Ali Mezouar

Nadia Saidi-Amroun

Virginie Griseri

Gilbert Teyssedre

Mohamed Saidi
email: nsaidi@usthb.dz

Correlation between chemical, thermal and dielectric properties of gamma-irradiated PTFE thin films

Keywords: PTFE films, Satellites, Gamma irradiation, Degradation, Thermal properties, Crystallinity

come

Correlation between chemical, thermal and dielectric

properties of gamma-irradiated PTFE thin films

Introduction

Polytetrafluoroethylene (PTFE) is probably the most wellknown polymer within the family of fluoro-plastics. Its structure (-C 2 -F 4 -) n has fluorine atoms bonded in a helix form around the carbon backbone [START_REF] Li | Chapter five -radiation degradation or modification of poly(tetrafluoroethylene) and natural polymers[END_REF]. Both C-C and C-F chemical bonds are extremely strong bonds (372 kJ/mol for C-C and 500 kJ/mol for C-F) [START_REF] Li | Chapter five -radiation degradation or modification of poly(tetrafluoroethylene) and natural polymers[END_REF][START_REF] Galante | Study on application of PTFE, FEP and PFA fluoropolymers on radiation dosimetry[END_REF][START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF]. The large size of fluorine atoms compared to the size of carbon atoms and the high electronegativity make the polymer with a long and rigid linear chain, which is the reason for the high thermal stability of the PTFE molecule (operating temperature up to 260 °C) [START_REF] Galante | Study on application of PTFE, FEP and PFA fluoropolymers on radiation dosimetry[END_REF][START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF][START_REF] Liu | Structural changes of polytetrafluoroethylene during irradiation in oxygen[END_REF], and high stability to most chemical solvents [START_REF] Lappan | Changes in the chemical structure of polytetrafluoroethylene induced by electron beam irradiation in the molten state[END_REF][START_REF] Tian | Surface modification of PTFE by 60Co γ-ray irradiation[END_REF].

PTFE is considered as one of the most important thermal insulation materials used around the main body of spacecrafts, in order to provide thermal insulation of the electronic components of the satellite [START_REF] Everett | Chemical alteration of poly(tetrafluoroethylene) teflon induced by exposure to vacuum ultraviolet radiation and comparison with exposure to hyperthermal atomic oxygen[END_REF][START_REF] Paulmier | Radiation-induced conductivity of space used polymers under high energy electron irradiation[END_REF][9]. It is also one of the most commonly used polymers for electrical insulation [START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF] due to its excellent dielectric strength, beyond 150 kV/mm under DC stress at 340 °C [START_REF] Huang | Morphological evolution of polytetrafluoroethylene (PTFE) during thermal-oxidative ageing above and below the melting temperature[END_REF].

The satellites are permanently exposed while in space to the gamma rays emitted from the sun [START_REF] Mohammadian-Kohol | Effect of gamma irradiation on the structural, mechanical and optical properties of polytetrafluoroethylene sheet[END_REF][START_REF] Ma | Change in surface microstructure and properties of PTFE after solar radiation and its mechanism[END_REF]. The degradation of PTFE films by gamma rays may lead to the reduction of satellite life in space operation. Therefore, the study of gamma irradiation of insulating materials and understanding their effects on the chemical structure and thermal properties, as well as the dielectric properties, are of great importance. PTFE is also used as insulation in a wide variety of wire and cables in applications requiring demanding thermal properties as with aircraft industry and in nuclear power plants. Radiation effects need therefore to be investigated even in aerobic conditions as the presence of oxygen decreases its radiation resistance [START_REF] Wyszkowska | Functional properties of poly (tetrafluoroethylene)(PTFE) gasket working in nuclear reactor conditions[END_REF].

The study of the effects of low dose gamma irradiation < 200 kGy on PTFE has been the subject of much research [START_REF] Serov | Double melting in polytetrafluoroethylene γ-irradiated above its melting point[END_REF]. Previous works on this topic have shown that PTFE undergoes degradation upon exposure to gamma ionizing radiation [START_REF] Oshima | Improvement of radiation resistance for polytetrafluoroethylene (PTFE) by radiation crosslinking[END_REF]. Therefore, even a low dose of irradiation either in air or under vacuum can lead to a significant decrease in its mechanical properties and molecular weight [START_REF] Su | Study on polytetrafluoroethylene aqueous dispersion irradiated by gamma ray[END_REF]. The radiation sensitivity of PTFE has been attributed to the presence of oxygen in the irradiation system, which participates in radiation-induced reactions [START_REF] Zhong | Radiation stability of PTFE irradiated under various conditions[END_REF][START_REF] Fischer | FTi.r Spectroscopy on electron irradiated polytetrafluoroethylene[END_REF][START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF].

Radiation sensitivity of PTFE can be exploited positively at the processing level. To reduce the cost of producing PTFE, many scientists and other professionals are concentrating on employing gamma or electron beam irradiation to improve crosslinking around the melting point [START_REF] Serov | Double melting in polytetrafluoroethylene γ-irradiated above its melting point[END_REF][START_REF] Zhong | Radiation stability of PTFE irradiated under various conditions[END_REF][START_REF] Fischer | FTi.r Spectroscopy on electron irradiated polytetrafluoroethylene[END_REF][START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF][START_REF] Lappan | Radiation-induced branching and crosslinking of poly(tetraf luoroethylene) (PTFE)[END_REF][START_REF] Oshima | Radiation-induced free radicals and their behaviour in crosslinked polytetrafluoroethylene (PTFE)[END_REF][START_REF] Oshima | Chemical structure and physical properties of radiation-induced crosslinking of polytetrafluoroethylene[END_REF].

Oshima et al. have studied the crosslinking of PTFE under gamma irradiation by electron spin resonance (ESR) spectroscopic analysis [START_REF] Oshima | Radiation-induced free radicals and their behaviour in crosslinked polytetrafluoroethylene (PTFE)[END_REF][START_REF] Oshima | Chemical structure and physical properties of radiation-induced crosslinking of polytetrafluoroethylene[END_REF]. During irradiation in oxygen, free radicals ( -CF 2 -ĊF -CF 2 -and-CF 2 -C Ḟ2 ) are formed in amorphous regions and then converted into peroxy radicals (

-CF 2 -CF 2 O Ȯ,-CF 2 -CF(O Ȯ) -CF 2 -)
by oxidation reaction at low temperature. Liu et al. were the first to report on lattice expansion and crystallite size increase in irradiated PTFE [START_REF] Liu | Structural changes of polytetrafluoroethylene during irradiation in oxygen[END_REF]. They explained this expansion by the formation of new groups during irradiation and their incorporation into the crystal lattice, leading to deformation of the lattice cell and ultimately to the expansion of the material.

Numerous studies have been carried out on the effect of gamma irradiation on mechanical properties. As an example, Mohammadian-Kohol et al. reported on an increase in stiffness with a decrease in tear strength of the PTFE sheet as the dose absorbed by the PTFE is increased [START_REF] Mohammadian-Kohol | Effect of gamma irradiation on the structural, mechanical and optical properties of polytetrafluoroethylene sheet[END_REF]. Furuta et al. also showed that as the gamma irradiation dose increases, the tensile strength of PTFE decreases significantly with decreasing crystal size [START_REF] Furuta | Molecular level analyses of mechanical properties of PTFE sterilized by Co-60 γ-ray irradiation for clinical use[END_REF][START_REF] Li | Study on structural evolution of polytetrafluoroethylene irradiated by electron beam under stretching using SAXS/WAXS[END_REF].

Though many researchers have studied the degradation of PTFE by gamma irradiation, a literature survey indicates that the effect of gamma irradiation on the dielectric properties of PTFE, in particular, dielectric permittivity and dissipation factor is still not completely understood. This study aims to report the correlation between structural, thermal, and dielectric properties of the gamma-irradiated PTFE thin films.

Materials and methods

Materials

The PTFE polymer films of 200 µm thickness were obtained from Goodfellow company and were used without any further treatment.

Gamma ray irradiation

The polymer samples of size 5 × 5 cm 2 were selected. Some samples were kept as virgin and the other samples were subjected to 60 Co gamma radiation source in air at room temperature with a dose rate of 22 Gy/min. Total doses of 30, 60, 100 and 200 kGy were adopted in this work.

Fourier-transform infra-red spectroscopy

FTIR spectra were recorded using a VERTEX 70 infrared spectrometer under transmission and attenuated total reflection (ATR) modes, designed for data acquisition in the region 400-4000 cm -1 with z resolution of 1 cm -1 . The purpose of using the reflection mode is to avoid peak saturation appearing in the transmission mode.

Differential scanning calorimetry

The Differential Scanning Calorimeter (DSC) method was used to investigate modifications in the thermal behaviour in virgin and irradiated PTFE. Our measurements were carried out using a TA Instruments Q2000 DSC. The calorimeter was previously calibrated with indium at the same heating/cooling rates employed in the analysis. Samples with a weight of 10 mg were enclosed in hermetic aluminium pans. A typical protocol consisting of two consecutive scans of cooling / heating thermal cycles has been followed. A temperature range from -100 up to 355 °C with a cooling/heating rate of 10 °C/min has been applied under a nitrogen atmosphere in order to prevent thermal degradation. Only the second scan is presented and analysed in this paper as the first scan is just used to erase the thermal history.

Dielectric spectroscopy

Dielectric spectroscopy measurements were performed using a Novocontrol Alpha-A broadband dielectric spectrometer under a nitrogen atmosphere with a temperature scan between -150 to 150 °C in steps of 20 °C, under a sinusoidal AC voltage Vrms value of 1 V.

The frequency range is from 1 Hz to 1 MHz. The temperature control and the resolution of the dissipation factor are given with precision of ± 0.1 °C and 5 × 10 -5 respectively.

The samples were provided with circular electrodes formed by sputtering a thin layer of gold (approximately 50 nm). The metallization diameter is directly linked to the sensitivity of the measuring bench where the capacitance is ideally equal to C 0 = 100 pF. Based on that, the electrode diameter for 200 µm-thick PTFE is D = 3 cm.

Results and discussion

Physicochemical analysis

The infrared transmission (FTIR) and reflection (ATR -FTIR) spectra of non-irradiated PTFE and of PTFE irradiated at different doses of gamma rays are shown in Figs. 1 and2 respectively. In the ATR mode (Fig. 2) it is possible to identify and analyse the peaks that were saturated in FTIR transmission mode (Fig. 1). The band at 2365 cm -1 was assigned to a combination of bands from two modes associated with the CF 2 backbone [START_REF] Liu | Structural changes of polytetrafluoroethylene during irradiation in oxygen[END_REF].

