A variational approach to necklaces formation in polyelectrolytes

It is well established that in a salt-free environment and under good or Θ-solvent conditions, the electrostatic interaction between the monomers of a single polyelectrolyte chain induces a rod-like conformation. In the pressence of of a small amount of salt, if the ions are homogeneously distributed such that their interaction with the chain is limited to screen the elctrostatic interactions, a rod-like conformation is also expected. Even in this simple model, the introduction of a a secondary interaction can lead to interesting structural modifications. Dobrynin et al. have presented a scaling argument and computing evidence that by turning on an attractive interaction between the monomers, the standard rod-like conformation, is abandon in favor of a “necklace” one. The necklace structure consists of highly stretched segments alternating with collapsed globules, the “beads” of the necklace. Another mean field calculation by Balazs et al. also produce similar results. In this paper we further explore this possibility, by means of a variational approach. The techniques we use here are adapted to the case of long (effectively infinite) weakly charged polymers. The necklace structure was first suggested by Kantor and Kardar in their study of polyampholytes. This was argued in the basis of a similarity with the Rayleigh instability that appears in charged fluids. Namely, a large charged droplet of fluid will split into smaller droplets to put the charges far away from each other and the splitting is only stopped by the increasing cost of surface energy. The droplets are kept together by cohesional molecular forces, which in the polymer case are replaced by the bonds between monomers.