Effect of Kapitza contact and consideration of tube-end transport on the effective conductivity in nanotube-based composites

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Effect of Kapitza contact and consideration of tube-end transport on the effective conductivity in nanotube-based composites

Tungyang Chen

^a兲

Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan George J. Weng

Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, New Jersey 08903

Wen-Ching Liu

Department of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan

共Received 15 July 2004; accepted 21 February 2005; published online 4 May 2005兲

Recent studies reported that the theoretical predictions of the effective thermal conductivity of nanotube-based composites by conventional micromechanical models are anomalously higher than those measured experimentally and suggested that the contact resistance on the interface could be the contributing factor to the lower measured value. We explore theoretically whether the large disagreement could be attributed to the effect of Kapitza contact resistance. Our simulations show that the thermal contact resistance on the lateral surfaces of the nanotubes could not be a major factor of this marked disparity. By contrast, the heat transport mechanisms at the ends of the nanotubes could be a significant factor to influence the value. We propose a few simple models to simulate the thermal conductivity at the ends of the nanotubes, similar to two springs in serial and/or in parallel. Under the propositions, we find that the experimental data can be better predicted than the conventional theory and that the tube-end transport is, in general, poor. © 2005 American Institute of Physics.

关DOI: 10.1063/1.1896094兴

I. INTRODUCTION

Carbon nanotubes have drawn enormous amount of ba- sic research because of the unique properties enabled by their nanoscale structure.

^1–3

Particularly, carbon nanotubes have unusually high thermal conductivities, axial Young’s modu- lus, and high strength. Among various applications, the de- velopment of nanocomposites based on carbon nanotubes is a promising direction in nanotechnology. For example, car- bon nanotubes have been embedded into polymers, epoxy resins, or silicone elastomer to get materials with good elec- tric and thermal transport properties.

^4–8

Recent experiments revealed that the effective thermal conductivity of carbon nanotube composites can be dramatically enhanced even with a very small amount

共less than 1 vol %兲 of nanotubes.^7,8

As such, it would be desirable to have a simple theoretical model to characterize the thermal transport behavior of car- bon nanotube composites. In a recent Letter, Choi et al.

⁷

studied the effective conductivity of a nanotube-fluid suspen- sion. They found experimentally that at 1 vol % fraction of nanotubes the effective conductivity of the system can be dramatically enhanced, k

^*

/ k

₁

= 2.6. With the thermal conduc- tivity of nanotubes k

₂

at 2000 W / m K and that of the oil

共the

matrix

兲 k1

at 0.1448 W / m K, the conventional theoretical predictions

^9–13

using spherical inclusions give the ratio k

^*

/ k

₁

= 1.03. This means that the measured data is much greater than that given by the theoretical models using spherical inclusions. Since nanotubes are not spherical and they are randomly oriented in the matrix Nan et al.

¹⁴

later

incorporated the effect of the aspect ratio of the nanotubes and their random orientations, and found that a conventional effective medium theory similar to Maxwell–Garnett ap- proximation could lead to a higher estimate and give reason- able agreement with the experiment if one adopts the con- ductivity ratio of k

₂

/ k

₁

= 500. However, if we adopted the phase conductivity data suggested by Choi et al.,

⁷

namely k

₂

/ k

₁

= 13800, the theoretical prediction

¹⁴

now gives k

^*

/ k

_m

= 47.041, a factor of

⬃18 larger than the value measured

experimentally. As remarked by Nan et al.

¹⁴

and others,

^15,16

the marked disparity between the predictions and the experi- mental observation may be due to the interfacial contact re- sistance between the matrix and the nanotubes, as the pres- ence of nanotube/matrix interfacial thermal resistance may result in an energy loss and subsequently lead to a degrada- tion in the effective thermal conductivity. The objective of this work is to assess theoretically whether the effect of ther- mal contact resistance is a major source of this large discrep- ancy and, if not, to propose a model that can better predict the effective conductivity of this high-contrast problem.

