Electrostatic repulsion charged spheres

In an aqueous system with large particles weU-separated by a distance, s, (D and 5 > t ) the electrostatic repulsion energy between two identical charged spheres may be approximated (1) ... 

Self-assembly of highly charged colloidal spheres can, under the correct conditions, lead to 3D crystalline structures. The highly charged spheres used are either polystyrene beads or silica spheres, which are laid down to give the ordered structures by evaporation from a solvent, by sedimentation or by electrostatic repulsion (Figure 5.34). The structures created with these materials do not show full photonic band gap, due to their comparitively low relative permittivity, although the voids can be in-filled with other materials to modify the relative permittivity. 

Electrostatic Model. A simple model that can account for the observed bond angles in a qualitative way comes from a consideration of electrostatic repulsions of electron pairs. Let us consider electron pairs around an atom as concentrations of charge placed on a more or less spherical surface, and let us assume that the electrons can move in pairs. Barring other forces, the most likely arrangement will be the one where the electron pairs exert the minimum repulsion on each other. This will be achieved when the electrons get as far away from each other as possible. Since the electrons are restricted by our assumption to a sphere, the maximum distance of separation corresponds to a maximum angle between their positions and the center of the sphere. 

For this purpose let us calculate by means of Coulomb s law the electrostatic energy which is set free when to a four-fold positively charged sphere with radius r a varying number of negative ions of the same size is added. As a result of the mutual repulsion of these latter ions they will arrange themselves in such a regular manner that this repulsion becomes a minimum. Thus a tetrahedral arrangement will have preference over a square. 

Glendinning, A.B. Russel, W.B. The electrostatic repulsion between charged spheres from exact solutions to the linearized Poisson-Boltzmann equation. J. Colloid Interface Sci. 1983, 93, 95-111 Carnie, S.L. Chan, D.Y.C. Interaction free energy between identical spherical colloidal... 

When both reactants in a redox reaction are kinetically inert, electron transfer must take place by a tunnelling or outer-sphere mechanism. For a reaction such as 25.46, AG° 0, but activation energy is needed to overcome electrostatic repulsion between ions of like charge, to stretch or shorten bonds so that they are equivalent in the transition state (see below), and to alter the solvent sphere around each complex. 

Modification of the surfaces of coUoidal silica spheres with silane coupling agents enables transfer of the particles to nonpolar solvents. With 3-methacryloxypropyltri-methoxysilane bonded to the surface, the particles have been transferred from water to the polymerizable monomer, methyl acrylate. Electrostatic repulsion due to a low level of residual charge on the particle surfaces cause the dilute dispersions of particles to form a non-close packed colloidal crystalline array (CCA). Polymerization of the methyl acrylate with 200 nm diameter silica spheres in a CC fixes the positions of the spheres in a plastic film by the reactions shown in Figure 11.14. The difriaction... 

An expression for the force of electrostatic repulsion between two charged crossed hemicylindrical surfaces is given in Equation (3.5) and in fact this turns out to be equivalent to interaction between a sphere and a flat plate. Integration of this expression leads directly to the potential energy of electrostatic repulsion, namely. 

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