Molecular modeling of electron traps in polymer insulators: Chemical defects and impurities

JOURNAL OF CHEMICAL PHYSICS VOLUME 115, NUMBER 6 8 AUGUST 2001 Molecular modeling of electron traps in polymer insulators: Chemical defects and impurities M. Meunier a) and N. Quirke b) Department of Chemistry, Imperial College of Science, Technology and Medicine, South Kensington SW7 2AY, United Kingdom A. Aslanides Electricité de France, DRD-Les Renardières 77818 Morêt sur Loing Cedex, France Received 26 February 2001; accepted 18 May 2001 The presence of space charge in the polymeric insulation of high-voltage cables is thought to be correlated with electric breakdown. However, a direct link between molecular properties, space charge formation and eventual breakdown has still to be established. It is clear that both physical e.g., conformational disorder and chemical defects e.g., broken bonds and impurities are present in insulating materials and that both may trap electrons. We have shown that by defining the defect energy in terms of the molecular electron affinity, a relationship is established between the electron trap and the molecular properties of the material. In a recent paper M. Meunier and N. Quirke, J. Chem. Phys. 113, 369 2000 we proposed methods that have made it possible to provide estimates of the energy, number and residence times of electrons in conformational traps in polyethylene. Typical physical trap energies are of the order of 0.15 ev and all are less than 0.3 ev. In the present paper we focus on the role of chemical defects, where we expect much deeper traps but at very low concentrations. Following the methodology used in our previous paper we have used density-functional theory to calculate trap energies for a set of chemical impurities and additives commonly found in polyethylene used for high-voltage cable insulation. In an extension of our approach we have estimated the effect of neighboring molecules on the trap energies of such defects. The resulting trap energy-trap density distribution reveals some very deep 1 ev traps presumably implicated in the formation of long-lived space charge in polymeric insulators and consequently to changes in the dielectric properties of the material. 2001 American Institute of Physics. DOI: 10.1063/1.1385160 I. INTRODUCTION Electrical supply companies use polymers, in particular polyethylene, 1 as insulators for high-voltage electric cables. However, over time, such insulators trap charge carriers electrons and/or ions to form a space charge that has been correlated with many undesired consequences such as electrical e.g., partial discharges, electrical trees ageing or degradation. There is now a vast literature concerned with the experimental characterization of space charge methods 2 include thermal step, 3 pulsed electro-acoustic, 4 thermal pulse, 5 pressure pulse, 6 photoconductivity, 7 and mirror image 8 and with phenomenological models of space charge formation and discharge. 9 In addition polymeric insulators suffer from chemical e.g., free radical formation leading to cross linking or bond breaking and physical e.g., structural relaxation above the glass temperature changes which may in turn be linked to and influence the space charge. In the long term the degradation of the polymer may lead to catastrophic electrical breakdown of the insulation. All of these a Current address is Molecular Simulation Inc., 230/250 The Quorum, Barnwell Road, Cambridge CB5 8RE, United Kingdom. b Author to whom correspondence should be addressed. Electronic mail: n.quirke@ic.ac.uk factors, when combined over time make the prediction of the probable working life of extruded cable very difficult. 10 Most current theories relate breakdown to charge accumulation and displacement of one type or another. For example in the space charge ageing model 9 the presence of both physical and chemical defects due to ageing or to the presence of additives leads to the accumulation of electrons in traps forming a relatively immobile space charge. The traps are thought to have a range of energies, from shallow 0.1 0.5 ev to deep a few ev. The resulting electrostatic and electromechanical forces are thought to lower the energy barriers to local conformational change producing new free volume and microvoids, eventually initiating failure via a variety of mechanisms such as impact ionization. An understanding of space charge accumulation and its link to dielectric breakdown would be a major scientific achievement as well as facilitating the prediction of cable lifetimes and the design of better cables. Experimental studies suffer from limitations due to resolution: Even if the charge density 11 and the mobility of the carriers can be evaluated, 12 no experimental technique is currently able to determine the exact nature at molecular level of the traps. Space charge accumulation and the related trapping-detrapping phenomena are still poorly understood, indeed it is unclear whether any trapping 0021-9606/2001/115(6)/2876/6/$18.00 2876 2001 American Institute of Physics

