Mar 3, 1997 - (Received 24 October 1996). We have studied conducting carbon-black (CB) composites with a percolation threshold pc of. 3 3 1024 in volume ...
VOLUME 78, NUMBER 9
PHYSICAL REVIEW LETTERS
3 MARCH 1997
High-Dilution Carbon-BlackyyPolymer Composites: Hierarchical Percolating Network Derived from Hz to THz ac Conductivity L. J. Adriaanse, J. A. Reedijk, P. A. A. Teunissen, and H. B. Brom Kamerlingh Onnes Laboratorium, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands
M. A. J. Michels* and J. C. M. Brokken-Zijp Shell Research and Technology Centre Amsterdam, P.O. Box 38000, 1030 BN Amsterdam, The Netherlands (Received 24 October 1996) We have studied conducting carbon-black (CB) composites with a percolation threshold s pc d of 3 3 1024 in volume fraction. The low pc is explained in terms of a fractal distribution of CB aggregates. To prove this model we measured the frequency dependence of the complex conductivity (up to 0.6 THz) for various CB concentrations. Two successive scaling regimes are found, the first dominated by the infinite percolating cluster and the second by finite clusters. We extend the anomalous diffusion model to explain this observation and show the parameters found to be consistent with the proposed structure. [S0031-9007(97)02471-X] PACS numbers: 72.80.Tm, 72.20.Dp, 77.84.Lf
In composites of a nonconducting matrix with a conducting filler material the conductivity often increases many orders of magnitude when the filler concentration spd becomes higher than a critical value. This percolation threshold spc d usually is of the order of 0.1–0.2 in volume fraction [1–3]. We have studied the transport properties in a series of conducting composites using a nonconducting thermoset polymer with carbon-black (CB) as filler material. The CB concentration was varied between 4 3 1024 , p , 1 3 1022 in volume fraction, the temperature T between 4 and 300 K. The CB particles are hollow semispheres of a few graphite layers, 1.0–1.5 nm in thickness, with a diameter of 30 nm. The p dependence of the dc conductivity follows the universal scaling law sdc ~ jp 2 pc jt [4], with an extremely low percolation threshold pc 3 3 1024 in volume fraction [5]. The exponent t 2.0 6 0.2 is, as expected for a percolating fractal structure, close to pc [1,2]. In this Letter we address the dependence of the conductivity on frequency svy2pd and concentration at various temperatures, its scaling properties and its relation to the underlying structure, and explain why pc is as low as 3 3 1024 volume fraction instead of 0.16 for percolation of hard spheres [6]. Experimental.— For the low-frequency regime (5 Hz–13 MHz), we used a HP4192A impedance analyzer, between 1 MHz and 1.8 GHz a HP4291A rf impedanceymaterial analyzer with a HP16453A test fixture and between 100 MHz and 20 GHz a HP8510B network analyzer. Between 30 GHz and 0.6 THz measurements were done with an ABmm vector network analyzer, extended with the ESA-1 option. At low frequencies (dc–150 MHz) we employed standard two-terminal and four-terminal sputtered Au contacts, between 0.1 and 20 GHz we used an open ended coaxial probe (HP85070B), and above 30 GHz a quasioptical configuration. Cooling was done in an OI-flow cryostat. 0031-9007y97y78(9)y1755(4)$10.00
Figure 1 gives the v dependence of s 0 sv, T d for CB13 as a typical example. By plotting s 0 sv, T dysdc sTd versus vysdc sT d the data taken at all temperatures fall on a single curve. The actual frequency range extends from a few Hz to 0.6 THz. It can be seen that after a frequencyindependent regime up to v0 , s 0 sv, T d becomes proportional to v 0.4360.03 ; above v1 the frequency dependence changes to v 0.7060.04 ; see inset. The first crossover frequency v0 is found by the intersection of the dc line with the extrapolated constant-slope part of the curve above the
FIG. 1. The frequency dependence of s 0 svd of CB13 sp 5 3 1024 d for 3.6 , T , 7.7 K. Normalizing the vertical and horizontal scale to sdc gives one single curve. The same curve even represents the room temperature data after multiplication of vysdc by a factor of 3. The straight lines a, b, and c correspond to the frequency exponents sy 0 d 0, 0.43, and 0.70, respectively. The inset shows y 0 svd in detail.
