Debye Layer Screening

FIGURE 11.32 Flow profiles in microchannels, (a) A pressure gradient, - AP, along a channel generates a parabolic or Poiseuille flow profile in the channel. The velocity of the flow varies across the entire cross-sectional area of the channel. On the right is an experimental measurement of the distortion of a volume of fluid in a Poiseuille flow. The frames show the state of the volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule, (b) In electroosmotic flow in a channel, motion is induced by an applied electric field E. The flow speed only varies within the so-called Debye screening layer, of thickness D. On the right is an experimental measurement of the distortion of a volume of fluid in an electroosmotic flow. The frames show the state of the fluorescent volume of fluid 0, 66, and 165 ms after the creation of a fluorescent molecule [165], Source http //www.niherst.gov.tt/scipop/sci-bits/microfluidics.htm (see Plate 12 for color version). 

It should be stressed that the negative slope of Uc for the EC rolls in Fig. 4.9a when approaching ft, cannot be described by the existing theories. It seems to be very plausible that at very small / the measured Uc has to be corrected by contributions due to alignment coatings on the confining electrode plates and from Debye screening layers at the boundaries. Some additional remarks are postponed to the next section. 

The above electroneutrality assumption holds everywhere except in the thin Debye screening layer next to the solid surface. The potential drop across the Debye layer can be significant even though it is only a few nanometers in thickness. It may be noted that ionic mass transport affects the current density, J, which influences the flow field by Lorentz body force. Therefore, one needs to solve simultaneously the full mathematical model consisting of continuity, N-S equation, Nernst-Planck equation, and the local electroneutrality conditions with the appropriate boundary condition. 

This is an inverse lengtli k is known as tire Debye screening lengtli (or double layer tliickness). As demonstrated below, it gives tire lengtli scale on which tire ion distribution near a surface decays to tire bulk value. Table C2.6.4 gives a few numerical examples. 

In the second group of models, the pc surface consists only of very small crystallites with a linear parameter y, whose sizes are comparable with the electrical double-layer parameters, i.e., with the effective Debye screening length in the bulk of the diffuse layer near the face j.262,263 In the case of such electrodes, inner layers at different monocrystalline areas are considered to be independent, but the diffuse layer is common for the entire surface of a pc electrode and depends on the average charge density <7pc = R ZjOjOj [Fig. 10(b)]. The capacitance Cj al is obtained by the equation... 

Daikhin, double layer capacitance of solid at rough electrodes, 52 Debye screening and diffuse layer near the surface, 50... 

Diffuse layer near the surface, Debye screening and, 50... 

The radial distribution function of polymer-counterion as a function of distance from the polyion surface exhibits two peaks. These peaks are interpreted as reflecting two populations of counterions a diffuse Debye-Htickel cloud at a distance characteristic of the Debye screening length, and the condensed layer near the polymer surface where the condensed counterions reside [43]. 

In concentrated suspensions many body interactions between the colloidal particles determine the effective colloid-colloid interaction. Beresford-Smith and Chan (1983) [37] showed that in that case the effective colloid-colloid interaction can nevertheless be described by an effective pair interaction energy to characterise the electrical double layer interaction. This pair interaction energy also has a screened Coulomb form just as in the classical DLVO theory but the Debye screening parameter k now depends on the intrinsic coxmterion concentration and the concentration of added electrolyte in the system. This makes the effective pair energy dependent on the volume fraction of the particles (see general discussion of the paper of Beresford-Smith and Chan [38]. 

The condition of electrical neutrality will not apply at the surface of a crystal, and since g+ is not equal to g there will be an excess of one of the defects. This effect, which is referred to in the early literature as the Frenkel-Lehovec space charge layer-results in an electric potential at the surface of the crystal [23-25]. In this instance, the surface will not simply be the external surface but will also include internal surfaces such as grain boundaries and dislocations. The effect decays away in moving from the surface to the bulk, and can be treated by classical Debye-Hiickel theory [26-29]. This leads to a Debye screening length, Lp, given by... 

FIG. 3 The partition function of the condensed counterion layer as a function of the distance r from the polyion of another counterion that has been brought from infinity to r. Debye screening length 30 A (0.01 M NaCl) polyion charge spacing 1.7 A. 

The charge density accumulation cannot be neglected for interphase layers such as the electrical double layer. Using a nondimensionalized form of Equation 2.2, it was shown that the electroneutrality condition is a direct result of the Debye screening length, where the ratio of the Debye length Xx, to the field length L is described as follows (Eq. 2.3) (Chu, 2005) ... 

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