University o Wollongong Research Online Faculty o Inormatics - Papers (Archive) Faculty o Engineering and Inormation Sciences 006 Mathematical modelling or a C60 caron nanotue oscillator Barry J. Cox University o Wollongong, arryc@uow.edu.au Ngamta Thamwattana University o Wollongong, ngamta@uow.edu.au James M. Hill University o Wollongong, jhill@uow.edu.au Pulication Details Cox, B. J., Thamwattana, N. & Hill, J. M. (006). Mathematical modelling or a C60 caron nanotue oscillator. In C. Jagadish & G. Lu (Eds.), Proceedings o 006 International Conerence On Nanoscience and Nanotechnology (pp. 80-83). USA: IEEE. Research Online is the open access institutional repository or the University o Wollongong. For urther inormation contact the UOW Lirary: research-pus@uow.edu.au
Mathematical modelling or a C60 caron nanotue oscillator Astract The discovery o ullerenes C 60 and caron nanotues has created an enormous impact on nanotechnology. Because o their unique mechanical and electronic properties, such as low weight, high strength, lexiility and thermal staility, ullerenes C 60 and caron nanotues are o considerale interest to researchers rom many scientiic areas. One prolem that has attracted much attention is the creation o gigahertz oscillators. While there are diiculties or micromechanical oscillators, or resonators, to reach a requency in the gigahertz range, it is possile or nanomechanical systems to achieve this. A numer o studies have ound that the sliding o the inner-shell inside the outer-shell o a multi-walled caron nanotue can generate oscillatory requencies up to several gigahertz. In addition, it has een oserved that the shorter the inner tue, the higher the requency, leading to the introduction o a C 60 -nanotue oscillator. Thus instead o multi-walled caron nanotues, high requencies can e generated using a ullerene C 60 oscillating inside a single-walled caron nanotue. In this paper, using the Lennard-Jones potential, we determine the potential or an oset C 60 molecule inside a single-walled caron nanotue, so as to determine its position with reerence to the crosssection o the caron nanotue. The condition or the C 60 initially at rest outside the caron nanotue to e sucked in and to start oscillating is also presented together with a mathematical model or the resulting oscillatory motion. This paper summarizes recent results otained y the present authors. Keywords Mathematical, modelling, or, C60, caron, nanotue, oscillator Disciplines Physical Sciences and Mathematics Pulication Details Cox, B. J., Thamwattana, N. & Hill, J. M. (006). Mathematical modelling or a C60 caron nanotue oscillator. In C. Jagadish & G. Lu (Eds.), Proceedings o 006 International Conerence On Nanoscience and Nanotechnology (pp. 80-83). USA: IEEE. This conerence paper is availale at Research Online: http://ro.uow.edu.au/inopapers/1635
Mathematical Modelling or a C 60 Caron Nanotue Oscillator Barry J. Cox, Ngamta Thamwattana* and James M. Hill Nanomechanics Group, School o Mathematics and Applied Statistics, University o Wollongong, Wollongong, NSW 5, Australia *Email: ngamta@uow.edu.au Astract The discovery o ullerenes C 60 and caron nanotues has created an enormous impact on nanotechnology. Because o their unique mechanical and electronic properties, such as low weight, high strength, lexiility and thermal staility, ullerenes C 60 and caron nanotues are o considerale interest to researchers rom many scientiic areas. One prolem that has attracted much attention is the creation o gigahertz oscillators. While there are diiculties or micromechanical oscillators, or resonators, to reach a requency in the gigahertz range, it is possile or nanomechanical systems to achieve this. A numer o studies have ound that the sliding o the inner-shell inside the outer-shell