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- 1. Journal of ELECTRONIC MATERIALS, Vol. 39, No. 3, 2010 DOI: 10.1007/s11664-009-1062-2 Ó 2010 TMS Modeling of Thermal Conductivity of Graphite Nanosheet Composites WEI LIN,1,2 RONGWEI ZHANG,1 and C.P. WONG1 1.—School of Material Science & Engineering, Georgia Institute of Technology, 771 Ferst Drive NW, Atlanta, GA 30332, USA. 2.—e-mail: wlin31@gatech.edu Recent experiments demonstrated a very high thermal conductivity in graphite nanosheet (GNS)/epoxy nanocomposites; however, theoretical anal- ysis is lacking. In this letter, an effective medium model has been used to analyze the effective thermal conductivity of the GNS/polymer nanocompos- ites and has shown good validity. Strong inﬂuences of the aspect ratio and the orientation of the GNS are evident. As expected, interfacial thermal resistance still plays a role in determining the overall thermal transport in the GNS/ polymer nanocomposites. In comparison with the interfacial thermal resis- tance between carbon nanotubes and polymers, the interfacial thermal resistance between GNS and polymers is about one order of magnitude lower, the reason for which is discussed. Key words: Graphene, graphite, nanosheet, carbon nanotube, modeling, thermal conductivity INTRODUCTION thermal conductivity. This great enhancement, hypothetically, resulted from the reduced GNS/ Graphene sheets are attracting increasing atten- polymer interfacial thermal resistance. However, tion due to their excellent electrical, thermal, and this hypothesis is somewhat contrary to the results mechanical properties.1 One potential application of of Hung et al.6 where a strong inﬂuence of interfa- graphene sheets is to incorporate them into polymer cial resistance was estimated. As such, intense matrices to prepare high-performance compos- studies on the thermal properties of GNS/polymer ites.1,2 However, manufacturing graphene-based and graphene/polymer nanocomposites are pre- composites has been very challenging due to the dicted; therefore, complete theoretical analysis of difﬁculties in large-scale production of graphene the thermal transport behavior of GNS/polymer sheets and their incorporation into polymer matri- nanocomposites is of great importance. ces.2 In comparison, exfoliated graphite—especially graphite nanosheets (GNS), stacks of a few graph- ene sheets with high aspect ratio—are advanta- THEORY AND MODEL geous in terms of preparation and dispersion.3–8 Effective medium theory (EMT) is used to analyze Recently, Sun and co-authors reported their prepa- the thermal conductivity of the GNS/EP composite ration of GNS with high aspect ratio and incorpo- in Ref. 8. EMT is commonly used to describe ration of the GNS into an epoxy (EP) matrix.8 microstructure–property relationships in micro- The as-prepared GNS/EP composite displayed an structurally heterogeneous materials, in particular extremely high in-plane thermal conductivity to calculate and/or predict the effective physical ($80 W mÀ1 KÀ1 at 33 vol.% GNS loading). Com- properties of a heterogeneous system. Following pared with carbon nanotube (CNT)/EP nanocom- Nan et al.,9 let us consider a composite material posites, there is no doubt that the GNS are with well-dispersed ﬁller embedded in a polymer considered more effective ﬁllers for improving matrix. Assuming a homogeneous and isotropic effective medium with property K0 and a perturba- (Received September 29, 2009; accepted December 14, 2009; tion in property, K¢(r), due to the presence of the published online January 13, 2010) ﬁller, the property of the heterogeneous medium at 268
