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A mobile ion whose charge has the opposite sign to a charged macromolecule, surface or other fixed species. Its local concentration can increase close to that fixed charge; counterion condensation distinguishes a bound fraction that remains localized in an infinite-dilution limit.

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.

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.

Chemical systems whose reactions, transport or equilibrium couple to electric charges and electric potential. An electrolyte carries mobile ions; a Poisson-Boltzmann equation can approximate their equilibrium distributions.

A medium with mobile ions capable of carrying electric charge. Dissolved salts typically supply both positive and negative species; a counterion-only model keeps only the mobile species required to oppose a fixed charge. Poisson-Boltzmann equations use ideal Boltzmann distributions for those species.

A chemical atom or molecular entity with nonzero net electric charge. An ion opposing a fixed charge is called a counterion.

Use the Gaussian units convention implicit in the $4\pi$ coefficient. For a uniform dielectric and a counterion-only electrolyte, the Boltzmann distribution and Poisson-Boltzmann equation areThe reference electric potential is $\varphi (a)=0$, so $n_0=n(a)$; it is not a prescribed nonzero bulk concentration at infinity. A salt reservoir with additional ion species would require their charge densities as well. With $u=\log (r/a)$, the cylindrical Laplacian is $r^{-2}{d}^2/d{u}^2$. Since $\beta q\varphi =\Phi +2u$ and $r^2=a^2{e}^{2u}$, the factors of $e^{2u}$ cancel:Here primes mean differentiation with respect to $u$, and the surface condition is $\Phi (0)=0$ in this coordinate.

Multiply by ${\Phi }^{\prime }$ to obtain a first integral:The two specified far-field limits set $K=0$. For a solution tending upwards to infinity, take the positive square root. Writing $b=\sqrt{C/2}$ gives $(e^{\Phi /2}{)}^{\prime }=b$, hence the cylindrical counterion-only Poisson-Boltzmann profileFor $b>0$, $\Phi \to \infty$ and ${\Phi }^{\prime }=2b/(1+bu)\to 0$. The corresponding number density is

Apply Gauss's law to a coaxial surface per unit cylinder length. Define the positive accumulated cylindrical screening charge byThe signed outward electric field is $E_r=2[-\lambda +Q(r)]/(ϵr)$, so its magnitude is $2|\lambda -Q(r)|/(ϵr)$. In the branch here, $Q(r)<\lambda$, the field is inward andIn this display, ${\varphi }^{\prime }$ is a radial derivative. The quantities $Q$ are electric charges per unit axial length, not total charges of an infinite cylinder.

At the surface, $a{\varphi }^{\prime }(a)=2\lambda /ϵ$. Define the charge-$q$ Bjerrum length ${\ell }_B=\beta q^2/ϵ$ and the Manning parameter $\eta =\lambda {\ell }_B/q$. Matching the derivative of the solution givesThus the condensed branch exists only for $\eta >1$, and its normalization isThe sign condition $b>0$ is essential: simply squaring $\eta -1$ would produce a spurious subcritical density. Direct integration using $rn(r)dr=n_0{a}^{2}du/(1+bu{)}^2$ yieldsEquivalently, $E(r)=2{\lambda }_c[1+b/(1+b\log (r/a))]/(ϵr)$: the remaining far-field line-charge magnitude is ${\lambda }_c$. The Manning condensed fraction is $1-1/\eta$.

Below threshold, no positive-density profile can satisfy those same far-field conditions. Indeed ${\Phi }^{\prime \prime }\le 0$, so a negative initial slope $2(\eta -1)$ cannot increase to zero. The physical interpretation is the infinite-dilution limit of cylindrical counterions: start with a neutral finite cylindrical cell and send its outer radius to infinity. For $\eta \le 1$, counterions escape to arbitrarily large radii, leaving $n_0\to 0$ and no finite condensed charge. The bare-cylinder Boltzmann distribution $n\propto r^{-2\eta }$ also shows why its radial normalization $\int r^{1-2\eta }dr$ diverges in this regime. In the local zero-density limit, $\Phi =2(\eta -1)u$; this does not obey the condensed branch's imposed limits. At $\eta =1$ the branch has zero amplitude and $\Phi =0$, rather than $\Phi \to \infty$.

Consequently the intended counterion condensation result, interpreted as a local infinite-dilution density, isEvery finite neutral cell still contains enough counterions to neutralize the cylinder; the missing residual charge in the local limiting profile is carried to infinity. Taking the infinite-cell limit before the charge integral differs from integrating the entire finite cell first. In SI units, the same reference-charge Bjerrum length is $q^2/(4\pi {ϵ}_{\mathrm{abs}}{k}_{B}T)$; the electrostatic prefactors must be changed consistently.

A mean-field coupling of the Poisson equation to an ideal Boltzmann distribution of mobile ions. In Gaussian units with uniform dielectric constant $ϵ$, ${\nabla }^2\varphi =-(4\pi /ϵ)[{\rho }_f+{\sum }_i{q}_i{n}_{i,0}{e}^{-\beta q_i\varphi }]$. The potential reference and chemical potentials determine the normalization factors. It neglects discrete-ion correlations and assumes local thermal equilibrium. A counterion-only model is not the same equation as a symmetric salt reservoir.