## Gravitation-dependent, thermally-induced self-diffraction in carbon nanotube solutions

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

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

## 2. Experiment: methods and results

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]

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]

*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]

*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, Δ

*T*(

*a*), is zero, and

*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.

*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 J

_{0}(

*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

^{-1}K

^{-1}. This value is slightly greater than the thermal conductivity for toluene (0.13 Wm

^{-1}K

^{-1}, Ref. [15

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

*that is*, the nanotube number, N, depends exponentially on the vertical height. This can be described by the Boltzmann distribution law,

_{tube}is the mass of the CNTs,

*g*is the gravitational constant,

*k*is the Boltzmann constant, N

_{B}_{0}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).

*x*for the relative change in the concentration, and we have ΔN/N

_{0}=-M

_{tube}

*g*Δ

*x*/(

*k*). This results in a variation in the absorption coefficient as Δ

_{B}T*α*

_{0}/

*α*

_{0}=ΔN/N

_{0}=-M

_{tube}

*g*Δ

*x*/(

*kBT*), which leads to a change in the temperature rise Δ

*T*=-M

_{g}_{tube}

*g*Δ

*x*Δ

*T*/(

*kBT*), where Δ

*T*is the temperature rise at the center. Interestingly, it has an opposite sign to Δ

*T*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.

_{tube}to be ~8×10

^{-16}g (or 1×10

^{7}

*Dalton*) if

*Δx*=ω=80 µm,

*λ*=800 nm,

*L*=1 mm,

*T*=293 K, and

*ΔT*=5 K (calculated by Eq. (2) and Eq. (3) for a laser power of 9 mW). For M

_{tube}<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.

*N*and

_{i}*σ*are the number and absorption cross section of the tubes that have a mass of

_{i}*M*, respectively. By substituting the Boltzmann-Maxwell law, Eq. (4) becomes

_{i}*N*(0) is the number of the tubes at

_{i}*x*=0. Thus, the heat flow equation has the form of :

*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

*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, M

_{tube}. It is important to note that all parameters involved in Eqs. (7) and (8) are either known or measurable except for M

_{tube}, 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 1.3×10

^{8}

*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

## Acknowledgments

## References and Links

1. | R. F. Service, “Superstrong nanotubes show they are smart, too,” Science |

2. | M. S. Dresslhaus, G. Dresslhuas, and P. C. Eklund, |

3. | M. Endo, S. Iijima, and M. S. Dresselhaus, |

4. | T. W. Ebbesen, |

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. |

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. |

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. |

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. |

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 |

10. | P. G. de Gennes and J. Prost, |

11. | I. C. Khoo and S. T. Wu, |

12. | S. Brugioni and R. Meucci, “Thermally induced nonlinear optical effects in an isotropic liquid crystal at 10.6 µm,” Appl. Opt. |

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. |

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. |

15. | H. El-Kashef, “Thermo-optical and dielectric constants of laser dye solvents,” Rev. Sci. Instrum. |

16. | M. S. Dresselhaus, G. Dresselhaus, and Ph. Avouris, Eds, |

17. | R. A. Serway, |

**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

- R. F. Service, "Superstrong nanotubes show they are smart, too," Science 281, 940-942 (1998). [CrossRef]
- M. S. Dresslhaus, G. Dresslhuas, and P. C. Eklund, Science of Fullerenes and Carbon Nanotubes, (Academic, New York, 1996).
- M. Endo, S. Iijima, and M. S. Dresselhaus, Carbon Nanotubes, (Pergamon, Oxford, 1996).
- T. W. Ebbesen, Carbon Nanotubes: Preparation and Properties, (CRC, Boca Raton, FL, 1997).
- 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]
- 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]
- 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]
- 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]
- 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]
- P. G. de Gennes and J. Prost, The Physics of Liquid Crystals, 2nd Edition, (Oxford Univ Press, Oxford, 1995).
- I. C. Khoo and S. T. Wu, Optics and Nonlinear Optics of Liquid Crystals, (World Scientific, Singapore, 1993).
- 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]
- 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]
- 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]
- H. El-Kashef, "Thermo-optical and dielectric constants of laser dye solvents," Rev. Sci. Instrum. 69, 1243-1245 (1998). [CrossRef]
- 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).
- R. A. Serway, Physics for Scientists and Engineers with Modern Physics, 4th Edition, (Saunders College Publishing, Philadelphia, 1996).

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