4.4 Calculation of Young’s Modulus
Enomoto et al. [36] have performed a combined transmission electron microscopy
(TEM) and Raman spectroscopy study in order to measure the apparent Young’s
modulus of different CNT samples synthesized using various methods. An
empirical relationship between the apparent Young’s modulus and the crystallinity
of the different samples was reported there. For this, the authors used the parameter,
I D /I G obtained from the spectroscopic data. The values of the experimentally
measured apparent Young’s modulus of different CNT samples using TEM were
plotted versus the I D /I G values obtained from the Raman spectra. A nonlinear fitting
was done to determine the relation between the Young’s modulus and I D /I G values.
As a result, for the apparent Young’s modulus E a , the empirical relation,
E a = 2.7 Â 10
4 (I D /I G )
−1.6 MPa with the crystallinity factor I D /I G was obtained. It is
obvious from the above equation that the apparent Young’s modulus of the CNTs is
inversely depending on the order of crystallinity.
For the present analysis, the I D /I G values of the SWCNT and MWCNT bundles
were calculated using the integrated areas under the D (I D ) and G-bands (I G ) found
in the Raman spectra recorded at 532 nm excitation wavelength. The I D /I G values
determined for the CarboLex and Aldrich SWCNT varieties and MWCNTs are
0.085, 0.097 and 0.34, respectively. Thus, the value of I D /I G is much smaller for the
SWCNTs compared to the MWCNTs. This finding points to an increased number
of defects in the MWCNTs. Hence, the SWCNTs with the smaller value of I D /I G
have a high crystallinity sp
2 network and, therefore, this structure also is the better
model system for a calculation of the Young’s modulus of CNTs theoretically.
Yakobson et al. [47] performed such a theoretical study using realistic many-body
potentials for the calculation. These authors suggested that the CNTs are resilient,
sustaining at extreme strain with no signs of brittleness, plasticity or atomic rearrangement. They predicted a theoretical value of the Young’s modulus, E a = 5.5
TPa, for properly chosen tube parameters. In our case, the values of the apparent
Young’s modulus calculated using the expression, E a = 2.7 Â 10
4 (I D /I G )
−1.6 MPa,
for SWCNTs (CarboLex), SWCNT (Aldrich) and MWCNTs turn out to be 1.39,
1.13 and 0.15 TPa, respectively. The order of magnitude of these values is in good
agreement with the theoretically calculated value given above [47].
Quite recently, a purity and defect density characterization of SWCNTs based on
the I G /I D ratio (inverse to our application) obtained from their Raman spectra was
investigated by Miyata et al. [48], which underlines the complexity of spectral
interpretation. In addition, different nanocarbon contributions to MWCNTs were
used by DiLeo et al. [49] for a calibration experiment aiming at a purity assessment
based on the I D /I G ratio of the corresponding Raman bands as a complement to
electron microscopy. Probably, several analytical methods are required for a unique
material characterization.
134
A. K. Ojha and H. M. Heise
Enomoto et al. [36] have performed a combined transmission electron microscopy
(TEM) and Raman spectroscopy study in order to measure the apparent Young’s
modulus of different CNT samples synthesized using various methods. An
empirical relationship between the apparent Young’s modulus and the crystallinity
of the different samples was reported there. For this, the authors used the parameter,
I D /I G obtained from the spectroscopic data. The values of the experimentally
measured apparent Young’s modulus of different CNT samples using TEM were
plotted versus the I D /I G values obtained from the Raman spectra. A nonlinear fitting
was done to determine the relation between the Young’s modulus and I D /I G values.
As a result, for the apparent Young’s modulus E a , the empirical relation,
E a = 2.7 Â 10
4 (I D /I G )
−1.6 MPa with the crystallinity factor I D /I G was obtained. It is
obvious from the above equation that the apparent Young’s modulus of the CNTs is
inversely depending on the order of crystallinity.
For the present analysis, the I D /I G values of the SWCNT and MWCNT bundles
were calculated using the integrated areas under the D (I D ) and G-bands (I G ) found
in the Raman spectra recorded at 532 nm excitation wavelength. The I D /I G values
determined for the CarboLex and Aldrich SWCNT varieties and MWCNTs are
0.085, 0.097 and 0.34, respectively. Thus, the value of I D /I G is much smaller for the
SWCNTs compared to the MWCNTs. This finding points to an increased number
of defects in the MWCNTs. Hence, the SWCNTs with the smaller value of I D /I G
have a high crystallinity sp
2 network and, therefore, this structure also is the better
model system for a calculation of the Young’s modulus of CNTs theoretically.
Yakobson et al. [47] performed such a theoretical study using realistic many-body
potentials for the calculation. These authors suggested that the CNTs are resilient,
sustaining at extreme strain with no signs of brittleness, plasticity or atomic rearrangement. They predicted a theoretical value of the Young’s modulus, E a = 5.5
TPa, for properly chosen tube parameters. In our case, the values of the apparent
Young’s modulus calculated using the expression, E a = 2.7 Â 10
4 (I D /I G )
−1.6 MPa,
for SWCNTs (CarboLex), SWCNT (Aldrich) and MWCNTs turn out to be 1.39,
1.13 and 0.15 TPa, respectively. The order of magnitude of these values is in good
agreement with the theoretically calculated value given above [47].
Quite recently, a purity and defect density characterization of SWCNTs based on
the I G /I D ratio (inverse to our application) obtained from their Raman spectra was
investigated by Miyata et al. [48], which underlines the complexity of spectral
interpretation. In addition, different nanocarbon contributions to MWCNTs were
used by DiLeo et al. [49] for a calibration experiment aiming at a purity assessment
based on the I D /I G ratio of the corresponding Raman bands as a complement to
electron microscopy. Probably, several analytical methods are required for a unique
material characterization.
134
A. K. Ojha and H. M. Heise