Zeta potential Smoluchowski theory

Like in the case of electrophoresis, the Smoluchowski equation is only valid for particles with thin double layers and negligible surface conductance (low zeta potentials). The theory was later generalized to arbitrary Ka values by Booth [43] for low zeta potentials, and was developed for arbitrary by Stigter [44], Considering the fact that rather concentrated suspensions are often used in sedimentation potential determinations, theories have also been elaborated to include these situations [45-47]. 

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,... 

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... 

According to the Helmholtz-Smoluchowski (H-S) theory, electroosmotic flow is directly proportional to the dielectric constant of the fluid, the zeta potential, and the voltage gradient, and inversely proportional to fluid viscosity. In general, the surfactant solution and the cosolvent have lower dielectric constants than water, and the surfactant solution may even have a much higher viscosity than that of water. Therefore, the addition of facilitating agents results in a reduction of electroosmotic flow as well as an increase in PAH solubility. 

Under these assumptions, the particle s velocity is linear in the applied electric field, V = hE, where the electrophoretic mobility b = e frj is given by the permittivity e and viscosity rj of the fluid and the zeta potential of the surface. In Smoluchowski s theory, the latter is equal to the voltage across the double layer, which is proportional to the surface charge at low voltage. 

The Debye thickness decreases with increasing electrolyte concentration and it decreases more for the high-valency electrolytes. The surface potential is estimated, sometimes indirectly, via electrokinetic experiments (see Section 10.6). Through these experiments we can measure the electrophoretic mobility, which for very small or very large particles can be related by theory to the so-called zeta potential (Hiickel and Smoluchowski equations. Equation 10.10). The zeta potential is approximately equal to the surface potential. 

In 1809, Reuss observed the electrokinetic phenomena when a direct current (DC) was applied to a clay-water mixture. Water moved through the capillary toward the cathode under the electric field. When the electric potential was removed, the flow of water immediately stopped. In 1861, Quincke found that the electric potential difference across a membrane resulted from streaming potential. Helmholtz first treated electroosmotic phenomena analytically in 1879, and provided a mathematical basis. Smoluchowski (1914) later modified it to also apply to electrophoretic velocity, also known as the Helmholtz-Smoluchowski (H-S) theory. The H-S theory describes under an apphed electric potential the migration velocity of one phase of material dispersed in another phase. The electroosmotic velocity of a fluid of certain viscosity and dielectric constant through a surface-charged porous medium of zeta or electrokinetic potential (0, under an electric gradient, E, is given by the H-S equation as follows ... 

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