Electrophoretic Smoluchowski equation

The electrophoretic mobility, /jl, can be converted to a zeta potential by using the Smoluchowski equation,... 

The mobility depends on both the particle properties (e.g., surface charge density and size) and solution properties (e.g., ionic strength, electric permittivity, and pH). For high ionic strengths, an approximate expression for the electrophoretic mobility, pc, is given by the Smoluchowski equation ... 

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. 

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. 

FIG. 12.4 The domain within which most investigations of aqueous colloidal systems lie in terms of particle radii and 1 1 electrolyte concentration. The diagonal lines indicate the limits of the Hiickel and the Helmholtz-Smoluchowski equations. (Redrawn with permission from J. Th. G., Overbeek, Quantitative Interpretation of the Electrophoretic Velocity of Colloids. In Advances in Colloid Science, Vol. 3 (H. Mark and E. J. W. Verwey, Eds.), Wiley, New York, 1950.)... 

Egorova, E. M. (1994). The validity ofthe Smoluchowski equation in electrophoretic studies of lipid membranes. Electrophoresis, 15, 1125-1131. 

A simple expression for the electrophoretic velocity of a uniformly charged nonconducting particle is the Smoluchowski equation [1]... 

Microelectrophoresis is used to measure the electrophoretic mobility or, in other words, the movement of liposomes under the influence of an electric field. From the electrophoretic mobility the electrical potential at the plane of shear or (zeta) potential can be determined (by the Helmoholtz-Smoluchowski equation). From the zeta potential values the surface charge density (o) can be calculated. 

For large Ka or a both Du s go to zero, meaning that then surface conduction may be neglected. For the electrophoretic mobility this means that the Helmholtz-Smoluchowski equation [4.3.41 also remains the correct limit for high Ka if surface conduction does occur. 

In the limits of small Ka (ku electrophoretic mobility, 17, to zeta potential. These are the Hiickel and Smoluchowski equations, respectively. They may be expressed by... 

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. 

Using SI units, the velocity of the electro-osmotic flow is expressed in meters per second (m/s) and the electric field in volts per meter (V/m). Consequently, in analogy to the electrophoretic mobility, the electro-osmotic mobility has the dimension square meters per volt per second. Because electro-osmotic and electrophoretic mobilities are converse manifestations of the same underlying phenomenon, the Hehnholtz-von Smoluchowski equation applies to electro-osmosis as well as to electrophoresis. In fact, when an electric field is applied to an ion, this moves relative to the electrolyte solution, whereas in the case of electro-osmosis, it is the mobile diffuse layer that moves under an appUed electric field, carrying the electrolyte solution with it. 

Enter the protocol zeta potential measurements with an initial pH point taken at the pH of the solution, in this case, pH 8.25, and points taken every 0.5 pH change until pH 3.7 is reached. Stir between each measure to insure the solution is homogeneous. Electrophoretic mobility is converted automatically to the potential, according to the Smoluchowski equation, by the system. 

The movement of colloidal particles in an electric field is termed electrophoresis. Most commercially available zetameters are designed to measure the electrophoretic mobility of particles. The potential is not measured directly, but is calculated from u. The Smoluchowski equation... 

FIGURE 2.3 potentials of spherical particles a = lOOnm at various KCl concentrations. The thick solid line represents apparent potentials calculated by means of the Smoluchowski equation, which produced identical results for three ionic strengths. Thin lines with points represent potentials calculated from the same experimental results by means of O Brien-White theory. With 0.001 M KCl, the highest and lowest electrophoretic mobilities exceeded the theoretical maximum/minimum, which corresponds to 4.73 X lO- m V-> s-> at 122mV. 

The action of external electric field on the free disperse system results in particle motion (electrophoresis). The electrophoretic velocity, vE, is not a function of ( -potential only, but also depends on the particle radius, r, and the type of electrolyte present in the system. However, it turns out (see fine print further down) that all of these factors can be simultaneously accounted for by the numerical coefficient, kt, introduced into the Helmholtz-Smoluchowski equation (V.26). If the particles are spherical, k, changes from 2/3 for particles smaller compared to the ionic atmosphere thickness (kt 1) to 1 for large particles ( kt 1). Consequently, the particle flux due to the applied electric... 

Henry Equation A relation expressing the proportionality between electrophoretic mobility and zeta potential for different values of the Debye length and size of the species. See also Electrophoresis, Hiickel Equation, Smoluchowski Equation. 

This is just the Helmholtz-Smoluchowski equation, as might have been expected, since electrophoresis is just the complement of electroosmosis. Its derivation shows that the electrophoretic velocity of a nonconducting particle is independent of the particle size and shape for a constant surface potential when the Debye length is everywhere small compared with the characteristic body dimension. Note that Eq. (7.2.6) differs from the Huckel large Debye length result (Eq. 7.2.2) only by the factor. ... 

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

Under conditions where the second term in the denominator is negligibly small, 1, Equation 10.32 is identical to the Helmholtz-Smoluchowski equation (10.13). On the other hand, for larger values of K and the electrophoretic mobility is lower than predicted by Equation 10.13. 

The double layer thickness is decisive for the retarding impact of double layer polarisation on the electrophoretic motion. Equation (2.42) yields simple linear relationships only for the two limiting cases of very thin double layers (ka oo) and infinitely thick ones (kg 0). The correction function/(/ca) then simplifies to a constant value of 3/2 and 1, respectively (Fig. 2.21 von Smoluchowski 1903 Debye and Hiickel 1924 O Brien and White 1978). 

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