Surface charge, description

The surface charge description of amphoteric mineral oxides has been given considerable attention in literature [see e.g. 22, 28-30], but, to some extent, it is still controversial. The controversy essentially boils down to the representation of the surface group(s) that act as proton adsorption site(s). In the classical [22, 30] and most common approach the surface is assumed to be monofunctional with a postulated surface oxygen group that can undergo two protonation steps, each governed by its own pKn value ("two-pKn model) ... 

Besides the experimental data mentioned above, the kinetic dependencies of oxide adsorption of various metals are also of great interest. These dependencies have been evaluated on the basis of the variation of sensitive element (film of zinc oxide) conductivity using tiie sensor method. The deduced dependencies and their experimental verification proved that for small occupation of the film surface by metal atoms the Boltzman statistics can be used to perform calculations concerning conductivity electrons of semiconductors, disregarding the surface charge effect as well as the effect of aggregation of adsorbed atoms in theoretical description of adsorption and ionization of adsorbed metal atoms. Considering the equilibrium vapour method, the study [32] shows that... 

The description of the sorption of charged molecules at a charged interface includes an electrostatic term, which is dependent upon the interfacial potential difference, Ai//(V). This term is in turn related to the surface charge density, electric double layer model. The surface charge density is calculated from the concentrations of charged molecules at the interface under the assumption that the membrane itself has a net zero charge, as is the case, for example, for membranes constructed from the zwitterionic lecithin. Moreover,... 

The surface complexation models used are only qualitatively correct at the molecular level, even though good quantitative description of titration data and adsorption isotherms and surface charge can be obtained by curve fitting techniques. Titration and adsorption experiments are not sensitive to the detailed structure of the interfacial region (Sposito, 1984) but the equilibrium constants given reflect - in a mean field statistical sense - quantitatively the extent of interaction. 

A polyelectrolyte solution contains the salt of a polyion, a polymer comprised of repeating ionized units. In dilute solutions, a substantial fraction of sodium ions are bound to polyacrylate at concentrations where sodium acetate exhibits only dissoci-atedions. Thus counterion binding plays a central role in polyelectrolyte solutions [1], Close approach of counterions to polyions results in mutual perturbation of the hydration layers and the description of the electrical potential around polyions is different to both the Debye-Huckel treatment for soluble ions and the Gouy-Chapman model for a surface charge distribution, with Manning condensation of ions around the polyelectrolyte. 

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

Based on the previous description of the double layer, it is logical to assume that a direct relationship between the absolute charge at the interface and the concentration of ions in the vicinity of the interface exists. Indeed, several models have been developed in the past that describe the ion concentration as a function of the actual surface charge at a specific distance jc from the interface. Furthermore, the famous Nemst equation, which is the basis for understanding many electrochemical reactions, proves to be helpful as it relates the ion concentration to a quantity called the electrical potential ( j/). The electrical potential is the work (W) required to move a unit charge (q) through the electrical field ... 

This is the fundamental equation for the description of electrocapillarity. Thereby a = Fa J2 is identified with the surface charge density, which is produced by electrons in the metal and compensated by ions in solution. This identification is generally doubtful, because the surface excesses T, depend on the position of the interface. If, however, the electrode is totally polarisable (no electrons are exchanged between the metal and the electrolyte), then the positioning of the interface is trivial and a represents the surface charge density. 

In the second step the charge arrives at the internal phase passing through the interface. The associated potential is known as the surface potential jump (also called surface potential, surface electrical potential, etc.). It is determined by dipoles aligned at the interface and by surface charges. It is not identical with the Volta potential difference (also sometimes called the surface potential) that has so far been used for the description of the electrical double layer. For the treatment of the electrical double layer, dipoles did not play a role. In particular in water, however, the aligned water molecules contribute substantially to the surface potential jump x- The Galvani potential, Volta potential, and surface potential jump are related by... 

