The interface effect of the effective electrical conductivity of carbon ...

IOP PUBLISHING

NANOTECHNOLOGY

Nanotechnology 18 (2007) 255705 (6pp)

doi:10.1088/0957-4484/18/25/255705

The interface effect of the effective electrical conductivity of carbon nanotube composites K Y Yan, Q Z Xue1 , Q B Zheng and L Z Hao College of Physics Science and Technology, China University of Petroleum, Dongying, Shandong 257061, People’s Republic of China E-mail: [email protected]

Received 19 November 2006, in final form 25 March 2007 Published 1 June 2007 Online at stacks.iop.org/Nano/18/255705 Abstract A model of the effective electrical conductivity for carbon nanotube (CNT) composites is presented by incorporating the interface effect with an average field theory. The dependence of the effective electrical conductivity on CNT length, diameter, concentration, and interface properties has been taken care of simultaneously in our treatment so that the model can describe well the interface effect of CNT composites. Predictions from the model are in good agreement with the experimental values of the effective electrical conductivity of CNT composites which the classical models have not been able to explain.

atomic bonding at the CNT-matrix [8]. Besides, people have found that the interface properties can tremendously affect the electrical transport properties of CNT composites. For example, the critical CNT content can be largely reduced by the surface treatment of CNTs, and the electrical conductivity of CNT composites can be affected largely by the surface treatment of CNTs [1, 9, 10]. There are interactions between the CNTs and the matrix, such as the attractive van der Waals interactions [11, 12] and chemical bonds. Therefore, an interface shell forms between the CNT and the matrix, whose properties differ from that of the CNT and the matrix and are determined by the properties and distributions of the matrix particulates and the CNTs and so on. Not considering the thickness of the interfaces between the CNTs and the matrix, several thermal conductivity models of CNT composites have been proposed [13–15]. For example, we have presented a model of the effective thermal conductivity for CNT composites. In the earlier model the dependence of the effective conductivity on nanotube length, diameter, concentration, and interface thermal resistance had been discussed. However, due to considering the interface effect as an interface thermal resistance with zero thickness, the model cannot explain the percolation phenomena in the electrical properties of the CNT composites. Recently, the experimental results have demonstrated that the thickness of

1. Introduction Carbon nanotubes (CNTs) have been investigated intensively for many potential applications, due to their unique structure and properties. In particular, CNT composites in which the CNTs are used as fillers have attracted much attention and are a promising direction in nanotechnology [1–5]. For example, people have produced conductive CNT composites at very low CNT filling concentration. Lower filling fractions imply smaller perturbations of bulk physical properties, such as optical transparency and strength, as well as lower cost [5]. Based on these findings, attempts have been made to explain the very low conductivity thresholds and the dependence of the electrical conductivity of CNT composites on the CNT loading by means of Monte Carlo simulations [6]. Recently, people have demonstrated that the interface properties of the CNT-matrix are very important for the transport properties of CNT composites and are of particular interest. Recently, Cahill and co-workers experimentally measured the interface thermal resistance ( RK ) and found that the thermal transport in the CNT composites would be limited largely by RK . Also, a very recent experiment has shown that a value of RK across the CNT-matrix is about 8.3 × −1 10−8 K m2 W [7]. Theoretical research has demonstrated that the RK in CNT composites was attributed to the weak 1 Author to whom any correspondence should be addressed.

0957-4484/07/255705+06$30.00

1

© 2007 IOP Publishing Ltd Printed in the UK

Nanotechnology 18 (2007) 255705

K Y Yan et al

interfacial layer r

nanotube E0

θ

X

a

part 2

t part 3

part 1

Figure 2. The cross-section picture of part 2, which is the CNT with the radial interfacial layer.

Figure 1. Schematic illustration of the ‘complex CNT’, which is a composite unit cell of a CNT with a thin interfacial layer.

