Anisotropic percolation in carbon black-polyvinylchloride composites

Solid State Communications, Vol. 47, No. 12, pp. 989-992, 1983. Printed in Great Britain.

0038-1098/83 $3.00 + .00 Pergamon Press Ltd.

ANISOTROPIC PERCOLATION IN CARBON BLACK-POLYVINYLCHLORIDE COMPOSITES I. Balberg The Racah Institute of Physics, The Hebrew University, Jerusalem 91904 Israel and N. Binenbaum and S. Bozowski RCA Laboratories, Princeton, NJ 08540 U.S.A.

(Received 29 April 1983 by S. Alexander) New features of the resistivity dependence on the melt-flow-distance of carbon black-polyvinylchloride are presented. These features are compared with features obtained from the first computer study on the resistance of a two-dimensional system of conducting sticks and its dependence on the system's anisotropy. The qualitative resemblance between the two dependences and the effect of the conducting material content on these dependences, indicate clearly that the conduction in the composites is well described by an anisotropic percolation process. Nevertheless, the fact that agreement is found with the major features of the resistivity indicates that we are concerned here with a composite in which anisotropic percolation takes place. The carbon black aggregates and the composites' preparation procedure have been described in detail previously [8]. Let us just mention that the composite is prepared by first mixing a desired weight percent (wt. %), co, of the conducting carbon black with the insulating PVC. This is followed by blending, compounding, and moulding the composite under compression [11 ]. In order to obtain the alignment of the aggregates in a given preferred direction, we have used the known technique [12, 13] for obtaining oriented fibers in plastics, i.e., by letting the molten composite flow under the compression. The larger the flow distance, r, the higher the degree of aggregate's alignment in the flow direction. While we do not have as yet a quantitative correlation between the degree of alignment and r, it is established [9, 13] that this degree increases monotonically with r. For the present study, r will be considered a measure of the alignment and thus a qualitative measure of the anisotropy of the composite. Following these considerations we have prepared samples with various loadings co. Each sample was 6 inches long, 1.5 mm thick and homogeneous with respect to w. One end of the sample, r = O, corresponds to the material which did not flow in the molten state, while the other end, r = 6", corresponds to the material of the largest flow distance. Hence the r = 0 end represents a macroscopically isotropic system (no preferred direction for aggregates alignment) while the r = 6" end represents

WHILE THERE ARE EXTENSIVE theoretical studies [1-3] of the anisotropic percolation problem there are very few experimental studies [4, 5] of corresponding systems. In particular there are no experimental studies on anisotropic composites. The simplest composite that one may consider for such a study is a system composed of elongated conducting particles (fibers) embedded in an insulating matrix. Although percolation in such systems has been studied before [6-8], the conducting fibers in all the systems studied, have been isotropically distributed. Hence, the results were always concerned with isotropic percolation. Very recently, it was shown [9] that, in a system composed of elongated carbon black aggregates embedded in an insulating plastic, there is an anisotropy in the resistivity which is associated with the preferred orientation of the aggregates in the polyvinylchloride (PVC). In this communication we would like to show that this resistivity anisotropy as well as other, new observations, on the carbon black-PVC composite, are well explained by anisotropic percolation in the system. This is done by comparing all these observations with results obtained by a computer simulation of the simplest system which resembles the composite, i.e., a two-dimensional conducting-sticks system. As far as we know there are no previous calculations of the resistance of such a system and only the percolation threshold for the isotropic system has been previously computed [10]. Since the composite is a threedimensional system and since sticks are only a crude approximation to the elongated aggregate structure [8], the comparison will be of a qualitative nature. 989

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the highest degree of macroscopic anisotropy. The resistivity of the samples was measured using the method which has been described previously [8, 9]. The computer simulation is carried out by "planting" randomly N sticks of a given length in a unit size square. This length, L, is expressed [10] in units of rs

