Aug 10, 2017 - Their theoretical studies based on the mechanical properties such as buckling strength and vibration frequency would give a useful reference for designing CNTs as .... coupled to the nanotube, via a voltage-controlled oscillator (VCO), ...... H.; Roundy, D.; Arias, T.A.; McEuen, P.L. A tunable carbon nanotube.
electronics Review
Carbon Nanotube-Based Nanomechanical Sensor: Theoretical Analysis of Mechanical and Vibrational Properties Toshiaki Natsuki 1,2, * 1 2
Faculty of Textile Science and Technology, Shinshu University, 3-15-1 Tokida, Ueda-shi 386-8567, Japan Institute of Carbon Science and Technology, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan
Received: 11 July 2017; Accepted: 4 August 2017; Published: 10 August 2017
Abstract: This paper reviews the recent research of carbon nanotubes (CNTs) used as nanomechanical sensing elements based mainly on theoretical models. CNTs have demonstrated considerable potential as nanomechanical mass sensor and atomic force microscope (AFM) tips. The mechanical and vibrational characteristics of CNTs are introduced to the readers. The effects of main parameters of CNTs, such as dimensions, layer number, and boundary conditions on the performance characteristics are investigated and discussed. It is hoped that this review provides knowledge on the application of CNTs as nanomechanical sensors and computational methods for predicting their properties. Their theoretical studies based on the mechanical properties such as buckling strength and vibration frequency would give a useful reference for designing CNTs as nanomechanical mass sensor and AFM probes. Keywords: carbon nanotube; nanomechanical sensor; vibration; beam theory; nonlocal elasticity
1. Introduction Since the discovery of carbon nanotubes (CNTs) in 1991 by Iijima [1], they have been recognized as one of the most promising nanomaterials and attracted great interest among researchers around the world because of the superior mechanical and physical properties. CNTs have diverse applications in various fields of science and engineering, such as electrical and electronic components, materials engineering, nano-sensor, energy storage and environment [2–6]. Many studies have revealed that the Young’s modulus of single-walled CNTs (SWCNTs) were about 1 TPa and multi-walled CNTs (MWCNTs) were tested to have a tensile strength ranged from 11 to 63 GPa, which are stronger and stiffer than any known substance [7–9]. Wei et al [10] reported that CNTs could carry high current densities up to 109 –1010 A/cm2 and remained stable for extended periods of time at high temperature. CNTs are the smallest scale nanomaterials that can be observed only by a transmission electron microscope (TEM). The excellent structural characteristics and the mechanical and physical properties make CNTs potential candidates for the next generation of nanoelectromechanical systems (NEMS) [11–15]. At present, many methods have been developed to synthesize CNTs, including arc-discharge, laser ablation, and chemical processes [16,17]. Chemical vapor deposition (CVD) is the most common chemical method, and is widely used for fabricating CNTs. This method is capable of controlling growth directions on a substrate and synthesizing a large quantity of carbon nanotubes. In the experimental operation, it is very difficult to operate and control a single CNT because of an extremely small size. Since the advanced CVD technology was developed, the CVD synthesis allowed CNT growth directly on desired substrates, and growth in a variety of forms such as powders or films, aligned or coiled, including to control the diameter and length of CNTs. New electronic products and the developments can be expected based on the fabrication and characterization of novel nano Electronics 2017, 6, 56; doi:10.3390/electronics6030056
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products and the developments can be expected based on the fabrication and characterization of carbon materials. studies on using CNTs as high-frequency circuit elements, actuators, novel nano carbonMany materials. Many studies on using CNTs as high-frequency circuitrotary elements, rotary nanosensors, transistors, etc. have been reported up to now [18–22]. Specifically, in CNT-based actuators, nanosensors, transistors, etc. have been reported up to now [18–22]. Specifically, in CNTresonators, the state-of-the-art highesthighest mass sensing resolution was achieved by using the based resonators, the state-of-the-art mass sensing resolution was achieved by CNTs using as CNTs resonating mass detector [23]. as the resonating mass detector [23]. Research Research and and development development on on application application of of CNTs CNTs used used in in nanosensors nanosensors is is now now an an important important subject for nano nanomeasurement. measurement.AsAs shown in Figure 1, research the research of NEMS is a collection of subject for shown in Figure 1, the of NEMS is a collection of various various technologies that include nano-materials and micro/nano processing, nanomechanics and technologies that include nano-materials and micro/nano processing, nanomechanics and nanoelectronic nanoelectronicdevices, devices,and andcomputational computationalsimulation simulationand andanalysis. analysis. With With the the enormous enormous development development of NEMS based nanosensors, the miniaturization and integration of electronic elements can be realized of NEMS based nanosensors, the miniaturization and integration of electronic elements can be using structural design anddesign manufacturing processes. Thus, nano-force mass measurement realized using structural and manufacturing processes. Thus,and nano-force and mass become possible, usingpossible, resonantusing nanosensor based systems.based systems. measurement become resonant nanosensor
Nano-materials and micro/nano processing
Structural analysis and computational science
Nanoelectromechanical systems (NEMS)
Nanomechanics and nanoelectronics devices
Structural design/Analysis/Manufacturing Figure Figure 1. 1. Components Components of of nanoelectromechanical nanoelectromechanical system. system.
At present, present, many many studies studies on on the the nanomechanical nanomechanical sensors sensors using using CNTs CNTs and and their their various various At applicationshave havebeen been reported. In review, the review, we on focus on the ofdesign of CNT-based sensors applications reported. In the we focus the design CNT-based sensors including including the nano-mass sensor and the sensor used as probe for atomic force microscope (AFM), the nano-mass sensor and the sensor used as probe for atomic force microscope (AFM), especially on especially the studies of their workingbased mechanisms based ontheoretical the available theoretical model. the studieson of their working mechanisms on the available model.
2. 2. CNT-Based CNT-BasedMass MassSensor Sensor 2.1. 2.1. Background Background of of Nanomechanical Nanomechanical Sensor Sensor Research Research Conventionally, Conventionally, silicon silicon based based materials materials can can be be used used for for detecting detecting aa tiny tiny amount amount of of mass mass at at the the nano-level. Professor Roukes of the California Institute of Technology (Caltech) and his colleagues nano-level. Professor Roukes of the California Institute of Technology (Caltech) and his colleagues used used silicon silicon carbide carbide beam beam for for the the nanomechanical nanomechanical resonator resonator and and made made it it possible possible to to measure measure the the − 21 minimum g) level level [24]. [24]. Ten Ten years years ago, ago, the the Caltech Caltech research research group minimummass massat atzeptogram zeptogram(10 (10−21 g) group developed developed aa mass mass sensor sensor by by using using silicon silicon instead instead of of silicon silicon carbide, carbide, and and achieved achieved its its high high mass mass resolution resolution with with − 18 attogram sensitivity[25]. [25].The The new device was developed, essentially in the same way as attogram (10 (10−18 g)g)sensitivity new device was developed, essentially in the same way as the the previous, but its smaller size and higher resonant frequency gave it a greater sensitively to added previous, but its smaller size and higher resonant frequency gave it a greater sensitively to added mass. mass. In In their their research, research, the the device device consisted consisted of of aa nanomechanical nanomechanical resonator resonator with with tiny tiny silicon silicon or or silicon silicon carbide nm wide. The flatflat beam was clamped at both ends, andand set carbidebeams beamsabout aboutaamicron micronlong longand and100 100 nm wide. The beam was clamped at both ends, oscillating at a frequency of 100 MHz or more. The principle of mass detection is based on frequency set oscillating at a frequency of 100 MHz or more. The principle of mass detection is based on shift when shift a tinywhen massaistiny attached the beam. mechanical mass sensor consisting of silicon frequency mass istoattached to In thethe beam. In the mechanical mass sensor consisting beam, it isbeam, important to know the of vibrationoffrequency for theshift attached nanomass. of silicon it is important to sensitivity know the sensitivity vibration shift frequency for the attached However, since the CNTs have very small diameter, higher stiffness and strength than silicon or silicon nanomass. However, since the CNTs have very small diameter, higher stiffness and strength than dioxide, thesilicon CNT-based resonator can maintain high resonance frequencies measure extremely tiny silicon or dioxide, the CNT-based resonator can maintain high to resonance frequencies to mass. The group of Professor Heer (Georgia Institute of Technology) initially proposed an idea of using measure extremely tiny mass. The group of Professor Heer (Georgia Institute of Technology) initially aproposed single CNT highly sensitive nanobalances [26]. sensitive As shownnanobalances in Figure 2, they an as idea of using a single CNT as highly [26].succeeded As shownin inattaching Figure 2, they succeeded in attaching a carbon particle to the end of a nanotube and setting the nanotube to a
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a carbonelectron particlemicroscope to the end of a nanotube and setting the nanotube to studied a specialtoelectron microscope special stage. The methods developed were well measure masses in stage. The methods developed were well studied to measure masses in the picogram-to-femtogram the picogram-to-femtogram range. The mass of this nanoparticle was determined from the resonance range. Theshift massofofa this wasunder determined from the electron resonance frequency(TEM) shift of a CNT frequency CNTnanoparticle nanocantilever a transmission microscope after the nanocantilever under a transmission electron microscope (TEM) after the mass was attached to the −15tip mass was attached to the tip of the CNT, and was measured to be about 22 fg (Femtogram: 10 g) −15 g) [26,27]. of the CNT, and was measured to be about 22 fg (Femtogram: 10 [26,27].