Two bands at 1200 and 1146 cm -1 due to the CF 2 asymmetric stretching in the amorphous region dominate the infrared spectra obtained by attenuated total reflection (ATR) [START_REF] Saidi-Amroun | Analysis of the Gamma Irradiation Effect on PTFE Films by FTIR and DSC[END_REF]. Its amplitude shows that the amorphous region is dominant. In addition, the presence of absorption bands in the region [700-800 cm -1 ] is also attributed to the CF 2 groups in the amorphous phase [START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF]. The bands at 507, 554, 625 and 638 cm -1 are associated to the CF 2 groups in the crystalline phase [START_REF] Legeay | AF fluoropolymer for optical use: Spectroscopic and surface energy studies; comparison with other fluoropolymers[END_REF]. According to this analysis, and confirmed by DSC analysis, the investigated PTFE has a semi-crystalline structure, where the amorphous phase is dominant.

Regarding the absorption bands located at 2927, 2853 and 2960 cm -1 , it has been attributed to the CH 2 -CH 3 elongation. Their presence would be due to impurities on the surface of the sample [START_REF] Mihály | FTIR and FT-Raman spectroscopic study on polymer based high pressure digestion vessels[END_REF].

The FTIR spectra in Fig. 1 show saturated bands (due to the thickness of PTFE sample) at 1200 cm -1 and from 400 to 700 cm -1 . These bands are clearly detected by ATR spectroscopy. The spectral bands assignment is reported in Table 1.

IR spectra in specific regions described above are plotted in Figs. 3 and4 for ATR and transmission modes, respectively.

In Fig. 3, the intensity and width of the 507 cm -1 band, which is attributed to the CF 2 mode in the crystalline phase, clearly increase as the absorbed dose is increased. The FTIR analysis illustrated in Fig. 4a indicates a decrease in the intensity of the bands associated with the CF 2 mode in the amorphous phase. Thus, the decrease in the height of the bands at 700-800 cm -1 (Fig. 4a) after gamma irradiation means also that the amorphous phase has decreased notably in the polymer. In the present case, the amorphous phase is therefore affected by the irradiation in favour of the crystalline phase. We can notice that the impact on the amorphous phase is quite important between 0 and 30 kGy, then the changes are milder (Fig. 4a). It can be concluded that the polymer with a high amorphous fraction undergoes a larger amount of molecular weight scission compared to the sample with a high crystalline content since the amorphous regions are considered to be the most oxygen-permeable. Consequently, a huge degree of molecular weight degradation occurs through enhanced chain-scission reactions in this zone.

According to IR data, gamma irradiation of PTFE in the presence of oxygen induces an increase in the degree of crystallinity, reflected by a decrease in the intensity of the amorphous phase absorption bands at 1200-1146 cm -1 (Fig. 3b), 700-800 cm -1 (Fig. 4a) and an increased absorption for the crystalline band at 507 cm -1 (Fig. 3a).

It should be noted that the wide band at 1792 cm -1 for non-irradiated PTFE (Fig. 4b) splits into two bands situated at 1792 cm -1 with a shoulder at 1810 cm -1 in the irradiated PTFE samples, which become very clear with the increase of irradiation dose. These two absorptions are attributed to the vibration of associated and isolated COOH groups [START_REF] Fischer | FTi.r Spectroscopy on electron irradiated polytetrafluoroethylene[END_REF][START_REF] Schierholz | Electron beam irradiation of polytetrafluoroethylene in air: investigations on the thermal behaviour[END_REF]. It confirms the occurrence of oxidation phenomena under irradiation since the band amplitude increases with the irradiation dose. At radiation doses above 30 kGy, we can observe an increase in the amplitude of the peak associated with C = O of the COF end groups in the range of 1884 cm -1 (Fig. 4b). It means that the scission of the chain is increased and therefore, a decrease in the average molecular weight of PTFE is expected.

Overall, it can be concluded that the high concentration of groups C = O (COOH free /association and COF) especially at higher doses reflects an increased degradation of PTFE and works to sweep free radicals produced by irradiation to produce stable end groups and lower molecular weight of the polymer [START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF].

Besides the above-mentioned data, it can be stated that the polymer morphology has a major impact on the diffusion of molecular oxygen within the polymer during irradiation. Diffusion is largely limited to amorphous regions because oxygen is unlikely to accommodate into the closely packed crystalline region [START_REF] Zhong | Radiation stability of PTFE irradiated under various conditions[END_REF][START_REF] Pasternak | Diffusion and permeation of oxygen, nitrogen, carbon dioxide, and nitrogen dioxide through polytetrafluoroethylene[END_REF]. This will result in degradation confined to the amorphous regions, which results principally from the cage effect for recombination of radicals in the crystalline region.

Fisher and Corelli have reported on such effects in the presence of oxygen [START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF]. In their research of the effect of ionizing radiation on PTFE, a resonant transformer cathode-ray accelerator was used as a radiation source.The cage effect, caused by the lack of oxygen in the crystalline zone, limits the mobility of the primary free radicals formed by chain scissions, which contributes to the restoration of the broken chains. According to the literature reviewed above, the originality of our work lies in the irradiation source, which was gamma irradiation, as well as the varying doses at which we achieved the same findings using a different irradiation source.

Thermal analysis

The DSC thermograms obtained during the second heating/cooling cycle of virgin PTFE are shown in Fig. 5. Three endothermic peaks are revealed as the temperature is increased. According to the previously reported behaviour of PTFE [START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF][START_REF] Villani | Calorimetric and diffractometric study of the lowest room-temperature transition of polytetrafluoroethylene[END_REF], the first peak located at around T 1 = 19 °C is attributed to the transition from the triclinic (Phase II) to the hexagonal (Phase IV) of PTFE. The second endothermic peak at T 2 = 27.3 °C corresponds to the transition from Phase IV to the pseudo hexagonal (Phase I) structure. These two peaks are observed during heating and cooling and correspond to reversible solid-solid crystalline phase transitions. The third endothermic peak at T m = 327.3 °C is attributed to the melting temperature [START_REF] Blumm | Characterization of PTFE using advanced thermal analysis techniques[END_REF]. During the cooling, an exothermic peak located at 312.4 °C is observed. It corresponds to the crystallization peak [START_REF] Fayolle | Radiation induced embrittlement of PTFE[END_REF]. It is followed by the solid-solid crystalline phase transitions described above. DSC measurements were performed on samples after different doses of gamma irradiation using the same protocol. Melting and crystallization peaks are shown and compared with data from a virgin sample. The final results are shown in Fig. 6.

After irradiation at 30 kGy, the amplitude of the melting peak is greatly increased and the melting temperature T m is also slightly increased. However, when the irradiation is continued from 60 to 200 kGy, the melting temperature tends to decrease.

During the cooling, the crystallization peak increases in amplitude after the irradiation under 30 kGy, but in this case, the crystallization temperature T c remained stable. The same response is detected with samples irradiated at 60 and 100 kGy. At 200 kGy, the peak amplitude rises again, and the T c decreases a little.

From the above results, it is shown that after exposure to irradiation, the heat of fusion rises quickly compared with the virgin sample. The heat of crystallization is also affected by the radiation dose. This often indicates that the polymer has undergone chain scissions under the effect of irradiation. Chain scissions decrease residual tension in the amorphous region [9, [START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF][START_REF] Saidi-Amroun | Analysis of the Gamma Irradiation Effect on PTFE Films by FTIR and DSC[END_REF][START_REF] Zhong | Radiationinduced crystal defects in PTFE[END_REF], which, in turn, leads to some increase in molecular motion and facilitates crystallization [START_REF] Su | Study on polytetrafluoroethylene aqueous dispersion irradiated by gamma ray[END_REF][START_REF] Saidi-Amroun | Analysis of the Gamma Irradiation Effect on PTFE Films by FTIR and DSC[END_REF]. In the melting and crystallization curves shown in Fig. 6, it is found that the increase in the absorbed dose up to 200 kGy shifts the melting and crystallization temperatures to lower values. This shift towards lower temperatures can be explained by a decrease in the polymer molecular weight caused by the scission of the macromolecular chains and a decrease in the size of the formed crystals [START_REF] Lappan | Radiation-induced branching and crosslinking of poly(tetraf luoroethylene) (PTFE)[END_REF][START_REF] Oshima | Radiation induced crosslinking of polytetrafluoroethylene[END_REF][START_REF] Kakigi | Study on chemical structures of poly (tetrafluoroethylene-co-perfluoroalkylvinylether) by soft-EB irradiation in solid and molten state[END_REF].

The information from DSC measurement for virgin and irradiated PTFE samples is summarized in Table 2. Melting temperature T m , and crystallization temperature T C , were determined as the maxima of relevant peaks of DSC curves. The enthalpy of melting (ΔH m ) and crystallization (ΔH C ) were calculated by the integration of the appro- priate peak area in the DSC thermogram at melting and crystallization, respectively. The degree of crystallinity was calculated from the following relationship [START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF]:

where ΔH 100 m is the theoretical melt enthalpy of 100% crystalline PTFE which equals 82 J/g [START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF].

In the present study, the number-average molecular weight M n was determined from the enthalpy of crystalliza- tion according to Suwa's method [START_REF] Suwa | Melting and crystallization behavior of poly(tetrafluoroethylene). New method for molecular weight measurement of poly(tetrafluoroethylene) using a differential scanning calorimeter[END_REF], it was estimated based on the following empirical relationship: where M n is the number-average molecular weight in g/mol. ΔH C is the enthalpy of crystallization in J/g. Through the results shown in Table 2, it appears that the temperatures of melting and crystallization are merely not affected by the radiation dose, with just a slight decrease for the highest dose. This indicates the stability of the thermal properties of PTFE under the influence of gamma irradiation.

From Table 2, it can be seen that the crystallinity of PTFE greatly increases with the dose of gamma irradiation. From the above FTIR discussion, it may be concluded that the crystallinity degree of PTFE is affected by the presence of oxygen in the irradiation system and the percentage of the amorphous fraction, as might be expected with the dose of gamma irradiation.

Figure 7 represents the degree of crystallization and the average molecular weight as a function of the absorbed radiation dose. A significant increase in crystallinity was found to be associated with a significant decrease in average molecular weight. This indicates that the mechanism of chain scission reactions is the dominant mechanism in PTFE during irradiation.

Dielectric properties

Materials were analysed by dielectric impedance spectroscopy method. The real permittivity versus the temperature is plotted in Fig. 8 at specific frequencies in the range [1 Hz -1 MHz]. In all the cases the real relative permittivity ε r ′ decreases continuously with the temperature from -150 °C to 150 °C. In addition, it is observed that as the frequency (2) M n = 2.1 × 10 10 × (4.184 × ΔH -5.16 C ) increases, the relative permittivity decreases (from 2.39 to 2.37 -150 °C). However at high temperature the permittivity tends to 2.304 for all the frequencies. A step in the real permittivity appears close to the room temperature between 10 and 20 °C for all the frequencies.

