Introduction
Polyethylene (PE) is the most widely used plastic, with an annual production of approximately 80 million metric tons. The predominant cause of failures in PE cable is water treeing, see in Fig1. Water treeing is a severe degradation phenomena in PE cables. Cable cure treatment is an economic life extension technology which increases the dielectric performance of dielectrics of PE cable. Simply speaking, the process of cable cure is inject the cable cure fluid into the conductor and let it react with PE. The byproducts of cure treatment act as impurities in PE which are believed to be the cause of loss tangent increase. One of my lab mate's previous work has calculated the effect of many impurities in PE(vinyl, Iodine,carbonyl etc. ). Similarly, we could adopt his method and use his results as a guide to study the effect of byproducts on PE after treatment. The objective is use DFT calculation to provide insight of the effect of byproducts at the atomic level on the physical and electronic structure of PE. Basically we will study the impurity formation energy, local physical disorder, effect on polyethylene electronic structure, trap depths and etc.
Fig1.water tree in polyethylene Picture from Cable cure brochure of Utlix[1]
Literature review
DFT approach in crystalline PE
Unlike the case in solid state and molecular physics, density functional approaches have not been widely used for polymers. Polyethylene has perhaps the simplest primary structure of any polymer. See in Fig2 the repeating unit of PE which forms a zigzag chain conformation. The first few DFT approaches were applied in crystalline PE.
Fig 2 The repeating unit and chains of polyethylene
We think of polymer crystallinity as the packing of molecular chains so that an ordered molecular array is produced. Crystal structures in PE can also be specified in terms of unit cells. Fig. 3 shows the unit cell for crystalline polyethylene and its relationship to the molecular chain structure; this unit cell has orthorhombic symmetry. In each unit cell, there are four (CH)2 groups situated in two polymer chains.
Many methods of DFT calculation have been conducted in crystalline PE. In general, it assumes periodic boundary in three dimensions. The optimum geometry was found by minimizing the forces on all atoms. The calculation was repeated for different chain lengths to minimize the energy as a function of all atomic positions. The electron-ion interaction is represented by ionic pseudo potentials for C and H. Different method may adopt different pseudo potentials and basis sets. Test different K points mesh to assure the convergence of the total energy in reciprocal space. Methodology of calculations is slightly different due to different approaches.
Figure 3. Crystalline structure of polyethylene.(a) Side-view of orthorhombic structure of polyethylene. (b) Projection of unit cell on the ab plane The larger circles represent C and the smaller circles represent H.
In table 1 and table 2[2], calculated structures for a single chain and lattice constant by different methods are listed and compared with experimental x-ray diffraction data. It shows that DFT calculations are in good agreements with experiment. One should note that LDA approximations underestimate the inter-chain distance, more precise result could adopt gradient –corrected approximation[3].
Table 1[2]. Measured structural parameters of crystalline PE and calculated structures for a single chain. Bond lengths are in Å, angles in degrees. Values marked with an asterisk (*) were assumed, and not measured or calculated directly
| rCC | rCH | αCCC | αHCH | |
| X-rays a | 1.53 | - | 112 | - |
| X-rays b | 1.534±0.006 | 1.09* | 112.0±0.3 | 109.5* |
| X-rays c | 1.533±0.022 | 1.07±0.022 | 111.9±1.8 | 107.0±1.8 |
| X-rays d | 1.527±0.007 | 1.091 * | 112±0.8 | - |
| Neutrons(4K) c | 1.578(5) | 1.06(1), 1.10(1) | 107.7(5) | 109.0(1.0) |
| Neutrons(90K) c | 1.574(5) | 1.07(1) | 108.1(5) | 110.0(1.0) |
| Hartree-Fock f | 1.536 | 1.083 | 112.7 | 106.8 |
| Hartree-Fock g | 1.541 | 1.087 | 112.5 | 107.2 |
| DF(PZ,XO) | 1.515, 1.534 | 1.10*,1.10* | 113.0, 113.0 | 109.7, 110.0 |
| LDA, GC | 1.512,1.536 | 1.111, 1.111 | 114.3, 113.8 | 105.2, 105.6 |
Table 2[2]. Comparison of measured (x-ray diffraction) and calculated (LDA) lattice constants and inter chain distances in PE. d(c..c) is the shortest distance between C atoms in different chains, and Φ is the angle between the ac-plane and the plane of the C atoms in the chain. Distances are in Å , angles in degrees.