Figure 2. Resonance vibrations of a nanotube loaded with a carbon nanoparticle. Adapted with Figure 2. Resonance vibrations of a nanotube loaded with a carbon nanoparticle. Adapted with permission from [26], Copyright AAAS, 1999. permission from [26], Copyright AAAS, 1999.
Subsequently, the research on an atomic-resolution nanomechanical mass sensor that can Subsequently, the research on an atomic-resolution nanomechanical mass sensor that can measure measure atomic mass level was reported [28]. Figure 3 shows the entire nanomechanical mass atomic mass level was reported [28]. Figure 3 shows the entire nanomechanical mass spectrometer apparatus. A TEM image of a nanomechanical mass spectrometer devicespectrometer constructed apparatus. A TEM image of a nanomechanical mass 3a. spectrometer device constructed a from a double-walled CNT (DWCNT) is shown in Figure The nanotube device was placedfrom at one double-walled CNT (DWCNT) is shown in Figure 3a. The nanotube device was placed at one end end of an ultra-high vacuum (UHV) chamber as shown in Figure 3b. In the UHV chamber, gold atoms of an evaporated ultra-high vacuum (UHV) chamber shown Figure 3b. Indevice. the UHV chamber, atoms were from a tungsten filament as away frominthe nanotube To control thegold number of were evaporated from a tungsten from the between nanotube the device. To controlsource the number of loaded atoms onto the device, afilament shutter away was inserted evaporation and the loaded atoms onto the device, a shutter was inserted between the evaporation source and the nanotube nanotube to interrupt the gold mass loading. A water-cooled quartz crystal microbalance (QCM) to interrupt gold mass A water-cooled crystal microbalance (QCM) provided an the alternate meansloading. of calibrating the systemquartz through measurement of mass flux. provided Figure 3c an alternate means of calibrating the system through measurement of mass flux. Figure 3c measure shows a shows a schematic of the nanomechanical resonance detection circuit, which was used to schematic of the nanomechanical resonance detection circuit, which was used to measure the resonant the resonant frequency of the nanotubes. The detection technique was based on a nanotube radio frequency of theand nanotubes. technique wasproperties based on aofnanotube radioexisted receivera design receiver design relied onThe thedetection unique field emission CNTs. There strong and relied on the unique field emission properties of CNTs. There existed a strong coupling between coupling between the field emission current from CNTs and their mechanical vibrations. Briefly, an the field emission current from CNTs and their mechanical vibrations. Briefly, an amplitude modulated amplitude modulated (AM), sweep generator were coupled to the nanotube, via a voltage-controlled (AM), sweep generator wereforcing coupled nanotube, a voltage-controlled oscillator RF oscillator (VCO), RF signal, it to to the resonate, and via consequently modulating the field(VCO), emission signal, forcing it to resonate, and consequently modulating the field emission current. The modulated current. The modulated field emission current was recovered by a lock-in amplifier and the resonance field current was recovered by a lock-in amplifier and the resonance peak was detected. peak emission was detected. Li and Zhu Li and Zhu [29] [29] proposed proposed an an optical optical detection detection technique technique for for weighing weighing aa single-atom single-atom by by using using surface plasmon and CNT resonator. The mass of a single atom could be determined by vibrational surface plasmon and CNT resonator. The mass of a single atom could be determined by vibrational frequency shift of of CNTs Figure 4 frequency shift CNTs while while the the atom atom attached attached to to the the nanotube nanotube surface. surface. Figure 4 shows shows schematic schematic of a mass sensor based on a doubly clamped CNT illuminated by a strong pump beam and aa weak weak of a mass sensor based on a doubly clamped CNT illuminated by a strong pump beam and signal A single single gold gold metal metal nanoparticle nanoparticle (MNP) (MNP) was was attached attached to to the the tip tip of of aa sharp sharp optical optical fiber fiber signal beam. beam. A and positioned at a distance R away from the CNT. Moreover, a single xenon atom was deposited and positioned at a distance R away from the CNT. Moreover, a single xenon atom was deposited onto The inset inset of of Figure Figure 44 shows shows the the energy energy levels levels of of aa localized localized exciton exciton in in CNTs CNTs onto the the CNT CNT surface. surface. The while dressing the thesurface surfaceplasmon plasmonand andthe the vibration CNT. AFM used to probe theand tip while dressing vibration of of thethe CNT. AFM waswas used to probe the tip and stabilize its distance to the CNT [30,31]. The mass of atoms attached to the CNT surface could stabilize its distance to the CNT [30,31]. The mass of atoms attached to the CNT surface could be be measured conveniently precisely through frequency shift in the absorption spectrum. Owing measured conveniently andand precisely through frequency shift in the absorption spectrum. Owing to to the ultralight mass and higher stiffness of the CNT, and spectral enhancement by using surface the ultralight mass and higher stiffness of the CNT, and spectral enhancement by using surface plasmon, a narrow linewidth with kHzkHz andand higher sensitivity, whichwhich was more plasmon, this thismethod methodresulted resultedinin a narrow linewidth with higher sensitivity, was sensitive than traditional electromechanical mass detection techniques. Especially, heating and energy more sensitive than traditional electromechanical mass detection techniques. Especially, heating and loss characteristics for electrical detection could becould avoided in all-optical technique. energy loss characteristics for electrical detection be avoided in all-optical technique.
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Figure3. 3. Nanomechanical Nanomechanical mass mass spectrometer spectrometer device TEM image of of a a Figure device and and schematics. schematics.(a) (a) TEM image Figure 3. Nanomechanical mass device spectrometer device and schematics. CNT (a) (DWCNT), TEM image of a nanomechanical mass spectrometer constructed from a double-walled (b) The nanomechanical mass spectrometer device constructed from a double-walled CNT (DWCNT), (b) The nanomechanical mass spectrometer device constructed from a double-walled CNT (DWCNT), (b) The nanotube devicewas wasplaced placedatatone one end end of of an (c)(c) Schematic of the nanotube device an ultra-high ultra-highvacuum vacuum(UHV) (UHV)chamber, chamber, Schematic of the nanotube device resonance was placeddetection at one end of an ultra-high vacuum (UHV) chamber, (c) Schematic of the nanomechanical circuit. Adapted Copyright Nature nanomechanical resonance detection circuit. Adaptedwith withpermission permissionfrom from[28], [28], Copyright Nature nanomechanical resonance detection circuit. Adapted with permission from [28], Copyright Nature Publishing Group, 2008. Publishing Group, 2008. Publishing Group, 2008.
Figure 4. Schematic of a mass sensor based on a doubly clamped carbon nanotube. Adapted with Figure 4. Schematic of a mass sensor based on a doubly clamped carbon nanotube. Adapted with permission from [29], IOP, 2012. Figure 4. Schematic of Copyright a mass sensor based on a doubly clamped carbon nanotube. Adapted with permission from [29], Copyright IOP, 2012. permission from [29], Copyright IOP, 2012.
Lassagne et al. [32] used extremely tiny CNT with 1 nm diameter as a nanomechanical resonator Lassagne et al.The [32] performances, used extremelywhich tiny CNT with 1 nm by diameter as a nanomechanical resonator for mass sensing. were tested measuring the mass of evaporated Lassagne et al. [32] used extremely tiny CNT with 1 nm diameter as a nanomechanical for mass sensing. The performances, which were tested by measuring the mass of evaporated chromium atoms, were exceptional. Figure 5 shows diagrams of (a) experimental setup and resonator (b) the chromium atoms, wereperformances, exceptional. Figure 5 shows diagrams (a) experimental setupof (b) the formeasurement mass sensing. The which were testedwere byofmeasuring the means mass evaporated circuitry. Nanomechanical CNT resonators fabricated by ofand standard measurement circuitry. Nanomechanical CNT resonators were fabricated by means of standard chromium atoms, were exceptional. Figure 5 shows diagrams of (a) experimental setup and (b)
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the measurementtechniques, circuitry. Nanomechanical CNT resonators werewere fabricated byout means of standard nanofabrication and nanomass sensing experiments carried by evaporating nanofabrication techniques, and nanomass sensing experiments were carried out by evaporating chromium atoms onto the CNT resonators. As shown in Figure 5b, to track the high-frequency charge chromium atoms onto the CNT resonators. As shown in Figure 5b, to track ac the high-frequency charge modulation, a mixing technique was employed by applying voltage V cos 2ft on the gate modulation, a mixing technique was employed by applying voltage Vgac gcos(2π f t) on the gate and and cos the source. Based on measurement of a current mixing Icurrent I mix on the VSD cos f the t on V source. Based on measurement of a mixing [2π ( f2+ δ ff )t]on SD mix on the drain at drain at frequency shifts of resonance wereas obtained as aoffunction of the f , the frequency change δ fchange , the shifts of resonance frequencyfrequency were obtained a function the deposited − 24 24 mass. The mass. resultsThe found that found the mass responsivity was 11 Hz/yg 1 yg = 10 g and the mass deposited results that the mass responsivity was 11 Hz yg 1 yg 10 g − 21 21 resolution is 25resolution zg 1 zg = When the CNT was cooled downwas to 5 cooled K in a 10 zg g 1at and the mass is 25 temperature. When the CNT zgroom 10temperature. g at room cryostat, the mass resolution increased and corresponded to a resolution of 1.4 zg because the signal down to 5 K in a cryostat, the mass resolution increased and corresponded to a resolution of 1.4 zg for the detection of for mechanical vibrations was improved. because the signal the detection of mechanical vibrations was improved.
