Nucleation critical clusters

It is widely recognized that the process of condensation is initiated by the formation and growth of self-nucleated or homogeneous nucleated critical clusters of the new phase resulting from local fluctuations in density and in the packing of the metastable film. The rate-determining step of the mechanism is incorpora-... 

Nucleation rate based on the classical nucleation theory The nucleation rate is the steady-state production of critical clusters, which equals the rate at which critical clusters are produced (actually the production rate of clusters with critical number of molecules plus 1). The growth rate of a cluster can be obtained from the transition state theory, in which the growth rate is proportional to the concentration of the activated complex that can attach to the cluster. This process requires activation energy. Using this approach, Becker and Coring (1935) obtained the following equation for the nucleation rate ... 

The mechanisms of droplet (or liquid germs) formation from a supersaturated vapour phase is still the subject of many investigations. After giving a brief account of the classical theory [64], which, as shown above, provides a simple method for estimating the energy barrier to overcome before effective nucleation is started, and permits the estimation of the critical cluster size, a complementary approach will be presented. 

Other computer simulations were made to test the classical theory. Recently, Ford and Vehkamaki, through a Monte-Carlo simulation, have identified fhe critical clusters (clusters of such a size that growth and decay probabilities become equal) [66]. The size and internal energy of the critical cluster, for different values of temperature and chemical potential, were used, together with nucleation theorems [66,67], to predict the behaviour of the nucleation rate as a function of these parameters. The plots for (i) the critical size as a function of chemical potential, (ii) the nucleation rate as a function of chemical potential and (iii) the nucleation rate as a function of temperature, suitably fit the predictions of classical theory [66]. 

The nucleation theorems have been used to obtain information about critical clusters from experimental data, see Ford IJ (1997) Phys Rev E 56 5615 Ford IJ (1996) J Chem Phys 105 8324. These theorems state that, from known size and internal energy of the critical cluster, the nucleation rate can be deduced as a function of the temperature and the supersaturation. 

As seal from Eq. (7.251), the value of TV depends upon several parameters of the system, e.g., the edge surface energy, . It also depends on the overpotential T, and one can see that the size of the critical cluster decreases with an increase in hjl. For 2D nucleation on quasi-perfect silver single crystals, the number of atoms in the minimum nucleus size at which AG begins to decrease with an increase in A varies from 25 to 67 atoms as t varies from -10 to -6 mV. 

All approaches are based either on the thermodynamical description of the gas-solid phase transition by classical nucleation theory or on a detailed discussion of the relevant chemical reactions leading finally to critical clusters (e.g. review by Gail, Sedlmayr, 1987d). We will refrain from a presentation of these various approaches but only list the basic molecules from which the primary condensates are likely to be formed ... 

When the supersaturation ratio S becomes greater than unit, the small liquid droplets (i.e. molecular clusters) commence to appear. Almost all the droplets are immediately destroyed due to evaporation and only small fraction of the droplets (critical clusters) with radii greater than a critical radius r have a chance to survive and grow by accretion of vapor molecules (monomers) onto their surface. It is assumed that macroscopic thermodynamics is applied to the critical clusters that are considered as liquid droplets containing the large number of monomers, that is nx>>i. The number of the critical clusters formed per unit time per unit volume is the nucleation rate J so that the number density of dust grains is Nd = JJdt. Expressions for calculation of the nucleation rate and other quantities can be found in the review paper by Draine (1981). 

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. 

Using Equations 3.3a and b, Englezos et al. (1987a) calculated the critical radius of methane hydrate to be 30-170 A. In comparison, critical cluster sizes using classical nucleation theory are estimated at around 32 A (Larson and Garside, 1986), while computer simulations predict critical sizes to be around 14.5 A (Baez and Clancy, 1994 Westacott and Rodger, 1998 Radhakrishnan and Trout, 2002). 

In Mullin s model, further molecular additions to the critical cluster results in nucleation. 

During the induction period this process continues on the surface of the nucleation site until the critical cluster has collected the next ion to be added triggers nucleation. Crystal growth then can follow. For barium sulfate, La Mer concluded that the slope of the line in Figure 8-1 is six and therefore the nucleation of barium sulfate is a seventh-order reaction overall. The critical cluster is then (Ba" " S04")3, and the addition of the seventh ion, either Ba " or 804 , constitutes the final step of the nucleation process. The question of the number of ions in the critical cluster, however, is by no means settled. Christiansen and Nielsen concluded that for barium sulfate the number is 8. Johnson and O Rourke also concluded that the nucleation rate of this salt is proportional to the fourth power of the concentration. The concept of a small critical nucleus is intuitively satisfying in that the nucleus then requires only a small number of steps for its formation. On the other hand, application of the... 

Becker-Doiing nucleation hypothesis indicates a much larger number, of the order of 100, for the critical cluster. Klein and Driy, in nucleation studies combining the drop method and homogeneous precipitation, found the rate of nucleation of strontium sulfate to depend on the 27th power of the concentration, indicating a nucleus containing 52 ions. 

According to eqs. (4.26a and b), the energy of formation of the critical cluster is also given by its volume Fcrit. This equation holds for both homgeneous and heterogeneous nucleation. If, according to Kaischew [4.4, 4.5], and FJ jj are... 

The rate of nucleation / is a probability process connected with the energy of formation of the critical cluster, AGo-ib which owing to the creation of the new crystal/solution and crystal/substrate interfaces, is always positive. The probability of a fluctuation connected with an increase of the Gibbs energy AG of a system is given in the case of nucleation, with AG = AGcdt, by... 

For a small number of Me atoms in a critical cluster, the strain of a 2D UPD Meads overlayer can be inherited by the nucleus. Using the atomistic approach [4.13], the rate of nucleation on top of a strongly compressed and internally strained 2D UPD Meads overlayer can be expressed by [4.54-4.57] (cf. Section 4.2) ... 

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