Electric Breakdown Model for Super-Thin Polyester Foil

Polyester has been more and more widely applied to the field of electrical insulation because of their excellent electrical and mechanical properties. One of the commonest concerns in industry is the reliability of materials. Therefore, the study of breakdown of solid dielectrics is extremely important in industrial applications. The increasing demanding for polyester films requires that the dielectric strength of polyester with various film thicknesses should be measured accurately. Lack of accurate data on dielectric strength could lead to design shortcomings: excessive insulation could lead to more expensive equipments, while inadequate insulation could lead to premature failure. The data of breakdown strength for polyester from several hundred microns to several millimeters can be directly obtained from the handbook of dielectric materials. However, there are few data of breakdown strength in micro-meter regime.


Introduction
Polyester has been more and more widely applied to the field of electrical insulation because of their excellent electrical and mechanical properties. One of the commonest concerns in industry is the reliability of materials. Therefore, the study of breakdown of solid dielectrics is extremely important in industrial applications. The increasing demanding for polyester films requires that the dielectric strength of polyester with various film thicknesses should be measured accurately. Lack of accurate data on dielectric strength could lead to design shortcomings: excessive insulation could lead to more expensive equipments, while inadequate insulation could lead to premature failure. The data of breakdown strength for polyester from several hundred microns to several millimeters can be directly obtained from the handbook of dielectric materials. However, there are few data of breakdown strength in micro-meter regime.
1. Intrinsic Breakdown --it is generally considered to be due to electrons in the insulator gain sufficient energy from the applied field to cross the forbidden energy gap from valence to the conduction band. Intrinsic breakdown is usually accomplished in the order of 10 -8 s (Seeger & Teller, 1939). 2. Streamer Breakdown --the mechanism is conceptually similar to the streamer mechanism in gases, namely when the avalanche exceeds certain critical size breakdown will occur. Transit time is usually short. 3. Thermal breakdown --when an insulator is stressed electrically, because of currents and dielectric losses due to polarization, heart is continuously generated within the insulator. If the rate of heat generation exceeds the rate of heat losses, then the insulator will undergo thermal breakdown. Such breakdown takes place usually in a much slower time scale. 4. Erosion Breakdown --insulators of contain voids or cavities within the material. These cavities are usually filled with gas or liquid medium of lower breakdown strength than the solid insulator. Accordingly, under normal working stress of the insulator the voltage across the cavities may exceed the breakdown value hence initiating breakdown in the cavities. 5. Breakdown due to tracking and other causes --Tracking is the formation of some permanent conducting paths, usually carbon, along the insulator surface due to degradation of the insulator or other causes. 6. Electromechanical breakdown -Electromechanical breakdown occurs in dielectric foils of low Young's modulus and easy deforming and it is because of the force of attraction between the surface charges. 7. Treeing breakdown --treeing occurs due to the erosion of material at the tips of the spark. Erosion results in the roughening of the surfaces, and hence becomes a source of dirt and contamination. This causes increased conductivity resulting either in the formation of conducting path bridging the electrodes or in a mechanical failure of the dielectric.
Electrical breakdown testing of polymers for insulator applications has long been a subject of interest. In order to improve insulator performance, it is essential to understand the mechanisms of electrical breakdown in solid dielectrics. However, unlike the case for gases, electrical breakdown and conduction mechanisms in polymeric solids are less understood. In solid dielectrics, the electrical transport phenomena include currents due to orientation and interfacial polarization in addition to electronic and ionic charge carriers (Barber et al., 2009). On account of the time of dielectric breakdown, this chapter mainly presents intrinsic breakdown and electromechanical breakdown.

