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The purpose of the preceding two chapters was to stress the link between the “chemical” nature of the first stages of formation of embryos, and the “phase” nature of their further growth. It has been shown, through one example of homogeneous nucleation of solid or liquid clusters from the vapor in Chap. 10, that the results of both chemical and phase approaches are identical, except for the constant contribution of B = −k B T In Q rep to the classical nucleation work, ΔG cl*, due to cluster mobility in these particular cases. While similar considerations concerning the influence of entropy in other specific cases of nucleation are scarce, an agreement exists as to the calculation of ΔG cl* which may be performed currently with the methods of classical thermodynamics. This will be the subject of the next two chapters*.

Condensation of a vapor on a liquid surface is a process occurring without any difficulty even at the smallest supersaturation. If condensation takes place on a solid of the same chemical species, crystal growth may be more or less hindered depending on the type of the face, as will be discussed in the fourth part of this book.

The physically adsorbed or physisorbed layersconsidered thus far are excellent model systems for understanding the thermodynamics of phase equilibria between two- and three-dimensional phases. The relatively weak van der Waals type of bindingof the adsorbed molecules among themselves and with the substrate, as in the cases of rare gases on graphite, results in a high exchange frequencywith the vapor and assures reversibility. The substrate is flat, due to the lamellar structure of graphite, and to the strong covalent bonds within the (0001)-lattice plane, which offers one type of adsorption site.

In Sect. 2.8 we have already discussed the way of calculating the thermodynamic parameters and state functions of surface phases, which are the adsorption layers. The Gibbs definition is, however, too general to be used in many cases of practical interest. For example, it supposes that the excess of matter in the surface phase comes from both phases touching at the interface. This is not the case, e.g., of adsorption without chemical reaction of a gas on a foreign solid substrate. Furthermore, according to the Gibbs definition, the adsorption layer is not limited in thickness, although its concentration must tend to that of the bulk phase with increasing distance from the surface. In fact, as we shall see in the case of the gas-solid surface at not too high a temperature, adsorption is limited to a monomolecular layer until its completion. Subsequently this condensed layer acts as a substrate, i.e. a new monomolecular layer can be formed on top of it, and so on. It should be pointed out, however, that in other systems, e.g. aqueous solutions, where the concentration of the adsorbate in the bulk phase could be much higher than that in a gas, adsorption layers can be thicker, their diffuse parts extending far beyond the molecular dimension.

It is hard to imagine that a crystal of macroscopic dimension, obtained in real growth conditions, could be perfect. Imperfections of different kinds, from point defects and stacking faults to edge and screw dislocations, are almost unavoidably present in a crystal lattice. They can be introduced by external factors, such as impurities or foreign particles present in the mother phase and engulfed in the crystal during growth, by thermal stress due to temperature differences or quenching, and may even be a consequence of the growth mechanism itself, as is the inclusion and condensation of lattice vacancies, the incoherent encounter of growth fronts, etc. [211, 220]. In rare cases, nearly perfect crystals are obtained by pulling material from the melt in precise crystallographic directions that favor the climb of the dislocations and their elimination at the ingot surface [221].

The condition for equilibrium in a heterogeneous system including an anisotropic solid and a vapor, may be written, by analogy with (8.1.1), in the most general form as $$ dG = {\mu ^{\left( g \right)}}dN + {\mu ^{\left( c \right)}}\frac{{dV}}{{{\upsilon _c}}} + \sum\limits_j {{\gamma _i}d{A_j}} = 0, $$ where V and v c are the volumes of the entire solid and of one molecule in the crystal lattice, respectively, while γ j and A j are the surface tension and the area of the j-th face. Capillarity approximation is implicitly admitted through the assumptions μ (c) = μ (c) ∞ and γ j = const. The expression is, therefore, valid for crystal sizes i 0 < i < ∞ (cf. Sect. 8.1).

Both Becker-Doring’s and Frenkel-Zeldovich’s treatments, considered thus far, used a theoretical artifice allowing the maintenance of a steady state, i.e. assuring a constant nucleation rate for an infinitely long period of time. In practice, when supersaturation is installed in a system, the size distribution of clusters first changes from equilibrium to that of a metastable state, which further breaks down because of the formation of overcritical nuclei. During their growth, nuclei consume matter from the mother phase, thus lowering the supersaturation. The nucleation rate decreases and finally falls to zero.

In the two preceding chapters we have shown the crucial role of surface free energy in the thermodynamic properties of large two-dimensional phases. This role is just as important, although quite different, in the case of small phases.

We shall limit our further considerations almost exclusively to the cases of growth and nucleation, from a vapor considered as an ideal gas, of a crystal considered as an Einstein solid in the high temperature range (k B T » hv) For this reason, we will focus here our attention on the equilibrium between those two large phases. Good comprehension of the phenomena in their simplest form should allow the interested reader to make the transposition to more complicated systems, for which also abundant specialized literature exists.

Although the idea of considering nucleation as a chemical reaction of polymerization followed almost immediately [12] upon the Volmer-Weber phase approach, the assimilation of polymer molecules to embryos of the new phase persists till now. The Gibbs thermodynamic analysis of the equilibria in heterogeneous systems, and the Wulff approach to the equilibrium shape of small anisotropic phases, have been too successful to be easily abandoned, even when the models, developed by Farkas [12], Kaischew and Stranski [10], and Becker and Doring [11] were similar to those for chemical chain reactions. This is perhaps the reason why the thermodynamics of these processes, developed mainly in the works of Frenkel [13] and Zeldovich [14] came later than the understanding of their kinetics. Such an anomaly had its perverse effects. As we shall see, Frenkel first found the right expression for the equilibrium size distribution of polymer molecules, and then replaced it by an erroneous one, in order to stick to the Becker-Doring kinetic result. One may regret that this error was at the origin of many lasting controversies concerning the role of entropy factors in nucleation kinetics, commonly known as Lothe-Pound replacement factors.

In Chap. 2 on thermodynamics we showed the difference between surface tension, γ, and surface free energy, f (s) [cf. (2.8.5)]. While γ represents pure mechanical work that must be done to increase the area of an interface, f (s) contains “chemical” terms, µ i Γ i , and its value at the equilibrium depends on the state of the bulk phases from both sides of the interface, via the chemical potentials. At the equilibrium between the F-face of a crystal and the vapor of only one chemical component, the chemical terms are due to the presence of admolecules and holes, as considered in Chap. 4, which build self-adsorbed layers with thickness increasing with temperature, up to a complete roughening of the surface. At temperatures much lower than that of the roughening transition, however, one can neglect in a first approximation the influence of self-adsorption on the surface free energy, and consider that the latter is identical to the surface tension. For this reason, the reader should not be confused if we use both terms to designate γ.