M-Q Hoang

S Le Roy

L Boudou

G Teyssèdre

Implementation of polarization processes in a charge transport model using time and frequency domain measurements on PEN films

The aim of this work is to highlight the polarization mechanisms in the time-and frequencydomains, and to implement the contribution of the polarization to the dielectric response into a charge transport model. To do so, Alternate Polarization Current (APC) and Dielectric Spectroscopy (DS) measurements have been performed on poly(ethylene naphthalene 2,6-dicarboxylate) (PEN), an aromatic polar material, providing information on polarization mechanisms in the time-domain and frequency-domain respectively. In the frequency-domain PEN exhibits 3 relaxation processes termed β, β * (sub-glass transitions) and α relaxations (glass transition) in increasing order of temperature. Conduction was also detected at high temperatures. Dielectric responses were treated using a simplified version of the Havriliak-Negami (HN) model (Cole-Cole (CC) model), using 3 parameters per relaxation process, function of the temperature. The time dependent permittivity obtained from the CC model is then added to a charge transport model. Simulated currents issued from the transport model implemented with the polarization are compared to the measured APC currents, showing a good consistency between experiments and simulations in a situation where the response is essentially from dipolar processes.

INTRODUCTION

An activity of modeling charge transport in insulators under electrical stress [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] was initiated by Le Roy et al ten years ago. A one-dimensional and bipolar model, able to describe transport, trapping and recombination of electronic carriers (electrons and holes) on the basis of experimental results has been developed. This model already gives good results on low density polyethylene (LDPE). However improvements are still possible, especially in order to predict the behavior of the space charge for different dielectric materials, particularly polar ones, under any electrical stress. The study of the variation of the material permittivity (ε) is necessary to take into account the polarization phenomena in the modeling. The permittivity depends on the experimental constraints (temperature, electrical field and polarization time) and its characterization can be done in the timedomain (Thermo-Stimulated Depolarization Current measurements -TSDC as an example) and in the frequency-domain (Impedance Spectroscopy measurements). Whatever the characterization domain, time or frequency, the treatment of the experimental data to obtain the material relative permittivity remains difficult [START_REF] Bravard | Dielectric Relaxation in Amorphous Poly(ethylene terephthalate) and Poly(ethylene 2,6-naphthalene dicarboxylate) and Their Copolymers[END_REF].

Poly(ethylene naphthalene 2,6-dicarboxylate) (PEN) has been chosen for this study as it is an aromatic polar polymer, and is known to give a significant response from dipolar mechanisms [3][4] for different measurements techniques. In our work, dielectric spectroscopy (DS) measurements, a wellknown technique for performing a complete characterization of the dielectric relaxations, were carried out in order to characterize the polar response of the material in the frequency-domain. This response is then fitted with the help of Havriliak-Negami functions and introduced into a charge transport model. The model results are validated at different temperatures by current measurements under a low field (Alternate Polarization Current -APC- [START_REF] Escribe-Filippini | Conduction characterization of polymer films using the alternate square wave method[END_REF]).

DS MEASUREMENTS AND DATA PROCESSING

Sample and measurement protocols

Semi-crystalline PEN films were commercial TEONEX® Q51 films provided by DuPont Teijin Films Co. The melting temperature is of 269°C and the glass transition temperature of 121°C [6]. Test films of 188 µm thickness were coated on each face with 16mmdiameter circular gold electrodes. DS measurements were carried out using a Novocontrol spectrometer over a frequency range going from 10 -1 to 10 6 Hz, and for a temperature range from -100 to 200°C by step of 5°C.

Experimental results

Dielectric results are shown in Fig. 1 as a function of frequency for different temperatures. The dielectric constant ε' (Fig. 1a) weakly decreases over the frequency range, while it increases with the temperature. Fig. 1b shows the imaginary part of permittivity, dielectric loss, ε" vs. log(frequency) as a function of temperature. Three relaxation peaks are observed. The first one, called β, is assigned to local fluctuations of ester groups (O-C=O) similar to the β-process in Poly(ethylene terephthalate) (PET) [START_REF] Schartel | Dielectric investigation on secondary relaxation of polyarylates: comparison of low molecular models and polymeric compounds[END_REF]- [START_REF] Coburn | Dielectric relaxation in poly(ethylene terephthalate)[END_REF]. The second peak, called β*, has been ascribed to the relative motion of the two naphthalene rings present in the polymer chain [START_REF] Dortliz | Molecular mobility in poly(ethylene-2,6-naphthalene dicarboxylate) as determined by means of deuteron NMR[END_REF]. The third peak, called α, has been related to the glass transition of PEN, involving long-range cooperative segments motion [START_REF] Belana | Tsc study of the polar and free charge peaks of amorphous polymers[END_REF]. The α-relaxation is naturally originating from a conformational collective rearrangement of the chains in the amorphous regions of the material. The amplitude of the α-relaxation peak decreases as a result of crystallization [START_REF] Bellomo | On some dielectric properties of PEN[END_REF]. Conduction was also detected after α-peak at 150°C in the low frequency range. β and β* peak maxima shift to higher frequencies when the temperature increases (Fig. 1). However, these three relaxations never appear simultaneously at a studied temperature. It is then necessary to model the global dielectric response of PEN, working on different temperature ranges for each relaxation.

Data processing by Cole-Cole functions

An empirical model of Havriliak and Negami [START_REF] Havriliak | A complex plane representation of dielectric and mechanical relaxation processes in some polymers[END_REF] has been used to describe the dielectric relaxations. The variation of the complex permittivity ε * (ω)= ε'(ω)-j ε"(ω) is given by:

* 1 2 ( ) (1 ( ) ) c c j ( 1
)
where ω=2πf is the electric field pulsation, Δε=ε s -ε ∞ is the dielectric dispersion of the relaxation, ε s and ε ∞ are the relaxed (ω=0) and unrelaxed (ω=∞) dielectric constant values, τ is the relaxation time associated with the relaxation, c1 and c2 (0≤c1, c2≤1) are shape parameters which describe the symmetric and the asymmetric broadening of the relaxation time distribution function, respectively. In the dielectric loss spectra of PEN, relaxation peaks are relatively symmetric. For this reason, we used the Cole-Cole function [START_REF] Cole | Dispersion and absorption in dielectrics: I. Alternating current characteristics[END_REF], a particular case of the Havriliak-Negami function with c2=1, to model the dielectric response.

Taking into consideration the different relaxation modes and adding a conduction term to ε * , a complete function is obtained:

* 1 0 ( ) 1 ( ) ( ) i n i c s i i j j ( 2
)
where c is similar to c1 in the Havriliak-Negami function, σ is ionic conduction term, ε 0 =8.85.10 -12 F.m -1 is the dielectric constant of vacuum, and s is a parameter related to the nature of the conduction mechanism (0≤s≤1). For PEN, 3 relaxations are observed (n=3), so there are 12 parameters that need to be fitted, and that are all temperature dependent. In order to fit the global dielectric loss response of the semi-crystalline PEN, we used an extrapolation method, that was first proposed by Coburn et al. [START_REF] Coburn | Dielectric relaxation in poly(ethylene terephthalate)[END_REF] for studying the relaxations of PET, and widely applied on PEN [START_REF] Bravard | Dielectric Relaxation in Amorphous Poly(ethylene terephthalate) and Poly(ethylene 2,6-naphthalene dicarboxylate) and Their Copolymers[END_REF], [START_REF] Nogales | Influence of Crystalline structure in the segmental mobility of semicrystalline polymers: poly(ethylene naphthalene-2,6-dicarboylate)[END_REF]. The fit by a Cole-Cole function was done for all the temperatures where the maximum of dielectric loss was well defined in the experimental frequency window. The dielectric losses vs. frequency in the temperature region of the β-relaxation peak (from -80°C to -20°C) are presented in Fig. 2. It is to note that for this relaxation, the peak has been attributed in the literature as being the sum of two contributions, β 1 and β 2 [START_REF] Bravard | Dielectric Relaxation in Amorphous Poly(ethylene terephthalate) and Poly(ethylene 2,6-naphthalene dicarboxylate) and Their Copolymers[END_REF], [START_REF] Hakme | Uniaxially stretched poly(ethylene naphthalene 2,6-dicarboxylate) films studied by broadband dielectric spectroscopy[END_REF]. However, for sake of simplification and as we already have a large number of parameters to extract, we fitted the β relaxation as a unique contribution. For this β relaxation, a set of Cole-Cole parameters able to fit the dielectric loss and permittivity vs. frequency, was obtained. An example of comparison between the fit (line) and the experimental dielectric loss (symbols) vs. frequency at -50°C is given in Fig. 3a. The fit has been performed around the peak maximum (between the two vertical dashed lines on Fig. 3a) in order to avoid the possible contributions of other relaxations. These variations are described by equations ( 3)-( 5): where T is the temperature, in K, k B =8.617.10 -5 eV.K -1 is the Boltzmann constant. The set of parameters is presented in Fig. 4 (open triangles). ∆ε and c follow nearly a linear variation with temperature, while ln(τ) linearly increases with the reciprocal absolute temperature following the Arrhenius law. For the β * -relaxation peak, the best approximations were found for temperatures ranging from 85 to 110°C where the maximum of dielectric loss was well defined in the experimental frequency window. However, it is necessary to take into account the overlapping β-process for high frequencies. This was done by adding to the contribution of β * -peak the extrapolation of the contribution of β-peak using equations (3)- [START_REF] Escribe-Filippini | Conduction characterization of polymer films using the alternate square wave method[END_REF] in the temperature region of β * -peak (filled triangles in Fig. 4). The experimental values of ε" in this region were fitted using the equation:

