PRISM formalism

The outline of the paper is as follows. In Sect. 2 we describe the basic RISM and PRISM formalisms, and the fundamental approximations invoked that render the polymer problem tractable. The predicticms of PRISM theory for the structure of polymer melts are described in Sect. 3 for a variety of single chain models, including a comparison of atomistic calculations for polyethylene melt with diffraction experiments. The general problem of calculating thermodynamic properties, and particularly the equation-of-state, within the PRISM formalism is described in Sect. 4. A detailed application to polyethylene fluids is summarized and compared with experiment. The develojanent of a density functional theory to treat polymer crystallization is briefly discussed in Sect. 5, and numerical predictions for polyethylene and polytetrafluoroethylene are summarized. 

The conditions required for accuracy of an incompressibility assumption at the level of the scattering functions and spinodal condition are easily derived within the PRISM formalism [67]. From Eq. (6.7) a small isothermal compressibility implies that — oCw -fOl P 1, which is generally true for any dense fluid. If the related wavevector-dependent condition... 

Polymer Alloys. Perturbation of meltlike conformation upon transfer to a multicomponent environment is not understood. The influence of proximity to phase boundaries, coupled density and concentration fluctuations, and mixture composition on both single-chain dimensions and miscibility are problems that have begun to be addressed within the PRISM formalism for the simple symmetric blend model by Singh and Schweizer and symmetric diblock copolymer model by David and Schweizer,"" and other more coarse-grained field-theoretic approaches. Comparisons with the few available simulations " have also... 

Constrained Polymers. The conformation of polymers constrained in various ways, for example, grafted to a flat surface ( brush ), adsorbed on a spherical colloidal particle, or tethered to a central branch point as in star polymers. All such problems involve potentially large nonideal conformational effects and also introduce additional complications associated with site inequivalence within the PRISM formalism. Progress for star polymers is briefly described in the next section. 

The above physical features imply that a fully self-consistent treatment of intramolecular and intermolecular pair correlations is more important for star polymers than linear chains, and the concept of ideality is expected to be of much less utility even at high melt densities. The treatment of star polymers within a self-consistent PRISM formalism has been very recently pursued by Grayce and Schweizer. Here we give a brief description of some of the essential theoretical modifications... 

The phosphido complex, U(PPP)4 [163823-64-9] (PPP = P(CH2CH2P(CH3)2), was prepared and fully characterized (216). This complex was one of the first actinide complexes containing exclusively metal-phosphoms bonds. The x-ray stmctural analysis iadicated a distorted bicapped triganol prism with 3—3-electron donor phosphides and 1—1-electron phosphide, suggesting a formally 24-electron complex. Similar to the amido system, this phosphido compound is also reactive toward iasertion reactions, especially with CO (216). 

Further oxidation of the nine-atom clusters to formal [Ge9] leads to linear polymers oi[-Ge9-] with two covalent intercluster bonds (Fig. 2i). Trimers [Ge9=Ge9=Ge9] (Fig. 6a) and tetramers [Ge9=Ge9=Ge9=Ge9] (Fig. 6b) occur via nonclassical bond formation between two neighboring atoms of the triangular prism basis planes of the c/oio-shaped clusters, which results in Ge-Ge-Ge bond angles of 90° and in considerably longer Ge-Ge contacts between the cluster units. Quantum-chemical calculations have shown that the exo-bonds participate in a delocalized electronic system that comprises the whole anion [204]. 

In this structure the (CuOz) layers are flat (z(Cu) - z(0) - 0.0). The coordination of the Cu atoms is square-planar while the atoms N = Ca0 ggSrQ are eight-coordinated and the coordination polyhedron is a square prism. In this compound, the copper is formally 2+. 

The trisulphides (and triselenides) of Ti, Zr, Hf, Nb and Ta crystallize in onedimensional structures formed by MSg trigonal prisms that share opposite faces. Metal atoms in these sulphides are formally in the quadrivalent state, and part of the sulphur exists as molecular anions, M S2 S . TaSj shows a metal-insulator transition of the Peierls type at low temperatures (Section 4.9). NbSj adopts a Peierls distorted insulating structure suggesting the possibility of a transformation to a metallic phase at high temperatures, but does not transform completely to the undistorted structure. Electronic properties and structural transitions of these sulphides have been reviewed (Rouxel et al, 1982 Meerschaut, 1982 Rouxel, 1992). 

Anyone who has seen the well-formed crystals of minerals in our museums must have been impressed by the great variety of shapes exhibited cubes and octahedra, prisms of various kinds, pyramids and double pyramids, flat plates of various shapes, rhombohedra and other less symmetrical parallelepipeda, and many other shapes less easy to describe in a word or two. These crystal shapes are extremely fascinating in themselves artists (notably Durer) have used crystal shapes for formal or symbolic purposes, while many a natural philosopher has been drawn to the attempt to understand first of all the geometry of crystal shapes considered simply as solid figures, and then the manner in which these shapes are formed by the anisotropic growth of atomic and molecular space-patterns. 

Dianion [Co8(CO)i8C)] is isoelcctronic with Rh8(CO)i9C but presents a very different stereochemistry as shown in structure XX (Fig. 8) (13). Its geometry can be described as a deformed square antiprism with a carbide atom in the center of the polyhedron. This structure can be derived from that of the bicapped trigonal prism by stretching the common edge of the two capped square faces, as indicated in Fig. 8 by the dotted line. In such a process the loss of an M—M bond formally generates a square face. The idealized symmetry of a tetragonal antiprism is Did. However,... 

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