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The anisotropy of the sticks system is determined by their alignments 0i with respect to a direction which will be called here the longitudinal (or y) direction. Correspondingly, the perpendicular direction will be called here the transverse (or x) direction. For the present study we have chosen 0i's which are randomly distributed within a selected interval such that, 0 u <~ Oi <~ 0 u . The smaller the 0 u the higher the degree of orientation. The isotropic distribution of the alignments which results in an isotropic system is obtained by setting 0 u = 90 °. We define the macroscopic anisotropy PII/P± of the system by considering the macroscopic longitudinal component of the sticks, PII = E~l L lcos Oil and the macroscopic transverse component of the sticks P± = E~l L [ sin 01[. The procedure of finding an intersection between two sticks as well as the determination of "longitudinal" or "transverse" percolation has been described previously [10, 14]. The percolation thresholds are defined as the smallest stick length, L e, which yields a longitudinal as well as a transverse percolation. Once the intersections and the percolating clusters are registered we attach a unit resistor to each intersection in a percolating cluster while assuming the sticks to have infinite conductance [15]. Applying the Kirchhoff junction theorem to the resistors' network [16] obtained, we have found both the longitudinal resistance, RII, and the transverse resistance, R±, of the system at hand, as a function of the anisotropy PII/P±. In view of equation (1) it is clear that increasing L while keeping N constant will yield the same qualitative behavior as increasing N while keeping L constant. Here, N = I00 and N = 500 sticks systems were used. Following the description of the carbon black-PVC composite and the sticks system used for the computer simulation, we are now in a position to compare the parameters which characterize the two systems. The longitudinal resistivity of the composite Oil (along the melt flow direction) is related to Rii while the composites' transverse resistivity/9± is related to R±. The amount of conducting material, co, is related to L (or AT) and the anisotropy parameter of the composite, r, is related to Pti/P±. The relationships between two corresponding quantities are not quantitatively known, -

Vol. 47, No. 12

ANISOTROP1C PRECOLATION IN CARBON BLACK-PVC COMPOSITES

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Fig. 1. The computed dependence of the percolation threshold on the anisotropy for three sticks ensembles (seeds). The dashed curve is the expected [14] dependence of both Lel I and Lea - for the two-dimensional infinite sticks system. but the following results show that proportionality is not a too bad approximation to describe these relationships. Turning to the results of the measurements and the computations, let us consider first the effect of anisotropy on the percolation threshold. The results of the computations, which are shown in Fig. 1, indicate clearly an increase of the critical stick length L e , with increasing anisotropy. Confirmation of this prediction is obtained by examination of the dependence of the composites' percolation threshold, we, on the corresponding anisotropy parameter r. Recalling that in the isotropic (r -> 0) composite we found [8] that the percolation threshold was 9.2 +- 0.3 wt. % and in view of the above prediction, we expect that 6oe will shift to higher values with increasing r. Alternatively, we expect that for a sample made of an 6o > 9.2 wt. %, percolation will be achieved for the low values of r but there will be a critical value o f r beyond which there will be no percolation. Indeed, the results presented in Fig. 2 for 10 wt. % show that the transverse conductivity o± decreases gradually with increasing r until at r = 2.9" there is a dramatic drop of the conductivity. This value o f r corresponds then to the degree of anisotropy for which the critical carbon loading is 6oc = 10 wt. %. Another conspicuous feature of the composite's

991

ANISOTROPIC PERCOLATION IN CARBON BLACK-PVC COMPOSITES

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Fig. 3. The computed transverse and longitudinal resistances, of an ensemble of 100 sticks as a function of the system's macroscopic anisotropy PII/P±. For comparison with the corresponding experimental results we reproduce in the insert the data presented in [9]. resistivity dependence on the anisotropy parameter r, has been presented in a previous publication [9]. It

was shown there that for a still larger 6o (11 wt.%) there is an increase of Pt, Ptl and P±/Pllwith increasing r. For computing the corresponding dependences, i.e., the dependences o f R 1 and RII on PIt/P±, we have examined a sticks system in which the sticks have the length L = 10 r s (the corresponding L e for the isotropic system is 4.2 rs). The results obtained in the computations are shown in Fig. 3. Indeed, the features exhibited by the experimental resttlts, which are reproduced in the insert, are similar to those derived in the computations. In particular, we see that R± > R II and that R±, RII and R~R It all increase with increasing PII/P±.The slight minimum in the dependence of Rib on PII/P±is due to the fact that, in the small sticks' ensemble used in the computations, all the sticks belong to a single percolating cluster for PII/P±< 5, and only for higher values of Pii/ei the connectedness of the sample begins to be important. The absence of this minimum in the results shown in the insert of Fig. 3 is probably due to the fact that for the three dimensional system with ~ = 11 wt. % this one-cluster condition is not fulfilled. This minimum, as well as the large differences in the resistance anisotropies shown in Fig. 3 is not due to computational fluctuations. This was concluded from the fact that

