Diffuse electric layer

It should be recalled that the term surface potential is used quite often in membranology in rather a different sense, i.e. for the potential difference in a diffuse electric layer on the surface of a membrane, see page 443.) It holds that 0 = 0 + X (this equation is the definition of the inner electrical potential 0). Equation (3.1.2) can then be written in the form... 

E. J. W. Verwey and K. F. Niessen described the electric double layer at ITIES using a simplifying assumption that consists of only two diffuse electric layers, each in one of the phases (see Fig. 4.1C). The overall potential difference between the two phases, Ao0, is thus given by the relationship... 

If the electric field E is applied to a system of colloidal particles in a closed cuvette where no streaming of the liquid can occur, the particles will move with velocity v. This phenomenon is termed electrophoresis. The force acting on a spherical colloidal particle with radius r in the electric field E is 4jrerE02 (for simplicity, the potential in the diffuse electric layer is identified with the electrokinetic potential). The resistance of the medium is given by the Stokes equation (2.6.2) and equals 6jtr]r. At a steady state of motion these two forces are equal and, to a first approximation, the electrophoretic mobility v/E is... 

For semiconductor electrodes and also for the interface between two immiscible electrolyte solutions (ITIES), the greatest part of the potential difference between the two phases is represented by the potentials of the diffuse electric layers in the two phases (see Eq. 4.5.18). The rate of the charge transfer across the compact part of the double layer then depends very little on the overall potential difference. The potential dependence of the charge transfer rate is connected with the change in concentration of the transferred species at the boundary resulting from the potentials in the diffuse layers (Eq. 4.3.5), which, of course, depend on the overall potential difference between the two phases. In the case of simple ion transfer across ITIES, the process is very rapid being, in fact, a sort of diffusion accompanied with a resolvation in the recipient phase. 

The effect of adsorption on the electroreduction of hydrogen ions, i.e. the Volmer reaction, is strongly affected by the potential difference in the diffuse electrical layer (Eq. 5.3.20). 

The first step is so fast that it can hardly be measured experimentally, while the second step is much slower (probably as a result of the repulsion of negatively charged species, R and R2-, in the negatively charged diffuse electric layer). The reduction of an isolated benzene ring is very difficult and can occur only indirectly with solvated electrons formed by emission from the electrode into solvents such as some amines (see Section 1.2.3). This is a completely different mechanism than the usual interaction of electrons from the electrode with an electroactive substance. 

As the membrane has a surface charge leading to formation of a diffuse electrical layer, the adsorption of ions on the BLM is affected by the potential difference in the diffuse layer on both outer sides of the membrane 02 (the term surface potential is often used for this value in biophysics). Figure 6.12 depicts the distribution of the electric potential in the membrane and its vicinity. It will be assumed that the concentration c of the transferred univalent cation is identical on both sides of the membrane and that adsorption obeys a linear isotherm. Its velocity on the p side of the membrane (see scheme 6.1.1) is then... 

At low electric fields [0(lV/cm)] the speed (U) of the particles is directly proportional to the applied field ( ) and hence we can define a parameter called the electromobility ( x) of the particles, given by U/E. Using the Poisson-Boltzmann theory of the diffuse electrical layer next to a charged surface, a simple relationship between p and V /o can be... 

Let us consider two parallel plates immersed in an electrolyte solution. As is well-known the ions form a diffuse electric layer, bearing a charge opposite in sign to that of the surface, sandwiched between the two plates. Because of the strong interactions between the ions and water molecules, the ions are hydrated and the simplifying assumption is made that their configuration is the same everwhere in the system. Of course, if the ion-surface interactions favor adsorption of dehydrated counterions,... 

The direct measurement of the various important parameters of foam films (thickness, capillary pressure, contact angles, etc.) makes it possible to derive information about the thermodynamic and kinetic properties of films (disjoining pressure isotherms, potential of the diffuse electric layer, molecular characteristics of foam bilayer, such as binding energy of molecules, linear tension, etc.). Along with it certain techniques employed to reveal foam film structure, being of particular importance for black foam films, are also considered here. These are FT-IR Spectroscopy, Fluorescence Recovery after Photobleaching (FRAP), X-ray reflectivity, measurement of the lateral electrical conductivity, measurement of foam film permeability, etc. 

When circular microscopic foam films (equilibrium or thinning) are studied it is necessary to know the pressure in the meniscus of the liquid being in contact with the film (see Fig. 2.2 A, B, C). In some cases it is very important to know the precise value of the capillary pressure, for example, in the calculation of low disjoining pressures n and the potential of the diffuse electric layer [Pg.50]

The development of the thermodynamics of thin films is related to the problem of stability of disperse systems. An important contribution to its solving are the works of the Russian scientists Derjaguin and Landau [1] and the Dutch scientists Verwey and Overbeek [2], known today as the DVLO theory. According to their concept the particular state of the thin liquid films is due to the change in the potential energy of molecular interaction in the film and the deformation of the diffuse electric layers. The thermodynamic characteristic of a state of the liquid in the thin film, as shown in Section 3.1, appears to be the dependence of disjoining pressure on film thickness, the n(/t) isotherm. The thermodynamic properties of... 

The authors have adopted the term diffuse electric layer for the diffuse part of the double electric layer other authors prefer to call it double diffuse layer (DDL). 

Obviously, this approximation looses meaning at almost complete overlapping of the diffuse electric layer and ym - y0. 

There are various ways to calculate nf/. The first expression for IL/ has been derived by Frumkin [20] in 1938 who calculated it as osmotic pressure. Derjaguin and Landau [1] in 1941 have calculated disjoining pressure as a change in film pressure. Some years later (1948) Verway and Overbeek [2] evaluated n<./ by the change in the energy of the diffuse electric layer. Scheludko [134] has determined Iin a very simple way as a deformation of the two opposite diffuse electric layers at the film surfaces. Later various correction to Y ei have been introduced [e.g. 135-143]. 

