A Charged Surface and Its Double Layer

Consider a semi-infinite expanse of ionic solution, bounded on one side by a planar solid surface where o is specified. Because of the condition of overall electroneutrality, the surface charge plus the bulk charge must sum to zero. From this and the Poisson and Boltzmann equations, one can derive (Israelachvili 1992) 

Here nsi is the density of species i in the solution next to the solid surface, and n j = o is the known density of species i in the bulk far from the surface, where = 0. For each species, n t can be obtained from t/tj, the value of fr at the solid surface, using the Boltzmann equation (2-41). Thus, Eq. (2-44) can be used to link the surface potential to the density a of charges bound to the solid surface. 

Consider, for example, a symmetric electrolyte which is one for which z = zi = —Z2-An example is NaCl for which zi = +1 for sodium and Z2 = — 1 for chloride. For a symmetric electrolyte, Eqs. (2-41) and (2-44) become 

Equation (2-45) gives the potential yjfs at the surface in terms of the charge a at the surface. Here ooi is the bulk concentration of the cation, such as Na. The potential at a distance X from the surface is obtained by integrating the Poisson-Boltzmann equation, yielding 

In this limit, the concentration of ions from the Boltzmann equation (2-41) can be expressed as 

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