Size, shape and diffusivity of a single Debye-H\"uckel polyelectrolyte chain in solution

Brownian dynamics simulations of a coarse-grained bead-spring chain model, with Debye-H\"uckel electrostatic interactions between the beads, are used to determine the root-mean-square end-to-end vector, the radius of gyration, and various shape functions (defined in terms of eigenvalues of the radius of gyration tensor) of a weakly-charged polyelectrolyte chain in solution, in the limit of low polymer concentration. The long-time diffusivity is calculated from the mean square displacement of the centre of mass of the chain, with hydrodynamic interactions taken into account through the incorporation of the Rotne-Prager-Yamakawa tensor. Simulation results are interpreted in the light of the OSFKK blob scaling theory (R. Everaers, A. Milchev, and V. Yamakov, Eur. Phys. J. E 8, 3 (2002)) which predicts that all solution properties are determined by just two scaling variables--the number of electrostatic blobs $X$, and the reduced Debye screening length, $Y$. We identify three broad regimes, the ideal chain regime at small values of $Y$, the blob-pole regime at large values of $Y$, and the crossover regime at intermediate values of $Y$, within which the mean size, shape, and diffusivity exhibit characteristic behaviours. In particular, when simulation results are recast in terms of blob scaling variables, universal behaviour independent of the choice of bead-spring chain parameters, and the number of blobs $X$, is observed in the ideal chain regime and in much of the crossover regime, while the existence of logarithmic corrections to scaling in the blob-pole regime leads to non-universal behaviour.