- Email: [email protected]
Model for thermal conductivity of carbon nanotube-based composites
ARTICLE IN PRESS
Physica B 368 (2005) 302–307 www.elsevier.com/locate/physb
Model for thermal conductivity of carbon nanotube-based composites Q.Z. Xue College of Physics Science and Technology, China University of Petroleum, Dongying, Shandong 257061, P.R. China Received 29 September 2004; received in revised form 28 July 2005; accepted 28 July 2005
Abstract Considering the carbon nanotubes’ (CNTs) orientation distribution a new model of effective thermal conductivity of CNTs-based composites is presented. Based on Maxwell theory, the formulae of calculating effective thermal conductivity of CNTs-based composites are given. The theoretical results on the effective thermal conductivity of CNTs/oil and CNTs/decene suspensions are in good agreement with the experimental data. The model is valid for the transport properties of the CNTs-based composites. r 2005 Elsevier B.V. All rights reserved. PACS: 44.30.+v; 44.10.+I; 67.57.Hi Keywords: Model; Thermal conductivity; Nanofluid
1. Introduction Carbon nanotubes (CNTs), due to their unique structure and remarkable physical properties, have attracted much attention in the past few years [1–4]. Recently, the rapid advance in the bulk synthesis of CNTs makes it possible to produce a host of CNTs-based composites [5]. Recent studies reveal that CNTs have anomalously high thermal
Tel.: +86 546 8392836; fax: +86 546 8392123.
E-mail address: [email protected].
conductivity. It can be expected that the CNTsbased suspensions can enhance thermal conductivity and the improved thermal performance would be applied to energy systems [6–8]. Recently, Chio et al. reported that the enhancement in thermal conductivity of suspension for 1.0 vol% CNTs in oil is 160% [8]. The anomalous thermal conductivity enhancement is theoretically interesting because the measured thermal conductivities have been considered to be much larger than the prediction from the existing models. The expressions of the conventional models of the effective thermal conductivity of two-phase
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.07.024
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
composites are briefly summarized as follows [8]: ke 3ða 1Þf ; ¼1þ ða þ 2Þ ða 1Þf km
Maxwell ½9,
ke a þ ðn 1Þ ðn 1Þð1 aÞf , ¼ a þ ðn 1Þ þ ð1 aÞf km Hamilton and Crosser ½10, ke 3b2 9b3 a þ 2 2 þ f 2, ¼ 1 þ 3bf þ 3b þ þ km 4 16 2a þ 3
303
theory, two formulae of calculating the effective thermal conductivity of CNTs-based composites are given. The model shows that the axial ratio and the space distribution of the CNTs can largely affect the effective thermal conductivity of CNTsbased composites so that the dispersion of quite a small amount of CNTs can result in a remarkable enhancement in the effective thermal conductivity of the composites that the existing models cannot understand.
Jeffery ½11, ke 3ða 1Þ ½f þ f ðaÞf 2 þ 0ðf 3 Þ, ¼1þ ða þ 2Þ ða 1Þf km Davis ½12, ke ¼ 1 þ af þ bf 2 ; km
Lu and Lin ½13,
where ke is the effective thermal conductivity of two-phase composites, km and kc are the thermal conductivities of the main phase and scattered phase, respectively, n is the shape factor of a particle given by n ¼ 3=j, where j is defined as the ratio of the surface area of a sphere (with a volume equal to that of the particle) to the surface area of the particle, f is the particle volume fraction of the scattered phase, and a ¼ kc =km , b ¼ ða 1Þ= ða þ 2Þ, a and b are pure fitting parameters [9–11]. All the existing models are only valid for the composites containing the spherical or rotational elliptical particles with small axial ratio M (M ¼ a=b), where a, b, c ( ¼ b) are the semi-radii along j-axis (j ¼ x; y; z). However, CNTs, in fact, can be regarded as rotational elliptical nanoparticles with a very large axial ratio M (M ¼ ða=bÞb1) so that the existing models cannot work on the CNTsbased composites. Besides, the existing models cannot describe the effect of the space distribution of the CNTs on the thermal conductivity. Due to the defects, the existing models fail to describe the experimental data, and there is really no way to make them work on the CNTs-based composites. Therefore, considering the very large axial ratio and the space distribution of the CNTs we built a new model for the effective thermal conductivity of CNTs-based composites. Based on Maxwell
