- Email: [email protected]
Stimulated percolation of electrical conductivity in heterogenous mixtures
Synthetic Metals 109 Ž2000. 295–299 www.elsevier.comrlocatersynmet
Stimulated percolation of electrical conductivity in heterogenous mixtures G. Wisniewski ´
)
Faculty of Technical Physics and Applied Mathematics, Technical UniÕersity of Gdansk, ´ ul. G. Narutowicza 11 r 12, 80-952 Gdansk, Poland Received 26 June 1999; received in revised form 15 July 1999; accepted 10 September 1999
Abstract In a composite of liquid dielectric with granular conducting dopant in a form of a micrometre particles of conductor, one observes the structure ordering process due to external electric field, manifested by the increase of observed electrical conductivity. The mechanism of such an ordering is similar to the phenomena occurring in the electrorheological liquids. By application of the conductor particles as an active polar dopant, the observed ordering effects take place at lower values of the electric field, compared with the ordering case of the granular dopants of an insulator. A simple theoretical description of the above mentioned phenomena was developed. They are attributed to the mutual interaction of conducting particles, having the field-induced dipole moments. The measured current increase is due to the power dissipation in the sample, related to the particle motion in viscous medium, as well as to the increase of sample capacitance. The ordering effect of such a medium consists in the mutual electrostatic interaction of the electric dipoles, induced in the dopant particles, as well as in the buildup of a linear or dendritic spatial structures with a symmetry determined by the direction of the external electric field. In the theoretical part of the work, the assumption on the coagulation of conductor particles into linear conducting structures has been used. One can search, however, the theoretical argumentation based on Flory’s concept with respect to the described dipole interactions and construct the statistical sum. By the application of the mathematical method, one can discuss topology and percolation effects in such mixtures. Apart from the effects of linear, regular ordering of the composite, one can observe also the nonlinear effects, when the conducting fractal structure is being created. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Polymer composites; Percolation; Electrical conductivity; Ordering processes
1. Introduction The electrical properties of composite materials are still the great hope of modern electronics. The superconducting ceramics and the polymer composites are good examples of such a tendency. The general rule is the dependence of their properties not only on the amount of individual components, but also on the technology of material preparation w1,2x. One of the factors that can influence the electrical conductivity of carbon black–polymer composites is the application of electric field during their preparation w1–4x. For the first time, this effect was observed for the carbon black–polyester composites, polymerized in the alternating electric field w1–4x. The application of alternating field was necessary because of high conductivity of the styrene
)
E-mail: [email protected]
monomers, polymerized radically in the presence of ion initiators. Despite the polymer gelation, in such a way, the electrical conductivity of composite could be increased by about 2 orders of magnitude. The resulting conductivity shows the resonance dependence on the frequency of applied field during the composite polymerization w4x. In order to investigate those phenomena more closely, the analogous experiments were carried out for the model mixture of liquid paraffin with carbon black, using the direct electric field. In this way, it was possible to establish the mechanism of the conductivity increase of mentioned composites. The chosen composite of liquid paraffin and carbon black can be frozen to the wax solid in the temperature of y278C. The dependence of its electrical conductivity on the carbon black concentration is typical for such solid composites, and the percolation threshold is observed at 22% content of the conducting dopant ŽFig. 1.. In a liquid state Žat room temperature, 208C., appreciable electrical conductivity of the mixture occurs for the
0379-6779r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 9 - 6 7 7 9 Ž 9 9 . 0 0 2 4 9 - 0
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´
296
much lower dopant concentration. The electrical current flowing in the sample grows gradually in time. For the amount of 5.5% of carbon black in liquid paraffin Ždiameter of the samples 10 mm, its thicknesss 1 mm., the dissipated power reaches the value exceeding the overcurrent protection of voltage supplier, causing its turn-off. The experimental current–time dependencies may be described by the formula I Ž t . s I0 Ž 1 y aEeb t .
yc
Ž 1.
