Introduction

Cross-linked polyethylene (XLPE) has been widely used in power cable insulation due to its excellent corrosion resistance, electrical properties, and chemical stability [1,2,3]. However, after a long period of operation under the high-voltage direct current (DC) field, it will accelerate ageing and damage because of the space charge accumulation and the electrical tree growth that cause irreversible changes in material properties and reduce insulation performance [4,5,6,7]. Many factors can induce the electrical treeing and the degradation of power cable insulation, such as the attack of hot electrons, local magnetic field effect produced by high current, the influence from water or temperature, and so on [8,9,10]. So suppressing the space charge accumulation and preventing the electrical tree growth are the key strategies to improve the lifetime of the power cable insulation. The previous studies focused on revealing the aging mechanism and developing methods to improve the electrical tree resistance [11,12,13,14,15]. Ieda et al. found that the breakdown property of polymer solids was governed by many complicated factors, such as volume effect, time effect, and defects [16]. To clarify the mechanism of electrical tree growth, Vaughan et al. studied the structures and chemical reaction processes of two electrical trees using confocal Raman microprobe spectroscopy, optical microscopy, and scanning electron microscopy [17]. Chen et al. investigated the electrical tree growth in XLPE cable insulation through an embedded needle electrode and found some branches which have conducting walls due to a layer of graphitic carbon, providing a means of early detection of prebreakdown phenomena in cable insulation [18].

In recent years, experimental studies indicated that nanosized particles are very promising fillers in power cable insulation to suppress the growth of electrical tree and prevent the degradation of polymer matrix [19,20,21]. Tanaka et al. conducted a comprehensive experimental investigation for XLPE and fumed silica (SiO2), and found that SiO2 nanocomposites additives could be applied extensively in the fields of extruded HV and EHV cables [22]. Zhang et al. used titanate coupling agent (TC9) and 3-(Methacryloyloxy)propyltrimethoxysilane (KH570) to modify the SiO2 surface, which greatly improved the DC conductivity, dielectric characteristics, and space charge properties in XLPE/SiO2 nanocomposites [23]. Li et al. proved that the addition of nano-Al2O3 fillers is also a good approach to raise once-lowered breakdown strength and the partial discharge resistance to protect the power cable [24]. Similarly, Ding and Varlow investigated treeing phenomena in epoxy-ZnO nanocomposites and found that the addition of a small amount (0.5 to 1 wt%) of zinc oxide particles in the epoxy insulation could extend the treeing time to breakdown [25]. Wang et al. explored the interface characteristics between SiC and XLPE using the pulsed-electro-acoustic (PEA) equipment and a dielectric analyzer and found that the best activity of electrical tree inhibition is achieved when the concentration of nanocomposite is 1 wt% [26]. Recently, Xiao et al. revealed the significant influence of TiO2 nanoparticles on the dielectric properties of TiO2/XLPE nanocomposites to suppress the space charge by PEA measurements [27]. Besides experimental studies, there are also many theoretical researches concentrating on this topic. By performing the quantum mechanics/molecular dynamics (QM/MD) simulations, Han et al. observed double electric layers around the SiO2 nanocluster and revealed the function of the SiO2 nanocluster as a stabilizer in trapping electrons [28]. More recently, a series of potential graphene-based particles candidates (N-doped single-vacancy graphene, graphene oxide, and B, N, Si, or P-doped graphene oxide) were predicted by the density functional theory method to be effective additives in power cables [29].

Although many experimental studies have proposed the promising SiO2 nanocomposites fillers, further theoretical study is desired to clarify the electronic interaction between the SiO2 fillers and the polymer matrix, as well as to design more efficient SiO2 particles with improved surface or bulk structures. The α-quartz (α-SiO2) is one of the most abundant and economical minerals with a wide range of applications, including high-frequency devices [30, 31], cellular and satellite network [32, 33], and quartz crystal microbalance [34,35,36]. Lots of experimental investigations showed that the α-SiO2 (001) surface is the most stable one [37,38,39,40], and the 1 × 1 pattern [41] and √84 × √84 reconstruction [42] are two kinds of popular crystallographic structures. Additionally, many models with the modified surfaces were constructed for increasing reaction rate and improving the adsorption effect, such as the dense model with a six-member ring and a triangle-like structure [39, 43], the hydroxylated model yielding silanol groups [44], the one with special point defects [45, 46], and the N or B-doped models [47,48,49,50]. Consequently, such abundant surface properties of α-SiO2 will greatly affect the interfacial interaction with XLPE, as well as the performance of XLPE/SiO2 nanocomposites in power cable insulation. With this expectation, we employed the first-principles calculations to investigate the electrical tree inhibiting mechanism of the SiO2 nanosized fillers with several different modified-surface structures by evaluating their ability of trapping hot electrons and restraining space charge. It is expected to provide valuable guidelines to develop potential candidates as additives in power cable insulation.

