Charge-density

For unmodified silica dispersions, the surface charge is the main stabilizing factor. Surface charge of colloidal siUca dispersions can be measured by cationic polyelectrolyte titration [54], The surface charge of two unmodified silica dispersions and the corresponding silane-modified siHca dispersions were made by titration of the said dispersions with a Polybrene solution, 4 g per liter. All silica dispersion had a concentration of 5 g SiO per liter. To compensate for the presence of soluble silicate species in the silica dispersions, the latter were quantitatively determined by using the ammonium heptamolybdate method [55]. 

The determination and interpretation of electronic charge distributions in molecular crystals have been reviewed by Koritsanszky and Coppens [179, 180]. Many of their formulations are used in the discussion that follows. 

X-rays are scattered predominantly by electrons rather than atomic nuclei. To determine atomic coordinates, electron densities are therefore assumed to be concentrated spherically around individual nuclei. This assumption ignores all possible effects that chemical bonding may have on electronic density in molecules. Such a hypothetical array of spherical atoms located at the nuclear positions of an actual molecule in a crystal is known as a promolecule. Molecular structures determined by routine crystallographic methods are invariably the structures of promolecules. 

To assess the effect of chemical bonding on the electron density it is assumed that the effect on core density is negligible, and that the total distortion will be due to valence-charge migration. The molecular charge density may hence be written as 

The total electron density is assumed to be a superposition of density units, each of which rigidly follows the motion of the nucleus that it is attached to. In terms of the scattering vector 

Deformation densities defined in this way typically show density accumulation in the bonds and lone pair regions. Exceptions were first observed [181] in standard X-ray electron-density mapping of a polycyclic molecule containing C, H, N and O atoms. A steady decrease in the order C-N C-0 N-N O-O, of deformation densities in bonds, was observed. The density along the 0-0 bond was found to be negative throughout. 

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The integral of p over all space gives the total excess charge in the solution, per unit area, and is equal in magnitude but opposite in sign to the surface charge density a ... 

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

One can write acid-base equilibrium constants for the species in the inner compact layer and ion pair association constants for the outer compact layer. In these constants, the concentration or activity of an ion is related to that in the bulk by a term e p(-erp/kT), where yp is the potential appropriate to the layer [25]. The charge density in both layers is given by the algebraic sum of the ions present per unit area, which is related to the number of ions removed from solution by, for example, a pH titration. If the capacity of the layers can be estimated, one has a relationship between the charge density and potential and thence to the experimentally measurable zeta potential [26]. 

The effect known either as electroosmosis or electroendosmosis is a complement to that of electrophoresis. In the latter case, when a field F is applied, the surface or particle is mobile and moves relative to the solvent, which is fixed (in laboratory coordinates). If, however, the surface is fixed, it is the mobile diffuse layer that moves under an applied field, carrying solution with it. If one has a tube of radius r whose walls possess a certain potential and charge density, then Eqs. V-35 and V-36 again apply, with v now being the velocity of the diffuse layer. For water at 25°C, a field of about 1500 V/cm is needed to produce a velocity of 1 cm/sec if f is 100 mV (see Problem V-14). 

Assume is -25 mV for a certain silica surface in contact with O.OOlAf aqueous NaCl at 25°C. Calculate, assuming simple Gouy-Chapman theory (a) at 200 A from the surface, (b) the concentrations of Na and of Cr ions 10 A from the surface, and (c) the surface charge density in electronic charges per unit area. 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

Alternative descriptions of quantum states based on a knowledge of the electronic charge density equation Al.3.14 have existed since the 1920s. For example, the Thomas-Femii description of atoms based on a knowledge of p (r)... 

Slater was one of the first to propose that one replace V m equation A 1.3.18 by a tenn that depends only on the cube root of the charge density [T7,18 and 19]. In analogy to equation A1.3.32, he suggested that V be replaced by... 

In a number of classic papers Hohenberg, Kohn and Sham established a theoretical framework for justifying the replacement of die many-body wavefiinction by one-electron orbitals [15, 20, 21]. In particular, they proposed that die charge density plays a central role in describing the electronic stnicture of matter. A key aspect of their work was the local density approximation (LDA). Within this approximation, one can express the exchange energy as... 

This equation is usually solved self-consistently . An approximate charge is assumed to estimate the exchange-correlation potential and to detennine the Flartree potential from equation Al.3.16. These approximate potentials are inserted in the Kolm-Sham equation and the total charge density is obtained from equation A 1.3.14. The output charge density is used to construct new exchange-correlation and Flartree potentials. The process is repeated nntil the input and output charge densities or potentials are identical to within some prescribed tolerance. 

This ionic potential is periodic. A translation of r to r + R can be acconnnodated by simply reordering the sunnnation. Since the valence charge density is also periodic, the total potential is periodic as the Hartree and exchange-correlation potentials are fiinctions of the charge density. In this situation, it can be shown that the wavefiinctions for crystalline matter can be written as... 

Reciprocal lattice vectors are usefiil in defining periodic fimctions. For example, the valence charge density, p (r), can be expressed as... 

Since and depend only on die valence charge densities, they can be detennined once the valence pseudo- wavefiinctions are known. Because the pseudo-wavefiinctions are nodeless, the resulting pseudopotential is well defined despite the last temi in equation Al.3.78. Once the pseudopotential has been constructed from the atom, it can be transferred to the condensed matter system of interest. For example, the ionic pseudopotential defined by equation Al.3.78 from an atomistic calculation can be transferred to condensed matter phases without any significant loss of accuracy. 

Figure Al.3.22. Spatial distributions or charge densities for carbon and silicon crystals in the diamond structure. The density is only for the valence electrons the core electrons are omitted. This charge density is from an ab initio pseudopotential calculation [27].

The exchange-repulsion energy is approximately proportional to the overlap of the charge densities of the interacting molecules [71, 72 and 73]... 

The electroneutrality condition can be expressed in temis of the integral of the charge density by recognizing the obvious fact that the total charge around an ion is equal in magnitude and opposite in sign to the charge on the central ion. This leads to the zeroth moment condition... 

The model used is the RPM. The average electrostatic potential ifr) at a distance r away from an ion / is related to tire charge density p.(r) by Poisson s equation... 

At finite positive and negative charge densities on the electrode, the counterion density profiles often exhibit significantly higher maxima, i.e. there is an overshoot, and the derived potential actually shows oscillations itself close to the electrode surface at concentrations above about 1 M. 

The surface charge density on each surface element is detennined by die boundary condition... 

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