Classical nucleation theory limitation

R. Although expressions for this parameter exist, they are derived by a hybrid of molecular mechanical and thermodynamic arguments which are not at present known to be consistent as droplet size decreases (8). An analysis of the size limitation of the validity of these arguments has, to our knowledge, never been attempted. Here we evaluate these expressions and others which are thought to be only asymptotically correct. Ve conclude, from the consistency of these apparently independent approaches, that the surface of tension, and, therefore, the surface tension, can be defined with sufficient certainty in the size regime of the critical embryo of classical nucleation theory. 

The classical nucleation theory embodied in Eq. (16) has a number of assumptions and physical properties that cannot be estimated accurately. Accordingly, empirical power-law relationships involving the concept of a metastable limit have been used to model primary nucleation kinetics ... 

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. 

Thompson and Spaepen have used classical nucleation theory to predict the homogeneous nucleation temperatures of binary alloys. The surface free energy they use has been given in Eq. (3.9), and some of its possible limitations... 

The classical nucleation theory can be used only when the droplet radius p is much larger than the interfacial width (which is of the same order as the correlation length Since %([Pg.215]

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]. 

In Fig. 13 we show the size of the critical bubble. Both at the binodal and the spinodal the size of the critical bubble diverges as expected from classical nucleation theory and Cahn-Hilliard theory, respectively. The insets show that both analytical approaches agree quantitatively with the SCF calculations in the limits x Xbin and X -> Xspin, respectively. Between binodal and spinodal the critical size has a minimum. At the minimum, the size of the critical bubble is only a few segment diameters. For the parameters used in the present calculations, the minimum corresponds to rather small nucleation barriers and is close to the regime of spinodal nucleation . For most practically relevant nucleation barriers the size of the critical bubble decreases upon increasing molar fraction. 

While the classical theory of nucleation is limited by the implicit assumptions in its derivation, it successfully predicts the nucleation behavior of a system. Inspection of the equation above clearly suggests that the nucleation rate can be experimentally controlled by the following parameters molecular or ionic transport, viscosity, supersaturation, solubility, solid-liquid interfacial tension, and temperature. 

The sharpness experimentally appears as a limiting superheat before which nothing happens, but beyond which very rapid boiling takes place. Calculating this rate is the province of nucleation theory, and the so-called classical theory is outlined below. 

Nowadays it is quite clear that it does not make sense to speak about two different theories of the nucleation rate the classical and the atomistic one. In reality, what we do have is a general nucleation theory comprising two limiting cases. The classical model describes the nuclei by means of macroscopic physical quantities and can be used to predict the size and to evaluate the nucleation work of sufficiently large critical clusters. The atomistic model is valid in the case of high supersaturations and very active substrates when the critical nuclei are very small. Therefore the quantitative interpretation of experimental data on the stationary nucleation rate based on the atomistic theory provides valuable information on the specific properties of clusters consisting of a few building units. 

Discussion of transient effects in the nucleation kinetics in polymers subjected to orienting stresses bases on a kinetic theory of nucleation proposed in [20-22] for the systems of oriented asymmetric elements. The theory converges to the classical approach in the limit of isotropic orientation. This fact explains apparent validity of the classical theory in the apphcations to unoriented polymers. 

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