Computational methods and models

Computational methods

The density functional theory (DFT) calculations were performed using the Vienna Ab-Initio Simulation Package (VASP) [51,52,53] with Perdew-Burke-Ernzerhof (PBE) functional [54]. The DFT-D2 method was employed to evaluate the van der Waals interaction in the geometry optimizations. The projected-augmented wave (PAW) method [55] was used to treat the interaction between valence electrons and ion cores. A 2 × 2 × 1 Monkhorst–Pack k-point mesh and a cut-off energy of 850 eV were selected for all calculations until the energy and forces of atoms were smaller than 1.0 × 10−5 eV and 0.01 eV/Å, respectively. These parameters were tested to make sure that the differences of absolute energies were well converged (within a few meV). The charge distribution was obtained by using a real-space Bader-charge analysis [56,57,58] based on so-called zero flux surfaces to divide atoms on which the charge density is a minimum perpendicular to the surface. The energy profiles along the reaction pathways were computed with the climbing image nudged elastic band (CI-NEB) method, then the transition state and reaction energy barrier were also determined [59, 60].

Theoretical models

Based on the α-SiO2 (001) surface, six types of models were constructed as listed in Fig. 1. The completely hydroxylated SiO2 (denoted as H-SiO2) has herringbone structure and a zigzag hydrogen bonded network with short and long hydrogen bonds. Each oxygen dangling bond at the bottom atomic layer was saturated with one hydrogen atom. The dense surface, i.e., reconstructed surface (denoted as R-SiO2), was built according to Rignanese’s prediction [39]. As we know, doping B or N could improve the material’s toughness, oxidation resistance, thermal stability, etc. by introducing active sites [48, 61]. So B-doped and N-doped SiO2 models were constructed and denoted as B-SiO2 and N-doped SiO2, respectively. Two types of oxygen vacancy defect models were built by removing one oxygen atom on the top layer and the one connecting two SiO2 tetrahedrons on the lower layer, which are denoted as E-SiO2 and V-SiO2, respectively. The 2 × 2 supercell with nine atomic layers was constructed to model the bulk region of the cell. For improving the computational efficiency, four atomic layers on the bottom were fixed during the optimization. A vacuum layer of 15 Å was set to minimize the periodic interaction. The 2-methylbutane C5H12 [i.e., CH3CH(CH3)CH2CH3], including one tertiary carbon to represent cross-linked point, was used to model XLPE and simulate its activity on the surface of the SiO2 nanoparticle. With this theoretical model, the local interfacial interaction between XLPE and SiO2 particles and the catalytic activity of SiO2 for XLPE were considered properly. To avoid any weak van der Waals interaction between the adsorbed molecules, we ensured that the nearest distance of two C5H12 molecules in the neighborhood lattice was more than 5.0 Å.

Fig. 1
figure 1

a~f The SiO2 models with optimized surface structures (side view). Top views of H-SiO2 and R-SiO2 are also given with labeling of the key bond lengths in Å and angles in degree (For clarity, only surface atoms are shown)

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Results and discussion

To reveal the mechanism of improving the anti-ageing properties and performance of XLPE by adding SiO2 particles, the electrical properties of a series of SiO2/XLPE nanocomposites were analyzed concerning the surface structures, interfacial interaction, H migration reaction activity, and space charge behavior.