Intrinsic breakdown
When an electric field is applied to an insulator, electrons gain sufficient energy from the applied field to cross the forbidden energy gap from the valence band to conduction band. When this process is repeated, more and more electrons become available in the conduction band, eventually leading to breakdown. Intrinsic breakdown usually occurs in the order of 10-8s. The initial density of free electrons is assumed to be large, and electron-electron collisions occur. The critical criterion of intrinsic breakdown is initial electrons formed by impact ionization. It consists of Hippel's low-energy criterion and Frohlich's high-energy criterion (Callen, 1949;Frohlich, 1939;Hippel & Alger, 1949;O'Dwyer, 1964;Seeger & Teller, 1939;Seitz, 1948;Whitehead, 1951).

Hippel low-energy criterion
Hippel low-energy criterion assumed that the energy E breakdown occurs when the applied field is high enough to accelerate electrons of any given energy in the conduction band to sufficient energy to further ionize the ions of the crystal by collision, thus leading to an exponential increase in the number of electrons in the conduction band. It may be easily seen that the applied field necessary to accelerate (on the average) an electron of given energy has a maximum when considered as a function of electron energy. In order to accelerate an electron, the applied field must compensate the energy lost by this electron to the lattice; but slow electrons have insufficient energy to excite the vibrational modes of the lattice, whereas very fast electrons interact for too short a time with the ions which they pass to transfer energy to them with appreciable probability. The breakdown strength, according to von Hippel's "low energy" criterion, therefore corresponds to the maximum, and an electron brought into the conduction band with low energy is immediately accelerated to the ionization energy and produces an additional conduction electron, the process thus building up in the form of an avalanche.

Frohlich high-energy criterion
Frohlich high-energy criterion is based on that an applied field sufficient only to accelerate electrons already having the ionization energy may lead to an instability and to electric breakdown (Callen, 1949;Cooper, 1966).
Eq. (1) shows the critical breakdown strength of intrinsic electron breakdown (Callen, 1949). For low energy breakdown theory 4 c u u     , and for high-energy breakdown theory, (1) Eq.
(1) may be written as B is the average rate of energy loss for a given electronic energy, and is expressed by Eq. (4), where u is the average effect of collisions on the electrons of energy. C is expressed by Eq.
(5).  is the relaxation time, e is charge of electron, e  is the effective charge per ion, m  is the effective mass of the electron, M is the reduced mass of the ions, a is the interionic distance,  is the natural angular frequency and t  is the natural angular frequency of the transverse mode.
Where e  is expressed in terms of  by Eq. (7) and  is expressed in terms of t  by Eq.
(8). We obtain Where εr and εop are the relative and optical values of the dielectric constant, T is the temperature of the lattice in K. In Table 1 the pertinent breakdown strength data for polymer are shown according to Eq. (10) and Eq. (11). The electron energy is 0.1～0.2 eV for Hippel's criterion and 5 eV for Frohlich's criterion. The theoretical values are compared with the experimental breakdown strengths in mm regime. The experimental breakdown strength EE which is given by Ref. (Wang et al. 1992) in mm regime is less than the breakdown strength of Hippel's criterion EBH and greater than the breakdown strength of Forhlich's criterion EBF. The critical breakdown strength of intrinsic breakdown model is calculated on the basis of Hippel's lowenergy criterion and Frohlich's high-energy criterion (Wang et al., 2008).

Electromechanical breakdown
Electromechanical breakdown is likely to happen in dielectric foils of low Young's modulus and easy deforming. When a dielectric foil is subjected to a high voltage, an electrostatic compressive force which will cause deformation of the dielectric foil is produced. If the electric stress exceeds the mechanical compressive strength of the dielectric foil, breakdown of the dielectric foil takes place. The electromechanical breakdown model is shown in Fig. 2.
If the thickness of the specimen is 0 d and is compressed to d under an applied voltage U , then the electrically developed compressive stress is in equilibrium Where Y is the Young's modulus.
A plot of thickness as a function of voltage on normalized scales is given in Fig. 3. It is obviously seen that there is a critical voltage above which the thickness goes to zero. This occurs Therefore, the critical breakdown strength of electromechanical breakdown is (Laghari & Sarjeant, 1992;Stark & Garton, 1955)   The electromechanical breakdown model is rather unrealistic as it assumed that the dielectric material somehow disappeared to an infinitesimal thickness at em U U  . Therefore, it may ignore several factors, such as possible earlier instability at microscopic areas of stress concentration, the dependence of Y on time and stress and plastic flow. Stack and Garton pointed out that the stress concentration at microscopic areas is instable, which may result in a localized thinning and breakdown. Blok and LeGrand considered that local regions subjected to higher-than-average electric fields and experienced a shear stress which tended to form an indentation. The indentation gives rise to an even more inhomogeneous field, produces a sharp depression and eventually causes the material to flow radially away (Blok & LeGrand, 1969).

