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Thermal vibration of a single-walled carbon nanotube predicted by semiquantum molecular dynamics Rumeng Liu and Lifeng Wang* Quantum effects should be considered in the thermal vibrations of carbon nanotubes (CNTs). To this end, molecular dynamics based on modified Langevin dynamics, which accounts for quantum statistics by introducing a quantum heat bath, is used to simulate the thermal vibration of a cantilevered singlewalled CNT (SWCNT). A nonlocal elastic Timoshenko beam model with quantum effects (TBQN), which can take the effect of microstructure into consideration, has been established to explain the resulting power spectral density of the SWCNT. The root of mean squared (RMS) amplitude of the thermal vibration of the SWCNT obtained from semiquantum molecular dynamics (SQMD) is lower than that obtained from classical molecular dynamics, especially at very low temperature and high-order modes. The natural frequencies of the SWCNT obtained from the Timoshenko beam model are closer to those obtained from molecular dynamics if the nonlocal effect is taken into consideration. However, the
Received 26th November 2014, Accepted 12th January 2015 DOI: 10.1039/c4cp05495d
nonlocal Timoshenko beam model with the law of energy equipartition (TBCN) can only predict the RMS amplitude of the SWCNT obtained from classical molecular dynamics without considering quantum effects. The RMS amplitude of the SWCNT obtained from SQMD and that obtained from TBQN coincide very well. These results indicate that quantum effects are important for the thermal vibration of the
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SWCNT in the case of high-order modes, short length and low temperature.
1. Introduction Thermal fluctuations are very closely related with the resonance properties of low-dimensional structures as they serve as nanoscale devices.1–4 The thermal vibration problems of a carbon nanotube (CNT), which can be used as a nanoelectronic component5,6 and an AFM tip,7,8 have attracted much research interest. Transmission electron microscopy is used to estimate Young’s modulus of the CNTs by measuring the amplitude of their intrinsic thermal vibrations.9,10 Recently, Moser et al. studied CNT mechanical resonators at cryostat temperatures of 1.2 K and 30 mK using an ultrasensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion.11,12 In addition to experiment methods, molecular dynamics (MD) and continuum models13–19 are very important tools to study thermal vibration in CNTs. However, in classical molecular dynamics (CMD), which does not allow for the description of quantum effects, each dynamical degree of freedom possesses the same average kinetic energy
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, P. R. China. E-mail: [email protected]; Fax: +86 25 8489 2003-8004; Tel: +86 25 8489 2003-8004
5194 | Phys. Chem. Chem. Phys., 2015, 17, 5194--5201
kBT/2 at thermal equilibrium, where kB is the Boltzmann constant and T is temperature. The pure quantum method is difficult to model in detail for the dynamics of many-body systems because of its complexity. However, some studies have shown that the quantum effects should be considered when the CNT vibrates at a low enough temperature.20,21 To overcome these obstacles, different semiclassical methods, which can include quantum effects into the dynamics of nanosystems, have been proposed.22–29 Dammak et al.26 presented semiquantum molecular dynamics (SQMD) that accounts for quantum statistics by introducing a quantum heat bath. In this approximation, both a dissipative force and a Gaussian random force having the power spectral density given by the quantum fluctuation-dissipation theorem30 are introduced in a Langevin-type approach. Very recently, Wang and Hu31 studied the thermal vibration of a single-walled CNT (SWCNT) using classical beam theory with quantum effects taken into consideration instead of the law of energy equipartition. It shows that the root of mean squared (RMS) amplitude of thermal vibration of a SWCNT predicted by the quantum theory is lower than that predicted by the law of energy equipartition. However, the effect of the microstructure of SWCNTs, which has a significant influence on the vibrations and wave propagation of SWCNTs,32 is not included in the beam models. In order to understand the influence of quantum effects to thermal vibrations
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of CNTs, a recent developed SQMD method is used to simulate free thermal vibration of a cantilevered SWCNT. The thermal vibrational spectrum of the SWCNT with quantum effects is presented first in this paper. Moreover, a nonlocal elastic beam theory with quantum effects taken into consideration is established to explain the resulting power spectral density of the SWCNT. This paper is organized as follows. In Section 2, the SQMD method based on Langevin dynamics, which will be used to calculate the RMS amplitude spectrum, is presented. In Section 3, the nonlocal Timoshenko beam model with quantum effects is established. Section 4 outlines a comparison between the resulting power spectral density of the SWCNT calculated by SQMD and CMD. The natural frequency and RMS amplitude of thermal vibration of the SWCNT obtained from the nonlocal elastic Timoshenko beam model with quantum effects (TBQN) and the nonlocal Timoshenko beam model with the law of energy equipartition (TBCN) are listed for analysis. Finally, the paper ends with some conclusions in Section 5.
