Influence of electro-mechanical stress on electrical properties of

IEEE Transactions on Dielectrics and Electrical Insulation

Vol. 12, No. 4; August 2005

791

Influence of Electro-mechanical Stress on Electrical Properties of Dielectric Polymers Jean-Pierre Crine Technology Consultant 575 Place Nicolet St-Bruno, QC J3V 1P6, Canada

ABSTRACT The influence of high electrical fields on water treeing, electrical treeing, relaxation, conductivity and charge mobility in various polymers is reviewed within the context of a molecular model. The real value of the ac field acting on water trees is questioned and it is shown that the strain induced by very large fields may affect the water tree growth in solutions with large dielectric constant. A model based on simple electrostatic and molecular parameters describes most water treeing results. Light emitted during electrical tree growth could be associated with the field-induced strain at the tip of the electrode. The strain in PET is also related to the Maxwell stress due to high dc field. The log of the charge stored in some polymers varies with the square of the field suggesting that there same phenomenon might influence charge formation under high fields. The significance of a constant mobility value is discussed in light of experiments showing that it is not constant at moderate fields in various polymers. It is shown that the only constant parameter under varying high fields is the activation volume typical of a given sample. Future work is pointed out, especially regarding the nature of the trapping sites and the influence of sample size on the activation volume. Index Terms — electromechanical stress, strain, aging, breakdown, water trees, electrical trees, mobility, space charge, activation volume, polymers

1 INTRODUCTION John Lewis was the first to draw a formal relationship between the field-induced electro-mechanical stress and some electrical properties (especially aging) of solid dielectric polymers [1-3]. The difference with the pioneer works of Stark and Garton [4] or Blok and Legrand [5], whose studies were mostly centered on very high field processes (such at breakdown [6]), is that he has shown that the relatively moderate fields used in many electrical aging studies (i.e. fields in the 10-50 kV/mm range) could generate significant degradation leading to the eventual dielectric failure [2]. Of course, mechanical aging is well known for generating bonds deformation, free radicals and rupture [7] in several polymers and also carbonyl groups in polyethylene (PE) [8]. The applied mechanical stresses being often well over the yield strength (typically, 15 MN/m2 for PE) of the polymer, it is not surprising that the molecular structure would be modified. When an electric field F is applied, it induces a Maxwell (electrostatic) stress described by ½ εoε’ F2 ,where εo and ε’ are the permittivity of free space and the relative dielectric constant of the dielectric, respectively. This stress is indeed moderate since for fields of 10-50 kV/mm applied at 220C to PE, it is in the 1 to 2.5x104 N/m2 range, i.e. 1,000 times

smaller than the yield strength of this polymer. Polymers with higher dielectric constant and smaller Young modulus may exhibit significant strains under low fields [9]. However, small stresses may generate significant molecular deformation, as demonstrated by several experimental measurements. It was shown by positron annihilation spectroscopy that the amorphous phase of several polymers [10] is compressed by dc fields and in PE the reduction could be in the 25-30% range at 20-25 kV/mm [10]. We have shown that the IR absorption doublet at 710-720 cm-1 in PE subjected to a field of 25 kV/mm applied for more than 30 mn is modified, suggesting a crystallinity (as deduced from the ratio of the two bands intensity) increase of ~10-15 % [11,12]. Sakamoto and Yahagi [13] have shown that the capacitance (that is, the sample thickness) of PE is modified by moderate fields and the deformation is permanent above ~ 100 kV/mm. A more indirect evidence based on small angle X-ray spectroscopy (SAXS) has shown that electrically-aged crosslinked PE (XLPE) samples had a different long period than unaged samples [14,15]; this result suggests that crystallinity had been modified but note that thermal aging might give similar results. Recently, Mamy et al. [16] have directly measured the strain in polyethylene terephtalate (PET) samples subjected to high fields. Thus, there is some evidence that the electro-

Manuscript received on 14 February 2005, in final form 23 June 2005.

1070-9878/05/$20.00 © 2005 IEEE

792

Jean-Pierre Crine: Influence of Electro-mechanical Stress on Electrical Properties of Dielectric Polymers

mechanical stress could modify the local molecular structure of the polymer. It is evident that intra-molecular bonds could not be broken by these moderate stresses and we have proposed that weak van der Waals bonds could be disrupted. Dissado, quite correctly, pointed out to us that the term “van der Waals bond” was inappropriate and the term “attraction bond” should be used [17]. We have proposed a model of electrical aging [11,18] and a model for water treeing [11,19] that are essentially molecular models, that is they do not refer to electronic theories to explain the aging process. Obviously, there are electrons in any dielectric and breakdown will be a typical electron avalanche phenomenon but it will last a few nanoseconds and this has little to do with the long preceding process of aging. We are also convinced that space charges are a consequence of aging- not a cause- and therefore the initiation of the degradation and charge transport processes is not affected by the space-charge field measured by various new techniques [20-25] after degradation has taken place. By the way, it is worth reminding the word of caution expressed recently by Boggs regarding the real significance of the sometime “glamorous output” of these new techniques [26]. If we are considering that stress and strain are important factors in the electrical properties of polymers, we should not forget to include the strain energy, i.e. the electromechanical energy [6], into the describing models. In this paper, we discuss the influence of the strain energy on dry and wet (water treeing) aging of PE, especially under large applied field. Simple calculations show that this is possibly not a negligible parameter in the case of water treeing. We also thought it might be useful to spend some time on what is, or what should be, the appropriate value of the actual field responsible for aging or charge displacement. For a thin flat film the applied field could be simply estimated as F equal to voltage/thickness and in a cable (or any cylindrical structure) as F = V/ri ln(ro/ri), where V is the voltage, ri and ro are the internal and external radius. For relatively thick samples the above relations do not describe well reality and a poorly known thickness effect must be included for reliable field distribution estimation. Note there is no formal theory yet although several empirical approximations are used. All that is assuming that the field is more or less homogeneously distributed at the electrodedielectric interface but we know that it is very rarely the case. Imperfections and asperities are the rule and therefore the “average” field based on voltage and thickness is, at best, a first order approximation. We have no solution to this formidable problem but we want to show that is very often difficult to determine the actual local field responsible for the observed phenomenon. We have used some well known water treeing data to illustrate our point but it can be easily transposed to several other phenomena where the interfaces are difficult to define with precision. We also explore the relations between aging and polarization, especially space charges concentration and mobility, of dielectric polymers from a molecular angle. We have therefore assumed that charge displacement is directly

dependent on the molecular deformation generated by the applied electric field. In other words, we will not discuss results in terms of energy levels or electron traps but in terms of molecular free activation energy and of activation (strained) volume. We do not pretend that this paper introduces a new theory and our intention is only to give a phenomenological interpretation of known experimental data based on molecular deformation induced by the Maxwell stress.

