The Electric Double-layer Around a Sphere

In many applications, the electric double layer around spherical particles is studied. If the radius of the particle, J p, is much larger than the Debye length, we can treat the double layer as planar. Otherwise, we have to consider the Poisson-Boltzmann equation for spherical symmetry. 

This linearized Poisson-Boltzmann equation is solved by [380, 385] 

For high potentials, one has to solve the full Poisson-Boltzmaim equation in radial coordinates. To our knowledge, no analytical solution of Eq. (4.25) has been reported. In analogy to Eq. (4.22), a good approximation for large xRp is [386] 

In many cases, we have an idea about the number of charged groups on a surface. Then, we might want to know the potential. The question is how are surface charge a and surface potential tj)o related. This question is also important because if we know a versus rj)o we can calculate do/drJ)Q. This is basically the capacitance of the double layer and can be measured. The measured capacitance can be compared with the theoretical result to verify the whole theory. 

Grahame derived an equation between a and t jQ based on the Gouy-Chapman theory. We can deduce the equation easily from the so-called electroneutrality condition. This condition demands that the total charge, that is, the surface charge plus the charge of the ions in the whole double layer, must be zero. The total charge in the double layer is g dx and we get [388] 

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