Helmholtz-Smoluchowski relation

Quantitatively, the convective liquid velocity from electroosmosis, Ueo, is given by the Helmholtz-Smoluchowski relation 49,50... 

IV. THE ELECTROCHEMICAL APPROACH TO ELECTRO-OSMOTIC DEWATERING HELMHOLTZ-SMOLUCHOWSKI RELATION... 

The theoretical approach generally used "in electro-osmotic dewatering is an electrochemical one in which the Helmholtz-Smoluchowski relation is used to relate the electro-osmotic convective liquid velocity to such parameters as the viscosity and permittivity of the solution, the zeta potential of the clay surface, and the strength of the applied field. Also, electrode kinetic effects are taken into account where the data point to the involvement of electrochemical reactions at the electrodes during the EOD process. " In combined pressure-electro-osmotic dewatering (CPEOD), the effect of pressure is interpreted in an empirical, ad-hoc manner without any attempt to develop a comprehensive theoretical framework that combines the two driving forces, namely, the pressure and the electric field. 

Here, A is the streaming potential difference developed between the ends of the capillary across which the applied pressure difference is Ap. Using the Helmholtz-Smoluchowski relation to replace by the electroosmotic velocity and Ohm s law to eliminate the conductivity and electric field we obtain the following expression connecting the streaming current, streaming potential, and electroosmotic velocity ... 

In a bid to calibrate the electrophoresis instrument independently, we carried out iso-electric pH (pi) measurements on gelatin solutions having concentrations 0.001, 0.01 and 0.1 % (w/v). The measured electrophoretic mobility (p) values could be converted to equivalent zeta potential (0 values through the Helmholtz-Smoluchowski relation p = scoC/t/. where the solvent dielectric constant and viscosity are given by b and rj respectively. The permittivity of vacuum is So = 8.85 X 10 C m and the value used for / 10 Pa s. Figure 2 depicts the dependence of C on solution pH. The measured pi = 5.5 0.5 value did not show concentration dependence within the limits of the experimental error. The value provides excellent matching with the nominal pi value cited by the manufacturer ( 5.0 0.2). This established the reproducibility, reliability and robustness of our measured electrophoretic mobility data. 

This equation, known as the Helmholtz-Smoluchowski equation, relates the potential at a planar bound surface region to an induced electro-osmosis fluid velocity 6. Recall that in the previous section surface charge was related to a potential in solution. In the following section surface charge will be related to the chemistry of the surface. A model for the development of surface charge in terms of acid-base dissociation of ionizable surface groups is introduced. 

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... 

Expression (V.25), referred to as the Helmholtz-Smoluchowski equation, relates the rate of relative phase displacement to some potential difference, Acp, within the electrical double layer. In order to understand the nature of this quantity, let us examine in detail the mutual phase displacement due to the external electric field acting parallel to the surface, taking into account the electrical double layer structure. Let us assume that the solid phase surface is stationary. Figure V-7 shows the distributions of the potential, cp(x ) (line 1), the rate of displacement of the liquid layers relative to the surface in the Helmholtz model, u(x) (line 1/), and the true distribution of the potential in the double layer (curve 2). 

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. 

With a finite-thickness double layer we may distinguish three effects that will alter the electrophoretic velocity from that given by the Helmholtz-Smoluchowski or Huckel relations. These effects, which in general are not mutually exclusive, are termed electrophoretic retardation, surface conductance, and relaxation (Shaw 1969). 

The excess of the volume charge in the diffuse layer causes the origin of the electric potential liquid solution. It is dependent on the distance y from the Helmholtz layer (Figure 8). The potential (f> is conventionally set zero at a big distance from the wall. The value of the potential in the diffuse layer in the closest vicinity to the Helmholtz layer (y = 0) is called the zeta potential, 4> 0) = When longitudinal driving electric field E is applied, the velocity fEOF of the plug-like EOF is related to the zeta potential by the Helmholtz-Smoluchowski equation ... 

A simple relation between the streaming potential and the zeta potential ( ) or electrical potential at the shear plane can be obtained by the Helmholtz-Smoluchowski equation [33] ... 

This relation is known as the Helmholtz-Smoluchowski equation. A sketch of the result from Equation 8.86 is included in Figure 8.14. The thickness of the skin is essentially the Debye length, which is dictated by the salt concentration. For aqueous solutions at room temperatures with the surface potential of 0.1 V and an electric field of 10 V/m, the terminal EOF velocity is about 10 nm/s, which is substantial in nanofluidics. 

The feedback mechanism of NDR is based on the electroosmotic velocity, u, being proportional to the value of zeta potential, at the velocity slip plane located adjacent to the nanopore surface. The Helmholtz-Smoluchowski equation relates the effective slip electroosmotic velocity to... 

The electrophorehc mobility, is related to the zeta potential, which is defined as the electric potenhal at the surface of shear of the particles and is therefore a measure of their total charge. Unfortunately, the electrophoretic mobility of dispersion particles does not depend solely on the zeta potential, but also in a complex way on particle size and on the ionic strength and viscosity of the aqueous phase [21]. It is only at the limits of very high and very low ionic strength that can be directly computed from the measured values (Helmholtz-Smoluchowski or Huckel approximations). 

According to the Helmholtz-Yon Smoluchowski [2,16-18,20,21] equation, the electroosmotic velocity, veof, is related to the potential in the following way ... 

---