Cyclization of Rouse chains at long- and short-time scales

We have investigated cyclization of a Rouse chain at long and short times by a Langevin dynamics simulation method. We measure St, the fraction of nonreacted chains, for polymerizations ranging from Z=5 to Z=800 and capture distances ranging from a=0.1b to a=8b where b is the bond length. Comparison is made with two theoretical approaches. The first is a decoupling approximation used by Wilemski and Fixman to close the relevant master equation [J. Chem. Phys. 58, 4009 (1973); 60, 866 (1974)]. The second approach is the renormalization group arguments of Friedman and O'Shaughnessy [Phys. Rev. Lett 60, 64 (1988); J. Phys. II 1, 471 (1991)]. We find that at long times St decays as a single exponential with rate k(infinity). The scaled decay rate K=k(infinity)tauR appears to approach a constant value independent of the capture distance for very large chains consistent with the predictions of both the renormalization group (RG) and Wilemski-Fixman closure approximation. We extract K*, the long chain limit of K, from the fixed point a=a* where K is independent of Z. K* is larger than both the RG and closure predictions but much closer to the RG result. More convincing evidence for the RG analysis is obtained by comparing the short-time decay of St to long-time results. The RG analysis predicts that dSdt should decay as a power law at early times and that the exponent in the power law is related to K by a simple expression with no free parameters. Our simulations find remarkable agreement with this parameter-free prediction even for relatively short chains. We discuss possible experimental consequences of our result.