Curved double layers

Figure 3.4 gives examples of real electrocapillary curves and differential capacity curves. Double layer models have to explain the shape of these curves. 

Higher-order corrections to Eq. [151] have also been derived " and used to investigate the elasticity of curved membranes and the adsorption of monovalent ions in thin, curved double layers. 

The repulsion between two double layers is important in determining the stability of colloidal particles against coagulation and in setting the thickness of a soap film (see Section VI-5B). The situation for two planar surfaces, separated by a distance 2d, is illustrated in Fig. V-4, where two versus x curves are shown along with the actual potential. 

The shape of the electrocapillary curve is easily calculated if it is assumed that the double layer acts as a condenser of constant capacity C. In this case, double integration of Eq. V-50 gives... 

Fig. V-12. Variation of the integral capacity of the double layer with potential for 1 N sodium sulfate , from differential capacity measurements 0, from the electrocapillary curves O, from direct measurements. (From Ref. 113.)...

Often the van der Waals attraction is balanced by electric double-layer repulsion. An important example occurs in the flocculation of aqueous colloids. A suspension of charged particles experiences both the double-layer repulsion and dispersion attraction, and the balance between these determines the ease and hence the rate with which particles aggregate. Verwey and Overbeek [44, 45] considered the case of two colloidal spheres and calculated the net potential energy versus distance curves of the type illustrated in Fig. VI-5 for the case of 0 = 25.6 mV (i.e., 0 = k.T/e at 25°C). At low ionic strength, as measured by K (see Section V-2), the double-layer repulsion is overwhelming except at very small separations, but as k is increased, a net attraction at all distances... 

Fig. VI-6. The force between two crossed cylinders coated with mica and carrying adsorbed bilayers of phosphatidylcholine lipids at 22°C. The solid symbols are for 1.2 mM salt while the open circles are for 10.9 roM salt. The solid curves are the DLVO theoretical calculations. The inset shows the effect of the van der Waals force at small separations the Hamaker constant is estimated from this to be 7 1 x 10 erg. In the absence of salt there is no double-layer force and the adhesive force is -1.0 mN/m. (From Ref. 66.)...

Fig. 3. Attraction—repulsion potentials as a function of distance between particle centers. Curve 1 represents the attractive potential caused by van der Waals forces, curve 2 is the repulsive potential caused by double-layer forces, and curve 3 is the resultant force experienced by the two particles.

This equation is a reasonable model of electrokinetic behavior, although for theoretical studies many possible corrections must be considered. Correction must always be made for electrokinetic effects at the wall of the cell, since this wall also carries a double layer. There are corrections for the motion of solvated ions through the medium, surface and bulk conductivity of the particles, nonspherical shape of the particles, etc. The parameter zeta, determined by measuring the particle velocity and substituting in the above equation, is a measure of the potential at the so-called surface of shear, ie, the surface dividing the moving particle and its adherent layer of solution from the stationary bulk of the solution. This surface of shear ties at an indeterrninate distance from the tme particle surface. Thus, the measured zeta potential can be related only semiquantitatively to the curves of Figure 3. 

It is possible on the basis of this model (arrangement O) to explain the constant capacitance region on the negative side of the C vs. E curve (Fig. 20.7), and why the capacitance in this region is independent of the nature of the cations in the solution. The model of the double layer is shown in Fig. 20.12 in which it can be seen that the surface of the electrode and the... 

Fig. 20.17 Potential energy-distance curves for a cathodic reaction showing how the potential energy barrier is lowered by when E < p,z.c. The barrier is assumed to be symmetrical so that /S => yi, where 5 is the distance of the O.H.P. from the surface of the electrode. Full curve—no field across double layer dashed curve-potential diflcrence is E and is negative...

The electrical double layer at a pc-Ag/aqueous solution interface has been discussed by LeiMs et al., Valette and Hamelin, and Beck et al. in many papers.24 63 67146 223 272363-368 Detailed reviews have been given by Hamelin63 and Vorotyntsev74 in this work only a few comments will be added. First, the diffuse-layer minimum in the C,E curve was obtained363... 

The electrical double-layer structure and fractal geometry of a pc-Ag electrode have been tested by Se vasty an ov et al.272 They found that the geometrical roughness of electrochemically polished pc-Ag electrodes is not very high (/pz 1.5 to 1.25), but the dependence of Chtr curves on cej, as well as on/pz, is remarkable (C, =30 to 80 fi cm-2 if/pz =1.5 to 1.0). 

