Analysis of Vibration Frequency of Carbon Nanotubes used as Nano-Force Sensors Considering Clamped Boundary Condition

Abstract

Carbon nanotubes (CNTs) can be used as atomic force microscope (AFM) probes since they are ideal tip materials with a small diameter, high aspect ratio, and stiffness. In this study, a model of CNTs clamped in an elastic medium is proposed as nanoscale force sensing AFM probes. The relationship between vibration frequency and axial force of the CNT probe clamped in an elastic medium is analyzed based on the Euler-Bernoulli beam model and the Whitney-Riley model. The clamped length of CNTs, and the elastic modulus of elastic medium affect largely on the vibration and the buckling stability of a CNT AFM probe. The result showed that the sensitivity to vibration increases as the applied loads increase. The critical load in which the vibration frequency decreases rapidly, moving to large ones with decreasing ratio of length to diameter of CNTs. The theoretical investigation on the vibration frequency of CNT loaded in the axial direction would give a useful reference for designing a CNT used as a nano-force sensor.

Review on Carbon Nanomaterials-Based Nano-Mass and Nano-Force Sensors by Theoretical Analysis of Vibration Behavior

“…Generally, perfect GSs are 2D materials and CNTs can be considered as tubes rolled from GSs, so both of CNTs and GSs are often analyzed based on classical plate/shell theories (e.g., [ 84 , 85 , 86 , 87 , 88 ]). Additionally, CNTs are 1D materials, so beam theories can be adopted for mechanical analysis of CNTs (e.g., [ 89 , 90 , 91 , 92 ]), which is used the most in theoretical evaluation of CNTs-based nano-mass and nano-force sensors. Moreover, graphene nanoribbons (GNRs) are special kinds of GSs owning 1D structures, so mechanical behavior of GNRs can be also analyzed by beam theories (e.g., [ 93 , 94 , 95 , 96 ]).…”

“…Similar to the nano-mass sensors, using the mechanism shown in Section 2.2 and the material properties determined from Section 3 , carbon nanomaterials-based nano-force sensors can also be investigated by continuum mechanical approaches. However, up to now, the studies of theoretical analysis about carbon nanomaterials-based nano-force sensors are only a few to our knowledge (e.g., [ 71 , 92 , 163 ]).…”

“…To interest more scholars in devoting themselves to the study of carbon nanomaterials-based nano-force sensors using theoretical analysis, we emphasize a previous study of CNTs-based nano-force sensors carried out by Natsuki and Urakami to show its feasibility [ 92 ]. This study was performed based on vibration analysis of CNTs using a continuum mechanical approach.…”

“… ( a ) A proposed carbon nanotubes-based nano-force sensor, and ( b ) the analytical model of the partial embedded carbon nanotubes resonator [ 92 ]. …”

On thermomechanics of multilayered beams

“…Analysis, modelling and assessment of elastic responses of small-scale structures has been a subject of great interest in the community of Engineering Science [2] due to the need to significantly design and optimize nanocomposites [3][4][5][6][7] and smaller and smaller technological devices [8][9][10][11][12][13][14][15]. So, to describe the deformation of advanced materials and new-generation structural systems, one needs to modify existing elasticity continuum methodologies, and non-local mechanics seems to be a promising choice to adequately capture technically important size effets [16][17][18][19][20][21] in comparison with computationally expensive atomistic strategies [22].…”