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Investigation of Double Walled Carbon Nanotubes for Mass Sensing
Available online at www.sciencedirect.com
ScienceDirect Procedia Technology 14 (2014) 290 – 294
2nd International Conference on Innovations in Automation and Mechatronics Engineering, ICIAME 2014
Investigation of Double walled Carbon Nanotubes for Mass Sensing Ajay M. Patela*, Anand Y. Joshib a
Associate Professor ,G. H. Patel College of Engineering & Technology, Vallabh Vidyanagar, Gujarat – 388120,India b Professor ,G. H. Patel College of Engineering & Technology, Vallabh Vidyanagar, Gujarat – 388120,India
Abstract This paper deals with studying the mass sensing characteristics of Double Walled carbon Nanotubes (DWCNT) using a continuum mechanics based approach. The interlayer separation in the form of Van der Waals interaction is modelled using characteristic spring element. The inner and outer walls of the nanotube are modelled as elastic beams with a spring element connecting the two layers. Two types of boundary conditions, i.e. cantilever and bridged are considered for the purpose of analysis. This analysis explores the resonant frequency shift of Double walled Carbon Nanotubes caused by the changes in the size in terms of length and masses. The results have shown that the dynamic characteristics are influenced by the change in length as well as masses attached to the centre and at the tip of DWCNT. The results also indicate that the mass sensing characteristics of DWCNT based nano balances can reach upto 0.1 Zeptogram. This investigation is further useful in the applications involving oscillators and sensors based on nano electromechanical devices vibrating at very high frequencies in the order of THz. © 2014 Elsevier The Authors. Published by Elsevier Ltd. under the CC BY-NC-ND license Ltd. This is an open access article and/or peer-review under responsibility of the Organizing Committee of ICIAME 2014. Selection (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of ICIAME 2014. Keywords: Double walled Carbon Nanotube, Finite Element Method, Resonators
1. Introduction Since controlled experiments at nanoscale are difficult and molecular dynamics simulations remain formidable for large scale systems, continuum elastic models have been widely used to study the mechanical behaviour of Single Walled and Multiwalled Carbon Nanotube (SWCNT and MWNT) [1–3]. ____________ *Ajay M. Patel. Tel.: +91-942-703-7011; fax:+0-269-223-6896. E-mail address:[email protected]
2212-0173 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of ICIAME 2014. doi:10.1016/j.protcy.2014.08.038
291
Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
Elastic continuum models have been widely used to study the vibrational behaviour of CNTs to avoid the difficulties encountered during experimental characterization of nanotubes [4].The resonant frequencies of fixed free and the bridged SWCNT have been investigated by Joshi et al. [5] the results showed the mass sensitivity of carbon nanotubes to different masses and different lengths. In general, the principle of mass detection using CNT-based sensor from a vibration analysis is based on the fact that the resonant frequency is sensitive to the attached mass [6]. Pradhan et al. [7] analyzed the vibration of orthotropic nanoplates by using nonlocal elasticity theory. To examine the small scale effect of nanoplate, they used nonlocal elasticity. Zhang et al. [8] represented a double elastic beam model that was developed to describe the transverse vibrations of double-walled carbon nanotubes (DWCNTs) under compressive axial load using Euler–Bernoulli beam theory. The vibration of DWCNT based mass sensor with inner and outer nanotubes of different lengths has been studied. Such DWCNTs with different wall lengths have been reported to have some specific