Physical and Numerical Modelling for Bipolar Charge Transport in Disorder Polyethylene Under High DC Voltage

Transcription

1 International Journal on Electrical Engineering and Informatics - Volume 2, Number 4, 21 Physical and Numerical Modelling for Bipolar Charge Transport in Disorder Polyethylene Under High DC Voltage I. Boukhris, E. Belgaroui, A. Kallel Laboratoire des matériaux composites céramiques et polymères (LaMaCoP) Faculté des sciences de Sfax BP 85 Sfax 3 Tunisie. imed_boukhris@yahoo.fr Abstract: This paper reports on validation of a numerical model for both transient and steady bipolar charge transport, trapping and recombination in polymeric insulators, such as low density polyethylene. The numerical model is based on the high precision Runge Kutta method for determining mobile and trapped charge local density within the sample, during the application of a DC voltage. In addition, we have applied the finite element method considered as the most general one, which can be applied to any sample shape and boundary conditions. It is applied to resolve Poisson s equation, thus providing the potential and its gradient within the sample and at the dielectric electrode interfaces. The principal results show the appearance of charge packets for the first time in modeling works although they have long been reported in experimental studies on polyethylene materials. The conditions for the appearance of such packets are discussed. For the net space charge density and the conduction current, the dynamics are in a good agreement with those observed in some experimental works during the transit time of the bipolar carriers. The model is already numerically validated for the space charge densities and the electric field distribution under low DC voltage. Keywords: Polymeric insulators; Bipolar charge transport; high DC voltage; model; space charge packets. 1. Introduction Insulating polymers have been widely used in various electric apparatus such as electrical cables because of their excellent, thermal and mechanical properties. The low density polyethylene (LDPE) is one of the principal polymers used for the cable insulation in the electrical transport [1, 2]. The existence of the space charges in polymeric insulating materials has been affirmed to cause very serious damages in electrical systems especially under high applied DC voltages [3, 4]. For these problems, different techniques for the space charge measurements were developed in order to understand and overcome their effects on the electrical properties such as electrical breakdown and conduction mechanism [5, 6]. During these last years, the theoretical modelling and simulation have been integrated for better understanding the space charge dynamics [7-12]. Previous works showed the existence of two charge dynamics corresponding to mobile and trapped carriers at high and low DC applied voltages, respectively [12, 13]. One of the important parameters in studying the space charge dynamics in insulators is the transient and steady state currents. Their evolution can determine the conduction mechanism involved in the insulators under the applied voltage. According to our knowledge, few numerical works are presented especially for the transient current of the bipolar charge transport in the insulating polyethylene. Recently, Baudoin et al. [13] have published a bipolar model for only the steady state current in LDPE sample. In this particular case, their results also show the existence of two aspects of charge accumulations under high and low voltages. Indeed, under low voltages the external current increases at the beginning of the injection and then decreases until it reaches a steady state, while under high voltage the current gradually increases without decreasing. These results constitute the limits of those obtained by our model, in the present work, at the end of the transient state. Received: April 1, 21. Accepted: December 16,

2 I. Boukhris, et al In this paper, we present new numerical results carried out by our model for both transient and steady state regimes. For high DC voltage, all the results of the space charge density and the electric field show the effects of the space charge packets. These packets, also called lobes, are previously shown on the net space charge densities in Alison s works [14, 15]. Principally, for asymmetrical bipolar transport, we show new aspects of space charge limited current between transient and steady states under high and low DC voltages. Indeed, the first aspect indicates the apparition of the space charge limited current peaks under low DC voltage. These peaks disappear under high DC voltage. The second aspect shows conduction current oscillations that occur during transient state under high DC voltage. These oscillations are also observed on the experimental conduction current profiles obtained by Alison [14, 15]. 2. Hypothesis and computational procedures A polyethylene film is assumed to be sandwiched between two electrodes under DC applied voltage, i.e. parallel-plane configuration. Thus we can admit that all physical mechanisms act mainly in the direction of sample thickness. The contact between electrode and polyethylene is supposed to be perfect, and the Schottky model is adopted for injection of current density. The polyethylene is kept under uniform temperature, and then the electronic carriers are considered to have an effective constant mobility. In addition, shallow and deep traps are distributed in the bulk of the polyethylene between the electrodes. According to Quirke et al. [16, 17], the shallow traps have a very weak residence time of about 1-12 s and the deep traps have an infinite residence time. For these reasons, we consider that the shallow traps are only integrated in transport mechanism and the deep traps contribute only to the charge accumulations. In this work, all the calculations were realized for a free additive polyethylene film (LDPE). 3. Physical equations, initial and boundary conditions 3.1 Equations for the electric and the space charge densities The physical model regroups Poisson's, the continuity and the transport equations. Poisson s equation, the initial and the boundary conditions are given as follows: 2 Vt ) ρ(x, + = 2 x ε < x < D (1) grad ( V ) = - E( x, (2) where V, E, and ρ(x, are the local potential, the electric field and the net charge density, respectively. The initial condition for free additive polyethylene film is: ρ ) = (3) The boundary conditions are writing as follows: ΔV = VC V A (4) V (, t > ) = V C (5) V ( D, t > ) = (6) D V A E dx =ΔV (7) V where V C and A are the potentials at the cathode and anode, respectively, and D is the dielectric thickness. 314