- 2. Modeling of Thermal Conductivity of Graphite Nanosheet Composites 269 position r is thus expressed as: K(r) = K0 + K¢(r). By A common mixing rule was used to calculate the using the Green’s function G for the homogeneous mass density (q) and the speciﬁc heat (Cp) of the medium and the transition matrix T for the entire nanocomposites, with composite medium, the resultant effective prop- erty of the composite is expressed as: Ke = K0 + q ¼ f qf þ ð1 À f Þqm ; (8) hTi (I + hGTi)À1, where I is the unit tensor and h i denotes spatial averaging. By neglecting the inter- Cp ¼ xCp;f þ ð1 À xÞCp;m ; (9) action between ﬁller units at relatively low ﬁller loading, the matrix T is simpliﬁed to where qm (0.93 g cmÀ3) (Appendix 1) and qf X X À Á (2.1 g cmÀ3) are the mass densities of the EP 0 0 À1 (EPONOL resin 53-BH-35) and the graphite, Tﬃ Tn ¼ Kn I À GKn : (1) n n respectively. Cp,f (speciﬁc heat of graphite) of 0.70 J gÀ1 KÀ1 is taken.11 x is the mass fraction of Taking the polymer matrix phase as the homoge- GNS in the nanocomposite sample. neous reference medium and assuming perfect It is presented in Ref. 8 that, at f = 0.33, ae11 and GNS/EP interfaces (the interfacial thermal resis- Ke11 are 35 mm2 sÀ1 and 80 W mÀ1 KÀ1, respec- tance will be modeled later), we obtain the effective tively. Based on these data and Eqs. 7–9, we cal- thermal conductivity of the composite in the culate Cp,m (the speciﬁc heat of the EP) to be Maxwell-Garnett form: Â À Á À ÁÃ 2 þ f b11 ð1 À L11 Þ 1 þ hcos2 hi þ b33 ð1 À L33 Þ 1 À hcos2 hi Ke11 ¼ Ke22 ¼ Km ; (2) 2 À f ½b11 L11 ð1 þ hcos2 hiÞ þ b33 L33 ð1 À hcos2 hiÞ Â À Á Ã 1 þ f b11 ð1 À L11 Þ 1 À hcos2 hi þ b33 ð1 À L33 Þhcos2 hi Ke33 ¼ Km ; (3) 1 À f ½b11 L11 ð1 À hcos2 hiÞ þ b33 L33 hcos2 hi 2.91 J gÀ1 KÀ1, and thus Km = 0.32 W mÀ1 KÀ1. Kfii À Km This Km of pure EP looks unexpectedly high bii ¼ À Á; (4) Km þ Lii Kfii À Km according to previous research experience of the authors’ group; however, in order to be consistent p2 p with the data in Ref. 8, let us just accept this value L11 ¼ L22 ¼ þ cosÀ1 p; (5) for the effective medium analysis in this study. 2ðp2 À 1Þ 2ð1 À p2 Þ3=2 Based on the data of ae11 in Ref. 8, Ke11 of the GNS/EP nanocomposites at various ﬁller loadings L33 ¼ 1 À 2L11 ; (6) are calculated. In our modeling, Kf11 is taken as 1500 W mÀ1 KÀ1 on the basis of the following con- where Ke11 (=Ke22) and Ke33 are the in-plane and siderations. The in-plane thermal conductivity of a the through-thickness effective thermal conductivi- single graphene sheet was estimated to be as high ties of the nanocomposite sample, respectively; f is as $5000 W mÀ1 KÀ1.12,13 Given the fact that a the volumetric ﬁller loading of the GNS; Km is the multiwalled carbon nanotube (MWNT), which is isotropic thermal conductivity of the EP matrix; Kf11 treated as a concentrically wrapped-up GNS,14 (=Kf22) and Kf33 are the in-plane and the through- possesses a similar longitudinal thermal conduc- thickness thermal conductivities of the GNS unit, tivity (>3000 W mÀ1 KÀ1)15 to a single-walled car- respectively; p reﬂects the aspect ratio of the GNS bon nanotube (SWNT) ($3500 W mÀ1 KÀ1),16 we (the thickness over the in-plane diameter of a GNS believe that a GNS possesses a high Kf11 of the same unit) and is, on average, 5/20008,10; hcos2hi reﬂects order of magnitude as Kf11 of a single graphene the statistical orientation of the GNS in the EP and sheet. Previous experimental data on CNTs also lies within [1/3, 1], where hcos2hi = 1/3 for a random showed that the longitudinal thermal conductivities dispersion, while hcos2hi = 1 for a fully parallel ori- of MWNT arrays were $80 W mÀ1 KÀ1.17–19 Since entation of the GNS plane relative to the surface the packing densities of these MWNT arrays are plane of the ﬁlm sample. $5% and tube–tube interactions degrade the effec- In Ref. 8, the effective thermal diffusivity (ae) of tive thermal conductivities,20 we estimate the the nanocomposites was directly obtained in the equivalent longitudinal thermal conductivity of an experimental measurement. The effective thermal individual MWNT to be at least 1600 W mÀ1 KÀ1. conductivity (Ke) was then calculated using the Therefore, the GNS, an unrolled form of a equation MWNT, should possess a Kf11 of approximately 1600 W mÀ1 KÀ1. Further proof comes from the Keii ¼ aeii qCp : (7) in-plane thermal conductivity of the commercial
