Evolution of Linear Viscoelasticity across the Critical Gelation Transition: Unification of Symmetry and Hyperscaling

In this work, we develop a rigorous theoretical framework for the evolution of linear viscoelastic properties across the sol–gel transition. More specifically, we derive general admissible expressions for the relaxation modulus and dynamic moduli as the critical gel state is approached from the pregel or the postgel side. These expressions possess a generalized multimode series representation and recover the critical gel power-law spectrum in the limit of vanishing distance from the gel point. We validate these expressions against the experimental data for various polymeric and colloidal systems. A central finding of the present work is the requirement of continuity of the dynamic moduli and their derivatives at the critical gel point, which imposes a profound physical constraint, necessitating the relaxation dynamics on both sides of the transition to be symmetric. This, in turn, leads to the hyperscaling relation, which is a theoretical requirement rather than an empirical proposal. We further show that the critical relaxation exponent ( n ) always remains above the relaxation scaling exponent (κ), establishing a previously unrecognized lower bound on n . We also analytically estimate, for the first time, the parameter C that characterizes the relative evolution of the storage modulus with respect to the loss modulus as the critical state is approached. These results reveal that the symmetry, scaling, and hyperscaling properties of the sol–gel transition are all consequences of a single unifying physical requirement originating from the continuity of the linear viscoelastic properties at the critical gel point.