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 Electron traps in polymers insulators 2877 FIG. 1. Schematic representation of impurity molecules investigated in the present study in their minimum energy conformations. a decane b 5-decanone c 5-decene d 4,6 decene e 5-decyne f Vinyl-5 nonane g 5-decanol h 5-decanal i propyl-4 heptane. Carbon atoms are represented in gray, hydrogen atoms in white, and oxygen atoms as spheres. site has been unequivocally identified in polymeric insulators. 13 Our work aims to increase our understanding of charge trapping in polymeric insulators by applying molecular modelling techniques to estimate the trapping ability of likely physical and chemical traps present in polyethylene. We have shown 14 that by defining the defect energy in terms of the molecular electron affinity, a relationship is established between the electron trap and the molecular properties of the material. In order to probe the contribution of conformational disorder to electron trapping in polyethylene using molecular modelling we considered the condensed phases of much smaller alkane molecules which have a band structure very similar to polyethylene. 15,16 We represented the local conformational disorder of segments of polyethylene in the amorphous regions by configurations taken from computer generated glassy phases of tridecane (n 13) wax. Our strategy has been to identify the local electron traps in polyethylene with the increase in the electron affinity of the more distorted n-c 13 H 28 molecules, conformationally trapped in the glass. This approach has made it possible to provide estimates of the energy, number, and residence times of conformational electron traps in polyethylene. Typical physical trap energies are of the order of 0.15 ev and all are less than 0.3 ev. In the present paper we focus on the role of chemical defects, where we expect much deeper traps but at very low concentrations. Following the methodology used in our previous work we have used density-functional theory to calculate trap energies for a set of chemical impurities and additives commonly found in polymeric insulators, such as polyethylene, used for high-voltage cable insulation. In an extension of our approach we have been able to estimate the effect of the presence of neighboring molecules on the trap energies of such defects. The resulting trap energy-trap density distribution reveals some very deep 1 ev traps presumably implicated in the formation of long-lived space charge in polymeric insulators and consequently to changes in the dielectric properties of the material. This trap distribution is the input to excess electron mobility studies to be reported in due course. 17 Chemical defects consist of additive molecules deliberately inserted during the cable manufacture as well as impurities and by-products of ageing. Chemical ageing, i.e., chemical changes within the material is mainly due to lowenergy electrons that dissociate molecules of the dielectric. The released fragments can then react with surrounding molecules. In polymers such as polyethylene, this will lead to both chain scission and cross-linking. The methodology and computational details are reported in Sec. II. The results of our calculations of the electron affinities of typical chemical defects are given in the following Sec. III. Finally, in Sec. IV, we discuss our results in

