Embryo critical radius

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. 

Free energy change during nucleation. The change of free energy, AG, increases with embryo size up to a critical radius, r. The critical free energy for nucleation is AG. ... 

The critical radius, r, is a radius above which an embryo will grow spontaneously. A plot of the critical radius as a function of the saturation ratio, S, is given in Figure 7.4. As the partial pressure ratio increases the critical size decreases for each value of the surface energy. 

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 ... 

The discussion above applies to heterogeneous nucleation on flat surfaces. For atmospheric particles the curvature of the surface complicates the situation. Fletcher (1958) showed that the free energy of formation AG of an embryo of critical radius r on a spherical nucleus of radius R is given by... 

It is interesting to calculate the size of the critical embryo involved in nucleation. From (4.28) at —40 °C the critical radius is 11 3 A, so that the embryo contains about 190 molecules. This result has interesting implications for the various flickering cluster theories of water structure. In order that the metastable supercooled state be maintained for reasonably long times at temperatures a few degrees above the nucleation threshold, it is necessary that... 

Fig. 8—Plot of the work required to generate an embryo as a function of its radius p at different temperatures. For T> Tc the work required is an increasing function of p so that an embryo of any radius, once generated, must ultimately shrink and disappear. For T < Tc, embryos whose radius is greater than a critical radius p can grow indefinitely until the entire sample has transformed to the ordered phase. Note that if the formation of embryos of radius p > p can be prevented, the sample can be supercooled.

Since the phenomenon of supercooling depends critically on the absence of embryos with radius p > p, it is of interest to estimate the relative probability A of thermally generating an embryo with radius p by heterophase fluctuations. We have, for T near... 

Given the composition of the critical nucleus, its radius has to be calculated. The Gibbs energy for the formation of an embryo, AG (Equation 1), reaches a maximum value AG for a particular value of R, the critical radius /f = -lyfAgly ). R ... 

Growth of the embryo does not induce a decrease in the free energy up to this critical radius r. Therefore, below this radius the embryo is unstable. It becomes a stable nucleus only above r. ... 

Figure 2 Change in Gibbs free energy, AG, generated by the creation of a solid spherical embryo in the bquid, as a function of its radius r. The critical radius r determines the limit between the domain of the unstable embryo (r < r ) and that of the stable nucleus...

Fignre 10.3 Schematic free energy-versus-embryo/nucleus-radius curves for two different temperatures. The critical free energy change (AG ) and critical nucleus radius (r ) are indicated for each temperature. 

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.

In this relationship, the excess free energy for the formation of clusters AG first increases then decreases with the radius of the particles, r. The critical radius, r, is associated with a maximum excess free energy, AG. Accordingly, there exists a critical number of atoms, , in cluster A at the radius of r. When r < r, the system can lower its free energy by dissolution of clusters. Thus, these clusters are not thermodynamically stable and dissolve quickly, whereas some new clusters form due to spontaneous collisions. These unstable particles (A < A ) are known as clusters or embryos, and their numbers follow the Boltzmann distribution and decrease exponentially with increases of AG as described in Equation 10.4. When the radius of a cluster is larger than the critical value (r> r ), it becomes stable and is referred to as a nucleus. Thus, the expressions of critical radius r and maximum excess free energy AG can be obtained mathematically when dAG,/dr is equal to zero [22, 23, 28] ... 

The embryo radius R has a critical size fhaf means fhaf all the embryos formed having a smaller size will dissolve and fhose having a higher size will grow and induce the freezing of fhe whole sample as far as pure water is concerned. As it will be shown later on, the situation is different when solutes are dissolved within the water. 

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