Spherical Geometry The Micelle Model

The final system we consider in this tutorial is that of a spherical charged polyelectrolyte which serves as a model for a colloidal particle, micelle, or globular protein. The primary characteristic that differs between modeling a micelle and a protein at this level of representation is the much greater surface charge density associated with a micelle. A secondary, but still important, consideration, is the pK, values of the protein ionization sites, which might require use of boundary condition BC3 instead of BC2 as assumed below. ° 

In a spherical coordinate system whose origin is at the center of a charged sphere of radius a (including a common electrolyte ion radius) and total charge QaSo, the cell model PB equation and boundary conditions are 

While no exact analytical solution to Eq. [292] is available, approximate nonlinear expressions corresponding to the PGC and NLDH solutions as well as the weak-field Debye-Hiickel solution are given below. 

The analytical approximation for the potential profile near a highly charged surface obtained earlier, given by Eqs. [147] and [149] or Eqs. [152] and [154], is applicable to a spherical micelle with Zc again representing the highest counterion valence and the screening constant. The potential at the surface, Eq. [153], then becomes 

Provided that the curvature correction term is small, this result is valid for symmetric salts and for mixed-salt divalent counterion concentrations satisfying 

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