Equation of State for Charged Films

A picture of the electrical lines of force is given in Fig. XV-12 [114] in the plane CD of the ionic groups, it will be a periodic field, whereas a little further into the solution the effect will be more that of a uniformly charged surface. The Donnan treatment is probably best justified if it is supposed that ions from solution penetrate into the region of CD itself and might in fact, lie between CD and AB. 

The Donnan effect acts to exclude like-charged substrate ions from a charged surface region, and this exclusion, as well as the concentration of oppositely charged ions, can be expressed in terms of a Donnan potential pD. Thus for a film of positively charged surfactant ions S one can write 

In the interfacial region, electroneutrality requires that c, = (5 ) + c, so that Eq. XV-4 becomes 

(S ) = lOOOr/r, where F is the surface excess in moles per square centimeter, or (5 ) = 1000 X IO /Nto, where a is in per molecule. With these substitutions, Eq. XV-6 may be solved for I/d to give 

At this point an interesting simplification can be made if it is assumed that r, as representing the depth in which the ion discrimination occurs, is taken to be just equal to 1/x, the ion atmosphere thickness given by Debye-Hiickel theory (see Section V-2). In the present case of a 1 1 electrolyte, k = (8ire V/1000eitr) / c /, and on making the substitution into Eq. XV-7 and inserting numbers (for the case of water at 20°C), one obtains, for t/ o in millivolts  

---