o a multi-walled caron nanotue can generate oscillatory requencies up to several gigahertz. In addition, it has een oserved that the shorter the inner tue, the higher the requency, leading to the introduction o a C 60 -nanotue oscillator. Thus instead o multi-walled caron nanotues, high requencies can e generated using a ullerene C 60 oscillating inside a single-walled caron nanotue. In this paper, using the Lennard-Jones potential, we determine the potential or an oset C 60 molecule inside a single-walled caron nanotue, so as to determine its position with reerence to the cross-section o the caron nanotue. The condition or the C 60 initially at rest outside the caron nanotue to e sucked in and to start oscillating is also presented together with a mathematical model or the resulting oscillatory motion. This paper summarizes recent results otained y the present authors. Keywords-caron nanotues; ullerenes C 60 ; gigahertz oscillators; Lennard-Jones potential I. INTRODUCTION Caron nanotues have many promising applications in new technological devices including in optical, mechanical and electrical systems. This is due to their size and their unique properties, such as high strength, low weight, lexiility and thermal staility. Recently, it has een ound that the oscillating o the inner tue o a multi-walled caron nanotue can generate the oscillatory requencies in the gigahertz range. This inding leads to many potential applications, such as ultraast optical ilters and nano-antennae. Cumings and Zettl [1] experiment on multi-walled caron nanotues, where they remove the cap rom one end o the outer-shell and attach a moveale nanomanipulator to the core in a high-resolution transmission electron microscope (TEM). By pulling the core out and pushing it ack into the outer-shell, they report an ultra low rictional orce against the intershell sliding. This result is also conirmed y Yu, Yakoson and Ruo []. Further, Cumings and Zettl [1] also oserve that the extruded core, ater release, quickly and ully retracts inside the outer-shell due to the restoring orce resulted rom the van der Waals interaction acting on the extruded core. These results led Zheng and Jiang [3] and Zheng, Liu and Jiang [4] to study the molecular gigahertz oscillators, where the sliding o the innershell inside the outer-shell o a multi-walled caron nanotue can generate oscillatory requencies up to several gigahertz. Based on Zheng et al. [4], which state that the shorter the inner tue, the higher the requency, instead o using the inner tue Liu, Zhang and Lu [5] ind that the high requency can e generated y the oscillating o a ullerene C 60 inside a singlewalled caron nanotue. A ullerene C 60 is a stale caron structure, where sixty caron atoms ond to orm an approximate sphere. For urther details o ullerenes, we reer the reader to Dresselhaus, Dresselhaus and Eklund [6]. While Liu et al. [5] and Qian, Liu and Ruo [7] study a C 60 -nanotue oscillator using molecular dynamical simulations, Cox, Thamwattana and Hill [8, 9] employ elementary mechanics and mathematical modelling techniques to investigate this prolem and this is the main contriution o the authors in the area. Further, the results otained in Cox et al. [8, 9] are in a good agreement with numerical results o Girialo, Hodak and Lee [10] and Hodak and Girialco [11], and molecular dynamical simulations o Liu et al. [5] and Qian et al. [7]. Here, we review Cox et al. [8, 9], and summarize the essential mechanisms o a C 60 -nanotue oscillator. For ull details o the mathematical derivations, we reer the reader to Cox et al. [8, 9]. II. POTENTIAL FUNCTION In the continuum approximation, caron atoms are assumed to e uniormly distriuted over the surace o the molecules. As a result, the nononded interaction energy etween two molecules is otained y integrating the interaction energy Φ(r) over the suraces o each entity, namely E = n n Φ r) dσ dσ, (1) g ( g where n g and n denote the mean surace density o atoms on a caron nanotue and a ullerene C 60, respectively, and r is the distance etween two typical surace elements d g and d on each molecule. In this paper, we adopt the well-known Lennard-Jones potential 6 1 + Φ( r ) = Ar Br, () 1-444-0453-3/06/$0.00 006 IEEE 80 ICONN 006
