JOURNAL OF APPLIED PHYSICS 106, 016103 共2009兲
Electron tunneling in conductor-insulator composites with spherical fillers G. Ambrosetti,1,2,a兲 N. Johner,1 C. Grimaldi,1,b兲 T. Maeder,1 P. Ryser,1 and A. Danani2 1
LPM, Ecole Polytechnique Fédérale de Lausanne, Station 17, CH-1015 Lausanne, Switzerland ICIMSI, University of Applied Sciences of Southern Switzerland, CH-6928 Manno, Switzerland
2
共Received 6 March 2009; accepted 2 June 2009; published online 7 July 2009兲 We report on our Monte Carlo calculations of the conductivity of monosized and conducting spherical particles dispersed in a homogeneous matrix, with interparticle transport mechanism given by electron tunneling. We show that our numerical results can be reproduced by a simple formula based on a critical path analysis and which gives also a practical way to estimate the characteristic tunneling length in real composites. We find that is about 1 nm for several low structure carbon black polymer composites, in agreement with the expected order of magnitude. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3159040兴 Polymers loaded with conductive fillers to increase the overall conductivity by several orders of magnitude have been widely used in the industry for applications such as antistatic materials, electromagnetic interference shielding, variable resistors, and Li-ion batteries. Carbonaceous fillers, especially carbon black 共CB兲, remain the most extensively used additives for conductive plastics.1 It is a well established fact that such random insulator-conductor mixtures undergo a quite abrupt transition of the overall conductivity when a critical concentration of the conductive phase is reached. Percolation theory2 has been often introduced to describe such behavior and identifies the transition with the formation of a global cluster of 共electrically兲 connected particles which spans the sample. The most accredited interparticle connectivity mechanism for carbon-polymer composites 共as well as for other conductor-polymer composites兲 is quantum-mechanical electron tunneling,3–6 where conduction is due to electrons tunneling through the 共insulating兲 matrix layer which separates the particles. Besides the amount of the conducting filler, and its critical density identifying the percolation threshold, an important microscopic parameter governing the overall composite conductivity is therefore the microscopic tunneling length , which measures the electron wave function decay distance within the polymer. In previous works, estimates of have been extracted from the temperature dependence of the conductivity but contradicting results were obtained depending on the model of low-temperature conduction used.6–8 Furthermore, information on the shape of the filler and on the general microstructure of the composites have been generally neglected, while they should be taken into account in order to correctly reproduce the observed percolation threshold values. In the following we describe a simple method to extract the tunneling factor directly from the 共room temperature兲 volume fraction dependence of the composite conductivity for homogeneous dispersions of spherical fillers of given diameter. By generalizing the critical path method of Refs. 9 and 10 to the case of particles with excluded volumes, we a兲
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provide a formula for in excellent agreement with Monte Carlo 共MC兲 results. When applied to low structured CBpolymer composites, for which the filler shape is well assimilated to that of a sphere, our formula provides values of of about 1 nm, with little dependence on the CB particle size. Let us consider a system of impenetrable conducting spheres of diameter D 共the filler兲 randomly distributed in a three-dimensional continuous insulating medium 共the matrix兲. We assume that the conducting spheres can connect via electron tunneling so that, given two spheres, the tunneling interparticle conductance takes the simplified form4 = 0e−2共r−D兲/ ,
共1兲
where r − D is the minimal distance between the surfaces of the two spheres whose centers are a distance r apart, and is the characteristic tunneling distance, which is of the order of few nanometers,4 while the prefactor 0 can be assimilated to the “contact” conductance. In our MC calculations we used a previously described simulation algorithm11,12 which allows to generate random distributions of nonoverlapping equally sized spheres inside a cubic cell 共with periodic boundary conditions兲 via random sequential addition 共RSA兲 and successive relaxation through MC runs. In this way we were able to generate distributions of spheres with volume fraction 共=D3 / 6, where is the number density兲 lower than the limit achievable through RSA, which is = 0.382.13 To investigate densities up to the metastable region of the hard sphere fluid we implemented a high density generation procedure which is somehow similar to that of Ref. 14. After having generated a spheres distribution close to the RSA limit, these are moved with MC sweeps, in which each sphere is randomly displaced by a trial move that is accepted if it does not lead to an overlap with any of its neighbors. At the same time, it is attempted to inflate the spheres by increasing their radius by a small amount and, again, accepted only if no overlap with neighboring particles is originating. Once the desired volume fraction is reached, the system is relaxed with several “regular” MC sweeps. To verify that equilibrium is attained, the radial distribution function was sampled and confronted with known results.14
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© 2009 American Institute of Physics
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FIG. 1. 共Color online兲 Tunneling conductivity in a system of spheres with different characteristic tunneling distance to spheres diameters ratio / D as a function of the volume fraction . The matrix intrinsic conductivity for m = 10−17 S / cm and the overall conductivity for the / D = 0.01 case are shown. Results from Eq. 共2兲 with 0 = 0.115 S / cm are displayed by dotted lines