Since PTFE is a non-polar polymer, the real permittivity decreases slightly as the temperature increases, as observed in Fig. 8. This is due to the fact that the polarization of PTFE is a purely electronic polarization [9,[START_REF] Aziz | Structure, dielectric properties and AC behavior of commercial polytetrafluroethylene (PTFE) polymer[END_REF]. According to the Clausius-Mossotti equation, the relative permittivity can be expressed as follows [9]: where N is the number of polarizable entities per unit volume, α e is the electronic polarizability, ρ is the mass density, ε o is the permittivity of free space, M is the molar weight.

The step of permittivity in the range [10-20 °C] can be associated to the solid-solid transition from the triclinic phase II under 19 °C to the hexagonal phase IV. This was already outlined in the DSC analysis section. However, the second transition to a pseudo-hexagonal crystal (phase I) localised at 27 °C does not provided any significant signature in the present spectra.

Figure 9 shows the relative permittivity (a) and dissipation factor, tan δ (b) versus frequency, at ambient temperature (20 °C), for the irradiated PTFE with different doses of gamma rays. It can be seen that the real permittivity of all samples (Fig. 9a) increases towards the lower frequencies.

In Fig. 9b it appears that the dissipation factor tan δ for irradiated PTFE versus frequency decreases rapidly with increasing frequency in the range 1 Hz-100 Hz. The decay rate becomes more important as the gamma irradiation dose is increased. At higher frequencies in the range 100 Hz -1 MHz, it is observed that whatever the dose of gamma irradiation of PTFE, tan δ remains constant.

The permittivity of virgin PTFE ranges between 2.37 and 2.33 from 1 Hz to 1 MHz at room temperature (see Fig. 8). It is important to notice that the dielectric permittivity of irradiated PTFE is modified as soon as the sample is irradiated. The PTFE irradiated with a dose of 200 kGy of gamma rays shows the highest dielectric permittivity value over the full frequency range, the real permittivity is estimated to have a value of 3.25 at 1 Hz. The main differences are always

(3) r = 3M 0 M o -N e
observed at low frequency. Clearly, the dielectric permittivity of PTFE samples increases with gamma irradiation at all frequencies.

As shown above, irradiation has strong effects on the dielectric properties, through a clear increase in the real permittivity and the appearance of low frequency features on both the real and imaginary part of the permittivity. These effects have to be put in relation to the structural analyses revealing that the gamma irradiation leads to the scission of the polymer chain in the amorphous regions and the formation of crystallized regions within it, which increases the heterogeneity of the structure. In addition, the formation of polar groups was revealed by FTIR.

In the context of irradiated PTFE, many mechanisms are possible drivers for the dielectric properties change: increase in conductivity due to radiation, ionic contributions to conduction and possible electrode polarization effects [START_REF] Ramesh | Dielectric behaviour of PVC-based polymer electrolytes[END_REF], orientation polarization effects and internal interface polarization due to material heterogeneity. Therefore, it is not straightforward to single out the effects.

There are existing reports showing a positive correlation between the permittivity of PTFE and the degree of crystallinity [9,[START_REF] Koizumi | Dielectric properties of polytetrafluoroethylene and tetrafluoroethylene-hexafluoropropylene copolymer[END_REF]. Crystallinity variations were obtained without changing the chemical structure, either by quenching or by annealing for different times. However, the reported changes in permittivity are mild, less than 2%, contrasting with present results. The introduction of polar groups in the material, either due to oxidation or to the imbalance of the fluorine atoms along the polymer chain, is possibly at the origin of the clear increase in real permittivity at intermediate to high frequency without appearance of losses.

High field electrical measurements realized on the same materials revealed a strong increase in the DC electrical conductivity even at low doses, and the formation of heterocharges that may be a fingerprint of ionic contributions [START_REF] Mezouar | Effect of Gamma-Irradiation on Creation and Dynamic of Space Charge in PTFE[END_REF]. Note that the increase of both the permittivity and the conductivity at low frequency cannot be explained by pure conduction effect only. It could be the reflect of charge accumulation at the electrodes. However, the time scale of the phenomena seems not compliant with space charge measurement results.

A positive correlation between the real permittivity of PTFE and the degree of crystallinity can be explained by impeded movement of charge carriers at the boundaries of the crystalline/amorphous interfaces, as was done for polyethylene glycol [START_REF] Alig | Ac conductivity and dielectric permittivity of poly(ethylene glycol) during crystallization: Percolation picture[END_REF]. This leads to the interfacial polarization of Maxwell-Wagner-Sillars (MWS) type. The interfacial polarization can be improved under the condition of increasing the crystalline phase amount, resulting in an increase in the real permittivity and of losses at low frequency. Not only the increase in crystallinity is produced by irradiation, but also a modification of the properties of the amorphous phase (chain scission). As a consequence, the contrast between the properties (permittivity, conductivity) of the two phases is increased and hence the MWS effect is strengthened.

Conclusion

In this study, correlation between the chemical, thermal, and dielectric properties of gamma-irradiated PTFE thin films was established. PTFE was irradiated with doses of 30, 60, 100, and 200 kGy in the presence of oxygen at room temperature. This work led to the following findings:

The results obtained from FTIR indicate that most of the polymer chain degradation occurs in the amorphous region. This is due to the oxygen permeability in the amorphous regions. Changes are revealed by the decrease in the intensity of amorphous phase-related absorption bands in the regions 1200-1146 cm -1 and 700-800 cm -1 and by the formation of COF and COOH groups (bands at 1800 cm -1 ) due to oxidation reactions. The FTIR analysis pointed out the increase in the crystalline phase amount. In addition, DSC provided quantitative data on the substantial increase of the crystallinity with irradiation, as well as the drop in the averaged molecular weight for irradiated PTFE samples.

The dielectric response of irradiated PTFE reveals an increase of the dissipation factor and of the real permittivity with increasing the dose of gamma irradiation, which is attributed to the enhancement of the crystalline phase. All these discussions about gamma irradiation of PTFE lead to the conclusion that the dielectric properties are closely correlated with structural properties.

Fig. 1 FTIR

 1 Fig.1FTIR spectra of virgin and irradiated PTFE

Fig. 3

 3 Fig.3ATR-FTIR spectra of virgin and irradiated PTFE with different doses of gamma rays (a) in the range of 540 -460 cm -1 , (b) in the range of 1300 -1050 cm-1

Fig. 4 Fig. 5

 45 Fig. 4 IR transmission spectra of virgin and irradiated PTFE with different doses of gamma rays (a) in the region of 800-700 cm -1 , (b) in the region of 1920-1730 cm -1

Fig. 6

 6 Fig. 6 DSC crystallization (a) and melting (b) curves for the virgin and irradiated PTFE at various doses. Scan rate of 10 °C/min

Fig. 7 Fig. 8

 78 Fig. 7 Degree of crystallinity (full triangle) and numberaveraged molecular weight (full circle) as a function of the absorbed dose calculated from DSC data on PTFE

9

  Dependence of relative permittivity (a) and dissipation factor tan (b) on the frequency at 20 °C of PTFE irradiated with different doses

Table 1

 1 Spectral bands assignment for PTFE[START_REF] Henri | Thermal ageing of PTFE in the melted state: Influence of interdiffusion on the physicochemical structure[END_REF][START_REF] Liu | Structural changes of polytetrafluoroethylene during irradiation in oxygen[END_REF][START_REF] Fischer | FTi.r Spectroscopy on electron irradiated polytetrafluoroethylene[END_REF][START_REF] Fisher | Effect of ionizing radiation on the chemical composition, crystallinity, and processibility of polytetrafluoroethylene[END_REF][START_REF] Legeay | AF fluoropolymer for optical use: Spectroscopic and surface energy studies; comparison with other fluoropolymers[END_REF][START_REF] Mihály | FTIR and FT-Raman spectroscopic study on polymer based high pressure digestion vessels[END_REF][START_REF] Schierholz | Electron beam irradiation of polytetrafluoroethylene in air: investigations on the thermal behaviour[END_REF]

	IR bands (cm -1 )	Assignment
	507	CF 2 wagging
	554	CF 2 bending
	625	CF 2 wagging
	638	CF 2 rocking
	700-800	Amorphous phase
	1200 -1146	CF 2 Symmetric stretching
	1546-1450	CF 2 Asymmetric stretch
	1792	C = O of COOH associated
	1810	C = O of COOH free
	1884	C = O of COF end group
	2365	A combination band of two
		modes associated with the CF 2 backbone
	2927-2853-2960	CH 2 and CH 3 stretching

Table 2

 2

	Evolution in thermal properties of virgin and gamma-	Dose (kGy)	( o C)	Δ ( ∕ )	(%)	( o C)	Δ ( ∕ )	( ∕	)
	irradiated PTFE	0	327.32	32.73	39.91	312.44	30.35	7.61 × 10 5
		30	328.68	68.81	83.91	312.98	68.08	1.17 × 10 4
		60	328.49	70.84	86.39	313.37	68.70	1.12 × 10 4
		100	328.23	75.05	91.52	312.95	73.73	7.80 × 10 3
		200	327.36	81.12	98.92	310.60	80.68	4.90 × 10 3

Acknowledgements This work is realized in the frame of the Partenariat Hubert Curien (Tassili research program) and the DGRSDT of Algeria. The authors are grateful to the Director and staff (especially M. R. Yefsah) of the Nuclear Research Centre of Algiers, for the provision of the irradiation facility.

Data availability Original data of the paper is available under reasonable request to the corresponding author.

Author contributions Conceptualization, SAIDI-AMROUN Nadia, GRISERI Virginie, and SAIDI Mohamed; Methodology, MEZOUAR Ali, SAIDI-AMROUN Nadia, GRISERI Virginie; Validation, SAIDI-AMROUN Nadia, GRISERI Virginie, TEYSSEDRE Gilbert, SAIDI Mohamed; Formal analysis and data interpretation, all; Investigation, MEZOUAR Ali; Resources, SAIDI-AMROUN Nadia, GRISERI Virginie; Writing-original draft preparation, MEZOUAR Ali; Writingreview and editing, SAIDI-AMROUN Nadia, GRISERI Virginie, and TEYSSEDRE Gilbert; All authors have read and agreed to the published version of the manuscript.

Declarations

Conflict of interest The authors declare no conflict of interest.