" " " * ( ) ( ) ( ) (6)
Fig. 3b shows the fit result of β * -peak at 95°C performed in the region of frequency from 1 to 10 6 Hz, where the influence of α-process contribution can be neglected. By doing so for all the temperatures from 85 to 110°C, a set of Cole-Cole parameters for the β * -peak was obtained. This set is presented in Fig. 4 (open circles), and fitted by equations ( 7)-( 9). Finally, the similar procedure was used to obtain the parameters related to the α-relaxation and the conduction. In this case we used the equation:

" " " " " * ( ) ( ) ( ) ( ) ( ) (10)
The extrapolation of the contribution of β-peak and β *peak was realized from equations ( 3)-( 5) and ( 7)-( 9) respectively in the temperature region of α-peak (filled triangles and filled circles in Fig. 4). Fig. 3c shows the fit result of dielectric loss at 150°C in full experimental frequency range with all contributions (β, β * , α and conduction).

IMPLEMENTATION OF POLARIZATION IN THE CHARGE TRANSPORT MODEL

Charge transport model and equations

The charge transport model used to reproduce the behavior of space charge in PEN has already been presented in the literature [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] for polyethylene. It takes ( , ) ( , ) ( , ) D x t J x t j x t x [START_REF] Havriliak | A complex plane representation of dielectric and mechanical relaxation processes in some polymers[END_REF] where ρ(x,t) is the net charge density, function of time and space, and j c (x,t) is the local conduction current. D(x,t) is the electrical displacement, which includes the polarization and is given in the frequency domain by [START_REF] Böttcher | Dielectrics in timedependent fields[END_REF]:

* 0 ( ) ( ) ( ) D E ( 13
)
where E is the electric field, ε 0 the vacuum permittivity and ε * (ω) the material permittivity defined by Cole-Cole functions:

* 1 1 ( ) ( ) 1 ( ) i n n i i i c i i i j (14)
Including equation ( 14) into [START_REF] Cole | Dispersion and absorption in dielectrics: I. Alternating current characteristics[END_REF] gives:

0 0 1 ( ) ( ) ( ) ( ) n i i i D E E ( 15
)
An inverse Fourier transform is then necessary to change the permittivity response from the frequencydomain to the time-domain. A particular attention should be paid during the data treatment on the sampling representativeness between frequency-and time-domain. Finally, applying the Laplace transform, taking into account E as a function of time and space, the electric displacement is obtained.

0 0 1 ( , ) ( , ) ( , ) * ( )
n i i i D x t E x t E x t t (16)

Current measurements at low fields (APC)

In order to validate the charge transport model, which now takes into account the time-dependent permittivity, alternate polarization current (APC) measurements [START_REF] Escribe-Filippini | Conduction characterization of polymer films using the alternate square wave method[END_REF] were carried out on PEN samples. The experimental protocol is presented in Fig. 5. Samples were polarized successively under electric fields E 0 and -E 0 during several half-periods T/2. In order to avoid charge injection into the dielectric, very low electric field E was applied. The polarity change between each step decreases clearly the scattering of consecutive current [START_REF] Adamec | On the determination of electrical conductivity in polyethylene[END_REF]. In our study, test films of thickness 188 or 75 µm were coated on each face with a 60mm-diameter circular gold electrode. They were polarized under low electric fields, 0.05 kV/mm during 5 half-periods of 1000s for a temperature range from ambient temperature to 90°C. At very low and high temperatures (-80, 130 and 150°C), a higher field was chosen (1.33 kV/mm during 2 half-periods of 1000s relative to 0.5 kV/mm) in order to decrease the noise of measured current.

In a previous communication [START_REF] Hoang | Electrical characterization of PEN films using TSDC and PEA measurements[END_REF], we indicated that charges were not injected in PEN samples bulk under 10 kV/mm if the temperature did not exceed 130°C. Fig. 6 shows the normalized current densities obtained in PEN as a function of time for different temperatures.

Depending on how many relaxation processes are at play, the polarization currents can be separated into 3 temperature regions: -80°C, from 28.6 to 71.4°C and from 91.1 to 130°C. The measured polarization currents correspond to the contributions of relaxations β, β * and α that appear consecutively when the temperature is increased. The current curve at 150°C differs from that at other temperatures. Conduction becomes dominant at high temperatures and APC measurements no longer reflect polarization processes only.

Comparison of simulation and experimental results

Simulations were performed for electric fields following the protocol used for APC measurements and for temperatures ranging from -80 to 50°C. In a first attempt to validate our model, the injection and transport contributions were not taken into account. At these low fields, and at relatively low temperatures, the conduction contribution should be low compared to polarization mechanisms and hence this hypothesis should hold. Hence, the external current in equation ( 12) is only due to the second part of the right hand side, i.e. to the variation of permittivity with time. Fig. 7 shows the comparison between simulated and experimental results (normalized currents) in the time-domain and for two temperatures: -80°C and 50°C. For both temperatures, at very low frequencies, the maximum of β-and β * -relaxation peaks were well defined in the studied frequency-window. We can see that the simulations are in good agreement with the measurements in time-domain for two different temperatures, qualitatively and quantitatively.

CONCLUSION

Fig. 1 .Fig. 2 .

 12 Fig. 1. Dielectric constant (a) and loss (b) vs. log(frequency) in PEN as a function of temperature

Fig. 3 .

 3 Fig. 3. Dielectric loss for PEN as a function of temperature. Experimental data and fit results at: a. -50°C; b. 95°C; c. 150°C.

1 )Fig. 4 .

 14 Fig. 4. Cole-Cole parameters vs. temperature obtained from fit method and used as extrapolated data. Open symbols correspond to fitting parameters; filled symbols correspond to extrapolated parameters. β-relaxation: triangles;; β *relaxation: circles.

Fig. 6 .Fig. 5 .

 65 Fig.6. APC profiles for PEN as a function of temperature.

Fig. 7 .

 7 Fig. 7. Comparison of normalized currents simulated by the charge transport model and APC normalized currents at -80 and 50°C.

ème conférence de la Société Française d'Électrostatique, 27-29 août 2014, Toulouse, France.


A Boularas
email: boularas@laplace.univ-tlse.fr

F Baudoin

G Teyssedre

C Villeneuve-Faure

S Clain

3D modelling of electrostatic force distance curve between the AFM probe and dielectric surface

Techniques derived from the near-field microscopies and particularly the Atomic Force Microscopy (AFM) are presented as alternative techniques for measurement of space charge compared to classical techniques due to their high sensitivity to the electrostatic force and a better resolution (≈ nm). One of the AFM derivative techniques which allow obtaining information on the charge state of the dielectric materials is based on the technique of Force Distance Curves (FDC) obtained by cycle of approach and retracts between the AFM probe and the dielectric. In this paper, three-Dimensional (3D) simulation results for the electrostatic force between an Atomic Force Microscope (AFM) tip and the surface of a dielectric are presented for different AFM tip geometries. The first aim is to analyse the effect of the geometric parameter on force-distance curves without contribution of the cantilever as AFM detection mode for electrostatic charges. At the second time the sensor is composed of a cantilever supporting a conical tip terminated by a spherical apex, we investigate in a second step is to see the effect of the cantilever on force distance curve. Simulation results are shown and compared with experimental data in order to validate our approach.

INTRODUCTION

The study of solid dielectric materials, especially injection phenomena and charge accumulation is still a very important challenge for the reliability of electrical systems. Several techniques for measuring space charge are available: pulsed electroacoustic -PEA-methods [START_REF] Maeno | High-resolution PEA charge distribution measurement system[END_REF], thermal methods (Laser Induced Modulation Method -LIMM-or Thermal Pulse -TP-) [START_REF] Lang | Laser intensity modulation method (LIMM): Experimental techniques, theory and solution of the integral equation[END_REF] but they are still limited in their spatial resolution (≈ micrometre), for the study of insulating films of small thickness as can be found in electronic components with thicknesses of the order 10-50 nm. Meanwhile, techniques derived from the near-field microscopy [START_REF] Binnig | Surface Studies by Scanning Tunneling Microscopy[END_REF] such as Atomic Force Microscopy (AFM) [START_REF] Binnig | Atomic Force Microscope[END_REF] and the derivative techniques such as Kelvin Force Microscopy (KFM) [START_REF] Rezende | Detection of charge distribution in insulator surfaces[END_REF][START_REF] Ishii | Static states and dynamic behaviour of charges: observation and control by scanning probe microscopy[END_REF][START_REF] Borowik | Calculating Kelvin force microscopy signals from static force fields[END_REF] and Force Distance Curve (FDC) [START_REF] Cappella | Force-distance curves by atomic force microscopy[END_REF][START_REF] Gomez | Electrostatic forces between sharp tip and metallic and dielectric surface[END_REF] have appeared. Our purpose here is to exploit near field techniques derived from Atomic Force Microscopy (AFM) which appears as invaluable tools for nanometer scale materials diagnostic [START_REF] Leach | The European nanotechnology landscape[END_REF]11].