- 3. 270 Lin, Zhang, and Wong pyrolytic graphite sheets (PGS) provided by interfacial resistance, RK, at the GNS edge is taken Panasonic, Inc. (Appendix 2). As the PGS den- into account, in analogy with the methodology by sity approaches the theoretical mass density of Nan et al.27 Nevertheless, the interfacial resistance graphite, the in-plane thermal conductivity of the along the through-thickness direction is neglected PGS becomes 1500 W mÀ1 KÀ1 to 1700 W mÀ1 KÀ1. based on the following factors: (1) the large aspect To be conservative, we take 1500 W mÀ1 KÀ1 in ratio and the ﬂat surface of GNS enhance GNS– our modeling. Accordingly, Kf33 is taken as polymer interaction,28 leading to a relatively negli- 15 W mÀ1 KÀ1. gible interfacial thermal resistance compared with that at the GNS edge; (2) since phonon acoustic RESULTS AND DISCUSSION mismatch is considered the main fundamental cause of interfacial thermal resistance between Figure 1 shows a comparison between Eq. 2 and GNS (or CNT) and polymer,23 and given that the the experimental data in Ref. 8. A strong inﬂuence soft-mode vibrations of GNS along its through- of the GNS orientation in the EP matrix on Ke11 is thickness direction, compared with the rigid-mode evident. hcos2hi = 0.91 is the value for which the vibrations in its in-plane direction, can be much calculated Ke33/Ke11 at 33 vol.% loading is between better coupled to the vibrations of the polymer 1/9 and 1/10, close to the actual average orientation matrix, it is reasonable to consider only the inter- of the GNS in the EP matrix in Ref. 8. The model facial thermal resistance at the GNS edge in such a predicts higher thermal conductivity enhancement highly oriented GNS/polymer composite. As such, than the experimental results. Moreover, a GNS/ the effective in-plane thermal conductivity of the polymer nanocomposite exhibits a very low perco- GNS can be expressed as lation threshold (fc < 0.64%),21 above which a con- tinuous GNS network forms in the nanocomposite. B Kf0 11 ¼ Kf0 22 ¼ ; (10) The ﬁller loadings under investigation are much 2RK þ B Kf 11 higher than fc; in this sense, the model used here has already underestimated Ke11.22 The mixture where B is the average lateral size (2 lm) of a GNS rule, percolation model, and Bruggeman models will unit. By replacing Kf11 and Kf22 with K¢11 and K¢22, f f predict even higher Ke11 than the Maxwell-Garnett respectively, in Eqs. 2 and 4, we get the modeling model at such high ﬁller loadings.23 Therefore, the results in Fig. 3. RK falls in the range of experimental data are far below predictions. The 1 9 10À9 m2 K WÀ1 to 6 9 10À9 m2 K WÀ1, one deviation observed between the theoretical predic- order of magnitude smaller than the RK of tion and the experimental results probably indicates 8 9 10À8 m2 K WÀ1 measured across CNT/polymer the inﬂuence of the GNS/EP interfacial resistance, interfaces.29 It is not difﬁcult to understand the consistent with the conclusion in Ref. 6. Therefore, relatively small RK of GNS/polymer interfaces. In GNS/polymer interface modiﬁcation is expected to polymer composites with SWNTs, due to the small further improve Ke, in analogy with CNT/polymer tube diameter (usually 10 nm), the probability of composites.24–26 attachment of the ends of the CNTs to the polymer Meanwhile, our modeling results show distinct matrix (the ends have to be somehow attached to the difference between CNT/polymer interface and polymer chain for effective thermal transport across GNS/polymer interface. Figure 2 shows a schematic the interface) is extremely low.30 In comparison, in illustration of a modiﬁed GNS unit after the GNS/polymer composites, the large lateral length (usually on the order of 1 lm) of the GNS results in a dramatically increased probability of attachment of the GNS edge to the polymer matrix. Figure 4 shows a strong inﬂuence of the aspect ratio on Ke11. The higher the aspect ratio of GNS, Fig. 2. Schematic illustration of a modiﬁed GNS unit after the inter- Fig. 1. A comparison between the effective medium model (with facial resistance, RK, at the GNS edge is taken into account. various GNS orientations) and the experimental results (solid dots) in A simple series model is used. B represents the lateral size of the Ref. 8. GNS.
