Close packed lattice structures

An inkjet printing of colloidal crystals was proposed by Frese et describing inkjet printing processes of monodispersed particles which are able to form two- or three-dimensional photonic crystals on the substrate surface by arranging in a closely packed lattice structure on the surface. The particle size was selected so that it will diffract light in the visible spectral region, i.e., particle size of 200-500 nanometers. In this work drop-on-demand inkjet printing techniques are utilized. 

Higher-order multipole moments enhance the forces between particles at short distances and their neglect is extremely questionable, especially if fine effects are looked at, as for instance the ground-state properties of close-packed lattice structures [244,246-251] or the viscosity To go beyond the point dipole approximation Klingenberg and co-workers [ 173,252] developed an empirical force expression for the interaction between two dielectric spheres in a uniform external field from the munerical solution of Laplace s equation [253]. Recently, Yu and co-workers [254,255] proposed a computationally efficient (approximate) dipole-induced-dipole model based on a multiple image method which accounts partially for multipolar interactions. 

Metallic elements are normally found forming close packed lattice structures of types body centered, face centered, or hexagonal close packed. All of them display high coordination numbers, either eight or twelve nearest neighbors. Furthermore metals show a number of other characteristic physical properties such as metallic luster and high thermal and electrical conductivities. Theories dealing with metals must therefore be able to explain these properties. 

The holes in the close-packed structure of a metal can be filled with smaller atoms to form alloys (alloys are described in more detail in Section 5.15). If a dip between three atoms is directly covered by another atom, we obtain a tetrahedral hole, because it is formed by four atoms at the corners of a regular tetrahedron (Fig. 5.30a). There are two tetrahedral holes per atom in a close-packed lattice. When a dip in a layer coincides with a dip in the next layer, we obtain an octahedral hole, because it is formed by six atoms at the corners of a regular octahedron (Fig. 5.30b). There is one octahedral hole for each atom in the lattice. Note that, because holes are formed by two adjacent layers and because neighboring close-packed layers have identical arrangements in hep and ccp, the numbers of holes are the same for both close-packed structures. 

When the radius ratio of an ionic compound is less than about 0.4, corresponding to cations that are significantly smaller than the anion, the small tetrahedral holes may be occupied. An example is the zinc-blende structure (which is also called the sphalerite structure), named after a form of the mineral ZnS (Fig. 5.43). This structure is based on an expanded cubic close-packed lattice of the big S2 anions, with the small Zn2+ cations occupying half the tetrahedral holes. Each Zn2+ ion is surrounded by four S2 ions, and each S2" ion is surrounded by four Zn2+ ions so the zinc-blende structure has (4,4)-coordination. 

Fig. 6.4 Layered structure of LixTiSa, showing the lithium ions between the TiSa sheets. This is an anion close-packed lattice in which alternate layers between the anion sheets are occupied by a redox-active titanium atom. Lithium inserts itself into the empty remaining layers. (Adapted from [68])...

Here, we have arranged the layers on a two-dimensional structure, even though the layers are arranged in three dimensional order. Note that only two crystallographic axes are indicated. We call this the natural stacking sequence because of the nature of the hexagonal close- packed lattice. 

The term crystal structure in essence covers all of the descriptive information, such as the crystal system, the space lattice, the symmetry class, the space group and the lattice parameters pertaining to the crystal under reference. Most metals are found to have relatively simple crystal structures body centered cubic (bcc), face centered cubic (fee) and hexagonal close packed (eph) structures. The majority of the metals exhibit one of these three crystal structures at room temperature. However, some metals do exhibit more complex crystal structures. 