| method | a0 | b0 | c0 | Φ | d(c..c) |
| X raysa | 7.42±0.01 | 4.945±0.01 | 2.546±0.004 | 41.4 | 4.14 |
| X raysb | 7.42±0.01 | 4.96±0.01 | … | 47.7 | 4.15 |
| X raysc | 7.388±0.008 | 4.929±0.004 | 2.539±0.002 | 45 | 4.59 |
| LDA | 6.73 | 4.53 | 2.52 | 41.3 | 3.67 |
Polyethylene was the first organic polymer for which band structures were reported, based on angle-resolved photoemission from oriented films of n-hexatriacontane [CH3(CH2)34CH3][5]. We show in Fig 4 [2] the energy bands calculated for a single chain using both the LD and GC approximations, and the overall form in both is the same as found in the measurements and previous band structure calculations. One should note that DFT calculation always underestimated the band gap, in the case of PE, it underestimated by about 25% relative to the experimental value of 8.8eV[2].
Fig 4[2] . Band structure of single chain of polyethylene calculated with LD (full curves) and GC (broken curves) approximations. Wave vector k in units of (2π/c0)
Core-Shell Model
Polyethylene usually has a semi crystalline structure. Crystallinity of LDPE is about 45%-65%. Since impurities are more likely to be incorporated into amorphous regions. We should better set a model represent the amorphous region of PE. Core-Shell model of PE provides a better representation of an amorphous environment than crystal bulk models commonly used in computational quantum mechanical studies of impurities in bulk material.
The rationale behind the Core-Shell model is to study chemical impurities in a PE chain which is sufficiently long to behave as infinite, while including the effect of neighboring chains in an amorphous like environment in which the chains have freedom to deform as a result of introduction of the impurity. The Core-Shell model consists of a Core chain, to which impurities are added, surrounded by six Shell chains in a super cell large enough to prevent interaction with chains in neighboring cells. Introduction of the impurities results in deformation of chains in the vicinity of the impurities. Our computations suggest that the Core-Shell model, which is discussed in greater detail in, represents an acceptable approximation to amorphous PE which can be implemented with a reasonable number of atoms and periodic boundary conditions. A computational quantum mechanical study of various chemical impurities in bulk PE has been undertaken using density functional theory (DFT) as implemented in the SIESTA code[4].
Figure5[4]. on the left Cristalline PE on the right Core-Shell model of PE
a) Impurity formation energy
The impurity formation energy is defined as 
E[PE+X] is the total energy of the bulk PE including the impurity X; E[PE] is the total energy of bulk PE without the impurity; ni indicates the number of atoms of type i that were added (ni positive) or removed (ni negative) from bulk PE to create the impurity; μi are the corresponding chemical potentials of these species; [6]
Table3 [4]Impurity formation energy of impurities in PE
| cell-content | carbonyl | Vinyl group | Double bond | Conjugated double bond |
| Formation energy | 1.63eV | -1.19eV | -2.47eV | -4.41eV |
b) Analysis of Density of States
Fig 6 represents the state density in a disordered dielectric material. Shallow and deep traps are related to physical and chemical disorder respectively. The physical disorder is due to the fact that polyethylene molecules can have different conformations, for example, the chain distortion around the impurity in core-shell model we introduced. The physical disorder has consequences in the energy band diagram, introducing localized states which has high density (order 1022cm-3) and they are located in an energy range of 10-2eV to 1eV from the extended states of the bands. These states assist charge transport. Transport can occur in these states easily by thermal activation of carriers to a band, leading to the concept of conduction through multiple trapping steps. It also refers as "shallow trap" which origins from this concept, because charges move from trap to trap experiencing each time a transition to the extended states of the band. [7]
Chemical impurities are believed to introduce localized states in PE band gap which can act as deep traps. Deep trap is linked with the presence of one or several atoms that are not part of the repeat unit that constitutes the backbone of the molecular chain. Deep traps have a much lower density than shallow traps as seen in Fig6. Deep traps can be analyzed through the compare of PDOS of PE model with impurities and without impurities.
