Polyethylene, dispersion curves

The dispersion curves of polyethylene have been the subject of intense study by several groups. The best available are those of Barnes and Franconi [8]. The model is a purely empirical ball-and-springs force-field with Williams-type potentials to include the intermolecular forces. The potentials have the form ... 

Fig. 10.3 Comparison of the dispersion curves [8] with the INS spectrum of polyethylene for the region 600-1600 cm. The horizontal lines show the van Hove singularities indicating where peaks in the INS spectrum are expected.

To understand the reasons for the differences between the experimental spectra and that predicted by the models, it is necessary to consider the data from which the models are built. The only way to directly measure the dispersion curves of a material is by coherent INS spectroscopy. For hydrogenous polyethylene, this method fails because the background caused by the incoherent scattering from hydrogen completely swamps the coherent signal ( 2.1.1). For perdeuterated polyethylene, the larger coherent and smaller incoherent cross sections of deuterium (see Appendix 1) has allowed the vs acoustic branch to be mapped by coherent INS [11]. 

Information on the dispersion curves of polyethylene has been obtained indirectly by studying the normal, unbranched, alkanes. The n-alkanes form a homologous series, are oligomers of polyethylene and have the same masses (-CH2- repeated n times). As more CH2 rmits are added they are constrained by the same forces in chains that simply grow longer. 

By combining the information drawn from ab initio (mode symmetry and frequency) and the infinite chain model (ky and frequency), the LAM modes above 200 cm can be unambiguously assigned [14]. Table 10.1 lists the assignments from pentane (n = 5) to pentacosane (n = 25). Fig. 10.8 shows the observed and calculated LAM modes, the dashed line is a polynomial fit and is the best available approximation to the V5 dispersion curve of polyethylene. The INS data for k > 0.7 is absent because it is not possible to assign the modes owing to the spectral... 

Fig. 10.8 The observed LAM modes ( ) and those calculated by DFT (o). The dashed line is a polynomial fit and is the best estimate of the V5 dispersion curve of polyethylene.

Early work on polyacetylene was only able to cover part of the spectral range and apparently suffers fi"om a calibration error [28]. Fig. 10.17a shows the INS spectrum of polyacetylene [29] recorded on TFXA. The spectrum in the region below 700 cm is remarkable in that it consists of a series of terraces, each terminating in a bandhead. This is reminiscent of the Vs mode of polyethylene and suggests that the modes are strongly dispersed. This is confirmed by the dispersion curves [30], and the resulting INS spectrum calculated from them. Fig. 10.17b and c. 

S. Hirata S. Iwata (1998). J. Chem. Phys., 108, 7901-7908. Density functional crystal orbital study on the normal vibrations and phonon dispersion curves of all trans-polyethylene. 

The internal modes of typical molecular crystals show little dispersion and a calculation of the frequencies at the F point in the Brillouin zone is usually sufficient for good agreement between observed and calculated spectra. However, there are many examples where significant dispersion is present the alkali metal hydrides ( 6.7.1), graphite ( 11.2.2) and polyethylene ( 10.1.1.1) being notable cases. In these instances, a calculation of the full dispersion curves are needed. In... 

Fig. 2. Vibrational dispersion curves for polyethylene plotted along Sc = Q,nf(x ...

Figure 3-3. Dispersion curves of crystalline orthorhombic polyethylene with two molecules per unit cell (from [49]). Comparison with Figure 3-1 shows the splitting of the frequency branches and the shape of the acoustical branches at p 0 (see text). Attention should be paid to the fact that the frequencies and the shape of the dispersion curves shown in Figure 3-1 and in this figure may differ because they have been calculated with two different force fields.

Figure 3-39. Dispersion curves of single-chain polyethylene, indicating the scheme of Fermi resonances which can occur between phonon of different branches.

Fig. 10. Variation of an. average drop diameter during compounding in an intermesliing, co-rotating twin screw extruder for 5 vol% polyethylene dispersed in polystyrene, extruded at three screw speeds, N = 150,200, and 250 rpm, at a throughput 0 = 5 kg/hr. The points are experimental, the curves computed from model-2.

