A Theoretical Study on the Influence of the Functional Group Electronic Effect on the Electron Mobility of Cross-Linked Polyethylene

Abstract

The effect of electron-donating and electron-withdrawing groups grafted onto polyethylene on electron mobility was studied using density functional theory. In order to ensure the accuracy of the calculation results, 17 basis sets from six methods were screened. 3-methylpentane was selected as the cross-linked polyethylene model. Compared with the experimental values, the theoretical calculation results show that wB97XD/6-311G(d,p) is more suitable for studying the electron mobility system. The roles of electron-donating and electron-withdrawing functional groups were studied. The results show that the electron mobility of grafting nitrobenzene (Ebnb) to polyethylene is the smallest among the studied molecules. As the ability of electron-donating groups increases, the electron mobility gradually increases, while the addition of the electron-withdrawing group reduces the electron mobility and the electron mobility gradually increases with increasing temperature. This investigation is expected to provide reliable information for the development of insulation materials for cables.

1. Introduction

With the rapid development of the economy and the strategic adjustment of energy structures, the development and utilization of renewable energy has become an important way to improve electricity use in daily life. At present, there are two major problems in the production and use of energy. One is that the distribution of energy production areas and major energy consumption areas is extremely uneven, as renewable energy is generally located far away from cities. There is an urgent need to build low-loss, large-capacity, long-distance power transmission channels. The other is limited by the absorption capacity of the traditional power system. Direct current (DC) transmission can be connected to the asynchronous power grid, which greatly reduces the difficulty of grid network dispatching. Thus, high-voltage (HV) and ultra-high-voltage (UHV) direct current transmission can solve the two problems of large-scale and long-distance transmission of electric energy and new energy consumption [1,2,3]. When transmission lines cross rivers and seas, or in densely populated urban areas, power cables need to go underground, so the outer layer needs to be surrounded by insulation materials. Cross-linked polyethylene (XLPE) insulation material has excellent electrical and mechanical properties, as well as low costs. It has become the most prominently used insulation material for high-voltage power cables. However, in the service process, space charges are accumulated locally, which distorts the electric field stress, leading to the breakdown of insulating materials, reducing the service life and failing to meet the growing social demand for the voltage level of high-voltage cables [4,5,6,7]. Therefore, in order to reinforce the breakdown strength of insulating material, researchers have carried out a lot of work. Ultra-clean technology is one of the effective methods, which can not only reduce impurities but also maintain material cleanliness. It has been practically applied in the cable manufacturing process, and the breakdown strength of materials has been significantly improved. However, this technology has a limitation due to the maximum breakdown strength of the material itself, and we need to find another way to further improve the breakdown strength [8].

In order to obtain XLPE insulated power cable with the ability to withstand a higher voltage level, the problem of insulation material breakdown caused by space charge accumulation is usually solved by adding a space charge inhibitor. By adding a voltage stabilizer, electrical aging caused by charge accumulation under strong electric fields of polymer insulation can be inhibited, and the electrical resistance of polymer insulation material can be improved [9]. Voltage stabilizers can withstand partial discharge, mitigate strong electric fields, inhibit macromolecular degradation and capture high-energy electrons [10]. A voltage stabilizer is usually a small molecular aromatic compound, which has poor compatibility with saturated aliphatic polymer. During the long-term operation of cables, the voltage stabilizer is easy to migrate and precipitate due to the high working temperature and large working field strength. The migration and precipitation of the voltage stabilizer will leave defects in the original place, which will reduce the electrical resistance of polymer insulation materials [11]. In order to fundamentally solve the problem of migration and precipitation of the voltage stabilizer, small molecular aromatic compounds with the function of voltage stabilizers can be grafted onto the polyethylene chain by a chemical reaction [9]. The issue of space charge is very complex, and its generation, transportation, and attenuation characteristics are related to the microstructure, impurities, temperature, electric field strength, and crystal state of the dielectric. Therefore, the issue of space charge has always been a hot topic in the field of insulation research.