Fig 6.Schematic representation of state density in a disordered dielectric material
Density of states (DOS) is the number of electronic states per unit energy and unit cell, as a function of energy. DFT calculation determines the electronic eigenvalues (i.e., energy states) and wave functions (i.e., eigenstates) from which various properties of the system can be determined, including the electron density, density of states (DOS), and the projected density of states (PDOS). The PDOS is the contribution of any orbital, atom or group of atoms to the DOS of the system. In this case, we could analyze contribution of hydrogen, carbon, impurity atoms to the density of states. Compare DOS of PE without the impurity and Core-Shell model with impurity, see the states introduced by impurity. From PDOS analysis of PE with impurity of carbonyl, see Fig 7 the extra states(shallow and deep traps) introduced by carbonal group[4].
Fig 7 [4]DOS of core-shell structure showing impurity states from carbonyl group.
c) Analysis of Charge density
Carrier transfer from one chain of the PE to another is very difficult, that is also why polyethylene is a good dielectric material. Inter-chain separation is a barrier to carrier transport in PE. See fig 8 the compare of the charge density contour lines in the vicinity of chemical impurities and PE chain without impurities[4]. In fig 8 to the left, it shows the charge density contour plot in the vicinity of the carbonyl impurity. To the right, it is the charge density away from the carbonyl impurity. We can see that the contour line connect around the carbonyl to some extent. The plot scale was chosen to demonstrate the effect of carbonyl impurity in increasing inter chain charge density. The compare of charge density indicate that the interaction between impurity and PE chains can link the polymer chains electronically, which facilitates charge transfer between chains and thereby increase conduction.
Fig 8[4] compare charge density contour around impurity
Proposal
We should be able to set a PE crystal model first, C-C and C-H bond length, the C-C-C and H-C-C bond angles, and the lattice constants of bulk orthorhombic crystalline PE should be reproduced to acceptable levels.
Small molecular impurities in PE in the core-shell model of PE should be calculated to get the same results by previous work of my lab mate. Same methodology could be applied which includes LDA approximaition, DZP(double zeta plus polarization basis set) and Troullier-Martins type Pseudopotentials. SIESTA code will be used.
Choose one of the byproducts of Cable Cure fluid which are are listed as follows:
Phenylmethyldimethoxy silane
C9H14O2Si Molecular weight 182.29
Trimethyl methoxy silane
C4H12OSi Molecular Weight 104.22
The dimer of Phenylmethyldimethoxysilane
End capped Phenylmethyldimethoxy silane
Hexamethyldisiloxane
C6H18OSi2 Molecular Weight 162.38
Since they are all large molecules, we could consider put it in parallel with PE chain in core-shell model we introduced before. Big physical disorder could be excepted due to the size of molecular of the byproducts. Localized states which act as "deep traps" could also be expected because of the foreign molecules introduced through PDOS analysis. Impurity formation energy could be calculated for each byproduct. In the long term, we hope it could give us insight of explanation of dielectric loss increase, and provide us from atomic view how it changes the electronic structure of PE and thus influence the conduction theory in polyethylene cables.
Reference
[1] Cable Cure Brochure of Utlix
[2] Density functional study of crystalline polyethylene B. Montanari, R.O. Jones
Chemical Physics Letters 272 (1997) 347-352
[3] Density functional study of molecular crystals: Polyethylene and a crystalline analog of bisphenol-A polycarbonate. B. Montanari, P. Ballone and R.O. Jones; Journal of chemical physics volume 108, number 16. 22 April 1998
[4] Quantum Mechanical Studies of Carbonyl Impurities in Dielectric Polyethylene. A.Huzayyin, S.Boggs, R.Ramprasad. (not published yet)
[5] Conformation and electronic structure of polyethylene: A density-functional approach; M.S.Miao, P.E. Van Camp etc. Physical review B; volume 54, Number 15, 15 October 1996-1
[6] First-principles calculations for defects and impurities: Application to Ⅲ-nitrides; Chris G. Van de Walle, Jorg Neugebauer; Journal of applied physics, volume 95, number 8; 15 April 2004
[7] Charge Transport Modeling in Insulating Polymers: From Molecular to Macroscopic Scale; G. Teyssedre and C. Laurent; IEEE Transactions on Dielectrics and Electrical Insulation; Vol.12,No.5; October 2005
