Free energy of the double layer

S. Fevine and A. Suddaby Simplified Forms for Free Energy of the Double-Layers of Two Plates in a Symmetrical Electrolyte. Proc. Phys. Soc. A 64, 287 (1951). 

Calculation of the Repulsive Force At constant surface charge, the part of the free energy of the double layer per unit surface area that enters in the calculation of the force is given by the expression1011... 

The above two contributions to the free energy of the double layer are both positive. The third term, the chemical free energy contribution, which is responsible for the spontaneous formation of the double layer, should be negative and larger than the sum of the previous two terms. 

The total free energy of the double layer is obtained by adding eqs 1, 3, and 7... 

Although this expression presents the basis of the free energy of the double layer, it is not truly expressive of the parameters that must be considered. According to Verwey and Overbeek and Ikeda, " the free energy of a system of reversible double layers is determined by the following ... 

The absolute value of the free energy of the double layer system, therefore, is generally less than. ( and only in the limiting case of very small potentials equal to t trJ/Q. 

We already mention in 4 of Chapter III that this decrease of the double layer charge may well be considered to be the most important feature in the interaction of two double layers. It follows from equation (23) that the free energy of the double layer increases (becomes less negative) with decreasing charge. Now an increase of the free energy is equivalent to a repulsion between the plates, so this decrease of the charge is the primary cause of the repulsion between two double layers. This point is dealt with in more detail in Chapter V. 

In these two expressions, eq. (35) and eq. (36), we dispose of the necessary equations for the calculation of the variation of the free energy of the double layer system as a consequence of the interaction. 

Applying the same principles as used in Part II we now have to calculate the change in free energy accompanying the approach of two colloidal particles. To this end we shall first consider the free energy of the double layer system for spherical particles, and add to it the free energy of the London-Van Der Waals attraction forces. From the curves of total free energy so constructed we shall derive the criteria for the stability of colloids. 

Whereas, when is large, we have at our disposal a direct, transformation of the energy of interaction of two plates into that of two spheres, such a simple transformation does not exist when xa is small. In that case we follow essentially the same method which we used to calculate the interaction of flat plates, viz. we first calculate the electric field in the double layer around, the particles, and after that the free energy of the double layers. This method was used by Levine and Dube and we can use several parts of their calculations, although our conclusions differ from, and are in part even opposite to theirs. 

It follows immediately from equations (63) and (64) that, in order to evaluate the free energy of the double layer system, it is necessary to express the charge of the particles as a function of the potential and the geometric configuration. To this end the charge Q of one particle is expressed with the aid of Gauss s theorem, i.e. 

On the basis of our considerations in Chapter III, we may describe this error (the use of field energy in stead of the free energy of the double layer) as consisting in the neglect of at least one of the two following effects (1) entropy effects due... 

The double layer itself seems a well defined concept, but as it can only exist in conjunction with two phases with a certain extension, the free energy of the double layer has to be defined as a part of the free energy of that whole system of two phases, and it is not selfevident. which part. 

The integral in this expression may be called the free energy of the double layer. In two cases the integration leads to a particularly simple result. If the total concentration of electrolytes is high the capacity of the double layer is practically equal to that of the Stern layer (c/. eq. (54)) and may be treated as a constant. (See> however, 5d, p. 156). For small concentrations of electrolyte and small potentials again the capacity is a constant, now proportional to x. Jn these two cases... 

For the diffuse double layer the result is somewhat less simple. The relation between a and 4 has been given in eq. (48), p. 130 and the free energy of the double layer becomes... 

There is another way of approach to the free energy of the double layer, which is more convenient for explicit calculations in some cases. (Interaction of two flat double layers, see chapter VI 3b, p. 252) and which shows clearly the dose connection between the activity of strong electrolytes according to Debye and Hucksl and the free energy of the double layer. 

An application of this second method of evaluating the free energy of the double layer will be given in chapter VI in the treatment of the interaction of two flat double layers. 

Application of the charging method of Debye and Huckel (c/ 4 h) is instructive in this case because the free energy of the double layer does not tend to 2 ero when the ionic charges do so at least not when the condition of constant surface potential is maintained ... 

Taking (86) and (87) together we find for the free energy of the double layer around a spherical particle... 

The interface between mercury and an aqueous solution is a very favourable case for studyir the properties of the electrical double layer By measuring the surface tension the free energy of the double layer is directly accessible The potential difference between the two phases can be altered within wide limits by applying an external potential difference From the relation between surface tension and potential the charge and the capacity of the double layer can be derived by differentiation (see 3, eq (20), 4, p 131) Moreover, these two magnitudes can also be directly determined by experiment So it is not to be wondered at that our best and most extensive data on the double layer are coming from the mercury water interface. 

In chapter IV the free energy of the double layer has been treated extensively. Two extreme cases have been considered there, one, the clcctrocapillary case (chapter IV, 3a,p. 119) where the double layer was formed by an external eturent and the other (chapter IV, 3b, p. 122), the reversible one, in which the double layer was formed by adsorption of ions (or electrons in the case of redox-equilibria) from the solution. 

By straightforward thermodynamic reasoning it has been proved in chapter IV (eq. (29) p. 123, cq. (70,71) p. 140) that the free energy of the double layer, or of a system of double layers, is equal to... 

---