2. Model for thermal conductivity of carbon nanotube-based composites We assume that ke is the effective thermal conductivity of the CNTs-based composites, kc and km are the thermal conductivities of the CNTs and the main media, respectively. ~ and heat flux ~ Let the temperature field E q be, respectively, defined as ~ ¼ rf, E
(1)
~ ~ q ¼ kE,
(2)
where f and k are the temperature distribution function and the thermal conductivity. The field factor is defined as the ratio between ~c in the CNT to the mean the temperature field E ~ temperature field E 0 in the composite. We assume that the applied field is along the j-axis (j ¼ x; y; z), according to Maxwell theory the field factor component along j-axis can be expressed as cj ¼
E c;j 1 , ¼ E 0;j 1 þ Bj kc =km 1
(3)
where Bj is the depolarization factor component of the CNT along j-axis, which is a parameter used to characterize the shape of the particle [14–17]. The CNTs can be considered as the same rotational elliptical nano-particles with very large axial ratio M (M ¼ a=bb1). So, the depolarization factor component is expressed as follows [15,16]: Bx ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 M ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q þ lnðM þ M 2 1Þ, 1 M2 2 3 ð1 M Þ
M41,
ð4Þ
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
304
By ¼ Bz ¼
1 Bx . 2
(5)
According to Eq. (4), when the length direction of CNTs is parallel (or vertical) to the x-axis direction, the value of Bx is equal to 0 (or 1/2). As we know, the CNTs are randomly dispersed in the main media. In other words, their space orientation is stochastic. According to Eq. (4) and the above discussion, when the applied field is along a stable direction, say x-axis, the value of Bx will range from 0 to 1/2, due to the angle, between the applied field and the length direction of the CNT, ranges from 0 to p=2. In other words, the value of the depolarization factor of CNTs, determined by their orientation distribution, ranges from 0 to 1/2. Therefore, in order to describe this orientation distribution, we can introduce a distribution function PðBj Þ of the depolarization factor. Therefore, the equivalent field factor will be expressed as ¯j ¼ c
hqj i ¼ hE j i
Z
1=2 0
1 PðBj Þ dBj , 1 þ Bj ðkc =km 1Þ (6)
where hqj i and hE j i are the spatial average of the heat flux component and the temperature field component along the j-axis, respectively. The distribution function PðBj Þ must satisfy the R 1=2 two conditions: 0obj o1=2 and 0 PðBj Þ dBj ¼ 1. Usually, the CNTs are randomly distributed in the nanofluid, a normal-like distribution function PðBj Þ can be adopted [14]: 2 PðBj Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . p Bj ð1 Bj Þ
(7)
For the nanofluid with randomly distributed CNTs, the equivalent field factor will be expressed as Z
hqj i fkc hE c;j i þ ð1 f Þkm hE 0;j i ¼ f hE c;j i þ ð1 f ÞhE 0;j i hE j i ¯ fkc cj þ ð1 f Þkm ¼ , ¯ j þ ð1 f Þ fc
ke;j ¼
ð9Þ
where f is the total volume fraction of the CNTs. Due to the assumption that the thermal conductivity of the CNTs-based composites is isotropy, the effective thermal conductivity components is isotropy. Substituting Eq. (8) into (9), we have the expression of the effective thermal conductivity of CNTsbased composites. pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f þ ð4f =pÞ kc =km arctg p=4 kc =km pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi . ke ¼ km 1 f þ ð4f =pÞ km =kc arctg p=4 kc =km (10) As discussed above, the CNTs’ distribution can affect effective thermal conductivity of CNTsbased composites. For comparison, we assume that the form of distribution function PðBj Þ can selected as PðBj Þ ¼ 2.
