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Statistical mechanics of dielectric breakdown
Physica A 261 (1998) 309–316
Statistical mechanics of dielectric breakdown J.L. Vicente a , A.C. Razzitte b , M.C. Cordero a , E.E. Mola a; ∗
a b
INIFTA, DivisiÃon QuÃmica TeÃorica, Sucursal 4, Casilla de Correo 16, La Plata 1900, Argentina LAFMACEL, Departamento de QuÃmica, Facultad de IngenierÃa, Universidad de Buenos Aires, Argentina Received 5 September 1997
Abstract A statistical picture of dielectric breakdown in solids is presented and discussed in this paper. The morphology selection mechanism is described by the stochastic model or model. The approach is formulated in terms of two functions () and (), which play the role of a free energy and that of an inverse temperature, respectively. Numerical evidence for a description of the probability distribution of branched structures grown on a square lattice is reported. c 1998 Elsevier Science B.V. All rights reserved.
PACS: 75.50.+p; 05.70.Ln; 02.50.+s Keywords: Dielectric breakdown; Branched structures; Electrical tree
Dielectric breakdown in solid insulators frequently occurs by means of narrow discharge channels that exhibit a strong tendency to branch into complicated stochastic patterns. The damaged structure has, in many cases, the form of trees. These “trees” are, at some level of approximation, fractal objets [1]. Branched discharges, or electrical trees, show a close structural similarity within a large class of discharge types. Even a qualitative classiÿcation of these structures is missing at the moment. The dielectric breakdown model (DBM) developed by Niemeyer et al. [2] to explain the formation of fractal tree structures has been extended [3] to incorporate nonoverlapping diagonal bonds (NDB). Tools of the thermodynamic formalism have proven to be extremely useful [4] in the analysis and characterization of fractal objets. This paper shows how the statistical mechanic methods can be applied to describe the distribution of electrical trees generated in a growth model. ∗
Corresponding author. Fax: +54 21 254 642.
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 1 6 - 1
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Fig. 1. An illustration of the two-dimensional model for simulating the dielectric breakdown model. The upper horizontal line represents the electrode and the perpendicular segment represents the tip where the discharge simulation begins. The other electrode is modeled by the lower horizontal line. The discharge pattern is indicated by the black dots connected by thick lines and it is considered equipotential ( = 0). The dashed bonds indicate all of the possible growth processes. The probability for each of these processes is proportional to the local electric ÿeld.
A two-dimensional square lattice (Fig. 1), in which two opposite sides represent the two electrodes, is considered. Breakdown starts at a point of a high local ÿeld and this enhancement is usually attributed to electrically conducting inclusions. The conducting inclusions are represented by electrode pins. Discharge simulation begins at one of the electrode pins where a short ÿlament represents the tip of one pin. The rules assumed for the growth of the discharge pattern (the electrical tree) are the following: (1) The electrical tree grows stepwise. The discharge structure has zero internal resistance, i.e. at each point of the structure the electric potential is = 0, whereas at the counterelectrode it is = 1. The discrete form of Laplace equation: i; k = 14 ( i+1; k + i−1; k + i; k−1 + i; k+1 )
(1)
is solved with the previous boundary conditions. (2) The probability that a bond will form between a point, which is already part of the electrical tree, and a new adjacent point is a function of the local ÿeld between the two points (i.e. the potential di erence between the two points). A power-law dependence with exponent is assumed and the probability associated with points i; k at the structure and i0 ; k 0 adjacent to but outside it is given by ( i0 k 0 ) : ( j; l)∈ ( j; l )
P(i; k → i0 ; k 0 ) = P
(2)
The sum in the denominator refers to all of the possible growth sites ( j; l) adjacent to the electrical tree, where is the set of possible candidates to be incorporate into the electrical tree.
J.L. Vicente et al. / Physica A 261 (1998) 309–316
311
Fig. 2. Two simulated tree patterns obtained with the model, with the same ÿeld exponent ( = 1). (a) a characteristic history pattern, (b) a pattern with a higher probability history.
