Multi-particle mass sensing based on a single-walled carbon nanotube resonator

doi:10.1007/s10483-026-3376-6

摘要/Abstract

Abstract:

Based on the Timoshenko beam theory, a model for mass resonator sensor to detect multiple particles is developed. For a beam made of single-walled carbon nanotube (SWCNT), the nonlocal effects are incorporated in the governing equations. The approximate analytical solution for the resonance frequency of the system is derived by assuming that the mass of the adsorbed particles is much smaller than that of the system. The mass and position parameters of the multiple adsorbed particles are decoupled to establish an efficient detection method utilizing resonant frequency shifts. The identification process for the doubly clamped beam is systematically analyzed in numerical simulations. In addition, the axial force arising from temperature changes is incorporated into the beam model. The robustness of the proposed particle detection method against noise is analyzed. The model and analytical framework presented in this study provide a theoretical guideline for the design of nanoscale mass resonator sensors and particle mass detection under thermomechanical coupling conditions.

Key words: mass identification, nonlocal effect, carbon nanotube (CNT), multiple particle

中图分类号:

O321

. [J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 883-904.

Jie WANG, Yin ZHANG. Multi-particle mass sensing based on a single-walled carbon nanotube resonator[J]. Applied Mathematics and Mechanics (English Edition), 2026, 47(4): 883-904.

图/表 14

Model	$L / d$	$\| S (ω 1) \| max$	$\| S (ω 2) \| max$	$\| S (ω 3) \| max$	$\| S (ω 4) \| max$
Timoshenko beam	10	26.916 2 $(− 2.88 %)$	59.437 9 $(− 13.14 %)$	103.518 6 $(− 22.45 %)$	147.722 9 $(− 31.64 %)$
	20	27.503 7 $(− 0.75 %)$	65.838 3 $(− 3.79 %)$	123.502 4 $(− 7.49 %)$	190.005 5 $(− 12.08 %)$
	30	27.618 0 $(− 0.34 %)$	67.244 5 $(− 1.73 %)$	128.756 3 $(− 3.55 %)$	203.207 4 $(− 5.97 %)$
	40	27.658 5 $(− 0.19 %)$	67.756 5 $(− 0.99 %)$	130.765 1 $(− 2.04 %)$	208.551 4 $(− 3.49 %)$
EBB	–	27.710 9	68.430 6	133.494 6	216.102 4

Particle	Result	Mass	$ξ 1$	$ξ 2$	$ξ 3$
One particle	Exact	$1 × 10 − 3$	0.3	–	–
One particle	Estimated	$1.000 6 × 10 − 3$ $(0.06 %)$	0.300 0	–	–
Two particles	Exact	$1 × 10 − 2$	0.4	–	–
	Estimated	$9.983 1 × 10 − 3$ $(− 0.17 %)$	0.401 1	–	–
	Exact	$1 × 10 − 3$	0.25	0.35	–
	Estimated	$9.987 2 × 10 − 4$ $(− 0.13 %)$	0.250 3	0.350 2	–
Three particles	Exact	$5 × 10 − 3$	0.3	0.4	–
	Estimated	$4.973 3 × 10 − 3$ $(− 0.53 %)$	0.301 3	0.401 8	–
	Exact	$1 × 10 − 3$	0.2	0.3	0.4
	Estimated	$9.957 4 × 10 − 4$ $(− 0.43 %)$	0.201 4	0.300 5	0.400 7
	Exact	$3 × 10 − 3$	0.3	0.4	0.45
	Estimated	$2.984 8 × 10 − 3$ $(− 0.51 %)$	0.303 8	0.396 5	0.457 3

Particle	Result	Mass	$ξ 1$	$ξ 2$	$ξ 3$
One particle	Exact	$1 × 10 − 3$	0.35	–	–
One particle	Estimated	$1.001 0 × 10 − 3$ $(0.10 %)$	0.349 9	–	–
Two particles	Exact	$6 × 10 − 3$	0.45	–	–
	Estimated	$6.002 8 × 10 − 3$ $(0.05 %)$	0.450 2	–	–
	Exact	$4 × 10 − 3$	0.25	0.45	–
	Estimated	$3.980 7 × 10 − 3$ $(− 0.48 %)$	0.251 4	0.451 6	–
Three particles	Exact	$5 × 10 − 3$	0.28	0.32	–
	Estimated	$4.969 1 × 10 − 3$ $(− 0.62 %)$	0.279 8	0.322 6	–
	Exact	$2 × 10 − 3$	0.23	0.33	0.42
	Estimated	$1.991 0 × 10 − 3$ $(− 0.45 %)$	0.231 3	0.330 7	0.421 3
	Exact	$3 × 10 − 3$	0.30	0.38	0.46
	Estimated	$2.985 1 × 10 − 3$ $(− 0.50 %)$	0.303 8	0.377 7	0.466 2