Neagu and co-worker measured the discharge current of polyester film using the isothermal relaxation current method and thermal stimulation current method and found that there was a polarity reversal of the discharge current in the untreated samples. It is believed that the discharge current is a result of the combined action of dipole spin off and space charge dissipation, and the fundamental reason for current polarity reversal is the space charge in the dielectric [12]. Roy and co-worker used a fluid model to simulate the charge transport process in LDPE, which was in good agreement with the electroacoustic method, discharge current and luminous intensity measurement results [13,14]. Boufayed and co-worker simulated the existence of a single level trap and exponential level trap in polyethylene and found significant differences in the space charge distribution and discharge current obtained from these two models [15]. Chen and co-worker used fluid simulation to simulate the process of charge trapping and detachment [16]. Zheng and co-worker simulated the dynamic change characteristics of the space charge of polyethylene under polarity reversal [17]. The value of electron affinity energy is usually used in theoretical calculations to determine the insulation performance of materials and follows the rule that the greater the electron affinity, the better the insulation performance [18]. However, Li and co-worker studied the space charge dissipation characteristics of polyethylene using the electroacoustic pulse method [19]. The results show that the space charge transfer rate and crystallinity of silicone oil cooled polyethylene are both large, and the electron affinity is also large. This result means that the electron mobility is a better criterion of the electrical resistance of materials. Yang and co-worker [20] developed a theoretical scheme to predict the carrier mobility of triphenylamine materials, which involves the Marcus electron transfer theory, a direct diabatic dimer model and the Brownian diffusion assumption. The design strategies of circular or linear dimers have been evaluated by the theoretical calculation results of carrier mobility. It was discovered that as for the triphenylamine dimer, because of the difference in the reorganization energy, macrocyclic dimers possess higher mobility than linear chain dimers. We will theoretically analyze polyethylene insulation materials from the perspective of electron mobility. Therefore, in this study, 3-Methylpentane (Pe) was selected as the model molecule for polyethylene. Density functional theory (DFT) was used to analyze the influence of electron-withdrawing and electron-donating groups on electron mobility at the atomic and molecular levels. wB97XD is an all electron exchange integration recovery (EX-I-R) method used to calculate the interaction energy and other physical quantum chemical quantities of complex molecules, such as the electronic structure, band states, and spectra. The wB97XD function is based on the wave function method and combines electron exchange and integration operations to achieve more accurate calculation results and can greatly reduce the time and cost of calculations [21]. The electron mobility can provide ideas and directions for the prediction of the electrical resistance of polyethylene insulation materials.

2. Computational Methods

In the present study, geometry optimization calculations of the stationary points on the ground state were performed using the DFT method [22] at the wB97XD/6-311G(d,p) level [21]. HOMO (E_HOMO) and LUMO (E_LUMO) are the single-particle eigenvalues of the Kohn–Sham equation in DFT. The relevant schematic formulae can be defined as follows: E_g = E_LUMO − E_HOMO, where E_g is used to denote the energy gaps between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). The π electron delocalization ability of the benzene ring is stronger than that of the σ electron (3-methylpentane). The entry of the benzene ring into the molecule is conducive to electron dissociation, because according to the Koopman theorem, the HOMO energy is high, so the ionization potential is small. It has a lower LUMO energy, so it has a larger electron affinity. IP(a) = E⁺(M⁺) − E(M) and EA(a) = E(M) − E⁻(M⁻); IP(a) and EA(a) are represented as the adiabatic ionization potentials and the adiabatic electron affinities, respectively, where E⁺(M⁺), E⁻(M⁻) and E(M) represent the energies of the cation, anion and neutral species in their lowest energy geometries, respectively. IP(v) = E⁺(M) − E(M) and EA(v) = E(M) − E⁻(M); IP(v) and EA(v) are donated as the vertical ionization potentials and the vertical electron affinities, respectively, while E⁺(M) refers to the energies of cation with neutral geometries; E⁻(M) is the energy of the anion with neutral geometries, where v and a represent vertical energy based on the geometry of the neutral molecule and adiabatic energy from the optimized structure for both the neutral and charged molecule, respectively. A schematic description of geometric coordinate modifications and energy changes is illustrated in Figure 1 [23]. They were all obtained based on the results of electronic structure calculations.