These techniques allow us to have information on the state of charge of insulation and their ability to store / dissipate these charges; the KFM is used to measured surface potential modification induced by charges. However vertical localization of charges seems difficult to attempt, EFDC (Electrostatic Force Distance Curve) follows electrostatic force evolution as function of tip-surface separation distance. This technique, based on FDC, appears promising because of its high resolution, sensitivity to charges localization and distance dependence. Measurements of EFDC and surface potential are available [START_REF] Villeneuve-Faure | Kelvin force microscopy characterization of charging effect in thin a-SiO x N y :H layers deposited in pulsed plasma enhanced chemical vapor deposition process by tuning the Silicon-environment[END_REF], but the quantitative interpretation of the results obtained requires the use of modelling. Modeling KFM electrostatic conditions remains difficult, due to the oscillating nature of the technique. Simplification of the problem is expected through direct study of the evolution of the electrostatic force induced by trapped charges. Therefore the objective of our study is to develop a numerical model to do a metrological analysis of the AFM probe with the force curves. To do this study, a three dimensional electrostatic model was developed with commercial software Comsol Multiphysics®. Several geometries of the AFM tip were tested: cone, pyramid and tetrahedron. These simulations help us to establish what geometry of the AFM tip will give us the results closest to that of the measurement. On the other hand, an analysis of the different parts of an AFM probe was performed to estimate the contribution of each element of the probe on the curve shape. A three-dimensional electromechanical model was also developed; the cantilever and tip are considered in this model. The coupling of the two studies, electrostatic and mechanical allows us to evaluate the effect of the contribution of the cantilever to the results of force curve. The force curve is obtained by gradually increasing the tip / sample distance, the force is estimated by deflection of the AFM probe.

The paper is structured as follows: in section 2 we describe the probe-thick insulator substrate system where we use to compute a force distance curve approximation between tip and sample with the three-dimensional electrostatic model. Poisson equation is solved using FEM to provide the electrical potential and electric field, the electrostatic force here is calculated using the Maxwell-stress-tensor. In section 3 we analyse how the kind of geometry of the AFM tip influance the tip-substrate electrostatic interaction and force distance curve. In section 4 we analyse the contribution of cantilever in the force distance curve by the three-dimensional electromechanical model, the electrostatic force here is estimate by the deflexion of cantilever ∆z. The result of this simulation result is compared each time with the experimental result. Finally in section 5 we summarize the main conclusions of the paper.

ELECTROSTATIC FORCE DISTANCE CURVE

The technique of Force Distance Curve (FDC) permits direct measurement of interaction forces between the AFM tip and surface of sample. It is possible to isolate the electrostatic interaction, it by to exploiting the sensitivity of electrostatic forces when the AFM tip is biased or in presence of space charges in the sample. This sensitivity has been demonstrated elsewhere [START_REF] Boularas | Multi-dimentional modelling of electrostatic force between atomic force microscopy tip and dielectric surface[END_REF][START_REF] Patil | Investigation in the electrostatic forces in scanning probe microscopy at low bias voltage[END_REF][START_REF] Kim | Dependance of approaching velocity on the Force-Distance curve in AFM[END_REF]. The electrostatic interactions are difficult to handle as they depend on the very geometry of the set-up and act over long distances: this leads to problems for building a realistic model. Therefore some simplified model system is used to describe the complex AFM setup. In general a sample is assumed to be flat and the geometry of the AFM tip is assumed as a conical tip with cylindrical symmetry. The most used shapes found in the literature are: A simple charge [START_REF] Hu | Scanning polarization force microscopy: A technique for imaging liquids and weakly adsorbed layers[END_REF] a sphere [START_REF] Terris | Contact electrification using force microscopy[END_REF] or a cone [START_REF] Beladi | Electrostatic forces acting on the tip in atomic force microscopy: Modelization and comparison with analytic expressions[END_REF] where a metallic tip and flat sample are assumed. The important case of dielectric is seldom treated due to its complexity, as image-charge method in thick dielectric film [START_REF] Gómez-Moñivas | Electrostatic force between sharp tips and metallic and dielectric sample[END_REF]. The numerical modelling of forces [START_REF] Colchero | Resolution enhancement and improved data interpretation in electrostatic force microscopy[END_REF][START_REF] Blaidi | Finite element simulations of the resolution in electrostatic force microscopy[END_REF][START_REF] Gros | Two-dimensional, electrostatic finite element study of tip-substrate interactions in electric force microscopy of high density interconnect structure[END_REF] is in accordance with experiments except for small tip to sample distances [START_REF] Makasheva | Dielectric Layers for RF-MEMS Switches: Design and Study of Appropriate structure Preventing Electrostatic Charging[END_REF]. Most of this study is mainly focused on the influence of experimental conditions (tip characteristic, velocity…) on the force. However, from the experimental point of view no systematic study of electrostatic force has been performed and theoretically, no 2D or 3D models are available in order to quantify the charges density. The EFDC method consists in exploiting electrostatic sensitivity of the FDC. For this, two approaches of the AFM tip and the surface of the sample are made. As shown on Fig. 1 (a) the first approach is with the bias applied on the tip and in the second one the tip is connected to the ground and it is considered as reference curve.Three main parts can be identified on each curve when the tip approaches the surface: 1-Tip approaches the surface, 2-Jump to contact (Van der Waals forces) and 3-Contact. As the electrostatic force is a long range force, the potential difference between tip and surface mainly modifies the approach behaviour (part 1 of the curve) without modifying the other steps. Thus, the electrostatic force can be extracted form the difference between force curve with and without potential.

THEORECAL MODELLING

To study the effect of the nanometre-scale parts of the AFM probe (the cone and the cantilever) on the electrostatic force interaction, we present in this paper several kinds of models. Let us consider the probe-sample system sketched in Fig. 2. In a first step we consider the simplified model with no consideration of the cantilever effect. The tip consists of a truncated cone of height L and cone aperture angle α ending with a semi-spherical apex radius R c , (all geometric parameters are given in table 1).

where ε is the space-dependent relative permittivity for the sample and the gas. Interface Γ is decomposed into constant potential boundary Γ 0 at the bottom of the insulating substrate (here taken as ground) and Γ 1 on the probe surface, while Neumann boundary conditions Γ 2 were used on the lateral sides and the upper side of the simulation box (see Fig. 2). The electric field is derived from the potential:

E grad U
The electric field Fig. 3 (b) is the crucial parameter governing the near-field microscopy [START_REF] Watanabe | Electrostatic force microscope imaging analyzed by the surface charge method[END_REF] and it is very sensitive to the radius of curvature of the AFM tip [START_REF] Boularas | Multi-dimentional modelling of electrostatic force between atomic force microscopy tip and dielectric surface[END_REF].

Therefore we are able to compute the electrostatic force acting on the probe function of the tip substrate separation distance by integration of the built-in Maxwell-stress-tensor (Mt) (3) aver all faces of the tip. We estimated the accuracy of our calculations to be better than 1% against the analytical expression of the electrostatic force acting on a conducting sphere in front of an infinite dielectric substrate [START_REF] Smythe | Static and dynamic electricity 5new[END_REF].

EFFECT OF TIP GEOMETRY ON THE ELECTRODTATIC FORCE SEPARATION DISTANCE CURVE

We are now going to present the results of the study of the Electrostatic Force Distance separation Curve EFDC. The aim here is to evaluate the impact of different geometrical forms on electrostatic interaction between the AFM tip and the thick insulator. The tip chosen for this study is the PtIr coated probe with radius curvature R c = 32 nm; other parameters are given in Tab.1. Fig. 3 shows the distribution of electric potential at the separation distance D = 100 nm, and the electric field recorded at the surface of the tip at the same distance D.

To simulate the force separation curve we varied the distance D between 0 and 400 nm. The development of the three-dimensional model allows studying the influence of the geometry of the AFM tip on the results of the force curve in order to come most close to the real shape of the AFM tip, Fig. 5. The most difficult task here is using three-dimensional (3D) solver with large geometries including nanometric features (here the apex).

FIG. 5 SEM image of the real AFM tip used in the experiment.

For this, three geometries are tested; the first one represents the tip with the cone and it is the most used in the literature. In the second step the tip is represented with the pyramidal shape and the last we used the tip with the tetrahedral shape. In Fig. 5 we present the comparison between the simulation results obtained by our FEM model on the different shape of the AFM tip with the experimental results of the force distance separation curve. The first observations show that the experimental results are in good correlation with simulation data obtained by the model with the tetrahedral shape of the AFM tip (green curve) Fig. 4.