where Ee is the external electric field intensity, t is the time, and a is the factor depending on the concentration of carbon black. The obtained values of b and c are: b s 2.1–2.5, c s 0.74–0.95. The example of such current–time dependence is shown in Fig. 2. For lower concentration of carbon black Žsmaller than 1%., the measured current transients exhibit another behaviour w8x. We have assumed that the mentioned increase of current flow is caused by the structural ordering of investigated composite. This is the result of interaction of the mobile conducting particles, having dipole moments induced by external field. The mechanism of such ordering is probably similar to this occurring in the electrorheological phenomena w3,5,6x, which is observed at higher values of electric field Žabout 10 6 Vrm.. With the use of conducting particles, the ordering effects can be observed at lower field intensities, even of the order of a few 10 4 Vrm. The microscopic observation of the samples gives some insight into the nature of proceeding phenomena. For the higher values of applied field Ž Ee ) 10 4 Vrm., one can observe the linear structures build of carbon black parti-
Fig. 2. Current–time dependences for the composites containing 5.5% of acetylene carbon black at different values of the applied electrical fields. The dashed lines are calculated from Eqs. Ž13. and Ž14..
cles, which regularity diminishes with the decrease of external field w8x.
2. Theoretical The obtained results indicate that, depending on the concentration of the particles and the electric field intensity, the conducting particles in viscous liquid tend to coagulation into linear structures or to spreading. The current oscillations are probably caused by the phenomena occurring in the transition region between these different phases. We shall begin the theoretical analysis of described effects from the consideration of interaction of two dipoles. We assume that they have parallelly aligned dipole moments p 0 and pN . The energy of its interaction is given by the formula: p 0 pN Us Ž 1 y 3cos 2 u . , Ž 2. 4p´ 0 ´ p r 3 where u is the angle between the axis of dipole and the radius r, connecting the centres of dipoles, and ´ 0 and ´ p are the dielectric constants of vacuum and liquid, respectively. The dipole moment p 0 of separate particle of diameter d and the dipole moment pN of linear particle agglomerate Žwith time-depending length N Ž t . d . w6,7x, are p0 s
Fig. 1. Dependence of electrical conductivity of the solid Žfrozen. paraffin–acetylene carbon black mixtures on the carbon black content at the temperature y278C.
pN s
1 2 1 4
p´ 0 ´ p d 3 Ee ,
Ž 3.
p´ 0 ´ p d 3 N 3 Ž t . Ee ,
Ž 4.
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´
The equation of motion of the dipole p 0 in the field of heavy dipole pN is given by m
d2 r 2
dt
qb
dr dt
gradU s 0.
Ž 6.
The above formula is in qualitative agreement with the experimental current–time dependencies, measured in the composites ŽFig. 2.. In order to determine the dimension of the spaces, influenced by the interaction of accumulating centre, one has to make some additional assumptions. We have to take into account the fact that the average energy of interacting dipoles
Ž 7.
²W : s y
Ž 5.
Choosing the following time and distance units t 0 s mrb , r0 s
ž
3p´ 0 ´ p d 3 Ee2 m 16 b
1r5
/
and considering a motion of a small dipole near the axis of the central dipole, we get d2 r dt2
dr q
q dt
N 3Ž t. r4
s0
Ž 8.
where r and t are now expressed in the units r 0 and t 0 . The spatial concentration nŽ t,r . of particles in the vicinity of central dipole is described by the continuity equation En Ž t ,r . Et
q div n Ž t ,r . Õ Ž t ,r . s 0.
Ž 9.
Assuming that the central dipole composed of N Ž t . particles is a ‘‘drain’’ for the ‘‘fluid’’ of particles, described by the function nŽ t,r ., one can write NŽ t. s
HÕ
n 0 y n Ž t ,r . d 3 r ,
Ž 10 .
where n 0 is the initial density of particles. The obtained equations can be solved in approximate way. The time dependence of the particle number N Ž t ., coagulated into linear structure with dipole moment pN is given by N Ž t . s exp
ž
4p n 0 a t 3d 2
/
Ž 11 .
I Ž t . s b Õ 2 Ž t ,r . n Ž t ,r . d 3 r .
