Crystalline solids, modelling

Students will describe crystalline-solid models as three-dimensional works of art and determine if they are solid works, linear works, or a combination of both. 

Describe the crystalline-solid models prepared in this activity as three-dimensional works of art that are solid (massive), linear (made mostly of thin, line-like parts), or both, explaining why they are solid or linear works of art. 

The present chapter is organized as follows. We focus first on a simple model of a nonuniform associating fluid with spherically symmetric associative forces between species. This model serves us to demonstrate the application of so-called first-order (singlet) and second-order (pair) integral equations for the density profile. Some examples of the solution of these equations for associating fluids in contact with structureless and crystalline solid surfaces are presented. Then we discuss one version of the density functional theory for a model of associating hard spheres. All aforementioned issues are discussed in Sec. II. 

The singlet-level theories have also been applied to more sophisticated models of the fluid-solid interactions. In particular, the structure of associating fluids near partially permeable surfaces has been studied in Ref. 70. On the other hand, extensive studies of adsorption of associating fluids in a slit-like [71-74] and in spherical pores [75], as well as on the surface of spherical colloidal particles [29], have been undertaken. We proceed with the application of the theory to more sophisticated impermeable surfaces, such as those of crystalline solids. 

Applications The general applications of XRD comprise routine phase identification, quantitative analysis, compositional studies of crystalline solid compounds, texture and residual stress analysis, high-and low-temperature studies, low-angle analysis, films, etc. Single-crystal X-ray diffraction has been used for detailed structural analysis of many pure polymer additives (antioxidants, flame retardants, plasticisers, fillers, pigments and dyes, etc.) and for conformational analysis. A variety of analytical techniques are used to identify and classify different crystal polymorphs, notably XRD, microscopy, DSC, FTIR and NIRS. A comprehensive review of the analytical techniques employed for the analysis of polymorphs has been compiled [324]. The Rietveld method has been used to model a mineral-filled PPS compound [325]. 

V. WATER SORPTION BY CRYSTALLINE SOLIDS A. General Model... 

Jantzen, C.M. and Plodinec, M.J. (1984). Thermodynamic model of natural, Medieval and nuclear waste glass durability. Journal of Non-Crystalline Solids 67 207-223. 

The period under review has seen a small, but apparently real, decrease in the annual number of publications in the field of the vibrational spectroscopy of transition metal carbonyls. Perhaps more important, and not unrelated, has been the change in perspective of the subject over the last few years. Although it continues to be widely used, the emphasis has moved from the simple method of v(CO) vibrational analysis first proposed by Cotton and Kraihanzel2 which itself is derived from an earlier model4 to more accurate analyses. One of the attractions of the Cotton-Kraihanzel model is its economy of parameters, making it appropriate if under-determination is to be avoided. Two developments have changed this situation. Firstly, the widespread availability of Raman facilities has made observable frequencies which previously were either only indirectly or uncertainly available. Not unfrequently, however, these additional Raman data have been obtained from studies on crystalline samples, a procedure which, in view of the additional spectral features which can occur with crystalline solids (vide infra), must be regarded as questionable. The second source of new information has been studies on isotopically-labelled species. 

The crystal field theory (CFT) was developed for crystalline solids by the physicist Hans Bethe in 1929. The model takes into account the distance separating the positively and... 

The model fundamental to all analyses of vibrational motion requires that the atoms in the system oscillate with small amplitude about some defined set of equilibrium positions. The Hamiltonian describing this motion is customarily taken to be quadratic in the atomic displacements, hence in principle a set of normal modes can be found in terms of these normal modes both the kinetic energy and the potential energy of the system are diagonal. The interaction of the system with electromagnetic radiation, i.e. excitation of specific normal modes of vibration, is then governed by selection rules which depend on features of the microscopic symmetry. It is well known that this model can be worked out in detail for small molecules and for crystalline solids. In some very favorable simple cases the effects of anharmonicity can be accounted for, provided they are not too large. 

Fig. 14.1 Model of a solid with cores at fixed lattice positions and valence electrons free to move throughout the crystalline solid.

