Cylindrical Distribution Function Research Articles

Local Structure of Atactic Polystyrene Investigated by Molecular Dynamics Method

Molecular dynamics computer simulation of three substances ethylbenzene (EB), pentastyrene (PS-5), and polystyrene-25 (PS-25) was performed to investigate the local order of the phenyl rings in monomers and side-chain polymers. Monomer molecules (EB) tend to be in T-configuration, which corresponds to isotropic local structure. Phenyl rings in chained molecules PS-5 and PS-25 partly cooperate in both parallel-displaced and “sandwich” configuration with π–π stacking. These configurations are locally anisotropic and lead to the increasing of Kerr constant K. Analysis of the local structure was performed by calculating the cylindrical distribution function.

Implementation of a cylindrical distribution function for the analysis of anisotropic molecular dynamics simulations.

The cylindrical distribution function (CDF) is a convenient anisotropic analogue of the radial distribution function, the difference being the use of cylindrical shells for binning. As such, CDF analysis can be a powerful tool for the analysis of positional correlations within anisotropic systems, such as liquid crystals. Here we describe a lightweight Python tool, cylindr, for the calculation of cylindrical distribution function, which is compatible with the output of a number of popular MD engines. We demonstrate the use of cylindr in computing the CDF of a number of exemplar materials: classical and ferroelectric nematics; lamellar and columnar liquid crystals.

Projection method for the Hankel transform

The numerical projection method for the zero-order Hankel transform inversion for the case of data given on a finite interval has been developed. The justification of the convergence of the projection method has been done for the general case that also includes the sine-Fourier transform inversion. An inequality on the norms of the zeroth-order Laguerre functions has been proved. The efficiency of the method was illustrated with the test data and for the problem of cylindrical distribution function calculation using the melt surface layer diffraction data.

Structure and dynamics in methanol and its lithium ion solution confined by carbon nanotubes

Molecular dynamics (MD) simulations have been carried out in the NVT ensemble to explore the particle dynamics and microscopic structure of the liquid methanol (MeOH) and infinitively diluted solution of Li + in MeOH confined in carbon nanotubes (CNTs) with diameters of 1.087 nm, 2.035 nm and 2.984 nm. The simulated local order of confined fluid defined in terms of radial distribution function (RDF) and running co-ordination numbers (r.c.n.) does not differ from bulk. Although the long-range structure defined in terms of cylindrical distribution function and distribution of cosines appears to be helix-like in the parietal area of the CNT. Translational self-diffusion coefficients inside the CNTs are about two times lower than those of bulk and at the same time depend slightly on the CNT diameter. It was clearly observed that diffusion coefficients of MeOH and Li + strongly depend on the distance from the CNT inner wall.

The short-range order in the surface layers of Cu-Au (Ge) melts by electron diffraction

An apparatus for measurement of electron beam diffraction from the surface of a melted droplet is described. The basic features of the method are discussed. The main results for the binary melts are presented. An effective algorithm is proposed to determine the cylindrical distribution function of the atoms in the melt surface by diffraction experiment data.

Current-voltage relationship in the auroral particle acceleration region

Abstract. The current-voltage relationship in the auroral particle acceleration region has been studied statistically by the Akebono (EXOS-D) satellite in terms of the charge carriers of the upward field-aligned current. The Akebono satellite often observed field-aligned currents which were significantly larger than the model value predicted by Knight (1973). We compared the upward field-aligned current estimated by three different methods, and found that low-energy electrons often play an important role as additional current carriers, together with the high-energy primary electrons which are expected from Knight's relation. Such additional currents have been observed especially at high and middle altitudes of the particle acceleration region. Some particular features of electron distribution functions, such as "cylindrical distribution functions" and "electron conics", have often been observed coinciding with the additional currents. They indicated time variability of the particle acceleration region. Therefore, we have concluded that the low-energy electrons within the "forbidden" region of electron phase space in the stationary model often contribute to charge carriers of the current because of the rapid time variability of the particle acceleration region. "Cylindrical distribution functions" are expected to be found below the time-varying potential difference. We statistically examined the locations of "cylindrical distribution function", and found that their altitudes are related to the location where the additional currents have been observed. This result is consistent with the idea that the low-energy electrons can also carry significant current when the acceleration region changes in time.

Interatomic correlation in partially oriented polymer fibers

AbstractThe atomic ordering in materials possessing cylindrical symmetry can be investigated by an approximation of the cylindrical distribution function in a given direction. The approximation gives correctly the extent of the interatomic correlation for a wide class of polymer fibers and reduces considerably the amount of the experimental and computational times.

