Fractional Viscosity Dependence of Reaction Kinetics in Glass-Forming Liquids

The diffusion of molecules in complex systems such as glasses and cell cytoplasm is slow, heterogeneous, and sometimes nonergodic. The effects of such intriguing diffusion on the kinetics of chemical and biological reactions remain elusive. In this Letter, we report that the kinetics of the polymer loop formation reaction in a Kob-Andersen (KA) glass forming liquid is influenced significantly by the dynamic heterogeneity. The diffusion coefficient $D$ of a KA liquid deviates from the Stokes-Einstein relation at low temperatures and $D$ shows a fractional dependence on the solvent viscosity ${\ensuremath{\eta}}_{s}$, i.e., $D\ensuremath{\sim}{\ensuremath{\eta}}_{s}^{\ensuremath{-}{\ensuremath{\xi}}_{D}}$ with ${\ensuremath{\xi}}_{D}=0.85$. The dynamic heterogeneity of a KA liquid affects the rate constant ${k}_{\mathrm{rxn}}$ of the loop formation and leads to the identical fractional dependence of ${k}_{\mathrm{rxn}}$ on ${\ensuremath{\eta}}_{s}$ with ${k}_{\mathrm{rxn}}\ensuremath{\sim}{\ensuremath{\eta}}_{s}^{\ensuremath{-}\ensuremath{\xi}}$ and $\ensuremath{\xi}={\ensuremath{\xi}}_{D}$, contrary to reactions in dynamically homogeneous solutions where ${k}_{\mathrm{rxn}}\ensuremath{\sim}{\ensuremath{\eta}}_{s}^{\ensuremath{-}1}$.