(3) A new bond (and point) is chosen randomly and added to the electrical tree. (4) With the new electrical tree and the new boundary conditions the process starts again. A couple of tree patterns obtained from the model are shown in Fig. 2 for a constant value ( = 1). According to the growth rule of the model, a collection of electrical trees CM of “mass M ” (number of bonds) is obtained and the branching structures CM give the state of damage as a function of “time” (number of bonds). In this context, a probability p(CM ; ) for each CM can be assigned to each value of . Based on a number of numerical simulations, the following expression is obtained: ln[p(CM ; )] = S(CM ) + A(CM ) () + (; M ) ;
(3)
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where () and (; M ) are two universal functions, A(CM ) plays the role of the energy of the electrical tree, and S(CM ) is a history degeneracy factor. The above expression for p(CM ; ) has the form of a Boltzmann weight if () is identiÿed with the inverse of the temperature and (; M ) with the free energy. Then, the following equation can be written as X exp[S(CM ) + A(CM ) ()] : (4) (; M ) = − ln |CM |=M
This paper is mainly devoted to providing evidence for Eq. (3) in the abovementioned electrode geometry. A history h is a sequence {s1 ; s2 ; : : : ; sM } of M sites in the interelectrode gap (included in Z × Z) so that for each m6M; sm = (im ; km ) is the node incorporated into the electrical tree at the step m. H (CM ) is the set of various possible histories that lead to the electrical tree CM . For each value of , the probability of any history h is deÿned by P(h; ) =
M Y m=1
P
im ; km
(i0 ; k 0 )∈ n (h)
i0 ; k 0
;
(5)
where for each m6M the potential distribution i; k on the mth outer boundary m (h) = ({s1 ; s2 ; : : : ; sm }) is determined. Moreover, p(CM ; ) is normalized over the set of all the electrical trees of mass M : X p(CM ; ) = 1 : (6) |CM |=M
Any history that maximizes the distribution of history probabilities P(p) is called a “characteristic history” of the electrical tree CM for a given . The M = 1000 tree pattern shown in Fig. 2a belongs to the set of characteristic history patterns. These characteristic histories are di erent from those occurring with the highest probabilities [5]. An M = 1000 tree pattern with a higher probability history is shown in Fig. 2b. A characteristic history pattern is generally more “symmetrical” (in respect to an axis containing the tip) than those with a higher probability history. To begin the study of the distribution of electrical trees various sets of 500 electrical trees of mass M = 100; 200; 300; 400; 500; 600; 700; 800; 900; and 1000, statistically generated with di erent values of the growth parameter = 0:5; 1:0; 1:5; 2:5; 3:0; 3:5; 4:0; 4:5 and 5:0, are computed. Fig. 3 shows the distribution of probability density histories p(CM ; ) for di erent values of the ÿeld exponent . The corresponding histogram is adjusted by means of a normal (Gaussian) distribution. The standard deviation () of the normal distribution increases when the ÿeld parameter is increased. By using the approximate saddle-point method: p(CM ; ) ≈ (CM )p[hc (CM ); ] ; where hc (CM ) is any characteristic history of CM .
(7)
J.L. Vicente et al. / Physica A 261 (1998) 309–316
313
Fig. 3. Histogram of the probability distribution of the p(CM ; ) values, with M and ÿxed ( = 0:5 and M = 300). The histogram is adjusted by means of a normal distribution.
Fig. 4. ln[p(CM ; )=p(CM ; ∗2 )] versus ln[p(CM ; ∗1 )=p(CM ; ∗2 )], as computed for a representative statistical sample of branched structures of mass M = 100. Data come from the saddle-point approximation. The straight lines are linear regression ÿts for the evaluation of b(; M ) and m().
