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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 20 — Oct. 2, 2006
  • pp: 8958–8966
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Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions

Wei Ji, Weizhe Chen, Sanhua Lim, Jianyi Lin, and Zhixin Guo  »View Author Affiliations


Optics Express, Vol. 14, Issue 20, pp. 8958-8966 (2006)
http://dx.doi.org/10.1364/OE.14.008958


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Abstract

We report the observation of thermally-induced self-diffraction in carbon nanotube (CNT) solutions under the influence of the gravity. We present a theoretical model in which CNTs are assumed to obey the Boltzmman distribution law. Under the approximations of small temperature rise and a very narrow distribution of CNT masses, the model simulation is consistent with the data measured at low laser powers. An immediate application of such a gravitation-dependent characteristic is the optical measurement for molecular weights of CNTs.

© 2006 Optical Society of America

1. Introduction

Recent years have witnessed an explosion of research into carbon nanotubes (CNTs) because of their unique nanostructures with remarkable mechanical, electronic, and optical properties that could lead to new applications in materials and devices. For example, CNTs are applicable to high-strength yet light-weight composite materials, electron field emission, hydrogen storage, and optical limiting [1

1. R. F. Service, “Superstrong nanotubes show they are smart, too,” Science 281, 940–942 (1998). [CrossRef]

9

9. J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, “Optical limiting properties of suspended and solubilized carbon nanotubes,” J. Phys. Chem. B 104, 7071–7076 (2000). [CrossRef]

]. To realize such applications, however, a number of outstanding problems have to be resolved. One of them is the requirement of fast analysis methods for industrial quality control applications. Transmission electron microscopy (TEM) or scanning electron microscopy (SEM) may be employed to characterize tube lengths, diameters, and other structural details for individual nanotubes. Yet, no investigation has been carried out for measuring the molecular weight and weight distribution of large-quantity CNTs. The molecular weight of CNTs is a fundamental characteristic which has a profound implication in many applications. Here we report, for the first time, the observation of gravitation-dependent, thermally-induced self-diffraction in CNT solutions. And we also present a demonstration how the observed nonlinear-optical effect can be exploited for directly determining the molecular weights of CNTs.

2. Experiment: methods and results

Obviously, finite-size CNTs are variable-length rods, which by virtue of shape and size can act as liquid crystals. Liquid crystals exhibit a variety of nonlinear-optical effects [10

10. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, (Oxford Univ Press, Oxford, 1995).

,11

11. I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).

]. One of them is self-diffraction due to thermal lensing, resulting from heat dissipation of absorbed light radiation to the surrounding medium [11

11. I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).

,12

12. S. Brugioni and R. Meucci, “Thermally induced nonlinear optical effects in an isotropic liquid crystal at 10.6 µm,” Appl. Opt. 41, 7627–7630 (2002). [CrossRef]

]. Therefore, we anticipate that CNTs possess similar behavior. And it has been confirmed by our experiment described as follows. We employed a continuous-wave laser beam from a double-frequency Nd:YAG (532 nm), He-Ne (632 nm), or Ti:Sapphire (780 nm) laser. The laser beam was focused onto a CNT solution. The CNTs used were recently developed octadecylamine (ODA)-modified multi-walled CNTs with an average length of a few microns [13

13. Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, “Large-scale preparation of solubilized carbon nanotubes,” Chem. Mater. 15, 3256–3260 (2003). [CrossRef]

]. These CNTs had high solubility (up to 13 g/L in toluene). In our experiment, the CNTs were dissolved in toluene and the solutions were contained in 1-mm-thick quartz cells. The ODA-modified CNT concentrations were in the range from 0.02 to 0.08 g/L (or 0.0014% to 0.006% in volume fraction). The CNT solution was placed at the focal point, as illustrated in Fig. 1(a). The experiment was conducted at room temperature (~295 K).

When the laser beam was normal incident onto the solution, we observed the spatial transverse variation of the transmitted irradiance at far field. If incident laser powers were low, nearly Guassian spatial profiles were recorded, as expected. As the incident laser power rose, however, the profile developed to a number of concentric rings, a typical diffraction pattern. The number of rings was increased as the incident laser power was increased. Figure 1(c) and Fig. 1(e) display the two photographs taken at 532 nm and 780 nm, respectively, with the set-up shown in Fig. 1(a). This phenomenon is similar to the one reported for liquid crystals [11

11. I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).

,12

12. S. Brugioni and R. Meucci, “Thermally induced nonlinear optical effects in an isotropic liquid crystal at 10.6 µm,” Appl. Opt. 41, 7627–7630 (2002). [CrossRef]

