Thickness-Dependent DC Electrical Breakdown of Polyimide Modulated by Charge Transport and Molecular Displacement

Article Thickness-Dependent DC Electrical Breakdown of Polyimide Modulated by Charge Transport and Molecular Displacement

Daomin Min 1,* , Yuwei Li 1, Chenyu Yan 1, Dongri Xie 1, Shengtao Li 1,*, Qingzhou Wu 2 and Zhaoliang Xing 3

1 State Key Laboratory of Electrical Insulation and Power Equipment, Xi’an Jiaotong University, Xi’an 710049, China; [email protected] (Y.L.); [email protected] (C.Y.); [email protected] (D.X.) 2 Institute of Fluid Physics, China Academy of Engineering Physics, Mianyang 621900, China; [email protected] 3 State Key Laboratory of Advanced Power Transmission Technology, Beijing 102209, China; [email protected] * Correspondence: [email protected] (D.M.); [email protected] (S.L.); Tel.: +86-186-8187-7702 (D.M.); +86-158-2993-9692 (S.L.) Received: 15 August 2018; Accepted: 8 September 2018; Published: 11 September 2018

Abstract: Polyimide has excellent electrical, thermal, and mechanical properties and is widely used as a dielectric material in electrical equipment and electronic devices. However, the influencing mechanism of sample thickness on electrical breakdown of polyimide has not been very clear until now. The direct current (DC) electrical breakdown properties of polyimide as a function of thickness were investigated by experiments and simulations of space charge modulated electrical breakdown (SCEB) model and charge transport and molecular displacement modulated (CTMD) model. The experimental results show that the electrical breakdown field decreases with an increase in the sample thickness in the form of an inverse power function, and the inverse power index is 0.324. Trap properties and carrier mobility were also measured for the simulations. Both the simulation results obtained by the SCEB model and the CTMD model have the inverse power forms of breakdown field as a function of thickness with the power indexes of 0.030 and 0.339. The outputs of the CTMD model were closer to the experiments. This indicates that the displacement of a molecular chain with occupied deep traps enlarging the free volume might be a main factor causing the DC electrical breakdown field of polyimide varying with sample thickness.

Keywords: charge transport; DC electrical breakdown; free volume; molecular chain displacement; sample thickness

1. Introduction Polyimide has excellent thermal stability, outstanding electrical properties, and high mechanical strength and is widely used as a dielectric material in power capacitors for energy storage [1–3], the solar array of satellite [4,5], and electronic devices [6], and so on. As the voltage level increases and the devices are required to be further miniaturized, dielectric materials with high electrical strength need to be developed. Insulation failures of solid dielectrics under electrical stresses can cause permanent damage to the materials and shorten the service life of electrical equipment or electronic devices. Therefore, the electrical breakdown mechanism of solid dielectrics is a research focus in the ﬁelds of dielectric energy storage and electrical insulation.

Polymers 2018, 10, 1012; doi:10.3390/polym10091012 www.mdpi.com/journal/polymers Polymers 2018, 10, 1012 2 of 18

The electrical breakdown field of dielectric materials depends on sample thickness. The electrical breakdown field of polyimide film under direct current (DC) voltage was measured at various sample thicknesses and various temperatures. The DC electrical breakdown field decreases with an increase in the sample thickness [7]. The relation between the DC electrical breakdown field and sample thickness obeys an inverse power law with power indexes of 0.16 and 0.25 at 300 ◦C and 400 ◦C, respectively. Regardless of the thickness of the film, the DC breakdown field decreases as the temperature increases in a range from 25 to 400 ◦C[7]. Similar inverse power law dependence of the DC electrical breakdown field on sample thickness was also observed on polytetrafluoroethylene [8] and an acrylic dielectric elastomer [9]. In addition, electrical breakdown properties of polyimide, polyethylene terephthalate, polyetheretherketone, and polyethersulfone were studied under alternating current (AC) voltage with a rising rate of 500 Vs−1, and it was found that the AC electrical breakdown field decreases with an increase in the sample thickness in an inverse power-law form for all the four polymers [10]. The electrical breakdown field under nanosecond pulses was also found to be a decreasing function of sample thickness on polyethylene, polytetrafluoroethylene, polymethyl methacrylate, and nylon [11]. There are several breakdown mechanisms such as electron avalanche breakdown [12,13], electromechanical breakdown [12,13], free volume breakdown, and space charge modulated electrical breakdown [14–16], to interpret the sample thickness effect on the electrical breakdown of polymers. In the electron avalanche breakdown [12,13], free electrons in the conduction band of a dielectric material could move a certain distance under the electric field, resulting in an energy gain of electrons. When the energy exceeds the band gap energy, electrons in the valence band might be excited to the conduction band, causing chemical bond scission. The released electrons further collided with and ionized other matrix atoms to cause an electronic avalanche effect, resulting in the multiplication of local current and eventually triggering electrical breakdown. Since a critical number of generations of electrons are required to be produced over the sample thickness by impact ionization, the electrical breakdown field decreases with an increase in the sample thickness [12,13]. The Stark–Garton model of electromechanical breakdown has been widely used to predict the breakdown strength of thermoplastics, and the electromechanical breakdown strength depends on Young’s modulus and dielectric constant for temperature-sensitive polymers [12,13]. Since both the electrostatic compressive stress generated by the electrostatic attraction of two electrodes and the opposing elastic stress depend on sample thickness, the electrical breakdown field is a decreasing function of sample thickness [12,13]. Moreover, in the free volume breakdown theory, it was assumed that the electrical breakdown field of a polymer depends on the longest average free path of electrons. Electrons are accelerated in free volume, and their average free path depended on the maximum length of free volume. When the electron obtained enough energy in free volume to overcome the potential barrier, the local current was multiplied to heat the material to an extremely high temperature, and eventually resulting in electrical breakdown [17]. The longest free path is a function of sample size from a statistical point of view, so the electrical breakdown strength is related to the sample thickness. However, the relations between DC electrical breakdown strength and sample thickness obtained by the models of electron avalanche breakdown, electromechanical breakdown, and free volume breakdown do not satisfy the inverse power law as observed in experiments [12,17]. It has demonstrated that charges are injected into and space charges are accumulated inside a dielectric material under the application of a high electric field [18–21]. The electric field inside the dielectric material will be distorted by the accumulated space charges, and electrical breakdown occurs when the maximum local electric field exceeds a threshold value [20,22]. Then, a model considering the formation of space charges and the distortion of electric field was established to simulate the DC electrical breakdown properties of low-density polyethylene, and it was found that the breakdown strength is an inverse power function of sample thickness [14]. A similar model was used to investigate the electrical breakdown properties of oil-impregnated paper [15] and low-density polyethylene-based nanocomposites [16]. The simulation results are in good agreement with experiments [15,16]. These results show that the electric field distortion caused by the space charge accumulation under a high field may play Polymers 2018, 10, 1012 3 of 18 an important role in the breakdown characteristics of polyimide. In addition, the macroscopic dielectric properties of polymers are under the influence of molecular chain motion [23]. A molecular chain with a deep trapped charge can move toward the electrode via the Coulomb force under the high electric field, resulting in the elongation of free volume [24–26]. The elongation of free volume caused by the displacement of the molecular chain associating with the accumulation of space charges and the distortion of electric field may determine the electrical breakdown behavior of polymeric materials. Therefore, in the present paper, two electrical breakdown models based on the space charge effect and molecular chain displacement dynamics are utilized to investigate sample thickness dependent the DC electrical breakdown properties of polyimide.

2. Experimental Procedure Polyimide films (type: 6051) with various thicknesses were purchased from Baoying County Seiko Insulation Materials Co., Ltd., Yangzhou, China. Firstly, polyamide acid was synthesized by polycondensation between phthalic acid dianhydride and diamino diphenyl ether in a high polar solution. Films were then prepared by the solution of polyamide acid through casting and stretching methods. Then, polyimide films were obtained via drying and imidization of polyamide acid at high temperature. The thickness of polyimide films ranges from 25 to 250 µm. Before the experiments, all the samples were heated in a chamber at about 100 ◦C for 12 h to eliminate the humidity inside. Au electrodes with a radius of 15 mm were sputtered by ion sputter coater on both sides of the sample. Then, a Broadband Dielectric Spectrometer (Concept 80, Novocontrol Technologies, Frankfurt, Germany) was used to measure the complex permittivity of polyimide with a thickness of 100 µm at room temperature. The voltage applied −2 5 on the sample was 1 Vrms and the frequency was from 10 to 10 Hz. A thermal stimulation depolarization current (TSDC, Concept 90, Novocontrol technologies, Frankfurt, Germany) was carried out on a polyimide sample with a thickness of 100 µm to investigate its trap distribution characteristics. The sample was placed in the vacuum chamber of TSDC device. The sample was first heated to 180 ◦C and then polarized for 30 min at an applied voltage of 250 V. After the polarization, the sample was cooled to −100 ◦C by liquid nitrogen and then short-circuited for 10 min. Thereafter, the sample was heated to 290 ◦C with a heating rate of 3 ◦Cmin−1. At the same time, the depolarization current was measured using a pico-ammeter (Keithley 6517B, Beaverton, OR, USA). Surface potential decay (SPD) measurement was performed on polyimide films with a thickness of 25 µm to investigate potential decay and carriers mobility characteristics. The SPD measurement includes two processes, which are the charging process and detection process. When a voltage is applied to the needle electrode, energetic charges emitted from the needle electrode will impact with air molecules and cause ionization, which is called corona discharging. During the corona discharging, a large amount of positive and negative charges are generated. Driven by the electric field of the grid electrode, charges will drift toward and be deposited on the sample surface. In this paper, voltages applied to the needle and gate electrodes were 12 and 8 kV, respectively, for positive corona discharging, and were −12 and −8 kV respectively for negative corona discharging. The charging time on polyimide sample was 2 min. After the charging process, the surface potential of samples was measured by a high-voltage electrostatic voltmeter (P0865, Trek, Lockport, NY, USA) with a non-contact probe (3453ST, Trek, Lockport, NY, USA). The DC electrical breakdown fields of polyimide films with different thicknesses from 25 to 250 µm were measured using a computer-controlled voltage breakdown test device (HJC-100kV, Huayang Instrument Co., Ltd., Yangzhong, China). The DC electrical breakdown experiments were carried out at 30 ◦C using spherical copper electrodes with a diameter of 25 mm in transformer oil (Kramai25#, China National Petroleum Corporation, Karamay, China). A DC voltage with a ramping rate of 1 kVs−1 was applied to the samples until the breakdown occurring. The breakdown voltages were recorded by a computer. The breakdown tests were repeated at least 15 times for each thickness of the samples. The average of all the data was taken as the electrical breakdown field for the sample with corresponding thickness. Polymers 2018, 10, 1012 4 of 18