3 Physical and Numerical Modelling for Bipolar Charge Transport The continuity equation, with trapping and recombination term sources, as well as the transport equation are given as follows: ρ ( e, h) μ j + t (x, = x ( e, h) S t( e, h) + S r ( e, h) (8) j where (x, = μ ρ E(x, (9) ( e, h) e, h ( e, h) μ μ e, h is the mobility of carrier, ρ eμ and ρ hμ j e, h) (x, ) S are the densities of the mobile electrons and holes, respectively. ( is the flux of the mobile electrons or holes. S t ( e, h ) t and r ( e, h) are the trapping and the recombination source terms for the electrons or the holes [12]. The net charge density ρ is locally composed of the mobile and the trapped carriers. This density is defined as follows: ρ = ρ + ρ ρ ρ (1) hμ ht eμ where ρet and ρ ht are the densities for the trapped electrons and holes, respectively. The boundary conditions for the injected charges are represented by the Schottky law: where 2 w (,) ei e eet j(, e = A T exp( ) exp( ) kt kt 4πε (11) 2 w (, ) hi e ee Dt j(d, h = A T exp( ) exp( ) kt kt 4πε (12) j e (, and j h (, are the fluxes of electrons and holes at the cathode and anode, respectively, T is the temperature of the sample, A is the Richardson constant and it is equal to A m -1 K -2. w ei and w hi are the injection barriers for the electrons and the holes, respectively Equations for the recombination rate, the current densities The recombination rate equations The total recombination rate equation is as follows: S t ) = S t ) + S t ) + S t ) (13) r r(e μ, h r(et, h μ ) r(et, h The recombination rate equation for mobile electron and trapped hole is written as: S t ) = s ρ (x,t ) ρ (x,t ) (14) r(e μ, h t ) 1 e μ h t where s 1 is the recombination coefficient for mobile electron and trapped hole. The recombination rate equation for trapped electron and mobile hole is represented by: S t ) = s ρ (x,t ) ρ (x,t ) (15) r(et, h μ ) 2 e t h μ et 315

4 I. Boukhris, et al where s 2 is the recombination coefficient for trapped electron and mobile hole. The recombination rate equation for trapped electron and trapped hole is as follows: S t ) = s ρ (x,t ) ρ (x,t ) (16) r( et, ht ) e t h t where s is the recombination term source for trapped electron and trapped hole Current densities The instantaneous local conduction current density for the mobile electron and hole is written as follows: jeμ, hμ(x, = ( μ e ρ eμ + μ h ρ hμ(x, ) E( x, (17) The instantaneous local displacement current density is: E(x, j(x, d = ε t The external current density, obtained by numerical integration, is: D J ( = ( je, hμ + j ) dx (18) μ d (19) Because of the stationary boundary equation (7), the following condition is always satisfied: D j dt ) dx = (2) All the model parameters used in this work are regrouped in table 1. Table 1. Parameters of the model Parameters Fixed values Parameters Fixed values Coefficients of trapping Traps density B e (electrons) s -1 dpe (electrons) 1 C m -3 B h (holes) s -1 dpt (holes) 1 C m -3 Coefficients of recombination Injection barriers S (trapped electron-trapped hole) m 3 C -1 s -1 w ei (electrons) 1.2 ev S 1 (mobile electron-trapped hole) m 3 C -1 s -1 W hi (holes) 1.2 ev S 2 (trapped electron-mobile hole) m 3 C -1 s -1 Temperature 25 C S 3 (mobile electron-mobile hole) Neglected (=) Applied voltage Δ V From 1 to 5 kv Mobilities Time step.1 s μ e (electron) m 2 V -1 s -1 Sample thickness 15 µm μ h m 2 V -1 s -1 Spatial discretization variable (hole) 316