In Fig. 4, ln[p(CM ; )=p(CM ; ∗2 )] versus ln[p(CM ; ∗1 )=p(CM ; ∗2 )] was plotted for two di erent trees of mass M = 100 and M = 250, respectively, for di erent values of the ÿeld exponent at the interval [0; 5]. Our choice of ∗1 = 1 and ∗2 = 2 is arbitrary. The following equation can be written from Fig. 4: ln[p(CM ; )] = ln[p(CM ; ∗2 )] + m() ln[p(CM ; ∗1 )=p(CM ; ∗2 )] + b(; M ) ;
(8)
where m() and b(; M ) (the slope and the value at the origin, respectively) are obtained by linear regression ÿts and are plotted on Figs. 5 and 6. By comparison with Eqs. (3) and (8), b(; M ) is identiÿed with (; M ) and m() with (). Whereas
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Fig. 5. Estimation of m() for M = 100 and M = 250.
Fig. 6. Estimation of b(; M )=M ln M for M = 100 and M = 250.
ln[p(CM ; ∗1 )=p(CM ; ∗2 )] is identiÿed with the branching structure energy A(CM ), and ln[p(CM ; ∗2 )] with the degeneracy factor S(CM ). In summary, the following conclusions can be drawn: (i) () is independent of M , even for M as small as M = 100. (ii) (; M ) scales as M ln M . (iii) Our choice for ∗1 ; ∗2 is arbitrary.
J.L. Vicente et al. / Physica A 261 (1998) 309–316
315
Fig. 7. Mean energy hAi of the branched structure versus , as estimated from a sample of 40 branched structures of mass M = 100 and 250, statistically generated with di erent values for the ÿeld exponent .
It is clear that () and (; M )=M ln M , as well as the degeneracy factor S(CM ) ≡ ln[p(CM ; ∗2 )], depend upon this choice, although they only change by additive and multiplicative constants. Both Figs. 5 and 6 clearly show the convergence of () and (; M )=M ln M towards two universal, mass-independent functions. To check the consistency of Eq. (3), let us compute the mean of the branched structure energy hAi for various sets of 40 branched structures of mass M = 100 and 250, statistically generated with di erent values for the ÿeld exponent, = 0:5; 1; 2; 3; 4 and 5. hAi = hln[p(CM ; ∗1 )=p(CM ; ∗2 )]i X = p(CM ; ) ln[p(CM ; ∗1 )=p(CM ; ∗2 )] |CM |=M
= − 0 (; M )= 0 () = −d (; M )=d () :
(9)
Data plotted in Fig. 7 were obtained from Eq. (4) by plotting (; M ) as a function of () from Figs. 5 and 6. From such a plot the derivative −d =d of Eq. (9) was evaluated and plotted in Fig. 7 for two di erent M values (M = 100 and 250). Numerical simulations of the model applied to a two-dimensional stochastic model of electrical treeing in solid dielectrics were carried out. It was found that simulated trees display remarkably similar behavior to that found experimentally. The probability distribution of these branched structures is given by a Boltzmann weight, Eq. (3), where the inverse temperature () only depends upon the growth parameter (see Fig. 5), while the free energy (; M ) scales as M ln M as an extensive thermodynamic
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J.L. Vicente et al. / Physica A 261 (1998) 309–316
quantity. The Boltzmann weight, Eq. (3), also involves a history entropy factor S(CM ) which is proportional to the logarithm of the total number of histories leading to CM , see Eq. (7). To the best of our knowledge this is the ÿrst evidence that electrical trees grown on a lattice with axial symmetry exhibit the statistical mechanic properties reported in this paper. This research project was ÿnancially supported by the Consejo Nacional de Investigaciones CientÃÿcas y TÃecnicas, the Comision de Investigaciones CientÃÿcas de la Provincia de Buenos Aires and the Universidad Nacional de La Plata. The authors are also indebted to FundaciÃon Antorchas for a grant. References [1] [2] [3] [4] [5]
B. Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977. L. Niemeyer, L. Pietronero, H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1033. J.L. Vicente, A.C. Razzitte, E.E. Mola, Proc. 5th. ICSD IEEE, Leicester, 1995. G. Radons, Phys. Rev. Lett. 75 (1995) 4719. T. Vicsek, F. Family, J. KerstÃez, D. Platt, Europhys. Lett. 2 (1989) 823.
