Order-disorder phenomena

Figure A2.5.16. The coexistence curve, = KI(2R) versus mole fraction v for a simple mixture. Also shown as an abscissa is the order parameter s, which makes the diagram equally applicable to order-disorder phenomena in solids and to ferromagnetism. The dotted curve is the spinodal.

The treatment of such order-disorder phenomena was initiated by Gorsky (1928) and generalized by Bragg and Williams (1934) [5], For simplicity we restrict the discussion to the synnnetrical situation where there are equal amounts of each component (x = 1/2). The lattice is divided into two superlattices a and p, like those in the figure, and a degree of order s is defined such that the mole fraction of component B on superlattice p is (1 +. s)/4 while that on superlattice a is (1 -. s)/4. Conservation conditions then yield the mole fraction of A on the two superlattices... 

The exponents apply not only to solid systems (e.g. order-disorder phenomena and simple magnetic systems), but also to fluid systems, regardless of the number of components. (As we have seen in section A2.5.6.4 it is necessary in multicomponent systems to choose carefully the variable to which the exponent is appropriate.)... 

Using the so-called "block copolymers (a block of Na A-monomers at one end is covalently bonded to a block of Nb B-monomers) one can also realize the analogy of order-disorder phenomena in metallic alloys with polymers one observes transitions from the disordered melt to mesophases with various types of long range order (lamellar, hexagonal, cubic, etc ). We shall not consider these phenomena here further, however... 

G. Foumet, Order-disorder phenomena in solid solutions, in. Phase Stability in Metals and Alloys", P S. 

In the second section a classification of the different kinds of polymorphism in polymers is made on the basis of idealized structural models and upon consideration of limiting models of the order-disorder phenomena which may occur at the molecular level. The determination of structural models and degree of order can be made appropriately through diffraction experiments. Polymorphism in polymers is, here, discussed only with reference to cases and models, for which long-range positional order is preserved at least in one dimension. 

Disordered structures belonging to the class (i) are interesting because, in some cases, they may be characterized by disorder which does not induce changes of the lattice dimensions and of the crystallinity, and a unit cell may still be defined. These particular disordered forms are generally not considered as mesomorphic modifications. A general concept is that in these cases the order-disorder phenomena can be described with reference to two ideal structures, limit-ordered and limit-disordered models, that is, ideal fully ordered or fully disordered models. 

Field emission microscopy was the first technique capable of imaging surfaces at resolution close to atomic dimensions. The pioneer in this area was E.W. Muller, who published the field emission microscope in 1936 and later the field ion microscope in 1951 [23]. Both techniques are limited to sharp tips of high melting metals (tungsten, rhenium, rhodium, iridium, and platinum), but have been extremely useful in exploring and understanding the properties of metal surfaces. We mention the structure of clean metal surfaces, defects, order/disorder phenomena,... 

While Eq. (2) models submonolayer order-disorder transitions and Eq. (60) model multilayer adsorption, it is of course possible to formulate a combined model which considers the competition between order-disorder phenomena in the first layer and adsorption of further layers . Then instead of Eqs. (2), (60) we write, for the simple cubic lattice,... 

In an ideal world, crystals would be perfect or stoichiometric with constant composition. But like people crystals are not exempt from imperfections or defects. Crystals with variable composition are termed non-stoichiometric crystals. The defect chemistry of oxides is enormously complex and is extremely vital to their properties. It has involved extensive research in many laboratories and is providing extraordinary insights into structural variations, the stability of structures and the formation of new structures. Here, we first define order-disorder phenomena that are commonly associated with oxides and describe our current understanding of them. The disorder or non-stoichiometry plays a crucial role in oxide applications including catalysis and it is therefore of paramount importance. 

CO/Rh(100) shows a very strong repulsion for CO molecules at nearest-neighbor positions and much weaker repulsion between molecules farther apart. The occurrence of different lateral interactions is typical for real systems and leads to complicated order-disorder phenomena, especially if there are different types of adsorbates. Thermal motion of the adsorbates may overcome some lateral interactions but not others. The phase diagram can be quite complex even if there is only one type of adsorbate, and the effect on the kinetics can be profound. 

When the defect interaction energy is much larger than the thermal energy, it can lead to an ordering of defects into superlattice structures and to the appearance of phases having ordered arrays of defects. Other interactions may also become important as the spin-dependent interactions between the d electrons in the Fj gS system (24). We shall not consider these order-disorder phenomena, since they are discussed by Wadsley (30). Some of the structural-consequences of ordering are considered below. 

