Cascade of Transitions of Polyelectrolytes in Poor Solvents

We develop a scaling model for the dilute solution conformation of a uniformly charged polymer in a poor solvent. We find that there is a range of temperatures and charge densities for which the polymer has a necklace-like shape with compact beads joined by narrow strings. The free energy of a polyelectrolyte in this conformation is lower than in a cylindrical globule because the length of the necklace is larger than that of a cylinder and is proportional to the total charge on the chain. With changing charge on the chain or temperature, the polyelectrolyte undergoes a cascade of abrupt transitions between necklaces with different numbers of beads. I. Introduction Charged polymers, called polyelectrolytes, have at-tracted much attention during the past three decades due to their unique properties and their technological importance. 1 One of the main technologically important properties of polyelectrolytes is that they dissolve in water. Water is a poor solvent for many polymers. In poor solvents, chains have negative second virial coef-ficients, corresponding to an effective attraction between monomers. This attraction causes chains without charged groups to collapse into spherical globules, 2,3 coalesce with each other, and precipitate from solution. The repulsion between charged groups significantly improves polymer solubility in solvents which are poor for uncharged backbones. In the present paper, we study the configuration of a polyelectrolyte chain in a poor solvent as a function of temperature (solvent quality) and the charge on the chain. The overall shape of a charged polymer in a poor solvent is determined by the balance of the electrostatic repulsion and the surface tension. It was suggested by Khokhlov 4 that, in order to optimize its energy, the polyelectrolyte chain takes the shape of an elongated cylindrical globule. The theory of Khokhlov was ex-tended by Raphael and Joanny 5 to the case of " mobile " charges on the chain and by Higgs and Raphael 6 to the case of screening of electrostatic interactions by added salt. The cylindrical globule 4-6 is unstable to capillary wave fluctuations similar to the ones that result in the splitting of a charged liquid droplet. 7,8 Kantor and Kardar 9 have recently proposed that a polymer with short-range attraction and long-range repulsion may form a necklace with compact beads joined by narrow strings. Below we extend this idea and develop a scaling theory that describes how, with varying solvent quality or charge on the chain, the polyelectrolyte in a poor solvent undergoes a cascade of abrupt transitions between necklace-like configurations with different numbers of beads. We find that the length of the necklace globule is proportional to the total polymer charge, in agreement with the prediction of Kantor and Kardar, 9 and is larger than the length of the cylindrical globule. Consequently, the free energy of the necklace globule is lower than that of the cylindrical one.