(b)
(a)
Figure Figure 5.5. Schematics Schematics of of the the experimental experimental setup setup and and the the measurement measurement circuitry. circuitry. Adapted Adapted with with permission from [32], Copyright ACS, 2008. permission from [32], Copyright ACS, 2008.
2.2. 2.2. Theoretical Theoretical Approach Approach of of Vibration Vibration Characteristics Characteristics As As described described above, above, the the detection detection principle principle of of aa CNT-based CNT-basedmass masssensor sensoris isbased based on on the the fact fact that that the resonant frequency is sensitive to the resonator mass, which includes the mass of the resonator the resonant frequency is sensitive to the resonator mass, which includes the mass of the resonator itself itself and and the the mass mass attached attached on on the the CNT CNT resonator. resonator. A A change change in in the the resonant resonant frequency frequency occurs occurs when when an additional mass is attached on the resonator. However, one of main problems on mass detection is an additional mass is attached on the resonator. However, one of main problems on mass detection how to to accurately quantify the tiny change is how accurately quantify the tiny changeininthe theresonant resonantfrequency frequencydue duetotothe theadded addedmass. mass. Some Some theoretical investigation, such as the molecular dynamics (MD) method and continuum mechanics, on theoretical investigation, such as the molecular dynamics (MD) method and continuum mechanics, mass sensing characteristics has been reported so far. Although MD is widely used in nanomaterial on mass sensing characteristics has been reported so far. Although MD is widely used in science and is ascience good method analyzing of CNT structure, the requested nanomaterial and is afor good methodproperties for analyzing properties of CNT structure,computational the requested power is limited to a fast computer with a high capacity hard disk. Moreover, the analysis using the computational power is limited to a fast computer with a high capacity hard disk. Moreover, the MD method is a cost-intensive and time-consuming process. In response to the issues, well established analysis using the MD method is a cost-intensive and time-consuming process. In response to the continuum model on a nanoscale is amodel challenging task. At present, the continuum mechanics issues, wellmechanics established continuum mechanics on a nanoscale is a challenging task. At present, has been successfully used to simulate static and dynamic properties of carbon nanotubes [33–36]. the continuum mechanics has been successfully used to simulate static and dynamic properties of Thenanotubes research group of Prof. Chou [37] reported initially that the CNT resonators were assumed to carbon [33–36]. be either cantilevered or bridged and were simulated based on structural mechanics The research group of Prof. Chou [37] reported initially thatthe themolecular CNT resonators were assumed method. In the research, the fundamental frequency of the CNT was obtained by simulating a SWCNT to be either cantilevered or bridged and were simulated based on the molecular structural mechanics as an equivalent space frame-like structure and solving of thethe motion of the system. For freea method. In the research, the fundamental frequency CNTequation was obtained by simulating vibration of the nanotube in its fundamental modes, the governing equation of motions was given by: SWCNT as an equivalent space frame-like structure and solving the motion equation of the system. For free vibration of the nanotube in its fundamental modes, the governing equation of motions was .. (1) [ M] y + [K ]{y} = {0}, given by: .. where [ M ] and [K ] were the global mass and the y K matrices, M stiffness y 0 , respectively, and y and {y} were (1) nodal displacement vector and acceleration vector, respectively. After condensation of the mass and where M and K were the global mass and stiffness matrices, respectively, and y and y were
the nodal displacement vector and acceleration vector, respectively. After condensation of the mass
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stiffness matrices, applying the static condensation method, the fundamental frequencies and mode and stiffness matrices, applying the static condensation method, the fundamental frequencies and shapes were obtained from the solution of the eigenproblem mode shapes were obtained from the solution of the eigenproblem
2 s yyp p = {00}, , 2[ M M s ω − [K]K ] s s
(2) (2)
was where K and M s were the condensed stiffness matrix and mass matrix, respectively. p where [K ]ss and [ M ]s were the condensed stiffness matrix and mass matrix, respectively. y p ywas the the displacement of carbon atoms, the angular frequency. was 2f the 2πf ) was displacement of carbon atoms, and and ω (ω = angular frequency. Based the changes in in resonant frequency duedue to Basedon onthe themethod methodofofmolecular molecularstructural structuralmechanics, mechanics, the changes resonant frequency the attached mass were calculated. Figure 6 shows the to the attached mass were calculated. Figure 6 shows thevariation variationofoffrequency frequencyshift shiftwith withthe theattached attached mass mass on on the the cantilevered cantilevered nanotube nanotube resonators. resonators. The The frequency frequency shifts shifts were were defined defined as as the the difference difference between frequencies of of aa nanotube nanotubewith withand andwithout withoutattached attachedmass. mass.The The effects between the the fundamental frequencies effects of of length diameter of CNTs on frequency the frequency investigated in detail. It was clear length andand diameter of CNTs on the shiftsshifts were were investigated in detail. It was clear that the that the frequency shift increased the of increase of attached and the sensitivity offrequency resonant frequency shift increased with the with increase attached mass, andmass, the sensitivity of resonant frequency shifts to length both tube and diameter had been demonstrated. The mass sensitivity shifts to both tube and length diameter had been demonstrated. The mass sensitivity of CNT-based −21 g, and a logarithmically linear relationship existed −21 g, reach of CNT-based nanobalances nanobalances could reach 10could and a10logarithmically linear relationship existed between the −20 g.than 10−20 g. between resonantand frequency and the attached mass when the massthan was 10 larger resonantthe frequency the attached mass when the mass was larger
(b)
(a)
Figure Figure 6.6. Resonant Resonant frequency frequency shifts shifts of of cantilevered cantilevered nanotubes nanotubes with with different different lengths lengths (a) (a) and and tube tube diameter vs. attached attachedmass. mass.Adapted Adapted with permission from Copyright AIP Publishing diameter (b) (b) vs. with permission from [37],[37], Copyright AIP Publishing LLC, LLC, 2004.2004.
Since DWCNTs and strength, thermal and chemical stability than singleSince DWCNTs have havehigher higherstiffness stiffness and strength, thermal and chemical stability than walled CNTs, the potential of DWCNTs as a nano mass sensor was also explored [38–40]. The inner single-walled CNTs, the potential of DWCNTs as a nano mass sensor was also explored [38–40]. andinner outerand walls of walls the DWCNT were modelled as finite [38] or[38] elastic beams [39,40], and The outer of the DWCNT were modelled aselements finite elements or elastic beams [39,40], the interaction force between the two walls was considered as a spring element connecting the two and the interaction force between the two walls was considered as a spring element connecting the two layersby byvan van Waals (vdW) interaction. The interlayer interaction is described by potential. the vdW layers derder Waals (vdW) interaction. The interlayer interaction is described by the vdW potential. The most common expression of potential Lennard–Jones potential is the utilized to express the The most common expression of Lennard–Jones is utilized to express interaction of carbon interaction of carbon atoms located on the two walls, given by atoms located on the two walls, given by
12 6 12 ρ 6 U)R= 4ε4 ρ − , U(R RR RR ,
(3) (3)
where atomic distance, ε isεthe depth of the and σand is the finite at which whereRRisisthe theinter inter atomic distance, is the depth ofpotential, the potential, σ is the distance finite distance at the interatomic potential is zero. The pressure caused by the vdW interaction can be described by which the interatomic potential is zero. The pressure caused by the vdW interaction can be described by
p = c ( u2 − u1 ), p cu2 u1 ,
(4) (4)
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where u1 and u2 are the vibrational deflection of inner and outer walls of DWCNTs, respectively. c is the vdW interaction coefficient between the walls. The vdW interaction coefficients can be estimated using Equation (5) [41] 320 × 2R j erg/cm2 c= , ∆ = 0.142 nm, (5) 0.16∆2 where R j is the center line radius of the jth nanotube, and ∆ is the length of carbon–carbon (C–C) bond. In the finite element model (FEM) [38], the element equation constructed by the global stiffness and mass matrices can be assembled. The governing coupled equations for the displacement are given by ..
K1 u1 + M1 u1 = c(u2 − u1 ),
(6)
..