The tip with the cone shape overestimates on the values of the force. The result obtained by tip with pyramidal shape is close to experimental result but the curvature of the EFDC curve is not respected. An offset of the electrostatic force appears between the simulated data and experience. This shift is due to the fact that this model does not take into account the effect of the cantilever in the calculation of electrostatic force. In summary we have shown that the result of force distance separation curve is sensitive to the shape of the AFM tip, and the tip with the tetrahedral shape shows results nearest to experimental results. The shift is due to the fact the model does not take into account the contribution of the cantilever. The cantilever is modeled as a beam as shown in Fig. 6 and it is positioned at the distance L (the height of tip) + D (separation distance between tip and sample) see table 1. The cantilever is characterized by several geometric parameters detailed in Table 2. It is also characterized by the k spring constant which can be calculated by the equation ( 5): where E is the Young's modulus, other parameters are taken from table 2. In order to solve numerically the electromecanic problem in domain Ψ, we use also the finite-element package COMSOL Multiphysics®. The domain Ψ is chosen large enough to neglect edge effects and obtain a relevant approximation of the problem. This domain is chosen with a cubic box shape in order to be able to mesh the domain with a regular mesh. Two important equations are solved in this model; the first one is Poisson equation ( 1) applied to the domain Ψ of boundary Γ detailed in Section 3. Interface Γ is decomposed into constant potential boundary Γ 0 on the bottom of the insulating substrate (here taken as ground) and Γ 1 on the surface of the cantilever, while Neumann boundary conditions Γ 2 were used on the lateral sides and the upper side of the box simulation (see Fig. 6). The second one is the mechanical equation which calculates the displacement force in the stationary case: Where σ is the stress tensor and F v is the forces exerted on the volume. The stress tensor must be continuous across a stationary boundary between two materials. where σ 1 and e σ 2 represent the stress tensor in materials 1 and 2, respectively, and n 1 is the normal pointing out from the domain containing material 1. The mechanical boundary conditions are presented with fixed constraint in the left side of the cantilever and the symmetry condition in the face of geometry (See Fig. 6).

EFFECT

z k F e .
Where k is the spring constant and ∆z is the displacement. At this step it is possible to estimate the influence of the cantilever in Electrostatic Force Distance separation Curve EFDC, by variying the distance D between 0 and 400 nm. The result of this curve is presented in Fig. 8 (a) it is compared to the result of force curve of the tip. Results show that the force created by the cantilever induces a very important force compared to the experimental EFCD. This feature is probably due to:

-the physical properties of the cantilever. Some studies show that the thickness of the cantilever, parameter T, plays an essential role on the force. Fig. 8 (b) shows the EFCD obtained with T = 3.5 µm instead of 2.75 µm as indicated by the data sheet. With this value, the EFCD is closed to the experimental data; -the shape of the cantilever. It would be interesting to simulate the force created by the cantilever with geometry without singularity, in this case the cantilever is not a perfect parallelepiped but with the curved angles. We note that the force curve due to the cantilever adds a constant force (red curve in Fig. 8). This contribution is added to the tip contribution, green curve in Fig. 8, in order to make a complete comparison of the force curve with the experimental results.

CONCLUSION

Two models have been developed in order to well understand the principle of electrostatic force measurement by AFM. This work focuses on the study of electrostatic force curve induced between two parts: the AFM probe and a flat dielectric sample. With the electrostatic model we could see the influence of the geometry of the tip on the curve electrostatic force. Electromecanical model has been used to evaluate the influence of the cantilever on the force curve. The results exhibit good qualitative and quantitative fit to experimental data, which allowed us to validate our approach. Results also highlight the fact that the radius of the tip plays an essential role on the value of the electrostatic force.

ACKNOWLEDGMENT

FIG. 1

 1 FIG.1 Electrostatic force vs. sample distance separation. A effect of applied voltage at the AFM tip; black curve represent the FDC without voltage at tip (reference) and green curve represent curve with voltage at tip. Subtracting the green curve and the reference curve provides the Electrostatic Force Distance separation Curve.

  Fig. 1 (b) shows the resulting electrostatic force as function of tip-sample distance for 25V applied bias on the conductive tip.

  FIG.2 Sketch of the geometric representation and parameters of the tip-thick insulator system used in electrostatic three-dimensional finite-element numerical calculation in domain Ω.

FIG. 3

 3 FIG.3 (a) Electric potential distributions created by the PtIr coated tip (R c = 32 nm α = 20°, D = 100 nm and e = 200 nm) in front of dielectric simulated by three-dimensional FEM COMSOL Multiphysics®. (b) Electric field recorded over the surface of the tip. The thick insulator substrate has a dielectric constant ε r = 4.35. The AFM tip is at 20 V while the backside of the substrate is at ground 0 V.

FIG. 4 .

 4 FIG.4. Electrostatic force versus distance separation between the AFM tip and sample for different tip geometry forms with the same geometry parameters, Apex radius R c = 32 nm and cone half angle α = 20°. Comparison between results obtained by finite element method and experimental results, 20 V bias at the surface of the AFM tip.

  FIG.6Sketch of the geometric representation and parameters of the cantilever-thick insulator system used in electromechanical three-dimensional finite-element numerical calculation in domain Ψ.

FIG. 7

 7 FIG.7 (a) Electric potential distributions and electric field vector created by the PtIr coated cantilever in front of dielectric simulated by three-dimensional FEM COMSOL Multiphysics®. (b) Displacement of the cantilever due to the deflection caused by the electrostatic force. The thick insulator substrate has a dielectric constant of ε r = 4.35. The AFM cantilever is at 20 V and D=100nm while the backside of the substrate is at ground 0 V.

FIG. 8

 8 FIG.8 Electrostatic force versus distance separation between the AFM tip and sample for different tip geometry forms with the same geometry parameters, Apex radius R c = 32 nm and cone half angle α = 20 °. Comparison between results obtained by finite element method FEM and experimental results, 20 V bias at the surface of the AFM cantilever, With (a) T=2.75 µm, (b) T=3.5 µm.

  This work was supported by Region Midi-Pyrénées (France), by FEDER Funds through COMPETE Program and by Fundação para a Ciência e a Tecnologia (Portugal), within the Project PEst-C/MAT/UI0013/2011.

TABLE 1 .

 1 ELECTRICAL MATERIAL PROPERTIES AND GEOMETRICAL PARAMETER USED IN FEM MODELS

	Term	Definition	Value
	ε r	Relative permittivity	4.35

TABLE 2 .

 2 GEOMETRICAL PARAMETER PROPERTIES OF CANTILEVER USED IN FEM MODELS (DATA SHEET)

	Probe	material k	T	Lc	W
			(N/m)	(µm)	(µm)	(µm)
	SCM-PIT	PtIr	2.8	2.75	225	28

ème conférence de la Société Française d'Electrostatique, 27-29 août 2014, Toulouse, France.


Thi Thu

Nga Vu
email: vu@laplace.univ-tlse.fr

Gilbert Teyssedre
email: gilbert.teyssedre@laplace.univ-tlse.fr

Bertrand Vissouvanadin
email: bertrand.vissouvanadin@laplace.univ-tlse.fr

Séverine Le Roy

Mohamed Mammeri
email: mohamed.mammeri@sileccable.com

Sèverine Leroy
email: severine.leroy@laplace.univ-tlse.fr

Christian Laurent
email: christian.laurent@laplace.univ-tlse.fr.

Investigating the effect of temperature gradient on field distribution in polymeric MV-HVDC model cable through simulation and space charge measurement

Keywords: Space charge, HVDC cable, electric field distribution, temperature gradient

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

With the advantages of using HVDC system compared to HVAC system, especially for long-distance transmissions [START_REF]An Introduction to High Voltage Direct Current (HVDC) Underground Cables[END_REF], HVDC cables are increasingly used and gradually replace HVAC cables for electrical energy transport. However, space charge accumulation in HVDC cables under various temperatures and applied fields is the major problem when in service. Indeed, when space charge density is sufficiently high, the local field strength may exceed the breakdown strength of the dielectric, leading to premature failure. This is why space charge has been extensively studied since over the last 20 years for the development of polymeric-type DC cable systems.

Amongst the different mechanisms of space charge accumulation the effect of temperature gradient on field distortion has been the subject of many studies. Interest on the subject begins with the work of McAllister et al. who considered space charge accumulation phenomenon from the macroscopic point of view using electromagnetic field theory [START_REF] Mcallister | Charge Accumulation in DC Cables: A Macroscopic Approach[END_REF]. In their work, charges accumulation was analysed as a consequence of a non-uniform conductivity. An expression of steady-state space charge distribution in polymeric DC cable has been proposed assuming that the conductivity increases exponentially with temperature and through a power law with field strength.

