Thermodynamics crystal formation

Figure 3.1 Dependence of protein crystallization on the thermodynamic conditions. The frequency of crystal formation is plotted against the osmotic second virial coefficient A2. Data are taken from a broad range of proteins and crystallizing solvent conditions (George et al1997). Reproduced from Prausnitz (2003) with permission.

A fluid under a pressure lower than its vapor pressure, and especially under a negative pressure, is unstable thermodynamically with respect to formation of a bubble filled with vapors of the substance and even (in the case of negative pressure) of an empty one. A fluid subjected to negative pressure is completely analogous in this respect to a supersaturated vapor, unstable with respect to formation of a condensate, or to a supercooled liquid, unstable with respect to crystal formation. 

The fact that liquids can be supercooled, that is, maintained for extended periods of time at temperatures below their thermodynamic phase transition, indicates that kinetics must play a significant role in crystal formation. The study of nucleation in liquids is aimed at understanding what factors prevent or encourage nucleation, and what rate of nucleation one can expect under a given set of circumstances. As we shall see, nucleation typically involves changes in clusters of molecules in the liquid, with from several tens to several hundreds of molecules taking part in the key steps. These numbers are intermediate between microscopic and macroscopic, so that methods of study based on small clusters and those based on the use of the thermodynamic limit both are useful but both also have limitations. 

When a polymerization is accompanied by phase transition, the overall thermodynamic parameters are the sum of parameters of the chemical reaction and phase transition (cf. p. 11). Thus, for instance, thermodynamics of polymerization in the crystalline state from a liquid monomer will be given by the thermodynamics of formation of the amorphous (condensed) polymer and polymer crystallization, provided, that polymerization proceeds in the solid state with monomer packing in the crystalline state simultaneously with propagation. 

As to the thermodynamically controlled formation of acyclic polymeric assemblies in solution, there are only a few examples in the literature. The main problems are due to a) competition with cyclization processes b) low association constants c) low solubility of the oligomers. In fact the first reported polymeric assembly in solution made of zinc 5-(4-pyridyl)-10,15,20-triphenylporphyrin (ZnPyTPP) units [47], actually consists, as successively demonstrated, of a cyclic tetramer (Sect. 3.3). The linear polymer, however, was unambiguously detected in the solid state by X-ray analysis of a ZnPyTPP single crystal [47]. This is one of the first examples in the literature of met-alloporphyrins, illustrating the fact that the stability of an assembly may be strongly dependent on the aggregation state. 

The thermodynamic model presented above only predicts when phase separation will occur. There are, however, two mechanisms by which phase separation can actually occur. The first mechanism is similar to that discussed in an earlier chapter for precipitation of crystals from a melt, where a nucleus is formed and then grows with time. By analogy, this mechanism is termed nucleation and growth. Many of the same factors which control crystal formation also affect phase separation by this mechanism. The second mechanism is termed spinodal decomposition. This mechanism involves a gradual change in composition of the two phases until they reach the immiscibility boundary. 

Many properties of alkalide crystals, powders, and films were measured. The original alkalide, Na (C222)Na , has been most thoroughly studied, in part because of the high stability of pure samples. In contrast to most alkalides, single crystals and vapor-deposited films of this sodide are stable in vacuo for many hours, even at room temperature. In addition to the crystal structure, we measured the thermodynamics of formation by an EMF method, optical... 

The science of zeolite and molecular sieve crystallization has to be located in the more general frame of crystallization science. All thermodynamic and kinetic principles which apply and can be used for rationalization of crystallization phenomena in general, are also valid in this specific area of crystal formation. 

Another aspect of the comparison between the calculated and experimental cohesive energy is important to recall here. It is related to the difference between the definition of cohesive energy and the crystal formation energy that is reported in thermodynamic tables, the main point probably being that quantum mechanical calculations refer to the static limit (T = 0 K and frozen nuclei), whereas experiments refer to some finite temperature. In fact, the... 

This simple model accommodates the existence of rod-like structures needed for liquid crystal formation, without recourse to anisotropy of individual molecules. The secretions are biphasic (mixed isotropic/liquid crystalline) over a narrow concentration range implying that they have the thermodynamic characteristics typical of an athermal liquid crystalline solution (Figure 12.10). 

The stepwise mechanism of co-crystal formation has been interpreted in terms of kinetic and thermodynamic co-crystallisation products, wherein the finite assemblies are the kinetic product whose formation is driven by the formation of stronger N I halogen bonds. The formation of the thermodynamic product is subsequently followed by the formation of enthalpically less favourable S I halogen bonds. 

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