Ionic atmosphere electrophoretic effect

When an external electric field is imposed on an electrolyte solution by electrodes dipped into the solution, the electric current produced is proportional to the potential difference between the electrodes. The proportionality coefficient is the resistance of the solution, and its reciprocal, the conductivity, is readily measured accurately with an alternating potential at a rate of 1 kHz in a virtually open circuit (zero current), in order to avoid electrolysis at the electrodes. The conductivity depends on the concentration of the ions, the carriers of the current, and can be determined per unit concentration as the molar conductivity Ae. At finite concentrations ion-ion interactions cause the conductivities of electrolytes to decrease, not only if ion pairs are formed (see Sect. 2.6.2) but also due to indirect causes. The molar conductivity Ae can be extrapolated to infinite dilution to yield Ae" by an appropriate theoretical expression. The modern theory, e.g., that of Fernandez-Prini (1969), takes into account the electrophoretic and ionic atmosphere relaxation effects. The molar conductivity of a completely dissociated electrolyte is ... 

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... 

Fig. 2.6 Electrophoretic effect. The ion moves in the opposite direction to the ionic atmosphere... 

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. 

In order to consider the influence of the ionic atmosphere on the electrophoretic mobility, the theoretical electrical charge of the ion q in Equation 6.14 is replaced by the smaller effective charge <2eff and the hydrodynamic radius r by the effective radius R of the ion, which includes its ionic atmosphere ... 

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. 

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 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 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. 

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. 

Non-ideality has been shown to be due to ionic interactions between the ions and consideration of these led to the concept of the ionic atmosphere (see Sections 10.3 and 10.5). These interactions must be taken into account in any theory of conductance. Most of the theories of electrolyte conduction use the Debye-Hiickel model, but this model has to be modified to take into account extra features resulting from the movement of the ions in the solvent under the applied field. This has proved to be a very difficult task and most of the modern work has attempted many refinements all of which are mathematically very complex. Most of this work has focused on two effects which the existence of the ionic atmosphere imposes on the movement and velocity of the ions in an electrolyte solution. These are the relaxation and electrophoretic effects. 

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. 

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. 

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. 

Cross term due to (i) An applied external field has an indirect effect on the solvent through the ions. The force exerted on the moving central ion by the solvent in between the ions of the ionic atmosphere is dependent on the ionic distribution in the ionic atmosphere. Hence the interaction between the ions and the solvent will be determined by the interactions between the ions themselves. If the symmetrical distribution is perturbed by the externally applied field this will have an effect on the interactions between an ion and the solvent, and this will result in an additional solvent flow about the ion. For a calculation of the electrophoretic effect this asymmetry should be considered, but in this derivation it is not and it is assumed that the symmetrical distribution given by Equation (12.3) can be used. This added effect is considered in more advanced treatments (see Section 12.10). 

Cross term due to (ii) The central ion of the asymmetric distribution is affected by the movement of the solvent around it due to the electrophoretic effect. Since any ion in the ionic atmosphere can be considered to be a central reference ion, the electrophoretic effect will affect all the ions. Because there are interactions between the ions and the solvent molecules, this will alter the asymmetry of the ionic atmosphere which would be set up due to relaxation in the absence of the electrophoretic effect. This has the consequence of an added perturbation on the asymmetry. For a calculation of the relaxation effect this extra... 

The equivalent conductance of salts or ions increases as the concentration decreases. This phenomenon is directly related to the interionic forces present in solution a given cation, for example, will have more anions in its vicinity than expected from a purely random distribution. This ionic atmosphere has two effects, electrophoretic and time of relaxation, both of which tend to decrease the ion s mobility. In the former effect, the solvent molecules associated with the ionic atmosphere are moving in a direction opposite to that of the central ion. In the latter, the ionic atmosphere moves slower than the central ion, causing a charge separation (electrostatic retarding force) on the central ion. 

As solutions become more dilute, the ionic atmosphere becomes weaker, with the result that both the electrophoretic and time-of-relaxation influences decrease approximately with the square root of the ionic strength of the solution. At infinite dilution there are no disturbing effects on the mobilities of the ions other than variations in solvent and temperature, and the equivalent conductance reaches its maximum value. Equation 5.6 may be written... 