© 1997 The American Physical Society
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first inclination point. The second crossover frequency v1 is analogously defined (see Fig. 1). Figure 2 shows conductivity plots for all samples, taken at room temperature. While sdc ~ sp 2 pc d2.0 , s 0 svd at high frequencies becomes linear in p. To emphasize this dependence s 0 svd is divided by p on the vertical axis. Table I gives the values for the crossover frequencies and the expo0 nent y 0 of s 0 svd ~ v y for v0 , v , v1 . Note that v0 shifts to higher values with increasing p (there is ambiguity in v1 at high p), see inset, and that the samples with p # 1.0 3 1023 have a similar v dependence as shown in Fig. 1. At high frequencies the data for all p fall on a single curve with slope 0.70 6 0.04. The data shown in Figs. 1 and 2 are corrected for a contribution from the polymer matrix (increasing from a few % at 1 GHz to about 80% at the highest frequency). Here we present only the real part of the conductivity. The imaginary part was always measured simultaneously and found to be consistent with the Kramers-Kronig relations. Percolation threshold.— In percolation theory the correlation length j is given by j j0 sf 2 fc d2n with the universal exponent n ø 0.88. For massive spheres the critical volume fraction fc is 0.16 [6]. The CB particles used here are hollow semispheres with shells of a few graphitic layers. The effective volume fraction at the percolation threshold is about ye pc with ye ø 5, still a factor 102 smaller than 0.16. To explain this difference, aggregation of the CB particles has to be taken into account. Aggregated structures are formed during the CB making in the gas [7] and dispersing in the liquid phase (clustercluster aggregation [2,8–10]), and likely have a fractal
FIG. 2. Frequency dependence of s 0 svd at room temperature for various p. To visualize the linear p dependence at high frequencies s 0 svdyp is plotted vs vy2p. The inset, where s 0 svd is normalized to sdc , clearly shows the p dependence of v0 .
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TABLE I. Crossover frequencies and frequency exponent y 0 for v0 , v , v1 . Above v1 , y 0 0.70 6 0.04 for all samples. p 4.5 5.0 1.0 2.5 4.0 5.5 1.0
3 3 3 3 3 3 3
1024 1024 1023 1023 1023 1023 1022
v0 sMHzd
v1 sGHzd
y0
0.50 6 0.08 0.60 6 0.08 4.0 6 0.5 s5.0 6 0.5d 3 101 s1.5 6 0.2d 3 102 s2.0 6 0.3d 3 102 s7 6 1d 3 102
0.40 6 0.06 0.40 6 0.06 1.0 6 0.2
0.48 6 0.03 0.43 6 0.03 0.39 6 0.03
structure. These new entities are randomly distributed in the polymer. Because of the fractal structure of the aggregate its effective volume fraction is f r 32df ye p. The scale enhancement factor r equals the ratio between the size of the new entity s2Re d and that of the CB particles s2R0 d: r Re yR0 . The fractal dimension df in the gas phase is measured from TEM pictures to be 1.85, which compares well with the values of Bourrat et al. [7]; for the cluster growth in the liquid phase a value of 1.6–1.7 seems appropriate [2,8,9]. Using an averaged value of 1.8 and taking fc yye pc ø 102 r 32df we obtain r , 40. In the system randomly built from the gas-liquid aggregates the correlation length jspd j0 sf 2 fc d2n with j0 , 2rR0 can now be written as j s1yyen d2R0 r 12s32df dn jp 2 pc j2n ,
(1)
