We describe a variational mean field study of polyelectrolyte expansion based on the application of the Gibbs-Bogoliubov inequality and a generalized Gaussian trial Hamiltonian. The screened electrostatic interactions among the charged beads on the polyion are approximated by a pairwise additive Yukawa potential while we treat the excluded volume effects in terms of the Dirac delta function in the way usual in studies of neutral polymers. Expressing the Hamiltonian in terms of Fourier components, the variational procedure yields a set of Euler equations that are analyzed by the method of dominant balance to study the scaling regimes in various limiting situations. The method predicts correct scaling laws for weakly screened polyelectrolytes, dominated by long-ranged Coulombic repulsions. At strong screening or low degrees of ionization, when the polymer resembles a self-avoiding walk, the calculations overestimate the scaling exponent, the value of approximately 4/3 replacing the Flory value, a deficiency known from earlier applications of the theory to nonionic macromolecules. The numerical solution to the Euler equations is used to calculate the mean square distances between monomer pairs in cyclic polyions as functions of the relative distance along the polymer backbone. Effects of the degree of polymerization and electrolyte screening are studied and the difficulties in providing a general numerical solution to the variational problem are discussed.
