Counter-Ion Condensation and System Dimensionality

Condensation of the counter-ions around a highly charged infinitely long cylindrical molecule, such as DNA, can be described in detail in terms of the solutions of the Poisson-Boltzmann (Gouy-Chapman) equation. By using the Alfrey-Berg-Morawetz (1951) solution of this equation one can show that a certain fraction of the counter-ions remain within finite distances of the poly-ion even when the volume of the system is expanded indefinitely; these ions can be appropriately called "condensed". The fraction of the macromolecule's charge represented by these ions is just 1-1/xi, where xi is the linear charge-density parameter of the macromolecule; this is also the value given by Manning's theory. The question arises: Is this property unique to the infinite cylinder? Using the same PB equation, we can consider the infinite charged plane and a large finite charged sphere for comparison. In the case of the plane all of the counter-ions are condensed in the above sense, not just a fraction, for any surface charge density of the plane. These ions form the classical Gouy double layer. On the other hand, none of the counter-ions of the charged sphere are condensed in the above sense, no matter how high the surface charge density. Thus the cylinder is a unique intermediate case in which a fraction of the counter-ions are condensed if the linear charge density is higher than the critical value of unity.