Large Debye Length

subscripts 2 and 3 correspond to the Debye layer at the entrance and exit of the cylindrical capillary. Substituting this in the momentum equation 

Using symmetry condition = 0 at r = 0, we have Q = 0. Integrating again, we have 

Integrating the velocity profile, u, for flow rate, Q, we can obtain 

Note Equation (6.118) shows that the ratio of EO flow to hydraulic flow is independent of the capillary radius. Hence, in terms of flow rates achievable, there is no particular advantage in using an electric field rather than a pressure gradient for large Debye length case. 

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In the limit of large Debye lengths (low electrolyte concentrations) the roughness would not bear on capacitance, which would thus obey Eq. (59). 

This is just the Helmholtz-Smoluchowski equation, as might have been expected, since electrophoresis is just the complement of electroosmosis. Its derivation shows that the electrophoretic velocity of a nonconducting particle is independent of the particle size and shape for a constant surface potential when the Debye length is everywhere small compared with the characteristic body dimension. Note that Eq. (7.2.6) differs from the Huckel large Debye length result (Eq. 7.2.2) only by the factor. ... 

For the limiting case of large Debye length, the entire capillary (pore) is within the double layer. If 1 and 4 designate locations in the external solution just outside of the double layer and 2 and 3 designate locations inside the pore entrance, then at equilibrium with no flow ( = 0) and no flux (j = 0) Ai = Z 4 = 0 and 1 2 = [Pg.396]

Another limit for which an analytic solution is readily obtained is that of large Debye length. In this case the ion concentrations are uniform across the pore and are given by the Boltzmann distribution of Eq. (6.5.15) that is. 

Here, e is the charge, r,y is the interionic distance between opposite charges and E is the dielectric constant. The value of AfH is negative at low salt concentration due to the tightness of ion pairs within the complex and the large Debye length, which implies a loose counterion cloud around the original polyelectrolytes. 

Let us now examine how we can obtain an estimate of /q from the measured electromobility of a colloidal particle. It turns out that we can obtain simple, analytic equations only for the cases of very large and very small particles. Thus, if a is the radius of an assumed spherical colloidal particle, then we can obtain direct relationships between electromobility and the surface potential, if either Kit > 100 or Kd < 0.1, where K" is the Debye length of the electrolyte solution. Let us first look at the case of small spheres (where Kd < 0.1), which leads to the Hiickel equation. 

For colloidal solutions, as a general rule, a barrier of 15-25kT is sufficient to give colloid stability, where the Debye length is also relatively large, say, greater than 20 nm. This electrostatic barrier is sufficient to... 

In addn, for an ionized gas to be called a plasma, it must have an equal number of pos and neg charges for, by definition, a plasma has no net charge. Regions termed "sheaths , having large (net charges) do develop at the plasma boundaries. Such sheaths are to the plasma what the surface is to a solid or liquid, and their thickness is of the order of the "Debye length ... 

PLASMA (Particle). 1 An assembly of ions, electrons, neutral atoms and molecules in which the motion of the particles is dominated by electromagnetic interactions. This condition occurs when the macroscopic electrostatic shielding distance (Debye length) is small compared to the dimensions of the plasma. Because of the large electrostatic potentials... 

Which thickness do we have to use This depends on the relevant parameter. If we are for instance, interested in the density of a water surface, a realistic thickness is in the order of 1 nm. Let us assume that a salt is dissolved in the water. Then the concentration of ions might vary over a much larger distance (characterized by the Debye length, see Section 4.2.2). With respect to the ion concentration, the thickness is thus much larger. In case of doubt, it is safer to choose a large value for the thickness. 

What are the correct values of the potentials In the metal the potential is the same everywhere and therefore 99 has one clearly defined value. In the electrolyte, the potential close to the surface depends on the distance. Directly at the surface it is different from the potential one Debye length away from it. Only at a large distance away from the surface is the potential constant. In contrast to the electric potential, the electroc/zmz caZpotential is the same everywhere in the liquid phase assuming that the system is in equilibrium. For this reason we use the potential and chemical potential far away from the interface. 

This formula is applied successively to obtain the integral over a large interval. To obtain the results which follow, p(l ) was evaluated at 20 discrete, logarithmically spaced values of l between 10 2and 10s Debye lengths. This was found to keep the relative error in 0dl(.v), due to numerical integration, below 0.1%. 

At constant surface charge, an increase in the Debye length implies an increase in the repulsion at any distance. However, this is not true for constant surface potential, since the function (Z/22)2 cosh2(Z/22) (for fixed Z) has a maximum at 1/2X 1.2. Consequently, an increase in the effective Debye length corresponds to an increase in repulsion only at large separations (Z > 2.47.) but to a decrease in repulsion at smaller separations. 

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