2 THE INFLUENCE OF HIGH FIELDS ON WATER TREEING 2.1 OUR RECENT MODEL INCLUDING THE STRAIN ENERGY The model is essentially an electro-fracture model based on Sletbak [27] and Filippini [28] earlier suggestions. If the stress (induced by the electric field F) applied on a liquid in the freevolume nanocavities is larger than the yield strength of the polymer, some weak attraction bonds will be broken and the nanocavities will grow further to form “sub-microcavities”. These cavities are more or less interconnected and when filled with water, they form a water tree. For each field cycle, a certain number n of sub-microcavities are then formed and this number will depend on the applied stress (i.e. ½ ε o ε’ҏ F2 ) and on the yield strength Y of the material. The criterion for water tree growth is that the energy dissipated in the liquid-filled cavities should be larger than the elastic energy stored in the treed volume. A water tree grows when the pressure exerted after N field cycles on an activation volume ΔV (i.e. the strained volume) filled by a liquid is larger than the yield strength of the polymer multiplied by the water-tree volume, i.e. for ½ ε o ε’ҏ F2 ΔVN ≥ Y x water-tree volume

(1)

The activation volume depends on the presence of liquid and we have assumed that it varies with the diffusion constant, i.e. ΔV varies with t 1/2. For the sake of simplicity, we have assumed that the water-tree volume would be a sphere of diameter L, which is the water-tree length; the assumed volume is then ~ L3/2. This is a gross simplification but it is reasonably close to reality and taking different shapes will not induce very different results [19]. Assuming a sphere, equation (1) becomes L ≈ (N ε o ε’ҏ ΔV t½ F2 / Y)1/3

(2)

We have shown [19] that equation (2) gives an excellent fit to existing data obtained with all sort of field, frequency, ions and testing cells provided that ΔV was adjusted for a given experimental condition. For results obtained with the Ashcraft cell, that is with water needle electrodes of radius r and distance d to the ground electrode, the field was calculated according to the well known equation popularized by Mason [29,30]

(3)

This may result in very high fields close to the breakdown strength of PE when a small needle radius is used. For example, a 7 kV voltage applied on a needle with a radius of 3 μm [29] should give 575 kV/mm. We will get back later on the degree of reliability of this value but let’s assume for the time being it is correct. At such high fields, the local strain would be very high, which means that the strain energy cannot be neglected [6]. The strain energy is given by σ2ΔV/2E , where σ is the stress (½ ε o ε’ҏ F2) and E is the Young modulus (~100-200 MN/m2 for PE at room temperature). Thus, equation (1) has to be modified to include this extra term and it becomes ½ εo ε’ҏF2ΔV[1- (½ εo ε’F2/2E)] N≥ Yxwater-tree volume (4) which gives a new relation for the water-tree length L ≈ (N ε o ε’ҏ ΔV t½ F2 [1-(½ ε o ε’ҏ F2 /2E)] / Y)1/3

(5)

Equation (5) is identical to equation (2) for stresses lower than ~10 % of the Young modulus, that is for fields lower than 200 kV/mm and for a ionic solution with a dielectric constant of 100. Increasing the dielectric constant would result in a lower field for the same stress. Therefore, the influence of the strain energy at high fields would be significant for fields above 200 kV/mm and for solutions with large dielectric constant. We may expect that solutions with large molar M, i.e. large ionic concentration, may have larger dielectric constant. On the other hand, pure water with its dielectric constant of ~ 80 would give a small strain energy value, unless extremely high fields are used. We have calculated the water-tree length with equations (2) and (5) as a function of field for trees grown in distilled water (ε’=80) [31]. The results shown in Figure 1 indicate that-as expected, the strain energy is too small to have any influence on water tree growth up to ~ 500 kV/mm; above that field value, there is a slight effect but we are getting close to the ultimate breakdown strength of PE. In Figure 2, we compare experimental results obtained with 0.1 M NaCl and various fields [28,29] with calculations made with equations (2) and (5). By trial and error, we have determined that ε’= 150 gave the best fit and we have used the same ΔV values for the two calculations. Quite clearly, equation (5) gives a better description of the experimental data and the surprising decrease of water-tree length at high fields can be explained by the influence of the strain energy. Another example of its influence at high fields is shown in Figure 3 where the results of Dissado et al. [29] obtained under different fields in 0.1M NaCl solution are compared with calculations made with equations (2) and (5). At 63 kV/mm, there is no difference between the values given by the two equations whereas the difference is very evident at 575 kV/mm. Thus, the strain energy must be included in the energy balance equation of processes involving high fields and very

793

large dielectric constants, such as in water treeing tests performed in the lab. Most electrical measurements in common dry polymers (which have low dielectric constant) are not concerned by this problem- unless measurements are performed in the MV/mm range. Of course, mechanical aging or breakdown usually performed under very large stresses and strains is affected by this property, but it is of little interest for most dielectricians.

exp

120

Eq.5

Eq. 2

100

L e n g th ( μ m )

F = 2V/r ln [1+(4 d/r)]


80 60 40 20 0 0

200

400 600 Field (kV/mm)

800

Figure 1. Length of water tree grown in PE with distilled-water needle electrodes [31] as a function of the field (equation (3)) at the tip of a 2.5μm radius needle. Solid line: calculation with equation 5; dashed line: equation 2. Dielectric constant: 80. Point-to-ground: 2 mm.

experim.