FIGURE 20.2 Forces acting on a charged particle. The particle is negatively charged and surrounded by a positively charged ionic atmosphere, indicated by the dashed circle. Fi is the electrical force, F2 is Stokes frictional drag, F3 is electrophoretic retardation, and F4 is the relaxation effect. 

Fig. 5.2.2. Schematic illustration of the electrophoretic and relaxation effects, a) No external electric field, b) the central cation moves under the influence of the external electric field X. The shaded areas depict the ionic atmosphere having an average excess negative charge. The small circles represent solvent molecules.

When the ions in solvent are forced to move by an external field two effects start to influence conductance. The ions of opposite charge move in opposite directions and their movement is slowed down by the collision of the ionic atmosphere with the solvent molecules. The symmetry of ion distributions is disturbed. These phenomena are called the electrophoretic effect and relaxation effect, respectively. The decrease in conductance resulting from both effects is the basis of the Debye-Hiickel-Onsager theory of conductance [31]... 

The most characteristic properties of ions are their abilities to move in solution in the direction of an electrical field gradient imposed externally. The conductivity of an electrolyte solution is readily measured accurately with a 1 kHz alternating potential in a virtually open circuit, in order to avoid electrolysis. The molar conductance of a completely dissociated electrolyte is A2 = A2°° - 2 + EC2 In C2 + J iR ) C2 — J" R")c2, where S, E, f, and f are explicit expressions, containing contributions from ionic atmosphere relaxation and electrophoretic effects, the latter two depending also on ion-distance parameters R. The infinite dilution can be split into the limiting molar ionic conductivities by using experimentally measured transport numbers extrapolated to infinite dilution, t+° and i °° = 1 - <+°°. For a binary electrolyte, Aa = 2+°° -I- and = i+ A2. Values of the limiting ionic molar conductivities in water at 298.15 K [1] are accurate to 0.01 S cm mol (S = Q ). 

The concentration dependence of ionic conductivity has been discussed briefly above. Due to ion-ion interactions, the conductivity per ion falls as the ion concentration is increased. Two specific interactions have been identified the electrophoretic effect due to the tendency of the ion atmosphere to move in the opposite direction of the ion and the relaxation effect due to the finite time required for the ion atmosphere to re-arrange itself due to the asymmetry imposed by the electric field. Onsager produced a limiting law that showed that the molar electrolytic conductivity fell with the square root of the ion concentration ... 

Electrophoretic Effect While the central ion moves in one direction, the ionic atmosphere which consists of ions of opposite charge move in the opposite direction. Thus, the central ions are forced to move against a stream of solvent. Their velocities are consequentiy reduced. 

At finite concentrations this formula needs modifying in two ways. In the first place, diffusion is governed by the osmotic pressure, or chemical potential, gradient (not, strictly, by the concentration gradient), so that the mean activity coefficient of the electrolyte must be taken into account. In the second place, ionic atmosphere effects must be allowed for. In diffusion, unlike conductance, the two ions are moving in the same direction, and the motion causes no disturbance of the symmetries of the ionic atmospheres there is therefore no relaxation effect. There is a small electrophoretic effect, however, the magnitude of which for dilute solutions has been worked out by Onsager, and the most accurate measurements support the extended formula based on these corrections. 

The electrophoretic effect of ionic atmosphere in diffusion was discussed by Onsager and Fuoss [24,25]. The effect of ionic atmosphere on the sedimenting velocity of a macro-ion in the presence of an added neutral salt may be discussed in a similar way. The forces ki acting on the ions must be balanced by other forces acting on the solvent molecules. Denoting bulk concentrations of the ions by rii, we have... 

So Schmidt and Erkkila experimenting on congo-sols and casein-sols found a rise in conductance of 6-30% for a frequency of about 10. In fields of very high tensions (100,000 V/cm) the velocity of the particles may be so large, that the particle is drawn out of its ionic atmosphere, so that both the time-of-relaxation effect and the electrophoretic retardation disappear. An example of this effect is found in Hartley s work on paraffin-chain salts. 

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