Surface structures

Figure 1 shows the optimized SiO2 models with key bond lengths and bond angles. For H-SiO2, one can see that all O–Si–O angles (~110°) are symmetry equivalent and all the silicon atoms are fully four-coordinated. The optimized lattice parameters (a = b = 5.05 Å and c = 5.50 Å) are well matched with experimental values (a = b = 4.92 Å and c = 5.41 Å) [62], and the calculated bond lengths of H–O and O–Si are also consistent with the values optimized with PW91 and B3LYP functional [37]. For R-SiO2, there are only four-coordinated Si and two-coordinated O atoms without any dangling bond. The predicted Si–O bond lengths (1.62 Å) and the Si–O–Si angles (122.9°) match well with previous results by Rignanese et al. [39] and Oleksandr et al. [63] using respective local density approximation (LDA) and PBE method. In B-SiO2 where one Si atom is substituted by B atom, the B atom still connects with four O atoms, keeping the BO4 center. In comparison, when one Si atom is replaced by N atom to form N-SiO2, only the NO3 skeleton is stabilized due to the rupture of one N–O bond. After removing one oxygen atom at the top layer of SiO2, there will be a three-coordinated Si atom to form E-SiO2. The bond length of Si–O bond (1.65 Å) is quite similar to that in H–SiO2 (1.66 Å), indicating that oxygen vacancy defect in E-SiO2 does not significantly change the geometrical structure. As the other point defect model, i.e., V–SiO2, the oxygen atom connecting two SiO2 tetrahedrons is absent and then an extra Si–Si bond is formed. The optimized Si–Si bond length with the value of 2.50 Å coincides with those obtained by employing the MP2 method [64] and B3LYP functional [65]. It is worth noting that for the models mentioned above, the obtained lattice parameters and key geometrical parameters are in good agreement with those in the literature with larger unit cell size, confirming that the present models are reliable. The electric dipole moments of different SiO2 cells were calculated and listed in Table 1. It is found that these values are very small, indicating that the doping or defect does not affect the property of polarization too much. Based on the discussion above, it is obvious to conclude that the doping, reconstruction, or defect can cause the surface structural change of SiO2 to different extents, and what is more, they will affect the electronic properties of surfaces by introducing many active sites and further influence the interfacial interaction with the XLPE chain.

Table 1 The dipole moment (D) of SiO2 surfaces, adsorption distances, adsorption energies (Eads) between C5H12 and SiO2, and the summarized Bader charge values at SiO2 surface
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Interfacial interaction

It is well known that the XLPE chain has a tendency of free movement, which probably accelerates the electrical tree generation with the attack of hot electrons. The SiO2 additives are capable of generating physical interaction with XLPE to constrain the movement of polyethylene chains. In this regard, accurate adsorption energy calculations will be of crucial importance to evaluate the intensity of physical interaction.

For each system, the adsorption energy (Eads) is calculated as follows:

$$ {\mathrm{E}}_{\mathrm{ads}}=\mathrm{E}\left({\mathrm{C}}_5{\mathrm{H}}_{12}+{\mathrm{SiO}}_2\right)-\mathrm{E}\left({\mathrm{C}}_5{\mathrm{H}}_{12}\right)-\mathrm{E}\left({\mathrm{SiO}}_2\right) $$

where E(C5H12 + SiO2) is the total energy of the SiO2 surface with the adsorbed C5H12, E(C5H12) and E(SiO2) are the energies of the C5H12 molecule and SiO2 surface, respectively.

The adsorption distances and adsorption energies between C5H12 and SiO2 surfaces in the optimized structures are collected in Table 1. As shown in Table 1, one can see that the C5H12 molecule usually locates 1.5~3.0 Å away from different SiO2 surfaces, and the adsorption energies vary from −0.10 to −0.41 eV, showing the pattern of van der Waals interaction between them and the characteristic of general physical adsorption. Among all the models, the N-SiO2 is the most active surface with the lowest Eads of −0.41 eV, while E-SiO2 is the most inactive one with the highest Eads of −0.10 eV. By contrast, the other H-SiO2, R-SiO2, B-SiO2, and V-SiO2 surfaces have moderate and almost similar adsorption ability for C5H12 with Eads of around −0.30 eV. In general, a larger adsorption intensity indicates a stronger capacity to constrain the movement of the polyethylene chain. Thus, the N-doped SiO2 is predicted to show the best performance among all the SiO2 additives mentioned above, and it is expected to be a potential candidate as effective additives in power cable insulation. Remarkably, the N-SiO2 plays a significant role in transferring charges from XLPE compared with others. By analyzing the Bader charge distribution of atoms belonging to SiO2 and C5H12, we obtained the transferred charge values as collected in Table 1. It shows that the summarized Bader charge value at the N-SiO2 surface is −0.56, which displays the strongest ability to attract the charge from the adsorbed C5H12 molecule among all SiO2 surfaces and then effectively inhibits the space charge accumulation of the XLPE. More details of space charge behavior, will be discussed in the “Space charge behavior” section.

As a summary, we come to a short conclusion that the SiO2 additives could restrict the movement of the polythene chain at different levels through van der Waals physical adsorption, and the N-doping is desired as a good strategy to modify the surface structure of SiO2 in order to strengthen the interfacial interaction.

