Arbitrary Potential Symmetrical Electrolyte

Now we solve the original nonlinear Poisson-Boltzmann equation (1.20). If the plate is immersed in a symmetrical electrolyte of valence z and bulk concentration n, then Eq. (1.20) becomes 

By taking into account Eq. (1.22), we find that constant =—2k so that Eq. 

Since y and dyidx are of opposite sign, we obtain from Eq. (1.33) 

FIGURE 1.4 Potential distribution y x) = ze J/ x)/kT around a positively charged plate with scaled surface potential yo = ze[j/JkT. Calculated foryo= U 2, and 4. Solid lines, exact solution (Eq. (1.37)) dashed lines, the Debye-Hiickel linearized solution (Eq. (1.25)). 

FIGURE 1.5 Concentrations of counterions (anions) (x) and coions (cations) n+ x) around a positively charged planar surface (arbitrary scale). Calculated from Eqs. (1.3) and (1.26)foryo = 2. 

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Field saturation. Consider a particle occupying a convex open domain CR3 (or ft2) with a smooth boundary du>, charged to the electric potential > 0, at equilibrium with an infinite solution of a symmetric electrolyte of a given average concentration. (Properties described below are directly generalizable to an arbitrary electrolyte or electrolyte mixture.)... 

Consider a charged plate with arbitrary surface potential ij/o in a symmetrical electrolyte solution of valence z and Debye-Hiickel parameter k. We take an x-axis perpendicular to the plate surface with its origin at the plate surface so that the region x > 0 corresponds to the solution phase while the region x < 0 to the plate interior. Equation (1.37) (or Eq. (1.38)) for the potential distribution ij/(x) around the surface in the region far from the surface, that is, at large kx, takes the form... 

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