Short time behavior and universal relations in polymer cyclization

This paper is a renormalization group study of the short time irreversible reaction rate for a single polymer with reactive groups attached to its end (cyclization). In melts we predict that for times less than the entanglement time the reaction rate k(t) scales as t -α where α is a nonuniversal power which depends on the molecular weight. This power tends to 1/4 as the chains become very long; the asymptotic behavior is “diffusion controlled” independently of the reactivity of the end groups. In dilute theta solvents we find marginal behavior in which k(t) depends logarithmically on t. For cyclization in good solvents the correlation hole plays a crucial role, causing asymptotic mean-field or “law of mass action” behavior, again independently of the chemistry of the reactive groups: k(t) scales as the equilibrium probability of chain end contact with weak time dependent corrections. Eliminating the nonuniversal elements between these results for k(t) and previously calculated long time reaction rates k ∞ allows the derivation of universal relations between the experimentally observable quantities k(t), k ∞ and the unperturbed longest polymer relaxation time τ.