Electrofracture breakdown
Zeller et al Zeller & Schneider, 1984) proposed electrofracture as an aging mechanism in which the growth of a filamentary crack through a dielectric was caused by electrostatic forces. The model is based on the concepts of fracture mechanics in which the change in electrostatic energy at growth is equal to the mechanical energy required for partial discharge channel formation.
In the electrofracture mechanism the local field at the crack tip is considered to be insufficiently high for the strain energy density em W contribution to be insignificant, i.e. electrostatic energy density es em W W  . So that the criterion for electrofracture is: The electrofracture breakdown strength at the end of the crack tip is given by EF E is the local field at the tip of the crack whereas the electromechanical breakdown field is the applied field.

Filamentary electromechanical breakdown
Fothergill proposed the filamentary electromechanical breakdown in which a filamentaryshaped crack propagates through a dielectric releasing both electrostatic energy and electromechanical strain energy stored in the material due to the applied electric field (Fothergill, 1991(Fothergill, , 1992. It operates at higher local fields.
The filamentary electromechanical breakdown model assumed electric field at the filament tip that are sufficiently high for em es W W  , so that the appropriate criterion is: The filamentary electromechanical breakdown strength is defined as In general, 2 f G Yr  , so that eq. (18) is written as (Hikita et al., 1987) A modified relationship between applied voltage and electric field at the tip of a filament is given by Eichhorn (Eichhorn, 1977) as: Where d is the distance from the filament to the counter electrode, and is assumed to be the full thickness of the specimen in the worst-case estimate. f r is the tubular crack of radius and depends on what initiates the filament. If it is assumed of / f E U r  , the radius of the filament can be predicted from eq. (20). Solving eq. (20) give 0.29 f r m   which is quite reasonable.
Both the electrofracture and filamentary electromechanical breakdown mechanisms assume a crack propagation process. The limiting speed of a crack is equal to that of a longitudinal elastic wave in the material.
According to the Young's modulus of different materials and Eq. (14), Eq. (16) and Eq. (19), the critical strengths of polyester, polyimide, polycarbonate, PE and polypropylene are obtained. The results are shown in Table 2, from which it is apparent that the critical strength of electric-mechanical breakdown is higher than the strength given by Ref. (Wang et al. 1992) in the mm regime. Therefore, the above results support the electric-mechanical breakdown mechanism for relatively thinner dielectric foils.