]. We attribute it to thermally-induced self-diffraction (or self-defocusing). Due to absorption of the laser energy by the CNTs and dissipation of the heat, there is a temperature rise in the solvent. The temperature rise is determined by the heat flow equation. Under the steady-state condition, the heat flow equation has the form of

k2T=Aexp(2ρ2ω2)
(1)

where T is the temperature, k is the thermal conductivity, ρ is the radial distance from the center of symmetry, ω is the laser beam waist, and A is a constant proportional to the absorption coefficient of the CNT solution or the input laser power. Gordon et al [14

14. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965). [CrossRef]

] have derived an expression for the temperature rise, shown as follows:

ΔT(ρ)=0.25α0Pk[En(2ρ2ω2)En(2a2ω2)2ln(ρa)]
(2)

where P is the input laser power, α 0 is the absorption coefficient of the CNT solution, a is the radial position at which the temperature rise, ΔT(a), is zero, and En(x)=xetdtt.

The temperature rise in the solvent induces a change in the refractive index by Δn(ρ)=dndT, with dndT being the thermo-optical coefficient of the solvent. As a result, different parts in the cross-section of the laser beam experience different phases, Δψ(ρ)=2πλΔn(ρ)L, where L is the sample thickness, and λ is the wavelength. Thus it gives rise to self-phase modulation. If the phase difference between the two points on the observation screen is mπ, m being an even or odd integer, constructive or destructive interference occurs, respectively, resulting in the appearance of diffraction pattern.

By the virtue of the Huygens principle under the cylindrical symmetry, one can calculate the electric field of the transmitted beam by

E(ρ')=E(0)0exp[i2πλΔn(ρ)Lρ2w2]J0(2πρρ'λz)ρdρ,
(3)

where ρ is the radial distance on the observation screen which is at a distance of z from the focal point, E(0) is the field at ρ =0, and J0(x) is the zeroth-order Bessel function. By employing both Eq. (2) and Eq. (3), we numerically simulate the irradiance distribution. Figure 2 shows excellent agreements if k=0.18 Wm-1K-1. This value is slightly greater than the thermal conductivity for toluene (0.13 Wm-1K-1, Ref. [15

15. H. El-Kashef, “Thermo-optical and dielectric constants of laser dye solvents,” Rev. Sci. Instrum. 69, 1243–1245 (1998). [CrossRef]

]). The larger thermal conduction in the toluene solution of CNTs is expected since multi-walled CNTs are known to have metallic properties and excellent thermal conductivity with their longitudinal thermal conductivity even exceeding the highest measured in-plane thermal conductivity of graphite [16

16. M. S. Dresselhaus, G. Dresselhaus, and Ph. Avouris, Eds, Carbon Nanotubes: Synthesis, Structure, Properties and Applications, Topics in Applied Physics, vol 80 (Springer, New York, 2000).

]. The Z-scan measurement in the inset of Fig. 2(a) is a piece of evidence that shows a negative sign for the nonlinear refraction, consistent with the thermo-optical property of the solvent. Figure 3 illustrates the linear dependence of the diffraction pattern on the CNT concentration with no diffraction pattern for the pure solvent. This indicates that CNTs play a crucial role in the absorption of light radiation and the heat transfer to the solution.

The photographs presented in Fig. 1(c) and Fig. 1(e) show nearly perfect circular rings for the diffraction pattern when they are observed with the sample lying horizontally and the laser beam propagating vertically, as displayed in Fig. 1(a). The multi-walled CNTs in the solution sample are a few microns in length, and comprise of over ten millions of carbon atoms in each CNT, which can be regarded as a “supermolecule”. Thus, the gravitation effect on these “giant-molecule” tubes should not be negligible. When the sample lies horizontally and the laser light travels vertically, there is no variation in the CNT concentration along the transverse directions of the laser beam; and, hence, the gravitational effect does not manifest itself in the diffraction patterns. However, it is a different case when the sample is erected in the vertical direction and the laser beam transverses across the sample in the horizontal direction, as shown in Fig. 1(b). Indeed, we have observed distorted diffraction patterns as shown in Fig. 1(d) and Fig. 1(f). The diffraction rings appear to be compressed in the top half of the rings and stretched in the bottom half.

