Statistical thermodynamics ordered structures

Recently there has emerged the beginning of a direct, operational link between quantum chemistry and statistical thermodynamic. The link is obtained by the ability to write E = V Vij—namely, to write the output of quantum-mechanical computations as the standard input for statistical computations, It seems very important that an operational link be found in order to connect the discrete description of matter (X-ray, nmr, quantum theory) with the continuous description of matter (boundary conditions, diffusion). The link, be it a transformation (probably not unitary) or other technique, should be such that the nonequilibrium concepts, the dissipative structure concepts, can be used not only as a language for everyday biologist, but also as a tool of quantitation value, with a direct, quantitative and operational link to the discrete description of matter. 

Statistical thermodynamics can provide explicit expressions for the phenomenological Gibbs energy functions discussed in the previous section. The statistical theory of point defects has been well covered in the literature [A. R. Allnatt, A. B. Lidiard (1993)]. Therefore, we introduce its basic framework essentially for completeness, for a better atomic understanding of the driving forces in kinetic theory, and also in order to point out the subtleties arising from the constraints due to the structural conditions of crystallography. 

In the following, we first describe (Section 13.3.1) a statistical mechanical formulation of Mayer and co-workers that anticipated certain features of thermodynamic geometry. We then outline (Section 13.3.2) the standard quantum statistical thermodynamic treatment of chemical equilibrium in the Gibbs canonical ensemble in order to trace the statistical origins of metric geometry in Boltzmann s probabilistic assumptions. In the concluding two sections, we illustrate how modern ab initio molecular calculations can be enlisted to predict thermodynamic properties of chemical reaction (Sections 13.3.3) and cluster equilibrium mixtures (Section 13.3.4), thereby relating chemical and phase thermodynamics to a modem ab initio electronic structure picture of molecular and supramolecular interactions. 

Statistical thermodynamic treatments of defect populations have lead to an explanation of existence of grossly nonstoichio-metric crystals in terms of microdomains of ordered structure. The model considers that the nonstoichiometric matrix is made up of a mosaic of small regions of ordered defect-free structures, the microdomains. To account for stoichiometric variation, one can postulate that at least two microdomains with different compositions occur. However, compositional change might simply arise at the surface of the domain. For example, if there are compositionally identical microdomains, one of which is bounded by an anion surface and one by a cation surface, variation in the two populations can give rise to compositional variation. In a strict sense, as each microdomain is ordered, the concept of a defect is redundant, except for... 

In order to assign the Raman bands and determine the absolute oceupancies of O2 and N2 molecules in the small and large cavities, we use the statistical thermodynamic expression derived by van der Waals and Platteeuw. " Let us consider an equilibrium state of the ice-hydrate-gas system. Then, the difference between the chemical potential of water molecules in ice, //h 0), and that in a hypothetical empty lattice of structure II hydrate, ju (h ), is given by... 

On the other hand, solids are characterized by a very ordered structure in which each ion or molecule is surrounded by a fixed number of neighbors whose nature and orientation are determined by the interparticle forces in the crystal. These may be chiefly ion-ion interactions, as in an ionic crystal, or intermolecular forces, as in a molecular crystal. Because of the high state of order in crystals it is a reasonably straightforward problem to calculate their thermodynamic properties on the basis of quite simple statistical mechanical models. 

Tbmer, J.S. Nonequilibrium thermodynamics, dissipative structures and biological order. In Schieve, W.C., Turner, J.S. (eds.) Lectures in Statistical Physics, Lecture Notes in Physics 28, Springer, Berlin (1974)... 

These conclusions have been strengthened by an analysis of suitable correlation functions and structure factors [99]. These results show (Fig. 31) that a cylindrical bottle brush is a quasi-lD object and, as expected for any kind of ID system, from basic principles of statistical thermodynamics, statistical fluctuations destroy any kind of long-range order in one dimension [108]. Thus, for instance, in the lamellar structure there cannot be a strict periodicity of local composition along the z-axis, rather there are fluctuations in the size of the A-rich and B-rich domains as one proceeds along the z-axis, these fluctuations are expected to add up in a random fashion. However, in the molecular dynamics simulations of Erukhimovich et al. [99] no attempt could be made to study such effects quantitatively because the backbone contour length L was not very large in comparison with the domain size of an A-rich (or B-rich, respectively) domain. 

