Ramified Lichtenberg figures caused by electrostatic microdischarges on the surface of polymeric polyethylene terephthalate have been studied. They occurred in consequence of the previous electret forming of the polymeric sheets and were initiated in the air gap between the grounded electrode and the polymeric sheets. Multifractal image analysis was used to determine the fractal dimensions of the Lichtenberg patterns in dependence on the loading voltages used for electret forming

Benešovský, P.

Ficker, T.

Kapička, V.

Macur, J.

Slavíček, P.

Acta Polytechnica

English

T. Ficker

V. Kapička

J. Macur

P. Slavíček

P. Benešovský

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Fractality of Electrostatic Microdischarges on the Surface of Polymers

CTU Open Journal Systems (Czech Technical University, Prague / České vysoké učení technické v Praze)

1 IntroductionElectrostatic erosive microdischarges accompany tribo-electric charging of various objects during their industrialproduction as well as during their use in everyday life. It isespecially polymeric solids that are liable to surface chargeaccumulation to such an extent that they can even cause elec-tric breakdown of an adjacent gas environment. Usually, thisphenomenon is undesirable because of its disruptive effects inindustrial production, laboratory research, and civil workingactivities. The study of these surface microdischarges is there-fore useful from various points of view.Surface microdischarges often assume the form ofbranched channel structures whose particular channels re-present streamer discharges propagated along the surface ofthe object. Such branched patterns can be visualised by usingphotographic or powder techniques. These visible tracks arecalled Lichtenberg figures and have often been used to studygaseous discharges on the surface of dielectrics [1–6].To visualise Lichtenberg figures, it was necessary to chosea transparent, highly resistive material in the form of thinlayers (foils). Polyethylene terephthalate (PET) seems to be aconvenient material for such a purpose. It is in broad use inindustrial, academic, as well as civil applications. PET is ahard, stiff, strong, dimensionally stable material that absorbsvery little water. It has good gas barrier properties and goodchemical resistance except to alkalis. Its crystallinity variesfrom amorphous to fairly high crystalline; it can be highlytransparent and colourless but thicker sections are usuallyopaque. It is widely known as a biaxially oriented and ther-mally stabilised film. It is produced in various modificationsaccording to its destination: Mylar® films are used for capaci-tors, graphics, photographic film base, recoding tapes, etc.Other modifications are used as fibres in a wide range oftextiles (Dacron®, Trevira®, Tyrelene®) or in the food indus-try (PET bottles) and the electrical industry (for insulation).Nevertheless, its probably most important feature, inves-tigated intensively in academic research, is the ability toconserve an electric charge for a very long time. This featureranks it among the high-quality electrets.The charging of electrets (forming electret states) is per-formed in various ways, the most frequent of which involvesinserting an electret foil between two parallel electrodesloaded by a sufficiently high electric field. The process ofelectret forming may be facilitated by increasing the ambienttemperature up to the point of glass transition of the polymer(~85 °C for PET). However, a good electret state can bereached at normal room temperature, too. As soon as thecharged electret foil is separated from the electrode system,electrostatic microdischarges may cross the air gap betweenthe charged foil and the grounded electrode to neutralise theelectret surface charge. In this way the latent Lichtenbergfigures are drawn into the background electret charge. Thesestructures may be visualised by using the powder techniqueand digital projection. After digital filtering of the back-ground electret charge, only the branched patterns of thestreamer channels remain on the surface (Figs. 1–3) and thesefiltered structures (true Lichtenberg figures) may by studiedfrom the morphological point of view. For example, seekingfor the relation between the fractal dimension of Lichtenbergfigures and the stage of initial electret state, it is possible toanswer the question whether the morphology of the Lichten-berg figures is a direct consequence of the initial electret stateor whether they are caused by a purely stochastic process.2 Experimental arrangementHighly resistive polymeric sheets of amorphous poly-ethylene terephthalate were carefully cleaned with ethanoland then inserted between two short-circuited copper platesfor 24 hours to eliminate surface charges. The PET sheets0.180 mm in thickness were pressed between flat bronzeelectrodes of diameters 1  20 mm and 2  40 mm. Thesmaller electrode was loaded with the negative electric poten-tial of 8.5 kV while the larger electrode was grounded. Thetime application of HV was chosen in the intervals of severalminutes up to hundreds of hours.3 Multifractal analysisAlthough the method of multifractal analysis