Jeroense and Morshuis [START_REF] Jeroense | Electric Fields in HVDC Paper-Insulated Cables[END_REF] have computed space charge and field distributions in the case of paper-insulated HVDC cables for different stages both for transient and steady state conditions. The insulating material conductivity was assumed to depend exponentially on temperature and field. By disregarding the field dependency of conductivity , an analytical expression of field distribution and resistance per meter cable was derived. In this particular case, the field distribution is mainly governed by the temperature drop across the cable radius. It has been shown that for a sufficiently large temperature drop, the field strength is inverted with respect to the capacitive field [START_REF] Jeroense | Electric Fields in HVDC Paper-Insulated Cables[END_REF]. Indeed, with the above hypotheses (temperature dependent ), using Maxwell's equations, it can be shown that in steady state conditions the field distribution inside the dielectric in cylindrical geometry is given by [START_REF] Fabiani | HVDC Cable Design and Space Charge Accumulation. Part 3: Effect of Temperature Gradient[END_REF] ) ( ) ( 0 0 0 r r r E r E    [START_REF]An Introduction to High Voltage Direct Current (HVDC) Underground Cables[END_REF] where E0 and 0 are the field and conductivity at reference position r0. The space charge density associated with a non-uniform conductivity is of the form:

dr r d r r E r tg ) ( ) ( ) ( 0         [2]
For (T) following an Arrhenius dependence with temperature with Ea as activation energy,

dr dT kT E r E r a r tg 2 0 ) ( ) (       [3]
where T is the temperature. It can be shown from Eq (1) that if the conductivity varies with radius faster than an hyperbolic law does, then the field increases from inner to outer screen. From Eq. 2 and 3, it can be seen that the sign of the space charge density associated with conductivity gradient is directly linked to d/dr: for a negative voltage applied to the conductor (E>0), a negative space charge is predicted in case of temperature gradient with conductivity increasing from inner to screen (Eq.3) whereas space charge is positive if the conductivity gradient is linked to a conductivity increasing with field (hence decreasing as r increases, cf. Eq.2): In a situation where the two processes are combined, the sign of the space charge is undetermined a priori. Boggs et al. [START_REF] Boggs | Effect of Insulation Properties on the Field Grading of Solid Dielectric DC Cable[END_REF] recently investigated the behaviour of the resistive field distribution under non-uniform temperature conditions for two categories of insulating polymers of which the conductivity has different thermal activation energy. They demonstrated the importance of using materials having low Ea and large field-dependence of conductivity to control the electric field distribution under conditions of large thermal gradients.

The field distribution in HVDC cable is controlled by the insulating material conductivity which is a strong function of temperature and can substantially vary with electric field. However, the dependency of conductivity with field and temperature is specific to each material. Besides, space charge accumulation processes may occur without necessarily being linked to the temperature gradient in the material. This further complicates the predicted field distribution. The purpose of this contribution is to compare predicted field distribution in cable models based on conductivity measurements taken on flat specimen to actual field distributions obtained by the pulsed electroacoustic method on cable samples.

In the purpose of investigating the evolution of conductivity of cable materials with field and temperature, current-voltage measurements have been carried out on XLPE plaque samples under various conditions of temperatures and voltages. Then, functional variation of conductivity with temperature and electric field is derived from the fit of the quasi-stationary charging currents (conduction currents). From the established semiempirical conductivity model, space charge distribution has been computed using Comsol® within XLPE MVmodel cable in presence or absence of a thermal gradient. Space charge measurements have been also carried out on MV-HVDC model cables and results are discussed and compared to those obtained with simulation.

EXPERIMENTAL Conductivity measurements

Conductivity measurements are performed in the form of plaques of cross-linked polyethylene (XLPE) with thicknesses varying from 500 µm to 600 µm. Samples were kept sealed in aluminium bags after cross-linking.

Both faces of samples were gold-metalized by sputtering to form 50 mm-diameter electrodes. A silicone ribbon was laid at the periphery of the electrodes to avoid edge phenomena. Measurements were performed in air at temperatures ranging from 0°C to 90°C by step of 10°. The temperature was controlled by a thermo stated sample holder. To avoid possible memory effects, a new sample has been used for each value of temperature.

Charging currents have been measured for 1 hour for 10 values of DC fields varying from 2 to 25 kV/mm under isothermal condition. Depolarization was applied for 1h after each volt-on step. Current values were recorded every 2 seconds all along the measurements.

Space charge measurement

Tested samples are sections of Medium Voltage -MV-XLPE cables, about 3 m long, with 4.5mm thick insulation (inner and outer radiuses are 5 mm and 9.5 mm, respectively). The conductor diameter is 8 mm. The inner and outer semicon (SC) thicknesses are 1 and 0.5 mm. Space charge measurements were realized using the pulse-electro-acoustic (PEA) method through the configuration represented in Fig. 1. The PEA-Cable device was provided by TechImp®, Italy. The cable is fixed to the PEA cell by means of a cable holder at the measuring point, which guarantees a good acoustic contact between outer SC and aluminium electrode of the measuring device. The pulse voltage is injected via the outer SC of the cable. The outer SC is removed over a length of 5 cm between the measuring location and the region where the pulse is applied. In this way, the cable itself acts as a decoupling capacitor. The DC voltage supplied by a high voltage generator is applied at inner conductor via a Rogowsky electrode avoiding corona discharges at the cable connection.

In this work, two different test conditions have been considered, being room temperature and 11°C temperature drop across the insulation. For the former condition, the cable was kept at ambient while for the second, temperatures at the inner and outer SC shields were set to 51°C and 40°C respectively. To achieve such a temperature drop across the cable insulation, the test sample was connected in a loop using a screw connector to allow an ac current induced by a transformer to flow in the cable core. The conductor was heated with a current of 200 A while the surface of the cable was kept in air. It is noteworthy that in this way the steady state temperature at the cable surface is influenced by the environment condition. In particular, the PEA cell tends to reduce the temperature at the cable surface since the aluminium shield of the PEA cell acts, besides as delay line, as a heat dissipater. Hence, the temperature drop is expected to be higher at the measuring location than somewhere else along the cable suspended in air. In our case, after 100 minutes of current injection, the cable surface temperature was in steady state at 40 o C. The heat flow over the cable radius being conservative under steady state, the temperature at the inner conductor could be estimated from the surface temperature (measured), the Joule losses (R.I

2 ) in the inner conductor and the thermal resistance of the cable insulation. A DC negative voltage of -40kV was applied at the inner conductor of the cable for 7 hours followed by 1 hour of depolarization. Pulse voltage was applied using a 5kV/30ns/10kHz pulse generator. For the present measurements, the pulse frequency was limited to 5kHz.

Raw PEA signals were recorded and averaged every 200 seconds all along the voltage cycle. They were subsequently treated by the deconvolution technique as detailed in the following subsection.

Deconvolution of PEA signals

Since the PEA method relies on the generation of electrostatic forces through interaction between embedded charges and the external pulse field [START_REF] Maeno | Measurement of Spatial Charge Distribution in Thick Dielectrics Using the Pulsed Electroacoustic Method[END_REF], acoustic wave propagation in the direction of the piezoelectric sensor mostly suffers from attenuation and dispersion especially when dealing with relatively thick insulating materials as for MV-cable samples. Furthermore both generation and propagation of waves are affected by the geometry of the cable. Especially, the contribution to the PEA signals of the electrostrictive force caused by the strain-induced variation of permittivitycan no longer be ignored as for plaque samples [START_REF] Holé | Influence of Divergent Electric Fields on Space-charge Distribution Measurements by Elastic Methods[END_REF]. The deconvolution method used in this work aims at correcting overall effects encountered in cable geometry, namely attenuation, dispersion, electrostriction and non-flat frequency response of the PEA set-up (sensor + voltage pulse). For sake of simplicity, the deconvolution is presented as a flow chart (see Fig. 2) and the different steps shall be briefly exposed. The reader may refer to [START_REF] Vissouvanadin | Deconvolution Techniques for Space Charge Recovery Using Pulsed Electroacoustic Method in Coaxial Geometry[END_REF] for more details about the method.

The deconvolution of raw PEA signal requires an accurate description of the full system which consists of the measurement set-up and the tested cable. Calibration procedure, which consists in applying a given DC voltage (Ucal) and acquiring the corresponding PEA signal, is used to derive the global system transfer function Hsyst (r,f). The latter is obtained by multiplying the position-dependant cable transfer function (g(r,f))) by the set-up transfer function (Hsetup). In the cable transfer function g(r,f)) are taken into consideration the effects of attenuation and dispersion of acoustic waves in the cable while Hsetup models the non flat frequency response of the detection and conditioning circuit. Material transfer function g(r,t) is computed from the frequency-dependent wave attenuation ((f)) and velocity (c(f)). These quantities are derived from the calibration signals spectra generated by the capacitive charges on outer and inner electrodes. Hsetup is obtained normalizing the spectrum of the first part of the calibration signal with the calibration voltage Ucal. The generated pressure distribution across the cable m(r) is computed from the Inverse Fourier Transform (IFFT) of the spectral ratio between the PEA signal VPEA(f) and the system transfer function Hsyst (r,f). Electric field E(r) and space charge (r) profiles are subsequently derived from the pressure distribution.