Fig. 1. Gravitation-dependant, thermally-induced self-diffraction in carbon nanotube solutions. (a) and (b) Schematic diagrams showing two experimental set-ups. (c) and (d) Diffraction patterns recorded at 532 nm with the set-ups shown in (a) and (b), respectively. (e) and (f) Diffraction patterns observed at 780 nm with the set-ups shown in (a) and (b), respectively. The input laser powers used are ~100 mW.
Fig. 2. Far-field distribution of the transmitted irradiance measured at 780 nm with various laser powers. The squares, triangles, circles, and stars are the experimental data obtained from the set-up in Fig. 1(a). The linear transmittance of the carbon nanotube solution is 85.2%. The half angle is defined as the ratio of the radial distance, ρ , on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (2) and (3) described in the text. The inset in A shows a Z-scan measured at 780 nm with an aperture in front of the detector.

To quantitatively describe the gravitational effect on the diffraction, we need a mathematical expression for the distribution of CNTs suspended or dissolved in a liquid under the influence of the gravity. We speculate that CNT solutions are similar to the Brownian suspension, in which the Brownian particles obey Maxwell-Boltzmann distribution, like ideal gas molecules. For an ideal gas in the gravitational field, the molecules distribute exponentially proportional to their altitude [17

17. R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th Edition, (Saunders College Publishing, Philadelphia, 1996).

]. Therefore, it is postulated that the CNTs arrange themselves in a similar way, that is, the nanotube number, N, depends exponentially on the vertical height. This can be described by the Boltzmann distribution law, N=N0exp(MtubegxkBT), where Mtube is the mass of the CNTs, g is the gravitational constant, kB is the Boltzmann constant, N0 is the CNT concentration at x=0, and the x-variable denotes the vertical distance from the center of symmetry for the laser beam. Because of this vertical gradient in the CNT concentration, there is the x-dependence of the diffraction. Thus, the diffraction pattern is distorted from cylindrically symmetrical rings, which is confirmed by our observation in Fig. 1(d) and Fig. 1(f).

Fig. 3. Far-field distribution of the transmitted irradiance recorded at 780 nm and an incident power of 9 mW with various carbon nanotube concentrations. The squares, triangles, circles, and stars are the experimental data obtained from the set-up in Fig. 1(a). The half angle is defined as the ratio of the radial distance, ρ , on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (2) and (3) described in the text.
Fig. 4. Far-field distribution of the transmitted irradiance measured at 780 nm with various laser powers. The squares, triangles, circles, and stars are the experimental data obtained from the set-up shown in Fig. 1(b). The transmittance of the carbon nanotube solution is 85.2%. The half angle is defined as the ratio of the x’-coordinate on the observation screen to the distance of z. The solid lines are the numerical simulations using Eqs. (7) and (8) with an approximation of very narrow weight distribution described in the text.

To estimate the change in the nonlinear refraction due to the gravity, we make the first-order derivative over x for the relative change in the concentration, and we have ΔN/N0=-Mtube g Δx/(kBT). This results in a variation in the absorption coefficient as Δα 0/α 0=ΔN/N0=-Mtube g Δx/(kBT), which leads to a change in the temperature rise ΔTg=-Mtube g Δx ΔT/(kBT), where ΔT is the temperature rise at the center. Interestingly, it has an opposite sign to ΔT for the top half of the diffraction rings, and the same sign for the bottom half, explaining why the top half is compressed (or it offset the temperature rise caused by the transverse variation of the laser irradiance) and the bottom half is stretched (or enhanced), respectively.

If the above temperature difference is responsible for the two adjacent rings, its phase difference, Δψ=2πλdndTΔTgL, should be π. Thus we estimate the minimum measurable Mtube to be ~8×10-16 g (or 1×107 Dalton) if Δx=ω=80 µm, λ=800 nm, L=1 mm, L=1mm,dndT=5.3×104K1, T=293 K, and ΔT=5 K (calculated by Eq. (2) and Eq. (3) for a laser power of 9 mW). For Mtube <8×10-16 g, no gravitational effect on the diffraction is seen. However, the CNT mass is computed to be 30×10-16 g, if we assume that the tube length is 10 µm, the radius of the inner tube is 10 nm, the spacing between the two adjacent concentric tubes is 0.34 nm and there are ten walls in the multi-walled CNT. Therefore, the gravitation effect is significantly pronounced.

