Carbon-Nanotube-Based Motor Driven by a Thermal Gradient - The

Article pubs.acs.org/JPCC

Carbon-Nanotube-Based Motor Driven by a Thermal Gradient I. Santamaría-Holek,*,† D. Reguera,*,‡ and J. M. Rubi*,‡ †

UMDI-Facultad de Ciencias, Universidad Nacional Autónoma de México, Campus Juriquilla, Querétaro 76230, México Departament de Física Fonamental, Universitat de Barcelona, 08028 Barcelona, Spain

‡

ABSTRACT: We present a model describing the dynamics of a thermally activated nanoelectromechanical device. The nanosystem consisted of two coaxial carbon nanotubes of disparate lengths. The presence of strong thermal inhomogeneities induced motion of the shorter nanotube along the track of the longer nanotube. A model combining the actions of frictional, van der Waals, and thermal forces and the effects of noise is proposed and used to reproduce the motions observed in experiments and simulations. The dynamics of the nanomotor reveals the existence of a rich variety of dynamical behaviors and a high sensitivity to noise and initial conditions.

1. INTRODUCTION The behavior of nanoelectromechanical systems (NEMS) is a relevant subject of study because of their versatility as parts of nano- and microdevices having highly efficient energy performance.1 Carbon nanotubes (CNTs) have become one of the most promising building blocks of NEMS1 because of their controllable electrical, mechanical, and surface properties, with applications ranging from force, chemical, and biological sensors to ultra-high-frequency resonators.2−4 The advantages of the tubular shape of CNTs have been exploited as lowdimensional tracks for allowing translational and rotational motions of cargoes and designing nanoscaled bearings and nanomachines.5−13 Multiwalled carbon nanotubes are particularly interesting in this respect because van der Waals interactions between coaxial nanotubes are strongly influenced by their chiralities, so they can be controlled with high precision.14 In particular, interactions between two coaxial double-walled nanotubes (DWNTs) of different lengths and their dependence on the corresponding pairs of chiral numbers was exhaustively examined in refs 14 and 15. Recent experiments have shown the possibility of generating translational and/or rotational motions of the outer wall of a DWNT with different combinations of chiral numbers of the two nanotubes by applying an electrical current.5 The translational motion always occurred from the midpoint toward the electrodes, and this directionality was not altered by inverting the direction of the current, thus eliminating electromigration effects as the driving mechanism. The rotational motion manifested preferentially in a stepwise way. The experiments and supporting simulations suggested that the translation and rotation of the external wall was due to the presence of strong thermal inhomogeneities that induced an inhomogeneous distribution of phonons in the inner nanotube.5,9 The relative intensity of these motions depended on the combination of chiralites,5 thus indicating that the coupling © 2013 American Chemical Society

between the temperature gradient and the van der Waals forces between nanotubes might control the direction of the motion along the low-energy pathways of the potential energy landscapes.5,14 In this article, we present a model that is able to reproduce the main features of the dynamics of the external nanotube as reported in experiments and computer simulations.5,9 The model is formulated in terms of a Langevin equation for the position of the external nanotube that includes the friction force, the van der Waals forces that depend on the chiralities of both nanotubes, the inhomogeneous distribution of temperature in the inner nanotube, and a random force due to thermal fluctuations. These ingredients reproduce the rich variety of observed behaviors and confirm the conjecture5 that the driving force that originates from the temperature inhomogeneities is responsible for the motion of the external nanotube. Unlike many biological molecular motors, these nanomotors operate in the inertial regime, which leads to novel properties and a rich dynamics that could be relevant for their optimal operation. The article is organized as follows: In section 2, we present the Langevin model, eqs 1 and 2, accounting for the dynamics of the thermally activated motor that incorporates the imposed temperature gradient and the presence of thermal fluctuations. Section 3 describes the formulation of an interpolation model accounting for the van der Waals interaction energy between nanotubes that reproduces with high precision the energy landscapes observed in the literature. In section 4, we present the results of the simulations performed using the Langevin model and compare them with experiments and molecular dynamics simulation results. In section 5, we show that the Joule heating effect due to the presence of an electric current Received: November 7, 2012 Revised: January 7, 2013 Published: January 9, 2013 3109