Francesco Mattiotti

Jérôme Dubail

David Hagenmüller

Johannes Schachenmayer

Jean-Philippe Brantut

Guido Pupillo

J.-P Brantut

Multifractality in the interacting disordered Tavis-Cummings model

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction. Multifractality is a generalization of the concept of a fractal object -usually characterized by a single fractal dimension -to scale-invariant probability distributions characterized by a continuous spectrum of fractal dimensions [1]. Multifractality plays an important role in a wide variety of situations including earth topography [2], turbulent flows [3], heartbeat dynamics [4], and financial markets [5]. Another example is the case of Anderson metal-insulator transitions in disordered quantum systems [6], where multifractality is known to be a meaningful feature of the critical wavefunctions connected to a characteristic (semi-Poissonian) distribution of consecutive energy levels and a multifractal spectrum that identifies the universality class of the transition [7][8][9][10]. In these systems, multifractality reflects the extended, yet non-ergodic, nature of the eigenstates, which is usually associated with finite, however limited conductivity. This is in contrast to good conductors and insulating phases where eigenstates are fully ergodic and fully non-ergodic, respectively, and energy levels are distributed according to different characteristic (Wigner-Dyson, Poissonian) distributions [7]. Beyond single-particle wavefunctions, multifractal behavior has been recently reported in strongly interacting disordered quantum systems in connection with many-body localization (MBL) [11][12][13][14][15], and in certain random matrix models [16][17][18][19] that serve as toy models for quantum chaos [10]. The precise connection between interactions, multifractality, level statistics and transport properties as well as experimental realizations of multifractal behavior in any interacting quantum many-body systems have so-far remained largely elusive.

In this work we demonstrate that multifractal eigenstates are found in interacting disordered quantum manybody spin systems coupled to a delocalized bosonic mode. We consider the Tavis-Cummings (TC) model of cavity quantum electrodynamics [20], that describes N two-level quantum emitters -or (pseudo) spins -coupled to a sin-gle cavity mode (Fig. 1a). We consider that model in the presence of disorder in the local Zeeman energies of the spins, a form of diagonal disorder that is common when the transition frequencies of the two-level systems are inhomogeneously broadened [21][22][23][24]. The total number of excitations M (number of excitons+photons) is conserved in the TC model; here we focus on half-filling (M = N/2) as a prototypical example of a finite density of excitations in the thermodynamic limit (N → ∞). We numerically demonstrate that, for all strengths of the light-matter interaction, the eigenstates of the TC Hamiltonian are multifractal in the computational basis. In contrast to the single-excitation case [8,25], multifractality in the many-body Hilbert space is associated to a qualitatively different energy level statistics: while Poisson level statistics is usually associated to fully localized eigenstates, here we simultaneously observe Poisson statistics and multifractal -extended -eigenstates [a similar phenomenon has been pointed out recently in certain power-law banded random matrix models [26]]. We numerically demonstrate that interactions qualitatively change the nature of the eigenstates with respect to the single-particle case, favoring delocalization in the Hilbert space. We also analyze the microcanonical Edwards-Anderson (EA) order parameter introduced in Ref. [27] which probes thermalization of single-spin excitations in the infinite-time limit. We find a non-vanishing EA parameter, signaling the absence of such thermalization, for all light-matter interaction strengths. Interestingly, our findings show that the disordered TC model possesses various known features of MBL phases, including Poisson statistics [28,29], lack of thermalization [30,31], and multifractality of the eigenstates [12,14,32]. However, unlike MBL, these features are found to be unstable towards a small breaking of integrability. We show this by adding a weak nearest-neighbor Hamiltonian hopping term between excitons, which turns the interacting multifractal phase into a perfect conductor for typical energies 4)] for the first-excited energy band (FEB) and for the central energy band (CB) for N/2 excitations in N sites (N indicated in the legend). The semi-Poisson value (dashed line) has been computed numerically [33], while the Poisson value (2 ln 2 -1, dash-dotted line) is known [34][35][36].

in the middle of the spectrum, while the low-energy states remain multifractal. This is in contrast to the single particle case, which is robust with respect to this perturbation [25]. We propose below a protocol to experimentally analyze this physics with ultra-cold atoms.

Model. The TC Hamiltonian for N quantum emitters coupled to a single bosonic mode in the rotating wave approximation reads ( = 1)

H TC = N n=1 ω n σ + n σ - n + g c √ N N n=1 a † σ - n + σ + n a , (1)
where ω n ∈ [-W/2, W/2] are random, uniformly distributed energies (shifted by the cavity frequency), g c is the collective coupling strength of the emitters to the cavity mode,

σ ± n are Pauli operators such that [σ - n , σ - m ] = [σ + n , σ + m ] = 0, [σ + n , σ - m ] = δ nm σ z n , and a † /a is the bo- son creation/annihilation operator with [a, a † ] = 1. The number of excitations M = N n=1 σ + n σ - n + a † a is con- served. The (non-normalized) eigenstates of H are of the form [37-42] |λ 1 , . . . , λ M = Σ † (λ 1 ) • • • Σ † (λ M ) |0 (2)
where the raising operators Σ

† (λ) = a † + 1 √ N N n=1 g c (λ- ω n ) -1 σ +
n act on the vacuum |0 (the state with no exciton and no photon), and where the coefficients λ 1 , . . . , λ M satisfy a system of M coupled non-linear equations,

- N λ i 2g 2 c + 1 2 N n=1 1 λ i -ω n = M j=1 j =i 1 λ i -λ j . (3)
[For M = 1, there is a single equation, equivalent to the secular equation of an arrowhead matrix, see Ref. [8].] Equations. ( 3) are called the 'Bethe equations', and they are a consequence of the underlying integrability of the model, see e.g. Refs. [37][38][START_REF] Tschirhart | From two Algebraic Bethe Ansätze to the dynamics of Dicke-Jaynes-Cummings-Gaudin quantum integrable models through eigenvalue-based determinants[END_REF][START_REF] Claeys | Richardson-Gaudin models and broken integrability[END_REF][START_REF] Tschirhart | [END_REF][START_REF] Tang | Integrability and quench dynamics in the spin-1 central spin XX model[END_REF].

Energy bands. For low excitation density (M N ), the r.h.s. in Eq. ( 3) is negligible, and the coefficients λ i approximately coincide with the eigenvalues of the M = 1 case [8]. For large coupling (g c W ), these correspond to the two polaritons λ i ±g c , or to one of the N -1 dark states λ i Ω n := (ω n+1 + ω n )/2, where we assume w.l.o.g. that

ω 1 < ω 2 < • • • < ω N . The corresponding operators are the polariton creation operators Σ † (±g c ) ≈ a † ± N n=1 σ + n / √ N and the dark state creation oper- ators Σ † (Ω n ) ≈ (g c / √ N ) N m=1 σ + m /(Ω n -ω m )
, with n = 1, . . . , N -1. For instance, for M = 2 excitations the approximate basis is formed by: (a) combinations of two polaritons, [Σ † (±g c )] 2 |0 and Σ † (g c )Σ † (-g c ) |0 ; (b) combinations of a polariton and a dark state,

Σ † (±g c )Σ † (Ω n ) |0 ; (c) combinations of two different dark states, Σ † (Ω n )Σ † (Ω n ) |0
. More generally, with M excitations at strong coupling, there are 2M + 1 energy bands centered at jg c , with j = -M, . . . , M . This prediction is compared to the numerical density of states in Fig. 1b,c for N = 12 (the dashed lines are for jg c ). Here, a low-density approximation works well for M = 2 (b), but fails for M = 6 at half filling (c). This discrepancy is expected, since the interaction effects are non-negligible at half filling. Nevertheless, the low-density description provides a general intuitive characterization of the many-body eigenstates for strong coupling. In particular, we focus our analysis on three sets of states that are good representatives of all the manybody states: the ground state (GS: [Σ † (-g c )] M |0 ), the first excited band (FEB, made of the 2 nd -to-N th excited states:

[Σ † (-g c )] (M -1) Σ † (Ω n ) |0
), and the central band (CB:

M j=1 [Σ † (Ω j )] |0 ).
For the CB we select a small energy window made of the N states in the middle of the spectrum. The results presented below depend on the energy band as long as they are well separated. At small coupling (g c W ) the energy bands are merged into one single larger band (Fig. 1b,c).

Level statistics. In order to study the transport properties of the TC model, as a first step we analyze the statistics of the energy levels, since it is usually related to localization in many-body interacting systems [35,36]. In order to analyze how the level statistics depends on the coupling strength, we compute the mean level spacing ratio [34][35][36]

r = min(E n+1 -E n , E n -E n-1 ) max(E n+1 -E n , E n -E n-1 ) , (4)
where E n are the sorted eigenenergies. In Fig. 1d the mean level spacing ratio is plotted against the coupling to the cavity (FEB: circles, CB: squares) for half filling (M = N/2). As we discuss below, the CB is representative of many other energy bands, while the FEB has peculiar features. As one can see, the level statistics is Poissonian for small coupling for both bands, while for large coupling (g c W ) the FEB reaches semi-Poissonian statistics and the CB remains Poissonian.

The different level statistics can be understood in terms of the previously introduced approximate characterization of the many-excitation eigenstates: indeed, the FEB energies are approximated as -(M -1)g c + Ω n and, therefore, keep the semi-Poisson statistics of the singleexcitation energies Ω n [8,25]. On the other hand, the energies in the CB are ≈ Ω 1 + Ω 2 + • • • + Ω M : since they result from the sum of random single-particle energies, they are independent random numbers, and thus show Poisson level statistics without level repulsion. Indeed, we have checked that also the other energy bands formed by more than one dark state show Poisson statistics.

Multifractality. We now proceed to analyze the delocalization of the many-body eigenfunctions in the Hilbert space by studying their multifractality. We define the generalized inverse participation ratio (IPR) in the manybody case (averaged in an energy window) as

I q = N M p=1 | p|E α | 2q (5)
where |p = (a † ) np ns j=1 σ + j |0 are the many-excitation product states (eigenbasis for g c = 0, with n p photons and n s spins, n s + n p = M ) and N M = M i=0 N i is the size of the Hilbert space subset with M excitations. The average I q is usually a power-law function of the Hilbert space size [7,8,12,15]

I q ∝ N -τq M , (6)
where the exponent τ q characterizes the delocalization. For instance, for fully extended (ergodic) eigenfunctions, one has | p|E α | 2 ∼ O(N -1 M ) and therefore τ q = q-1. On M . b,d Fitted exponent τq as a function of q for the ground state (GS), FEB and CB, for gc/W = 10 (b) and gc = 0.01 (d). GS' and FEB's data are overlapping in b, and all curves overlap in d. Black dashed line: fully extended case (τq = q -1), dotted line: single-excitation case (τq = min(2q -1, 0) [8]). Inset in (d): τ2, averaged over the full spectrum, as a function of the coupling strength.

the other hand, localized states feature one or few components | p|E α | 2 ∼ O(1), implying τ q = 0. Any non-linear dependence on q identifies the system as multifractal [7].

