Negative electrostatic contribution to the bending rigidity of charged membranes and polyelectrolytes screened by multivalent counterions

Bending rigidity of a charged membrane or polyelectrolyte screened by monovalent counterions is known to be enhanced by electrostatic effects. We show that in the case of screening by multivalent counterions the electrostatic effects reduce the bending rigidity. This inversion of the sign of the electrostatic contribution is related to the formation of two-dimensional strongly correlated liquids (SCL) of counterions at the charged surface due to strong lateral repulsion between them. When a membrane or a polyelectrolyte is bent, SCL is compressed on one side and stretched on the other so that thermodynamic properties of SCL contribute to the bending rigidity. Thermodynamic properties of SCL are similar to those of Wigner crystal and are anomalous in the sense that the pressure, compressibility and screening radius of SCL are negative. This brings about substantial negative correction to the bending rigidity. For the case of DNA this effect qualitatively agrees with experiment.