3. Experimental Results Figure1 shows the real and imaginary parts of relative complex permittivity, ε’ and ε”, as a function of frequency in semi-logarithmic coordinates obtained from the polyimide sample at room temperature. It can be seen that the real part of the relative complex permittivity increases slightly from 3.5 to 3.6 with a decrease in frequency. The imaginary part of the relative complex permittivity is lower than 3.6 × 10−3 in the frequency range from 10−2 to 105 Hz. The dielectric relaxation strength of the relaxation peak around 30 Hz is very small, which means the dipolar moment is very low. The low dielectric loss ε”/ε’ indicates that Joule heating generated by the orientation of dipoles during the DC electrical breakdown experiments at room temperature can be neglected.

Figure 1. The real and imaginary parts of relative complex permittivity of polyimide as a function of frequency at room temperature.

Figure2 demonstrates the TSDC experimental results of polyimide. Thermally stimulated relaxation processes can be observed at the temperature range from 10 to 170 ◦C. One obvious relaxation peak is around 69 ◦C, while another relaxation peak may locate around 135 ◦C. There may be other relaxation processes between 69 and 135 ◦C. In order to reveal the thermally stimulated processes and their activation energies, the experimental results in Figure2 were analyzed by the classical TSDC theory [27]:

Z T0 EA 1 EA jTSC(T) = B exp − − exp − dT (1) kBT βτ0 T0 kBT

−2 −2 where jTSC(T) is the thermally stimulated depolarization current density in Am , B is a constant in Am , EA is the activation energy of the relaxation process in eV, τ0 is the relaxation time constant in s, β is ◦ −1 the heating rate in Cs , kB is the Boltzmann constant, T0 is the initial temperature of the sample at the beginning of the heating process in ◦C, and T is the temperature of the sample after heating in ◦C.

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FigureFigure 2. Thermal 2. Thermal stimulation stimulation depolarization depolarization current current (TSDC) (TSDC) experimental experimental andand fitting fitting results results of of polyimide.polyimide. Symbols Symbols and and solid solid curves curves represent represent experimental experimental and and fitting fitting results, results, respectively. respectively. The The fitting curvesfitting are calculatedcurves are calculated by Equation by Equation (1). (1).

The TSDCThe TSDC experimental experimental results results were were ﬁtted fitted by Equation by Equation (1), and (1), four and relaxation four relaxation peak components peak couldcomponents be obtained. could As be shown obtained. in Figure As shown2, it in can Figure be seen 2, it can that be the seen ﬁtting that the results fitting are results in good are in agreement good agreement with the experiments. We can determine the peak temperature, activation energy and with the experiments. We can determine the peak temperature, activation energy and relaxation time relaxation time for the four relaxation processes listed in Table 1. The activation energies of four peaks for theat 69, four 87, relaxation 109 and 135.5 processes °C are 0.60, listed 0.65, in 0.70 Table and1. 0.83 The eV, activation respectively. energies As the of temperature four peaks at at the 69, 87, ◦ 109 andrelaxation 135.5 peakC are increase, 0.60,0.65, the corresponding 0.70 and 0.83 activati eV, respectively.on energy increases. As the The temperature three peaks atat 69, the 87 relaxation and ◦ peak109 increase, °C may the correspond corresponding to shallow activation traps that energy assist carriers increases. hopping The process three peaksin polyimide, at 69, 87while and the 109 C maypeak correspond at 135.5 °C to shallowmay correspond traps that to deep assist traps carriers that can hopping capture processmobile carriers in polyimide, and accumulate while the space peak at 135.5charges.◦C may The correspond energy of todeep deep traps traps is consistent that can capturewith the mobileresults obtained carriers from and accumulatethe Arrhenius space relation charges. The energybetween of conductivity deep traps and is consistent temperature with [28]. the The results activati obtainedon energy from 0.83 eV the of Arrhenius relaxation peak relation at 135.5 between conductivity°C was used and as temperature deep trap energy [28]. parameter The activation in the simulation energy 0.83 of electrical eV of relaxation breakdown. peak at 135.5 ◦C was used as deep trap energy parameter in the simulation of electrical breakdown. Table 1. Parameters extracted from TSDC experimental results by Equation (1).

Table 1. ParametersPeak Temperature extracted (°C) from TSDCEA (eV) experimental B (Am−2) resultsτ0 by(s) Equation (1). 69 0.60 2.63 × 10−4 7.50 × 10−7 ◦ −2 Peak Temperature87 ( C) EA (eV) 0.65 B 2.01(Am × 10)−4 4.23 ×τ 100 (s)−7 69109 0.60 0.70 2.63 1.80× ×10 10−−44 3.237.50 × 10× 10−7 −7 −4 −7 87135.5 0.65 0.83 2.01 1.60× ×10 10−4 3.094.23 × 10× 10−8 109 0.70 1.80 × 10−4 3.23 × 10−7 135.5 0.83 1.60 × 10−4 3.09 × 10−8 Figure 3 shows the surface potential experimental results of samples charged by negative corona discharging and positive corona discharging as a function of time. During the charging process, Figurenegative3 showscharges the and surface positive potential charges are experimental deposited on results the surface of samples of polyimide charged and by electric negative fields corona dischargingare built-up and positiveinside the corona material. discharging After the charging, as a function surface of time.charges During inject theinto chargingand move process, toward negativethe chargesgrounded and positive electrode charges in the are bulk deposited of material. on the The surface migration of polyimide of charges and leads electric to the fields decay are of built-up surface inside the material.potential. After The thedecay charging, rate of surface surface potential charges varies inject intobefore and and move after toward the injected the grounded charge carriers electrode flow in the out of the dielectric material [29,30], as shown in Figure 3. The time when the front charge carriers bulk of material. The migration of charges leads to the decay of surface potential. The decay rate of surface arrives at the grounded electrode is defined as transit time tT. The transit point existing at the potential varies before and after the injected charge carriers flow out of the dielectric material [29,30], beginning of the potential decay curve can represent the mobility of carriers controlled by shallow as showntraps in[29,30]. Figure In3. order The time to obtain when the fronttransit charge time t carriersT, the potential arrives atdecay the groundedresults were electrode fitted by is definedan as transitexponential time tT .function The transit [31], point and existingthen we at obtained the beginning the relation of the potentialbetween decaytdφs/d curvet and cant. The represent time the mobilitycorresponding of carriers to controlled the peaksby can shallow be regarded traps as [29 transit,30]. Intime order tT, and to obtainis used theto calculate transit time carriertT ,mobility the potential decaycontrolled results were by shallow fitted traps by an according exponential to the function following [31 equation], and then [29], we obtained the relation between tdϕs/dt and t. The time corresponding to the peaks can be regarded as transit time tT, and is used to calculate carrier mobility controlled by shallow traps according to the following equation [29],

2 µ0(e,h) = d /φs0tT (2) Polymers 2018, 10, x FOR PEER REVIEW 6 of 18 Polymers 2018, 10, x FOR PEER REVIEW 6 of 18 μφ= 2 Polymers 2018, 10, 1012 0e,h() dts0 T 6 of(2) 18 μφ= 2 0e,h() dts0 T (2) Here μ0 is carrier mobility controlled by shallow traps in m2V−1s−1, d is the thickness of sample in Here µ is carrier mobility controlled by shallow traps in m2V−1s−1, d is the thickness of sample m, φs0Here represents μ00 is carrier the initial mobility surface controlled potential by inshallow V. The traps subscripts in m2V have−1s−1, thed is following the thickness meaning: of sample (e) for in in m, ϕ represents the initial surface potential in V. The subscripts have the following meaning: (e) electronsm, φs0 representss0 and (h) forthe holes. initial By surface calculation, potential the in hole V. andThe electronsubscripts mobilities have the controlled following by meaning: shallow (e) traps for forareelectrons electrons1.80 × and10−14 and (h) m 2for (h)V−1 sholes. for−1 and holes. By 3.67 calculation, By × 10 calculation,−14 m2 Vthe−1s hole−1 the respectively. holeand electron and electron mobilities mobilities controlled controlled by shallow by shallow traps −14 2 −1 −1 −14 2 −1 −1 trapsare 1.80 are × 1.80 10−14× m102V−1s−1m andV 3.67s ×and 10−14 3.67 m2V×−110s−1 respectively.m V s respectively.

