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Dielectric Breakdown of Polycrystalline Alumina: A Weakest-Link FailureAnalysis Benjamin Block, Youngjin Kim and Dinesh K. Shetty † Materials Science and Engineering, University of Utah, Salt Lake City, Utah 84112 The effects of varying electrode geometry (ball and ring) andsize (radius), dielectric liquid (castor oil and Diala oil), andspecimen thickness on the dielectric breakdown of a commer-cial-grade alumina were investigated. The breakdown strengthwas expressed in terms of the maximum electric field in the cera-mic calculated by finite element analysis (FEA) at the break-down voltage. The breakdown strength decreased systematicallywith increasing electrode radius and specimen thickness, andthe strength was higher in the Diala oil (dielectric constant, e r = 2.3 0.06) as compared to the castor oil ( e r = 4.6 0.13). These effects of the electrode geometry, specimen thick-ness, and of the dielectric liquid on the breakdown strength of the alumina were analyzed with a weakest-link failure modelemploying Laplace and Weibull distributions for a population of defects in the material. The measured size or scaling effects of the electrodes, specimen thickness, and of the dielectric liquid onbreakdown strength were in better agreement with the Laplacedistribution for a population of surface defects. The dielectricbreakdown is likely initiated at surface pits produced by grainpullout. The measured area concentration of surface pits agreedwith the defect density analyzed from the weakest-link failuretheory. FEA of specimens containing surface and subsurfacecavities revealed that electric field concentrations were alwaysgreater for surface pits as compared to subsurface cavities.There is, in fact, no electric field concentration at a subsurfacecavity located more than about 100 – 800 l m below the surfacedepending on the top electrode size. I. Introduction P OLYCRYSTALLINE alumina is widely used as insulators inhigh-voltage devices. 1 Alumina is attractive for theseapplications because of its high dielectric breakdownstrength, dimensional stability, and low cost. However, thebreakdown strengths reported in the literature for alumina atroom-temperature range from less than 10 kV/mm to greaterthan 100 kV/mm for bulk specimens, 2 – 5 and as high as1100 kV/mm for thin films. 4,6,7 Part of this variation in thebreakdown strength can be attributed to variations in themicrostructure of the alumina, specifically, porosity, grainsize, and composition. 2,8 – 10 Extrinsic variables, such as thegeometry and size of the electrodes employed in breakdowntesting, the liquid dielectric medium employed to suppressflashover, and the thickness of the test specimens, affect thebreakdown strength more than the intrinsic material vari-ables. This is a relatively unexplored area with little datareported in the open literature.American Society for Testing and Materials (ASTM) Stan-dard D-149-97a, 11 which is commonly used for measuringdielectric breakdown strengths of solid electrical insulatingmaterials, recommends fixed cylindrical electrodes with flatends or fixed cylindrical electrodes with hemispherical ends.Fixed electrodes, in general, are unsatisfactory due to the factthat damage produced on the electrode surface during abreakdown test can affect the results of the subsequent testswith the same electrode. It is also a common practice toexpress the breakdown strength as the average breakdownvoltage ( V b ) divided by the thickness of the specimen ( t ). 11 The use of this nominal thickness averaged electric field toexpress the dielectric breakdown strength masks importanteffects of field intensifications on the specimen surface due tothe electrode/specimen contact geometry and the surroundingdielectric liquid medium. With flat electrodes, the maximumelectric field occurs near the three-phase (electrode/specimen/dielectric liquid) boundary and is sensitive to the edge radius.With spherical electrodes, the maximum electric field occursat the contact point on the specimen surface. Liquids withlow dielectric constants enhance the electric fields in the cera-mic, whereas liquids of high dielectric constants suppress theelectric fields. As analytical solutions of electric field varia-tions are available for only simple electrode and specimengeometries, the effects of field intensification on dielectricbreakdown are often neglected.This article reports on the effects of (a) electrode geometryand size, (b) dielectric constant of the oil, and (c) specimenthickness on the electric field variations on the surface and inthe bulk of disk specimens of alumina calculated by