Charge density: diffuse layer solution

The Stern model (1924) may be regarded as a synthesis of the Helmholz model of a layer of ions in contact with the electrode (Fig. 20.2) and the Gouy-Chapman diffuse model (Fig. 20.10), and it follows that the net charge density on the solution side of the interphase is now given by... 

The diffuse charge density in the solution side of the double layer is obtained by adding up all countercharge... 

The effect known either as electroosmosis or electroendosmosis is a complement to that of electrophoresis. In the latter case, when a field F is applied, the surface or particle is mobile and moves relative to the solvent, which is fixed (in laboratory coordinates). If, however, the surface is fixed, it is the mobile diffuse layer that moves under an applied field, carrying solution with it. If one has a tube of radius r whose walls possess a certain potential and charge density, then Eqs. V-35 and V-36 again apply, with v now being the velocity of the diffuse layer. For water at 25°C, a field of about 1500 V/cm is needed to produce a velocity of 1 cm/sec if f is 100 mV (see Problem V-14). 

An electroosmotic flux is formed as a result of the effect of the electric field in the direction normal to the pores in the membrane, delectric diffuse layer in the pore with a charge density p. The charges move in the direction of the x axis (i.e. in the direction of the field), together with the whole solution with velocity v. At steady state... 

The interfacial capacitance increases with the DDTC concentration added. The relationship among potential difference t/ of diffusion layer, the electric charge density q on the surface of an electrode and the concentration c of a solution according to Gouy, Chapman and Stem model theory is as follows. 

The charge in the diffuse layer can be considered equivalent to the Gouy charge density qd placed at a distance K-1 from the OHP. This gives rise to a parallel-plate condenser model. The potential at one plate—deep in the solution side—is taken at zero, while the potential at the other plate—which coincides with the OHP—is [f0. This latter potential is often referred to in the study of electrokinetic phenomena as the zeta ( ) potential. Thus,... 

The charge density, Volta potential, etc., are calculated for the diffuse double layer formed by adsorption of a strong 1 1 electrolyte from aqueous solution onto solid particles. The experimental isotherm can be resolved into individual isotherms without the common monolayer assumption. That for the electrolyte permits relating Guggenheim-Adam surface excess, double layer properties, and equilibrium concentrations. The ratio u0/T2N declines from two at zero potential toward unity with rising potential. Unity is closely reached near kT/e = 10 for spheres of 1000 A. radius but is still about 1.3 for plates. In dispersions of Sterling FTG in aqueous sodium ff-naphthalene sulfonate a maximum potential of kT/e = 7 (170 mv.) is reached at 4 X 10 3M electrolyte. The results are useful in interpretation of the stability of the dispersions. 

The micellar surface has a high charge density and the stability of the aggregate is heavily dependent on the binding of counterions to the surface. From the solution of the Poisson-Boltzmann equation one finds that a large fraction (0.4—0.7) of the counterions is in the nearest vicinity of the micellar surface300. These ions could be associated with the Stern layer, but it seems simpler not to make a distinction between the ions of the Stern layer and those more diffusely bound. They are all part of the counterions and their distribution is primarily determined by electrostatic effects. 

The formal potential, E0/, contains useful information about the ease of oxidation of the redox centers within the supramolecular assembly. For example, a shift in E0/ towards more positive potentials upon surface confinement indicates that oxidation is thermodynamically more difficult, thus suggesting a lower electron density on the redox center. Typically, for redox centers located close to the film/solution interface, e.g. on the external surface of a monolayer, the E0 is within 100 mV of that found for the same molecule in solution. This observation is consistent with the local solvation and dielectric constant being similar to that found for the reactant freely diffusing in solution. The formal potential can shift markedly as the redox center is incorporated within a thicker layer. For example, E0/ shifts in a positive potential direction when buried within the hydrocarbon domain of a alkane thiol self-assembled monolayer (SAM). The direction of the shift is consistent with destabilization of the more highly charged oxidation state. 

H. Ohshima, Diffuse double layer interaction between two spherical particles with constant surface charge density in an electrolyte solution, Colloid Polymer Sci. 263, 158-163 (1975). 

With FITEQL numeric procedure Hayes et al. fitted edl parameters to the three models of electric double layer DLM (diffuse layer model), CCM (constant capacity model) and TLM (three layer model) for the following oxides a-FeOOH, AI2O3 and TiC>2 in NaNC>3 solutions [51]. The fitting was performed for surface reaction constants, edl capacity and the densities of the hydroxyl groups on the surface of the oxides. The quality of the fitting was evaluated by the minimization of the function of the sum of the square deviations of the calculated value from the standard error of measured charge. The lower value of the function the better was the fit... 

The Poisson-Boltzman (P-B) equation commonly serves as the basis from which electrostatic interactions between suspended clay particles in solution are described ([23], see Sec.II. A. 2). In aqueous environments, both inner and outer-sphere complexes may form, and these complexes along with the intrinsic surface charge density are included in the net particle surface charge density (crp, 4). When clay mineral particles are suspended in water, a diffuse double layer (DDL) of ion charge is structured with an associated volumetric charge density (p ) if av 0. Given that the entire system must remain electrically neutral, ap then must equal — f p dx. In its simplest form, the DDL may be described, with the help of the P-B equation, by the traditional Gouy-Chapman [23-27] model, which describes the inner potential variation as a function of distance from the particle surface [23]. 

Gouy length — The width of the diffuse double layer at an electrode depends on a number of factors among which the charge density q on the surface of the metal and the concentration c of electrolyte in the solution are paramount. Roughly speaking, the charge in, and the potential of, the double layer falls off exponentially as one proceeds into the solution from the interface. 

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]... 

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