Percolation threshold and electrical conductivity of a ...

JOURNAL OF APPLIED PHYSICS 110, 123715 (2011)

Percolation threshold and electrical conductivity of a two-phase composite containing randomly oriented ellipsoidal inclusions Y. Pan,1 G. J. Weng,1,a) S. A. Meguid,2 W. S. Bao,2,3 Z.-H. Zhu,3 and A. M. S. Hamouda4 1

Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, New Jersey, 08903, USA 2 Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada 3 Department of Earth and Space Science and Engineering, York University, Toronto, Ontario M3J 1P3, Canada 4 Department of Mechanical and Industrial Engineering, Qatar University, Doha, Qatar

(Received 15 October 2011; accepted 14 November 2011; published online 27 December 2011) An explicit, analytical theory for the percolation threshold, percolation saturation, and effective conductivity of a two-component system involving randomly oriented ellipsoidal inclusions is proposed. The ellipsoids may take the shape of a needle, prolate or oblate spheroid, sphere, or disk. This theory is based upon consideration of Ponte Casta~neda–Willis [P. Ponte Casta~neda and J. R. Willis, J. Mech. Phys. Solids 43, 1919 (1995)] microstructure in conjunction with Hashin–Shtrikman [Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962)] upper bound. Two critical volume concentrations, c* and c**, that represent the respective percolation threshold at which the conductive network begins to develop, and the percolation saturation, are identified. During this very short range of concentration, the electrical conductivity of the composite is found to exhibit a very sharp increase, while over the entire range, the calcutilated conductivity exhibits the widely reported sigmoidal shape. Comparison with measurement on a multi-walled carbon nanotube/alumina composite indicates that the theory could capture the major features of the experimentally observed trends C 2011 American Institute of Physics. [doi:10.1063/1.3671675] sufficiently well. V I. INTRODUCTION

Polymer composites containing conductive fillers are of great interests due to various potential applications such as radar absorption, electronic packaging, high charge storage, sensors, and others that require a combination of electrical, thermal, and mechanical properties. It has been widely reported that, within a very narrow range of loading of additive fillers, the electrical or thermal conductivity of the resulting composite increases sharply from nearly zero to several orders of magnitude. This phenomenon is directly tied to the issue of percolation threshold and increased connectedness as the filler concentration continues to increase. This process has a profound effect on the overall conductivity of the composite systems. The issues of percolation are complex, and so are the predictions of effective conductivity. Both are highly dependent on the microstructures. Complex networks and Bethe lattice are frequently called upon to build the percolation threshold,1–7 while Monte Carlo and other numerical methods are commonly adopted to compute the effective conductivities.8,9 Our objective here is to develop an analytical, homogenization approach to predict the percolation threshold and the effective conductivity of a two-phase composite containing randomly oriented ellipsoidal inclusions. The ellipsoids may take the shape of a needle, prolate or oblate spheroid, sphere, or disk. Along this line, some models have been developed to study the effective conductivity a)

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of two phase composites containing high aspect-ratio fillers.10–16 These models have provided significant insights into the effects of filler concentration and shape, with or without a perfect interfacial condition. The issue of percolation threshold, however, has remained beyond the reach of these approaches. The simple and analytical nature of homogenization methods are appealing, because, unlike the Monte Carlo or numerical methods, they could be readily used to predict a host of properties without resorting to heavy computations. But unless the issue of percolation can be resolved within the framework of homogenization process, the theory will not be able to capture the sharp increase of conductivity over a very narrow range of concentration the origins of which are the percolation process. It is with this view that we set out to undertake this study. The proposed approach is based upon consideration of the Ponte Casta~neda–Willis microstructure17 in conjunction with the Hashin–Shtrikman upper bound.18 II. THE THEORY

