Strong-coupling electrostatic theory of polymer counterions close to planar charges

Strong-coupling phenomena, such as like-charge macroion attraction, opposite-charged macroion repulsion, charge renormalization, and charge inversion, are known to be mediated by multivalent counterions. Most theories treat the counterions as point charges and describe the system by a single coupling parameter that measures the strength of the Coulomb interactions. In many biological systems, the counterions are highly charged and have finite sizes and can be well-described by polyelectrolytes. The shapes and orientations of these polymer counterions play a major role in the thermodynamics of these systems. In this work we apply a field-theoretic description in the strong-coupling regime to the polymer counterions in the presence of a fixed charge distribution. We work out the special cases of rodlike polymer counterions confined by one, and two charged walls, respectively. The effects of the geometry of the rodlike counterions and the excluded volume of the walls on the density, pressure, and free energy of the rodlike counterions are discussed.

Figure 1

Schematic diagram showing the volume available to the charged rods close to the oppositely charged wall. The parametrizations in the figure corresponds to the excluded volume interactions $\tilde{\mathrm{\Omega }}(\tilde{z},u_z)$ defined in Eq. (30).

Figure 2

One wall: (a) Zeroth-order density for the rods calculated from Eq. (36). On the wall ${\tilde{\rho }}_0\left(\tilde{z}\right)$ vanishes as the wall is impenetrable. The point-particle case is also shown. (b) Leading-order density term for different lengths of the rods and the corresponding term for the point-particle case.

Figure 3

Two walls: (a) Zeroth-order density calculated from Eq. (37) for different lengths of the rods confined between two walls. (b) The corresponding leading-order density term for different lengths of the rods and the point particle.

Figure 4

Interwall pressure vs the interwall distance for various lengths of the rods.