As is seen from the figure, at film thickness less than 20 nm, there is a maximum in the curve which indicates that either the theory about Ylei or flvw or both are not correct. Lyklema and Mysels [168] attributed this discrepancy to nei though their studies were performed at counterion concentration from 10 2 to 10 1 mol dm"3 the maximum in the figure corresponds to this concentration range at which the modem theory of the diffuse electric layer does not require considerable corrections of the classical DLVO-theory. There exist several other experimental facts that indicate deviation from the DLVO-theory at foam films of thicknesses less than 20 nm (see Section 3.4.1.3). 

Potential of the diffuse electric layer at solution/air interface... 

Under certain conditions aqueous electrolyte solutions form foam films of equilibrium thickness. For a microscopic horizontal film this thickness is determined by the positive component of disjoining pressure (FU) which depends on the potential of the diffuse electric layer at the foam film/air(gas) interface. 

Fig. 3.21. -potential of the diffuse electric layer vs. surfactant concentration at aqueous solution/air... 

Table 3.2 presents the potential values at the surfactant solution/air interface, corresponding to the plateau in the h(C) curves and the values of the charge density Ob of the diffuse electric layer, calculated according to the following formula [161,169]... 

Fig. 3.25 presents the aqueous solutions in the absence of a surfactant at constant ionic strength (HC1 + KC1) [186,197], It can be seen that at pH > 5.5, op-potential becomes constant and equal to about 30 mV. At pH < 5.5 the potential sharply decreases and becomes zero at pH 4.5, i.e. an isoelectric state at the solution surface is reached. As it is known, the isoelectric point corresponds to a pH value at which the electrokinetic phenomena are not observed. Since in the absence of the potential of the diffuse electric layer, the electrokinetic potential (zeta-potential) should also be equal to zero, the isoelectric point can be used to determine pH value at which isoelectric state is controlled by the change in pH. This is very interesting, for it means that the charge at the surface of the aqueous solutions is mainly due to the adsorption of H+ and OH" ions. Estimation of the adsorption potential of these ions in the Stem layer (under the assumption that the amounts of both ions absorbed are equal) showed that the adsorption potential of OH" ions is higher. It follows that ( -potential at the solution/air interface appears as a result of adsorption of OH" ions. 

Fig. 3.30. (a) Diffuse electric layer potential and (b) surface charge density as a function of pH ... 

The effect of pH on the diffuse electric layer potential and the surface charge density that was found for foam films from the zwitterionic biosurfactant lyso PC provides new evidence of the mechanism of formation of electrostatic interactions in the case of non-ionic surface active agent. 

In conclusion it is worth noting that the method of equilibrium foam film proved to be very appropriate for the determination of the equilibrium diffuse electric layer potential at the solution/air interface. Though it is an indirect experimental technique, it provides reliable results about the appearance of a negative surface charge in the case of surfactant-free solutions as well as in the case of non-ionic surfactant solutions. The existence of an isoeletric point and the re-charging of the interface can be considered as a direct evidence. 

The determination of the ( -potential from the directly measured disjoining pressure isotherms will be treated in Section 3.4. Thus, the (po(h) dependence can be followed along with understanding the charge-potential relationship of interacting diffuse electric layers in foam films. 

Let us first outline the theoretical background of the evaluation of both the charge and potential of two interacting diffuse electric layers. It is well known that the charge and potential distribution in the diffuse layer can be represented with a sufficient degree of accuracy using the Poisson-Boltzman (PB) approximation [e.g. 246]. For a planar film from aqueous symmetrical electrical electrolyte of valence z, the respective equation can be written in dimensionless form as... 

Making use of Eq. (3.91) it is easy to note that if Ym and Xd are known, there no need to assume constant charge or constant potential conditions. As a matter of fact, self consistent values of >o and Q0 can be generated at each integration of Eq. (3.90) carried out with the given Ym and Xd. Thus, the response of Y0 and Q0 to the overlap of the two diffuse electric layers can be obtained without any additional approximations except those inherent to the PB approximation itself. 

It appears that the data obtained in the above manner prove to be reliable for inferring the charge-potential relationship. Therefore, Fig 3.45 provides convincing evidence that in the case considered double layer repulsive interaction under the conditions of constant charge of the diffuse electric layer is operative. If so, the first integration of Eq. (3.90) predicts that... 

Qo and Y0 when cosh Ym = 1 (i.e. at infinite separation where there is no overlap of the diffuse electric layers). 

Eq. (3.92) implies a linear relation between cosh Y0 and cosh Ym. Therefore, values for go., and Fo, can be assessed by extrapolating this relation to cosh Ym = 1. This is demonstrated in Fig. 3.46 where a plot of cosh Yg versus cosh Ym is shown. Making use of least-squares analysis we obtain a slope of 1.01 0.01 and an intercept of 1.80 0.4. Furthermore, in conformity with Eq. (3.92) the intercept yields Fo, = 119 0.03 or, respectively, (po = 30.5 0.7 mV and <70i = 1.29 0.04 mC m 2 for the diffuse electric layer potential and diffuse double layer charge density at infinite separation. These values are in... 

The approach demonstrated above seems to provide a sufficiently rigorous basis for evaluating the charge and potential of interacting diffuse electric layers in thin liquid films from electrolytes. This approach seems to function whenever reliable experimental data for the film mid-plane potential and film thickness are available. 

There is no doubt that the FE(/i) isotherms of foam films from non-ionic surfactants plotted at various pH provide reliable information. The quantitative analysis in this case requires account of the role of the Stem-layer and of the charge-potential relationship of interacting diffuse electric layers. This is an object of further research. 

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