Capillarity approximation

Wolynes P G 1997 Folding nucleus and energy landscapes of larger proteins within the capillarity approximation Proc. Natl Acad. Sci. (USA) 94 6170-5... 

Katz JL (1992) Pnre Appl Chem 64 1661 and references therein The classical theory is linked to the capillarity approximation, taking into acconnt that, in this approach, the free energy change to form the germ is the snm of a bnlk and a snrface contribntion, respectively proportional to the volnme and to the snrface of a spherical clnster. 

Figure 1, Composition of the critical cluster (bubble of critical size), Xgas, thermodynamic driving force of critical bubble formation, CJ,radius of the critical bubble, Rc, and work of critical bubble formation, [J Gc, computed for the case of boiling in binary liquid-gas solutions in dependence on supersturation here expressed via the density of the liquid puq (for the details see Ref 21). By the number (1), the results are shown computed via the classical Gibbs approach employing the capillarity approximation, number (2) refers to computations via the generalized Gibbs approach and number (3) to computations via the van der Waals square gradient density functional method.

It is evident that the question of determination of the size of the critical cluster is a rather non-trivial problem, it depends qualitatively on the definition of the size, i.e., for which of the dividing surfaces the parameter has to be computed and which assumptions are employed in its determination. Employing the classical Gibbs method and the capillarity approximation, we arrive at curve 1 in Fig.lc. 

A homogeneous metastable phase is always stable with respect to the formation of infinitesimal droplets, provided the surface tension a is positive. Between this extreme and the other thermodynamic equilibrium state, which is inhomogeneous and consists of two coexisting phases, a critical size droplet state exists, which is in unstable equilibrium. In the classical theory, one makes the capillarity approximation the critical droplet is assumed homogeneous up to the boundary separating it from the metastable background and is assumed to be the same as the new phase in the bulk. Then the work of formation W R) of such a droplet of arbitrary radius R is the sum of the... 

The surface tension a in (11.25) is taken as that of the bulk liquid monomer thus it is assumed that clusters of a small number of molecules exhibit the same surface tension as the bulk liquid. This is the major assumption underlying classical nucleation theory and it has been given the name of the capillarity approximation. 

The classical theory of homogeneous nucleation dates back to pioneering work by Volmer and Weber (1926), Farkas (1927), Becker and Doring (1935), Frenkel (1955), and Zeldovich (1942). The expression for the constrained equilibrium concentration of clusters (11.57) dates back to Frenkel. The classical theory is based on a blend of statistical and thermodynamic arguments and can be approached from a kinetic viewpoint (Section 11.1) or that of constrained equilibrium cluster distributions (Section 11.2). In either case, the defining crux of the classical thoery is reliance on the capillarity approximation wherein bulk thermodynamic properties are used for clusters of all sizes. 

The principal limitation to the classical theory is seen as the capillarity approximation, the attribution of bulk properties to the critical cluster (see Figure 11.8). Most modifications to the classical theory retain the basic capillarity approximation but introduce correction factors to the model [e.g., see Hale (1986), Dillmann and Meier (1989, 1991), and Delale and Meier (1993)]. 

For small / and very large /, (10.57) departs from reality. When / = 1, N[ should be identical to N], but (10.57) does not produce this identity. The failure to approach the proper limit as / 1 is a consequence of the capillarity approximation and the specific... 

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]. 

We show a few selected experimental results in Fig. 6.10. The measured nucleation mercury-drop flux (in drops per cm per second) is plotted as a function of the vapor supersaturation in the nucleation zone of maximal supersaturation. Fig. 6.10 also shows two estimates of the nucleation rate. One estimate is based on the classical theory employing the capillarity approximation and macroscopic values of the surface... 

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).

G. L. Griffin and R. P. Andres, Microscopic capillarity approximation Free energies of small clusters, J. Chem. Phys. 71 ... 

The nucleation theory just described is referred to as classical nucleation theory. It relies on the capillarity approximation, in which crystallites of microscopic size are treated as if they are macroscopic, and in which the kinetics is described as the stepwise attachment of single molecules across the crystal-melt interface. In fact, this approximation may not be valid under realistic conditions. A small crystallite may not achieve bulk properties at its center, and its interface may be so strongly curved that the planar value Ysl no longer applies. The interface may be diffuse rather than sharp, so that the description of the kinetics as resulting from addition of solid particles one after another may not be valid instead, a collective fluctuation may result in the simultaneous incorporation of a larger number of molecules in a loosely structured crystallite. 

---