100 90 80

L (μ m)


Eq. 5 Eq. 2

70 60 50 40 30 20 10 0 0

100

200

300

400

500

600

Field at the tip of needle (kV/m m ) Figure 2. Water tree length in PE (0.1 M NaCl solution) as a function field (equation (3)) showing the influence of the strain energy (solid line) at very high fields. Different needle radii, point-to-ground: 2.5 mm. Dashed line: equation 2; solid line: equation 5 with E= 1.2x108 N/m2, ε’ = 150. The calculated result for 575 kV/mm was added for comparison purpose with the experimental results shown in Figure 3.

2.2 WHAT IS THE ACTUAL FIELD DURING WATER TREE GROWTH ? As discussed above, the field values calculated with equation (3) could be very high and one may wonder if PE could sustain 575


kV/mm for 200 hours (see Figure 3) without breakdown. Another troubling question is whether it is realistic to imagine that the field at the tip of the tree would be identical to the field at the tip of the electrode. In other word, the water tree would grow without affecting the surrounding polymer. As noted by Dissado [29], this is an ideal limit that might not represent the real situation. We have then gone to another ideal limit and assumed that the conical cavity was conductive although several authors have pointed out that this is doubtful [28,29,32]. For that purpose, we have calculated the field at the top of the water tree using equation (6) [29] F = V (1+R/D) 1/2/ R tanh-1 [1/(1+R/D)1/2]

(6)

2000 1500

.5N,200

.5N,368

1N,368

Nitta,50H

.1N,63,2kHz

.1N,575

.1N,203

1000 500 0 0.E+00

2.E-16

4.E-16 3

1/2

6.E-16

8.E-16

1/2

600

L Y/Nt

500

Figure 4. Field at the top of the water tree (equation 6) as a function of water tree length measured by Yoshimura and Fan [31,33,34] in various solutions. Distilled water results (x): 200 Hz, 5 kV; all other results obtained under 50 Hz and 8 kV.

400

300

63kV/mm,500Hz 63kV/mm,calc. 203kV/mm,1.5kHz

200

203kV/mm,Eq.5 203kV/mm,Eq.2 575kV/mm,2khz

100

575kV/mmEq.2 575kV/mm,Eq. 5 0 0

200

400

600

800

1000

t (h)

Figure 3. Water tree length in 0.1 M NaCl solution as a function of time for fields of 63, 203 and 575 kV/mm, as calculated with equation (3) [29]. Symbols: experiment, solid line; calculation with equation (5) including the strain energy; dashed line: calculation with equation 2, ε’ taken as 150 and E =1.2x108 N/m2. Note that for the 63 kV/mm case, equations (2) and (5) gave identical results.

distilled,200Hz Fie ld a t tre e top (k V /m m )

Length (microns)

Sq u ar e o f Fie ld (k V /m m ) 2

794

(Nm/s )

.13M,NaCl

tap,8kV

400

600

AgNO3

150 100 50 0 0

200

800

Water-Tree Length (μ m)

where R= r + L and D = d – L; r being the needle radius, L the water-tree length and d the point-to-plane distance.

Figure 5. Square of field at the top of water trees calculated with equation (6) as a function of L3Y/Nt1/2 (see equation 2) for various results [29,32,35]; open symbols: 0.1 M NaCl; filled symbols: 0.2 to 1 M NaCl. The open symbols results are those shown in Figure 3.

A typical relation between the calculated fields is shown in Figure 4 for some results obtained by Yoshimura and Fan [33,34] using different ionic solutions. The fields are obviously much smaller than those calculated at the tip of the needle cavity using equation (3) and they are rapidly as low as ~10 kV/mm. In order to confirm that the assumption of a conductive cavity is not correct, we have plotted in Figure 5 the square of the field (from equation 6) vs. L3Y/Nt1/2. Equation (2) predicts that this should give straight lines with slopes equal to εoε’ΔV. We have included in Figure 5 the results (open symbols) shown in Figure 3 that were well fitted by equations (2) or (5). Clearly, straight lines are rarely observed and for the results of Nitta [35], the slope has a negative slope, contrarily to our expectation. Therefore, the assumption of a conductive point does not work, which does not mean that the field should be constant within the water tree.

It is always possible to claim that a space charge can be generated at the top of the water tree, as indeed observed by Ohki et al. [22], and the local field would be enhanced by the space charge. Although we agree that space charges can be generated in water trees (after all, weak bonds have been broken and there is some local degradation, which should indeed facilitate charge accumulation), we dislike the concept of explaining every mysterious phenomenon by the presence of space charges. We believe that degradation occurs first and then space charges are formed. Said differently, the space charge field is posterior to the generation of defects and measuring a field, hours after the degradation had occurred is not very helpful. For example, Ohki et al. [22] grow water trees in samples under ac field for a certain time and then they put these samples into a space-charge measurement set-up to finally perform charge measurement under



a dc field. How can we be sure that the measured charge (and the induced field) was due to the water treeing part of the experiment and not from the charge measurement part of the experiment? Of course, trapped charges (generated under ac) could remain present for a very long time and could then be detected after the application of a dc field. But how much and how long would they remain trapped? We know that charged PE samples could lose a large fraction of their charge in a few minutes and with the limited sensitivity of the charge measurement set-ups, it is difficult to believe that sufficient charges generated under ac could still be detected hours later. In addition, the results of Ohki et al. [22] show unequivocally that the charge polarity in water trees is the polarity of the injecting dc electrode. What would happen to the charges of the other polarity generated by the previous ac field? When all these arguments are considered, it seems difficult to assume that space charges measured after treeing experiments are the only cause of water trees growth. This discussion on the role of space charges in water treeing should not be restricted to water treeing only and it should be considered in the broader perspective of the relation between molecular deformation and charge accumulation in polymer dielectrics (see also [36]). Finally, can we really assume that the stress generated in the liquid at the tip of the metal electrode is 100% transmitted to the tip of the water needle cavity (a few mm away from the metal electrode) where the tree grows? Would not it be more realistic to imagine that some fraction of this stress is lost ? In that case, what is the fraction of “efficient” stress actually applied at the tip of the water needle? It would not be surprising if the size of the cavity and the volume of liquid as well as the nature of the liquid would affect the transmission of stress from the metal to the polymer. Presently, there is no model able to consider all these potential parameters. Considering all the unknowns surrounding the simple question about the real value of the applied field, it would be worth to spend some time on ways and means to make more reliable estimation of the strength of the actual field involved in many electrical phenomena. Space charge measurements performed after the degradation leading to the accumulation of charges is not the best solution, at least in its present form. What is needed are both a theory and a technique measuring in situ and in real time the spatial and time evolution of the applied stress following the application of a high electrical field on a dielectric.

