Polyethylene band structure

Propylene content in EPM rubber can be determined with the help of IR spectra. A propylene band near 1155 cm 1 has been widely used [79] for EPM analysis, frequently in combination with the polyethylene band at 721 cm"1. Tacticity is important in EPM rubber, and the bands at 1229 and 1252 cm"1 are characteristic of syndiotactic and isotactic structures, respectively, (both bands are present in atactic polypropylene as well). Polymer structure may vary in the relative tactic placement of adjacent head to tail propylene units and in the sequence distribution of base units along the chain. Some of them can be identified [80] by infrared spectra, such as isolated or head to tail propylene units ... 

Experimental information on the valence levels comes essentially from photoemission XPS and UPS measure densities of states (DOSs) convoluted with absorption cross sections, and these DOS values can be compared with those computed from VEH valence-band structures [195]. This has now been done for several CPs and the agreement is good. It would be more instructive to compare the actual band structure to angle-resolved (ARUPS) measurements, but this has never been done. What comes nearest is an ARUPS study of a series of long alkanes taken as models for polyethylene, a nonconjugated polymer [196]. 

Fig. 4.13 Theoretical band structures for polyethylene, solid lines filled states (valence bands), dashed lines empty states (conduction bands) for details see text. Reproduced with permission of the American Institute of Physics from Stile et al. (2000).

Recent calculations of the band structure of polyethylene have employed variations of the ab initio method incorporating electron correlation. Sun and Bartlett (1996) utilised many-body perturbation theory to encompass electron correlation in the ab initio framework. Siile et al. (2000) and Serra et al. (2000) have employed variants of DFT. These calculations involved the optimisation of local effective potentials and a local-density approximation respectively. Figure 4.13 shows a comparison of the band structure obtained by Siile et al., Fig. 4.13(d), with those obtained by other ab initio DFT calculations using the Hartree-Fock (HF), Fig 4.13(a), and Slater approaches, Figs. 4.13(b) and (c). 

Results of photoemission studies of polyethylene have shown definite evidence for wide energy bands among deep valence orbitals ( ), but the nature of the fundamental absorption edge has not been resolved. Band structure calculations predict direct interband excitations to occur above 12.6 eV (.8) whereas the absorption threshold is at 7.2 eV and a strong peak in e occurs at 9.0 eV. The momentum dependence of the absorption threshold indicates that the threshold is of excitonic origin, i.e. the excitation is localized by the strong electron-hole or configuration inter-... 

CNDO/2 calculated band structure of all-trans-polyethylene. 

Figure I. Electronic structure of all-trans polyethylene (A) valence band structure (B) density of states histogram and its integration curve (C) experimental ( ) and theoretical (-----------------------) XPS spectra.

Polymer Conformation and Crystallinity. Beyond the stereoregularity and tacticity, the geometrical conformation of the polymer chain in the solid material could influence its electronic structure, through a modification of its valence band molecular orbitals. Indeed, a few years ago, very characteristic band structures were calculated for T, G, TG, and TGTG polyethylenes ( ). More recently. Extended Huckel crystal orbital calculations showed that for isotactic polypropylene, a zig-zag planar or a helical conformation resulted in significant changes in the theoretical valence band spectra, supporting the idea that conformation effects could be detected experimentally by the XPS method ( ). 

Figure 2 shows the HF, two DFT variants (LDA and gradient corrected BLYP), and MBPT(2) band structures of polyethylene calculated with the 6-31G basis set [66]. The MBPT(2) bands are above the HF and below the DFT ones. The correlation shift is different at different points in the bands, being around 2 eV for the first two bands. For the third band, the shift is about 5 eV at 0 and 3 eV at n/a. Table 3 list the calculated and measured peaks, the positions of which are not sensitive to the distribution of the incident radiation in the spectra [66]. The HF values are much larger than the measured data while DFT results are too small. As expected, MBPT(2) greatly reduces the... 

A preferentially and a sheaf-like aggregation with random in-plane orientation are observed for the thinner films (thicknesses of 0.1, 0.2 and 0.4 pm in panels a-c). By contrast thick films (0.6 pm and thicker, panel d) show a morphology that resembles the well known (bulk) spherulitic form with a banded structure, characteristic of linear polyethylene crystallized from the melt at moderately high undercooling. 

Theoretical interpretations rapidly followed to support the analysis of the ESCA core levels [36]. The theoretical analysis of ESCA valence spectra has been less direct. The ESCA spectmm and a semi-empirical band structure of polyethylene devised by Wood et al. are sketched in Fig. 36.4. It shows that band stmcture is not directly measurable and that transformations have to be applied such that calculated data is in a form readily comparable to experiments. 