K2 u2 + M2 u2 = c(u1 − u2 ),
(7)
where K1 , K2 are stiffness matrices, M1 , M2 are mass matrices, and the wall deflections of DWCNT are given as u j = Yj eiωt , (8) and Yj , j = 1, 2, are the vibrational amplitudes and ω is the vibrational frequency. According to Equations (6)–(8), the element equation for global system can be written as "
K1 + c −c
−c K2 + c
#(
Y1 Y2
)
"
−ω
2
M1 0
0 M2
#(
Y1 Y2
)
= 0.
(9)
The vibrational frequency of DWCNTs can be obtained by a nontrivial solution of Equation (9). For the continuum mechanics, Patel and Joshi [39] derived a second order governing differential equation of Euler beam model and considered nonlinear vdW interaction between nanotubes, by using a single-mode Galerkin approximation. The dynamic response of the DWCNT-based mass sensor was analyzed based on time response, Poincaré maps, and fast Fourier transformation. The results showed that the appearance of regions of periodic, subharmonic and chaotic behavior were strongly dependent on mass, inner and outer nanotubes and the geometric imperfections of DWCNTs. Shen et al. [40] analyzed the vibration frequency of DWCNT-based mass sensor using nonlocal Timoshenko beam theory. The natural frequencies of a nonlocal cantilever beam attached to a tip mass were computed using the transfer function method. They assumed that the inner and outer walls of DWCNTs had different lengths, and the atom-resolution mass was attached to the free end of the inner nanotube. Effects of the attached mass and outer-to-inner tube length ratio on the natural frequencies were discussed. The results showed that the mass sensor based on DWCNTs with different wall lengths had a noticeable advantage over that based on SWCNTs. The mass sensitivity of DWCNT-based sensor could be enhanced when short DWCNTs were used, especially for larger attached mass. The nonlocal Timoshenko beam model was more adequate than the nonlocal Euler–Bernoulli beam model for short DWCNT sensors. The mass sensing characteristics of DWCNT based nanobalances can reach up to 0.1 Zeptograms [37]. It is obvious that mass sensitivity of nano-mass detection system can be improved by using smaller sized CNTs since CNT-based resonator results high oscillation frequency. In order to increase sensitivity of the resonators, the author Natsuki and co-workers proposed using a double end fixed CNT with applied tensile loads and investigated the possibility of application [42,43]. Figure 7 shows a schematic diagram of the clamped CNT beam, subjected to an axial force and carrying an attached concentrated mass mc . Based on the continuum model with the Euler–Bernoulli beam, the motion equations of a freely vibrating CNT under the axial tensile load N can be given as [42]
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7. Schematic diagram of clamped carbon nanotube (CNT) beam model with attached mass Figure 7. under tensile load.
4Y x, t
N 2Y x, t
A 2 2Y x, t
∂2 Y1 ( x,1 t2) ρAω2 ∂2 Y1 (1 x,2 t) 0 ∂4 Y1 ( x, t1 )4 N =0 − x EI 2x − EI t EI EI ∂x ∂t2 ∂x4 2 ∂N2 Y2(2x, 4tY)2 x, t N Y2 tx), t ρAω Y22(x, x, tt) A22∂2 Y ∂4 Y2 ( x, =0 0 − − 4 2 2 EI t 2 EI ∂t ∂x4 x EI EI ∂x2x
0 xa , a, 0≤x≤
(10) (10)
a a x≤ Lx ≤ L, ,
(11) (11)
where vibration amplitudes of the deflection, ω is the circular frequency, j (j j= where YY arethethe vibration amplitudes offlexural the flexural deflection, is the circular j 1,1, 22) are EI is the flexural stiffness of CNT, A is the cross-sectional area, ρ is the mass density, L is the length of frequency, EI is the flexural stiffness of CNT, A is the cross-sectional area, is the mass density, CNT, a is the position in which concentrated mass is attached, x is the axial coordinate and t is the time. L is the length of CNT, a is the position in which concentrated mass is attached, x is the axial According to corresponding boundary conditions such as bound states and continuum states, the coordinate and t is the time. resonant frequency (ω ) of the CNT resonators with an attached concentrated mass can be derived According to corresponding boundary conditions such as bound states and continuum states, from Equations (10) and (11), and given by the following matrix equation: the resonant frequency of the CNT resonators with an attached concentrated mass can be derived from Equations (10) and (11), and by)|the following matrix equation: (mc , ω | Mgiven 8×8 = 0.
(12)
M mthe 0 equation of CNT bending moment(12) c, Based on Rayleigh’s energy method [42], differential can 88 . be given by Px M of CNT bending moment can Based on Rayleigh’s energy method [42], the differential equation y( x ) = A cosh(αx ) + Bsinh(αx ) + + 0, (13) be given by 2N N where P is mass load acted on the middle of the CNT, and M0 is theMmoment at the left clamped end. Px A cosh(ofxA) and yxconstants B sinh( x)be determined 0 from the clamped boundary The moment M0 , the integration B, can (13) 2N N , conditions: 0 y (0) = 0. (14) (0CNT, ) = y0 and ( L/2)M= where P is mass load acted on the middle of ythe 0 is the moment at the left clamped end. The Moreover, moment M the integration andbeam B, can determined the kineticconstants energy ofof theACNT canbeaccordingly be from given the by clamped 0 , maximum boundary conditions: L/2 Z . 1 y02m 2 CNT y 0 yLy2 (14) Tmax = (15) ( x ) 0 .dx, 2 L 0
Moreover, the maximum kinetic energy of the CNT beam can accordingly be given by where mCNT (mCNT = ρAL) is the mass of theL CNT resonator. 2 mCNTaccording 1 (15)2and Substituting Equation (13) into Equation and y x 2 dx to the relationship between force(15) Tmax 2 L deflection, the equivalent stiffness of CNT beam can be obtained 0 , P 2Nα where mCNT mCNT AL k eq =is the mass = of the CNT resonator. (16) 2 αL αL αL ymax 2tanh 4(15) sinh + αL Substituting Equation (13) into Equation and4 according to the 2 − sinh 2 relationship between force and deflection, the equivalent stiffness of CNT beam can be obtained q N and α = EI . P 2 N keq of the CNT resonators carrying concentrated mass (mc ) yields Thus, the resonant frequency L L L L ymax (16) 2 tanh ssinh 2 sinh 4 4 2 2 k eq 1 ω= , (17) 2π φρAL + mc N and .
EI
Thus, the resonant frequency of the CNT resonators carrying concentrated mass mc yields
where
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3 2 L 2 2 L2 4 2 4 2 L 2 sinh 2 2 12 L 2 2 2 cosh
where and
and
φ=
β=
3β2 2
−
L 2 L sinh l sinh 2 2 2L L 2 2
αβγL 2
2
2
9(18) of 20
2 −4β2 α2 γ2 L2 + 2βγ + 4γ αL sinh αL 12 2 β2 + γ2 2βγ γ2 2 αL αL 2 + 2αL sinhαl − αL sinh 2 − 2
+
− 2γ2l cosh tanh
2
(18)
1
4 , L L L L L L 2 L αl L 2 tanh sinh 2 1 sinh 2 tanh sinhtanh 4 sinh 4 4 2 2 4 4 2 2 , γ = . .
(19)
(19) 2 αL 2 αL αL αL αL αL αL sinh + − sinh sinh + − sinh 2tanh αL 2tanh 4 4 2 2 4 4 2 2 Figure 8 shows a comparison of the resonant frequency between the Euler–Bernoulli beam Figure 8 shows a comparison of the resonant frequency betweenmass. the Euler–Bernoulli beam model model and the Rayleigh’s energy method as a function of attached The result obtained using and the Rayleigh’s energy method as a function of attached mass. The result obtained using Rayleigh’s Rayleigh’s energy method was a little higher than that obtained by the continuum elastic method energythe method a little higher obtained by the continuum elastic method when the mass when mass was of the sensor wasthan lessthat than 10 3 zg (the error range being within 2.4%). It was of the sensor was less than 10−3 zg (the error range being within 2.4%). It was observed that the two observed that the two results were very consistent with each other for increasing attached mass. The results were very consistent with each other for increasing attached mass. The error range was within 1 zg . The error range when was within only 0.06% thetoattached reached to 10models two analytical only 0.06% the attached mass when reached 10−1 zg. mass The two analytical were proven to be models were proven to be successful andagreement effective owing to athem. good agreement between them. successful and effective owing to a good between Based Based on on the the Euler–Bernoulli Euler–Bernoulli beam beam model, model, Figure Figure 99 shows shows the the resonant resonant frequency frequency shifts shifts vs. vs. attached concentrated mass for the CNT beam with axial force and without axial force. The attached concentrated mass for the CNT beam with axial force and without axial force. The results results show show that that the the frequency frequency shift shift of of the the CNT CNT resonator resonator was was increased increased by by axial axial tensile tensile loads, loads, and and the the difference in the frequency shifts increased with the increase of attached mass. The high tensile difference in the frequency shifts increased with the increase of attached mass. The high tensile strength strength of CNT 63 suggested GP [44]) suggested that higher mass sensitivity could beby obtained by of CNT (up to 63(up GP to [44]) that higher mass sensitivity could be obtained increasing increasing tensile force acting on the CNTs. The frequency shift of the CNT resonator for an attached tensile force acting on the CNTs. The frequency shift of the CNT resonator for an attached mass larger −1 zgthan mass larger found upan toaxial 28% tensile when load an axial load of 20 nN, 10 1 zgtowas than 10 was found increase uptotoincrease 28% when of 20tensile nN, corresponding to corresponding to of the tensile of about 19 GPa, applied on the showed CNTs. The showed the tensile stress about 19 stress GPa, was applied on thewas CNTs. The results thatresults the sensitivity 1 − 1 of CNT-based sensor under the tensile canthe reach at least which demonstrates 10 zg , higher that the sensitivity of CNT-based sensorload under tensile load10 can zg, reach at least which accuracy than the 1 zg sensitivity reported by Li and Chou [37]. demonstrates higher accuracy than the 1 zg sensitivity reported by Li and Chou [37].