(U ) and the corresponding boundary conditions are as follows: ∂U 1 ∂ 1 ∂ 2U r + 2 ∇ 2U = = 0, (2) r ∂r ∂r r ∂θ 2

the interface has an important effect on the critical content of the electrical percolation of CNT composites [10]. Therefore, the existing models cannot describe the interface effect on the electrical properties of CNT composites exactly. In this paper, we present a model of the effective electrical conductivity for CNT composites by incorporating the interface effect with an average field theory. The dependence of the electrical conductivity on CNT length, diameter, concentration, and interface properties has been taken care of simultaneously in our treatment so that the model can describe well the interface effect of CNT composites.

Uc |r=0 = constant, Um |r→∞ = −E 0r cos θ,

Us |r=a+t = U f |r=a+t , dUs dUm −σs = −σf , dr r=a+t dr r=a+t

(3a ) (3b) (3c) (3d ) (3e) (3 f )

where Uc , Um , and Us are the electrical potentials of the CNT, the matrix, and the interfacial layer, respectively. Solving equation (2), we obtain ∞ B1n n A1n r + n (C1n cos nθ + D1n sin nθ ), Uc = r n=1

Since interface conductivity plays an important role in the electrical transport properties of CNT composites, a CNT coated with a thin interfacial layer should be regarded as a ‘complex CNT’ (as shown in figure 1). As we know, the quantum effect of the CNT is important for the electrical conductivity of CNT composites. In our treatment, the quantum effect can also be regarded as a kind of interfacial effect, which affects the electrical conductivity of the interfacial layer and affects the electrical conductivity of the CNT composites accordingly. Therefore, the interface effect should include the quantum effect, which is determined by the separation of the CNTs and the interaction between the CNT and the matrix particulate in this model. We consider that the complex CNT consists of three parts: the left interface (1), the CNT with the radial interface (2), and the right interface (3) (as shown in figure 1). In this paper, σc , σm , σs , and σe are the electrical conductivities of the CNT, the matrix, the interfacial layer and the CNT composite, respectively. The longitudinal equivalent electrical conductivity σ233 of part 2, the CNT (length L , diameter d ) coated with the radial interfacial layer (thickness t ), can be obtained directly from the mixture rule for a simple parallel model of the radial interface/CNT, i.e.

σc d 2 + σs (4dt + 4t 2 ) . (d + 2t)2

at large distance,

Uc |r=a = Us |r=a , dUc dUs −σc = −σ , s dr r=a dr r=a

2. Model of the effective conductivity of CNT composites

σ233 =

at the CNT centre,

r < a, ∞ B2n n A2n r + n (C2n cos nθ + D2n sin nθ ), Us = r n=1

(4)

a < r < a + t, ∞ B3n A3n r n + n (C3n cos nθ + D3n sin nθ ), Um = r n=1 r > a + t. Substituting the boundary conditions into equation (4), we have Uc = Ar cos θ,

0 r < a,

Us = Br cos θ + Cr −1 cos θ, Um = D

cos θ − E 0r cos θ, r

a < r < a + t,

(5)

a + t r → ∞,

where

A = 2 Fσs , B = F(σs + σc ),

(1)

C = a 2 F(σs − σc ),

The transverse equivalent electrical conductivity σ211 of part 2 should be obtained by using Maxwell’s theory. We apply an electrical field, say E 0 , on a CNT with the radial interface along a radial axis (say x axis), as shown in figure 2. According to Maxwell’s equations, Laplace’s equation for the potential

D = (a + t)2 F(σs + σc ) + a 2 F(σs − σc ) + E 0 (a + t)2 , F= 2

2σm E 0 . a 2 (σs − σc )(σs − σm ) a+t − (σs + σm )(σs + σc )


K Y Yan et al

j = σ E,

We can obtain the electrical field in the CNT along the x -axis ( E c,x ) and the electrical field along the x -axis in the interfacial layer ( E s,x ) as follows: dUc = −2 Fσs , dx dUs =− = −F (σs + σc ) + a 2 (σs − σc ) dx sin2 θ − cos2 θ . × r2