where A and B denote the attractive and repulsive constants respectively. Here, we use A = 17.4 evå 6 and B = 9 10 3 evå 1 or the interaction etween C 60 -graphene (Girialco et al. [10]). III. INTERACTION OF A C 60 FULLERENE LOCATED ON THE AXIS OF A SINGLE-WALLED CARBON NANOTUBE By using (1) and (), we ind the potential o an atom on the caron nanotue o radius a interacting with all atoms o the spherical ullerene o radius to e given y P( ρ) = n n πa ( ρ + ) 4 πb ( ρ + ) ( ρ ) 10 4 ( ρ ) 10 (ρ) (5ρ), where ρ is the distance etween the centre o the C 60 molecule and an atom on the caron nanotue, as shown in Fig. 1. Figure 1. Geometry o a ullerene entering a caron nanotue The van der Waals interaction orce etween the ullerene molecule and an atom on the tue is given y F vdw = P, thus rom Fig. 1 the axial orce is o the orm (3) Figure. Force experienced y a C 60 molecule due to van der Waals interaction with a semi-ininite caron nanotue The integral o F tot z (Z) represents the work imparted to the ullerene and equates directly to the kinetic energy. Thereore, tot integral o F z rom Z = to Z 0 represents the acceptance energy (W a ) or the system and must e positive or a nanotue to accept a ullerene y suction orce alone. I W a is negative, then the magnitude o W a represents the initial kinetic energy needed y ullerene in the orm o the inound initial velocity or it to e accepted into the nanotue. In Fig. 3, we show the acceptance energy W a or a ullerene and a nanotue o radii in the range 6.1 < a < 6.5 Å. From the igure, W a = 0 when a = 6.338 Å and nanotues which are smaller than this will not accept C 60 ullerenes y suction orce alone. This implies that a (10, 10) nanotue (a = 6.784 Å) will accept a C 60 ullerene rom rest, however a (9, 9) nanotue (a = 6.106 Å) will not. The result o this model is well in agreement with Hodak and Girialco [11] and Okada, Saito and Oshiyama [1]. F z ( Z z) dp = (4). ρ dρ As a result, the total axial orce F z tot (Z) o the entire caron nanotue (0 z < ) interacting with the ullerene can e otained y perorming surace integral o (4) over the tue, which upon simpliying, we have 8π n n a tot g Fz ( Z) = 1 4 3 A + λ λ B 80 336 51 56 5 + + + +, 5 6 3 3 4 λ λ λ λ λ where λ = (a + Z )/. In Fig., we plot (5) and ind that equation F z tot (Z) = 0 admits at most two real roots, Z = ± Z 0, when the radius a is less than some critical value a 0, where is given. In the case o a C 60 ullerene ( = 3.55 Å) then a 0 = 6.509 Å. We note rom the orm o (5) that Z only appears in actors o Z. Thereore, the orce is symmetrical aout Z = 0. (5) Figure 3. Acceptance energy threshold or a C 60 molecule to e sucked into a caron nanotue The suction energy W or a ullerene, which is the total work perormed y the van der Waals interactions on a C 60 molecule entering a caron nanotue, can e determined similar to W a ut the integral is perormed over the entire range, rom Z = to. In Fig. 4, we plot W or a C 60 molecule entering a nanotue with radii in the range 6 < a < 10 Å. We note that W is positive when a > 6.7 Å and has a maximum value o W = 3.4 ev at a = a max = 6.783 Å. Accordingly, a (10, 10) caron nanotue with a = 6.784 Å is 81
