Surface States and Other Complications

Unlike the case illustrated in Fig. 10, changes in the solution redox potential have been observed to cause no change in the magnitude of Vsc- This situation is termed Fermi level pinning in other words, the band edge positions are unpinned in these cases so that the movement of iiredox accommodated by Vh rather than by Vsc- As mentioned earlier, it appears [37] that surface state densities as low as 10 cm ( 1% of a mono-layer) suffice to induce complete Fermi level pinning in certain cases. Of course, intermediate situations are possible. Thus, the ideal case manifests a slope of 1 in a plot of Vsc (or an equivalent parameter) 

p is the so-called Schottky barrier height, is the metal work function, and 5 is a dimensionless parameter. Attempts have been made to relate S to semiconductor properties [44-48]. 

To further complicate matters, the nonideal behavior of semiconductor-electrolyte interfaces as noted earher is exacerbated when the latter are irradiated. Changes in the occupancy of these states cause further changes in Vh, so that the semiconductor surface band edge positions are different in the dark and under illumination. These complications are considered later. The surface states as considered earher are shallow (with respect to the band edge positions) and can essentially be considered as completely ionized at room temperature. However, for many oxide semiconductors, the trap states may be deep and thus are only partially ionized. Specifically, they may be disposed with respect to the semiconductor Fermi level such that they are ionized only to a depth that is small relative to W [49]. The physical manifestation of such deep traps as observed in the AC impedance behavior of semiconductor-electrolyte interfaces has been discussed [14, 49]. 

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The problem is to prevent instability, not only to maintain the appearance of the emulsion, but so that the characteristics of the emulsion and of medicaments dissolved in the emulsion are as little changed on ageing as possible. As an example, ageing might alter the absorption of heparin from O/W emulsions where absorption of heparin appears to be directly related to the particle size and total surface area of the oil droplets [11]. Fat emulsions are used extensively in intravenous feeding [12] where it is vital that particles remain below 1 in diameter to avoid thrombophlebitis and other complications, but the state of the art is exemplified by the statement [13], that the emulsions must be stored in a refrigerator and no antibiotics, vitamins or potassium supplements added because they may break the emulsions . Lynn [14] reports some experiments on the addition of disodium carbenicillin and sodium cloxacillin to intravenous lipid emulsions which verify this statement. The special case of intravenous emulsions is dealt with in Section 8.7.2. 

So far we have assumed that the electronic structure of the crystal consists of one band derived, in our approximation, from a single atomic state. In general, this will not be a realistic picture. The metals, for example, have a complicated system of overlapping bands derived, in our approximation, from several atomic states. This means that more than one atomic orbital has to be associated with each crystal atom. When this is done, it turns out that even the equations for the one-dimensional crystal cannot be solved directly. However, the mathematical technique developed by Baldock (2) and Koster and Slater (S) can be applied (8) and a formal solution obtained. Even so, the question of the existence of otherwise of surface states in real crystals is diflBcult to answer from theoretical considerations. For the simplest metals, i.e., the alkali metals, for which a one-band model is a fair approximation, the problem is still difficult. The nature of the difficulty can be seen within the framework of our simple model. In the first place, the effective one-electron Hamiltonian operator is really different for each electron. If we overlook this complication and use some sort of mean value for this operator, the operator still contains terms representing the interaction of the considered electron with all other electrons in the crystal. The Coulomb part of this interaction acts in such a way as to reduce the effect of the perturbation introduced by the existence of a free surface. A self-consistent calculation is therefore essential, and the various parameters in our theory would have to be chosen in conformity with the results of such a calculation. 

The results obtained for the stochastic model show that surface reactions are well-suited for a description in terms of the master equations. Since this infinite set of equations cannot be solved analytically, numerical methods must be used for solving it. In previous Sections we have studied the catalytic oxidation of CO over a metal surface with the help of a similar stochastic model. The results are in good agreement with MC and CA simulations. In this Section we have introduced a much more complex system which takes into account the state of catalyst sites and the diffusion of H atoms. Due to this complicated model, MC and in some respect CA simulations cannot be used to study this system in detail because of the tremendous amount of required computer time. However, the stochastic ansatz permits to study very complex systems including the distribution of special surface sites and correlated initial conditions for the surface and the coverages of particles. This model can be easily extended to more realistic models by introducing more aspects of the reaction mechanism. Moreover, other systems can be represented by this ansatz. Therefore, this stochastic model represents an elegant alternative to the simulation of surface reaction systems via MC or CA simulations. 

Under field conditions water evaporation normally occurs from the same surfaces as evaporation of pesticide. Any temperature effect caused by the former will therefore affect the latter, and in this repect the state of affairs is simpler. On the other hand, leaves can restrict water evaporation, and the soil surface is very complex hence, other complications appear under field conditions. All these factors, however, will make the pesticide evaporation rate (if the pesticide is fully exposed on an outer surface) faster than that calculated from the water rate. 

As stated previously, another distinction usually made is between slurry and supported catalyst reactors. In slurry photocatalytic reactors the catalyst is present in the form of small particles suspended in the water being treated. These reactors generally tend to be more efficient than supported catalyst reactors, because the semiconductor particles provide a larger contact surface area per unit mass. In fact, the state of the photocatalyst is important both to increase contaminant adsorption and to improve the distribution of absorbed radiation. In a slurry unit the photocatalyst has a better contact with the dissolved molecules and is allowed to absorb radiation in a more homogeneous manner over the reaction volume. Using suspended catalyst has been the usual practice in PTC, CPC, and other types of tubular reactors. The drawback of this reactor design is the requirement for separation and recovery of the very small particles at the end of the water treatment process. This may eventually complicate and slow down the water throughput. 

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