Universal relaxation equation for disordered and complex systems*

Slower-than-exponential relaxations often occur in disordered systems such as stress relaxation in polymers.  Lacking a microscopic theory, these are commonly fit to the empirical Kohlrausch-Williams-Watts (KWW) stretched exponential or the Curie-von Schweidler (CvS) power-law algebraic decay. In this work an anomalous-diffusion limited, mixed second-order reaction equation is used to unify the above relaxation laws as different limits of the same overall behavior. Here, relaxation is modeled as a mixed second-order reaction between a concentration of reactants that undergo anomalous diffusion. The resulting general expression uses four parameters: the minority-to-majority reactant ratio 0 < m < 1, the anomalous power-law exponent 0 < β < 1, the characteristic relaxation time τ, and the relaxation amplitude f_Δ. The m = 0 and m = 1 limits correspond to the KWW and CvS expressions, respectively, and the intermediate m values represent a new class of previously unrecognized relaxation functions. A fitting algorithm is introduced that identifies confidence intervals for each of the four experimental parameters. Prominent examples of disordered systems from polymer stress relaxation, biomechanics, energy storage, and dielectric relaxation show excellent fits to the proposed relaxation expression.