Motion ordered/disordered

That is, S —> 0 as T - 0. The perfect crystal part of this statement of the third law refers to a substance in which all the atoms are in a perfectly orderly array, and so there is no positional disorder. The T— 0 part of the statement implies the absence of thermal motion-—thermal disorder vanishes as the temperature approaches zero. As the temperature of a substance is raised from zero, more orientations become available to the molecules and their thermal disorder increases. Thus we can expect the entropy of any substance to he greater than zero above T = 0. 

Liquid-solid transitions in suspensions are especially complicated to study since they are accompanied by additional phenomena such as order-disorder transition of particulates [98,106,107], anisotropy [108], particle-particle interactions [109], Brownian motion, and sedimentation-particle convection [109], Furthermore, the size, size distribution, and shape of the filler particles strongly influence the rheological properties [108,110]. More comprehensive reviews on the rheology of suspensions and rubber modified polymer melts were presented by Metzner [111] and Masuda et al. [112], respectively. 

Squaric acid (H2SQ) has been chosen as a first test compound because it has a very simple molecular structure. Planar sheets of the squarate (C4O4) groups are linked to each other in a two-dimensional network through O - H...0 bonds (Fig. 1) with weak van der Waals forces [52,53]. The protons perform an order/disorder motion above the antiferroelectric phase transi-... 

Figure 2 shows the schematic structure in the paraelectric (T > Tn) and an-tiferroelectric (T < Tn) phases, hi the paraelectric phase the time-averaged position of the H atoms hes in the middle of an O - H...0 bond, whereas in the antiferroelectric phase, the protons locahze close to one or the other O atom. Prior to the recent NMR work [20-25], the largely accepted model of the phase transition was that the phase transition involved only the ordering of the H atoms in the O - H...0 bonds, and no changes in the electronic structure of the C4 moieties were considered to take place. The NMR results show that, in addition to the order/disorder motion of the H atoms, the transition also involves a change in the electronic charge distribution and symmetry of the C4 squares. 

For a pure order-disorder transition it is justifiable to use a relaxation-type equation of motion for the order parameter dynamics ... 

Usually it is assumed that tc is the only temperature-dependent variable in Eq. 9. This might be the case for an order-disorder type rigid lattice model, where the only motion is the intra-bond hopping of the protons, since the hopping distance is assumed to be constant and therefore also A and A2 are constant. This holds, however, only for symmetric bonds. Below Tc the hydrogen bonds become asymmetric and the mean square fluctuation amplitudes are reduced by the so-called depopulation factor (l - and become in this way temperature-dependent also. The temperature dependence of tc in this model is given by Eq. 8, i.e. r would be zero at Tc, proportional to (T - Tc) above Tc and proportional to (Tc - T) below Tc. 

However, in the 1980s some doubts arose about the validity of this model. Results of several experiments indicated that a model based on order-disorder motion of PO4 groups could be more appropriate to describe the phase transition [3,4]. Within such a type of a model, the isotope effect on Tc could be explained in terms of a geometric isotope effect. More precisely. 

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. 

Finally, ordered water molecules were added to the model where unexplained electron-density was present in chemically feasible locations for water molecules. Temperature factors for these molecules (treated as oxygen atoms) were allowed to refine individually. If refinement moved these molecules into unrealistic positions or increased their temperature factors excessively, the molecules were deleted from the model. Occupancies were constrained to 1.0 throughout the refinement. This means that B values reflect both thermal motion and disorder (Section II.C). Because all B values fall into a reasonable range, the variation in B can be attributed to thermal motion. Table 8.2 shows the progress of the refinement. 

As mentioned above, polymorphs may also be related by order-disorder transitions, e.g. the onset of free rotation of a group of atoms, or local tumbling in semi-plastic or plastic phases. This may be due to random orientation of the molecules or ions, but is also diagnostic of the onset of a reorientational motion. Roughly spherical molecules and ions are likely to show order-disorder phase transitions to a plastic state. In the cases of co-crystals or of crystalline salts this process may affect only one of the components, leading to semi-plastic crystals (an example will be discussed below). Order-disorder phase transitions have often... 

Monte Carlo techniques were first applied to colloidal dispersions by van Megen and Snook (1975). Included in their analysis was Brownian motion as well as van der Waals and double-layer forces, although hydrodynamic interactions were not incorporated in this first study. Order-disorder transitions, arising from the existence of these forces, were calculated. Approximate methods, such as first-order perturbation theory for the disordered state and the so-called cell model for the ordered state, were used to calculate the latter transition, exhibiting relatively good agreement with the exact Monte Carlo computations. Other quantities of interest, such as the radial distribution function and the excess pressure, were also calculated. This type of approach appears attractive for future studies of suspension properties. 

There is, however, an alternative (but still indirect) way to view these molecules. It involves studies of crystalline solids and the use of the phenomenon of diffraction. The radiation used is either X rays, with a wavelength on the order of 10 cm, or neutrons of similar wavelengths. The result of analyses by these diffraction techniques, described in this volume, is a complete three-dimensional elucidation of the arrangement of atoms in the crystal under study. The information is obtained as atomic positional coordinates and atomic displacement parameters. The coordinates indicate the position of each atom in a repeat unit within the crystal, while the displacement parameters indicate the extent of atomic motion or disorder in the molecule. From atomic coordinates, it is possible to calculate, with high precision, interatomic distances and angles of the atomic components of the crystal and to learn about the shape (conformation) of molecules in the crystalline state. 

Proteins, also, can vibrate in part or as a whole in the crystalline state. In contrast to crystals of small molecules, crystals of proteins almost invariably contain large numbers of molecules of solvent of crystallization, often corresponding to 50% or more of the unit cell volume. The extent to which these water molecules are ordered varies dramatically. Near the surface of the protein the water molecules may be well ordered. Beyond the first layer, water molecules typically show increasing levels of disorder. In addition, because of the high solvent content, there may be considerable motion and disorder in the protein molecule, particularly in the orientations of side chains. As a result, the number of independent Bragg reflections that can be measured is reduced, and this effectively reduces the resolution of the electron-density map. 

Thermal motion and disorder are two of the main factors limiting our ability to obtain accurate distances and angles by X-ray diffraction methods. As quoted earlier in this Chapter, if only the atoms kept still it would make the task much easier. An understanding of the effects of thermal motion and disorder is necessary so that they may be corrected for when determining bond distances. In order to obtain the most accurate set of bond distances, the X-ray diffraction experiment should be performed at as low a temperature as possible, thus reducing all atomic motion. 

Order-disorder phase transitions are especially common in crystalline 7T-donor acceptor complexes between planar polycyclic aromatic hydrocarbons and other organic compounds. The disordered phase can sometimes be characterized in terms of either a static- or a dynamic-disorder model, as shown in Figure 13.2. The dynamic-disorder model consists of disordered components in motion within the confines of a broad well in the potential energy curve, whereas the static disorder model requires that the disordered components be localized in two or more sites in the asymmetric unit, within one or another of the wells of a multiwell-potential energy curve. When the possible sites for static disorder are not resolved by the effective resolution of the data set, it is difficult to choose between these models. This turned out to be true for an anthracene-tetracyanobenzene complex studied at several temperatures above and below the transition temperature Tj, of 206 It was found ° ° ... 

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