practical applications [9]. Elishakoff et al. [10] studied the vibrations of double walled carbon nanotubes (DWCNTs) cantilever with attached bacterium on the tip. Shen et al. [11] that a DWCNT based mass sensor with a length ratio (L2/ L1) is more sensitive that a SWCNT with the length ratio equal to zero and with shorter outer tube DWCNT is more sensitive. Molecular structural mechanics approach has been used Joshi et al [12] for investigating the dynamic responses of chiral single walled carbon nanotubes based nano biosensors. An accurate analysis of the resonant frequency of DWCNTs with free-fixed end conditions was recommended by Natsuki et al. [13] considering both in-phase and anti-phase modes of DWCNTs. As a result of the fact that the classical continuum approach is scale-independent [14] the classical continuum theory is applied to analyze the mechanical behaviour of nanostructures such as CNTs. Patel and Joshi[15]analyzed vibrational characteristics of double walled carbon Nanotube (DWCNT) modelled using spring elements and lumped masses. The effect of waviness along the axis has been evaluated by subjecting the nanotube to different boundary conditions namely bridged, cantilever and simply supported and the vibration responses of straight and wavy DWCNTs are investigated by Patel and Joshi [16]. The current study focuses on evaluating the mass sensing characteristics of DWCNT using the dynamics of a DWCNT based resonators with different attached masses. Cantilever and bridged boundary conditions are used in DWCNT. The inner and outer walls of carbon nanotube are modeled as two individual elastic beams interacting each other by Van der Waals forces. To simulate the interlayer interactions and describe the Van der Waals potentials between carbon atoms on different layers appropriate spring elements are utilized. Simulations are carried out using a Finite Element Procedure. 2. Finite Element Procedure In this paper, for utilizing the finite element procedure potential energy is used to evaluate linear spring stiffness. The total force is the sum of the force generated by the electrons and the electrostatics force between positive charges. The general formula for the potential energy is
U U r Uθ Uφ Uω U vdw
(1)
Where, Ur is the energy due to bond stretch interaction, Uθ the energy due to bending (bond angle variation), UΦ the energy due to dihedral angle torsion, Uω the energy due to out-of plane torsion and Uvdw the energy due to nonbonded Vander Waals interaction. 2 1 2 k r ( r r0 ) k r ( r ) 2 2 1 2 1 2 Uθ kθ (θ θ 0 ) kθ ( θ ) 2 2 Ur
1
1 2 Uτ Uφ U ω kτ ( φ ) 2
(2) (3) (4)
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Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
Where kr, kθ, and kT are the bond stretching, bond bending and torsional resistance force constants, respectively, while Δr, Δθ and Δø represent bond stretching increment, bond angle variation and angle variation of bond twisting, respectively. The interlayer interaction is described by the van der Waals potential. The Lenard-Jones 6-12 potential is utilized to express the interaction of carbon atoms located on the different walls
ρ 12 ρ 6 U ( R ) 4ε , R R
(5)
Where R is the inter atomic distance and ε=3.8655×10-13 N nm and σ=0.34 nm, respectively [17]. In order to evaluate the vibration description of DWCNTs, equations are developed that describe the dynamic equilibrium of the Finite Element model. The element equation constructed by the global stiffness and mass matrices can be assembled. The dynamics equation hence becomes
c (u2 u1 ) K1u1 M1u1 c (u2 u1 ) K 2u2 M 2u2 Where K1 ,
(6) (7)
K 2 are stiffness matrices, M 1 , M 2 are mass matrices.
iωt iωt (8) u1 Y1e u2 Y2 e Where u1 and u2 are the deflection of inner and outer tube,Y1 and Y2 are the vibrational amplitude and ω is the vibrational frequency.
