Determination of the Bending Modulus of an Individual Multiwall Carbon Nanotube using an Electric Harmonic Detection of Resonance Technique

Abstract

We report a new method of measuring the amplitude and phase of oscillations of individual multiwall carbon nanotubes (MWNTs). As in many other experiments, we excite the oscillations electrostatically, but we show that we can detect the amplitude and phase of the resulting oscillation electrically. As an example, we present measurements of the fundamental and first two overtones of the diving board resonance of a MWNT at 0.339, 2.42, and 5.31 MHz in ambient conditions. The corresponding quality factors were 67, 36, and 25.

The Very‐Low Shear Modulus of Multi‐Walled Carbon Nanotubes Determined Simultaneously with the Axial Young's Modulus via in situ Experiments

“…; L is the length and D o and D i are the outer and inner diameters of the cylinder (in our case the CNT); E is the axial Young's modulus and r ¼ 2.1 g cm À3 [9] is the mass density per unit volume. With known D o , D i and L, which can be directly measured using transmission electron microscopy (TEM) and/or scanning electron microscopy (SEM), the Young's modulus of a CNT can be calculated using Equation 1, from f i measured in resonance experiments.…”

“…In Equation 2, E is the axial Young's modulus; r ¼ 2.1g cm À3 [9] is the mass density of the CNT per unit volume; y is the transverse displacement; x is the axial coordinate; t is the time;…”

“…[4] A novel method has recently been developed to precisely measure the Young's modulus of single-walled CNTs (SWCNTs) by studying the diameter dependence of the buckling wavelengths of the SWCNTs on elastomeric substrates. [5] Of all the available methods, the electricfield-induced resonance method is widely used to study the mechanical properties of nanomaterials, including nanotubes, [2,[6][7][8][9] nanowires [10] and nanobelts. [11] In carrying out resonance experiments, carbon nanotubes or other nanowires are fixed at one end or at both ends.…”

Coupling between flexural modes in free vibration of single-walled carbon nanotubes

“…Thermodynamic behavior in nanoscale structures is fundamental to understanding the resonance properties of the next generation of nanoelectronic, photovoltaic, and thermoelectric devices. [2][3][4][5][6][7][8][9][10][11] As an ideal structure for nano-electromechanical systems, carbon nanotubes (CNTs) provide a unique platform to test mechanics models at the nanoscale. The thermal vibration of CNTs, which is crucial to their application as a mechanical resonator, has stimulated a substantial body of research.…”

“…The thermal vibration of CNTs, which is crucial to their application as a mechanical resonator, has stimulated a substantial body of research. [2][3][4][5]9,12 Transmission electron microscopy has been used to measure the thermal vibration of CNTs to obtain their Young's modulus. [12][13][14] The amplitude of CNTs at thermal equilibrium was predicted by beam models together with the law of energy equipartition and molecular dynamics (MD).…”