Remarks on polyelectrolyte conformation

We discuss the conformations of linear polyions assuming that a) the corresponding uncharged chain is flexible ; b) electrostatic forces dominate the monomer-monomer interactions; c) no salt is added. 1) For the dilute case (non overlapping chains) correcting a recent self-consistent calculation by Richmond [1a], we find an overall polyion size R = Nd which is a linear function of the polymerization index N in agreement with the early work of Hermans and Overbeek, [1b], Kuhn, Kunzle, and Katchalsky [1c]. 2) There is a range of very low concentration c (c** < c < c*) where the chains do not overlap (c < c*) but where the electrostatic interactions between polyions are much larger than thermal energies ( c > c**) : here we expect that the polyions build up a 3-dimensional periodic lattice ; however, the detection of such an extremely dilute lattice appears difficult. 3) Practically all experiments on salt-free polyelectrolytes have been performed at concentrations c > c* where different chains overlap each other. To discuss this regime we restrict our attention to cases where the charge per unit length is near (or above) the condensation threshold : then a single length ξ( c) characterizes the correlation; in 3 dimensions ξ scales like the Debye radius associated with the counter ions. We consider several possible conformations : a) hexagonal lattice of rigid rods ; b) cubic lattice of rigid rods ; c) isotropic phase of partially flexible chains. The various rigid rod structures appear to have very similar electrostatic energies. This suggests that the isotropic phase might possibly be the most favorable. We analyse this latter phase using the same scaling methods which have recently been helpful for neutral polymer solutions (2). In the isotropic model each chain behaves like a succession of segments of size. Inside one segment electrostatic effects are important and similar to case (1) above. Between segments the interactions are screened out, and tach chain is ideal on a large scale, with radius R (c) ∼ c-1/4 N1/2 . If we (tentatively) assume that the dynamical effects of entanglements are weak, we are than led to a viscosity ηsp/c ∼ Nc-1/2.