Bending rigidity of stiff polyelectrolyte chains: A single chain and a bundle of multichains

We study the bending rigidity of highly charged stiff polyelectrolytes, for both a single chain and many chains forming a bundle. A theory is developed to account for the interplay between competitive binding of counterions and charge correlations in softening the polyelectrolyte (PE) chains. The presence of even a small concentration of multivalent counterions leads to a dramatic reduction in the bending rigidity of the chains that are nominally stiffened by the repulsion between their backbone charges. The variation of the bending rigidity as a function of f(0), the fraction of charged monomers on the chain, does not exhibit simple scaling behavior; it grows with increasing f(0) below a critical value of f(0). Beyond the critical value, however, the chain becomes softer as f(0) increases. The bending rigidity also exhibits intriguing dependence on the concentration of multivalent counterion n(2); for highly charged PEs, the bending rigidity decreases as n(2) increases from zero, while it increases with increasing n(2) beyond a certain value of n(2), When polyelectrolyte chains form a N-loop condensate (e.g., a toroidal bundle formed by N turns (winds) of the chain), the interloop coupling further softens the condensate, resulting in the bending free energy of the condensate that scales as N for large N.