Hard-sphere electrostatic model

As early as 1929, Linus Pauling, > o is arguably the most influential chemist of the twentieth century, developed a set of five principles that can be used to rationalize the structures of many ionic solids. Pauling s rules, which follow, are based on a completely ionic, hard-spheres electrostatic model. 

The simplest way to treat the solvent molecules of an electrolyte explicitly is to represent them as hard spheres, whereas the electrostatic contribution of the solvent is expressed implicitly by a uniform dielectric medium in which charged hard-sphere ions interact. A schematic representation is shown in Figure 2(a) for the case of an idealized situation in which the cations, anions, and solvent have the same diameters. This is the solvent primitive model (SPM), first named by Davis and coworkers [15,16] but appearing earlier in other studies [17]. As shown in Figure 2(b), the interaction potential of a pair of particles (ions or solvent molecule), i and j, in the SPM are ... 

More realistic treatment of the electrostatic interactions of the solvent can be made. The dipolar hard-sphere model is a simple representation of the polar nature of the solvent and has been adopted in studies of bulk electrolyte and electrolyte interfaces [35-39], Recently, it was found that this model gives rise to phase behavior that does not exist in experiments [40,41] and that the Stockmeyer potential [41,42] with soft cores should be better to avoid artifacts. Representation of higher-order multipoles are given in several popular models of water, namely, the simple point charge (SPC) model [43] and its extension (SPC/E) [44], the transferable interaction potential (T1PS)[45], and other central force models [46-48], Models have also been proposed to treat the polarizability of water [49],... 

The hole correction of the electrostatic energy is a nonlocal mechanism just like the excluded volume effect in the GvdW theory of simple fluids. We shall now consider the charge density around a hard sphere ion in an electrolyte solution still represented in the symmetrical primitive model. In order to account for this fact in the simplest way we shall assume that the charge density p,(r) around an ion of type i maintains its simple exponential form as obtained in the usual Debye-Hiickel theory, i.e.,... 

Any fundamental study of the rheology of concentrated suspensions necessitates the use of simple systems of well-defined geometry and where the surface characteristics of the particles are well established. For that purpose well-characterized polymer particles of narrow size distribution are used in aqueous or non-aqueous systems. For interpretation of the rheological results, the inter-particle pair-potential must be well-defined and theories must be available for its calculation. The simplest system to consider is that where the pair potential may be represented by a hard sphere model. This, for example, is the case for polystyrene latex dispersions in organic solvents such as benzyl alcohol or cresol, whereby electrostatic interactions are well screened (1). Concentrated dispersions in non-polar media in which the particles are stabilized by a "built-in" stabilizer layer, may also be used, since the pair-potential can be represented by a hard-sphere interaction, where the hard sphere radius is given by the particles radius plus the adsorbed layer thickness. Systems of this type have been recently studied by Croucher and coworkers. (10,11) and Strivens (12). 

In the following, we focus on the soft-sphere method since this really is the workhorse of the DPMs. The reason is that it can in principle handle any situation (dense regimes, multiple contacts), and also additional interaction forces—such as van der Waals or electrostatic forces—are easily incorporated. The main drawback is that it can be less efficient than the hard-sphere model. 

Fig. 6-21. Charge distribution profile across a metal/aqueous solution interface (M/S) (a) the hard sphere model of aqueous solution and the jellium model of metal (the jellium-sphere model), (b) the effective image plane (IMP) and the effective excess charge plane x, (c) reduction in distance /lxd,p to the closest approach of water molecules due to electrostatic pressure, o, = differential excess charge on the solution side og = total excess charge on the solution side Oy = total excess charge on the metal side.

Molecular Mechanics. Molecular mechanics (MM), or empirical force field methods (EFF), are so called because they are a model based on equations from Newtonian mechanics. This model assumes that atoms are hard spheres attached by networks of springs, with discrete force constants. The force constants in the equations are adjusted empirically to repro duce experimental observations. The net result is a model which relates the "mechanical" forces within a structure to its properties. Force fields are made up of sets of equations each of which represents an element of the decomposition of the total energy of a system (not a quantum mechanical eneigy, but a classical mechanical one). The sum of the components is called the force field eneigy, or steric energy, which also routinely includes the electrostatic eneigy components. Typically, the steric energy is expressed as... 

More realistic approaches should, of course, comprise solvent models that give rise to electrostatic interactions. Shelley and Patey [273] used grand canonical MC simulations to investigate the demixing transition in model ionic solutions where the solvent is explicitly included. Charged hard-sphere ions in neutral, dipolar, and quadrupolar hard-sphere solvents were consi-... 

This conclusion is further strengthened considerably by the theoretical calculation of CBE originally performed by Pearson and Gray (102) and later on somewhat modified by Pearson and Mawby (8). Values of CBE are calculated according to three models, viz. the hard sphere model, the polarizable ion model and the localized molecular orbital model. Only the last one, treating the bonds as covalent, is able to account in a satisfactory way for the values found experimentally for such halides as HgCl2 and CdCl2. For LiCl and NaCl, on the other hand, an acceptable fit with the experimental values is obtained already by the hard sphere model, which certainly indicates a predominantly electrostatic interaction. 

The wave character of the particles plays no part in the bonding between ions since we are concerned in this case with heavy particles. A simple treatment, based on the classical laws of electrostatics, does in fact lead to satisfactory results, in which the ions are considered as charged, polarizable, almost hard spheres (Kossel, Van Arkel and De Boer). Calculations can thus be carried out for the ionic bond from which general rules can be readily deduced. The domain, in which these rules are found to be valid, is very extensive. They are even found to hold in cases where the model of ionic bonding employed certainly cannot be considered as the correct approximation to the constitution. The ionic bond is of paramount importance especially for the solid state. 

This section on concentrated suspensions discusses the rheological behavior of sj tems which are colloidally stable and colloidally unstable suspensions. For stable sj tems, the rheology of sterically stabilized and electrostatically stabilized systems wiU be considered. For sterically stabilized suspensions, a hard sphere (or hard particle) model has been successfid. Concentrated suspensions in some cases behave rheologically like concentrated polymer solutions. For this reason, a discussion of the viscosity of concentrated polymer solutions is discussed next before a discussion of concentrated ceramic suspensions. 

Ionic crystals may be viewed quite simply in terms of an electrostatic model of lattices of hard-sphere ions of opposing charges. Although conceptually simple, this model is not completely adequate, and we have seen that modifications must be made in it. First, the bonding is not completely ionic with compounds ranging from the alkali halides, for which complete ionicity is a very good approximation, to compounds for which the assumption of the presence of ions is rather poor. Secondly, the assumption of a perfect, infinite mathematical lattice with no defects is an oversimplification. As with all models, the use of the ionic model does not necessarily imply that it is true , merely that it is convenient and useful, and if proper caution is taken and adjustments are made, it proves to be a fruitful approach. 

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