almost exactly the optimal size to maximize W and thereore have a C 60 ullerene accelerate to a maximum velocity upon entering the tue. Our model predicts that C 60 molecule in a (10, 10) caron nanotue will accelerate to a velocity o 93 m/s; this result is in a reasonale agreement with a molecular dynamical simulation o Qian et al. [7], which predict the velocity o 840 m/s or C 60 molecule entering the tue. π ( n + 1 / ) I n = ( α β cosθ ) dθ, (7) π and α = a + ε and β = aε. We note that the integrals I n can e evaluated in terms o elliptic integrals or in terms o hypergeometric unctions (see Cox et al. [9]). Figure 4. Suction energy or a C 60 molecule entering a caron nanotue IV. PREFERRED POSITION OF A C 60 MOLECULE INSIDE A SINGLE-WALLED CARBON NANOTUBE The preerred position o a C 60 molecule inside a singlewalled caron nanotue is where the molecule admits the minimum potential energy. Here, we determine this location with reerence to the cross-section o a caron nanotue. In axially symmetric cylindrical polar coordinates, we assume that the ullerene C 60 o radius is located at (ε, 0, 0) as shown in Fig. 5, and in a caron nanotue o ininite extent with a parametric equation (acosθ, asinθ, z). We note that ε is the distance etween the centre o the oset C 60 molecule and the central axis o the tue, a is the tue radius, π θ π and < z <. Figure 6. The potential o an oset C 60 molecule inside a (10, 10) and a (16, 16) caron nanotues, with respect to the radial distance ε rom the tue axis In Fig. 6, the potential energy E is plotted with respect to the distance ε. It can e seen that the preerred position or the C 60 molecule inside a (10, 10) caron nanotue is where the centre o the C 60 molecule lies on the tue axis (ε = 0). For a (16, 16) caron nanotue, we ind ε = 4.314 Å. Further, we oserve that as the tue radius gets larger, the location where the minimum energy occurs tends to e closer the nanotue wall. These results agree with the indings o Girialco et al. [10]. V. OSCILLATION OF A C 60 MOLECULE INSIDE A SINGLE- WALLED CARBON NANOTUBE In axially symmetric cylindrical polar coordinates (r, z), we assume a C 60 molecule is located inside a caron nanotue o length L, centred around the z-axis and o radius a. As shown in Fig. 7, we assume that the centre o the C 60 molecule lies on the z-axis. This assumption is valid or a (10, 10) caron nanotue, since the centre o the C 60 molecule is likely to e on the tue axis due to the minimum potential energy. Figure 5. An oset C 60 molecule inside a caron nanotue From Fig. 5, ρ = (a + ε aεcosθ + z ) 1/, thus y perorming surace integral o (3) over the entire caron nanotue, we otain the potential energy E or the oset C 60 molecule inside the caron nanotue, namely A B 105 E = 4π a n ng (3I + 5 I3 ) + ( I5 8 5 18 1155 9009 4 6435 6 1155 8 (6) + I6 + I7 + I8 + I9 ), 64 18 64 56 where the integrals I n are deined y Figure 7. Geometry o the C 60 molecule oscillation Here we adopt Newton s second law to descrie the oscillatory motion o the molecule inside a single-walled caron nanotue, namely d Z m = F ( Z), vdw (8) dt 8
where Z is the distance etween the centres o the C 60 molecule and the caron nanotue, m is the total mass o the ullerene, F vdw (Z) is the van der Waals restoring orce, which generates oscillatory motion o the C 60 molecule. We note that here we neglect the rictional eect, which is reasonale or certain chiralities and diameter o the tue. For example, inside the caron nanotue (10, 10), the C 60 molecule tends to move along the axial direction and tends not to suer the rocking motion due to its preerred location is on the z-axis. From the symmetry o the prolem, we only consider the total axial orce in the range L z L, as such F vdw (Z) in the right hand side o (8) can e replaced y F tot z (Z), tot Fz ( Z) = πang [ P( ρ ) P( ρ1) ], (9) where P(ρ) is deined y (3), ρ 1 = [a + (Z + L) ] 1/ and ρ = [a + (Z L) ] 1/. In Fig. 8, we plot F tot z (Z) as given in (9) or a (10, 