Abstract
We solve the nonlinear Poisson–Boltzmann (P–B) equation of statistical thermodynamics for the external electrostatic potential of a uniformly charged flat plate immersed in an unbounded strong aqueous electrolyte. Our rather general variational formulation yields new solutions for the external potential derived from both the classical Boltzmann distribution and its heuristic Eigen–Wicke modification for concentrated symmetric electrolytes. Electrostatic potentials of these mean-field solutions satisfy a homogeneous condition at a free boundary plane parallel to the electrically conducting plate. The preferred position of this plane, characterizing the outer limit of the charged electrolyte, is determined by minimizing electrostatic free energy of the electrolyte. For a given uniform density of surface charge exceeding a well-defined and experimentally accessible threshold, we show that the generalized nonlinear P–B equation predicts a unique sharp interface separating a charged boundary layer or double layer from electroneutral bulk electrolyte. Sharp electric boundary layers are shown to be an essentially nonlinear phenomenon. In the super-threshold regime, the diffuse Gouy–Chapman solution is inapplicable and thus the Derjaguin–Landau–Verwey–Overbeek analysis, predicting electrostatic repulsion between two sufficiently separated and identically charged parallel plates must be rejected. Similar limitations restrict the applicability of the Grahame equation relating surface charge density to surface potential.
Highlights
Electrostatic fields adjacent to electrically charged surfaces immersed in an electrolyte are important in a wide variety of applications, extending from electrochemistry, through colloid science to the biophysics of ionic atmospheres [1,2]
The classical nonlinear mean-field Poisson–Boltzmann (P–B) equation is widely accepted as a continuum model for such fields, which are described by a non-dimensional electrostatic potential function ψ(x)
The Boltzmann distribution of mobile ions in a dielectric continuum in conjunction with the Poisson equation of classical electrostatics leads to a meanfield approximation to the charge density of electrolytic ions
Summary
Electrostatic fields adjacent to electrically charged surfaces immersed in an electrolyte are important in a wide variety of applications, extending from electrochemistry, through colloid science to the biophysics of ionic atmospheres [1,2]. For a uniform density of surface charge above a well-defined threshold, the functional is minimized at this extremal We call such a solution a sharp or finite boundary-layer solution. We apply standard physical interface conditions [13] for electrostatic fields at the fixed surface x = 0 and at the interface represented by the free boundary plane at x = X This leads to finite boundary-layer solutions that satisfy the physical requirements of charge conservation, global electroneutrality and electrochemical equilibrium, as discussed in appendices B and D. Numerical examples illustrating the super-threshold regimes of the P–B and E–W models exhibit unusual external electric-field profiles, with sharp cut-offs, for charged surfaces immersed in aqueous electrolytes This remarkable behaviour suggests a number of simple observations and experiments with outcomes at variance with predictions based on the traditional assumption of diffuse ionic atmospheres
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