The Degree of Ordering

Some forms of disorder common in adsorbed layers are islands of clustered adsorbates leaving patches of bare substrate domains in which different patches of the overlayer have identical structure but do not match at their junction because of an error in registry, i.e., an error in relative positioning parallel to the surface periodicity errors, in which individual adsorbates do not fit in the periodic arrangement of the surrounding 

While perfect order is never present, perfect disorder also does not exist at surfaces. 

In the liquid (or the gaseous) state of overlayers on surfaces, the adsorbates cannot pass through each other this gives rise to a limited amount of short-range order. Additionally, there is always some non-zero parallel component of the substrate-adsorbate interaction that will make the adsorbates spend more of their time at one type of location than at others this also is a form of ordering. 

As an example of the analysis of the ordering of an overlayer of adsorbates, we may take the question of detecting island formation. LEED provides a means for identifying when island formation takes place although it does not always give a definitive answer. To monitor island formation the presence of adsorbate-induced extra spots in the diffraction is necessary. Thus the adsorbate must produce a superlattice and we assume this case in the following discussion. 

To recognize island formation one takes advantage of the difference between coherent and incoherent diffraction from a set of N identical scatterers. If the waves scattered off the individual scatterers are incoherent in their phases, the observed intensity will be proportional to N (addition of intensities). If, however, these scattered waves are coherent, the intensity will be proportional to N (addition of amplitudes). Incoherence occurs either when the incident wave arrives with incoherent phases at different scatterers, which occurs in practice for scatterers separated by at least the coherence length of the incident beam, or when the scatterers themselves are located incoherently, i.e. are disordered. 

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In what follows the degree of order of the hexagonal phases in Cases 1-6 will be compared to see whether there are basic differences between them. For this purpose the heats of fusion for the hexagonal phases... 

The degree of ordering of the microspheres was estimated by using the radial distribution function g(D) of the P4VP cores of the microspheres (Fig. 11). As previously described, for hexagonal packed spheres, the ratio of the peaks of the distances between the centers of the cores would be For the film at r = 0.5, the... 

Fig. 6. Same as in Fig. 5 but with random fluctuations introduced after the second quench (see text) near the straight APB. at following t (a) too, (b) 500, (c) 1000, and (d) 2000. The grey level varies linearly with the degree of order. from the completely dark for the most disordered region, to the coinplet.ely white for the most Ordered one. 

It should be noted that studying states of order in thermal equilibrium as a function of temperature yields the possibility of measuring the degree of order of a system in values of corresponding equilibrium temperature . This way, the results of residual resistometry are independent of the detailed formalism between state of order and electrical resistivity... 

It will be recalled that in Chapter 7, in dealing with the anions of weak acids, we found that we were able to obtain good qualitative agreement between theory and experiment, if we replaced Jmv by the electrostatic Jti. As has been pointed out more than once, the use of simple electrostatic theory necessarily implies that there is an increment in the degree of order in the solvent subject to the electrostatic field. In the -coefficients of Table 24, however, we have found that for 6 out of the... 

Recently Blake et al.153) made such studies in the case of human (HL) and tortoise egg-white (TEWL) lysozyme based on crystallographic refinements at 1,5 and 1,6 A resolution, respectively. By these investigations they attempted to obtain information on the perturbations of water structure in the hydration shell by neighboured protein molecules and by high salt concentrations as well as on the degree of order of the bound water. The authors came to the conclusion that the number of ordered water molecules are 128 in TEWL and 140 in HL, whereas the overall content is made up of 650 and 350 water molecules per lysozyme molecule. 

Numerous experimental studies have revealed the degree of order and... 

Even when complete miscibility is possible in the solid state, ordered structures will be favored at suitable compositions if the atoms have different sizes. For example copper atoms are smaller than gold atoms (radii 127.8 and 144.2 pm) copper and gold form mixed crystals of any composition, but ordered alloys are formed with the compositions AuCu and AuCu3 (Fig. 15.1). The degree of order is temperature dependent with increasing temperatures the order decreases continuously. Therefore, there is no phase transition with a well-defined transition temperature. This can be seen in the temperature dependence of the specific heat (Fig. 15.2). Because of the form of the curve, this kind of order-disorder transformation is also called a A type transformation it is observed in many solid-state transformations. 

A non-electrochemical technique which has been employed to alter the physical characteristics of a number of polymers is that of stress orientation [26, 27], in which the material is stressed whilst being converted to the desired form. This has the effect of aligning the polymer chains and increasing the degree of order in the material, and is obviously most applicable to materials which can be produced via a precursor polymer. With Durham polyacetylene (Section 4.2.1) increases in length in excess of a factor of twenty have been achieved, with concomitant increases in order, as shown by X-ray diffraction and by measurements of the anisotropy of the electrical conductivity perpendicular and parallel to the stretch direction. 

It is important to emphasise, however, that here—as in the cases above—it is probably the effect of differing charge distributions in the anions on their patterns of solvation, i.e. on the TAS° term relating to the degree of ordering induced locally in the assembly of solvent molecules, that is responsible for the observed differences in pKa. 

Comparison of these expressions indicates that the activation entropy is related to the steric factor for the reaction. One may interpret the steric factor in terms of the degree of order of molecular configurations required to bring about the reaction, and this viewpoint is generally regarded as more satisfactory from an intellectual viewpoint than is that which regards Ps as an a posteriori correction factor necessary to obtain agreement between theory and experiment. 

It is important to consider why H depends on the degree of order of the crystalline phase. Three different types of diffusion process act during the nucleation process. They are diffusion within the melt, within the interface between the melt and a nucleus (or crystal), and within the nucleus. It is obvious that the diffusion of chains within the melt can not be related to the dependence of H on the degree of order of the crystalline phase within the nucleus (or crystal). Therefore, the phase dependence of H should arise from... 

It is concluded that H increases with an increase of the degree of order of the crystalline phases,... 

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