Figure 3. Surface potentials of polyimide charged by negative corona discharging and positive corona Figuredischarging 3. Surface as a function potentials of oftime polyimide at room charged temperature by negative . corona discharging and positive corona discharging asas aa functionfunction ofof timetime atat roomroom temperature.temperature. Figure 4 shows the experimental results of the DC electrical breakdown field of the polyimide film, FigureFb, as 4a 4 showsfunction shows the the of experimental thickness,experimental d results, at results room of the temperature.of DC the electrical DC electrical As breakdown shown breakdown in field Figure of field the 4a, polyimide theof the DC polyimide electrical film, Fb, asbreakdownfilm, a function Fb, as a fieldof function thickness, of polyimide of dthickness,, at room films temperature. d decreases, at room temperature.with As shown an increase in FigureAs inshown 4samplea, the in DC Figurethickness. electrical 4a, Forthe breakdown DCexample, electrical field the ofelectricalbreakdown polyimide breakdown field films of decreases polyimide field is with 639 films V anμm increasedecreases−1 at a sample in samplewith thicknessanthickness. increase of in25 For sample μ example,m, 371 thickness. Vμ them− electrical1 at For100 example,μ breakdownm, and 307the −1 −1 −1 fieldVelectricalμm is−1 at 639 250breakdown V µμmm. Inat addition, field a sample is 639 the thickness V derivativeμm−1 at of a 25sample ofµ DCm, 371electricalthickness Vµm breakdown ofat 25 100 μm,µ m,371 field and Vμ withm 307−1 at Vrespect µ100m μm, atto and 250sample µ307m. InthicknessVμ addition,m−1 at 250dF the bμ/dm.d derivative decreasesIn addition, of with DC the electricalderivativethe increase breakdown of DCin sample electrical field thickness. with breakdown respect The tofield relation sample with thicknessrespectbetween to thedsampleFb /dDCd decreaseselectricalthickness breakdown withdFb/d thed decreases increase field inand with sample sample the thickness. increase thickness Thein looks sample relation like thickness.betweenan inverse the The power DC relation electrical function. between breakdown Accordingly, thefield DC andweelectrical change sample breakdown the thickness linear looks coordinatesfield likeand an sample inversein Figure thickness power 4a to function. doublelooks like Accordingly,logarithmic an inverse coordinates we power change function. the in linearFigure Accordingly, coordinates 4b. It can inbewe Figure seenchange from4a the to Figure doublelinear 4bcoordinates logarithmic that the DC in coordinates Figure electrical 4a to inbreakdown double Figure4 logarithmicb. field It can of bepolyimide coordinates seen from is Figurelinearin Figure 4withb that 4b. sample It the can DCthicknessbe seen electrical from under breakdownFigure double 4b that fieldlogarithmic the of polyimideDC electricalcoordinates. is linear breakdown The with electrical sample field of thickness breakdown polyimide under isfield linear double decreases with logarithmic sample with coordinates.increasingthickness undersample The electrical double thickness breakdownlogarithmic in an inverse field coordinates. decreases power function with The increasing electrical form, Fb sample =breakdown kd−n. thicknessThe power field in index andecreases inverse calculated power with −n functionfromincreasing the form, experimental sampleFb = kdthickness .results The powerin isan 0.324. inverse index Similar calculated power results function from were the form, experimental also F bobserved = kd−n. resultsThe by power Diaham is 0.324. index Similaret al. calculated [7], results and werethefrom power alsothe observedexperimental index of by sample Diaham results thickness et is al. 0.324. [7], dependent and Similar the power results DC electrical index were of also sample breakdown observed thickness field by dependent Diahamis in a range et DC al. of electrical[7], 0.16 and to breakdown0.25.the power index field isof in sample a range thickness of 0.16 to dependent 0.25. DC electrical breakdown field is in a range of 0.16 to 0.25. 700 800 (a) ) (b) )

-1 700 700 -1 800 m m

μ (a) ) 600 (b) μ

) 600

-1 700 -1 m m

μ 600 n=0.324 μ 500600 500 500 n=0.324 500 400 400 400 400 300 300 300 300 200 Electrical breakdwon field (V (V field breakdwon Electrical 200 0 50 100 150 200 250 10 100 1000 200 Electrical breakdwon field (V Sample thickness (μm) (V field breakdwon Electrical 200 Sample thickness (μm) 0 50 100 150 200 250 10 100 1000 μ μ Figure 4. ExperimentalSample resultsresults thickness ofof (DC DCm) electrical electrical breakdown breakd own ﬁeld field of of polyimide polyimideSample thickness at at various various ( m) thicknesses thicknesses in linearinFigure linear coordinates 4. coordinates Experimental (a) ( anda) resultsand in doublein doubleof DC logarithmic electrical logarithmic breakd coordinates coordinatesown field (b). (b of). polyimide at various thicknesses in linear coordinates (a) and in double logarithmic coordinates (b).

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4.4. NumericalNumerical AnalysisAnalysis ofof SampleSample Thickness-DependentThickness-Dependent DCDC ElectricalElectrical BreakdownBreakdown AtAt aa high high electric electric field, field, charges charges in an in electrode an electrod cane be can injected be injected into a dielectricinto a dielectric material viamaterial Schottky via thermionicSchottky thermionic emission or emission quantum or mechanical quantum mechanic tunneling.al Thetunneling. accumulation The accumulation of space charges of space under charges high voltageunder high causes voltage electric causes field distortionelectric field inside distortion the dielectric inside material the dielectric [14–16 ,32material,33]. The [14–16,32,33]. maximum local The electricmaximum field localFmax electricwould befield much Fmax higherwould than be much the applied higher electric than the field. applied Electrical electric breakdown field. Electrical would occurbreakdown when Fwouldmax exceeds occur awhen critical Fmax value exceedsFc. This a critical is named value as F thec. This space is chargenamed modulated as the space electrical charge breakdownmodulated electrical model (SCEB) breakdown or bipolar mode chargel (SCEB) transport or bipolar model charge [14– transp16]. Inort addition, model [14–16]. a charge In transport addition, anda charge molecular transport displacement and molecular modulated displacement electrical mo breakdowndulated electrical model (CTMD) breakdown is utilized model to investigate(CTMD) is theutilized sample to thicknessinvestigate dependent the sample DC thickness electrical dependent breakdown DC of polyimideelectrical breakdown [26]. Since freeof polyimide volume exists [26]. inSince polymers, free volume it may exists be enlarged in polymers, by the displacementit may be enlarged of molecular by the chainsdisplacement with trapped of molecular charges bychains the Coulombwith trapped force charges under the by high the electricCoulomb field force [24,25 under]. Electrons the high are acceleratedelectric field in the[24,25]. free volumeElectrons under are theaccelerated effect of thein the electric free volume field to obtainunder the energy. effect The of energythe electricw of field electrons to obtain gained energy. from theTheelectric energy fieldw of dependselectrons on gained the local from electric the electric field Ffieldand depends the length on of the the local free electric volume fieldλ, namely F and thew = lengtheFλ. The of the energy free ofvolume electrons λ, namely increases w = witheFλ. The an increase energy of in electrons the local electricincreases field with and an theincrease enlargement in the local of freeelectric volume field causedand the by enlargement the displacement of free of volume molecular caused chains. by th Whene displacement the maximum of molecular energy of chains. electrons Whenwmax theis max T highermaximum than energy the deep of trapselectrons energy w ET ,is electrons higher than have the enough deep energytraps energy to jump E over, electrons the potential have enough barrier, whichenergy will to jump result over in electrical the potential breakdown barrier, [ 26which]. will result in electrical breakdown [26].

4.1.4.1. NumericalNumerical AnalysisAnalysis byby SpaceSpace ChargeCharge ModulatedModulated ElectricalElectrical BreakdownBreakdown (SCEB)(SCEB) ModelModel

4.1.1.4.1.1. SCEBSCEB ModelModel FigureFigure5 5 shows shows the the schematic schematic diagram diagram of of the the SCEB SCEB model model for for a a dielectric dielectric material material under under DC DC voltagevoltage [[14–16].14–16]. TheThe cathodecathode andand anodeanode areare locatedlocated at atx x= = 00 andand xx = d, respectively. respectively. AfterAfter aa highhigh electricelectric ﬁeldfield isis appliedapplied onon thethe material, charges are injected from metal electrodes into the dielectric material. ElectronsElectrons andand holesholes onon metalmetal electrodeselectrodes havehave toto overcomeovercome effectiveeffective potentialpotential barriersbarriers betweenbetween thethe electrodeselectrodes andand thethe dielectricdielectric material.material. AfterAfter thethe chargescharges areare injectedinjected intointo thethe dielectricdielectric material,material, theythey willwill migratemigrate inin thethe extendedextended statesstates oror shallowshallow trapstraps aboveabove thethe mobility mobility edge edge of of electrons electronsE Eµμ(e)(e) andand belowbelow thethe mobilitymobility edgeedge ofof holesholesE Eµμ(h)(h),, which which are are named named as as free free electrons and free holes.holes. In addition,addition, deepdeep trapstraps conﬁnedconfined in in a a single single energy energy level, level,E ET(e)T(e) andand EET(h)T(h),, are are considered considered in the simulation modelmodel forfor electronselectrons andand holes,holes, respectively.respectively. WhenWhen freefree electronselectrons andand holesholes trappedtrapped byby deepdeep traps,traps, spacespace chargecharge accumulationaccumulation isis formed formed gradually. gradually. The The trapped trapped charges charges can becan released be released from deepfrom traps deep by traps the thermal by the activationthermal activation process when process they when get enough they get energy. enough When energy. free When electrons free and electrons holes encounterand holes eachencounter other ineach the other material, in the charge material, recombination charge recombination will occur. Thewill mathematicaloccur. The mathematical equations of equations the SCEB of model the SCEB are givenmodel in are the given following in the section. following section.

