Surface counterion concentration

At finite positive and negative charge densities on the electrode, the counterion density profiles often exhibit significantly higher maxima, i.e. there is an overshoot, and the derived potential actually shows oscillations itself close to the electrode surface at concentrations above about 1 M. 

FIGURE 15.9 The counterion concentration as a function of the distance from a surface according to the G-C theory for three different hulk concentrations of a 1 1 electrolyte 1, 50, and 500mM, respectively. The surface charge density is kept constant to -0.1 C/m. ... 

At high levels of i (i.e., [M+] )ex[/]), Eq. 11-55 indicates that the organic counterion concentrations must asymptotically approach a constant value set by the total surface charge density (the cation exchange capacity of the clay) as long as CEC Fvic[co-ion], ex ... 

Secondary Micelles. These micelles are formed only by dihydroxy bile salts in the presence of increased counterions. Secondary micelles are probably formed by the aggregation of primary micelles. Since the surface available for hydrophobic interaction is expended in the formation of the primary micelles, the bonding that takes place is probably between some of the hydrophilic parts of the bile salts. It is suggested (Figure 11c) that in the presence of increased counterion concentrations... 

Figure 4.2 Left Potential-versus-distance for a surface potential of )/>o = 50 mV and different concentrations of a monovalent salt in water. Right Local co- and counterion concentrations are shown for a monovalent salt at a bulk concentration of 0.1 M and a surface potential of 50 mV. In addition, the total concentration of ions, that is the sum of the co- and counterion concentrations, is plotted.

Counterion binding is not a well defined quantity, with various experimental techniques weighing the ion distribution slightly differently. Thermodynamic methods (e.g. ion activities or osmotic coefficients) monitor the free counterion concentration, transport methods (e.g. ion self diffusion or conductivity) the counterions diffusing with the micelle, and spectroscopic methods (e.g. NMR) the counterions in close contact with the micelle surface. Measurement of the effect of Na+ counterions on the symmetric S-O stretching modes would also be expected to be highly dependent on the distance of the counterion from the micelle surface (similar to the NMR method). 

When the flow conditions are kept constant the boundary layer will be constant and the current density will reach a maximum value independent of the applied electrical potential gradient if the counterion concentration and thus the salt concentration at the membrane surface become 0. The maximum current density is referred to as the limiting current density. Thus is i — t im for mCf —> 0 and... 

Biological surfaces, as well as metal oxides, acquire an electrostatic charge in water through the dissociation of acidic or basic groups which lie near the solid/ aqueous interface. As two such surfaces, separated by an aqueous electrolyte, are brought together the volume between them decreases and consequently the counterion concentration of the solution must tend to increase in order to preserve overall electro-... 

Electrostatic interactions in solutions containing charged particles and ions can be described using the Poisson-Boltzmann equation. A charged surface attracts counterions into a double layer of thickness defined by the Debye length, which depends on counterion concentration and solvent dielectric constant. From simplified theories, expressions can be derived for the attractive interaction potential between charged spheres. 

It has to be emphasized that the nature of the electrolyte ions influences the surface potential value because the effective surface charge concentration is reduced if shght hydrophobic, adsorbophilic electrolytic counterions are included in the eluent. 

Figure 4. Schematic plots of electrical potential, counterion concentration, and co-ion concentration in the diffuse layer versus distance from a charged surface.

Surface complexation models can be extended to account explicitly for electrostatic sorption by calculating excess counterion concentrations in the diffuse layer in addition to specific sorption. Counterions in the diffuse layer (e.g., Ca ) can then be treated as distinct from those in bulk solution (e.g., Ca2+) and those that are specifically sorbed (e.g., =Sp-Caf). The total sorption is given by the sum of the concentrations of specifically sorbed and electrostatically sorbed species ... 

In the acidic route (with pH < 2), both kinetic and thermodynamic controlling factors need to be considered. First, the acid catalysis speeds up the hydrolysis of silicon alkoxides. Second, the silica species in solution are positively charged as =SiOH2 (denoted as I+). Finally, the siloxane bond condensation rate is kinetically promoted near the micelle surface. The surfactant (S+)-silica interaction in S+X 11 is mediated by the counterion X-. The micelle-counterion interaction is in thermodynamic equilibrium. Thus the factors involved in determining the total rate of nanostructure formation are the counterion adsorption equilibrium of X on the micellar surface, surface-enhanced concentration of I+, and proton-catalysed silica condensation near the micellar surface. From consideration of the surfactant, the surfactants first form micelles as a combination of the S+X assemblies, which then form a liquid crystal with molecular silicate species, and finally the mesoporous material is formed through inorganic polymerization and condensation of the silicate species. In the S+X I+ model, the surfactant-to-counteranion... 

The volume excluded by the ions becomes important in relatively thin films, insofar as the counterion concentration is markedly higher in the vicinity of a charged surface. This effect was taken into account by means of the Bikerman equation ... 

To express counterion distributions more quantitatively, counterion concentration c+ profiles for a 64 base-pair DNA at various polymer concentrations are plotted in Figure 4 as functions of the radial coordinate r measured from the axis of the DNA cylinder at its center and in Figure 5 as functions of the z coordinate along the surface of the cylinder. The very high counterion concentration ( 3 M) on the surface of the polyion rapidly decreases in both radial and longitudinal directions, and dilution of the polymer concentration has the slightest effect on these profiles. 

FIG. 5 Counterion concentration c+ profiles for a 64 base-pair DNA at various polymer concentrations as functions of the z coordinate along the surface of the cylinder. 

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