Homogeneous Gas Phase Nucleation

All of the expressions discussed above are sometimes used in the nuclea-tion literature. In the next section we will proceed actually to calculate homogeneous gas phase nucleation rates in terms using Eq. (13) and will compare our results with experiment. 

In the previous section, we have shown that one must know the capture rates Ci and the equilibrium cluster concentrations h,- in order to calculate the homogeneous gas phase nucleation rate. In this section we will discuss first the capture rates, which are not a major issue in nucleation theory, and then the equilibrium concentrations. We will show that accurate calculation of the equilibrium concentrations is an exceedingly difficult problem that must be solved very accurately to calculate nucleation behavior. Therefore, various approaches to this problem will be discussed and their advantages and disadvantages will be pointed out. Finally, we will propose a new approach to calculating the equilibrium concentrations, which we believe holds great promise. 

As a first approximation, it is reasonable to consider a cluster of i molecules as a sphere of radius r, and surface area A-jirf. The capture rate can be estimated simply as the frequency of collisions between the cluster and the monomer background gas times the probability a,- that a molecule that collides with the cluster will stick. The collision frequency is the surface area of the cluster, 4 7rrf, times the ideal gas collision rate per unit area,.  

There are two problems with this picture. First a cluster is not really spherical it can actually be distorted from spherical (Fig. 5). Second, we do not know much about the sticking coefficient a,-, although it is generally assumed to be about 1.0 and independent of cluster size. Errors in the cluster area might increase c, from the value in Eq. (30) by as much as a factor of five or ten. Similarly if a, is actually 0.01 this would reduce the capture rate, and hence the nucleation rate, by a factor of 100. These errors in Eq. (30) for the capture rate will, of course, affect the nucleation rate. However, as we will see, these errors are insignificant compared with the uncertainties in calculating the equilibrium cluster concentrations, which we will examine below. 

Equilibrium Ouster Concentrations—Statistical-Mechanical Considerations 

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Another strategy to synthesize the particle from volatile organometallic precursors is achieved by promoting homogeneous gas-phase nucleation, the so-called chemical vapor synthesis (CVS). " In this case, the precursor is evaporated using a carrier gas and reacted with a co-substrate (e.g., O2) to produce the desired material, which can be collected as powder. The typical experimental setup is assembled according to Figure... 

There exists an extensive literature on nucleation theory. A great diversity of problems have been discussed. They range from homogeneous gas phase nucleation,to condensation of the primodial vapor in the solar system to form meteorites, and to formation of voids in nuclear reactor materials. Several collections of review articles " as well as a book have been published recently devoted solely to nucleation problems. In these works, each author has advocated his own particular approach to nucleation theory or dealt solely with his own pet nucleation problem. 

Initially we will examine homogeneous gas phase nucleation. This is the classic test problem of nucleation theory. It is believed that condensation of a gas is the simplest nucleation problem. This problem provides a test of the conceptual usefulness of nucleation theory. The lessons learned in discovering how to solve this as yet unsolved problem will hopefully provide a guide to more complex nucleation problems. 

In Section 2 we will examine very carefully the mathematical formalism for calculating nucleation rates. The expressions obtained in this section will be appropriate for homogeneous gas phase nucleation. Although the formal solutions appropriate to this problem are well known, they are worth examining closely as they must be thoroughly understood in order to treat complex nucleation phenomena. 

In Section 3 we will discuss actual homogeneous gas phase nucleation rate calculations. Extensive reliable experimental data exist and the predictions of various nucleation theories will be compared with experiment. The principle purpose of this section will be to undemtand fully what must be reliably known in order to predict nucleation behavior accurately. As we will show, the most serious problem in homogeneous nucleation theory is the... 

In the particular case of homogeneous gas phase nucleation with the rate equations expressed as in Eq. (16), it is actually straightforward to obtain an analytic expression for rig. 

Therefore the method introduced here, when applied to homogeneous gas phase nucleation, gains nothing over what we already knew. We have obtained a solution to this problem by an alternative method. The power of this method is seen when complex nucleation processes are studied. 

This equation for the nucleation rate is identical to that of Eq. (17) for the case of homogeneous gas phase nucleation. However, here we have made no assumptions about the coefficients in the rate equations. 