10) caron nanotue o length L = 19 Å, and it can e seen that the orce is close to zero everywhere except at the tue extremities. The pulse-like orce at the tue ends operates to attract the C 60 molecule ack towards the centre o the tue. shows the variation o the oscillatory requency with respect to the nanotue length. This result is in good agreement with the molecular dynamical simulation o Liu et al. [5], which conirms their inding that the shorter the caron nanotue, the higher the oscillatory requency. Figure 9. The variation o the oscillatory requency o a C 60 molecule with respect to the length o a (10, 10) caron nanotue ACKNOWLEDGMENT The authors are grateul to the Australian Research Council or support through the Discovery Project Scheme. Figure 8. The total axial orce or a C 60 molecule oscillating inside a (10,10) caron nanotue For < a << L, we ind that F tot z (Z) can e estimated y the Dirac delta unction. As a result, (8) can e reduced to give d Z m = W [ δ ( Z + L) δ ( Z L) ], (10) dt where W is the suction energy. By utilizing the Heaviside step unction, (10) can e integrated to give 1 / v = dz dt = W m + v, 0 (11) or L Z L, where v 0 is the initial velocity that the C 60 molecule is ired on the z-axis towards the open end o the tue in the positive z-direction. We note that the initial velocity v 0 is introduced or the case where the C 60 molecule is not sucked into the tue solely y the suction orce due to the strong repulsion. From (11), it implies that the C 60 ullerene travels inside the caron nanotue at the constant speed v. As shown in Cox et al. [9], upon using (11) we otain the velocity v = 93 m/s or the case o the C 60 molecule initially at rest outside the caron nanotue (10, 10) and the C 60 molecule gets sucked into the tue due to the attractive orce alone. As such, we otain the requency = v/(4l) = 36.13 GHz. Figure 9 REFERENCES [1] J. Cumings and A. Zettl, Low-riction nanoscale linear earing realized rom multi-walled caron nanotues, Science, vol. 89, pp. 60 604, 000. [] M. F. Yu, B. I. Yakoson and R. S. Ruo, Controlled sliding and pullout o nested shells in individual multiwalled caron nanotues, J. Phys. Chem. B, vol. 104, pp. 8764-8767, 000. [3] Q. Zheng and Q. Jiang, Multiwalled caron nanotues as gigahertz oscillators, Phys. Rev. Lett., vol. 88, 045503, 00. [4] Q. Zheng, J. Z. Liu and Q. Jiang, Excess van der Waals interaction energy o a multiwalled caron nanotue with an extruded core and the induced core oscillation, Phys. Rev. B, vol. 65, 45409, 00. [5] P. Liu, Y. W. Zhang and C. Lu, Oscillatory ehavior o C60-nanotue oscillators: a molecular-dynamics study, J. Appl. Phys., vol. 97, 094313, 005. [6] M. S. Dresselhaus, G. Dresselhaus and P. C. Eklund, Science o Fullerenes and Caron nanotues, San Diego: Academic Press, 1996. [7] D. Qian, W. K Liu, and R. S. Ruo, Mechanics o C 60 in nanotues, J. Phys. Chem. B, vol. 105, pp. 10753-10758, 001. [8] B. J. Cox, N. Thamwattana and J. M. Hill, Mechanics o atoms and ullerenes in single-walled caron nanotues. I. Acceptance and suction energies, Proc. R. Soc. Lond., Ser. A, accepted or pulication. [9] B. J. Cox, N. Thamwattana and J. M. Hill, Mechanics o atoms and ullerenes in single-walled caron nanotues. II. Oscillatory ehaviour, Proc. R. Soc. Lond., Ser. A, accepted or pulication. [10] L. A. Girialco, M. Hodak and R. S. Lee, Caron nanotues, uckyalls, ropes, and a universal graphitic potential, Phys. Rev. B, vol. 6, pp. 13104-13110, 000. [11] M. Hodak and L. A. Girialco, Fullerenes inside caron nanotues and multi-walled caron nanotues: optimum and maximum sizes, Chem. Phys. Lett., vol. 350, pp. 405-411, 001. [1] S. Okada, S. Saito and A. Oshiyama, Energetics and electronic structures o encapsulated C 60 in a caron nanotue, Phys. Rev. Lett., vol. 86, pp. 3835-3838, 001. 83