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. 

In order to verify which of the above nucleation mechanisms accurately represents hydrate nucleation, it is clear that experimental validation is required. This can then lead to such qualitative models being quantified. However, to date, there is very limited experimental verification of the above hypotheses (labile cluster or local structuring model, or some combination of both models), due to both their stochastic and microscopic nature, and the timescale resolution of most experimental techniques. Without experimental validation, these hypotheses should be considered as only conceptual aids. While the resolution of a nucleation theory is uncertain, the next step of hydrate growth has proved more tenable for experimental evidence, as discussed in Section 3.2. 

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. 

At the present time, quantum computations are quite demanding, and, thus, the size of a system that can be treated using computational quantum methods is limited. A perfectly logical question, "Why the application of quantum methods can systematically improve the nucleation theory " leads to very simple answers ... 

In 1968, Roe [19] developed the SE limiting case equations for particle number from the homogeneous nucleation theory. He showed that the SE equation... 

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. 

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... 

Figure 4. Boundary of limiting superheats of acetone and water solutions in acetone 1 - acetone, 2 - acetone + 10 % water, 3 - acetone + 30 % water. Solid line - line of liquid-acetone vapor phase equilibrium, C - critical point, dashed line - calculation by homogeneous nucleation theory for J = 10 s m (acetone). ...

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 nucleation theories for potentiodynamic conditions [44] lead mainly an instantaneous or progressive nucleation also for reversible and irreversible limiting conditions. The following dependences of 7p, Ep, and AE /2 with v can be obtained for four limiting cases. 

Mirabel and Clavelin (1978) ha ve derived the limiting behavior of binary homogeneous nucleation theory when the concentration of one of the vapor species becomes very small. For a low concentration of one species, say, B, the preexponential factor C simplifies from (11.90) but one has to distinguish two cases ... 

Mirabel and Clavelin (1978) have derived the limiting behavior of binary homogeneous nucleation theory when the concentration of one of the vapor species becomes very small. 

Nucleation theory has been advanced for vaporization of pure substances1 and for nucleation of bubbles from solutions containing dissolved gas.2 Bubbles of a critical radius and larger grow while bubbles having radii less than this dimension tend to decay. The result of nucleation theory is the prediction of the maximum attainable limit of supersaturation. Two equations are sufficient for this... 

This type of equation is also encountered in other areas, such as nonlinear waves, nucleation theory, and phase field models of phase transitions, where it is known as the damped nonlinear Klein-Gordon equation, see for example [165, 355, 366]. In the (singular) limit r 0, (2.15) goes to the reaction-diffusion equation (2.3). Front propagation in HRDEs has been studied analytically and numerically in [149, 150, 152, 151, 374]. The use of HRDEs in applications is problematic. Such equations are obtained indeed very much in an ad hoc manner for reacting and dispersing particle systems, and they can be derived neither from phenomenological thermodynamic equations nor from more microscopic equations, see below. 

The limit thexmodynamlc nonequilibrium corresponds to the attainment of liquid superheats at which intensive spontaneous boiling is observed on nucleus bubbles of fluctuation nature. The physical definiteness of this boundary is conditioned by a very shazi> dependence of the nucleation rate J(T. P ) on the Gibbs number G- = A T where is the work of formation of a critical, nucleus [4 5j. is the Boltzmaim constant. By the homogeneous nucleation theory the value of J is calculated making use of thermodynamic parameters. Thus, for water at atmospheric pressux T have J ... 

The crystallization of an amorphous material proceeds by the processes of nucleation and growth, and the crystallization rate is suppressed by reducing either (or both) of these processes. We discuss first the kinetics of nucleation. Turnbull and Cohen, using nucleation theory and a number of assumptions, arrive at Eq. (1.1) (see Turnbull (1969)) as a reasonable upper limit to the nucleation frequency as a function of undercooling. 

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]. 

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