Surfaces, charged spherical

There is a limit to the charge that the surface of a droplet can sustain, for the electrical stress generated by the surface charges can balance the surface tension forces. At that point the droplet becomes unstable and breaks up. Lord Rayleigh (1882) analyzed the criterion of instability for a conducting droplet using spherical harmonics to describe the modes of oscillation. The natural frequency of the nth mode of oscillation of a droplet was found to be given by... 

G.R. McVey, "Shock Pressure at and Close ro the Surface of Spherical Pentolite Charges Inferred from Optical Measurements , BRL Rept 917(1954) 30) T. Sakurai,... 

Calculation of Coagulation Rate. Here we discuss an interaction potential of two charged particles in a liquid within a framework of DLVO theory. Following this theory, the overall interaction potential U, of charged spherical particles of the same radius R and surface distance d is a sum of a coulombic repulsive force of charged particles and a van der Waals attractive force given by the equation (28) ... 

Before we proceed to the Gouy-Chapman theory of electrical double layers, it is worthwhile to note that relations similar to Equations (45) and (47) can also be derived for double layers surrounding spherical particles. The equation for surface charge density takes the form... 

Develop the equivalent of Equation (47) for the surface charge density of a spherical particle (i.e., the relation between a and (d j//dr) evaluated at the surface). 

As an example, we consider the motion of a spherical particle. C, is the potential at a distance 6 from the particle surface. The distance U I S is the hydrodynamic radius of the particle. It can be larger than the particle radius due to the binding of liquid molecules or ions. Up to this distance the surface charge density amounts to ag. The entire charge of the particle is Q = 47t(R + 5)2ag 4TtR2ag. An electric field of strength E causes a force QE. At constant drift velocity, v0, the force is compensated by the friction force... 

Please note that the electrostatic double-layer force is fundamentally different from the Coulomb force. For example, if we consider two identical spherical particles of radius R you cannot take Eq. (6.1), insert the total surface charge as Qi and Q2, use the dielectric permittivity of water and expect to get a reasonable result. The main differences are the free charges (ions) in solution. They screen the electrostatic field emanating from the surfaces. 

Gibbs free energy of double layer around a spherical particle. We can use an equation similar to Eq. (4.35), we only have to use the total charge of the particle instead of the surface charge density. The total charge of the particle is given by Eq. (5.39). 

The analysis of the first effect can be made by comparing a spherical and an infinite rod-shaped aggregate with the same radius, surface charge density and amphiphile concentration. The infinite rod really corresponds to the condition in thenor-... 

Fig. 6.3. The calculated reduced surface potential = e(ri)/kT versus the logarithm of the amphiphile concentration C (M) with no salt added for a spherical, cylindrical and planar aggregate. The surface charge density has been chosen as fixed at a8 = 0.228 Cm- 2. The radii of the sphere and the cylinder are 1.8 nm...

In Equation (2.183) new surface charges, qex, have been introduced these charges can be described as the response of the solvent to the external field (static or oscillating) when the volume representing the molecular cavity has been created in the bulk of the solvent. We note that the effects of qex in the limit of a spherical cavity coincide with that of the cavity field factors historically introduced to take into account the changes induced by the solvent molecules on the average macroscopic field at each local position inside the medium more details on this equivalence will be given in Section 2.7.4. 

For a given (spherical) surface of atom A with surface area SAa, a (still constant but atom-specific) surface charge density, oA, can be defined such that ... 

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