In Fig. 2, τ q is computed for strong (g c /W = 10, see a,b) and weak coupling (g c /W = 0.01, see c,d) by fitting I q against N M at half filling, for different energy bands. The CB is shown in a,c, as an example, showing power-law relations between I q and N M for any q. Similar power-law trends are observed also for the other energy bands, so that we determine τ q by numerical fitting. The computed τ q vs. q is shown in panel b,d for the GS, the FEB (whose results coincide with the GS) and the CB. Strong deviations from the ergodic case (dashed line) appear in all cases, indicating a clear multifractal behaviour both for strong (b) and weak (d) coupling, and for all the energy bands. For strong coupling, τ q grows monotonically with q, similarly to what is observed in power-law banded random matrix models [7]. As the coupling decreases, τ q approaches the single-excitation (non-interacting) case [8] (dotted line) for all the energy bands, with τ q ≈ 0 for q > 1/2: such behaviour usually characterizes a transition to a 'frozen phase', that combines properties of localized and critical states [7]. However, here the multifractal character varies smoothly as a function of the coupling strength, with no clear sign of a phase transition. This is shown in the inset of Fig. 2d, where τ 2 (related to the standard IPR, I 2 ) averaged over the full spectrum is shown as a function of g c /W . As the coupling increases, τ 2 grows and saturates to a finite value τ 2 ≈ 0.5, indicating that the eigenstates are multifractal for any coupling strength. This is in contrast to models showing many-body localization, where τ 2 is finite and non-unity only for strong disorder [12]. Interestingly, here the eigenfunctions are more delocalized than the corresponding non-interacting case [8].

Transport. So far our analysis has revealed that the TC model has Poisson level statistics, yet its eigenstates are not localized in the Hilbert space (see Ref. [26] for similar findings in a matrix model). To further characterize the transport properties of the model, we analyze the long-time survival probability, or microcanonical EA parameter, introduced in Ref. [27],

S = 1 N M E 1 N N n=1 | E|σ z n |E | 2 . ( 7
)
This parameter has a simple physical interpretation [27].

The system is initialized in an infinite temperature state with a small magnetization of spin n, σ z n 0 = Tr(ρ 0 σ z n ) = , where the density matrix is ρ 0 = (I + σ z n )/N M . Then, the long-time local magnetization is σ z n ∞ = lim t→∞ Tr(e -iHt ρ 0 e iHt σ z n ) = E | E|σ z n |E | 2 /N M . Therefore, Eq. ( 7) represents the ratio of the long-time local magnetization over the initial one, σ z n ∞ / σ z n 0 , averaged over all the spins. Interestingly, in Ref. [27] it has been found that in the Richardson model -a related many-body spin model without a cavity mode and with all-to-all spin couplings -for any coupling strength, S remains finite in the thermodynamic limit. Such absence of thermalization has been related to the nonergodicity of the many-body eigenfunctions. We perform the same analysis in the TC model, see Fig. 3a and observe that S saturates to a finite value when increasing N , for any finite coupling strength. Such value is close to unity for small coupling, indicating a nearly perfect insulator, and it decreases with g c . However, in contrast to the Richardson model where S vanishes at infinite coupling strength [27], here we observe that S ≈ [1 + 3.3(g c /W ) 2 ]/[1 + 37.4(g c /W ) 2 ], reaching a positive constant when g c W . Integrability-breaking perturbations. So far we have focused on the TC Hamiltonian (1), known to be integrable, and we found Poisson level statistics, multifractality, and absence of thermalization signaled by the nonvanishing EA parameter. We now ask: Do these features survive integrability-breaking perturbations? To check this, we add a small nearest-neighbour hopping term

H NI = H TC + J N -1 n=1 σ + n σ - n+1 + σ - n σ + n+1 , (8)
with J W, g c . We find that even a very small hopping (J/W = 0.1 with g c /W = 10) is enough to change all the properties of the CB: the level statistics changes from Poisson to Wigner-Dyson (see Fig. 3c, with level spacing ratio r = 0.549 ± 0.007), the survival probability tends to zero once extrapolated to the thermodynamic limit (see inset of Fig. 3a, non-integrable NI compared to the integrable TC model), and the eigen-states become ergodic (see Fig. 3b). The multifractality of the GS and FEB and the semi-Poisson statistics of the FEB remain unaffected by nearest-neighbor hopping, see Fig. 3b,d, respectively. The same robustness to nearest-neighbour hopping has been observed in the noninteracting case [25], and it is consistent with the fact that the FEB reflects 'single-particle physics', since its eigenstates are constituted by one dark state and M -1 robust polaritons.

Proposal for an experimental realization with cold atoms. We propose to implement the model using hyperfine states of alkali atoms trapped in a high finesse cavity. Disorder would be imprinted using a laser with randomly distributed intensity, inducing a vector lightshift of the ground hyperfine manifold. This can be achieved with the lowest spontaneous emission for the heaviest alkalis such as Rb and Cs. Considering for concreteness 87 Rb as in [START_REF] Baghdad | Spectral Engineering of Cavity-Protected Polaritons in an Atomic Ensemble with Controlled Disorder[END_REF], the two-level system can be encoded using the |F = 1, m F = 1 and |F = 2, m F = 2 levels. Circularly polarized light incident from the side at the wavelength of 795 nm, either forming an incommensurate lattice like in [START_REF] Sauerwein | Engineering random spin models with atoms in a high-finesse cavity[END_REF] or a speckle pattern, will induce a random effective magnetic field with tunable strength. Raman coupling involving cavity photons in one arm yields then a realization of the random Tavis-Cummings model, with an effective tunable g c and a long lifetime, limited only by the spontaneous emission [START_REF] Bohnet | [END_REF].

The atoms can be prepared into a coherent spin state at the equator of the collective Bloch sphere, realizing a coherent superposition of excitations close to half-filling. The time-evolution of the spin polarization can then be monitored in time and even in space if the spins are individually addressable [46,[START_REF] Liu | Realization of strong coupling between deterministic single-atom arrays and a highfinesse miniature optical cavity[END_REF]. As shown in [START_REF] Sauerwein | Engineering random spin models with atoms in a high-finesse cavity[END_REF], the participation ratio in the linear regime can be estimated from response functions, and it would be interesting to explore possible extensions to the interacting case.

Last, the open character of cavity QED platforms will lead to dissipation. The finite photon lifetime will set an upper bound to the experimentally accessible timescales during which the system possesses a fixed number of excitations. Very narrow cavities with Rb atoms in the strong coupling regime have been demonstrated with photon lifetimes up to 35 µs [START_REF] Wolke | [END_REF], which could be leveraged to ensure the largest intrinsic lifetime. Beyond this timescale, continuous pumping would then be needed to stabilize the system around an average number of excitations. The use of long-lived atomic states will ensure that dissipation is predominantly due to photons leaking from the cavity, which can then be detected. In particular, the super-radiant emission expected for fully-delocalized excitations may bear the signature of the multifractal character of the excitations when disorder is increased.

Discussion and outlook. The results above imply that the eigenfunctions are partially delocalized even if the level statistics suggests otherwise, thus favouring exciton transport. Moreover, cavity-mediated interactions be-tween the excitons favor delocalization of the wave functions both in the integrable and non-integrable models (see Figs. 2 and3b). In the former, this is associated with a change in the exponent τ q which is positive for q > 1, as opposed to the frozen phase observed in the non-interacting case (see Fig. 2b). As a consequence, the many-body interactions seem to make the eigenfunctions more delocalized, i.e. more similar to those of a good conductor. The effects of interactions are also striking in the non-integrable case as they appear to turn the system into a perfect conductor at finite energy. This is in stark contrast to the non-interacting single-excitation model, where multifractality is robust to nearest-neighbor hopping perturbations at all energies [25].

Those described above are examples of qualitative modifications in the many-particle eigenfunctions due to collective light-matter coupling of spins to the photon field, resulting in new transport properties and phases of matter -a result relevant to the growing field of lightmodified quantum matter [49,50]. They might find direct applications in improving exciton conductivity in, e.g., disordered molecular systems, where it has recently been observed that a strong collective light-matter coupling to a cavity field strongly boosts the diffusion coefficient, even inducing ballistic transport [51,52].

FIG. 1 .

 1 FIG. 1. a The Tavis-Cummings (TC) model [Eq. (1)]: N twolevel system (or pseudo-spins), with disorder in their transition energies ωn, are coupled to the same mode of an optical cavity, with collective coupling strength gc. Multiple excitations can be present, either in the two-level systems (blue circles) or as photons in the cavity. b,c Density of states of the TC model (1) for N = 12 as a function of the energy E and of the coupling gc, in units of the disorder strength W . Each panel represents different numbers of excitations: M = 2 (b) and M = 6 (c). The dashed lines represent the low-excitation-density approximation (M N ): jgc, with j = -M, . . . , M . d Mean level spacing ratio [Eq. (4)] for the first-excited energy band (FEB) and for the central energy band (CB) for N/2 excitations in N sites (N indicated in the legend). The semi-Poisson value (dashed line) has been computed numerically[33], while the Poisson value (2 ln 2 -1, dash-dotted line) is known[34][35][36].

FIG. 2 .

 2 FIG. 2. a,c Average generalized IPR Iq [Eq. (5)] within the CB for different q values (see legends) as a function of the M -excitation Hilbert subspace size NM (corresponding to N = 4, 6, 8, 10, 12, 14, 16 respectively, with M = N/2), for gc/W = 10 (a) and gc = 0.01 (c). Lines: power-law fits ∝ N -τq

10 - 2 FIG. 3 .

 23 FIG.3. a Long-time survival probability S [Eq.(7)], against the collective coupling strength gc. The dashed line is the best fit with [1 + A(gc/W ) 2 ]/[1 + B(gc/W ) 2 ], where A = 3.3 and B = 37.4. Inset: extrapolation of S to the thermodynamic limit (dashed lines) by the finite-N scaling S(N ) = S + c1/N + c2/N 2 (continuous lines, adapted from Ref.[27]), both for the integrable case [TC: Eq. (1), gc/W = 10] and for the non-integrable case [NI: Eq. (8), gc/W = 10 and J/W = 0.1]. b Fitted exponent τq as a function of q in the non-integrable Hamiltonian (8) for different energy bands (see legend), with gc/W = 10 and J/W = 0.1. GS's and FEB's data are overlapping. Black dashed line: fully extended case (τq = q -1), dotted line: single-excitation case (τq = min(2q -1, 0)). c,d Probability density function P (s) of the normalized level spacing s at half filling (N = 14, M = 7) and strong coupling (gc/W = 10) for the CB (c) and FEB (d), for the integrable (TC) and non-integrable (NI) case (J/W = 0.1). "W.-D.": Wigner-Dyson distribution.