Fig. 36.4. Sketch of the theoretical band structure and experimental density of states of polyethylene.

Finally, we show in Figure 12 the experimental band structures together with the calculated ones for polyethylene.62 These density-functional results (with a local-density approximation) demonstrate a very good agreement between theory and experiment, but it should be stressed that the experimental data have been shifted rigidly about 2 eV upwards in energy, which is consistent with the results above. We add that band structures from Hartree-Fock calculations were in general too wide,63 which is a common deficiency of Hartree-Fock calculations. 

Figure 12 Experimental (points) and theoretical (solid curves) band structures for polyethylene...

There are also QP (quasi particle correlated band structure) calculations for polyethylene (PE)66,67 and polytetrafluorethylene (teflon).67 In the PE case a G-31G and dementi s double basis,68 respectively, was applied. In both calculations66,67 a full geometry optimization was performed. With the G-31G basis a gap of 10.3 eV was obtained, it increased, however, with the poorer double basis of dementi to 11.6 eV, ev max(0 (= —JP) lies at — 8.2 eV while the experimental values of the ionization potential are at 7.6-8.8 eV.69 On the other hand the gap value estimated on the basis of experiment is at 8.8 eV,69... 

Most polymers (typified by polystyrene and polyethylene) are electrically insulating and have conductivities doped with iodine to become electrically conducting (values have now been reported up to olO Scm ) represented a pivotal discovery in polymer science that ultimately resulted in the award of the Nobel Prize for Chemistry in 2000 [4]. The study of electrically conducting polymers is now well advanced and two extremes in the continuum of transport mechanisms exist. If the charge carriers are present in delocalized orbitals that form a band structure along the polymer backbone, they conduct by a delocalization mechanism. In contrast, isolated groups in a polymer can function as acceptors or donors of electrons and can permit... 

Experimental results on the band dispersion in o-bond polymers are very limited due to difficulty in preparing thin films with oriented chains [20, 31, 32, 62]. Here, we introduce the band dispersion of quasi-one-dimensional polymer polyethylene. Early work on the band structure study was carried out on systems with alkyl chains and was aimed at understanding the electronic structure of polyethylene, in particular, the possible existence of one-dimensional band structure in thin films where molecular chains assemble via weak interchain interactions. There is renewed interest in the band dispersions as they determine carrier transport properties in nanoscale molecular electronics [63]. 

The third part of Chapter 10 describes the calculation of various mechanical properties (such as bulk modules and shear modules) of some simple polymers based on their band structure but corrected for correlation. In the case of polyethylene and some of its halogenated derivatives the theoretical results are compared with available experimental data. 

After a number of semiempirical (extended Huckel, CNDO/2,< > INDO,< and MINDO< >) calculations in 1970, Andre performed the first ab initio band-structure computation of an infinite polyethylene chain. Since this polymer is mass-produced in the plastics industry, the theoretical study of its properties (especially of its mechanical properties, see Chapter 10) is of great practical interest, which explains the large... 

After a number of semiempirical crystal-orbital (CO) calculations (extended Hvickel and CNDO/2 CO calculations for further references see Table III of Andre ) Otto et performed the first ab initio band-structure calculations on different fluorinated polyethylenes. The calculations were conducted for all six different polyfluoroethylenes, namely (CFH-CHj), (CF2-CH2), (CFH-CFH),, (CFH-CFH), (CF2-CFH)jt, and (CF2), which can be obtained from polyethylene through fluorine substitution. 

Inspection of the five water clusters (see Figure 7.1) around a cytidine unit shows, however, that most of the water molecules are situated in the immediate neighborhood of the cytosine molecules (forming the larger part of their first hydration shell) while only some of the water molecules are further apart than in the case of a cytosine stack. Therefore, the error is less than one would expect without looking at the details. When the motivation of this model calculation is understood in the sense discussed above, it can be regarded as the next step after the work of Clementi, who calculated the band structure of polyethylene in the field of periodic point charges situated around it in different ways. 

As the next example we take the (CH ), polyethylene (PE) chain. In a subsequent calculation Suhai also applied the intermediate exciton theory with QP one-particle energies to this chain. Using again a 6-31G basis he calculated the HF and subsequently the QP band structures of this system. In the latter case he obtained for the fundamental gap a value of 10.3 eV, while the experimental value is 8.8 eV.< The reasons for this 1.5 eV discrepancy have already been discussed (see Sections 5.3 and 8.2.1.1). 

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