Universal Crossover Dynamics of a Semi-Flexible Polymer in Two Dimensions

We present a unified scaling theory for the dynamics of monomers for dilute solutions of semiflexible polymers under good solvent conditions in the free draining limit. Our theory encompasses the well-known regime of mean square displacements (MSDs) of stiff chains growing like $t^{3/4}$ with time (R. Granek, J. Phys. II (Paris) {\bf 7}, 1767 (1997); E. Farge and A. C. Maggs, Macromolecules {\bf 26}, 5041 (1993)) due to bending motions, and the Rouse regime $t^{2 \nu / (1+ 2\nu)}$ where $\nu$ is the Flory exponent describing the radius $R$ of a swollen flexible coil. We identify how the prefactors of these laws scale with the persistence length $\ell_p$, and show that a crossover from stiff to flexible behavior occurs at a MSD of order $\ell^2_p$ (at a time proportional to $\ell^3_p$), a second crossover (to diffusive motion) occurs when the MSD is of order $R^2$. We also provide compelling evidence for the theory by carrying out large scale Molecular Dynamics simulations in $d=2$ dimensions.