3 HIGH FIELD EFFECTS IN ELECTRICAL TREEING Although the influence of the strain energy is unlikely in dry polymers, the very high fields at the point of a needle, as in electrical treeing, generate significant local strain and deformation. In a recent paper, Lee [37] has shown that large stress enhancement factors can be induced in small asperities or defects at the semicon/dielectric interface of an extruded cable. In Figure 6, we show the electrostatic energy calculated for two different needle-like defects in PE. According to our model [11,18], a nanocavity can be generated when the electrostatic energy, i.e. ½ ε ΔV F2, dissipated in the polymer is larger than half the energy for hole creation, Eh. The concept of hole refers to Eyring [38] and

795

Adam and Gibbs [39] theories, where holes (in the sense of cavities) of a given size typical of the polymer have to be formed to allow the movement of molecular chains. These nanocavities (or “holes”) are different from the microcavities generated during the polymer processing not only by their much smaller size but also by the fact that they are essentially local expansion of the polymer matrix resulting in points of lower density and in empty spaces filling the free volume between molecular chains (see [11] for more detail). The Eh value determined at the glass transition temperature varies between ~4 x10-21 to 3-4x10-20 J depending of the polymer [40]. For PE, which is in the rubbery condition at room temperature, Eh is ~3.2x10-21 J at 22oC. We see in Figure 6 that the electrostatic energy is rapidly larger than such a Eh value, whatever the needle size, for moderate voltages. It means that holes can be generated or free-volume cavities can be crushed by the electrostatic fields induced by such relatively moderate voltages. This will create some molecular deformation and eventually it may lead to what Zeller has called electro-fracture [41]. In fact, the electrostatic energy near a 10 μm-radius needle whose point is 5 mm from the ground is in the 5 eV (8x10-19 J) range at 10-12 kV. It is enough to break the strong covalent and intra-molecular bonds. In Figure 6, we have also added the kinetic energy of electrons accelerated by the field in the holes whose length was the diameter of the supposedly spherical activation volume; it is very dubious that the cavities would be spherical but it is much simpler to make this assumption. We see that electrons injected from the tip of a needle with short implantation into the polymer has a much lower kinetic energy (for the same voltage) and most of the degradation will come from the electrostatic energy. However, an electron injected from a 10 μm-radius needle 5 mm away from the ground has a kinetic larger than 5 eV for voltages above ~15 kV. Therefore, it might well be possible that electrical trees are growing when submicro-defects are induced by the electromechanical stress near the needle electrode. Later on, electrons will be injected into these “open” spaces and an avalanche leading to breakdown will be eventually initiated but it should be kept in mind that the first step was molecular deformation by the Maxwell stress. Of course, we may interpret the same phenomenon using a space charge field [36] but our approach seems to give a good phenomenological description without having to rely on this parameter, which is fairly difficult to measure in a needle-plane configuration. In order to support even further our point of view, let us look at some recent data on photon emission during an electrical treeing experiment [42]. It is well known that light is emitted from the tip of a needle subjected to sufficiently high fields. We will not get here into the discussion of whether it is due to fast electron collision or electron-hole recombination because we are interested by the origin of the phenomenon not by the electronic process arriving later. If Lewis contention that a deformation varying with the square of field is correct and if the number of emitted photons depends on the concentration of deformed molecules, it would mean there should be some relation between the square of the field and the number of emitted photons. This purely mechanistic approach does not negate the presence of space charges; it simply


electrostatic energy

7 6 5 4 3 2 1 0 0

5

10

E kin

1000

15

20

Photon Counts (cps)

En erg y (eV)

796

100

Voltage (kV) Figure 6. Electrostatic energy (solid lines) and kinetic energy (dashed lines) calculated for two different sizes of needle. Filled symbols: point-to-ground distance 5 mm, point radius: 10 μm, ΔV= 10-24 m3 (size of cavity: 128 μm). Open symbols: point-to-ground distance 2.5mm, point radius: 5 μm, ΔV= 10-25 m3 (size of cavity: 60 μm).

states that molecular deformation is the first and essential step leading to eventual electronic effects. We have proposed that the concentration of deformed molecules at constant time varies as [11] n = no exp(1/2 εoε’ ΔV F2/kT)

(7)

where no is an adjustable parameter. Equation (7) means that a plot of log n vs. F2 should yield a straight line of slope 1/2εoε’ ΔV/kT. We have taken the number of photons emitted from a semicon point into polybutadiene as reported by Zheng and Boggs [42] but contrarily to these authors, we plotted the results on a semi-log graph (Figure 7), in agreement with equation (7). As Zheng and Boggs, we can deduce there are two different processes but Figure 7 tells us that the change of process occurs at higher voltage than predicted from the linear plot used by Zheng and Boggs [42]. From the slope at the higher fields, we may deduce an activation volume of 1.73x10-27 m3, assuming a dielectric constant of 2.5. More interesting, we may calculate the field of the change of process if we assume (see [11,18]) that this corresponds to the “holes” creation or disappearance, i.e. when the electrostatic energy is equal to Eh /2. The estimated value of Eh for polybutadiene at its glass transition Tg = 190 K varies from 4.5 to 7.5x10-21 J [40] and we have taken an average value of 6x10-21 J. The customary approximation to estimate the value of Eh at a temperature T is to suppose that Eh (T) = Eh (Tg) – k (T-Tg)

(8)

where k is the Boltzmann constant. Although not precise, equation (8) gives reasonably correct values. Thus, a rough estimate of Eh for polybutadiene at 295 K would be ~4.5x10-21 J. Substituting this value into equation (9) gives the critical field Fc [11,18], which is the onset of photon emission (and of strong molecular deformation)

10 0

100000 200000 300000 400000 500000 Square of Field (kV/mm)2

Figure 7. Photon counts emitted from a semicon electrodes (radius: 1μm, point-to-ground: 2mm) into polybutadiene [42] as a function of the square of the field (calculated with equation (3)).

