Free energy of a double layer system

To this end we shall consider, in the present chapter, the general problem of the free energy of a double layer system. 

As stated above, the free energy is the amount of work to be performed in building up, by some reversible and isothermal process, the double layers of the system. The formation of these double layers occurs spontaneously when the wall and the solution are brought into contact. Hence we can state, a priori, that the free energy of a double layer system must be a negatme quantity (work js gained by the formation). 

It is well known that Verwey and Overbeek and Derjaguin and Landau " " (DLVO) have given a quantitative treatment of the interaction of electrical double layers. According to them, the free energy of a double layer may be expressed as a difference between the surface energy G, of the system in its equilibrium state and the surface free energy G of a standard state in which no double layer is present ... 

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. 

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. 

For this case a method has been used by Verwey and Over-BEEK consisting in a calculation of the distribution of charge and potential around two spherical particles and a derivation of the free energy of the double layer system by application of eq. (1) ... 

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 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. 

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... 

In the DLVO framework, the free energy of a system of two overlapped double layers is composed of an electrostatic energy, an entropic contribution due to the ions in the double layer, and a chemical term, where applicable.4... 

The determination of the free energy of a system of double layers may foUow a number of different ways. We shall here consider two methods. 

The common point in the work of C. and R. and in that of L. and D. is that they consider the total energy of the electric field of the double layer system, instead of its free energy, as the energy quantity determining the work associated with a variation of the distance of the plates or particles. 

FREE ENERGY OF A SYSTEM OF ELECTROCHEMICAL DOUBLE LAYERS... 

Free energy of a system of electrochemical double layers. ... 

Referring to Section V-2, the double-layer system associated with a surface whose potential is some value j/o requires for its formation a free energy per unit area or a t of... 

Since the equations of state of the system are summarized by the curves in Figure 2, all interesting thermodynamic properties of the interface will have a simple representation in such a diagram. We shall consider the free energy of formation of a single charged surface and the interaction free energy due to the overlap of two identical planar double layers. 

Equation 37 gives the analytical form of the free surface energy of the diffuse part of the double layer and has been derived by a number of authors [8,9, 40] for 1 1 ionic surfactants. For systems with mixed valences, the integral in Eq. 34 is usually not available in close analytical expressions and numerical integration is often required. 

The total free energy of the system is composed of several contributions an electrostatic energy due to the distribution of charges, an energy due to the additional interactions of ions (not included in the mean potential), an entropic term due to the mobile ions, and a chemical free energy which is responsible for the formation of the double layer. 

For a neutral gas, the adsorption (which is proportional to the free energy of the system, see Eq. (8)) can be calculated easily. For charged particles one should account in the Gibbs adsorption equation for the adsorption of all particles of the system (including those responsible for the charging of the surface [23]). Therefore, one should first identify the mechanism of formation of the double layer. In this case,... 

The free energy F of the system contains a polymer (Fp), a double layer (Fa), and an adsorption (Fa) contribution. 

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