H migration reaction activity

All the models mentioned above have completely hydroxylated surface structure except R-SiO2. However, in the real experimental operations, the SiO2 nanoparticles with incompletely hydroxylated surfaces are very common. So we constructed an incompletely hydroxylated SiO2 model by removing one H atom on the top layer of H-SiO2 surface, and investigated its adsorption activity for C5H12. It was found that the H atom of C5H12 could migrate easily to the unsaturated O atom, leading to the breaking of the C–H bond and the formation of a new H–O bond. The calculated adsorption energy (−1.80 eV) is much lower than those of other models studied above. The typical chemical adsorption activity will destroy the structure of XLPE and finally induce the electrical tree growth. Therefore, the SiO2 nanoparticles with incompletely hydroxylated surfaces are not beneficial for protecting XLPE. This provides important information for the experimental treatment of the SiO2 additives. On the other hand, it inspires us to investigate the H migration reaction activity of other SiO2 additives, even though they could stabilize the C5H12 with a certain adsorption distance.

We then employed the CI-NEB method to determine the H migration reaction pathways from C5H12 to the surfaces mentioned above. For H-SiO2, N-SiO2, V-SiO2, and R-SiO2, we found that no stable H migration products could be obtained, suggesting that H migration reactions cannot happen between these surfaces and the C5H12 molecule. Despite that we intended to elongate the C–H distance of C5H12 and set the H atom approaching to the surface as the initial structure, the geometrical optimization for the system still converged to the previous stable C5H12 adsorbed structure, i.e., the H atom went back to the C5H12 part again. The possible reason is that there is no dangling bond or active sites on these surfaces which can accept an extra hydrogen atom. In the cases of B-SiO2 and E-SiO2 systems, however, the situations are quite different. The reaction pathways for H migration from C5H12 to the B-SiO2 or E-SiO2 have been successfully determined and illustrated in Figs. 2 and 3, respectively. As shown in Fig. 2, the H atom in C5H12 migrates gradually to the hydroxyl connecting with B atom in B-SiO2 to yield one H2O molecule, and meanwhile the B–O bond is broken. It is a barrierless and exothermic reaction process with reaction energy of −1.02 eV, showing a very favorable process in energy. It is interesting to know why H migration can happen easily on this surface, but not on others. We believe that the B-doping effect may play a key role in it. The B-doping could keep the main skeleton of SiO2 where the B atom is four-coordinated. However, the B–O bonds are elongated (~1.48 Å) in comparison with those in the three-coordinated B atom (~1.34 Å), as denoted in Fig. 2, which shows that the formers are weakened. By accepting the transferred H atom, this B–O bond is broken and a water molecule is formed, resulting in a more stable planar BO3 center. With respect to the H migration reaction on oxygen vacancy E-SiO2 surface, as shown in Fig. 3, when the C–H bond is elongated to be 1.50 Å, the transition state (TS) is reached, in which the distance between H and Si is shortened to be 1.66 Å. In the product, the distance between H and Si is 1.47 Å, indicating that a new H–Si bond is formed. This process needs to overcome an activation barrier of 0.46 eV with the reaction energy of 0.04 eV, indicating that it may happen at room temperature. Thus, it can be concluded that B-SiO2 and E-SiO2 are not appropriate additives in power cable insulation due to the potential damage to XLPE.

Fig. 2
figure 2

The H migration reaction pathway on B-SiO2 surface. The key bond distances in Å are denoted at the bottom by showing related atoms at top layers

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Fig. 3
figure 3

The H migration reaction pathway on E-SiO2 surface. The key bond distances in Å are denoted at the bottom by showing related atoms at top layers

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Chemical activity is also a key factor to evaluate the performance of additives in power cable insulation. If the H atom of XLPE could migrate easily to the SiO2 surface to form a more stable structure, it will lead to the formation of a carbon center radical. This radical could either cross-link with other radicals to affect the stability of the XLPE chain, or react with other impurity substances to induce the growth of electrical treeing. According to the analysis of H migration reaction activity above, we screen out the H-SiO2, N-SiO2, and V-SiO2 with completely hydroxylated surfaces, as well as the reconstructed R-SiO2 as effective additives in power cable insulation.

