Deviations from SI Behavior

The excess chemical potential, due to deviations from symmetrical ideal behavior, is thus 

both the excess chemical potential, and the activity coefficient measure the deviations from similarity of the two quantities. This is fundamentally different from the deviations from the ideal-gas behavior (i.e., total lack of interactions), discussed in section 6.1. Here the limiting behavior of the activity coefficient is 

The general case (6.16) is not very useful since in general, we do not know the dependence of Aab on the composition. However, if the two components 

Note that since here we exclude the case of an ideal-gas mixture, pT is finite, and the condition (6.18) is essentially a condition on Aab. If we further assume that PtAas is independent of the composition, then we can integrate (6.16) to obtain the first-order deviations from SI behavior, namely 

In the phenomenological characterization of small deviations from SI solutions, the concepts of regular and athermal solutions were introduced. Normally, the theoretical treatment of these two cases was discussed within the lattice theories of solutions. Here, we discuss only the very general conditions for these two deviations to occur. First, when Pt ab does not depend on temperature, we can differentiate (6.19) with respect to T to obtain 

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Figure 6.1 shows such curves for mixtures of carbon disulphide and acetone, and the second for mixtures of chloroform and acetone. The first shows positive deviations from SI behavior in the entire range of compositions the second shows negative deviations from SI solution. 

In this appendix, we resolve the conflict by showing that the two measures of the deviations from SI behavior, pAAB and y 1 (or PA/Pfi), are not, in general, equivalent. We start with the first-order expression for the partial pressure, say of water, in the mixture... 

Clearly, Aab = 0 is a necessary and sufficient condition for SI behavior (Ben-Naim 2006). Therefore, any finite Aab value represents a measure of the extent of deviation from SI behavior. More precisely, in the general case we write... 

Figure 2.9 shows the excess Gibbs energy of the mixture for the same set of molecular parameters at e = -1 we have a SI solution and = 0. As lei either increases or decreases, we find positive deviations from SI behavior. 

We note, first, that the activity coefficients in (4.144)-(4.146) differ fundamentally from the activity coefficient introduced in Section 4.7. To stress this difference, we have used the superscript D to denote deviations from DI behavior, whereas in Section 4.7, we used the superscript S to denote deviations from SI behavior. (In the next section, we shall elaborate on an example for which three kinds of ideality can be distinguished in a very simple and explicit manner.) Furthermore, each of the activity coefficients defined in (4.144)-(4.146) depends on the thermodynamic variables, say T and or T and P, etc. This has been stressed in the notation. In practical applications, however, one usually knows which variables have been chosen, in which case one can drop the arguments in the notation for 7 . 

Here we obtain A/U > 0 whenever + sBB > 2sAB and Aab < 0 whenever saa + sbb < 2sab. It is tempting to conclude that in these two cases we shall obtain stable and unstable mixtures, respectively. However, we must remember that in the limit of 0, Pt = PP and Pt -ab = P2PVint[eAA + sBB — 2 /U must be small. Therefore, one cannot predict the behavior at large values of Pt ab Thus, from the above discussion one can predict the occurrence of positive or negative deviations from SI solutions. But since in this limit P — 0, and Pt 0 we also expect Pt -ab to be small, therefore we must have miscibility in the entire region of compositions. 

The SI behavior is realized by a variety of two-component systems of two similar species. Deviations from this behavior may be expressed by introducing either an activity coefficient yf or an excess function. These are defined as... 

The difficulties of using the standard analysis for a-Si H were demonstrated by Snell et al (1979) Their results are shown in Fig. 9 for barriers on a-Si H with different doping levels. First, the extrapolation was difficult because the forward-bias capacitance could be measured to only 0.2 V. Adding to this complication was the fact that the 1 /C plots did not exhibit a straight-line dependence. For the depletion layer approximation the slope of curve yielded the donor density. Thus the observed deviation from linear behavior suggests that the charge density in the depletion region is not uniform. 

Experimental thermodynamic data for ternary Cr-Fe-Si alloys are limited. [1968Che] report the activity of silicon in Cr-Fe-Si melts along sections with 12, 18 and 25 % Cr at 1600°C. It was shown that the activity exhibits a negative deviation from ideal behavior. The dependence of logioTsi on chromium concentration is linear, and extrapolation of the relationship to zero chromium content yields a value of about 0.0027. 

There are essentially three significant quantities that can be derived from the inversion of the KB theory. The first is a measure of the extent of deviation from symmetrical ideal (SI) solution behavior, A b. defined below in the next section. It also provides a necessary and sufficient condition for SI solution. The second is a measure of the extent of preferential solvation (PS) around each molecule. In a binary system of A and B, there are only two independent PS quantities these measure the preference of, say, molecule A to be solvated by either A or B molecules. Deviations from SI solution behavior can be expressed in terms of either the sum or difference of these PS quantities. Finally, the Kirkwood-Buff integrals (KBIs) may be obtained from the inversion of the KB theory. These provide information on the affinities between any two species for instance, PaGaa measures the excess of the average number of A particles around A relative to the average number of A particles in the same region chosen at a random location in the mixture. All these quantities can be obtained from the KB integrals. 

In contrast, methyl-for-chlorine substitution is decidedly nonlinear, a feature also displayed by the lighter Group 14 compounds. This curvature is not an artifact of the BAC-MP4 predictions, since it is observable in the (admittedly limited) experimental data for these compounds (Fig. 10). In fact, the deviations from linearity are even greater in the experimental data. Such behavior is also observed in the analogous Si compounds and is related to the negative hypercongugation (anomeric) effect, in which electron density from... 

If the analogy that is drawn between the Si=Si dimer on the Si(100)-2 x 1 surface and an alkene group is reasonable, then certain parallels might be expected to exist between cycloaddition reactions in organic chemistry and reactions that occur between alkenes or dienes and the silicon surface. In other words, cycloaddition products should be observed on the Si(100)-2 x 1 surface. Indeed, this prediction has been borne out in a number of studies of cycloaddition reactions on Si(100)-2x1 [14], as well as on the related surfaces of Ge(100)-2 x 1 (see Section 6.2.1) and C(100)-2 x 1 [192-195]. On the other hand, because the double-bonded description is only an approximation, deviations from the simple picture are expected. A number of studies have shown that the behavior differs from that of a double bond, and the asymmetric character of the dimer will be seen to play an important role. For example, departures from the symmetry selection rules developed for organic reactions are observed at the surface. Several review articles address cycloaddition and related chemistry at the Si(100)-2 x 1 surface the reader is referred to Refs. [10-18] for additional detail. 

A long time ago, Hong, Noolandi, and Street [16] investigated geminate electron-hole recombination in amorphous semiconductors. In their model they included the effects of tunneling, Coulomb interaction, and diffusion. Combination of tunneling and diffusion leads to an S(t) oc t 1/2 behavior. However, when the Coulomb interactions are included in the theory, deviations from the universal t /2 law are observed—for example, in the analysis of photoluminescence decay in amorphous Si H, as a function of temperature. 

In a Tafel plot, the logarithm of the current is plotted against t], as illustrated in Figure 1. Note that the slope is equal to —ccnF/2.2RT and the intercept corresponds to logj o. From these values, k° can be determined with Eq. 11. Tafel plots are often employed in corrosion studies, since k° is usually small and the condition Co(0, t) Si Cq can be accomplished by simply stirring the solution. Deviations from the idealized Tafel behavior are seen at large t], where Cq(0,/) becomes significantly smaller than Cq. 

The third quantity is a measure of the deviation of the mixture from SI solution behavior (see Section 1.3.7 in Chapter 1). This quantity is defined as... 

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