Dielectric Breakdown Model For Conductor-Loaded and Insulator-Loaded Composite Materials
Dielectric Breakdown Model For Conductor-Loaded and Insulator-Loaded Composite Materials
Dielectric Breakdown Model For Conductor-Loaded and Insulator-Loaded Composite Materials
I. INTRODUCTION duction the DBM has been broadly studied to describe ex-
perimental results 关7–13兴, though the physical origin of such
The dielectric breakdown phenomenon in solid materials stochastic fluctuations is beyond the DBM and cannot be
has been widely studied both theoretically and experimen- explained by it 关14兴.
tally due to its importance in the electrical industry. The The present paper addresses the problem of dielectric
design of insulators bearing high electric strength is highly breakdown in composite materials. While breakdown phe-
desirable, and in the past years composite materials such as nomena in conductor-loaded dielectric have been studied in
resin matrix filled by fibers or strong particles have been recent years from the standpoint of percolation theory
widely used with such purpose in many industrial applica- 关15,16兴, insulator-loaded dielectric materials have received
tions 关1– 4兴. For example, high density polyethylene is one of much less attention. Theoretical efforts have concentrated on
the most widely used materials for the production of insula- lattice models in an attempt to see whether the basic physical
tors, and composites containing carbon black and titanium mechanism of breakdown can be identified in these materi-
dioxide have recently been tested experimentally 关5兴 to de- als. Some efforts have focused on the breakdown of fuse
termine the influence of such particles on the dielectric prop- networks, while others have concentrated on dielectric break-
erties of the material. It has been shown that dielectric break- down in networks.
down still produces branching structures, such as those in
Our interest, however, is mainly focused on the study of
homogeneous materials but with an extension of damage and
fractal dielectric trees. Considering that percolation condi-
a distribution of breakdown times dependent on the concen-
tions impose a limit to the breakdown processes we are in-
tration and electrical characteristics of the filler. We note that
while both a low failure probability and a small damage are terested in the study of the breakdown phenomenon below
desirable, our results suggest that this goal is not always that limit, i.e., how the concentration of particles in the com-
attainable. posite material modifies the characteristics of the dielectric
From the theoretical point of view, dielectric breakdown trees. In a previous paper 关17,18兴 we applied the DBM to
in homogeneous materials has been described as a stochastic describe dielectric breakdown patterns in conductor-loaded
process producing fractal structures that are called electrical composites and in this work we generalize the DBM to de-
trees. The most widely used model is the dielectric break- scribe dielectric breakdown patterns in insulator-loaded com-
down model 共DBM兲, first introduced by Niemeyer, Pietron- posites. Insulating particles are distributed at random in the
ero, and Wiesmann 关6兴, which assumes that the dielectric is resin matrix, and the dielectric breakdown propagates ac-
homogeneous, i.e., the electrical tree propagates in a dielec- cording to new rules to take into account electrical properties
tric medium without inhomogeneities. The main feature in and particle sizes. In this way we extend the DBM to take
the DBM is the dependence of the breakdown probability on into account material inhomogeneities from the point of view
the local electric field in the material, a fact that attempts to of electrical properties.
consider the basic mechanism underlying breakdown in real The extension of the DBM model presented in this paper
materials. Stochastic fluctuations produce breakdown chan- also allows us to describe dielectric breakdown patterns by
nels that damage the material increasing the local electric means of their fractal dimension and their Weibull distribu-
field and eventually producing new channels. Since its intro- tion parameters 关19兴.
In Sec. II we present a description of the DBM, and in
Sec. III the model generalization is introduced and a brief
*Author to whom correspondence should be addressed. Mailing description of the model for conductor-loaded composites is
address: Casilla de Correo 314, 共1900兲 La Plata, Argentina. FAX: also included for comparison. Results are presented in Sec.
0054 221 4254642. Email address: eemola@inifta.unlp.edu.ar IV, and our conclusions are summarized in Sec. V.
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DIELECTRIC BREAKDOWN MODEL FOR CONDUCTOR- . . . PHYSICAL REVIEW E 69, 016123 共2004兲
P 共 i,k→i ⬘ ,k ⬘ 兲 ⬀ 共 i ⬘ ,k ⬘ 兲 , 共5a兲
IV. RESULTS
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FIG. 3. Dependence of Weibull distribution parameters 共a兲 the characteristic time ␣ and 共b兲 the shape factor  , on the fraction of
conducting particles, p and , calculated from a set of 100 electrical trees by employing the SCMC.
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DIELECTRIC BREAKDOWN MODEL FOR CONDUCTOR- . . . PHYSICAL REVIEW E 69, 016123 共2004兲
A numerical study comparing both the SCMC and the (  ⬃1). In the SCMI this behavior indicates a large propor-
SCMI was performed. Electrical trees are characterized by tion of breakdowns at short times. It is important to note that
their fractal dimension D and the ␣ and  parameters of the although the inclusion of insulator filler in the composite
Weibull distribution. Their dependencies on both the concen- increases the average failure time and ␣ 共which is a desirable
tration of particles p and the exponent were studied. property兲, it also produces a strong reduction of  and a
In the SCMI a critical concentration exists (p * ⫽0.42 broad distribution of failure time that in practical applica-
⫾0.03) beyond which dielectric breakdown does not occur. tions means a loss in the reliability of the material.