We acknowledge useful discussions with Jakob Reichel, Romain Long, Dmitry Abanin, Alexandre Faribault, Ivan Khaymovich and Felipe Herrera. This work has benefited from state grants managed by the French National Research Agency under the Investments of the Future Program with the reference ANR-21-ESRE-0032, and by the ANR project CLIMAQS. We acknowledge support from ECOS-CONICYT through project nr. C20E01 and the CNRS through the IEA 2020 campaign. Computing time was provided by the High Performance Computing Center of the University of Strasbourg. JPB acknowledges funding from the Swiss National Science Foundation (grant No 184654) and the Swiss State Secretariat for Education, Research and Innovation (grant MB22.00063).


Yushun Zhao

Yufan Xu

Gilbert Teyssèdre

Jianyi Xue

Yanning Zhao

Bin Du

Methyl and carbon-carbon double bond in anhydride molecules modulated surface charge trap depth of epoxy materials

Keywords: Epoxy resin, acid anhydride, chemical group, quantum chemical calculation, trap depth

Epoxy insulators in gas insulated switchgear and gas insulated transmission lines tend to accumulate surface charges, leading to insulation flashover. Improving the surface trap characteristics of epoxy materials, which can accelerate the surface charge dissipation of epoxy insulators, is a promising method to improve the surface insulation performance. The surface trap characteristics of epoxy materials are strongly influenced by the chemical groups in the acid anhydride molecules. In this work, by quantum chemical calculations and isothermal surface potential decay tests, taking six organic anhydrides that differ only in the methyl and carbon-carbon double bonds, we find the modulation laws of methyl and carboncarbon double bonds on the charge trap depth within and between molecular chains. The regulation mechanism is revealed from the microscopic perspectives of electron energy structure and electron cloud offset. The changes of surface charge trap depth of epoxy materials are primarily attributed to the changes in the spatial distribution of the electron cloud density between and on the valence bonds caused by the interaction between the electron-donating methyl group and the electron-absorbing carbon-carbon double bond.

INTRODUCTION

S a polymer dielectric with excellent electrical properties, epoxy resin is widely used in gasinsulated switchgear (GIS) and gas-insulated transmission lines (GIL) [START_REF] Ruan | Enhanced DC surface insulation of Al2O3/ER brought by the permeability of fluorine-containing release agent[END_REF][START_REF] Shimakawa | One-dimensional modeling of charge transport in epoxy for DC-GIS insulating spacer[END_REF]. However, after long-term operation under DC voltage, the surface of epoxy materials tends to accumulate a large amount of charge, leading to distorted electric field distribution and insulation flashover. Improving the surface charge trap characteristics of epoxy materials can accelerate the surface charge dissipation and suppress the surface electric field distortion, thus improving the DC insulation performance [START_REF] Wang | Effect of surface charges on flashover voltage-an examination considering charge decay rates[END_REF][START_REF] Yadam | Understanding charge trap characteristics of epoxy nanocomposite under steep fronted lightning impulse voltage[END_REF][START_REF] Du | Effects of DC and pulse voltage combination on surface charge dynamic behaviors of epoxy resin[END_REF]. It has been shown that the surface charge trap characteristics of polymer dielectrics are strongly affected by the chemical groups in the polymer molecular chains [START_REF] Li | Molecular structure modulated trap distribution and carrier migration in fluorinated epoxy resin[END_REF][START_REF] Akram | Charge transport and trapping of surface modified stator coil insulation of motors[END_REF][START_REF] El-Shahat | Effect of chemical impurities on charge injection barriers at the interface of copper and polyethylene[END_REF][START_REF] Mai | Modulation of epoxy polymer trapping energy levels by fluorinated diluents to improve insulation properties[END_REF][START_REF] Du | Doping C60(OH) to regulate the crosslink network and energy band structures of epoxy resin and allow for electronic directional drive in C60(OH)[END_REF]. The chemical groups introduce local state traps into the band gap of polymer dielectrics, which affect the trap depth and DC insulation performance of polymer dielectrics. The trap characteristics in insulating polymers in previous investigations determined by quantum chemical calculations were concerned with chemical defects in polyethylene [START_REF] Meunier | Molecular modeling of electron traps in polymer insulators: chemical defects and impurities[END_REF]. Over the years, the calculation method was refined by considering different types of chemical defects and crosslinking residues nanocomposites [START_REF] Chen | Electronic structure of polyethylene: role of chemical, morphological and interfacial complexity[END_REF][START_REF] Kubyshkina | Communication: band bending at the interface in polyethylene-MgO nanocomposite dielectric[END_REF]. The prediction issued from molecular simulations were compared with experimental results, for example the kinetics of positive charge propagation in polyethylene [START_REF] Takada | Determination of charge-trapping sites in saturated and aromatic polymers by quantum chemical calculation[END_REF]. Interesting aspects also concern the change in electronic properties due to ageing: density functional theory (DFT) calculations were used to compare the trap energies of charge traps produced by chemical groups, including the produced carbon-carbon double bonds, conjugated double bonds, and carbonyl groups during thermal degradation [START_REF] Zhu | Determination of trap energy in polyethylene with different aging status by molecular dynamics and density function theory[END_REF]. Afterwards, the technique was applied to design the insulating polymers. For example, quantum chemical calculations coupled to isothermal surface potential decay (ISPD) were used to design the epoxy resin molecules with investigating the effect of grafting trifluoromethyl (-CF3) groups. The results showed that -CF3 introduced electron traps in the epoxy material, thus hindering the dissipation of electrons on the surface of the epoxy material [START_REF] Yang | Enhancing dielectric strength of epoxy polymers by constructing interface charge traps[END_REF].

Organic anhydrides can form cross-linked networks with epoxy resins which have excellent dielectric and insulating properties, making them the most commonly used curing agents in the preparation of epoxy materials [START_REF] Chen | Synthesis and application of thermal latent initiators of epoxy resins: a review[END_REF][START_REF] Xie | Micro-structure and thermomechanical properties of crosslinked epoxy composite modified by nano-SiO2: a molecular dynamics simulation[END_REF]. Among them, hexahydrophthalic anhydride (HHPA), tetrahydrophthalic anhydride (THPA), phthalic anhydride (PA), methyl hexahydrophthalic anhydride (Me-HHPA), methyl tetrahydrophthalic anhydride (Me-THPA), and methyl nadic anhydride (MNA) are the six most common organic anhydride curing agents available in the market. These anhydride curing agents only differ in the chemical groups of methyl and carboncarbon double bonds. Therefore, it can be easily concluded that the chemical groups methyl and carbon-carbon double bonds in anhydride molecules will modulate the chemical structure of epoxy resin/anhydride cross-linked molecules, and potentially the surface charge trap depth of the epoxy materials.

In this paper, by comparing the distributed energy levels of above six geometrically optimized epoxy resin/anhydride cross-linked molecules, we found the modulation laws of A methyl and carbon-carbon double bonds on the intramolecular charge trap depth in epoxy materials. In addition, by analyzing the ISPD test data, we found the modulation laws of methyl and carbon-carbon double bonds on the intermolecular charge trap depth in epoxy materials. This paper focuses on explaining these regulation laws from the microscopic perspectives of electron energy structure and electron cloud offset, in order to provide theoretical references for the structural design and surface charge trap modulation of polymer materials.

QUANTUM CHEMICAL CALCULATIONS

Six epoxy resin/anhydride molecules

Epoxy polymer dielectrics are usually obtained from the cross-linking curing process of epoxy resins and curing agents. Here, taking bisphenol-A type epoxy resin (DGEBA) as an example, we considered the minimum periodic structure of the cross-linked network. It was formed by cross-linking of one DGEBA molecule and two acid anhydride molecules, as shown in Fig. 1a, called DGEBA/AH molecule. In Fig. 1a, Rn is the anhydride molecule after opening the five-membered ring. The methyl and carbon-carbon double bond in anhydride molecule are an electron-donating group and an electron-absorbing group, respectively. After they are introduced into a certain molecular segment, the electron interaction between them changes the electron cloud density between and on valence bonds in the segment, which may cause the change of the lowest unoccupied molecular orbital (LUMO) energy level and the highest occupied molecular orbital (HOMO) energy level of the segment. In other words, the change magnitude in the LUMO and HOMO energy levels of the segment may be related to the interaction intensity between the electrondonating group methyl and the electron-absorbing group carbon-carbon double bond. The changes in the LUMO and HOMO energy levels of adjacent segments determine the changes in the charge trap depth within the molecular chain [START_REF] Li | The energy distribution of trapped charges in polymers based on isothermal surface potential decay model[END_REF]. The quantification of energy level changes is obtained using quantum chemical calculations, as shown in the following section.

Charge trap depth of the DGEBA/AH molecule

Below we analyzed the effect of methyl and carbon-carbon double bond on the charge trap depths based on ideal molecular groups, ignoring factors such as pending bonds and defects. The geometries of each oligomer were optimized by DFT using Gaussian 09 software, and their energy level distribution and molecular orbital information were obtained. The B3LYP hybrid functional method and 6-31 G(d) basis function set, widely adopted to simulate low molecular weight polymers, were used in the calculations.

Fig. 2 shows the energy band structure and distributed energy levels of DGEBA/THPA molecule. The zero level in the band diagram is the vacuum energy level. Fig. 2(b) shows the energy band structure of the DGEBA/THPA molecule. In Fig. 2(b), if we focus on the molecular orbital corresponding to an energy level, the molecular orbital is locally distributed along the molecular chain. Therefore, we can obtain the distributed energy levels of DGEBA/THPA molecules on different segments. The DGEBA/THPA molecule is divided into two anhydride segments and one resin segment, and its distributed energy level values and numbers are shown in Fig. 2(c). In three dimensions, the spatial arrangement of the same type of atoms in segments A and C is not perfectly symmetrical, therefore, the energy levels of segments A and C are different.

Comparing Fig. 2b and Fig. 2c, it can be seen that the LUMO energy level of segment C is the LUMO energy level of the whole DGEBA/THPA molecule, and the HOMO energy level of segment B is the HOMO energy level of the whole DGEBA/THPA molecule. The molecular orbital distribution corresponding to the LUMO and HOMO energy levels of the DGEBA/THPA molecule are shown in Fig. 2c. If extra electrons captured at the LUMO of a molecular segment drift toward its neighboring segment under an electric field, the captured electrons need to overcome an energy barrier, which means that the LUMO of the segment is an electron trap site, and the energy barrier height is the depth of the electron trap created by that site [START_REF] Takada | Determination of charge-trapping sites in saturated and aromatic polymers by quantum chemical calculation[END_REF]. Similarly, if extra holes captured at the HOMO of a molecular segment drifts toward its neighboring segment under an electric field, the captured holes need to overcome an energy barrier, which means that the HOMO of the segment is a hole trap site, and the energy barrier height is the depth of the hole trap created by that site. As shown in Fig. 2c, the LUMO energy level of segment C is an electron trap site, which produces an electron trap depth φte of 0.408 eV; the HOMO energy level of segment B is a hole trap site, which produces a hole trap depth φth of 1.286 eV.

