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240 Mott-Schottky approximation is
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The Mott-Schottky plot following from Eqs (4.5.12) and (4.5.14) is the relationship 2 C"2 = 5- p + constant (4.5.15) which can be used to determine the concentration of electron donors It has so far been assumed that the semiconductor-electrolyte interphase does not contain either ions adsorbed specifically from the electrolyte or electrons corresponding to an additional system of electron levels. These surface states of electrons are formed either through adsorption (the Shockley levels) or through defects in the crystal lattice of the semiconductor (the Tamm levels). In this case analogously as for specific adsorption on metal electrodes three capacitors in series cannot be used to characterize the semiconductor-electrolyte interphase system and Eq. (4.5.6) must include a term describing the potential difference for surface states. The presence of surface states results in a certain compression of the space charge region and leads to a decrease in the band bending. The situation of the electric double layer at a semiconductor/electrolyte solution interface affected by light radiation will be dealt with in Section 5.10. 4.5.2 Interfaces between two electrolytes The electrical double layer has also been investigated at the interface between two immiscible electrolyte solutions and at the solid electrolyteelectrolyte solution interface. Under certain conditions, the interface between two immiscible electrolyte solutions (ITIES) has the properties of an ideally polarized interphase. The dissolved electrolyte must have the following properties: (a) The electrolyte in the aqueous phase, BXAX (assumed to dissociate completely to form cation B ^ and anion A ^ ) , is strongly hydrophilic, while the electrolyte in the organic phase, B2A2 (completely dissociated to cation B 2 + and anion A2~), is strongly hydrophobic. These properties can be expressed quantitatively by the conditions for the distribution coefficients between water and the organic phase k^ Ax and k^ Al: <oAl l, ^ A 2 1 (b) The equilibria of the exchange reactions Bi+(w) + B2+(o) ^ Bj+(o) + B2+(w) Ar(w) + A 2 -(o) Ar(o) + A2~(w) (4.5.16)
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241 are shifted to the left. In other words, the standard potential difference between water and the organic phase for the transfer of the cation of the aqueous phase and the anion of the organic phase must be as positive as possible and the corresponding potential difference for the cation of the organic phase and the anion of the aqueous phase must be as negative as possible. In addition, there must be a sufficiently large difference between the less positive value of A 0 for the first pair and the less negative value for the second pair (cf. Section 3.2.8). Under these conditions, the potential difference between the two phases AQ0 can be changed by charge injection from an external electric current source. The appropriate experimental arrangement is shown in Fig. 5.18. E. J. W. Verwey and K. F. Niessen described the electric double layer at ITIES using a simplifying assumption that consists of only two diffuse electric layers, each in one of the phases (see Fig. 4.1C). The overall potential difference between the two phases, A^^, is thus given by the relationship A:0 = 0 2 (o)-0 2 (w) (4.5.18) where the potential differences in the diffuse layers, 02(o) and 02(w), are defined such that the potential in the bulk of the phase is subtracted from the potential at the interface. The terms <j)2{o) and $2(w) are given by Eq. (4.3.14), where the appropriate permittivity, e(w) or e(o), is substituted for each phase, along with the electrolyte concentrations c(w) and c(o). As the surface charges a(w) and a(o) are equal except for the sign (cf. Eq. 4.1.1), Eqs (4.5.18) and (4.3.14) can be used to calculate the dependence of $2(o) and 02(w) on A ^ . The electrical double layer has, of course, a similar structure when a single electrolyte is present in the distribution equilibrium in the system. It is interesting that the experimentally measured zero-charge potential is practically identical with the value of A 0 = 0, calculated using the TATB assumption (3.2.64). This fact helps to justify the use of this assumption. The electrical double layer at the solid electrolyte-electrolyte solution interface has been studied primarily in colloid suspensions, especially for silver halides. The potential difference between the solid and liquid phases can be changed by changing the concentration of the 'potential-determining ions', i.e. either the silver or halide ions. In solid oxide suspensions, hydrogen ions act as potential-determining ions. The zero-charge potential of this system can be found from the dependence of the electrophoretic mobility (see Section 4.5.4) on the concentration of potential-determining ions, i.e. it corresponds to zero electrophoretic mobility. With this type of interface the structure of the electrical double layer depends to a marked degree on the preparation and thus also on the final structure of the solid phase. Two cases are most often observed: (a) Ions are adsorbed from solution on the surface of the solid phase and counter ions form a diffuse layer.