For a given distribution of spheres, we associated to every particle pair a bond with conductance given by Eq. 共1兲 and, through a bond decimation procedure,15 we extracted the system conductivity.12 To reduce computational times to manageable limits, a cutoff distance is introduced in order to reject negligibly small bond conductances. We choose the cutoff distance to be equal to twice the spheres diameter, which is equivalent to a cutoff on the interparticle conductance of 0e−60 for / D = 1 / 15 case 共and even less for smaller / D values兲. The cutoff distance implies in turn an artificial geometrical percolation threshold of the system at ⯝ 0.012,12 and a lower bound on the volume fractions has to be introduced to avoid its effect. In Fig. 1 we report the calculated composite conductivity 共symbols兲 as a function of the volume fraction for several values of / D for a system size of ⬃900 spheres. The prefactor in Eq. 共1兲 was set equal to 0 = 1 S and each symbol is the outcome of 300 realizations of the system. Due to the exponential dependence of Eq. 共1兲, the distribution of the calculated conductivities was approximately of log-normal form so that the average of the logarithm of is reported in the figure. To establish a connection between our MC results and the observed conductivity in real CB-polymer composites we have to take into account the finite conductivity m of the polymer, which falls typically in the range m ⯝ 10−13 – 10−18 S / cm at room temperature. This may be
done by considering that the overall conductivity of the composite will be given approximately by the sum of the tunneling contribution and the matrix intrinsic conductivity m. As an example, we plot in Fig. 1 with a thick curve the soobtained conductivity for / D = 0.01 and m = 10−17 S / cm, showing a behavior similar to the typical s-shaped percolation curve obtained from the experiments. In analogy with the common usage, we identify the percolation threshold c with the value of at which the conductivity contribution due to the filler overtakes the conductivity due to the polymer matrix and becomes dominant. For the particular example of Fig. 1 we have therefore c ⯝ 0.2. We are now going to show that the MC results of Fig. 1 can be very well reproduced by a simple formula obtained by extending the critical path method of Ambegaokar et al.9 to our system of impenetrable spheres. Following Ref. 9 we then label the interparticle bond conductances arising in our system from the largest to the smallest and stop the procedure once the already labeled bonds entail the formation of a conductive path that spans the sample. If we then take the lowest value of this subset of conductances, c, and substitute it to all the 共larger兲 bond conductances of the subnetwork, we obtain a lower bound on the overall system conductance of the form of ⬀ c. In their work,10 Seager and Pike applied this method to random distributions of pointlike particles with tunneling interparticle conductance of the form of Eq. 共1兲 obtaining excellent agreement with MC results. They also realized that for their system the critical lowest conductance was of the form c = 0e−2␦/, and thus ⬀ 0e−2␦/, where ␦ is the diameter needed to have 共geometrical兲 percolation for an equivalent system of fully penetrable spheres. By applying this concept to our system of impenetrable spheres, we then find
⯝ 0e−2␦共兲/ ,
共2兲
where 0 is a prefactor proportional to the contact conductance 0 and ␦共兲 is now the critical distance between the closest surfaces of two spheres of diameter D and volume fraction as obtained from the semipenetrable sphere percolation model.11,12,16 In this model each sphere is surrounded by a penetrable concentric shell of width ␦ / 2 and two given spheres are considered connected if their shells overlap. In other words, two particles are connected if the distance between their centers is lower than D + ␦. For a given , ␦共兲 / 2 is then the minimal shell width for which a percolating cluster of overlapping semipenetrable spheres exists and its value is obtained by recording the probability of having a cluster spanning the opposite ends of the simulation cell and by using finite size analysis.12 Calculated values of ␦共兲 are reported in Fig. 2 from Ref. 16 共squares兲 together with few values evaluated by us 共stars兲. The solid line is an interpolation formula fitting the numerical data in the whole range of considered. When compared with the MC results of Fig. 1, Eq. 共2兲 共dotted lines兲 is in excellent agreement with our numerical calculations, showing that the Seager and Pike conductivity estimate can be extended to the case of finite size hard spheres.
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FIG. 2. 共Color online兲 Geometrical percolation threshold of the hard-coresoft-shell model from algorithm of present work and from results of Ref. 16. Shown is the critical interaction distance to sphere diameter ratio ␦ / D as a function of the hard spheres volume fraction .
Equation 共2兲 has an interesting practical consequence. Indeed, at the percolation threshold c, where the matrix and tunneling conductivities are equal, we find
冉 冊
␦ 共 c兲 1 0 = ln , 2 m
共3兲
which is an equation permitting to estimate the decay length directly from conductivity curves of composites with a known diameter D of the filler particles. This can be realized by plugging into Eq. 共3兲 the measured value of m and, knowing c and D, the corresponding value of ␦共c兲 obtained from Fig. 2. The precise value of 0 has little effect on the final result because of the logarithmic dependence on the 0 / m ratio in Eq. 共3兲. Since by construction 0 is proportional to the tunneling conductivity of the spheres at contact, we may use the 共asymptotic兲 value of measured at large volume fractions. We applied this method to low structure CB-polymer composites, which are morphologically the closest to random distribution of independent spheres, and one spherical graphite-polymer composite. The results are reported in Table I, showing that all the CB, irrespective of the particle size and of 0 / m, have a characteristic tunneling length of about 1 nm, which is the expected order of magnitude.4 However, the graphite case shows an overestimation of . This may be originating from a lowered crossTABLE I. Characteristic tunneling distance as extrapolated from experimental results on low structure CB and graphite-polymer composites.