finite ele-ment analysis (FEA). The results of the FEA were employedto calculate breakdown strengths from breakdown voltages interms of the maximum electric fields ( E max ) sustained by thespecimens before breakdown in a variety of testing schemes.The breakdown strength decreased systematically withincreasing electrode radius, specimen thickness, and dielectricconstant of the oil. The effects of these extrinsic variables onthe breakdown strength of the alumina were analyzed with aweakest-link failure model employing Laplace and Weibulldistributions for a population of defects in the material. Themeasured size or scaling effects of the electrodes, specimenthickness, and of the liquid medium on breakdown strengthwere in better agreement with the Laplace distribution for apopulation of surface defects. The dielectric breakdown islikely initiated at surface pits produced by grain pullout. Themeasured area concentration of surface pits agreed with thedefect concentration analyzed from the weakest-link failuretheory. FEA of specimens containing surface and subsurfacecavities revealed that electric field concentrations were alwaysgreater for surface pits as compared to subsurface cavities.There is, in fact, no electric field concentration at a subsurfacecavity located more than 100 – 800 l m below the surfacedepending on the top electrode size. II. Materials and Experimental Procedures (1) Polycrystalline Alumina Table I lists some properties of the commercial-grade alu-mina (Grade AD-94; CoorsTek, Golden, CO) employed in C. M. Landis—contributing editorManuscript No. 32577. Received January 3, 2013; approved June 5, 2013. † Author to whom correspondence should be addressed. e-mail:
[email protected] 3430 J. Am. Ceram. Soc., 96 [11] 3430–3439 (2013)DOI: 10.1111/jace.12492 © 2013 The American Ceramic Society Journal this study. The breakdown strength listed in Table I wasmeasured in accordance with ASTM Standard D 149-97aand is the thickness average electric field at breakdown calcu-lated using the following equation: E b ¼ V b t (1)In Eq. (1), E b is the nominal breakdown strength, V b is theaverage breakdown voltage, and t is the average thickness of the test specimens. The reported density of the alumina is3.7 g/cc, but the theoretical density is not known preciselybecause of the multiple crystalline and glass phases due toliquid-phase sintering additives. The ceramic does not exhibitwater absorption or gas permeability, thus indicating noopen porosity. The volume fraction of closed porosity is notknown.Sintered, ground, and annealed disks of alumina, 76.2 mmin diameter and 4 mm in thickness, were purchased from thematerial supplier. Sintering was done at 1645 ° C over a 52-hour cycle, ground to the final thickness of 4 mm using dia-mond grinding wheels, and annealed at 1500 ° C for 30 min.To investigate the effect of specimen thickness on breakdownstrength, additional disks were procured with thickness of 2,1, 0.5, and 0.25 mm. These disks were prepared in the samemanner as the 4 mm disks.A typical composition of the alumina is 93.3 wt% Al 2 O 3 ,4.1 wt% SiO 2 , 0.7 wt% MgO, 0.8 wt% BaO, with theremaining made up of ZrO 2 , CaO, etc. The average grain sizeis 12 l m. The dielectric constant is reported to be 9.1. (2) Dielectric Breakdown Test Figure 1 shows a schematic of the ceramic test specimen, theelectrodes and the test chamber employed in the breakdowntests. The electrodes were made of copper with replaceablestainless steel ball bearings in contact with the specimen atthe center. The diameters of the ball bearings could be variedto change the distribution of the electric field on the surfaceand in the bulk of the test specimen. The stainless steel ballswere rotated after each test or replaced to ensure freshsurface in contact with the test specimen. In this study,breakdown tests were conducted with steel balls of fixedradius, R 2 = 12.7 mm, for the bottom electrode. The radiusof the ball in the top electrode ( R 1 ) was varied between1.5875 and 12.7 mm. The bottom electrode, the test specimen,and part of the top electrode were immersed in a dielectricliquid of sufficiently high breakdown strength to avoid surfaceflashover. Two different oils, castor and Diala , were used asthe surrounding dielectric liquid.To investigate the effects of spatially distributed electricfields, ring electrodes were used in this study (Fig. 2). Thecontact circle radius of the electrodes ( R c ) was fixed at19.05 mm, whereas the ring cross-sectional radius ( R e ) waseither 1.5875 or 12.7 mm. These electrodes were also madeof stainless steel.A power supply (Model #950; Hipotronics, Inc., Brewster,NY) with a