Surface of shear

The 2eta potential (Fig. 8) is essentially the potential that can be measured at the surface of shear that forms if the sohd was to be moved relative to the surrounding ionic medium. Techniques for the measurement of the 2eta potentials of particles of various si2es are collectively known as electrokinetic potential measurement methods and include microelectrophoresis, streaming potential, sedimentation potential, and electro osmosis (19). A numerical value for 2eta potential from microelectrophoresis can be obtained to a first approximation from equation 2, where Tf = viscosity of the liquid, e = dielectric constant of the medium within the electrical double layer, = electrophoretic velocity, and E = electric field. 

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. 

Not all of the ions in the diffuse layer are necessarily mobile. Sometimes the distinction is made between the location of the tme interface, an intermediate interface called the Stem layer (5) where there are immobilized diffuse layer ions, and a surface of shear where the bulk fluid begins to move freely. The potential at the surface of shear is called the zeta potential. The only methods available to measure the zeta potential involve moving the surface relative to the bulk. Because the zeta potential is defined as the potential at the surface where the bulk fluid may move under shear, this is by definition the potential that is measured by these techniques (3). 

The charge on the surface of colloid particles is an important parameter, and DNA/cation self-assembled complexes are no exception. It can be measured experimentally as the -potential or electrokinetic potential (the potential at the surface of shear be-... 

In order to describe the effects of the double layer on the particle motion, the Poisson equation is used. The Poisson equation relates the electrostatic potential field to the charge density in the double layer, and this gives rise to the concepts of zeta-potential and surface of shear. Using extensions of the double-layer theory, Debye and Huckel, Smoluchowski,... 

Figure 9. Formation of Stern plane and diffuse layer on particle surface ( I 0 = surface or Nernst potential, = potential of inner Flelmholtz plane, I 5 = Stern potential, l = thickness of Stern plane, ZP = zeta potential at surface of shear, d = distance from particle surface).

The electroosmotic velocity as defined in Eq. (1) is directly proportional to the , potential at the surface of shear defined as... 

The Stern surface is drawn through the ions that are assumed to be adsorbed on the charged wall. (This surface is also known as the inner Helmholtz plane [IHP], and the surface running parallel to the IHP, through the surface of shear (see Chapter 12) shown in Figure 11.9, is called the outer Helmholtz plane [OHP]. Notice that the diffuse part of the ionic cloud beyond the OHP is the diffuse double layer, which is also known as the Gouy-Chapman... 

FIG. 11.9 Schematic illustration of the variation of potential with distance from a charged wall in the presence of a Stern layer. See Chapter 12 for discussion of surfaces of shear and zeta potential. 

Next, let us consider the application of Equation (21) to a particle migrating in an electric field. We recall from Chapter 4 that the layer of liquid immediately adjacent to a particle moves with the same velocity as the surface that is, whatever the relative velocity between the particle and the fluid may be some distance from the surface, it is zero at the surface. What is not clear is the actual distance from the surface at which the relative motion sets in between the immobilized layer and the mobile fluid. This boundary is known as the surface of shear. Although the precise location of the surface of shear is not known, it is presumably within a couple of molecular diameters of the actual particle surface for smooth particles. Ideas about adsorption from solution (e.g., Section 7.7) in general and about the Stern layer (Section 11.8) in particular give a molecular interpretation to the stationary layer and lend plausibility to the statement about its thickness. What is most important here is the realization that the surface of shear occurs well within the double layer, probably at a location roughly equivalent to the Stern surface. Rather than identify the Stern surface as the surface of shear, we define the potential at the surface of shear to be the zeta potential f. It is probably fairly close to the... 

Distances within the double layer are considered large or small, depending on their magnitude relative to k-1. Thus in dilute electrolyte solutions, in which k is large, the surface of shear —which is close to the particle surface even in absolute units —may be safely regarded as coinciding with the surface in units relative to the double-layer thickness. Therefore, in the case for which k is large (or k small), Equation (21) becomes... 

The resulting expression is easily integrated again with the following limits (a) at the surface of shear = T and v = 0 (b) at the outside edge of the double layer J/ = 0 and v equals the observed velocity of particle migration. Therefore... 

In all the sections of this chapter until now we have focused attention on electrophoresis. We have seen that the potential at the surface of shear can be measured from electrophoretic mobility measurements, provided the system complies with the assumptions of a manageable model. One feature that has been conspicuously lacking from our discussions is any comparison between electrophoretically determined values of f and potential values determined by another method. The reasons for this are twofold ... 

In this section we describe electroosmosis and in the following section the streaming potential. These two electrokinetic techniques also permit the evaluation of f, but are subject to objection 1. In Section 12.8 we examine in greater detail the location of the surface of shear, which is the essence of objection 2 above. 

