Free energy defective crystal

As an example, the space charge properties of Ti02 were analyzed by Ikeda and Chiang [16]. Assuming cation Frenkel disorder - that is, cation vacancies and interstitials are the predominant defects - the free energy change due to the introduction of such defects into a perfect crystal is given by... 

There was proposed a classification of one-sided phases, the composition of a phase at T=0 being chosen as a main criterion. "Genuine NS phases" remain one-sided in the entire temperature range, even at T—>0. The formation of such phases can be connected with a decrease in the crystal free energy under deviation from stoichiometiy due to an increase in the contribution of the electron subsystem to the crystal total energy and defect ordering. 

The most direct effect of defects on tire properties of a material usually derive from altered ionic conductivity and diffusion properties. So-called superionic conductors materials which have an ionic conductivity comparable to that of molten salts. This h conductivity is due to the presence of defects, which can be introduced thermally or the presence of impurities. Diffusion affects important processes such as corrosion z catalysis. The specific heat capacity is also affected near the melting temperature the h capacity of a defective material is higher than for the equivalent ideal crystal. This refle the fact that the creation of defects is enthalpically unfavourable but is more than comp sated for by the increase in entropy, so leading to an overall decrease in the free energy... 

The effect of different types of comonomers on varies. VDC—MA copolymers mote closely obey Flory s melting-point depression theory than do copolymers with VC or AN. Studies have shown that, for the copolymers of VDC with MA, Flory s theory needs modification to include both lamella thickness and surface free energy (69). The VDC—VC and VDC—AN copolymers typically have severe composition drift, therefore most of the comonomer units do not belong to crystallizing chains. Hence, they neither enter the crystal as defects nor cause lamellar thickness to decrease, so the depression of the melting temperature is less than expected. 

A very different model of tubules with tilt variations was developed by Selinger et al.132,186 Instead of thermal fluctuations, these authors consider the possibility of systematic modulations in the molecular tilt direction. The concept of systematic modulations in tubules is motivated by modulated structures in chiral liquid crystals. Bulk chiral liquid crystals form cholesteric phases, with a helical twist in the molecular director, and thin films of chiral smectic-C liquid crystals form striped phases, with periodic arrays of defect lines.176 To determine whether tubules can form analogous structures, these authors generalize the free-energy of Eq. (5) to consider the expression... 

The papers of Wagner and Schottky contained the first statistical treatment of defect-containing crystals. The point defects were assumed to form an ideal solution in the sense that they are supposed not to interact with each other. The equilibrium number of intrinsic point defects was found by minimizing the Gibbs free energy with respect to the numbers of defects at constant pressure, temperature, and chemical composition. The equilibrium between the crystal of a binary compound and its components was recognized to be a statistical one instead of being uniquely fixed. 

In these equations gv is the change in Gibbs free energy on taking one atom from a normal lattice site to the surface of the crystal and (gt + gv) the change when an atom is taken from a normal lattice site to an interstitial site, both at constant temperature and pressure. cr denotes a site fraction of species r on its sublattice, and is the chemical potential of a normal lattice ion in the defect-free crystal. 

In considering the equilibrium of the crystal with a second phase the Gibbs chemical potentials are required and we therefore express these in terms of the defect chemical potentials so far discussed. The Gibbs free energy of the system is given by... 

We consider first the activity coefficients. The contribution of the defects, N cation vacancies and N divalent ions, to the Gibbs free energy of the doped crystal is... 

The solubility and the hydrolysis constants enable the concentration of iron that will be in equilibrium with an iron oxide to be calculated. This value may be underestimated if solubility is enhanced by other processes such as complexation and reduction. Solubility is also influenced by ionic strength, temperature, particle size and by crystal defects in the oxide. In alkaline media, the solubility of Fe oxides increases with rising temperature, whereas in acidic media, the reverse occurs. Blesa et al., (1994) calculated log Kso values for Fe oxides over the temperature range 25-300 °C from the free energies of formation for hematite, log iCso fell from 0.44 at 25 °C to -10.62 at300°C. 

The defect concentration comes from thermodynamics. While we will discuss thermodynamics of solids in more detail in Chapter 2, it is useful to introduce some of the concepts here to help us determine the defect concentrations in Eq. (1.43). The free energy of the disordered crystal, AG, can be written as the free energy of the perfect crystal, AGq, plus the free energy change necessary to create n interstitials and vacancies (n, =n-o = n), Ag, less the entropy increase in creating the interstitials ASc at a temperature T ... 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

Solid-solid reactions are as a rule exothermic, and the driving force of the reaction is the difference between the free energies of formation of the products and the reactants. A quantitative understanding of the mechanism of solid-solid reactions is possible only if reactions are studied under well-defined conditions, keeping the number of variables to a minimum. This requires single-crystal reactants and careful control of the chemical potential of the components. In addition, a knowledge of point-defect equilibria in the product phase would be useful. 

Wherever there is a defect in a crystal lattice, interatomic forces will remain unbalanced and the free energy will be less negative than elsewhere in the crystal, although generally the lattice will deform locally to smooth this out. Nevertheless, defect sites (especially of the extended variety) tend to be more chemically reactive than the bulk crystal and tend to be active sites for crystal growth, dissolution, corrosion, and catalytic activity. 

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