Order-disorder theory limiting cases

The lattice gas approach is valid within certain limits for typical metallic hydrides, binaries as well as ternaries. Deviation from this idealized picture indicates that metallic hydrides are not pure host-guest systems, but real chemical compounds. An important difference between the model of hydrogen as a lattice gas, liquid, or solid and real metal hydrides lies in the nature of the phase transitions. Whereas the crystallization of a material is a first-order transition according to Landau s theory, an order-disorder transition in a hydride can be of first or second order. The structural relationships between ordered and disordered phases of metal hydrides have been proven in many cases by crystallographic group-subgroup relationships, which suggests the possibility of second-order (continuous) phase transitions. However, in many cases hints for a transition of first order were found due to a surface contamination of the sample that kinetically hinders the transition to proceed. 

Most studies of disordered solids have been based on simple tight binding Hamiltonians of the kind described in Section 3.3. While this approach is of limited validity, it is at least susceptible to a certain amount of rigorous mathematical analysis. Other Hamiltonians, such as pseudopotential Hamiltonians, which might be more desirable in a given context, pose many more difficulties in a disordered system unless simple lowest-order perturbation theory happens to be adequate, as in the case of the Ziman theory of liquid metals, which is quite successful for the simple metals. 

I believe that empirical observations only attain the status of laws when they are accompanied by theories that support their generality or, sometimes, show their limitations. The second law of thermodynamics, with which I began this chapter, is surely a case in point. The claim that spontaneous processes will always increase the entropy of the universe is something that one can accept as a law primarily because of the statistical argument that there are more ways for things to be disordered than ordered. Certainly, one wants experimental studies to bolster the claim, but such studies by themselves would not be convincing because of the general inductive problem of empirical science—one could never rule out the possibility that some particular set(s) of conditions not studied in the experimental tests would lead to an exception to the... 

The theory developed must be apphed with care if the adsorbate is disintegrated into separate molecules occupying fixed sites in the adsorbent. This may happen in extremely microporous media (like those formed by cavities in crystalline structures) or on structured heterogeneous surfaces. The reason for such a limitation is that adsorbate is described by means of a thermodynamic equation of state. This assumes that adsorbate behaves more or less like a continuous medium and that interactions between its molecules are the same as in a bulk phase. On the contrary, a pore in a microporous medium may often contain only few molecules of the adsorbate, so that statistical fluctuations from a thermodynamic equihbrium state become significant. It may be expected that errors associated with these fluctuations cancel out in disordered media. However, a more ordered adsorbate might possess some properties introduced by the order. In this case, the above-described localized approach, the semi-analytical approach by Aranovich and Donohue [111], or direct molecular simulations become more suitable. 

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