(11)
In this case, the equivalent field factor will be expressed as Z 1=2 1 dBj jj ¼ 2 k 1 þ B j c =km 1 0 2km ð12Þ ¼ ln kc þ km =2km . kc km The corresponding expression of the effective thermal conductivity of CNTs-based composites can be deduced as ke ¼ km
1 f þ 2f 1 f þ 2f
kc kc km km kc km
ln kc2kþkmm þkm ln kc2k m
.
(13)
1=2
2 1 dBj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 þ B ð1 B Þ p B j c =km 1 0 j j sffiffiffiffiffiffi sffiffiffiffiffiffi! 4 km p kc ¼ arctg . ð8Þ p kc 4 km
¯j ¼ c
The effective thermal conductivity component ke;j can be obtained as
3. Numerical calculations and discussion Using Eqs. (10) and (13) we calculated the thermal conductivity of the CNTs–oil and
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
CNTs–decene suspensions, respectively. In the calculation, the thermal conductivity of CNTs is taken as 600–3000 W/mK, and the thermal conductivities of the oil and decene are taken as 0.1448 W/mK [8] and 0.14 W/mK [7], respectively. As shown in Fig. 1a, when the thermal conductivity of CNTs is taken as 600 W/mK the theoretical results given by Eq. (10) is in good agreement with the experimental data, whereas
those theoretical results given by the existing models are much smaller than the experimental data, even if we select the thermal conductivity of CNTs 3000 W/mK (Fig. 1b). For example, the measured enhancement in thermal conductivity for 1 vol% CNTs in oil is 160%, whereas the enhancement predicted by Hamilton–Crosser model is not more than 10% (Fig. 1b) and the enhancements predicted by the other four models are almost identical, about 3%.
1.2 Thermal conductivity ratio (Ke/Km)
Thermal conductivity ratio (Ke/Km)
5 kc=600 W/mK 4
kc=900 W/mK kc=1500 W/mK kc=2000 W/mK
3
Experimental data
2
(a)
0.2
0.4 0.6 0.8 CNTs volume fraction (%)
1.1
0.0
1.0
(a) 1.2
1.06
Thermal conductivity ratio (Ke/Km)
1.08 Thermal conductivity ratio (Ke/Km)
Kc = 3000 W/mK Kc = 2000 W/mK Kc = 1500 W/mK Kc =1000 W/mK Experimental data
1.0 1 0.0
Maxwell Davis Jeffery Lu-Lin Hamiton-Crosser
1.04
1.02
1.00 0.0 (b)
305
0.2 0.4 0.6 0.8 CNTs volume fraction (%)
Fig. 1. (a) Comparison between experimental data of CNTs–oil suspensions and the calculated values using Eq. (10). (b) Calculated thermal conductivity of CNTs–oil suspensions by the conventional models. In the calculation, n ¼ 3 for the Hamiton–Crosser model [10]; f ðaÞ ¼ 0:5 for Davis model [12]; a ¼ 3 and b ¼ 4:51 for Lu–Lin model [13].
(b)
1.0
Maxwell Davis Jeffery Lu-Lin Hamiton-Crosser Experimental data
1.1
1.0 0.0
1.0
0.2 0.4 0.6 0.8 CNTs volume fraction %
0.2
0.4 0.6 0.8 CNTs volume fraction %
1.0
Fig. 2. (a) Comparison between experimental data of CNTs–decene suspensions and the calculated values using Eq. (13). (b) Calculated thermal conductivity of CNTs–decene suspensions by the conventional models. In the calculation, n ¼ 3 for the Hamiton–Crosser model [10]; f ðaÞ ¼ 0:5 for Davis model [12]; a ¼ 3 and b ¼ 4:51 for Lu–Lin model [13].