E cx = − E sx

(6)

where · · · denotes an average over the system volume. To find the effective electrical conductivity, one has to know the distribution of the electrical field and electrical current density in the system when an electrical filed E0 is applied. The internal field in CNTs and matrix particles averaged over all orientations are equal to, respectively [15], 1 E in,com = E0, 1 + Bcom,k (σcom,k − σe,k )/σc,k k=x,y,z (15) 1 E in,m = E0, 1 + Bm,k (σm,k − σe,k )/σe,k k=x,y,z

We can introduce the transverse equivalent electrical conductivity as follows:

jx = σ211 E x

(7)

where jx and E x are the spatial average of the electrical current density and the electrical field along the x -axis respectively, i.e.

1 jx = jx dv, V v (8)

1 E x dv. E x = V v

where Bm,k and Bcom,k are the depolarization factors of the matrix particles and the complex CNTs, respectively, which are only determined by the shape of the matrix particle and the complex CNT along the k -axis (k = x, y, z ), respectively. The electrical current density in the CNTs and matrix particles averaged over all orientations are equal to, respectively,

The electrical current density in the CNT and the interfacial layer along the x -axis, respectively, are

jc,x = σc E c,x , js,x = σs E s,x .

jin,com = σc E in,com , jin,m = σm E in,m .

(9)

d 2 σc σs + 2σs (σc + σs )(t 2 + 2at) (10) . d 2 σs + 2(σc + σs )(t 2 + 2at) The transverse and longitudinal equivalent electrical 11 33 and σcom , of the composite unit cell of a conductivities, σcom CNT (length L , diameter d ) coated with a thin interfacial layer (thickness t ) can be obtained directly from the mixture rule for a simple parallel (series) model of this part 1/part 2/part 3 11 = (σs 2t + σ211 L)/(L + 2t), (as shown in figure 1), i.e. σcom 33 11 33 and (L + 2t)/σcom = 2t/σs + L/σ233 . σcom and σcom can be expressed respectively as 2 d σc + 2(σc + σs )(t 2 + 2at) σs 11 L 2 σcom = + 2t , L + 2t d σs + 2(σc + σs )(t 2 + 2at) =

where f is the volume fraction of the CNTs that is filled, 2 2 L L α = π(d+π2t)d 2 (L+ = (d+2t)d 2 (L+ , which is the ratio between 2t) 2t ) the volume of the CNT and the volume of the complex CNT. Therefore, f /α is the volume fraction of the complex CNTs filled. Because the matrix particles are irregular and their size is much smaller than the CNT length, for simplicity all the matrix particles can be regarded as balls, and all the CNTs can be regarded as the same rotationally prolate ellipsoids because their aspect ratio M = Ld 1, we have [17]

Bm,x = Bm,y = Bm,z = 1/3,

(L + 2t)σs [σc d 2 + σs (4dt + 4t 2 )] = . 2tσc d 2 + 2tσs (4dt + 4t 2 ) + σs L(d + 2t)2

1 − Bc,z . 2 If so, equation (12) reduces as 33 σe − σm σe − σcom f f + 3 1− 33 α 2σe + σm 3α σe + Bc,z (σcom − σe ) 11 σe − σcom = 0, +4 11 − σ ) 2σe + (1 − Bc,z )(σcom e

Bc,x = Bc,y =

(11) We turn now to discuss the effective electrical conductivity of the CNT composites. The whole composite can be regarded as a system in which the complex CNTs are randomly embedded in the matrix particles. Let the electrical field E and electrical current density j be defined respectively by

E = −∇φ,

(16)

Substituting equations (15) and (16) into equation (14), we obtain the equation for the effective electrical conductivity of the CNT composites [15, 16]: σe,k − σm,k f 1− α σ + Bm,k (σm,k − σe,k ) e,k k=x,y,z f σe,k − σcom,k + (17) = 0, α σe,k + Bcom,k (σcom,k − σe,k ) k=x,y,z

Substituting equations (8) and (9) into equations (6) and (7), and making use of equation (5), we have a+t a σc E c 2πr dr + a σs E s 2πr dr j x dv σ211 = v = 0 a a+t E s 2πr dr v E x dv 0 E c 2πr dr + a

33 σcom

(13)

where φ is the electrical potential and σ is the electrical conductivity. According to the average polarization theory, the effective electrical conductivity of the CNT composite is determined by the definition j = σe E, (14)

(18) (19)

(20)

11 33 and σcom , respectively, are the transverse and where σcom longitudinal electrical conductivities of the complex CNT.