Recent development in polymer breakdown theory
Many studies have been published on high-field electrical conduction and breakdown of insulating polymers and there are also many good reviews and books which deal with this subject (Bartnikas & Eichhorn, 1983;Ieda, 1980O'Dwyer, 1973). However, the dielectric breakdown processes of polymers are complicated due to their highly complex structures. Therefore, much work has been devoted to clarify the correlation between the breakdown characteristics and the properties inherent to polymers. Theories on the breakdown mechanisms are usually divided into three categories: electronic, thermal (Hap & Raju, 2006) and mechanical processes.
The different degrees of ordering of macromolecules and the presence of regions of varying densities gives rise to different conductivities under an applied field. A non-uniform voltage drop occurs within the polymer when a current begins to flow, leading to variable distribution of electric field. A detailed analysis of breakdown processes in polymers has been discussed by Ieda et al (Ieda, 1980). Artbauer proposed the basic idea of Free Volume Theory (Artbauer, 1996), which was taken in explaining the breakdown phenomena of polymer around the glass transition temperature. The amorphous phase in polymers with unoccupied volume consisting of holes of molecular order presents a free path for the electrons to accelerate under an external field and gain energy. Job et al. improved dielectric strength of polymers by chemical modifications. They have increased the breakdown strength of polyethylene terephthalate (PET) films by insitu polymerization of a layer of polyaniline (PANI). A 30% increase in dielectric strength was observed when the non-conductive PANI filled the voids of PET. And they proposed the breakdown was the electron avalanche process (Job et al., 2003). Breakdown in polymers with saturated bonds, such as polyethylene and polypropylene, can not be explained by the electron avalanche mechanism as they have a short free path length. As a result, the electrons cannot gain sufficient energy by acceleration to cause avalanche. In these polymers, electrical breakdown is the last step in a process of polymer degradation that results in the formation of conducting channels. The formation of these conducting channels depends on the intensity and time of the applied external field. The lifetime of these polymers decreases exponentially with applied electric field (Zakrevski et al., 2003). Overall, a general consensus has emerged that the reduction in dielectric breakdown strength of polymers is closely related to the dissociation (bond scission) of polymer molecules under an applied external field. The dissociation process is considered to be the initiation of polymeric breakdown. However, the exact mechanism of dissociation is unclear. One approach to explain this phenomenon is the non-uniform distribution of electric field intensity within a polymer due to electrode defects and inherent differences within the structure of polymer (Zakrevski et al., 2003).
The dielectric breakdown field of polyimide (PI) thin films has been studied across the temperature range from 25 to 400 °C under dc conditions by Diaham et al. it appeared that both the breakdown field value and β-scale parameters decrease with increasing electrode area following the area extrapolation law. It was found that the β-parameter has presented an unexpected increase with increasing the thickness thanks to percolation theories (Diaham et al., 2010).

Peruani et al described the dielectric breakdown model in conductor-loaded composites.
Conducting particles are distributed at random in the insulating matrix, and the dielectric breakdown propagates according to new rules to take into account electrical properties and particle size. Dielectric breakdown patterns are characterized by their fractal dimension and the parameters of the Weibull distribution. Studies are carried out as a function of the fraction of conducting in homogeneities. The fractal dimension of electrical trees approaches the fractal dimension of a percolation cluster when the fraction of conducting particles approximates the percolation limit (Peruani et al., 2003). DammigQuiña et al developed a capacitive dielectric breakdown model (CDBM) of electrical tree growth to simulate dielectric breakdown in polymer materials. Dependence of the fractal dimension on the applied voltage, the breakdown process has its thermodynamic origin in the interplay between energy injection and space charge formation processes. The model is capable of predicting electrical trees with fractal dimensions and qualitatively reproducing the temporal evolution of the breakdown process (DammigQuiña et al., 2008).
Kim and Shi investigated the dielectric strength of an interlevel low relative permittivity dielectric for various film thicknesses and temperatures by using I-V measurements with metalinsulator-semiconductor (MIS) structures. It is found that the dielectric breakdown mechanism is electromechanical in origin for relatively thick films (thickness >500 nm). The dielectric strength is proportional to the square root of Young's modulus of the films (Kim & Shi, 2001).