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
306
Thermal conductivity ratio (Ke/Km)
As shown in Fig. 2a, when the thermal conductivity of CNTs is taken as 3000 W/mK, the theoretical results given by Eq. (13) is in good agreement with the experimental data, whereas the theoretical results given by the existing models are much smaller than the experimental data, even if we select the thermal conductivity of CNTs 3000 W/mK (Fig. 2b). For example, the measured enhancement in thermal conductivity for 1 vol% CNTs in decene is 19.6%, whereas the enhancement predicted by Hamilton–Crosser model is not more than 10% (Fig. 2b) and the enhancements predicted by the other four models are almost identical, about 3%. As we noted, Choi et al. reported that the thermal conductivity enhancement for suspension with 1.0 vol% CNTs in oil (km ¼ 0.1448 W/mK) was 160%. However, Xie et al. reported that the thermal conductivity enhancement for nanofluid with 1.0 vol% CNTs in decene (km ¼ 0.14 W/mK) was only 19.6%. We believe that this discrepancy would be attributed to the difference of the distribution state of the CNTs in the nanofluid. As shown in our model, the distribution state of the CNTs in the nanofluid has an important effect on the thermal conductivity of the nanofluid. Therefore, in this model we introduce two forms of the distribution function PðBj Þ of the CNTs. In Fig. 3, using Eq. (13) we calculated the thermal
2.5
Kc = 3000 W/mK Kc = 2000 W/mK Kc =1500 W/mK Experimental data
1.5
0.0
0.2 0.4 0.6 0.8 CNTs volume fraction %
4. Conclusion In summary, considering the very large axial ratio and the space distribution of the CNTs, a simple model of effective thermal conductivity of CNTs-based composites is presented. Based on Maxwell theory, two formulae of calculating the effective thermal conductivity of CNTs-based composites are given. The model shows that dispersion of quite a small amount of CNTs can result in a remarkable enhancement in the effective thermal conductivity of the composites that the existing models cannot understand. By comparing with the experiment data of the CNTs-contained suspensions, we can find that the observed anomalous thermal conductivity enhancement lies within reasonable range covered by our model.
References
2.0
1.0
conductivity of the CNTs–oil suspensions again. We find that the theoretical results are still smaller than the experimental data, even though the theoretical results are much larger that that of conventional models. From the discussion above, the distribution state of the CNTs has an important effect on the thermal conductivity of the nanofluid. Because we have considered the effect of the distribution of the CNTs in our model, the thermal conductivity enhancement in the composites with dilute CNTs is not anomalously beyond our model predictions.
1.0
Fig. 3. Comparison between experimental data of CNTs–oil suspensions and the calculated values using Eq. (13).
[1] S. Iijima, Nature (London) 354 (1991) 56. [2] P.M. Ajayan, O. Stephen, C. Cplliex, D. Trauth, Science 265 (1994) 1212. [3] W.A. de Heer, W.S. Bacsa, A. Chatelain, R.H. Baker, L. Forro, D. Ugarte, Science 268 (1995) 845. [4] H. Dai, E.W. Wong, C.M. Lieber, Science 272 (1996) 523. [5] R. Ramasubramaniam, J. Chen, H.Y. Liu, Appl. Phys. Lett. 83 (2003) 2928. [6] M.J. Biercuk, M.C. Liaguno, M. Radosavljevic, J.K. Hyun, A.T. Johnson, J.E. Fischer, Appl. Phys. Lett. 80 (2002) 2767. [7] H.Q. Xie, Hohyun Lee, Wonjin Youn, M. Choi, J. Appl. Phys. 94 (2003) 4967. [8] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Appl. Phys. Lett. 79 (2001) 2252.
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307 [9] J.C. Maxwell, A Treatise on Electricity and Magnetism, second ed., Oxford University Press, Cambridge, UK, 1904, p. 435. [10] R.L. Hamilton, O.K. Crosser, IEC Fundamentals 2 (1962) 187. [11] D.J. Jeffrey, Proc. Roy. Soc. London, Ser. A 335 (1973) 355.
307
[12] R.H. Davis, Int. J. Thermopyhs. 7 (1986) 609. [13] S. Lu, H. Lin, J. Appl. Phys. 79 (1996) 6761. [14] L. Gao, J.Z. Gu, J. Phys. D: Appl. Phys. 35 (2002) 267. [15] Q.Z. Xue, Eur. Polym. J. 40 (2004) 323. [16] W.T. Doyle, I.S. Jacobs, J. Appl. Phys. 71 (1992) 3926. [17] Q.Z. Xue, Phys. Lett. A 307 (2003) 313.