(12) 3


K Y Yan et al

Figure 3. The dependence of the effective conductivities on CNT length L with different volume fraction f .

Figure 4. The dependence of the effective conductivities on CNT diameter d with different volume fraction f , with σm = 10−9 S cm−1 , σs = 1.85 S cm−1 , σc = 1850 S cm−1 , L = 2.5 × 104 nm, t = 2.5 nm.

The value of Bc,z can be obtained by another method. When M > 15, the percolation threshold ( f c ) of the CNT composite is inversely proportional to the CNT aspect ratio M [18], and has f c = 0.7/M = 0.7 Ld , which is found by Monte Carlo simulation [19–21]. In the general case of a CNT composite, equation (21) gives the relation between the percolation threshold and the longitudinal depolarization factor of the CNT [19]:

f c = (5 − 3 Bc,z )Bc,z /(1 + 9 Bc,z ).

(21)

For CNTs, M 1 and Bc,z 1, so, from equation (21) we have d fc Bc,z = = 0.14 . (22) 5 L Substituting equation (22) into equation (20), we have equation (23) to study the electrical effective conductivity of CNT composites with interfacial layers: 33 f f σe − σm σe − σcom 3 1− + d 33 − σ ) α 2σe + σm 3α σe + 0.14 L (σcom e 11 σe − σcom = 0. (23) +2 d/L) 11 − σ ) σe + (1−0.14 (σcom e 2 When d L , equation (23) reduces as 33 σe − σm σe − σcom f f + 3 1− d 33 − σ ) α 2σe + σm 3α σe + 0.14 L (σcom e 11 σe − σcom = 0. +4 11 σe + σcom

Figure 5. The dependence of the effective conductivities on CNT conductivity with different volume fractions f , with σm = 10−9 S cm−1 , σs = 1.85 S cm−1 , L = 2.5 × 104 nm, d = 20 nm, t = 2.5 nm.

those by other methods. Therefore, in order to increase the electrical conductivity of the composite, longer CNTs obtained by chemical vapour deposition can be chosen. The dependence of the effective electrical conductivity of the CNT composites on CNT diameter is shown in figure 4. The percolation threshold of the composite decreases with increasing CNT diameter, which is caused by the fact that the larger CNT diameter leads to fewer conductive passages at the same CNT concentration. Smaller diameter leading to better transport properties indicates a good way to improve the effective electrical conductivity of CNT composites. As we know, the electrical conductivities of CNTs obtained by different methods are different from each other, whose effect on the effective electrical conductivity of the CNT composite is distinguished. As shown in figure 5, the CNT composites containing CNTs with the same shape and different electrical conductivities have the same percolation threshold and have different electrical conductivities. In other words, the effective electrical conductivity is enhanced obviously by the increasing electrical conductivities of CNTs. This may be applicable in nano-semiconductor technology. The effect of interface thickness on the effective electrical conductivity of CNT composites is shown in figure 6. It can

(24)

3. Results and discussion To investigate the interface effect on the effective electrical conductivity of CNT composites, using equation (23) several simulations were carried out. The effect of the CNT length on the electrical conductivity enhancement in CNT composites is shown in figure 3. It is easily found that the effective electrical conductivity increases rapidly and the percolation threshold decreases with increasing CNT length with the same CNT concentration. That is to say, the transport properties of the CNT composite will be better with larger length. It is well known that the CNTs obtained by chemical vapour deposition are much longer than 4


K Y Yan et al

Figure 6. Effect of the thickness of the interface on the electrical conductivity of CNT composites. In the calculation, σm = 1 × 10−9 S cm−1 , σc = 1850 S cm−1 , σs = 1.85 S cm−1 , d = 10 nm and L = 25 μm.