2878 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 Meunier, Quirke, and Aslanides terms of trap distribution and its possible influence on the insulating properties of the material. II. METHODOLOGY Chemical defects can be divided into two groups. The first group or impurities includes the chemical defects found on a polymeric chain, i.e., hydroxyl and ketone functions, double or triple carbon carbon bonds, branching, and so on. This group of chemical defects is modeled as a chemical modification of an alkane chain. 14 The second group consists of the by-products of the dicumyl peroxide, an additive commonly used in commercial cable manufacture as a crosslinking agent. Other chemical defects in this second group originate from the by-products of antioxidant agents i.e., Santanox, Irganox used in commercial cable manufacture. These have not been included in the present work. In addition, water, methane, and other small molecules can also be found in industrial cables. We define the defect energy in terms of the molecular electron affinity EA EA E R e E R e, 1 where E(R e ) is the total energy of the neutral species in its equilibrium geometry R e and E (R e ) is the total energy of the anion in its equilibrium geometry. For the case where the anion is not allowed to relax, R e R e and we have a vertical electron affinity. The total energies are calculated using density-functional theory DFT, as implemented in the code DMOL. 18 We have recently demonstrated 14,19 that DMOL predicts electron affinities consistent with the experimental data for a range of atoms and molecules including alkanes. The trap energy, E (trap), is defined as the energy difference between the electron affinity of the system with and without the defect, so that E trap EA defect EA reference, 2 where the reference electron affinity EA (reference) is taken as the vertical electron affinity of, e.g., the all-trans n-c 13 H 28 molecule, representing the conduction band of the molecular crystal, and by implication that of polyethylene, so that n-c E trap EA defect EA 13 H 28 ) all-trans. 3 A positive trap energy means that the electron affinity of the system containing the defect is greater than the electron affinity of the reference system. This suggests that an excess electron present in the dielectric would lower its total energy by leaving the conduction band and localizing on the defect. In order to calculate the vertical electron affinity we need E(R e ), the total energy of the neutral species in its equilibrium geometry R e. For the reference molecule, n-c 13 H 28, the equilibrium geometry is known to be planar all-trans. 20 When the chain is modified through the creation of a chemical defect, the minimum energy conformation may change. In addition some impurity molecules are relatively large and have complex potential energy surfaces with many local minima. In order to be sure of obtaining a good estimate of E(R e ), it is necessary to explore thoroughly the potential energy surface and to compare the energy of many different minima to identify the lowest. In the present work, computational limitations restrict our conformation space searches to those carried out using classical forcefield simulations. The aim is to get a low-energy subset for each molecule studied before performing accurate ab initio calculations. The following procedure has been adopted: i Construct the molecule in vacuum see Figs. 1 and 2 for impurities and decomposition products ; ii molecular-dynamics simulations in the canonical ensemble at 500 K for 20 picoseconds ps, with a time step of 1 femtosecond fs, using the COMPASS 21 forcefield implemented in the code Discover. 22 COM- PASS represents both bonded bond stretch, angular distortion, etc. and nonbonded or dispersion interactions; iii geometry optimization using successively the steepest descent and conjugate gradient algorithms at the end of the molecular-dynamics runs; iv repeat steps ii and iii 1500 times. The lowest energy conformation of small linear alkane chains (C n H 2n 2 ) is believed to be the all-trans planar conformation for n 18. 20 Using this result we have tested our simulated annealing procedure for n 6, 8, 10, and 13. Our procedure finds the all-trans conformation for alkane chains with n 13, for these cases the minimum is found rather quickly i.e., after 327 iterations for n 10. For the tridecane chain (n 13) the lowest minimum energy conformation is found only twice after 1047 iterations and does not correspond to the all-trans conformation. If we assume only three degrees of freedom per dihedral angle (g,t,g ) in the chain, the total number of conformations for n-c 13 H 28 is 3 10 59 049, too large to be searched using the procedure described above. In view of this limitation we have used the decane molecule (n-c 10 H 22 ) rather than tridecane to model the chemical impurity defects found in polyethylene. With n 10 we are confident that our scheme samples thoroughly the conformational space of the modified alkanes. Since we are interested in energy differences at fixed n this approximation should not introduce significant error in the calculation of trap energies. Note that for the compounds appearing in Table I, the decane molecule is the only one for which the minimum energy conformation all-trans is known. 