2. Molecular dynamics model Here, the CMD and the SQMD methods used to study thermal vibrations of SWCNTs are briefly presented. The dynamic equation of each atom in the molecular dynamics simulation is based on the Langevin equation. The dynamics equation of the n-th atom with mass Mn and position rn is : Mn¨rn = Fn Mngrn + Nn, (1) where Fn is the force caused by the interaction with all the other atoms, g is the effective frictional coefficient, and Xn = {xan}a=13 are random forces with the Gaussian distribution. In CMD, Nn and g are related by the fluctuation-dissipation theorem at temperature T hxan(t)xbm(0)i = 2MngkBTdabdnmd(t),
(2)
where dab and dnm are the Kronecker symbols, while d(t) is the Dirac delta function. To include quantum features into molecular dynamics, for an oscillator with frequency u at temperature T, the power spectral density of the random forces is given by the quantum fluctuation-dissipation theorem30 hxanxbmiu = 2MngkBTdabdnmp(u,T),
(3)
where pðu; TÞ ¼
1 hu hu=kB T : þ 2 kB T eðhu=kB T Þ 1
(4)
Different from the white noise generated by eqn (2) in CMD, the stochastic force spectrum is a color noise.33–35 In this paper, a
Fig. 1
technique proposed by Savin et al.35 is used to generate random forces. The dimensionless power spectral density in eqn (4) can be expressed as u 1 pðuÞ ¼ u þ u ; 2 e 1
(5)
where u = hu/kBT is a dimensionless frequency. The dimensionless random vector functions Sn(t) = {Sa,n}a=13 = S0n(t) + S1n(t) of the dimensionless time t = tkBT/h are constructed to generate the color noise. The power spectral density of random forces in eqn (3) can be expressed as: hxanxbmiu = 2MngkBThSanSbmiu,
(6)
hSanSbmiu = hS0anS0bmiu + hS1anS1bmiu.
(7)
and
Here, the random functions S0n(t) and S1n(t) are uncorrelated, and S0n(t) will generate the power spectra 12u, while S1n(t) will u in eqn (5). The first term in eqn (7) gives the generate u e 1 contribution of the zero-point oscillations to the power spectral density of random forces, and it does not need to be taken into account during the CNT thermal-vibration simulations. The random function S1an(t) can be approximated by the sum of two random functions with relatively narrow frequency spectra S1an(t) = c1zan1(t) + c2zan2(t).
(8)
In this sum, the dimensionless random functions zani(t), i = 1, 2, satisfy the equations of motion as i2zani(t) G izani 0 (t) zani00 (t) = Zani(t) O
(9)
where Zani(t) are d-correlated white-noise functions hZani(t)Zbkj(t)i = 2Gidabdnkdijd(t),
(10)
1 = 2.7189, O 2 = 1.2223, G1 = where c1 = 1.8315, c2 = 0.3429, O 2 = 3.2974 are dimensionless parameters. San(t) 5.0142, and G can be generated by solving numerically eqn (8), Nn = {xa,n}a=13 in eqn (1) can be obtained by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xan ðtÞ ¼ xan ð ht=kB T Þ ¼ kB T 2Mn g= hS1an ðtÞ: (11) For a single-walled armchair (5, 5) CNT shown in Fig. 1, the first four rings of atoms at one end of the tube are fixed to simulate cantilever boundary conditions. To avoid the possible boundary effect of the last layer at the free end, the tip position u is the average position of the third last ring of 10 atoms. The molecular dynamics simulations are carried out based on Brenner’s secondgeneration reactive empirical bond order (REBO) potential,36 which has been widely used in many studies on the mechanical behavior of carbon materials. The long-range van der Waals
Molecular structure model of an armchair (5, 5) SWCNT. Atoms in red are fixed, trajectories of the blue ones are recorded during simulations.