II. THE KAPITZA CONTACT ON THE LATERAL SURFACE

To develop such a model, we have performed a theoret- ical analysis which incorporates the mechanism of Kapitza thermal contact resistance

¹⁷

along the interfaces of the phases. We assume that both phases

共the nanotubes and the

matrix

兲 obey the Fourier’s law of heat conduction, in which

the thermal conductivities of the matrix

共phase 1兲 and the

nanotube

共phase 2兲 are represented as k1

and k

₂

respectively.

a兲Electronic mail: [email protected]

JOURNAL OF APPLIED PHYSICS 97, 104312共2005兲

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The mechanism of Kapitza thermal contact resistance as- sumes that the normal component of the heat flux is continu- ous across the interfaces, while the temperature T undergoes a discontinuity which is proportional to the normal compo- nent of the heat flux through an interface parameter

␤

k

₁⳵

^T

1

⳵

ⁿ ^{= k}

²

⳵

^T

2

⳵

ⁿ ⁼

兩␤共T1

− T

₂兲兩_⌫

^,

共1兲

where n is the unit normal on the interface

⌫ defined by the

inclusion to the matrix. Specifically, the thermal contact re- sistance can be constructed by a limiting process by assum- ing a thin interphase between the two phases with low conductivity.

¹⁸

The constant

␤

is non-negative and has the dimension of conductivity divided by length. The case of

␤

→0 is usually referred to as the “adiabatic” boundary con- dition, while

␤

→⬁ represents the perfect bonding condition.

We now consider a nanotube-based composite with carbon nanotubes randomly dispersed in a matrix. In analyzing the thermal transport behavior, an effective medium theory based on dilute approximation was employed to determine the ef- fective thermal conductivity. This theory, which simulates the composite as a single inclusion embedded in an infinite medium, is well suited in modeling the nanotube-based com- posite, particularly when the volume concentration of the inclusion phase is very small. To proceed, let us introduce a thermal intensity influence tensor T, which is defined as the average of the thermal intensity of the nanotubes versus the uniform intensity H

⁰_j

applied on the boundary. Also, let us define a jump factor J in the following

1 V

₂冕_⌫^共T²

^{− T}

¹^兲nⁱ

^ds

^{⬅ J}^ij

^H

⁰^j

^.

^共2兲

The geometry of the nanotube is simulated as a prolate spheroid characterized by an aspect ratio. As in microme- chanical estimates, the corresponding dilute influence func- tion can be written in terms of some elementary functions for a prolate spheroid.

¹⁹

The effective thermal conductivity k

^*

of the composite, with thermal contact resistance on the inter- faces, can then be derived in the form

²⁰

k

^*

= k

₁

+ c

具共k2

− k

₁兲T典 − c具k1

J

典, 共3兲

where c

共⬅V2

/ V

兲 is the volume fraction of the nanotubes

and the angular bracket

具·典 stands for an average for all ori-

entations of the nanotubes. Typically, nanotubes

⁷

have a mean diameter of

⬃25 nm and a length of ⬃50␮

^{m, which} is nearly equal to an aspect ratio of 2000. Now, upon substi- tution of formulas of T and J into

共3兲, the effective conduc-

tivity of the nanotube-based composite can be derived as

k

^*

= k

₁

+ c

冉

^3a

␤共1 + k

^8a

^␤2

/k ^k

¹₁兲 + 6k2

− 5 3 k

₁

+ 1

3 k

₂冊

^,

^共4兲

where the constant a denotes the radius of the nanotube. To check the obtained formula, we first reduce the formula

共4兲

to the perfect bonding situation by letting

␤

→⬁ and k

₂Ⰷk1

. Interestingly, this exactly recovers Eq.

共9兲 of

Nan et al.

¹⁴

k

^*

k

₁

= 1 + ck

₂

3k

₁

.

