Using Langevin dynamics simulations, we have investigated the distribution of counterions around a flexible polyelectrolyte chain as a function of polymer concentration (C_p), salt concentration (C_s), and valency of the counterion from the added salt. In the present simulations, the aqueous solutions are at room temperatures and polymer concentrations are either below or comparable to overlap concentrations. The net polymer charge and the radius of gyration (R_g) of a labeled chain are found to decrease with an increase in either C_p or C_s. We present details of the distribution of monovalent and divalent counterions inside the counterion worm surrounding a polymer chain, when a salt-free solution of polyelectrolytes with monovalent counterions is challenged by a salt with divalent counterions. The simulation results for the dependence of R_g on chain length (N), C_p and C_s are compared with the theory of Muthukumar [J. Chem. Phys. 86, 7230 (1987); 105m 5183 (1996)] which contains two parameters, viz., degree of ionization (alpha) and strength (w) of excluded volume interaction. Using the values of alpha and w as determined by simulations, there is a very good agreement between theoretical predictions and simulations for monovalent counterions. For the case of divalent counterions there is evidence for significant bridging between polymer segments mediated by counterions. This bridging leads to an enhanced shrinkage of polymer size beyond expected from averaged electrostatic screening. A mean-field counting of the bridging effect as an effective two-body attraction leads to a good agreement between theory and simulations.
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