Triplet correlation indirect

Most integral equations are based on the Ornstein-Zernike (OZ) equation [3-5]. The idea behind the OZ equation is to divide the total correlation function h ri2) iiito a direct correlation function (DCF) c r 12) that describes the fact that molecules 1 and 2 can be directly correlated, and an indirect correlation function 7( 12), that describes the correlation of molecule 1 with the other molecules that are also correlated with molecule 2. At low densities, when only direct correlations are possible, 7(r) = 0. At higher densities, where only triplet correlations are possible, we can write... 

In this chapter we discuss three-site systems. We extend the three models treated in Chapter 4 direct correlation, indirect correlation mediated through the adsorbent molecule, and indirect correlation mediated by a chain of communicating subimits. Here, we discuss separately two possible structures of the system, a linear and a triangle arrangement of the sites (Fig. 5.1). Two fundamentally new features are discussed in considerable detail the nonadditivity of the triplet correlation and the possibility of long-range correlations. 

In all these cases the triplet correlation is expressible in terms of the pair correlations and the temperature dependence of the correlation is predictable, knowing the ligand-ligand interactions. This is, in general, not the case for systems with indirect correlations, discussed in the following sections. 

We have seen in Section 4.5 that the conditions for having indirect pair and triplet correlations are the same as those for conformational changes induced by the binding process. As in Sections 3.4 and 4.5, the change in the mole fraction of the L form is the same also in this model, i.e.. 

The general expressions for the indirect correlations are fairly complicated. Nevertheless, we can find conditions under which there exist pair and triplet correlations by factoring out the factors that determined the sign of the indirect correlations. In terms of the parameters h, h, h, and T), T, the pair correlations may be written as... 

The triplet correlation in this system is quite complicated, even when abc is assumed to be independent of the conformation of the subunits. Nevertheless, we can make the following statements regarding the indirect triplet correlations ... 

When one of is unity, it makes two of the indirect pair correlations unity, but the indirect triplet correlation becomes equal to the third pair correlation. For example, when = 1, we have... 

Throughout this chapter, we use very simple models to introduce a number of concepts that are frequently used in the context of the theory of fluids. Examples are the pair-correlation function, direct and indirect correlations, potential of average force, nonadditivity of the triplet correlation function, and so on. All these will be introduced again in Chapter 5. However, it is easier to grasp these concepts within the simple models. This should facilitate understanding them in more complex systems. 

Figure5.3. The triplet indirect correlation y(l, 1,1) as a function of for various values of A, indicated next to each curve.

Whenever two of the As are unity, there is neither pair nor triplet indirect correlation. 

When only one of r or is unity, the triplet indirect correlation may differ... 

Table 6.3 shows the pair, triplet, and quadruplet correlations at six temperatures. It is quite clear that the correlations do not change monotonically with temperature. For instance, the pair correlation initially increases with temperature (which is indicative of the dominance of indirect correlation), then decreases up to 30 °C, and again increases. This is also true for the corresponding free energies. 

Recall, however, that if /z = 1, there is no indirect correlation for both pairs and triplets of ligands i.e., we already know that... 

From the above analysis we conclude that whenever there are indirect correlations, there also exists nonadditivity of the potential of average force. This conclusion is probably true also for the triplet potential of average force in liquids. 

---