Energy of nucleation

Nucleation is initiated by local fluctuations of concentration within a metastable region. The activation energy of nucleation depends on the value of the interface energy required to create a nucleus. The droplet grows by diffusion of macromolecules into the nucleate domains. The natural form of the phase separation through NG mechanism is the sea-island type. 

Rc is found by setting d AG/dR = 0. Putting the result into Eq. 12.29 then yields the critical free energy of nucleation... 

Figure 5.1 Gibbs energy of nucleation of water drops at different water vapor oversaturations of at 25°C. Curve 1 is plotted for oversaturation p/poo = 8.1 = 0.5...

It would be beneficial to have a convenient way of unambiguously determining nucleation induction times when seeking to understand the effects of varying composition and processing conditions on nucleation, and it is essential if the induction times are used in mathematical models such the Fisher-Tumbull equation (5). In the Fisher-Tumbull model, activation energies of nucleation are calculated from nucleation induction times. The usual assumption is that the experimental... 

Exp. data Ref. Overpot. interval r] [mV] Energy of nucleation [J X 1020] Nucleus size [atoms] Overpot. interval I [mV] Nucleus size [atoms]... 

As the amount of metal cation or anion sorbed on a surface (surface coverage or loading, which is affected by the pH at which sorption occurs) increases, sorption can proceed from mononuclear adsorption to surface precipitation (a three-dimensional phase). There are several thermodynamic reasons for surface precipitate formation (1) the solid surface may lower the energy of nucleation by providing sterically similar sites (McBride, 1991) (2) the activity of the surface precipitate is less than 1 (Sposito, 1986) and (3) the solubility of the surface precipitate is lowered because the dielectric constant of the solution near the surface is less than that of the bulk solution (O Day et al., 1994). There are... 

The rate of crystallite nucleation /(/ ) is described by a (6.54)-type expression, where N and AG ff are replaced by the critical crystalline nucleus size, N, and the free energy of nucleation, AG. ... 

Fig. 3.1. Plot of free energy of nucleation, AGn, against cluster size, r. The free energy of nucleation is given by the difference between surface and bulk energy terms. Only above a critical size, r, corresponding to AGn, does nucleation become energetically favourable...

Fig. 3.9 A, B. Schematieai representation of the free energy states and the activation energy barriers for mineralization. Pathway A, crystallization of a mineral phase from pure and impure solution (with no major structural modifications) pathway B, formation of a crystalline mineral from intermediate phases of different crystal structure. AGn, free energy of nucleation AGg, free energy of growth AGx, free, energy of phase transformation...

Figure 7.2 Gibbs energy of nucleation as a function of size, is the critical nucleus or the size where the nucleus becomes stable.

The Gibbs energy of nucleation as a function of the size of the nuclei is shown in Figure 7.2. Very small clusters are instable. The formation in this size region is a statistical process. A maximum marks the critical size of the nucleus and the critical Gibbs energy of nucleation between instability and stable growths. 

In the initial phase of nucleation, the positive surface term prevails. The addition of new monomers to the cluster causes an increase in the free energy. As the volume grows faster than the surface, the favorable volume term starts to dominate the free energy expression when the nuclei exceed a certain critical size (oc)-The free energy of the nuclei with the critical size is called the activation free energy of nucleation. From this point stable growth of the cluster occurs. Every cluster below this size can be dissolved because the interactions are not enough to hold the molecules in the cluster. 

Expressions for the energy requirement of two-dimensional nucleation and the critical size of a two-dimensional nucleus may be derived in a similar manner to those for homogeneous three-dimensional nucleation (section 5.1.1). The overall excess free energy of nucleation may be written... 

The Gibbs free energy of nucleation of the shear transformation under stress... 

SAXS measurements are normally performed in situ and therefore require access to synchrotron radiation [21, 23, 27, 46]. SAXS is probably one of the most used in situ techniques for the study of zeolite synthesis. In situ cells, similar to those for WAXS or XRD measurements, can easily cope with elevated synthesis temperatures and pressures. It has been used to determine the activation energy of nucleation, where two independent studies show that it is around 70-85 kj mol [47, 48]. Also, three distinct particle sizes were observed in the reaction mixtures, being primary units of approximately 2.8 nm, their aggregates, and the actual zeolite crystals [47]. Other mechanistic studies include the effect of the precursor molecule, the chemical nature of which appears to be important in... 

The apparent agreement with theory has, however, to be taken with caution. All experiments up to now have been made in a relatively small overpotential interval and no more than one cusp, i.e., two slopes, of the log J/rfc curve has been observed. The determination of the number of atoms in the critical nucleus is also quite uninformative as a criterion for the validity of the small cluster model equation. Furthermore it does not lead to any conclusions about the free energy of nucleation or forces of interaction between the cluster atoms themselves, and eventually between them and the substrate which would be the ultimate goal of a nucleation rate study. 

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