Electric double-layer surface potential calculation

Double integration with respect to EA yields the surface excess rB+ however, the calculation requires that the value of this excess be known, along with the value of the first differential 3TB+/3EA for a definite potential. This value can be found, for example, by measuring the interfacial tension, especially at the potential of the electrocapillary maximum. The surface excess is often found for solutions of the alkali metals on the basis of the assumption that, at potentials sufficiently more negative than the zero-charge potential, the electrode double layer has a diffuse character without specific adsorption of any component of the electrolyte. The theory of diffuse electrical double layer is then used to determine TB+ and dTB+/3EA (see Section 4.3.1). 

The description of the sorption of charged molecules at a charged interface includes an electrostatic term, which is dependent upon the interfacial potential difference, Ai//(V). This term is in turn related to the surface charge density, electric double layer model. The surface charge density is calculated from the concentrations of charged molecules at the interface under the assumption that the membrane itself has a net zero charge, as is the case, for example, for membranes constructed from the zwitterionic lecithin. Moreover,... 

Fig. 1. Variation of the eiectric potential near a surface in the presence of an electrolyte solution, (a) Electrical double layer at the surface of a solid positively charged, in contact with an electrolyte solution, (b) The variation of the electrical potential when the measurement is made at an increasing distance from the surface, and when the liquid phase is mobile at a given flow rate. The zeta potential [) can be calculated from the streaming potential, which can be measured according to the method described by Thubikar et al. [4].

This is the important Poisson-Boltzmann (PB) equation and the model used to derive it is usually called the Gouy-Chapman (GC) theory. It is the basic equation for calculating all electrical double-layer problems, for flat surfaces. In deriving it we have, however, assumed that all ions are point charges and that the potentials at each plane x are uniformly smeared out along that plane. These are usually reasonable assumptions. 

In the years 1910-1917 Gouy2 and Chapman3 went a step further. They took into account a thermal motion of the ions. Thermal fluctuations tend to drive the counterions away form the surface. They lead to the formation of a diffuse layer, which is more extended than a molecular layer. For the simple case of a planar, negatively charged plane this is illustrated in Fig. 4.1. Gouy and Chapman applied their theory on the electric double layer to planar surfaces [54-56], Later, Debye and Hiickel calculated the potential and ion distribution around spherical surfaces [57],... 

For an electrophysiological experiment you form an electrode from a 5 cm long platinum wire (0.4 mm diameter) by bending it in the shape of a spiral. Calculate the total capacitance of the diffuse electric double layer for aqueous solutions of a monovalent salt at concentrations of 0.1 and 0.001 M. Assume a low surface potential. 

There are various techniques to measure different properties of electric double layers. A wide range of information was obtained from electrocapillary experiments. In an electrocapillary experiment the surface tension versus potential of a metallic surface is measured. From this the capacitance and the surface charge can be calculated. For technical reasons this is routinely only possible for mercury. 

The external field which in the present case is generated only by the surface, is considered non-zero only in the first layer of molecules, i.e. it represents the effect of an adsorption potential for the water molecules. Of course, in general E is not zero in the liquid, for instance because of the occurrence of electrical double layers. We assume, however, that for small values of ft the double layer field is small compared with the polarization field. Experiment [3] as well as more detailed calculations presented in the more extended version of this paper show that, indeed, when the hydration forces are important, the effect of double layers can be neglected. 

A quantitative treatment of the effects of electrolytes on colloid stability has been independently developed by Deryagen and Landau and by Verwey and Over-beek (DLVO), who considered the additive of the interaction forces, mainly electrostatic repulsive and van der Waals attractive forces as the particles approach each other. Repulsive forces between particles arise from the overlapping of the diffuse layer in the electrical double layer of two approaching particles. No simple analytical expression can be given for these repulsive interaction forces. Under certain assumptions, the surface potential is small and remains constant the thickness of the double layer is large and the overlap of the electrical double layer is small. The repulsive energy (VR) between two spherical particles of equal size can be calculated by ... 

As seen from Equations 1.54-1.56, the intrinsic stability constants of surface reactions are dependent on two factors a chemical and an electric contribution. The chemical contribution is taken into consideration by the mass balance the electric contribution is treated by the charge balance. There are several surface complexation models that mainly differ in the description of the electric double layer that is used to calculate the surface potential, which is done by different double-layer models. These models have been mentioned previously in this chapter. Since, however, the terminology usually used in electrochemistry, colloid chemistry and, especially, in the discussions of surface complexation models is different, they are repeated again ... 

In another mode of presentation of experimentally determined surface charging data, values of parameters of a certain model of an electrical double layer, adjusted to the experimentally determined results, are reported rather than the PZC. Usually, this information is sufficient to calculate the PZC, and the result of such a calculation (rounded to the nearest one-tenth of pH unit) is used in the present compilation when the PZC is not explicitly reported in the original publication. A few studies report the results (usually electrokinetic potentials) for... 

Instead of using the middle potential F(o) and the potential (l), it is useful to perform the calculations with the dimensionless potentials u and u, and a dimensionless surface charge a. A solution to the problem of pressure due to overlapping electrical double layers in a thin liquid film was first given by Langmuir in form of an approximation... 

The Smith-Ewart mechanism does not take into account any polymerization in the aqueous phase. This may be true for monomers that are quite insoluble in water, like styrene, but appears unlikely for more hydrophilic ones like methyl methacrylate or vinyl acetate. In addition, it was calculated by Flory that there is insufficient time for a typical cation radical (like a sulfate ion radical) to add to a dissolved molecule of monomer like styrene before it becomes captured by a micelle. This was argued against, however, on the ground that Flory s calculations fail to consider the potential energy barrier at the micelle surfaces from the electrical double layer. This barrier would reduce the rate of diffusion of the radical ions into the micelles. ... 

What will be the effect of the ions that are present in the electrical double layer As already described, Eq. (9) predicts that for a KCl electrolyte of bulk concentration 0.15 M and a membrane surface of potential — 50 mV, the concentration of ions at the membrane surface will be around 1 molar. For a cubic lattice of ions of concentration C moles per liter, the average interionic distance r may be calculated from the equation... 

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