Space charge behavior

When the cable insulation ages because of corrosion, high temperature, and dampness, there will be space charges accumulation. The space charge with high energy is usually called “hot electron”. It was generally observed that the increase of space charge accumulation in XLPE produces a deteriorative effect on the long-term operation and reliability of insulation materials. In this section, the ability of capturing hot electrons of SiO2 was investigated based on the Bader charge analysis. One or two extra electrons are injected respectively into the system to model the existence of hot electrons, denoted as ionic states. The charge distributions were obtained through summarizing the charge values of atoms belonging to C5H12 or SiO2 substrates, respectively.

Figure 4 depicted Bader charge results of neutral state and ionic states. For neutral state, only B-SiO2 and N-SiO2 surfaces have a relatively strong ability to attract electrons from C5H12, resulting in the distribution of negative charges on their surfaces (−0.14 for B-SiO2, and −0.56 for N-SiO2), as listed in Table 1. When one electron is injected to the system, one can see that the electron density mainly locates on the SiO2 additive, but not at C5H12. In particular, the charge values of B-SiO2, E-SiO2, and N-SiO2 are up to −0.92, −0.92, and − 1.17, respectively. When injecting two electrons, SiO2 still carries negative charges in large quantity. Especially for N-SiO2, the charge distribution on the surface reaches −1.63, indicating the strongest ability to capture hot electrons. Therefore, it can be inferred that the negative charges will mainly accumulated on SiO2 additives instead of XLPE. In this way, the XLPE was protected to weaken or avoid the attack of hot electrons. Additionally, we can see from Fig. 4 that part of the negative charges gather in the vacuum layer from −0.11 to −0.39 when excess electrons exist in the system. It means the charge could also be trapped in the middle of the matrix rather than the XLPE chain. As suggested in the experimental measurement for space charge distribution of TiO2/XLPE nanocomposites, the trapped negative charge in the interface region could shorten the effective distance of “solitary waves” migration, reduce the carrier mobility, weaken the impurity ionization, and finally make the heterocharge disappear [27].

Fig. 4
figure 4

a~f The Bader charge distributions of neutral state (in green) and ionic states with one electron injection (in blue) and two electrons injection (in red) for different SiO2 models

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In this way, the interfacial interaction helps the charge transferring out of the XLPE materials.

It is worth noting that all six SiO2 patterns we studied have the strong capability to capture hot electrons. It means the effect of doping or defect SiO2 nanoparticles on the space charge distribution is trivial. Similarly, we predict that other various oxygen vacancies of SiO2, such as nonbridging oxygen hole center (NBOHC), oxygen-deficiency related center (ODC), and silanone groups (SGs) [45], as well as another oxides, such as MgO, Al2O3, and TiO2 etc., will likely have similar properties to trap hot electrons to suppress the growth of electrical tree. It essentially relates to the energy band structures of metal-oxide semi-conductors. Taking SiO2 as an example, it was well studied in literature that the bottom of the conductive bands is mainly contributed from the 3s and 3p orbitals of the Si atom [49]. Similarly, we calculated the charge density of valence band (VB) and conduction band (CB) for the adsorption models of C5H12 with different SiO2 surfaces. As shown in Fig. 5, one can see that the VB orbital is around the C5H12 molecule, while the CB orbital mainly locates at the H-SiO2 or R-SiO2 substrates. This means that the hot electrons will take precedence to enter into the CB orbital and be trapped by the SiO2 additives. In the cases of B-SiO2, N-SiO2, E-SiO2, and V-SiO2, as shown in Suppl. Fig. S1, a similar conclusion could be reached since the dopant or vacancy produces more active sites in the additives to facilitate the attraction for extra electrons. Therefore, the orbital analysis could help us further understand why the additives could effectively trap hot electrons to suppress the electrical tree growth.

Fig. 5
figure 5

a~d The charge density of valence band (VB) and conduction band (CB) for the adsorption models of C5H12 with H-SiO2 and R-SiO2

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Conclusions

The SiO2 additives can effectively suppress the space charge accumulation and inhibit electrical tree growth of power cable insulation by constraining the movement of cross-linked polyethylene and trapping hot electrons. We performed density functional theory calculations to screen a series of SiO2 additives through studying the surface structures, interfacial interaction, H migration reaction activity, and space charge behavior. The SiO2 with incompletely hydroxylated or boron-doped surfaces are not good candidates as potential additives because they could facilitate the H migration reaction and destroy the XLPE chain. The N-doped SiO2 with a completely hydroxylated surface is predicted to be the most promising additive among the patterns we studied due to the results of its strongest abilities of adsorption to XLPE and transferring charge, as well as the weakest chemical activity. This information is useful for the experimental treatment of SiO2 additives and the design of other potential surface-modified SiO2 additives.