Figure Figure 8. 8. Comparison Comparison of of the the resonant resonant frequency frequency between between the the Euler–Bernoulli Euler–Bernoulli beam beam model model and and the the Raleigh’s energy method. Adapted with permission from [42], Copyright Springer, 2014. Raleigh’s energy Adapted with
Furthermore, in Ref. [43], Natsuki et al. proposed using the nonlocal Euler–Bernoulli beam model for investigating the resonant frequency of CNT resonators. The nonlocal elasticity theory was developed to incorporate the size effect by introducing an intrinsic length scale, which gave the information about the forces between atoms. Nonlocal beam theory has been widely used to study the vibration properties of CNTs and has been proven to be more effective than other theoretical approaches such as the Euler beam model. In the case of nanostructures, the nonlocal effects considered
developed to incorporate the size effect by introducing an intrinsic length scale, which gave the information about the forces between atoms. Nonlocal beam theory has been widely used to study the vibration properties of CNTs and has been proven to be more effective than other theoretical approaches such as the Euler beam model. In the case of nanostructures, the nonlocal effects considered in the nonlocal elasticity theory played an important role in the vibration analysis and Electronics 2017, 6, 56 10 of 20 determined by the magnitude of nonlocal parameters [45]. For the theoretical approach considering nonlocal parameter e0a L ( e0 is material constant, in the theory played important in the vibration analysis and determined by a thenonlocal internalelasticity characteristic length an and L is therole external characteristic length), the resonant the magnitude of nonlocal parameters [45]. frequency of the CNT resonators based on Rayleigh’s energy method can be obtained as the following: For the theoretical approach considering nonlocal parameter e0 a/L (e0 is material constant, a the k eq internal characteristic length and L is the external characteristic length), the resonant frequency of the 1 (20) CNT resonators based on Rayleigh’s energy method can be obtained as the following: 2 AL mc , s 1 definedk eqas Equation (18). The parameter is given where k eq is defined as Equation (16), and ω= , (20) 2π (φ + µ)ρAL + mc by
2 2 (16), and φ defined as Equation (18). The parameter where k eq is defined as Equation µ is given by L sinh L cosh L 4L cosh L 2 e a 2 2 0 α ( β2 + γ2 ) L (21) Le a 2 αL 2 2 2 2 sinhαL − cosh αL + 4αβγL cosh L 3 L 0 2 2 2 µ= . (21) 4 L sinh 3α2 L2 γ2 − β2 3L (2 ) L . 2 + − 4αγ2 Lsinh αL − 3αβγL
2
2
Figure 10 10 shows shows the the variation variation of of the the resonant resonant frequency frequency of of the the clamped clamped CNT CNT with with the the attached attached Figure mass. The Theinfluence influenceofof nonlocal parameter the frequency was investigated, a L mass. thethe nonlocal parameter e0 a/Le0on theon frequency shift wasshift investigated, showing showingwith increase with increasing attached mass, especially larger than 1.0mass zg. The mass sensitivity increase increasing attached mass, especially for largerfor than 1.0 zg. The sensitivity of CNT of CNT resonator usually depends on the variation of the frequency shift with attached mass. resonator usually depends on the variation of the frequency shift with attached mass. Therefore, Therefore, the results indicated that the nanomass in the order of 0.001–1.0 zg had higher sensitivity the results indicated that the nanomass in the order of 0.001–1.0 zg had higher sensitivity to the to theresonator. CNT resonator. CNT
Figure 9. Resonant frequency shifts vs. attached concentrated mass for the CNT beam with axial force Figure 9. Resonant frequency shifts vs. permission attached concentrated mass for the CNT beam and without axial force. Adapted with from [42], Copyright Springer, 2014.with axial force and without axial force. Adapted with permission from [42], Copyright Springer, 2014.
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Figure 10. Effect of the nonlocal coefficienton on the the resonant resonant frequency of of thethe CNTs under attached Figure 10. Effect of the nonlocal coefficient frequency CNTs under attached different nanomasses. Adapted with permissionfrom from [43], [43], Copyright 2015. different nanomasses. Adapted with permission CopyrightSpringer, Springer, 2015. Figure 10. Effect of the nonlocal coefficient on the resonant frequency of the CNTs under attached
different nanomasses. Adapted permission from [43], Copyright Springer, 2015. 3. CNT Probes for Atomic Force with Microscope
3. CNT Probes for Atomic Force Microscope
3. CNT Probes and for Atomic Force Microscope 3.1. Fabrication Structures of CNT Tip
3.1. Fabrication and Structures of CNT Tip
Atomic Force Microscope (AFM) was a very useful instrument for imaging, measuring and
3.1. Fabrication Structures of(AFM) CNT Tip Atomic Forceand Microscope a verytheuseful forsilicon imaging, measuring characterizing nanoscale features [46].was However, poorlyinstrument characterized and nitride probe and Force Microscope (AFM) was verythe useful instrument for imaging, andprobe characterizing nanoscale features [46]. poorly characterized silicon and can nitride tips Atomic currently employed in AFM limitHowever, somea applications since conventional siliconmeasuring tips easily characterizing nanoscale features [46].some However, the poorly characterized andfor nitride tips currently employed inon AFM limit applications since conventional silicon tips probe can easily break during an impact the scanned surface. CNTs are potentially idealsilicon materials AFM probe currently employed in AFM limit surface. some properties applications since conventional siliconsuch tipsfor can application to their robust mechanical and well-defined geometry as AFM aeasily small breaktips during andue impact on the scanned CNTs are potentially ideal materials probe break during an impact on the scanned surface. CNTs are potentially ideal materials for AFM probe diameter and high aspect ratio [47–49]. There were some reports on the use of CNT probes tip increase application due to their robust mechanical properties and well-defined geometry such as a small application due to the their robust mechanical properties andgroup well-defined geometry such as a small the sensitivity in the AFM.were The research of Dai first attached individual diameter and high and aspect resolution ratio [47–49]. There some reports onProf. the use of CNT probes tip increase diameter and high aspect ratio [47–49]. There were some reports on the use of CNT probes tip increase CNTs with several micrometers to the silicon microcantilever sensing tips of an AFM [50]. They the sensitivity and the resolution in the AFM. The research group of Prof. Dai first attached individual the sensitivity and in theresistant AFM. The of Prof. Daibecause first attached demonstrated thatthe theresolution CNT tips were toresearch damage group from tip crashes of theirindividual flexibility CNTsCNTs withwith several micrometers totothe silicon microcantilever sensing tips of an AFM [50]. several micrometers the silicon microcantilever sensing tips of an AFM [50]. They They and stiffness. Figure 11 shows the SEM image of a CNT scanning probe (a), and a schematic showing demonstrated that the CNT tips were resistant to damage from tip crashes because of their flexibility demonstrated thatCNT the CNT were totrenches damage with from deep tip crashes because of their(b). flexibility the ability of the tip totips trace theresistant profile of and narrow features Due to and stiffness. Figure 11 11 shows the image of a CNT scanning probe a probe, schematic showing and stiffness. Figure theSEM SEM imagewith scanning probe (a),(a), andand a schematic showing the large aspect ratio ofshows CNT, AFM, coupled aCNT high-aspect-ratio CNT scanning exhibited the ability of the CNT tip to trace the profile of trenches with deep and narrow features (b). Due toto the the ability of the CNT tip to trace the profile of trenches with deep and narrow features (b). Due the ability to resolve features. Their exceptional mechanical strength and the ability to buckle the large aspect ratio of CNT, AFM, coupled with a high-aspect-ratio CNT scanning probe, exhibited large reversibly aspect ratio of CNT, AFM, coupled with high-aspect-ratio CNT scanning probe, exhibited the enabled the resolution of steep andadeep nanoscale features. thetoability to features. resolve features. Their exceptional mechanical strength and the ability to buckle ability resolve Their exceptional mechanical strength and the ability to buckle reversibly reversibly enabled the resolution of steep and deep nanoscale features. enabled the resolution of steep and deep nanoscale features.