μ(e) FigureFigure 5.5. SchematicSchematic diagramdiagram ofof spacespacecharge charge modulated modulated electrical electrical breakdown breakdown model. model. Here, Here,E Eµ(e) andand EEµμ(h) areare the mobility edges of electronselectrons andand holes,holes, EET(e)T(e) and E T(h) areare the the energy of deep trapstraps forfor electronselectrons andand holes,holes,E EFF isis thethe FermiFermi level,level, andand EEgg is the band gap.

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Electrons and holes are assumed to be injected into the dielectric material from cathode and anode, respectively, by Schottky thermionic emission. The current densities of charge injections are determined by effective injection barriers between the sample and its electrodes, electric ﬁelds at the interfaces, and temperature [14–16,32,33].

" p # 2 Ein(e) eF(0, t)/4πε0εr jin(e)(t) = AT exp − exp (3) kBT kBT " p # 2 Ein(h) eF(d, t)/4πε0εr jin(h)(t) = AT exp − exp (4) kBT kBT

Here, jin(e) and jin(h) represent the current densities caused by electrons injected from the cathode −2 and holes injected from the anode into the dielectric material, respectively, in Am ; Ein(e) and Ein(h) are the effective injection barriers for electrons and holes, respectively, in eV; A is the Richardson 6 −2 −2 constant (=1.20 × 10 Am K ); e is the elementary charge in C; ε0 is the vacuum permittivity in −1 Fm ; εr is the relative permittivity of a dielectric material; and t is the time after applying a voltage in s. Furthermore, F(0, t) and F(d, t) are the electric ﬁelds at the interfaces of x = 0 and x = d, respectively. The injected charges migrate in the dielectric material, and the current densities of electrons and holes are expressed as follows [14–16,32,33]:

jc(e)(x, t) = qfree(e)(x, t)µ0(e)F(x, t) (5)

jc(h)(x, t) = qfree(h)(x, t)µ0(h)F(x, t) (6) where, jc(e) and jc(h) are the conduction current densities formed by electrons and holes, respectively, −2 −3 in Am ; qfree(e) and qfree(h) are the density of free electrons and free holes, respectively, in Cm ; and F is the electric ﬁeld in Vµm−1. The variations of charges in the dielectric material satisfy the charge continuity equations with the consideration of charge trapping, detrapping, and recombination processes. Traps in this model are assumed to be with single trap energy when charges are trapping and detrapping, and recombination occurs when electrons and holes encounter each other. The following equations demonstrate such a series of processes [14–16,32,33]:

∂qfree(e)(x,t) ∂jc(e)(x,t) ∂t + ∂t = (7) qtrap(e) −P ( )q ( ) 1 − + P ( )q ( ) − Reµ,hµq ( )q ( ) − Reµ,htq ( )q ( ) tr e free e eNT(e) de e trap e free e free h free e trap h ! ∂qtrap(e)(x, t) qtrap(e) = Ptr(e)qfree(e) 1 − − Pde(e)qtrap(e) − Ret,hµqtrap(e)qfree(h) (8) ∂t eNT(e)

∂qfree(h)(x,t) ∂jc(h)(x,t) ∂t + ∂t = (9) qtrap(h) −P ( )q ( ) 1 − + P ( )q ( ) − Reµ,hµq ( )q ( ) − Ret,hµq ( )q ( ) tr h free h eNT(h) de h trap h free e free h trap e free h ! ∂qtrap(h)(x, t) qtrap(h) = Ptr(h)qfree(h) 1 − − Pde(h)qtrap(h) − Reµ,htqfree(e)qtrap(h) (10) ∂t eNT(h)

Here, qtrap(e) and qtrap(h) are the density of trapped electrons and trapped holes, respectively, −3 3 −1 −1 in Cm ; while Reµ,hµ, Ret,ht, and Ret,hµ are the recombination coefﬁcients in m C s ; with subscripts having the following meanings: eµ for mobile electrons, et for trapped electrons, hµ −1 for mobile holes, and ht for trapped holes. In addition, Ptr is the trapping probability in s ; Pde −1 −3 represents detrapping probability in s ; and NT is the density of deep traps in m . Polymers 2018, 10, 1012 9 of 18

The trapping probability of a mobile charge into a deep trap has a positive correlation with carrier mobility and the density of deep traps, and is inversely proportional to permittivity, namely, Ptr(e,h) = eNT(e,h)/ε0εr [34]. The detrapping probability of a trapped charge at a deep trap center is determined by trap energy and temperature, Pde(e,h) = υATEexp(−ET(e,h)/kBT)[14–16,32,33]. −1 Here, υATE represents the attempt-to-escape frequency in s . Based on the Langevin recombination model, the recombination coefficient between free electrons and free holes can be expressed as, Reµ,hµ = (µ0(e) + µ0(h))/ε0εr [35]. According to the Shockley–Read–Hall model [36], trap-assisted recombination coefficients between free electrons and trapped holes and that between trapped-electron and free-hole can be expressed as, Reµ,ht = µ0(e)/ε0εr and Ret,hµ = µ0(h)/ε0εr, respectively. The units of recombination coefficients are m3C−1s−1. After space charges are accumulated in the polymeric material, the electric potential and electric field inside the material are distorted. The relation between electric potential ϕ and space charge is determined by Poisson’s equation [14–16,32,33]:

∂2φ(x, t) q ( )(x, t) + q ( )(x, t) + q ( )(x, t) + q ( )(x, t) = − free e trap e free h trap h 2 (11) ∂x ε0εr

When we solve Poisson’s equation, a boundary condition should be given. During the DC electrical breakdown experiment, a positive ramping voltage with a rising rate of kramp is applied to the sample until electrical breakdown occurs. Accordingly, the electric potential is set to be zero on the cathode and equals the external applied voltage on the anode. In addition, the electric potential on the anode or the external applied voltage is proportional to time after the application of voltage. The boundary condition is expressed as follows:

φ(d, t) = Vappl(t) = kramptramp (12)

−1 Here Vappl is the external applied voltage in V; kramp is the ramp rate of voltage source in kVs ; and tramp is the applied time of ramping voltage in s. After obtaining the electric potential in the polymeric material, we can calculate the electric field by the definition of electric field, F = −∇ϕ.

4.1.2. Parameters for SCEB Model The thickness range of polyimide samples used in the simulations was similar to the experiments, namely from 25 to 250 µm. The simulation temperature was the same as that during the breakdown measurements, which was around 303 K. The energy of deep traps was extracted from the TSDC experimental results. The density and energy of deep traps for electrons and holes were assumed to be the same value, which were 6.25 × 1020 m−3 and 0.83 eV, respectively. The effective injection barriers of electrons and holes were 0.98 eV. The carrier mobilities of electrons and holes controlled by shallow traps in polyimide were 3.67 × 10−14 m2V−1s−1 and 1.80 × 10−14 m2V−1s−1, respectively, which were obtained by the SPD measurements. The relative permittivity of polyimide used in the simulations was 3.5.

4.1.3. Simulation Results of SCEB Model In the SCEB model, it is assumed that electrical breakdown occurs when the maximum local electric field Fmax inside the material exceeds a threshold Fc. If the space charge effect is not considered, the internal electric field distributes evenly or is equal everywhere. Consequently, the electric field inside the sample increases linearly with an increase in the application time of a ramp voltage. The breakdown time tb is proportional to the sample thickness d, namely, tb = Fcd/Vramp. Based on the assumption of no space charge accumulated in the dielectric material, the electrical breakdown strength will not vary with the sample thickness, namely, Fb = Vramptb/d = Fc. In other words, the breakdown strength is independent of material thickness. Obviously, this ideal situation does not match the experimental results. Polymers 2018, 10, x FOR PEER REVIEW 10 of 18 strength will not vary with the sample thickness, namely, Fb = Vramptb/d = Fc. In other words, the Polymers 2018, 10, 1012 10 of 18 breakdown strength is independent of material thickness. Obviously, this ideal situation does not match the experimental results. The SCEB model is then used to simulate the DC electricalelectrical breakdown process of polyimide under the influenceinfluence of spacespace chargecharge accumulationaccumulation andand electricelectric fieldfield distortion.distortion. After charges are injected into the dielectric material from the electrodes, they may be captured by traps formed by carbonyl groups [37,38] [37,38] or by the energy level diff differenceerence between pyromellitic dianhydride and and 4.4’ 4.4’ diaminodiphenyl ether [[39].39]. Then, space charges are accumulated inside the material, resulting in the distortion ofof electricelectric fields.fields. FigureFigure6 6shows shows how how the the distribution distribution of of space space charges charges and and electric electric fields fields in ain polyimide a polyimide film film with with a thicknessa thickness of of 100 100µm μm depend depend on on the the time timet elapsed t elapsed after after the the application application of of a rampa ramp voltage voltageV rampVramp.. It canIt can be be seen seen from from Figure Figure6 that 6 that the the amount amount of space of space charges charges accumulated accumulated in the in materialthe material increases increases and and the distortionthe distortion of the of electricthe electric field field becomes becomes serious serious with with the elapsed the elapsed time aftertime theafter application the application of voltage. of voltage. As shown As shown in Figure in Figure6a, at 6a, the at initial the initial moment, moment, the applied the applied voltage voltage is very is low,very solow, there so there is little is little charge charge injection. injection. The The internal internal electric electric field field is determined is determined by by the the combination combination of theof the applied applied electric electric field field and and the the accumulated accumulated space space charges. charges. At At the the beginning,beginning, thethe internalinternal electric fieldfield is almost determined by the applied voltage, resulting in a uniform distribution, as shown in Figure6 6b.b. AsAs increasingincreasing thethe appliedapplied voltage,voltage, chargescharges areare injectedinjected continuouslycontinuously intointo thethe materialmaterial andand the space space charge charge accumulation accumulation leads leads to to el electricectric field field distortion. distortion. For For example, example, at x at = x10= μ 10m µnearbym nearby the thecathode, cathode, the space the space charge charge density density is −148.8 is − Cm148.8−3 at Cm t =− 3150at s,t =and 150 the s, space and the charge space density charge is density −408.3 −3 −3 isCm−408.3 at tb = Cm 403 s justat tb before= 403 the s just occurrence before the of electrical occurrence breakdown. of electrical The breakdown. electric field The distortion electric factor field distortionα that is defined factor α asthat the is ratio defined between as the the ratio maximum between lo thecal maximum electric field local and electric the average field and electric the average field, electricα = Fmax/ field,Fappl, isα 1.2= F atmax t /=F 150appl s,, isand 1.2 1.3 at tat= tb 150 = 403 s, ands. Since 1.3 atthetb mobility= 403 s. of Since holes the is mobilitysmaller than of holes that of is smallerelectrons, than more that positive of electrons, charges more accumulate positive charges near the accumulate anode than near negative the anode charges than near negative the cathode. charges nearAccording the cathode. to Poisson’s According equation, to Poisson’s the derivative equation, of the the derivative electric offield the with electric respect field withto position respect tois positionproportional is proportional to the density to the of accumulated density of accumulated charges. Therefore, charges. the Therefore, position the at positionwhich the at maximum which the maximumelectric field electric occurs field is biased occurs toward is biased the toward anode. theThe anode. maximum The maximumlocal electric local field electric increases field with increases time withas shown time in as Figure shown 7, in and Figure electrical7, and breakdown electrical breakdown occurs when occurs the maximum when the local maximum electric localfield electricexceeds field525 V exceedsμm−1. The 525 simulation Vµm−1. Theresults simulation indicate resultsthat the indicate DC electrical that the breakdown DC electrical starts breakdown from the middle starts fromof the thematerial, middle which of the is material,consistent which with the is consistent observation with by thepulsed observation electroacoustic by pulsed apparatus electroacoustic [20]. The apparatuseffect of space [20]. charge The effect accumulation of space charge on electrical accumulation breakdown on electrical was also breakdown observed was in alsopoly(vinylidene observed in poly(vinylidenefluoride-trifluoroethylene-chlorofluoroethylene) fluoride-trifluoroethylene-chlorofluoroethylene) terpolymer. It terpolymer.was found Itthat was foundthe electrical that the electricalbreakdown breakdown field decreases field decreaseswith an increase with an in increase the charge in the injection charge injectionfrom electr fromodes electrodes into the material into the material[40]. [40].