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(b) The electrical double layer has a structure similar to ITIES, but the diffuse layer in the solid becomes a simple Helmholtz layer because of the high concentration of ions. It should, however, be noted that the electrical double layer at the metal-fused electrolyte interface does not have this character, in spite of the ion concentration being high. In this system, the space charge includes several ion layers at the interface. 4.5.3 Electrokinetic phenomena The adsorption of ions at insulator surfaces or ionization of surface groups can lead to the formation of an electrical double layer with the diffuse layer present in solution. The ions contained in the diffuse layer are mobile while the layer of adsorbed ions is immobile. The presence of this mobile space charge is the source of the electrokinetic phenomena.^ Electrokinetic phenomena are typical for insulator systems or for a poorly conductive electrolyte containing a suspension or an emulsion, but they can also occur at metal-electrolyte solution interfaces. Consider a solid surface in contact with a dilute electrolyte solution. The plane where motion of the liquid can commence is parallel to the outer Helmholtz plane but shifted in the direction into the bulk of the solution. The electric potential in this plane with respect to the solution is termed the
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Of the four electrokinetic phenomena, two (electroosmotic flow and the streaming potential) fall into the region of membrane phenomena and will thus be considered in 6. This section will deal with the electrophoresis and sedimentation potentials. If the electric field E is applied to a system of colloidal particles in a closed cuvette where no streaming of the liquid can occur, the particles will move with velocity v. This phenomenon is termed electrophoresis. The force acting on a spherical colloidal particle with radius r in the electric field E is 4^rerE02 (for simplicity, the potential in the diffuse electric layer is identified with the electrokinetic potential). The resistance of the medium is given by the Stokes equation (2.6.2) and equals 6jrr/rv. At a steady state of motion these two forces are equal and, to a first approximation, the electrophoretic mobility v/E is
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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
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t This term must be distinguished from the concept of electrochemical kinetics, discussed in 5.
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solution, on the properties of the surface of the particles and also on the charge of the particles themselves. If ampholytic particles are involved, then it also depends markedly on the pH, as the particles obtain a charge through dissociation that is dependent on the pH. This fact was utilized by A. Tiselius to develop an electrophoretic method that is especially useful for proteins. At a given pH various proteins are ionized to a different degree and also have different mobilities. The original single sharp boundary between a solution of a protein mixture in a suitable buffer and the pure buffer separates into several boundaries in an electric field, corresponding to differently mobile components. Thus electrophoresis permits analysis of a mixture of proteins without destructive chemical reactions. The experimental methods for determining the boundary position in either classical or free electrophoresis are the same as in the study of diffusion (see Section 2.5.5). In addition, electrophoresis can be carried out by saturating a suitable porous carrier, e.g. filter paper, with a pure buffer and applying the studied solution as spots or bands. The evaluation methods are analogous to those used in paper chromatography. Electrophoresis can be used preparatively to separate the components of a mixture, to concentrate fine suspensions in solution, etc. A further electrokinetic phenomenon is the inverse of the former according to the Le Chatelier-Brown principle: if motion occurs under the influence of an electric field, then an electric field must be formed by motion (in the presence of an electrokinetic potential). During the motion of particles bearing an electrical double layer in an electrolyte solution (e.g. as a result of a gravitational or centrifugal field), a potential difference is formed between the top and the bottom of the solution, called the
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