Counterion condensation
Counterion condensation is a phenomenon described by Manning's
theory, which assumes that counterions can condense
onto polyions until the charged density between neighboring monomer charges
along the polyion chain is reduced below a certain critical value. In the model the
real polyion chain is replaced by an idealized line charge, where the polyion
is represented by a uniformly charged thread of zero radius, infinite
length and finite charge density, and the condensed counterion layer is
assumed to be in physical equilibrium with the ionic atmosphere surrounding
the polyion. The uncondensed mobile ions in the ionic atmosphere are treated
within the Debye–Hückel approximation. The
phenomenon of counterion condensation now takes place when the dimensionless
Coulomb coupling strength
where represents the Bjerrum length and
the distance between neighboring charged monomers.
In this case the Coulomb interactions dominate over
the thermal interactions and counterion condensation is favored. For many standard
polyelectrolytes, this phenomenon is relevant, since the
distance between neighboring monomer charges typically ranges between 2 and 3 Å and
7 Å in water.
The Manning theory states that the fraction of "condensed" counter ions is, where "condensed" means that the counter ions are located within the Manning radius.
At infinite dilution the Manning radius diverges and the actual concentration of ions close to the charged rod is reduced.