The electrical double layer at pc-Zn/fyO interfaces has been studied in many works,154 190 613-629 but the situation is somewhat ambiguous and complex. The polycrystalline Zn electrode was found to be ideally polarizable for sufficiently wide negative polarizations.622"627 With pc-Zn/H20, the value of Eg was found at -1.15 V (SCE)615 628 (Table 14). The values of nun are in reasonable agreement with the data of Caswell et al.623,624 Practically the same value of Eff was obtained by the scrape method in NaC104 + HjO solution (pH = 7.0).190 Later it was shown154,259,625,628 that the determination of Eo=0 by direct observation of Emin on C,E curves in dilute surface-inactive electrolyte solutions is not possible in the case of Zn because Zn belongs to the group of metals for which E -o is close to the reversible standard potential in aqueous solution. 

The dependence of the electrical double-layer parameters of pc-Cd on temperature (0 to 85 °C) has been studied647,648 in H2O + KF solutions. The Emin depends slightly on T, the temperature coefficient BEmiri/dT being 0.15 mV K"1. C,-at cr < -0.09 C m 2has been found to decrease as the temperature increases. C, rises if a decreases and at a = 0 the inner-layer temperature coefficient BCt/BT is equal to 0.05 8//F cm-2 K 1. It has been pointed out that the intersection point of Ch a curves at various temperatures lies at a less negative a than the charge at which the C a curves have the minimum value. The same is the case with pc-Pb electrodes,649 but for Hg/H20 the opposite is observed.305... 

Mishuk et a/.675,676 have applied the modified amplitude demodulation method to electrochemically polished pc-Bi in aqueous NaF solution. The curves of the real component of the nonlinear impedance Z" as a function of the electrode potential, unlike pc-Cd and pc-Pb, intersect for various cNaF at E - -0.62 V (SCE),674 i.e., at Ea=0 for pc-Bi, as obtained by impedance.666-672 The different behavior of pc-Bi from pc-Cd and pc-Pb at a > 0 has been explained by the semimetallic nature of pc-Bi electrodes. A comparison of inner-layer nonlinear parameter values for Hg, Cd, and Bi electrodes at a < 0 shows that the electrical double-layer structure at negative charges is independent of the metal.675,676... 

In the region of a very good correspondence has been found between experimental and calculated C,E curves and this has been taken to indicate that the electrical double-layer structure conforms to the GCSG theory. Comparison of the ChE curves for Hg/TMU and Fe/TMU shows that the dependence of Cf on E is less pronounced for an Fe electrode than for Hg/TMU, and the values of Cf for Fe are remarkably lower than for Hg. The same is the case for Fe/DMF, DMAA, MPF, and HMPA interfaces.732-736... 

Various pc electrode models have been tested.827 Using the independent diffuse layer electrode model74,262 the value of E n = -0.88 V (SCE) can be simulated for Cd + Pb alloys with 63% Pb if bulk and surface compositions coincide. However, large deviations of calculated and experimental C,E curves are observed at a 0. Better correspondence between experimental and calculated C,E curves was obtained with the common diffuse-layer electrode model,262 if the Pb percentage in the solid phase is taken as 20%. However, the calculated C, at a Ois noticeably lower than the experimental one. It has been concluded that Pb is the surface-active component in Cd + Pb alloys, but there are noticeable deviations from electrical double-layer models for composite electrodes.827... 

One of the most problematic issues, still to be fully resolved, is the dependence of the degree of oxidation on potential, as measured most commonly by cyclic voltammetry at low scan rates. There is currently no accepted model to describe the shape of the curve and the hysteresis between anodic and cathodic scans. The debate over whether the charge has a significant component due to a polymer/solution double layer is still not fully resolved. 

The application of the overpotential t] can be considered to be equivalent to the displacement of the potential energy curves by the amount 7]F with respect to each other. The high field is now applied across the double layer between the electrode and the ions at the plane of closest approach. It is apparent from Fig. 12 that the energy of activation in the favoured direction will be diminished by etrjF while that in the reverse direction will be increased by (1 — ac)r]F where the simplest interpretation of a is in terms of the slopes of the potential energy curves (a = mi/ mi+m )) at the points of intersection electrode processes indeed are the classical example of linear free energy relations. 

Measurement of the differential capacitance C = d /dE of the electrode/solution interface as a function of the electrode potential E results in a curve representing the influence of E on the value of C. The curves show an absolute minimum at E indicating a maximum in the effective thickness of the double layer as assumed in the simple model of a condenser [39Fru]. C is related to the electrocapillary curve and the surface tension according to C = d y/dE. Certain conditions have to be met in order to allow the measured capacity of the electrochemical double to be identified with the differential capacity (see [69Per]). In dilute electrolyte solutions this is generally the case. 

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