IV. On the interaction of colloidal particles.—III. Some theorems concerning the free energy of double layers

Abstract

Making use of the Poisson-Boltzmann equation, a number of theorems relating to the free energy of the electric double layers of colloidal particles in dilute sols are developed. A general proof is given for the equivalence of two methods for determining this free energy, viz. by (i) using the fictitious process of charging the excess ions in the double layers subject to the condition of constant surface density of ions and (ii) imagining that the particles are charged by bringing ions from the interior of the dispersion medium to the surface. In the case of two infinite parallel plates, the difficulty that arises due to the infinity of the surface potential, when all the ions are supposed completely discharged, is satisfactorily explained. The present work is an extension of some of the results obtained by Verwey and Overbeek in their recent book on the theory of the stability of lyophobic colloids. Our theorem of the equivalence of the two methods of evaluating the free energy is more general than a similar theorem proved by Casimir in this book. We are able to derive the Casimir theorem by introducing special conditions.