X-ray studies on the fine structure of high strength polyacrylonitrile fiber.

The fine structure of a high-strength homopolymer-type polyacrylonitrile (PAN) fiber prepared by additional heat drawing after usual wet spinning was investigated using wide (WAXD) and small (SAXS) angle X-ray scattering in comparison with those of ordinary homo- and copolymer-type PAN fibers prepared by usual wet spinning method. WAXD data were analyzed by a cylindrical distribution function (CDF) method. The meridional section of CDF showed that the crystal chain has a fiber periodicity of 2.5A in all these three kinds of fibers. The periodicity deteriorated at 12A and 10A in the high-strength homopolymer fiber-type and homo-and copolymer-type fibers. The CDF equatorial section of the high strength homopolymer-type fiber showed peaks at 6.5A, 11.8A and 16.4A. For investigating the long spacing, the meridional SAXS intensity of the fibers was measured before and after thermo-oxidative treatment at 200°C-300°C. Both ordinary homo- and copolymer-type fibers showed SAXS peaks due to the long spacing after oxidative heat-treatment. In the case of high strength homopolymer-type fiber, however, no SAXS peak appeared even after thermo-oxidation. This is the most important structural character of the high strength homopolymer-type fiber.

X-ray CDF study on the behaviour of the amorphous chain under extension

The behaviour of the amorphous chain in vulcanized natural rubber (NR) and styrene-butadiene rubber (SBR) under extension is studied by means of the cylindrical distribution function (CDF) derived from wide angle X-ray scattering (WAXS). In the case of the highly stretched vulcanized NR where the orientation-induced crystallization took place, the observed WAXS intensity was separated into a crystalline peak component and a broad amorphous one from which the CDF was obtained. The CDF calculated from the broad component shows that some of the amorphous segments of the highly stretched vulcanized NR are extended as fully as the crystalline chains and that the fraction of the segments increases with increasing extension. In the case of SBR, such fully stretched segments as in the case of the vulcanized NR show higher degree of an interchain packing order than those in the vulcanized NR.

Structure of polystyrene glasses

Wide-angle X-ray scattering from both unoriented and axially deformed glassy polystyrenes (atactic and quenched isotactic) is compared with that calculated for isolated molecules in different conformations. No satisfactory fit is obtained. It is apparent that the scattering beyond s=1.0 A ̊ −1 is very similar to that from benzene and styrene, having a large contribution from contacts between phenyl groups attached to neighbouring molecules which are not represented in a single chain model. The only significant difference between the scattering of the polymer and the two low molecular weight liquids is that there is a small peak at s=0.75 A ̊ −1 (0.62 Å −1 for the isotactic glass) which forms on polymerization and was first reported by Katz in 1927. For drawn samples the peak intensifies on the equator and apparently represents interchain correlations. However, in comparison with those of other non-crystalline polymers this interchain peak is weak, at a surprisingly low scattering angle in relation to the expected spacing of the chains and shows a very significant increase in intensity with increasing temperature. A model is proposed in which the phenyl groups segregate on a molecular scale to form stacks; there are fewer stacks than chain backbones and they have a low electron density core which expands considerably in relation to its small diameter as the sample temperature increases. The model accounts for the position and temperature sensitivity of the interchain ‘polymerization’ peak. It also shows some similarities to the organization of isotactic molecules in the crystalline state. The stacks of phenyl groups contribute to the X-ray pattern as if they were flexible superchains. The cylindrical distribution function derived from the scattering pattern of the oriented polymer indicates that the phenyl groups are in register in all directions over distances of the order of several chain diameters.

Structural analysis of an oriented liquid crystalline copolyester

Wide angle X-ray scattering from an aligned sample of a random copolyester is presented, together with an analysis of the derived cylindrical distribution function. High local orientational correlation of chains is observed, although the global orientation is less marked. There is some evidence of lateral biaxiality, which is compared with that predicted for theoretical studies of liquid crystalline structures.

