Layers extended double

A brief description of the various states shown in Figure 9.10 is given as follows [33, 34] States (a)-(c) correspond to a suspension that is stable in the colloid sense, with stability being obtained as a result of net repulsion due to the presence of extended double layers (i.e., at low electrolyte concentration), the result of steric... 

In one respect Derjag.uin s method is superior-to Levine and. Dube s. It is possible to. apply the principle of Der-j a g u i n s method to the complete Gouy-Chapman equation, whereas in Levine s and Dube s treatment it is unavoidable to introduce the linear approximation of Debye and Huekel. The theory for large spherical particles, surrounded by a thin double layer may therefore be made almost as exact as that of flat plates, but in the case of small particles with an extended double layer the situation is less favourable. 

Nevertheless this does not constitute a serious defect in the theory. As explained earlier (cf. Chapter II, 4), precisely in the case of small particles with an extended double layer, the application of the linear approximation may be allowed and gives reliable results, even for relatively high potentials. 

Two main interactions can cause gel formation with particulate materials (1) Long-range repulsion between the partides, e.g. using extended double layers or steric repulsion, using adsorbed surfactant or polymer layers. (2) Van der Waals attraction between the partides (flocculation), which can produce three-dimensional gel networks in the continuous phase. 

Ionic strength of medium, surface structure and non-spherical particles can affect diffusion speed of particles. The thickness of the electric double layer (Debye length) changes with the ions in the medium and the total ionic concentration. An extended double layer of ions around the particle results from a low conductivity medium. So, the diffusion speed reduces and hydrodynamic diameter increases. The diffusion speed can be affected with a change in surface area. The diffusion speed will reduce with an adsorbed polymer layer. Polymer conformation can alter the apparent size. 

This produces a double layer of charge, one localized on the surface of the plane and the other developed in a diffuse region extending into solution. 

As the reaction proceeds, the diffusion layer extends into the bulk of the solution outside the double layer. When the diffusion-layer thickness increases much more than the autocorrelation distance of the asymmetrical nonequilibrium fluctuation, a steady state emerges. In contrast to Eq. (103), in this case the following condition holds,... 

If only the three-phase-boundaries (tpb) were electrocatalytically active one would expect Cd values of the order of 10 pF/cm2. The thus measured high Cd values also provide evidence that the charge transfer zone is extended over the entire gas-exposed electrode surface, i.e. that an effective double layer is formed over the entire gas exposed electrode surface. 

Overbeek and Booth [284] have extended the Henry model to include the effects of double-layer distortion by the relaxation effect. Since the double-layer charge is opposite to the particle charge, the fluid in the layer tends to move in the direction opposite to the particle. This distorts the symmetry of the flow and concentration profiles around the particle. Diffusion and electrical conductance tend to restore this symmetry however, it takes time for this to occur. This is known as the relaxation effect. The relaxation effect is not significant for zeta-potentials of less than 25 mV i.e., the Overbeek and Booth equations reduce to the Henry equation for zeta-potentials less than 25 mV [284]. For an electrophoretic mobility of approximately 10 X 10 " cm A -sec, the corresponding zeta potential is 20 mV at 25°C. Mobilities of up to 20 X 10 " cmW-s, i.e., zeta-potentials of 40 mV, are not uncommon for proteins at temperatures of 20-30°C, and thus relaxation may be important for some proteins. 

Gorin has extended this analysis to include (1) the effects of the finite size of the counterions in the double layer of spherical particles [137], and (2) the effects of geometry, i.e. for cylindrical particles [2]. The former is known as the Debye-Huckel-Henry-Gorin (DHHG) model. Stigter and coworkers [348,369-374] considered the electrophoretic mobility of polyelectrolytes with applications to the determination of the mobility of nucleic acids. 

Work in this area has been conducted in many laboratories since the early 1980s. The electrodes to be used in such a double-layer capacitor should be ideally polarizable (i.e., all charges supplied should be expended), exclusively for the change of charge density in the double layer [not for any electrochemical (faradaic) reactions]. Ideal polarizability can be found in certain metal electrodes in contact with elelctrolyte solutions free of substances that could become involved in electrochemical reactions, and extends over a certain interval of electrode potentials. Beyond these limits ideal polarizability is lost, owing to the onset of reactions involving the solvent or other solution components. 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

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