The four-parameter model is very simple and often a reasonable first-order model for polymer crystalline solids and polymeric fluids near the transition temperature. The model requires two spring constants, a viscosity for the fluid component and a viscosity for the solid structured component. The time-dependent creep strain is the summation of the three time-dependent elements (the Voigt element acts as a single time-dependent element) ... 

The interest in efficient optical frequency doubling has stimulated a search for new nonlinear materials. Kurtz 316) has reported a systematic approach for finding nonlinear crystalline solids, based on the use of the anharmonic oscillator model in conjunction with Miller s rule to estimate the SHG and electro optic coefficients of a material. This empirical rule states that the ratio of the nonlinear optical susceptibility to the product of the linear susceptibilities is a parameter which is nearly constant for a wide variety of inorganic solids. Using this empirical fact, one can arrive at an expression for the nonlinear coefficients that involves only the linear susceptibilities and known material constants. 

The quasi-chemical model was derived by Guggenheim for application to organic fluid mixtures. Applying it to crystalline solids is not immediate, because it necessitates conceptual modifications of operative parameters, such as the above-mentioned contact factor. Empirical methods of derivation of the above parameters, based on structural data, are available in the literature (Green, 1970 Sax-ena, 1972). We will not treat this model, because it is of scanty application in geochemistry. More exhaustive treatment can be found in Guggenheim (1952) and Ganguly and Saxena (1987). 

Molecular conformation is highly related to functional properties. Since the conformation of the crystalline solids can be precisely determined by diffraction methods, molecular modeling is most important for interpreting molecular structures in solution. This is, however, even more difficult for theoreticians. While carbohydrates dissolve in a variety of solvents, the important solvent for biological systems is water and this solvent deserves special emphasis. 

Among the several transition alumina phases, y-Al203 is the most important and most studied phase for catalysis [57, 58]. However, even nowadays, several aspects of its structural and surface chemistry are still not well understood, since y-Al203 is a poorly crystalline solid, showing some variation in its structural stoichiometry and a wide range of defects. In the last 50 years, several empirical models for y-AI2O3 surface have been reported, trying to explain the complexity of this surface... 

Doremus, R. H. 2000. Diffusion of water in rhyolite glass Diffusion-reaction model. Journal of Non-Crystalline Solids, 261, 101-107. 

The spring model suggests that the symmetry of the crystal lattice determines the different forms of the dielectric tensor that is, they are related to the seven types of crystalline solid (amorphous solids and most liquids are isotropic). This is summarized as follows ... 

One of the simplest cases of phase behavior modeling is that of solid—fluid equilibria for crystalline solids, in which the solubility of the fluid in the solid phase is negligible. Thermodynamic models are based on the principle that the fugacities (escaping tendencies) of component iyf y are equal for all phases at equilibrium under constant temperature and pressure (51). The solid-phase fugacity, ff, can be represented by the following expression at temperature T ... 

An early method of describing electrons in crystals was the method of nearly free electrons we shall refer to it as the NFE model. In this the potential energy V(x, y, z) in (6) is treated as small compared with the electron s total energy . This is, of course, never the case in real crystals the potential energy near the atomic core is always large enough to produce major deviations from the free-electron form. Therefore, until the introduction of the concept of a pseudopotential , it was thought that the NFE model was not relevant to real crystalline solids. 

We turn now to the interaction energy e2/r12 between electrons and consider first its effect on the Fermi surface. The theory outlined until this point has been based on the Hartree-Fock approximation in which each electron moves in the average field of all the other electrons. A striking feature of this theory is that all states are full up to a limiting value of the energy denoted by F and called the Fermi energy. This is true for non-crystalline as well as for crystalline solids for the latter, in addition, occupied states in fc-space are separated from unoccupied states by the "Fermi surface . Both of these features of the simple model, in which the interaction between electrons is neglected, are exact properties of the many-electron wave function the Fermi surface is a real physical quantity, which can be determined experimentally in several ways. 

R.L. Coble. Sintering crystalline solids I. Intermediate and final state diffusion models. J. Appl. Phys., 32(5) 787, 1961. 

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