(a)
(b)
(a) CNT (MWCNT) tip and(b) Figure 11. SEM image of a multi-walled schematic of the CNT tip to trace the profile of trenches. (a) SEM image of a CNT scanning probe, (b) the ability of the CNT tip to trace the Figure 11. SEM image of of a multi-walled tipand andschematic schematic of the CNT to trace Figure 11. SEM image a multi-walledCNT CNT (MWCNT) (MWCNT) tip of the CNT tip totip trace the the profile of trenches. (a) SEM image of a CNT scanning probe, (b) the ability of the CNT tip to trace profile of trenches. (a) SEM image of a CNT scanning probe, (b) the ability of the CNT tip to trace the the profile of trenches with deep and narrow features. Adapted with permission from [51], Copyright AIP Publishing LLC, 2002.
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profile of trenches with deep and narrow features. Adapted with permission from [51], Copyright AIP2017, Publishing LLC, 2002. Electronics 6, 56 Electronics 2017, 6, 56
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The investigation and development of CNT AFM probes includes the fabrication process of CNT profile of trenches with deep and narrow features. Adapted with permission from [51], Copyright Theand investigation and development AFM probes process of CNT probes their mechanical stability of inCNT structures. CNT includes probes the can fabrication be provided by manual AIP Publishing LLC, 2002. probes and their mechanical stability in structures. CNT probes can be provided by manual attachment attachment of a CNT to the tip of the AFM cantilever or by growing a CNT from the ends of the of a CNT to tip of the AFM cantilever orCNT byvapor growing a CNT from theprocess. ends ofThe the silicon tip of the investigation and development of AFMdeposition probes includes the fabrication process of CNT silicon tipThe ofthe the AFM probe based on chemical (CVD) welding method AFM probe based on chemical vapor deposition (CVD) process. The welding method was developed probes and their mechanical stability in structures. CNT probes can be provided by manual was developed for fixing CNT to the silicon probe end [52,53]. CVD methods reported by Prof. Hafner attachment of the a CNT to the tip ofend theof AFM cantilever or by growing CNT from the ends for fixing CNT to silicon probe [52,53]. CVD methods reported by Prof. Hafner etofal.the [54,55] et al. [54,55] allowed aligned growth CNTs on the silicon probe asurface, and the diameter and silicon tip of the AFM probe based on chemical vapor deposition (CVD) process. The welding method allowed growth of CNTs on silicon the probe surface, diameter and length of CNT length ofaligned CNT could be controlled bythe adjusting growth timeand andthe its growth temperatures (Figure was for fixing CNT to silicon time probeand end [52,53]. CVD methods reported by Prof. Hafner could be developed controlled thethe growth growth (Figure [56]. Hafner 12) [56]. Hafner etby al.adjusting successfully fabricated CNTitsAFM tipstemperatures using ethylene and12)iron catalysis et al. [54,55] allowed aligned growth of CNTs on the silicon probe surface, and the diameter and et al. successfully fabricatedsilicon-cantilever-tip CNT AFM tips using assemblies. ethylene andIndividual iron catalysis deposited on commercial deposited on commercial CNTs grew from the pores length of CNT could be controlled by adjusting the growth time and its growth temperatures (Figure silicon-cantilever-tip assemblies. Individual CNTs grew from the pores etchediron in the flattened area at etched in the flattened area at the tip after reaction of the electrodeposited catalytic tips with 12) [56]. Hafner et al. successfully fabricated CNT AFM tips using ethylene and iron catalysis the tip after reaction of the electrodeposited iron catalytic tips with ethylene at elevated temperature. ethylene at elevated temperature. SEM images assemblies. demonstrated that these tips deposited on commercial silicon-cantilever-tip Individual CNTsnanotube grew from thegrew poresin an SEM images that tips12a). grew in an idealand orientation forfabricated AFM imaging idealetched orientation for AFM imaging (see Zettl [57] and indemonstrated the flattened area at these the tipnanotube after Figure reaction of the Cumings electrodeposited iron catalytic tips with (see Figure 12a). Cumings and Zettl [57] fabricated and demonstrated the controlled and reversible ethylene atthe elevated temperature. SEM images demonstrated that these nanotube tips 13 grew in an TEM demonstrated controlled and reversible telescopic extension of CNTs. Figure shows telescopic offor CNTs. 13 shows TEM images of aatelescoped which and hasupon nine ideal orientation AFMFigure imaging (see 12a). Cumings and Zettlnanotube, [57]was fabricated images of extension a telescoped nanotube, which hasFigure nine shells and four-shell core extracted demonstrated controlled and reversible telescopic extension of CNTs. Figure 13 shows TEM had shells and a four-shell core was extracted telescoping. Their promise results suggested that MWCNTs telescoping. Theirthe results suggested thatupon MWCNTs had great for the nanomechanical or images of a telescoped nanotube, which has nine shells and a four-shell core was extracted upon great promise for the nanomechanical or NEMS applications. The preparation of DWCNTs with a short NEMS applications. The preparation of DWCNTs with a short outer wall could be carried out by the telescoping. Their resultsout suggested that MWCNTs had greatouter promise the nanomechanical or [58]. outer wall be the carried by using the method of burning wallfor using an electricDWCNTs current method of could burning outer wall an electric currentthe [58]. Compared to SWCNTs, or NEMS applications. The preparation of DWCNTs with a short outer wall could be carried out by the Compared to SWCNTs, DWCNTs or tube MWCNT with probes a protruding tube asdue AFM would be MWCNT with a protruding inner as AFM wouldinner be better to probes the high spatial method of burning the outer wall using an electric current [58]. Compared to SWCNTs, DWCNTs or better due toUsing the high spatial resolution. Using DWCNT, with a shorter outer wall, the CNTto probes resolution. with a shorter thewould CNT probes expected have were high MWCNT withDWCNT, a protruding inner tube asouter AFMwall, probes be betterwere due to the high spatial expected to have high resolution and high stability due to thick-walled structures and small probe resolution and high stability due to thick-walled structures and small probe tips. resolution. Using DWCNT, with a shorter outer wall, the CNT probes were expected to have high tips. The resolution of AFM is due mainly determined by the probe’s shape resolution AFM mainly resolution and high stability to thick-walled structures and small probeand tips.mechanical properties, especially the dimension probe end tip.tip. A fundamental issue of the application forproperties, CNTs as AFM ofofAFM is probe mainly determined by the probe’s shape especiallyThe theresolution dimension ofthe the end A fundamental issue ofand themechanical application for CNTs as probes is to how and of obtain the buckling of CNTs with awith small tip diameter. By dimension the probe end Astability fundamental issue of the application for as now, AFMespecially probes isthe todesign how design and obtain thetip. buckling stability of CNTs a small tip CNTs diameter. By probes is to how design and obtain theinstability buckling stability of CNTs with a smallwhen tip diameter. By axial there are several methods to investigate the properties of CNTs when subjected to an now,AFM there are several methods to investigate the instability properties of CNTs subjected to an now, there are several methods to investigate the instability properties of CNTs when subjected to an force. instability phenomenon of CNTs under compressive loading was observed axial The force. The instability phenomenon of CNTs under compressive loading experimentally was observed axial[59–62]. force. The instability phenomenon of CNTs under compressive loading was observedby in by TEM Figure shows the direct14observation nanotube buckling characterization experimentally by TEM14[59–62]. Figure shows theofdirect observation of nanotube buckling experimentally by TEM [59–62]. Figure 14 shows the direct observation of nanotube buckling situ transmission by TEM. The transmission TEM images of Figure series of deformation processes characterization in situ TEM. The14a–f TEMclarified imagesaof Figure 14a-f clarified a seriesfor of characterization by in situ transmission TEM. The TEM images of Figure 14a-f clarified a series of CNTs under compression deformation processes forforce. CNTs under compression force. deformation processes for CNTs under compression force.
Figure 12. Electron microscopy of chemical vapor deposition (CVD) nanotube tips. (a) SEM image of
Figure 12. Electron microscopy of chemical vapor deposition (CVD) nanotube tips. (a) SEM image of Figure Electron microscopy of chemical vapor deposition nanotube image of an 12. MWCNT grown from a silicon-cantilever-tip assemblies; (b)(CVD) TEM image of the tips. CNT(a) tip.SEM Adapted an MWCNT grown from a silicon-cantilever-tip assemblies; (b) TEM image of the CNT tip. Adapted with permission Copyright PNAS, 2000. an MWCNT grown from from[55], a silicon-cantilever-tip assemblies; (b) TEM image of the CNT tip. Adapted with permission from [55], Copyright PNAS, 2000. with permission from [55], Copyright PNAS, 2000.
Figure 13. TEM image of a telescoped nanotube. Adapted with permission from [57], Copyright AAAS, 2000.
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Electronics 2017, 6, 56 13 of 20 Figure 13. TEM image of a telescoped nanotube. Adapted with permission from [57], Copyright
AAAS, 2000.
However, the challenge because of However, the dynamics dynamicsexperiments experimentsofofnanotube nanotubebuckling bucklingremains remainsa major a major challenge because the nanometric scale of CNTs. Until now, the experimental investigation is very difficult to accurately of the nanometric scale of CNTs. Until now, the experimental investigation is very difficult to measure themeasure bucklingthe load. Thus, the theoretical is mainly usedisfor predicting loads accurately buckling load. Thus, thestudy theoretical study mainly used the for buckling predicting the and buckling modes of CNT structures. buckling loads and buckling modes of CNT structures.
