Correlation forces between helical macro-ions in the weak coupling limit

When correlation effects are relatively weak, electrostatic interaction forces between cylindrical macro-ions may be divided into two contributions (Lee 2010 J. Phys.: Condens. Matter 22 414101). Firstly, there is a mean field contribution, described by the theory of Kornyshev and Leikin (1997 J. Chem. Phys. 107 3656) at large separations. Secondly, we have correlation forces, which we analyze by performing an expansion in the number density of condensed ions. We see three distinct contributions, for which analytical expressions are given for both general and helical contributions. Firstly, there is a term (of leading order in the expansion) that is a change in the solvation energies of uncondensed counter-ions due to two macro-molecular interfaces. Secondly, we have a contribution that comes from fluctuations in the condensed ion charge density being repelled by their 'images' in the other molecule. Both of these contributions are repulsive. Lastly, there exists an attractive Oosawa contribution that arises from fluctuations in the condensed ions about one molecule correlating with those about the other molecule. The first two forces do not depend on the orientation of the molecules about their long axes. However, the Oosawa force may do so, depending on the pattern of bound and fixed charges. For a DNA like charge distribution, we see that the strength of this dependence is governed by the relative proportion of bound ions, between two positions that represent the DNA groove centers. We see that, at a Debye screening length equivalent to physiological salt concentrations, the correlation forces can be neglected for univalent ions. For divalent ions, they contribute a small, albeit significant, correction. Our calculations suggest that increasing the salt concentration reduces the size of these forces.