Critical nucleus volume

Classic nucleation theory must be modified for nucleation near a critical point. Observed supercooling and superheating far exceeds that predicted by conventional theory and McGraw and Reiss [36] pointed out that if a usually neglected excluded volume term is retained the free energy of the critical nucleus increases considerably. As noted by Derjaguin [37], a similar problem occurs in the theory of cavitation. In binary systems the composition of the nuclei will differ from that of the bulk... 

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 classical model is an obvious approximation since the interface may be significantly diffuse and occupy a substantial fraction of the cluster volume. A clean separation between bulk and interfacial energies therefore becomes problematic. The classical model for the critical nucleus may be expected to be a reasonable... 

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... 

We are assuming As/ and Ahf don t vary much with temperature. Remember that this is just the bulk free energy (per unit volume) of the crystal, it does not include any additional free energy from those segments at the surfaces. We can now return to our expression for the critical nucleus size (thickness), which was given previously in Equation 10-25, and substituting for Ag, we obtain Equation 10-30 ... 

Here the first term is the surface free energy, proportional to the surface area of the crystallite, while the second is the bulk free energy, proportional to its volume. The first term is always positive because of the work that must be done to create an interface, while the second is negative in the supercooled liquid, because the solid has the lower free energy under these circumstances. It is clear from the form of this equation that the free energy of the crystallites increases up to a size i, which is called the critical nucleus and represents a barrier that must be surmounted on the path from pure liquid to pure crystal. 

W is the Gibbs energy required to form a critical nucleus and AG d is the activation energy for rearrangement. Vm is the molar free volume of the crystal nv is the... 

Analogously to the previous case, one can discuss the process of the separation of the solid or liquid phase (of molar volume Vm) out of a solution with supersaturation a=clc0, where c and c0 are the concentrations of the supersaturated and saturated solutions, respectively. In the case of the ideal solution, the expression for the work of critical nucleus formation can be written as... 

One may expect (a more detailed derivation will be given below) that eq.(IV. 15) remains valid for a heterogeneous nucleus, i.e. the work of a critical nucleus formation is proportional to the nucleus volume. Then the work of heterogeneous formation of critical nucleus, Wcrhe equals the work of homogeneous formation of a critical nucleus, Wcrhom, multiplied by the ratio of nuclei volumes, i.e. by the value of/(0), namely... 

Consequently, one can regard the pre-exponential factor in eq. (IV. 18), J0, as the quantity determined by the ratio of the number of molecules per unit volume in the metastable phase, n0, to the critical nucleus life time, lcr... 

Ti-wadeite, but structural similarity in the case of Zr-wadeite. These differences/ similarities influence the energetics of nucleation, with the result that it is more difficult to form a critical nucleus of Ti-wadeite. Vessal and Dickinson (1994) have undertaken both constant volume and constant pressure molecular dynamics simulations to test this hypothesis. 

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 ...

Alternatively, the volume of the critical nucleus may favour the amorphous state. The number of ions in the critical nucleus will be of the order 10-1000 resulting in a cluster size 5-20 A. If the stable critical nucleus is smaller than the dimensions of the unit cell of a potential crystalline phase, for example, the large unit cell of hydroxyapatite, then since the nucleus grows without reference to the basic structural unit of the crystalline phase the less-ordered atomic arrangement of the amorphous phase may be favoured. 

Nucleation is a rate phenomenon therefore, even though the condition oc > c is necessary, it is nonetheless insufficient to bring about nucleus formation. An additional requirement must be satisfied. That is, in order for cluster formation to proceed to completion, the monomers that represent potential candidates for a given critical nucleus must be retained for a sufficiently long period of time within a volume of molecular dimensions. If intermicellar exchange proceeds extremely rapidly, then the candidate monomers will be redistributed before nucleation can occur. Accordingly, it may be deduced that an increase in the communication rate will decrease the probability of nucleus formation. The expected trend of number of nuclei versus the rate of intermicellar exchange (A ex) is illustrated schematically in Fig. 5d. 

With the volume of the critical nucleus = (47r/3)r j one can also calculate the number of atoms in the critical nucleus by dividing the volume of Np critical nuclei Avogadro constant) through the molar volume of the metal V. The result is... 

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