2500 800 tramp=5s (Vappl=5kV) tramp=5s (Vappl=5kV) tramp=10s (Vappl=10kV) )

-3 2000 700 tramp=20s (Vappl=20kV) tramp=30s (Vappl=30kV) tramp=10s (Vappl=10kV)

) tramp=40.5s (Vappl=40.5kV) 1500 tramp=20s (Vappl=20kV) -1 600 m

t =30s (V =30kV) μ (b) 1000 ramp appl 500 tramp=40.5s (Vappl=40.5kV) 500 400 0 300 -500 200 Electric field (V -1000 Space charge density (Cm charge density Space 100 -1500 (a) 0 0 20406080100 0 20406080100 Position (μm) Position (μm) Figure 6. Numerical results of space charge and electric fieldfield profilesprofiles as a function of position. (a) The distribution of space charges (b) and electric fieldsfields at variousvarious times.times. The ramping rate of the applied −1 voltage is 11 kVskVs−1, ,the the cathode cathode and and the the anode anode are are located atat x = 0 and 100 µμm,m, respectively,respectively,t ramptramp is the elapsed time of voltage, and Vappl representsrepresents the the instantaneous instantaneous voltage voltage applied applied to to the the sample. sample.

Polymers 2018, 10, x FOR PEER REVIEW 11 of 18

Polymers 2018, 10, 1012 11 of 18 Polymers 2018, 10, x FOR PEER REVIEW 11 of 18

Figure 7. Numerical results of the maximum local electric field as a function of time in polyimide samples with various thicknesses ranging from 25 to 250 μm.

Figure 8 demonstrates the DC electrical breakdown field of polyimide as a function of sample thickness obtained from the calculations by the SCEB model. It is obvious that the DC electrical breakdown field decreases with an increase in sample thickness. As shown in Figure 8a, the DC electrical breakdown field is 416 Vμm−1 at the sample thickness of 25 μm, 405 Vμm−1 at 100 μm, and 388 Vμm−1 at 250 μm. The DC electrical breakdown field at the sample thickness of 25 μm is 28 Vμm−1 higher than that at 250 μm. Chen et al. obtained similar results on low-density polyethylene [14]. The −1 −1 DC electricalFigure 7. breakdownNumericalNumerical results results field ofis of 419the the maximum Vμm at localthe local sample electric electric thicknessfield field as a function of 25 μm, of 406time V in μ m polyimide at 100 μm, −1 and 399samples Vμm with at various 250 μ m.thicknesses The difference ranging between from 25 to the 250 DC µμm. electrical breakdown field at the sample thicknesses of 25 μm and that at 250 μm is 20 Vμm−1 [14]. Figure 888b demonstratesdemonstrates shows the DC thethe electrical DCDC electricalelectrical breakdown breakdownbreakdow fieldn as fieldfield a function ofof polyimidepolyimide of sample asas aa functionthicknessfunction of ofin sample sampledouble thicknesslogarithmic obtained coordinates. from from the The calculations numerical by results the SCEBobtained model.model. from It It isthe is obvious obvious SCEB modelthat that the show DC electricalthat the breakdown strengthfield field decreases Fb and samplewith an thickness increase d in satisfies samplesample an thickness. inverse powerAs As shown shown function, in in Figure Fb = kd 88a,a,−n, theandthe DC DCthe − − electricalinverse power breakdown breakdown index nfield field equals is is 416 416to V0.030. Vμµmm−1 However,at1 atthe the sample sample the inversthickness thicknesse power of 25 of functionμ 25m,µ m,405 405Vcurveμm V−µ 1obtained mat 1001 at μ 100m, by and µthem, − 388andSCEB V 388μ simulationm− V1 µatm 2501 μatism. far 250 The fromµ m.DC the Theelectrical experime DC electrical breakdownntal results. breakdown fiel Thed at inversethe field sample power at the thickness sampleindex n of thicknessof 25 SCEB μm is simulation 28 of V 25μmµm−1 higheris much 28 V thanµ lowerm− that1 higherthan at 250 that than μ m.obtained Chen that at etfrom 250al. obtained experiments,µm. Chen similar etwhich al.results obtainedis 0.324 on low-density assimilar shown in results polyethylene Figure on 4b. low-density Therefore, [14]. The − DCpolyethylenethe electricalCTMD model breakdown [14]. Thethat DC considers field electrical is 419 the breakdownV μdisplacementm−1 at the field sample isof 419the thickness Vmolecularµm 1 at of the 25chain sampleμm, with406 thickness V trappedμm−1 atof 100charges 25 μµm,m, − − and406causing V399µm Vanμ1 matenlargement− 1001 at µ250m, andμm. in 399The free V differenceµ mvolume1 at 250 betweenisµ usedm. The theto difference furtherDC electrical study between breakdownthe the relationship DC electrical field atbetween the breakdown sample the − thicknessesfieldbreakdown at the sample ofstrength 25 μm thicknesses andand samplethat at of 250 thickness 25 µμmm andis 20of that Vpolyimide.μm at−1 250 [14].µ m is 20 Vµm 1 [14]. Figure 8b shows the DC electrical breakdown field as a function of sample thickness in double 420 800 logarithmic coordinates. The numerical results(a) obtained(b) from the SCEB model show that the ) ) -1 -1 −n m breakdownm strength Fb and sample thickness d satisfies an inverse power function, Fb = kd , and the μ μ 600 inverse power index n equals to 0.030. However, the inverse power function curve obtained by the SCEB simulation is far from the experimental results. The inverse power indexn=0.030 n of SCEB simulation is much lower400 than that obtained from experiments, which400 is 0.324 as shown in Figure 4b. Therefore, the CTMD model that considers the displacement of the molecular chain with trapped charges causing an enlargement in free volume is used to further study the relationship between the breakdown strength and sample thickness of polyimide.

200 380 breakdwonElectrical field (V Electrical breakdwon field (V 420 0 50 100 150 200 250 80010 100 1000 (a) (b) ) ) μ μ -1 -1 Sample thickness ( m) Sample thickness ( m) m m μ μ 600 Figure 8. Numerical results of the DC electrical breakdown ﬁeldfield as a function of sample thickness of polyimide inin the the linear linear coordinates coordinates (a) and(a) theand double the double logarithmic logarithmic coordinates coordinates (nb=0.030). Symbol (b). Symbolrepresents ■ therepresents simulation400 the simulation results of the results space of charge the space modulated charge electricalmodulated400 breakdown electrical (SCEB)breakdown model, (SCEB) while model,--- is awhile ﬁtting - - curve- is a fitting calculated curve by calculated an inverse by power an inverse function. power function.

Figure8b shows the DC electrical breakdown ﬁeld as a function of sample thickness in double logarithmic coordinates. The numerical results obtained from the SCEB model show that the 200 −n 380 breakdwonElectrical field (V breakdownElectrical breakdwon field (V 0 strength 50Fb 100and sample 150 thickness 200 250d satisﬁes an10 inverse power function, 100Fb = kd 1000, and the inverse power indexSamplen equals thickness to ( 0.030.μm) However, the inverse powerSample functionthickness (μ curvem) obtained by the SCEB simulation is far from the experimental results. The inverse power index n of SCEB Figure 8. Numerical results of the DC electrical breakdown field as a function of sample thickness of simulation is much lower than that obtained from experiments, which is 0.324 as shown in Figure4b. polyimide in the linear coordinates (a) and the double logarithmic coordinates (b). Symbol ■ Therefore, the CTMD model that considers the displacement of the molecular chain with trapped represents the simulation results of the space charge modulated electrical breakdown (SCEB) model, charges causing an enlargement in free volume is used to further study the relationship between the while - - - is a fitting curve calculated by an inverse power function. breakdown strength and sample thickness of polyimide.

