Clusters partition function

In the derivation of equation (4.42), it has been tacitly assumed that the cluster partition function Zjy remains unaffected by the flux of clusters to higher classes, i.e., Z] f = Z. A correction for the depletion of the cluster population due to this flux can be introduced by the factor r, known as the Zeldovich or non-equilibrium factor [4.16]. 

The simple stochastic model bypasses the need for an evaluation of cluster partition functions and dynamics that have previously been employed ° to study the density dejjendence of kf. The density dependence of the recombination rate is the simplest example of the rollover predicted by the Kramers theory for passage over a barrier. In the theory of this chapter the barrier arises from entropy considerations in a free energy surface rather than from a potential energy surface. 

Lothe and Pound first recognized the importance of the difference between Eqs. (41) and (36) and also of the translation and rotation contribution to the cluster partition function. They added these directly onto the drop model, which they assumed correctly described the internal properties of a small cluster. Then in Eq. (37) they wrote... 

It is possible to make further progress with the analysis of the results by writing N , the equilibrium number of clusters of size n, in terms of a cluster partition function, Q [17,18]... 

Q is the cluster partition function for a single physical cluster of size n on a lattice with a total of M sites and at temperature T. The summation is over all possible connected physical clusters and 17" is the energy of each such cluster as given by Eq. (1). It should be noted that the summation in Q includes all the M placings of each distinct cluster on the lattice. The result (4) only applies at sufficiently low coneentrations that the clusters do not interact (i.e., the system is ideal) and we assume for the purposes of this further analysis that this is an acceptable approximation up to concentrations of the order of... 

As with the other perturbatiou theories discussed in this chapter, the key quantity in the theory is the clnster partition function which enumerates the number of associated states for which an isolated associated cluster cau exist. For this case the cluster partition function is simplified as follows ... 

Let us first derive the expression for the cell-cluster partition function. Let us consider I particles (1, 2. .. Q sharing a cell cluster formed by cells a,. .. A. In this cell cluster the molecules move imder the influence of ... 

This theory has not been very successful it does well for some substances but fails badly for others. Certain semiempirical calculations have been more successful, but such calculations lack the appeal of theories developed from first-principles. As a result, much recent interest has centered upon microscopic approaches to the study of the clusters in which the details of the structure or dynamics of the nucleation micro-clusters are considered. These microscopic studies have predominantly used two basic approaches the statistical mechanical calculation of cluster partition functions, and Monte Carlo and molecular dynamics simulations of supersaturated systems. 

The additional factor of Qi(V, T) in Eq. (21) makes the leading term in the sum unity, as suggested by the usual expression for the cluster expansion in terms of the grand canonical partition function. Note that i in the summand of Eq. (20) is not explicitly written in Eq. (21). It has been absorbed in the n , but its presense is reflected in the fact that the population is enhanced by one in the partition function numerator that appears in the summand. Equation (21) adopts precisely the form of a grand canonical average if we discover a factor of (9(n, V, T) in the summand for the population weight. Thus... 

Once the cluster expansion of the partition function has been made the remaining thermodynamic functions can be obtained as cluster expansions by taking suitable derivatives. Of particular interest are the expressions for the equilibrium concentrations of intrinsic point defects for the various types of lattice disorder. Since the partition function is a function of Nx, N2, V, and T, it is convenient for the derivation of these expressions to introduce defect chemical potentials for each of the species in the set (Nj + N2) defined, by analogy with ordinary Gibbs chemical potentials (cf. Section I), by the relation... 

In the equations developed by Reilly and Wood (15) from the cluster Integral model (1 6), y+ is calculated in complex solutions from excess properties of single salt solutions. Note that the cluster Integral approach 1s based upon terms which represent the contributions of pair-wise ion interactions 1n various types of clusters to the potential interaction energy. Then, the partition function and the excess properties of the solution can be evaluated. The procedure is akin to the vlrial expansion 1n terms of clusters. 

Just as in our abbreviated descriptions of the lattice and cell models, we shall not be concerned with details of the approximations required to evaluate the partition function for the cluster model, nor with ways in which the model might be improved. It is sufficient to remark that with the use of two adjustable parameters (related to the frequency of librational motion of a cluster and to the shifts of the free cluster vibrational frequencies induced by the environment) Scheraga and co-workers can fit the thermodynamic functions of the liquid rather well (see Figs. 21-24). Note that the free energy is fit best, and the heat capacity worst (recall the similar difficulty in the WR results). Of more interest to us, the cluster model predicts there are very few monomeric molecules at any temperature in the normal liquid range, that the mole fraction of hydrogen bonds decreases only slowly with temperature, from 0.47 at 273 K to 0.43 at 373 K, and that the low... 

The key to a treatment of molecular clusters in situations of thermal equilibrium are the N-particle partition functions. Specifically, the classical two-particle partition function, Z2(T), is given by [183, 184, 377]... 

Each supramolecular cluster species W is associated with a partition function qh which may be factored into translational, rotational, vibrational, and electronic contributions in the usual manner [cf. (13.63)]. The equilibrium condition corresponding to (13.84) may now be expressed as [cf. (13.65)]... 

The simplest QCE model incorporates environmental effects of cluster-cluster interactions by (1) approximate evaluation of the excluded-volume effect on the translational partition function >trans (neglected in Section 13.3.3) and (2) explicit inclusion of a correction A oenv) for environmental interactions in the electronic partition function qiQiec. Secondary environmental corrections on rotational and vibrational partition functions may also be considered, but are beyond the scope of the present treatment. 

The traditional apparatus of statistical physics employed to construct models of physico-chemical processes is the method of calculating the partition function [17,19,26]. The alternative method of correlation functions or distribution functions [75] is more flexible. It is now the main method in the theory of the condensed state both for solid and liquid phases [76,77]. This method has also found an application for lattice systems [78,79]. A new variant of the method of correlation functions - the cluster approach was treated in the book [80]. The cluster approach provides a procedure for the self-consistent calculation of the complete set of probabilities of particle configurations on a cluster being considered. This makes it possible to take account of the local inhomogeneities of a lattice in the equilibrium and non-equilibrium states of a system of interacting particles. In this section the kinetic equations for wide atomic-molecular processes within the gas-solid systems were constructed. 

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