- Email: [email protected]
Elastic properties of single and multilayered nanotubes
Pergamon
PII: S0022-3697(97)00045-0
J. Phys. Chem Solids Vo158, No. 11, pp. 1649-1652, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/97 $17.00 + 0.00
ELASTIC PROPERTIES OF SINGLE AND MULTILAYERED NANOTUBES JIAN P I N G LU# Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, CA 27599, USA Abstract--Elastic properties of single- and multilayered nanotubes are calculated using an empirical model. It is predicted that the nanotubes are the strongest materials. The Young's modulus and the shear modulus are comparable to that of the diamond, but the bulk modulus is almost twice as large. Elastic moduli are shown to be insensitive to details of nanotube structure such as helicity, tube radius and number of layers. These unusual properties ensure that nanotubes will have a wide range of potential applications. © 1997 Elsevier Science Ltd.
Keywords: nanotubes, Young's modulus 1. INTRODUCTION
2. THE LATTICE DYNAMICS MODEL
potentials between atoms. For the most successful lattice dynamics model of graphite, interactions up to fourthneighbor in-plane and out-of-plane interactions are included [7]. The force constants are empirically determined by fitting to measured elastic constants and phonon frequencies. The local structure of a nanotube layer can be constructed from the conformal mapping of the graphitic sheet on to a cylindrical surface. For a typical nanotube of a few nm in radius, the curvature is small enough that one expects short-range atomic interactions to be the same as that in graphite. Thus, we adopt the same set of parameters that was used for intraplan interactions in graphite [7] for intralayer interactions in all nanotubes. The different layers in a multilayered nanotube are not as well registered as they are in the single-crystal graphite. Thus, one cannot adopt the same set of parameters for interlayer interactions. Instead, we model the interlayer interactions by the summation of all pair-wise radial harmonic potentials up to a certain cutoff distance dcu,. The spring constant k(r) is assumed to scale as the van de Waals interaction, k(r) = ko(ro/r) 6, where ro is the interlayer distance in the graphite. The constant ko is determined by fitting to the elastic constant c33 of the single-crystal graphite. Different choices of dcut are explored. It is found that our results are insensitive to dcut. Table 1 lists the complete set of final parameters used in our calculations. We will try to answer three important questions. (a) How elastic properties of single-layer nanotubes depend on the structural details, such as size and helicity? (b) How interlayer interactions affect elastic properties? (c) How elastic properties of nanotubes compare with those of graphite and diamond.
In an empirical lattice dynamics model the atomic interactions are approximated by a sum of pair-wise harmonic
3. SINGLE-LAYER NANOTUBES
The discovery of carbon nanotubes [1 ] stimulates a great interest in these novel materials. Their electronic [2] and magnetic properties [3] depend sensitively on the structural details such as tube radius and helicity. It has been speculated that nanotubes may also possess novel mechanical properties. Recent measurements have inferred a Young's modulus that is several times that of diamond [4]. This calls for accurate theoretical calculations. The mechanical properties of small single-layer nanotubes have been studied by several groups using the empirical-force-potential molecular dynamics simulations [5, 6]. A Young's modulus that is four times that of the diamond were predicted. However, those calculations were based on single-layer small nanotubes of several ~, in radius. While most samples of nanotubes are multilayered and of several nm in radius. A practical method of investigating elastic properties is to use the empirical lattice dynamics model. The phonon spectrum and elastic properties of the graphite has been successfully calculated using such models [7]. The similarity in local structure between the graphite and the nanotubes ensure that a similar model is applicable for nanotubes. The great advantage of such a mode is that it can be easily applied to nanotubes of different size, helicity, and number of layers. One such model has been used to predict the phonon spectrum of small single-layer nanotubes [8]. Here we present results of applying a similar model to calculate elastic properties of singleand multilayered nanotubes of various size and geometry.
#E-mail: [email protected].