Ž 12 .
HÕ
In the cases considered above, corresponding to initial and final stages of mixture ordering, one can obtain IŽ t. A
3r5
1 y Ž Drl . exp Ž 3tr2 k .
exp
3t
ž / 2k
Ž 13 .
where
ky1 s
4p´ 0 ´ p n 0 d 4 Ez2 12 b
.
Ž 15 .
4p´ 0 ´ p r 3
RD s
ž
p 02 4p´ 0 ´ p kT
1r3
/
Ž 16 .
The considered interaction orders the conductor particles in the spaces less than R D . We can now estimate the number M of aggregation processes in the plane-parallel sample with the base area S and the thickness D. If the probability of finding the conductor particle at the distance making possible the coagulation is determined by ps
RD
H0
3r 2
l30
3
exp y Ž rrl . d r s 1 y exp y Ž R D rl .
3
Ž 17 . where
ž
4 3
y1 r3
p n0
/
Ž 18 .
is the initial distance between particles then
where a depends on the external electric field. In the course of ordering process one observes the flow of electrical current in the external circuit. The movement of particles in viscous medium causes the power dissipation, amounting b Õ per particle. Thus, one can write the formula for the time dependence of the current I Ž t . flowing in the mixture:
E
p 0 pN
must be greater than the thermal energy kT Žwith k the Boltzmann constant and T the temperature.. Such a reasoning introduces into our considerations the counterpart of Debye radius
ls ,
297
Ž 14 .
½
M s SDn 0 p s SDn 0 1 y exp y Ž R D rl .
3
5
Ž 19 .
and the mean distance between such agglomeration centres may be expressed in the form
½
L s l 1 y exp y Ž R D rl .
3
y1 r3
5
Ž 20 .
If the arising macrodipole assumes the form of the linear structure of length Nd Žwe shall show this in the discussion. and such structures stand up on average at distances L, the minimum requirement for the possibility of reaching the interelectrode connections is that the number of particles in the sphere of radius L cannot be smaller than the diameter of the sphere ‘effective interaction range’. Here, the assumption on the coagulation of conductor particles into linear conducting structures has been used. This is justified by the microscopic observations. One can search, however, the theoretical argumentation based on the Flory’s w7x concept with respect to the described here
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´
298
significant difficulties to explain the creation of the neuron-similar network of conducting particles and the current oscillations at low electrical fields. We think that the first effect is very similar to the phenomenon described by Matsushita w5x. His experiment consisted on the electrolytical deposition of the dendritic zinc crystals on the surface of a solid. In this work, the build up of fractal structures was explained by the dominant role of Brown thermal motion of the particles. We think that the dendritic structures of carbon black, created in the weak external field, are also the fractals. We were not able to determine the fractal dimension of those structures and we know only that it is an increasing function of the applied field Ee . This behaviour is different than in the phenomena studied by Matsushita w5x and Stanley and Meakin w6x, and needs further research.
Fig. 3. Dependence of the time t corresponding to the current singularity for the two paraffin–carbon black composites as a function of the reciprocal of applied electric field.
dipole interactions. For such interactions we can construct the statistical sum of the form:
ž
Z s P Ž r . exp y
²W : kT
/
Ž 21 .
where ²W : is given by Eq. Ž15. and P Ž r . is the probability of volume distribution of particles Žused in Eq. Ž17.., given by 3r 2
3
exp y Ž Rrl . Ž 22 . l3 Calculating the free energy minimum, we get the ‘‘effective interaction range’’ in the form PŽ r. s
RN s
ž
1r3
3 p 02 8p´ 0 ´ p kT
/
N f R D N.
Ž 23 .
The interaction extents thus grow proportionally to the buildup of macrodipole dimensions, which means that the macrodipole buildup proceeds in an linear manner. If we assume that percolation threshold is reached when L f RN, then we can calculate the build-time t of such ‘‘connecting’’ cluster
ts
1 4a n 0 E
2
ln
ž
kTl d n0 E 2
/
.