The elemental equation for global system become
K1 c c Y1 2 M1 0 Y1 c K c ωn 0 M 0 Y2 2 2 Y2
(9)
For the Finite Element Model of DWCNT, the Van der Waals force field between the interfacial layers is represented by a spring element COMBIN40, and attached mass is joined with outer nanotube using element SOLID186. The spring stiffness coefficient of Equations (2)-(4) are taken to be equal to kr =6.52×10-7 N nm-1,kθ = 8.76× 10-10 N nm rad-2 and
kτ = 2.78×10-10 N nm rad-2 [18]. To model the interlayer interactions and describe the
van der waals potentials between carbon atoms on different layers COMBIN40 linear spring element is utilized. 3. Result and Discussions The model that has been developed for the purpose of analysis is as under: Fig.1 shows fixed free DWCNTs and bridged DWCNTs having attached mass at the end of their outer wall and at the centre of the outer wall respectively with the variation in the mass from 10-2 zg to 104 zg. The inner diameter D1= 2R1 =0.7nm and outer tube diameter D2= 2R2 =1.4nm, where R1 and R2 are the inner and outer tube radius of the centre line. The DWCNT were assumed to have an elastic modulus of E= 1TPa (with Poisson ratio =0.25) and a density of = 2.3g/cm3. The effective thickness of DWCNT t=0.3nm. Fig.2 (a) & (b) display the finite element model and mode shape of bridged DWCNTs. Fig.3(a) shows the results of fundamental
Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
frequency of Bridged DWCNT Resonator vs. attached mass. It is observed from simulated results that resonant frequencies are reducing with increase in the attached mass. A clearly indication of mass sensing is found from a mass value of 1 zg for Length of double walled carbon nanotubes are 10 and 14 nm and 0.1 zg for a length of 7 nm. This shows that shorter length DWCNTs are more sensitive to mass change and better suited as a mass sensor. Also it can be specified here that as for a mass change of 0.1 zg, there is a change in frequency, hence the mass sensing capability of the device is 0.1 zg. Fig.3 (b) displays the results of fundamental frequency of Cantilever DWCNT vs. Mode shape for attached mass 10-2 to 104 Zg and without mass.
Figure 1. (a) Cantilevered (b) bridged DWCNTs with the variation of mass attach outer nanotubes
Figure 2. (a) Finite element model DWCNTs (b) Mode shape of bridged DWCNTs
Figure 3(a) Fundamental Frequency of Bridged DWCNT Resonator vs. attached mass (b) Fundamental frequency of Cantilever DWCNT Vs Mode shape for different values of mass .
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Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
4. Conclusion
Mass sensing ability of DWCNT has been checked using a continuum mechanics and Finite Element approach. It has been observed that mass sensing characteristics of DWCNTs can reach up to 0.1 zg. Shorter length DWCNTs are more sensitive to attached mass and hence more suitable for mass sensing. An increasing trend in the frequency is observed with increasing the number of modes for different values of attached masses. Resonant frequencies are reducing double walled carbon nano tube with increase attached mass (zg).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Goh, C. S., Wei, J., Lee, L. C., and Gupta, M., (2006) Simultaneous Enhancementin Strength and Ductility by Reinforcing Magnesium with Carbon Nano-tubes, Mater. Sci. Eng., A, 423, pp. 153–156. Ru, C. Q. (2004), Elastic Models for Carbon Nanotubes, Encyclopedia of Nanoscience and Nanotechnology, Vol. 2, H. S. Nalwa ed., American Scientific,Stevenson Ranch, CA, pp. 731–744. Zhang, Y. Q., Liu, G. R., and Xie, X. Y., (2005) Free Transverse Vibrations of Double-Walled Carbon Nanotubes Using a Theory of Nonlocal Elasticity, Phys. Rev. B, 71, p. 195404. Yoon J., Ru C.Q., Mioduchowski A., Comput. Sci. Technology. 