Energy critical radius

Figure 1.9 The balance of endothermic surface energy and the exothermic formation of the stable condensed phase during nucleation from the vapour phase. The critical radius, above which the nuclei become stable, is where the resultant Gibbs energy change has zero slope...

Figure 15. Electrocapillary energy for the formation of a breakthrough pore in a thin surface oxide film on metals as a function of pore radius.7 AE E - Epzc, where Epzc is the potential-of-zero charge of the film-free metal. Al is the activation banier for the formation of a breakthrough pore and r is its critical radius. M, metal OX, oxide film EL, electrolyte solution, h a 2 x I O 9 m, am = 0.41 J m-2, a = 0.01 J m-2, ACj= 1 F m"2. a, AE=0.89 V b, AE=0.9 V c,A = 1.0 V. (From N. Sato, J. Electmckem. Soc. 129,255,1982, Fig. 2. Reproduced by permission of The Electrochemical Society, Inc.)...

The efficiency of resonance transfer is often given in terms of a critical radius R0. If R0 is the distance such that the rate of energy transfer is equal to the sum of all other donor deactivation rates... 

The process of formation of a bubble having a critical radius, can be computed using a semiclassical approximation. The procedure is rather straightforward. First one computes, using the well known Wentzel-Kramers-Brillouin (WKB) approximation, the ground state energy Eq and the oscillation frequency //() of the virtual QM drop in the potential well U JV). Then it is possible to calculate in a relativistic framework the probability of tunneling as (Iida Sato 1997)... 

The sixth power dependence explains why resonance energy transfer is most sensitive to the donor-acceptor distance when this distance is comparable to the Forster critical radius. 

The critical radius diverges inversely with supercooling. Substitution of this result for into Eq. (1.38) leads to the free energy of formation of critical spherical nucleus as... 

Note that the critical radius is when the free energy of the cluster is at maximum, not when AGr = 0. The latter occurs when the cluster radius is 1.5r. ... 

Recall that this equation could be minimized with respect to particle radius to determine the critical particle size, r, as given by Eq. (3.35). This critical radius could then be used to determine the height of the free energy activation energy barrier, AG, as given by Eq. (3.36). A similar derivation can be performed for a cubic particle with edge length, a. 

The critical radius, R is the size where the embryo (nucleus) has a 50 50 chance of either redissolving or growing into a stable nucleus it is determined by the balance between the surface energy required to form the embryo,... 

This balance is shown in Fig. 1.1. The typical size of Rc is about 100 molecules— between 1 and 2 nm in diameter. Solvent molecules can adsorb on the embryos and change their surface energy the critical radius will therefore depend not only on the material of the nucleating phase but also on the solution phase. 

Fig. 1.1 Energetics of nucleation. The critical radius, Rc, depends on the balance between surface and volume energies of the growing particle.

The critical radius given by Eq. 19.92 is equal to the critical radius for homogeneous nudeation in the bulk liquid. This is the expected result because 7LM = ySM (so that the liquid/solid interface makes an angle of 90° with the mold) and the inward pressure on the interface due to curvature, AP = 2-yLS/R (Eq. 12.4), is then exactly balanced by the change in bulk free energy across the interface, jjphase trans — gB (Eq. 12.1). Substitution of Eq. 19.92 into Eq. 19.91 yields the critical free energy for nudeation ... 

When a nucleus has reached a radius of r, farther growth lowers the Gibbs energy of the total system. To a first-order approximation, AGB depends linearly on temperature. As the temperature decreases below the equilibrium value, the absolute value of AGB increases and the critical radius decreases. 

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