Potential in the diffuse layer

For semiconductor electrodes and also for the interface between two immiscible electrolyte solutions (ITIES), the greatest part of the potential difference between the two phases is represented by the potentials of the diffuse electric layers in the two phases (see Eq. 4.5.18). The rate of the charge transfer across the compact part of the double layer then depends very little on the overall potential difference. The potential dependence of the charge transfer rate is connected with the change in concentration of the transferred species at the boundary resulting from the potentials in the diffuse layers (Eq. 4.3.5), which, of course, depend on the overall potential difference between the two phases. In the case of simple ion transfer across ITIES, the process is very rapid being, in fact, a sort of diffusion accompanied with a resolvation in the recipient phase. 

The electric potential in the diffuse layer of a planar surface decays exponentially... 

The potential in the diffuse layer decreases exponentially with the distance to zero (from the Stem plane). The potential changes are affected by the characteristics of the diffuse layer and particularly by the type and number of ions in the bulk solution. In many systems, the electrical double layer originates from the adsorption of potential-determining ions such as surface-active ions. The addition of an inert electrolyte decreases the thickness of the electrical double layer (i.e., compressing the double layer) and thus the potential decays to zero in a short distance. As the surface potential remains constant upon addition of an inert electrolyte, the zeta potential decreases. When two similarly charged particles approach each other, the two particles are repelled due to their electrostatic interactions. The increase in the electrolyte concentration in a bulk solution helps to lower this repulsive interaction. This principle is widely used to destabilize many colloidal systems. 

The Boltzmann part of the model involves development of an expression for the charge density p. For any ion i, its electrochemical potential in the diffuse layer must be equal to that in the bulk of the solution, that is,... 

The potential in the diffuse layer is determined as a function of distance from the oHp by integrating the field as given by equation (10.6.12) or (10.6.15). An analytical expression is only obtained for the case of symmetrical electrolytes. The derivation presented here is for 1-1 electrolytes for which the result obtained below is most often applied. 

In a similar way we may understand another remarkable point ro be derived from M ii 11 e r s tables. For a flat double layer we found that the form of the electric potential curve is radically changed by an increase in the ionic charge. M ii 11 e r s data however, show, that for a spherical particle (with jco 1) the valency of the ions in the solution has only a minor influence upon the decline of the electric potential in the diffuse layer. Hence, also in this respect the Debyc-Hiickel theory is a much better approximation for the spherical double layer field than for the flat double layer, once we wish to apply this theory to cases where the potential is no longer small. 

It has long been recognized that the "Potential has quite a different character from that of the potential (the total potential difference of the double layer), s is usually considerably smaller than it reacts strongly to the addition of indifferent electrolytes, which leave Po unaltered. Although c is perhaps not identical with the potential (the potential in the diffuse layer according to Stern s picture), it is felt that will resemble ps much more than Pq. ... 

The first point impHes a complication of the theory the second point is in a certain respect an advantage, as it eliminates the very high potentials in the diffuse layer for which the theory becomes less reliable (application of the Boltzmann factor in eq. (2) vide the discussion of this equation in the introductory part of chapter II). 

The excess of the volume charge in the diffuse layer causes the origin of the electric potential liquid solution. It is dependent on the distance y from the Helmholtz layer (Figure 8). The potential (f> is conventionally set zero at a big distance from the wall. The value of the potential in the diffuse layer in the closest vicinity to the Helmholtz layer (y = 0) is called the zeta potential, 4> 0) = When longitudinal driving electric field E is applied, the velocity fEOF of the plug-like EOF is related to the zeta potential by the Helmholtz-Smoluchowski equation ... 

Figure 8 Course of electric potential in the diffuse layer of 10moldm NaCI. The zeta potential is supposed to be 100 mV. The curve is a solution of the Poisson-Boltzmann equation and is approximately exponential.

The integration of (3.83) allows us to calculate the variation of the potential as a function of the distance from the surface. To simplify the integration, we set sinh(x) = X. The result (3.91) shows that the potential in the diffuse layer varies exponentially. 

Figure 7.6 Variation in the potential in the diffuse layer, as a function of distance, calculated from the OHP (weak potential approximation). Influence of (a) electrolyte concentration (1 1) and (b) charges of ions in the electrolyte (z z). Marker indicates the distance Data from [18]...

It has been mentioned in chapter VI, 8, p, 263 that Stern s correction results in a lower potential drop in the diffuse double layer (9 instead of The potential in the diffuse layer, instead of being oply dependent on the amount of potential-determining ions, now also depends upon the total electrolyte concentration and is lower, the higher the electrolyte content. This explains why in several cases a sort of critical u-potential has been found and it shows how in the refinements of the double layer theory of stability, conceptions of the older theories (like discharge by adsorption of counter ions) again play a role. 

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