Phase diagrams cluster models

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

The relationship between the phase diagram (Fig. 1.82) and the cluster models (Fig. 1.86) has not yet been clarified. It is important that detailed in-situ X-ray and neutron diffraction studies are carried out to study the structure as a function of temperature and non-stoichiometry (more detailed and recent results are reviewed in refs 52, p. 61 and 59, p. 699). 

Figure 13.4 Low-level 18-cluster QCE model (RHF/3-21G level) of the water phase diagram, showing (above) the dominant W24 clathrate-type cluster of the ice-like solid phase, and (below) the overall phase diagram near the triple point (with a triangle marking the actual triple point). Note that numerous other clusters in the W2o-W26 range were included in the mixture, but only that shown (with optimal proton ordering) acquired a significant population.

Extension of the model Hamiltonian to include higher-order interactions results in the Cu-Au phase diagram shown in fig. 6.24. In particular, the cluster expansion for the total energy in this case includes four-site interactions between those sites... 

There is much to be admired in the analyses presented here. In my opinion, this serves as a shining example of many different elements in the quest to build effective theories of material response. First, it is to be noted again that the key structural features of these complex materials have been mapped onto a corresponding two-dimensional model, itself already an intriguing and useful idea. In addition, the construction of the cluster expansion illustrates the way in which first-principles calculations for a relatively complex material can be used to inform a more tractable (and enlightening) model. Finally, the tools of statistical mechanics can be used to explore the phase diagrams of these systems which helps explain observed features of these problems but also suggests further complexity as can be seen in fig. 3 of de Fontaine et al. (1990). 

Bral] Brauwers, M., Occurrence ofthe a Phase Computed from a Cluster Model , J. Phys. F Met Phys., 7(6), 921-927 (1977) (Crys. Sfructure, Phase Diagram, Phase Relations, Thermodyn., Calculation, 17)... 

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