The Ponte Casta~neda–Willis (PCW) theory was originally developed for the estimate of effective elastic moduli of a two-phase composite. In this microstructure, the randomly oriented ellipsoidal inclusions take an isotropic spatial distribution that is conceptually described by a spherical outer shell, each containing an ellipsoidal inclusion as an inner core. Such a microstructure is depicted in Fig. 1(a), where for ease of exposition, the ellipsoidal inclusions are sketched as long needles. This microgeometry was later discovered by Hu and Weng19 to be directly connected to the

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netization factors. For a spheroidal inclusion with a symmetric axis 3, its components are (see for instance Ref. 26)

S11 ¼ S22 ¼

8 < :

i aða2 1Þ1=2 cosh1 a ; a > 1 h i 1 a að1 a2 Þ1=2 ; a < 1 3=2 cos

a 2ða2 1Þ3=2 a 2ð1a2 Þ

h

(3)

FIG. 1. (Color online) (a) The original Ponte Casta~ neda–Willis (1995) (Ref. 17) microstructure of an isotropic composite, in which conductive fillers do not touch the outer sphere and thus no percolation, (b) percolation threshold at c ¼ c*, (c) percolated state at c* < c < c**, and (d) approaching percolation saturation as c ! c**.

double-inclusion model of Hori and Nemat-Nasser.20 Based on this observation, the effective moduli derived by PCW were successfully recovered from the double-inclusion model. The PCW estimate and the Mori–Tanaka estimate21–23 were recently proved to possess a dynamical basis.24 The original PCW formulation was only intended to the condition that the inner ellipsoidal inclusions do not touch the outer sphere so that all inclusions are well separated. This microgeometry was shown to give rise to the effective elastic moduli tensor, Le, of a two-phase composite [see Eq. (31) of Ref. 24] 1 hT1 i; Le ¼ L0 þ c I chT1 iSd L1 0

c ¼ 1=a2 ; and c ¼ a;

(4)

(1)

where L0 is the elastic moduli tensor of the matrix (phase 0), c is the volume concentration of the ellipsoidal inclusions, Sd is Eshelby’s S-tensor25 representing the spatial distribution (in this case a sphere for the outer shell), and the angle brackets < . > represent the orientational average of the enclosed quantity. Tensor T1 is given by 1 T1 ¼ ½ðL1 L0 Þ1 þ SL1 0 ;

where a is the aspect ratio (length-to-diameter ratio) and S33 ¼ 1 2S11 . With a sphere, it reduces to S11 ¼ S22 ¼ S33 ¼ 1/3, for an infinitely long fiber, S33 ¼ 0, S11 ¼ S22 ¼ 1/2, and for an extremely thin flake, S33 ¼ 1, S11 ¼ S22 ¼ 0. While the original PCW theory was intended only for the condition that ellipsoidal inclusions never touch the outer sphere, the idea of separable double inclusions can be extended to determine the condition of percolation threshold at which conduction paths for electric current begin to develop. Using this double inclusion model, a conductive path can only occur when the inner ellipsoidal filler touches the outer sphere; indeed without both tips of an inner inclusion reaching to the outer sphere, a conductive path would never be established. This is illustrated in Fig. 1(b). Here we define the volume concentration of inclusions at which the inner ellipsoids are inscribed in the outer sphere as the percolation threshold, c*, of the isotropic composite. This condition characterizes the onset of the development of conducting paths throughout the composite. The percolation threshold can be obtained immediately by computing the volume ratio of the inner ellipsoidal and the outer sphere at this inscribing condition. Denoting ra ¼ rb, and rc as the semi-axes of the spheroid and R the radius of the outer sphere, this condition occurs at rc ¼ R for prolate and ra ¼ R for oblate inclusions, leading to the percolation threshold of the composite containing prolate and oblate inclusions,