Determination of local environment of particular atoms in a solid. Applicable to 130 isotopes (e.g., in mineralogy, studies of Al applied to order-disorder phenomena studies of As, Sb, and Bi isotopes in sulfosalts applied to structural investigations)... 

P is the key function characterizing a percolation process, and here it plays the role of the order parameter used to describe order-disorder phenomena and phase transitions. Its behavior for a square lattice is show in Fig. 39. 

In this chapter, the theory of conformation-dependent polymer-solvent interactions, which was developed in detail by Schweizer (20-22) for soluble TT-conjugated polymers, will be used to explain both qualitatively and quantitatively a large body of observations on the polysilylenes (23, 24). The same theory has been used recently to interpret qualitatively order-disorder phenomena and the electronic thermochromism of TT-conjugated-polymer solutions and films (25, 26). The study presented in this chapter represents part of an ongoing effort to understand in a unified fashion both the optical properties (27-30) and order-disorder transitions (20-24) of flexible, conjugated-polymer solutions. 

Cregg PJ, Crothers DSF, Wickstead AW (1994) An approximate formula for the relaxation time of a single domain ferromagnetic particle with nniaxial anisotropy and collinear field. J Appl Phys 76 4900-4902 Dang MZ, Rancourt DG (1996) Simnltaneons magnetic and chemical order-disorder phenomena in FesNi, FeNi, and FeNij. Phys Rev B 53 2291-2301... 

As it was pointed out in the Introduction, the problem of the coexistence of displacive and order-disorder phenomena at the ferroelectric phase transitions of BaTiOs has met growing interest in recent time. Strong support of the order-disorder model comes 30 years ago from EPR measurements performed on Mn" " "-, Cr -, and Fe -doped BaTiOs [218-222] because in the low-temperature rhombohedral phase it was observed that Mn" " ", which substitutes isovalent Ti" " " sites, is displaced off-centre by 0.14 A along <111> directions with a reorientational hopping with correlation times 10 -10 s. 

Those theories of ordering that assume that the thermodynamics of ordering can be explained by Q alone are termed mean-field theories. Here two such theories are explored and compared that of Bragg and Williams (1934) and that of Landau (1937). While some attempt has been made to consider the pressure-dependence of order-disorder phenomena in minerals (Hazen and Navrotsky 1996), here I shall limit the discussion of these phenomena to their temperature-dependence alone. 

Wood BJ, Kirkpatrick RJ, Montez B (1986) Order-disorder phenomena in MgAl204 spinel. Am Mineral 71 999-1006... 

Potential dependent halide adsorption, including order-disorder phenomena, is well known [134, 246, 247], and it is reasonable to expect that the breakdown of the PEG-C1 blocking layer might also be potential dependent. In a similar fashion, the composition and structure of thiols and disulfides adsorbed on gold from simple electrolytes have been shown to be potential dependent [282]. In the present example, it is also possible that the approach of SPS to the electrode surfaces is screened by complexation with the potential dependent concentration of Cu+ that is generated at the electrode. Importantly, the equilibrium Cu+ concentration in the additive-free system (i.e. Equations 2.1 and 2.2) is of the order of 400 (tmol L 1 and Cu+ is known to form complexes with all the additives under consideration [239, 279, 280, 283-285]. Furthermore, the equilibrium Cu+ concentration decreases exponentially with potential, that is, 60 mV per decade of concentration [283-285]. Thus, the increasing rate of SPS adsorption with... 

Elcock, E. W. Order-Disorder Phenomena. London Methuen New York Wiley. 1956. 

The effects of temperature on LEED intensities cannot be considered here. Heating the crystal (or cooling it) gives important information about thermal vibrations, order-disorder phenomena, and surface melting (44, 146, 147, 147a, 193). 

To eliminate some of the present ambiguity concerning these structures, we have examined the behavior of a number of oligomers and the polymer of adenylic acid, as well as the polymer of A6-hydroxyethyladenylic acid (poly HEA). We have been concerned with both the explanation of their optical properties in terms of proposed models and the thermodynamics of the melting process. It is hoped that the information so obtained will help to clarify the importance of order-disorder phenomena in biology and questions concerning the structure of such biologically important polynucleotides as RNA. 

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