dx.doi.org/10.1021/jp311028e | J. Phys. Chem. C 2013, 117, 3109−3113

The Journal of Physical Chemistry C

Article

Reasonable estimates can be made for the different parameters and functions involved in eqs 1 and 2. For the frictions, we assume that ζrot = ζr2, which implies that the translational, τ = M/ζ, and rotational, τrot = I/ζrot, relaxation times are approximately the same, as was verified for (4,4) @(9,9) DWNTs in refs 18 and 19. As an estimate of the relaxation time, we use the value τ = 350 ps reported in ref 13. For an outer nanotube with 103 carbon atoms and the characteristic velocities reported in simulations for telescopic motions, v ≈ 100 m/s, the order of magnitude of the friction forces is f f ≈ 10 pN. Simulation results indicate that the thermal diffusivity κzz in eq 1 depends on the chiral numbers of the DWNTs.13 A model expression depending on the chiral factors Smn and Cmn that is able to account for all possible behaviors of the magnitude of the force associated with the temperature gradient as a function of the chiral combination is not easy to obtain. Hence, we consider the thermal diffusivity κzz to be a constant

gives rise to a symmetric temperature gradient along the inner nanotube, as observed in the experiments. Finally, conclusions are presented in section 6.

2. LANGEVIN MODEL OF A THERMALLY-DRIVEN DWNT MOTOR Consider two coaxial nanotubes with lengths Lin and Lout (Lout ≪ Lin). The ends of the inner nanotube (in) are fixed, whereas the external nanotube (out) is allowed to perform translational and rotational motions characterized by the axial z(t) and azimuthal θ(t) degrees of freedom. Neglecting motions along the radial direction, the equations of motion for the position of the center of mass and the rotational angle of the outer nanotube are ⎛ dT ⎞ Mz(̈ t ) = −ζz(̇ t ) + κzz⎜ ⎟ + FzvdW + FzR (t ) ⎝ dz ⎠ z(t )

(1)

and Iθ(̈ t ) = −ζrotθ(̇ t ) + rFθvdW + MθR (t )

FT = κzz

(2)

3 acc nout 2 + noutmout + mout 2 2π

(3)

with acc ≈ 0.142 nm being the interatomic-bond distance between two carbon atoms.16 The first terms on the right-hand sides of eqs 1 and 2 represent the translational and rotational friction forces, respectively, experienced by the external nanotube during its motion, where ζ and ζrot are the corresponding friction coefficients. This assumption of a linear relation agrees with experimental reports on the telescoping motions of multiwalled carbon nanotubes.7,8 The second force on the right-hand side in eq 1 results from the inhomogeneous temperature distribution along the inner nanotube that can be created by imposing different temperatures at the ends of the inner tube or by applying an electric current. This force introduces the thermal diffusivity, κzz, which couples the translational motion with the temperature gradient. Similar couplings have been used to analyze water polarization under high temperature gradients.17 The third and second terms on the right-hand side in eqs 1 and 2, respectively, account for the van der Waals force, FzvdW, and torque, rFθvdW, between the inner and outer nanotubes, which are present even in the absence of temperature gradients. Because of the symmetries of these forces, they essentially maintain the radial distance between the tubes constant.7,14 Finally, the last terms in eqs 1 and 2 account for the random force, FzR, and torque, MθR, respectively, due to thermal fluctuations. We assume that both are Gaussian-distributed with vanishing mean and that they obey the fluctuation− dissipation relations ⟨FzR (t ) FzR (0)⟩ = 2kBTζδ(t )

(4)

⟨MθR (t ) MθR (0)⟩ = 2kBTζrotδ(t )

(5)

(6)

The values of κzz for two particular combinations of chiral numbers were calculated in ref 13 by fitting the simulated data of the force FT for different temperature gradients ΔT/Lin using eq 6 (see Figure 4 of ref 13). The values obtained for the thermal diffusivities were κzz ≈ 1.5 × 10−21 J/K for the combination (8,2)@(17,2) and κzz ≈ 2.29 × 10−21 J/K for (6,4) @(14,5). Therefore, for temperature gradients in the range of 3−10 K nm−1,5,13 these values predict forces in the range of FT ≈ 4.5−15 pN, which, compared with the estimates of the friction, f f ≈ 10 pN, and the effective van der Waals forces, FvdW 0 ≈ 10 pN, support the conjecture that the temperature gradient is the driving force for the motion of the external nanotube.

