Polyelectrolyte-mediated bridging interactions: columnar macromolecular phases.

We present a mean-field theory for charged polymer chains in an external electrostatic field in the weak and strong coupling limits. We apply the theory to describe the statistical mechanics of flexible polyelectrolyte chains in a hexagonal columnar lattice of stiff cylindrical macroions, such as DNA, in a bathing solution of a uni-univalent salt (e.g. NaCl). The salt effects are first described in the Debye-Hückel framework. This yields the macroion electrostatic field in the screened Coulomb form, which we take to represent the mean field into which the chains are immersed. We introduce the Green's function for the polyelectrolyte chains and derive the corresponding Edwards equation which we solve numerically in the Wigner-Seitz cylindrical cell using the ground state dominance ansatz. The solutions indicate the presence of polyelectrolyte bridging, which results in a like-charge attraction between stiff macroions. Then we reformulate the Edwards theory for the strong coupling case and use the standard Poisson-Boltzmann picture to describe the salt solution. We begin with the free energy which we minimize to obtain the Euler-Lagrange equations. The solutions yield self-consistently determined monomer density and electrostatic fields. We furthermore calculate the free energy density as well as the total osmotic pressure in the system. We again show that bridging implicates like-charge attractions of entropic origin between stiff cylindrical macroions. By analyzing the osmotic pressure we demonstrate that, in certain parts of the parameter space, a phase transition occurs between two phases of the same hexagonal symmetry.