参考文献 55

[1]	IIJIMA, S. Helical microtubules of graphitic carbon. nature, 354(6348), 56–58 (1991)
[2]	ARASH, B. and WANG, Q. A review on the application of nonlocal elastic models in modeling of carbon nanotubes and graphenes. Computational Materials Science, 51(1), 303–313 (2012)
[3]	POPOV, V. N., VAN DOREN, V. E., and BALKANSKI, M. Elastic properties of single-walled carbon nanotubes. Physical Review B, 61(4), 3078–3084 (2000)
[4]	MARTEL, R., DERYCKE, V., LAVOIE, C., APPENZELLER, J., CHAN, K. K., TERSOFF, J., and AVOURIS, P. Ambipolar electrical transport in semiconducting single-wall carbon nanotubes. Physical Review Letters, 87(25), 256805 (2001)
[5]	WONG, E. W., SHEEHAN, P. E., and LIEBER, C. M. Nanobeam mechanics: elasticity, strength, and toughness of nanorods and nanotubes. Science, 277(5334), 1971–1975 (1997)
[6]	LI, J. J. and ZHU, K. D. Weighing a single atom using a coupled plasmon-carbon nanotube system. Science and Technology of Advanced Materials, 13(2), 025006 (2012)
[7]	ZHANG, Y. and ZHAO, Y. P. Mass and force sensing of an adsorbate on a string resonator. Sensors and Actuators B: Chemical, 221, 305–311 (2015)
[8]	SAZONOVA, V., YAISH, Y., ÜSTÜNEL, H., ROUNDY, D., ARIAS, T. A., and MCEUEN, P. L. A tunable carbon nanotube electromechanical oscillator. nature, 431(7006), 284–287 (2004)
[9]	HÜTTEL, A. K., STEELE, G. A., WITKAMP, B., POOT, M., KOUWENHOVEN, L. P., and VAN DER ZANT, H. S. J. Carbon nanotubes as ultrahigh quality factor mechanical resonators. Nano Letters, 9(7), 2547–2552 (2009)
[10]	CHASTE, J., EICHLER, A., MOSER, J., CEBALLOS, G., RURALI, R., and BACHTOLD, A. A nanomechanical mass sensor with yoctogram resolution. Nature Nanotechnology, 7(5), 301–304 (2012)
[11]	GEORGANTZINOS, S. K. and ANIFANTIS, N. K. Carbon nanotube-based resonant nanomechanical sensors: a computational investigation of their behavior. Physica E: Low-dimensional Systems and Nanostructures, 42(5), 1795–1801 (2010)
[12]	HSU, J. C., LEE, H. L., and CHANG, W. J. Thermal buckling of double-walled carbon nanotubes. Journal of Applied Physics, 105(10), 103512 (2009)
[13]	KHANIKI, H. B. On vibrations of nanobeam systems. International Journal of Engineering Science, 124, 85–103 (2018)
[14]	ZHANG, Y. and ZHAO, Y. P. Measuring the nonlocal effects of a micro/nanobeam by the shifts of resonant frequencies. International Journal of Solids and Structures, 102-103, 259–266 (2016)
[15]	İNCE, A. and ERKOÇ, Ş. Molecular-dynamics simulations of silicene nanoribbons under strain. Physica Status Solidi B: Basic Research, 249, 74–81 (2012)
[16]	HEIDARY, Z., RAMEZANI, S. R., and MOJRA, A. Exploring the benefits of functionally graded carbon nanotubes (FG-CNTs) as a platform for targeted drug delivery systems. Computer Methods and Programs in Biomedicine, 238, 107603 (2023)
[17]	MIKHASEV, G., RADI, E., and MISNIK, V. Modeling pull-in instability of CNT nanotweezers under electrostatic and van der Waals attractions based on the nonlocal theory of elasticity. International Journal of Engineering Science, 195, 104012 (2024)
[18]	ANH, V. T. T., DAT, N. D., NGUYEN, P. D., and DUC, N. D. A nonlocal higher-order shear deformation approach for nonlinear static analysis of magneto-electro-elastic sandwich micro/nano-plates with FG-CNT core in hygrothermal environment. Aerospace Science and Technology, 147, 109069 (2024)
[19]	THAI, H. T. A nonlocal beam theory for bending, buckling, and vibration of nanobeams. International Journal of Engineering Science, 52, 56–64 (2012)