When an electric field is applied across the working capillary, the double-layer ions begin to migrate and soon reach the steady-state velocity. In the steady state, electrical and viscous forces balance one another. The forces exerted on the ions by the medium are equal and opposite to the forces exerted on the medium by the ions consequently, the liquid also attains a stationary-state velocity. The tangential displacement of the fluid relative to the wall defines a surface of shear at which the potential equals f. 

Next a change of variable is helpful. We replace r by a distance measured from the surface of shear x, where... 

The objective of comparing values of f determined from electrophoresis with those determined by other electrokinetic methods was stated at the beginning of Section 12.6. Enough experiments have been conducted in which at least two of the electrokinetic methods we have discussed are compared to leave no doubt as to the self-consistency of f as determined by these different methods. There is no guarantee, however, that self-consistent f potentials are correct. Consistency means only that f has been extracted from experimental quantities by a self-consistent set of approximations. It should be emphasized, however, that the existence of a potential at the surface of shear —which is the common component in all the electrokinetic analyses we have discussed —is more than amply confirmed by these observations. 

Two conditions must be met to justify comparisons between f values determined by different electrokinetic measurements (a) the effects of relaxation and surface conductivity must be either negligible or taken into account and (b) the surface of shear must divide comparable double layers in all cases being compared. This second limitation is really no problem when electroosmosis and streaming potential are compared since, in principle, the same capillary can be used for both experiments. However, obtaining a capillary and a migrating particle wiih identical surfaces may not be as readily accomplished. One means by which particles and capillaries may be compared is to coat both with a layer of adsorbed protein. It is an experimental fact that this procedure levels off differences between substrates The surface characteristics of each are totally determined by the adsorbed protein. This technique also permits the use of microelectrophoresis for proteins since adsorbed and dissolved proteins have been shown to have nearly identical mobilities. 

The surface of shear is the location within the electrical double layer at which the various electrokinetic phenomena measure the potential. We saw in Chapter 11 how the double layer extends outward from a charged wall. The potential at any particular distance from the wall can, in principle, be expressed in terms of the potential at the wall and the electrolyte content of the solution. In terms of electrokinetic phenomena, the question is How far from the interface is the surface of shear situated and what implications does this have on the relation between measured zeta potential and the surface potential ... 

First, the very existence of a surface of shear implies some interesting behavior within the fluid phase of the system under consideration. In our discussion of all electrokinetic phenom-... 

At this point, it is convenient to recall Figure 7.13 and the discussion of it. In that context we observed that there is generally a variation of properties in the vicinity of an interface from the values that characterize one of the adjoining phases to those that characterize the other. This variation occurs over a distance r measured perpendicular to the interface. In the present discussion viscosity is the property of interest and the surface of shear —rather than the interface per se —is the boundary of interest. The model we have considered until now has implied an infinite jump in viscosity, occurring so sharply that r is essentially zero. From a molecular point of view such an abrupt transition is highly unrealistic. A gradual variation in rj over a distance comparable to molecular dimensions is a far more realistic model. 

How must the expressions derived in the sections above be modified to take into account the variation in rj and the finite distance over which it increases The answer is that rj — the viscosity within the double layer —must be written as a function of location. Our objective in discussing this variation is not to examine in detail the efforts that have been directed along these lines. Instead, it is to arrive at a better understanding of the relationship between f and the potential at the inner limit of the diffuse double layer and a better appreciation of the physical significance of the surface of shear. 

Note, further, that this leveling off occurs at progressively lower potentials as the concentration of electrolyte increases. Increasing both the potential and the electrolyte concentration tends to increase the field in the double layer (see Table 12.1), which in turn increases the viscosity of solvent in the double layer. As the effective viscosity of the medium increases, the surface of shear occurs progressively further from the surface. This accounts for the fact that hs falls behind J/0 as [/0 increases. These conclusions are consistent with the experimental observation that HS for Agl becomes independent of the concentration of the potentialdetermining Ag + and I ions once the concentrations of these ions are well removed from the conditions at which the particles are uncharged. 

There are several minor corrections that tend to reduce the discrepancy between the two curves, for example, corrections for relaxation and finite ion size. It should also be remembered that electrophoresis measures the net charge inside the surface of shear. To the extent that this diverges from the surface of the molecule, the two techniques may very properly... 

Electrokinetic phenomena are only directly related to the nature of the mobile part of the electric double layer and may, therefore, be interpreted only in terms of the zeta potential or the charge density at the surface of shear. No direct information is given about the potentials tf/0 and charge density at the surface of the material in question. 

The zeta potential is the resultant potential at the surface of shear due to the charges + QE of the electrokinetic unit and -Qe of the mobile part of the double layer - i.e. 

Let s be the potential difference developed between the ends of a capillary tube of radius a and length / for an applied pressure difference p. Assuming laminar flow, the liquid velocity vx at a distance x measured from the surface of shear and along a radius of the capillary is given by Poiseuille s equation, which can be written in the form... 

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