(2)

where L1 is the elastic moduli tensor of the ellipsoidal inclusions (phase 1) and S is the Eshelby tensor associated with the shape of the ellipsoid. Equation (1) can also be applied to estimate the effective electrical or thermal conductivity, dielectric permittivity and magnetic susceptibility, and a host of other transport properties of a two-component system. To put it in the context of electrical conductivity, tensors Le, L0, and L1 are to be identified as the effective conductivity of the composite, and the conductivities of the matrix and inclusions, respectively, and tensor S is identified with the well-known shape or demag-

respectively, where a ¼ rc/ra. With the ellipsoidal shape, this condition is reached when rc(> rb > ra) ¼ R, and c* ¼ 1/a1a2, with a1 ¼ rc/ra and a2 ¼ rc/rb. Equation (4) immediately suggests that the percolation threshold is lower with prolate inclusions as compared to that with oblate inclusions the aspect ratios of which are reciprocal to each other. This implies that, in general, a needletype network is easier to develop than a disk-type, and the former will result in a higher effective conductivity at the same volume concentration. This feature is consistent with experimental observations.27 Now let us consider the case that the matrix is isotropic, with the electrical conductivity, r0, and the spheroidal inclusions transversely isotropic, with the conductivities r3, and r1 ¼ r2 in the axial and transverse directions, respectively. In this case, Eq. (1) can be simplified to re cT (5) ¼1þ ; r0 1 cT=3 where the diagonal components of the orientation average hT1 i is given by T r0, with


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1 n1 1 n2 1 n3 1 þ þ T¼ ; 3 1 þ ðn1 1ÞS11 1 þ ðn2 1ÞS22 1 þ ðn3 1ÞS33 (6) and ni ¼ ri/r0, i ¼ 1,2,3, representing the contrast ratios of the conductivity in the three directions. It can be seen that for a highly conductive filler in a nonconductive matrix, such that ni in each direction is much greater than 1, as in the case of carbon nanotubes in a polymer matrix, this average reduces to T ¼ ð1=3Þð1=S11 þ 1=S22 þ 1=S33 Þ; even in the presence of anisotropy, r1 ¼ r2 6¼ r3 . With such high conducting fillers, the effective conductivity is mainly dominated by the shape of inclusions. The result with isotropic inclusions follows readily by setting n1 ¼ n2 ¼ n3 ¼ n, under which n1 1 1 1 : þ þ T¼ 3 1 þ ðn 1ÞS11 1 þ ðn 1ÞS22 1 þ ðn 1ÞS33 (7) Combination of Eqs. (5) and (7) will make the theory particularly useful. When the inclusions are spherical, the result leads to the well-known Maxwell formula28 re cðr1 r0 Þ ¼1þ ; r0 ð1=3Þð1 cÞðr1 r0 Þ þ r0

(8)

where r1 is the isotropic conductivity of the fillers. Equation (8) also turns out to be the Hashin–Shtrikman (HS) lower bound if the inclusions are more conducting than the matrix. This expression can also be used to construct the HS upper bound by switching the more conducting phase to serve as the matrix and the less conducting one as spherical inclusions. This is given by rHSðþÞ ð1 cÞðr0 r1 Þ e ¼1þ ; r1 ð1=3Þcðr0 r1 Þ þ r1

(9)

FIG. 2. (Color online) The calculated percolation threshold, c*, and percolation saturation, c**, and effective electrical conductivity, with filler aspect ratios of 0.1, 0.2, 5, and 10 and contrast ratio n ¼ 108. The HS upper and lower bounds are also shown.

bound would serve as a guideline for the evolution of the overall conductivity. The volume concentration at which the two curves intersect thus represents another critical volume concentration, to be denoted as c**. By equating Eq. (5) to the HS upper bound, it is determined to be c ¼

3ð2n2 n 1 3nTÞ : 2 2ðn 1Þ T

(10)