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interaction is calculated by the Lennard-Jones 12–6 potential.37 The velocity verlet formulation of the BBK integrator38 with a time step of 1 fs is used during the simulations. The canonical ensemble is often used as initial conditions for trajectories with constant energy dynamics18,19 to calculate the RMS amplitude of the thermal vibration of the CNT. This method can be used in the case in which the heat bath follows the law of equipartition. However, for the system in the color-noise heat bath, the energy will be transferred from the low-frequency modes to the high-frequency ones in the micro-canonical ensemble.35 To avoid this problem, the systems are simulated by Langevin equations with very small friction coefficients for 10 ns at a constant temperature to generate thermal-equilibrium states. The tip displacement u(t) is sampled at 10 fs intervals during the last 2 ns. The RMS amplitude spectrum of the free tip of the SWCNT is pffiffiffi 2 uRMS ðuÞ ¼ uðuÞ; (12) 2 where u(u) is the Fourier transform of u(t), and the one-sided spectrum type is selected during the transform. To increase the accuracy of the calculations, the final result is obtained by averaging 60 independent simulations.
3. Nonlocal Timoshenko beam with quantum effects To predict the mechanical behavior of CNTs with small-scale effects, nonlocal continuum theory, such as the nonlocal Timoshenko beam model has been widely used to study wave propagation and vibration of CNTs.39–43 This section starts with the dynamic equations of a nonlocal Timoshenko beam of uniform cross-section with length L placed along direction x in the (x, y, z) coordinate system. The dynamic equations of the beam are @2 @2w @j @ 2 w rA 1 r0 2 2 ¼ 0; (13a) þ bAG @x @t2 @x @x2 @2 @2j @w @2j rI 1 r0 2 2 þ bGA j EI 2 ¼ 0; 2 @x @t @x @x
d r0 ¼ pffiffiffiffiffi; 12
(14)
where d is the axial distance between two particles in the materials. For the armchair SWCNT, d is the axial distance between two rings of carbon atoms. The boundary conditions of a cantilever beam are wð0; tÞ ¼ 0;
@ 2 wðL; tÞ ¼ 0; @x2
jð0; tÞ ¼ 0;
@ 2 jðL; tÞ ¼ 0: @x2 (15)
The dynamic deflection and slope can be given by ^ jot ; w ¼ we
^ e jot ; j¼j
(16)
ˆ represents the deflection amplitude of the beam, j ^ is where w the slope amplitude of the beam due to bending deformation pffiffiffiffiffiffiffi alone, and j ¼ 1. Let x = x/L.
