Zeldovich factor

Calculate the value of the Zeldovich factor for water at 20°C if the vapor is 5% supersaturated. 

Equation 19.17 may be interpreted in a simple way. If the equilibrium concentration of critical clusters of size Afc were present, and if every critical cluster that grew beyond size Mc continued to grow without decaying back to a smaller size, the nucleation rate would be equal to J = (3CNexp[-AQc/(kT)]. However, the actual concentration of clusters of size Mc is smaller than the equilibrium concentration, and many supercritical clusters decay back to smaller sizes. The actual nucleation rate is therefore smaller and is given by Eq. 19.17, where the first term (Z) corrects for these effects. This dimensionless term is often called the Zeldovich factor and has a magnitude typically near 10-1. 

The Zeldovich factor r depends on the geometrical form of the cluster [4.17, 4.18], and for liquid droplets has the value... 

The dimensionless Zeldovich factor is always less then unity, and has an order of... 

From an experimental point of view, however, it seems that little progress has been made since the evaluation of the Volmer-Weber [4.11] equation in 1926. The experimentalist is bound to use eqs. (4.32) and (4.33), disregarding the small uninformative dependence of A on 77 derived from the Zeldovich factor F or the attachment probability Watt.yVcnt More significant progress seems to have been made with the development of the small cluster model based on the atomistic approach of Becker and Doering. 

The time lapse ro.s can be determined at different amplitudes of the nucleation pulse, so that the/- 7 dependence is easily found. Fig. 5.8 shows a In /vs. 1/171 plot obtained in this way. The relation is linear in this representation, as expected from eq. (5.2). The contributions of the overvoltage dependence of the Zeldovich factor Tand the attachment fi-equency < att,ycrit obviously small enough to produce an appreciable effect on the general In/- 1/171 relation. From the slope, the specific edge... 

The Zeldovich factor generally has a value ranging from 0.1 to 1 and thus has only a slight effect on the rate of nucleation. 

According to CNT, the pre-exponential factor T in Equation 10.9— related to the frequency and growth probability for a polymorph clnster at a certain snpersatnra-tion (Auer and Frenkel 2004)—is a fnnction of the nnmber density of sites for nncle-ation pi, the attachment rate of particles to the critical cluster //, and the Zeldovich factor z (Equation 10.20a) ... 

The Zeldovich factor represents a thermodynamic correction parameter and takes into account the fact that a cluster having reached the critical size does not necessarily nucleate, but could fluctuate in size back into the sub-critical region (Schmelzer 2003). The Zeldovich factor for nncleation rate on a polymeric membrane is a function of porosity and contact angle. According to Equation 10.1 ... 

Table 10.3 summarizes the form of the Zeldovich factor for different cases homogeneous nucleation (e = 0, 0 = 180°), heterogeneous nucleation on a nonporous support (e = 0), and heterogeneous nucleation on a microporous membrane. 

Expressions of the Zeldovich Factor (z) for Nucleation on Different Membrane Surfaces... 

FIGURE 10.16 Zeldovich factor for heterogeneous and homogeneous nucleation versus contact angle at different membrane porosity. (Adapted from Journal of Crystal Growth, 310, Curcio, E., Di Profio, G., and Drioli, E., Prohahilistic aspects of polymorph selection by heterogeneous nucleation on microporous hydrophobic membrane surfaces, 5364-5369, Copyright (2008), with permission from Elsevier.)... 

Ter Horst, J.H. and Jansens, P.J. 2005. Nucleus size and Zeldovich factor in two-dimensional nucleation at the Kossel crystal (0 0 1) surface. Surf. Sci. 574 77-88. 

Determining the flux of monomers to the critical nuclei (/, molecules/L sec) is a challenge. This flux has been modeled using various statistical mechanics approaches. These models estimate the flux of molecules from the solution to the surface of the critical nuclei but they include various adjustments such as the Zeldovich factor (Markov, 2003), which accounts for the possibility that some critical-sized clusters will dissolve to a smaller size instead of growing into crystallites. Presently there is no widely accepted model for J, so Eq. (9.28) is usually recast into a semi-empirical form, which combines the monomer flux and cluster surface area into a pre-exponential term (Q, crystallites/L sec). 

Doring [4] provided an analytical solution considering a system of constant composition in which supercritical clusters are reintroduced into the system as the equivalent amount of discrete units. A steady state expression of the nucleation rate follows, Jg = Zam Cm y where am is the net probability of addition of an atom per unit time from a critical cluster of size m, and Cm, the equilibrium concentration of critical clusters, is related to the monomer concentration through the Bolztmann equation, Cm = Cl exp(—AGm /kT) and the nondimensional Zeldovich factor Z, which accounts for the fact that the steady state concentration at m is only 1 /2 of the concentration at equilibrium, and that critical clusters may still decay [5]. 

The comparison of this equation with Eq. (62) shows that the contribution of the omitted terms in the derivation of Eq. (62) can be accounted for by the introduction of the factor F known as the Zeldovich nonequilibrium factor. The factor is always less than unity, having an order of magnitude of 10 and accounts for the depletion of the cluster population due to the process of growth.It seems that the Zeldovich factor depends on the geometrical form of the cluster (see Refs. 72 and 78) and for a liquid droplet has the value r = (AGc/37rfc7TV ) /l... 

Comparing equations (2.19) and (2.23) it is not difficult to see that the only difference between / and / , is the Zeldovich factor F. That being the case, it appears that this quantity should be associated with the difference between the quasi equilibrium and the actual, steady state number of critical nuclei. The considerations that follow shed additional light on this subject. 

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