Theory disorder

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

Nix and Shockley [6] gave a detailed review of the status of order-disorder theory and experiment up to 1938, with emphasis on analytic improvements to the original Bragg-Williams theory, some of which will be... 

Nearly all experimental eoexistenee eurves, whether from liquid-gas equilibrium, liquid mixtures, order-disorder in alloys, or in ferromagnetie materials, are far from parabolie, and more nearly eubie, even far below the eritieal temperature. This was known for fluid systems, at least to some experimentalists, more than one hundred years ago. Versehaflfelt (1900), from a eareflil analysis of data (pressure-volume and densities) on isopentane, eoneluded that the best fit was with p = 0.34 and 8 = 4.26, far from the elassieal values. Van Laar apparently rejeeted this eonelusion, believing that, at least very elose to the eritieal temperature, the eoexistenee eurve must beeome parabolie. Even earlier, van der Waals, who had derived a elassieal theory of eapillarity with a surfaee-tension exponent of 3/2, found (1893)... 

Figure A3,12.2(a) illnstrates the lifetime distribution of RRKM theory and shows random transitions among all states at some energy high enongh for eventual reaction (toward the right). In reality, transitions between quantum states (though coupled) are not equally probable some are more likely than others. Therefore, transitions between states mnst be snfficiently rapid and disorderly for the RRKM assumption to be mimicked, as qualitatively depicted in figure A3.12.2(b). The situation depicted in these figures, where a microcanonical ensemble exists at t = 0 and rapid IVR maintains its existence during the decomposition, is called intrinsic RRKM behaviour [9].

The present paper is devoted to the theoretical formulation and numerical implementation of the NDCPA. The dynamical CPA is a one-site approximation in which variation of a site local environment (due to the presence, for example, of phonons with dispersion) is ignored. It is known from the coherent potential theory for disordered solids [21], that one can account in some extension the variation of a site local environment through an introduction of a nonlocal cohcn-cnt potential which depends on the difference between site... 

These excitations are widely used to characterize electronic states and excitations, to test theories about electronic phenomena, and to delineate the nature of local sites in glasses, disordered soHds, intercalates, etc. However, this aspect of changing color with pressure is so general as to be hardly satisfactory for defining piezochromism. 

In Sec. 3 our presentation is focused on the most important results obtained by different authors in the framework of the rephca Ornstein-Zernike (ROZ) integral equations and by simulations of simple fluids in microporous matrices. For illustrative purposes, we discuss some original results obtained recently in our laboratory. Those allow us to show the application of the ROZ equations to the structure and thermodynamics of fluids adsorbed in disordered porous media. In particular, we present a solution of the ROZ equations for a hard sphere mixture that is highly asymmetric by size, adsorbed in a matrix of hard spheres. This example is relevant in describing the structure of colloidal dispersions in a disordered microporous medium. On the other hand, we present some of the results for the adsorption of a hard sphere fluid in a disordered medium of spherical permeable membranes. The theory developed for the description of this model agrees well with computer simulation data. Finally, in this section we demonstrate the applications of the ROZ theory and present simulation data for adsorption of a hard sphere fluid in a matrix of short chain molecules. This example serves to show the relevance of the theory of Wertheim to chemical association for a set of problems focused on adsorption of fluids and mixtures in disordered microporous matrices prepared by polymerization of species. 

Sec. 4 is concerned with the development of the theory of inhomogeneous partly quenched systems. The theory involves the inhomogeneous, or second-order, replica OZ equations and the Born-Green-Yvon equation for the density profile of adsorbed fluid in disordered media. Some computer simulation results are also given. 

Adsorption of hard sphere fluid mixtures in disordered hard sphere matrices has not been studied profoundly and the accuracy of the ROZ-type theory in the description of the structure and thermodynamics of simple mixtures is difficult to discuss. Adsorption of mixtures consisting of argon with ethane and methane in a matrix mimicking silica xerogel has been simulated by Kaminsky and Monson [42,43] in the framework of the Lennard-Jones model. A comparison with experimentally measured properties has also been performed. However, we are not aware of similar studies for simpler hard sphere mixtures, but the work from our laboratory has focused on a two-dimensional partly quenched model of hard discs [44]. That makes it impossible to judge the accuracy of theoretical approaches even for simple binary mixtures in disordered microporous media. 

We conclude, from the results given above, that both the ROZ-PY and ROZ-HNC theories are sufficiently successful for the description of the pair distribution functions of fluid particles in different disordered matrices. It seems that at a low adsorbed density the PY closure is preferable, whereas... 

To the best of our knowledge, there was only one attempt to consider inhomogeneous fluids adsorbed in disordered porous media [31] before our recent studies [32,33]. Inhomogeneous rephca Ornstein-Zernike equations, complemented by either the Born-Green-Yvon (BGY) or the Lovett-Mou-Buff-Wertheim (LMBW) equation for density profiles, have been proposed to study adsorption of a fluid near a plane boundary of a disordered matrix, which has been assumed uniform in a half-space [31]. However, the theory has not been complemented by any numerical solution. Our main goal is to consider a simple model for adsorption of a simple fluid in confined porous media and to solve it. In this section we follow our previously reported work [32,33]. 

From the theoretical point of view, a density functional type theory for systems confined to microporous media is lacking. This seems to be one of the reasons why the problem of crystallization of fluids in disordered media has not been solved so far. Further work in future is needed, however, to solve this and relevant problems. Our expectation is that a combined application of theoretical methods and simulation would provide faster progress in studies of fluids and mixtures in microporous media. At present, the models studied in theory and simulations are quite far from the systems of experimental focus. Hopefully, favorable changes will occur in future. 

In the early days of alloy theory, Lifshitz argued on intuitive grounds that one can calculate the properties of a disordered alloy from one sufficiently large sample," and referred to this property of a large sample as self-averaging. It can be seen most easily in exact... 

A different approach is adopted here. Within the LMTO-ASA method, it is possible to vary the atomic radii in such a way that the net charges are non-random while preserving the total volume of the system . The basic assumption of a single-site theory of electronic structure of disordered alloys, namely that the potential at any site R depends only on the occupation of this site by atom A or B, and is completely independent of the occupation of other sites, is fulfilled, if the net charges... 

We have developed a theory that allows to determine the effective cluster interactions for surfaces of disordered alloys. It is based on the selfconsistent electronic structure of surfaces and includes the charge redistribution at the metal/vacuum interface. It can yield effective cluster interactions for any concentration profile and permits to determine the surface concentration profile from first principles in a selfconsistent manner, by... 

SPR-KKR can. straightforwardly be combined with the CPA alloy theory to deal with disordered alloys. 

Another way of looking at it is that Shannon information is a formal equivalent of thermodynamic entroi)y, or the degree of disorder in a physical system. As such it essentially measures how much information is missing about the individual constituents of a system. In contrast, a measure of complexity ought to (1) refer to individual states and not ensembles, and (2) reflect how mnc h is known about a system vice what is not. One approach that satisfies both of these requirements is algorithmic complexity theory. 

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