Debye–Hückel approximation ($({\nabla }^2-{\kappa }_D^2)\varphi =-{\rho }_{\mathrm{fixed}}/\epsilon$)

The Debye–Hückel approximation linearizes mobile-ion Boltzmann densities about a neutral bulk electrolyte. It is valid for ionic potential energies small compared with $k_{B}T$. The inverse Debye–Hückel screening length satisfies ${\kappa }_D^2={\sum }_i{n}_i{z}_i^2{e}^2/(\epsilon k_{B}T)$. The quadratic screened charging free energy includes both electric field energy and the linearized ionic-response term $\epsilon {\kappa }_D^2{\varphi }^2/2$.

For a plane bounding a symmetric electrolyte in $x>0$, assume no displacement field on the other side, bulk potential zero, and $|ze\psi |\ll k_{B}T$. The Debye–Hückel approximation gives ${\psi }^{\prime \prime }={\kappa }^2\psi$, whose decaying solution is ${\psi }_0{e}^{-\kappa x}$. The electrostatic interface boundary conditions give $\sigma =-\epsilon {\psi }^{\prime }(0^{+})=\epsilon \kappa {\psi }_0$. Equivalently, its diffuse charge density is ${\rho }_e=-\epsilon {\kappa }^2\psi$ and its integral is $-\sigma$. Here $\epsilon ={\epsilon }_0{\epsilon }_r$ in SI units and ${\kappa }^2=2z^2{e}^2{n}_{\infty }/(\epsilon k_{B}T)$ when $n_{\infty }$ counts each ionic species. An isolated sheet with identical electrolyte on both sides has two diffuse layers, so the total sheet charge is $2\epsilon \kappa {\psi }_0$ instead. A surface geometry and the field on its other side must therefore be specified.

For a uniformly charged sphere of radius $a$, total electric charge $Q$, and an exterior dilute electrolyte of permittivity $\epsilon$, the Debye–Hückel approximation gives $(r^2{\varphi }^{\prime }{)}^{\prime }={\kappa }^2{r}^2\varphi$. Decay at infinity and Gauss's law, $-\epsilon {\varphi }^{\prime }(a)=Q/(4\pi a^2)$, select the displayed electric potential. Substituting $u=r\varphi$ reduces the proof to $u^{\prime \prime }={\kappa }^{2}u$. The screening parameter is ${\kappa }^2={\sum }_i{z}_i^2{e}^2{n}_{i,\infty }/(\epsilon k_{B}T)$, and the ion profiles are $n_i\simeq n_{i,\infty }(1-z_{i}e\varphi /(k_{B}T))$. The validity condition is $|z_{i}e\varphi (a)|\ll k_{B}T$ for every relevant species. Integrating the diffuse charge density using the radial equation gives total screening charge $-Q$.

A lateral sinusoidal surface charge density with wave number $k$ in the Debye–Hückel approximation generates vertical exponential or hyperbolic functions with inverse length $p=\sqrt{k^2+{\kappa }_D^2}$. In a homogeneous electrolyte filling all space, an isolated sheet at $z_s$ produces $\varphi ={\sigma }_k(x)e^{-p|z-z_s|}/(2\epsilon p)$. Boundary conditions determine the normalization when the electrolyte occupies only a gap.

Two equally modulated charge sheets in identical full-space electrolyte have charging energy per area ${\alpha }^2[1+e^{-pd}\cos \theta ]/(4\epsilon p)$, where $p=\sqrt{k^2+{\kappa }_D^2}$. Its phase-dependent interaction favors a half-wavelength offset, with opposite charge patches facing each other. For an electrolyte gap with negligible exterior displacement response, the energy is instead ${\alpha }^2[\coth (pd)+\cos \theta /\sinh (pd)]/(2\epsilon p)$, with the same minimizing phase. Absolute energies require specified exterior electrostatic interface boundary conditions.