20 For each molecule investigated, the lowest potential energy configuration found by simulated annealing is used as the initial configuration for DFT calculations of geometry optimization and electron affinity. 14,19 The resulting minimum energy conformations are displayed in Figs. 1 and 2. We have investigated the effect of the polarization of neighboring molecules on the electron affinity, and subsequently on the trap depth of chemical defects. Clusters of 3 and 5 molecules containing one impurity defect have been evaluated. As for the isolated molecule, the trap energy is defined as the energy difference between the electron affinity of the cluster with and without the defect. Due to the increased calculation time for clusters the chain length of the alkane studied has been reduced to 6 compared to 10 for the impurities. Since we are interested in differences with respect to the isolated chain (n 6) this reduction should not

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 Electron traps in polymers insulators 2879 FIG. 2. Schematic representation of the decomposition scheme of the dicumyl peroxide (C 18 H 22 O 2 ). The chemical reaction of free radicals originating from the decomposition of the dicumyl peroxide gives either the acetophenone which is stable or the cumylic alcohol that can react to form the alpha-methyl styrene and cumene. Carbon atoms are represented in gray, hydrogen atoms in white, and oxygen atoms as spheres. significantly change the results. Clusters of 3 molecules consisted of 3C 6 H x O y x 12 or 14, y 1 or0 molecules with the backbone atoms the carbon atoms in a plane. Clusters of 5 molecules consisted of the cluster of 3 molecules plus two molecules above and below the middle chain, in two perpendicular planes. The defect, when introduced, was always situated on the chain occupying the central position of the cluster. The distance between the carbon atoms on neighboring alkane chains was set to 5 0.1 Å, close to the experimental value of 4.928 Å for polyethylene. 23 Two chemical impurities have been studied using this methodology, impurity 1, 3-hexanone C 6 H 12 O, and impurity 2, 3-hexanol C 6 H 14 O. TABLE I. Electron affinity and trap depth Eqs. 1 3 of molecules presented in Figs. 1 and 2. All values are in electron volts. Molecules a to i are chemical impurities while molecules j to m are decomposition products. III. RESULTS An experimental study of polyethylene thin films has identified some of the chemical impurity groups present in the material. 24 From these data we have selected 13 impurities and decomposition products for study. These molecules possess common defects such as unsaturated bonds, branches, and oxygen atoms. In this section we report the results for the minimum energy conformations, electron affinities, and trap energies of these defects. A. Chemical impurities The electron affinity and consequently the trap depth with respect to the reference molecule n-c 10 H 22 for impurities, n-c 13 H 28 for physical defects 14 are given in Table I. Clearly some of these molecules have trap energies that are considerably deeper than physical defects in agreement with the expectation that chemical defects may constitute significant electron traps. Molecule Electron affinity Trap depth a decane (n-c 10 H 22 ) 1.366 a 0 b 5-decanone (C 10 H 20 O) 0.913 0.453 c 5-decene (C 10 H 20 ) 1.244 0.122 d 4,6-decene (C 10 H 18 ) 0.923 0.443 e 5-decyne (C 10 H 18 ) 1.325 0.041 f 5-vinyl nonane (C 10 H 20 ) 1.209 0.157 g 5-decanol (C 10 H 21 O) 1.180 0.186 h 5-decanal (C 10 H 20 O) 0.921 0.445 i 4-propyl heptane (C 10 H 22 ) 1.245 0.121 j alpha-methylstyrene, C 9 H 10 0.23 1.53 k Cumylalcohol, C 9 H 12 O 1.02 0.28 l Acetophenone, C 8 H 8 O 0.40 0.90 m Cumene, C 9 H 12 1.26 0.04 a Reference 19. B. Decomposition products In this section, some of the by-products of the crosslinking agent dicumyl peroxide found in industrial polyethylene are examined. The dicumyl peroxide compound dissociates Fig. 2 when added to the polymer resin, so that only the by-products of this dissociation are found in the manufactured cable. In addition, the quantities involved are relatively small. We have considered molecules j, k, l, and m in Fig. 2 for which the experimental signal is above the detection threshold 1 ppm. The electron affinities and trap depths are summarized in Table I.

