Page 155 - Carbon Nanotubes

Mechanical and thermal properties of carbon nanotubes 145

Fig. lb. HRTEM of a bent tube (not the same as la) showing the strain in the region of the kinks, in-
cluding a stress fracture; note the compression of the layers at the kinks and their expansion in the regions
between kinks.

ing SWNTs. Alignment of tubes in a composite matrix cantilevered SWNT of length 1 pm with the bending
caused by slicing of the matrix has indicated that the force constant. The fundamental vibrational fre-
thinner MW tubes are also quite flexible[l3]. quency in this case is about 12 MHz, in a range that
Considering a nanotube to be a graphite cylinder is easily observable by electrical methods. This range
means that the extremely high elastic constant of in- suggests a possible means of measuring the mechani-
plane graphite (C,, = 1060 GPa) can be used as the cal properties when individual isolated tubes are
Young’s modulus for calculating both the elastic bend- cantilever-mounted to a larger body and can be readily
ing and the extension of NTs. Thus, one can use the manipulated.
standard beam deflection formula[ 141 to calculate the The mechanical properties of the NTs have not as
bending of a tube under an applied force. For exam- yet been experimentally studied because the difficulty
ple, the deflection of a cantilever beam of length I with of getting pure samples free of amorphous, graphitic,
a force f exerted at its free end is given by and polyhedral carbon particles and the need to char-
acterize the tubules (e.g., their size and number of lay-
d = f13/(3EI) (2) ers). However, rapid progress is being made on the
production, purification, and isolation of nanotubes
where E is the Young’s modulus and I is the areal mo- so that it is likely that some definitive measurements
ment of inertia of the cross-section of the tube about will appear in the near future. Recent demonstrations
its central axis, I = n(r; - rf)/4. For a typical of alignment of nanotubes using polymer matrices are
10-layer MWNT with an inner diameter of 3 nm, an showing promise as a method for alignment and sep-
outer diameter of 6.5 nm, and length of 1 pm, the de- aration and may provide a means to investigate the
flection would be 2.3 nm/pdyne. This calculation as- mechanical properties of individual, as well as assem-
sumes that the 10 SWNTs that make up this MWNT blies of, SWNTs and MWNTs[13,16].
act as a single, uniform, homogeneous medium. Work on the production and oxidation of SWNT
Overney et al. [ 151 calculated the rigidity of short samples at SRI and other laboratories has led to the
SW tubes using ab initio local density calculations to observation of very long bundles of these tubes, as can
determine the parameters in a Keating potential. The be seen in Fig. 2. In the cleanup and removal of the
Young’s modulus resulting from this calculation is amorphous carbon in the original sample, the SWNTs
about 1500 GPa, which is in very good agreement with self-assemble into aligned cable structures due to van
the continuum value of 1060 GPa. Again, it appears der Waals forces. These structures are akin to the SW
that use of the continuum model of MWNTs and nanotube crystals discussed by Tersoff and Ruoff;
SWNTs based on the properties of the graphene sheet they show that van der Waals forces can flatten tubes
is well justified. It is important to recognize that in cal- of diameter larger than 2.5 nm into a hexagonal cross-
culating the moment of inertia of a single walled tube, sectional lattice or honeycomb structure[ 171.
one must consider the wall thickness of the tube to be Since most SWNTs have diameters in the range of
0.34 nm (i.e., the normal graphite layer separation). 1-2 nm, we can expect them to remain cylindrical when
Thus, a typical 1 pm long single wall tube with a di- they form cables. The stiffness constant of the cable
ameter of 1.1 nm will deflect 16 nmhdyne; indeed, structures will then be the sum of the stiffness con-
SWNTs are much more flexible than the thicker stants of the SWNTs. However, just as with MWNTs,
MWNTs, an observation that is well documented by the van der Waals binding between the tubes limits ten-
the TEM photos of these tubes. sile strength unless the ends of all the tubes can be
One can calculate the vibrational frequency of a fused to a load. In the case of bending, a more exact