We present a model for the ion distribution near a charged surface, based on the response of the ions to the presence of a single test particle. Near an infinite planar surface this model produces the exact density profile in the limits of weak and strong coupling, which correspond to zero and infinite values of the dimensionless coupling parameter. At intermediate values of the coupling parameter our approach leads to approximate density profiles that agree qualitatively with Monte Carlo simulation. For large values of the coupling parameter our model predicts a crossover from exponential to algebraic decay at large distance from the charged plate. Based on the test-charge approach we argue that the exact density profile is described, in this regime, by a modified mean-field equation, which takes into account the interaction of an ion with the ions close to the charged plate.
Test-charge theory for the electric double layer
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