The current deconvolution method is potentially applicable in any type of cable. As illustrations, we show in Fig. 3 results obtained with two types of XLPE insulated cables: medium voltage (MV) and mini-cable of which insulating thicknesses are respectively 4.5 mm and 1.5 mm. For the MV cable (Fig. 3.a), measurements were carried out at room temperature with successive polarizationdepolarization steps in which voltages applied to the core conductor were -40 and -80 kV. Polarization / depolarization lasted for 3 / 1 h. For mini cables (Fig. 3.b), measurements were carried out in presence of 10°C temperature gradient. The cable core was heated to 70°C while the external cable surface was at 60°C. One cycle of polarization/depolarization were applied. The polarization stage was -55 kV for 4 h; depolarization was for 4 h.

The spatial evolution of space charge density is represented in form of cartography. This is a convenient way to follow the processes of charges accumulation and relaxation in the material. In this representation, the time is given on X-axis; the radius is along the vertical axis (anode to the top, cathode to the bottom) and colour scales represent charge density, negative and positive polarity being represented by cool and warm colours respectively. In the MV cable, the cartography of space charge density (Fig. 3.a) highlights the slow transit of negative charges from the inner to the outer electrode. In the mini cable (Fig. 3.b), it has been observed that a fast transit of negative charges front occurs during the first 20 min of polarization. Positive charges injection occurred at the anode; however these charges stay trapped close to the electrode. Space charge pattern also reveals that the bulk of mini cable is mainly occupied by negative charges arising from injection process at the cathode.

SIMULATION RESULTS BASED ON CONDUCTIVITY DATA

DC conductivity of insulating polymers is generally temperature and electric field-dependent. Such a coupling between field and conductivity makes the assessment of the electric field distribution across the cable radius rather complex analytically even with assuming steady state condition. This difficulty is overcome when using numerical solver to compute the field distribution under temperature gradient condition. Another advantage of using numerical solver is the possibility of obtaining solutions both under transient and steady state conditions.

Fig. 4 shows the field dependence of quasi steady-state current obtained on XLPE plaque samples for different temperatures. It can be observed that for all temperatures, the current increases with field with a slope substantially higher than 1, even at low temperature. Therefore, the conductivity seems always field-dependent. The current increases by over two orders of magnitude for the 50°C temperature variation shown here.

Through fitting of these current measurement data, an expression of conductivity vs. temperature and electric field has been established. It is expressed as [START_REF] Vu | Electric field profile measurement and modeling in multi-dielectrics for HVDC application[END_REF]:

  E .E T B T k -E A T,E B a ). ) ( ).sinh( .exp( = ) ( [4]
where A and  are constants, Ea is the thermal activation energy, kB is the Boltzmann's constant, and B = aT+b is a temperature-dependent coefficient to account for the change in threshold field versus temperature. Coefficients for equation ( 4) are as follows (with  in S/m, E in V/m, T in K): A=0.8 (SI), Ea=1 eV, a=0 and b= 10 -2 (T<313K), a= 1.3 10 -9 and b=3.36 10 -7 (T≥313K).

The above semi-empirical expression of conductivity has been implemented in COMSOL to compute space charge and electric field distribution in MV-cable in absence or presence of a thermal gradient. We consider the case of -40 kV applied to the core conductor of the cable (inner electrode). The outer electrode (outer SC) is set to 0V. We have represented in Fig. 5 the predicted change in field distribution across the cable radius during 7 h of voltage-on in ambient condition (22°C). In these conditions, the electric field does not considerably change in time; the distribution stays similar to that of the Laplace field (with constant conductivity). The field magnitude roughly decreases and increases in time at the inner and the outer electrodes, respectively. This is due to a higher value of the field in the vicinity of the inner conductor at the beginning of polarization, resulting in a higher conductivity (owing to eq. 4). To converge to the steady state regime, in which (r).E(r).r is constant due to current flux conservation, the field strength at this location decreases and so does the conductivity. Conversely, close to the outer electrode, since the conductivity of the insulation is initially lower (the initial field being lower) the field is increasing in time. After 7 h of polarization, the field values were estimated to 11.8 kV/mm and 6.8 kV/mm at inner and outer SC respectively. For the present conditions, space charge density (negative, not shown here) was very low; the highest valuefound at the inner SC after 7 h of chargingwas of the order of -10 mC/m 3 .

Fig. 6 shows space charge and electric field profilesobtained from simulationat various times under -40 kV in presence of 11°C thermal gradient (40°C at the outer SC and 51°C at the inner SC). From Fig. 6a, it can be observed that negative charges are rapidly accumulating adjacent to inner conductor during the first hour of polarization. After 3 h, their density varies from -40 mC/m 3 to -16 mC/m 3 along the cable radius. Then, the space charge density begins to slightly decline at inner electrode while at outer electrode it is still increasing. As a consequence, the electric field decreases and increases in time respectively at the inner and outer conductor and the field distribution is inverted (in respect to the initial Laplacian field) after about 3 h of polarization (see Fig. 6b). Furthermore, it can be noticed that there is a position in the insulation bulk (located at about 2 mm from the inner conductor) where the field magnitude remains nearly unchanged during the polarization.

Hence, the presence of 10°C temperature drop within the XLPE insulation of the cable induces a drastic change compared with the previous situation (homogeneous temperature). Indeed, a higher amount of negative charges is expected to accumulate in the bulk of the insulating polymer when temperature gradient takes place between the inner and outer conductor. This is because the temperature drop within the insulation results in a higher gradient of conductivity along the cable radius leading to higher distortion of field compared to the situation where the temperature is homogeneous for which only the field contributes to the change in the conductivity along the cable radius.

SPACE CHARGE MEASUREMENTS

Space charge and electric field distributions obtained under -40 kV at room temperature are shown in Fig. 7. Space charge profiles (Fig. 7a) show positive charges accumulated in the bulk of the sample at 1-2 mm from anode and heterocharges build up at the close vicinity of the anode immediately after voltage application. These charge densities are small, being less than 80 mC/m³, and the steady state of space charge accumulation process seems reached after 5 h of poling. During polarization, the electric field profiles slightly deviate from the initial Laplacian field distribution due to space charge accumulation (Fig. 7b). After 7 h of polarization, the electric fields to the inner and outer electrodes are found to be approximately 13 kV/mm and 2 kV/mm. The respective field values short after voltage application were 12.8 kV/mm and 4 kV/mm so that in this case the maximum stress remains located to the inner SC. Fig. 8 shows the results when a temperature gradient of 11°C is applied across the cable insulation. In Fig. 8a, we observe that negative charges are accumulating in the middle of the cable insulation during the first hour of poling. Then, negative charges tend to move progressively towards the anode forming heterocharges after 7 h of polarization. The density of negative charges in insulation bulk is approximately -80 mC/m 3 . Fig. 8b reports changes in field profile during voltage-on step. It clearly shows the respective decrease and increase of field strength at the inner and outer SC due to the build-up of negative charges in the bulk of the insulation. After 7 h of polarization, the highest stress is located at the outer electrode and is estimated to about 10.5 kV/mm: the field is approximately multiplied by a factor 2 during the polarization stage at the outer electrode. Conversely, at the inner electrode, the field strength is approximately halved in respect to its initial value. Finally, the lowest field strength appears to be located at the middle of insulation. This is due to the configuration of the negative charges, which are accumulated adjacent to the anode (Fig 8a ), and the diverging nature of field in coaxial geometry.

Comparing experimental results obtained in absence and in presence of a temperature gradient across the insulation of the cable, it is clear that the thermal gradient has a strong effect on charges accumulation and field distribution in the cable. Without thermal gradient, a small quantity of positive charges are accumulating in the bulk of cable insulation while in presence of temperature gradient negative charges are building up adjacent to the anode forming heterocharges.

DISCUSSION

Questions that can be raised from the present approach concern the capability of conductivity measurements to predict field distortion inside HVDC cables. Deviations between model and experiments may arise for different reasons: First, conductivity measurements taken on flat specimen, with gold electrodes may not be representative of cable conditions as processing/crosslinking conditions of samples are different, the outgassing degree is different and electrodes are of different nature. Second, only some processes of charge build up are considered, i.e. the nonhomogeneous conductivity due to field and temperature gradients encountered in the cable geometry. In flat geometry as example it is well know that space charge builds up, which would not be expected with only a temperature and field dependent conductivity: this is due to space charge processes linked for example to ionic species formation and drift or to space charge limited conduction. It is therefore interesting to evaluate to what extent the modelled field distribution is verified by experiments issued from space charge distribution measurements.

In absence of thermal gradient, both simulation and measurement show a low amount of accumulated space charge and therefore a weak distortion of electric field. However, it can be noted that the amount and polarity of charges are quite different. Experimental charge density was about twice that predicted from simulation. This difference indicates that other mechanisms of space charge generation such as injection or field-assisted impurity dissociation (which are not taken into account in the simulation) actually take place in addition to charge accumulation due to the conductivity gradient.