PolymersPolymers2018 2018,,10 10,, 1012 x FOR PEER REVIEW 1212 of of 18 18

4.2. Numerical Analysis by Charge Transport and Molecular Displacement Modulated (CTMD) Model 4.2. Numerical Analysis by Charge Transport and Molecular Displacement Modulated (CTMD) Model 4.2.1. Dynamics of Molecular Displacement under Electric Field 4.2.1. Dynamics of Molecular Displacement under Electric Field The magnitude of molecular displacement depends on temperature, electric field, and The magnitude of molecular displacement depends on temperature, electric field, and mechanical mechanical strength. The motion of molecular chains will be accelerated by increasing the strength. The motion of molecular chains will be accelerated by increasing the temperature, electric temperature, electric field, or mechanical force [23,41]. In the experiment of DC electrical breakdown, field, or mechanical force [23,41]. In the experiment of DC electrical breakdown, the electric field is the electric field is high enough to facilitate the motion of molecular chains. On the one hand, if a high enough to facilitate the motion of molecular chains. On the one hand, if a molecular chain has molecular chain has a dipole moment, the chain will rotate under the external electric field. On the a dipole moment, the chain will rotate under the external electric field. On the other hand, when a other hand, when a molecular chain with deep traps formed by chemical defects captured charge molecular chain with deep traps formed by chemical defects captured charge carriers, a Coulomb force carriers, a Coulomb force due to the external electric field will act on the chain [24,25]. As shown in due to the external electric field will act on the chain [24,25]. As shown in Figure9, molecular chains Figure 9, molecular chains with negative carriers will move toward the anode, while those with with negative carriers will move toward the anode, while those with positive carriers will move toward positive carriers will move toward the cathode. The displacement of a molecular chain is an integral the cathode. The displacement of a molecular chain is an integral of time, so it depends on the retention of time, so it depends on the retention time of charges in trap centers or the actuation duration of the time of charges in trap centers or the actuation duration of the Coulomb force on the molecular chain. Coulomb force on the molecular chain. Since the retention time of charges in trap centers increases Since the retention time of charges in trap centers increases exponentially with an increase in trap exponentially with an increase in trap energy, the charges reside longer in the deep traps, which energy, the charges reside longer in the deep traps, which means that the Coulomb force acts a longer means that the Coulomb force acts a longer period on molecular chains with occupied deep traps period on molecular chains with occupied deep traps compared to molecular chains with occupied compared to molecular chains with occupied shallow traps. The displacement of a molecular chain shallow traps. The displacement of a molecular chain with trapped charges in the dielectric material with trapped charges in the dielectric material will enlarge the local free volume. Electrical will enlarge the local free volume. Electrical breakdown occurs when the energy of electrons gained breakdown occurs when the energy of electrons gained from electric field in the enlarged free volume from electric field in the enlarged free volume is greater than the energy of deep traps. It can be is greater than the energy of deep traps. It can be concluded that the displacement of a molecular concluded that the displacement of a molecular chain with occupied deep traps would determine the chain with occupied deep traps would determine the electrical breakdown strength of the material. electrical breakdown strength of the material. Consequently, we will focus on the molecular chains Consequently, we will focus on the molecular chains with deep traps in the following numerical with deep traps in the following numerical simulations. The velocity equation for the motion of a simulations. The velocity equation for the motion of a molecular chain with trapped charges is molecular chain with trapped charges is expressed as [24,26], expressed as [24,26],

dλ/dλλτt = =−µmolμ F − λ/τmol (13) ddt mol F mol (13) wherewhere λλ isis thethe displacementdisplacement ofof aa molecular molecular chainchain inin m; m;µ μmolmol isis thethe mobilitymobility ofof molecularmolecular chainschains inin 22 −−11 −1−1 mm VV ss ; and; and τmolτmol is theis the relaxation relaxation time time constant constant of of molecular molecular chains chains in in s. s. The The mobility mobility of molecularmolecular chainschains isis determineddetermined byby thethe shallowshallow traptrap controlledcontrolled carriercarrier mobilitymobility and thethe energyenergy ofof deepdeep traps,traps, namelynamelyµ μmolmol = µμ0Pde/(/(PPtr tr+ +Pde P).de The). The relaxation relaxation time time constant constant of of molecular molecular chains chains equals the retention timetime ofof chargescharges inin deepdeep traps,traps, namelynamelyτ τmolmol = ττ00exp(exp(EETT/k/BkTB).T ). InIn orderorder toto calculatecalculate thethe displacementdisplacement ofof molecularmolecular chains,chains, wewe needneed toto determinedetermine electricelectric ﬁeldfield distributiondistribution inin thethe materialmaterial ﬁrstly.firstly. Then,Then, wewe cancan obtainobtain thethe velocityvelocity andand displacementdisplacement ofof thethe molecularmolecular chainchain byby solvingsolving EquationEquation (13).(13). Therefore,Therefore, wewe usedused aa CTMDCTMD toto simulatesimulate thethe spacespace chargecharge distributiondistribution andand molecularmolecular chainchain motionmotion inin polyimide.polyimide.

FigureFigure 9.9. SchematicSchematic diagramdiagram ofof thethe displacementdisplacement ofof molecularmolecular chainschains withwith occupiedoccupied deepdeep trapstraps underunder anan electricelectric ﬁeld.field. TheThe molecularmolecular chainschains withwith positivepositive andand negativenegative trappedtrapped chargescharges movemove towardtoward thethe cathodecathode andand anode,anode, respectively.respectively.

Polymers 2018, 10, 1012 13 of 18 Polymers 2018, 10, x FOR PEER REVIEW 13 of 18

4.2.2. Simulation Results of CTMD Model Figure 1010 shows shows how how the electricthe electric field distributionsfield distributions and molecular and molecular chain displacement chain displacement distributions dependdistributions on the depend time tramp onelapsed the time after tramp the elapsed application after of the aramp application voltage of at thea ramp sample voltage thickness at the of sample 100 µm forthickness example, of according100 μm for to example, the calculations according by the to CTMDthe calculations model. As by the the applied CTMD voltage model.V Asappl therises, applied more andvoltage more V spaceappl rises, charges more are and injected more and space accumulated charges are inside injected the sample, and accumulated which can distort inside the the electric sample, field. Aswhich shown can indistort Figure the 10 a,electric the electric field. fieldAs shown has a convexin Figure shape 10a, with the electric a maximum field valuehas a inconvex the middle shape of with the samplea maximum in the value case of in homo the middle space charges.of the sample The maximum in the case local of homo electric space field increasescharges. non-linearlyThe maximum with local the − − elapsedelectric field time ofincreases voltage. non-linearly For example, with the maximum the elapse locald time electric of voltage. field is For 107 example, Vµm 1 at the 10 maximum s, 262 Vµm local1 at − 20electric s, and field 520 Visµ 107m 1Vatμm 40.2−1 at s. 10 Molecular s, 262 Vμ chainsm−1 at with 20 s, negative and 520trapped Vμm−1 at charges 40.2 s. move Molecular toward chains the anode, with whilenegative those trapped with positive charges trapped move toward charges the move anode, toward while the those cathode. with When positive the displacement trapped charges is not move very longtoward or thethe relaxation cathode. timeWhen constant the displacement of molecular is chains not very is very long large, or thethe secondrelaxation term time on the constant right side of canmolecular be neglected chains accordingis very large, to the the molecular second term chain on displacement the right side Equation can be (13),neglected and the according velocity to of the molecular chainchain with displacement trapped charges Equation is proportional (13), and the to thevelocity electric of field.the molecular Therefore, chain the molecular with trapped chain displacementcharges is proportional curve has the to same the electric tendency field. as the Th electricerefore, field the curve molecular as shown chain in Figure displacement 10b. The maximumcurve has ofthe molecular same tendency chain displacement as the electricλmax fieldappears curve in as the shown middle in of Figure the sample, 10b. The while maximum the displacement of molecular value nearchain the displacement electrodes is λ relativelymax appears small. in the The middle maximum of the local sample, molecular while the displacement displacement is 0.077 value nm near at 10 the s, 0.35electrodes nm at 20is relatively s, and 1.59 small. nm at The 40.2 maximum s. local molecular displacement is 0.077 nm at 10 s, 0.35 nm at 20 Figures, and 1.5911a showsnm at 40.2 the maximums. energy of electrons gained from electric field for the samples at variousFigure thicknesses 11a shows accordingthe maximum to the energy calculations of electrons by the gained CTMD from model. electric The field molecular for the chainsamples with at occupiedvarious thicknesses deep traps according will move to under the calculations the electric by field, the resultingCTMD model. in the The extension molecular of free chain volume with insideoccupied the deep sample. traps In will other move words, under the the length electric of fi freeeld, volume resulting in in a dielectricthe extension material of free increases volume inside as the molecularthe sample. chain In other displacement words, increases.the length It of is assumedfree volume that in the alength dielectric of free material volume increases is very small as the in themolecular initial state,chain which displacement can be neglected increases. for It polyimideis assumed at that room the temperature length of free which volume is much is very lower small than in itsthe glass initial transition state, which temperature. can be neglected Therefore, for polyim the lengthide at of room free volumetemperatureλ is equalwhich to is themuch displacement lower than valueits glass of transition molecular temperature. chains under Therefore, the applied the electriclength of field. free volume The molecular λ is equal chain to the displacement displacement or thevalue length of molecular of free volume chains under increase the with applied the appliedelectric field. voltage The increases. molecular The chain increases displacement in both or local the electriclength of field free and volume molecular increase chain with displacement the applied leadvoltage tothe increases. increase The in theincreases energy in of both electrons local electric gained fromfield and the electricmolecular field chain in thedisplacement free volume, lead as to shown the increase in Figure in the 11 a.energy When of theelectrons maximum gained energy from ofthe electrons electric fieldwmax in= the (eF freeλ)max volume,exceeds as theshown trap in energy Figure level11a. WhenET, the the local maximum currents energy would of electrons multiply andwmax = the (eF temperatureλ)max exceeds risesthe trap suddenly, energy level causing ET, the electrical local currents breakdown would eventually. multiply and The the temperature influence of molecularrises suddenly, chain displacementcausing electrical on electrical breakdown breakdown eventually. field was The also influence evidenced of inmolecular poly(vinylidene chain fluoride-hexafluoropropylene)displacement on electrical breakdown copolymer [field42]. Inwa addition,s also theevidenced simulation in resultspoly(vinylidene of the CTMD fluoride- model indicatehexafluoropropylene) that the breakdown copolymer occurs [42] at. In the addition, position the where simulation the electric results field of isthe the CTMD highest model as shown indicate in Figurethat the 10 breakdown. occurs at the position where the electric field is the highest as shown in Figure 10.

