Critical nucleus/radius

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... 

The critical nucleus corresponds to p = 1 = m, and the global free-energy minimum corresponds to infinitely large dimensions of the cylinder in both length and radius. 

Expanding the free energy of formation of a spherical nucleus of radius r [given by Eq. (1.38)] around that (F ) for a critical nucleus, we get... 

However, the dislocation is practically infinitely long compared to the size of any realistic critical nucleus. If the nucleus were of uniform radius along a long length of the dislocation, AQc would be very large. A critical nucleus will form from a local fluctuation in the form of a bulge of the cylinder associated with the metastable state A, as illustrated in Fig. 19.16. The problem is thus to find the particular bulged-out shape that corresponds to a minimum activation barrier for nucleation. 

The critical nucleus radius, Rc, is found by setting = 0, with the result... 

Englezos et al. (1987a) andEnglezos and Bishnoi (1988) determined an expression for the radius of the hydrate critical nucleus using the Gibbs free energy per unit volume of hydrate formed (Agv) in a modification of Equations 3.2a and b as... 

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. 

This is a general form of the well-known Gibbs-Thomson (Lord Kelvin) equation applied to the case of electrochemical metal deposition. It gives the size of the critical nucleus and its equilibrium form in terms of the normal distances of the equilibrium form faces from the Wulff point, hi, as a function of the overvoltage. When this form is a regular polyhedron (cr,- = const.), the size of the nucleus can be given by the radius of the inscribed sphere, perit = h, so that... 

Figure 1. Free energy required for the stability of a critical nucleus. At the lowest temperature both the energy barrier and the critical radius for a nucleus is the smallest, and both increase with increasing temperatme. At some temperature To, the critical radius is too large for a stable nucleus to form. This corresponds to a system that is not supersaturated. After Raghaven and Cohen (1975).

The position of a maximum on the W r) curve, corresponding to the work required for the formation of a critical nucleus, Wcn can be obtained by setting the derivative of the work with respect to the radius equal to zero, i.e... 

Fig. IV-2. The influence of the extent of metastability, -Ap, on the profile of the energy barrier, radius rcr, and the work of the critical nucleus formation, Wcr...

The conditions of heterogeneous nucleation, as well as the work of formation and the curvature radius of the critical nucleus, may be significantly influenced by the linear tension of the perimeter of wetting, x if x<0, the formation of heterogeneous nuclei becomes easier... 

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