Two sets of methods for computer simulations of molecular fluids have been developed Monte Carlo (MC) and Molecular Dynamics (MD). In both cases the simulations are performed on a relatively small number of particles (atoms, ions, and/or molecules) of the order of 100simulation supercell. The interparticle interactions are represented by pair potentials, and it is generally assumed that the total potential energy of the system can be described as a sum of these pair interactions. Very large numbers of particle configurations are generated on a computer in both methods, and, with the help of statistical mechanics, many useful thermodynamic and structural properties of the fluid (pressure, temperature, internal energy, heat capacity, radial distribution functions, etc.) can then be directly calculated from this microscopic information about instantaneous atomic positions and velocities. 

The melting, or dissolution, of long chain molecules at high dilution is a natural consequence of phase equilibrium. The dissolution process results in the separation of the solute molecules and is usually accompanied by a change in the molecular conformation of the chain from an ordered structure to a statistical coil. However, it is also possible for the individual polymer molecules to maintain the conformation in solution that is typical of the crystalline state. This is particularly true if the steric requirements that favor the perpetuation of a preferred bond orientation or the ordered crystalline structure can be maintained by intramolecular bonding, such as hydrogen bonds. Further alterations in the thermodynamic environment can cause a structural transformation in the individual molecules. Each molecule is then... 

When individual, isolated molecules exist in helical, or other ordered forms, environmental changes, either in the temperature or solvent composition, can disrupt the ordered structure and transform the chain to a statistical coil. This conformational change takes place within a small range of an intensive thermodynamic variable and is indicative of a highly cooperative process. This reversible intramolecular order-disorder transformation is popularly called the helix-coil transition. It is an elementary, one-dimensional, manifestation of polymer melting and crystallization. 

The present volume was suggested and stimulated by the aforementioned thoughts. We shall be concerned here with the phenomena and problems associated with the participation of macromolecules in phase transitions. The term crystallization arises from the fact that ordered structures are involved in at least one of the phases. The book is composed of three major portions which, however, are of unequal length. After a deliberately brief introduction into the nature of high polymers, the equilibrium aspects of the subject are treated from the point of view of thermodynamics and statistical mechanics, with recourse to a large amount of experimental observation. The second major topic discussed is the kinetics of crystallization. The treatment is intentionally very formal and allows for the deduction... 

After having considered the structural behavior of single chains we turn now to the collective properties of polymers in bulk phases and discuss in this chapter liquid states of order. Liquid polymers are in thermal equilibrium, so that statistical thermodynamics can be applied. At first view one might think that theoretical analysis presents a formidable problem since each polymer may interact with many other chains. This multitude of interactions of course can create a complex situation, however, cases also exist, where conditions allow for a facilitated treatment. Important representatives for simpler behavior are melts and liquid polymer mixtures, and the basic reason is easy to see As here each monomer encounters, on average, the same surroundings, the chain as a whole experiences in summary a mean field , thus fulfilling the requirements for an application of a well established theoretical scheme, the mean-field treatment . We shall deal with this approach in the second part of this chapter, when discussing the properties of polymer mixtures. 

The symmetry and structure of the smectic-A liquid crystals are reviewed the natural order parameters are identified. The relationship of the smectic-A phase to the nematic (or cholesteric) and isotropic phases in homologous series is also examined. The McMillan form of the single molecule potential function is then deduced starting from the Kabayashi form of the potentiaP and using the formal development presented earlier. The derivation of the statistical thermodynamics then follows, along with a presentation of McMillan s numerical results and a comparison with experiment. Improvements in the theory introduced by Lee et al are also considered. In the last section, the important question of whether the smectic-A to nematic (cholesteric) phase transition can ever be second order is examined. 

Arg unent supportii the presence of ordered structure in ionic solutions was also advaiaed, albeit in an indirect manner, from thermodynamic properties. In 1959, Frank and Thompson [15] pointed out on the basis of statistical-mechanical considerations that the Debye-Huckel formalism [16] of simfde electrolyte solutions breaks down at a concentration mo, where the Debye... 

NNs over the entire range of film compositions investigated. This preference is partially reversed for a distance of 2 NN, where a small local maximum is visible for the surface with 25% Pd, indicating a preference for Pd atoms separated by 2 NN units, as they are present in the (2 x 2) structure on the (111) plane of ordered CujPd bulk alloys. In an SRO analysis, such a (2 x 2) structure would generate values of a(2) = l and 0 (r) = —1/3 for r 2. In the STM-based data, however, the variations of a(r) are much less enhanced, and for r>2, they lie within the margins of the statistical uncertainty. Even prolonged anneahng at lower T was shown to not yield any periodic superstmcture [57]. Chapter 11 will include a thermodynamic description of the lateral atom distribution, and it wiU show why ordered structures are experimentally inaccessible to most surface alloys. 

---