Analytical Solution of Vibration Analysis on Fixed-Free Single-Walled Carbon Nanotube-Based Mass Sensor

Author(s)
Thin-Lin Horng

ABSTRACT

Fixed-free single-walled carbon nanotubes (SWCNTs) have attracted a lot of interest in recent years due to their suitability for a wide range of applications, such as field emission and vacuum microelectronic devices, nanosensors, and nanoactuators. Based on a cantilever beam-bending model with a rigid mass at the free end and mode analysis, an analytical solution is developed in the present study to deal with the resonant frequency and mode shapes of a SWCNT- based mass sensor. The resonant frequency shift and mode shape of the fixed-free SWCNTs caused by the addition of a nanoscale particle to the beam tip are examined in order to explore the suitability of SWCNTs as a mass detector device. The simulation results reveal that the volume of the added particle has little effect on the first resonant frequency. In contrast, the second resonant frequency decreases with increasing the volume of the added particle. Furthermore, the resonant frequency shift of the first mode is very obvious for the amount of added mass, and the second resonant frequency decreases rapidly with increasing volume of added particle. Therefore, the first and second resonant frequencies can be used in the measurement of the mass of added particle and its volume, respectively.

Fixed-free single-walled carbon nanotubes (SWCNTs) have attracted a lot of interest in recent years due to their suitability for a wide range of applications, such as field emission and vacuum microelectronic devices, nanosensors, and nanoactuators. Based on a cantilever beam-bending model with a rigid mass at the free end and mode analysis, an analytical solution is developed in the present study to deal with the resonant frequency and mode shapes of a SWCNT- based mass sensor. The resonant frequency shift and mode shape of the fixed-free SWCNTs caused by the addition of a nanoscale particle to the beam tip are examined in order to explore the suitability of SWCNTs as a mass detector device. The simulation results reveal that the volume of the added particle has little effect on the first resonant frequency. In contrast, the second resonant frequency decreases with increasing the volume of the added particle. Furthermore, the resonant frequency shift of the first mode is very obvious for the amount of added mass, and the second resonant frequency decreases rapidly with increasing volume of added particle. Therefore, the first and second resonant frequencies can be used in the measurement of the mass of added particle and its volume, respectively.

Cite this paper

T. Horng, "Analytical Solution of Vibration Analysis on Fixed-Free Single-Walled Carbon Nanotube-Based Mass Sensor,"*Journal of Surface Engineered Materials and Advanced Technology*, Vol. 2 No. 1, 2012, pp. 47-52. doi: 10.4236/jsemat.2012.21009.

T. Horng, "Analytical Solution of Vibration Analysis on Fixed-Free Single-Walled Carbon Nanotube-Based Mass Sensor,"

References

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[8] H. J. Qi, K. B. K. Teo, K. K. S. Lau, M. C. Boyce, W. I. Milne, J. Roberson and K. K. Gleason, “Determination of Mechanical Properties of Carbon Nanotubes and Vertically Aligned Carbon Nanotube Forests Using Nanoindentation,” Journal of the Mechanics and Physics of Solids, Vol. 51, No. 11-12, 2003, pp. 2213-2237. doi:10.1016/j.jmps.2003.09.015

[9] G. W. Wang, Y. P. Zhao and G. T. Yang, “The Stability of a Vertical Single-Walled Carbon Nanotube under Its Own Weight,” Materials & Design, Vol. 25, No. 6, 2004, pp. 453-457. doi:10.1016/j.matdes.2004.01.003

[10] R. B. Chen, C. H. Lee, C. P. Chang and M. F. Lin, “Electronic and Optical Properties of Finite Carbon Nanotubes in an Electric Field,” Nanotechnology, Vol. 18, No. 7, 2007, Article ID 075704. doi:10.1088/0957-4484/18/7/075704

[11] J. C. Hsu, R. P. Chang and W. J. Chang, “Resonance Frequency of Chiral Sin-gle-Walled Carbon Nanotubes Using Timoshenko Beam Theory,” Physics Letters A, Vol. 372, No. 16, 2008, pp. 2757-2759. doi:10.1016/j.physleta.2008.01.007