2880 J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 Meunier, Quirke, and Aslanides FIG. 3. The energy trap density distribution per cm 3 predicted by our calculations for model PE. Thin line: Physical defects Ref. 14, triangles: Chemical defects 100, the line is a guide to the eye. IV. DISCUSSION Table I suggests that chemical defects at the higher end of the range of trap energies from 0.04 to 1.53 ev are likely to be responsible for the deep trapping of electrons in polyethylene. Chemical defects containing carbonyl functions compounds b, h, and l or conjugated double bonds compounds d and j constitute the deeper traps, whereas chemical defects originating from hydroxyl compounds g and k and nonconjugated double or triple carbon-carbon bonds compounds c, e, and f present shallower traps. From the density of traps Fig. 3 the residence time of the electrons inside the traps can be estimated using a two-potential well model 25 in which the two-well minima are separated by a barrier of height E t corresponding to the trap energy, with separation d 1/ AB 4 where AB 0 exp E t e/kt, 5 AB represents the hopping probability from well A to B, d is the average trap separation, 0 is the attempt frequency, e is the elementary charge, k is the Boltzmann constant, and T is FIG. 4. Log plot of the density of traps per cm 3 as a function of the electronic residence time. Thick line, physical defects Ref. 14 ; triangles, chemical defects, the line is a guide to the eye. FIG. 5. The variation of electron affinity with the number of carbon atoms n for a an alkane chain Ref. 14, small filled circles; as well as C 6 clusters, open triangles b these data as fitted using Eq. 7 of the text, full line c for Impurity 1, filled triangles d Impurity 1 as fitted using Eq. 7, long dashed line e Impurity 2, filled squares, and f Impurity 2 as fitted using Eq. 7, dashed line. the temperature in Kelvin. The attempt frequency 0 can be taken from the result for the infinitely deep square well 26 0 h/8m e d 2 4.168 10 13 s 1. 6 The resulting residence times are plotted in Fig. 4 from which it can be seen that the residence time of an electron in a chemical trap may be greater than in physical traps by several orders of magnitude. This strongly suggests that the electrons trapped in chemical defects are responsible for the long lived space charge thought to degrade the performance of a polyethylene insulator over time. 25 The electron affinity calculations have been performed on isolated molecules in their ground state. In nature, the environment neighboring molecules has an effect on the electronic properties, e.g., on the electron affinity, of the molecules. To estimate the effect of neighboring molecules on the value of the electron affinity, we have performed calculations using a cluster model consisting of alkane chains surrounding the chemical defect as described in Sec. II. We consider initially the variation of the electron affinity of the alkane n-c 6 H 14 without a defect. We have shown previously 14,17 that the electron affinity of linear saturated alkanes increases with the chain size. For an infinite chain length corresponding to the polymer this computed value is 0.65 ev 14 consistent with the experimental data of 0.5 0.5 ev 27 for polyethylene and with 0.3 0.5 ev 28 for hexatriacontane (n-c 36 H 74 ). In our cluster calculations, the electron affinity also increases with the system size from 1.589 ev for a single hexane molecule, to 1.035 for three, to 0.77 ev for five, consistent with the results for increasing single chain length. Indeed the fit to the electron affinity as a function of carbon atoms present in the alkane molecule (n-c n H 2n 2 ), Eq. 7 E n 0.6499 7.7961 1/n 11.897 1/n 2, 7 also represents the effect of adding carbon atoms to the alkane cluster. The data are plotted together with the fit in Fig. 5. Figure 5 also contains the electron affinity of two chemical impurities impurity 1, 3-hexanone n-c 6 H 12 O and impurity 2, 3-hexanonol n-c 6 H 14 O in an alkane cluster as a func-

