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...

A wide variety of two- and three-dimensional physical-chemical systems display domain patterns in equilibrium. The phenomenology of these patterns, and of the shapes of their constituent domains, is reviewed here from a point of view that interprets these patterns as a manifestation of modulated phases. These phases are stabilized by competing interactions and are characterized by periodic spatial variations of the pertinent order parameter, the corresponding modulation period generally displaying a dependence on temperature and other external fields. This simple picture provides a unifying framework to account for striking and substantial similarities revealed in the prevalent "stripe" and "bubble" morphologies as well as in commonly observed, characteristic domain-shape instabilities. Several areas of particular current interest are discussed.

Chain-like macromolecules (polymers) show characteristic adsorption properties due to their flexibility and internal degrees of freedom, when attracted to surfaces and interfaces. In this review we discuss concepts and features that are relevant to the adsorption of neutral and charged polymers at equilibrium, including the type of polymer/surface interaction, the solvent quality, the characteristics of the surface, and the polymer structure. We pay special attention to the case of charged polymers (polyelectrolytes) that have a special importance due to their water solubility. We present a summary of recent progress in this rapidly evolving field. Because many experimental studies are performed with rather stiff biopolymers, we discuss in detail the case of semi-flexible polymers in addition to flexible ones. We first review the behavior of neutral and charged chains in solution. Then, the adsorption of a single polymer chain is considered. Next, the adsorption and depletion processes in the many-chain case are reviewed. Profiles, changes in the surface tension and polymer surface excess are presented. Mean-field and corrections due to fluctuations and lateral correlations are discussed. The force of interaction between two adsorbed layers, which is important in understanding colloidal stability, is characterized. The behavior of grafted polymers is also reviewed, both for neutral and charged polymer brushes.