Following the notation of White et al. [91 each singlelayer nanotube is indexed by a pair of integers (n l, n2), 1649
1650
JIANG PING LU
Table 1. Force constant parameters: q~n),~i~,),ck[~)are intralayer force constants respectively for nth near-neighbor radial, inplane tangential, and out-of-plane tangential components (see Ref. [7]) Intralayer force constants q~ = 31.252 t#~ = 26.748 ~ = 12.092 q~ =-6.3731 t#~ = 2.7978 t#3 = 1.9 t#~ = -2.5508 ~b4 = 0.9488 Interlayer force parameters ko = -0.0707 dcut= 5.5,~
~o = 8.6545 4~2o=-0.93122 q~o = 1.2695 q~t4o= -0.54984 ro = 3.35.~
ko, dcu,, ro are interlayer force parameters. All spring constants are in units of 104 dyne/cm 2. This set of parameters reproduce the measured elastic moduli of the graphite (see Table 2). corresponding to a lattice vector L = nlal + n2a2 on the graphite plane, where al, a2 are the graphite plane unit cell vectors. The structure of the nanotube is obtained by the conformal mapping of a graphite strip onto a cylindrical surface. The nanotube radius is given by R = ao W/3(n~ + n 2 + n In 2)/27r, where ao = 1.42 ,~ is the carbon-carbon bond length. The helicity is defined as the angle between the at and L. In principle, force constants depend on the size of the nanotube as overlaps of r orbitals can depend on curvatures [10]. Such dependence is very weak. In this paper, we neglect this effect and concentrate on the dependence of elastic properties on the geometry and interlayer interactions. Thus, the same set of parameters listed in Table 1 are used for all nanotubes. The tensile stiffness as measured by Young's modulus is defined as the stress/strain ratio when a material is axially strained. For most materials, the radial dimension is reduced when it is axially elongated. The ratio of the reduction in radial dimension to the axial elongation defines the Poisson ratio ~,. We first calculate the Poisson ratio by minimizing the strain energy with respect to both the radial compression and the axial extension. The Young's modulus Y is then calculated from the second derivative of the strain energy density with respect to the axial strain at the fixed u. The axial symmetry of nanotubes leads to five independent elastic constants. Choosing the tube axis as the z-directions, these constants are C . = C 22, C 33, C44 ~-C55 , C 6 6 , and Cj3 = C23. They are calculated from the second derivatives of the strain energy density with respect to various strains. (For a reference on the elastic properties of materials see Ref. [13].) The overall hardness of a materials is measured by the bulk modulus. For axially symmetric materials this is given by [l 1] B = 4I'(CII - C66) + 2(1 - z')Cl3 + C33 3(1 +2u)
(1)
Table 2 lists the bulk, Young's and shear modulus (referred to as the torsional shear) calculated for selective examples of single-layer nanotubes. An important
Table 2. Elastic moduli of selective single-layer nanotubes (n i,n 2)
R
(5,5) 0.34 (6,4) 0.34 (7,3) 0.35 (8,2) 0.36 (9,1 ) 0.37 (10,0) 0.39 ( 10,10) 0.68 (50,50) 3.39 (100,100) 6.78 (200,200) 1 3 . 5 6 Graphite a Graphite b Diamond c
B
Y
M
v
0.7504 0.7503 0.7500 0.7495 0.7489 0.7483 0.7445 0.7429 0.7428 0.7428 0.0083 0.0083 0.442
0.9680 0.9680 0.9680 0.9681 0.9681 0.9682 0.9685 0.9686 0.9686 0.9686 1.02 0.0365 1.063
0.4340 0.4340 0.4412 0.4466 0.4503 0.4518 0.4517 0.4573 0.4575 0.4575 0.44 0.004 0.5758
0.2850 0.2850 0.2849 0.2847 0.2846 0.2844 0.2832 0.2827 0.2827 0.2827 0.16 0.012 0.1041
