Journal of ELECTRONIC MATERIALS, Vol. 39, No. 3, 2010
Ó 2010 TMS
Modeling of Thermal Conductivity of Graphite Nanosheet
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: email@example.com
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,
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
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
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
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,
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.
Modeling of Thermal Conductivity of Graphite Nanosheet Composites 271
Appendix I. http://www.resins.com/Products/Techni
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-
Items 100 70 25
(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
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)
the higher Ke11. Therefore, further exfoliation of the
GNS to produce higher-aspect-ratio ﬁllers would be
expected to further enhance the thermal conduc-
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