[12] R. Mateiu, Z. J. Davis, D. N. Madsen, K. M?lhave, P. B?ggild, A. M. Rassmusen, M. Brorson, C. J. H. Jacobsen and A. Boisen, “An Approach to a Multi-Walled Carbon Nanotube-Based Mass Sensor,” Micro-electronic Engineering, Vol. 73-74, 2004, pp. 670-674. doi:10.1016/S0167-9317(04)00181-9

[13] D. H. Wu, W. T. Chien, C. S. Chen and H. H. Chen, “Resonant Frequency Analysis of Fixed-Free Single- Walled Carbon Nanotube-Based Mass Sensor,” Sensors and Actuators A: Physical, Vol. 126, No. 1, 2006, pp. 117-121. doi:10.1016/j.sna.2005.10.005

[14] E. Saether, S. J. V. Frankland and R. B. Pipes, “Transverse Me-chanical Properties of Single-Walled Carbon Nanotube Crystals. Part I. Determination of Elastic Moduli,” Composites Science and Technology, Vol. 63, No. 11, 2003, pp. 1543-1550. doi:10.1016/S0266-3538(03)00056-3

[15] T.-L. Horng, “Ana-lytical Solution of Flexural Vibration Responses on Nanoscale Processing Using Atomic Force Microscopy,” Journal of Mate-rials Processing Technology, Vol. 209, No. 6, 2009, pp. 2940-2945. doi:10.1016/j.jmatprotec.2008.06.059

[16] T.-L. Horng, “Analyses of Vibration Responses on Nanoscale Processing in a Liquid Using Tapping-Mode Atomic Force Microscopy,” Applied Surface Science, Vol. 256, No. 1, 2009, pp. 311-317. doi:10.1016/j.apsusc.2009.08.021

[17] K. Sohlberg, B. G. Sumpter, R. E. Tuzun and D. W. Noid, “Continuum Methods of Mechanics as a Simplified Approach to Structural Engineering of Nanostructures,” Nanotechnology, Vol. 9, No. 1, 1998, pp. 30-36. doi:10.1088/0957-4484/9/1/004

[18] Y. J. Liu and X. L. Chen, “Continuum Models of Carbon Nanotube-Based Composites Using the Boundary Element Method,” Electronic Journal of Boundary Elements, Vol. 1, No. 2, 2003, pp. 316-335.

[19] S. P. Timoshenko and J. M. Gere, “Theory of Elastic Stability,” McGraw-Hill, New York, 1961.

[1] A. Bianco, K. Kostarelos and M. Prato, “Applications of Carbon Nanotubes in Drug Delivery,” Current Opinion in Chemical Biology, Vol. 9, No. 6, 2005, pp. 674-679. doi:10.1016/j.cbpa.2005.10.005

[2] R. S. Ruoff and D. C. Lorents, “Mechanical and Thermal Properties of Carbon Nano-tubes,” Carbon, Vol. 33, No. 7, 1995, pp. 925-930. doi:10.1016/0008-6223(95)00021-5

[3] D. Srivastava, M. Menon and K. Cho, “Computational Nanotechnology with Carbon Nanotubes and Fullerenes,” Computing in Science and Engineering, Vol. 3, No. 4, 2001, pp. 42-55. doi:10.1109/5992.931903

[4] R. Saito, G. Dresselhaus and M. S. Dresselhaus, “Physical Properties of Carbon Nanotubes,” Imperial College, London, 1998. doi:10.1142/9781860943799

[5] P. J. F. Harris, “Carbon Nanotubes and Related Structures,” Cambridge University Press, Cambridge, 1999. doi:10.1017/CBO9780511605819

[6] H. Dai, J. H. Hafner, A. G. Rinzler, D. T. Colbert and R. E. Smalley, “Nanotubes as Nanoprobes in Scanning Probe Microscopy,” Letters to Nature, Vol. 384, 1996, pp. 147-150. doi:10.1038/384147a0