(17)
Substituting eqn (16) and (17) into eqn (13), one obtains ^L b2 s2 w
^ rr0 2 o2 @ 2 w ^ ^ @2w @j ¼ 0; þ 2 bG @x2 @x @x
(18a)
^ rIr0 2 o2 @ 2 j ^ ^ 1 @w @2j ^þ ¼ 0; þ s2 2 2 b2 r2 s2 1 j @x L bGA @x2 L @x
(18b)
where b2 ¼
rAL4 o2 ; EI
r2 ¼
I ; AL2
s2 ¼
EI : bAGL2
(19)
^ , eqn (18) becomes Eliminating j BC
@2w ^ ^ @4w ^ ¼ 0; þ BD þ L2 þ b2 s2 C þ b2 s2 Dw 4 @x @x2
(20)
where rr0 2 o2 1 bG rIr0 2 o2 2 2 C¼ L s bGA D ¼ L2 b2 r2 s2 1 :
B ¼
(13b)
where w(x, t) is the displacement of section x of the beam in direction y at moment t, j is the slope of the deflection curve of the beam when the shearing force is neglected, A is the cross Ð section area of the beam, I = y2dA is the moment of inertia for the cross section of the beam, b is the form factor of shear depending on the shape of the cross-section. E, r, and G are Young’s modulus, mass density and the shear modulus of the beam, respectively. r0 is a small-scale parameter with a length unit describing the effects of the microstructure on elastic behavior. In many research studies, it has been revealed that the small-scale effect is significant on wave propagation and thermal vibration in CNTs.44–49 However, no experiments have been conducted to predict the magnitude of r0 for CNTs. Wang
5196 | Phys. Chem. Chem. Phys., 2015, 17, 5194--5201
and Hu obtained r0 by a comparative study of the gradient method with atomic lattice dynamics and it yields32
(21)
In the case of
BD þ L2 þ b2 s2 C þ BD þ L2 þ b2 s2 C 2 4BCDb2 s2 1=2 2BC 0; ˆ and j ^ of eqn (20) read50 the solutions w ˆ = C1 cosh a1x + C2 sinh a1x + C3 cos a2x + C4 sin a2x, w (22a) 0
0
0
0
^ ¼ C1 sin ha1 x þ C2 cosh a1 x þ C3 sin a2 x þ C4 cos a2 x; j (22b)
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where 8
> @fjn ðxÞ rLon 2 r0 2 @fwn ðxÞ > > > > > dx > þI : 2 L2 0 A ; @x @x
Ð
(36) In the case of hn { kBT, eqn (36) becomes kB T 9: 2 i rLon 2 Ð 1 h > 2 > dxþ A ð f ð x Þ Þ þ I f ð x Þ > w jn n > 0 = 2 " # 2 2 1 > > @fjn ðxÞ 2 @fwn ðxÞ 2 > > > rLon r0 > > þI dx > : 2 L2 0 A ; @x @x
Dn 2 ¼ 8 > > > >
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Cite this: Phys. Chem. Chem. Phys., 2015, 17, 5194
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Thermal vibration of a single-walled carbon nanotube predicted by semiquantum molecular dynamics Rumeng Liu and Lifeng Wang* Quantum effects should be considered in the thermal vibrations of carbon nanotubes (CNTs). To this end, molecular dynamics based on modified Langevin dynamics, which accounts for quantum statistics by introducing a quantum heat bath, is used to simulate the thermal vibration of a cantilevered singlewalled CNT (SWCNT). A nonlocal elastic Timoshenko beam model with quantum effects (TBQN), which can take the effect of microstructure into consideration, has been established to explain the resulting power spectral density of the SWCNT. The root of mean squared (RMS) amplitude of the thermal vibration of the SWCNT obtained from semiquantum molecular dynamics (SQMD) is lower than that obtained from classical molecular dynamics, especially at very low temperature and high-order modes. The natural frequencies of the SWCNT obtained from the Timoshenko beam model are closer to those obtained from molecular dynamics if the nonlocal effect is taken into consideration. However, the
Received 26th November 2014, Accepted 12th January 2015 DOI: 10.1039/c4cp05495d
nonlocal Timoshenko beam model with the law of energy equipartition (TBCN) can only predict the RMS amplitude of the SWCNT obtained from classical molecular dynamics without considering quantum effects. The RMS amplitude of the SWCNT obtained from SQMD and that obtained from TBQN coincide very well. These results indicate that quantum effects are important for the thermal vibration of the
www.rsc.org/pccp
SWCNT in the case of high-order modes, short length and low temperature.
