Lattice hierarchical

Although the folding of short proteins has been simulated at the atomic level of detail [159,160], a simplified protein representation is often applied. Simplifications include using one or a few interaction centers per residue [161] as well as a lattice representation of a protein [162]. Some methods are hierarchical in that they begin with a simplified lattice representation and end up with an atomistic detailed molecular dynamics simulation [163]. 

The approaches considered allow modeling of the primary texture of PS and the processes, limited by individual PBUs that mainly correspond to level III and partially to level IV in the hierarchical system of models (see Section 9.6.3). PBUs are identical in regular PSs, and simulation of numerous processes may be reduced to analysis of a process in a single PBU/C or PBU/P. An accurate modeling of the processes in irregular PSs requires the studies of the properties of structure and properties of the ensembles (clusters) of particles and pores (level IV of the system of models) and the lattices of such clusters (levels V to VII of the system of models). Let us consider the composition of clusters on the basis of fractal [127], and the lattices on the basis of percolation [8] theories. 

Eqs. (31) and (32) becomes more complicated. The dimensionality of the set of equations, however, coincides with that of the system in the QCA. A more exact description is obtained with the correlators of greater dimensionality m>2 (see, e.g., Refs. [90,91]). Of special interest are the one-dimensional systems with s — 2. Exact solutions for the migrating adspecies on the one-dimensional lattice have been obtained for a small number of sites [92]. In Refs. [93,94] the procedure of numerical analysis of the hierarchical system of equations has been elaborated, which is applicable not only to the one-dimensional [94,95] but also to the two-dimensional lattices [95,96], as well, the interaction with the second neighbors being taken into account (d — 1) [97]. Also, it should be noted that the expansions (virial or diagrammatic) [98] similar to the common expansion in the equilibrium theory of condensed systems [77] are used for closing the kinetic equations. 

The first application of hierarchical SA for parameter estimation included refinement of the pre-exponentials in a surface kinetics mechanism of CO oxidation on Pt (a lattice KMC model with parameters) (Raimondeau et al., 2003). A second example entailed parameter estimation of a dual site 3D lattice KMC model for the benzene/faujasite zeolite system where benzene-benzene interactions, equilibrium constants for adsorption/desorption of benzene on different types of sites, and diffusion parameters of benzene (a total of 15 parameters) were determined (Snyder and Vlachos, 2004). While this approach appears promising, the development of accurate but inexpensive surfaces (reduced models) deserves further attention to fully understand its success and limitation. 

In this work, we describe the results of theoretical studies of Poisson s ratio in disordered structures composed of two phases of disparate elastic properties applying a renormalization group approach to a model of percolation on a hierarchical cubic lattice. Although this approach has been described in detail elsewhere [160], we present it briefly here, for completeness. At the percolation... 

Figs. 1 and 2 depict the result of hierarchical training of a KMC model of benzene in the NaX lattice by simultaneously fitting self-diffusivity data at different loadings 0 (number of benzene molecules per cage) and adsorption isotherms, respectivelyThe optimized... 

The materials which have been mentioned here so far are predominantly shaped in planar films of hierarchical order. However, the synthesis of hierarchically structured particles is also highly desirable, as they might be further processed and used for the preparation of composite porous materials. Wu et al. showed the synthesis of raspberry-like hollow silica spheres with a hierarchically structured, porous shell, using individual PS particles as sacrificial template [134]. In another intriguing approach by Li et al. [135], mesoporous cubes and near-spherical particles (Fig. 10) were formed by controlled disassembly of a hierarchically structured colloidal crystal, which itself was fabricated via PMMA latex and nonionic surfactant templating. The two different particle types concurrently generated by this method derive from the shape of the octahedral and tetrahedral voids, which are present in the template crystal with fee lattice symmetry. 

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