λ_e is referred to as the electron reorganization energy, and it can be calculated from the following Formulae (1) found in reference [23], where E(M⁻) are the energies of neutral species with the geometries of the anion.

λ_e = [E(M⁻) − E(M)] + [E⁻(M) − E⁻(M⁻)]

(1)

The Marcus-type expression is highly suitable for predicting carrier mobility ($\mu$) [24]. In the absence of an electric field, the carrier mobility is evaluated by using the Einstein–Smoloshosky relationship (2) [25]:

$$\mu ={\displaystyle \frac{\mathrm{e}D}{{\mathrm{K}}_{\mathrm{B}}T}}$$

(2)

where e represents the electron charge, k_B is the Boltzmann constant, T refers to the temperature in kelvin, and D is the diffusion coefficient (3) in Supplementary Materials and can be expressed as [26]:

$$D={\displaystyle \frac{1}{2\mathrm{n}}}\sum _i{d}_i^2{K}_i{P}_i$$

(3)

where n refers to the spatial dimension (for the crystal, n is 3), i refers to a specific hopping pathway, and d_i and k_i are the electron hopping centroid to centroid distance and the hopping rate, respectively in Supplementary Materials. P_i is the relative probability (4) for the charge carrier hopping to a particular pathway.

$$P_i={\displaystyle \frac{K_i}{\sum _i{K}_i}}$$

(4)

Based on the Marcus theory, the electron hopping rate (k_i) can be evaluated (5) by [27]

$$K_i={\displaystyle \frac{4{\mathsf{\pi }}^2{V}_{\mathrm{A}\mathrm{B}}^2}{h}}{\displaystyle \frac{1}{\sqrt{4\mathsf{\pi }{\lambda }_e{\mathrm{K}}_{\mathrm{B}}T}}}exp\left(-{\displaystyle \frac{{\lambda }_e}{4{\mathrm{K}}_{\mathrm{B}}T}}\right)$$

(5)

where h denotes the Planck constant and V_AB is the electron coupling energy (6) in Supplementary Materials. F is the Fock operator of the dimer.

$$V_{\mathrm{A}\mathrm{B}}=〈{\phi }_i^{\mathrm{H}\mathrm{O}\mathrm{M}\mathrm{O}/\mathrm{L}\mathrm{U}\mathrm{M}\mathrm{O}}\left|F\right|{\phi }_f^{\mathrm{H}\mathrm{O}\mathrm{M}\mathrm{O}/\mathrm{L}\mathrm{U}\mathrm{M}\mathrm{O}}〉$$

(6)

where the orbital of the electron hopping between two molecules is MO_e [24]. All the electronic structure calculations were performed by the GAUSSIAN 09 program package [28].

3. Results and Discussion

3.1. Theoretical Method Screening and Stationary Point Geometries

The molecular abbreviations, names and molecular formulas involved in this article are shown in

Table 1. The molecular name, molecular formula, and corresponding abbreviation (ab.) of the studied molecules.

Molecular Formula	Molecular Name	ab.	Molecular Formula	Molecular Name	ab.
	3-Methylpentane	Pe		3-Nitromethylpentane	Nmp
	3-Aminobenzoic acid	Aba		3-Benzylchloropentane	Bcp
	3-dimethylaminopentane	Dap		3-Ethyl-4-butylaniline	Eba
	3-aminomethylpentane	Amp		3-Ethyl-4-butyldimethylaminobenzene	Ebdb
	3-Ethylbutylbenzene	Ebb		3-Ethyl-4-butylnitrobenzene	Ebnb
	3-Ethyl-4-butyl-3-aminobenzoic acid	Ebaa

. In order to screen out the optimal calculation level, 3-methylpentane was selected as the model molecule in this study, and the IP values of the respective experimental data are taken as a reference, respectively. The method with the calculation results closest to the experimental values is identified as the optimal method and base group settings in this study. According to the calculation results in

Table 2. Calculated values of EA(a) and IP(a) (eV) of 3-Methylpentane under different functional and basis settings, as well as the experimental value.