Following the same division of the segments as above, we obtained the charge trap depths for the other five DGEBA/AH molecules. The charge trap depths of all DGEBA/AH molecules are listed in Table . I.

Modulation of charge trap depths within molecules by methyl

As shown in Table I, the electron and hole trap depths of DGEBA/Me-HHPA molecule are almost unchanged, compared with that of DGEBA/HHPA molecule; the electron and hole trap depths of DGEBA/MNA molecule are both slightly decreased, compared with those of DGEBA/THPA molecule. It can be found that when there is no carbon-carbon double bond in the original anhydride molecule, the introduction of methyl has almost no effect on the charge trap depths of DGEBA/AH molecule; when there is a carbon-carbon double bond in the original anhydride molecule, the introduction of methyl causes the charge trap depths of DGEBA/AH molecule to show a decreasing trend, which may be related to the interaction between the methyl and carbon-carbon double bond. A detailed discussion is described below.

Fig. 3 shows the energy band structure and distributed energy levels of DGEBA/Me-THPA molecule. The variation of the charge trap depths within molecules is determined by the variation of the distributed energy levels. Comparing Fig. 2c and Fig. 3c, it can be seen that the distributed energy levels of DGEBA/Me-THPA molecule do not change significantly compared to those of DGEBA/THPA molecule, except for a small increase in the HOMO energy level of segment A and the HOMO energy level of segment C. We tried to explain these variations.

If an electron-absorbing group is introduced into a molecular segment, due to electrostatic interactions, the group will attract the orbital electrons between and on the valence bonds in the segment. When the orbital electrons between the valence bonds are attracted by the group, the orbital electrons shift towards the positively charged center of the group, and the electron cloud density between the positively charged center and the atom whose orbital electrons have shifted increases. This makes the electron orbital system between the valence bonds in the segment more stable, so the energy of the orbital electrons (conduction band electrons) moving between the valence bonds decreases, the LUMO energy level of the segment decreases [START_REF] Zhao | Introducing chlorine into epoxy resin to modulate charge trap depth in the material[END_REF]. When the orbital electrons on the valence bonds are attracted by the electron-absorbing group, the orbital electrons shift towards the positively charged center of the group. In this way, the electron cloud density on the valence bonds decreases, and the overlap of the electron cloud is reduced. This makes the covalent bonds formed between the bonding atoms more unstable, so the energy of the orbital electrons (valence band electrons) moving on the valence bonds increases, the HOMO energy level of the segment increases [START_REF] Zhao | Introducing chlorine into epoxy resin to modulate charge trap depth in the material[END_REF].

In the anhydride segment of the DGEBA/THPA molecule, there is an electron-absorbing group carbon-carbon double bond. In contrast to that of DGEBA/THPA molecule, one of the carbon atoms of carbon-carbon double bond in DGEBA/Me-THPA molecule connects an electron-donating group methyl through a covalent bond. The carbon-carbon double bond attracts orbital electrons on the covalent bonds connected to the carbon atom in the methyl, resulting in a decrease in the electron cloud density on the valence bonds. From the above analysis, the HOMO energy levels of the anhydride segments will be elevated. The spatial configuration of the three atoms connected to the carbon atom of the carbon-carbon double bond is planar triangular type, which makes a long spatial distance between the carbon-carbon double bond and the orbital electrons between the valence bonds provided by methyl, and their electrostatic attraction is weak. There is no obvious change in the electron cloud density between the original valence bonds, therefore, the introduction of methyl does not cause any significant variation in the LUMO energy levels of the anhydride segments. Similarly, the introduction of methyl leads to an increase in the HOMO energy level of the anhydride segment, therefore, the hole trap depth of DGEBA/MNA molecule also shows a small decrease. Differently, the methyl in DGEBA/MNA molecule is attached to a bridge ring, which is easily attracted to the carbon-carbon double bond and moves to a position near it in the three-dimensional space. In this way, the attraction of the carbon-carbon double bond to the orbital electrons between the original valence bonds is weakened, and the electron cloud density between the original valence bonds is reduced. This causes an increase in LUMO energy level of the anhydride segment, so the electron trap depth of the DGEBA/MNA molecule shows a small decrease.

Modulation of charge trap depths within molecules by carbon-carbon double bond

Compared with that of DGEBA/Me-HHPA molecule, the electron trap depth of DGEBA/Me-THPA molecule decreases slightly, and the hole trap depth shows a small reduction; compared with that of DGEBA/HHPA molecule, the electron trap depth of DGEBA/THPA molecule decreases slightly, while the hole trap depth does not change; compared with that of DGEBA/HHPA molecule, the electron trap depth of DGEBA/PA molecule increases substantially, and the hole trap depth increases slightly. The changes in charge trap depth of distributed energy levels are compared. The carbon atoms in carbon-carbon single bonds are sp 3 hybridized, and carbon atoms in carbon-carbon double bonds are sp 2 hybridized. The s-orbital electrons move close to the nucleus and have a weak shielding effect on the nucleus. Therefore, the larger the s-orbital component contained in the hybridized orbital of a bonding atom, the stronger the attraction of the bonding atom to its surrounding electrons. This makes the attraction of the carbon atoms to its surrounding electrons stronger when the carbon-carbon single bond changes to a carbon-carbon double bond.

In the anhydride segment of DGEBA/Me-HHPA molecule, there is an electron-absorbing group ester group, as shown in Fig. 4a. The ester group attracts the orbital electrons between the valence bonds in anhydride segment, which increases the electron cloud density near the positively charged center of it. After the introduction of the para carbon-carbon double bond, as shown in Fig. 3a, it will have a slight attraction to the orbital electrons. This causes the average motion radius of the orbital electrons to increase and the electron cloud density near the positively charged center to decrease. Therefore, the LUMO energy level of the anhydride segment of DGEBA/Me-THPA molecule tends to be slightly elevated. In addition, the introduced carbon-carbon double bond is directly connected to the methyl through a covalent bond. The carbon-carbon double bond attracts the orbital electrons on valence bonds in the anhydride segment, which makes the electron cloud density on valence bonds show a small decrease. As a result, the HOMO energy level of anhydride segment of the DGEBA/Me-THPA molecule shows a small elevation. In the resin segment of DGEBA/Me-HHPA molecule, the 2p orbital of the oxygen atom in the ether oxygen group connected with the benzene ring forms a p-π conjugation with the large π orbitals of the benzene rings, which makes the electron cloud density between the conjugated atoms lower and the internal energy of the conjugated system lower, thus forming a more stable conjugated system [START_REF] Zhao | Introducing chlorine into epoxy resin to modulate charge trap depth in the material[END_REF]. In contrast to those of the DGEBA/Me-HHPA molecule, the anhydride segments of the DGEBA/Me-THPA molecule are curled toward the resin segment in three-dimensional space, as can be seen in Fig. 3c. Under the electrostatic effect of the electronabsorbing group ester group in anhydride segment, the degree between the p-orbital and the large π-orbital in resin segment is reduced, which makes the p-π conjugated system more unstable. As a result, the HOMO energy level of resin segment of the DGEBA/Me-THPA molecule shows a small increase.

After the introduction of carbon-carbon double bond, the variation trends of distributed energy levels of DGEBA/THPA molecule are consistent with those of the DGEBA/Me-THPA molecule, which will not be discussed here.

In the anhydride segment of the DGEBA/PA molecule, the 2p orbitals of the carbon atoms in the three carbon-carbon double bonds introduced are parallel to each other and overlap, forming a highly delocalized large π orbital perpendicular to the benzene ring plane. The delocalized large π orbital electrons between the valence bonds in anhydride segment are readily attracted to the ester group, which results in a substantial increase in the electron cloud density near the positively charged center of the ester group. Therefore, compared to those of the DGEBA/HHPA molecule, the LUMO energy level of anhydride segment of the DGEBA/PA molecule is substantially lower and the electron trap depth of the DGEBA/m molecule is significantly increased. The ester group in the curled anhydride segment makes the p-π conjugated system in the resin segment more unstable, as described earlier. This leads to a small increase in the HOMO energy level of the resin segment and a small increase in the hole trap depth of the DGEBA/PA molecule.

In the above we analyzed the modulation laws on charge trap depth within the molecular chains. However, in general, the excess electrons would be localized at the AH ends, then, it is what happens at chain end that controls the actual transport. In other words, the interchain charge transport influenced by the chain-end groups controls the actual charge transport. Different chain-end groups produced different interchain charge transport situations, which may lead to different ISPD results. The following sections analyze the modulation of electron trap depth between molecular chains through ISPD experimental results.

ISOTHERMAL SURFACE POTENTIAL DECAY TESTS

Experimental raw materials

The bisphenol-A type epoxy resin is branded as DER331. It is chemically pure, with an epoxy value of 0.535 mol/100g, purchased from Shanghai Kaiyin Chemical Co., Ltd. Anhydride curing agent: HHPA, chemically pure; THPA, chemically pure; Me-HHPA, chemically pure; Me-THPA, chemically pure; MNA, chemically pure. These anhydride curing agents were purchased from Shanghai Macklin Biochemical Co., Ltd, and their molecular structures were shown in Fig. 1. Curing accelerator: N, N-dimethylbenzylamine (BDMA), purchased from Shanghai Aladdin Biochemical Technology Co., Ltd.

Sample preparation

Equation ( 1) is the formula for calculating the anhydride mass per 100 g of epoxy resin, in where Wh is the relative molecular weight of the anhydride, Ev is the epoxy value of the epoxy resin, k is the correction coefficient. II shows the curing formulations of six DGEBA/AH systems. The mass of BDMA added was 0.6% (0.6 g) of the mass of epoxy resin in the curing systems. The resin and curing agents were weighed and poured into the reaction kettle in turn. They were evenly dispersed with a high-speed disperser at a constant temperature of 353.15 K, and a vacuum degassing treatment was performed simultaneously. After mixing thoroughly, the BDMA was added to the reaction kettle and the high-speed dispersion and vacuum degassing treatment were continued. Afterwards, the resin castings were poured into the mold (the mold should be preheated in a constant temperature oven at 353.15 K for at least 3 h before casting). After putting the mold into a constant temperature oven, the oven was vacuumed for 6 min. The resin castings were cured at 353.15 K for 4 h and at 393.15 K for 12 h to obtain the cured epoxy samples. The cured samples are circular-shaped with a diameter of 100 mm and a thickness of 1 mm. Before the ISPD tests, all samples were cleaned by anhydrous alcohol and deionized water in an ultrasonic cleaner, and then dried in a baking oven at 333.15 K for 24 h, in order to eliminate the humidity and the residual charges inside the materials. The cured samples are circular-shaped with a diameter of 100 mm and a thickness of 1 mm. Before the ISPD tests, all samples were cleaned with anhydrous alcohol and deionized water in an ultrasonic cleaner, and then dried in a baking oven at 333.15K for 24 h, in order to eliminate the humidity and the residual charges inside the materials.