(a) (b) 700 t =5s (V =5kV) t =10s (V =10kV) tramp=5s (Vappl=5kV) tramp=10s (Vappl=10kV) 2.0 ramp appl ramp appl t =20s (V =20kV) t =30s (V =30kV) 600 tramp=20s (Vappl=20kV) tramp=30s (Vappl=30kV) ramp appl ramp appl t =40.2s (V =40.2kV) ) tramp=40.2s (Vappl=40.2kV) ramp appl -1 1.5 m 500 μ 400 1.0 300

200 0.5 Electric field (V Electric field 100 Molecular displacement (nm)

0 0.0 0 20406080100 0 20406080100 Position (μm) Position (μm) Figure 10.10. NumericalNumerical results results of of the the electric electric field field and and molecular molecular chain chain displacement displacement as a function as a function of position. of Theposition. distributions The distribution of electrics fieldof electric (a) and field molecular (a) and chain molecular displacement chain displacement (b) at various (b times.) at various The ramping times. −1 rateThe oframping voltage rate is 1 kVsof voltage, the cathodeis 1 kVs and−1, the the cathode anode are and located the anode at x = 0are and located 100 µm, atrespectively, x = 0 and 100tramp μm,is therespectively, elapsed time tramp of is ramp the elapsed voltage, time and Vofappl ramprepresents voltage, the and instantaneous Vappl represents voltage theapplied instantaneous to the sample. voltage applied to the sample.

Polymers 2018, 10, x FOR PEER REVIEW 14 of 18 Polymers 2018, 10, 1012 14 of 18 PolymersWhen 2018 ,the 10, xmolecular FOR PEER REVIEW displacement is not very large, Equation (13) can be simplified to be 14dλ of/d 18t = μmolF. Replacing F by krampt/d and considering the initial condition λ(t = 0) = 0, we can obtain λ = When the the molecular molecular displacement displacement is isnot not very very larg large,e, Equation Equation (13) (13)can be can simplified be simplified to be d toλ/d bet (krampμmol/2d)t2. The molecular displacement or the length of free volume is a function of the reciprocal =d λμ/dmoltF=. Replacingµ F. Replacing F by krampF tby/d kand consideringt/d and considering the initial thecondition initial conditionλ(t = 0) = 0,λ( twe= 0)can = obtain 0, we canλ = of sample molthickness and the squareramp of time. Generally, the elapsed time after the application of a ramp (obtainkrampμmolλ/2=d ()kt2. Theµ molecular/2d)t2. The displacement molecular or displacement the length of orfree the volume length is of a freefuncti volumeon of the is areciprocal function voltage untilramp the occurrencemol of breakdown in a dielectric material is proportional to the sample of sample the reciprocal thickness of and sample the square thickness of time. and theGenerall squarey, the of time.elapsed Generally, time after the the elapsed application time of after a ramp the thickness. Since the variation in the square of time is faster than that of sample thickness, the voltageapplication until of the a ramp occurrence voltage untilof breakdown the occurrence in a ofdiel breakdownectric material in a dielectric is proportional material isto proportionalthe sample molecular displacement or the length of free volume increases with the increase in sample thickness thickness.to the sample Since thickness. the variation Since the in variationthe square in the of squaretime is of faster time isthan faster that than of that sample of sample thickness, thickness, the as shown in Figure 11b. Nevertheless, the derivative of molecular displacement with the respect to molecularthe molecular displacement displacement or the or the length length of offree free volume volume increases increases with with the the increase increase in in sample sample thickness thickness sample thickness decreases with the increase in sample thickness. In addition, electrical breakdown as shown in Figure 1111b.b. Nevertheless, the derivati derivativeve of molecular displacement with the respect to occurs when the maximum energy of electrons that is a product of the maximum length of free sample thickness decreases with the increase in sample thickness. In addition, electrical breakdown volume and the maximum electric field exceeds the trap energy level; therefore, the maximum electric occurs whenwhen thethe maximum maximum energy energy of electronsof electrons that th isat a productis a product of the of maximum the maximum length oflength free volumeof free field at pre-breakdown decreases with an increase in sample thickness as depicted in Figure 11b. volumeand the and maximum the maximum electric ﬁeldelectric exceeds field exceeds the trap the energy trap level;energy therefore, level; therefore, the maximum the maximum electric electric ﬁeld at pre-breakdownfield at pre-breakdown decreases decreases with an with increase an increase in sample in2.2 thicknesssample thickness as depicted as depicted in Figure in 11 Figureb. 800 11b. )

2.0 -1 2.2 800 700 m μ

1.8 )

2.0 -1

600700 m 1.6 μ 1.8 1.4 1.6 500600 1.2 1.4 400500 1.0 1.2 (b) 0.8 300400 Maximum electric field (V 1.00 50 100 150 200 250 Maximum molecular displacement(nm) μ (b)

Sample thickness ( m) Maximum electric field (V 0.8 300 0 50 100 150 200 250 Maximum molecular displacement(nm) Figure 11. Numerical results of time-dependent maximum energy ofSample electrons thickness gained (μm) from the electric field for the samples at various thicknesses (a) and sample thickness-dependent maximum local Figure 11. NumericalNumerical results results of of time-dependent time-dependent maximum maximum energy of electrons gained from the electric molecular displacement and maximum local electric field at pre-breakdown (b). fieldﬁeld for the samples at various thicknesses (a) andand samplesample thickness-dependentthickness-dependent maximum local molecular displacement and maximum local electric ﬁeld at pre-breakdown (b). Figuremolecular 12 displacementdemonstrates and the maximum DC electrical local electric breakd fieldown at field pre-breakdown of polyimide (b). as a function of sample thicknessFigure obtained 12 demonstrates from the the calculations DC electrical by breakdown CTMD model. field ofAs polyimide shown in as Figure a function 12a, of the sample DC Figure 12 demonstrates the DC electrical breakdown field of polyimide as a function of sample breakdownthickness obtained field decreases from the with calculations an increase by CTMD in samp model.le thickness. As shown The in DC Figure electrical 12a, the breakdown DC breakdown field thickness obtained from the calculations by CTMD model. As shown in Figure 12a, the DC isfield 643 decreases Vμm−1 at withthe sample an increase thickness in sample of 25 thickness. μm, 402 V Theμm− DC1 at electrical100 μm, and breakdown 297 Vμm field−1 at is 250 643 μ Vm.µ mThe−1 breakdown field decreases with an increase in sample thickness. The DC electrical breakdown field DCat the electrical sample breakdown thickness of field 25 µ m,at the 402 sample Vµm−1 thicknessat 100 µm, of and 25 μ 297m is V µ346m− V1 μatm 250−1 higherµm. The than DC that electrical at 250 is 643 Vμm−1 at the sample thickness of 25 μm, 402 Vμm−1 at 100 μm, and 297 Vμm−1 at 250 μm. The μbreakdownm. It can be field seen at the from sample Figure thickness 12b that of 25thereµm is 346a linear Vµm −relationship1 higher than between that at 250 theµ m.DC It breakdown can be seen DC electrical breakdown field at the sample thickness of 25 μm is 346 Vμm−1 higher than that at 250 strength and the sample thickness in the log-log coordinates, which can be expressed in the form of μfromm. It Figure can be 12 seenb that from there Figure is a linear 12b relationshipthat there is between a linear the relationship DC breakdown between strength the DC and breakdown the sample an inverse power function Fb = kd−n. The inverse power index n equals 0.339, which is close toF the strengththickness and in the the log-log sample coordinates, thickness whichin the canlog-log be expressed coordinates, in the which form ofcan an be inverse expressed power in function the formb of= experimentalkd−n. The inverse result power n = 0.324. index n equals 0.339, which is close to the experimental result n = 0.324. an inverse power function Fb = kd−n. The inverse power index n equals 0.339, which is close to the experimental700 result n = 0.324. 800 ) (a) ) (b)

-1 700 -1 m 700 m 800 μ 600 μ 600 ) (a) ) (b)

-1 700 -1 m m 500 n=0.339 μ 500600 μ 600 400500 n=0.339 400500 300400 300400 300 200300

Electrical breakdwonElectrical field (V 200 0 50 100 150 200 250 Electrical breakdwon (V field 10 100 1000 μ Sample thickness (μm) 200 Sample thickness ( m)

Electrical breakdwonElectrical field (V 200 0 50 100 150 200 250 Electrical breakdwon (V field 10 100 1000 Figure 12. NumericalNumericalSample results results thickness of the the (μ m)DC DC electrical electrical breakdow breakdownn field ﬁeld as a Sample function thickness of sample (μm) thickness of polyimide in linear coordinates (a) andand inin doubledouble logarithmiclogarithmic coordinatescoordinates ((bb).). SymbolSymbol ■ represents Figure 12. Numerical results of the DC electrical breakdown field as a function of sample thickness of thethe simulation simulation results results of of the the SCEB SCEB model, model, while while ------is a fittingis a ﬁtting curve curve calculated calculated by an inverse by an inverse power polyimide in linear coordinates (a) and in double logarithmic coordinates (b). Symbol ■ represents function.power function. the simulation results of the SCEB model, while - - - is a fitting curve calculated by an inverse power

function.

