Long-and Short-Range Order

The detailed descriptions of the structures of calamitic phases allow us to classify the mesomorphic liquid crystal state, and to place this state in context with the crystalline and amorphous liquid states [3]. Table 1 describes the relationship between ordered crystals, disordered or soft crystals, liquid crystals and the isotropic liquid. 

As noted in the introductory section, disordered crystals have long-range positional order whereas liquid crystals have short-range periodic order. The extent of the bond orientational order differentiates the hexatic and non-hexatic liquid crystal phases [41-43], The positional order can be described in terms of an ideal average lattice structure 

Ordered crystals Disordered crystals Liquid crystals Isotropic liquid 

Correlation function Two-dimensional System Weakly coupled 2-D system liquid crystal 

Goodby, G. W. Gray, H.-W. Spiess, V. Vill Copyright WILEY-VCH Verlag GmbH. 1998 

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No investigation of a solid, such as the electrode in its interface with the electrolyte, can be considered complete without information on the physical structure of that solid, i.e. the arrangement of the atoms in the material with respect to each other. STM provides some information of this kind, with respect to the 2-dimensional array of the surface atoms, but what of the 3-dimensional structure of the electrode surface or the structure of a thick layer on an electrode, such as an under-potential deposited (upd) metal At the beginning of this chapter, electrocapillarity was employed to test and prove the theories of the double layer, a role it fulfilled admirably within its limitations as a somewhat indirect probe. The question arises, is it possible to see the double layer, to determine the location of the ions in solution with respect to the electrode, and to probe the double layer as the techniques above have probed adsorption Can the crystal structure of a upd metal layer be determined In essence, a technique is required that is able to investigate long- and short-range order in matter. 

The Inden model [20] is frequently used to describe second-order magnetic order-disorder transitions. Inden assumed that the heat capacity varied as a logarithmic function of temperature and used separate expressions above and below the magnetic order-disorder transition temperature (TtIS) in order to treat the effects of both long- and short-range order. Thus for z = (T/TtIS) < 1 ... 

De Jong E. W., et al. (1987) X-ray diffraction and Si magic angle spinning NMR of opals incoherent long- and short-range order in opal CT. Am. Min. 72, 1195-1203. 

In even more realistic studies of fluids in porous systems, the details of the long and short range ordering of the pores have been examined. Gubbins and co-workers and Snook and co-workers have carried out a substantial amount of work in this direction for both ordered and amorphous systems. Snook and... 

Fig. 13.09. Long- and short-range order in a two-dimensional binary alloy of composition MN (a) the fully ordered structure (s = 1) (b, c) structures with no long-range order (s = o). The structure (c) differs from (b) in that it displays a high degree of short-range order and consists of two anti-phase domains in each of which long-range order is perfect.

For certain lipid systems, tc is independent of hydration, salt concentration, lipid identity, changes in the periodicity of the transforming phases and in long-and short-range order but is dependent on the magnitude and direction of the T-jump [16, 18] and the identity of the small molecule additive [117a]. 

Depending on the system, temporal correlations of changes in long- and short-range order may [142] or may not [16, 129] exist. 

Another useful complementarity between electrons and X-rays for the study of environmental materials is their difference of structural sensitivity to long and short range order. Electron diffraction is quite sensitive to the superstructure of minerals. This capability was used to determine the layer structure of triclinic and hexagonal bimessite... 

One or more crystalline lamellar (Lc) phases may be formed by all phospholipids at low temperature and/or low levels of hydration. When long and short range order is found in three dimensions then the result is a 3D lamellar crystal, which is a true crystal. The three dimensional crystalline order results from the close packing of two dimensional phospholipid crystalline sheets (Figure 1.90). In all crystalline and ordered states, phospholipid close packing and molecular conhguration is dehned in terms of a number of parameters. These parameters are ct—the mean cross sectional area of a fatty acid alkyl chain perpendicular to the chain axis, 4>—the tilt angle of the chain with respect to bilayer plane, —the thickness of the... 

Among the most important requirements in the theory of chemical bonds is the development of a unified method for the description of the chemical interaction between atoms, which would be based on the structure of the atomic electron shells and in which one would utilize the wave functions and the electron density distributions calculated for isolated (free) ions on the basis of the data contained in Mendeleev s periodic table of elements. This unified approach should make it possible to elucidate the interrelationship between the various physical properties and the relationship between the equilibrium and the excited energy states in crystals. In contrast to the study of chemical bonds in a molecule, an analysis of the atomic interaction in crystals must make allowances for the presence of many coordination spheres, the long- and short-range symmetry, the long- and short-range order, and other special features of large crystalline ensembles. As mentioned already, the band theory is intimately related to the chemi-... 

These materials belong to the group of disordered ferroelectrics where long- and short-range orders coexist. In the spirit of preceding discussion this means that they are in mixed ferroglass phase. As these substances in the mixed phase are very rich in physical properties, they attract much attention from the point of view of their possible nanoelectronics applications. Let us describe briefly some characteristic features of these materials. 

Incompatible occupancy values may result from stoichiometry deviation. Coexistence of long- and short-range order. 

Relation between long and short range order during deformation of polyethylene... 

These mesophases are defined and characterized by maity physical parameters such as long- and short-range order orientational distribution functions, and so oa They are explained in greater detail in the following chapters. Here we continue to use the rigid-rod model and pictorially describe these phases in terms of their molecular arrangement... 

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