  Light scattering and viscometric measurements on weak polyelectrolytes show
two important aspects of counterion condensation, namely, non-monotonic
variation in the polyelectrolyte size with the increase in the electrostatic
strength, and, monovalent counterion selectivity in determining the nature of
collapse transition at high electrostatic strengths. Here, we present a
self-consistent variational theory for weak polyelectrolytes which includes the
effects of the polarizability of monovalent counterions. Our theory reproduces
several experimental findings including non-monotonic conformational size with
the variation in the electrostatic strength and a shift from a continuous to a
discontinuous collapse transition with the increase in the dipole strength of
condensed ions. At low dipole strength and high electrostatic strength, our
theory predicts a series of solvent quality driven size transitions spanning
the re-entrant poor, theta and good solvent regimes. At high dipole strength,
the size remains that of a compact globule independent of solvent quality. The
dipole strength of the ion-pair formed due to counterion condensation, which
depends on the size and polarizability of the monovalent counterions, is found
to be an important molecular parameter in determining the nature of collapse
transition, and the size of the collapsed state at high electrostatic strength.