To obtain a more rigorous solution, we have to take the distribution of tube masses into account for the absorption coefficient. For a dilute solution or suspension, its absorption coefficient may be described by

α=iσiNi
(4)

where Ni and σi are the number and absorption cross section of the tubes that have a mass of Mi, respectively. By substituting the Boltzmann-Maxwell law, Eq. (4) becomes

α=iσiNi(0)exp(MigxkBT)
(5)

where Ni(0) is the number of the tubes at x=0. Thus, the heat flow equation has the form of :

k2T=exp[2(x2+y2)ω2]iAiexp(MigxkBT)
(6)

Eq. (6) is a nonlinear equation. For small values of the temperature rise (ΔT<5 K) or low laser powers (a few mW), we take the first-order approximation by substituting a constant background temperature, T 0, into the right-hand side of Eq. (6). Then, the solution has the form of

ΔT=i0.25qiσiNi(0)Pk[En(2ρi2ω2)En(2a2ω2)2ln(ρi2a)]
(7)

with ρi=(x+Mtgω22kBT0)2+y2 and qi=exp(Mi2g2ω28kB2T02) . Combining Eq. (7) with the following Huygens principle,

E(x',y')=E(0,0)exp(i2πλdndTΔTLx2+y2ω2)exp[i2πλz(xx'+yy')]dxdy,
(8)

where x and y are the two Cartesian coordinates on the observation screen, we numerically compute the profiles of the transmitted beam along the x -axis, |E(x ,0)|2. To simplify the numerical calculation, we assume that all the CNTs have a very narrow distribution of masses, representing by a single, average mass, Mtube. It is important to note that all parameters involved in Eqs. (7) and (8) are either known or measurable except for Mtube, which has been taken as an adjustable one in the computer simulation. Figure 4 shows good theoretical fits with the molecular mass being 8×10-15 g, or 1.3×108 Dalton, for input laser powers below 10 mW. The discrepancy is found for input laser powers greater than 10 mW. It is not surprising because the first-order approximation taken in solving Eq. (6) becomes invalid for higher temperature rises. In addition, we neglect the molecular weight distribution of the CNTs by assuming that all the CNTs have an average mass.

3. Conclusion

In conclusion, we have presented the observation of thermally-induced self-diffraction in CNT solutions. We have also described a thermally-induced, self-defocusing theory, which is in excellent agreement with the experimental data. We have found that the observed self-diffraction depends on the gravitation because of heavy molecular weights of CNTs. We have proposed a theoretical model in which CNTs are assumed to distribute themselves exponentially along the height. Under the approximations of small temperature rise and a narrow distribution of CNT masses, the model is consistent with the measurements at low laser powers. Such a gravitation-dependent characteristic opens up a new avenue to investigate giant molecular weights.

Acknowledgments

This work is supported by the Special FY2003 Academic Research Fund for the Materials Science and Engineering Initiatives Project, National University of Singapore (R-114-000-110-112).

References and Links

1.

R. F. Service, “Superstrong nanotubes show they are smart, too,” Science 281, 940–942 (1998). [CrossRef]

2.

M. S. Dresslhaus, G. Dresslhuas, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes, (Academic, New York, 1996).

3.

M. Endo, S. Iijima, and M. S. Dresselhaus, Carbon Nanotubes, (Pergamon, Oxford, 1996).

4.

T. W. Ebbesen, Carbon Nanotubes: Preparation and Properties, (CRC, Boca Raton, FL, 1997).

5.

X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, “Broadband optical limiting with multiwalled carbon nanotubes,” Appl. Phys. Lett. 73, 3632–3634 (1998). [CrossRef]

6.

P. Chen, X. Wu, X. Sun, J. Lin, W Ji, and K. L. Tan, “Electronic structure and optical limiting behavior of carbon nanotubes,” Phys. Rev. Lett. 82, 2548–2551 (1999). [CrossRef]

7.

L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, “Single-wall carbon nanotubes for optical limiting,” Chem. Phys. Lett. 307, 317–319 (1999). [CrossRef]

8.

S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, “Optical limiting in single-walled carbon nanotube suspensions,” Chem. Phys. Lett. 317, 510–514 (2000). [CrossRef]

9.

J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, “Optical limiting properties of suspended and solubilized carbon nanotubes,” J. Phys. Chem. B 104, 7071–7076 (2000). [CrossRef]

10.

P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, (Oxford Univ Press, Oxford, 1995).

11.

I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).

12.

S. Brugioni and R. Meucci, “Thermally induced nonlinear optical effects in an isotropic liquid crystal at 10.6 µm,” Appl. Opt. 41, 7627–7630 (2002). [CrossRef]

13.

Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, “Large-scale preparation of solubilized carbon nanotubes,” Chem. Mater. 15, 3256–3260 (2003). [CrossRef]

14.