has been de-scribed in our preceding paper in this issue, yet, we would liketo repeat main features of the method.Multifractal calculations [7–11] have represented a funda-mental aid for geometrical investigations of complex disor-dered systems since the early eighties. Multifractal analysis –in the form of box-counting method – assumes the studiedcomplex system to be embedded in E-dimensional Euclideanspace. This Euclidean space is partitioned into a grid with the22 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 4/2004Fractality of ElectrostaticMicrodischarges on the Surface ofPolymersT. Ficker, V. Kapička, J. Macur, P. Slavíček, P. BenešovskýRamified Lichtenberg figures caused by electrostatic microdischarges on the surface of polymeric polyethylene terephthalate have beenstudied. They occurred in consequence of the previous electret forming of the polymeric sheets and were initiated in the air gap between thegrounded electrode and the polymeric sheets. Multifractal image analysis was used to determine the fractal dimensions of the Lichtenbergpatterns in dependence on the loading voltages used for electret forming.Keywords: Electrostatic microdischarges, Lichtenberg figures, multifractal formalism, fractal dimension, electret saturated states.same dimension E. The basic cell of the grid is of a linear size. Using probability moments pi a partition sum mq is definedon the gridm q piqiI( , ) ( ) 1, pnNii( )  	 ,piiI( ) 11 , q   ( , ) (1)where ni is a number of points in the i-th cell and N representsa number of all points of the complex system.The spectrum of generalised dimensions Dq is usuallyused for the purpose of the analysisDm qqq q qlimln ( , )( * ) ln*  1. (2)The complex systems under investigation are in the formof graphical bitmap files that were created by means of a scan-ner. In this way the object is converted into a set of graphicalpixels that are covered with a two-dimensional grid with thebasic cell . On this grid the partition sum (1) is determined.To avoid a dependence on the orientation of the grid, severalpositions of the grid in the Euclidean space are realised andan average partition sum M qq  ( , ) is calculatedM m qq jj( ) ( , )1212. (3)The averages Mq( ) are determined for a series of -gridsand the slopes within the coordinate system (ln , ln ( )Mq  )are estimated using the linear regression. The slopes dividedby (q 1) give the generalised dimensions Dq. An object whichproduces identical Dq for all q is called ‘monofractal’ while dif-ferent Dq–values indicate a ‘multifractal’ object.3 Results and discussionIn this study 150 PET samples were investigated. Threedifferent forming voltages (8 kV, 4 kV, 2 kV) and ten differentforming intervals (480 min., 240 min., 120 min., 60 min.,30 min., 15 min., 5 min., 1 min., 0.5 min., 0.25 min.) wereused to create corresponding electret states each of whichwas represented mostly by five PET samples. However,some of the samples were removed from the statistical ensem-ble because their Lichtenberg structures were not developedor because of other experimental problems. The remaining110 samples were analysed and their fractal dimensions de-©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 23Acta Polytechnica Vol. 44  No. 4/2004Dimensions Dt/min. 8 kV 4 kV 2 kV480 – 1.384 1.253240 – 1.387 1.404120 1.404 1.296 1.20560 1.505 – 1.26030 1.522 1.214 1.13315 1.613 1.175 1.2145 1.514 1.267 1.0981 1.494 1.268 dischargesdisappear0.5 1.290 1.204– 1.085Average dimensions Ds of saturated states and theirprobability errors1.477  0.026 1.253  0.022 1.224  0.025Table 1: Results of experimentsA) B)Fig. 1: Lichtenberg figures generated at 8 kV and 30 min., (D  1.486, sample no. 27)termined. The resulting Hausdorf-Besicovitch dimensions ofthe Lichtenberg figures generated from the saturated electretstates are listed in Table 1. The experiments were performedat normal atmospheric conditions with small variations(less than 5 %) around the average values: p2 = 102.19 kPa,0  24.31 °C, 0  69.7 % RH. The processing of one typicalsample (cleaning, conditioning, forming, separating, devel-oping, digitising, filtering, analysing) took about 30 hours.Comparing our experimental conditions (forming electricfields and times) with those of normally used [12], it is possi-ble to assume that for all our combinations of voltages andtimes (Table 1) the saturated electret states were reached.These saturated electret states are dependent only on form-ing voltages U and not on times t, since the chosen times(t  0.25min.) are sufficiently long for reaching electret satura-tion with PET samples at 2 kV and higher voltages (U  2 kV).Therefore, if the fractal dimension D of the Lichtenbergfigures risen from the saturated electret states is dependent onthe stage of saturation, then the fractal dimension D should bedependent on the forming voltages U and has to show an in-creasing dependence D(U). This hypothesis has proved to besound and is illustrated by Figs. 4–6 in which the increasinglinear dependence D aU b  is well recognisable. In addi-tion, if our forming times are sufficiently long for reachingsaturation states, the average dimensions Ds calculated fromthose belonging to the same columns of Table 1, i.e. for thevoltages of 8 kV, 4 kV, and 2 kV (dimensions 1.477, 1.253and 1.223), also have to be an increasing function of voltage24 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 4/2004A) B)Fig. 2: Lichtenberg figures generated at 4 kV and 30 min., (D  1.246 , sample no. 86 )A) B)Fig. 3: Lichtenberg figures generated at 2 kV and 60 min., (D  1.225 , sample no. 131 )Ds(U), which is documented in Fig. 7. In addition, this trendof branching structures with increasing voltages is visible inFigs. 