Fc = (Eh / εoε’ ΔV) 1/2

(9)

The calculated value with a ΔV of 1.73x10 m is 3.43x108 V/m, corresponding to a voltage of 1543 V (from equation 3), which is slightly higher than the ~ 900 V suggested by Zheng and Boggs [42]. Note that their electrode has not exactly the shape of a needle and therefore using equation (3) to calculate the field may yield some error. However, the main point here is to show that plotting the results according to our model gives the expected straight line (i.e. the activation volume) and from this activation volume it is possible to determine the change of regime. The value thus calculated, 1543 V, is in excellent agreement with the change of slope in Figure 7, which is around 3.5x108 V/m, i.e. for ~1550 V. Once again, we did not have to consider the influence of the space charge to predict the lower voltage limit of the light emission process, that clearly appears related to the molecular deformation induced by the Maxwell stress. -27

3

4 POLARIZATION AND CHARGE CONCENTRATION UNDER HIGH FIELDS AND PARALLEL-PLANE ELECTRODES Polarization by space charges is a phenomenon typical of dielectrics occurring above a given high field. It is our contention that space charges are a consequence of aging, i.e. charges can be injected (with the proper electrodes) only when molecules have been deformed by the fieldinduced strain. The best experimental evidence supporting this contention is that indeed no one has ever been able to

detect any charge injection in PE below 10-15 kV/mm, i.e. below Fc, the onset of strong molecular deformation. Above Fc , electrons (or holes) could then be injected in sites associated with the local deformation. The exact nature of these sites remain to be clearly identified and a formal theory (without the pitfalls of Schottky theory- see [11] for a brief discussion) needs to be developed to describe charge injection into these sites. However, there exists experimental facts strongly suggesting that some charging processes in several polymers are definitely related to the field-induced molecular deformation. We have already shown in [11] that the charge concentration measured by Montanari et al. [25] in various XLPE samples varies with F2 above ~ 9-12 kV/mm, in agreement with equation (7). A more recent example are the results of Mamy et al. [16] who measured the strain in 25 μm PET films subjected to high dc fields. When the measured strain is plotted as a function of the square of the field, a straight line is observed, as expected, at high fields (Figure 8). The authors had also observed a direct link between the strain and the current measured in the same samples [16], which is another evidence of the influence of molecular deformation on electrical properties of polymers. However, from a theoretical point of view, it is not evident to directly relate current and strain . This deserves further work. Let us give another demonstration by analyzing some thermally stimulated currents (TSC) measurements on high density PE [43]. Fraile et al. recorded the β and α relaxations of HDPE by TSC and they integrated the area under the curves of the α relaxation obtained under various polarizing fields at 46 oC [43]. They assumed that it would be proportional to the stored space charge and they reported the results (in arbitrary units) as a function of the applied field. We have taken these results and plotted them as log charge vs. the square of the field and, as expected, it gives the straight line shown in Figure 9. Some may argue that the fit is not perfect but it should be kept in mind that the charge integration is rarely a precise calculation and that we had to extract the results from a small graph. Thus, errors are likely to have occurred and the exact charge values may be slightly different from the ones shown in Figure 9 but we are more interested by the general trend than by a precise fit. It is quite clear that the charge varies with the square of the field and not with a higher exponent, as expected with the space charge theory [36]. All these experimental results of various properties of different polymers obtained with very different measurement techniques are a clear evidence that the Maxwell stress induced by the field has a significant influence on many electrical properties of polymers. Considering the good fit between the experimental change of processes (i.e. change of slope or injection, etc.) and the values predicted by our tentative model based on molecular properties, it appears that molecular deformation might have a more important role than often assumed.

C h a rg e (a rb itra ry u n its)


797

10 Tpolarization: 46oC

1 0

20000 40000 60000 80000 Square of Field (kV/mm)2

100000

Figure 8. Strain measured in PET films [16] plotted as the square of the field.

S train


0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 0

40000

80000

120000

Square of Field (kV/mm)2 Figure 9. Stored charge (given in arbitrary units by [43]) in the α relaxation of HDPE measured by TSC as a function of the square of the field.

5 CHARGE MOBILITY UNDER HIGH FIELDS Charge mobility can be easily deduced when the conductivity and the charge concentration are known. For example, the mobility of electrons in PE or XLPE at 22oC is often quoted as being 3x10-16 m2/Vs [20] at low/moderate fields and around 1-2x10-14 m2/Vs [21,44,45] in the wave packets regime. However, a steady state current under constant field is rarely (if ever) achieved in conductivity measurements and the charge concentration evolves with time [20-25], which means that the above value for mobility has a limited sense unless the time of field application is reported. A constant mobility value also implies that it is not sensitive to high fields when both charge and currents are rapidly increasing above ~10-12 kV/mm [25,45,46]. Another problem is that the nature of the charge carriers is rarely well known and when it is established they are electrons and holes, they have often different mobilities. Most measurement techniques require several seconds to give reliable results (charge content or current), implying that only the mobility of the less mobile species is estimated. It has