which gives a factor 4 shorter correlation length than would follow from simply writing j 2R0 jp 2 pc j2n . Including j, at least four length scales are present: the size of the CB particle (2R0 , 30 nm diam), of the fractal gas-phase (typically lG ø 100 nm [7]) and liquid-phase aggregates slL 2Re 2rR0 ø 80R0 d, and of the fractal heterogeneity in the percolating network formed by the gas-liquid aggregates as building blocks fjspdg. The size of the gas-phase aggregates lG will be independent of p. For the liquid-phase aggregates a slight p dependence is to be expected. We note that very recently in another type of CB-polymer composite the presence of two fractal structures was nicely demonstrated with an electric force microscopic [3]. s 0 sv, Td in the fractal regime.—The conductivity on a fractal structure is the sum of the conductivity on the infinite percolating cluster ss` d with carrier density n` and on finite-size clusters ssf d with density nf (n n` 1 nf is the total carrier density). Using the Einstein diffusion equation for a three-dimensional system with diffusion constant D, the contribution to s 0 sv, T d of the infinite cluster can be written as µ ∂µ ∂ n` e2 D n` e2 a2 Rsvd 22dw 0 , (2) s` sv, Td 6kB T 6kB T t a in which a is the random-walk step length, tsT d the temperature-dependent associated time constant, and
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R 2 std ~ t 2ydw the mean square displacement of the carrier on the infinite cluster [2]. The carrier concentration n` nsjyj0 d2byn with b 0.41 [1,2] and j ! j0 for p ! 1. In general various frequency regimes can be distinguished: For v # vj , a distance R $ j (j is the fractal correlation length) is traveled in a time t $ 1yvj , and for frequencies vj , v , vt , the response is sensitive to the fractal structure in the sample, since Rsvd , j. The latter range is bordered by vt t 21 . Around vt , the distribution in t starts to affect the frequency dependence of s 0 svd. With sRyaddw tyt svyvt d21 , Eq. (2) gives for vj , v , vt µ ∂ µ ∂122ydw v n` e2 a2 s`0 sv, T d . (3) 6kB T t vt For v , vj , s`0 svd becomes µ ∂µ ∂ n` e2 a2 j 22dw 0 s` sv, Td sdc sT d . 6kB T t a
(4)
The expressions for the contribution sf0 sv, Td of the finite-size clusters (size j or smaller) to s 0 sv, Td are as in Eqs. (2) and (3), but with nf replacing n` and dw0 for 0 sv, T d is given by dw . For vj , v , vt s1 µ ∂ µ ∂122ydw0 nf e 2 a 2 v 0 sf sv, T d . (5) 6kB T t vt 0
R 2 std ~ t 2ydw is now an average over random walks on all finite-size clusters [2]; dw0 is related to dw by dw s1 2 by2nddw0 . From Eqs. (3) and (4) it follows that in the absence of finite-size clusters s 0 sv, T d would become frequency dependent above a crossover frequency vj vt sjyad2dw ; the p dependence of vj is given by vj ~ sp 2 pc dndw . Let us define vj0 as the frequency at which the contribution of the finite-size clusters starts to dominate s 0 svd. In standard percolation theory j0 equals a, and hence vj vj0 . In the presence of internal structure of the building blocks, j0 might be larger than a and Eqs. (3)–(5) may lead to vj0 . vj with a ratio depending on the precise structure within the building block. Also, if large finite-size clusters would be less abundant than expected for randomly packed systems, s 0 svd ~ v 122ydw for v . vj , and only at the higher frequency vj0 the exponent becomes 1 2 2ydw0 . In this case the ratio between vj and vj0 will depend on the degree of suppression of the larger cluster sizes. Apart from the anomalous diffusion within a cluster as considered above, the frequency dependence of the effective conductivity of a random mixture results from polarization effects between clusters inside the mixture [11]. For the latter process close to the percolation threshold percolation-scaling theory predicts s 0 svd ~ v tyt1s and vj ~ sp 2 pc dt1s [11–13]. The three-dimensional values for the universal scaling exponents t and s are t ø 2.0 and s ø 0.7 [1,2]. Discussion.—The experiment shows two crossover frequencies v0 and v1 and two values of the frequency
