Nucleation classical rate theory

A few of the many contributors to the classical rate theory of boiling nucleation are Volmer (VI), Becker and Doring (B2), Frenkel (F7), Fisher (F3), and Bernath (B4). All agree that a prime requirement for nucleation to occur in a liquid is that the liquid must be superheated. The bubbles formed are cooler than the liquid therefore nucleation is strictly irreversible. Because of the superheat, a temperature driving force exists between liquid and bubble. However, because surface tension forces are immense for tiny bubbles, a collapsing tendency exists which may counteract the tendency of a bubble to grow by absorbing heat. One problem faced by any theory of nucleation is to explain the formation of a bubble which will not collapse. 

As with nucleation, classical theories of crystal growth 3 20 2135 40-421 have not led to working relationships, and rates of crystallisation are usually expressed in terms of the supersaturation by empirical relationships. In essence, overall mass deposition rates, which can be measured in laboratory fluidised beds or agitated vessels, are needed for crystalliser design, and growth rates of individual crystal faces under different conditions are required for the specification of operating conditions. 

The work of Reiss and co-workers puts the question of the equilibrium distribution of liquid embryos in dilute supercooled vapors on sound conceptual ground. However, having to calculate embryo free energies by simulation rules out the use of such an approach in practical applications. To overcome this limitation, Weakliem and Reiss [67] developed a modified liquid drop theory that combines elements of the physically consistent cluster with the conventional capillarity approximation. These same authors have also developed a rate theory which allows the calculation of nucleation rates in supercooled vapors [68]. The dependence of the predicted rates on supersaturation agree with classical nucleation theory, but the temperature dependence shows systematic deviations, in accordance with scaling arguments [54]. 

Fig. 6.10. Logarithmic plot of the homogeneous nucleation rate for mercury as a function of saturation ratio S = p lPsat -15°C and 39 °C (Martens et al., 1987). Short solid line segments represent experimental results. Dot-dashed lines indicate predictions of the classical BDZ theory in the capillarity approximation dashed lines represent classical theory extended by including changes of cohesive energy with particle size (McClurg et al., 1997).

According to classical crystallization theory, the final particle size depends mainly on the ratio of nucleation to growth rates. A slow nucleation leads to a low number of nuclei, which can grow and reach large polydisperse sizes. On the contrary, if nucleation is quick, a large number of nuclei are formed so that the final particle size will be small and monodisperse. 

Transfomation from a meta-stable phase, such as supersaturated solution, to a thermodynamically more favorable phase requires first the crystal nucleation of a germ of the new phase. According to the classical nucleation theory, the volume nucleation rate J (cm" sec ), describing the number of nuclei(i.e., a critical germ) formed per volume per time, is given by ... 

It is remarkable that the predictions of classical nucleation theory without any consideration of polymer connectivity are borne out in experiments. At higher supercooling, deviations are expected because of temperature dependence of the nucleation rate prefactor. 

Nucleation rate based on the classical nucleation theory The nucleation rate is the steady-state production of critical clusters, which equals the rate at which critical clusters are produced (actually the production rate of clusters with critical number of molecules plus 1). The growth rate of a cluster can be obtained from the transition state theory, in which the growth rate is proportional to the concentration of the activated complex that can attach to the cluster. This process requires activation energy. Using this approach, Becker and Coring (1935) obtained the following equation for the nucleation rate ... 

Failure of the Classical Nucleation Theory There are several suggested explanations for the failure of the classical nucleation theory to quantitatively predict the nucleation rate, including the following ... 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

The atmospheric situation is complicated by varying conditions of temperature, relative humidity, and concentrations of other gases such as NH3 which can enhance nucleation rates over those expected for a well-mixed air mass at a fixed temperature and RH (e.g., see Nilsson and Kulmala, 1998). However, there is a general consensus that the observed rates of nucleation of H2S04 often, indeed usually, exceed those expected from classical binary homogeneous nucleation theory. (Note that this is not always the case. For example, Pirjola et al. (1998) reported that the measured formation of nuclei in the Arctic boundary layer... 

Homogeneous nucleation may be described by assuming that critical-size nuclei will be formed from ideal vapor (water or air) at a rate, I, given by classical nucleation theory [4]. The equation is... 

The kinetics of nucleation of one-component gas hydrates in aqueous solution have been analyzed by Kashchiev and Firoozabadi (2002b). Expressions were derived for the stationary rate of hydrate nucleation,./, for heterogeneous nucleation at the solution-gas interface or on solid substrates, and also for the special case of homogeneous nucleation. Kashchiev and Firoozabadi s work on the kinetics of hydrate nucleation provides a detailed examination of the mechanisms and kinetic expressions for hydrate nucleation, which are based on classical nucleation theory. Kashchiev and Firoozabadi s (2002b) work is only briefly summarized here, and for more details the reader is referred to the original references. 

Classical nucleation theory is the basis for understanding condensation and it predicts the dependencies correctly. Unfortunately, quantitatively the predictions often do not agree with experimental results [28,29], Theory predicts too low nucleation rates at low temperatures. At high temperatures the calculated rates are too high. Empirical correction functions can be used and then very good agreement is achieved [30], Ref. [31] reviews experimental methods. General overviews are Refs. [32-34],... 

In spite of the widespread recognition of the theoretical inadequacies of classical nucleation theories, attempts to formulate more realistic theories have met with limited success, in part because nucleation rate measurements are notoriously difficult to make. Consequently, the available data base with which to evaluate various theories is inadequate. Molecular level approaches would seem to hold promise of providing more rigorously acceptable theories without resorting to the use of uncertain bulk properties in treating clusters that are intrinsically molecular. Furthermore, new experimental techniques, such as molecular beams and cluster spectroscopy, make the properties of small clusters amenable to investigation at the molecular level. 

Theoretical approaches to nucleation go back almost 80 years to the development of Classical Nucleation Theory (CNT) by Volmer and Weber, Becker and Doring and Zeldovich [9,10,17-20]. CNT is an approximate nucleation model based on continuum thermodynamics, which views nucleation embryos as tiny liquid drops of molecular dimension. In CNT, the steady-state nucleation rate /, can be written in the form / a where jS, is the monomer condensation... 

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