maximum capacity of 75 kV was used to applyac voltage at 60 Hz at a ramp rate of 2 kV/s. The powersupply automatically shut down the test at breakdowndetected by a surge in current, and captured the voltage atbreakdown, V b . These test conditions are consistent with thespecifications of ASTM Standard D149-97a. 11 The break-down of the ceramic appeared as a puncture or perforationthrough the thickness of the specimen. Figure 3 shows atypical dielectric puncture produced on the surface of thealumina specimen in a dielectric breakdown test. The surfacearound the puncture showed evidence of local heating andmelting of the ceramic. (3) Measurements of Dielectric Constants of the Oils and of the Ceramic Both the maximum electric field and the spatial variation inthe electric field in the ceramic are functions of the dielectricconstant of the ceramic and of the surrounding oil. It was,therefore, necessary to measure the dielectric constants Table I. Properties of the Commercial-Grade Alumina Property Average value Composition (wt% Al 2 O 3 ) 93.3Grain size ( l m) 12Dielectric constant, e r 9.1Dielectric strength, E b (kV/mm) 8.3Electrical conductivity, r (S/cm) at 25 ° C < 10 14 Thermal conductivity, j (W/mK) at 20 ° C 22.4 Fig. 1. A schematic of the test specimen, electrodes, and liquidcontainer used in dielectric breakdown tests of alumina. Fig. 2. Ring electrodes used to expand the stressed area andvolume of the ceramic specimen. November 2013 Dielectric Breakdown of Alumina 3431 accurately for input to the FEA. The dielectric constant, orrelative permittivity, e r , was calculated from the measuredcapacitance of a disk specimen using the following equation: e r ¼ CtA e 0 (2)In Eq. (2), C is the capacitance of the disk specimen, t is itsthickness, and A is its area; e 0 is the permittivity of free spacewith the value 8.8542 9 10 12 F/m. 12 To measure the capaci-tance of dielectric liquids, a liquid specimen holder, shown inFig. 4, was used. It consisted of a brass cup, a Teflon ring(DuPont Co., Wilmington, DE), and a brass lid. The brasscup and the brass lid acted as the electrodes, whereas theTeflon ring separated the electrodes and contained theliquid specimen. Capacitance was measured using a time domaindielectric spectrometer (Model TDDS-1; Imass, Inc., Accord,MA) in the frequency range, 1 – 10,000 Hz, whereas an LCRmeter (Model 4284A; Hewlett-Packard Co., Palo Alto, CA) wasemployed in the 20 Hz – 1 MHz range. The Teflon ring contrib-uted to the measured capacitance of the dielectric liquid. A cor-rection was made by measuring the capacitance of the ringwithout the dielectric liquid ( C 0 ) and subtracting it from thetotal capacitance, C : C l ¼ð C C 0 Þ (3)The dielectric constant was, then, calculated using Eq. (2)with C l replacing C .The dielectric constant of the alumina was also calculatedfrom the measured capacitance of a disk specimen. In thiscase, however, there was no need for the Teflon ring andcorrection of the capacitance. All capacitance measurementswere made inside an environmental chamber where thetemperature could be controlled in the range, 20 ° C – 45 ° C. (4) Finite Element Analysis Figure 5 shows the mesh in the vicinity of the top ball elec-trode used in FEA of the electric fields using COMSOL with an axisymmetric model. The mesh was refined near thecontact point between the top electrode and the ceramic, theregion of high electric fields and field gradients. The meshrefinement was established by a convergence study such thatthe electric fields calculated at a given location, for example,at the top electrode contact point, in two successive meshrefinements, differed by less than 0.01 kV/mm. The top elec-trode surface was maintained at a potential of 75 kV,whereas the bottom electrode was at zero potential (ground).The validity of the FEA was established by calculating themaximum electric field in a dielectric plate placed between asphere and a plane electrode and comparing the results withvalues reported in the literature. 