a
Ref.
c
0 共S/cm兲
m 共S/cm兲
D 共nm兲
共nm兲
17 18 19 20 21 22a
0.35 0.40 0.20 0.25 0.37 0.26
0.1–1 0.1–1 10−2 10−3 0.1 10−2
5 ⫻ 10−13 10−18 10−18 5 ⫻ 10−15 10−17 10−18
300 320 90 150 250–300 5100
1.57–1.44 0.78–0.74 0.97 1.59 0.80–0.96 35.45
Spherical graphite.
over threshold induced by the large amount of submicrometric debris which is often found together with the main graphite particles in the composite. In principle Eq. 共3兲 may be used to extract values also from conductivity curves of nonspherical or multidispersed size fillers. To this end, however, one should evaluate the dependence of the critical distance ␦共兲 for the particular morphology at hand. This is, nevertheless, a much simpler calculation than a full numerical solution of the conductivity problem because it concerns only the geometrical connectivity. In summary, we have evaluated numerically the conductivity of a dispersion of spherical particles of finite diameter carrying current through interparticle tunneling. We have shown that the resulting conductivity can be expressed by a simple formula obtained from a generalization of the critical path method of Refs. 9 and 10. When applied to real composites with fillers of spherical shape, this formula provides a simple tool to evaluate the characteristic tunneling length directly from the dependence of the measured conductivity on the filler volume fraction. This study was supported by the Swiss Commission for Technological Innovation 共CTI兲 through project GraPoly, 共CTI Grant No. 8597.2兲, a joint collaboration led by TIMCAL Graphite and Carbon SA. Discussions with E. Grivei were greatly appreciated. Carbon Black Polymer Composites, edited by E. Sichel 共Dekker, New York, 1982兲. 2 D. Stauffer and A. Aharony, Introduction to Percolation Theory 共Taylor & Francis, London, 1994兲. 3 G. R. Ruschau, S. Yoshikawa, and R. E. Newnham, J. Appl. Phys. 72, 953 共1992兲. 4 I. Balberg, Phys. Rev. Lett. 59, 1305 共1987兲. 5 R. D. Sherman, L. M. Middleman, and S. M. Jacobs, Polym. Eng. Sci. 23, 36 共1983兲. 6 P. Sheng, E. K. Sichel, and J. I. Gittleman, Phys. Rev. Lett. 40, 1197 共1978兲. 7 D. van der Putten, J. T. Moonen, H. B. Brom, J. C. M. Brokken-Zijp, and M. A. J. Michels, Phys. Rev. Lett. 69, 494 共1992兲. 8 A. Aharony, A. B. Harris, and O. Entin-Wohlman, Phys. Rev. Lett. 70, 4160 共1993兲. 9 V. Ambegaokar, B. I. Halperin, and J. S. Langer, Phys. Rev. B 4, 2612 共1971兲. 10 C. H. Seager and G. E. Pike, Phys. Rev. B 10, 1435 共1974兲. 11 G. Ambrosetti, N. Johner, C. Grimaldi, A. Danani, and P. Ryser, Phys. Rev. E 78, 061126 共2008兲. 12 N. Johner, C. Grimaldi, I. Balberg, and P. Ryser, Phys. Rev. B 77, 174204 共2008兲. 13 J. D. Sherwood, J. Phys. A 30, L839 共1997兲. 14 C. A. Miller and S. Torquato, J. Appl. Phys. 68, 5486 共1990兲. 15 R. Fogelholm, J. Phys. C 13, L571 共1980兲. 16 D. M. Heyes, M. Cass, and A. C. Brańca, Mol. Phys. 104, 3137 共2006兲. 17 L. Flandin, A. Chang, S. Nazarenko, A. Hiltner, and E. Baer, J. Appl. Polym. Sci. 76, 894 共2000兲. 18 Z. Rubin, S. A. Sunshine, M. B. Heaney, I. Bloom, and I. Balberg, Phys. Rev. B 59, 12196 共1999兲. 19 S. Nakamura, K. Saito, G. Sawa, and K. Kitagawa, Jpn. J. Appl. Phys., Part 1 36, 5163 共1997兲. 20 L. Karásek, B. Meissner, S. Asai, and M. Sumita, Polym. J. 共Tokyo, Jpn.兲 28, 121 共1996兲. 21 N. Probst, Carbon Black: Science and Technology, edited b by J. B. Donnet, R. C. Bansal, and M. J. Wang 共Dekker, New York, 1993兲, Chap. 8. 22 K. Nagata, H. Iwabuki, and H. Nigo, Compos. Interfaces 6, 483 共1999兲. 1
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