  Counterion distribution around an isolated flexible polyelectrolyte in the
presence of a divalent salt is evaluated using the adsorption model [M.
Muthukumar, J. Chem. Phys. {\bf 120}, 9343 (2004)] that considers Bjerrum
length, salt concentration, and local dielectric heterogeneity as physical
variables in the system. Self consistent calculations of effective charge and
size of polymer show that divalent counterions replace condensed monovalent
counterions in competitive adsorption. The theory further predicts that at
modest physical conditions, polymer charge is compensated and reversed with
increasing divalent salt. Consequently, the polyelectrolyte collapses and
reswells, respectively. Lower temperatures and higher degrees of dielectric
heterogeneity enhance condensation of all species of ions. Complete diagram of
states for the effective charge calculated as functions of Coulomb strength and
salt concentration suggest that (a) overcharging requires a minimum Coulomb
strenth, and (b) progressively higher presence of salt recharges the polymer
due to either electrostatic screening (low Coulomb strength) or negative coion
condensation (high Coulomb strength). A simple theory of ion-bridging is also
presented which predicts a first-order collapse of polyelectrolytes. The
theoretical predictions are in agreement with generic results from experiments
and simulations.