Method	EA(a)	IP(a)	Method	EA(a)	IP(a)
HF/6-311G(d,p)	−4.08	9.12	M062X/6-311G(d,p)	−2.72	10.40
HF/6-311+G(d,p)	−1.93	9.89	M062X/6-311+G(d,p)	−1.34	10.40
MP2/6-311G(d,p)	−3.13	10.53	M062X/6-311G(3df,2p)	−2.42	10.37
MP2/6-311+G(d,p)	−1.52	10.38	wB97XD/6-311G*	−3.14	10.31
PBE/6-311G(d,p)	−2.37	9.45	wB97XD/6-311+G*	−1.72	10.30
PBE/6-311+G(d,p)	−1.06	9.46	wB97XD/6-311G(d,p)	−3.14	10.13
B3LYP/6-311G(d,p)	−2.54	9.62	wB97XD/6-311+G(d,p)	−1.72	10.26
B3LYP/6-311+G(d,p)	−1.14	10.01	wB97XD/6-311G(3df,2p)	−2.92	10.23
B3LYP/6-311G(3df,2p)	−2.28	9.58	Experimental data	--	10.08 [29]

, it is found that wB97XD/6-311G(d,p) is more suitable for the study of the electron mobility system, and the calculated IP values of 3-methylpentane are closest to the experimental values, IP_Pe (10.13 eV) ≈ IP_PeEXP (10.08 eV) [29]. Meanwhile, we selected two sets of models with different carbon atom numbers here, ruling out the possibility of errors in the wB97XD/6-311G(d,p) level due to differences in carbon chain length. The optimized structure of the stable point of the studied molecule has been completed at the wB97XD/6-311G(d,p) level, as shown in Figure 2.

3.2. Frontier MOs and Electron Mobility

The electron affinity energy and ionization potential of molecules are the important parameters to estimate the ability of oxidation and reduction. The calculated values of the adiabatic EA(a) and IP(a) at the wB97XD/6-311G(d,p) level and the corresponding experimental data [30] are presented in

Table 3. The λ_e, E_g, EA and IP of studied molecules calculated as well as the corresponding experimental data in the bracket (in eV).

Molecular Formula	λe	Eg	EA(a)	IP(a)	EA(v)	IP(v)
	0.16	13.77	−3.14	10.13 (10.08) [29]	−3.21	11.04
	0.65	4.47 (4.60) [32]	0.16	7.78	−0.16	8.10
	0.19	10.94	−2.91	7.27	−3.01	8.04
	0.22	11.56	−2.82	8.27	−2.93	9.11
	0.52	10.36	−1.51	8.47	−1.75	8.67
	1.55	10.57	−0.02	10.36	−0.67	10.90
	7.05	12.97	0.29	10.11	−2.88	10.64
	0.73	8.42	−0.20	7.37	−0.57	7.81
	0.66	8.92	−1.47	6.74	−1.80	6.95
	0.51	9.22	−1.55	7.10	−1.80	7.47
	0.69	8.95	0.72	9.99	0.36	10.70

, as well as the calculated HOMO-LUMO energy gap (E_g). From Table 3 and

Table 4. Calculated electron mobility (m² V⁻¹ s⁻¹) of the studied molecule at a wide temperature range.

