The adsorption of large ions from solution to a charged surface is investigated theoretically. A generalized Poisson-Boltzmann equation, which takes into account the finite size of the ions is presented. We obtain analytical expressions for the electrostatic potential and ion concentrations at the surface, leading to a modified Grahame equation. At high surface charge densities the ionic concentration saturates to its maximum value. Our results are in agreement with recent experiments.PACS numbers: 61.20. Qg,82.65.Dp,82.60.Lf The interaction between charged objects (interfaces, colloidal particles, membranes, etc) in solution is strongly affected by the presence of an electrolyte (salt) and is of great importance in biological systems and industrial applications [1,2]. The main effect is screening of the Coulomb interaction characterized by the so-called Debye-Hückel screening length [3], which depends on the ionic strength of the solution. The Deryaguin-LandauVerwey-Overbeek theory, based on the competition between screened Coulomb and attractive van der Waals interactions, has been very successful in explaining the stabilization of charged colloidal particles [4].One of the most widely used analytical method to describe electrolyte solutions is the Poisson-Boltzmann (PB) approach [5]. For low electrostatic potentials (less than 25 mV), the PB equation can be linearized and yields the Debye-Hückel theory [3]. The PB is a continuum mean-field like approach assuming point-like ions in thermodynamic equilibrium and neglecting statistical correlations. This theory has been successful in predicting ionic profiles close to planar and curved surfaces and the resulting forces. However, it is known to strongly overestimate ionic concentrations close to charged surfaces. In particular, this shortcoming of the PB theory is pronounced for highly charged surfaces and multivalent ions.Since the PB equation does not take into account the finite size of the adsorbing ions, the ionic concentration close to the surface can easily exceed the maximal allowed coverage by orders of magnitude. Several attempts have been proposed to include the steric repulsion in order to improve upon the PB approach [6,7]. One of the first attempts to incorporate steric effects is the Stern layer modification [6,8] of the PB approach. Steric effects are introduced by excluding the ions from the first molecular layer close to the surface. However, it seems difficult to improve on this method in a systematic way. More recent modifications [6,7,[9][10][11] rely either on Monte Carlo computer simulations or on numerical solutions of integral equations (the "hypernetted chain" equation [9]). These approaches involve elaborate numerical calculations and lack the simplicity of the original PB approach.In this Letter, we propose a simple way to include steric effects in the original PB approach. This modified PB equation clearly shows how ionic saturation takes place close to a charged surface. The equation is derived for 1:z asymmetric and z:z symmetric...

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We formulate the exact non-linear field theory for a fluctuating counter-ion distribution in the presence of a fixed, arbitrary charge distribution. The Poisson-Boltzmann equation is obtained as the saddle-point of the field-theoretic action, and the effects of counter-ion fluctuations are included by a loop-wise expansion around this saddle point. The Poisson equation is obeyed at each order in this loop expansion. We explicitly give the expansion of the Gibbs potential up to two loops. We then apply our field-theoretic formalism to the case of a single impenetrable wall with counter ions only (in the absence of salt ions). We obtain the fluctuation corrections to the electrostatic potential and the counter-ion density to one-loop order without further approximations. The relative importance of fluctuation corrections is controlled by a single parameter, which is proportional to the cube of the counter-ion valency and to the surface charge density. The effective interactions and correlation functions between charged particles close to the charged wall are obtained on the one-loop level. PACS. 82.70.-y Disperse systems -61.20.Qg Structure of associated liquids: electrolytes, molten salts, etc. -82.45.+z Electrochemistry and electrophoresis