Statistical mechanics of dielectric breakdown J.L. Vicente a , A.C. Razzitte b , M.C. Cordero a , E.E. Mola a; ∗
a b
INIFTA, DivisiÃon QuÃmica TeÃorica, Sucursal 4, Casilla de Correo 16, La Plata 1900, Argentina LAFMACEL, Departamento de QuÃmica, Facultad de IngenierÃa, Universidad de Buenos Aires, Argentina Received 5 September 1997
Abstract A statistical picture of dielectric breakdown in solids is presented and discussed in this paper. The morphology selection mechanism is described by the stochastic model or model. The approach is formulated in terms of two functions () and (), which play the role of a free energy and that of an inverse temperature, respectively. Numerical evidence for a description of the probability distribution of branched structures grown on a square lattice is reported. c 1998 Elsevier Science B.V. All rights reserved.
PACS: 75.50.+p; 05.70.Ln; 02.50.+s Keywords: Dielectric breakdown; Branched structures; Electrical tree
Dielectric breakdown in solid insulators frequently occurs by means of narrow discharge channels that exhibit a strong tendency to branch into complicated stochastic patterns. The damaged structure has, in many cases, the form of trees. These “trees” are, at some level of approximation, fractal objets [1]. Branched discharges, or electrical trees, show a close structural similarity within a large class of discharge types. Even a qualitative classiÿcation of these structures is missing at the moment. The dielectric breakdown model (DBM) developed by Niemeyer et al. [2] to explain the formation of fractal tree structures has been extended [3] to incorporate nonoverlapping diagonal bonds (NDB). Tools of the thermodynamic formalism have proven to be extremely useful [4] in the analysis and characterization of fractal objets. This paper shows how the statistical mechanic methods can be applied to describe the distribution of electrical trees generated in a growth model. ∗
Corresponding author. Fax: +54 21 254 642.
c 1998 Elsevier Science B.V. All rights reserved. 0378-4371/98/$ – see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 9 8 ) 0 0 3 1 6 - 1
310
J.L. Vicente et al. / Physica A 261 (1998) 309–316
Fig. 1. An illustration of the two-dimensional model for simulating the dielectric breakdown model. The upper horizontal line represents the electrode and the perpendicular segment represents the tip where the discharge simulation begins. The other electrode is modeled by the lower horizontal line. The discharge pattern is indicated by the black dots connected by thick lines and it is considered equipotential ( = 0). The dashed bonds indicate all of the possible growth processes. The probability for each of these processes is proportional to the local electric ÿeld.
A two-dimensional square lattice (Fig. 1), in which two opposite sides represent the two electrodes, is considered. Breakdown starts at a point of a high local ÿeld and this enhancement is usually attributed to electrically conducting inclusions. The conducting inclusions are represented by electrode pins. Discharge simulation begins at one of the electrode pins where a short ÿlament represents the tip of one pin. The rules assumed for the growth of the discharge pattern (the electrical tree) are the following: (1) The electrical tree grows stepwise. The discharge structure has zero internal resistance, i.e. at each point of the structure the electric potential is = 0, whereas at the counterelectrode it is = 1. The discrete form of Laplace equation: i; k = 14 ( i+1; k + i−1; k + i; k−1 + i; k+1 )
(1)
is solved with the previous boundary conditions. (2) The probability that a bond will form between a point, which is already part of the electrical tree, and a new adjacent point is a function of the local ÿeld between the two points (i.e. the potential di erence between the two points). A power-law dependence with exponent is assumed and the probability associated with points i; k at the structure and i0 ; k 0 adjacent to but outside it is given by ( i0 k 0 ) : ( j; l)∈ ( j; l )
P(i; k → i0 ; k 0 ) = P
(2)
The sum in the denominator refers to all of the possible growth sites ( j; l) adjacent to the electrical tree, where is the set of possible candidates to be incorporate into the electrical tree.
J.L. Vicente et al. / Physica A 261 (1998) 309–316
311
Fig. 2. Two simulated tree patterns obtained with the model, with the same ÿeld exponent ( = 1). (a) a characteristic history pattern, (b) a pattern with a higher probability history.