Molecular Formula	233 K	263 K	273 K	300 K	313 K	333 K	363 K
	9.93 × 10−8	1.16 × 10−7	1.22 × 10−7	1.35 × 10−7 (1.7 × 10−7) [34]	1.41 × 10−7	1.48 × 10−7	1.59 × 10−7
	7.17 × 10−23	1.72 × 10−22	2.17 × 10−22	3.91 × 10−22	4.88 × 10−22	6.86 × 10−22	1.04 × 10−21
	8.79 × 10−12	1.08 × 10−11	1.15 × 10−11	1.31 × 10−11	1.39 × 10−11	1.49 × 10−11	1.64 × 10−11
	7.43 × 10−13	9.52 × 10−13	1.02 × 10−12	1.20 × 10−12	1.29 × 10−12	1.41 × 10−12	1.58 × 10−12
	2.78 × 10−16	5.42 × 10−16	6.60 × 10−16	9.85 × 10−16	1.24 × 10−15	1.60 × 10−15	2.23 × 10−15
	4.62 × 10−21	3.85 × 10−20	7.24 × 10−20	2.99 × 10−19	5.56 × 10−19	1.26 × 10−18	3.75 × 10−18
	6.92 × 10−50	1.29 × 10−45	2.41 × 10−44	1.76 × 10−41	3.19 × 10−40	1.43 × 10−38	2.37 × 10−36
	1.41 × 10−22	3.70 × 10−22	4.93 × 10−22	9.36 × 10−22	1.24 × 10−21	1.79 × 10−21	2.92 × 10−21
	3.58 × 10−19	8.53 × 10−19	1.10 × 10−18	1.96 × 10−18	2.52 × 10−18	3.51 × 10−18	5.44 × 10−18
	6.23 × 10−20	1.20 × 10−19	1.45 × 10−19	2.23 × 10−19	2.70 × 10−19	3.45 × 10−19	4.78 × 10−19
	3.39 × 10−48	8.40 × 10−48	1.10 × 10−47	2.01 × 10−47	2.62 × 10−47	3.70 × 10−47	5.86 × 10−47

, it can be seen the EA(a) value of the polyethylene material grafted with the electron-donating group or the electron-withdrawing group is greater than that of the cross-linked polyethylene itself, because the cross-linked polyethylene grafted with other groups is equivalent to introducing a deep trap in the cross-linked polyethylene, which improves its electrical resistance. At the same time, we also found that the introduction of electron-withdrawing groups will lead to a sharp increase in electron affinity and increase the electrical tolerance. However, compared to the grafted electron-donating groups, it can also lead to a larger HOMO-LUMO energy gap, making it more difficult for electrons to be excited, resulting in a decrease in electron mobility. However, structures with lower energy gaps are more likely to be ionized by high-energy electrons, thereby playing a role in electron buffering. Therefore, a structure with both electron-withdrawing and electron-donating groups is our preferred choice, which is consistent with the results of Chen and co-worker’s [31] study on the enhanced electrical resistance of 3-Aminobenzoic acid (Aba) blended with polyethylene, supporting our calculation results. It is not difficult to see from Table 3 and Table 4 that introducing functional groups with a benzene ring structure will reduce the energy gap between HOMO-LUMO, increase electron affinity, and reduce electron mobility, such as E_gDap (10.94 eV) > E_gEbdb (8.92 eV), EA(a)_Dap (−2.91 eV) < EA(a) _Ebdb (−1.47 eV), μ_Dap (1.31 × 10⁻¹¹ m² V⁻¹ s⁻¹) > μ_Ebdb (1.96 × 10⁻¹⁸ m² V⁻¹ s⁻¹); E_gAmp (11.56 eV) > E_gEba (9.22 eV), EA(a)_Amp (−2.82 eV) < EA(a)_Eba (−1.55 eV), μ_Amp (1.20 × 10⁻¹² m² V⁻¹ s⁻¹) > μ_Eba (2.23 × 10⁻¹⁹ m² V⁻¹ s⁻¹); E_gNmp (10.57 eV) > E_gEbnb (8.95 eV), EA(a)_Nmp (−0.02 eV) < EA(a) _Ebnb (0.72 eV), μ_Nmp (2.99 × 10⁻¹⁹ m² V⁻¹ s⁻¹) > μ_Ebnb (2.01 × 10⁻⁴⁷ m² V⁻¹ s⁻¹). This is because the benzene ring is both an electron-withdrawing and electron-donating group, resulting in molecules having both electron-donating and electron-withdrawing functional groups. It forms a large electron delocalization structure with the benzene ring, which can attract electrons to oscillate around the atomic nucleus, leading to energy dissipation, reducing electron mobility, and reducing the impact on polymer molecular chains. Therefore, when the stabilizer is uniformly dispersed in the polymer, buffering high-energy electrons can effectively suppress the initiation and growth process of the electrical tree in XLPE, which is consistent with the results of Chen and co-worker [32,33]. At the same time, we also found a negative correlation between electron affinity and electron mobility, that is, the higher the electron affinity, the lower the electron mobility, and the better the electrical resistance of insulation materials. This is consistent with the research results of Fu and co-worker [18], but we found an interesting phenomenon in Table 4. We can see that the research objects without benzene ring structures strictly follow the rule that the greater the electron affinity energy, the smaller the electron mobility. However, after the introduction of a benzene ring, it changed. The larger the electron affinity energy, the greater the electron mobility. EA(a)_Ebb (−1.51 eV) > EA(a)_Eba (−1.55 eV), μ_Ebb (9.85 × 10⁻¹⁶ m²V⁻¹s⁻¹) > μ_Eba (2.23 × 10⁻¹⁹ m²V⁻¹s⁻¹); this might due to the introduction of strong electron-donating groups in Eba, which increases the electron cloud density on the benzene ring, weakens its ability to accept electrons, and results in a relative decrease in EA(a) value. Meanwhile, the introduction of strong electron-donating carriers increases the depth and density of traps compared to Ebb and slows down the electron transport rate. In addition, the introduction of electron-withdrawing groups not only reduces the electron density on the benzene ring but also increases the EA(a) value, making it more capable of accepting high-energy hot electrons. It can also increase the depth and density of traps in the structure, thereby increasing the electrical resistance.