共5兲

We mention that if one assumes that the inclusion is of spherical shape, the present dilute approximation gives k

^*

/ k

₁

= 1 + 3c, which also exactly agrees with the Maxwell–

Garnett formula

¹⁴

and also agrees with Maxwell,

⁹

Jeffrey,

¹⁰

Hamilton and Crosser,

¹¹

Davis,

¹²

and Lu and Lin,

¹³

for k

₂ Ⰷk1

to the order of O共c兲.

Back to

共4兲, to examine the effect of thermal contact

resistance we have made numerical calculations for k

^*

for various

␤

ranging from

⬁ 共perfect bonding兲 to zero 共debond-

ing

兲. To our surprise, the effect of thermal contact resistance

is not significant. Specifically, both the cases of

␤

^{= 0 and}

␤

→⬁ yield the same value 共5兲, but the value k

^*

varies in between with no more than 6% for different values of

␤

^and for different ratios of k

₂

/ k

₁

. In particular, we notice that for a spheroidal tube with an aspect ratio greater than 100 the estimated conductivity

共3兲 is nearly the same with that of a

system consisting of infinitely long nanotubes. This, of course, is not realistic as the nanotubes are, in fact, sur- rounded by the matrix, though the nanotubes have very large aspect ratios. In real situations, heat could actually transfer from the ends of the nanotubes into the matrix, and con- versely. In fact, under the micromechanical models the effec- tive thermal conductivity in the tube direction for an aligned composite

共see Fig. 1, case 1兲, designated as k储

, is estimated to be nearly

k

_储

=

共1 − c兲k1

+ ck

₂⬅ kV

,

共6兲

where k

_V

also stands for Voigt’s estimate. Note that for k

₂ Ⰷk1

, k

_储

= ck

₂

, thus the behavior of k

_储

is similar to that of two springs in parallel, while in the transverse direction it is ap- proximate to k

_⬜

= k

₁共1+2c兲. By taking the orientational av-

erages over all possible directions of k

_储

and k

_⬜

,

共5兲 can be

reconstructed. The value of k

_储

suggests that along the tube direction the heat transport is transmitted perfectly between the tube ends and the matrix; i.e., the thermal intensities in the nanotubes and in the matrix are the same. We conjecture that the large discrepancy between the theoretical predictions and the experimental data is mainly due to the fact that the value of k

_储

is overestimated in the conventional microme- chanical models. These observations also reflect that the marked disparity is not due to the thermal contact resistance

FIG. 1. A schematic illustration of the conduction mechanism at the ends of nanotubes. Case 1 is close to that simulated by micromechanical models;

case 2 is the proposed opposite extremity which is similar to two springs in serial.

104312-2 Chen, Weng, and Liu J. Appl. Phys. 97, 104312共2005兲

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共i.e., the␤

^effect

兲 on the lateral interface but could likely be

attributed to the transport energy loss on the tube ends that has not been accounted for in the conventional microme- chanical models.

III. PROPOSITIONS FOR THE TUBE-END TRANSPORT A more realistic modeling in simulating the transport behavior of the tube ends is to consider the nanotube as a cylindrical inclusion with a finite length in which the tube ends are covered by hemisphere caps, and then examine the

␤

effect on the tube ends. But to solve such a boundary-value problem, it would have to call for some complex mathemati- cal formulation. For the simplicity of studying the transport characteristics of the tube ends and to provide a simple jus- tification to support our conjecture, we now assume that the conduction behavior along the tube direction is similar to two springs in serial; that is, the heat flux along the tube direction is the same in the nanotubes as well as in the matrix

共see Fig. 1, case 2兲. This is the opposite extremity in contrast

to that simulated by the micromechanical models

共Fig. 1,

case 1

兲. Under this assumption, the effective thermal conduc-

tivity along the tube direction for an aligned composite

共Fig.

1, case 2

兲 is approximated as

k

_储

=

关共1 − c兲k1

−1

+ ck

₂⁻¹兴⁻¹⬅ kR

,

共7兲

where k

_R

also stands for Reuss’s estimate. This tube-end con- ductivity

共7兲 allows us to determine the 33 components of

the thermal intensity influence tensor T

₃₃

as

T

₃₃

= k

₁

ck

₁

+

共1 − c兲k2

.

