Electrophoretic relaxation effect

L is Avagadro s constant and k is defined above. It can be seen that there are indeed two corrections to the conductivity at infinite dilution tire first corresponds to the relaxation effect, and is correct in (A2.4.72) only under the assumption of a zero ionic radius. For a finite ionic radius, a, the first tenn needs to be modified Falkenliagen [8] originally showed that simply dividing by a temr (1 -t kiTq) gives a first-order correction, and more complex corrections have been reviewed by Pitts etal [14], who show that, to a second order, the relaxation temr in (A2.4.72) should be divided by (1 + KOfiH I + KUn, . The electrophoretic effect should also... 

A finite time is required to reestabUsh the ion atmosphere at any new location. Thus the ion atmosphere produces a drag on the ions in motion and restricts their freedom of movement. This is termed a relaxation effect. When a negative ion moves under the influence of an electric field, it travels against the flow of positive ions and solvent moving in the opposite direction. This is termed an electrophoretic effect. The Debye-Huckel theory combines both effects to calculate the behavior of electrolytes. The theory predicts the behavior of dilute (<0.05 molal) solutions but does not portray accurately the behavior of concentrated solutions found in practical batteries. 

Overbeek and Booth [284] have extended the Henry model to include the effects of double-layer distortion by the relaxation effect. Since the double-layer charge is opposite to the particle charge, the fluid in the layer tends to move in the direction opposite to the particle. This distorts the symmetry of the flow and concentration profiles around the particle. Diffusion and electrical conductance tend to restore this symmetry however, it takes time for this to occur. This is known as the relaxation effect. The relaxation effect is not significant for zeta-potentials of less than 25 mV i.e., the Overbeek and Booth equations reduce to the Henry equation for zeta-potentials less than 25 mV [284]. For an electrophoretic mobility of approximately 10 X 10 " cm A -sec, the corresponding zeta potential is 20 mV at 25°C. Mobilities of up to 20 X 10 " cmW-s, i.e., zeta-potentials of 40 mV, are not uncommon for proteins at temperatures of 20-30°C, and thus relaxation may be important for some proteins. 

Ideas concerning the ionic atmosphere can be used for a theoretical interpretation of these phenomena. There are at least two effects associated with the ionic atmosphere, the electrophoretic effect and the relaxation effect, both lowering the ionic mobilities. Formally, this can be written as... 

An analysis of Eq. (7.49) shows that the electrophoretic effect accounts for about 60 to 70% of the decrease in solution conductivity, and the relaxation effect for the remaining 40 to 30%. 

Debye and Falkenhagen predicted that the ionic atmosphere would not be able to adopt an asymmetric configuration corresponding to a moving central ion if the ion were oscillating in response to an applied electrical field and if the frequency of the applied field were comparable to the reciprocal of the relaxation time of the ionic atmosphere. This was found to be the case at frequencies over 5 MHz where the molar conductivity approaches a value somewhat higher than A0. This increase of conductivity is caused by the disappearance of the time-of-relaxation effect, while the electrophoretic effect remains in full force. 

These rules are based on the theory of conductivity of strong electrolytes accounting for the electrophoretic effect only (the relaxation effect terms outbalance each other). 

In closer approximations, correction must be made for conductivity effects (relaxation and electrophoretic) and for the real shape of the particles. Thus, the velocity of electrophoretic motion depends on the composition of the... 

As soon as the concentration of the solute becomes finite, the coulombic forces between the ions begin to play a role and we obtain both the well-known relaxation effect and an electrophoretic effect in the expression for the conductivity. In Section V, we first briefly recall the semi-phenomenological theory of Debye-Onsager-Falkenhagen, and we then show how a combination of the ideas developed in the previous sections, namely the treatment of long-range forces as given in Section III and the Brownian model of Section IV, allows us to study various microscopic... 

When in motion, the diffnse electrical donble-layer aronnd the particle is no longer symmetrical and this canses a rednction in the speed of the particle compared with that of an imaginary charged particle with no donble-layer. This rednction in speed is cansed by both the electric dipole field set np which acts in opposition to the applied field (the relaxation effect) and an increased viscons drag dne to the motion of the ions in the donble-layer which drag liqnid with them (the electrophoretic retardation effect). The resnlting combination of electrostatic and hydrodynamic forces leads to rather complicated eqnations which, nntil recently, conld only be solved approximately. In 1978, White and O Brien developed a clever method of nnmerical solntion and obtained detailed cnrves over the fnll range of Ka valnes (0 °°)... 

