Zeta potential Helmholtz-Smoluchowski equation

We have now reached the position of having two expressions —Equations (27) and (39) —to describe the relationship between the mobility of a particle (an experimental quantity) and the zeta potential (a quantity of considerable theoretical interest). The situation may be summarized by noting that both the Huckel and the Helmholtz-Smoluchowski equations may be written as... 

FIG. 12.8 Plot of rju/e versus f/0, that is, the zeta potential according to the Helmholtz-Smoluchowski equation, Equation (39), versus the potential at the inner limit of the diffuse part of the double layer. Curves are drawn for various concentrations of 1 1 electrolyte with / = 10 15 V-2 m2. (Redrawn with permission from J. Lyklema and J. Th. G. Overbeek, J. Colloid Sci., 16, 501 (1961).)... 

Criticize or defend the following proposition Zeta potentials for three different polystyrene latex preparations were calculated by the Helmholtz-Smoluchowski equation from electrophoresis measurements made in different concentrations of KCl.f... 

These zeta potentials are inaccurate because the range of kRs values exceeds the range of validity for the Helmholtz-Smoluchowski equation. The nature of the error is such as to make the estimated values of f too low. 

Besides there is uncertainty in hydrodynamics inside adsorbed layer, which can be (or not) permeable for flux. Hydrodynamic profile can deviate from parabolic Pois-seuille profile, that s why in this case the Helmholtz-Smoluchowski equation does not correctly transform the streaming potential into the zeta-potential values. So we indicate calculated f potential as apparent zeta-potential ( ). 

The most common method for determining the zeta-potential is the microelectrophoretic procedure in which the movements of individual particles under the influence of a known electric field are followed microscopically. The zeta potential can be calculated from the electrophoretic velocity of the particles using the Helmholtz-Smoluchowski equation. 

It should be emphasized that EOD is most attractive when the water is trapped between fine clay particles (i.e., small pores or water transport chaimels) and caimot be further removed efficiently by the application of pressure or vacuum, etc. If the ionic concentration of the trapped water is low, the thickness of the electrochemical double layerio i can become comparable to or even exceed the pore size, thus requiring corrections to the zeta potential approach based on the simple Helmholtz-Smoluchowski equation such corrections cannot, however, be carried out accurately despite many attempts. " ... 

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. 

This formula for the electroosmotic velocity past a plane charged surface is known as the Helmholtz-Smoluchowski equation. Note that within this picture, where the double layer thickness is very small compared with the characteristic length, say alX t> 100, the fluid moves as in plug flow. Thus the velocity slips at the wall that is, it goes from U to zero discontinuously. For a finite-thickness diffuse layer the actual velocity profile has a behavior similar to that shown in Fig. 6.5.1, where the velocity drops continuously across the layer to zero at the wall. The constant electroosmotic velocity therefore represents the velocity at the edge of the diffuse layer. A typical zeta potential is about 0.1 V. Thus for = 10 V m" with viscosity that of water, the electroosmotic velocity U 10 " ms, a very small value. 

Streaming potential was measured using a Brookhaven Instruments Corp. (Holtsville, NY, USA) BI-EKA commercial instrument which has a crossflow slit geometry. Childress and Elimelech (1996) and Elimelech et al. (1994) described the measuring cell and the principle in detail. For comparison, another surface potential analyser was used as constructed and described by Pihlajamaki (1998). The streaming potential, from which the zeta potential can be calculated with the Helmholtz-Smoluchowski equation, was measured in the presence of 10 mM NaCl, 1 mM NaHCO s and 0.5 mM CaCh, unless otherwise... 

Zeta potential is determined Trom electrophoretic mobility (v) with the Helmholtz-Smoluchowski equation (Eq. 13). 

The above equation is a different form of the Helmholtz-Smoluchowski equation. Equation 14 shows that if the average electroosmotic velocity is determined from the experimentally measured current-time relationship, the zeta potential can be calculated. Introducing the electroosmotic mobility concept, Peo = u -vJEx, Eq. 14 can be further simplified in the following format ... 

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 ... 

Particle trajectories are determined by a combination of fluid-flow, electrophoresis and DEP. The fluid-flow is driven by electroosmosis for the case of interest here. For the thin Debye layer approximation, electroosmotic flow may be simply modeled with a slip velocity adjacent to the channel walls that is proportional to the tangential component of the local electric field, as shown by the Helmholtz-Smoluchowski equation 12). Here, the proportionality constant between the velocity and field is called the electroosmotic mobility, tigo- Fluid-flow in microchannels becomes even simpler for ideal flow conditions where the zeta potential, and hence /ieo, is uniform over all walls and where there are no pressure gradients. For these conditions, it can be shown that the fluid velocity at all points in the fluid domain is given by the product of the local electric field and// o(/i). 

Figure 5. Zeta potential of pristine PTFE (pristine) and PTFE with carbon flash evaporated layer from distances 2, 4 and 7 cm (C2, C4 and C7). Black columns represent data obtained by streaming current method (Helmholtz-Smoluchowski equation), the orange ones by streaming potential (Fairbrother-Mastins equation) [41].

The influence of membrane effective fixed charge, Xf, on the transport of ions is estimated by determining the ion / transport number or fraction of the total electric current transported by ion i (TO, that is f = Ij/Ix since Zi h = 1, for single salts L + t. = 1. However, electrical characterization of membrane-surface/electrolyte interface is usually carried out by TSP measurements (A( ) gt), which allows the determination of zeta potential (Q, the electrical potential at the shear plane, by using the Helmholtz-Smoluchowski equation [33] ... 

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] ... 

The electrophoretic mobilities of the minerals were measured with a Zeta-Meter (Pen-Kem Lazar Zee Meter, type 501, USA), the mobilities being converted into zeta potentials by means of the Helmholtz-Smoluchowski equation. The filler concentration of the suspension was 20 mg/1. The ionic strength was kept constant by adding 1 X 10 mol/1 KCl. Measurements were carried out after a conditioning and pH regulating time of 30 min in a nitrogen atmosphere at pH 8.5. 

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 Helmholtz—Von Smoluchowski equation relates the electroosmotic velocity f eof to the zeta potential in the following way ... 

The dependence of the velocity of the EOF (Ve,) on the zeta potential is expressed by the Helmholtz-von Smoluchowski equation [13] ... 

The Helmholtz-von Smoluchowski equation indicates that under constant composition of the electrolyte solution, the EOF depends on the magnitude of the zeta potential, which is determined by various factors inhuencing the formation of the electric double layer, discussed above. Each of these factors depends on several variables, such as pH, specihc adsorption of ionic species in the compact region of the double layer, ionic strength, and temperature. 

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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