Long range ordering problems

Long Range Ordering Problems 11.3.1 Polytypism in Layered Structures... 

We have used here that X is either CO or a vacant site and (CO) + ( ) = . The main problem with this mean-field approximation is that at temperatures below the order-disorder transition there is a strong correlation between the occupations of sites. As will be shown in Section 4.3 (pco,co for neighboring sites is about 24 kJ/mol, which corresponds to a thermal energy at 2880 K. This means that neighboring sites will not be occupied simultaneously at any realistic temperatures, and a more sophisticated approach is needed that describes this well. For weak interactions eqn. (11) may be acceptable, but one should be aware that there is some correlation in the occupation of sites even if there is no long-range order. 

This volume covers the structural relations between thermotropic and lyotropic liquid crystals (Chapters 1 and 2) and compares them with the micellar systems (Chapter 3). The interfacial aspects and the accompanying stability problems are covered in Chapters 5 and 6. The molecular dynamics in liquid crystals, the importance of water structure and of counter-ion binding for their stability are three essential factors for long range order systems, which are treated in Chapters 7, 8, and 9. The final chapter by E. J. Ambrose illustrates the change of order in a biological system under malignant conditions. 

The second drawback of the mean-field approximation is neglect of electron-phonon interaction, which should contribute attractively to the gi contribution and lead to a lattice modulation (Peierls instability). However, a ground state of such a nature is not stabilized in the TM2X family. The absence of long-range order is deeply rooted in the one-dimensional problem and arises when the effect of interactions on the different response functions is calculated perturbatively [26]. 

Percolation. Application of local interactions to the problem of generating long-range order, including phase transitions and morphogenesis, to large-scale discrete models. 

A further set of problems which obviously follows from the above discussion concerns the mechanism of shear-plane formation, although we should emphasize that the considerations involved here are quite separate from the thermodynamic ones discussed above. We discuss these mechanistic problems in Section 4 after considering a second structural feature in shear-plane systems, viz. the remarkable long-range ordering that commonly occurs in oxides containing these defects. 

The theoretical calculation of the thermodynamic properties of liquids presents an even more difficult problem than that of solids, for in a liquid the molecules are in a state of disorder intermediate between the random motion of a gas, and the long range order of a solid. We must therefore be content with an even cruder approximation than for solids. ) A model which leads to a useful semi-quantitative correlation is based upon the idea of the free volume. 

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