= Solution
\b[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> requires
$$
\mu_i=\mu_i^0+k_BT\ln n_i(\mathbf r)+z_i e\psi(\mathbf r),\qquad
\boxed{n_i(\mathbf r)=n_i^\infty e^{-z_ie\psi(\mathbf r)/(k_BT)}}.
$$
The 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.
\b[Self-consistent screening.] Insert these densities into <Poisson's equation>, with solvent permittivity $\epsilon$ and fixed charge density $\rho_f$:
$$
-\epsilon\nabla^2\psi=\rho_f+\sum_i z_ie n_i^\infty
e^{-z_ie\psi/(k_BT)}.
$$
This is the <Poisson-Boltzmann equation>. For a symmetric monovalent salt with bulk density $n_s$ of \i[each] sign, outside the fixed charge it becomes
$$
\nabla^2\psi=\frac{2en_s}{\epsilon}\sinh\left(\frac{e\psi}{k_BT}\right).
$$
For $|z_i e\psi|\ll k_BT$, linearizing and using bulk neutrality gives the <Debye–Hückel approximation>
$$
\nabla^2\psi=\kappa^2\psi,\qquad
\boxed{\kappa^2=\frac{e^2}{\epsilon k_BT}\sum_i z_i^2n_i^\infty,
\quad\lambda_D=\kappa^{-1}}.
$$
Thus 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 solution
$$
\psi(r)=\frac{Q}{4\pi\epsilon(1+\kappa a)}\frac{e^{-\kappa(r-a)}}r,
\qquad r\ge a,
$$
assuming fixed uniform surface charge and no ions inside. It follows from the decaying spherical solution and $-\epsilon\psi'(a)=Q/(4\pi a^2)$. 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.
\b[Biophysical scales and limitations.] In <water> at room <temperature>, the monovalent formula is approximately $\lambda_D=0.304\,\mathrm{nm}/\sqrt{I/(\mathrm{mol\,L^{-1}})}$, with $I$ the <ionic strength>. It gives about $0.79\,\mathrm{nm}$ at $0.15\,\mathrm{mol\,L^{-1}}$ and $9.6\,\mathrm{nm}$ at $1\,\mathrm{mmol\,L^{-1}}$. 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.
Back to article page