992

ANISOTROPIC PRECOLATION IN CARBON BLACK-PVC COMPOSITES

agreement between the dependences shown in Fig, 4 and Fig. 5 becomes quite impressive when a detailed comparison is made: the dependences of both R± and P//PJ.rnin on the corresponding anisotropy parameters are exponential-like for the lower concentrations of the conducting material and they approach linearity for the higher concentrations of this material. In conclusion, we have shown that the resistivity dependence on the melt-flow-distance of the carbon black-PVC composite can be explained by an an±sotropic percolation process. In particular, this dependence resembles the corresponding dependence obtained by a computer simulation study of percolation in a two-dimensional system of conducting sticks.

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Vol. 47, No. 12

REFERENCES I0

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Fig. 5. The dependence of the relative transverse resistivity Pz/Pimm, of three composites, on their flow distance in the molten state, r. results obtained for an N = 500 sticks ensemble have shown the same features. The above differences between the computational and experimental results appear then to be due to real differences between the two- and threedimensional systems. Let us turn now to the effect of the sticks length variation on the resistance of the sticks system and compare it with the effect of carbon loading variations on the measured resistivity. For example, we present here the transverse resistance, R±, and the transverse resisitivity Pi. The results for RjI and PU which yield similar conclusions will be discussed elsewhere. In Fig. 4 we show the computed dependence of Rl on PII/P± for three values of L, all of which correspond to the far-above-percolation regime. As expected, it is found that the larger the L the lower the resistance for a given anisotropy. It is further observed, however, that the dependence on the anisotropy becomes weaker with increasing L. These qualitative features are also exhibited by the experimental results shown in Fig. 5. It is seen that the transverse resistivity (which is presented in units of its value P±min at r = 1") increases with r and decreases with co. The qualitative

7. 8. 9. 10. 11. 12. 13. 14. 15.

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J.P. Straley, J. Phys. C13,4335 (1980). H. Nakanishi, P.J. Reynolds & S. Redner, J. Phys. AI4, 855 (1981). C.J. Lobb, D.J. Frank & M. Tinkham, Phys. Rev. B23, 2262 (1981). L.N. Smith & C.J. Lobb, Phys. Rev. B20, 3653 (1979). K.S. Mendelson & F.G. Karioris, J. Phys. C13, 6197 (1980). F. Carmona, F. Barreau, P. Delhaes & R. Canet, J. Phys. Lett. (Paris] 41, L531 (1980). These authors study the effect of the aspect ratio on the resistivity but describe it as "anisotropy". Their samples are, however, isotropic (they do not report any macroscopic anisotropy or resistivity anisotropy). Hence, their study is concerned with isotropic percolation in a system where the microscopic particles are anisotropic but their evenly distributed alignments form an isotropically macroscopic system. D.M. Bigg, Polym. Eng. ScL 19, 1188 (1979). I. Balberg & S. Bozowski, Solid State Commun. 44,551 (1982). I. Balberg & P.J. Zanzucchi, Appl. Phys. Lett. 40, 1022 (1982). G.E. Pike & C.H. Seager, Phys. Rev. B10, 1421 (1974). R.J. Ryan, RCA Rev. 39, 87 (1978). A. Okagawa, R.G. Cox & S.G. Mason, J. Coll. Int. Sci. 71, 11 (1979). P.F. Bright & W.M. Darlington, Plastics and Rubber Processing and Application 1, 139 (1981). I. Balberg & N. Binenbaum,Phys. Rev. B in press. This is justified for the present system since it was established that the microscopic resistors in this composite are determined by the tunneling between the carbon aggregates rather than by the resistance of the aggregates. See, for example, E.K. Sichel, J.I. Gittleman & Ping Sheng, Phys. Rev. BI8, 5712 (1978). S. Kirkpatrick, Rev. Mod. Phys. 45,574 (1973).