Physica B 368 (2005) 302–307 www.elsevier.com/locate/physb
Model for thermal conductivity of carbon nanotube-based composites Q.Z. Xue College of Physics Science and Technology, China University of Petroleum, Dongying, Shandong 257061, P.R. China Received 29 September 2004; received in revised form 28 July 2005; accepted 28 July 2005
Abstract Considering the carbon nanotubes’ (CNTs) orientation distribution a new model of effective thermal conductivity of CNTs-based composites is presented. Based on Maxwell theory, the formulae of calculating effective thermal conductivity of CNTs-based composites are given. The theoretical results on the effective thermal conductivity of CNTs/oil and CNTs/decene suspensions are in good agreement with the experimental data. The model is valid for the transport properties of the CNTs-based composites. r 2005 Elsevier B.V. All rights reserved. PACS: 44.30.+v; 44.10.+I; 67.57.Hi Keywords: Model; Thermal conductivity; Nanofluid
1. Introduction Carbon nanotubes (CNTs), due to their unique structure and remarkable physical properties, have attracted much attention in the past few years [1–4]. Recently, the rapid advance in the bulk synthesis of CNTs makes it possible to produce a host of CNTs-based composites [5]. Recent studies reveal that CNTs have anomalously high thermal
Tel.: +86 546 8392836; fax: +86 546 8392123.
E-mail address: [email protected].
conductivity. It can be expected that the CNTsbased suspensions can enhance thermal conductivity and the improved thermal performance would be applied to energy systems [6–8]. Recently, Chio et al. reported that the enhancement in thermal conductivity of suspension for 1.0 vol% CNTs in oil is 160% [8]. The anomalous thermal conductivity enhancement is theoretically interesting because the measured thermal conductivities have been considered to be much larger than the prediction from the existing models. The expressions of the conventional models of the effective thermal conductivity of two-phase
0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.07.024
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
composites are briefly summarized as follows [8]: ke 3ða 1Þf ; ¼1þ ða þ 2Þ ða 1Þf km
Maxwell ½9,
ke a þ ðn 1Þ ðn 1Þð1 aÞf , ¼ a þ ðn 1Þ þ ð1 aÞf km Hamilton and Crosser ½10, ke 3b2 9b3 a þ 2 2 þ f 2, ¼ 1 þ 3bf þ 3b þ þ km 4 16 2a þ 3
303
theory, two formulae of calculating the effective thermal conductivity of CNTs-based composites are given. The model shows that the axial ratio and the space distribution of the CNTs can largely affect the effective thermal conductivity of CNTsbased composites so that the dispersion of quite a small amount of CNTs can result in a remarkable enhancement in the effective thermal conductivity of the composites that the existing models cannot understand.
Jeffery ½11, ke 3ða 1Þ ½f þ f ðaÞf 2 þ 0ðf 3 Þ, ¼1þ ða þ 2Þ ða 1Þf km Davis ½12, ke ¼ 1 þ af þ bf 2 ; km
Lu and Lin ½13,
where ke is the effective thermal conductivity of two-phase composites, km and kc are the thermal conductivities of the main phase and scattered phase, respectively, n is the shape factor of a particle given by n ¼ 3=j, where j is defined as the ratio of the surface area of a sphere (with a volume equal to that of the particle) to the surface area of the particle, f is the particle volume fraction of the scattered phase, and a ¼ kc =km , b ¼ ða 1Þ= ða þ 2Þ, a and b are pure fitting parameters [9–11]. All the existing models are only valid for the composites containing the spherical or rotational elliptical particles with small axial ratio M (M ¼ a=b), where a, b, c ( ¼ b) are the semi-radii along j-axis (j ¼ x; y; z). However, CNTs, in fact, can be regarded as rotational elliptical nanoparticles with a very large axial ratio M (M ¼ ða=bÞb1) so that the existing models cannot work on the CNTsbased composites. Besides, the existing models cannot describe the effect of the space distribution of the CNTs on the thermal conductivity. Due to the defects, the existing models fail to describe the experimental data, and there is really no way to make them work on the CNTs-based composites. Therefore, considering the very large axial ratio and the space distribution of the CNTs we built a new model for the effective thermal conductivity of CNTs-based composites. Based on Maxwell