共8兲

While in the transverse direction, the components of T and J are still extracted from the conventional micromechanical approach

²⁰

as those for k

^*

. Upon substitution of these back

共3兲, carrying out the orientational average, we can derive the

averaged effective conductivity as

k

^∧

=

冉

^{1 −} ⁴ ³ ^c

冊

^k

¹

^{+ c}

冉

^3a

␤共1 + k

^8a

^␤2

/k ^k

²₁兲 + 6k2

+ 1 3

k

₁共k2

− k

₁兲

ck

₁

+

共1 − c兲k2冊

^.

^共9兲

Here, in distinction with the conventional micromechanical estimate k

^*

,

共4兲, we have designated this value as k^∧

. Note that for both k

^*

and k

^∧

, the Kapitza contact was not consid- ered in the tube ends but only on the lateral surface. This is due to the reasoning that the micromechanical models treat a high-aspect-ratio—say, larger than 100—spheroidal inclu- sion as an infinite long cylinder, and thus the

␤

effect could not be incorporated in the formulation. To unravel the impli- cations of

共4兲 and 共9兲, we have made numerical calculations

for k

^*

and k

^∧

for various

␤

ranging from

⬁ 共perfect bonding兲

to zero

共debonding兲. As anticipated, the values of k^*

and k

^∧

do differ a lot for each

␤

value, while, by contrast, the effect of thermal contact resistance on the lateral surface of the nanotubes is rather minor. That is, for different

␤

, the values of either k

^*

or k

^∧

do not vary much

共within 6%兲 for different

ratios of k

₂

/ k

₁

. Since both k

^*

and k

^∧

are relatively insensitive

to

␤

, we simplify

共4兲 and 共9兲 by letting ␤

→⬁ for conve- nience of application and making use of the fact that k

₂ Ⰷk1

. Under the simplications,

共4兲 will exactly recover Eq.

共9兲 of Nan et al.,¹⁴

while

共9兲 is simply given by

k

^*

k

₁

= 1 + ck

₂

3k

₁

, k

^∧

k

₁⬟ 1 +

5 3 c.

共10兲

We note that we did not claim that the estimate of k

^∧

is a better prediction nor a better lower bound, but rather that the estimate of k

^∧

simply serves here as a justification of our conjecture that the real tube-end transport phenomena is not perfect and is, in fact, rather poor.

Since the conventional micromechanical estimate for the axial conductivity along the tube ends k

_储

is analogous to that of Voigt’s estimate, while the present series model for k

_储

is like Reuss’s assumption in that direction, this suggests that the real effective behavior must lie somewhere between these two predictions. To further explore the significance of k

_储

, we now follow a recent idea proposed by Chiang and Weng

²¹

in their examination of Hill’s

²²

theory of composite elasticity and simulate the value of k

_储

by the arithmetic mean, the geometric mean, and the reciprocal mean based on the Voigt’s and Reuss’s estimates of k

₁

and k

₂

, i.e.,

k

_储^A

= 1

2

共kV

+ k

_R兲, k储G

=

冑

k

_V

k

_R

, k

_储^Re

= 2

冉

^k ¹

V

+ 1

k

_R冊⁻¹

^.

^共11兲

As for k

^∧

, the components of T and J in the transverse di- rection, counting the effect of

␤

, are still extracted from the conventional micromechanical approach and the effective conductivity is derived by incorporating the orientational av- erages, while the component of T

₃₃

follows the step as in

共8兲.