J. Chem. Phys., Vol. 115, No. 6, 8 August 2001 Electron traps in polymers insulators 2881 tion of the number of carbon atoms in each cluster. The data follow the same trend as hexane and have been fitted using the same functional form by changing the value of the constant term in E(n). The root-mean-square rms deviation of the fit to, for example, the results for impurity 2 is 0.02 ev the same as for the fit to the alkanes. Rather than performing cluster calculations with ever increasing numbers of neighbors which would quickly become computationally prohibitive and raise new questions concerning the details of exactly how the environment should be constructed, we use the fits in Fig. 5 to estimate the trap energy of the chemical impurity defects in the presence of neighboring alkane chains. Since the n dependent terms are identical, the trap energy is the difference in the constant terms. For impurity 1, the trap energy changes from 0.69 ev for an isolated molecule, to 0.59 ev for a molecule in an alkane wax due to the greater delocalization of the excess charge over the alkane molecules surrounding the defect. For impurity 2, however, the neighbors have no significant effect the trap energy changes from 0.21 to 0.22 ev. While the effect of neighboring molecules on the electron affinity is significant, the effect on the trap energy, in this calculation, is small. The current work represent only a preliminary estimate of the effect of delocalization on trap energies and further work is in progress to obtain a more accurate picture of the role of the environment in determining trap energies in alkane waxes and polyethylene. V. CONCLUSIONS Ab initio calculations have been carried out to characterize the charge carrier trapping properties of several relevant chemical defects in a model polymeric insulator. It is found that some of these defects represent deep traps for the electron, leading to large residence times for the charge carrier. Work is in progress to investigate the influence of these traps on the conduction and transport properties of model polymers. 17 1 The first polyethylene insulated cable was a mile of submarine cable between the Isle of Wight and the mainland of England installed by Dean s company in 1938, see J. A. Allen, Studies in innovation in the steel and chemical industries Manchester University Press, 1967. 2 N. H. Ahmed and N. N. Srivinas, IEEE Trans. Dielectr. Electr. Insul. 4, 644 1997. 3 A. Toureille, Rev. Gen. Electr. 8, 15 1991. 4 Y. Li, T. Takada, and N. Takasu, J. Phys. D 26, 986 1993. 5 R. E. Collins, J. Appl. Phys. 51, 2973 1980. 6 J. Lewiner, IEEE Trans. Electr. Insul. 21, 351 1986. 7 A. Dias Tavares, J. Chem. Phys. 59, 2154 1973. 8 C. Le Gressus, F. Valin, M. Henriot, M. Gautier, J-P. Duraud, T. S. Sudarshan, R. G. Bommakandi, and G. Blaise, J. Appl. Phys. 69, 6325 1991. 9 L. A. Dissado, G. Mazzanti, and G. C. Montanari, IEEE Trans. Dielectr. Electr. Insul. 2, 1147 1995 ; 4, 496 1997 ; see also L. A. Dissado, abstracts CSC 3 Electric charge in solid insulators, 141 1998. 10 L. A. Dissado and J. C. Fothergill, Electrical Degradation and Breakdown in Polymers Peter Peregrinus Ltd., London, UK, 1992. 11 G. C. Montanari, D. Fabiani, L. Bencivenni, B. Garros, and C. Audry, Proceedings of the CEIDP 38 1999. 12 N. Hozumi, Y. Muramoto, and M. Nagao, Proceedings of the CEIDP 350 1999. 13 H. J. Wintle, abstracts CSC 3 Electric charge in solid insulators, 49 1998. 14 M. Meunier and N. Quirke, J. Chem. Phys. 113, 369 2000. 15 C. Laurent and C. Mayoux, Electronic Processes in Electrical Ageing Arpeggio Seminar, 1994. 16 J. J. Pireaux, Phys. Rev. A 14, 2133 1976. 17 J. Anta, L. Marcelli, M. Meunier, and N. Quirke in preparation. 18 DMOL User Guide, Molecular Simulations Inc. San Diego 1997. 19 M. Meunier, N. Quirke, and D. Binesti, Mol. Simul. 23, 109 1999. 20 G. M. Goodman, Chemical Applications of Molecular Modelling RSC, 1998. 21 H. Sun, J. Phys. Chem. 102, 7338 1998. 22 DISCOVER User Guide, Molecular Simulations Inc. San Diego 1996. 23 S. Kavesh and J. M. Schultz, J. Polym. Sci., Part A-2 8, 243 1970. 24 G. Rochas, Borealis private communication. 25 R. Bartnikas, IEEE Trans. Dielectr. Electr. Insul. 4, 544 1997. 26 P. W. Atkins, Physical Chemistry, 4th ed. Oxford University Press, New York, 1990. 27 K. J. Less and E. G. Wilson, J. Phys. C 6, 3110 1973. 28 R. Dudde and B. Reihl, Chem. Phys. Lett. 196, 91 1992.