Polyester foil
The polyester foil studied here is compound of biaxially-oriented polyethylene terephthalate, as shown in Fig. 4. Biaxially-oriented Polyethylene terephthalate, commonly abbreviated BOPET, is made from stretched polyethylene terephthalate (PET) and used for its high tensile strength, chemical and dimensional stability, transparency, gas and aroma barrier properties and electrical insulation ( Sarker et al., 2001).
The end properties of the BOPET are direct consequence of their super molecular structure. The final structure of the BOPET depends mainly by the melt spinning conditions and by the subsequent heat-mechanical modifications, too. The desirable super molecular structure and the needed final morphologies and properties of the PET can be obtained by controlling of the forming process parameters as well as the subsequent treatment. For example, BOPET can be metallized by vapor deposition of a thin film of evaporated aluminum, gold, or other metal onto it. Metallized BOPET film, along with other plastic films, is used as a dielectric in foil capacitor. Four film thicknesses of 9µm, 12µm, 15µm and 18 µm were selected to study the breakdown strength.

Experimental setup
Dielectric breakdown field measurements have been performed using a dc voltage supplied by a 15 kV voltage source connected to the sample. The electrical schematic diagram is shown in Fig. 5. In Fig. 5, 1, 2 and 6 are transmission lines. 3 is the polymer-foil switch. 4, 5 are the charging resistance. The breakdown field was identified when the polymer-foil switch is self-breakdown, and output a rectangular pulse. The maximum voltage supplied was also recorded using a divider connected to the generator and was considered as the breakdown voltage (Wang et al., 2004(Wang et al., , 2006. 1,2,6 transmission line; 3 polymer-foil switch; 4,5 charging resistance  6 is the experimental arrangement of polyester-foil switch. The high voltage electrode (HE) and ground electrode (GE) are planes made of copper, and between HE and GE is the polyester foil, whose thickness varies among 9µm, 12µm, 5µm and 18µm. The selfbreakdown voltage is attained by a resistive divider. The measurement is accomplished by a 500MHz digital storage oscilloscope.

Statistical analysis
Although a number of samples of the same material may be tested, each sample might exhibit breakdown at a different voltage due to random structural and measurement differences. The breakdown voltage can be considered as a random variable, which necessitates a statistical analysis of the breakdown data. Electrical breakdown can be considered as a system failure under the condition where a weak link in the system fails. The Weibull distribution, which is a type of extreme value distribution, is the most common for such applications (Weibull, 1951;Wu & Vollertsen, 2002).
Where ( ) F x is the cumulative probability of failure,  is the scale parameter (V/cm) corresponding to a probability of failure ( ) F  =63.2%,  is the shape parameter. A high value is related to a low scattering of the data. The location (or threshold) parameter  has been set to zero.
The experimental data have been ranking using the median rank approximation given by (Chauvet & Laurent, 1993;Fothergill, 1990): Where, i and n are the rank of a failed sample and the total number of tested samples, respectively. For plotting the Weibull distribution law, the transformation of equ. (21) into equ. (23) has been realized: 10 10 10 The Weibull parameters have been extracted considering a confidence interval of 90 %. Both the maximum likelihood and least square fit methods have been also applied leading to similar α-and β-parameter values (Diaham et al., 2010;Laihonen et al., 2007aLaihonen et al., , 2007b.