where M is the mass of the nanotube, I = Mr2 is its moment of inertia, and dots over a variable indicate a time derivative. The radius of the nanotube is given by the combination of chiral numbers (mout,nout) r=

dT dz

3. POTENTIAL ENERGY LANDSCAPE AND EFFECTIVE FORCES FOR DWNTS The effective force FvdW(z,θ) experienced by the outer nanotube originates from the van der Waals interactions between carbon atoms and is characterized by an energy landscape, ϕmn(θ,z), showing patterns that depend in nontrivial way on the chiralities (nin,min)@(nout,mout).14 No simple formula exists to represent this potential energy for an arbitrary combination of nanotubes and chiralities. However, for armchair (nin,nin)@(nout,nout) and zigzag (nin,0) @(nout,0) nanotubes, this landscape is well interpolated by the function15,20 U (z , θ ) = U0 −

⎛ 2π ⎞ ΔUθ ⎛ 2π ⎞ ΔUz cos⎜ z⎟ − cos⎜ θ ⎟ 2 2 ⎝ δθ ⎠ ⎝ δz ⎠

(7)

where U0 is the mean energy of interwall interactions. The energy landscape represented by eq 7 was fitted to the simulation results obtained by molecular dynamics calculations for different armchair and zigzag combinations, and the corresponding values of the parameters ΔUθ, ΔUz, δθ, and δz are listed in refs 15 and 20. We used those values for the combinations (5,5)@(10,10) and (13,0)@(22,0) in our simulations. The obtained potentials are shown in Figure 1, and the corresponding force is just FvdW(z,θ) = −∇U(z,θ). The lack of a general analytical expression to describe the potential energy for arbitrary-chirality DWNTs led us to propose here an empirical interpolation formula that allows

where δ(t) is the Dirac delta function. 3110



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Using FvdW(z,θ) = −∇ϕmn(z,θ), the components of the van der Waals force are given by ⎡n ⎛ z rθ ⎞⎤ in ⎢ + FzvdW = FzvdW C S sin ⎜ ⎟⎥ 0 mn ⎢⎣ nout ⎝ mn acc acc ⎠⎥⎦ ⎡n ⎛ z rθ ⎞⎤ in + Smn ⎟⎥ FθvdW = FθvdW ⎜Cmn 0 sin⎢ ⎢⎣ nout ⎝ acc acc ⎠⎥⎦

(9)

Estimates of the magnitudes of these two forces = FzvdW 0

ϕ0 ⎛ n in ⎞ ⎜ ⎟Cmn acc ⎝ nout ⎠

FθvdW 0 =

ϕ0 ⎛ n in ⎞ ⎜ ⎟Smn acc ⎝ nout ⎠

(10)

can be made using a typical value of ϕ0 ≈ 10 meV, thus obtaining FvdW ≈ 10 pN for both components. It was shown in 0 refs 22−24 that the magnitude of the effective friction force is mainly determined by the corrugation of the potential energy surface.

4. RESULTS Using all of these estimates, we solved eqs 1 and 2 numerically, using a stochastic version of Euler’s algorithm. Figure 2 shows plots of the results for the position z(t) and the orientation θ(t) of the center of mass of the external nanotube in terms of time. The numerical results correspond to (5,5)@(10,10), (13,0) @(22,0), and (8,2)@(17,2) DWNTs subjected to a temperature gradient of ΔT/Lin = 8 K nm−1, mimicking the conditions reported in ref 13. The parameters used in eq 1 were T(0) = 500 K, τ = 350 ps, and κzz = 2 × 10−21 J/K. The (5,5)@(10,10) armchair tube performs a translational motion with small angular oscillations due to the thermal fluctuations. In contrast, the (13,0)@(22,0) zigzag tube in this case rotates clockwise with no net axial displacement, whereas the (8,2)@(17,2) performs a helical motion, following the valleys of the energy landscape (see Figure 1c). It is important to emphasize that the trajectories found depend on the initial conditions and, because of the thermal forces and torques, are different for different realizations of the noise. At sufficiently high temperatures, this thermal force promotes an irregular oscillatory motion of the external nanotube that overcomes the van der Waals potential energy barriers. The results of the present model agree with the simulations reported in ref 13. Figure 3 presents the results obtained from eqs 1 and 2 for (8,2)@(17,2) and (17,0)@(26,0) DWNTs under the same conditions as used for the simulations reported in ref 5: ΔT/Lin = 8 K nm−1. Once again, the model reproduces the trajectories found in the simulations. The different behaviors found for the (8,2)@(17,2) combination in Figures 2 and 3 are clear evidence of the influence of noise on the trajectories.