Unlike c*, this critical concentration also depends on the contrast ratio of the constituent properties. As the contrast increases from n ¼ 10 to 103, the results are shown in Fig. 3. It is evident that, as n increases beyond 102, the variations gradually reach an asymptotic state, implying that c** becomes property-independent. This limiting case can be

which we will need later. III. RESULTS AND DISCUSSION

We now examine the results obtained from the theory. We first found that as the volume concentration increases beyond c*, the effective conductivity given by Eq. (5) increases sharply. With the aspect ratios of 10 and 0.1, and 5 and 0.2, to represent the prolate and oblate shape, the normalized effective conductivity is shown in Fig. 2, along with the HS upper and lower bounds, for the contrast ratio n ¼ r1 =r0 ¼ 108 . The corresponding percolation thresholds are marked by the cross x signs. For c < c*, which is the original PCW range, the curves all reside between the HS bounds, regardless of the aspect ratio. As the volume concentration increases beyond c*, the effective conductivities are seen to exhibit rapid increase, eventually reaching the HS upper bound. The microgeometry in this transitional stage involves significant development of percolation paths as depicted from Figs. 1(b) to 1(c) and to 1(d). Of course any calculated results above the HS upper bound must be discarded as they are not useful, so from there on, the HS upper

FIG. 3. (Color online) The saturation percolation, c**, against the filler aspect ratio, a, as the contrast ratio increases. All curves asymptotically converge to c n!1 .


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FIG. 4. (Color online) Comparison of the predicted percolation thresholds with experimental data for various composite systems with prolate or oblate fillers. Note that a ¼ 103 was used in Stankovich (Ref. 41).

obtained by setting n ! 1, and the asymptotic value is found to be c n!1 ¼

9 : 1=S11 þ 1=S22 þ 1=S33

(11)

This value, like c*, depends only on the aspect ratio of the inclusions. This form is particularly useful for the study of electrical conductivity with high conducting fillers because the contrast ratio is usually very high, sometimes to the order of 101220 (see, for instance, Ref. 29). The HS upper bound calculated at c** corresponds to the idealized condition that the conducting phase serves as the matrix and the less conducting one as spherical inclusions. A possible microgeometry approaching such a state is depicted in Fig. 1(d), where the white grids can be imagined as ideally spherical. From c* to c**, the network connectedness undergoes significant growth, and the overall conductivity increases dramatically. The predicted c* and c** are plotted against the filler aspect ratio in Fig. 4, where the experimentally reported percolation data for various composite materials are also marked.27,30–42 Note that this is a double-log plot, so that it could cover a wider range of aspect ratio and finer range of percolation threshold and saturation. Here the lower solid line gives the c* variation and the upper dashed line the c** dependence. The range between these two critical volume concentrations is seen to be very narrow, and, despite its narrowness, it contains most of the experimental data. Indeed most data points are expected to lie above c*, which in our definition is the onset point, while experimental data are usually taken at conditions with significant percolation paths. The comparison appears to be better for the oblate type than for the prolate one. In the latter case at the aspect ratios of 103-104, some data are found to lie outside the range. We believe that formation of fibers bundles and/or fiber waviness could potentially reduce the effective aspect ratio of the fillers.29–31 Such an effect is indicated by the horizontal arrow. Finally we applied Eq. (5) and the HS upper bound to a multi-walled carbon nanotube/alumina composite. The

FIG. 5. (Color online) Comparison between the predicted effective conductivity and experimental data for a multi-walled CNT/alumina composite.

obtained results are shown as the solid line in Fig. 5, along with the measured data.43 In this curve, the percolation threshold, c*, and percolation saturation, c**, are also indicated. At the aspect ratio a ¼ 102, the range from c ¼ 0 to c*—the original PCW range—is very short. From c* to c**, during which the percolation paths continue to build up, the overall conductivity is seen to exhibit a very steep increase. This is the region that is commonly referred to as percolation transition. After c**, with every filler already participating in the percolation networks, any increase of volume concentration will not result in significant change of percolation paths, and thus the increase of conductivity will be gradual. This is reflected by the HS upper-bound curve. Indeed, it is the sharp increase associated with the c*– c** range that is the figure of merit. The calculated overall conductivity is seen to display a sigmoidal shape. This trend is also evident from the test data. This is a feature that homogenization theories without a percolation threshold could not deliver. We also note that all the data points lie below the HS upper bound; this feature displays the consistency between the bound and the experiment. IV. CONCLUSIONS

