Surfaces Cylinders and Spheres

Bulk Model Perturbed Gouy-Chapman Approximation 

Consider now the one-dimensional bulk PB equation (in d dimensions) that corresponds to the Debye-Hiickel equation 

Inserting this assumed solution into Eq. [136] and retaining terms of order lower than 1/r, we find 

We note that for the spherical case of d = 3, the only approximation in Eq. [139] is use of Eq. [140], Thus, the solution developed below will be almost exact for a sphere, numerically as well as analytically, and we will see that the asymptotic form of the solution exactly matches the spherical Debye-Hiickel potential. For the cylindrical case, however, Eq. [139] can be justified only on numerical grounds since the asymptotic solution should behave as a sum of modified Bessel functions. To put Eq. [139] into a form more amenable to solution, we temporarily change variables from r to [ 0 and introduce the following substitutions 

The surface potential is found using a generalization of Grahame s equation [25] obtained by applying Eq. [15]  

---