[20]	ERINGEN, A. C. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics, 54(9), 4703–4710 (1983)
[21]	WANG, Q. Y. and ZHANG, Z. L. Chaotic vibration of a curved CNT conveying magnetic fluid in the thermo-magnetic field considering the surface effects. Thin-Walled Structures, 202, 112047 (2024)
[22]	ZENG, Z., LU, K., WANG, X. F., and HU, R. C. Stochastic analysis for the embedded single-walled carbon nanotube under random vibrations. International Journal of Structural Stability and Dynamics, 25(8), 2550083 (2025)
[23]	MAWPHLANG, B. R. K. L. L. and PATRA, P. K. Study of the large bending behavior of CNTs using LDTM and nonlocal elasticity theory. International Journal of Non-Linear Mechanics, 166, 104828 (2024)
[24]	LI, C., CHEN, R. J., LI, C., and QING, H. Two-phase nonlocal integral model with bi-Helmholtz kernel for free vibration analysis of multi-walled carbon nanotubes considering size-dependent van der Waals forces. Applied Mathematics and Mechanics (English Edition), 46(11), 2095–2114 (2025) https://doi.org/10.1007/s10483-025-3313-8
[25]	KANG, D. K., YANG, H. I., and KIM, C. W. Geometrically nonlinear dynamic behavior on detection sensitivity of carbon nanotube-based mass sensor using finite element method. Finite Elements in Analysis and Design, 126, 39–49 (2017)
[26]	ALI-AKBARI, H. R., CEBALLES, S., and ABDELKEFI, A. Nonlinear performance analysis of forced carbon nanotube-based bio-mass sensors. International Journal of Mechanics and Materials in Design, 15(2), 291–315 (2019)
[27]	CEBALLES, S., SAUNDERS, B. E., and ABDELKEFI, A. Nonlocal Timoshenko modeling effectiveness for carbon nanotube-based mass sensors. European Journal of Mechanics-A/Solids, 92, 104462 (2022)
[28]	ZHANG, Y. and ZHAO, Y. P. Detecting the mass and position of an adsorbate on a drum resonator. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 470(2170), 20140418 (2014)
[29]	HEINRICH, S. M. and DUFOUR, I. Toward higher-order mass detection: influence of an adsorbate’s rotational inertia and eccentricity on the resonant response of a Bernoulli-Euler cantilever beam. Sensors, 15(11), 29209–29232 (2015)
[30]	LI, X. F., TANG, G. J., SHEN, Z. B., and LEE, K. Y. Resonance frequency and mass identification of zeptogram-scale nanosensor based on the nonlocal beam theory. Ultrasonics, 55, 75–84 (2015)
[31]	YAO, L. Z., WANG, T., JIANG, C. L., ZHAO, Q., SUI, Y., LU, Y., WANG, Y. K., SUN, Y., CONG, Z. C., and DONG, T. J. Multi-particle sorting using signals from particles trapped by single optical fiber tweezers. Optical Fiber Technology, 88, 103994 (2024)
[32]	ZHAO, L., WANG, F., ZHANG, Y. L., and ZHAO, X. Z. Theoretical study on the dynamic behavior of a plate-like micro-cantilever with multiple particles attached. PLoS One, 11(3), e0151821 (2016)
[33]	DOHN, S., SCHMID, S., AMIOT, F., and BOISEN, A. Position and mass determination of multiple particles using cantilever based mass sensors. Applied Physics Letters, 97(4), (2010)
[34]	WEI, C. X. and ZHANG, Y. Mass identification of multiple particles on a doubly clamped resonator. Sensors and Actuators B: Chemical, 360, 131682 (2022)
[35]	MA, S. J., LI, M. X., WANG, S. L., LIU, H., WANG, H., REN, L., HUANG, M. H., and ZHANG, X. W. Multiple particle identification by sequential frequency-shift measurement of a micro-plate. International Journal of Mechanical Sciences, 231, 107587 (2022)
[36]	JENSEN, K., KIM, K., and ZETTL, A. An atomic-resolution nanomechanical mass sensor. Nature Nanotechnology, 3(9), 533–537 (2008)
