Order-disorder problem

The establishment of a formula for the a functions essentially involves solving the order-disorder problem in a suitable notation. Mayer s method is similar to that discussed by Domb and Hiley201 following earlier work of Rushbrooke and Scoins78 and Fournet20 . We shall not discuss it in detail, but it may help to clarify the difference between the expansions of Section IV and that above by considering the evaluation of (cf. Eq. (78))... 

For these phases, MAS Si and Al-NMR data could be very useful to obtain a better understanding of this order/disorder problem. 

Detonation, Free Volume Theory of Multi-component Fluid Mixtures. The free volume theory of the liq state is extended to multi-component fluid mixts by using the method of moments in the treatment of the order-disorder problem. The results of this extension are given in the article by Z.W. 

The order-disorder problem reqd for calculation of the pseudopotential has been solved approx by three different methods ... 

The variation of 5(7) near the N-I phase transition will be measured in this experiment and will be compared with the behavior predicted by Landau theory, " " which is a variant of the mean-field theory first introduced for magnetic order-disorder systems. In this theory, local variations in the environment of each molecule are ignored and interactions with neighbors are represented by an average. This type of theory for order-disorder phase transitions is a very useful approximate treatment that retains the essential features of the transition behavior. Its simplicity arises from the suppression of many complex details that make the statistical mechanical solution of 3-D order-disorder problems impossible to solve exactly. 

Approximate solution of the cooperative order-disorder problem We define the degree of positional order as q = 22 —, where J is the fraction of molecules in the -sites, and the degree of orientational order as s = 25 —1, where is the fraction of molecules in 1-orientations. If N is the total number of molecules in the system, there are evidentiy... 

By tradition, the order parameter in any order-disorder problem is always taken such that it is unity in the perfectly ordered phase and vanishes for the completely disordered phase. Examination of the average values described above shows that the proper order parameter for the nematic liquid crystal is... 

Introduction 53. 2 Strictly Regular Solutions 55 3. Quasi-Chemical Approximation 59. 4 General Remarks concerning the Order-Disorder Problem 64 5 Mole-... 

Let us now consider more closely the order-disorder problem inherent in the evaluation of the combinatorial factor g(iVji, Nb, Nab)-... 

There exist a certain number of interesting methods for the solution of the order-disorder problem introduced by departures of randomness of mixing due to a non-vanishing w. We shall enumerate some of them in 4. Here we shall briefly describe the so called quasi-chemical approximation due essentially to Guggenheim. If we could assume that the number of configurations corresponding to a given value of Nas (and so of Naa, may be calculated as if the various types of pairs did not interfere with one another, we would have... 

General Remarks Conceming the Order-Disorder Problem... 

In 3 we used the quasi chemical approach for the order-disorder problem. We have chosen this particular method because of the intuitive character of the quasi-chemical equation which shows in a clear way how the "ordering" parameter ze affects the local order as expressed by the number of couples Nab- However the quasi-chemical treatment being based on a guess of the form on the combinatorial factor g NA, Nb, Nab) it is necessary to test it by comparison with exact treatments. 

The second exact treatment of the order-disorder problem is based on a powerful method due to Kramers and Wannier [1941] and has permitted to Onsager [1944], Houtappel [1950], Wannier [1950], Kac and Ward [1952] and Ter Haar [1954] to obtain closed solutions for the order-disorder problem in two dimensional lattices. It is outside... 

GENERAL REMARKS CONCERNING THE ORDER-DISORDER PROBLEM 65... 

The contribution of theoretical studies is rapidly growing, especially the use of self-consistent schemes associated with experimental studies [118-121]. Many of the calculations are relative to the dynamics of the copolymer chains, to order-disorder problems and to diffusion problems [122,123]. The simulation in block copolymer studies will rapidly develop in the near future, both in static and dynamic analyses [124,125]. 

The standard analytic treatment of the Ising model is due to Landau (1937). Here we follow the presentation by Landau and Lifschitz [H], which casts the problem in temis of the order-disorder solid, but this is substantially the same as the magnetic problem if the vectors are replaced by scalars (as the Ising model assumes). The themiodynamic... 

This is strong evidence for assuming that dispersions of ideal hard spheres would be expected to show a transition in the viscous behaviour between

short time selfdiffusion coefficient Z)s. This still shows a significant value after the order-disorder transition. The problem faced by the rheologist in interpreting hard sphere systems is that at high concentrations there is... 

In some thermodynamic models there are also potential minima associated with different site occupations, even though the composition may not vary, e.g., a phase with an order/disorder transformation. This must be handled in a somewhat different fashion and the variation in Gibbs energy as a function of site fraction occupation must be examined. Although this is not, perhaps, traditionally recognised as a miscibility gap, there are a number of similarities in dealing with the problem. In this case, however, it is the occupation of sites which govern the local minima and not the overall composition, per se. 

Only two high-temperature homogeneous reactions have been investigated in detail for their kinetics by geochemists. One is the Fe-Mg order-disorder reaction in orthopyroxene, and the other is the hydrous species interconversion reaction in rhyolitic melt. The two reactions have been applied as geospeedometers in various geochemical and meteoritic problems. Because they are often encountered in geochemical kinetics literature, the two reactions are discussed in depth below. 

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. 

A fully microscopic interpretation of the temperature dependence of the absorption maximum, even well above any order-disorder transition temperature, is a formidable task because of the potential importance of many complicated physical factors (27-30). As a first attack on this problem, we have adopted a simple mean-field (or effective-medium) approach (28-30) with the assumption that the absorption peak (to) is linearly perturbed from its limiting dl -trans value ((Orod) by the presence of bond rotational defects (free energy of formation, e)... 

From a conceptual viewpoint the primary theoretiotl problem yet to be solved is the stress transfer mechanism in polymer solids. As noted earlier, polymers have statistical structures v4ien in the glassy state and a rather broad spectrum of order-disorder when in the crystalline state. Detailed analysis of stress transfer throi a glassy structure requires comprehensive analysis of chain conformation in the (nonequilibrium) glass which in turn requires an imderstanding of both the intramolecular and intermolecular energetics. 

As it was pointed out in the Introduction, the problem of the coexistence of displacive and order-disorder phenomena at the ferroelectric phase transitions of BaTiOs has met growing interest in recent time. Strong support of the order-disorder model comes 30 years ago from EPR measurements performed on Mn" " "-, Cr -, and Fe -doped BaTiOs [218-222] because in the low-temperature rhombohedral phase it was observed that Mn" " ", which substitutes isovalent Ti" " " sites, is displaced off-centre by 0.14 A along <111> directions with a reorientational hopping with correlation times 10 -10 s. 

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