Results and discussion
Experiments were performed to measure the self-breakdown voltage of 10µm level foil. Fig.  7 and Fig. 8 show self-breakdown voltage and deviation thickness-varied polyester foils under positive voltage, respectively. The dashed lines through the data points represent the average self-breakdown voltage of the foils. From Fig. 7, it is clearly seen that the average self-breakdown voltage under positive voltage for 9µm, 12µm, 15µm, 18µm foils are 4.2, 5.7, 7.2 and 9.3 kV respectively, and the corresponding standard deviations are 0.6, 0.5, 0.8 and 0.9 kV. As shown in Fig. 8, the breakdown strength for 9, 12, 15, 18µm foils under positive voltage are 467, 475, 480 and 517kV/mm, and the corresponding standard deviations are 67, 42, 51 and 50 kV/mm, respectively.
Compared to the dielectric strength of thickness in mm regime, it can obviously that the breakdown strength for 10µm level foil is far greater than that breakdown strength in mm regime. That is named 'film strengthenization effect'. Nevertheless, as far as the breakdown strength for those four kinds of thickness foils are concerned, it exhibits a slight increase with increasing thickness for our results. It seems to be contrary to the film strengthenization effect. The reasons for this phenomenon are surface roughness conditions of electrodes, thickness deviation, defects and treatment processing, and so on. This 'curious' behavior also can be interpreted with the percolation theory. It based on a priori supposed that when the probability to find defects into the bulk increases, an increase in the scattering of the breakdown and a decrease in the shape parameter value ought to be observed (Diaham et al., 2010;Essam, 1980;Helgee&Bjellheim, 1991). Fig. 10 show self-breakdown voltage and breakdown strength for thicknessvaried polyester foils under negative voltage, respectively. The average self-breakdown voltage under negative voltage for 9, 12, 15, 18µm foils are 3.8, 5.4, 7.4, and 9.0kV respectively, and the corresponding standard deviations are 0.5, 0.7, 0.4 and 0.9 kV. The breakdown strength for 9, 12, 15, 18µm foils are 444, 450, 490 and 500kV/mm, and the corresponding standard deviations are 53, 58, 28 and 49 kV/mm. It is also found that the breakdown voltage and dielectric strength of polyester film under negative voltage is smaller than that one under positive voltage. In other words, the dielectric strength of polyester film is evidently dependent of the voltage polarity. It can be interpreted by the space charge effect in relation to polyester solid structure (Ieda, 1987).  The influence of the thickness on the Weibull parameters was also investigated. Fig. 11 shows the α-parameter values in a plot versus the polyester film thickness. The solid line and dash line represent the linear fits. This result is not in good agreement with typical value found in the literature for polymers and for such a litter variation of the thickness (Diaham et al., 2010;Helgee&Bjellheim, 1991).

Fig. 9 and
In comparison with the theoretical results, the experimental results show that the critical breakdown strength of the polyester film in the intrinsic breakdown model is far lower than (a) 9µm (b) 12µm (c) 15µm (d) 18µm the experimental results and the critical breakdown strength of the polyester film in the electromechanical breakdown model and filamentary electromechanical breakdown model almost agrees with the experimental data of the polyester film. Therefore, it is concluded that the dielectric breakdown mechanism for thickness in micro-meter range is dominantly controlled by the electromechanical breakdown mechanism. Figure 11. Changes in the scale parameter versus the polyester-foil thickness

Conclusion
The theoretical and experimental results of dielectric strength of polyester films in micrometer regime are presented. Theoretical analysis was carried out based on the intrinsic breakdown model and the electromechanical breakdown model. The critical breakdown strength of intrinsic breakdown model is estimated on the basis of Hippel's low-energy criterion and Frohlich's high-energy criterion. The electromechanical breakdown model, electrofracture and filamentary electromechanical breakdown model are also introduced. The thickness dependence of dielectric strength of the polyester films was also investigated in a plane-plane electrode arrangement with different polar DC voltage. The thickness of the polyester films varied among 9µm, 12µm, 15µm and 18µm. Experimental results show that there is greatly increasing on the dielectric strength in micro-meter regime, compared to the dielectric strength of thickness in mm regime. Nevertheless, it exhibits a slight decrease with increasing thickness for our results. This 'curious' behavior is discussed on the basis of the percolation theory. It is also found that the dielectric strength of polyester film under negative voltage is smaller than that one under positive voltage. In other words, the dielectric strength of polyester film is evidently dependent of the voltage polarity. It can be interpreted by the space charge effect in relation to polyester solid structure. The comparison between the theoretical analysis and the experimental results is also presented.
The results show that the critical breakdown strength of the polyester film in the intrinsic breakdown model is far lower than the experimental results, and the critical breakdown strength of the polyester film in the electromechanical breakdown model almost agrees with the experimental data of the polyester film. Therefore, it is concluded that the dielectric breakdown mechanism for thickness in micro-meter range is dominantly controlled by the electromechanical breakdown mechanism.