Structural lattice stability

One of the most important parameters that defines the structure and stability of inorganic crystals is their stoichiometry - the quantitative relationship between the anions and the cations [134]. Oxygen and fluorine ions, O2 and F, have very similar ionic radii of 1.36 and 1.33 A, respectively. The steric similarity enables isomorphic substitution of oxygen and fluorine ions in the anionic sub-lattice as well as the combination of complex fluoride, oxyfluoride and some oxide compounds in the same system. On the other hand, tantalum or niobium, which are the central atoms in the fluoride and oxyfluoride complexes, have identical ionic radii equal to 0.66 A. Several other cations of transition metals are also sterically similar or even identical to tantalum and niobium, which allows for certain isomorphic substitutions in the cation sublattice. 

In this context it is interesting to note that archaea, which possess S-layers as exclusive cell wall components outside the cytoplasmic membrane (Fig. 14), exist under extreme environmental conditions (e.g., high temperatures, hydrostatic pressure, and salt concentrations, low pH values). Thus, it is obvious one should study the effect of proteinaceous S-layer lattices on the fluidity, integrity, structure, and stability of lipid membranes. This section focuses on the generation and characterization of composite structures that mimic the supramolecular assembly of archaeal cell envelope structures composed of a cytoplasmic membrane and a closely associated S-layer. In this biomimetic structure, either a tetraether... 

It should be remembered that the CALPHAD approach is based on the hypothesis that, for all the phases and structures existing across the complete alloy system, entire Gibbs energy vs. composition curves may be constructed even by extrapolation into regions where they are unstable or metastable. A particular case concerns the pure component elements for which the relative Gibbs energy for the different crystal structures (the so-called lattice stabilities) must also be established and defined as a function of temperature (and pressure). 

Element Temperature Stability range/°C Pressure/GPa Crystal structure Lattice parameters/pm Atomic volume pm3/106 Molar volume/ cm3/mol Density/ g/cm3... 

Lattice stabilities of Ca, Sr andYb disilicides have been reported by Brutti etal. (2006). Structural stabilities of six different prototype lattices have been investigated and discussed. 

Chapter 6 therefore deals in detail with this issue, including the latest attempts to obtain a resolution for a long-standing controversy between the values obtained by thermochemical and first-principle routes for so-called lattice stabilities . This chapter also examines (i) the role of the pressure variable on lattice stability, (ii) the prediction of the values of interaction coefficients for solid phases, (iii) the relative stability of compounds of the same stoichiometry but different crystal structures and (iv) the relative merits of empirical and first-principles routes. 

This ensures that thermochemical (TC) lattice stabilities are firmly anchored to the available experimental evidence. Although the liquid phase might be considered a common denominator, this raises many problems because the structure of liquids is difficult to define it is certainly not as constant as popularly imagined. It is therefore best to anchor the framework for lattice stabilities in the solid state. 

It is clearly desirable to see if the total curve can be de-convoluted into parts that can be identified with a specific physical property so that trends can be established for the many cases where data for metastable structures are not experimentally accessible. In principle, the TC lattice stability of an element in a specified crystal structure 0 relative to the standard state a can be comprehensively expressed as follows (Kaufman and Bernstein 1970) ... 

With the exception of a few allotropic elements, the necessary input parameters to Eqs (6.1) or (6.2) are not available to establish the lattice stabilities of metastable structures. Therefore an alternative solution has to be found in order to achieve the desired goal. This has evolved into a standard format where the reference or ground state Gibbs energy is expressed in the form of genera] polynomials which reproduce assessed experimental Cp data as closely as possible. An example of such a standard formula is given below (Dinsdale 1991) ... 

The increasing availability of electron energy calculations for lattice stabilities has produced alternative values for enthalpy differences between allotropes at 0 K which do not rely on the various TC assumptions and extrapolations. Such calculations can also provide values for other properties such as the Debye temperature for metastable structures, and this in turn may allow the development of more physically appropriate non-linear models to describe low-temperature Gibbs energy curves. 

Comparison between FP and TC lattice stabilities Despite the variety of assumptions that have been used, some general trends for die resultant lattice stabilities have been obtained for various crystal structures across the periodic table. The mean values of such (FP) lattice stabilities can therefore be compared with the equivalent values determined by thermochemical (TC) methods. Such a comparison shows the following irrqiortant features (Miodownik 1986, Watson et al. 1986, Saunders et al. 1988, Miodownik 1992) ... 

A sinusoidal variation of the 0 K energy difference between b.c.c. and close-packed structures is predicted across the transition metal series, in agreement with that obtained by TC methods (Saunders et al. 1988). For the most part magnitudes are in reasonable agreement, but for some elements FP lattice stabilities are as much as 3-10 times larger than those obtained by any TC methods (Fig. 6.6). 

While this concept of mechanical instability offers a potential explanation for the large discrepancies between FP and TC lattice stabilities for some elements, the calculations of Craeivich et al. (1994) showed that such instabilities also occur in many other transition elements where, in fact, FP and TC values show relatively little disagreement. The key issue is therefore a need to distinguish between permissible and non-permissible mechanical instability. Using the value of the elastic constant C as a measure of mechanical instability, Craievich and Sanchez (1995) have found that the difference between the calculated elastic constant C for f.c.c. and b.c.c. structures of the transition elements is directly proportional to the FP value of(Fig. 6.9(a)). 

At one end of the spectrum are first-principles methods where the only input requirements are the atomic numbers Za, Zb,. .. the relevant mole fractions and a specified crystal structure. This is a simple extension to the methods used to determine the lattice stability of the elements themselves. Having specified the atomic numbers, and some specific approximation for the interaction of the relevant wave functions, there is no need for any further specification of attractive and repulsive terms. Other properties, such as the equilibrium atomic volumes, elastic moduli and charge transfer, result automatically from the global minimisation of... 

Allotrope Stability range, °C Crystal structure Lattice parameters, pm Reference temperature, °C Atoms per cell, Z Density, calculated, g/ cm3 Average M—M distance, pm... 

As a rule, the distortion of the water lattice that is found in water without a solute (1) can easily take place in cooperation with the accompanying cation except in the cases of potassium, rubidium, and cesium. These ions are large enough to fill the cavities of the water lattice and to attenuate the lattice vibrations, thus preventing a local collapse of the structure and an increase in the number of interstitial water molecules. The normal water structure is essentially retained, and the lattice, stabilized by cations of the proper size, rejects the complex nonfitting ion (2). 

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