Helmholtz-Smoluchowski equation electroosmotic velocity

As we turn to the discussion of the cross-processes, it would be worth pointing out that when kt 1, the mutual displacement of dispersion medium layers occurs only within a thin layer of liquid in a direct vicinity to the wall. Consequently, the velocity distribution in the medium inside the capillary has the profile shown in Fig. V-13, b. The electroosmotic flux of the medium, QE, is thus equal to the product between the capillary cross-section and the net electrioosmotic phase displacement velocity, % described by the Helmholtz -Smoluchowski equation (V.26), i.e. ... 

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. 

As pointed out in the previous section of this chapter, attention should be paid to electromigration rather than electroosmosis to establish the removal mechanisms of mixed heavy metals. However, high concentrations of multiple metal contaminants, especially divalent cations, can affect the electroosmotic flow in electrokinetic remediation, which is a factor that should be taken into consideration with regard to the removal mechanisms of mixed heavy metals. The electroosmotic velocity, V, is given by the Helmholtz-Smoluchowski equation (Acar and Alshawabkeh, 1993 Mitchell, 1993) ... 

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 ... 

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). 

In the majority of microfluidic cases where 1/k is much smaller than the chaimel height the Helmholtz-Smoluchowski equation provides a reasonable estimate of the flow velocity at the edge of the double layer field. As such when modelling two and three dimensional flow systems it is common to apply this equation as a shp boundary condition on the bulk flow field. Since beyond the double layer by definition pe = 0, the flow equation reduces to (assuming pure electroosmotic flow) to a simple Lapla-... 

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... 

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 ... 

Using the SI units, the velocity of the EOF is expressed in meters/second (m s ) and the electric held in volts/meter (V m ). Consequently, the electroosmotic mobility has the dimension of m V s. Since electroosmotic and electrophoretic mobility are converse manifestations of the same underlying phenomena, the Helmholtz-von Smoluchowski equation applies to electroosmosis, as well as to electrophoresis (see below). In fact, it describes the motion of a solution in contact with a charged surface or the motion of ions relative to a solution, both under the action of an electric held, in the case of electroosmosis and electrophoresis, respectively. 

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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