Figure 8. Comparison between the experimental results [10] of electrical conductivity of CNT-epoxy composites and the values calculated by the model. L = 25 000 nm, and d = 13 nm. σm = 5 × 10−15 S cm−1 , σc = 1850 S cm−1 , σs = 1.85 × 10−5 S cm−1 . Inset: effect of the interfacial layer on the electrical conductivity of CNT composite around the threshold.

carbon content of about 95% and treated by 40% HNO3 (pH = 2.5) at 100 ◦ C for about 1 h so that the interfaces of the CNTs are oxidized [10]. The experimental parameters are as follows: d = 10–20 nm, L = 10 000–50 000 nm, σc = 1850 S cm−1 , and σm = 5 × 10−15 S cm−1 . For comparison, we let L = 25 000 nm, d = 13 nm, σm = 5 × 10−15 S cm−1 , and σc = 1850 S cm−1 in the calculation, respectively [10]. The thickness and conductivity of the interfacial layer are fitted as t = 2.5 nm and σs = 1.85 × 10−5 S cm−1 , respectively. As shown, considering the interface effect, the predictions from equation (24) are in excellent agreement with the experimentally observed values of the effective electrical conductivity of CNT composites in the whole CNT content which the traditional model cannot understand. Without considering the interface effect, the disagreement of the fitted curve with the reported data is also obviously shown in figure 8. According to equation (24), the percolation threshold decreases with increasing interfacial layer thickness so that, when the volume fraction of the CNTs is smaller than the percolation threshold, the effective electrical conductivity of CNT composites increases rapidly with increasing interfacial layer thickness (as shown in the inset of figure 8), in accord with the results in figure 7. However, when the volume fraction of the CNTs is larger than the percolation threshold, the effective electrical conductivity of CNT composites is mainly decided by the electrical conductivity of the complex CNTs. The electrical conductivity of the complex CNTs decreases rapidly with increasing interfacial layer thickness so, when the volume fraction of the CNTs is larger than the percolation threshold, the effective electrical conductivity of CNT composites decreases rapidly with increasing interfacial layer thickness. There is another comparison between the experimental data and the theoretical results of the electrical conductivity of CNT-epoxy resin composites [22]. In this work, the CNTs (d = 2–3 nm, M = 5000–10 000, σm = 7.9 × 10−16 S cm−1 ), more than 80% with one or two walls, were synthesized by catalytic chemical vapour deposition, and then were treated by

Figure 7. Effect of the electrical conductivity of the interface on the electrical conductivity of CNT composites. In the calculation, σm = 2.5 × 10−9 S cm−1 , σc = 1850 S cm−1 , L = 25 000 nm, and d = 20 nm.

easily be seen from figure 6 that the percolation threshold decreases with increasing interfacial layer thickness. Besides, the effective electrical conductivity of CNT composites increases rapidly with increasing interfacial layer thickness within the threshold region. The effect of the electrical conductivity of the interfacial layer (σs ) on the effective electrical conductivity of CNT composites is shown in figure 7. We can find that the effective electrical conductivity increases rapidly with increasing σs after the percolation threshold, whereas the effective electrical conductivity increases very slowly with increasing σs before the percolation threshold.

4. Comparison with the experiment Figure 8 shows the comparison between the experimental data and the theoretical results of the electrical conductivity of CNT-epoxy composites, in which the multiwalled CNTs are synthesized by a chemical vapour deposition method with 5


K Y Yan et al

the experimentally observed values of the effective electrical conductivity of CNT composites in the whole CNT content which the traditional model cannot understand.

Acknowledgments This work was supported by the Key Project of the Chinese Ministry of Education under contract no. 106036, the Shandong Natural Science Fund under contract no. Y2005A10 and the CNPC Innovation Fund.