3.3. Wide Temperature Range Electron Mobility

The theoretical calculations in this study follow Marcus’ law [27]. It is easy to see that the value of electron mobility is closely related to temperature through the formula. Therefore, we set five temperature gradients in the actual working temperature range of cross-linked polyethylene and explored the relationship between temperature and the electron mobility of cross-linked polyethylene on the level of theoretical calculation, so as to judge the impact of temperature on cross-linked polyethylene insulation. Table 4 presents the calculated electron mobility of the studied molecules at seven temperature gradients of 233 K, 263 K, 273 K, 300 K, 313 K, 333 K and 363 K. From Table 4, we can see that all molecules follow the same pattern, that is, the electron mobility increases with the temperature, which is consistent with the results of Yang co-worker [20] and. At the same time, we also found that the electron mobility of Ebnb is the lowest, and the temperature change has the smallest impact on it, which is consistent with our results as mentioned above. In addition, this is shown in Table 4, where the electron mobility of molecules other than Bcp varies by within 3 orders of magnitude over a wide temperature range, while Bcp has a change of 14 orders of magnitude. This may be caused by the large number of electron layers and relative atomic mass of the grafting group, which weakens its ability to bind electrons during the reheating process. Therefore, there is a significant order of magnitude change in electron mobility.

4. Conclusions

A theoretical investigation on the effect of electron-withdrawing and electron-donating groups grafting cross-linked polyethylene on electron mobility has been carried out. In the cross-linked polyethylene model (3-methylpentane) system, wB97XD/6-311G(d,p) is the most suitable method after comparing the experimental value of IP with different methods and base group calculations. The grafting of different electron-withdrawing and electron-donating groups significantly improves the electrical properties of XLPE. Among them, the electron-withdrawing group nitrobenzene (Ebnb) had the lowest electron mobility and the smallest change in mobility under different temperatures. The electrical information revealed by the electron affinity and electron mobility values will contribute to the rational design of additive molecules in practical applications.