In presence of temperature gradient, the polarity of charges are similar for both experiments and simulation. However, simulation predicts a quasi-uniform distribution of charges within the insulation bulk with an average density of about -30 mC/m 3 while experimental results show that charges accumulate preferentially adjacent to the anode with a higher density (about -80 mC/m 3 ) which leads to a difference in the distribution of field. For simulation, the field increases monotonically from the inner to the outer conductor after 7 hours of polarization under -40 kV while experimental results indicate that the minimum of field occurs in the bulk of insulation. Assumption made on the nature of electrodes is likely to be at the origin of this difference. With regard to simulation, charges are computed assuming perfect ohmic contacts meaning that charges can freely enter or leave the insulating material. However, the formation of heterocharges observed experimentally seems to indicate the possible presence of a potential barrier toward charge extraction at the external SC. Nonetheless, one can obtain a rather good estimation of electrode fields during steady state using simulation based on space chargeinduced conductivity gradient.

A last comment concerns the kinetics to approach the steady state regime. In the configuration of temperature gradient, the time constant for charge redistribution (=/) is of the order of 3h at 50°C (inner SC) and 6.7h at 40°C (outer SC) (both are given for a field of 10kV/mm). Referring to the field distribution of Fig. 8b, it seems that effectively the field distribution tends to a stable distribution after polarization for 7h.

CONCLUSION

An expression of conductivity versus temperature and electric field has been derived based on charging current measurements on crosslinked polyethylene constituting the insulation of a MV-HVDC cable. The time dependence of the field strength distribution in the MV cable insulation has been predicted in absence or presence of a temperature gradient. Simulated charge and field distributions are compared to those obtained experimentally by PEA method under -40kV applied to the conductor in absence and presence of a thermal gradient. The results show that with temperature gradient across the radius of the cable insulation, space charge builds up within the insulation due to the non-homogeneity of the insulation conductivity. The field redistribution is very significant, the maximum in local field moving to the outer SC screen, as expected from the simulation. It is shown that the field distortion is ever pronounced than predicted due to heterocharge build close to the outer SC screen.

Fig. 1 :

 1 Fig. 1: Schematic representation of the PEA Cable system.

Fig. 2 :

 2 Fig. 2: Flow chart of the used deconvolution technique

Fig. 4 :

 4 Fig. 4: Experimental current-field plots of XLPE at different temperatures. Solid lines are fits to Eq.4.

Fig. 5 :

 5 Fig. 5: Computed field distribution at different times in model cable stressed under -40kV at 22°C.

Fig. 6 :

 6 Computed space charge (a) and field (b) distribution at different times in MV cable under-40kV in presence of thermal gradient.

Fig. 7 :

 7 Space charge (a) and electric field (b) profiles vs. time in MV cable under -40kV at 22°C.

Fig. 8 :

 8 Space charge (a) and electric field (b) distribution in model cable at different times under -40kV in presence of thermal gradient.


Séverine Le Roy
email: severine.leroy@laplace.univ-tlse.fr

Thi Thu

Nga Vu
email: vu@laplace.univ-tlse.fr

Christian Laurent
email: christian.laurent@laplace.univ-tlse.fr

Gilbert Teyssedre
email: gilbert.teyssedre@laplace.univ-tlse.fr

Modelling charge generation and transport in solid organic dielectrics under DC stress

Keywords: Model, space charge, transport, plane parallel geometry, cable geometry, mobility, hopping, temperature and field dependent phenomena

Models describing the behavior of organic dielectric under electrical and thermal stress would be extremely useful in the development of more compact, reliable and ecofriendly systems for conversion and transport of electric energy. With information on conductivity and permittivity, it is possible to predict the behavior of a dielectric as a function of temperature and field. These data are however not sufficient and space charges need to be taken into account. Hence, models including charge generation and transport function of time and space need to be developed. It is particularly the case when the object under study is a cable, as the electric field and temperature are no more homogeneous within the insulation. In the present paper, simulations have been performed using a one-dimensional cylindrical model, featuring charge injection, transport, trapping, detrapping and recombination, in order to highlight the effects of geometry, temperature and temperature gradients on the material behavior.

INTRODUCTION

Challenges in the field of solid insulating materials are linked to the development of more compact, reliable and eco-friendly systems for conversion and transport of electric energy. Polyethylene has advantageously replaced oil-impregnated paper as insulation in high voltage AC cables, and is now the favorite insulating material for HVDC technologies for electrical energy transport. HVDC cables are used in critical applications such as long distance transmissions, interconnexion of asynchronous networks, power link for isolated regions (islands, offshore platforms). For these large-scale electrical networks, the reliability needs to be maximal. It is however well known that space charge may build up in the bulk of the insulation under various temperatures and electric fields, due to charge injection at electrodes or to dissociation of residues and impurities inside the bulk. These charges may affect the reliability of the dielectric, by enhancement of the electric field as an example. The presence of cable joints, and hence interfaces between insulating materials having different electrical properties is also likely to favor charge accumulation at the interfaces. Hence, the presence of charges remains the major problem in the increase of reliability of such electrical systems. Models would be extremely useful to predict the material behavior, and to answer to the main industrial issues. The easiest way to simulate the insulating material behavior is to model its macroscopic behavior with the help of its conductivity, which is function of temperature and field. This kind of model, based on conductivity measurements, is really useful, as it is able to describe the space charge accumulation at an interface between two materials having conductivities and permittivities that vary differently with temperature and field [START_REF] Jeroense | Electric Fields in HVDC Paper-Insulated Cables[END_REF][START_REF] Vu | Electric field profile measurement and modeling in multi-dielectrics for HVDC application[END_REF]. The simulated results using such models are consistent when compared to accumulated space charge measured on plane parallel samples [START_REF] Vu | Electric field profile measurement and modeling in multi-dielectrics for HVDC application[END_REF]. However, the simulated results deviate from the experimental space charge behavior when considering cable geometry and taking into account the real temperature gradient inside the dielectric [START_REF] Vu | Investigating the effect of temperature gradient on field distribution in polymeric MV-HVDC model cable through simulation and space charge measurement[END_REF]. This could be due to the fact that processes such as charge generation and conduction are not taken into account. Hence, more complex models also need to be developed, that take into account charge generation and transport mechanisms. These models generally consider the generation of charges at the electrodes, transport and trapping inside the bulk and recombination. They are one dimensional, function of the thickness of the plane parallel sample, and time dependent. Only few attempts have been made to develop these transport models for cable geometry [START_REF] Roy | Simulation of Space Charge Distribution in XLPE insulated Cable with a Temperature Gradient[END_REF]. This work focuses on modeling the material electrical behavior in a cylindrical configuration, and to investigate the evolution of measurable variables such as space charge with temperature and field gradient, which are present in a cable system.

CHARGE TRANSPORT MODEL

The physical model is based on the scheme represented in Figure 1 [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF]. Electronic carriers are provided by injection at the electrodes. Conduction takes place via a hoping mobility, traducing the transport of carriers through shallow levels that are related to the structural disorder of the polymer. Deep trapping is described using a unique level of deep traps for each kind of carriers, from where charges can escape by thermal activation. Recombination processes involving mobile and trapped carriers are taken into account. For sake of simplification, internal charge generation (dissociation) is not included in the model.

Under service conditions, the Laplacien field inside the dielectric is not uniform, due to the cylindrical geometry. Moreover, the current flowing in the conductor produces an increase in the temperature at the vicinity of the inner electrode, leading to a non-uniformity of the temperature inside the dielectric. Figure 2 shows an example of field and temperature distribution in a 1.5 mm cable (minicable) submitted to an applied field of 40 kV/mm, where the inner temperature is at 60°C, whereas the outer electrode is at 20°C (the geometry will have a non negligible effect on the mechanisms taken into account in the simulations). A schematic representation of the cylindrical system used for the discretization of the dielectric is given in Figure 3.

Based on this representation, the time and space dependent equations describing the behaviour of charge carriers in cylindrical geometry are the following, neglecting diffusion:

) , ( ) , ( 1 ) , ( t r s r t r j r t t r n a a a = ∂ ∂ + ∂ ∂ (1) ε ρ ) , ( ) , ( 1 t r r t r V r r r - =       ∂ ∂ ∂ ∂ ( 2
) ) , ( ). , ( ). , ( ) , ( t r E t r n t r t r j a a a µ = (3)
where na represents the carrier density (electrons and holes, trapped or mobiles), t is the time, r is the radius. ja is the current density referring to the mobile carriers, and sa(r,t) is the source term, representing all the mechanisms inducing a variation of space charge, and not linked to transport. V is the voltage, ρ is the net charge density, ε is the permittivity. µa is the mobility of the mobile carrier, and E is the electric field.

Transport is described using a hoping type mobility of the form, for each carrier:

                = ) ( 2 ) , ( sinh ) ( exp ) , ( 2 ) , ( , , r T k d t r eE r T k ew t r E d t r B B h e h e µ µ ν µ ( 4
)
where d is the distance between traps, and e the elementary charge. ν is the attempt to escape frequency, kB the Boltzmann constant, wµe and wµh the hopping barrier height for electrons and holes respectively. T is the temperature, function of the radius.