63 (2003) 1533. Joshi A. Y., Harsha S. P., and Sharma, S. C., (2010), Vibration Signature Analysis of Single Walled Carbon Nanotube Based Nanomechanical Sensors, Physica E, 42, pp. 2115–2123. Li, C. Y., and Chou, T. W. (2004), Mass Detection Using Carbon Nanotube Based Nanomechanical Resonators, Appl. Phys. Lett. 84, pp. 5246–5248. Pradhan, S. C., Phadikar, J. K. and Karthik,G., Vibration analysis of multilayered orthotropic nanoplates using nonlocal elasticity theory, IE(I) Journal-MM(2009)90:16-23. Zhang Y., Liu G., Han X., Phys. Lett. A 340 (2005) 258. Natsuki T., Lei, X. W., Ni, Q. Q., and Endo, M., 2010, “Vibrational Analysis of Double-Walled Carbon Nanotubes with Inner and Outer Nanotubes of Different Lengths”, Phys. Lett. A, 374, pp. 4684–4689. Elishakoff, I., Versaci, C., and Muscolino, G., 2011, “Clamped-Free DoubleWalled Carbon Nanotube-Based Mass Sensor,” Acta Mech., 219(5), pp. 29–43. Zhi-bin shen, Bin Deng, Xian-Fang Li, Guo-Jin Tang, “Vibration of Double –walled Carbon Nanotube-Based Mass Sensor via Nonlocal Timoshenko Beam Theory” Journal of Nanotechnology in Engineering and Medicine by ASME, (2012),vol.2 Joshi A. Y., Satish C Sharma, S. P. Harsha, “Zeptogram scale mass sensing using single walled carbon nanotube based bio sensors”, Sensors & Actuators A. Physical,168, (2), 2011, 275-280. T. Natsuki, X.W. Lei, Q.Q. Ni, M. Endo, Phys. Lett. A 374 (2010) 2670. T. Murmu, S.C. Pradhan, Comput. Mater. Sci. 46 (2009) 854. Patel Ajay M., Joshi A.Y., 2013, “Vibration Analysis of Double Wall Carbon Nanotube based Resonators for Zeptogram level Mass Recognition”, Computational Materials Science, Elsevier Publishers, Volume 79, , Pp 230–238. Patel Ajay M., Joshi A.Y. 2014 “Effect of Waviness on the Dynamic Characteristics of Double Walled Carbon Nanotubes” Nanoscience Nano technology Letter, vol- 6, 1-9 Li C, Chou TW. Vibrational behaviors of multiwalled-carbon- nanotube-based nanomechanical resonators. Appl Phys Lett (2004) 84(1):121–123. Cornell W.D., Bally C.I., Gould I.R., Merz K.M. Jr., Feguson D.M., Spellmeyer D.C., Fox T., Caldwell J.W., Kollman P.A., (1995),Journal of American Chemical Society 1175179-5197.
ScienceDirect Procedia Technology 14 (2014) 290 – 294
2nd International Conference on Innovations in Automation and Mechatronics Engineering, ICIAME 2014
Investigation of Double walled Carbon Nanotubes for Mass Sensing Ajay M. Patela*, Anand Y. Joshib a
Associate Professor ,G. H. Patel College of Engineering & Technology, Vallabh Vidyanagar, Gujarat – 388120,India b Professor ,G. H. Patel College of Engineering & Technology, Vallabh Vidyanagar, Gujarat – 388120,India
Abstract This paper deals with studying the mass sensing characteristics of Double Walled carbon Nanotubes (DWCNT) using a continuum mechanics based approach. The interlayer separation in the form of Van der Waals interaction is modelled using characteristic spring element. The inner and outer walls of the nanotube are modelled as elastic beams with a spring element connecting the two layers. Two types of boundary conditions, i.e. cantilever and bridged are considered for the purpose of analysis. This analysis explores the resonant frequency shift of Double walled Carbon Nanotubes caused by the changes in the size in terms of length and masses. The results have shown that the dynamic characteristics are influenced by the change in length as well as masses attached to the centre and at the tip of DWCNT. The results also indicate that the mass sensing characteristics of DWCNT based nano balances can reach upto 0.1 Zeptogram. This investigation is further useful in the applications involving oscillators and sensors based on nano electromechanical devices vibrating at very high frequencies in the order of THz. © 2014 Elsevier The Authors. Published by Elsevier Ltd. under the CC BY-NC-ND license Ltd. This is an open access article and/or peer-review under responsibility of the Organizing Committee of ICIAME 2014. Selection (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of ICIAME 2014. Keywords: Double walled Carbon Nanotube, Finite Element Method, Resonators
1. Introduction Since controlled experiments at nanoscale are difficult and molecular dynamics simulations remain formidable for large scale systems, continuum elastic models have been widely used to study the mechanical behaviour of Single Walled and Multiwalled Carbon Nanotube (SWCNT and MWNT) [1–3]. ____________ *Ajay M. Patel. Tel.: +91-942-703-7011; fax:+0-269-223-6896. E-mail address:[email protected]
2212-0173 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of the Organizing Committee of ICIAME 2014. doi:10.1016/j.protcy.2014.08.038
291
Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