2. Model for thermal conductivity of carbon nanotube-based composites We assume that ke is the effective thermal conductivity of the CNTs-based composites, kc and km are the thermal conductivities of the CNTs and the main media, respectively. ~ and heat flux ~ Let the temperature field E q be, respectively, defined as ~ ¼ rf, E
(1)
~ ~ q ¼ kE,
(2)
where f and k are the temperature distribution function and the thermal conductivity. The field factor is defined as the ratio between ~c in the CNT to the mean the temperature field E ~ temperature field E 0 in the composite. We assume that the applied field is along the j-axis (j ¼ x; y; z), according to Maxwell theory the field factor component along j-axis can be expressed as cj ¼
E c;j 1 , ¼ E 0;j 1 þ Bj kc =km 1
(3)
where Bj is the depolarization factor component of the CNT along j-axis, which is a parameter used to characterize the shape of the particle [14–17]. The CNTs can be considered as the same rotational elliptical nano-particles with very large axial ratio M (M ¼ a=bb1). So, the depolarization factor component is expressed as follows [15,16]: Bx ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 M ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q þ lnðM þ M 2 1Þ, 1 M2 2 3 ð1 M Þ
M41,
ð4Þ
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
304
By ¼ Bz ¼
1 Bx . 2
(5)
According to Eq. (4), when the length direction of CNTs is parallel (or vertical) to the x-axis direction, the value of Bx is equal to 0 (or 1/2). As we know, the CNTs are randomly dispersed in the main media. In other words, their space orientation is stochastic. According to Eq. (4) and the above discussion, when the applied field is along a stable direction, say x-axis, the value of Bx will range from 0 to 1/2, due to the angle, between the applied field and the length direction of the CNT, ranges from 0 to p=2. In other words, the value of the depolarization factor of CNTs, determined by their orientation distribution, ranges from 0 to 1/2. Therefore, in order to describe this orientation distribution, we can introduce a distribution function PðBj Þ of the depolarization factor. Therefore, the equivalent field factor will be expressed as ¯j ¼ c
hqj i ¼ hE j i
Z
1=2 0
1 PðBj Þ dBj , 1 þ Bj ðkc =km 1Þ (6)
where hqj i and hE j i are the spatial average of the heat flux component and the temperature field component along the j-axis, respectively. The distribution function PðBj Þ must satisfy the R 1=2 two conditions: 0obj o1=2 and 0 PðBj Þ dBj ¼ 1. Usually, the CNTs are randomly distributed in the nanofluid, a normal-like distribution function PðBj Þ can be adopted [14]: 2 PðBj Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . p Bj ð1 Bj Þ
(7)
For the nanofluid with randomly distributed CNTs, the equivalent field factor will be expressed as Z
hqj i fkc hE c;j i þ ð1 f Þkm hE 0;j i ¼ f hE c;j i þ ð1 f ÞhE 0;j i hE j i ¯ fkc cj þ ð1 f Þkm ¼ , ¯ j þ ð1 f Þ fc
ke;j ¼
ð9Þ
where f is the total volume fraction of the CNTs. Due to the assumption that the thermal conductivity of the CNTs-based composites is isotropy, the effective thermal conductivity components is isotropy. Substituting Eq. (8) into (9), we have the expression of the effective thermal conductivity of CNTsbased composites. pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 f þ ð4f =pÞ kc =km arctg p=4 kc =km pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi . ke ¼ km 1 f þ ð4f =pÞ km =kc arctg p=4 kc =km (10) As discussed above, the CNTs’ distribution can affect effective thermal conductivity of CNTsbased composites. For comparison, we assume that the form of distribution function PðBj Þ can selected as PðBj Þ ¼ 2.
(11)
In this case, the equivalent field factor will be expressed as Z 1=2 1 dBj jj ¼ 2 k 1 þ B j c =km 1 0 2km ð12Þ ¼ ln kc þ km =2km . kc km The corresponding expression of the effective thermal conductivity of CNTs-based composites can be deduced as ke ¼ km
1 f þ 2f 1 f þ 2f
kc kc km km kc km
ln kc2kþkmm þkm ln kc2k m
.