Deep trapping is described through a single trapping level for each kind of carrier. Charge carriers have a given probability to escape from traps by overcoming a potential barrier, wtr. This is accounted for by the detrapping coefficients Di of the form:

        - = ) ( exp . ) , ( , , r T k ew t r D B trh tre h e ν (5)
The recombination of carriers is accounted for considering different coefficients S i for the several electron-hole pairs. The only source of charges is the injection of electronic carriers at each electrode, which is function of the sign of the electric field at the electrode. Prior to electric field application, we consider that the Fermi level of the metal has the same energy value as the Fermi level of the dielectric, i.e. that both materials are in thermodynamic equilibrium. This means that prior to electric field application, there is a current flowing from the metal towards the dielectric, counterbalanced by a current flowing from the dielectric towards the metal. When the electric field is applied, this 'reverse current' is still present, even if the net current is clearly directed towards the dielectric. In order to include this mechanism in the modeling, a modified Schottky law has been used for charge injection at each electrode, of the form:

        -                 - = 1 4 ) , ( ) ( exp ) ( exp ) ( ) , ( 0 , 2 , r B B h e h e t r eE r T k e r T k ew r AT t r j ε πε (6)
where r=ri for the anode and r=ro for the cathode.It should be noted that there is no extraction barrier at the electrodes, so the extraction fluxes for holes at the cathode and for electrons at the anode follow the transport equation.

In the modeling, the temperature distribution is of the form:

( )

        -         - = i o o i i i r r r T r T r r r T r T ln ) ( ) ( ln ) ( ) ( (7)
where T(ri) and T(ro) are the temperature at the inner and the outer electrode, respectively.

RESULTS AND DICUSSION

In this section, simulations have been performed on a cable, the inner radius (ri) being 1.9 mm, the outer radius (ro) being 3.4 mm, providing an insulation thickness of 1.5 mm. Note that the anode is the inner electrode, and the cathode is the outer electrode. Hence, a positive voltage of 60 kV is applied at the inner electrode. The parameters used for the simulations are given in Table 1, and are the ones that have been optimized from measurements achieved on space charge, current and electroluminescence, for a plane parallel LDPE [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF].

Geometry dependence

Figure 4 compares the simulated net charge density at the beginning and after one hour of polarization, for a plane parallel geometry [START_REF] Roy | Description of charge transport in polyethylene using a fluid model with a constant mobility: fitting model and experiments[END_REF] and a cylindrical geometry, at 60°C, for the parameters listed in Table 1. Note that the influence charges (capacitive + image charges) have not been added to the simulated profiles in order to better compare the different results. After voltage application (10s), positive charges are detected at the vicinity of the anode in the plane parallel geometry, whereas positive charges have already penetrated deep inside the material for the cylindrical geometry.

Negative charges have also been injected at the cathode side for both cases, and remain at the vicinity of the electrode. After one hour of polarization, the plane parallel sample shows positive charges in the first half of the dielectric, and negative charges in the second half, these charges being considered as homocharges. Positive homocharges, enhanced by the electric field nonuniformity, dominate the space charge profile for the cylindrical geometry at the end of the polarization. Negative homochages injected at the cathode are observed only at the vicinity of this electrode. The net charge density for both the plane parallel and the cylindrical geometry are of the same amount, and the profiles at the end of the polarization remain close. This is particularly surprising as the electric field distribution is completely different at the beginning of the polarization (Figure 5 for planar geometry and Figure 6 for cylindrical geometry). However, the electric field profiles are comparable at the end of the polarization, as if the space charge distribution inside the dielectric counterbalances the electric field distortion due to cylindrical geometry.

m 3 .C -1 .s -1 m 3 .C -1 .s -1 m 3 .C -1 .s -1 m 3 .C -1 .

Temperature dependence

Simulations have been performed for a cylindrical geometry in isothermal conditions, in order to illustrate the temperature dependence of charge transport and injection in a cable. Figure 7 shows the comparison of the space charge profiles obtained at room temperature (25°C) and at 60°C for different times. At t=10s, it is possible to observe a higher injection of positive charges at the anode for 60°C compared to 25°C, due to the temperature dependence of the Schottky law. Positive charges remain close to the anode at 25°C, whereas they have already reached the middle of the dielectric for a higher temperature. The charge penetration is easier to observe on the electric field profiles (Figure 6 for 60°C and Figure 8 for 25°C).

Thermal gradient

A temperature gradient has been added to the modelling, on the basis of equation ( 7), and the outputs are compared to an isothermal case. The temperature gradient along the insulation is considered here as being in the steady state. Parameters used for these simulations are those of Table 1. Whatever the temperature gradient, form 0 to ∆T=35°C, only positive charges are observed in the sample bulk after one hour of polarization (Figure 9). These positive charges, injected at the inner electrode, transport through the dielectric and are extracted at the cathode. The higher is the temperature gradient, the higher is the amount of positive charge inside the bulk. This is due to the temperature dependence of charge injection. When the temperature is low at the outer electrode (∆T=35°C), a small amount of negative charges are injected compared to the positive charges injected at the anode. These negative charges are also slowed down from spreading inside the bulk material due to the low electric field. However, the space charge profiles for each temperature gradient remain close. This is not the case for the electric field distributions (Figure 10), which vary to a large extent depending on the temperature gradient. After one hour of polarization, the electric field close to the inner electrode remains low, disabling charge injection from this side. On the contrary, positive charges, already injected at the anode and distributed in the sample, imply an increase of the electric field at the outer electrode, of the order of 1.5 compared to the Laplacien field. Actually, the temperature values and temperature gradients applied here are by no way exaggerated situations in respect to practical ones: inner temperature of the order of 90°C and gradients of the order of 50°C are commonly encountered, depending on the size of the HVDC cable. The isothermal case leads to the situation where the stress is the most homogeneous along the radius of the cable. For the two other cases (∆T=20 and 35°C), the cumulative effect of a high electric field and a high temperature at the inner electrode, implies a higher charge injection and mobility of the positive charges compared to the negative ones. It can be seen that the field at the outer SC may even exceed the Laplacien field at the inner SC. It must be stressed out that the field and charge distribution after one hour are not necessarily representative of a steady state. A simulation of a negative applied voltage at the inner electrode would lead to the same kind of result, i.e. a dominance of the negative charges inside the dielectric, and a global increase of the electric field at the outer electrode, although the optimized trapping depth for hoping transport is lower in the case of holes compared to the one for electrons. Modeling the material behavior under electrical and thermal stress, with a particular geometry such as a cable system is not an easy task, and the simulated results obtained in the present paper may not account for all physical processes at play in real situation of XLPE insulated cables (presence of residues that may affect charge transport and/or could be a source of internal charge generation). However, when modeling the macroscopic behavior of the dielectric in the same conditions, using the field and temperature dependence of the conductivity and permittivity of the material, a field strength decreasing with increasing cable radius is always obtained, whatever the polarization time [START_REF] Vu | Investigating the effect of temperature gradient on field distribution in polymeric MV-HVDC model cable through simulation and space charge measurement[END_REF]. This is not the case experimentally, and the electric field profiles simulated with a more complex transport model are closer to the experimental observations.

CONCLUSION

A plane parallel one dimensional model featuring charge injection and transport has been modified to take into account the cylindrical geometry of a cable specimen, in order to see the effect of the geometry, the temperature and the temperature gradients on the simulated space charge profiles. In the absence of temperature gradient, the space charge tends to counterbalance the nonhomogeneity of the electric field. However, the presence of a temperature gradient clearly shows a difference in the electric field profiles, leading to a dramatic field enhancement of the electric field with the temperature gradient. The space charge patterns are however comparable, whatever the case under study (isothermal or not isothermal conditions).

The parameters of the model have not been optimized so as to fit experimental data on XLPE insulated cables. It is still noticeable that such a transport model remains necessary when trying to properly describe the behavior of an insulating material in a cable system under electrical stress, as the simulated results in the case of a transport model are closer to reality compared to a macroscopic model featuring only the field and temperature dependence of the material conductivity and permittivity. However, both descriptions remain useful, as macroscopic models are powerful when describing the charge behavior in a sandwiched system, where more than one dielectric exists. Transport models are still difficult and rather long to implement and parameterize in this case.

Figure 1 :

 1 Figure 1: Schematic representation of the charge transport model.

Figure 2 :

 2 Figure 2: temperature and field dependence upon the cylindrical geometry

1 Figure 3 :

 13 Figure 3: schematic representation of the onedimensional cylindrical system used in the simulations.

Table 1 .Figure 4 :

 14 Figure 4: Net charge density function of the distance from electrode for a) a planar geometry, and b) for a cylindrical geometry. Applied voltage=60kV, T=60°C.

Figure 5 :

 5 Figure 5: Electric field for different simulation time for a planar geometry. Applied voltage=60kV, T=60°C.

Figure 6 :

 6 Figure 6: Electric field for different simulation time for a cylindrical geometry. Applied voltage=60kV, T=60°C.

Figure 7 :

 7 Figure 7: Comparison of the simulated space charge profiles for a cylindrical geometry at 25°C and at 60°C.

Figure 8 :

 8 Figure 8: electric field for different times for a cylindrical geometry. Applied voltage=60kV, T=25°C.

Figure 9 :

 9 Figure 9: space charge profiles for different temperature gradients after one hour of polarization.

Figure 10 :

 10 Figure 10: electric field profiles for different temperature gradients after one hour of polarization.