Elastic continuum models have been widely used to study the vibrational behaviour of CNTs to avoid the difficulties encountered during experimental characterization of nanotubes [4].The resonant frequencies of fixed free and the bridged SWCNT have been investigated by Joshi et al. [5] the results showed the mass sensitivity of carbon nanotubes to different masses and different lengths. In general, the principle of mass detection using CNT-based sensor from a vibration analysis is based on the fact that the resonant frequency is sensitive to the attached mass [6]. Pradhan et al. [7] analyzed the vibration of orthotropic nanoplates by using nonlocal elasticity theory. To examine the small scale effect of nanoplate, they used nonlocal elasticity. Zhang et al. [8] represented a double elastic beam model that was developed to describe the transverse vibrations of double-walled carbon nanotubes (DWCNTs) under compressive axial load using Euler–Bernoulli beam theory. The vibration of DWCNT based mass sensor with inner and outer nanotubes of different lengths has been studied. Such DWCNTs with different wall lengths have been reported to have some specific practical applications [9]. Elishakoff et al. [10] studied the vibrations of double walled carbon nanotubes (DWCNTs) cantilever with attached bacterium on the tip. Shen et al. [11] that a DWCNT based mass sensor with a length ratio (L2/ L1) is more sensitive that a SWCNT with the length ratio equal to zero and with shorter outer tube DWCNT is more sensitive. Molecular structural mechanics approach has been used Joshi et al [12] for investigating the dynamic responses of chiral single walled carbon nanotubes based nano biosensors. An accurate analysis of the resonant frequency of DWCNTs with free-fixed end conditions was recommended by Natsuki et al. [13] considering both in-phase and anti-phase modes of DWCNTs. As a result of the fact that the classical continuum approach is scale-independent [14] the classical continuum theory is applied to analyze the mechanical behaviour of nanostructures such as CNTs. Patel and Joshi[15]analyzed vibrational characteristics of double walled carbon Nanotube (DWCNT) modelled using spring elements and lumped masses. The effect of waviness along the axis has been evaluated by subjecting the nanotube to different boundary conditions namely bridged, cantilever and simply supported and the vibration responses of straight and wavy DWCNTs are investigated by Patel and Joshi [16]. The current study focuses on evaluating the mass sensing characteristics of DWCNT using the dynamics of a DWCNT based resonators with different attached masses. Cantilever and bridged boundary conditions are used in DWCNT. The inner and outer walls of carbon nanotube are modeled as two individual elastic beams interacting each other by Van der Waals forces. To simulate the interlayer interactions and describe the Van der Waals potentials between carbon atoms on different layers appropriate spring elements are utilized. Simulations are carried out using a Finite Element Procedure. 2. Finite Element Procedure In this paper, for utilizing the finite element procedure potential energy is used to evaluate linear spring stiffness. The total force is the sum of the force generated by the electrons and the electrostatics force between positive charges. The general formula for the potential energy is
U U r Uθ Uφ Uω U vdw
(1)
Where, Ur is the energy due to bond stretch interaction, Uθ the energy due to bending (bond angle variation), UΦ the energy due to dihedral angle torsion, Uω the energy due to out-of plane torsion and Uvdw the energy due to nonbonded Vander Waals interaction. 2 1 2 k r ( r r0 ) k r ( r ) 2 2 1 2 1 2 Uθ kθ (θ θ 0 ) kθ ( θ ) 2 2 Ur
1
1 2 Uτ Uφ U ω kτ ( φ ) 2
(2) (3) (4)
292
Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
Where kr, kθ, and kT are the bond stretching, bond bending and torsional resistance force constants, respectively, while Δr, Δθ and Δø represent bond stretching increment, bond angle variation and angle variation of bond twisting, respectively. The interlayer interaction is described by the van der Waals potential. The Lenard-Jones 6-12 potential is utilized to express the interaction of carbon atoms located on the different walls
ρ 12 ρ 6 U ( R ) 4ε , R R
(5)
Where R is the inter atomic distance and ε=3.8655×10-13 N nm and σ=0.34 nm, respectively [17]. In order to evaluate the vibration description of DWCNTs, equations are developed that describe the dynamic equilibrium of the Finite Element model. The element equation constructed by the global stiffness and mass matrices can be assembled. The dynamics equation hence becomes
c (u2 u1 ) K1u1 M1u1 c (u2 u1 ) K 2u2 M 2u2 Where K1 ,
(6) (7)
K 2 are stiffness matrices, M 1 , M 2 are mass matrices.