J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, “Long-Transient Effects in Lasers with Inserted Liquid Samples,” J. Appl. Phys. 36, 3–8 (1965). [CrossRef]

15.

H. El-Kashef, “Thermo-optical and dielectric constants of laser dye solvents,” Rev. Sci. Instrum. 69, 1243–1245 (1998). [CrossRef]

16.

M. S. Dresselhaus, G. Dresselhaus, and Ph. Avouris, Eds, Carbon Nanotubes: Synthesis, Structure, Properties and Applications, Topics in Applied Physics, vol 80 (Springer, New York, 2000).

17.

R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th Edition, (Saunders College Publishing, Philadelphia, 1996).

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(120.4640) Instrumentation, measurement, and metrology : Optical instruments
(160.4760) Materials : Optical properties

ToC Category:
Diffraction and Gratings

History
Original Manuscript: April 6, 2006
Revised Manuscript: June 28, 2006
Manuscript Accepted: July 12, 2006
Published: October 2, 2006

Citation
Wei Ji, Weizhe Chen, Sanhua Lim, Jianyi Lin, and Zhixin Guo, "Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions," Opt. Express 14, 8958-8966 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-20-8958


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References

  1. R. F. Service, "Superstrong nanotubes show they are smart, too," Science 281, 940-942 (1998). [CrossRef]
  2. M. S. Dresslhaus, G. Dresslhuas, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes, (Academic, New York, 1996).
  3. M. Endo, S. Iijima, and M. S. Dresselhaus, Carbon Nanotubes, (Pergamon, Oxford, 1996).
  4. T. W. Ebbesen, Carbon Nanotubes: Preparation and Properties, (CRC, Boca Raton, FL, 1997).
  5. X. Sun, R. Q. Yu, G. Q. Xu, T. S. A. Hor, and W. Ji, "Broadband optical limiting with multiwalled carbon nanotubes," Appl. Phys. Lett. 73, 3632-3634 (1998). [CrossRef]
  6. P. Chen, X. Wu, X. Sun, J. Lin, W. Ji, and K. L. Tan, "Electronic structure and optical limiting behavior of carbon nanotubes," Phys. Rev. Lett. 82, 2548-2551 (1999). [CrossRef]
  7. L. Vivien, E. Anglaret, D. Riehl, F. Bacou, C. Journet, C. Goze, M. Andrieux, M. Brunet, F. Lafonta, P. Bernier, and F. Hache, "Single-wall carbon nanotubes for optical limiting," Chem. Phys. Lett. 307, 317-319 (1999). [CrossRef]
  8. S. R. Mishra, H. S. Rawat, S. C. Mehendale, K. C. Rustagi, A. K. Sood, R. Bandyopadhyay, A. Govindaraj, and C. N. R. Rao, "Optical limiting in single-walled carbon nanotube suspensions," Chem. Phys. Lett. 317, 510-514 (2000). [CrossRef]
  9. J. E. Riggs, D. B. Walker, D. L. Carroll, and Y.-P Sun, "Optical limiting properties of suspended and solubilized carbon nanotubes," J. Phys. Chem. B 104, 7071-7076 (2000). [CrossRef]
  10. P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, (Oxford Univ Press, Oxford, 1995).
  11. I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).
  12. S. Brugioni and R. Meucci, "Thermally induced nonlinear optical effects in an isotropic liquid crystal at 10.6 µm," Appl. Opt. 41,7627-7630 (2002). [CrossRef]
  13. Y. J. Qin, L. Q. Liu, J. H. Shi, W. Wu, J. Zhang, Z. X. Guo, Y. F. Li, and D. B. Zhu, "Large-scale preparation of solubilized carbon nanotubes," Chem. Mater. 15, 3256-3260 (2003). [CrossRef]
  14. J. P. Gordon, R. C. C. Leite, R. S. Moore, S. P. S. Porto, and J. R. Whinnery, "Long-Transient Effects in Lasers with Inserted Liquid Samples," J. Appl. Phys. 36, 3-8 (1965). [CrossRef]
  15. H. El-Kashef, "Thermo-optical and dielectric constants of laser dye solvents," Rev. Sci. Instrum. 69, 1243-1245 (1998). [CrossRef]
  16. M. S. Dresselhaus, G. Dresselhaus, and Ph. Avouris, Eds, Carbon Nanotubes: Synthesis, Structure, Properties and Applications, Topics in Applied Physics, vol 80 (Springer, New York, 2000).
  17. R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th Edition, (Saunders College Publishing, Philadelphia, 1996).

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