Constant charge density

Fig. 7. Maps of the electronic charge density in the (110) planes In the ordered twin with (111) APB type displacement. The hatched areas correspond to the charge density higher than 0.03 electrons per cubic Bohr. The charge density differences between two successive contours of the constant charge density are 0.005 electrons per cubic Bohr. Atoms in the two successive (1 10) planes are denoted as Til, All, and T12, A12, respectively, (a) Structure calculated using the Finnis-Sinclair type potential, (b) Structure calculated using the full-potential LMTO method.

The combination of state-of-the-art first-principles calculations of the electronic structure with the Tersoff-Hamann method [38] to simulate STM images provides a successful approach to interpret the STM images from oxide surfaces at the atomic scale. Typically, the local energy-resolved density of states (DOS) is evaluated and isosurfaces of constant charge density are determined. The comparison between simulated and measured high-resolution STM images at different tunneling... 

Fig. 4.15. Contours of constant charge density for Si(lll). The occupied portion of the dangling-bond surface state on Si(lll) is shown. Dots locate nuclei of surface atoms, the vacuum is above, and the charge density is in a.u.X lOL (Reproduced from Appelbaum and Hamann, 1976, with permission.)...

Fig. 10. Silicon (001) surface saturated by hydrogen but for a single location. This location, the dangling bond, is marked by a red arrow (left). A constant charge density contour shows the local extent of the dangling bond, which covers an area of about 1 nm x 1 nm (center). A constant charge density contour has a different shape than the current contour, the apparent height of the dangling bond at a bias voltage of — 2 V and a current value of 50pA is about 1.5 A (right).

Therefore, the present treatment of the double layer interaction leads to the same results for the interaction free energy as the imaginary charging approach for systems of arbitrary shapes and constant surfece potential or constant charge density and to the same results as the Langmuir equation for parallel plates and arbitrary surface conditions. It can be, however, used for systems of any shape and any surfece conditions, since it does not imply any of the above restrictions. 

When the surface charge is generated through dissociation equilibrium, the difference between the forces calculated using the new model and the Gouy-Chapman theory is much larger than for constant charge density or constant surface potential. 

If the surfaces of the plates are kept at constant charge density, the corresponding boundary condition becomes... 

This derivative is equal to zero in the absence of specific adsorption. For anion adsorption, and constant charge density, the point of zero charge moves in the negative direction in order to counterbalance adsorption. For cations, Ez moves in the positive direction, assuming constant charge density. In aqueous solution, specific adsorption only occurs close to Ez far from Ez, solvent molecules are attracted so strongly that it is difficult to push them out of the way. 

This simple and appealing result shows that, for H 1 /k, the sphere-wall interaction depends linearly on the charge densities of each surface, and decays exponentially with the separation distance. The result does not depend on whether the surfaces are considered to be constant charge density or constant potential, because the potentials of an isolated wall and sphere were used in its derivation. Phillips [13] has compared Eq. (24) with a numerical solution of the linear Poisson-Boltzmann equation, and shows that it errs by less than about 10% for xh>3 when 0.5 [Pg.257]

Note that, unlike Eqs. (33) and (34), it is nearly always impossible to satisfy Eqs. (39) and (40) with a solution to Eq. (31) as Kh 0 (the exception being when o-w = —o-s), because the solution to Eq. (31) is linear in that limit, and a line cannot have two different slopes. For this reason, the restriction that Kh 1 /Ka discussed above is of more consequence in the constant charge density case than in the constant potential case. 

Interestingly, Carnie et al. [17] point out that the linear Derjaguin approximation for constant charge density surfaces is attractive for all kH only when other case, the force becomes repulsive at very small separations, because the need for counterions to balance the extra charge leads to high osmotic stresses. Conversely, the linear Derjaguin approximation for constant potential surfaces is repulsive for all k i only when... 

FIG. 3 Comparison of the linear Derjaguin approximation with a numerical solution of the linear Poisson-Boltzmann equation for (a) constant potential and (b) constant charge density boundary conditions. (From Ref. 13.)... 

Mathematically, this problem bears some resemblance to those considered above. The governing partial differential equation is still Eq. (6), and on the surfaces boundary conditions of constant potential, constant charge density or linear regulation [i.e., Eq. (45)] must be imposed. However, a further constraint arises from the need to satisfy mechanical equilibrium at the interface, and it is this new condition that provides the mathematical relation needed to calculate the interface shape. The equation is the normal component of the surface stress balance, and it is given by [12]... 

The conditions considered in the calculation were a) constant potential constant charge density Ob at... 

According to the muffin-tin approximation the potential V(f) inside the muffin-tin spheres is replaced by its spherical mean V(r), and outside the atomic spheres V(f) is approximated by a constant Vq. The electron-electron interaction in this region is calculated from the constant charge density pg defined by ... 

Plots of the potential drop across the diffuse layer are shown as a function of electrode charge density in fig. 10.18 for a 1-1 electrolyte with concentrations in the range 0.01-1 M. As electrolyte concentration increases, the absolute value of decreases for constant charge density a. This demonstrates that the screening ability of the electrolyte increases with the increasing concentration. 

Another problem arises in the case of weak or moderate adsorption [46]. When the sign of the potential drop across the diffuse layer changes with solution composition or electrode charge density, there is a change in the nature of the predominant ion at the oHp. Since different ions have different sizes, the position of the oHp also changes. Ionic size effects are not considered in the GC model of the diffuse layer. Thus, use of the model based on equation (10.8.10) must consider the possibility that and K d vary with adsorbed charge density Oad for constant charge density 0 on the electrode [46]. 

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