4. Conclusions In the composite of liquid dielectric and granular conducting dopant Žcarbon black. the ordering phenomena are observed after the application of an electric field. The ordering of internal composite structure and the growth of flowing current are caused by the interaction of the conducting particles, having dipole moments induced by electric field. This effect is similar to that occurring in the electrorheological fluids. As described above, the network of conducting particles in insulating medium may be modified by external electric field. In this way, one can change the structure of the network and the number of its connections. Therefore, it is possible ‘to learn’ the composite to conduct the electric current. We have thus an important applied method of the change of composite electrical properties. In particular, it is possible to make the resistors and the capacitors having almost arbitrary parameters.
Acknowledgements The author would like to thank Dr. W. Tomaszewicz for his helpful contribution to this work.
Ž 24 .
The dependence of time t on the external electric field corresponds quite well with experimental data ŽFig. 3..
3. Discussion The presented formulas describe well the behaviour of composite in the strong electric field. However, there exist
References w1x G. Wisniewski, B. Jachym, Ordering Processes of Heterogenous ´ Mixtures, in: International Conference on Carbon Black, Mulhouse, France, 1993. w2x G. Wisniewski, W. Tomaszewicz, B. Jachym, in preparation. ´ w3x T.C. Halsey, Electrorheological fluids, Science 258 Ž1992. 761–766.
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´ w4x B. Jachym, M. Garbacki, I. Witort, G. Wisniewski, T. Andrzejewski, ´ Polish patent No 148308. w5x M. Matsushita, in: D. Avnir ŽEd.., The Fractal Approach to Heterogenous Chemistry: Surfaces, Colloids, Polymers, Wiley, Chichester, 1989. w6x H.E. Stanley, P. Meakin, Nature 335 Ž1988. 405–409.
299
w7x P.J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, 1953. w8x G. Wisniewski, Ordering processes of heterogenous mixtures, in ´ International Conference on Solid State Crystals ’98, Proceedings of SPIE, 3724 Ž1999. 130–137.
Stimulated percolation of electrical conductivity in heterogenous mixtures G. Wisniewski ´
)
Faculty of Technical Physics and Applied Mathematics, Technical UniÕersity of Gdansk, ´ ul. G. Narutowicza 11 r 12, 80-952 Gdansk, Poland Received 26 June 1999; received in revised form 15 July 1999; accepted 10 September 1999
Abstract In a composite of liquid dielectric with granular conducting dopant in a form of a micrometre particles of conductor, one observes the structure ordering process due to external electric field, manifested by the increase of observed electrical conductivity. The mechanism of such an ordering is similar to the phenomena occurring in the electrorheological liquids. By application of the conductor particles as an active polar dopant, the observed ordering effects take place at lower values of the electric field, compared with the ordering case of the granular dopants of an insulator. A simple theoretical description of the above mentioned phenomena was developed. They are attributed to the mutual interaction of conducting particles, having the field-induced dipole moments. The measured current increase is due to the power dissipation in the sample, related to the particle motion in viscous medium, as well as to the increase of sample capacitance. The ordering effect of such a medium consists in the mutual electrostatic interaction of the electric dipoles, induced in the dopant particles, as well as in the buildup of a linear or dendritic spatial structures with a symmetry determined by the direction of the external electric field. In the theoretical part of the work, the assumption on the coagulation of conductor particles into linear conducting structures has been used. One can search, however, the theoretical argumentation based on Flory’s concept with respect to the described dipole interactions and construct the statistical sum. By the application of the mathematical method, one can discuss topology and percolation effects in such mixtures. Apart from the effects of linear, regular ordering of the composite, one can observe also the nonlinear effects, when the conducting fractal structure is being created. q 2000 Elsevier Science S.A. All rights reserved. Keywords: Polymer composites; Percolation; Electrical conductivity; Ordering processes