In this article, we have proposed an explicit, analytical approach to determine the percolation threshold, percolation saturation, and effective conductivity of an isotropic composite that contains randomly oriented ellipsoidal inclusions. The model was built upon the Ponte Casta~neda–Willis microstructure in conjunction with the Hashin–Shtrikman upper bound. It provides two critical volume concentrations, c* and c**, that represent the percolation threshold and percolation saturation, respectively, in terms of aspect ratio of inclusions. It was within this very narrow range of concentrations that most reported experimental data on percolation threshold exist, and it is also within this range that the electrical conductivity shows a remarkable jump. Before the filler concentration reaches c*, the fillers are well separated, and the conductivity of the composite is low. After passing c*, percolation paths build up progressively until it


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approaches c**, at which all fillers participate in the percolation process. Subsequent increase of filler concentration can only improve the conductivity gradually, as guided by the Hashin–Shtrikman upper bound. This bound in reality can never be reached as it requires the conducting fillers to form a homogeneous matrix and the less-conducting phase to exist as spherical inclusions. As such, the HS upper bound can only serve as a guide. But the curve formed by the three segments—the original PCW estimate from 0 to c*, the relaxed PCW model from c* to c**, and the HS upper bound—shows the remarkable feature of a sigmoidal shape, and the test data after c** were also very close to the bound. This threesegment model is simple and analytical with no adjustable parameters. It could well be applied to study other transport phenomena of similar microstructures. ACKNOWLEDGMENTS

This work has been supported by Qatar National Research Fund NPRP 09-508-2-192, U.S. National Science Foundation MSM 05-10409, and Natural Sciences and Engineering Research Council of Canada. 1

S. Kirkpatrick, Rev. Mod. Phys. 45, 574 (1973). R. Zallen, The Physics of Amorphous Solids (Wiley, New York, 1983). 3 I. Balberg and N. Binenbaum, Phys. Rev. B 28, 3799 (1983). 4 D. Stauffer and A. Aharony, Introduction to Percolation Theory (Taylor and Francis, Philadelphia, 1991). 5 G. Grimmett, Percolation, 2nd ed. (Springer, New York, 1999). 6 C.-W. Nan, Y. Shen, and J. Ma, Annu. Rev. Mater. Res. 40, 131 (2010). 7 A. P. Chatterjee, J. Chem. Phys., 132, 224905 (2010); J. Phys: Condens. Matter 23, 375101 (2011). 8 C. Li and T.-W. Chou, Appl. Phys. Lett. 90, 174108 (2007). 9 W. J. Kim, M. Taya, and M. N. Nguyen, Mech Mater. 41, 1116 (2009). 10 H. Hatta and M. Taya, J. Appl. Phys. 58, 2478 (1985). 11 Y. Benveniste and T. Miloh, Int. J. Eng. Sci. 24, 1537 (1986); J. Appl. Phys. 69, 1337 (1991). 12 D. P. H. Hasselman and L. F. Johnson, J. Compos. Mater. 21, 508 (1987). 13 C. W. Nan, R. Berringer, D. R. Clarke, and H. Gleiter, J. Appl. Phys. 81, 6692 (1997). 14 C. W. Nan, G. Liu, Y. H. Lin, and M. Li, Appl. Phys. Lett. 85, 3549 (2004). 2