5 Physical and Numerical Modelling for Bipolar Charge Transport 4. Results and discussions 4.1. Results for the charge densities Figures (1.a, 1.b) show the net space charge densities in the bulk of the sample for different instants, and under 3 kv and 5 kv high DC voltages, respectively. Before the steady state at 2 s, all profiles show the appearance of the charge lobe aspects which are previously observed by Alison [14, 15] and are also called charge packets. A general observation shows that the lobes of the electrons and holes, produced in the vicinity of cathode and the anode respectively, propagate one towards the other until the middle of the bulk where they are recombined. During their propagation, the intensity of each packets decreases gradually under the effect of the deep trapping, and their velocities increase with the applied DC voltage. Another particular important observation indicates the appearance of the alternative space charge zones in the middle of the sample bulk, with negative and positive signs, at 11 s and 7 s under 3 and 5 kv respectively. These zones expand with the increase of the DC voltages as it is seen in figure (1. b) at 7 s. They will constitute hetero-charge zones for the sample as it is known in the literature [5, 6]. However, at the steady state, the space charge packets disappear as it is indicated in the space charge density profiles of figures (1.a, 1.b) at 2 s. Net charge density (C/m 3 ) s 3 s 6 s 11 s 2 s 3 kv x 1-4 Figure 1.a. Net charge density profiles at different times under 3 kv DC voltage. 3 Net charge density (C/m 3 ) s 2 s 3 s 7 s 2 s 5 kv (b) x 1-4 Figure 1.b. Net charge density profiles at different times under 5 kv DC voltage. 317

6 I. Boukhris, et al To more explain the previous aspects, we have illustrated in figure (2. a) the instantaneous profiles of the absolute values for the mobile electron densities under 5 kv. It is clearly observed that the intensification occurs close to the cathode (depth=) on the profile at the instant 1 s. In figure (2. b), we show that, under high DC voltage, the mobile charge densities are dominant compared to those of the trapped charges. Indeed, we can indicate that the ratio between the mobile and trapped electrons densities can reach a factor 5 at 3 s. Mobile electron density (C/m 3 ) s 4 s 1 s 25 s 3 s 5 kv x 1-4 Figure 2.a. Mobile electron density profiles at different times under 5 kv DC voltage. Trapped electron density (C/m 3 ) s 2s 3s 25s 3s 5 kv (b) x 1-4 Figure 2.b. Trapped electron density profiles at different times under 5 kv DC voltag 4.2. The recombination rate density under high and low DC voltage Figures (3.a, 3.b) represent the instantaneous profiles of the recombination rate densities in the sample bulk under 4 and 15 kv, respectively. At the steady state, the profiles change their concavities according to the intensity of the applied DC voltage. Indeed, under high DC voltage (4 kv), the profiles turn their concavities toward the high absolute values of the recombination rate density as for example for the profile at 5 s in figure (3.a). Under low DC voltage, the concavities for the steady state profiles turn toward the low absolute values as 318

7 Physical and Numerical Modelling for Bipolar Charge Transport shown in figure (3.b). These results are in a good agreement with those obtained by the steady model of Baudoin et al. [13]. During the transient state, our model allows us to follow the transient profiles, and it can especially indicate the effect of the presence of the space charge packets on the shape of the recombination rate profile. This effect induces the existence of two peaks as seen at 9 s under 4 kv. These peaks are attributed to the mobile charge densities which are considered as dominants in the total recombination rate between electrons and holes. At steady state, the recombination rate is highly confined close to the electrodes under high DC voltage, whereas it is greatly confined in the middle of the sample when the low DC voltage is applied, as seen in figures (3.a) and (3.b), respectively. Recombination rate density (C.m -3.s -1 ) kv 6 s s 2 s 5 s x 1-4 Figure 3.a. Recombination rate density profiles at different times under 4 kv DC voltag Recombination rate density (C.m -3.s -1 ) (b) 15 kv 2s 6s 8s 1s x 1-4 Figure 3.b. Recombination rate density profiles at different times under 15 kv DC voltage 319