(13)
1=2
2 1 dBj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 1 þ B ð1 B Þ p B j c =km 1 0 j j sffiffiffiffiffiffi sffiffiffiffiffiffi! 4 km p kc ¼ arctg . ð8Þ p kc 4 km
¯j ¼ c
The effective thermal conductivity component ke;j can be obtained as
3. Numerical calculations and discussion Using Eqs. (10) and (13) we calculated the thermal conductivity of the CNTs–oil and
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
CNTs–decene suspensions, respectively. In the calculation, the thermal conductivity of CNTs is taken as 600–3000 W/mK, and the thermal conductivities of the oil and decene are taken as 0.1448 W/mK [8] and 0.14 W/mK [7], respectively. As shown in Fig. 1a, when the thermal conductivity of CNTs is taken as 600 W/mK the theoretical results given by Eq. (10) is in good agreement with the experimental data, whereas
those theoretical results given by the existing models are much smaller than the experimental data, even if we select the thermal conductivity of CNTs 3000 W/mK (Fig. 1b). For example, the measured enhancement in thermal conductivity for 1 vol% CNTs in oil is 160%, whereas the enhancement predicted by Hamilton–Crosser model is not more than 10% (Fig. 1b) and the enhancements predicted by the other four models are almost identical, about 3%.
1.2 Thermal conductivity ratio (Ke/Km)
Thermal conductivity ratio (Ke/Km)
5 kc=600 W/mK 4
kc=900 W/mK kc=1500 W/mK kc=2000 W/mK
3
Experimental data
2
(a)
0.2
0.4 0.6 0.8 CNTs volume fraction (%)
1.1
0.0
1.0
(a) 1.2
1.06
Thermal conductivity ratio (Ke/Km)
1.08 Thermal conductivity ratio (Ke/Km)
Kc = 3000 W/mK Kc = 2000 W/mK Kc = 1500 W/mK Kc =1000 W/mK Experimental data
1.0 1 0.0
Maxwell Davis Jeffery Lu-Lin Hamiton-Crosser
1.04
1.02
1.00 0.0 (b)
305
0.2 0.4 0.6 0.8 CNTs volume fraction (%)
Fig. 1. (a) Comparison between experimental data of CNTs–oil suspensions and the calculated values using Eq. (10). (b) Calculated thermal conductivity of CNTs–oil suspensions by the conventional models. In the calculation, n ¼ 3 for the Hamiton–Crosser model [10]; f ðaÞ ¼ 0:5 for Davis model [12]; a ¼ 3 and b ¼ 4:51 for Lu–Lin model [13].
(b)
1.0
Maxwell Davis Jeffery Lu-Lin Hamiton-Crosser Experimental data
1.1
1.0 0.0
1.0
0.2 0.4 0.6 0.8 CNTs volume fraction %
0.2
0.4 0.6 0.8 CNTs volume fraction %
1.0
Fig. 2. (a) Comparison between experimental data of CNTs–decene suspensions and the calculated values using Eq. (13). (b) Calculated thermal conductivity of CNTs–decene suspensions by the conventional models. In the calculation, n ¼ 3 for the Hamiton–Crosser model [10]; f ðaÞ ¼ 0:5 for Davis model [12]; a ¼ 3 and b ¼ 4:51 for Lu–Lin model [13].