J. Appl. Phys. 110, 123715 (2011) 15

T. Chen, G. J. Weng, and W. C. Liu, J. Appl. Phys. 97, 104312 (2005). Y. Ngabonziza, J. Li, and C. F. Barry, Acta Mech. 220, 289 (2011). 17 P. Ponte Casta~ neda and J. R. Willis, J. Mech. Phys. Solids 43, 1919 (1995). 18 Z. Hashin and S. Shtrikman, J. Appl. Phys. 33, 3125 (1962). 19 G. K. Hu and G. J. Weng, Mech. Mater. 32, 495 (2000); Acta Mech. 140, 31 (2000). 20 M. Hori and S. Nemat-Nasser, Mech. Mater. 14, 189 (1993). 21 G. J. Weng, Int. J. Eng. Sci. 22, 845 (1984). 22 Y. Benveniste, Mech. Mater. 6, 147 (1987). 23 G. J. Weng, Int. J. Eng. Sci. 28, 1111 (1990). 24 G. J. Weng, Mech. Mater. 42, 886 (2010). 25 J. D. Eshelby, Proc. R. Soc. London, Ser. A 241, 376 (1957). 26 L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media, 2nd ed. (Elsevier, Amsterdam, 1984). 27 W. Zheng, S.-Ch. Wong, and H. J. Sue, Polymer 43, 6767 (2002). 28 J. C. Maxwell, A Treatise on Electricity and Magnetism, 3rd ed. (Clarendon Press, Oxford, 1982), Vol. 1, Chap. 9, pp. 435–441. 29 F. Deng and Q.-S. Zheng, Appl. Phys. Lett. 92, 071902 (2008). 30 C. Li, E. T. Thostenson and T.-W. Chou, Comp. Sci. Tech. 68, 1445 (2008). 31 N. Grossiord, J. Loos, L. van Laake, M. Maugey, C. Zakri, C. E. Koning, and A. J. Hart, Adv. Funct. Mater. 18, 3226 (2008). 32 A. E. Eken, E. J. Tozzi, D. J. Klingenberg, and W. Bauhofer, J. Appl. Phys. 109, 084342 (2011). 33 ¨ . Pekcan, J. Colloid Interface Sci. 344, S. Kara, E. Arda, F. Dolastir, and O 395 (2010). 34 J. Z. Kovacs, R. E. Mandjarov, T. Blisnjuk, K. Prehn, M. Sussiek, J. Mu¨ller, K. Schulte, and W. Bauhofer, Nanotechnology 20, 155703 (2009). 35 Z. Ounaies, C. Park, K. E. Wise, E. J. Siochi, and J. S. Harrison, Comp. Sci. Tech. 63, 1637 (2003). 36 F. Dalmas, R. Dendievel, L. Chazeau, J. Cavaille, and C. Gauthier, Acta Mater. 54, 2923 (2006). 37 Y. R. Hernandez, A. Gryson, F. M. Blighe, M. Cadek, V. Nicolosi, W. J. Blau, Y. K. Gun’ko, and J. N. Coleman, Scripta Mater. 58, 69 (2008). 38 Y. J. Kim, T. S. Shin, H. D. Choi, J. H. Kwon, Y.-C. Chung, and H. G. Yoon, Carbon 43, 23 (2005). 39 J. K. W. Sandler, J. E. Kirk, I. A. Kinloch, M. S. P. Shaffer, and A. H. Windle, Polymer 44, 5893 (2003). 40 R. K. Goyal, S. D. Samant, A. K. Thakar, and A. Kadam, J. Phys. D 43, 365404 (2010). 41 S. Stankovich, D. A. Dikin, G. H. B. Dommett, K. M. Kohlhaas, E. J. Zimney, E. A. Stach, R. D. Piner, S. T. Nguyen, and R. S. Ruoff, Nature 442, 282 (2006). 42 G. Chen, W. Weng, D. Wu, and C. Wu, Eur. Polymer J. 39, 2329 (2003). 43 K. Ahmad, W. Pan, and S.-L. Shi, Appl. Phys. Lett. 89, 133122 (2006). 16
