Electrophoresis relaxation effect

In this section, we shall first give a brief review of the phenomenological theory of these effects.5 -6 26 We shall then show how the methods we have discussed in the previous sections may be extended to derive a microscopic theory of the relaxation effect the microscopic theory of electrophoresis will be considered in the next section. 

The relaxation term was calculated as in the earlier theory with electrophoresis and relaxation being treated independently. Other severe approximations were made in the treatments of the electrophoretic and relaxation effects, but this was done for mathematical ease. This resulted in a (1 + sfc) term in the denominator. 

For theories later than the Fuoss-Onsager 1932 treatment it is useful to express the effects of electrophoresis, relaxation and other contributions in a form showing how they modify the external field under which the ions are migrating. 

Henry s equation (2.6) assumes that is low, in which case the double layer remains spherically symmetrical during electrophoresis. For high zeta potentials, the double layer is no longer spherically symmetrical. This effect is called the relaxation elfect. Henry s equation (2.6) does not take into account the relaxation effect, and thus this equation is correct to the first order of Ohshima et al. [19] derived an accurate analytic mobility expression correct to order 1/ka in a symmetrical electrolyte of valence z and bulk concentration (number density) n with the relative error less than 1% for 10 < Ka < 00, which is... 

For the case where the hxed charges are not uniformly distributed in the polyelectrolyte layer and that the relative permittivity in the poly electrolyte layer does not take the same value as that in the bulk solution phase, the above theory must be modihed as discussed by Ohshima and Kondo [52] and Hsu et al. [54]. Tseng et al. [55] considered the effects of charge regulation on the mobility in the polyelectrolyte layer. The case where the polyelectrolyte layer is not fully ion-penetrable is considered in Ref. [56]. Varoqui [57] considered the case where electrically neutral polymers are adsorbed with an exponential segment density distribution onto the particle surface with a charge density. Ohshima [58] extended Varoqui s theory [57] to the case where adsorbed polymers are charged. Saville [59] and Hill et al. [60] considered the relaxation effects of soft particles in electrophoresis. 

In the case of electrophoresis, the retardation and relaxation effects are influenced by surface conduction. In Equations 10.10 and 10.25, K p, the specific electric conductivity of the bulk solution should be replaced by a term accounting for the excess conduction at the surface as well. Taking the streaming potential as an example, the expression for Equation 10.24, should be modified to contain a contribution from the surface. Hence,... 

As we will see in the next section, when field-inversion electrophoresis is used to separate nucleic acids in certain size ranges, intra-tube relaxation effects take place. Of course, as the angle between the two fields increases from 90 to 180 , we expect that the crossed field effects discussed earlier will be modified by more and more of the intra-tube effects that sometimes dominate in field-inversion electrophoresis. It is not the goal of this article to discuss the angles between 90 to 180 . The next section presents a discussion of field-inversion electrophoresis as well as the ways to optimize the reorientation-induced separation obtained with this experimental arrangement. 

B In Huckel s and Henry s treatment of electrophoresis the reader who is familiar with the theory of the conductivity of strong electrolytes will have missed the so-cailed time-of-relaxation effect This effect, originating in the deformation of the double layer also has a retarding influence on the electrophoresis. In the applied field the charge of the double layer is displaced in a direction opposite to the movement of the particle Not only does this charge retard the electrophoresis by its movement (electrophoretic retardation see 6a), but also by the dissymmetry of the double layer resulting from this displacement a retarding potential difference is set up. 

The problem of the influence of the time-of-relaxation effect on electrophoresis has been attacked by various authors They treat it in different ways and different approximations. Owing to the mathematical intricacies their results are far from satisfying. More recently Overbeek Booth and Henry independently have given new treatments of this effect. Booth and Overbeek treated the relocation effect for spherical insulating particles surrounded by a Gcuy double layer. Booth and... 

The equivalence of electrophoresis and electro-osmosis has also been repeatedly tested It has been explained in 6b that reliable values of the ( -potential can only be calculated from electroph.oretic measurements if the time-of-relaxation effect can be neglected. If is not very small this is only realised in the case of large particles with a thin double layer. It follows from Henry s considerations (c/ 6a) that just in this case the electrophotelic velocity is equal to the velocity of electro-osmosis, both obeying the equation of Helmholtz Smoluchow ski (4, 26). This equality can be demonstrated very clearly by the ultramicroscopic method for the determination of the electrophoretic velocity. 

A quantitative relation between the D.C. and the double layer (third effect) is very difficult to give because the increase of the D.C. rests completely upon the time-of-relaxation effect, and we know already from the electrophoresis how difficult calculations of this effect are. 

A related observation is that fully relaxed supercoiled DNA/dye complexes are somehow different from nicked circular DNA/dye complexes in the presence of the same concentration of free dye, where the binding ratios should be the same. This is readily seen in gel electrophoresis in the presence of sufficient dye concentration so that at least one, but not all, of the topoisomers is positively supercoiled. The slowest moving, presumably fully relaxed, topoisomer migrates significantly faster than the nicked circle, and this difference increases with the amount of dye present. This is not observed with chloroquine, perhaps because the effect is too small. However, it is readily apparent in the original gels of Keller0 61) in which ethidium was used to unwind the topoisomers. We have confirmed this effect for ethidium and have observed similar behavior for proflavine, 9-aminoacridine, and quinacrine. 

There are two experiments which beautifully illustrate the correctness of the idea of an ionic atmosphere and its manifestation in terms of the relaxation and electrophoresis which occur when the ion moves under the influence of an external field. These are the Debye-Falkenhagen effect and the Wien effect. 

The early conductance theories given by Debye and Hiickel in 1926, Onsager in 1927 and Fuoss and Onsager in 1932 used a model which assumed all the postulates of the Debye-Hiickel theory (see Section 10.3). The factors which have to be considered in addition are the effects of the asymmetric ionic atmosphere, i.e. relaxation and electrophoresis, and viscous drag due to the frictional effects of the solvent on the movement of an ion under an applied external field. These effects result in a decreased ionic velocity and decreased ionic molar conductivity and become greater as the concentration increases. 

The progressive reduction in mobility as the concentration increases is due to interionic interactions of a coulombic nature. The dominant effects are due to relaxation, electrophoresis and their cross terms. 

Electrophoresis and relaxation were taken to be totally independent phenomena, whereas they are not. As a result the derivation neglected, (i) the effect of the asymmetry of the ionic atmosphere on the electrophoretic effect, and (ii) the effect of electrophoresis on the movement of the ion in an asymmetrical ionic distribution. These are cross terms described below. 

In aU of these modifications no account was taken of the need to consider cross terms arising from the effect of relaxation on the electrophoretic effect, and from the effect of electrophoresis on relaxation, but they did hint at the form of the conductance theory put forward later by Fuoss and Onsager. 

Pay particular note the direct effect of considering both relaxation, AXreiaxation, and electrophoresis, AXeiectrophoresis, Contributes a term in i.e. they give the S /c term in the Debye-Hiickel-Onsager equation. When the effects of the cross terms are considered they have no term in and thus do not contribute to the Sy/c term. But all five corrections contribute to the other terms in the final Fuoss-Onsager equation. 

Under non-ideal conditions, the external field is modified by electrophoresis and relaxation, and this modified field is (X — AX). As the concentration decreases, AX also decreases, and in the limit as c — 0, i.e. ideal conditions, AX 0. But, the velocity with which the ion migrates under this modified field is still defined in terms of the external field, even for the non-ideal case where the actual effective field under which the ion migrates is (X AX), i.e. ... 

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