Statistical mechanics was originally formulated to describe the properties of systems of identical particles such as atoms or small molecules. However, many materials of industrial and commercial importance do not fit neatly into this framework. For example, the particles in a colloidal suspension are never strictly identical to one another, but have a range of radii (and possibly surface charges, shapes, etc.). This dependence of the particle properties on one or more continuous parameters is known as polydispersity. One can regard a polydisperse fluid as a mixture of an infinite number of distinct particle species. If we label each species according to the value of its polydisperse attribute, a, the state of a polydisperse system entails specification of a density distribution p(a), rather than a finite number of density variables. It is usual to identify two distinct types of polydispersity variable and fixed. Variable polydispersity pertains to systems such as ionic micelles or oil-water emulsions, where the degree of polydispersity (as measured by the form of p(a)) can change under the influence of external factors. A more common situation is fixed polydispersity, appropriate for the description of systems such as colloidal dispersions, liquid crystals, and polymers. Here the form of p(cr) is determined by the synthesis of the fluid. 

A more sophisticated description of the solvent is achieved using an Apparent Surface Charge (ASC) [1,3] placed on the surface of a cavity containing the solute. This cavity, usually of molecular shape, is dug into a polarizable continuum medium and the proper electrostatic problem is solved on the cavity boundary, taking into account the mutual polarization of the solute and solvent. The Polarizable Continuum Model (PCM) [1,3,7] belongs to this class of ASC implicit solvent models. 

Despite the simple form of Equation (1.83), the detailed formulation of an extended Lagrangian for CPCM is not a straightforward matter and its implementation remains challenging from the technical point of view. Nevertheless, is has been attempted with some success by Senn and co-workers [31] for the COSMO-ASC model in the framework of the Car-Parrinello ab initio MD method. They were able to ensure the continuity of the cavity discretization with respect to the atomic positions, but they stopped short of providing a truly continuous description of the polarization surface charge as suggested,... 

In the previous contributions to this book, it has been shown that by adopting a polarizable continuum description of the solvent, the solute-solvent electrostatic interactions can be described in terms of a solvent reaction potential, Va expressed as the electrostatic interaction between an apparent surface charge (ASC) density a on the cavity surface which describes the solvent polarization in the presence of the solute nuclei and electrons. In the computational practice a boundary-element method (BEM) is applied by partitioning the cavity surface into Nts discrete elements and by replacing the apparent surface charge density cr by a collection of point charges qk, placed at the centre of each element sk. We thus obtain ... 

A field theoretical description of counterion fluctuations has also been formulated. In this approach, fluctuating electrostatic potential is introduced in which the surface charge density is treated as a variational parameter [54]. The methodology captures the nonlinearity of the counterion distributions of highly charged systems. 

After an overview over the experimental techniques and results from the literature (Sect. 7.2) and some words about technical aspects and our experience concerning problems with some materials (Sect. 7.3), the experiments of the authors can be outlined as follows first, measurements of ohmic and capacitive currents in the contact mode are described (Sect. 7.4), followed by a description of some surface charge measurements in the non-contact mode (Sect. 7.5). The chapter closes with some experiments to probe electro-mechanical properties by the use of piezo response microscopy (Sect. 7.6) with its own brief literature overview. All three experimental parts are opened by a short introduction to the SFM techniques implemented in our lab. 

Madrid, L. and Diaz-Barrientos, E., Description of Titration curves of mixed materials with variable and permanent surface charge by a mathematical model. 1. Theory. 2. Application to mixtures of lepidocrocite and montmorillonite, J. Soil Sci., 39, 215, 1988. 

We shall start from methods similar to that previously described, characterized by the use of the apparent surface charge (ASC) description of the electrostatic interaction term Ve/, passing then to consider other continuum methods, which use a different description of Ve/.To complete the exposition we shall introduce, where appropriate, methods not based on the solution of a Schrodinger equation, and hence not belonging to the category of continuum effective Hamiltonian methods. We shall pass then to a selection of methods based on mixed continuum-discrete representation of the solvent, to end up with the indication of some approaches based on a full discrete representation of the solvent. 

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