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307
306
Thermal conductivity ratio (Ke/Km)
As shown in Fig. 2a, when the thermal conductivity of CNTs is taken as 3000 W/mK, the theoretical results given by Eq. (13) is in good agreement with the experimental data, whereas the theoretical results given by the existing models are much smaller than the experimental data, even if we select the thermal conductivity of CNTs 3000 W/mK (Fig. 2b). For example, the measured enhancement in thermal conductivity for 1 vol% CNTs in decene is 19.6%, whereas the enhancement predicted by Hamilton–Crosser model is not more than 10% (Fig. 2b) and the enhancements predicted by the other four models are almost identical, about 3%. As we noted, Choi et al. reported that the thermal conductivity enhancement for suspension with 1.0 vol% CNTs in oil (km ¼ 0.1448 W/mK) was 160%. However, Xie et al. reported that the thermal conductivity enhancement for nanofluid with 1.0 vol% CNTs in decene (km ¼ 0.14 W/mK) was only 19.6%. We believe that this discrepancy would be attributed to the difference of the distribution state of the CNTs in the nanofluid. As shown in our model, the distribution state of the CNTs in the nanofluid has an important effect on the thermal conductivity of the nanofluid. Therefore, in this model we introduce two forms of the distribution function PðBj Þ of the CNTs. In Fig. 3, using Eq. (13) we calculated the thermal
2.5
Kc = 3000 W/mK Kc = 2000 W/mK Kc =1500 W/mK Experimental data
1.5
0.0
0.2 0.4 0.6 0.8 CNTs volume fraction %
4. Conclusion In summary, considering the very large axial ratio and the space distribution of the CNTs, a simple model of effective thermal conductivity of CNTs-based composites is presented. Based on Maxwell theory, two formulae of calculating the effective thermal conductivity of CNTs-based composites are given. The model shows that dispersion of quite a small amount of CNTs can result in a remarkable enhancement in the effective thermal conductivity of the composites that the existing models cannot understand. By comparing with the experiment data of the CNTs-contained suspensions, we can find that the observed anomalous thermal conductivity enhancement lies within reasonable range covered by our model.
References
2.0
1.0
conductivity of the CNTs–oil suspensions again. We find that the theoretical results are still smaller than the experimental data, even though the theoretical results are much larger that that of conventional models. From the discussion above, the distribution state of the CNTs has an important effect on the thermal conductivity of the nanofluid. Because we have considered the effect of the distribution of the CNTs in our model, the thermal conductivity enhancement in the composites with dilute CNTs is not anomalously beyond our model predictions.
1.0
Fig. 3. Comparison between experimental data of CNTs–oil suspensions and the calculated values using Eq. (13).
[1] S. Iijima, Nature (London) 354 (1991) 56. [2] P.M. Ajayan, O. Stephen, C. Cplliex, D. Trauth, Science 265 (1994) 1212. [3] W.A. de Heer, W.S. Bacsa, A. Chatelain, R.H. Baker, L. Forro, D. Ugarte, Science 268 (1995) 845. [4] H. Dai, E.W. Wong, C.M. Lieber, Science 272 (1996) 523. [5] R. Ramasubramaniam, J. Chen, H.Y. Liu, Appl. Phys. Lett. 83 (2003) 2928. [6] M.J. Biercuk, M.C. Liaguno, M. Radosavljevic, J.K. Hyun, A.T. Johnson, J.E. Fischer, Appl. Phys. Lett. 80 (2002) 2767. [7] H.Q. Xie, Hohyun Lee, Wonjin Youn, M. Choi, J. Appl. Phys. 94 (2003) 4967. [8] S.U.S. Choi, Z.G. Zhang, W. Yu, F.E. Lockwood, E.A. Grulke, Appl. Phys. Lett. 79 (2001) 2252.
ARTICLE IN PRESS Q.Z. Xue / Physica B 368 (2005) 302–307 [9] J.C. Maxwell, A Treatise on Electricity and Magnetism, second ed., Oxford University Press, Cambridge, UK, 1904, p. 435. [10] R.L. Hamilton, O.K. Crosser, IEC Fundamentals 2 (1962) 187. [11] D.J. Jeffrey, Proc. Roy. Soc. London, Ser. A 335 (1973) 355.
307
[12] R.H. Davis, Int. J. Thermopyhs. 7 (1986) 609. [13] S. Lu, H. Lin, J. Appl. Phys. 79 (1996) 6761. [14] L. Gao, J.Z. Gu, J. Phys. D: Appl. Phys. 35 (2002) 267. [15] Q.Z. Xue, Eur. Polym. J. 40 (2004) 323. [16] W.T. Doyle, I.S. Jacobs, J. Appl. Phys. 71 (1992) 3926. [17] Q.Z. Xue, Phys. Lett. A 307 (2003) 313.