Figure 14. Series of TEM images of deformation processes for MWNTs initiated by applying Figure 14. Series of TEM images of deformation processes for MWNTs initiated by applying compressive force in the sample direction. Adapted with permission from [62], Copyright MDPI, 2012. compressive force in the sample direction. Adapted with permission from [62], Copyright MDPI, 2012.
3.2. Theoretical Mode and Approach of Buckling Properties 3.2. Theoretical Mode and Approach of Buckling Properties Under axial compression, CNTs are not so strong compared with tension loads due to the high Under axial compression, CNTs are not so strong compared with tension loads due to the high aspect ratio. Therefore, the buckling property of CNTs is very important in order to understand the aspect ratio. Therefore, the buckling property of CNTs is very important in order to understand strength and failure of CNTs used as CNT probes of AFM. To investigate buckling behavior of the the strength and failure of CNTs used as CNT probes of AFM. To investigate buckling behavior CNTs, some theoretical research of buckling analysis, such as compression buckling [63–65], torsional of the CNTs, some theoretical research of buckling analysis, such as compression buckling [63–65], buckling [66], bending buckling [67], and radial buckling [68], has been carried out in the last decade. torsional buckling [66], bending buckling [67], and radial buckling [68], has been carried out in the The first detailed analysis on the buckling of SWCNTs under axial compression was reported by last decade. The first detailed analysis on the buckling of SWCNTs under axial compression was Yakobson et al. [69] based on both MD method and continuum shell model. They found that CNTs reported by Yakobson et al. [69] based on both MD method and continuum shell model. They found could be reversibly switched into different morphological patterns when subjected to large that CNTs could be reversibly switched into different morphological patterns when subjected to large deformations. As shown in Figure 15, the buckling phenomenon of CNT subjected to axial deformations. As shown in Figure 15, the buckling phenomenon of CNT subjected to axial compression compression was simulated. Figure 15a displays four singularities corresponding to shape changes was simulated. Figure 15a displays four singularities corresponding to shape changes of the CNT with of the CNT with a length of 6 nm and diameter of 1 nm. The presence of four singularities at higher a length of 6 nm and diameter of 1 nm. The presence of four singularities at higher strains was quite a strains was quite a striking feature, and the patterns shown in Figure 15b–e illustrate the striking feature, and the patterns shown in Figure 15b–e illustrate the corresponding morphological corresponding morphological changes. The results showed that CNTs were remarkably resilient, changes. The results showed that CNTs were remarkably resilient, sustaining extreme strain without sustaining extreme strain without brittleness, plasticity, or atomic rearrangement. They also indicated brittleness, plasticity, or atomic rearrangement. They also indicated that the peculiar behavior beyond that the peculiar behavior beyond Hooke’s law could be well described by a continuum shell model. Hooke’s law could be well described by a continuum shell model. Comparing the results obtained by Comparing the results obtained by continuum shell model with those from MD simulations, continuum shell model with those from MD simulations, Yakobson et al. found that the results were in Yakobson et al. found that the results were in good agreement with one another. good agreement with one another.
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Figure Figure 15. 15. Series SeriesofofTEM TEMimages imagesofofdeformation deformationprocesses processesfor for MWNTs MWNTs initiated initiated by by applying applying compressive force in the sample direction. Adapted with permission from [69], Copyright compressive force in the sample direction. Adapted with permission from [69], CopyrightAmerican American Physical PhysicalSociety, Society, 1996. 1996.
The Donnell shell models (DSM) has been widely used to evaluate the buckling properties of The Donnell shell models (DSM) has been widely used to evaluate the buckling properties of CNTs CNTs under compression, bending and torsion. Silvestre [70] presented a study on the buckling under compression, bending and torsion. Silvestre [70] presented a study on the buckling behaviors of behaviors of SWCNTs under torsion by using DSM. Straightforward analytical expressions to SWCNTs under torsion by using DSM. Straightforward analytical expressions to calculate the critical calculate the critical angle of twist for CNTs were derived. Despite its simplicity, the procedure angle of twist for CNTs were derived. Despite its simplicity, the procedure presented was shown to give presented was shown to give accurate simulation results for a wide range of CNT length and accurate simulation results for a wide range of CNT length and diameter. Hu et al. [71] investigated diameter. Hu et al. [71] investigated the effects of CNT microstructure on the wave propagation of the effects of CNT microstructure on the wave propagation of both SWCNTs and DWCNTs, based on a both SWCNTs and DWCNTs, based on a nonlocal shell model. The nonlocal elastic shell theory was nonlocal shell model. The nonlocal elastic shell theory was showed to provide better results for the wave showed to provide better results for the wave dispersion than the classical shell theory. There is good dispersion than the classical shell theory. There is good agreement between the elastic shell theory and agreement between the elastic shell theory and the MD method. The nonlocal parameters could be the MD method. The nonlocal parameters could be suggested by MD-based estimation. suggested by MD-based estimation. Based on the successful application of the cylindrical shell model for SWCNT [69], the Based on the successful application of the cylindrical shell model for SWCNT [69], the multiplemultiple-shell model [70–74] and the beam model [75,76] were proposed and developed for simulating shell model [70–74] and the beam model [75,76] were proposed and developed for simulating buckling properties of MWCNTs under axial compression. The adjacent nanotubes for MWCNTs were buckling properties of MWCNTs under axial compression. The adjacent nanotubes for MWCNTs coupled to each other by the vdW interaction. The vdW interaction force pk(k+1) between tube k and p k k 1 between were to each other the The vdW interaction forcedeflection k + 1 coupled of an N-walled CNT (by k= 1, vdW 2, · · ·interaction. , N − 1) is proportional to the radial − wkk) (wk+1tube between thean two neighboring and k+1 of N-walled CNTnanotubes, k 1, 2,,given N 1 asis proportional to the radial deflection wk 1 wk between the two neighboring nanotubes, given as p k ( k +1) = c k ( k +1 ) · ( w k +1 − w k ) ,
p k k 1 ck k 1 wk 1 wk
where the vdW interaction coefficient is given by [74]
,
by [74] where the vdW interaction coefficient is given πε Rk Rk+1 σ6 1001σ6 7 1120 13 c k ( k +1) = Hk(k+1) − Hk(k+1) 4 Rk∆Rk 1 6 10013 6 7 11209 13 ck k 1 H k k 1 H k k 1 9 4 3 and and m Hk(k+1) = ( Rk + Rk+1 )−m
dθ
Z π/2 0
H kmk 1 Rk Rk 1 m
(22)
(23) (23)
4Rk Rk+1 , (24) m/2 (m = 7, 13), Kk(k+1) = ( R k + R k +1 )2 12 − Kk(k+1d)cos2 θ m 7, 13 , K k k 1 4Rk Rk 1 2 (24) 2 m2 Rk Rk 1 , 1 K k k 1 cos
0
(22)
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where is the carbon–carbon bond length, and R is the radius of nanotube k, and are the where ∆ is the carbon–carbon bond length, and Rk isk the radius of nanotube k, σ and ε are the vdW vdW radius and the well depth of the Lennard–Jones potential, respectively. radius and the well depth of the Lennard–Jones potential, respectively. Based on continuum models, the author Natsuki and co-investigators proposed DWCNT tips Based on continuum models, the author Natsuki and co-investigators proposed DWCNT tips with shorter outer wall [77]. The CNT structures are able to increase the stability of AFM nanoprobes with shorter outer wall [77]. The CNT structures are able to increase the stability of AFM nanoprobes and to obtain higher resolution due to small tip diameter. This makes the buckling investigation of and to obtain higher resolution due to small tip diameter. This makes the buckling investigation of DWCNTs with different wall lengths more significant from both theoretical and practical DWCNTs with different wall lengths more significant from both theoretical and practical perspectives. perspectives. Figure 16 shows an analytical model of cantilevered DWCNTs with different inner and Figure 16 shows an analytical model of cantilevered DWCNTs with different inner and outer nanotube outer nanotube lengths, and L2 and L1 are the inner and outer lengths, respectively. lengths, and L2 and L1 are the inner and outer lengths, respectively.