already been postulated that charges (whatever their nature) could drift in the insulating material attached to a molecule (similar to a kink displacement along a molecule length) and/or jumping from one molecular segment to another one when they are sufficiently close [47]. This old idea might need to be revisited. In addition, mobility decreases with time [45] which makes the notion of constant mobility even more fuzzy. In other words, the physical significance of many charge mobility values in polymers could be questioned. It is customary [44,45] to associate all these behaviors to space charge fields that are supposed to increase the local internal field (i.e. increase the mobility) and at the same time to saturate the charge concentration at high fields (an example is shown in Figure 10). Obviously, there is something strange in this paradoxical interpretation and the objective of this last section is to bring forward some arguments demonstrating that it is possibly time to review either the definition or the significance of charge mobility in polymers. For that, we have taken the results of Montanari et al. [25,46] who have measured conductivity and charge concentration in two XLPE samples that appear to be very similar (120-150 μm thick, fields applied up to 60,000 s for conductivity and 10,000 s for charge measurements). The mobility was then calculated from conductivity and charge measurements obtained after 10,000 s and results are shown in Figure 10. We have also taken the mobility results of Chen et al. [20] deduced from the time of flight of charge packets between the two gold electrodes during charge measurements on 200 μm LDPE (voltage applied for 30 mn). Note that aluminum electrodes appeared to block charge injection and gave different charge concentration [20]. Both sets of charge results were obtained with the PEA technique, which should allow a direct comparison. The first conclusion to be drawn from Figure 10 is that the results of Chen et al. and of Montanari et al. are very similar although the samples, the electrodes and the polarization time were different. This could be fortuitous but their mobility values are in the expected range. The next conclusion is that indeed a value of 3x10-16 m2/Vs is reasonable in the ohmic regime (below ~ 10 kV/mm) provided measurements are performed in the 10,000-60,000 s range (3 to 16 h). It is almost certain that for longer time (as can be expected in any industrial application), the charge would be significantly larger and the conductivity slightly smaller, which means a smaller mobility. But the biggest problem illustrated in Figure 10 is the rapid increase of mobility with increasing field above 12 kV/mm followed by a quasi saturation above 40 kV/mm. Interestingly, Hozumi et al. [21] have measured a decreasing mobility between 20 and 60 kV/mm in EVA. What is then the significance of a constant mobility in the 10-40 kV/mm range? The only constant parameter in that field range is the activation volume, that is the slope in a log μ vs. F2 plot (not represented here). In order to further validate our molecular approach, we have calculated the critical field (equation (9)) which is the onset of significant molecular deformation and the field Fbb where van der Waals attraction bonds are broken given by [11,18] Fbb = (Ecoh/1/2 εo ε ’ ΔV) ½

(10)

unaged film

1.E-14 M obility (m 2 /Vs)

798

unaged cable

Chen,Au

1.E-15

1.E-16 0

20

40 Field (kV/mm)

60

80

Figure 10- Mobility in XLPE deduced from charge and conductivity measurements ([25,46], 120-150 μm thick, circles) and from the time of flight of charges moving through 200μm LDPE samples with gold electrodes during charge measurements ([20], x). The arrows indicate Fc= 12.5 kV/mm (calculated from equation (9)) and Fbb= 39.6 kV/mm (calculated from equation 10) with ΔV= 1.05x10-24 m3 deduced from charge measurements [25] plotted as log n vs. F2.

where Ecoh is the energy of cohesion of the polymer (~1.6x10-20 J for PE at room temperature); the value of ΔV=1.05x10-24 m3 was deduced from the slope of log n vs. F2 plots for the charge results of Montanari et al. [25]. The calculated values were Fc= 12.5 kV/mm and Fbb = 39.6 kV/mm and they are represented by arrows in Figure 10. Quite clearly, they bound very well the limits of the rapidly increasing mobility giving further support to our claim that some electrical properties of polymers are related to molecular deformation. The saturated mobility above ~40 kV/mm (for those specific samples) therefore corresponds to the onset of wave packets, as predicted elsewhere [11,18]. This also agrees with the observation of Chen et al. [20] who reported wave packets for fields as low as 30 kV/mm in the samples used for the mobility results shown in Figure 10. It is important to keep in mind that for aging [11,18] the activation volume varies with the sample size and the exact relation remains to be developed for other phenomena. Thus, it is recommended to compare results obtained on similar samples or the ΔV values should be determined for each specific condition.

6 CONCLUSION The influence of electromechanical stresses induced by the electrical field applied on various polymers was reviewed and they appear to have a strong influence on several electrical properties. In our opinion, this suggests that molecular deformation is the key process and a molecular approach was tentatively applied to describe several phenomena. It was shown that light emitted during electrical tree growth was directly related to the molecular deformation of the polymer (polybutadiene). The measured strain in PET subjected to fields larger than 160 kV/mm was also directly related to the field-induced Maxwell stress. The charge stored during the α relaxation of PE measured by TSC was also shown to depend on the molecular deformation of the polymer. Finally, the notion of a constant mobility value in PE was disputed since



this occurs only at low (< 10 kV/mm) and high (wave packets regime) fields. It was shown that the only constant parameter in that field range is the activation volume, which depends on sample size, nature and possibly on its morphology. Note that the values of the activation volumes deduced for the various properties and materials discussed here are ranging between 1.5x10-27 and 2x10-24 m3 and, assuming spherical volumes, this means radii in the 0.8 to 8 nm range, which is typical of the size of molecular segments. It was also shown that water treeing under very high fields could be sensitive to the strain energy, especially for ionic solutions with large dielectric constant. The real value of the field at the tip of water trees is still unclear and it is not evident that a space charge field could describe existing data.

REFERENCES [1]

[2]

[3]

[4] [5] [6]

One evidence in favor of an apparent influence of the Maxwell stress on several dielectric and electrical properties of solid dielectrics is the fact that it is possible to predict the field corresponding to change of processes (such as charge injection or transport) from our simple molecular model. Considering the number of agreements between theoretical predictions and experimental facts and the variety of materials and properties, it seems difficult to call these agreements purely fortuitous. Among the observed agreement between theory and experiment, our model was able to predict the correct critical field (charge injection) and the onset of wave packets in polyethylene. It also predicts that the upper limit of the ohmic regime in various polymers corresponds to the onset of molecular deformation, that is when the electrostatic energy is equal to half the energy for hole creation. It was also pointed out that wave packets and saturation in mobility and charge concentration measurements occur when the electrostatic energy is equal to the cohesion energy of the polymer. This results in weak attraction bonds breaking and possibly in radicals formation. At this point, the induced deformation is then partly irreversible and removing the field will not bring back the stressed molecules in their original state. Thus, the basic ideas put forward in this paper are in a fair agreement with a wide variety of data and it might be worth spending more time to verify more in depth their validity and their implications. Many things are still unclear (both on the theoretical and on the experimental level) and require more work: the physical nature of the sites where charges could be injected above Fc, the relation between ΔV and the sample size, the role of morphology are just some of them. However, it appears that molecular deformation contributes very significantly to the electrical properties of polymers, in direct agreement with the pioneer work of John Lewis.