Structure of oriented polystyrene monofilaments and its relationship to brittle‐to‐ductile transition

AbstractOriented atactic polystyrene monofilaments show brittle‐to‐ductile transition in the vicinity of a degree of birefringence Δn = −2 × 10−3 (at a temperature of 20°C and at the stretching rate of 100%/min) independently of the various spinning conditions. The amorphous orientation of a cylindrically symmetric system was investigated by wide‐angle x‐ray scattering experiments. The orientation of polystyrene monofilaments in real space is denoted by P2(cos α) within the precision of measurement. The coefficient D2(r) of the second term in an expansion of cylindrical distribution function in spherical harmonics has two main peaks, near r = 1.6 Å and r = 4 ∼ 5 Å. The negative peak near r = 1.6 Å indicates orientation of phenyl groups in a plane perpendicular to the fiber axis. The positive peak near r = 5 Å (brittle region) or near r = 4 Å (ductile region) indicates the piling up of phenyl groups for the direction of the fiber axis. It is most probable that the amorphous state of atactic polystyrene consists of an appropriate cohesion of planar zigzag chains of syndiotactic polystyrene. The lower spacing (r = 4 Å) of the positive second peak in ductile region suggests that there are interchain phenyl groups near r = 5 ∼ 6 Å in a plane perpendicular to the fiber axis and that the molecular chains are extended parallel to the direction of the fiber axis. This parallel packing of chain segments along the fiber axis suppresses the formation and growth of cracks of polystyrene monofilaments resulting in ductile fracture.

On the Calculation of the Molecular Cylindrical Distribution Function and the Order Parameter from X-Ray Diffraction Data of Liquid Crystals

A general formula is derived for the x-ray diffraction from an assembly of elongated molecules, and it is shown that this formula is too complex to be solved. To enable the calculation of the molecular cylindrical distribution function (mcdf) and the order parameter from the x-ray data, the approach taken in the literature has been to assume that only molecules approximately perpendicular to the scattering vector contribute significantly to the scattered radiation. Calculations on a model indicate, however, that this ``literature approximation'' is totally unjustified in the derivation of the mcdf, and, consequently, that the mcdf's obtained from x-ray data must be regarded as of no value. For the order parameter, the model calculations show that the errors made in using the ``literature approximation'' are not as large as in the case of the mcdf, and it appears that under favorable conditions the x-ray data may yield approximate values for the order parameter.

Structure of Oriented Sodium Metaphosphate Glass Fibers

The structure of oriented sodium metaphosphate glass fibers has been investigated employing cylindrical distribution functions obtained from the x-ray scattering pattern. Although this method has rarely been used, it is the only sound one available. The difficulties in interpreting the functions were eased somewhat by the use of difference, rather than complete, cylindrical distribution functions. The former are usually less cluttered with spurious peaks than are the latter. The results indicate that the NaPO3 fibers are made up of long chains of PO4 tetrahedra, and that the axes of these chains have a strong preference for lying along the fiber axis direction. Both the general appearance of the distribution functions and the results of normal distribution function calculations agree with the existence of a very high degree of orientation in the fibers. The locations of sharp peaks on the functions suggest that the individual PO4 tetrahedra have shapes similar to that found in crystalline RbPO3. In the latter material the average in-chain P–O distance is 1.62 Å and the average out-of-chain distance is 1.46 Å. Best agreement with the experimental distribution functions is obtained with a puckered version of a planar chain, in which each tetrahedron is tilted relative to the chain axis through two mutually perpendicular angles of 20° to 30° and the intratetrahedral angles are near 110°. The high effective radial resolution obtained in the difference cylindrical distribution functions, as a result of the angular resolution, suggests that the simplified cylindrical distribution function will be a useful way to study the structures of orientated amorphous polymer fibers.

Application of the Simplified Cylindrical Distribution Function: Orientation in Linear Polyethylene

The simplified cylindrical distribution function has been applied to the determination of the degree of orientation in a moderately oriented linear polyethylene fiber. The function has been calculated using the first two and ten lines of the scattering pattern and, though poorly resolved, yields useful orientation information. It is concluded that the greater the knowledge of the molecular structure of the material under investigation the less the amount of intensity data required to produce a useable function. Although the technique is applicable to both crystalline and noncrystalline materials, the standard methods, applicable only to crystalline materials, are simpler. Therefore, the method can be considered to apply mainly to moderately oriented noncrystalline fibers.

A Simplified Cylindrical Distribution Function

A simplified cylindrical distribution function is proposed, which yields a considerable amount of structural information concerning moderately oriented noncrystalline fibers. This function can be obtained with only a small fraction of the labor required for the complete cylindrical distribution function. It is the transform of the difference between the scattered intensity and its average over the polar angle in reciprocal space, and is calculated by means of an expansion of the Fourier-Bessel integral in spherical harmonics. Because all isotropic contributions to the scattered intensity are automatically omitted from the calculation, no consideration need be given to independent scattering, air scattering or, as a rule, the contribution of extraneous wavelengths. The resulting function is the difference between the cylindrical and radial distribution functions on an arbitrary scale, measured from an unknown baseline. Although these features limit the information obtained, that which is obtained is of considerable use in elucidating fiber structures.