Primitive cluster model

The second model is a simplified version of a model by Lovett and Ben-Naim published in 1969. It is referred to as the primitive cluster model for water. This model produces the same results for water as the primitive model. However, it has two advantages. First, one can follow the molecular mechanism underlying the characteristic behavior of liquid water. Second, it... 

Fig. 2.9 (a) The primitive model and (b) the primitive cluster model for water-like particles in one dimension. In the latter, each sequence of bonded molecules is viewed as a different component. ... 

In the next section, we shall examine in more detail the structural changes that occur in the system using the primitive cluster model. The information relevant to understanding the behavior of water is the same in the two models. The only... 

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 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. 

Fig. 2.28 The pressure dependence of the volume for the primitive cluster model as in Fig. 2.13 but for different choices of the force constant k in Eq. (2.5.22).

Thus, in order to achieve a stabilization of clusters in the primitive model, we have to introduce three-body potentials. The addition of three-body potentials in the primitive model would have rendered the solution of the 1-D partition function impossible. Instead, the primitive cluster model takes into account all the hydrogen bondings as part of the internal function of the cluster. The very design of the primitive cluster model (Fig. 3.21b) is such that when a solute penetrates in between the hydrogen-bonded molecule it leaves the hydrogen bond intact. Therefore, a solute penetrating a hole within a cluster is expected to stabilize the cluster, i.e. it will shift the equilibrium towards formation of more hydrogen bonds. 

Application of the primitive cluster model for dilute solutions of inert solutes... 

In this section, we extend the application of the primitive cluster model discussed in Sec. 2.5.4 to examine the solvation thermodynamics of simple solutes in the water-like solvent. The model is schematically described in Fig. 3.21b. The only new feature that is added here compared with the model discussed in Sec. 2.5.4 is that clusters of water-like particles contain holes in which solute molecules can be accommodated. This feature is the analog of the cavities formed by the network of hydrogen-bonded molecules in real water. 

Results for HR solute in dilute solutions of the primitive cluster model... 

We begin with some results on the effect of the solute on the distribution of cluster size. In all of the following discussions, the HB solvent refers to the primitive cluster model for water. 

It is now possible to establish a correspondence between the functions co(l) and Q 1) and the accepted model of short-range hierarchy order. From Fig. 69, the primitive cluster comprises 7 particles, the next one at the 2-nd level 72 = 49 particles, and the 1th level cluster ll particles (here l is the level number). Obviously, for a level l comprising N = ll particles, the level number may be defined as... 

Structural changes induced by the change in the temperature. It is clear that these minima also occur in the ranges of T and P at which we have observed the anomalous temperature dependence of the volume, namely 2 < T < 3 and 4 < P < 9. (See Sec. 2.5.3. Note, however, that in Sec. 2.5.3 we studied the primitive model, but as we have pointed out the cluster model exhibits the same behavior as the primitive model). Once we increase the pressure beyond P 10, the system is highly compressed and the quantity 9(Xhb)/9T becomes positive, i.e. increasing the temperature causes an expansion from the close-packed structure into the open, HBed structure. 

These two different points of view were explained in Sec. 2.5 in connection with the primitive and cluster-primitive onedimensional models for water. In the primitive model single water molecules are defined through their pair potential. The structure formed by these water molecules — clusters of HBed molecules — is a result of the specific pair potential. On the other hand, in the cluster primitive model, the structures — clusters of HBed molecules — are assumed to be a part of the description of the model, and the HBs are now part of the internal description of the clusters. 

In Sec. 2.5, we introduced two 1-D models for water. The two models are almost equivalent in their capacity to unveil the molecular reasons for the outstanding properties of liquid water. Extending the application of these two models for aqueous solutions shows that while the primitive model fails to show large negative anomalous entropy and enthalpy of solvation of inert solutes, the primitive cluster succeeds. The reason is that the entropy and enthalpy of solvation of a solute in water are due to the capability of the solute to induce structural changes in the solute. In the TD primitive model, one could not achieve that effect, not because of any deficiency of the model but because of the assumption of nearest-neighbor interactions only. 