13 – 15 III. Experimental and FEA Results (1) Dielectric Constants Table II lists the averages and standard deviations of themeasured dielectric constants for the alumina, castor oil, andDiala oil samples at 60 Hz and 21 ° C. The value measuredfor alumina ( e r = 9.3 0.12) was close to the value reportedby the vendor (see Table I). The dielectric constants for thecastor oil ( e r = 4.6 0.13) and the Diala oil ( e r = 2.3 0.06)were lower than that of alumina. The dielectric constants of the oils showed only a modest change with temperature. Forexample, the dielectric constant of castor oil decreased from4.6 at 21 ° C to 4.28 at 45 ° C. (2) Electric Field Variations-FEA Results To validate the FEA, electric fields on the top surface of adielectric plate ( e d ) in contact with a spherical electrode onthe top surface and immersed in a dielectric liquid ( e l ) werecalculated for the case of the top electrode surface at a con-stant potential, V , whereas the bottom surface of the dielec-tric plate was at zero potential (ground). This particularproblem was selected for validation of the FEA modelbecause Binns and Randall, 13 Takuma and Kawamoto, 14 and Poli 15 have calculated electric fields on the surface of thesolid dielectric by other numerical methods, such as the finitedifference method 13 and the charge simulation method. 14 – 16 Table III lists the values of the normalized maximum electricfields, ( E max R 1 / V ), on the surface of the solid dielectric plate,for several values of e s = ( e d / e l ) and R 1 = t , where R 1 is theradius of the top electrode and t is the thickness of thedielectric plate. For all three values of the relative dielectricconstant, e s , the maximum electric fields calculated by FEAwere in good agreement with the corresponding valuesreported in the literature. It is useful to note that Binns andRandall, 13 Takuma and Kawamoto, 14 and Poli 15 considereda dielectric plate infinite in in-plane dimensions, whereasthe FEA was conducted with a finite disk with R = 30 mmand R 1 = t = 10 mm. Additional FEA calculations for R = 60 mm and R = 120 mm indicated that there was no sig-nificant change in the maximum electric field. This result wasuseful because it indicated that in-plane boundary conditionsof the ceramic specimen did not affect either the maximumelectric field or the radial variation in the electric field as longas R >> t . Based on this result, multiple tests were conductedon the ceramic disk specimens.Figure 6 shows plots of the normalized electric field, E ð r Þ = E , versus the normalized radial position, r / R , on thetop surface of the alumina disk in castor and Diala oilswith the smallest and the largest top electrode balls. E is thenominal thickness-averaged electric field, V / t , where V is theapplied voltage and t is the thickness of the alumina disk. R is the radius of the alumina disk. The FEA calculations wereperformed for V = 75 kV, R = 38.1 mm, and t = 4.0 mm.The maximum electric field in the ceramic occurs at the Fig. 3. A typical puncture produced on the surface of the aluminaceramic by dielectric breakdown. Fig. 4. Specimen holder used to measure capacitance and dielectricconstants of liquids. 3432 Journal of the American Ceramic Society—Block et al. Vol. 96, No. 11 contact point between the top ball electrode and the aluminadisk. The electric field drops along a radius. The maximumelectric field was higher and the field dropped more rapidlyfor small top electrode radius and Diala oil. The maximumfield-intensification factor, E ð r ¼ 0 Þ = E , increased from 1.21for R 1 = 12.7 mm to 4.39 for R 1 = 1.5875 mm in castor oil.The corresponding values for Diala oil were 1.42 and 6.79.In all cases, the bottom electrode ball radius was fixed at R 2 = 12.7 mm. Electric fields were also analyzed in the diskvolume and on the bottom surface of the alumina disk. Forthe 12.7 mm electrodes, the electric field variation throughthe thickness was symmetrical with respect to the mid-thick-ness plane, with lower value in the mid-thickness plane thanthe surfaces. For smaller electrodes at the top, the electricfields were asymmetrical with respect to the mid-thicknessplane. The electric field variations on the surface and in thebulk were also analyzed for top electrode ball radius of 3.18,4.76, 6.35, 7.94, 9.53, and 11.11 mm. The trends were similarand the magnitudes of the fields were between the twoextremes shown in Fig. 6.The electric field variations on the top surface of an alu-mina disk ( R = 38.1 mm, t = 4 mm) in Diala oil for twocases of the ring electrodes ( R c = 19.05 mm, R e = 1.5875 and12.7 mm) at 75 kV are shown in Fig. 7. The general trendsin the electric fields with the ring electrodes were similar tothose obtained with the ball electrodes. However, the valuesof the peak electric fields along the contact circles of therings were lower than those calculated for the balls. Fig. 5. Finite element mesh used to calculate electrical fields in the alumina ceramic with spherical electrodes and dielectric liquid. Table II. Measured Dielectric Constants of Alumina, CastorOil, and Diala Oil at 60 Hz and 21 ° C Material e r Alumina (AD-94) 9.30 0.11Castor oil 4.59 0.06Diala oil 2.30 0.12 Table III. Normalized Maximum Electric Field, ( E max R 1 / V ),on the Surface of A Dielectric Plate in Contact with