The electrophoretic effect

The ions are moving in the solvent, and the effect of the solvent on the movement of the ion and its ionic atmosphere under the applied field must also be considered. This is discussed under the heading of the electrophoretic effect. 

When an ion moves it carries its ionic atmosphere with it. This ionic atmosphere is made up, in part, of ions which have the same charge as the central ion and will move in the same direction as the central ion under an external field. It also contains ions which will move in the opposite direction. An ion will drag the solvent molecules in its vicinity along with it. Since cations and anions move in opposite directions as a result of the externally applied electric field, each ion moves against a stream of solvent molecules. 

If the central reference ion and an ion of opposite charge in the ionic atmosphere are passing each other, solvent will be pulled along with each ion but in opposite directions. So the central reference ion of the pair wiU, in effect, see solvent streaming past itself in the opposite direction, and this wdl exert a viscous drag on the central reference ion, slowing it down. 

These effects are covered by the general term electrophoretic effect , and their net effect is always to slow the ion down, resulting in a lower ionic conductivity than would be expected if there were no ionic atmosphere. 

3 Conductance equations for strong electrolytes taking non-ideality into consideration early conductance theory 

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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... 

Removal of Cake by Mass Forces. This method of limiting cake growth employs mass or electrophoretic forces on particles, acting tangentially to or away from the filter medium. Only mass forces are considered here because the electrophoretic effects have been discussed previously. 

The Dehye-Hbckel theory of electrolytes based on the electric field surrounding each ion forms the basis for modern concepts of electrolyte behavior (16,17). The two components of the theory are the relaxation and the electrophoretic effect. Each ion has an ion atmosphere of equal opposite charge surrounding it. During movement the ion may not be exacdy in the center of its ion atmosphere, thereby producing a retarding electrical force on the ion. 

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%. 

The influence of interionic fores on ion mobilities is twofold. The electrophoretic effect (occurring also in the case of the electrophoretic motion of charged colloidal particles in an electric field, cf. p. 242) is caused by the simultaneous movement of the ion in the direction of the applied... 

In the ideal case, the ionic conductivity is given by the product z,Ft/ . Because of the electrophoretic effect, the real ionic mobility differs from the ideal by A[/, and equals U° + At/,. Further, in real systems the electric field is not given by the external field E alone, but also by the relaxation field AE, and thus equals E + AE. Thus the conductivity (related to the unit external field E) is increased by the factor E + AE)/E. Consideration of both these effects leads to the following expressions for the equivalent ionic conductivity (cf. Eq. 2.4.9) ... 

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). 

It is well-known that the electrophoretic effect involves the hydrodynamical properties of the solvent in a very crucial way for this reason, the theory of this effect is rather difficult. However, using a Brownian approximation for the ions, we have been able to obtain recently a microscopic description of this effect. This problem, together with the more general question of long-range hydrodynamical correlations, is discussed in Section VI. 

As was announced at the beginning of this section, both the relaxation and the electrophoretic effects are proportional to V C. More precisely, if we use the definitions (113) and the definition of the current per unit volume ... 

However, the present formulation has the advantage of furnishing a mathematically rigorous foundation to the classical theory and is readily extended to other physical situations, like plasmas32 and semiconductors. Also, it allows us to give a microscopic foundation to the theory of the electrophoretic effect, which is much more delicate because it involves the difficult question of long-range hydrodynamical correlations this point will be the object of Section VI. 

Our discussion of Section V has indicated that the electrophoretic effect has to be found in the Ta term defined in Eq. (301) (see also Eq. (312)) moreover we have already found a diagram (Fig. 14a) in which the solvent is transmitting the wave number —k from ion /S to ion a, as we expect to find from the classical theory. This term was not calculated in Section V because it gives a contribution of order ei to while the relaxation term is of order e6 it will be considered presently. 

In order to calculate the contribution of the electrophoretic effect to the limiting law, as given by the graph of Fig. 22, the simplest way is to use the above-mentioned analogy between... 

Equation (6.313) indicates that electrophoretic mobility is independent of the shape of the particles. Suppose, however, that the particle is spherical. Then one could arrive at the electrophoretic mobility in a completely different way and in a maimer used to calculate the electrophoretic effect in conduction (see Section 4.6.4). One starts with Stokes law (see Section 4.4.8)... 

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. 

In any case, in sulphuric acid the electrophoretic term is very small because of the high viscosity of sulphuric acid (ij = 24 5 centipoise at 25°). Thus the electrophoretic effect could only decrease the conductivity by 0 3%, even at c = 1. Substituting, e = 100, T — 298°K and — 10 [4, 5] we have... 