Outer tube Inner tube
P vdW force
L1 L2
Figure16. 16.Schematic Schematicdiagram diagramof ofclamped clampedCNT CNTwith withdifferent differenttube tubelength. length. Figure
As shown in Figure 16, the governing differential equations of the inner and outer nanotubes, in As shown in Figure 16, the governing differential equations of the inner and outer nanotubes, in which adjacent walls are coupled to each other by the vdW interaction forces, can be given by which adjacent walls are coupled to each other by the vdW interaction forces, can be given by d 4 w1 d 2 w1 dEI (25) P c12 w2 w1 0 x L1 , 2 4 w1 4 d w 2 + P 21dx= c12 · (w2 − w1 ) 0 ≤ x ≤ L1 , (25) ( EI )1 41 dx dx dx 4
( EI )2
dEI4 w22 d=w42c ·c21(w w− w2 1 w ) dx4 dx
21
1
2
0 xL 0 ≤ x ≤1 , L1 ,
(26) (26)
d4 w34 d2 w d + w3 P d232 w= (27) 3 (27) EIdx 3 3 Pdx 23 0 0L1L1≤ xx ≤LL2 ,2 , dx dx where w j ( j = 1, 2) are the transverse deflections of the inner and outer nanotubes, respectively, w3 are the transverse deflections of the inner and outer w j j 1, 2 deflection iswhere the transverse of the inner nanotube between L1 and L2nanotubes, , and ( EI ) respectively, ( j = 1, 2, 3) isw3theis
( EI )3
j
flexural stiffnessdeflection of the nanotubes. the transverse of the inner nanotube between L1 and L2 , and EI j j 1, 2, 3 is the In order to obtain the solution of Equations (25)–(26), the corresponding boundary conditions, flexural stiffness of the nanotubes. considering a cantilevered DWCNT subjected to an axial loading P, are given as follows: In order to obtain the solution of Equations (25)–(26), the corresponding boundary conditions, considering a cantilevered subjected to an axial loading P, are given as follows: (a) For the inner and outerDWCNT nanotubes with fixed ends: (a) For the inner and outer nanotubes with fixed ends: dw1 ( x ) dw2 ( x ) w1 ( 0 ) = = w 0 = = 0; 2( ) dw dw2 x dx x=0 dx 1 x x =0 w1 0 w2 0 0 dx x0 dx x0 ; (b) For the inner and outer nanotubes with free ends:
(28) (28)
and outer (b) For the inner nanotubes with free ends: d 2 w2 ( x ) d 3 w2 ( x ) d 2 w3 ( x ) d 3 w3 ( x ) dw3 ( x ) = 0; = = = 0, EI + P (29) ( ) 2 3 2 3 3 dx dx 2 x = L dx3 dx dx x = L2 x= L x= L x= L d 2 w3 x 2 d 3 w3 x 2 dw3 x d w2 x 1 d w2 x 1 (269 0 , EI 3 P 0 dx and dx 2 x L conditions dx 3 xat dx 2 L x,the dx 3of displacement ) x L (c) For the continuous position relationships force 2 L L x L 1 2 1 1 2 ; between the transverse deflections w1 and w3 are given as:
(c)
For the continuous conditions at position L1 , the relationships of displacement and force between the transverse deflections
w1 and w3 are given as:
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w1 L1 w3 L1 ,
dw x dw1 x d 2 w3 x d 2 w1 x 3 , , dx x L1 dx x L1 dx 2 x L dx 2 x L 1
16 of 20
1
(30) x d 3 w x 2 dw x 2 d w1 x dw 3 3 1 dw x dw x d w x d w x ( ) ( ) ( ) ( ) 3EI 1P 1 P = dx = dx32 , w1 ( LEI 3 1 ) 1= w33( L1 ), dx 3, dx2 dx dx x L x = L1x L1 x =dx L1 x = Ldx x = L1 1 x L1 (30) x L 3 1 1 d3 w x 3 ( .x ) 1 (x) + P dwdx + P dwdx = ( EI )3 d wdx33(x) . ( EI )1 dx13( ) x = L1 x = L1 x = L1 x = L1 Substituting the deflection functions of the inner and outer nanotubes, w1 , w2 and w3 , into 3
the above boundary buckling loadinner obtained from eigenvalue Pc can Substituting theconditions, deflection the functions of the andbe outer nanotubes, w , w andanalysis. w , into the 1
2
3
Asboundary shown inconditions, Figure 17,the thebuckling bucklingload instability of obtained CNTs can occur in different mode shapes above Pc can be from eigenvalue analysis. basedAs onshown the proposed theoretical model. The fundamental mode 1, among three modes, has the in Figure 17, the buckling instability of CNTs can occur in different mode shapes based maximum instability, whosemodel. criticalThe buckling strain was 0.0051. The three buckling strains with on the proposed theoretical fundamental mode 1, among modes, has of theCNTs maximum the mode 2 and 3 were 0.024 and 0.07, respectively. Although the only inner nanotube was subjected instability, whose critical buckling strain was 0.0051. The buckling strains of CNTs with the mode 2 and to an axial load, deflection of theAlthough outer nanotube similar to that of the inner 3 were 0.024 andthe 0.07, respectively. the onlyhappens inner nanotube was subjected to annanotubes axial load, through the vdW interaction force between adjacent Figure shows the buckling the deflection of the outer nanotube happens similar nanotubes. to that of the inner18 nanotubes through thestress vdW of the DWCNT as a function of length ratio for the fundamental mode, associated with L L interaction force between adjacent nanotubes. Figure 18 shows the buckling stress of the DWCNT as a Outer Inner the lowestofbuckling load. In the 18, the curve (A) stands the fixedwith innerthe tube and buckling variable function length ratio LOuter /LFigure the fundamental mode,for associated lowest Inner for outer tube, the18, curve (B) for(A) the fixed for outer inner tube.outer The influences of load. In the and Figure the curve stands the tube fixed and innervariable tube and variable tube, and the structural parameter the tube length aspect on inner the buckling stress of DWCNT AFM probes were curve (B) for the fixedofouter and variable tube. The influences of structural parameter of significant based on the buckling length mismatch outer nanotubes. It based was found that the the length aspect stress of between DWCNTinner AFMand probes were significant on the length DWCNT probe with showed Itincreasing stabilityAFM as the outer nanotube mismatchAFM between inner andtype outer(A) nanotubes. was foundbuckling that the DWCNT probe with type (A) elongates. The buckling stress of theasCNT probenanotube was about three times than the of curve (B), showed increasing buckling stability the outer elongates. The larger buckling stress the CNT where the SWCNT or times the inner nanotube fixed. Due tothe theSWCNT reinforcement effectnanotube of the outer probe was about three larger than thewas curve (B), where or the inner was nanotube, to obtain better stability when using the CNT structure covered the fixed. Dueittowas the possible reinforcement effect of the outer nanotube, it was possible to obtain betterwith stability outer probe. when layer usingfor thethe CNT structure covered with the outer layer for the probe.
F
cr 0.0051 Mode 1
F
cr 0.024 Mode 2
F Mode 3
cr 0.070
Figure , 00)@ , 00) Figure 17. 17. Schematic Schematicdiagram diagramdescribing describingthe the buckling buckling instability instabilityofof cantilevered cantilevered (16 16, @(26 26, DWCNTs, with the theinner innertube tubeofof2020nm nm and length ratio of outer to inner tubes 20. Adapted DWCNTs, with and thethe length ratio of outer to inner tubes of 20.ofAdapted with with permission [77], Copyright Elsevier, permission from from [77], Copyright Elsevier, 2011. 2011.
Electronics 2017, 6, 56 Electronics 2017, 6, 56
17 of 20 17 of 20
Figure Figure 18. 18.Variation Variation of ofthe thebuckling bucklingstress stress of ofDWCNTs DWCNTs with with the the length length ratio ratio for forthe thefirst firstbending bending mode mode 1. Adapted with permission from [77], Copyright Elsevier, 2011. 1. Adapted with permission from [77], Copyright Elsevier, 2011.
4. Conclusions 4. Conclusions The discovery of CNTs in the early 1990s has stimulated ever-broader science and engineering The discovery of CNTs in the early 1990s has stimulated ever-broader science and engineering devoted to production and application of various CNTs. Due to their special structures, exceptional devoted to production and application of various CNTs. Due to their special structures, exceptional mechanical and physical properties, CNTs hold many potential applications in the nanotechnology mechanical and physical properties, CNTs hold many potential applications in the nanotechnology industry. Among the mechanical behaviors, the vibration and buckling instability have a significant industry. Among the mechanical behaviors, the vibration and buckling instability have a significant impact on the performance of CNT-based nanosensor or nanodevices. Thus, it is crucial for us to impact on the performance of CNT-based nanosensor or nanodevices. Thus, it is crucial for us to understand the mechanical behaviors of the unique CNTs. The review discusses the application of understand the mechanical behaviors of the unique CNTs. The review discusses the application of CNTs used as nanomechanical sensors, and their performance predicted by theoretical model and CNTs used as nanomechanical sensors, and their performance predicted by theoretical model and analysis. Actually, the technology and experimental work on CNT-based sensor is still in its infancy analysis. Actually, the technology and experimental work on CNT-based sensor is still in its infancy stage. Owing to a lack of an efficient or accurate characterization technique, a quantitative stage. Owing to a lack of an efficient or accurate characterization technique, a quantitative experimental experimental study is not feasible for controlling extremely small size of CNTs. With the study is not feasible for controlling extremely small size of CNTs. With the development of CNT development of CNT materials with specific structures and the micro processing technology [78,79], materials with specific structures and the micro processing technology [78,79], it is expected that it is expected that high-performance of CNT-based sensors and nanodevices will be realized in the high-performance of CNT-based sensors and nanodevices will be realized in the near future. near future. Conflicts of Interest: The author declares no conflict of interest. Conflicts of Interest: The author declares no conflict of interest.
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