Molecular confirmation and crystal structure of the β form of poly(ethylene oxybenzoate)

The crystal structure of the β form of poly(ethylene oxybenzoate) was analyzed by x-ray diffraction. Four nearly extended molecular chains pass through a unit cell with parameters a = 8.19 Å, b = 11.07 Å, c (fiber axis) = 19.05 Å, β = 114.8°, and the space group P2₁/n-C. The structural difference between the α and β forms is mainly due to the internal rotation angles for the virtual bond and the ? CH₂? CH₂? bond. They are essentially in trans confirmation in the β form, while the α form contains cis and gauche (? CH₂? CH₂? ) conformations.     

A nontrivial algebraic cycle in the Jacobian variety of the Klein quartic

We prove some value of the harmonic volume for the Klein quartic C is nonzero modulo , using special values of the generalized hypergeometric function ₃ F ₂. This result tells us the algebraic cycle C − C ⁻ is not algebraically equivalent to zero in the Jacobian variety J(C).      

X-ray crystallographic study of solid-state polymerization of trioxane and tetraoxymethylene

An X-ray study has been applied to clarify the relationship between crystal structures and crystal orientations of tetraoxymethylene and polyoxymethylene obtained by solid- state polymerization of tetraoxymethylene. The polyoxymethylene crystal obtained by the solidstate polymerization of a tetraoxymethylene single crystal was found to be definitely oriented with respect to the original tetraoxymethylene single crystal without any aftertreatment; i.e., the polymer chain (the c-axis) is parallel to the b-axis of tetraoxymethylene and the a-axis of polyoxymethylene coincides to the c-axis of tetraoxymethylene.

In addition to the main orientation, other preferred crystalline orientations (“twin structure”) of polyoxymethylene were observed in polymers polymerized at lower temperatures (60-80°C); i.e., the polymer chains are parallel to either the [100], [001], or [101] direction of the original tetraoxymethylene crystal, though the amount of such oriented crystallites is small. The twin structure is different from that of polyoxymethylene obtained from trioxane. This difference is due to the crystal structures of these original materials; tetraoxymethylene occurs in the monoclinic system, while trioxane belongs to the trigonal system, as does polyoxymethylene. These characteristic orientations are explained in terms of the molecular arrangements in the crystals of the original materials.     

Physical Characteristics and Sintering Behavior of MgO-Doped ZrO2nanoparticles

Nanosized particles of 13mol% MgO-doped ZrO₂ with a narrow distribution of pore sizes were prepared by the coprecipitation technique using optimized parameters of synthesis. Transmission electron microscopy analysis of the calcined powder reveals that the majority of the particles have grain sizes in the 10–20nm range. From nitrogen adsorption analysis an average particle size of 13nm was estimated, which is similar to the average pore size diameter (12nm). Besides the unimodal distribution of pore sizes, the linear shrinkage curve of a powder compact exhibits several inflexions indicating different rates of densification up to 1600°C. After sintering at 1600°C for 2h, the microstructure features of a compact are characteristics of the intermediate stage with interconnected porosity preferentially observed at grain boundaries. These results are explained as a size effect of nanoparticles of magnesia-doped zirconia during sintering.     

Structures of three crystal modifications of poly(3,3-dimethyloxacyclobutane)

Three crystal modifications of poly(3,3-dimethyloxacyclobutane) [? CH₂C(CH₃)₂CH₂O? ]_n were found and their structures were analyzed by x-ray diffraction. Modification I is obtained only under tension and disappears on relaxing the tension. From the fiber period of 4.83 Å, the molecular structure seems to be planar zigzag. In modification II, two chains in T₃GT₃? conformation pass through a monoclinic cell with parameters a = 8.93 Å, b = 7.48 Å, c (fiber axis) = 8.35 Å, β = 97.9°, and the space group P2₁/c-C. In modification III, two (T₂G₂)₂ chains pass through an orthorhombic cell with parameters a = 15.60 Å, b = 5.74 Å, c (fiber axis) = 6.51 Å, and the space group, C222₁–D. Molecular conformations of the three crystal modifications correspond to those of polyoxacyclobutane.     

Structural studies of polyethers. IV. Structure analysis of the polyoxymethylene molecule by three-dimensional fourier syntheses

Detailed structure analysis of the polyoxymethylene molecule was carried out by the method of three-dimensional Fourier synthesis. The diffraction data were measured on a highly ordered polyoxymethylene sample prepared from a single crystal of tetraoxane by solid-state radiation polymerization. By assuming the anisotropic thermal factors for the oxygen and carbon atoms and taking into account the contribution of the hydrogen atoms, good agreement between the observed and calculated structure factors was obtained; the R factor is 8.8%. The distance between the oxygen atom and the helix axis r(O) = 0.671 A. is slightly shorter than r(C) = 0.691 A. The bond distance C? O = 1.42₁ A., the angle ∠COC = 112°24′, and ∠OCO = 110°49′. All the internal rotation angles of the skeletal bonds are 78°13′. It is deduced from the experimental results that each crystallite of polyoxymethylene is composed of only one type of helix, right-handed or left-handed.