(n i,n2), index; R, radius in nm; B, Y,M, are bulk, Young's and shear moduli in units of TPa (10 t3 dyne/cm2). ~, is the Poisson ratio. Experimental values for graphite and diamond are listed for comparison. aGraphite along the basal plane [12]. bGraphite along the C axis [12]. CDiamond along the cube axis [13]. quantity in determining the values of elastic constants is the thickness h of nanotubes. Previous calculations have taken h = 0.66 ,~ for single-layer nanotubes, which leads to the unusually large Young's modulus predicted [6]. For multilayered nanotubes, all experiments indicate that the interlayer distance is the same as that in the graphite, h = 3.4 .~,. Thus, it is clear that the proper value for the layer thickness is the interlayer distance. It makes sense the one uses the same value as the layer thickness for all single-layer nanotubes. This enables us to compare results across nanotubes of different size and number of layers. For comparison, elastic moduli of the graphite [ 12] and the diamond [13] are also listed in Table 2. We conclude that: (1) elastic moduli are insensitive to size and helicity; (2) the Young's and shear moduli of nanotubes are comparable to that of diamond; (3) the bulk- modulus is almost twice that of graphite.
4. MULTILAYERED NANOTUBES The interlayer distance in all experimentally observed multilayered nanotubes is comparable to that in graphite. This puts a strong constraint on possible combinations of using single layers to form multilayered nanotubes. We have calculated elastic moduli for many different combinations. It is found that elastic properties are insensitive to different combinations as long as the constraint is satisfied. Because of this insensitivity we will use the results for one series of multilayered nanotubes to illustrate our main points. The series chosen is constructed from (5n, 5n), n = 1,2,3...single-layer tubes. This is one of the most likely structures for multilayered nanotubes, as the interlayer distance, d = 3.4 ,~, is very close to that actually observed [14].
Elastic properties of single and multilayered nanotubes Table 3. Elastic moduli of multilayered nanotubes constructed from the (5n,5n) series of single-layer tubes N
R
B
Y
M
v
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.34 0.68 1.02 1.36 1.70 2.03 2.37 2.71 3.05 3.39 3.73 4.07 4.41 4.75 5.09
0.7504 0.7365 0.7317 0.7295 0.7281 0.7273 0.7267 0.7262 0.7259 0.7256 0.7254 0.7252 0.7251 0.7250 0.7248
0.9680 0.9700 0.9707 0.9710 0.9712 0.9714 0.9715 0.9715 0.9716 0.9716 0.9717 0.9717 0.9717 0.9718 0.9718
0.4340 0.4501 0.4542 0.4559 0.4568 0.4573 0.4576 0.4578 0.4580 0.4581 0.4582 0.4582 0.4583 0.4583 0.4584
0.2850 0.2805 0.2789 0.2781 0.2777 0.2774 0.2772 0.2770 0.2769 0.2768 0.2768 0.2767 0.2766 0.2766 0.2766
N, number of layers; R, radius of the out-most layer in nm; B, Y, M are bulk, Young's and shear moduli in TPa. ~, is Poisson ratio.
Table 3 lists the bulk, Young's, and shear moduli for this series of nanotubes up to 15 layers. Table 4 lists all five independent elastic coefficients together with those for the graphite and the diamond (the latter has only three independent elastic coefficients). From these two tables one observes that there are only small changes in moduli for the first few layers. Once the radius of the out-most layer is greater than one nm, the elastic moduli become essentially independent of the number of layers. The same is true for all other multilayered nanotubes we have calculated. Thus, we arrive at an important conclusion: the elastic property is the samefor all nanotubes with a radius larger than 1 nm. Comparing Table 3 with Table 2 one finds: the interlayer van de Waals interactions have a negligible contribution to both the tensile and shear stiffness of multilayered
1651
nanotubes. This is not surprising as the interlayer interactions are two orders of magnitude smaller than the intralayer interactions. The Young's modulus of multilayered nanotubes has been deduced recently by Treacy et al. [4] by measuring the thermal vibrations of individual tubes. Their values range from 0.4 to 4 TPa with the average values of 1.6 TPa. These results are substantial larger than our predicted values of l TPa. Clearly, the large uncertainty in experiments needed to be reduced in order to compare quantitatively with our calculations [161.
