Poisson-Boltzmann equation

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.

For an ideal planar electric double layer at fixed surface charge density, let $v=e{\varphi }_0/(k_{B}T)$. Adding bulk divalent chloride concentration $\delta c$ adds two monovalent coions per counterion. Differentiating the squared Grahame equation gives the displayed expression, with all denominator concentrations evaluated at the surface. The numerator concentrations are bulk increments. Dropping depleted coions at strongly negative potential gives the leading counterion approximation. A large finite increment instead requires solving the fixed-charge equation at the new composition; exponentiating a first-order voltage estimate does not make that estimate exact.

A negatively charged plane with monovalent positive counterions in $z>0$ and no salt reservoir has the mean-field profile displayed above, where $b=e/(2\pi {\ell }_B|\sigma |)$ and Bjerrum length ${\ell }_B=e^2/(4\pi \epsilon k_{B}T)$. Its potential relative to the wall is $2k_{B}T\log (1+z/b)/e$. The positive-ion charge density integrates to $|\sigma |$, neutralizing the plane; the field decays algebraically. The potential has no finite zero-at-infinity reference, unlike the screened symmetric-reservoir problem.

For a symmetric monovalent point-ion electrolyte next to a plane, define $u=e\psi /(k_{B}T)$ and ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$, with $n_{\infty }$ the density of each sign. The Poisson-Boltzmann equation becomes $u^{\prime \prime }={\kappa }_D^2\sinh u$. Multiplying by $u^{\prime }$ and matching zero field and potential in the bulk gives $(u^{\prime }{)}^2=2{\kappa }_D^2(\cosh u-1)$. The decaying branch integrates to the displayed profile, reducing to $u_0{e}^{-{\kappa }_{D}z}$ at weak potential. This solution assumes a salt reservoir and is distinct from a counterion-only diffuse layer.

A strong negative surface potential creates exponentially enhanced positive counterion density and exponentially suppressed negative co-ion density. The co-ion population remains positive, unlike an extrapolated linear approximation. Finite ion size and correlations eventually limit the applicability of point-ion mean-field theory.

For $|e\psi |\ll k_{B}T$, $\psi \prime \prime ={\kappa }^2\psi$ with ${\kappa }^2=2e^2{n}_0/(\epsilon k_{B}T)$ for a symmetric monovalent salt with per-species reservoir density $n_0$. The half-space solution with $\psi (0)=\zeta$ and vanishing bulk potential is $\zeta e^{-\kappa x}$. If density denotes the sum of species, the written factor of two changes.

For point ions of charge $\pm e$ in potential $\psi$, equilibrium electrochemical potentials give $n_{\pm }=n_0\exp [\mp e\psi /(k_{B}T)]$, with $n_0$ the reservoir density of each species. The charge density is $-2e{n}_0\sinh [e\psi /(k_{B}T)]$. A negative wall enriches positive counterions and depletes negative co-ions.

For a symmetric electrolyte with mobile charges $\pm q$, bulk density $n_0$ of each sign, and constant permittivity $\epsilon$, the Poisson-Boltzmann equation gives $n_{\pm }=n_0{e}^{\mp q\psi /(k_{B}T)}$. In planar geometry let $E=-{\psi }^{\prime }$ and $\rho =q(n_{+}-n_{-})$. Then $k_{B}T(n_{+}+n_{-}{)}^{\prime }=\rho E$ and $\epsilon E{E}^{\prime }=\rho E$ by Gauss's law, proving that the displayed bulk-subtracted normal stress is constant. At the midpoint of identical equally charged plates, symmetry gives $E=0$ and $\Pi =2n_0{k}_{B}T[\cosh (q{\psi }_{\mathrm{mid}}/(k_{B}T))-1]$. Its ionic osmotic pressure and electrostatic contributions together transmit the force; bare surface-charge repulsion alone omits essential terms.

For a planar surface contacting a symmetric monovalent ideal electrolyte on one side, the Poisson-Boltzmann equation is ${\psi }^{\prime \prime }=(2e{n}_{\infty }/\epsilon )\sinh (e\psi /(k_{B}T))$. Multiplication by ${\psi }^{\prime }$ and integration using the zero field and zero potential in the bulk givesThe solution decays towards the bulk, and the boundary condition $\sigma =-\epsilon {\psi }^{\prime }(0^{+})$ therefore yields the displayed charge-potential relation, with ${\kappa }^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Its small-potential limit is the planar Debye-Hückel double layer. It is nonlinear but still a mean-field model with point-like ions and constant permittivity; it does not include a compact interfacial layer or finite-ion-size saturation.

Differentiate the Grahame equation for a planar symmetric monovalent electrolyte with respect to surface potential at fixed reservoir concentration and temperature. The result is the displayed capacitance per unit area. It approaches $\epsilon {\kappa }_D$ in the Debye–Hückel approximation. A compact surface layer can contribute a separate series capacitance; point-ion packing assumptions also fail at sufficiently high near-surface concentration.^[ref]

Confine the counterions of a uniformly charged infinite cylinder within a coaxial cylinder of radius $R$, with total mobile line charge $\lambda$. The electric field vanishes at $R$, while its surface value at $a$ is fixed by the bare line charge. This gives $a{\varphi }^{\prime }(a)=2\lambda /ϵ$ and $R{\varphi }^{\prime }(R)=0$ in Gaussian units. Sending $R/a\to \infty$ defines the infinite-dilution limit of cylindrical counterions.

In a neutral counterion-only cylindrical cell, send the outer radius to infinity at fixed cylinder charge. Below the Manning parameter threshold, number density tends to zero at each fixed radius. Above threshold, the cylindrical counterion-only Poisson-Boltzmann profile retains line charge $\lambda -q/{\ell }_B$ locally, while the rest recedes to infinity. Therefore the limit of the full finite-cell charge integral is not the charge integral of the pointwise limiting density.

For $u=\log (r/a)$ and $\Phi =\beta q\varphi -2u$, a counterion-only Poisson-Boltzmann equation reduces to ${\Phi }^{\prime \prime }=-C{e}^{-\Phi }$. Conditions $\Phi (0)=0$, $\Phi \to \infty$ and ${\Phi }^{\prime }\to 0$ give $\Phi =2\log (1+bu)$, $b=\sqrt{C/2}>0$. Matching surface Gauss's law gives $b=\eta -1$, so this branch requires the Manning parameter $\eta >1$. Its number density is $n(r)=(\eta -1{)}^2(a/r{)}^2/[2\pi {\ell }_B{a}^2(1+(\eta -1)\log (r/a){)}^2]$. It leaves residual line-charge magnitude $q/{\ell }_B$.

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 $z\lambda {\ell }_B/e$.