Free Energy of Double-Layer Interaction

In this section, we calculate the interaction energy V from the difference between the Helmholtz free energy F of the system of two interacting particles at a given separation and those at infinite separation (Fig. 8.10), namely. 

The form of the Helmholtz free energy F depends on the type of the origin of surface charges on the interacting particles. The following two types of interaction, that is (i) interaction at constant surface charge density and (ii) interaction at constants surface potential are most frequently considered. We denote the free energy F for the constant surface potential case by F and that for the constant surface 

FIGURE 8.10 Difference between the free energy F R) of two interacting particles at separation R and that for infinite separation gives the potential energy of the double- 

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ALTERNATIVE EXPRESSION FOR THE ELECTRIC PART OF THE FREE ENERGY OF DOUBLE-LAYER INTERACTION... 

The electric part Eei of the free energy of double-layer interaction (Eq. (5.4)) can be expressed in a different way on the basis of the Debye charging process in which all the ions and the particles are charged simultaneously from zero to their full charge. Let /I be a parameter that expresses a stage of the charging process and varies from 0 to 1. Then F x can be expressed by... 

If the dissociation of the ionizable groups on the particle surface is not complete, or the configurational entropy Sc of adsorbed potential-determining ions depends on N, then neither of ij/o nor of cr remain constant during interaction. This type of double--layer interaction is called charge regulation model. In this model, we should use Eqs. (8.35) and (5.44) for the double-layer free energy [ 11-13]. 

Application of DLVO Theory. Our approach to determine the contribution of double-layer interaction and van der Waals potentials to AGads involves comparing differences in the magnitudes of AGads found on the same solid but with different solution conditions, to potentials (U), or theoretical free energy components, evaluated from the DLVO-Lifshitz theory of colloid stability. 

The same graphical method can also be used to illustrate the nature of the double layer interaction free energy and to bring out a simple physical result which can be used to check numerical algorithms commonly used to calculate the interaction free energy. 

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. 

What is the Gibbs free energy of an electric double layer The energy of an electric double layer plays a central role in colloid science, for instance to describe the properties of charged polymers (polyelectrolytes) or the interaction between colloidal particles. Here, we only give results for diffuse layers because it is simpler and in most applications only the diffuse layer is relevant. The formalism is, however, applicable to other double layers as well. 

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. 

Therefore, the present treatment of the double layer interaction leads to the same results for the interaction free energy as the imaginary charging approach for systems of arbitrary shapes and constant surfece potential or constant charge density and to the same results as the Langmuir equation for parallel plates and arbitrary surface conditions. It can be, however, used for systems of any shape and any surfece conditions, since it does not imply any of the above restrictions. 

The present approach reduces to the traditional ones within their range of application (imaginary charging processes for double layer interactions between systems of arbitrary shape and interactions either at constant surface potential or at constant surface charge density, and the procedure based on Langmuir equation for interactions between planar, parallel plates and arbitrary surface conditions). It can be, however, employed to calculate the interaction free energy between systems of arbitrary shape and any surface conditions, for which the traditional approaches cannot be used. 

The purpose of this article is to present a model and to calculate on its basis the metastable equilibrium thicknesses of the film as a function of the applied pressure. In section II, the interaction energy of the film was calculated, assuming planar interfaces free of thermal fluctuations. The double layer interaction was calculated by accounting for the charge recombination at the surface with increasing electrolyte concentration. An approximate... 

Once the system of eqs 3 and 4 is solved under the boundary conditions (7a—d), the total free energy of the system can be calculated by adding the van der Waals interactions between the surfaces to the double layer free energy composed of electrostatic, entropic, and chemical contributions,11 and the stability ratio can be calculated in the usual manner.18... 

In summary, the polarization model represents an extended Poisson-Boltzmann approach, in which the hydration and the double layer are not independent interactions, but are intimately coupled to each other, via an electrostatic coupling between the fields ip(z) and m(z). These fields can be calculated by solving Eqs. (2) (3) and (4), and the total free energy of the (5b) system can be obtained by summing up the terms provided by... 

For the ensuing discussion we take i,j = 1,2 i / j, The potential energy V of the double layer interaction is given by the equality V = F-F0,4 where F0 is the free energy for two single spheres. If the densities [Pg.118]

G. M. Bell, S. Levine, and L. N. McCartney, Approximate Method of Determining the Double-Layer Free Energy of Interaction between Two Charged Colloidal Spheres, J. Colloid Interface Sci. 33 (3), 335-359 (1970). 

To get the interaction potential we must first evaluate the free energy of formation of the electrical double layer between two charged bodies. This is defined as the work done in charging up the surfaces. The process by which uniformly charged surfaces are charged up from a neutral reference state has been discussed by Yerwey and Overbeek [4], who have shown that the electrostatic work of charging a surface is given by the simple formula... 

In the presence of the double layer the free Gibbs energy of the analyte interaction with the surface will be... 

The potential energy V Qi) of the double-layer interaction per unit area between two parallel plates 1 and 2 at separation h is given by Eq. (9.66). The double-layer free energy F (h) of two parallel plates 1 and 2 at constant surface potential ij/o is given by Eq. (8.54), namely. 

An additional unique feature of electrosorption is that the coverage is a function of potential, at constant concentration in solution. Thus, we can discuss two types of isotherms those yielding 0 as a function of C and those describing the dependence of 0 on E. This is not a result of faradaic charge transfer. Neither is it due to electrostatic interactions of the adsorbed species with the field inside Ihc compact part of the double layer, since a potential dependence is observed even for neutral organic species having no permanent dipole moment. As we shall see, it turns out that the potential dependence of 0 is due to the dependence of the free energy of adsorption of water molecules on potential. 

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