5. CONCLUSIONS In conclusion, we have investigated elastic properties of single and multilayered nanotubes using an empirical model. The simplicity of the model enables us to explore the dependence of elastic moduli on the nanotube geometry such as the size, the helicity, and the number of layers. It is concluded that elastic properties are the same for all nanotubes with a radius larger than 1 nm. The predicted Young's modulus ( ~ 1 TPa) and shear modulus ( - 0 . 4 5 TPa) are comparable to that of diamond, an order of magnitude larger than that of typical carbon fibers [15]. The bulk modulus ( - 0 . 7 4 TPa) is almost twice as large as that of diamond, making the nanotubes the hardest of all known materials. Nanotubes should hold great promise for making light super-strong materials.
Acknowledgements--Theauthor thanks L. McNeil, R. Superfine and S. Washburn for discussions, and R. Smiley for communicating experimental results. This work is supported by a grant from US Department of Energy, and in part by a grant from The Petroleum Research Foundation.
Table 4. Elastic coefficients (in TPa) of multilayered nanotubes constructed from the (5n,5n) series n
R
CII
C33
C44
C66
Ci3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Graphite a Diamond b
0.34 0.68 1.02 1.36 1.70 2.03 2.37 2.71 3.05 3.39 3.73 4.07 4.41 4.75 5.09
0.3952 0.4057 0.4094 0.4113 0.4125 0.4132 0.4137 0.4141 0.4144 0.4147 0.4149 0.4151 0.4152 0.4153 0.4154 1.06 1.076
1.0528 1.0545 1.0551 1.0554 1.0556 1.0557 1.0558 1.0559 1.0559 1.0560 1.0560 1.0560 1.0561 1.0561 1.0561 0.0365 1.076
0.1893 0.1914 0.1921 0.1925 0.1928 0.1929 0.1930 0.1931 0.1932 0.1932 0.1933 0.1933 0.1933 0.1934 0.1934 0.0044 0.5758
0.1347 0.1373 0.1382 0.1387 0.1390 0.1392 0.1393 0.1394 0.1395 0.1396 0.1396 0.1397 0.1397 0.1398 0.1398 0.44 0.5758
0.1487 0.1507 0.1513 0.1517 0.1519 0.1520 0.1521 0.1522 0.1523 0.1523 0.1524 0.1524 0.1524 0.1524 0.1525 0.015 0.125
Values for the graphite and the diamond are listed for comparison. 'Ref. [12]. bRef. [ 13].
1652
JIANG PING LU REFERENCES
1. Iijima, S., Nature, 1991, 354, 56; Ebbesen, T.W. and Ajayan, P.M., Nature, 1992, 358, 220. 2. Hamada, N. et al., Phys Rev. Lett., 1992, 68, 1579; Wang, L. et al., Phys. Rev. B, 1992, 46, 7175; Saito et al., ibid, 1992, 6, 1804. 3. Lu, J.P., Phys Rev Lett, 1995, 74, 1123. 4. Treacy, M.M., Ebbesen, T.W. and Gibson, J.M., Nature, 1996, 371, 678. 5. Robertson, D.H., Brenner, D.W. and Mintmire, J.W., Phys. Rev. B, 1992, 45, 12592. 6. L Yakobson, B., Brabec, C.J. and Bemholc, J., Phys. Rev. Lett., 1995, 76, 2511. 7. A1-Jishi,R. and Dresselhaus, G., Phys. Rev. B, 1982, 26, 4514.