References [1] Liu Y Q and Gao L 2005 Carbon 43 47 [2] Grunlan J C, Mehrabi A R, Bannon M V and Bahr J L 2004 Adv. Mater. 16 150 [3] Sandler J K W, Kirk J E, Kinloch I A, Shaffer M S P and Windle A H 2003 Polymer 44 5893 [4] Yang Y L, Gupta M C, Dudley K L and Lawrence R W 2005 Nano Lett. 5 2131 [5] Bryning M B, Islam M F, Kikkawa J M and Yodh A G 2005 Adv. Mater. 17 1186 [6] Foygel M, Morris R D, Anes D, French S and Sobolev V L 2005 Phys. Rev. B 71 104201 [7] Huxtable S et al 2003 Nat. Mater. 2 731 [8] Xue L, Keblinski P, Philpot S R, Choi S U S and Eastman J A 2003 J. Chem. Phys. 118 337 [9] Shin D H, Yoon K H, Kwon O H, Min B G and Hwang C I 2006 J. Appl. Polym. Sci. 99 900 [10] Kim Y J, Shin T S, Choi H D, Kwon J H, Chung Y C and Yoon H G 2005 Carbon 43 23 [11] Zheng Q B, Xue Q Z, Yan K Y, Hao L Z, Gao X L and Li Q 2007 J. Phys. Chem. C 111 4628 [12] Steuerman D W, Star A, Narizzano R, Choi H, Ries R S, Nicolini C, Stoddart J F and Heath J R 2002 J. Phys. Chem. B 106 3124 [13] Garboczi E J, Snyder K A, Douglas J F and Thorpe M 1995 Phys. Rev. E 52 819 [14] Nan C W, Liu G, Lin Y H and Li M 2004 Appl. Phys. Lett. 85 3549 [15] Xue Q Z 2006 Nanotechnology 17 1655 [16] Xue Q and Xu W M 2005 Mater. Chem. Phys. 90 298 [17] Doyle W T and Jacobs I S 1992 J. Appl. Phys. 71 3926 [18] Balberg I, Binenbaum N and Wagner N 1984 Phys. Rev. Lett. 52 1465 [19] Lagarkov A N and Sarychev A K 1996 Phys. Rev. B 53 6318 [20] Celzard A, McRae E, Deleuze C, Dufort M, Furdin G and Mareche J F 1996 Phys. Rev. B 53 6209 [21] Balberg I 1986 Phys. Rev. B 33 3618 [22] Barrau S, Demont P, Perez E, Peigney A, Laurent C and Lacabanne C 2003 Macromolecules 36 9678

Figure 9. Comparison between the experimental results [22] of electrical conductivity of CNT-epoxy resin composites and the values calculated by the model. σm = 7.9 × 10−15 S cm−1 , σc = 1850 S cm−1 , L = 16 × 103 nm, d = 35 nm, t = 2.5 nm. (This figure is in colour only in the electronic version)

palmitic acid so that the interfaces of the CNTs were modified. In order to consider that the CNTs intersect with each other in the experiment, it is supposed that the CNT length value is reduced due to the CNTs intersecting. In the calculation, when σm = 7.9 × 10−15 S cm−1 , σs = 5 × 10−6 S cm−1 , σc = 1850 S cm−1 , L = 1.6 × 103 nm, d = 2.5 nm, and t = 1.5 nm, the calculated curve can reflect the experimental data trend properly, as shown in figure 9.

5. Conclusion In summary, we present a model of the effective electrical conductivity of CNT composites by incorporating the interface effect with an average field theory. The dependence of the effective electrical conductivity on CNT length, diameter, concentration, and interface properties has been taken care of simultaneously in our treatment, so that the model can describe well the interface effect of CNT composites. The model predicts that the percolation threshold decreases with increasing interface thickness. Also, when the CNT content is larger than the percolation threshold, the electrical conductivity of the interface has a large effect on the effective electrical conductivity of CNT composites. Predictions from the model are in excellent agreement with

6