Helmholtz-Smoluchowski theory

The Helmholtz-Smoluchowski theory equation describes the electroosmotic flow (qe) as it is shown in Eqs. 3 and 4. 

It should also be noted that in the limit of kRs - 0, Equation (47a) reduces to the Hiickel equation, and in the limit of kRs - oo, it reduces to the Helmholtz-Smoluchowski equation. Thus the general theory confirms the idea introduced in connection with the discussion of Figure 12.1, that the amount of distortion of the field surrounding the particles will be totally different in the case of large and small particles. The two values of C in Equation (40) are a direct consequence of this difference. Figure 12.5a shows how the constant C varies with kRs (shown on a logarithmic scale) according to Henry s equation. 

In a number of instances a decision had to be taken as to what to call "fundamentals and what "advanced". In the case of electric double layers, this decision related to the classical Gouy-Stern theory versus modem statistical theories. For pragmatic reasons we decided to emphasize the former the equations are simple and analytical, and can account for the great majority of situations met in practice. However, a section is Included to give an impression of more a priori statistical approaches. In the domain of electrokinetics the decision was between simple theories on the level of Helmholtz-Smoluchowski (HS), that may apply to perhaps 30-50% of all systems studied in practice, or on... 

It was derived by Chen eind Keh and applies to the Helmholtz-Smoluchowski range. The authors considered pair interactions only, but did so in some detail, also discussing differences in size, -potential and alignment of the pair with respect to the applied field. Equation [4.6.571 was confirmed by Anderson So, in contradistinction to the work by Kozak and Davis, and by Shilov et al., these theories predict a finite, but minor dependence on sol concentration. Eventually experiments have to decide. 

Ill) slopes at the te.p. The slopes (du/3pH) p decrease with Increasing c. The mobility u may be converted into f, using the Helmholtz-Smoluchowski equation if xa is large enough (no double layer polarization and no influence of surface conduction close to the zero point). Then, at low electrol3rte concentration / 9pH may approach 59 mV per pH unit at 25 , as would be the case for ilf° If the Nernst equation applies. However, such a steep slope persists only close to the zero point mostly it is much lower. Let us assume absence of specific adsorption (zeroth-order Stem theory, see flg. 3.17a) then we may write... 

Velocity profiles across the capillary have a Poisseuille shaped flow and the expression predicts that the electroosmotic coefficient of permeability should vary with the square of the radius. In practice, it is found generally that this law is not as satisfactory as the Helmholtz-Smoluchowski approach for predicting electroosmotic behavior in soils. The failure of small pore theory may be because most clays have an aggregate structure with the flow determined by the larger pores [6], Another theoretical approach is referred to as the Spiegler Friction theory [25,6]. Its assumption, that the medium for electroosmosis is a perfect permselective membrane, is obviously not valid for soils, where the pore fluid comprises dilute electrol d e. An expression is derived for the net electroosmotic flow, Q, in moles/Faraday,... 

In 1879, Helmholtz introduced one of the first theories concerning electroosmo-sis,andSmoluchowskimodifieditinl914.AccordingtotheHelmholtz-Smoluchowski theory (H-S theory), the electro-osmotic flow velocity (Veo) is directly proportional... 

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 ... 

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. 

It has been pointed out above that electroosmotic and electrophoretic mobilities are converse manifestations of the same underlying phenomena therefore the Helmholtz-von Smoluchowski equation based on the Debye-Huckel theory developed for electroosmosis applies to electrophoresis as well. In the case of electrophoresis, is the potential at the plane of share between a single ion and its counterions and the surrounding solution. 

Helmholtz and v. Smoluchowski obtained the same equation as (21) using more rigorous hydrodynamic theory than tliat involved in equation (18). 

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