Debye Layer Overlap

In this section, we discuss what happens if the Debye length X]) becomes comparable with the transverse length scale a of the channel and the Debye layer from various parts of the wall overlaps at the center of the channel. Eor a standard electrolyte with concentration about 

1 mol/m in water, this will happen for a cylindrical channel with radius a of the order 10 nm. With modern nanotechnology, it is in fact possible to make such channels intentionally. Such dimensions actually occur in nature for some porous materials. 

In the earlier section, we have seen how a nonequilibrium EO flow can be generated by applying an electrical potential difference, Acp, along a channel where an equilibrium Debye layer exists. For an ideal EO flow without any external pressure gradient, the expression for flow rate, has been derived. In this section, we study the flow rate variation as a function of externally applied pressure difference, Ap. This is analogous to studying the capability of an EO microchannel as a micropump. 

The equation can also be solved by superimposing an ED flow u ir) and a standard PoiseuiUe flow Up r), with opposite sign for Ap. Note that the superposition procedure works because of the linearity of the N-S equation due to negligible inertial term (V V) V  

The governing equation for the pressure gradient-driven flow is 

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For relatively wide channels with negligible electrical double-layer overlap (r/8 > 10), a nearly flat flow profile is expected. It has often been stated that when the channel size and the Debye length are of similar dimensions (r 8), complete electrical double-layer overlap occurs and the EOF is negligible. However, when r 8, a significant EOF can still be created the EOF velocity in the central part of the channel is approximately 20% of that in an infinitely wide channel. Only at conditions where r/8 1 is the EOF fully inhibited by double-layer overlap [25], It should be noted here that the approximations made by using the Rice and Whitehead theory at r/8 < 10 may lead to significant errors in the calculation of the velocity distribution and magnitude of the EOF [17] compared to more sophisticated models. 

Figure 36. Defect concentration and conductance effects for three different thicknesses Li L2 Lj. The mesoscale effect on defect concentration (l.h.s.) discussed in the text, when L < 4J, is also mirrored in the dependence of the conductance on thickness (r.h.s.). If the boundary layers overlap , the interfacial effect previously hidden in the intercept is now resolved. It is presupposed that surface concentration and Debye length do not depend on L. (Both can be violated, c , at sufficiently small L because of interaction effects and exhaustibility of bulk concentrations.)36 94 (Reprinted from J. Maier, Defect chemistry and ion transport in nanostructured materials. Part II. Aspects of nanoionics. Solid State Ionics, 157, 327-334. Copyright 2003 with permission from Elsevier.)...

Debye length is somehow a measurement of the diffuse layer thickne.ss. which i.s directly linked to the distance at which electrical repulsion between two approaching interfaces begins. In effect, the pressure (force per unit imerfacial area) produced by two diffuse layers overlapping may be expressed as follows in the case of approaching flat surfaces separated by a distance x ... 

Disjoining Pressure, Fig. 5 and 2 are electrical potentials of charged surfaces. and 2 are both negative, (a) The distance between two negatively charged surfaces, h, is bigger than the thickness of the Debye layers, R. Electrical double layers do not overlap and there is no electrostatic interaction between these surfaces, (b) The... 

As can be seen, the use of Debye s approximation beyond the limits of its applicability results in serious errors. Another important conclusion that can be drawn from Fig. 4 is that when double layers overlap, both the surface charge density and surface potential become dependent on pore size. 

The electroviscous effect present with solid particles suspended in ionic liquids, to increase the viscosity over that of the bulk liquid. The primary effect caused by the shear field distorting the electrical double layer surrounding the solid particles in suspension. The secondary effect results from the overlap of the electrical double layers of neighboring particles. The tertiary effect arises from changes in size and shape of the particles caused by the shear field. The primary electroviscous effect has been the subject of much study and has been shown to depend on (a) the size of the Debye length of the electrical double layer compared to the size of the suspended particle (b) the potential at the slipping plane between the particle and the bulk fluid (c) the Peclet number, i.e., diffusive to hydrodynamic forces (d) the Hartmarm number, i.e. electrical to hydrodynamic forces and (e) variations in the Stern layer around the particle (Garcia-Salinas et al. 2000). 

Electrostatic forces, acting when the electric double layers of two drops overlap, play an important role. As mentioned above, oil drops are often negatively charged because anions dissolve in oil somewhat better than cations. Thus, the addition of salt increases the negative charge of the oil drops (thus their electrostatic repulsion). At the same time it reduces the Debye length and weakens the electrostatic force. For this reason, emulsion stability can exhibit a maximum depending on the salt concentration. 

This potential is valid for arbitrarily widely separated spheres, but it breaks down when the Debye double layers begin to overlap significantly (Russel et al. 1989). However, for small separations, kD < 2, the Derjaguin approximation may still be used, as long as D < a-, that is, Ka 2. An analytic expression can then be obtained if the potential is small enough that the Poisson-Boltzmann equation can be linearized (Russel et al. 1989, p. 117). For small separations between particles, a choice must be made between a constant-potential or a constant-charge boundary condition. For a constant-potential boundary condition, one can write the approximate expression... 

When charged colloidal particles in a dispersion approach each other such that the double layers begin to overlap (when particle separation becomes less than twice the double layer extension), then repulsion will occur. The individual double layers can no longer develop unrestrictedly, as the limited space does not allow complete potential decay [10, 11]. The potential v j2 half-way between the plates is no longer zero (as would be the case for isolated particles at 00). For two spherical particles of radius R and surface potential and condition x i <3 (where k is the reciprocal Debye length), the expression for the electrical double layer repulsive interaction is given by Deryaguin and Landau [10] and Verwey and Overbeek [11],... 

As the electrolyte concentration is low and the Debye radius many times exceeds this distance an identification of the potential in this zone with the potential of the adsorbing ions is reasonable. At high electrolyte concentration the diffuse layer thickness can be comparable to this distance. Even if the counterions are indifferent their distribution in this layer cannot be neglected because it decreases the electrostatic component of surfactant ion adsorption., i.e. enhance its adsorption. In this case we have to consider a discreteness of charges must, the formation of a counter ion atmosphere around the adsorbed ions and their overlap with the neighbour adsorbed ions. 

The electric double layers formed at the microchannel walls do not overlap, and the Debye-Huckel linearization principle remains as valid. 

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