Equilibrium properties molecular crystals

When very high pressures (> 1 GPa) are applied to liquid phases, glasses, or molecular crystals, mobility is reduced and steric effects become more important both in equilibrium and in kinetic aspects. Equations (9) and (14) are still valid, but equilibria and kinetics of chemical reactions must take into account the energetic, structural, and dynamic properties of the environment as well. 

The statistical thermodynamic method discussed here provides a bridge between the molecular crystal structures of Chapter 2 and the macroscopic thermodynamic properties of Chapter 4. It also affords a comprehensive means of correlation and prediction of all of the hydrate equilibrium regions of the phase diagram, without separate prediction schemes for two-, three-, and four-phase regions, inhibition, and so forth as in Chapter 4. However, for a qualitative understanding of trends and an approximation (or a check) of prediction schemes in this chapter, the previous chapter is a valuable tool. 

Leaving aside this dynamical problem, we can make some further remarks about the equilibrium structure of Van der Waals molecules. Some attempts have been made to predict this structure from the molecular properties, multipole moments, polarizabilities, which are reflected in the long range interactions (electrostatic, dispersion). Other authors have assumed that the equilibrium structure of Van der Waals dimers resembles the structure of nearest neighbour pairs in molecular crystals. The latter approach could possibly be justified by packing considerations (short range repulsion). An example of the first approach is the prediction of a T-shaped (0 = 90°, 0 = < ) = 0°) equilibrium structure for the Nj-dimer, mainly... 

The extensive properties of the overall system that is not in equilibrium, such as volume or energy, are simply the sums of the (almost) equilibrium properties of the subsystems. This simple division of a sample into its subsystems is the type of treatment needed for the description of irreversible processes, as are discussed in Sect. 2.4. Furthermore, there is a natural limit to the subdivision of a system. It is reached when the subsystems are so small that the inhomogeneity caused by the molecular structure becomes of concern. Naturally, for such small subsystems any macroscopic description breaks down, and one must turn to a microscopic description as is used, for example, in the molecular dynamics simulations. For macromolecules, particularly of the flexible class, one frequently finds that a single macromolecule may be part of more than one subsystem. Partially crystalhzed, linear macromolecules often traverse several crystals and surrounding liquid regions, causing difficulties in the description of the macromolecular properties, as is discussed in Sect. 2.5 when nanophases are described. The phases become interdependent in this case, and care must be taken so that a thermodynamic description based on separate subsystems is still valid. 

Even if not directly observable, intermolecular forces influence the microscopic and bulk properties of matter, being responsible for a variety of interesting phenomena such as the equilibrium and transport properties of real fluids, the structure and properties of liquids and molecular crystals, the structure and binding of Van der Waals (VdW) molecules (which can be observed under high resolution rotational spectroscopy [5-8] or molecular beam electric resonance spectroscopy [9]), the shape of reaction paths and the structure of transition states determining chemical reactions [10]. 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

In the field of polymer science, the crystallization behavior of polymer blends represents a key issue for the analysis of structure-properties relationships of macro-molecular systems. The presence of the second polymer component, either in the melt or in the solid state, can infiuence the whole crystallization process of the polymer phases, thus the morphology, phase behavior, and physical/mechanical properties. The crystallization processes are controlled by several factors, which are related to equilibrium thermodynamics, kinetic aspects, thermal conditions, melt rheology, as well as chain structure and polymer/polymer interactions. In the present chapter, an overview of the thermodynamic conditions, accompanied by a description of main morphological features of blends containing one or both crystallizable components, is reported. 

One can conclude that stepped isotherms and low pressure hysteresis loops may only be observed on monodispersed and large crystals of network types of molecular sieves. The hysteresis loop was examined in detail by scanning the region between the adsorption and the desorption branch of the hysteresis. The points betweoi the two branches were stable for many hours, thus, the low pressure hysteresis is controlled by equilibrium properties and not by kinetical effects. Previous studies provided clear evidence that the stepped isotherms of argon and nitrogen on MFI-crystals can be rationally explained by localised adsorptive molecules at the channel walls and intersections [10]. 

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