Polymers 2018, 10, 1012 15 of 18

4.3. Discussion When the space charge effect is considered as demonstrated by the SCEB model, space charges accumulate inside a dielectric material with an increase in the application time of external voltage and the internal electric field is distorted, which would cause a drop in breakdown strength. Homo space charges accumulate near the interfaces between the dielectric material and its electrodes [18]. Consequently, the electric field near the electrode is weakened, and conversely the electric field at the middle of the material is enhanced. The more homo space charges that accumulate inside the material, the more severely the internal electric field will be distorted or the larger the maximum electric field Fmax will be, which will lead to a decrease in the electrical breakdown field [20]. Generally, when the electrodes and the sample are the same, the accumulation of space charges will increase with an increase in the application time of external voltage, which results in a larger Fmax and a lower breakdown strength. With the sample thickness increasing, the time required for the internal electric field to reach the breakdown threshold becomes longer under the same ramp voltage. This means that the breakdown time of thicker samples is longer than that of thinner samples. The accumulation of space charges inside the thicker sample is greater, and the distortion of the electric field is more severe, which makes the electrical breakdown field of a relatively thicker sample lower than that of the relatively thinner sample [14]. Therefore, the breakdown strength decreases as the thickness increases. In the CTMD model, molecular chains with occupied deep traps displace under the external electric field, leading to an increase in the length of free volume [24,26]. The voltage applied to the sample increases with a ramping rate; consequently, both the local electric field and the molecular chain displacement or the length of free volume increase with time. Accordingly, electrons will be accelerated to higher energy in the free volume at a relatively longer duration after the application of voltage. When electrons gain enough energy to jump over the potential barrier, the sample will be broken by the accelerated electrons [12,43]. The displacement of a molecular chain with trapped charges under an electric field is the integral of elapsed time after the application of voltage. As the thickness of the sample increases, the time required to reach the breakdown threshold ET is prolonged. Consequently, the displacement value of a molecular chain increases with the increase in sample thickness. According to the breakdown criterion wmax > ET, the increase in the free volume length will cause the decrease in the electric field F required for the electron to be accelerated to the energy ET. Therefore, the electrical breakdown field is inversely related to the sample thickness. Figure 13 shows a comparison of DC electrical breakdown field as a function of sample thickness calculated by the SCEB model and the CTMD model with the experimental results. It can be seen from the comparison that the simulation results obtained by the CTMD model are in good agreement with the experimental results. However, the breakdown results calculated by the SCEB model deviate significantly from the experimental values. It indicates that DC electrical breakdown is a cumulative process. Therefore, compared with the SCEB breakdown model considering the space charge effect, the CTMD model considering the charge transport and molecular chain displacement can give a better explanation for the DC electrical breakdown characteristics of polyimide. Polymers 2018, 10, x FOR PEER REVIEW 15 of 18

4.3. Discussion When the space charge effect is considered as demonstrated by the SCEB model, space charges accumulate inside a dielectric material with an increase in the application time of external voltage and the internal electric field is distorted, which would cause a drop in breakdown strength. Homo space charges accumulate near the interfaces between the dielectric material and its electrodes [18]. Consequently, the electric field near the electrode is weakened, and conversely the electric field at the middle of the material is enhanced. The more homo space charges that accumulate inside the material, the more severely the internal electric field will be distorted or the larger the maximum electric field Fmax will be, which will lead to a decrease in the electrical breakdown field [20]. Generally, when the electrodes and the sample are the same, the accumulation of space charges will increase with an increase in the application time of external voltage, which results in a larger Fmax and a lower breakdown strength. With the sample thickness increasing, the time required for the internal electric field to reach the breakdown threshold becomes longer under the same ramp voltage. This means that the breakdown time of thicker samples is longer than that of thinner samples. The accumulation of space charges inside the thicker sample is greater, and the distortion of the electric field is more severe, which makes the electrical breakdown field of a relatively thicker sample lower than that of the relatively thinner sample [14]. Therefore, the breakdown strength decreases as the thickness increases. In the CTMD model, molecular chains with occupied deep traps displace under the external electric field, leading to an increase in the length of free volume [24,26]. The voltage applied to the sample increases with a ramping rate; consequently, both the local electric field and the molecular chain displacement or the length of free volume increase with time. Accordingly, electrons will be accelerated to higher energy in the free volume at a relatively longer duration after the application of voltage. When electrons gain enough energy to jump over the potential barrier, the sample will be broken by the accelerated electrons [12,43]. The displacement of a molecular chain with trapped charges under an electric field is the integral of elapsed time after the application of voltage. As the thickness of the sample increases, the time required to reach the breakdown threshold ET is prolonged. Consequently, the displacement value of a molecular chain increases with the increase in sample thickness. According to the breakdown criterion wmax > ET, the increase in the free volume length will cause the decrease in the electric field F required for the electron to be accelerated to the energy ET. Therefore, the electrical breakdown field is inversely related to the sample thickness. Figure 13 shows a comparison of DC electrical breakdown field as a function of sample thickness calculated by the SCEB model and the CTMD model with the experimental results. It can be seen from the comparison that the simulation results obtained by the CTMD model are in good agreement with the experimental results. However, the breakdown results calculated by the SCEB model deviate significantly from the experimental values. It indicates that DC electrical breakdown is a cumulative process. Therefore, compared with the SCEB breakdown model considering the space charge effect, the CTMD model considering the charge transport and molecular chain displacement can give a Polymers 2018, 10, 1012 16 of 18 better explanation for the DC electrical breakdown characteristics of polyimide.

800

) 700 Experiment -1

m SCEB model

μ 600 CTMD model 500

400

300

200 Electrical breakdwon field (V 10 100 1000 Sample thickness (μm) Figure 13. Comparison of experimental results and simulation results obtained by SCEB and charge transport and molecular displacement modulated (CTMD) models on DC electrical breakdown ﬁeld as a function of sample thickness.

5. Conclusions The thickness-dependent DC electrical breakdown properties of polyimide were investigated by experiments and simulations of the SCEB model and the CTMD model. The experimental results show that the electrical breakdown field decreases with an increase in sample thickness in the form of an inverse power function, and the inverse power index is n = 0.324. The trap properties, carrier mobilities, and complex permittivity were also measured for further investigation into the influencing mechanism of thickness on the electrical breakdown characteristics of polyimide. Then, the SCEB model was used to simulate the electrical breakdown properties of polyimide. Charges are injected into and transport inside the dielectric material under external electric field. During the migration of mobile charges in shallow traps, some of them are captured by deep traps and form space charges. The electric field in the middle of the sample is then enhanced by the accumulation of space charges. When the maximum local electric field exceeds a threshold value, the material is broken. More space charges are accumulated in thicker samples and the electric field is distorted more severely, resulting in the electrical breakdown field decreasing as the sample thickness increases. In the CTMD simulations, both the charge transport and molecular chain displacement dynamics processes were considered. After mobile charges are captured by deep traps formed by chemical defects on molecular chains, the molecular chain with occupied deep traps moves toward the electrode driven by the Coulomb force, which enlarged the free volume. When electrons gain enough energy in the free volume from the electric field, electrical breakdown occurs in polyimide. Moreover, the elapsed time after the application of ramp voltage until the occurrence of electrical breakdown increases with an increase in sample thickness, and the length of free volume is proportional to the square of time; accordingly, the free volume at pre-breakdown is larger at the relatively thicker sample. Since the threshold electron energy is a product of the length of free volume and electric field, the electrical breakdown field decreases with an increase in sample thickness following an inverse power function. The comparison between the experimental results and simulations indicates that the simulation results obtained by the CTMD model are closer to the experiments and the electrical breakdown is modulated by both charge transport and molecular displacement dynamics.

Author Contributions: Conceptualization, D.M. and S.L.; Data curation, Y.L., C.Y., D.X., Q.W. and Z.X.; Formal analysis, Q.W.; Funding acquisition, D.M.; Investigation, Y.L. and D.X.; Resources, Z.X.; Supervision, D.M. and S.L.; Writing—original draft, D.M. and Y.L.; Writing—review and editing, D.M. Funding: This work was supported by the National Natural Science Foundation of China (grant No.: 51507124), the National Basic Research Program of China (grant No.: 2015CB251003), the State Key Laboratory of Advanced Power Transmission Technology (grant No.: SGGR0000DWJS1800562), the State Key Laboratory of Power Systems of Tsinghua University (grant No.: SKLD16KZ04), the China Postdoctoral Science Foundation (grant No.: 2014M552449), and State Grid Corporation of China Science Project (grant No.: 522722160001). Conﬂicts of Interest: The authors declare no conﬂict of interest. Polymers 2018, 10, 1012 17 of 18

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