Recently, Lyubattsev and Ben-Naim (unpublished) have carried out Monte Carlo simulations on the primitive model, where next nearest neighbor interactions are included. In this model, the entropy of solvation is indeed much lower than in the cluster model reported in this section. 

This deficiency of the primitive model was eliminated in the cluster model (see Sec. 2.5). Similarly, in Widom s model a solute molecule entering between two solvent molecules not only allows an HB across the solute but actually enhances the formation of HBs. 

The reason I have doubts as to the relevance of the above results to the problem of hydrophobic interactions is the following. In Sec. 3.9,1 have compared the results of solvation of a simple solute in the primitive and cluster models. We have seen that in the former we could not obtain the characteristic... 

In this work we start with the primitive jellium model, as appropriate for alkaline metals. In the jellium model for metal clusters a fundamental input is the size-dependent ionic density. Fortunately, when one of us started this calculation in 1984 [3], some experimental data about the size dependence of the nearest-neighbor distance were available from EXAFS (extended X-ray absorption fine structure) measurements [19]. Except for fine details the size dependence is very weak. This means that in a first approximation the bulk density of the metal can be used as input for a cluster calculation. A second question is the size dependence of the shape. Since electron micrographs very often show a spherical shape, at least for the larger clusters, a spherical shape will be assumed for all cluster sizes. This means that for monovalent systems the radius R of the jellium cluster is determined by its bulk density... 

For the purposes of this discussion, the primitive cavity model and the polaron models will both be considered as one and referred to as the cavity model. However, we shall point out at various points instances where a certain property depends sensitively on the detailed difference between the two types of cavity model. One can loosely say that those properties of the solutions which seem to be insensitive to the nature of the metal support the cavity model while those which depend on the metal provide support to the cluster model. However, a truly consistent explanation of all the observed properties requires the unified model described in Part (2-E). The importance of the unified model will become clear as we proceed successively with the explanation of the properties listed in Section I. 

The considered CNDO method for periodic systems formally corresponds to the model of an infinite crystal or its main region consisting of L primitive cells. This semiempirical scheme was also apphed for the cychc-cluster model of a crystal allowing the BZ summation to be removed from the two-electron part of matrix elements. In the next section we consider ZDO methods for the model of a cyclic cluster. 

The models with periodic boundary conditions (the supercell and the cyclic-cluster models) allow calculation of the one-electron states of perfect and defective crystals at the same level of approximation. The supercell model (SCM) and cychc-cluster model (COM) have both similarities and discrepancies. One similarity is that in both models not a standard primitive unit cell but an extended unit cell (supercell or large unit cell) is considered. The discrepancy is that the periodic boundary conditions in the SCM are introduced for the infinite crystal or its main region, but in the CCM model - for the extended unit cell itself. 

The second necessary ingredient in the primitive quasichemical formulation is the excess chemical potential of the metal-water clusters and of water by itself. These quantities p Wm — can typically be obtained from widely available computational packages for molecular simulation [52], In hydration problems where electrostatic interactions dominate, dielectric models of those hydration free energies are usually satisfactory. The combination /t xWm — m//, wx is typically insensitive to computational approximations because the water molecules coat the surface of the awm complex, and computational errors can compensate between the bound and free ligands. 

The unrestricted form of the primitive model (UPM) becomes important for more complex fluid systems. Stell argued that symmetry breaking in the UPM may play an important role in determining critical behavior [17]. In spite of this potential utility, the UPM is rarely explored. In MC simulations of the cluster structure in the UPM, Camp and Patey [259] compared results for asymmetrical charges Xq = z+/z = 1,2,4 at the diameter ratio Xa = vapor phase contains, above all, neutral clusters such as trimers for Xq — 2 and tetrahedral pentamers for Xq = 4, as well as higher clusters. At Xq = 4 asymmetry effects not covered by simple theories seem to play a role. 

---