[37]	ZHANG, Y. Frequency spectra of nonlocal Timoshenko beams and an effective method of determining nonlocal effect. International Journal of Mechanical Sciences, 128-129, 572–582 (2017)
[38]	GARCIA-SANCHEZ, D., SAN PAULO, A., ESPLANDIU, M. J., PEREZ-MURANO, F., FORRÓ, L., AGUASCA, A., and BACHTOLD, A. Mechanical detection of carbon nanotube resonator vibrations. Physical Review Letters, 99(8), 085501 (2007)
[39]	REDDY, J. N. Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 45(2), 288–307 (2007)
[40]	WANG, Q. and WANG, C. M. The constitutive relation and small scale parameter of nonlocal continuum mechanics for modelling carbon nanotubes. Nanotechnology, 18(7), 075702 (2007)
[41]	ZHANG, Y. and LIU, Y. Detecting both the mass and position of an accreted particle by a micro/nano-mechanical resonator. Sensor, 14(9), 16296–16310 (2014)
[42]	ZHANG, Y. Detecting the stiffness and mass of biochemical adsorbates by a resonator sensor. Sensors and Actuators B: Chemical, 202, 286–293 (2014)
[43]	BENZAIR, A., TOUNSI, A., BESSEGHIER, A., HEIRECHE, H., MOULAY, N., and BOUMIA, L. The thermal effect on vibration of single-walled carbon nanotubes using nonlocal Timoshenko beam theory. Journal of Physics D: Applied Physics, 41(22), 225404 (2008)
[44]	AVSEC, J. and OBLAK, M. Thermal vibrational analysis for simply supported beam and clamped beam. Journal of Sound and Vibration, 308(3), 514–525 (2007)
[45]	YAO, X. H. and HAN, Q. Investigation of axially compressed buckling of a multi-walled carbon nanotube under temperature field. Composites Science and Technology, 67(1), 125–134 (2007)
[46]	WANG, J. and ZHANG, Y. Mass detection based on the 3:1 internal resonance in a piezoelectric laminated microbeam resonator sensor. Nonlinear Dynamics, 113(17), 22625–22649 (2025)
[47]	ZHANG, Y. Eigenfrequency computation of beam/plate carrying concentrated mass/spring. Journal of Vibration and Acoustics, 133(2), 021006 (2011)
[48]	RAMOS, D., TAMAYO, J., MERTENS, J., CALLEJA, M., and ZABALLOS, A. Origin of the response of nanomechanical resonators to bacteria adsorption. Journal of Applied Physics, 100(10), 106105 (2006)
[49]	MCCAIG, H. C., MYERS, E., LEWIS, N. S., and ROUKES, M. L. Vapor sensing characteristics of nanoelectromechanical chemical sensors functionalized using surface-initiated polymerization. Nano Letters, 14(7), 3728–3732 (2014)
[50]	DUAN, W. H., WANG, C. M., and ZHANG, Y. Y. Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. Journal of Applied Physics, 101(2), 024305 (2007)
[51]	WANG, C. M., ZHANG, Y. Y., and HE, X. Q. Vibration of nonlocal Timoshenko beams. Nanotechnology, 18(10), 105401 (2007)
[52]	ZHANG, Y. Y., WANG, C. M., and TAN, V. B. C. Assessment of Timoshenko beam models for vibrational behavior of single-walled carbon nanotubes using molecular dynamics. Advances in Applied Mathematics and Mechanics, 1(1), 89–106 (2009)
[53]	CEBALLES, S. and ABDELKEFI, A. Uncertainty analysis and stochastic characterization of carbon nanotube-based mass sensor with multiple deposited nanoparticles. Sensors and Actuators A: Physical, 332, 113182 (2021)
[54]	HUA, M. J. and WU, Y. Bifurcation in most probable phase portraits for a bistable kinetic model with coupling Gaussian and non-Gaussian noises. Applied Mathematics and Mechanics (English Edition), 42(12), 1759–1770 (2021) https://doi.org/10.1007/s10483-021-2804-8
[55]	LIU, Z., SONG, X. M., and ZHANG, M. A packet-dropping fusion Kalman filter algorithm based on non-Gaussian noise estimation. Mechanical Systems and Signal Processing, 228, 112457 (2025)