[7]

[8] [9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17] [18]

[19] [20]

ACKNOWLEDGMENT Stimulating comments and suggestions from the reviewers were greatly appreciated.

799

[21]

T.J. Lewis, “The Microphysics of Charge in Dielectrics”, Proc. 3rd Int. Conf. Elec. Charge in Solid Insulators, Société Française du Vide, Paris, pp. 16-24, 1998. T.J. Lewis, J.P. Llewellyn and M.J. van der Sluijs, “Electrokinetic properties of Metal-Dielectric Interfaces”, IEE Proc. A, Vol. 140, pp. 385-392, 1993; see also: T.J. Lewis, J.P. Llewelin, M.J. van der Sluijs, J. Freestone and N. Hampton, “A New Model for Electrical Ageing and Breakdown in Dielectrics”, 7th IEE Conf. Diel. Meas. Appl., pp. 220225, 1996 and IEEE CEIDP, pp. 328-333, 1996. J.P. Jones, T.J. Lewis and J.P. Llewellyn, “Field-induced Changes in the Viscoelastic Properties of Polyethylene”, IEEE ICSD, pp. 284-287, 2004. K.H. Stark and C.G. Garton, “Electric Strength of Irradiated Polythene”, Nature, Vol. 176, pp. 1225-1226, 1955. J. Blok and D.G. LeGrand, “Dielectric Breakdown of Polymer Films”, J. Appl. Phys., Vol. 40, pp. 288-293, 1969. J.C. Fothergill, “Filamentary Electromechanical Breakdown in Polymers”, IEEE ICSD, pp. 323-327, 1992; see also: L.A. Dissado and J.C. Fothergill, Electrical Degradation and Breakdown in Polymers, especially chapter 11, Peter Peregrinus Ltd, London, 1992. S.N. Zhurkov, V.A. Zakrevskyi, V.E. Korsukov and V.S. Kuksenko, “Mechanism of Submicrocrack Generation in Stressed Polymers”, J. Polymer Sci. A2, Vol. 10, pp. 1509-1520, 1972; see also: S.N. Zhurkov, Int. J. Fracture, Vol. 1, pp. 311-20, 1965. B.M. Fanconi,“ Chain Scission and Mechanical Failure of PE ”, J. Appl. Phys., Vol. 54, pp. 5577-5583, 1983. T. Tanaka,M. Sato and M. Kazako, “High Field Maxwell Stress-Strain Characteristics of Conventional Polymers as Actuators”, IEEE ICSD04, pp. 364-367, 2004. W. Brandt and J. Wilkenfeld, “Electric Field Dependence of Positronium Formation in Condensed Matter”, Phys. Rev. B, Vol. 12, pp. 2579-2587, 1975. J.P. Crine, “On the Interpretation of some Electrical Aging and Relaxation Phenomena in Solid Dielectrics”, to be published in IEEE Trans. DEI, 2005 and also IEEE CEIDP, pp. 1-16, 2004. M. Meunier and J.P. Crine, “In Situ Infra Red Spectroscopy of Electric Field Induced Micro-Structural Changes in Polyethylene”, IEEE Conf. Elec. Insul. Diel. Phenom., pp. 377-381, 1985. M. Sakamoto and K. Yahagi, “Influence of High Electric Fields on Capacitance Measurements in PE”, Japn. J. Appl. Phys., Vol. 19, pp. 253-259, 1980. J.C. Fothergill, “Aging of Extruded Insulation in Transmission Cables Key Issues”, EPRI Workshop on Aging of Extruded Insulation for Transmission Class Cables, Detroit, February 27- March 1, 2002. J.C. Fothergill, GC. Montanari, G. Stevens, C. Laurent, G. Teyssedre, L.A. Dissado, U. Nilsson and G. Platbrood, “Electrical, Microstructural, Physical and Chemical Characterization of HV XLP Peelings for an Electrical Ageing Diagnostic Database”, IEEE Trans. Dielectr. Electr. Insul., Vol. 10, pp. 514-527, 2003 and references therein. P.R. Mamy, L. Boudou and J.L. Martinez-Vega, “Correlation Between Electrical Conduction and Mechanical Deformation Induced by an Electric Field in PET”, IEEE ICSD, pp. 615-618, 2004. L.A. Dissado, personal communication at the ICSD, 2004. J.P. Crine, “Aging and Polarization of Polyethylene Under High Electrical DC Fields”, IEEE Trans. Dielectr. Electr. Insul., Vol. 9, pp. 697-703, 2002. J.P. Crine and J. Jow, “A Water Treeing Model”, IEEE Trans. Dielectr. Electr. Insul., Vol. 12, to be published, 2005. G. Chen, T.Y.G. Tay, A.E. Davies, Y. Tanaka and T. Takada, “Electrodes and Charge Injection in Low-Density Polyethylene”, IEEE Trans. Dielectr. Electr. Insul., Vol. 8, pp. 867-873, 2001. N. Hozumi, Y. Muramoto and M. Nagao, “Estimation of Carrier Mobility Using Space Charge Measurement Technique”, IEEE CEIDP, pp. 350-353, 1999.