Figure 1. Potential energy landscapes for three different combinations of chiralities: (a) (5,5)@(10,10); (b) (13,0)@(22,0), obtained from eq 7; and (c) (8,2)@(17,2) obtained using eq 8, with A = −9 and B = 4.

many of them to be mimicked. The proposed effective interaction potential ϕmn(θ,z) is ⎧ ⎫ ⎡n ⎛ ⎪ rθ ⎞⎤⎪ z ⎢ in ⎜Cmn ⎥ ⎬ ϕmn = ϕ0⎨ + + 1 cos S ⎟ mn ⎪ ⎢⎣ nout ⎝ acc acc ⎠⎥⎦⎪ ⎭ ⎩

(8)

5. TEMPERATURE DISTRIBUTION AND GRADIENTS GENERATED BY THE JOULE HEATING EFFECT Finally, we discuss the origin of the thermal gradients that drive the motion of these DWNT motors. In practice, two strategies can be followed to produce an inhomogeneous distribution of phonons along the inner nanotube by means of a temperature gradient. In simulations, the temperature gradient can be obtained by keeping the two ends of the inner nanotube at

where ϕ0 is the mean energy, Cmn = A cos(εnm), and Smn = B sin(εnm) with εnm = (π/4)(min/nin + mout/nout). The parameters A and B can be tuned to reproduce the amplitude, periodicity, and chirality of the energy landscape of several tube combinations with great accuracy. In particular, it reproduced the energy landscape of the (8,2)@(17,2) combination reported in ref 6 and used in our simulations; see Figure 1.21 3111



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Figure 3. (a) Position z(t) and (b) angle θ(t) as functions of time for a single trajectory corresponding to an (8,2)@(17,2) and a (17,0) @(26,0) DWNT, using the same conditions as reported in ref 5.

where ρ is the mass density of the nanotube and cp is its specific heat. The last term represents the Joule heat effect per unit volume. λ and 9 are the thermal conductivity and the electrical resistivity, respectively, which, at T ≈ 300 K, have typical values of λexp ≈ 7 × 103 W m−1 K−1 and Rexp ≈ 104 Ω.27,28 Equation 11 predicts a temperature change of ΔT ≈ 50−5000 K in a (5,5) nanotube of length Lin ≈ 103 nm if a current of I ≈ 0.01− 0.1 mA is applied. This range of values is compatible with temperature differences observed in experiments.5 Equation 11 also predicts the stationary temperature profile T (z) = T0 + az + b(L in − z)z

(12)

where a = (T1 − T0)/Lin, b = 9 i /(2λ). The symmetric temperature profile when T(0) = T(Lin) = 300 K is shown in Figure 4 for four values of the applied current. The profiles predicted are in excellent agreement with those recently measured in similar systems.29 2

6. CONCLUSIONS We have formulated a model that is able to quantitatively describe the dynamics of a nanomotor consisting of two coaxial carbon nanotubes that is activated by a temperature gradient.

Figure 2. Theoretical results for the position z(t) and the dependence of the angle θ(t) as a function of time for (a) a (5,5)@(10,10) system, (b) a (13,0)@(22,0) DWNT, and (c) an (8,2)@(17,2) system.

different temperatures with two heat baths.5,9 In experiments, the temperature gradient can be produced by applying an electrical current (Joule’s effect).5 Theoretically, we assume that the ends of the nanotube are kept at different constant temperatures [T(0) = T0 and T(Lin) = T1] and that an electrical current per unit area i is also applied. The stationary temperature profile is then determined by performing a nonequilibrium thermodynamic analysis.25 Neglecting the coupling between thermal and electrical effects26 and taking into account the symmetry of the problem, this analysis yields the heat equation ρc p

∂ ⎛⎜ ∂T ⎞⎟ ∂T = λ + 9i 2 ∂t ∂z ⎝ ∂z ⎠

Figure 4. Temperature as a function of position obtained from eq 12 for a (5,5) nanotube with Lin = 103 nm, T(0) = T(L) = 300 K, λ = 6.6 × 103 W m−1 K−1 and ℛ = 3.4 × 104 Ω. When a temperature gradient of 0.3 K/nm is imposed through the boundaries, the symmetry is broken (black dotted curve).

