The Electrical Double Layer Gouy-Chapman Theory

6 THE ELECTRICAL DOUBLE LAYER GOUY-CHAPMAN THEORY 

The theoretical inconsistencies inherent in the Poisson-Boltzmann equation were shown in Section 11.4 to vanish in the limit of very small potentials. It may also be shown that errors arising from this inconsistency will not be too serious under the conditions that prevail in many colloidal dispersions, even though the potential itself may no longer be small. Accordingly, we return to the Poisson-Boltzmann equation as it applies to a planar interface, Equation (29), to develop the Gouy-Chapman result without the limitations of the Debye-Hiickel approximation. 

The left-hand side of this equation is the derivative of (dip/dx)2 therefore 

The integration constant in this expression is easily evaluated if we define the potential in the solution at x = oo to be zero. At the same limit, dip/dx also equals zero. In view of these conventions, Equation (53) becomes 

This result may be integrated further if we restrict the electrolyte in solution to the symmetrical z z type. In that case, Equation (54) can be written as 

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