Zeta Potential Thin Electrical Double Layers

4 ZETA POTENTIAL THIN ELECTRICAL DOUBLE LAYERS 

In these equations v is the relative velocity between the particle and the surrounding medium. The difference between Equations (28) and (29) therefore equals the net viscous force on the volume element  

3 Location of a volume element of solution adjacent to a planar wall. 

Under stationary-state conditions an equal and opposite force is exerted on the volume element by the electric field acting on the ions contained in the volume element. The force on the ions is given by the product of the field strength times the total charge. The latter equals the charge density p times the volume of the element therefore 

The Poisson equation for a planar surface (Equation (11.26)) may now be used as a substitution for p to yield 

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As stated above, the results can be interpreted in terms of potential at the plane of shear, termed the zeta potential ( ). Since the exact location of the shear plane is generally not known, the zeta potential is usually taken to be approximately equal to the potential at the Stern plane. Two simple relations can be used to calculate zeta potentials in limiting cases, one for small particles with thick electric double layers, and one for large particles with thin electric double layers. 

Smoluchowski Equation (Electrophoresis) A relation expressing the proportionality between electrophoretic mobility and zeta potential for the limiting case of a species that can be considered to be large and having a thin electric double layer. Also termed Helmholtz—Smoluchowski Equation. See also Electrophoresis, Henry Equation, Hiickel Equation. 

First we consider the electric potential in the conducting liquid. It is assumed that the electric charge density is not affected by the external electric field due to the thin electrical double layers (EDLs) and small fluid velocity therefore the charge convection can be ignored and the electric field equation and the fluid fiow equation are decoupled [5]. Based on the assumption of local thermodynamic equilibrium, for small zeta potential, the electric potential due to the charged wall is described by the lin-... 

Our next task is to relate this mobility to the zeta potential. This requires a number of assumptions, and we focus on the most important of these. We derive the equations for thick electrical double layers (Section 12.3) and for thin double layers (Section 12.4) first and then examine how intermediate cases can be studied (Section 12.5). 

To obtain Eqs. 11, 12, 13, and 14, it was assumed that the concentration of each ionic species within the electric double layer is related to the electric potential energy by a Boltzmann distribution. A comparison of Eq. 11 with the numerical results obtained by Prieve and Roman [2] shows that the thin-layer polarization model is quite good over a wide range of zeta potentials when Ka > 20. If 1(1 is small and Ka is large, the interaction between the diffuse counterions and the particle surface is weak and the polarization of the double layer is also weak. In the limit of... 

We consider a micro- or nanochannel having a uniform cross-section as shown in Fig. la. When the channel is in contact with an electrolyte, its surface is charged with usually negative ions. The counter-ions in the liquid are then attracted onto the surface while the co-ions are repelled away from the wall. The thin layer near the surface where the counter-ions are thus highly concentrated is called the electric double layer (EDL). The amount of accumulation of the counter-ions in the EDL is determined in such a way that the electric potential difference induced between the wall surface and the bulk equals the zeta potential, which is an intrinsic property of the interface. When an external electric field is applied along the channel, it exerts the Coulomb force to the ions thereby driving the fluid flow. This kind of fluid motion is called electroosmo-sis (e. g. [1]), the working principle of the electroosmotic purrq). 

An analytical study using the method of reflections was conducted by Chen and Keh [9] to investigate the electrophoretic motion of two freely suspended nonconducting spherical particles with infinitely thin double layer. The particles may differ in size and zeta potential, and they are oriented arbitrarily with respect to the imposed electric field. The resulting translational and angular electrophoretic velocities are given by... 

A detailed analysis of the theories of electroosmotic flow in porous media was presented earlier [22] of the theories by Overbeek [23-25] and Dukhin and his co-workers [26-30], Overbeek extended von Smoluchowski s work to packed capillaries under conditions of low electric field strength. The model can be applied to porous or nonporous packing particles of any shape, and the particles can be assumed to be nonconducting, have uniform zeta potential, and a thin double layer. The average EOF velocity in a column for CEC can be expressed as... 

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