(3) A new bond (and point) is chosen randomly and added to the electrical tree. (4) With the new electrical tree and the new boundary conditions the process starts again. A couple of tree patterns obtained from the model are shown in Fig. 2 for a constant value ( = 1). According to the growth rule of the model, a collection of electrical trees CM of “mass M ” (number of bonds) is obtained and the branching structures CM give the state of damage as a function of “time” (number of bonds). In this context, a probability p(CM ; ) for each CM can be assigned to each value of . Based on a number of numerical simulations, the following expression is obtained: ln[p(CM ; )] = S(CM ) + A(CM ) () + (; M ) ;
(3)
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J.L. Vicente et al. / Physica A 261 (1998) 309–316
where () and (; M ) are two universal functions, A(CM ) plays the role of the energy of the electrical tree, and S(CM ) is a history degeneracy factor. The above expression for p(CM ; ) has the form of a Boltzmann weight if () is identiÿed with the inverse of the temperature and (; M ) with the free energy. Then, the following equation can be written as X exp[S(CM ) + A(CM ) ()] : (4) (; M ) = − ln |CM |=M
This paper is mainly devoted to providing evidence for Eq. (3) in the abovementioned electrode geometry. A history h is a sequence {s1 ; s2 ; : : : ; sM } of M sites in the interelectrode gap (included in Z × Z) so that for each m6M; sm = (im ; km ) is the node incorporated into the electrical tree at the step m. H (CM ) is the set of various possible histories that lead to the electrical tree CM . For each value of , the probability of any history h is deÿned by P(h; ) =
M Y m=1
P
im ; km
(i0 ; k 0 )∈ n (h)
i0 ; k 0
;
(5)
where for each m6M the potential distribution i; k on the mth outer boundary m (h) = ({s1 ; s2 ; : : : ; sm }) is determined. Moreover, p(CM ; ) is normalized over the set of all the electrical trees of mass M : X p(CM ; ) = 1 : (6) |CM |=M
Any history that maximizes the distribution of history probabilities P(p) is called a “characteristic history” of the electrical tree CM for a given . The M = 1000 tree pattern shown in Fig. 2a belongs to the set of characteristic history patterns. These characteristic histories are di erent from those occurring with the highest probabilities [5]. An M = 1000 tree pattern with a higher probability history is shown in Fig. 2b. A characteristic history pattern is generally more “symmetrical” (in respect to an axis containing the tip) than those with a higher probability history. To begin the study of the distribution of electrical trees various sets of 500 electrical trees of mass M = 100; 200; 300; 400; 500; 600; 700; 800; 900; and 1000, statistically generated with di erent values of the growth parameter = 0:5; 1:0; 1:5; 2:5; 3:0; 3:5; 4:0; 4:5 and 5:0, are computed. Fig. 3 shows the distribution of probability density histories p(CM ; ) for di erent values of the ÿeld exponent . The corresponding histogram is adjusted by means of a normal (Gaussian) distribution. The standard deviation () of the normal distribution increases when the ÿeld parameter is increased. By using the approximate saddle-point method: p(CM ; ) ≈ (CM )p[hc (CM ); ] ; where hc (CM ) is any characteristic history of CM .
(7)
J.L. Vicente et al. / Physica A 261 (1998) 309–316
313
Fig. 3. Histogram of the probability distribution of the p(CM ; ) values, with M and ÿxed ( = 0:5 and M = 300). The histogram is adjusted by means of a normal distribution.
Fig. 4. ln[p(CM ; )=p(CM ; ∗2 )] versus ln[p(CM ; ∗1 )=p(CM ; ∗2 )], as computed for a representative statistical sample of branched structures of mass M = 100. Data come from the saddle-point approximation. The straight lines are linear regression ÿts for the evaluation of b(; M ) and m().