ASpherical Electrode R 1 / t e s FEA Reference 13 Reference 14 Reference 15 1 1 1.75 1.77 1.82 1.771 2 2.36 2.4 2.38 2.401 4 3.50 3.45 3.68 3.53 Fig. 6. Normalized electric field plotted as function of normalizedposition along a radius on the top surface of alumina disk for twodifferent top ball electrode radii in castor and Diala oils. Fig. 7. Normalized electric field plotted as function of normalizedposition along a radius on the top surface of alumina disk fordifferent ring electrode radii in Diala oil. November 2013 Dielectric Breakdown of Alumina 3433 Figure 8 illustrates the effect of specimen thickness on theelectric field variations on the surface of alumina disks( R = 38.1 mm) in Diala oil with the same spherical elec-trodes ( R 1 = 3.175 mm, R 2 = 12.7 mm) at 75 kV. One inter-esting trend in Fig. 8 is that the normalized peak electricfield at the contact points of the ball electrodes approachesthe nominal electric field, E ¼ V = t , as the thickness of thespecimen is reduced. For t = 0.25 mm, for example, E max ¼ 1 : 14 E . For thin film specimens, the maximum electricfields can be reasonably approximated by the nominal elec-tric fields.Results of the FEA, such as those shown in Figs. 6, 7,and 8, were used to calculate the breakdown strengths interms of E bmax , the maximum electric field at the top electrodecontact point at the breakdown voltage using the followingequation: E bmax ¼ E max E V b t (4)In Eq. (4), E max = E is the maximum field-intensification factorcalculated by FEA as shown in Figs. 6, 7, and 8, V b is thebreakdown voltage recorded in the test and t is the thicknessof the specimen employed in the test. It should be noted that E max occurs at r = 0 for the ball electrodes and at r = R c forthe ring electrodes. (3) Dielectric Breakdown Fields of Alumina Figure 9 shows plots of the mean maximum electric field atbreakdown, E bmax , as function of the top electrode ball radiusfor alumina disks ( t = 4 mm) tested in castor and Diala oilswith a bottom ball electrode radius of 12.7 mm. The errorbars represent two standard deviations. The means and thestandard deviations were based on 20 tests in each test serieswith the same electrodes and the oil. The maximum field atbreakdown decreased with increasing radius of the top ballelectrode. The breakdown fields were higher in Diala oilthan those in castor oil, particularly for small top electrodes.The maximum electric fields at breakdown measured withthe ring electrodes are also plotted in Fig. 9. The breakdownfields were lower with the ring electrodes as compared tothose measured with the ball electrodes for the same elec-trode radius. The difference was especially obvious for thesmall electrode radii, R 1 = R e = 1.5895 mm.The effect of specimen thickness on the breakdown field isillustrated in Fig. 10. The breakdown field increased withdecreasing specimen thickness. This increase was gradual inthe thickness range from 4 to 0.5 mm. The increase in thebreakdown field was more pronounced for the 0.25 mmdisks. IV. Discussion A trend that can be discerned from the data of Figs. 9 and10 is that the breakdown field is higher when the electric fielddistribution in the ceramic is spatially localized. This is thecase when the breakdown tests are conducted with small ballelectrodes, Diala oil, and thin specimens. This scaling effectof the electrically stressed ceramic surface area or volumesuggested that dielectric breakdown of alumina might beexhibiting the characteristics of a weakest-link failurephenomenon analogous to brittle fracture. 17,18 Weakest-linkfailure theory has been used in the past to rationalize thevariations of dielectric breakdown field with electrode sizeand shape in paper capacitors, 19 casting resins, 20 epoxy resin, 21 metal-oxide films, 22 transformer oil, 23 liquid nitrogen, 24,25 andalumina. 26,27 The theoretical basis of weakest-link failure is the theory of extreme values, or more specifically, the mathematical rela-tion between a population distribution and the distribution of Fig. 8. Normalized electric field plotted as function of normalizedposition along a radius on the top surface of alumina disks with threedifferent thickness obtained with ball electrodes ( R 1 = 3.175 mm, R 2 = 12.7 mm) in Diala oil. Fig. 9. Variations of the maximum electric fields at breakdown of alumina with electrode radius in tests in castor and Diala oils withball and ring electrodes. Fig. 10. Variation in the maximum electric field at breakdown of alumina with specimen thickness in tests in Diala oil with ballelectrodes ( R 1 = 3.175 mm, R 2 = 12.7 mm). 3434 Journal of the American Ceramic Society—Block et al. Vol. 96, No. 11