The electronic ink is supplied by E ink corporation (Comiskey 1998). The film consists of electrophoretic microcapsules in a polymer binder, coated on to a 25 pm polyester/indium tin oxide sheet (Fig. 14.10). Optical contrast is achieved by moving black and white sub-micron particles with opposite charge in a transparent fluid within a microcapsule. Depending on which sub-micron particles are closest to the viewer, light is scattered back (white state) or absorbed (black state). The electrophoretic effect is multi-stable - without any electric field the microcapsules keep their switching state. This greatly reduces the power consumption of the display (Ritter 2001). 

Depending on s, /i"1 may vary from 3-1000 A, or beyond. For example, in 10 2 M NaCl, A"1 = 30 A. The important point is that double layer thickness h 1 varies as 1/ 1/2 and thus also with overall electrolyte concentration c as l/c1/2. Consequently, retardation by the electrophoretic effect, which we indicated to be greatest for thin double layers (small A 1), increases with electrolyte concentration. 

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 ... 

Electrodecantation — Application of the electrophoretic effect (- electrophoresis) to concentrate a suspension of particles in a liquid by collecting the particles near one of the two electrodes used to apply a voltage across the cell containing the suspension. 

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-... 

A Second Braking Effect of the Ionic Cloud on the Central Ion The Electrophoretic Effect... 

Another factor which tends to retard the motion of an ion in solution is the tendency of the applied potential to move the ionic atmosphere, with its associated solvent molecules, in a direction opposite to that in which the central ion, with its solvent molecules (cf. p. 114), is moving. An additional retarding influence, equivalent to an increase in the viscous resistance of the solvent, is thus exerted on the moving ion this is known as the electrophoretic effect, since it is analogous to the resistance acting against the movement of a colloidal particle in an electrical field (cf. p. 530). 

The ion cloud with the solvation shells of its ions moves in a direction opposite to the central ion. Therefore, the central ion does not move relative to a resting medium but rather against a solvent flow. The resulting reduction in mobility is called the electrophoretic effect. Both effects become more important with increasing electrolyte concentration and result in a decrease in conductivity. 

Q.22.8 What is the electrophoretic effect Does it increase linearly with concentration Assuming everything else could be kept constant, how would increasing the ion s charge impact this effect Is assuming everything is constant a good assumption ... 

The influence of the interionic forces is due to two phenomena, namely, the electrophoretic effect and the time-of-relaxation effect. The net ionic atmosphere around a given ion carries the opposite charge and therefore moves in a direction opposite to the central ion. The final result is an increase in the local viscosity, and retardation of the central ion. This is called the electrophoretic effect. The time-of-relaxation effect is also related to the fact that the ionic atmosphere around a given ion is moving and therefore disrupted from its equilibrium configuration. It follows that the ionic atmosphere must constantly be re-formed from new counter ions as the ion under observation moves through the solution. The net effect is that the electrical force on each ion is reduced so that the net forward velocity is smaller. 

The derivation of an expression for the electrophoretic effect is based on the Debye-Hiickel theory described earlier in section 3.8. On the basis of the Boltzmann distribution law, the local concentration of ion i at a distance r from a central reference ion j is... 

The limiting ionic mobility for ion i, is reduced by an amount Am, (relx) as a result of the field relaxation effect. Thus, ignoring the electrophoretic effect one may write... 

The Electrophoretic Effect. According to the Debye-Hiickel theory an ion is surrounded by an ionic atmosphere distributed with radial symmetry around the ion as center. This ion atmosphere, it will be recalled, is due to the fact that interionic attractions and repulsions tend to produce a slight preponderance of negative ions in the vicinity of a positive ion, and vice versa. Although the ion atmosphere is treated as a reality in mathematical discussions it actually is the result of a time average of a distribution of the ions. Each ion serves as a center of an ion atmosphere, and the relative position of each ion with respect to the other charged bodies in the solution influences the atmospheres of all the other ions. 

The factors determining the velocity V are, so far, unknown. The velocity, vx9 however, determines the limiting conductance, Ao, of an ion. The velocity which has an opposite sign to that of V is the retardation due to the electrophoretic effect. 

As already mentioned it has been customary to deal with the ionic atmosphere as if it were a reality, and the derivation just given assumes that an electric force acting on the ion atmosphere will produce a motion of the solvent. However, the effect of a potential gradient cannot be directly on the solvent, but must have its influence indirectly through the ions. The fundamental explanation of the electrophoretic effect must therefore be sought in a modification of inter-reactions between ions and solvent produced by the ion atmosphere, and the latter is, as we have seen, due in turn to a time average of the distribution of the ions. 

It is of interest to estimate the influence of the electrophoretic effect on the equivalent conductance of typical electrolytes. From equation (11) for a singly charged ion... 

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