Open three-dimensional lattice

Figure 6.7 Schematic drawing of the crystal structure of glycolate oxidase. Each disc is an octamer, and these octamers are twisted relative to each other to form an open three-dimensional lattice. The image is adapted from [1].

Crystallization of 5 in the open air from an initially aprotic solvent (N,N-dimethyl-acetamide) led to a non-layered structure which is characterized by a three-dimensional lattice of loosely-packed host species interspaced by channel-type zones accommodating the solvent guest components (Fig. 9). 

Zeolites have an open, three-dimensional framework structure with pores (channels) and interconnecting cavities in the alumosilicate lattice. In Table 4.8, the basic characteristics of the most important zeolite species of commercial use are presented. 

Zeolite is a crystalline, porous aluminosilicate mineral with a unique interconnecting lattice structure. This lattice structure is arranged to form a honeycomb framework of consistent diameter interconnecting channels and pores. Negatively charged alumina and neutrally charged silica tetrahedral building blocks are stacked to produce the open three-dimensional honeycomb framework. 

The percolation probability (q) for the lattice models is defined as the probability that a given site (or bond) belongs to an infinite open cluster (47). It is fundamental to percolation theory that there exists a critical value qc of q such that 9(q) = 0 3t q < qc, and (q) > 0 if > qc. The value qc is called the critical probability or the percolation threshold. Mathematical methods of calculating this threshold are so far restricted to two dimensions, consistent with the experience in the field of phase transitions that three-dimensional problems in general cannot be solved exactly (12,13). Almost all quantitative information available on the percolation properties of specific lattices has come from Monte Carlo calculations on finite specimens (8,11,12). In particular. Table I summarizes exactly and approximately known percolation thresholds for the most important two- and three-dimensional lattices. For the bond problem, the data presented in Table I support the following well-known empirical invariant (8)... 

The three-dimensional lattice is more open than that of the felspars and in the hydrated form the cations are not firmly held but are free to migrate within the lattice and can be readily exchanged. For... 

This chapter opens with a description of the packing of polymer molecules in crystals. X-ray diffi-action gives an extremely precise description of this, since the polymer lattice diffracts X-rays as does any other three-dimensional lattice. The shape and mutual arrangement of the minute crystals are then described. The crystals are separated one from another amorphous regions whose dimensions are comparable with those of the crystals. The information on these matters stems from electron and light microscopy, techniques which are less precise than X-ray diffiractiotL Crystalline synthetic polymers are invariably partly ciystalUne and partly amorphous the term crystalline polymer always implies partially crystalline. 

Owing to the simphcity and versatility of surface-initiated ATRP, the above-mentioned AuNP work may be extended to other particles for their two- or three-dimensionally ordered assemblies with a wide controllabiUty of lattice parameters. In fact, a dispersion of monodisperse SiPs coated with high-density PMMA brushes showed an iridescent color, in organic solvents (e.g., toluene), suggesting the formation of a colloidal crystal [108]. To clarify this phenomenon, the direct observation of the concentrated dispersion of a rhodamine-labeled SiP coated with a high-density polymer brush was carried out by confocal laser scanning microscopy. As shown in Fig. 23, the experiment revealed that the hybrid particles formed a wide range of three-dimensional array with a periodic structure. This will open up a new route to the fabrication of colloidal crystals. 

The first attempt towards three-dimensional diamondoid, tetrahedral networks was made with adamantane-l,3,5,7-tetracarboxylic acid. Interpenetration of five different networks, however, precluded the formation of internal chambers. A more open structure with large chambers was obtained from the tetra-2-pyridone methane called tecton . Its diamondoid lattice entrapped butyric acid (Figure l.lTf. ... 

Beyne and Froment [ref. 28] applied percolation theory to reaction and deactivation in the real three-dimensional ZSH-5 lattice. The structure of the catalyst enters in the equation for the reduced accessibility of active sites caused by blockage, P in (22) and this quantity is related to the percolation probability for this structure, P It is generally accepted that in zBH-5 the reactions take place at the channel intersections The probability that an intersection of channels (the origin in a network] is connected with an infinite number of open intersections is the percolation probability. It decreases as a growing number of intersections becomes blocked and drops to zero well before they are all blocked One way of relating P to the probability that an intersection is blocked, q, is Honte Carlo simulation. Based upon work by Gaunt and Sykes [ref. 29] on the percolation probability and threshold in diamond, Beyne and Froment derived a polynomial expression for P, However, the probability that a site is... 

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