3 MARCH 1997
exponent y 0 ; see Table I. In the anomalous diffusion approach s 0 svd, if dominated by the infinite cluster, see Eq. (3), is related to dw , 3.75: s 0 svd ~ v 122ydw v 0.47 . This exponent compares well to the experimental values 0.39–0.48; see Table I. A similar value 0.50 6 0.05 has been observed in porous silicon [14], where the results were analyzed neglecting the presence of finitesize clusters. Because of the good agreement of the experimental and theoretical values, we identify v0 with vj . The experimental exponent 0.70 6 0.04 above v1 , see Table I, is close to the value of the exponent 1 2 2ydw0 , 0.6 0.7 from the anomalous diffusion approach or tyst 1 sd ø 0.74 of the scaling theory. From the ratio between v1 and v0 of 102 103 for p # 1.0 3 1023 the length scale probed by the carriers at v1 can be estimated. Let us consider the data on CB13 as a representative example. With pc 3 3 1024 , sp 2 pc dypc ø 0.7, 2n and j 0.4R0 pc jsp 2 pc dypc j2n [see Eq. (1)], j equals 7 3 102 R0 or 11 mm (in 10–100 mm thick specimens of CB15 with p , 1.3pc inhomogeneities become visible, as expected from the scaling of j). At room temperature the crossover frequencies v0 and v1 are, respectively, of the order of 1 MHz and 1 GHz; see Fig. 1 and Table I. Using Eqs. (2) and (3) the following approximate relation holds: lsv1 dyj ø fs 0 sv1 dys 0 sv0 dg1ys22dw d sv0 yv1 d1ydw . For CB13 this relation gives lsv1 d , 0.1jsCB13d , 70R0 . This value is similar to the size of the gas-liquid aggregate lL , as deduced from the value of pc . The observation that for length scales above lL the infinite cluster dominates s 0 svd, gives an indication that the growth process influences the cluster distribution. As is clear from Fig. 2, only for p # 1.0 3 1023 v1 can be defined. The onset frequency v0 is visible in all samples and is seen to shift to higher frequencies with increasing p: v0 ~ sp 2 pc da with a 2.0 6 0.3. This behavior is qualitatively as expected. With increasing p the correlation length jspd becomes shorter and hence is probed at higher frequencies. In the diffusion model vj ~ sp 2 pc dndw . Percolation-scaling theory predicts vj ~ sp 2 pc dt1s . Both exponents are larger than experimentally observed and indicate that not all aspects of the system can be understood within the models used. Also the p dependence of the zero-frequency limit of the real part of the dielectric constant e 0 s0d, which is predicted to diverge when p ! pc as e 0 s0d ~ jp 2 pc j2s , is not found in our experiment (not shown). Other carbon-black composites show a similar behavior [13]. Such a behavior can be explained if large finite clusters have less weight than theoretically anticipated. For v ¿ vj0 , the length scales over which the carriers travel are well inside the aggregated entities; see above. Because the structure of the gas-liquid aggregated entity is supposed to be (almost) independent of p, s 0 svd will depend linearly on p [for not too large p values, nf and 1757
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hence sf0 are proportional to p [1,2], and hence also Eq. (5) leads to s 0 svd ~ p]. Going to lower frequencies the p dependence of s 0 svd will gradually change from linear to the quadratic dependence observed in the dc conductivity ssdc ~ jp 2 pc j2.0 d. The p dependence in Fig. 2 nicely illustrates this expected behavior. If we associate the random step length with the distance between CB particles (30 nm), the corresponding frequency vt is calculated to be above 1 THz, which is too high to be observed in this experiment. Even a lower cross-over frequency might be difficult to see, because y 0 for v . vt is expected to be in the range 0.7–0.9 [15– 17], which is close to value of y 0 0.7 for v . v1 due to the finite-size clusters. T-v scaling.