Theory of polymer cyclization

The simplest example of a reacting polymer system is studied, that of a single chain with reacting end groups. The long-time cyclization rate is calculated systematically using renormalization-group methods. Several types of single-chain dynamics are treated, with and without hydrodynamical and excluded-volume interactions. The hydrodynamical cases (``non-free-draining'') describe cyclization in dilute polymer solutions. For free-draining polymers we find ``diffusion-controlled'' behavior for long chains: The ratio of the reaction rate to the longest relaxation rate of the unperturbed polymer is a universal constant independent of the chemistry of the reacting ends. For non-free-draining models the analogous ratio vanishes to order {ε} ({ε}=4-d). In good solvents (hydrodynamics plus excluded volume) we predict that a ``mass action'' law holds asymptotically: The molecular weight dependence of the reaction rate is that of the probability that the chain ends are together in equilibrium. In all cases, the asymptotic behavior, either diffusion controlled or mass action, is determined by the class of dynamics and is independent of the reactivity of the end groups.