For clarity, we designate the resulting effective conductivity of nanotube-based composites based on these three different mean values for k

_储

, as k

_A

, k

_G

, and k

_Re

, respectively. The ex- plicit forms for k

_A

, k

_G

, and k

_Re

can be derived as

k

_A

= k

₁

+ c

共k2

− k

₁兲冉

² ³ ^a

␤共1 + k

^2a

2

/k

^␤₁兲 + 2k2

+ 1 3 A

冊

^{+ D,}

k

_G

= k

₁

+ c

共k2

− k

₁兲冉

² ³ ^a

␤共1 + k

^2a

2

/k

^␤₁兲 + 2k2

+ 1

3 B

冊

^{+ D,}

^共12兲

k

_Re

= k

₁

+ c

共k2

− k

₁兲冉

² ³ ^a

␤共1 + k

^2a

2

/k

^␤₁兲 + 2k2

+ 1 3 C

冊

^{+ D,}

where the constants A, B, C, and D are defined as

A = 1 2

ck

₁

+ k

₁

+ k

₂

− ck

₂

k

₂

− ck

₂

+ ck

₁

,

B =

冑^{共共1 − c兲k}¹

^{+ ck}

²^兲共1 − c兲k

^k

¹

^k

2²

+ ck

₁

− k

₁

c

共k2

− k

₁兲

,

共13兲

C =

2

冉共1 − c兲k

¹

1

+ ck

₂

+

共1 − c兲k2

+ ck

₁

k

₁

k

₂ 冊⁻¹

^{− k}

¹

c

共k2

− k

₁兲

,

(4)

D = 2

3 c

₂

k

₁冉

^a

␤

^2a

共1 + k^␤^{共1 + k}2

/k

₁²兲 + 2k

^/k

¹^兲 2

− 2

冊

^.

The variations for these predictions are demonstrated in Fig. 2 which shows a comparison for

共10兲 and 共12兲 and the

real experimental data from Choi et al.

⁷

on nanotube suspen- sions in which the phase conductivities is suggested as k

₂

/ k

₁

= 13800. Again, we found that the

␤

effect is rather minor and, as expected, the experimental data

⁷

are indeed bounded between the values of k

^*

and k

^∧

. Particularly, we observed that the reciprocal model k

_Re

is closer to the experi- mental data than the others and could possibly serve as a good estimate. This suggests that the tube-end transport is, in general, poor, which confirms our earlier conjecture, and thus the conventional micromechanical model as suggested by Nan et al.

¹⁴

could give an unexpected high estimate.

IV. CONCLUDING REMARKS

In summary, conventional micromechanical estimates k

^*

predict higher thermal conductivity in nanotube-based com- posite than what is actually measured in experiments. We have demonstrated that the thermal contact resistance along the lateral surfaces of the nanotubes could not be a major factor in this. By contrast, depending on the thermal transfer mechanisms at the ends of the nanotubes, the effective con- ductivity could vary significantly. In this work we simulated the conduction behavior at the ends of the nanotubes by a two-spring model in serial. A simple formula for the effective

conductivity k

^∧

based on this proposition has been derived. It turns out that the experimental data fall into the range be- tween k

^∧

and k

^*

but are much closer to k

^∧

. The present study suggests that conventional models may still work in nanos- caled composites, but the conduction phenomena around the tube caps

共ends兲 need to be properly simulated. The present

modified micromechanical approach offers a simple method to characterize the tube-end transport behavior qualitatively and to explain the much lower experimental data as com- pared to the estimate of the conventional micromechanical models. Of course, the proposed modifications still do not have a rigorous physical basis, but our analyses strongly point to the implications that the tube-end transport is poor.

We hope that the present study may interest further investi- gations on the likely transport mechanisms on the tube caps experimentally, or from the standpoint of atomic or molecu- lar dynamic simulation.

ACKNOWLEDGMENTS

T.C. was supported by the National Science Council, Taiwan, under NSC Contract No. 93-2211-E006-005. G.J.W.

was supported by the National Science Foundation under CMS Grant No. 01-14801.

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FIG. 2. Numerical calculations for k^*, k^∧, k_A, k_G, and k_Re in which the thermal conductivity of the nanotubes k2is taken as 2000 W / m K and that of the oil共the matrix兲 k1as 0.1448 W / m K. The dots indicate experimental data recorded from Ref. 7.

Effect of Kapitza contact and consideration of tube-end transport on the effective conductivity in nanotube-based composites