8 I. Boukhris, et al 4.3. The external current densities under low and high DC voltages In figures 4-(d), the profiles show the instantaneous external current density J under 1-4 kv applied DC voltages. These profiles indicate the existence of two evolutions of the external current under low and high DC voltages, as it is observed in figures 4-(a, b) and 4-(c, d), respectively. Under low DC voltages, the external current density increases rapidly until its maximum and then decreases before the steady state establishment. As it is shown in figure 6, the maximum of the current density is composed of two peaks which indicate the bipolar character of the charge transport. Indeed, the first and the second peaks (P1 and P2) occur at the end of the transient times of the holes and the electrons, respectively. In this figure 6, the mobilities of the holes and the electrons are equals to and m 2 V -1 s -1, respectively. (c) Figures (5.a, 5.b) show the electric field distribution at different instants under 3 and 5 kv, respectively. The first observation of the profiles shows the existence of an alternative shape in the bulk of the sample at 11 s and 7 s under 3 and 5 kv DC voltage, respectively. These shapes are associated to the alternative space charge zones previously observed on the net space charge densities of figures (1.a, 1.b). The second observation concerns the appearance of the interfacial electric field intensification, as it is clearly shown especially on figure (5. a) after 6 s. Indeed, just after the application of the high DC voltage, the interfacial electric field decreases, in its absolute value, until 6 s under 3 kv (3 s under 5 kv), and thereafter it increases until the steady state. This increase induces an intensification of the current as it is observed on figures (4. c) and (4. d). (b) (d) Figure 4. External current density J versus time under: 1 kv dc stress, (b) 15 kv dc stress, (c) 3 kv dc stress and (d) 4 kv dc stress. Figures (5.a, 5.b) show the electric field distribution at different instants under 3 and 5 kv, respectively. The first observation of the profiles shows the existence of an alternative shape in the bulk of the sample at 11 s and 7 s under 3 and 5 kv DC voltage, respectively. These shapes are associated to the alternative space charge zones previously observed on the net space charge densities of figures (1.a, 1.b). The second observation concerns the appearance of the interfacial electric field intensification, as it is clearly shown especially on figure (5. a) after 6 s. Indeed, just after the application of the high DC voltage, the interfacial electric field decreases, in its absolute value, until 6 s under 3 kv (3 s under 5 kv), and thereafter it increases until the steady state. This increase induces an intensification of the current as it is observed on figures (4. c) and (4. d). 32

9 Physical and Numerical Modelling for Bipolar Charge Transport Electric field (V/m) -1.4 x kv 1s 6s 11s 5s x 1-4 Figure 5.a. Electric field profiles at different times under 3 kv DC voltage -1.5 x s 3 s 7 s 15 s Electric field (C/m 3 ) kv x 1-4 Figure 5.b. Electric field profiles at different times under 5 kv DC voltage. 321

10 I. Boukhris, et al 1 x 1-1 P1 P2 15 kv 9.5 External current density (A.m -2 ) Time (s) Figure 6. External current density versus time under 15 kv dc stress. e,h The transient time t r (e,h) of the electrons and holes is calculated from the position x f of e,h the mobile charge front equations [18]: x f ( t ) = α e,h ( μe, h ΔV t ), where α e, h is the coefficient of proportionality. From these equations, the transient times of holes and electrons are, respectively, equals to and s as it is shown in figures (7.a, 7.b). 1.5 x x 1-4 Position of mobile holes front (m) kv tr = s Position of mobile electrons front (m) kv tr = s (b) Time (s) Time (s) Figure 7. Hole position fronts versus time under 15 kv DC voltage. (b) Electron position fronts versus time under 15 DC voltage. Under high DC voltages (figures 4-c and 4-d), the external current density increases gradually until the establishment of the steady state. The decrease of the current do not occurs because the injection is very important under high DC voltage. Indeed, previously in section 4.1, we have indicated that the high voltage induces an increase of the interfacial electric field. This increase intensifies the current densities injected, as it can be shown in the Schottky s laws (equations 11 and 12). This intensification of the injection can largely compensate the 322