- 4. Modeling of Thermal Conductivity of Graphite Nanosheet Composites 271 APPENDIX Appendix I. http://www.resins.com/Products/Techni calDataSheet.aspx?id=4061 Sales Speciﬁcation Test Method/ Property Units Value Standard Viscosity at 25°C cP U-Z2 ASTM D1544 Color Gardner 6 ASTM D1544 Solids %m/m 34–37.5 ASTM D1259 Fig. 3. The inﬂuence of GNS/polymer interfacial thermal resistance on the overall effective thermal conductivity of the nanocomposites (hcos2hi = 0.91). Appendix II. http://industrial.panasonic.com/www- ctlg/ctlg/qAYA0000_WW.html PGS (lm) Items 100 70 25 Thermal conductivity (W mÀ1 KÀ1) X, Y direction 600–800 750–950 1500–1700 Z direction 15 15 (15) Thermal diffusivity 9–10 9–10 9–10 (cm2 sÀ1) Density (g cmÀ3) 0.85 1.10 2.10 Speciﬁc heat (at 50°C) 0.85 0.85 0.85 (J gÀ1 KÀ1) Heat resistance 400 400 400 (in air) (°C) Pull strength (MPa) X, Y direction 19.6 22.0 30.0 Z direction 0.4 0.4 0.1 Fig. 4. Inﬂuence of the aspect ratio of the GNS on the overall Bending test (times) 30000 30000 30000 effective thermal conductivity of the nanocomposites (hcos2hi = R5 180° 0.91). Electric conductivity 10000 10000 (20000) (S cmÀ1) the higher Ke11. Therefore, further exfoliation of the GNS to produce higher-aspect-ratio ﬁllers would be expected to further enhance the thermal conduc- REFERENCES tivity of GNS/polymer composites. In this case, however, interfacial thermal resistance will proba- 1. A.K. Geim and K.S. Novoselov, Nat. Mater. 6, 183 (2007). 2. S. Stankovich, D.A. Dikin, G.H.B. Dommett, K.M. Kohlhaas, bly become more inﬂuential. E.J. Zimney, E.A. Stach, R.D. Piner, S.T. Nguyen, and R.S. Ruoff, Nature 442, 282 (2006). CONCLUSIONS 3. B. Debelak and K. Lafdi, Carbon 45, 1727 (2007). 4. H. Fukushima, L.T. Drzal, B.P. Rook, and M.J. Rich, Therm. A conventional effective medium model shows Anal. Calorim. 85, 235 (2006). good validity in predicting the anisotropic effective 5. S. Ganguli, A.K. Roy, and D.P. Anderson, Carbon 46, 806 thermal conductivity of GNS/polymer nanocompos- (2008). ites. The model also indicates the inﬂuence of GNS/ 6. M.T. Hung, O. Choi, Y.S. Ju, and H.T. Hahn, Appl. Phys. Lett. 89, 023117 (2006). polymer interfacial thermal resistance on the over- 7. A.P. Yu, P. Ramesh, M.E. Itkis, E. Bekyarova, and R.C. all thermal transport. Haddon, J. Phys. Chem. C 111, 7565 (2007). 8. L.M. Veca, M.J. Meziani, W. Wang, X. Wang, F. Lu, P. ACKNOWLEDGEMENTS Zhang, Y. Lin, R. Fee, J.W. Connell, and Y.P. Sun, Adv. Mater. 21, 2088 (2009). The authors acknowledge NSF (#0621115) for 9. C.W. Nan, R. Birringer, D.R. Clarke, and H. Gleiter, J. Appl. ﬁnancial support and Dr. Wei Wang and Prof. Phys. 81, 6692 (1997). Ya-ping Sun at Clemson University for helpful 10. L.M. Veca, F. Lu, M.J. Meziani, L. Cao, P. Zhang, G. Qi, L. discussion. Qu, M. Shrestha, and Y.P. Sun, Chem. Commun. 2565 (2009).
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