Purely attractive model

As we showed in Chapter 8 when we were dealing with the repulsive case, Flory s theory has no sound theoretical basis. For reasons still unclear (1990), it leads to interesting results. This is why we shall examine here the predictions of this theory in the attractive case. 

For d = 3, Flory s equation which gives the swelling of an isolated chain reads (see (8.1.34), 

Which solution must be chosen when z 0 In order to answer this question, let us come back to the expression of F(X) given by (8.1.33). For d = 3, we have 

We can see that for z 0, F(X) goes to — go when i - 0. Thus, for z 0, F(i) has no minimum and the model predicts a complete collapse of the chain (i = 0). This conclusion is in agreement with the more general comments made in the preceding sections. 

In order to avoid this catastrophe, we must use a more sophisticated model. In fact, Flory and de Gennes7 showed that the introduction of three-body forces was sufficient to obtain reasonable results the corresponding model will now be studied. 

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This discussion shows that a purely attractive continuous chain reduces to a point, and this result, established here in a rather intuitive way, is confirmed by more precise calculations which will be given later. Finally, we must admit that a purely attractive model is unrealistic. Thus, the results obtained with the continuous model corresponding to the weight (14.2.5) cannot be (analytically) continued for b < 0. 

In principle, such a model would be valid if the forces between polymers were such that 4 > i > 3. In fact, this is not true for neutral polymers, the forces with the longest range are van der Waals forces for which i = 6. We see that in this case z — 0 when S-+ 0. Thus, it looks as if this attractive interaction should behave like a short range interaction and should mingle with the repulsive interaction therefore, it seems that we come back to the purely repulsive or purely attractive model which was rejected for reasons expounded in Section 3. 

Of course, it is quite clear from section 4.1 that a pure statistical model is an oversimplification that will adversely affect the accuracy of prediction despite its attractive ease of implementation. Therefore, mixed models are also used that take at least some regional information into consideration and can be seen as statistical models split into compartments. Within the compartments solely statistical features are considered, but promoter organization is somewhat reflected by the arrangement of the compartments which represent different promoter regions. 

Purely attractive or repulsive chains existence of continuous models... 

Such interactions are shown in Figure 7.5 in order to describe the purely attractive case, as well as the weak and strong repulsions at large distances. In order to understand how clustering can emerge from this type of interaction, we now treat the direct correlation function associated with this model in the mean spherical approximation fashion, which amounts to setting... 

Agreement between pjj calculated from the test particle method and from Eq. [41] was nearly exact. Garde et al. also found that the contribution of the attractive part of the solute-water interaction energy to p, is reasonably well represented by even a very crude model and that this contribution shows little temperature dependence. These observations suggest that purely repulsive models capture many of the essential features of hydrophobicity for small solute molecules. Note, however, that this conclusion does not hold for large solutes or for interfaces. ... 

It is interesting to consider another approximate derivation, which uses the implicit solvent models discussed earlier (Sect. 12.4). Indeed, we can decompose the binding reaction into the steps shown in Fig. 12.5 [94] first, the ligand charges are switched off in pure solvent, leaving a nonpolar solute second, the attractive... 

The modes of addition shown in Figure 6.3 are similar to those shown in Figure 6.2 and are consistent with extant mechanistic work [6,9] they accurately predict the identity of the slower reacting enantiomer. It should be noted, however, that variations in the observed levels of selectivity as a function of the steric and electronic nature of substituents and the ring size cannot be predicted based on these models alone more subtle factors are clearly at work. In spite of such mechanistic questions, the metal-catalyzed resolution protocol provides an attractive option in asymmetric synthesis. This is because, although the maximum possible yield is 40 %, catalytic resolution requires easily accessible racemic starting materials and conversion levels can be manipulated so that truly pure samples of substrate enantiomers are obtained. 

Although the coefficients of determination and the correlation coefficients are conceptually simple and attractive, and are frequently used as a measure of how well a model fits a set of data, they are not, by themselves, a good measure of the effectiveness of the factors as they appear in the model, primarily because they do not take into account the degrees of freedom. Thus, the value of R can usually be increased by adding another parameter to the model (until p =J), but this increased R value does not necessarily mean that the expanded model offers a significantly better fit. It should also be noted that the coefficient of determination gives no indication of whether the lack of perfect prediction is caused by an inadequate model or by purely experimental uncertainty. 

Figure 10 shows the (111) substrate and the neighboring sites around a central bridge-bonded sulfate (in black). We model the interactions between the adsorbates in two different ways. First, we consider a shell of purely hard interactions, in which the simultaneous bonding of two anions to neighboring sites is simply excluded. These excluded neighboring sites are displayed in white in Figure 10. Next, we consider a second shell of neighboring sites with either finite attractive or finite repulsive interactions. These are displayed in grey in Figure 10.

The principle has been demonstrated with CaLB in simple model transesterifications as well as in the enantioselective acylation of 1-phenylethanol, in batchwise and continuous procedures. The high operational stability of CaLB, which contrasts with the generally rapid deactivation in pure scC02, is one of the attractive aspects of this approach. The reaction rate was approximately eight times better than that in pure scC02 under otherwise identical conditions [96]. 

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