[7] P. Poncharal, Z. L. Wang, D. Ugarte and W. A. D. Heer, “Elec-trostatic Deflections and Electro-Mechanical Resonances of Carbon Nanotubes,” Science, Vol. 283, No. 5407, 1999, pp. 1513-1516. doi:10.1126/science.283.5407.1513

[8] H. J. Qi, K. B. K. Teo, K. K. S. Lau, M. C. Boyce, W. I. Milne, J. Roberson and K. K. Gleason, “Determination of Mechanical Properties of Carbon Nanotubes and Vertically Aligned Carbon Nanotube Forests Using Nanoindentation,” Journal of the Mechanics and Physics of Solids, Vol. 51, No. 11-12, 2003, pp. 2213-2237. doi:10.1016/j.jmps.2003.09.015

[9] G. W. Wang, Y. P. Zhao and G. T. Yang, “The Stability of a Vertical Single-Walled Carbon Nanotube under Its Own Weight,” Materials & Design, Vol. 25, No. 6, 2004, pp. 453-457. doi:10.1016/j.matdes.2004.01.003

[10] R. B. Chen, C. H. Lee, C. P. Chang and M. F. Lin, “Electronic and Optical Properties of Finite Carbon Nanotubes in an Electric Field,” Nanotechnology, Vol. 18, No. 7, 2007, Article ID 075704. doi:10.1088/0957-4484/18/7/075704

[11] J. C. Hsu, R. P. Chang and W. J. Chang, “Resonance Frequency of Chiral Sin-gle-Walled Carbon Nanotubes Using Timoshenko Beam Theory,” Physics Letters A, Vol. 372, No. 16, 2008, pp. 2757-2759. doi:10.1016/j.physleta.2008.01.007

[12] R. Mateiu, Z. J. Davis, D. N. Madsen, K. M?lhave, P. B?ggild, A. M. Rassmusen, M. Brorson, C. J. H. Jacobsen and A. Boisen, “An Approach to a Multi-Walled Carbon Nanotube-Based Mass Sensor,” Micro-electronic Engineering, Vol. 73-74, 2004, pp. 670-674. doi:10.1016/S0167-9317(04)00181-9

[13] D. H. Wu, W. T. Chien, C. S. Chen and H. H. Chen, “Resonant Frequency Analysis of Fixed-Free Single- Walled Carbon Nanotube-Based Mass Sensor,” Sensors and Actuators A: Physical, Vol. 126, No. 1, 2006, pp. 117-121. doi:10.1016/j.sna.2005.10.005

[14] E. Saether, S. J. V. Frankland and R. B. Pipes, “Transverse Me-chanical Properties of Single-Walled Carbon Nanotube Crystals. Part I. Determination of Elastic Moduli,” Composites Science and Technology, Vol. 63, No. 11, 2003, pp. 1543-1550. doi:10.1016/S0266-3538(03)00056-3

[15] T.-L. Horng, “Ana-lytical Solution of Flexural Vibration Responses on Nanoscale Processing Using Atomic Force Microscopy,” Journal of Mate-rials Processing Technology, Vol. 209, No. 6, 2009, pp. 2940-2945. doi:10.1016/j.jmatprotec.2008.06.059

[16] T.-L. Horng, “Analyses of Vibration Responses on Nanoscale Processing in a Liquid Using Tapping-Mode Atomic Force Microscopy,” Applied Surface Science, Vol. 256, No. 1, 2009, pp. 311-317. doi:10.1016/j.apsusc.2009.08.021

[17] K. Sohlberg, B. G. Sumpter, R. E. Tuzun and D. W. Noid, “Continuum Methods of Mechanics as a Simplified Approach to Structural Engineering of Nanostructures,” Nanotechnology, Vol. 9, No. 1, 1998, pp. 30-36. doi:10.1088/0957-4484/9/1/004

[18] Y. J. Liu and X. L. Chen, “Continuum Models of Carbon Nanotube-Based Composites Using the Boundary Element Method,” Electronic Journal of Boundary Elements, Vol. 1, No. 2, 2003, pp. 316-335.

[19] S. P. Timoshenko and J. M. Gere, “Theory of Elastic Stability,” McGraw-Hill, New York, 1961.