Homogeneous nucleation critical radius

For homogeneous nucleation, critical radius of a stable solid particle nucleus... 

When the nucleus is a liquid, the angle 6 is called tire wetting angle. It can be seen that the critical radius in heterogeneous nucleation is given by the same equation as tlrat for homogeneous nucleation, but the radius now refers... 

If we compare eqns (7.11) and (7.3) we see that the expressions for the critical radius are identical for both homogeneous and heterogeneous nucleation. But the expressions for the volume of the critical nucleus are not volume is... 

Nucleation of solids from liquids critical radius for homogeneous and heterogeneous nucleation... 

Using the properties of water Li and Cheng (2004) computed from the classical kinetics of nucleation the homogeneous nucleation temperature and the critical nu-cleation radius ra. The values are 7s,b = 303.7 °C and r nt = 3.5 nm. However, the nucleation temperatures of water in heat transfer experiments in micro-channels carried out by Qu and Mudawar (2002), and Hetsroni et al. (2002b, 2003, 2005) were considerably less that the homogeneous nucleation temperature of 7s,b = 303.7 °C. The nucleation temperature of a liquid may be considerably decreased because of the following effects dissolved gas in liquid, existence of corners in a micro-channel, surface roughness. 

The critical radius given by Eq. 19.97 is equal to the critical radius for homogeneous nucleation in the bulk liquid. This result is similar to that obtained in Exercise 19.7... 

F.M. d Heurle evaluated a specific thickness of the layers (an analogue of the critical radius of nuclei in a homogeneous system for more detail, see Ref. 31) for compounds of the Ni-Si binary system. For Ni2Si, its value was found to be 0.15 nm, i.e. the nucleus does not contain even one lattice unit. Although higher values were obtained for other nickel silicides, they never exceeded 1 nm. Therefore, the nucleation process can hardly play any significant role in the formation of most transition-metal silicides, except in some special cases. This conclusion is likely to be valid for any other chemical compound layer. It should be noted, however, that there is also a different viewpoint.38 132... 

As a particular example, one can consider the homogeneous nucleation in the pure water vapor at 25° C. The surface tension coefficient of water is a = 71.96 N/m at this temperature. Table 5.1 shows some characteristics of the new phase. When the oversaturation is p/p =8.1, the critical nucleus of 0.5 nm radius is seen to comprise 18 water molecules. The equihbrium pressure of such nuclei is not high (approximately 10 bar). Since the water vapor pressure in real clouds is usually no more than 0.1% over that of the saturated vapor, it is unrealistic to expect in the rea sonable time scale the homogeneous formation of water drops in Earth s atmosphere. 

FIGURE 7.6 Critical nudeus radius of silicon calculated by homogeneous nucleation theory for 0.2 atm and 0.7 atm. Atomic radius of silicon is 0.146 run. From Sawano [18]. 

Fig. 1. Calculated homogeneous nucleation rate and critical nuclei radius as a function of supersaturation...

Statistical molecular fluctuations within a liquid can lead to the formation of microscopic vaporlike regions that have a radius greater than the critical one for bubble growth as given by Eq. 15.15. The formation of such nuclei and their subsequent growth is referred to as homogeneous nucleation. There have been two main approaches to the calculation of the temperature required for homogeneous nucleation. These are, respectively, determination of the thermodynamic limit and the kinetic limit. 

The Kinetic Limit. To initiate homogeneous nucleation, a vapor embryo is required that has the critical radius r. In principle, the formation of just one such embryo would be sufficient to initiate the nucleation process, but in practice it is found that conditions must be such that J, the number of vapor embryos formed in a unit volume per unit time, has a high value (typically J > 1012). Carey [4] derives the following expression for J ... 

Figure 6 The volume free energy of homogeneous nucleation of undercooled water at —40°C as a function of the cluster radius at —40°C (see Equation (2)). According to Equation (3),r = 1.85 nm. The critical nucleus then contains ca. 200 molecules. The value for a was obtained from the nucleation data of Michelmore Franks ...

For homogeneous nucleation (e = 1 and 0 = 180°), as well as for a solid surface (e = 0), the value of R becomes equal to the value of critical radius expressed by Equation 10.7a in agreement with provisions of the CNT. 

Nucleation by ideal nucleants should be energetically favorable in comparison with homogeneous nucleation in the polymer matrix and heterogeneous nucleation by other additives or contaminants. This favorable energetic balance could be found in surfaces with weak interactions with the polymer matrix and with radii of curvature larger than the radius of the critical nucleus in CNT. 

Therefore, the maximum of the function is found for the same critical radius as for homogeneous nucleation ... 

The nucleation can be homogeneous in the glass or nonhomogeneous if it occurs around some impurities or from the surface. A seed, which can be defined by its radius r, evolves with time. It can either grow if its size is over the critical size, or disappear otherwise. Indeed, in the case of a homogeneous nucleation, the seed free energy AGseed can be described by two terms ... 

Figure 10.6 Schematic free-energy-versus-embryo/nucleus-radius plot on which are presented curves for both homogeneous and heterogeneous nucleation. Critical free energies and the critical radius are also shown.

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