Thickness of the double layer

The repulsion between oil droplets will be more effective in preventing flocculation Ae greater the thickness of the diffuse layer and the greater the value of 0. the surface potential. These two quantities depend oppositely on the electrolyte concentration, however. The total surface potential should increase with electrolyte concentration, since the absolute excess of anions over cations in the oil phase should increase. On the other hand, the half-thickness of the double layer decreases with increasing electrolyte concentration. The plot of emulsion stability versus electrolyte concentration may thus go through a maximum. 

Mobility is affected by the dielectric constant and viscosity of the suspending fluid, as indicated in Eq. (22-28). The ionic strength of the fluid has a strong effect on the thickness of the double layer and hence on As a rule, mobility varies inversely as the square root of ionic strength [Overbeek, Adv. Colloid Sci., 3, 97 (1950)b... 

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. 

Calculate the surface charge a [C rrr2] and the surface potential and the "thickness" of the double layer, k1. ... 

In seawater the thickness" of the double layer as given by k1 (Eq. 3.9) is a few Angstroms, equal approximately to a hydrated ion. In other words, the double layer is compressed and hydrophobic colloids, unless stabilized by specific adsorption or by polymers, should coagulate. Some of this coagulation is observed in the estuaries where river water becomes progressively enriched with electrolytes (Fig. 7.14a). That these colloids exist in seawater for reasonable time periods is caused... 

The zeta potential and the thickness of the double layer (1/k) decrease rapidly with an increase in ionic strength or the valence of the electrolytes in the capillary (Equations (5) and (6)). Therefore, the ionic strength and the nature of the ions in the electrolyte solution are very important parameters determining the strength of the EOF. Careful control of the ionic... 

Figure 4.7. Variation of the potential with distance (in the diffuse double layer) for different concentrations and thicknesses of the double layer, d. ...

One possible, although speculative explanation of the effect of the addition of sulfamic acid or sodium sulfate may be based on Eq. (4.9). According to this equation, the variation in the concentration c of a nonreacting electrolyte changes the thickness of the metal-solution interphase, the double-layer thickness It appears that as the thickness of the double layer, decreases, the coercivity of the Co(P) deposited decreases as well. 

The average thickness of the double layer, lk, that is, the Debye-Huckel length, is given as (Chattoraj and Birdi, 1984) ... 

From Helmholtz s equation it is possible to calculate the equivalent thickness of the double layer, S, as well as the electric moment M, i.e. the distance to which a proton and an electron must be separated, in vacuo, to give the same electric moment. From a knowledge of V and F we are in a position to calculate the electric moment of each adsorbed molecule, a few of these are given in the following table ... 

The transverse voltage is the drop of potential across the double layer and is of the order of 0-01 to 0 05 volt. It will be noted that the amount of liquid transported is dependent on the nature of the liquid and on the current and is independent of the diameters or lengths of the tubes of the diaphragm. Somewhat divergent views are held as to the actual thickness of the double layer (Lamb, Phil. Mag. xxv. 52, 1888) a point which we have 7 referred to. 

Colloids The Thickness of the Double Layer and the Bulk Dimensions Are of the Same Order. The sizes of the phases forming the electrified interface have not quantitatively entered the picture so far. There has been a certain extravagance with dimensions. If, for instance, the metal in contact with the electrolyte was a sphere (e.g., a mercury drop), its radius was assumed to be infinitely large compared with any dimensions characteristic of the double layer, e.g., the thickness K-1 of the Gouy region. Such large metal spheres, dropped into a solution, sink to the bottom of the vessel and lie there stable and immobile. 

The quantity /is a numerical factor that depends on the ratio a/K, where a is the radius of the spherical or cylindrical particle. In other words, / depends on the ratio of the radius of the particle to the effective thickness K 1 of the diffuse layer. When alKX is large, (the particle large in comparison with the diffuse-charge thickness), the numerical factor is always equal to irrespective of the shape of the particle. When the particle is small compared with the thickness of the double layer,/is i for cylindrical particles parallel to the field and for spherical particles (Fig. 6.140). 

Consider a simple interfacial region at a mercury/solution interface. The electrolyte is 0.01 M NaF and the charge on the electrode is 10 iC negative to the pzc. The zeta potential is -10 mV on the same scale. What is the capacitance of the Helmholtz layer and that of the diffuse layer Galculate the capacitance of the interfaces. Take the thickness of the double layer as the distance between the center of the mercury atoms and that of hydrated K+in contact with the electrode through its water layer. (Bockris)... 

One of the most important quantities to emerge from the Debye-Huckel approximation is the parameter k. This quantity appears throughout double-layer discussions and not merely at this level of approximation. Since the exponent kx in Equation (37) is dimensionless, k must have units of reciprocal length. This means that k has units of length. This last quantity is often (imprecisely) called the thickness of the double layer. All distances within the double layer are judged large or small relative to this length. Note that the exponent kx may be written x/k a form that emphasizes the notion that distances are measured relative to k in the double layer. 

The Derjaguin approximation illustrated in the above example is suitable when kR > 10, that is, when the radius of curvature of the surface, denoted by the radius R, is much larger than the thickness of the double layer, denoted by k 1. (Note that for a spherical particle R = Rs, the radius of the particle.) Other approaches are required for thick double layers, and Verwey and Overbeek (1948) have tabulated results for this case. The results can be approximated by the following expression when the Debye-Hiickel approximation holds ... 

The above discussions illustrate that the interactions between overlapping electrical double layers depend on a number of considerations, such as the magnitude of the surface potential, the thickness of the double layer, and the type of electrolyte, among others. Moreover, the expressions that have been obtained here (and others that are available in the literature) depend on additional conditions that are determined by the approximations made in deriving the expressions. 

We stipulate the electrode to be smooth (though not necessarily flat) and of constant area A. By smooth we mean that any undulations in the electrode surface should not exceed the thickness of the double layer. For an electrode that is less smooth than this, the concept of electrode area is somewhat vague and the effective electrode area may change with time. By prescribing a constant electrode area, we exclude one of the most practical electrodes the dropping mercury electrode treated in Chap. 5. 

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