Diffuse double layer charge

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 diffuse double layer charge over a surface of area S can be taken by integration of the charge density over the surface (defined by S2). 

The treatment in the case of a plane charged surface and the resulting diffuse double layer is due mainly to Gouy and Qiapman. Here may be replaced by d /dx since is now only a function of distance normal to the surface. It is convenient to define the quantities y and yo as... 

The quantity 1 /k is thus the distance at which the potential has reached the 1 je fraction of its value at the surface and coincides with the center of action of the space charge. The plane at a = l//c is therefore taken as the effective thickness of the diffuse double layer. As an example, 1/x = 30 A in the case of 0.01 M uni-univalent electrolyte at 25°C. 

Charged surface plus diffuse double layer of ions —>... 

In the case of a charged particle, the total charge is not known, but if the diffuse double layer up to the plane of shear may be regarded as the equivalent of a parallel-plate condenser, one may write... 

Derive the general equation for the differential capacity of the diffuse double layer from the Gouy-Chapman equations. Make a plot of surface charge density tr versus this capacity. Show under what conditions your expressions reduce to the simple Helmholtz formula of Eq. V-17. 

F r d ic Current. The double layer is a leaky capacitor because Faradaic current flows around it. This leaky nature can be represented by a voltage-dependent resistance placed in parallel and called the charge-transfer resistance. Basically, the electrochemical reaction at the electrode surface consists of four thermodynamically defined states, two each on either side of a transition state. These are (11) (/) oxidized species beyond the diffuse double layer and n electrons in the electrode and (2) oxidized species within the outer Helmholtz plane and n electrons in the electrode, on one side of the transition state and (J) reduced species within the outer Helmholtz plane and (4) reduced species beyond the diffuse double layer, on the other. 

Figure 39. Current-time variation in nickel pitting dissolution in NaCl solution.89,91 1, double-layer charging current 2, fluctuation-diffusion current 3, minimum dissolution current 4, pit-growth current (Reprinted from M. Asanuma andR. Aogaki, Nonequilibrium fluctuation theory on pitting dissolution. II. Determination of surface coverage of nickel passive film, J. Chem. Phys. 106, 9938, 1997, Fig. 2. Copyright 1997, American Institute of Physics.)...

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. 

FIG. 8 Inverse differential capacity at the zero surface charge vs. inverse capacity Cj of the diffuse double layer for the water-nitrobenzene (O) and water-1,2-dichloroethane (, ), interface. The diffuse layer capacity was evaluated by the GC ( ) or the MPB (0,)> theory. (From Ref. 22.)... 

To evaluate the contribution of the SHG active oriented cation complexes to the ISE potential, the SHG responses were analyzed on the basis of a space-charge model [30,31]. This model, which was proposed to explain the permselectivity behavior of electrically neutral ionophore-based liquid membranes, assumes that a space charge region exists at the membrane boundary the primary function of lipophilic ionophores is to solubilize cations in the boundary region of the membrane, whereas hydrophilic counteranions are excluded from the membrane phase. Theoretical treatments of this model reported so far were essentially based on the assumption of a double-diffuse layer at the organic-aqueous solution interface and used a description of the diffuse double layer based on the classical Gouy-Chapman theory [31,34]. 

It is also possible for ions in the water, especially positively charged ions, or cations, to be attracted to the negatively charged surfaces. This leads to a zone of water and ions surrounding the clay particles, known as the diffuse double layer. 

A related technique is the current-step method The current is zero for t < 0, and then a constant current density j is applied for a certain time, and the transient of the overpotential 77(f) is recorded. The correction for the IRq drop is trivial, since I is constant, but the charging of the double layer takes longer than in the potential step method, and is never complete because 77 increases continuously. The superposition of the charge-transfer reaction and double-layer charging creates rather complex boundary conditions for the diffusion equation only for the case of a simple redox reaction and the range of small overpotentials 77 [Pg.177]

To obtain the dipole moment we set o = —o% in Eq. (18.14) so that the diffuse double layer is free of excess charge (see Section 4.3). 

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