Counterion condensation

Localization of a finite fraction of oppositely charged ions near a charged object even as the confining volume grows without bound. In the salt-free infinite-cylinder mean-field model, the Manning parameter must exceed one for a nonzero limiting local density. The condensed charge does not include ions that escape to infinity; finite-cell charge neutrality and the local infinite-volume limit must not be interchanged.

In the mean-field infinite-dilution cylinder model, the fraction of neutralizing charge that remains localized is $f_c=\max (1-1/\eta ,0)$, where $\eta$ is the Manning parameter. For $\eta >1$, integrating the cylindrical counterion-only Poisson-Boltzmann profile gives $Q_c=\lambda -q/{\ell }_B$. Below threshold, the local limiting density vanishes even though every finite neutral cell contains the full counterion charge.

The positive mobile charge per unit axial length accumulated from cylinder radius $a$ to $r$ is $Q(r)=2\pi {\int }_a^{r}s\rho (s)ds$. For fixed negative line charge $-\lambda$, Gauss's law gives signed radial electric field $E_r=2[-\lambda +Q(r)]/(ϵr)$ in Gaussian units. Its absolute value is the field magnitude. An integral over a local infinite-dilution profile measures condensed charge, rather than every ion in a finite neutral cell.

The dimensionless line-charge coupling $\eta =\lambda {\ell }_B/q$, with negative cylinder line charge $-\lambda$ and counterions of charge $q>0$. Here the Bjerrum length uses the same reference charge $q$. The mean-field infinite-dilution threshold is $\eta =1$. If ${\ell }_B$ is instead tabulated for elementary charge $e$ and counterions have charge $ze$, the corresponding coupling is $z\lambda {\ell }_B/e$.