8. 9. 10. 11. 12. 13. 14. 15. 16.
Al-Jishi, R. et al., Z Chem. Phys., 1993, 2t)9, 77. White, C.T. et al., Phys. Rev. B, 1993, 47, 5485. Lu, J.P. and Yang, W., Phys. Rev. B, 1994, 49, 11421. Lu, J.P., Phys. Rev. Lett., 1997, 79, 1297. Blakslee, O.L. et al., J. Appl. Phys., 1970, 41, 3373; Seldin, E.J. and Nezbeda, C.W., J. Appl. Phys., 1970, 41, 3389. Huntington, J.B., In Solid State Physics 7, ed. F. Seitz and D. Tumbull. Academic Press, New york, 1958. Ruoff, R.S. et al., Nature, 1993, 364, 514. Jacobson, R.L. et al., Carbon, 1995, 33, 1217. More recent experiments by R. Smiley's group has yielded a Young's modulus which is in agreement with our prediction. R. Smiley, private communications.
PII: S0022-3697(97)00045-0
J. Phys. Chem Solids Vo158, No. 11, pp. 1649-1652, 1997 © 1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0022-3697/97 $17.00 + 0.00
ELASTIC PROPERTIES OF SINGLE AND MULTILAYERED NANOTUBES JIAN P I N G LU# Department of Physics and Astronomy, University of North Carolina at Chapel Hill, Chapel Hill, CA 27599, USA Abstract--Elastic properties of single- and multilayered nanotubes are calculated using an empirical model. It is predicted that the nanotubes are the strongest materials. The Young's modulus and the shear modulus are comparable to that of the diamond, but the bulk modulus is almost twice as large. Elastic moduli are shown to be insensitive to details of nanotube structure such as helicity, tube radius and number of layers. These unusual properties ensure that nanotubes will have a wide range of potential applications. © 1997 Elsevier Science Ltd.
Keywords: nanotubes, Young's modulus 1. INTRODUCTION
2. THE LATTICE DYNAMICS MODEL
potentials between atoms. For the most successful lattice dynamics model of graphite, interactions up to fourthneighbor in-plane and out-of-plane interactions are included [7]. The force constants are empirically determined by fitting to measured elastic constants and phonon frequencies. The local structure of a nanotube layer can be constructed from the conformal mapping of the graphitic sheet on to a cylindrical surface. For a typical nanotube of a few nm in radius, the curvature is small enough that one expects short-range atomic interactions to be the same as that in graphite. Thus, we adopt the same set of parameters that was used for intraplan interactions in graphite [7] for intralayer interactions in all nanotubes. The different layers in a multilayered nanotube are not as well registered as they are in the single-crystal graphite. Thus, one cannot adopt the same set of parameters for interlayer interactions. Instead, we model the interlayer interactions by the summation of all pair-wise radial harmonic potentials up to a certain cutoff distance dcu,. The spring constant k(r) is assumed to scale as the van de Waals interaction, k(r) = ko(ro/r) 6, where ro is the interlayer distance in the graphite. The constant ko is determined by fitting to the elastic constant c33 of the single-crystal graphite. Different choices of dcut are explored. It is found that our results are insensitive to dcut. Table 1 lists the complete set of final parameters used in our calculations. We will try to answer three important questions. (a) How elastic properties of single-layer nanotubes depend on the structural details, such as size and helicity? (b) How interlayer interactions affect elastic properties? (c) How elastic properties of nanotubes compare with those of graphite and diamond.