This effect is called the relaxation effect. Second, in the presence of the ionic atmosphere, a viscous drag is enhanced than in its absence because the atmosphere moves in an opposite direction to the moving ion. This retarding effect is called the electrophoretic effect. In Eq. (7.1), the Ah°°-term corresponds to the relaxation effect, while the E-term corresponds to the electrophoretic effect. For details, see textbooks of physical chemistry or electrochemistry. 

Investigations of the electrophoretic behaviour of monodispersed carboxylated polystyrene latex dispersions as a function of particle size and electrolyte concentration by Shaw and Ottewill191 have confirmed, at least qualitatively, the existence of tea and relaxation effects. 

As the dependency does not include any specific property of the ion (in particular its chemical identity) but only its charge the explanation of this dependency invokes properties of the ionic cloud around the ion. In a similar approach the Debye-Huckel-Onsager theory attempts to explain the observed relationship of the conductivity on c1/2. It takes into account the - electrophoretic effect (interactions between ionic clouds of the oppositely moving ions) and the relaxation effect (the displacement of the central ion with respect to the center of the ionic cloud because of the slightly faster field-induced movement of the central ion, - Debye-Falkenhagen effect). The obtained equation gives the Kohlrausch constant ... 

Electrophoretic effect — A moving ion driven by an electric field in a viscous medium (e.g., an electrolyte solution) is influenced in its movement by the - relaxation effect and the electrophoretic effect. The latter effect is caused by the countermovement of ions of opposite charge and their solvation clouds. Thus an ion is not moving through a stagnant medium but through a medium which is moving opposite to its own direction. This slows down ionic movement. See also -> Debye-... 

The interesting point is that when the ionic cloud moves, it tries to carry along its entire baggage the ions and the solvent molecules constituting the cloud plus the central ion. Thus, not only does the moving central ion attract and try to keep its cloud (the relaxation effect), but the moving cloud also attracts and tries to k p its central ion by means of a force which is then termed the electrophoretic force F. ... 

At infinite dilution, neither relaxation nor electrophoretic effects are operative on the drift of ions both these effects depend for their existence on a finite-sized ionic cloud. Under these special conditions, the infinite-dilution mobility can be considered to be given by the Stokes mobility... 

For a cylindrical particle oriented at an arbitrary angle between its axis and the applied electric field, its electrophoretic mobility averaged over a random distribution of orientation is given by pav = /i///3 + 2fjLj3 [9]. The above expressions for the electrophoretic mobility are correct to the first order of ( so that these equations are applicable only when C is low. The readers should be referred to Ref. [4-8, 10-13, 22, 27, 29] for the case of particles with arbitrary zeta potential, in which case the relaxation effects (i.e., the effects of the deformation of the electrical double layer around particles) become appreciable. 

It will be observed that the values of AA tend towards a limit at very high potentials the relaxation and electrophoretic effects are... 

Calculation of the Zeta Potential. The conversion of electrophoretic mobility to zeta potential is complicated somewhat by the existence of the electrophoretic relaxation effect. Figure 5 shows a schematic diagram of this effect. As we expose an emulsion droplet and its surrounding double layer to an electric field, the double layer distorts to the shape shown in the figure. This distorted double layer now creates its own electric field that... 

Figure 5. A schematic diagram of the electrophoretic relaxation effect. The distorted ion cloud around the particle generates its own electric field that opposes the motion of the particle.

OyBrien and White (10), taking into account the relaxation effect, derived the relationship between electrophoretic mobility and zeta potential. The results of the theoretical calculation are shown in Figure 6. Figure 6 shows the relationship between electrophoretic mobility, U, and zeta potential for two values of Ka, 114 and 285. Each value of Ka has its own relation-... 

The lower the dielectric constant e of the solvent, the stronger the interionic interactions and, thus, the relaxation and electrophoretic effects. For the latter, solvent viscosity also plays a decisive role. When both the relaxation and the electrophoretic constants are known, the conductivity coefficient is obtained ... 

---