This phenomenon happens when all possible growth sites for In the SCMC the reduction of  is accompanied by a
the tree are occupied with insulator particles. Below p * the reduction of ␣ and is a consequence of the instantaneous
propagation time distribution follows a Weibull distribution incorporation of the conducting particles to the tree.
with ␣ and  parameters depending on p and . Also, the For practical purposes the extension of damage also has
fractal dimension of electrical trees, D, depends on p and , an economic impact, and in this sense, the variation of D
below p * . Qualitatively, similar behaviors of ␣ ,  , and D as must also be considered.
a function of p are obtained for all values of , except Finally, it is interesting to note that in the SCMC and due
⫽1. to the presence of conducting particles, electrical damage
The fractal dimension D behaves monotonically with p must be distinguished from mechanical damage. For ex-
共see Fig. 4兲 when ⬎1. Whereas for ⫽1, D presents a ample, the reduction of ␣ with the increase of p indicates
minimum. Electrical trees simulated with ⫽1 are charac- that the material rapidly becomes a conductor but in fact, it
terized by a higher degree of branching compared with those also shows that the number of breakdown channels is very
simulated with greater values of . Insulating particles basi- small. Since the conducting particles are incorporated instan-
cally act as obstacles inhibiting possible paths for branching, taneously to the tree, they are not included in the calculation
and therefore D decreases in the interval 0⭐ p⬍0.35. While of the propagation time; note, however, that D measures the
the branching capacity of trees with ⫽1 is enough to find a extension of the structure including conducting particles.
path to reach the counterelectrode, D will decrease only ac- Since the direction of the electrical tree propagation is
counting for the reduction of growing paths. In the interval known, such property can be useful to detect small mechani-
0.35⭐ p⬍p * the reduction in the number of breakdown cal failures in the material.
channels is so strong that the electrical trees have to increase The model presented in this paper mimics quite well fill-
their degree of branching to reach the counterelectrode and D ers with very high permittivity and mechanical strength, a
increases. Accordingly, ␣ also exhibits a nonmonotonic be- combination that makes it extremely difficult for them to be
havior for ⫽1 diminishing even below its value for p penetrated by an electrical tree. Treeing breakdowns there-
⫽0. fore avoid the fillers whenever possible, even to the extent of
On the other hand, insulating particles in electrical trees adopting extremely low field tortuous paths. Calculations
simulated with ⬎1 have the effect of increasing the degree show that for materials whose dielectric breakdown is de-
of branching in the whole interval 0⭐p⭐ p * , because di- scribed by values greater than one, the onset of tortuosity
electric trees themselves have a low capacity for branching. defines the smallest p value (⬇0.30) for effective breakdown
Breakdown structures now must explore more alternative inhibition. This can be followed by looking either at the de-
paths as p is increased and, consequently, ␣ increases with p pendence of 共Fig. 4兲 on p or at the dependence of ␣ 共Fig.
for ⬎1. 2兲 on p.
As p approaches p * (p→p*) the values of ␣ seem to In practical insulation systems the quality of the polymer-
merge together for ⬎1. ␣ roughly represents the branching insulation interface will clearly assume a major importance.
degree of the trees, which diminishes as increases. One Poor binding or contributory mechanical stresses will cause
could speculate then about the existence of a common value the interface to fail rapidly and hence facilitate the path of
for ␣ in p⫽p* 共and for Ⰷ1) because the number of pos- the breakdown around the filler particle thereby reducing its
sible growing paths for the tree decrease strongly. effectiveness.
The main difference in the breakdown process between a
polymer matrix with conducting or insulating filler is in the
ACKNOWLEDGMENTS
behavior of the characteristic propagation time ␣ . While in
the SCMC a reduction in ␣ is observed when the fraction of This research project was financially supported by the
conducting particles, p, is increased 关Fig. 3共a兲兴, in the SCMI Consejo Nacional de Investigaciones Cientı́ficas y Técnicas,
␣ increases 共for ⬎1) with the fraction of insulating par- the Comisión de Investigaciones Cientı́ficas de la Provincia
ticles, p 关Fig. 2共a兲兴. de Buenos Aires, and by the Universidades Nacionales de La
The dependence of  which is the shape parameter of the Plata 共UNLP兲 and Buenos Aires 共UBA兲. E.E.M. acknowl-
Weibull distribution, on p and is qualitatively similar for edges the useful discussions held with Professor L. A. Dis-
both the SCMI and the SCMC. As p increases, the distribu- sado, University of Leicester, United Kingdom, during his
tion becomes broad approaching an exponential distribution visits at that University.
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