ISPD measurement

The ISPD was tested with a needle-mesh grid-plate electrode system. Fig. 5 shows the schematic diagram of the experimental setup. The needle electrode is 2 cm from the mesh grid electrode, and the mesh grid electrode is 1 cm from the grounded electrode. During the test, the needle electrode and mesh grid electrode are applied with -10 kV and -6kV DC high voltage, respectively. After charged for 10 min, the samples are moved to the Kelvin probe of electrostatic voltmeter in 1 s via a rotatable platform. The distance between the probe and the sample surface is kept as 2 mm. The samples with smooth surface are selected for good contact with the grounded electrode. All experiments are carried out in an airtight chamber with the measurement temperature keeping constant as 323.15 K.

TEST RESULTS AND DISCUSSIONS

Surface potential decay results

Fig. 6 shows the normalized ISPD curves of DGEBA/AH cured samples after negatively corona charging at the temperature of 323.15 K. These curves were obtained by fitting the ISPD data with a double exponential decay equation. The double exponential decay equation represents the simultaneous occurrence of charge decay processes controlled by shallow and deep traps, so the normalized potential can directly reflect the characteristics of the potential decay [START_REF] Li | The energy distribution of trapped charges in polymers based on isothermal surface potential decay model[END_REF]. As can be seen in Fig. 6, the normalized surface potentials of the DGEBA/PA and DGEBA/HHPA samples decay much faster in the early stages than in the later stages. This indicates that there are more shallow electron traps in the DGEBA/PA and DGEBA/HHPA samples. More deep electron traps are present in the other four DGEBA/AH samples. In addition, after a long decay time, the normalized surface potentials of the samples are still high except for the DGEBA/PA sample, indicating that substantial deep trapped electrons need a longer time and a higher temperature to be excited.

Surface trap energy distribution results

The relationship between the surface trap depth (Et) and the surface trap density (Nt) are shown in ( 2) and ( 3), where k is the Boltzmann constant, T is the absolute temperature, v is the attempt-to escape frequency of trapped electron, 4.17×10 13 s -1 , t is the decay time, ε0 is the vacuum dielectric constant, εr is the relative dielectric constant, qe is the electron charge, L is the sample thickness, V is the surface potential [START_REF] Li | The energy distribution of trapped charges in polymers based on isothermal surface potential decay model[END_REF].

t ln( ) E kT v t  (2) 0r t e dV Nt q L dt     (3)
According to ( 2) and ( 3), the surface trap distribution of the six DGEBA/AH cured samples can be obtained, as shown in Fig. 7. DGEBA/PA and DGEBA/HHPA samples have higher peak densities of shallow electron traps than deep electron traps. The other four DGEBA/AH samples are the opposite. These are consistent with the results obtained from the normalized surface potential decay curves. It is worth noting that DGEBA/Me-HHPA, DGEBA/THPA, and DGEBA/Me-THPA samples do not show a significant shallow trap density peak. The differences in the distribution of shallow and deep traps indicate the structural differences in the polymer materials. The effect of methyl and carbon-carbon double bond on the aggregation structure between the molecular chains may have led to the structural differences. The wave function of the conduction band edge has inter-chain characteristics. The conduction electrons in the lowest energy state are naturally channeled out of the dense regions, and they tend to move to the low-density As a result, the extra electrons tend to move into the low-density regions where many electron traps are formed. In DGEBA/AH crosslinked molecules, the methyl and the carbon-carbon double bond contributed by the anhydride molecule are located at the chain ends. Due to the electrostatic attraction, the methyl and the carbon-carbon double bond at the chain ends will lead to a more condensed interchain distance and less free volume. The extra electrons are localized in small free volume spaces and have difficulty moving freely, which makes the small free volume spaces behave as deep electron traps. THPA, Me-HHPA, Me-THPA, and MNA molecules all contain methyl or carbon-carbon double bond, so the resin samples cured by these anhydrides contain more inter-chain deep electron traps. In addition, the chain-terminal carbon-carbon double bond provides a stronger attraction between the extra electrons and the surrounding macromolecular chains. This attraction will further localize the electrons and deepen the inter-chain electron trap depth. Therefore, the resin samples cured by THPA, Me-THPA, and MNA have deeper inter-chain electron traps, as shown in Fig. 7.

The HHPA molecule has neither methyl nor carbon-carbon double bond. Therefore, the samples cured by HHPA have expanded inter-chain distances and larger free volume spaces, forming shallower electron traps. This is consistent with the results in Fig. 7.

Compared to the HHPA-cured samples, the PA-cured samples have a higher density of shallow and deep electron traps. This is due to a highly delocalized large π orbital in the benzene ring of the PA molecule. On the one hand, the large π orbital electrons of the chain-terminal benzene ring of adjacent DGEBA/PA molecules strongly repel each other, resulting in a large free volume space in the chain end regions, where form a high density of shallow electron traps. On the other hand, the strong repulsive effect causes the obvious internal bending and folding of the molecular chains, leading to a small free volume space in the internal folding regions, where form a high density of deep electron traps. Both the depth and density of the traps affect the charge transport. The shallower traps allow charges to transport in hopping through multiple detrapping and trapping steps, which facilitates charge transport. The higher trap density reflects the smaller trap distance, which may lead to the overlap of trap wave-functions. This allows charges to tunnel from one localized state to another without transition to extended state, facilitating the charge transport. The surface potential decay in the early stage is related to the electron transport in the shallow traps. Although the electron trap of PAcured sample is not the shallowest, its shallow electron trap density is much higher than that of other samples. This facilitates the tunneling transport of electrons, resulting in the fastest initial decay rate.

CONCLUSION

This study investigates the modulation mechanism of surface charge trap depth in epoxy materials by methyl and carbon-carbon double bonds in anhydride molecules, based on quantum chemical calculations and ISPD tests. The conclusions are summarized as follows:

(1) When there is no carbon-carbon double bond in the original anhydride molecule, the introduction of methyl has almost no effect on the charge trap depths of the DGEBA/AH molecule. When there is a carbon-carbon double bond in the original anhydride molecule, the introduction of methyl causes the charge trap depths of the DGEBA/AH molecule to show a decreasing trend.

(2) The introduction of one carbon-carbon double bond results in a slight decrease in the charge trap depth of the DGEBA/AH molecule, while the introduction of three carbon-carbon double bonds results in a significant increase in that. The interaction between the methyl group and carbon-carbon double bond changed the spatial distribution of the electron cloud density between and on the valence bonds, which subsequently changed the charge trap depth of the DGEBA/AH molecule.

(3) The PA-cured samples have a much higher density of shallow electron traps than that of other samples. This facilitates the tunneling transport of electrons, leading to the fastest initial decay rate.

Fig. 1 .

 1 Fig. 1. Molecular structure diagrams.

Fig. 1b shows

  Fig. 1b shows the molecular structures of six common anhydride curing agents, which are HHPA, THPA, PA, Me-HHPA, Me-THPA and MNA. In contrast to HHPA molecule, the Me-HHPA molecule has a methyl group introduced at the meta-position of the anhydride group. Compared to THPA molecule, the Me-THPA molecule introduces a methyl group at the meta-position of the anhydride group. In contrast to THPA molecule, the MNA molecule introduces a methyl group by being attached to a bridge ring. Compared to HHPA molecule, the THPA molecule has a carbon-carbon double bond at the para-position of the anhydride group. In contrast to Me-HHPA molecule, the Me-THPA molecule has a carbon-carbon double bond at the para-position of the anhydride group. Compared to HHPA molecule, the PA molecule introduces a carbon-carbon double bond at both the para and adjacent positions of the anhydride group.The methyl and carbon-carbon double bond in anhydride molecule are an electron-donating group and an electron-

Fig. 2 .

 2 Fig. 2. The energy band structure and distributed energy levels of DGEBA/THPA molecule .As shown in Fig.2c, the LUMO energy level of segment C is an electron trap site, which produces an electron trap depth φte of 0.408 eV; the HOMO energy level of segment B is a hole trap site, which produces a hole trap depth φth of 1.286 eV.Following the same division of the segments as above, we obtained the charge trap depths for the other five DGEBA/AH molecules. The charge trap depths of all DGEBA/AH molecules are listed in Table.I.

Fig. 3 .

 3 Fig. 3. The energy band structure and distributed energy levels of DGEBA/Me-THPA molecule.

Fig. 4

 4 shows the energy band structure and distributed energy levels of DGEBA/Me-HHPA molecule. In contrast to those of DGEBA/Me-HHPA molecule, the LUMO energy level of anhydride segment of DGEBA/Me-THPA molecule tends to increase slightly, and the HOMO energy level of anhydride segment shows a small increase; the LUMO energy level of resin segment of DGEBA/Me-THPA molecule does not change, while the HOMO energy level of resin segment shows a small increase.

Fig. 4 .

 4 Fig. 4. The energy band structure and distributed energy levels of DGEBA/Me-HHPA molecule.

Fig. 5 .

 5 Fig. 5. Experimental setup for ISPD measurement.

Fig. 6 .

 6 Fig. 6. Normalized surface potential decay of the DGEBA/AH cured samples.

Fig. 7 .

 7 Fig. 7. Surface Trap Energy Distribution of the DGEBA/AH cured samples.

TABLE I CALCULATED

 I CHARGE TRAP DEPTH PARAMETERS

	Molecule types	φte (eV)	φth (eV)
	DGEBA/HHPA	0.460	1.286
	DGEBA/THPA	0.408	1.286
	DGEBA/PA	1.804	1.749
	DGEBA/Me-HHPA	0.464	1.280
	DGEBA/Me-THPA	0.402	1.034
	DGEBA/MNA	0.159	0.937

TABLE II FORMULATIONS

 II OF DGEBA/AH SYSTEMS

	Anhydride type	Relative molecular mass	AH/g	DGEBA/g
	HHPA	154.16	1.286	100
	THPA	152.15	1.286	100
	PA	148.12	1.749	100
	Me-HHPA	168.19	1.280	100
	Me-THPA	166.18	1.034	100
	MNA	178.18	0.937	100