Asymptotic analysis of primitive model electrolytes and the electrical double layer

Abstract

The multicomponent primitive model electrolyte is analyzed using the Ornstein-Zernike equation and the asymptotic behavior of the direct correlation function. An approximation for the screening length is derived from the second-moment condition \ensuremath{\kappa}=${\mathrm{\ensuremath{\kappa}}}_{\mathit{D}}$/ \ensuremath{\surd}1-(${\mathrm{\ensuremath{\kappa}}}_{\mathit{D}}$d${)}^{2}$/2+(${\mathrm{\ensuremath{\kappa}}}_{\mathit{D}}$d${)}^{3/6}$ , where ${\mathrm{\ensuremath{\kappa}}}_{\mathit{D}}^{\mathrm{\ensuremath{-}}1}$ is the Debye length and d is the ion diameter. This is accurate up to 1M concentration for monovalent aqueous electrolytes and considerably extends the range of validity of the classical Debye-H\"ukel theory. The asymptotic behavior of the ionic pair correlation functions is formally analyzed, and exact expressions are given for the decay length and the effective charge on the ions in terms of the direct correlation function. Three different regimes are identified: monotonic exponential, for ${\mathrm{\ensuremath{\kappa}}}_{\mathit{D}}$d\ensuremath{\lesssim}\ensuremath{\surd}2, and two types of damped oscillatory, electrostatic dominated at intermediate concentrations and core dominated at high concentrations distinguished by whether or not the oscillations are in charge or in number density. The electrical double layer is also analyzed and it is shown that the asymptotic behavior of the density profiles and the interaction pressure is the same as for the bulk correlation functions. The hypernetted chain closure (with and without bridge functions) is used to obtain numerical results for binary symmetric aqueous electrolytes (monovalent with d=4 and 5 \AA{}, and divalent with d=4 \AA{}), and the three asymptotic regimes are explored.