—In many hopping models [15–17] the characteristic frequency dependence of the response s 0 svd to the applied ac field will scale with sdc for v sufficiently below the phonon frequency sø1013 Hzd. Taking the extended pair approximation (EPA) [18] as an example, this scaling law has the form s 0 sv, Tdysdc sT d 2 1 ~ fvyssdc sT dTdg0.725 . Although there are many materials in which this relation is reasonably obeyed, there are also cases [16], like the CB composites—see Fig. 1, where vysdc sT d works better as scaling parameter than vysdc sTdT . In the diffusion approach taken here, combination of Eqs. (3) and (4) leads to ssv, T dysdc sTd ~ fvnykB Tsdc sT dg122ydw . The T-independent scaling found experimentally is obtained, if nykB T is T independent, e.g., as in the case of a flat density of states, where kB T determines the number of states involved. In summary, the frequency dependence of the conductivity in the studied CB composites is well described by an extension of the anomalous diffusion model for hopping on a fractal. The data at lower concentrations show a clear crossover (at v v0 ) from a uniform to a fractal regime, where the infinite percolating cluster dominates the frequency dependence of s 0 svd. The presence of finite-size clusters becomes dominant above the second crossover frequency v1 . These findings support the explanation given for the extremely low percolation threshold. Furthermore, the formulas used lead naturally to the experimentally observed scaling law of ssv, T dysdc sT d vs vysdc sT d.
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The discussions with Jos de Jongh and the participation of Irvin-Paul Faneyte, Bart Smits, and Hubert Martens in the experiments are gratefully acknowledged. This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which is sponsored by the Nederlandse Orgnaisatie voor Wetenschappelijk Onderzoek (NWO).
*Also at Vakgroep Theoretische Natuurkunde, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [1] D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor & Francis, London, 1992). [2] Fractals and Disordered Systems, edited by A. Bunde and S. Havlin (Springer Verlag, Berlin, 1991). [3] R. Viswanathan and M. B. Heaney, Phys. Rev. Lett. 75, 4433 (1995). [4] D. van der Putten et al., Phys. Rev. Lett. 69, 494 (1992); A. Aharony et al., ibid. 70, 4160 (1993); M. A. J. Michels et al., ibid. 70, 4161 (1993). [5] European Patent No. EP0370586A2. [6] R. Zallen, The Physics of Amorphous Solids (Wiley and Sons, New York, 1983), p. 170. [7] X. Bourrat et al., Carbon 26, 100 (1988). [8] H. G. E. Hentschel, J. M. Deutch, and P. Meakin, J. Chem. Phys. 81, 2496 (1984). [9] L. Salomé and F. Carmona, Carbon 24, 599 (1991); L. Salomé, J. Phys. II (France) 3, 1647 (1993). [10] R. Schueler, J. Petermann, K. Schulte, and H-P. Wentzel, Macromol. Symp. 104, 261 (1996). [11] Y. Song, T. W. Noh, S. I. Lee, and J. R. Gaines, Phys. Rev. B 33, 904 (1986). [12] D. Stroud and D. J. Bergman, Phys. Rev. B 25, 2061 (1982). [13] L. Benguigui, J. Yacubowicz, and M. Narkis, J. Polym. Sci. B 25, 127 (1987). [14] M. Ben-Chorin et al., Phys. Rev. B 51, 2199 (1995). [15] S. R. Elliott, Adv. Phys. 36, 135 (1987). [16] A. R. Long, in Transport in Solids, edited by M. Pollak and B. Shklovskii (Elsevier Science Publ., New York, 1991). [17] M. P. J. van Staveren, H. B. Brom, and L. J. de Jongh, Phys. Rep. 208, 1 (1991). [18] S. Summerfield, Philos. Mag. B 52, 9 (1985).