Cubed compound, in PVC siding manufacture, 25 685 Cube lattice, 8 114t Cubic boron nitride, 1 8 4 654 grinding wheels, 1 21 hardness in various scales, l 3t physical properties of, 4 653t Cubic close-packed (CCP) structure, of spinel ferrites, 11 60 Cubic ferrites, 11 55-57 Cubic geometry, for metal coordination numbers, 7 574, 575t. See also Cubic structure Cubic symmetry Cubic silsesquioxanes (CSS), 13 539 Cubic structure, of ferroelectric crystals, 11 94-95, 96 Cubic symmetry, 8 114t Cubitron sol-gel abrasives, 1 7 Cucurbituril inclusion compounds,... 

Compounds of stoichiometry AX2 with six-coordinated A require (according to eqn (11.1)) that X be three coordinate. Since none of the close packed lattices have cage points with three coordination, these structures are less simple. The rutile (202240) and anatase (202242) forms of Ti02 are based on FICP and FCC lattices of Ti respectively, but fitting the ions into positions of three coordination results in distortions that lower the symmetry. An alternative derivation of these structures is described in Section 11.2.2.4 below. 

Not all structures are based on close packed lattices. Ions that are large and soft often adopt structures based on a primitive or body centred cubic lattice as found in CsCl (22173) and a-AgI (200108). Others, such as perovskite, ABO3 (Fig. 10.4), are based on close packed lattices that comprise both anions and large cations. The larger and softer the ions, the more variations appear, but the lattice packing principle can still be used. Santoro et al. (1999,2000) have shown how close-packing considerations combined with the use of bond valences can give a quantitative prediction of the structure of BaRuOs (10253). 

In this review, we introduce another approach to study the multiscale structures of polymer materials based on a lattice model. We first show the development of a Helmholtz energy model of mixing for polymers based on close-packed lattice model by combining molecular simulation with statistical mechanics. Then, holes are introduced to account for the effect of pressure. Combined with WDA, this model of Helmholtz energy is further applied to develop a new lattice DFT to calculate the adsorption of polymers at solid-liquid interface. Finally, we develop a framework based on the strong segregation limit (SSL) theory to predict the morphologies of micro-phase separation of diblock copolymers confined in curved surfaces. 

Monodisperse spherical colloids and most of the applications derived from these materials are still in an early stage of technical development. Many issues still need to be addressed before these materials can reach their potential in industrial applications. For example, the diversity of materials must be greatly expanded to include every major class of functional materials. At the moment, only silica and a few organic polymers (e.g., polystyrene and polymethylmethacrylate) can be prepared as truly monodispersed spherical colloids. These materials, unfortunately, do not exhibit any particularly interesting optical, nonlinear optical or electro-optical functionality. In this regard, it is necessary to develop new methods to either dope currently existing spherical colloids with functional components or to directly deal with the synthesis of other functional materials. Second, formation of complex crystal structures other than closely packed lattices has been met with limited success. As a major limitation to the self-assembly procedures described in this chapter, all of them seem to lack the ability to form 3D lattices with arbitrary structures. Recent demonstrations based on optical trapping method may provide a potential solution to this problem, albeit this approach seems to be too slow to be useful in practice.181-184 Third, the density of defects in the crystalline lattices of spherical colloids must be well-characterized and kept below... 

The rhombohedral unit cells for rhodium and iridium trifluorides (44) contain two formula units. The structure can be related to the first structure type by considering anion positions, which here correspond to a hexagonal, close-packed array. There are no vacant anion sites and the cations occupy one-third of the octahedral holes. This leads to M—F—M angles of 132°, characteristic for filling adjacent, octahedral holes in a hexagonal close-packed lattice. Alternatively, the structure can be described as a linking of octahedra through all corners, but the octahedra are now tilted with respect to each other. 

The relationship between cubic close-packed (ccp) structures and ionic compounds of type B1 is obvious. Interstitial sites with respect to metal positions are at fractional coordinates of the type 00 and equivalent to the ionic sites in Bl. The Madelung constant of Al type metals with interstitially localized free electrons is therefore the same as that of rocksalt structures. It is noted that the interstitial sites define the same face-centred lattice as the metal ions. 

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