The primitive cluster model for water and its partition function

The primitive cluster model for water and its partition function 

In this section, we introduce what is essentially an equivalent model to the one described in Sec. 2.5.2. This model, referred to as the primitive cluster model, has several features that make it more useful in the study of the molecular mechanism underlying the anomalous behavior of liquid water and aqueous solutions. For water, as we shall see below, the two models provide essentially the same results. However, with the cluster model, we can get a deeper insight into the mechanism underlyingthe anomalies of water, namely the structural changes (here, essentially the change in the cluster-size distribution) in the liquid that lead to the anomalous behavior. As we shall see in Sec. 3.9, the cluster model is also more convenient for the study of some of the most outstanding properties of aqueous solutions of simple solutes. 

The model is a simplified version of a model first developed by Lovett and Ben-Naim (1969). The idea is to define a sequence of n HBed particles as an n-cluster (mimicking the ice-like clusters of HBed molecules in liquid water). In doing so, we include all the HBing in the cluster as part of the internal partition function of the -cluster. The interaction potential between any pair of clusters (including the 1-cluster, i.e. the monomers, or non-HBed particle), is now the hard-rod HR) potential. (In the original model, this part was chosen to be a square-well potential. As with the primitive model discussed in Sec. 2.5.2, the 

The primitive cluster model is essentially the same as the primitive model as depicted in Fig. 2.9a. We could have left all the interactions between the particles exactly the same as in the primitive models, in which case the models would be identical except for the two different views of the system, the one-component and the MM views. As we have seen in Sec. 2.3, these two views are completely equivalent. In Fig. 2.9b the primitive cluster model is depicted. Clearly, the only difference is in renaming a sequence of HBed molecules as a cluster. For computational simplicity, we also assume that the interactions between all non-HBed particles are simply the HR types of potential. 

We present a brief derivation of the partition function for the cluster model. It is basically the same as the one derived by Lovett and Ben-Naim (1969), except for the simplification of the pair interactions. In the following, we treat the case of pure water. We shall revert to this model for the study of aqueous solutions in Sec. 3.9. 

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