In Fig. 4, ln[p(CM ; )=p(CM ; ∗2 )] versus ln[p(CM ; ∗1 )=p(CM ; ∗2 )] was plotted for two di erent trees of mass M = 100 and M = 250, respectively, for di erent values of the ÿeld exponent at the interval [0; 5]. Our choice of ∗1 = 1 and ∗2 = 2 is arbitrary. The following equation can be written from Fig. 4: ln[p(CM ; )] = ln[p(CM ; ∗2 )] + m() ln[p(CM ; ∗1 )=p(CM ; ∗2 )] + b(; M ) ;
(8)
where m() and b(; M ) (the slope and the value at the origin, respectively) are obtained by linear regression ÿts and are plotted on Figs. 5 and 6. By comparison with Eqs. (3) and (8), b(; M ) is identiÿed with (; M ) and m() with (). Whereas
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Fig. 5. Estimation of m() for M = 100 and M = 250.
Fig. 6. Estimation of b(; M )=M ln M for M = 100 and M = 250.
ln[p(CM ; ∗1 )=p(CM ; ∗2 )] is identiÿed with the branching structure energy A(CM ), and ln[p(CM ; ∗2 )] with the degeneracy factor S(CM ). In summary, the following conclusions can be drawn: (i) () is independent of M , even for M as small as M = 100. (ii) (; M ) scales as M ln M . (iii) Our choice for ∗1 ; ∗2 is arbitrary.
J.L. Vicente et al. / Physica A 261 (1998) 309–316
315
Fig. 7. Mean energy hAi of the branched structure versus , as estimated from a sample of 40 branched structures of mass M = 100 and 250, statistically generated with di erent values for the ÿeld exponent .
It is clear that () and (; M )=M ln M , as well as the degeneracy factor S(CM ) ≡ ln[p(CM ; ∗2 )], depend upon this choice, although they only change by additive and multiplicative constants. Both Figs. 5 and 6 clearly show the convergence of () and (; M )=M ln M towards two universal, mass-independent functions. To check the consistency of Eq. (3), let us compute the mean of the branched structure energy hAi for various sets of 40 branched structures of mass M = 100 and 250, statistically generated with di erent values for the ÿeld exponent, = 0:5; 1; 2; 3; 4 and 5. hAi = hln[p(CM ; ∗1 )=p(CM ; ∗2 )]i X = p(CM ; ) ln[p(CM ; ∗1 )=p(CM ; ∗2 )] |CM |=M
= − 0 (; M )= 0 () = −d (; M )=d () :
(9)
Data plotted in Fig. 7 were obtained from Eq. (4) by plotting (; M ) as a function of () from Figs. 5 and 6. From such a plot the derivative −d =d of Eq. (9) was evaluated and plotted in Fig. 7 for two di erent M values (M = 100 and 250). Numerical simulations of the model applied to a two-dimensional stochastic model of electrical treeing in solid dielectrics were carried out. It was found that simulated trees display remarkably similar behavior to that found experimentally. The probability distribution of these branched structures is given by a Boltzmann weight, Eq. (3), where the inverse temperature () only depends upon the growth parameter (see Fig. 5), while the free energy (; M ) scales as M ln M as an extensive thermodynamic
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J.L. Vicente et al. / Physica A 261 (1998) 309–316
quantity. The Boltzmann weight, Eq. (3), also involves a history entropy factor S(CM ) which is proportional to the logarithm of the total number of histories leading to CM , see Eq. (7). To the best of our knowledge this is the ÿrst evidence that electrical trees grown on a lattice with axial symmetry exhibit the statistical mechanic properties reported in this paper. This research project was ÿnancially supported by the Consejo Nacional de Investigaciones CientÃÿcas y TÃecnicas, the Comision de Investigaciones CientÃÿcas de la Provincia de Buenos Aires and the Universidad Nacional de La Plata. The authors are also indebted to FundaciÃon Antorchas for a grant. References [1] [2] [3] [4] [5]
B. Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1977. L. Niemeyer, L. Pietronero, H.J. Wiesmann, Phys. Rev. Lett. 52 (1984) 1033. J.L. Vicente, A.C. Razzitte, E.E. Mola, Proc. 5th. ICSD IEEE, Leicester, 1995. G. Radons, Phys. Rev. Lett. 75 (1995) 4719. T. Vicsek, F. Family, J. KerstÃez, D. Platt, Europhys. Lett. 2 (1989) 823.