Fig. IILl. Frequency-dispersion curves of the isolated chain of polyethylene. From Tasumi, Shimanouchi, and Miyazawa (1962)...

The frequency distribution of the isolated polyethylene chain was calculated first by Wunderlich (1962b) Irom the frequency dispersion curves of Tasumi, Shimanouchi, and Miyazawa (1962). However, neutron scattering cross sections depend on vibrational displacements of scattering nuclei as well as vibrational frequencies. Accordingly, the frequency distribution of the polyethylene chain weighted with squared amplitudes of hydrogen nuclei was calculated as shown in Fig. III.3 by Kitagawa and Miyazawa (1967 b). 

The frequency-dispersion curves of the acoustic branches (Vj and Vg) of polyethylene are shown in Fig. III.1. For the phase difference of 90°, the vibrational frequencies reach maxima and give rise to the frequency-distribution peaks near 550 and 200cm . These peaks were in fact observed by Danner, Safford, Boutin, and Berger (1964), Myers, Donovan, and King (1965) [to be discussed in detail in Section VII]. The vibrational modes of the two acoustic branches for the phase difference of 90° are schematically shown in Fig. III.4. 

The dynamical matrix D iS) of orthorhombic polyethylene was constructed by Kitagawa (1968), Kitagawa and Miyazawa (1968a, 1970b) with 36 Cartesian symmetry coordinates S S) (12 for carbon and 24 for hydrogen atoms). The dynamical matrix of a fc-group may be factorized into submatrices of symmetry species and dispersion curves of different species may intersect each other, but those of the same species do not. Frequency-dispersion curves of the eight vibrational branches below 600 cm are shown in Fig. V.4(a) 4(h). 

Fig. V.4(a) 4(h). Frequency-dispersion curves of the orthorhombic polyethylene crystal. From Kitagawa (1969), Kitagawa and Miyazawa (1968 a)...

Fig. VII.S. Frequency-dispersion relation of the orthorhombic crystal of perdeuterated polyethylene. Open circles are for experimental data of Feldkamp, Venkataraman, and King (1968) and solid lines are dispersion curves (and symmetry assigments) calculated by...

Fig. 1 Dispersion curves of the molecular vibration of planar zigzag polyethylene chains.

Fig. 1. Dispersion curves ot an infinite polymethylene chain (solid Unes) and an orthorhomtw crystal of polyethylene (the dt iied Unes show vflnations perpendicular to the a-axis)L From Tasumi and Krimm, Ref. [6]...

Field-cycling NMR relaxometry is a versatile and powerful method for investigating molecular dynamics over a large range of timescales. It has been applied to manifold materials which show broad distributions of molecular motions, for example proteins, liquid crystals, synthetic polymers, and liquids confined in porous materials. Figure 7A represents an example for the investigation of polymer dynamics. The T -dispersion curve in the double-logarithmic scale shows the typical slopes observed in polyethylene oxide melts above the critical molecular... 

Fig. 2.28. The dispersion curve for an infinite i.solated chain of methylene units (crystalline polyethylene).

MGr (for polyethylene, for example) a slight increase of dielectric permeation at frequencies from 10 kHz to 60 MHz is observed s = 121 for non-radiated and s = 2.29 for polymer radiated by 640 kGr dose [163], This indicates the predominant influence of the anomalous dispersion curve run on the refractive index change n =/(/ ) and the possibility of at least one more part of increment and saturation tSn at high radiation doses of polymeric materials, isotactic PP, in particular, compared with doses indicated in this paragraph. 

A universal calibration curve was established by plotting the product of the limiting viscosity numbers and molecular weight, Mw[iy], vs. the elution volume, EV, for a variety of characterized polymers. The major usefulness of the universal calibration curve was to validate individual molecular-weight values and to provide extended molecular-weight calibration at the ends of the calibration curve where fractions of narrow dispersion of the polymer being analyzed are not available. The calibration curve was monitored daily with polystyrene fractions certified by Pressure Chemicals. The relationship between the polyethylene fractions and polystyrene fractions was determined using the universal calibration curve. 

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