Sphere with surface charge

Here p(A) is the Fourier transform of p(R). This Born ion is considered as a conducting sphere with its charge Q being smeared over the surface of its cavity p(R) = (Q/4Tra2)8(R - a), p(k) = Qsm(ka)/ka. Outside the cavity the electrostatic field created by this charge is fully equivalent to the field due to the point charge Q considered earlier. By this means for R > a... 

FIGURE 15.3 Interaction between two soft spheres 1 and 2 at separation H. Spheres 1 and 2 are covered with surface charge layers of thicknesses di and CI2, respectively. The core radii of spheres 1 and 2 are and a2, respectively. 

Fig. 25. Snapshot of the simulation with surface charge (7 =+9.9(j.Ccm . Na+ small spheres, Cl" large spheres. The liquid slab is confined by the metal surface (bottom) and by a free surface (top).

Number of lines of flux passing through the surface of a sphere with a charge +q at its center (Figure 5A.3)... 

The approximate calculation of the solvent reorganization energy for the charge transfer between spherical reactants was carried out by Marcus [2]. Under the assumption that the distance between the centres of reactants R is much larger than their radii a and b and that the reactants can be described by non-polarizable spheres with the charges strictly and uniformly distributed over their surfaces, the expression for Eg obtained in Ref. [2] has the form... 

Figure 34 Various potential profiles as a function of distance from the surface for a charged sphere of radius 20 A with surface charge densities of -0.002 eo/A and —0.01 eo/A in a mixed 1 1-2 1 electrolyte with concentration 0.1-0.02 M. Exact PB (circles according to Eq. [389]), DH (triangles Eq. [307]), and Levine (stars Eq. [302]) solutions are compared with the PGC (solid lines s. [152]) and NLDH (dashed lines Eqs. [253] with approximate G values) solutions and their ADH//PCG (dotted lines Eq. [ 5]) and ADH//NLDH (dotted-dashed lines Eq. [308] with either exact or approximate G values) potentials.

Figure 52 compares the counterion concentration profile for a bulk 0.05 M 1 1 electrolyte with 1- or 2-A hard-sphere ions near a charged cylinder of radius 10 A with surface charge density = 0.01 cq/A. Surface concentrations obtained from Metropolis Monte Carlo simulations (circles) are seen to be about 15% larger than those predicted by the PB equation without including activity corrections. Using the hard-sphere activities of Eqs. [429] and [430] results in much better agreement for 1-A ions the improvement for 2-A ions is less impressive but still noticeable. Similar results for a bulk 0.05... 

Fig. 4. Three macromolecular models with hydrophobic (black) and cationic surface groups (grey). Left A nonpolar sphere with positively charged patches. Middle Lysozyme coarse grained to the amino acid level.Right Lysozyme in atomistic detail (hydrogens not shown).

In view of this equation the effect of the ionic atmosphere on the potential of the central ion is equivalent to the effect of a charge of the same magnitude (that is — zke) distributed over the surface of a sphere with a radius of a + LD around the central ion. In very dilute solutions, LD a in more concentrated solutions, the Debye length LD is comparable to or even smaller than a. The radius of the ionic atmosphere calculated from the centre of the central ion is then LD + a. 

The dashed line in the complex in (4.21) and (4.22) indicates an outer-sphere (o.s.) surface complex, Kos stands for the outer-sphere complex formation constant and kads [M 1 s 1] refers to the intrinsic adsorption rate constant at zero surface charge (Wehrli et al., 1990). Kos can be calculated with the help of a relation from Gouy Chapman theory (Appendix Chapter 3). 

The presence of an electrostatic field subjects a material to a stress. Such fields can arise from the presence of free charges in the material or from the action of an external field. A number of authors (Hogan, H12, H13 Graf, G8 Doyle, Moffett, and Vonnegut, D6 and others) give the outward pressure on a liquid sphere with a uniformly distributed surface charge as... 

For nonconducting equal spheres with equal surface distribution of charge, the following equation can be derived, which will give either (1) the time for... 

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