Helmholtz-Smoluchowsky equation

The equations of the electrokinetic processes were derived in 1903 by the Polish physicist Maryan Ritter von Smoluchowski on the basis of ideas concerning the function of EDL in these processes that had been developed by H. Helmholtz in 1879. These equations are often called the Helmholtz-Smoluchowski equations. 

The dimentionaless ica is a measure of the ratio between the particle radius and the thickness of the ionic double layer. In the limit ica (the double layer is very thin compared with particle radius f,(ica) = 3/2 and the result is the Helmholtz-Smoluchowski equation. In the limit ica + 0, f,(ica) 1 and Huckel s result is obtained. 

Equation (39) is known as the Helmholtz-Smoluchowski equation. No assumptions are made in its derivation as to the actual structure of the double layer, only that the Poisson equation applies and that bulk values of rj and e apply within the double layer. It has been shown that this result is valid for values of kR larger than about 100 (i.e., kRs > 100 for spherical particles). 

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... 

A major remaining problem is that many systems of interest in colloid chemistry do not correspond to either of these two limiting cases. The situation is summarized in Figure 12.4, which maps the particle radii Rs and 1 1 electrolyte concentrations that correspond to various kRs values. Clearly, there is a significant domain of particle size and/or electrolyte concentration for which neither the Huckel nor the Helmholtz-Smoluchowski equations can be used to evaluate f from experimental mobility values. The relationship between f and u for intermediate values of kRs is the topic of the following section. 

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.)... 

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. 

Now suppose we reexamine the derivation of the Helmholtz-Smoluchowski equation as given in Section 12.4. Returning to Equation (37), we note that the relationship between u and f is given by... 

Under conditions in which the second term is negligibly small, Equation (73) becomes identical to Equation (39), the Helmholtz-Smoluchowski result. On the other hand, when the concentration and f increase, the value of f that would be associated with an observed mobility is larger than the Helmholtz-Smoluchowski equation would indicate. 

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).)... 

What is the Helmholtz-Smoluchowski equation How is it different from the Hiickel equation ... 

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. 

Where l is the length of the capillary, a is the radius of the capillary, Jo, is a Bessel function of the first kind and k = sj / r /poS) = 8, is the viscous skin depth, which is the distance at which the amplitude of the vorti-city (transverse) wave has attenuated by a factor of the natural logarithm e . Inserting (3) into (1) and using Poisson s equation for the charges distribution we can solve for the FDSP Helmholtz-Smoluchowski equation [Reppert et al., 2001],... 

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 ( ). 

A few corrections have been proposed to modify the Helmholtz-Smoluchowski equation (3, 16, 28). Under certain experimental conditions many of these corrections become too small to be considered (6). However, these corrections are not as important as those that have to be introduced when potentials are calculated from mobility data (4, 5, 20, 21, 32, 33). 

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. 

As a later generalization we present a more rigorous derivation of the Helmholtz-Smoluchowski equation for high yq, in which the curvature of the field is explicitly taken into account. This method, given by Anderson and Prleve l, may be considered as an elaboration of the Smoluchowski theorem for the case of electrophoresis of large spheres. Surface conduction is still Ignored. 

Figure 4.5. Identification of symbols in the derivation of the Helmholtz-Smoluchowski equation for spherical particles at high xa r " - a =...

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. 

According to general experience the Influence of sol concentration Is absent when the volume fraction remains of the order of a few percent. This may be concluded from older experiments and from recent electroacoustic studies, discussed in sec. 4.5d. Experiments involving mlcro-electrophoresls are not suited to stud3dng the volume fraction effect because the required extreme dilution may readily lead to spurious adsorption on the particles. Reed and Morrison studied theoretically the hydrodynamic interactions between pairs of different particles in electrophoresis they corrected the Helmholtz-Smoluchowski equation for various distances between the (spherical) particles and values of the electroklnetlc potentials of the particles, and... 

I) Particles with very large ko the Helmholtz-Smoluchowski equation 4.3.4] applies and remains an acceptable approximation when the particles are not spherical. The applicability limit can be inferred from fig. 4.26,... 

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... 

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. 

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