In an empirical lattice dynamics model the atomic interactions are approximated by a sum of pair-wise harmonic
3. SINGLE-LAYER NANOTUBES
The discovery of carbon nanotubes [1 ] stimulates a great interest in these novel materials. Their electronic [2] and magnetic properties [3] depend sensitively on the structural details such as tube radius and helicity. It has been speculated that nanotubes may also possess novel mechanical properties. Recent measurements have inferred a Young's modulus that is several times that of diamond [4]. This calls for accurate theoretical calculations. The mechanical properties of small single-layer nanotubes have been studied by several groups using the empirical-force-potential molecular dynamics simulations [5, 6]. A Young's modulus that is four times that of the diamond were predicted. However, those calculations were based on single-layer small nanotubes of several ~, in radius. While most samples of nanotubes are multilayered and of several nm in radius. A practical method of investigating elastic properties is to use the empirical lattice dynamics model. The phonon spectrum and elastic properties of the graphite has been successfully calculated using such models [7]. The similarity in local structure between the graphite and the nanotubes ensure that a similar model is applicable for nanotubes. The great advantage of such a mode is that it can be easily applied to nanotubes of different size, helicity, and number of layers. One such model has been used to predict the phonon spectrum of small single-layer nanotubes [8]. Here we present results of applying a similar model to calculate elastic properties of singleand multilayered nanotubes of various size and geometry.
#E-mail: [email protected].
Following the notation of White et al. [91 each singlelayer nanotube is indexed by a pair of integers (n l, n2), 1649
1650
JIANG PING LU
Table 1. Force constant parameters: q~n),~i~,),ck[~)are intralayer force constants respectively for nth near-neighbor radial, inplane tangential, and out-of-plane tangential components (see Ref. [7]) Intralayer force constants q~ = 31.252 t#~ = 26.748 ~ = 12.092 q~ =-6.3731 t#~ = 2.7978 t#3 = 1.9 t#~ = -2.5508 ~b4 = 0.9488 Interlayer force parameters ko = -0.0707 dcut= 5.5,~
~o = 8.6545 4~2o=-0.93122 q~o = 1.2695 q~t4o= -0.54984 ro = 3.35.~
ko, dcu,, ro are interlayer force parameters. All spring constants are in units of 104 dyne/cm 2. This set of parameters reproduce the measured elastic moduli of the graphite (see Table 2). corresponding to a lattice vector L = nlal + n2a2 on the graphite plane, where al, a2 are the graphite plane unit cell vectors. The structure of the nanotube is obtained by the conformal mapping of a graphite strip onto a cylindrical surface. The nanotube radius is given by R = ao W/3(n~ + n 2 + n In 2)/27r, where ao = 1.42 ,~ is the carbon-carbon bond length. The helicity is defined as the angle between the at and L. In principle, force constants depend on the size of the nanotube as overlaps of r orbitals can depend on curvatures [10]. Such dependence is very weak. In this paper, we neglect this effect and concentrate on the dependence of elastic properties on the geometry and interlayer interactions. Thus, the same set of parameters listed in Table 1 are used for all nanotubes. The tensile stiffness as measured by Young's modulus is defined as the stress/strain ratio when a material is axially strained. For most materials, the radial dimension is reduced when it is axially elongated. The ratio of the reduction in radial dimension to the axial elongation defines the Poisson ratio ~,. We first calculate the Poisson ratio by minimizing the strain energy with respect to both the radial compression and the axial extension. The Young's modulus Y is then calculated from the second derivative of the strain energy density with respect to the axial strain at the fixed u. The axial symmetry of nanotubes leads to five independent elastic constants. Choosing the tube axis as the z-directions, these constants are C . = C 22, C 33, C44 ~-C55 , C 6 6 , and Cj3 = C23. They are calculated from the second derivatives of the strain energy density with respect to various strains. (For a reference on the elastic properties of materials see Ref. [13].) The overall hardness of a materials is measured by the bulk modulus. For axially symmetric materials this is given by [l 1] B = 4I'(CII - C66) + 2(1 - z')Cl3 + C33 3(1 +2u)
