Page 153 - Carbon Nanotubes

MECHANICAL AND THERMAL PROPERTIES
OF CARBON NANOTUBES

RODNEY S. RUOFF and DONALD C. LORENTS
Molecular Physics Laboratory, SRI International, Menlo Park, CA 94025, U.S.A.
(Received 10 January 1995; accepted 10 February 1995)
Abstract-This chapter discusses some aspects of the mechanical and thermal properties of carbon nano-
tubes. The tensile and bending stiffness constants of ideal multi-walled and single-walled carbon nano-
tubes are derived in terms of the known elastic properties of graphite. Tensile strengths are estimated by
scaling the 20 GPa tensile strength of Bacon’s graphite whiskers. The natural resonance (fundamental vi-
brational frequency) of a cantilevered single-wall nanotube of length 1 micron is shown to be about 12 MHz.
It is suggested that the thermal expansion of carbon nanotubes will be essentially isotropic, which can be
contrasted with the strongly anisotropic expansion in “conventional” (large diameter) carbon fibers and
in graphite. In contrast, the thermal conductivity may be highly anisotropic and (along the long axis) per-
haps higher than any other material. A short discussion of topological constraints to surface chemistry
in idealized multi-walled nanotubes is presented, and the importance of a strong interface between nano-
tube and matrix for formation of high strength nanotube-reinforced composites is highlighted.
Key Words-Nanotubes, mechanical properties, thermal properties, fiber-reinforced composites, stiffness
constant, natural resonance.

1. INTRODUCTION 2. MECHANICAL PROPERTIES
The discovery of multi-walled carbon nanotubes 2.1 Tensile strength and yield strength
(MWNTs), with their nearly perfect cylindrical struc- Tersoff[4] has argued convincingly that the elastic
ture of seamless graphite, together with the equally re- properties of the graphene sheet can be used to pre-
markable high aspect ratio single-walled nanotubes dict the stain energy of fullerenes and nanotubes. In-
(SWNTs) has led to intense interest in these remark-
able structures[l]. Work is progressing rapidly on the deed, the elastic strain energy that results from simple
production and isolation of pure bulk quantities of calculations based on continuum elastic deformation
of a planar sheet compares very favorably with the
both MWNT and SWNT, which will soon enable their more sophisticated ab initio results. The result has
mechanical, thermal, and electrical properties to be been confirmed by ab initio calculations of Mintmire
measured[2]. Until that happens, we can speculate et al. [6] This suggests that the mechanical properties
about the properties of these unique one-dimensional of nanotubes can be predicted with some confidence
carbon structures. A preview of the mechanical prop- from the known properties of single crystal graphite.
erties that might be expected from such structures was
We consider the case of defect-free nanotubes, both
established in the 1960s by Bacon[3], who grew car- single-walled and multi-walled (SWNT and MWNT).
bon fibers with a scroll structure that had nearly the The stiffness constant for a SWNT can be calculated
tensile mechanical properties expected from ideal in a straightforward way by using the elastic moduli
graphene sheets. of graphite[7] because the mechanical properties of
The mechanical and thermal properties of nanotubes single-crystal graphite are well understood. To good
(NTs) have not yet been measured, mainly because of approximation, the in-plane elastic modulus of graph-
the difficulties of obtaining pure homogeneous and ite, C1 1, which is 1060 GPa, gives directly the on-axis
uniform samples of tubes. As a result we must rely, Young’s modulus for a homogeneous SWNT. To ob-
for the moment, on ab initio calculations or on con- tain the stiffness constant, one must scale the Young’s
tinuum calculations based on the known properties of modulus with the cross-sectional area of the tube,
graphite. Fortunately, several theoretical investigations which gives the scaling relation
already indicate that the classical continuum theory
applied to nanotubes is quite reliable for predicting the
mechanical and some thermal properties of these
tubes[4,5]. Of course, care must be taken in using such
approximations in the limit of very small tubes or where A,, is the cross-sectional area of the nanotube,
when quantum effects are likely to be important. The and A is the cross-sectional area of the hole. Because
fact that both MWNTs and SWNTs are simple single we derive the tensile stiffness constant from the ma-
or multilayered cylinders of graphene sheets gives con- terial properties of graphite, each cylinder has a wall
fidence that the in-plane properties of the graphene thickness equivalent to that of a single graphene sheet
sheet can be used to predict thermal and mechanical in graphite, namely, 0.34 nm. We can, thus, use this
properties of these tubes. relationship to calculate the tensile stiffness of a SWNT,
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