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An ionic atmosphere. A negatively charged molecule attracts positive counterions and repels negative co-ions; signs reverse for a positive molecule. The resulting electric double layer partially screens the bare electric field. Far from an isolated molecule, the electrolyte recovers its neutral bulk composition. At thermodynamic equilibrium, spatial uniformity of each species' electrochemical potential requiresThe logarithm implicitly uses a common reference density. This result for Boltzmann ion densities in a symmetric electrolyte neglects ion correlations, finite size and nonideal activities, but shows immediately why equal concentrations of the two signs do not persist close to a charged surface.

Self-consistent screening. Insert these densities into Poisson's equation, with solvent permittivity $ϵ$ and fixed charge density ${\rho }_f$:This is the Poisson-Boltzmann equation. For a symmetric monovalent salt with bulk density $n_s$ of each sign, outside the fixed charge it becomesFor $|z_{i}e\psi |\ll k_{B}T$, linearizing and using bulk neutrality gives the Debye–Hückel approximationThus added salt shortens the Debye length: electrostatic screening becomes stronger over a shorter distance. The factor of two in a $1:1$ salt comes from two ionic species, not from double-counting the density of salt pairs.

For example, a charged sphere of radius $a$ and total charge $Q$ has the exterior linearized solutionassuming fixed uniform surface charge and no ions inside. It follows from the decaying spherical solution and $-ϵ{\psi }^{\prime }(a)=Q/(4\pi a^2)$. At zero salt it reduces to the Coulomb potential; at finite salt it decays exponentially as well as geometrically. The excess mobile charge integrates to $-Q$ in an infinite neutralizing reservoir, although a finite sphere surrounding the molecule generally contains only part of that ionic atmosphere.

Biophysical scales and limitations. In water at room temperature, the monovalent formula is approximately ${\lambda }_D=0.304\mathrm{nm}/\sqrt{I/(\mathrm{mol}{\mathrm{L}}^{-1})}$, with $I$ the ionic strength. It gives about $0.79\mathrm{nm}$ at $0.15\mathrm{mol}{\mathrm{L}}^{-1}$ and $9.6\mathrm{nm}$ at $1\mathrm{mmol}{\mathrm{L}}^{-1}$. Consequently the same charged protein can exert very different long-range interactions mediated by electrostatics at different salt concentrations. Close to strongly charged DNA, the linear approximation fails; nonlinear screening, counterion condensation, ion-specific adsorption and changes of surface ionization through charge regulation may matter. Mean-field theory also misses strong multivalent-ion correlations and finite-size saturation: the exponential law cannot be extrapolated to arbitrarily large near-surface concentrations.

DLVO theory adds attractive Van der Waals forces and electric double layer repulsion to obtain an effective colloid pair potential. For two identical spheres of radius $b$ at a surface gap $h\ll b$, a useful schematic form isHere $A_H$ is the Hamaker constant. For a symmetric monovalent electrolyte with number density $n$ of each ionic species, ${\kappa }_D^2=2n{e}^2/(ϵk_{B}T)$. The exponential is a simplified screened repulsion; its amplitude depends on electric potential at the surface, charge regulation and geometry. Constant-charge and constant-potential boundary conditions need not produce the same close-approach law.

A primary attractive minimum near contact may be separated from dispersed configurations by an energy barrier; a shallower secondary minimum can produce loose reversible aggregation. A barrier much larger than $k_{B}T$ can give long-lived kinetic stabilization without making the dispersed state the global free-energy minimum. Adding salt shortens the Debye–Hückel screening length and can lower the barrier, allowing colloidal coagulation. Changing pH can alter surface ionization, so screening alone does not determine stability.

One can vary salt concentration while holding particle concentration, temperature and pH fixed, then follow particle aggregation using light scattering or microscopy. A solvent-only control and an independently measured surface-charge indicator help distinguish a change of screening from adsorption or charge reversal. Classical DLVO theory omits steric stabilization, hydration forces and strong ionic correlations; the formal attraction diverging at contact is cut off by molecular-scale physics.

Use SI units, uniform absolute permittivity $\epsilon$ and a neutral bulk reservoir. Local equilibrium of ideal mobile ions gives $n_i=n_{i,\infty }{e}^{-z_{i}e\psi /(k_{B}T)}$. Combining their charge with the Poisson equation yieldsThe 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 $u=e\psi /(k_{B}T)$, ${\kappa }_D^2=2e^2{n}_{\infty }/(\epsilon k_{B}T)$. Outside fixed sources,For $|u|\ll 1$, the Debye-Hückel approximation is ${\nabla }^{2}u={\kappa }_D^{2}u$. A planar wall bounding electrolyte in $z>0$ has $\psi ={\psi }_0{e}^{-{\kappa }_{D}z}$; assuming zero field on the other side, its charge is $\sigma =\epsilon {\kappa }_D{\psi }_0$. A sphere of radius $a$ and total charge $Q$ instead givesThe 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 $u^{\prime \prime }={\kappa }_D^2\sinh u$ by $u^{\prime }$ and use $u,u^{\prime }\to 0$ at infinity:The sign selects a profile decaying to the bulk for either sign of surface potential. Integration givesFor 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: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.^[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: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: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.