Random structures, modeling

Bryan, N.D., Robinson, VJ., Livens, F.R., Hesketh, N., Jones, M.N. and Lead, J.R. (1997) Metal-humic interactions a random structural modelling approach. Geochim. 

Chaotic fractal sets on rectangular lattices have been used to the define the effective conductivity of the composite material. The effective conductivity of the composite material is defined using the fractal random structure model of a composite and the iteration method of averaging. Comparison of the calculation with experimental data is also given. 

J.D. Bryngelson, When is a potential accurate enough for structure prediction Theory and application to a random heteropolymer model of protein folding, J. Ghem. Phys. 100 (1994), 6038-6045. 

Wood and Hill consider that the role of fluoride in these glasses is uncertain. Phase-separation studies suggest that the structure of the glass might relate to the crystalline species formed, in which case a microcrystallite glass model is appropriate. But other evidence cited above on the structure-breaking role of fluoride is compatible with a random network model. 

All of the studies published so far have been aiming at the reconstruction of the total electron density in the crystal by redistribution of all electrons, under the constraints imposed by the MaxEnt requirement and the experimental data. After the acceptance of this paper, the authors became aware of valence-only MaxEnt reconstructions contained in the doctoral thesis of Garry Smith [58], The authors usually invoke the MaxEnt principle of Jaynes [23-26], although the underlying connection with the structural model, known under the name of random scatterer model, is seldom explicitly mentioned. 

When it is employed to specify an ensemble of random structures, in the sense mentioned above, the MaxEnt distribution of scatterers is the one which rules out the smallest number of structures, while at the same time reproducing the experimental observations for the structure factor amplitudes as expectation values over the ensemble. Thus, provided that the random scatterer model is adequate, deviations from the prior prejudice (see below) are enforced by the fit to the experimental data, while the MaxEnt principle ensures that no unwarranted detail is introduced. 

The error-free likelihood gain, V,( /i Z2) gives the probability distribution for the structure factor amplitude as calculated from the random scatterer model (and from the model error estimates for any known substructure). To collect values of the likelihood gain from all values of R around Rohs, A, is weighted with P(R) ... 

At each stage during the structure determination process, the current structural model gives an estimate of the prediction variance Z2 to be associated with the calculated amplitude. The contribution of the random part of the structure to this prediction variance decreases while the structure determination proceeds, and uncertainty is removed by the fit to the observations. Rescaling of Z2 would be needed during the optimisation of the Bayesian score. 

It is important to realize that the random-chain model need not imply an absence of residual structure in the unfolded population. Formative articles—many of them appearing on the pages of Advances in Protein Chemistry—recognized this fact. Kauzmann s famous review raised the central question about structure in the unfolded state (Kauzmann, 1959) ... 

When the random-walk model is expanded to take into account the real structures of solids, it becomes apparent that diffusion in crystals is dependent upon point defect populations. To give a simple example, imagine a crystal such as that of a metal in which all of the atom sites are occupied. Inherently, diffusion from one normally occupied site to another would be impossible in such a crystal and a random walk cannot occur at all. However, diffusion can occur if a population of defects such as vacancies exists. In this case, atoms can jump from a normal site into a neighboring vacancy and so gradually move through the crystal. Movement of a diffusing atom into a vacant site corresponds to movement of the vacancy in the other direction (Fig. 5.7). In practice, it is often very convenient, in problems where vacancy diffusion occurs, to ignore atom movement and to focus attention upon the diffusion of the vacancies as if they were real particles. This process is therefore frequently referred to as vacancy diffusion... 

As was noted above, the structure and behavior of Nation polymers were intensively investigated and structural models were suggested. It is obvious that during the transformation from precursor (nonionic form) to ionic form the internal structure of the polymer is reorganized into the ordered type. But this order has a random nature, and all earlier investigations were peformed with such randomly ordered films. 

For reasons which will become clear, we examine first the case of high temperature H20(as). Two random network models relevant to our hypothesis have been described in the literature. Both are based on distortions from a single locally tetrahedral structure that is like ice Ih. Kell s model 77> is much too small to be very useful. Nevertheless, its successful construction, just as for the case of Ge(as) 78>, Si02(as) 79>, and others, shows the viability of the random network concept. 

Alben and Boutron suggest that the peak in the X-ray and neutron scattering functions at 1.7 A-1 is indicative of an anisotropic layer structure extending over at least 15 A in Polk type continuous random network models. To show this better Fig. 52 displays the radial distribution function of the Alben-Boutron modified... 

The reader should recall that the fitting of a structure to diffraction data is not unique. We have shown that both the constructed modified random network model of Polk, as well as the network simulated by allowing Gaussian distributions of atom-atom distances can fit the observed structure functions for low density H20(as), and the latter, with modification to include small OOO... 

The model just described does not conform in detail to the random network model proposed earlier. In particular, the use of a continuum outside the nearest-neighbor tetrahedral structure removes some of the correlations inherent in a continuous network. Nevertheless, given the existence of a broad OOO angular distribution, coupled to an 00 distance distribution much broader than in H20(as), it is unlikely that this assumption introduces any features in serious disagreement with those characteristic of a random network model. 

Clearly, any measurement that differentiates between the properties of high and low temperature forms of H20(as), and/or delineates the relationship between H20(as) and liquid H20, can be used to test the hypotheses advanced vis a vis their structures. These and the experimental tests suggested, together with the construction of continuous random network models more sophisticated than that for Ge(as), the increased use of computer simulation, and exploitation of the available experimental information to guide the choice of appproximations in a statistical mechanical theory should increase our understanding of H20(as) and, uitimately, liquid H20. 

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