Volmer-Weber equation

The nucleation rate constant (Eq. (5.8.5)) k depends on the size of the critical crystallization nucleus according to the Volmer-Weber equation... 

From an experimental point of view, however, it seems that little progress has been made since the evaluation of the Volmer-Weber equation in 1926. The experimentalist is bound to use Eq. (62), disregarding the small noninfor-mative dependence of ki from rjc derived from the Zeldovich factor or the attachment frequency Wa,c ... 

Vii ial equation of state in two dimensions, 931 Virial isotherm, 936 Visible radiation, 797 Volcanoes, in electrocatalysis, 1284 Volmcr, Max, 1048,1474 Volmer. Weber, electrodeposition. 1303. 1306 Volta, 1423, 1455 Volta potential difference, 822 Voltammetry. 1432 1434 cyclic, 1422 1423 diffusion control reactions, 1426 electron transfer reaction, 1424... 

The pre-exponential factor A 1p is independent of potential as long as Ncrjt is potential independent, and the factor 3 depends on the mechanism of attachment. Equation [7] reduces to a classical Volmer-Weber type model for nucleation if Ncrjt 3- The total potential dependence of the nucleation rate in an overpotential range where Ncrit is constant according to the atomistic model is thus given by ... 

Thermodynamic and mechanical equilibrium on a curved vapor-liquid interface requires a certain degree of superheat in order to maintain a given curvature. Characteristics of homogeneous and heterogeneous nucleation can be estimated in the frame of classical theory of kinetics of nucleation (Volmer and Weber 1926 Earkas 1927 Becker and Doring 1935 Zel dovich 1943). The vapor temperature in the bubble Ts.b can be computed from equations (Bankoff and Flaute 1957 Cole 1974 Blander and Katz 1975 Li and Cheng 2004) for homogeneous nucleation in superheated liquids... 

Careful examination reveals that the modified Stem-Volmer equation is mathematically identical to the original nonlinear model developed by Ryan and Weber (22). Fluorescence quenching curves for Cu -FA and application of the modified Stem-Volmer data treatment to the experimental information are shown in figure 2. Since the nonlinear data treatment and the modified Stem-Volmer equations are algebraically identical, their ability to fit experimental data and provide meaningful parameters is the same. 

The above kinetic equations, developed based on the thermodynamic approach of Gibbs (1928), Volmer and Weber (1926), and Becker and Doring (1935), belong to the so-called classical nucleation theories. They have been criticized for the use of surface energy (interfacial tension), cr, which is probably of little physical significance when applied to small molecular assemblies of the size of critical nucleus. 

Volmer and Weber derived an equation for the nucleation rate. ... 

In 1926, Volmer and Weber found that the nucleation rate shows a negative exponential dependence on critical free energy barrier (Volmer and Weber 1926). Becker and Doting further proposed that the activation energy for the short-distance diffusion of molecules molecules to enter the crystalline phase should be considered as well (Becker and Doring 1935). Turnbull and Fisher derived the prefactor for the rate equation of crystal nucleation (Turnbull and Fisher 1949). The rate of polymer crystal nucleation i with the change of critical free energy barrier can be expressed as... 

The nucleation rate is dominated by two factors. One is the critical free energy barrier of nucleation. Its exponential dependence was first proposed by Volmer and Weber (1926). The other is the diffusion energy barrier for molecules crossing over the liquid-solid interfaces. Its exponential dependence was first proposed by Becker and Doring (1935). The quantitative expression of the prefactor in the kinetic equation of the nucleation rate is given by Turnbull and Fisher (1949) as... 

Volmer and Weber ( ) and Becker and D5ring ( ), by theoretical treatment of the energetics and kinetics of nucleus formation derive the equation... 

Jhe general expression for the nucleation work, equation (1.32), says that AG(n) has a minimal value for clusters formed with a minimal excess energy ( ). In terms of the classical nucleation theory developed in the pioneering works of Gibbs [1.1], Volmer [1.11], Volmer and Weber [1.16], Kossel[1.17], Stranski [1.12, 1.13, 1.18, 1.19], Farkas [1.20], Stranski and... 

Volmer and Weber [2.14], Farkas [2.15] and Kaischew and Stranski [2.16-2.18] were the first who examined the stationary nucleation kinetics and derived theoretical expressions for the stationary nucleation rate. However, in this Chapter we shall present the results of the more rigorous treatments of Becker and Doring [2.6] and ofZeldovich [2.19] and Frenkel [2.20] who laid the foundations of the contemporary classical nucleation theory (see also [2.V-2.9] and [2.21-2.24]). For the sake of simplicity we shall neglect both the line tension effects (equations (1.42) and (1.70)) and the dependence of the specific free surface energy on the size of the clusters (equation (1.43). 

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