., a percolated network structure).