In the previous sections we have obtained an expression for the steady state homogeneous gas phase nucleation rate. Two approaches were examined. The first, that of McDonald, yields an anal)dic result in a straightforward way. The second approach, which is less direct, is more suited to numerical solutions to complex nucleation problems. It yields the same result for homogeneous nucleation as obtained with McDonald s approach. We will now examine a few alternative forms of Eq. (13) for the homogeneous gas phase nucleation rate. These forms are often referred to in the literature and it is useful to be familiar with them, even though they will not be discussed in this paper. 

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. 

We will now use this expression to calculate homogeneous gas phase nucleation rates. The result we will get is known as classical nucleation theory. 

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.

TtUfle I. Critical Supersaturations S for Homogeneous Gas Phase Nucleation-Comparison of Experiment with the Predictions of the Classical Theory... 

In the past two sections we have discussed at some length the progress that has been made and the questions that remain in developing a theory of homogeneous gas phase nucleation. We have found that the biggest problem is the description of the free energy of a small cluster. Homogeneous gas phase nucleation is widely studied not for its practical importance but rather as a test problem intended to refine and develop concepts to apply to more complex and important problems. 

All of the above discussion is strictly applicable only to homogeneous gas phase reactions. Usually the above considerations do apply reasonably well to non-polar liquids and nonpolar solutions, although normal Z values may be an order of magnitude less than for gas reactions. Reactions in solids are often much more complex, since they are usually heterogeneous, involve catalytic effects, reactions at preferential sites (dislocations, etc), and nucleation phenomena. These complicated processes are quite beyond the scope of the present article. For some description of these phenomena, and further references, the reader should consult Refs 9, 10 11... 

In spite of these studies and results, the relative importance of the gas-phase nucleation compared to the surface nucleation is unclear as yet. In fact, the number of diamond particles collected from the gas phase is very small compared to the typical surface nucleation densities, thus the homogeneous nucleation mechanism cannot account for the large variety of nucleation densities observed on different substrate materials and from different surface pretreatments. It is speculated and also supported by a recent experimentl l that the nuclei formed in the gas phase may reach the growing surface and increase the surface nucleation density. However, how the diamond particles formed in the gas phase could serve as seeds on the substrate surface for the subsequent growth of a diamond film remains unknown. 

The remaining steps include chemical/physical surface processes, such as homogenous or heterogeneous solid-solid interactions, solid-phase nucleation and diffusion into graphite heterogeneous gas-solid interactions, that is, adsorption/desorption and reaction of molecules with the wall to form atoms homogeneous gas-phase reactions and processes by which analyte leaves the furnance. 

This reaction proceeds in the gas phase at temperatures above about 600°C and is usually applied either in chemical vapor deposition of carbide layers on solid substrates or to produce carbide powders with very fine (submicron) grain sizes. In the latter case the nucleation of the carbide must proceed in the homogeneous gas phase. 

A gas phase precipitation can only be carried out in this way when care is taken to suppress nucleation in the gas phase. This means that rapid homogeneous gas phase reactions should be avoided. Several ways to achieve ttiis are conceivable ... 

Assuming that the rate of gas-to-particle conversion of any condensable species is greater than its rate of formation in the gas phase (which is the case for heten eneous nucleation predominant in the atmosphere, but may not be valid for homogeneous nucleation in clean-air smog-chamber studies) ... 

It appears that the increase in particle density is exponential and that the best conditions are reached at 323 K, where a great number of small particles are produced per surface unit. However, at 353 K large agglomerates are produced by accretion of 10-50 nm particles, which is consistent with homogeneous nucleation of particles in the gas phase. 

Although Eq. 27 appears to be the most likely initiation reaction, we cannot rule out a process in which water vapor and DMTC react, based on the ab initio results described in Sect. 4.6. If this does occur, however, it apparently does not lead to homogeneous nucleation of particles, since anecdotal evidence from the glass industry indicates that DMTC and water vapor can be premixed prior APCVD of tin oxide without substantial buildup of solids in delivery lines. Perhaps this is due to significant kinetic barriers to the decomposition of the tin-water complexes that initially form, so that further gas-phase reaction does not occur until the reactants enter the heated boundary layer above the substrate. 

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