Simplified Mean Field Theory for Polyelectrolyte Phase Behaviour

Simplified mean field theory for the phase behaviour of polyelectrolytes in solution is described in detail. It can be viewed as the polyelectrolyte analogue of Flory-Huggins theory. A basic model is analyzed, then extended to look at the effects of multivalent ions (specifically divalent coions), of distinguishable counterions, and of an ionization equilibrium between counterions and polyelectrolyte. Analytic transformation of the free energy to reduce the number of extensive variables facilitates the calculation of binodal curves, tielines, spinodal curves, and the spinodal instability directions. Typical salting out behaviour and salt partitioning are seen. Unexpectedly, a small region of three phase coexistence is found: it is examined in detail for the basic model. In addition, the mean field spinodal is found to be identical to that occurring in common applications of the random phase approximation, and to have an interpretation in terms of electrostatic excluded volume, including for the ionizable polyelectrolyte case where fluctuations in the ionization equilibrium give an additional contribution to the Debye-Hückel screening length. The reasons for this are elucidated in an Appendix