Diffuse double layer free energy

We recall that the first integral in Equation 23a represents the change in electrical free energy in forming the diffuse double layer. This contribution to f, the free energy of formation of the charged interface, is positive and hence represents an unfavourable component which opposes the formation of the charged interface. 

Figure 6.1. Free-energy change for the general electrochemical reaction, Eq. (6.6) initial state, Ox, in the bulk of the solution, outside the diffusion double layer final state. Red, in the bulk of the solution outside the diffuse double layer.

The main, currently used, surface complexation models (SCMs) are the constant capacitance, the diffuse double layer (DDL) or two layer, the triple layer, the four layer and the CD-MUSIC models. These models differ mainly in their descriptions of the electrical double layer at the oxide/solution interface and, in particular, in the locations of the various adsorbing species. As a result, the electrostatic equations which are used to relate surface potential to surface charge, i. e. the way the free energy of adsorption is divided into its chemical and electrostatic components, are different for each model. A further difference is the method by which the weakly bound (non specifically adsorbing see below) ions are treated. The CD-MUSIC model differs from all the others in that it attempts to take into account the nature and arrangement of the surface functional groups of the adsorbent. These models, which are fully described in a number of reviews (Westall and Hohl, 1980 Westall, 1986, 1987 James and Parks, 1982 Sparks, 1986 Schindler and Stumm, 1987 Davis and Kent, 1990 Hiemstra and Van Riemsdijk, 1996 Venema et al., 1996) are summarised here. 

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. 

In a third step the counterions are released from the surface. Stimulated by thermal fluctuations, they partially diffuse away from the surface and form the diffuse double layer. The entropy and, at the same time, the energy increases. One can show that both terms compensate, so that in the third step no contribution to the Gibbs free energy results. 

The contribution of the free energy of the diffuse double layer, AGadb to the interfacial adhesion may be estimated as follows (36) ... 

Grahame, D. C. 1950. Effects of dielectric saturation upon the diffuse double layer and the free energy of hydration of ion. J. Chem. Phys. 18 903-909. 

This result is Identical to (3.2.3], for purely diffuse double layers. In words, the Gibbs energy for a double layer with a charge-free inner layer, but vdth specific adsorption, is the same as that for a purely diffuse layer, the quantitative difference being that, at given y/°, a° is lower. No additional terms are needed for the charge-free layer because all ions are diffuse and, hence, do not contribute. Equation [3.6.58 or 59] can in this approximation also be written as... 

Stability of Microemulsions. The first attempt to describe the microemulsion stability in terms of different free energy components was made by Ruckenstein and Chi (55) who evaluated the enthalpic (Van der Waals potential, interfacial free energy and the potential due to the compression of the diffuse double layer) and entropic... 

In the earlier paper, the free energy of activation for the rate process in which a vesicle approached the synaptic membrane was treated as an electrostatic interaction between a positive point charge and the field of the membrane. It would be more appropriate to consider the interaction between two diffuse double layers. Such a problem was considered in detail by Verwey and Overbeek, who obtained the following expression for the potential energy per unit area between two planar double layers having identical surface potentials and separated by a distance d ... 

In this case, as indicated above, the colloid stability is controlled by the form of the interaction free-energy curve as a function of particle separation. The DLVO theory in its original form considers just two contributions to this energy, namely the attractive van der Waals potential and the repulsive potential that arises when the diffuse double layers round the two particles overlap. To put this in a quantitative form we need to examine more closely the origin of the curves shown in Figures 3.6 and 3.7. 

The situation is still more complex in the presence of surfactants. Recently, a self-consistent electrostatic theory has been presented to predict disjoining pressure isotherms of aqueous thin-liquid films, surface tension, and potentials of air bubbles immersed in electrolyte solutions with nonionic surfactants [53], The proposed model combines specific adsorption of hydroxide ions at the interface with image charge and dispersion forces on ions in the diffuse double layer. These two additional ion interaction free energies are incorporated into the Boltzmann equation, and a simple model for the specific adsorption of the hydroxide ions is used for achieving the description of the ion distribution. Then, by combining this distribution with the Poisson equation for the electrostatic potential, an MPB nonlinear differential equation appears. 

D. C. Grahame, J. Chem. Phys., 18, 903 (1950). Effects of Dielectric Saturation upon the Diffuse Double Layer and the Free Energy of Hydration of Ions. 

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... 

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. 

For one-electron transfer reactions occurring via outer-sphere mechanisms, wp and ws can be estimated on the basis of electrostatic double-layer models. Thus, if the reaction site lies at the outer Helmholtz plane (o.H.p.), wp = ZFd and ws = (Z - 1 )Fcharge number of the oxidized species and (j>d is the potential across the diffuse layer. Rewriting eqn. (7) in terms of rate constants rather than free energies yields the familiar Frumkin equation [8]... 

The stability of inverse micelles has been treated by Eicke (8,9) and by Muller (10) for nonaqueous systems, while Adamson (1) and later Levine (11) calculated the electric field gradient in an inverse micelle for a solution in equilibrium with an aqueous solution. Ruckenstein (5) later gave a more complete treatment of the stability of such systems taking both enthalpic (Van der Waals (VdW) interparticle potential, the first component of the interfacial free energy and the interparticle contribution of the repulsion energy from the compression of the diffuse part of the electric double layer) and entropic contributions into consideration. His calculations also were performed for the equilibrium between two liquid solutions—one aqueous, the other hydrocarbon. 

The discussion of the relative stability of solutions with inverse micelles and of liquid crystals containing electrolytes may be limited to the enthalpic contributions to the total free energy. The experimentally determined entropy differences between an inverse micellar phase and a lamellar liquid crystalline phase are small (12). The interparticle interaction from the Van der Waals forces is small (5) it is obvious that changes in them owing to added electrolyte may be neglected. The contribution from the compression of the diffuse electric double layer is also small in a nonaqueous medium (II) and their modification owing to added electrolyte may be considered less important. It appears justified to limit the discussion to modifications of the intramicellar forces. 

The earlier concepts of microemulsion stability stressed a negative interfacial tension and the ratio of interfacial tensions towards the water and oil part of the system, but these are insuflBcient to explain stability (13). The interfacial free energy, the repulsive energy from the compression of the diffuse electric double layer, and the rise of entropy in the dispersion process give contributions comparable with the free energy, and hence, a positive interfacial free energy is permitted. 

In order to describe the stability of fine disperse systems stabilized by diffuse ionic layers, one has to use the total free energy of interaction between particles, instead of the energy per unit film area, and compare the barrier height,, to the thermal energy, kT. For us to be able to use the solution derived for the case of plane-parallel surfaces, let us introduce some effective area of particle contact, Se[. Then the potential barrier height for the particles can be expressed as = A5 max St(. When diffuse part of electrical double... 

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