Semi-infinite geometry

In semi-infinite planar geometry, the electrode occupies the x = 0 plane and transport occurs perpendicularly to that plane from a limitless unimpeded medium as shown in Fig. 29. To prevent radial diffusion to the edge of the electrode, it is necessary to have walls of some kind to constrain the transport direction to be normal to the electrode. Because of this requirement, electrodes with precise semi-infinite planar geometry are difficult to fabricate and are, in fact, rather rare in practice. Nevertheless, because theoretical derivations are simplest for this geometry and because many practical geometries closely approximate the semi-infinite planar one as a limiting case, the geometry of this section is of paramount importance. 

We shall be concerned almost exclusively with the reaction 

In Sect. 5.6, we briefly describe modifictions needed when the semiinfinite geometry is spherical rather than planar. 

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In general, the peculiarities of the surface effects in thin semiconductors, for which application of semi-infinite geometry becomes incorrect were examined in numerous papers. As it has been shown in studies [101, 113, 121 - 123] the thickness of semiconductor adsorbent becomes one of important parameters in this case. Thus, in paper [121] the relationship was deduced for the change in conductivity and work function of a thin semiconductor with weakly ionized dopes when the surface charge was available. Paper [122] examined the effect of the charge on the temperature dependence of the work function and conductivity of substantially thin adsorbents. Papers [101, 123] focused on the dependence of the surface conductivity and value of the surface charge as functions of the thickness of semiconductor and value of the surface band bending caused by adsorption and application of external field. 

A > Cs) and dissolved (loading equal to or less than saturation, A < C,). For the latter, solutions to Fick s second law (Eq. 4.2) are well known, and the particular expression for semi-infinite geometry is... 

Of course, the semi-infinite geometry (D—>°°) is a special case of this treatment, and hence we first demonstrate that the solution obtained by Fredrickson [12] is a special case of Eqs. (53)-(57). Being interested in the solution near z=-D/2, it is clear that the second term in Eq. (53) dominates, and the first term may be neglected. In terms of a shifted coordinate z =z+D/2 this yields... 

Fig. 53. Schematic isotherms (density p versus chemical potential pi) corresponding to the gas-liquid condensation in capillaries of thickness D, for the case without (a) and with (b) prewetting, and adsorption isotherm (c) for a semi-infinite system, where the surface excess density pjs is plotted vs. pi. Full curves in (a) and (b) plot the density p vs. pi for a bulk system, phase coexistence occurs there between p,p, (bulk gas) and pn, (bulk liquid), while in the capillary due to the adsorption of fluid at the walls the transition is shifted from paKX to a smaller value rc(D, 7) (with pic(7>, T) 1 /D, the Kelvin equation ), and the density jump (from ps D) to pt D)) is reduced. Note also that in the ease where a semi-infinite system exhibits a first-order wetting transition 7W, for 7 > 7W one may cross a line of (first-order) prewetting transitions (fig. 54) where the density in the capillary jumps from p to p>+ or in the semi-infinite geometry, the surface excess density jumps from p to p +, cf. (c), which means that a transition occurs from a thin adsorbed liquid film to a thick adsorbed film. As pi the thickness of the adsorhed liquid film in the semi-infinite...

The foregoing considerations for finite geometry are now extended to semi-infinite geometry. 

This geometric factor takes into account any deviations from the semi-infinite geometry and can in most cases be described as a product of independent correction factors. 

Before discussing the actual semi-infinite geometry which is relevant for surface effects, let us first consider a system in which all atoms are equally coordinated - a bulk material or some regular clusters, such as a square, a cube, etc., - and let us prove that the equilibrium inter-atomic distance Rq is an increasing function of the coordination number Z. 

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