11 Physical and Numerical Modelling for Bipolar Charge Transport reduction of the mobile charges due to the recombination and the extraction charges at the electrodes. The consequence of the mobile charge compensation indicates that the transient external current reaches the steady state without decreasing Conduction and displacement currents under dc voltage Figures (8.a, 8.b) show the instantaneous profiles of the conduction and the displacement currents, respectively, under 3 kv. Just after the application of this high DC voltage, all the profiles for both conduction and displacement currents show the oscillation during the transient evolution of these currents in the sample bulk. When the steady state is established, all oscillations disappear. At this moment, the conduction current tends to a maximum constant value A/m 2 as seen in figure (8.a), and the displacement current vanishes in order to satisfy the physical condition represented by the equation (2). 1.8 x Conduction current density (A.m -2 ) s 5 s 8 s 1 s x 1-4 Figure 8.a. Conduction current density profiles at different times under 3 kv dc stress. 6 x 1-5 Displacement current density (A.m -2 ) (b) 2 s 5 s 8 s 1 s Depth(m) x 1-4 Figure 8.b. Displacement current density profiles at different times under 3 kv dc stress. 323

12 I. Boukhris, et al Conclusion The symmetrical model confirms the existence of two dynamics of charges which depend on the applied voltage. For the high voltages, dynamic implicates the mobile charges, whereas for the weak voltages, the trapped charges control this dynamics. More particularly, for the high voltages, the model brings to light lobes of charges observed for the first time in modelling although they were already shown in experiments. These packets of charges generate the appearance of zones of alternation of electrons and holes in the centre of the sample and they are increasingly significant that the tension applied increases. These zones of alternations also appear in several experiments on polymers and are called zones of hetero charges. We chose to show in the second validation the effect of the voltage applied to measurable quantities such as the current. Indeed, the principal results show that our model reproduced two significant aspects already observed in several former works. The first aspect concern to the space charge limited current which shows in the bipolar case the appearance of two peaks on the current which are associated respectively with the two types of carriers (rapid and slow). Under high voltage, the peaks disappear and the current continues to increase until reaching its stationary state. References [1] L. Deschamp, C. Caillot, M. Paris, J. Perret, Rev. Gene. Elect. 5 (1983) [2] H. Auclair, B. Dhuicq, E. Favrier, Rev. Gene. Elect. 3 (1988) [3] E. Ridder and O.B. Monsen, Jicable 99 (1999) [4] A. Jenni, U. Rengel, A. Meier, J. Osley, Jicable 99 (1999) [5] L. A. Dissado, J.C. Fothergill, Electrical Degradation and Breakdown in Polymers (Peter Peregrinas Ltd. London). [6] R. Coelho, B. Aladenize, Les diélectrique, Ed. Hermès, Paris, [7] J. M. Alison, R. Hill, J. Phys. D: Appl. Phys. 27 (1994) [8] S. Le Roy, P. Segur, G. Teyssedre, C. Laurent, J. Phys. D: Appl. Phys. 37 (24) [9] G. Hen, S. Han Loi, Mater. Res. Soc. Symp. 889 (26) W8-6-1/6 [1] K. Kaneko, Y. Suzuoki, T. Mizutani, IEEE Trans. Dielectr. Electr. Insul. 6 (1999) [11] M. Fukuma, M. Nago, M. Kosaki, Pro. 4th Int Conf. on Properties and Applications of Dielectric Materials, Brisbane, 1994, pp [12] E. Belgaroui, I. Boukhris, A. Kallel, G. Teyssedre, C. Laurent, J. Phys. D: Appl. Phys. 4 (27) [13] F. Baudoin, S. Le Roy, G. Teyssedre, C. Laurent, J. Phys. D: Appl. Phys. 41 (28) 2536 (1pp). [14] J. M. Alison, Meas. Sci. Technol. 9 (1998) [15] J. M. Alison, CSC 3 Pro. 3th Int Conf. on Electric Charge in Solid Insulator, France, [16] M. Meunier, N. Quirke, J. Chem, Phys 113 (2) [17] M. Meunier, N. Quirke, J. Chem. Phys 115 (21) [18] Sworakowski J, Janas K, Nespurek S and Vala M, IEEE Trans. Dielectr. Electr. Insul , (26). Imed Boukhris was born in Jerba, Tunisia on 22 February, He got his master degree in physics in 28. He is a staff in Laboratoire des matériaux composites céramiques et polymères (LaMaCoP) Faculté des sciences de Sfax BP 85 Sfax 3 Tunisie and can be reach at imed_boukhris@yahoo.fr. 324