1. Introduction The electrical properties of composite materials are still the great hope of modern electronics. The superconducting ceramics and the polymer composites are good examples of such a tendency. The general rule is the dependence of their properties not only on the amount of individual components, but also on the technology of material preparation w1,2x. One of the factors that can influence the electrical conductivity of carbon black–polymer composites is the application of electric field during their preparation w1–4x. For the first time, this effect was observed for the carbon black–polyester composites, polymerized in the alternating electric field w1–4x. The application of alternating field was necessary because of high conductivity of the styrene
)
E-mail: [email protected]
monomers, polymerized radically in the presence of ion initiators. Despite the polymer gelation, in such a way, the electrical conductivity of composite could be increased by about 2 orders of magnitude. The resulting conductivity shows the resonance dependence on the frequency of applied field during the composite polymerization w4x. In order to investigate those phenomena more closely, the analogous experiments were carried out for the model mixture of liquid paraffin with carbon black, using the direct electric field. In this way, it was possible to establish the mechanism of the conductivity increase of mentioned composites. The chosen composite of liquid paraffin and carbon black can be frozen to the wax solid in the temperature of y278C. The dependence of its electrical conductivity on the carbon black concentration is typical for such solid composites, and the percolation threshold is observed at 22% content of the conducting dopant ŽFig. 1.. In a liquid state Žat room temperature, 208C., appreciable electrical conductivity of the mixture occurs for the
0379-6779r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 7 9 - 6 7 7 9 Ž 9 9 . 0 0 2 4 9 - 0
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´
296
much lower dopant concentration. The electrical current flowing in the sample grows gradually in time. For the amount of 5.5% of carbon black in liquid paraffin Ždiameter of the samples 10 mm, its thicknesss 1 mm., the dissipated power reaches the value exceeding the overcurrent protection of voltage supplier, causing its turn-off. The experimental current–time dependencies may be described by the formula I Ž t . s I0 Ž 1 y aEeb t .
yc
Ž 1.
where Ee is the external electric field intensity, t is the time, and a is the factor depending on the concentration of carbon black. The obtained values of b and c are: b s 2.1–2.5, c s 0.74–0.95. The example of such current–time dependence is shown in Fig. 2. For lower concentration of carbon black Žsmaller than 1%., the measured current transients exhibit another behaviour w8x. We have assumed that the mentioned increase of current flow is caused by the structural ordering of investigated composite. This is the result of interaction of the mobile conducting particles, having dipole moments induced by external field. The mechanism of such ordering is probably similar to this occurring in the electrorheological phenomena w3,5,6x, which is observed at higher values of electric field Žabout 10 6 Vrm.. With the use of conducting particles, the ordering effects can be observed at lower field intensities, even of the order of a few 10 4 Vrm. The microscopic observation of the samples gives some insight into the nature of proceeding phenomena. For the higher values of applied field Ž Ee ) 10 4 Vrm., one can observe the linear structures build of carbon black parti-
Fig. 2. Current–time dependences for the composites containing 5.5% of acetylene carbon black at different values of the applied electrical fields. The dashed lines are calculated from Eqs. Ž13. and Ž14..
cles, which regularity diminishes with the decrease of external field w8x.
2. Theoretical The obtained results indicate that, depending on the concentration of the particles and the electric field intensity, the conducting particles in viscous liquid tend to coagulation into linear structures or to spreading. The current oscillations are probably caused by the phenomena occurring in the transition region between these different phases. We shall begin the theoretical analysis of described effects from the consideration of interaction of two dipoles. We assume that they have parallelly aligned dipole moments p 0 and pN . The energy of its interaction is given by the formula: p 0 pN Us Ž 1 y 3cos 2 u . , Ž 2. 4p´ 0 ´ p r 3 where u is the angle between the axis of dipole and the radius r, connecting the centres of dipoles, and ´ 0 and ´ p are the dielectric constants of vacuum and liquid, respectively. The dipole moment p 0 of separate particle of diameter d and the dipole moment pN of linear particle agglomerate Žwith time-depending length N Ž t . d . w6,7x, are p0 s
Fig. 1. Dependence of electrical conductivity of the solid Žfrozen. paraffin–acetylene carbon black mixtures on the carbon black content at the temperature y278C.
pN s
1 2 1 4
p´ 0 ´ p d 3 Ee ,
Ž 3.
p´ 0 ´ p d 3 N 3 Ž t . Ee ,
Ž 4.