(1)
Table 2 lists the bulk, Young's and shear modulus (referred to as the torsional shear) calculated for selective examples of single-layer nanotubes. An important
Table 2. Elastic moduli of selective single-layer nanotubes (n i,n 2)
R
(5,5) 0.34 (6,4) 0.34 (7,3) 0.35 (8,2) 0.36 (9,1 ) 0.37 (10,0) 0.39 ( 10,10) 0.68 (50,50) 3.39 (100,100) 6.78 (200,200) 1 3 . 5 6 Graphite a Graphite b Diamond c
B
Y
M
v
0.7504 0.7503 0.7500 0.7495 0.7489 0.7483 0.7445 0.7429 0.7428 0.7428 0.0083 0.0083 0.442
0.9680 0.9680 0.9680 0.9681 0.9681 0.9682 0.9685 0.9686 0.9686 0.9686 1.02 0.0365 1.063
0.4340 0.4340 0.4412 0.4466 0.4503 0.4518 0.4517 0.4573 0.4575 0.4575 0.44 0.004 0.5758
0.2850 0.2850 0.2849 0.2847 0.2846 0.2844 0.2832 0.2827 0.2827 0.2827 0.16 0.012 0.1041
(n i,n2), index; R, radius in nm; B, Y,M, are bulk, Young's and shear moduli in units of TPa (10 t3 dyne/cm2). ~, is the Poisson ratio. Experimental values for graphite and diamond are listed for comparison. aGraphite along the basal plane [12]. bGraphite along the C axis [12]. CDiamond along the cube axis [13]. quantity in determining the values of elastic constants is the thickness h of nanotubes. Previous calculations have taken h = 0.66 ,~ for single-layer nanotubes, which leads to the unusually large Young's modulus predicted [6]. For multilayered nanotubes, all experiments indicate that the interlayer distance is the same as that in the graphite, h = 3.4 .~,. Thus, it is clear that the proper value for the layer thickness is the interlayer distance. It makes sense the one uses the same value as the layer thickness for all single-layer nanotubes. This enables us to compare results across nanotubes of different size and number of layers. For comparison, elastic moduli of the graphite [ 12] and the diamond [13] are also listed in Table 2. We conclude that: (1) elastic moduli are insensitive to size and helicity; (2) the Young's and shear moduli of nanotubes are comparable to that of diamond; (3) the bulk- modulus is almost twice that of graphite.
4. MULTILAYERED NANOTUBES The interlayer distance in all experimentally observed multilayered nanotubes is comparable to that in graphite. This puts a strong constraint on possible combinations of using single layers to form multilayered nanotubes. We have calculated elastic moduli for many different combinations. It is found that elastic properties are insensitive to different combinations as long as the constraint is satisfied. Because of this insensitivity we will use the results for one series of multilayered nanotubes to illustrate our main points. The series chosen is constructed from (5n, 5n), n = 1,2,3...single-layer tubes. This is one of the most likely structures for multilayered nanotubes, as the interlayer distance, d = 3.4 ,~, is very close to that actually observed [14].
Elastic properties of single and multilayered nanotubes Table 3. Elastic moduli of multilayered nanotubes constructed from the (5n,5n) series of single-layer tubes N
R
B
Y
M
v
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.34 0.68 1.02 1.36 1.70 2.03 2.37 2.71 3.05 3.39 3.73 4.07 4.41 4.75 5.09
0.7504 0.7365 0.7317 0.7295 0.7281 0.7273 0.7267 0.7262 0.7259 0.7256 0.7254 0.7252 0.7251 0.7250 0.7248
0.9680 0.9700 0.9707 0.9710 0.9712 0.9714 0.9715 0.9715 0.9716 0.9716 0.9717 0.9717 0.9717 0.9718 0.9718
0.4340 0.4501 0.4542 0.4559 0.4568 0.4573 0.4576 0.4578 0.4580 0.4581 0.4582 0.4582 0.4583 0.4583 0.4584
0.2850 0.2805 0.2789 0.2781 0.2777 0.2774 0.2772 0.2770 0.2769 0.2768 0.2768 0.2767 0.2766 0.2766 0.2766
N, number of layers; R, radius of the out-most layer in nm; B, Y, M are bulk, Young's and shear moduli in TPa. ~, is Poisson ratio.
