Electrostatic interaction between two identical surfaces

An important case is the interaction between two identical parallel surfaces of two infinitely extended solids. It is, for instance, important to understand the coagulation of sols. We can use the resulting symmetry of the electric potential to simplify the calculation. For identical solids the surface potential -ipo on both surfaces is equal. In between, the potential decreases (Fig. 6.9). In the middle the gradient must be zero because of the symmetry, i.e. d f(f x/ 2)/df = 0. Therefore, the disjoining pressure in the center is given only by the osmotic pressure. Towards the two surfaces, the osmotic pressure increases. This increase is, however, compensated by a decrease in the Maxwell stress term. Since in equilibrium the pressure must be the same everywhere, we have  

For low potentials we can further simplify this expression. Therefore we write the exponential functions in a series and neglect all terms higher than the quadratic one  

It remains to find ipm. If the electric double layers of the two opposing surfaces overlap only slightly (x XD), then we can approximate 

In order to calculate the Gibbs free interaction energy per unit area we still have to integrate  

If we use expression (4.22) for ip, which is also valid at higher potentials, we get 

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