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´
The equation of motion of the dipole p 0 in the field of heavy dipole pN is given by m
d2 r 2
dt
qb
dr dt
gradU s 0.
Ž 6.
The above formula is in qualitative agreement with the experimental current–time dependencies, measured in the composites ŽFig. 2.. In order to determine the dimension of the spaces, influenced by the interaction of accumulating centre, one has to make some additional assumptions. We have to take into account the fact that the average energy of interacting dipoles
Ž 7.
²W : s y
Ž 5.
Choosing the following time and distance units t 0 s mrb , r0 s
ž
3p´ 0 ´ p d 3 Ee2 m 16 b
1r5
/
and considering a motion of a small dipole near the axis of the central dipole, we get d2 r dt2
dr q
q dt
N 3Ž t. r4
s0
Ž 8.
where r and t are now expressed in the units r 0 and t 0 . The spatial concentration nŽ t,r . of particles in the vicinity of central dipole is described by the continuity equation En Ž t ,r . Et
q div n Ž t ,r . Õ Ž t ,r . s 0.
Ž 9.
Assuming that the central dipole composed of N Ž t . particles is a ‘‘drain’’ for the ‘‘fluid’’ of particles, described by the function nŽ t,r ., one can write NŽ t. s
HÕ
n 0 y n Ž t ,r . d 3 r ,
Ž 10 .
where n 0 is the initial density of particles. The obtained equations can be solved in approximate way. The time dependence of the particle number N Ž t ., coagulated into linear structure with dipole moment pN is given by N Ž t . s exp
ž
4p n 0 a t 3d 2
/
Ž 11 .
I Ž t . s b Õ 2 Ž t ,r . n Ž t ,r . d 3 r .
Ž 12 .
HÕ
In the cases considered above, corresponding to initial and final stages of mixture ordering, one can obtain IŽ t. A
3r5
1 y Ž Drl . exp Ž 3tr2 k .
exp
3t
ž / 2k
Ž 13 .
where
ky1 s
4p´ 0 ´ p n 0 d 4 Ez2 12 b
.
Ž 15 .
4p´ 0 ´ p r 3
RD s
ž
p 02 4p´ 0 ´ p kT
1r3
/
Ž 16 .
The considered interaction orders the conductor particles in the spaces less than R D . We can now estimate the number M of aggregation processes in the plane-parallel sample with the base area S and the thickness D. If the probability of finding the conductor particle at the distance making possible the coagulation is determined by ps
RD
H0
3r 2
l30
3
exp y Ž rrl . d r s 1 y exp y Ž R D rl .
3
Ž 17 . where
ž
4 3
y1 r3
p n0
/
Ž 18 .
is the initial distance between particles then
where a depends on the external electric field. In the course of ordering process one observes the flow of electrical current in the external circuit. The movement of particles in viscous medium causes the power dissipation, amounting b Õ per particle. Thus, one can write the formula for the time dependence of the current I Ž t . flowing in the mixture:
E
p 0 pN
must be greater than the thermal energy kT Žwith k the Boltzmann constant and T the temperature.. Such a reasoning introduces into our considerations the counterpart of Debye radius
ls ,
297
Ž 14 .
½
M s SDn 0 p s SDn 0 1 y exp y Ž R D rl .
3
5
Ž 19 .
and the mean distance between such agglomeration centres may be expressed in the form
½
L s l 1 y exp y Ž R D rl .
3
y1 r3
5
Ž 20 .