Table 3 lists the bulk, Young's, and shear moduli for this series of nanotubes up to 15 layers. Table 4 lists all five independent elastic coefficients together with those for the graphite and the diamond (the latter has only three independent elastic coefficients). From these two tables one observes that there are only small changes in moduli for the first few layers. Once the radius of the out-most layer is greater than one nm, the elastic moduli become essentially independent of the number of layers. The same is true for all other multilayered nanotubes we have calculated. Thus, we arrive at an important conclusion: the elastic property is the samefor all nanotubes with a radius larger than 1 nm. Comparing Table 3 with Table 2 one finds: the interlayer van de Waals interactions have a negligible contribution to both the tensile and shear stiffness of multilayered
1651
nanotubes. This is not surprising as the interlayer interactions are two orders of magnitude smaller than the intralayer interactions. The Young's modulus of multilayered nanotubes has been deduced recently by Treacy et al. [4] by measuring the thermal vibrations of individual tubes. Their values range from 0.4 to 4 TPa with the average values of 1.6 TPa. These results are substantial larger than our predicted values of l TPa. Clearly, the large uncertainty in experiments needed to be reduced in order to compare quantitatively with our calculations [161.
5. CONCLUSIONS In conclusion, we have investigated elastic properties of single and multilayered nanotubes using an empirical model. The simplicity of the model enables us to explore the dependence of elastic moduli on the nanotube geometry such as the size, the helicity, and the number of layers. It is concluded that elastic properties are the same for all nanotubes with a radius larger than 1 nm. The predicted Young's modulus ( ~ 1 TPa) and shear modulus ( - 0 . 4 5 TPa) are comparable to that of diamond, an order of magnitude larger than that of typical carbon fibers [15]. The bulk modulus ( - 0 . 7 4 TPa) is almost twice as large as that of diamond, making the nanotubes the hardest of all known materials. Nanotubes should hold great promise for making light super-strong materials.
Acknowledgements--Theauthor thanks L. McNeil, R. Superfine and S. Washburn for discussions, and R. Smiley for communicating experimental results. This work is supported by a grant from US Department of Energy, and in part by a grant from The Petroleum Research Foundation.
Table 4. Elastic coefficients (in TPa) of multilayered nanotubes constructed from the (5n,5n) series n
R
CII
C33
C44
C66
Ci3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Graphite a Diamond b
0.34 0.68 1.02 1.36 1.70 2.03 2.37 2.71 3.05 3.39 3.73 4.07 4.41 4.75 5.09
0.3952 0.4057 0.4094 0.4113 0.4125 0.4132 0.4137 0.4141 0.4144 0.4147 0.4149 0.4151 0.4152 0.4153 0.4154 1.06 1.076
1.0528 1.0545 1.0551 1.0554 1.0556 1.0557 1.0558 1.0559 1.0559 1.0560 1.0560 1.0560 1.0561 1.0561 1.0561 0.0365 1.076
0.1893 0.1914 0.1921 0.1925 0.1928 0.1929 0.1930 0.1931 0.1932 0.1932 0.1933 0.1933 0.1933 0.1934 0.1934 0.0044 0.5758
0.1347 0.1373 0.1382 0.1387 0.1390 0.1392 0.1393 0.1394 0.1395 0.1396 0.1396 0.1397 0.1397 0.1398 0.1398 0.44 0.5758
0.1487 0.1507 0.1513 0.1517 0.1519 0.1520 0.1521 0.1522 0.1523 0.1523 0.1524 0.1524 0.1524 0.1524 0.1525 0.015 0.125
Values for the graphite and the diamond are listed for comparison. 'Ref. [12]. bRef. [ 13].
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