(11) 3112



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(13) Hou, Q.-W.; Cao, B.-Y.; Guo, Z.-Y. Thermal gradient induced actuation in double-walled carbon nanotubes. Nanotechnology 2009, 20, 495503. (14) Saito, R.; Matsuo, R.; Kimura, T.; Dresselhaus, G.; Dresselhaus, M. S. Anomalous potential barrier of double-wall carbon nanotube. Chem. Phys. Lett. 2001, 348, 187−193. (15) Belikov, A. V.; Lozovik, Y. E.; Nikolaev, A. G.; Popov, A. M. Double-wall nanotubes: Classification and barriers to walls relative rotation, sliding and screwlike motion. Chem. Phys. Lett. 2004, 385, 72−78. (16) Dresselhaus, M. S.; Dresselhaus, G.; Saito, R. Physics of carbon nanotubes. Carbon 1995, 33, 883−891. (17) Bresme, F.; Lervik, A.; Bedeaux, D.; Kjelstrup, S. Water Polarization under Thermal Gradients. Phys. Rev. Lett. 2008, 101, 020602. (18) Servantie, J.; Gaspard, P. Translational dynamics and friction in double-walled carbon nanotubes. Phys. Rev. B 2006, 73, 125428. (19) Servantie, J.; Gaspard, P. Rotational dynamics and friction in double walled carbon nanotubes. Phys. Rev. Lett. 2006, 97, 186106. (20) Popov, A. M.; Lozovik, Yu. E.; Sobennikov, A. S.; Knizhnik, A. A. J. Exp. Theor. Phys. 2006, 108, 621−628. (21) Equation 8 can also be used to interpolate the potential energy of armchair and zigzag DWNTs with great accuracy. (22) Kolmogorov, A. N.; Crespi, V. H. The smoothest bearings: Interlayer sliding in multiwalled carbon nanotubes. Phys. Rev. Lett. 2000, 85, 4727. (23) Zhao, X.; Cummings, P. T. Molecular dynamics study of carbon nanotube oscillators revisited. J. Chem. Phys. 2006, 124, 134705. (24) Zhang, S.; Liu, W. K.; Ruoff, R. S. Atomistic Simulations of Double-Walled Carbon Nanotubes (DWCNTs) as Rotational Bearings. Nano Lett. 2004, 4, 293−297. (25) de Groot, S. R.; Mazur, P. Nonequilibrium Thermodynamics; Dover: New York, 1980. (26) These cross effects are characterized by the thermoelectric power, εexp ≈ 10−6 V K−1. (27) Berber, S.; Kwon, Y.-K.; Tomanek, D. Unusually High Thermal Conductivity of Carbon Nanotubes. Phys. Rev. Lett. 2000, 84, 4613. (28) Hepplestone, S. P.; Srivastava, G. P. Low-temperature mean-free path of phonons in carbon nanotubes. J. Phys: Conf. Ser. 2007, 92, 012076. (29) Deshpande, V. V.; Hsieh, S.; Bushmaker, A. W.; Bockrath, M.; Cronin, S. B. Spatially Resolved Temperature Measurements of Electrically Heated Carbon Nanotubes. Phys. Rev. Lett. 2009, 102, 105501. (30) Zambrano, H. A.; Walther, J. H.; Jaffe, R. L. Thermally driven molecular linear motors: A molecular dynamics study. J. Chem. Phys. 2009, 131, 241104. (31) Somada, H.; Hirahara, K.; Akita, S.; Nakayama, Y. A. Molecular Linear Motor Consisting of Carbon Nanotubes. Nano Lett. 2009, 9, 62−65. (32) Rurali, R.; Hernández, E. R. Thermally induced directed motion of fullerene clusters encapsulated in carbon nanotubes. Chem. Phys. Lett. 2010, 497, 62−65.

The model accounts for the observed nonequilibrium temperature profile and contains a simple expression for van der Waals forces between nanotubes that reproduces with high accuracy some of the chirality-dependent energy patterns obtained by simulations. The numerical solutions of the model reproduce the trajectories found in molecular dynamics simulations and show that the thermal gradient combined with thermal noise is responsible for the motions observed in experiments and simulations. The model proposed also predicts a series of singular behaviors that could improve the performance of these motors as parts of NEMS. It can also be used to describe the dynamics of other combinations of CNTs, like in the case where the shorter nanotube is inside the longer one.30−32

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (I.S.-H.), [email protected] (D.R.), [email protected] (J.M.R.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the MINECO of Spain through the I3 Program and Grant FIS2008-01299 cofinanced by European Union’s FEDER funds and by UNAM DGAPA ID 100112-2. J.M.R. thanks the Generalitat of Catalunya for the Icrea Academia Award.

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REFERENCES

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Carbon-Nanotube-Based Motor Driven by a Thermal Gradient - The

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