1–3 which present three different Lichtenberg figures independence on voltage (the forming times were chosen suffi-ciently long to reach saturated electret states for the respectivevoltages). All these experimental findings lead us to the opin-ion that the morphology of Lichtenberg figures (i.e. theirfractal dimension) is not a consequence of purely stochasticprocesses but rather it is dependent on the stage of theelectret saturation which is determined solely by the formingvoltages U, provided sufficiently long forming times havebeen used.4 ConclusionsThe performed experiments have shown that thebranching of Lichtenberg figures caused by electrostaticmicrodischarges on the surface of polyethylene terephthalateis not quite a stochastic phenomenon but rather a processthat is governed by the stage of electret forming. Since theattainability of a saturated electret state is dependent on theforming voltage U and time t, the fractal dimension D of thecorresponding Lichtenberg figures should be a function ofthese parameters, i.e. D(U, t). Nevertheless, for sufficientlylong times t > tmin the dimension D is a function of voltageonly, i.e. D(U). It seems to be natural that the D(U) functioncannot increase its values to infinity but rather it should berestricted by some maximum Dmax and minimum Dmin valuescharacterising a particular electret (0  Dmin < Dmax  2).However, a verification of this idea would require furtherexperiments.AcknowledgmentThis work was supported by the Grant Agency of theCzech Republic under the Grant No. 202/03/0011.References[1] Morris A. T.: “Heat developed and powder Lichtenbergfigures and the ionization of dielectric surfaces producedby electrical impulses.” Br. J. Appl. Phys., Vol. 2 (1951),p. 98–109.[2] Bertein H.: “Charges on insulators generated by break-down of gas.” J. Phys D, Vol. 6 (1973), p. 1910–1916.©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/ 25Acta Polytechnica Vol. 44  No. 4/2004Fig. 4: The fractal dimesion D in dependence on forming volt-ages U for forming time t  5 min.Fig. 5: The fractal dimension D in dependence on forming volt-ages U for forming time t  15 min.Fig. 6: The fractal dimension D in dependence on forming volt-ages U for forming time t  30 min.Fig. 7: The average fractal dimension Ds in dependence on form-ing voltages U[3] Murooka Y., Koyama S.: “A nanosecond surface dis-charge study in low pressures.” J. Appl. Phys., Vol. 50(1979), p. 6200–6206.[4] Takahashi Y., Fujii H., Wakabayashi S., Hirano T.,Kobayashi S.: “Discharges due to separation of a Co-rona-charged Insulating sheet from a grounded metalcylinder.” IEEE Trans. El. Insul., Vol. 24 (1989),p. 573–580.[5] Niemeyer L.: A fractal Lichtenberg figure. 7th Interna-tional Symp. on High Voltage Engineering, Dresden,1991, p. 937–938.[6] Femia N., Lupo G., Tucci V.: Fractal characterization ofLichtenberg figures: a numerical approach. 7th Interna-tional Symp. on High Voltage Engineering, Dresden,1991, p.921–923.[7] Kudo K.: “Fractal analysis of electrical trees.” IEEETrans. Diel. El. Insul., Vol. 5 (1998), p. 713–727.[8] Feder J.: Fractals. Plenum. New York, 1988.[9] Ficker T., Druckműller M., Martišek D.: “Unconven-tional Multifractal Formalism and Image Analysis ofNatural Fractals.” Czech. J. Phys., Vol. 49 (1999),p.1445–1459.[10] Ficker T.: “Normalized Multifractal Spectra within theBox-Counting Method.” Czech. J. Phys., Vol. 50 (2000),p. 389–403.[11] Losa G. A., Nonnenmacher T. F., Weibel E. R.: Fractals inBiology and Medicine. Birkhäuser, Basel, 1993.[12] Belana J., Mudarra M., Calaf J., Canadas J. C., Menén-dez E.: “TSC study of the polar and free charge peaks ofamorphous polymers.” IEEE Trans. on El. Insul., Vol. 28(1993), p. 287–293.Assoc. Prof. RNDr. Tomáš Ficker, DrSc.phone: + 420 541 147 661e-mail: fyfic@fce.vutbr.czDepartment of PhysicsAssoc. Prof. RNDr. Jiří Macur, CSc.phone: +420 541 147 249e-mail: macur.j@fce.vutbr.czDepartment of InformaticsFaculty of Civil EngineeringUniversity of TechnologyŽižkova 17662 37 Brno, Czech RepublicProf. RNDr. Vratislav Kapička, DrSc.RNDr. Pavel Slavíček, Ph.D.phone: +420 549 496 830Department of Physical ElectronicsMasaryk University of BrnoFaculty of ScienceKotlářská 2616 00 Brno, Czech RepublicMgr. Petr BenešovskýDepartment of PhysicsFaculty of Civil EngineeringUniversity of TechnologyŽižkova 17662 37 Brno, Czech Republic26 ©  Czech Technical University Publishing House http://ctn.cvut.cz/ap/Acta Polytechnica Vol. 44  No. 4/2004

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Fractality of Electrostatic Microdischarges on the Surface of Polymers

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