If the arising macrodipole assumes the form of the linear structure of length Nd Žwe shall show this in the discussion. and such structures stand up on average at distances L, the minimum requirement for the possibility of reaching the interelectrode connections is that the number of particles in the sphere of radius L cannot be smaller than the diameter of the sphere ‘effective interaction range’. Here, the assumption on the coagulation of conductor particles into linear conducting structures has been used. This is justified by the microscopic observations. One can search, however, the theoretical argumentation based on the Flory’s w7x concept with respect to the described here
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´
298
significant difficulties to explain the creation of the neuron-similar network of conducting particles and the current oscillations at low electrical fields. We think that the first effect is very similar to the phenomenon described by Matsushita w5x. His experiment consisted on the electrolytical deposition of the dendritic zinc crystals on the surface of a solid. In this work, the build up of fractal structures was explained by the dominant role of Brown thermal motion of the particles. We think that the dendritic structures of carbon black, created in the weak external field, are also the fractals. We were not able to determine the fractal dimension of those structures and we know only that it is an increasing function of the applied field Ee . This behaviour is different than in the phenomena studied by Matsushita w5x and Stanley and Meakin w6x, and needs further research.
Fig. 3. Dependence of the time t corresponding to the current singularity for the two paraffin–carbon black composites as a function of the reciprocal of applied electric field.
dipole interactions. For such interactions we can construct the statistical sum of the form:
ž
Z s P Ž r . exp y
²W : kT
/
Ž 21 .
where ²W : is given by Eq. Ž15. and P Ž r . is the probability of volume distribution of particles Žused in Eq. Ž17.., given by 3r 2
3
exp y Ž Rrl . Ž 22 . l3 Calculating the free energy minimum, we get the ‘‘effective interaction range’’ in the form PŽ r. s
RN s
ž
1r3
3 p 02 8p´ 0 ´ p kT
/
N f R D N.
Ž 23 .
The interaction extents thus grow proportionally to the buildup of macrodipole dimensions, which means that the macrodipole buildup proceeds in an linear manner. If we assume that percolation threshold is reached when L f RN, then we can calculate the build-time t of such ‘‘connecting’’ cluster
ts
1 4a n 0 E
2
ln
ž
kTl d n0 E 2
/
.
4. Conclusions In the composite of liquid dielectric and granular conducting dopant Žcarbon black. the ordering phenomena are observed after the application of an electric field. The ordering of internal composite structure and the growth of flowing current are caused by the interaction of the conducting particles, having dipole moments induced by electric field. This effect is similar to that occurring in the electrorheological fluids. As described above, the network of conducting particles in insulating medium may be modified by external electric field. In this way, one can change the structure of the network and the number of its connections. Therefore, it is possible ‘to learn’ the composite to conduct the electric current. We have thus an important applied method of the change of composite electrical properties. In particular, it is possible to make the resistors and the capacitors having almost arbitrary parameters.
Acknowledgements The author would like to thank Dr. W. Tomaszewicz for his helpful contribution to this work.
Ž 24 .
The dependence of time t on the external electric field corresponds quite well with experimental data ŽFig. 3..
3. Discussion The presented formulas describe well the behaviour of composite in the strong electric field. However, there exist
References w1x G. Wisniewski, B. Jachym, Ordering Processes of Heterogenous ´ Mixtures, in: International Conference on Carbon Black, Mulhouse, France, 1993. w2x G. Wisniewski, W. Tomaszewicz, B. Jachym, in preparation. ´ w3x T.C. Halsey, Electrorheological fluids, Science 258 Ž1992. 761–766.
G. Wisniewskir Synthetic Metals 109 (2000) 295–299 ´ w4x B. Jachym, M. Garbacki, I. Witort, G. Wisniewski, T. Andrzejewski, ´ Polish patent No 148308. w5x M. Matsushita, in: D. Avnir ŽEd.., The Fractal Approach to Heterogenous Chemistry: Surfaces, Colloids, Polymers, Wiley, Chichester, 1989. w6x H.E. Stanley, P. Meakin, Nature 335 Ž1988. 405–409.
299
w7x P.J. Flory, Principles of Polymer Chemistry, Cornell Univ. Press, Ithaca, 1953. w8x G. Wisniewski, Ordering processes of heterogenous mixtures, in ´ International Conference on Solid State Crystals ’98, Proceedings of SPIE, 3724 Ž1999. 130–137.