= Solution
Use SI units, uniform absolute <permittivity> $\varepsilon$ and a neutral bulk reservoir. Local equilibrium of ideal mobile <ions> gives $n_i=n_{i,\infty}e^{-z_ie\psi/(k_BT)}$. Combining their charge with the <Poisson equation> yields
$$
\boxed{\nabla^2\psi=-\frac1\varepsilon\left[\rho_f+\sum_i z_ie n_{i,\infty}e^{-z_ie\psi/(k_BT)}\right].}
$$
The reference is $\psi=0$ in the bulk. Boundary data specify fixed charge, fixed potential or a surface reaction/<charge regulation> law. These are physically different problems. The <mean-field theory> assumes point <ions>, <thermal equilibrium> and a continuum <solvent>; ion-size packing, dielectric structure and strong electrostatic correlations can invalidate it.
For a symmetric monovalent electrolyte let $n_\infty$ count each sign and define $u=e\psi/(k_BT)$, $\kappa_D^2=2e^2n_\infty/(\varepsilon k_BT)$. Outside fixed sources,
$$
\nabla^2u=\kappa_D^2\sinh u.
$$
For $|u|\ll1$, the Debye-Hückel approximation is $\nabla^2u=\kappa_D^2u$. A planar wall bounding electrolyte in $z>0$ has $\psi=\psi_0e^{-\kappa_Dz}$; assuming zero field on the other side, its charge is $\sigma=\varepsilon\kappa_D\psi_0$. A sphere of <radius> $a$ and total charge $Q$ instead gives
$$
\psi(r)=\frac{Qe^{-\kappa_D(r-a)}}{4\pi\varepsilon r(1+\kappa_Da)},\qquad r\geq a.
$$
The factors arise from geometry and <boundary conditions>; planar screening cannot be substituted unchanged for a small sphere. The weak-potential condition must hold where the potential is largest, particularly at the surface.
The <nonlinear planar Poisson-Boltzmann profile> is also exactly integrable. Multiply $u''=\kappa_D^2\sinh u$ by $u'$ and use $u,u'\to0$ at infinity:
$$
(u')^2=2\kappa_D^2(\cosh u-1),\qquad
u'=-2\kappa_D\sinh(u/2).
$$
The sign selects a profile decaying to the bulk for either sign of surface potential. Integration gives
$$
\boxed{u(z)=4\operatorname{artanh}\left[\gamma e^{-\kappa_Dz}\right],\qquad
\gamma=\tanh(u_0/4).}
$$
For weak $u_0$ this reduces to the linear solution. Strong surface potentials have a bounded far-field amplitude $4\gamma$, while the near-wall <ion> <density> still grows strongly. There is no need to permit the negative <ion> <densities> produced by an invalid extrapolation of a linearized <density> formula.
The same first integral yields the <Grahame equation> and its <diffuse-layer differential capacitance> per unit <area>:
$$
\boxed{\sigma=\frac{2\varepsilon\kappa_Dk_BT}{e}\sinh(u_0/2),\qquad
C_D=\frac{d\sigma}{d\psi_0}=\varepsilon\kappa_D\cosh(u_0/2).}
$$
These describe the <diffuse layer>. A compact interfacial layer adds a voltage drop and can place its <capacitance> in series with $C_D$; a measured electrode <capacitance> need not equal the ideal diffuse contribution.<The Electrical Double Layer and the Theory of Electrocapillarity by David C. Grahame>[section VI, printed pp. 470–474, especially equations (31)–(48); diffuse charge and differential versus integral capacitance]{ref}
For comparison, a <planar counterion-only double layer> next to a negative plane has positive <counterions> and no salt reservoir. Its profile is algebraic:
$$
\psi(z)-\psi(0)=\frac{2k_BT}{e}\ln(1+z/b),\qquad
n(z)=\frac1{2\pi\ell_B(z+b)^2},\qquad
b=\frac{e}{2\pi\ell_B|\sigma|},\quad
\ell_B=\frac{e^2}{4\pi\varepsilon k_BT}.
$$
Integrating $en(z)$ over $z>0$ gives $|\sigma|$, checking charge neutrality. This problem cannot be obtained by putting a finite reservoir <density> into the symmetric-electrolyte answer without changing its <boundary conditions>.
Applications include screening around synthetic <colloids>, diffuse-layer electrode <capacitance> and <forces> between charged plates. In planar geometry, the bulk-subtracted normal <stress> is constant:
$$
\Pi=k_BT(n_++n_--2n_\infty)-\frac{\varepsilon E^2}{2}.
$$
At the midpoint of identical plates, symmetry sets $E=0$, so the remaining excess ionic <osmotic pressure> determines their repulsion. A comparison with <force> or <capacitance> measurements should vary salt and surface potential independently, test <charge regulation> and <adsorption>, and distinguish compact-layer effects from failures of diffuse screening. Nonlinear Poisson-Boltzmann theory retains finite-potential Boltzmann populations, but still omits correlations and finite-ion-volume saturation.
\Image[/past-exam-of-the-soft-matter-and-biological-physics-course-of-the-university-of-cambridge/2013/paper-1-planar-screening.png]
{title=Original linear and nonlinear planar screening profiles and diffuse-layer capacitance}
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