Embryo free energy

The work of Reiss and co-workers puts the question of the equilibrium distribution of liquid embryos in dilute supercooled vapors on sound conceptual ground. However, having to calculate embryo free energies by simulation rules out the use of such an approach in practical applications. To overcome this limitation, Weakliem and Reiss [67] developed a modified liquid drop theory that combines elements of the physically consistent cluster with the conventional capillarity approximation. These same authors have also developed a rate theory which allows the calculation of nucleation rates in supercooled vapors [68]. The dependence of the predicted rates on supersaturation agree with classical nucleation theory, but the temperature dependence shows systematic deviations, in accordance with scaling arguments [54]. 

Later on Cahn and Hilliard presented some thermodynamic estimates for the nucleation of liquid in vapour. Values of AO and the composition profiles c(r) of the embryos have been estimated using the mean-field and gradient expansion approximations for the free energy functional F c(7 ). A number of qualitative features in variation... 

After certain manipulations, the rhs of Eq. (25) can be reduced to the free energy F c for a certain uouuniform alloy and can be calculated using the methods mentioned in Sec. 6. Similar microscopic treatment can be performed for the embryo diffusion kinetics in the a-space. 

A central assertion of homogeneous nucleation theory is that interfacial free energy costs induce a spherical symmetry in the phase embryo. However, these simulation studies indicate that inter molecular interactions may not permit the development of spherical symmetry when these interactions are strong and highly asymmetric. 

Any surfactant adsorption will lower the oil-water interfacial tension, but these calculations show that effective oil recovery depends on virtually eliminating y. That microemulsion formulations are pertinent to this may be seen by reexamining Figure 8.11. Whether we look at microemulsions from the emulsion or the micellar perspective, we conclude that the oil-water interfacial free energy must be very low in these systems. From the emulsion perspective, we are led to this conclusion from the spontaneous formation and stability of microemulsions. From a micellar point of view, a pseudophase is close to an embryo phase and, as such, has no meaningful y value. 

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. ... 

Colloidal particles are formed from a homogeneous medium by the clustering of smaller units to form "embryos" of various sizes. In the case of polymers in solution the aggregates may be of repeat units of the same or different molecules. The specific surface area of such embryos is very great, and its creation requires the expenditure of an amount of energy equal to the area, A, times the interfacial free energy, y ... 

AGv and AG, both expressed per unit volume of polymer. For isotropic, amorphous polymer particles of radius r, the free energy of formation of an embryo will be... 

Consider the energy balance of a nucleating (or condensing) drop. As the droplet (or embryo) is formed, its surface free energy goes from 0 to nrd2y, where d is the diameter of the drop and y is the liquid surface tension. If the free-energy potential per molecule is < >a in the vapor... 

Using these free energy concepts, the equilibrium number density of embryos of size r is given by... 

Since the total change in Gibbs free energy in this process of embryo creation is... 

Here, if o-23 is the interfacial free energy between the foreign substrate and the embryo phase, one has an expression ... 

Figure 4-3 Contributions to Gibbs free energy for homogeneous embryo formation.

Fig. 6.1. Free energies of crystalline (curves 1) and non-crystalline (curves 2) nuclei vs the number of atoms in the nucleus for glass-forming (a, b) and nonglass-forming melts (c). N and IV 1 are the critical sizes of embryos...

Embryos. The excess free energy of a spherical embryo of radius r over that of the same volume of phase a is given by... 

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