Order-disorder transitions averaging

Fig. 8 Temperature dependence of the four NMR peaks of squaric acid [20]. Note how the four peaks coalesce to one above the phase transition, but that the average of the peak positions does not stay constant, as required for a pure order/disorder transition. It increases around the transition temperature, emphasizing an additional displacive component, coexisting with the order/disorder one...

All the SB in crystals are thus of some kind of order-disorder transitions with JT or PJT origin of the ordering distortion units. Order-disorder transitions in crystals with rigid dipole molecules may be considered as an extreme case of such SB when two or more possible positions of the molecule in the lattice are regarded as due to corresponding PJT distortions from their averaged high-symmetry hypothetical formation (similar to enantiomers, see below). 

The predicted intrinsic width of the order-disorder transition of a mono-disperse, flnite-molecular-weight polymer solution was also tested. The average molecular weights of dialkyl-substituted polysilylenes are in the order of 6 X 10, which implies that N is 3000-5000 silicon atoms. With equation 9, the theory predicts that ATq/Tc is 0.004-0,006, which for Tc = -30 corresponds to an intrinsic width of roughly 1 or 2 C. This result is in good agreement with the experimental observations summarized in Table II. 

Fig.8. Variation with temperature of the average segregant concentration at the Ll2(100) surface (solid lines) and at the first underlayer (dotted lines) in AB3 model alloy calculated in the FCEM approximation for different segregation/order factors r (marked near the plots). Arrows indicate order-disorder transition temperatures (for r =3.5, Ts=Tb).

In addition to the segregation/order factor, and depending on its magnitude, the crystal structure and surface orientation can strongly affect the surface composition in ordered alloys. For example, unlike the case of the equiatomic bulk truncated composition of Ll2(100), LRO tends to maintain the Ll2(lll) surface with nominal bulk concentration (0.25). Therefore, the two ordered surfaces are expected to exhibit quite different segregation characteristics for the same r value (Fig. 10). Moreover, SRO causes pronounced changes of surface sublattice and average compositions associated with a considerable reduction of the order-disorder transition temperature (especially in fee alloys). 

The thermochromism of Ag2[HgI4] is due to an order-disorder transition which involves no less than three phases. According to Ketalaar (33), both the yellow low-temperature 0 modification and the red high-temperature oc form contain iodide ions which are cubic close-packed, while the silver and mercury ions occupy some of the tetrahedral holes. The 0 form has tetragonal symmetry, with the mercury ion situated at the corners of a cubic unit cell and the silver ions at the midpoints of the vertical faces. As the temperature is increased it becomes possible for the silver and mercury ions to occupy each others lattice sites and also the two extra lattices sites Hop and bottom face centers of the unit cubel which were unoccupied at lower temperatures. Above 52°C. the mercury and silver ions are completely disordered. The a modification has. therefore, averaged face-centered cubic symmetry. More recently, magnetic (39) and dielectric polarization (37, 39) measurements confirm the presence of a third phase, the 0 modification. With an increase... 

If the noise term is turned off, the system is driven towards the nearest saddle point. Therefore, the same set of equations can be used to find and test mean-field solutions. The complex Langevin method was first applied to dense melts of copolymers [74], and later to mixtures of homopolymers and copolymers [80] and to diluted polymers confined in a slit under good solvent conditions [77]. Figure 2 shows examples of average density configurations (p ) for a ternary block copolymer/homopolymer system above and below the order/disorder transition. 

Fig. 2 Averaged densities across the order-disorder transition in a two-dimensional ternary system with A, B homopolymers and A-B copolymers (20% homopolymer volume fraction), as obtained from Complex Langevin simulation runs...

Unfortunately, this approach does not give a deeper insight into a structure of surface films at the molecular level. The theory involves a concept of a certain averaging effects connected with heterogeneity of sohd surfaces. Moreover, molecular interactions are usually described in terms of a mean field approximation. As a consequence, the integral equation approach cannot elucidate many experimental findings. In particular, various phase transitions in adsorbed layers, such as the order-disorder transition, cannot be explained in the fiamework of this theory. 

Surface coverage diagram identifying the average coverage at a given temperature and pressure for the 0-Pt(321) system in equilibrium with an 62 reservoir. Order-disorder transitions are not shown. 

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