The Drop Model

First we will employ the classical drop model of a small cluster, which is by far the most widely known and used cluster model. We will use this model to calculate nucleation rates and will compare the predicted nucleation rates with experimental results. Then we will discuss various deficiencies in the drop model. Next we will briefly recount two inspired but unsuccessful attempts to amend the drop model. This will be followed by discussion of two additional models, which are, in some ways, related to the drop model. After that, we will discuss a purely atomistic approach to calculating F) - iFy, and the predictions of this approach will be compared with experiment. Finally, we will point out the deficiencies in this atomistic approach and suggest a new approach to calculating Fi — iFb- This new approach takes into account both the deficiencies in the drop model and the atomistic model. It is readily suited to calculating homogeneous gas phase nucleation rates for any system for which the thermodynamic properties of both gas phase and the condensed phase are known. 

The drop model of microcluster thermodynamic properties rests on a simple and appealing physical picture of the cluster. It is assumed that the 

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Not all properties of the nuclei can be explained by the shell model. For calculation of binding energies and the description of nuclear reactions, in particular nuclear fission, the drop model of the nucleus has proved to be very useful. In this model it is assiuned that the nucleus behaves hke a drop of a liquid, in which the nucleons correspond to the molecules. Characteristic properties of such a drop are cohesive forces, surface tension, and the tendency to split if the drop becomes too big. 

In order to calculate the binding energy E ) of the nuclei, Weizsacker developed a semi-empirical formula based on the drop model ... 

The drop model of nuclei (section 2.4) proved to be useful to explain fission (Bohr and Wheeler, 1939) due to the surface tension of a liquid, a droplet assumes a spherical shape. If energy is supplied, the droplet begins to oscillate between spherical and elongated shapes. With increasing distortion, elongation passes a threshold and the droplet splits into two parts. In nuclei, the repulsive Coulomb forces, which... 

The half-lives of the longest-lived isotopes of transuranium elements (Fig. 14.8) show a continuous exponential decrease with increasing atomic number Z. Whereas up to element 103 the half-life is mainly determined by a decay, the influence of spontaneous fission seems to become predominant for elements with Z > 106. The drop model of nuclei predicts a continuous decrease of the fission barrier from about 6 MeV for uranium to about zero for element 110. That means that according to the drop model, elements with Z > 110 are not expected to exist, because normal vibrations of the nuclei should lead to fission. 

Fig. 9. Critical supersaturation for homogeneous gas phase nucleation of n-butylbenzene vapor as a function of temperature. The predictions of the drop model (dashed curve) are in good agreement with experiment (solid curve) although a systematic difference is apparent.

Gritidsins of the Drop Model and Oasacal Nucleation Ihewy... 

It is clear from Table 1 and Fig. 9 that classical nucleation theory, based on the drop model, does not yield complete agreement with experiment. Before proceeding to the various attempts to improve on classical nucleation theory, it is desirable to understand why it fails. There are two basic types of criticisms, which we will now discuss. 

The argument used in arriving at the drop model expression for the cluster free energy [Eq. (38)] did not include any mention of rotation and translation. 

Tolman also noted another problem with the drop model. It is really derived from continuous thermodynamics and so implicitly assumes that the... 

Abraham and Dave have found a striking prediction of the drop model, which demonstrates that it really should not be applicable to a small cluster. In Eq. (38) of the drop model, we interpreted Ff — iFb of Eq. (37) as the area of a cluster times the surface free energy per unit area in bulk material. It could just as well be interpreted as the number of molecules on the cluster surface times the surface free energy per surface atom in bulk material. When this is done, one can calculate the number of surface molecules in a cluster of i molecules. Abraham pointed out that the drop model then regards a 10-molecule cluster as having 20 surface molecules ... 

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... 

The first term on the right-hand side of Eq. (44) is, of course, just the surface free energy term of the drop model. The second term, the translational free energy, is obtained from... 

Reiss and his co-workers took a similar approach to that of Lothe and Pound. However, they do not agree with his treatment of the rotational partition function, which they contend is already implicitly included in the drop model. They also disagree in the treatment of the replacement factor. 

Typically, the Lx>the-Pound treatment predicts nucleation rates about 10 larger than predicted by the drop model. As a result, it does well for materials such as C6H6 for which the classical nucleation theory fails badly however, it therefore gives very bad results for materials that are adequately described by the drop model, such as water. Reiss treatment seems to predict nucleation rates close to those predicted by classical nucleation theoryIt also cannot give agreement with all of the experimental data. 

Kiang and co-workers " have taken an approach similar to that of Abraham. However, rather than starting their calculations from the drop model that we have used until now, they started from a general form for the excess free energy due to Fisher ... 

Here a is the microscopic surface free energy of the cluster, ai the area of a cluster of i molecules, and t and c constants. [Note that if x is, as for spherical particles, for suitable values of r and c Eq. (48) looks like either the classical drop model or the drop model modified to include rotation, translation, etc. Therefore, Eq. (48) is, in a sense, a generalization of the classical drop model.] The Fisher drop model, Eq. (48), was originally developed to describe properties of gases very near the critical point, x and r can be obtained from critical-point indices and are found to be x = and t = 2.333. Hamill showed that c could be obtained from the density of the gas and cr from the second virial coefficient. Using the same equation for the free energy of a... 

Fig. 12. Surface energy B (i) of water microclusters (Briant and Burton ). According to the drop model E i)/i should be a constant. It clearly is not.

The work of Briant and Burton and Binder and Stauffer suggest that Eq. (50) is a better general form of the surface energy of a microcluster than that of the drop model. Therefore, we suggest that a nucleation theory be developed using, in Eq. (36),... 

Now Eq. (52) for a droplet containing an ion is subject to all of the criticisms of the drop model itself. It neglects rotation and translation of the drop. It is based on macroscopic continuum thermodynamics and so there is no reason to expect that it should apply to small drops. It contains no consideration of the structure of a small drop. In addition, it does not consider that the ion itself may perturb the configurations of the molecules in the drop. 

Not surprisingly, the attempts to improve on Eq. (52) took the same direction as those to improve on the drop model itself. First Russell " discussed statistical-mechanical corrections to Eq. (52). His corrections were very similar to the corrections of Lothe and Pound to the drop model. Then various workers, " following the success of Burton s microscopic approach to calculating homogeneous nucleation rates (Section 3.8), started trying to calculate properties of water clusters containing ions from a purely microscopic point of view. We will not review all of this work here but will only summarize that of Briant and Burton, which appears to be the most extensive study to date. None of the workers on this problem have reached a point where they expect to be able reliably to calculate nucleation rates of on ions in the near future. Our principal purpose will be to see how bad Eq. (52) is and what are the prospects for improving on it. 

Fig. 15. Free energy change AF(i) for condensation of water on Cs" and F ions at 300 K at 5 = 1 (Briant and Burton. The exact results are compared with the predictions of the drop model (anooth curve). The agreement is very poor.

Fig. 16. Average energies of clusters of eight water molecules on various ions at 300°K (Briant and Burton . The data show that the cluster energy depends on the sign of the charge on the ion, contrary to the predictions of the drop model, and is roughly a linear function of the ion radius ri.

Based on the liquid concentration model, proposed by Ya.Frenkel [80] and developed for the supereooled vitreseent substances in papers [124-126], V.lrzhak, proposed an original alternative of the drop model for oligomer systems [127], This model allowed us to determine the dependenee of the aggregates distribution fimction on the intermolecular interaction and... 

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