Three-dimensionally ordered crystals

The first summation in (xiii) is the same as the intensity from a three dimensionally ordered crystal in which every unit cell would contribute the same amplitude proportional to T, and indicates that there will be a Bragg diffraction pattern corresponding to this "average" or "statistical" crystal. The second summation in (xiii) contains the squared amplitudes of... 

It was, however, observed that such systems under appropriate conditions of concentration, solvent, molecular weight, temperature, etc. form a liquid crystalline solution. Perhaps a little digression is in order here to say a few words about liquid crystals. A liquid crystal has a structure intermediate between a three-dimensionally ordered crystal and a disordered isotropic liquid. There are two main classes of liquid crystals lyotropic and thermotropic. Lyotropic liquid crystals are obtained from low viscosity polymer solutions in a critical concentration range while thermotropic liquid crystals are obtained from polymer melts where a low viscosity phase forms over a certain temperature range. Aromatic polyamides and aramid type fibers are lyotropic liquid crystal polymers. These polymers have a melting point that is high and close to their decomposition temperature. One must therefore spin these from a solution in an appropriate solvent such as sulfuric acid. Aromatic polyesters, on the other hand, are thermotropic liquid crystal polymers. These can be injection molded, extruded or melt spun. 

Heat-treated PBZO fibers have a three-dimensionally ordered crystal structure with limited long-range order (94,210). The structure exhibits a relatively... 

For rigid-chain crystallizable polymers, spontaneous transition into the nematic phase is accompanied by crystallization intermolecular interactions should lead to the formation of a three-dimensional ordered crystalline phase. 

Let us consider a structural limiting model, in which the polymer molecules, presenting a periodic conformation, are packed in a crystal lattice with a perfect three-dimensional order. Besides this limiting ordered model, it is possible to consider models of disordered structures having a substantially identical lattice geometry. 

In a crystal atoms are joined to form a larger network with a periodical order in three dimensions. The spatial order of the atoms is called the crystal structure. When we connect the periodically repeated atoms of one kind in three space directions to a three-dimensional grid, we obtain the crystal lattice. The crystal lattice represents a three-dimensional order of points all points of the lattice are completely equivalent and have the same surroundings. We can think of the crystal lattice as generated by periodically repeating a small parallelepiped in three dimensions without gaps (Fig. 2.4 parallelepiped = body limited by six faces that are parallel in pairs). The parallelepiped is called the unit cell. 

Sadakane, M., Asanuma, T., Kubo, J. et al. (2005) Facile procedure to prepare three-dimensionally ordered macroporous (3DOM) perovskite-type mixed metal oxides by colloidal crystal templating method, Chem. Mater. 17, 3546. 

Considering the crystal imperfections that are typically found in all crystals, the crystal quality of organic pigments is a major concern. The external surface of any crystal exhibits a number of defects, which expose portions of the crystal surface to the surrounding molecules. Impurities and voids permeate the entire interior structure of the crystal. Stress, brought about by factors such as applied shear, may change the cell constants (distances between atoms, crystalline angles). It is also possible for the three dimensional order to be incomplete or limited to one or two dimensions only (dislocations, inclusions). 

The porous membrane templates described above do exhibit three-dimensionality, but with limited interconnectedness between the discrete tubelike structures. Porous structures with more integrated pore—solid architectures can be designed using templates assembled from discrete solid objects or su-pramolecular structures. One class of such structures are three-dimensionally ordered macroporous (or 3-DOM) solids, which are a class of inverse opal structures. The design of 3-DOM structures is based on the initial formation of a colloidal crystal composed of monodisperse polymer or silica spheres assembled in a close-packed arrangement. The interconnected void spaces of the template, 26 vol % for a face-centered-cubic array, are subsequently infiltrated with the desired material. 

Note 4 At one time, a number of mesophases were identified as smectic on the basis of their optical textures, but they are in fact soft crystals characterised by very low yield stresses. Hence, these three-dimensionally ordered phases should no longer be called smectic mesophases. They are akin to plastic crystals with some elementary long-range order and are referred to by the letters E, J, G, H, and K. 

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. 

Liquid Crystalline Polymers. One class of polymers that requires some special attention from a structural standpoint is liquid crystalline polymers, or LCPs. Liquid crystalline polymers are nonisotropic materials that are composed of long molecules parallel to each other in large clusters and that have properties intermediate between those of crystalline solids and liquids. Because they are neither completely liquids nor solids, LCPs are called mesophase (intermediate phase) materials. These mesophase materials have liquid-like properties, so that they can flow but under certain conditions, they also have long-range order and crystal structures. Because they are liquid-like, LCPs have a translational degree of freedom that most solid crystals we have described so far do not have. That is, crystals have three-dimensional order, whereas LCPs have only one- or two-dimensional order. Nevertheless, they are called crystals, and we shall treat them as such in this section. 

Crystallinity. Crystals are an ordered, regular arrangement of units in a repeating, three-dimensional lattice structure. Small molecules, which in the liquid state have three-dimensional mobility, crystallize readily when cooled. It is not so easy for polymers, because a repeating unit cannot move independently of its neighbors in the chain. Nevertheless, some polymers can and do crystallize, though never completely. 

Many globular proteins can be crystallized to achieve three-dimensional order. Study of the crystals of a globular protein can lead to a complete determination of its three-dimensional structure. 

Crystals suitable for protein x-ray studies may be grown by a variety of techniques, which generally depend on solvent perturbation methods for rendering proteins insoluble in a structurally intact state. The trick is to induce the molecules to associate with each other in a specific fashion to produce a three-dimensionally ordered array. A typical protein crystal useful for diffraction work is about 0.5 mm on a side and contains about 1012 protein molecules (an array 104 molecules long along each crystal edge). Note especially that, because protein crystals are from 20 to 70% solvent by volume, crystalline protein is in an environment that is not substantially different from free solution. 

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