1. Introduction Thermal fluctuations are very closely related with the resonance properties of low-dimensional structures as they serve as nanoscale devices.1–4 The thermal vibration problems of a carbon nanotube (CNT), which can be used as a nanoelectronic component5,6 and an AFM tip,7,8 have attracted much research interest. Transmission electron microscopy is used to estimate Young’s modulus of the CNTs by measuring the amplitude of their intrinsic thermal vibrations.9,10 Recently, Moser et al. studied CNT mechanical resonators at cryostat temperatures of 1.2 K and 30 mK using an ultrasensitive method based on cross-correlated electrical noise measurements, in combination with parametric downconversion.11,12 In addition to experiment methods, molecular dynamics (MD) and continuum models13–19 are very important tools to study thermal vibration in CNTs. However, in classical molecular dynamics (CMD), which does not allow for the description of quantum effects, each dynamical degree of freedom possesses the same average kinetic energy
State Key Laboratory of Mechanics and Control of Mechanical Structures, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, P. R. China. E-mail: [email protected]; Fax: +86 25 8489 2003-8004; Tel: +86 25 8489 2003-8004
5194 | Phys. Chem. Chem. Phys., 2015, 17, 5194--5201
kBT/2 at thermal equilibrium, where kB is the Boltzmann constant and T is temperature. The pure quantum method is difficult to model in detail for the dynamics of many-body systems because of its complexity. However, some studies have shown that the quantum effects should be considered when the CNT vibrates at a low enough temperature.20,21 To overcome these obstacles, different semiclassical methods, which can include quantum effects into the dynamics of nanosystems, have been proposed.22–29 Dammak et al.26 presented semiquantum molecular dynamics (SQMD) that accounts for quantum statistics by introducing a quantum heat bath. In this approximation, both a dissipative force and a Gaussian random force having the power spectral density given by the quantum fluctuation-dissipation theorem30 are introduced in a Langevin-type approach. Very recently, Wang and Hu31 studied the thermal vibration of a single-walled CNT (SWCNT) using classical beam theory with quantum effects taken into consideration instead of the law of energy equipartition. It shows that the root of mean squared (RMS) amplitude of thermal vibration of a SWCNT predicted by the quantum theory is lower than that predicted by the law of energy equipartition. However, the effect of the microstructure of SWCNTs, which has a significant influence on the vibrations and wave propagation of SWCNTs,32 is not included in the beam models. In order to understand the influence of quantum effects to thermal vibrations
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of CNTs, a recent developed SQMD method is used to simulate free thermal vibration of a cantilevered SWCNT. The thermal vibrational spectrum of the SWCNT with quantum effects is presented first in this paper. Moreover, a nonlocal elastic beam theory with quantum effects taken into consideration is established to explain the resulting power spectral density of the SWCNT. This paper is organized as follows. In Section 2, the SQMD method based on Langevin dynamics, which will be used to calculate the RMS amplitude spectrum, is presented. In Section 3, the nonlocal Timoshenko beam model with quantum effects is established. Section 4 outlines a comparison between the resulting power spectral density of the SWCNT calculated by SQMD and CMD. The natural frequency and RMS amplitude of thermal vibration of the SWCNT obtained from the nonlocal elastic Timoshenko beam model with quantum effects (TBQN) and the nonlocal Timoshenko beam model with the law of energy equipartition (TBCN) are listed for analysis. Finally, the paper ends with some conclusions in Section 5.
2. Molecular dynamics model Here, the CMD and the SQMD methods used to study thermal vibrations of SWCNTs are briefly presented. The dynamic equation of each atom in the molecular dynamics simulation is based on the Langevin equation. The dynamics equation of the n-th atom with mass Mn and position rn is : Mn¨rn = Fn Mngrn + Nn, (1) where Fn is the force caused by the interaction with all the other atoms, g is the effective frictional coefficient, and Xn = {xan}a=13 are random forces with the Gaussian distribution. In CMD, Nn and g are related by the fluctuation-dissipation theorem at temperature T hxan(t)xbm(0)i = 2MngkBTdabdnmd(t),
(2)
where dab and dnm are the Kronecker symbols, while d(t) is the Dirac delta function. To include quantum features into molecular dynamics, for an oscillator with frequency u at temperature T, the power spectral density of the random forces is given by the quantum fluctuation-dissipation theorem30 hxanxbmiu = 2MngkBTdabdnmp(u,T),
(3)
where pðu; TÞ ¼
1 hu hu=kB T : þ 2 kB T eðhu=kB T Þ 1
(4)
Different from the white noise generated by eqn (2) in CMD, the stochastic force spectrum is a color noise.33–35 In this paper, a
Fig. 1
technique proposed by Savin et al.35 is used to generate random forces. The dimensionless power spectral density in eqn (4) can be expressed as u 1 pðuÞ ¼ u þ u ; 2 e 1
(5)
where u = hu/kBT is a dimensionless frequency. The dimensionless random vector functions Sn(t) = {Sa,n}a=13 = S0n(t) + S1n(t) of the dimensionless time t = tkBT/h are constructed to generate the color noise. The power spectral density of random forces in eqn (3) can be expressed as: hxanxbmiu = 2MngkBThSanSbmiu,
(6)
hSanSbmiu = hS0anS0bmiu + hS1anS1bmiu.