iωt iωt (8) u1 Y1e u2 Y2 e Where u1 and u2 are the deflection of inner and outer tube,Y1 and Y2 are the vibrational amplitude and ω is the vibrational frequency.
The elemental equation for global system become
K1 c c Y1 2 M1 0 Y1 c K c ωn 0 M 0 Y2 2 2 Y2
(9)
For the Finite Element Model of DWCNT, the Van der Waals force field between the interfacial layers is represented by a spring element COMBIN40, and attached mass is joined with outer nanotube using element SOLID186. The spring stiffness coefficient of Equations (2)-(4) are taken to be equal to kr =6.52×10-7 N nm-1,kθ = 8.76× 10-10 N nm rad-2 and
kτ = 2.78×10-10 N nm rad-2 [18]. To model the interlayer interactions and describe the
van der waals potentials between carbon atoms on different layers COMBIN40 linear spring element is utilized. 3. Result and Discussions The model that has been developed for the purpose of analysis is as under: Fig.1 shows fixed free DWCNTs and bridged DWCNTs having attached mass at the end of their outer wall and at the centre of the outer wall respectively with the variation in the mass from 10-2 zg to 104 zg. The inner diameter D1= 2R1 =0.7nm and outer tube diameter D2= 2R2 =1.4nm, where R1 and R2 are the inner and outer tube radius of the centre line. The DWCNT were assumed to have an elastic modulus of E= 1TPa (with Poisson ratio =0.25) and a density of = 2.3g/cm3. The effective thickness of DWCNT t=0.3nm. Fig.2 (a) & (b) display the finite element model and mode shape of bridged DWCNTs. Fig.3(a) shows the results of fundamental
Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
frequency of Bridged DWCNT Resonator vs. attached mass. It is observed from simulated results that resonant frequencies are reducing with increase in the attached mass. A clearly indication of mass sensing is found from a mass value of 1 zg for Length of double walled carbon nanotubes are 10 and 14 nm and 0.1 zg for a length of 7 nm. This shows that shorter length DWCNTs are more sensitive to mass change and better suited as a mass sensor. Also it can be specified here that as for a mass change of 0.1 zg, there is a change in frequency, hence the mass sensing capability of the device is 0.1 zg. Fig.3 (b) displays the results of fundamental frequency of Cantilever DWCNT vs. Mode shape for attached mass 10-2 to 104 Zg and without mass.
Figure 1. (a) Cantilevered (b) bridged DWCNTs with the variation of mass attach outer nanotubes
Figure 2. (a) Finite element model DWCNTs (b) Mode shape of bridged DWCNTs
Figure 3(a) Fundamental Frequency of Bridged DWCNT Resonator vs. attached mass (b) Fundamental frequency of Cantilever DWCNT Vs Mode shape for different values of mass .
