Past exam of the quantum condensed matter course of the University of Cambridge / 2006 / Section A / A3 / d / Solution

The Debye–Hückel approximation linearizes the hyperbolic sine, giving $({\nabla }^2-{\kappa }^2)\psi =0$ withThe planar Debye-Hückel double layer has $\psi (s)={\psi }_0{e}^{-\kappa s}$. For an isolated sphere of radius $a$, the Debye-Hückel potential of a charged sphere is $\psi (r)={\psi }_0(a/r)e^{-\kappa (r-a)}$ outside the sphere; the factor $a/r$ accounts for spherical spreading as well as screening.

The Debye–Hückel screening length is the characteristic distance over which the ion cloud suppresses a weak surface potential. Adding salt increases ionic strength and therefore increases $\kappa$, shortening the interaction range. Linearization requires $|q\psi |\ll k_{B}T$, particularly near the surface where the potential is largest. It means the ionic electrostatic energy is small compared with thermal energy; for monovalent ions near room temperature, $k_{B}T/e\simeq 26\mathrm{mV}$. This condition alone does not remove other limitations of point-ion mean-field theory at high concentration or strong correlations.