Potential distribution around spherical surfaces

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

For studying the stability of colloidal particles in suspension (Chapter 13) or for determining the potential at the surface of particles (Chapter 12), one often needs expressions for potential distributions around small particles that have curved surfaces. Solving the Poisson-Boltzmann equation for curved geometries is not a simple matter, and one often needs elaborate numerical methods. The linearized Poisson-Boltzmann equation (i.e., the Poisson-Boltzmann equation in the Debye-Hiickel approximation) can, however, be solved for spherical electrical double layers relatively easily (see Section 12.3a), and one obtains, in place of Equation (37),... 

Follow Debye and Hiickel [380] and calculate the potential distribution around a spherical particle with a low surface potential tpo. Use polar coordinates. For this geometry, only the dependency in r remains and the Poisson equation reads... 

The general behavior of the numerical results of Eq. (20) for the potential distribution between the particle surfaces is shown in Fig. 3 for the constant surface potential interaction between two spherical particles with radii of 20 nm and surface potentials of 50 mV. The effect of the interaction is to distort the potential profiles around the particles. At a separation distance of about 10-particle radii ( 200 nm), the potential profiles are almost identical to those around the isolated particles. However, the distortion of the potential profiles becomes stronger with decreasing the separation distance between the surfaces. The strong distortion of the potential profiles increases the repulsive interaction between the particles as shown below. 

Provided that the bulk concentrations remain constant, the flux J of Eq. (9-3) is also constant. Consider a spherical distribution of potential reactants B around a particular molecule of A. The surface area of a sphere at a distance r from A is 4irr2. Thus, the expression for the flow of B toward A is... 

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