(7)
and
Here, the random functions S0n(t) and S1n(t) are uncorrelated, and S0n(t) will generate the power spectra 12u, while S1n(t) will u in eqn (5). The first term in eqn (7) gives the generate u e 1 contribution of the zero-point oscillations to the power spectral density of random forces, and it does not need to be taken into account during the CNT thermal-vibration simulations. The random function S1an(t) can be approximated by the sum of two random functions with relatively narrow frequency spectra S1an(t) = c1zan1(t) + c2zan2(t).
(8)
In this sum, the dimensionless random functions zani(t), i = 1, 2, satisfy the equations of motion as i2zani(t) G izani 0 (t) zani00 (t) = Zani(t) O
(9)
where Zani(t) are d-correlated white-noise functions hZani(t)Zbkj(t)i = 2Gidabdnkdijd(t),
(10)
1 = 2.7189, O 2 = 1.2223, G1 = where c1 = 1.8315, c2 = 0.3429, O 2 = 3.2974 are dimensionless parameters. San(t) 5.0142, and G can be generated by solving numerically eqn (8), Nn = {xa,n}a=13 in eqn (1) can be obtained by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xan ðtÞ ¼ xan ð ht=kB T Þ ¼ kB T 2Mn g= hS1an ðtÞ: (11) For a single-walled armchair (5, 5) CNT shown in Fig. 1, the first four rings of atoms at one end of the tube are fixed to simulate cantilever boundary conditions. To avoid the possible boundary effect of the last layer at the free end, the tip position u is the average position of the third last ring of 10 atoms. The molecular dynamics simulations are carried out based on Brenner’s secondgeneration reactive empirical bond order (REBO) potential,36 which has been widely used in many studies on the mechanical behavior of carbon materials. The long-range van der Waals
Molecular structure model of an armchair (5, 5) SWCNT. Atoms in red are fixed, trajectories of the blue ones are recorded during simulations.
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interaction is calculated by the Lennard-Jones 12–6 potential.37 The velocity verlet formulation of the BBK integrator38 with a time step of 1 fs is used during the simulations. The canonical ensemble is often used as initial conditions for trajectories with constant energy dynamics18,19 to calculate the RMS amplitude of the thermal vibration of the CNT. This method can be used in the case in which the heat bath follows the law of equipartition. However, for the system in the color-noise heat bath, the energy will be transferred from the low-frequency modes to the high-frequency ones in the micro-canonical ensemble.35 To avoid this problem, the systems are simulated by Langevin equations with very small friction coefficients for 10 ns at a constant temperature to generate thermal-equilibrium states. The tip displacement u(t) is sampled at 10 fs intervals during the last 2 ns. The RMS amplitude spectrum of the free tip of the SWCNT is pffiffiffi 2 uRMS ðuÞ ¼ uðuÞ; (12) 2 where u(u) is the Fourier transform of u(t), and the one-sided spectrum type is selected during the transform. To increase the accuracy of the calculations, the final result is obtained by averaging 60 independent simulations.