800


[22] Y. Ohki, Y. Ebinuma and S. Katakai, “Space Charge Formation in Water-treed Insulation”, IEEE Trans. Dielectr. Electr. Insul., Vol. 5, pp. 707-712, 1998; also Vol. 4, pp. 52-57, 1997 and Vol. 8, pp. 111-116, 2001. [23] Y. Zhang, J. Lewiner, C. Alquié and N. Hampton, “Evidence of Strong Correlation Between Space-Charge Buildup and Breakdown in Cable Insulation”, IEEE Trans. Dielectr. Electr. Insul., Vol. 3, pp. 778-783, 1996. [24] P. Nothinger, S. Agnel and A. Toureille, “ Thermal Step Method for Space Charge Measurements under Applied dc Field”, IEEE Trans. Dielectr. Electr. Insul., Vol.. 8, pp. 985-94, 2001. [25] G.C. Montanari, D. Fabiani, L. Bencivenni, B. Garros and C. Audry, “Space Charge and Conduction Current measurements for the Evaluation of Aging of Insulating Materials for DC Applications”, IEEE CEIDP, pp. 38-42, 1999. [26] S.A. Boggs, “A Rational Consideration of Space Charge”, IEEE Electr. Insul. Mag., Vol. 20, No 4, pp. 22-27, 2004. [27] J. Sletbak, “A Theory of Water Tree Initiation and Growth,” IEEE Trans. Power Appts. Syst., Vol. 98, pp. 1358-1368, 1979. [28] C.T. Meyer, J.C. Filippini and N. Felici, “Water Tree Propagation in Relation to Mechanical Properties of Polyethylene”, IEEE CEIDP, pp. 374-381, 1978; see also: J.C. Filippini and C.T. Meyer, “Effect of Frequency on the Growth of Water Trees in PE”, IEEE Trans. Electr. Insul., Vol. 17, pp. 554-559, 1982 and Vol. 23, pp. 275-278, 1988. [29] L.A. Dissado, S.V. Wolfe, J.C. Filippini, C.T. Meyer and J.C. Fothergill, “An Analysis of Field-Dependent Water Tree Growth Models,” IEEE Trans. Electr. Insul., Vol. 23, pp. 345-356, 1988. [30] J.H. Mason, “Dielectric Breakdown in Solid Insulation”, Chapter 1, Progress in Dielectrics, pp. 1-58, Heywood and Co., London, 1959. [31] N. Yoshimura, F. Noto and K. Kikuchi, “Growth of Water Trees in Polyethylene and in Silicone Rubber by Water Electrodes”, IEEE Trans. Electr. Insul., Vol. 12, pp. 411-416, 1977. [32] J.C. Filippini, J.Y. Koo and J.L. Chen, “Electrical Behavior of Water Trees under High Frequencies”, IEEE Trans. Electr. Insul., Vol. 24, pp. 75-80, 1989. [33] Z.H. Fan and N. Yoshimura, “Silver Tree”, IEEE Trans. Dielectr. Electr. Insul., Vol. 3, pp. 131-135, 1996. [34] Z.H. Fan and N. Yoshimura, “The Influence of Crystalline Morphology on the Growth of Water Trees in PE”, IEEE Trans. Dielectr. Electr. Insul., Vol. 3, pp. 848-859, 1996. [35] Y. Nitta, “A Possible Mechanism for Propagation of Water Trees from Electrodes”, IEEE Trans. Electr. Insul., Vol. 9, pp. 109-112, 1974. [36] L.A. Dissado,G. Mazzanti and G.C. Montanari, “The Role of Space Charges in the Electrical Aging of Insulation Materials”, IEEE Trans. Dielectr. Electr. Insul., Vol. 4, pp. 496-505, 1997. [37] W.K. Lee, “A Study of Electric Stress Enhancement. Implication in Power Cable Design”, IEEE Trans. Dielectr. Electr. Insul., Vol. 11, pp. 976-982, 2004.

[38] N. Hirai and H. Eyring, “Bulk Viscosity of Liquids”, J. Appl. Phys., Vol. 29, pp. 810-816, 1958 and; J. Polym. Sci., Vol. 37, pp. 51-60, 1959. [39] J.H. Gibbs and E.A. DiMarzio, “Nature of the Glass Transition and the Glassy State”, J. Chem. Phys., Vol. 28, pp. 373-383, 1958. [40] Y. Havlicek, V. Vojta, M. Ilavsky and J. Hrouz, “Molecular Parameters of Polymers Obtained from the Gibbs-DiMarzio Theory of Glass Formation”, Macromolecules, Vol, 13, pp. 357-362, 1980. [41] H.R. Zeller and W.R. Schneider. “Electrofracture Mechanics of Dielectric Aging”, J. Appl. Phys., Vol. 56, pp. 455-459, 1984. [42] Z. Zheng and S.A. Boggs, “Defect Tolerance of Solid Dielectric Transmission Class Cable”, IEEE Electr. Insul. Mag., Vol. 21, No. 1, pp. 35-41, 2005. [43] J. Fraile, A. Torres and J. Jimenez, “Space Charge Relaxation in HDPE”, J. Mater. Sci., Vol. 24, pp. 1323-1326, 1989. [44] S. Leroy, G. Teyssedre, C. Laurent and P. Segur, “Modelling of Space Charge, Electroluminescence and Current in LDPE Under dc and ac Field”, IEEE CEIDP, pp. 29-32, 2004. [45] J.M. Alison, G. Mazzanti, G.C. Montanari and F. Palmieri, “Mobility Estimation in Polymeric Insulation Through Space Charge profiles Derived by PEA Measurements”, IEEE CEIDP, pp. 35-38, 2002. [46] B. Garros, C. Audry, H. Schadlich, G.C. Montanari, L. Bencivenni, “Evaluation of Insulation Degradation of Stressed XLPE Cables”, Proc. Jicable 99, pp. 441-445, 1999. [47] T. Miyamoto and K. Shibayama, “Free Volume Model for Ionic Conductivity of Polymers”, J. Appl. Phys., Vol. 44, pp. 5372-5376, 1973.

Jean-Pierre Crine (SM’85, F’94) was graduated with a B.Sc. and a M.Sc. (Energy) degrees from the University of Quebec in Montreal and obtained the Ph.D. (Engineering Physics) degree from Ecole Polytechnique (Montreal) in 1978. For the following twenty years, he was with the Research Institute (IREQ) of Hydro-Québec as a research scientist and then as a manager of the Cables and Insulation department. During these years, he was active in the study of the electrical and aging properties of the insulating materials (especially solids and liquids) used in high voltage equipment: transformers, cables, capacitors, etc. He is now a consultant for the electrical industry and the material suppliers. He is also an Associate professor at the Ecole Polytechnique and the Ecole de Technologie Supérieure (both in Montreal) where he gives courses on cable engineering, R&D management and technology project management. He was awarded the Whitehead Memorial Lecture by the IEEE Dielectrics and Electrical Insulation Society in 2004.