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Ajay M. Patel and Anand Y. Joshi / Procedia Technology 14 (2014) 290 – 294
4. Conclusion
Mass sensing ability of DWCNT has been checked using a continuum mechanics and Finite Element approach. It has been observed that mass sensing characteristics of DWCNTs can reach up to 0.1 zg. Shorter length DWCNTs are more sensitive to attached mass and hence more suitable for mass sensing. An increasing trend in the frequency is observed with increasing the number of modes for different values of attached masses. Resonant frequencies are reducing double walled carbon nano tube with increase attached mass (zg).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
Goh, C. S., Wei, J., Lee, L. C., and Gupta, M., (2006) Simultaneous Enhancementin Strength and Ductility by Reinforcing Magnesium with Carbon Nano-tubes, Mater. Sci. Eng., A, 423, pp. 153–156. Ru, C. Q. (2004), Elastic Models for Carbon Nanotubes, Encyclopedia of Nanoscience and Nanotechnology, Vol. 2, H. S. Nalwa ed., American Scientific,Stevenson Ranch, CA, pp. 731–744. Zhang, Y. Q., Liu, G. R., and Xie, X. Y., (2005) Free Transverse Vibrations of Double-Walled Carbon Nanotubes Using a Theory of Nonlocal Elasticity, Phys. Rev. B, 71, p. 195404. Yoon J., Ru C.Q., Mioduchowski A., Comput. Sci. Technology. 63 (2003) 1533. Joshi A. Y., Harsha S. P., and Sharma, S. C., (2010), Vibration Signature Analysis of Single Walled Carbon Nanotube Based Nanomechanical Sensors, Physica E, 42, pp. 2115–2123. Li, C. Y., and Chou, T. W. (2004), Mass Detection Using Carbon Nanotube Based Nanomechanical Resonators, Appl. Phys. Lett. 84, pp. 5246–5248. Pradhan, S. C., Phadikar, J. K. and Karthik,G., Vibration analysis of multilayered orthotropic nanoplates using nonlocal elasticity theory, IE(I) Journal-MM(2009)90:16-23. Zhang Y., Liu G., Han X., Phys. Lett. A 340 (2005) 258. Natsuki T., Lei, X. W., Ni, Q. Q., and Endo, M., 2010, “Vibrational Analysis of Double-Walled Carbon Nanotubes with Inner and Outer Nanotubes of Different Lengths”, Phys. Lett. A, 374, pp. 4684–4689. Elishakoff, I., Versaci, C., and Muscolino, G., 2011, “Clamped-Free DoubleWalled Carbon Nanotube-Based Mass Sensor,” Acta Mech., 219(5), pp. 29–43. Zhi-bin shen, Bin Deng, Xian-Fang Li, Guo-Jin Tang, “Vibration of Double –walled Carbon Nanotube-Based Mass Sensor via Nonlocal Timoshenko Beam Theory” Journal of Nanotechnology in Engineering and Medicine by ASME, (2012),vol.2 Joshi A. Y., Satish C Sharma, S. P. Harsha, “Zeptogram scale mass sensing using single walled carbon nanotube based bio sensors”, Sensors & Actuators A. Physical,168, (2), 2011, 275-280. T. Natsuki, X.W. Lei, Q.Q. Ni, M. Endo, Phys. Lett. A 374 (2010) 2670. T. Murmu, S.C. Pradhan, Comput. Mater. Sci. 46 (2009) 854. Patel Ajay M., Joshi A.Y., 2013, “Vibration Analysis of Double Wall Carbon Nanotube based Resonators for Zeptogram level Mass Recognition”, Computational Materials Science, Elsevier Publishers, Volume 79, , Pp 230–238. Patel Ajay M., Joshi A.Y. 2014 “Effect of Waviness on the Dynamic Characteristics of Double Walled Carbon Nanotubes” Nanoscience Nano technology Letter, vol- 6, 1-9 Li C, Chou TW. Vibrational behaviors of multiwalled-carbon- nanotube-based nanomechanical resonators. Appl Phys Lett (2004) 84(1):121–123. Cornell W.D., Bally C.I., Gould I.R., Merz K.M. Jr., Feguson D.M., Spellmeyer D.C., Fox T., Caldwell J.W., Kollman P.A., (1995),Journal of American Chemical Society 1175179-5197.