3. Nonlocal Timoshenko beam with quantum effects To predict the mechanical behavior of CNTs with small-scale effects, nonlocal continuum theory, such as the nonlocal Timoshenko beam model has been widely used to study wave propagation and vibration of CNTs.39–43 This section starts with the dynamic equations of a nonlocal Timoshenko beam of uniform cross-section with length L placed along direction x in the (x, y, z) coordinate system. The dynamic equations of the beam are @2 @2w @j @ 2 w rA 1 r0 2 2 ¼ 0; (13a) þ bAG @x @t2 @x @x2 @2 @2j @w @2j rI 1 r0 2 2 þ bGA j EI 2 ¼ 0; 2 @x @t @x @x
d r0 ¼ pffiffiffiffiffi; 12
(14)
where d is the axial distance between two particles in the materials. For the armchair SWCNT, d is the axial distance between two rings of carbon atoms. The boundary conditions of a cantilever beam are wð0; tÞ ¼ 0;
@ 2 wðL; tÞ ¼ 0; @x2
jð0; tÞ ¼ 0;
@ 2 jðL; tÞ ¼ 0: @x2 (15)
The dynamic deflection and slope can be given by ^ jot ; w ¼ we
^ e jot ; j¼j
(16)
ˆ represents the deflection amplitude of the beam, j ^ is where w the slope amplitude of the beam due to bending deformation pffiffiffiffiffiffiffi alone, and j ¼ 1. Let x = x/L.
(17)
Substituting eqn (16) and (17) into eqn (13), one obtains ^L b2 s2 w
^ rr0 2 o2 @ 2 w ^ ^ @2w @j ¼ 0; þ 2 bG @x2 @x @x
(18a)
^ rIr0 2 o2 @ 2 j ^ ^ 1 @w @2j ^þ ¼ 0; þ s2 2 2 b2 r2 s2 1 j @x L bGA @x2 L @x
(18b)
where b2 ¼
rAL4 o2 ; EI
r2 ¼
I ; AL2
s2 ¼
EI : bAGL2
(19)
^ , eqn (18) becomes Eliminating j BC
@2w ^ ^ @4w ^ ¼ 0; þ BD þ L2 þ b2 s2 C þ b2 s2 Dw 4 @x @x2
(20)
where rr0 2 o2 1 bG rIr0 2 o2 2 2 C¼ L s bGA D ¼ L2 b2 r2 s2 1 :
B ¼
(13b)
where w(x, t) is the displacement of section x of the beam in direction y at moment t, j is the slope of the deflection curve of the beam when the shearing force is neglected, A is the cross Ð section area of the beam, I = y2dA is the moment of inertia for the cross section of the beam, b is the form factor of shear depending on the shape of the cross-section. E, r, and G are Young’s modulus, mass density and the shear modulus of the beam, respectively. r0 is a small-scale parameter with a length unit describing the effects of the microstructure on elastic behavior. In many research studies, it has been revealed that the small-scale effect is significant on wave propagation and thermal vibration in CNTs.44–49 However, no experiments have been conducted to predict the magnitude of r0 for CNTs. Wang
5196 | Phys. Chem. Chem. Phys., 2015, 17, 5194--5201
and Hu obtained r0 by a comparative study of the gradient method with atomic lattice dynamics and it yields32
(21)
In the case of
BD þ L2 þ b2 s2 C þ BD þ L2 þ b2 s2 C 2 4BCDb2 s2 1=2 2BC 0; ˆ and j ^ of eqn (20) read50 the solutions w ˆ = C1 cosh a1x + C2 sinh a1x + C3 cos a2x + C4 sin a2x, w (22a) 0
0
0
0
^ ¼ C1 sin ha1 x þ C2 cosh a1 x þ C3 sin a2 x þ C4 cos a2 x; j (22b)
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where 8
> @fjn ðxÞ rLon 2 r0 2 @fwn ðxÞ > > > > > dx > þI : 2 L2 0 A ; @x @x
Ð
(36) In the case of hn { kBT, eqn (36) becomes kB T 9: 2 i rLon 2 Ð 1 h > 2 > dxþ A ð f ð x Þ Þ þ I f ð x Þ > w jn n > 0 = 2 " # 2 2 1 > > @fjn ðxÞ 2 @fwn ðxÞ 2 > > > rLon r0 > > þI dx > : 2 L2 0 A ; @x @x
Dn 2 ¼ 8 > > > >
