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Atomistic and continuum studies of carbon nanotubes under pressure
Computational Materials Science 24 (2002) 159–162 www.elsevier.com/locate/commatsci
Atomistic and continuum studies of carbon nanotubes under pressure P.S. Das a, L.T. Wille b
b,*
a Col Tec Inc., 1250 Washington St., Columbus, IN 47201, USA Department of Physics, Florida Atlantic University, 777, Glades Road, 33431 Boca Raton, FL, USA
Abstract The results of finite element simulations of carbon nanotubes under pressure are presented and compared with atomistic studies. The dominant failure mode is shell buckling, characterized by diamond shaped bulges. Critical pressures are calculated for two types of boundary conditions. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Since their discovery in 1991, carbon nanotubes have continued to receive a great deal of attention, both theoretical and experimental [1]. Their remarkable physical and chemical properties have led to suggestions of numerous applications. Although problems remain with controlled growth and high volume production, it is generally believed that of all the fullerenes, nanotubes will have the largest technological impact. One area that has received a great deal of scrutiny involves the elastic properties of these materials. On the experimental side, single-walled nanotubes (SWNTs) [2] multi-walled tubes [3] and SWNT ropes [4] have all been studied intensively with regards to their behavior under stress. Theoretically, various parameters, such as Young’s moduli and Poisson ratios, as well as deformation modes,
*
Corresponding author. Tel.: +1-561-297-3379; fax: +1-561297-2662. E-mail address: [email protected] (L.T. Wille).
have been determined by a number of atomistic techniques, including classical molecular dynamics (MD) with empirical potentials [5–7] or a tightbinding description [8], as well as several types of first-principles calculations, such as density-functional based pseudopotential theory [9] and Hartree–Fock theory [10]. Attention has also been given to continuum descriptions of nanotubes, based on the theory of shells [11]. Yakobson et al. [12] wrote down analytical expressions for the energy of a shell in terms of local stresses and deformation. The parameters entering this expression were derived from atomistic simulations. The various deformation modes and critical strains were calculated analytically and found to be in good agreement with the results of the microscopic studies. Govindjee and Sackman [13] used Bernoulli–Euler beam bending theory to investigate the applicability of continuum theory to an atomic system such as a nanotube. They emphasize that the tube should be broken down in a number of parallel crosssections, rather than described as a rigid sheet or rod as had often been assumed. A similar point
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P.S. Das, L.T. Wille / Computational Materials Science 24 (2002) 159–162
was made by Ru [14] who emphasized that the ‘‘straight normal postulate’’ of continuum shell theory (infinite subdivisibility of the shell without interlayer slips) evidently breaks down once atomic dimensions are reached. Thus, attempts to reconcile continuum theory with atomistic simulations [5] had to use parameters, such as the effective thickness of the shell, whose physical origin was doubtful at best. Similar expressions are often used to interpret experimental observations (see, e.g., Ref. [15] and references therein) and may therefore also be questioned. Finally, a clear breakdown of the continuum theory occurs when the tubes undergo plastic deformation, since the continuum model is insensitive to the chirality of the wrapping index, while tight-binding MD simulations [16] show a dependence of the elastic limit on chirality. Clearly, the continuum approximation can be very valuable and may be the only feasible approach for large and complex systems, but its applicability deserves further scrutiny. Here we propose to study the use of finite element (FE) calculations to elucidate the elastic behavior of SWNTs. The canonical problem of relating and reconciling atomic and continuum descriptions has received a great deal of attention lately, particularly in the context of multiscale computational methods. A general scheme for passing between the two limits, applicable to systems with reduced dimensionality such as films and rods, was recently described by Friesecke and James [17]. A coarsegrained MD method which provides a seamless coupling cross length scales was developed by Rudd and Broughton [18]. Alternatively, Tadmor et al. [19] describe a FE method designed to incorporate atomic energy functionals, assuming that such expressions are known for the system under study. An adaptive FE scheme, particularly well suited to study systems with grains or internal interfaces, was developed by Shenoy et al. [20] Although our eventual aim is to develop a combined FE/MD code, as an essential and non-trivial intermediate step, we present here pure FE simulations and compare them to the results of earlier MD work [6,7]. In this study, a cylindrical nanotube with wall thickness 1.0 a.u., length 30.0 a.u. and radius 12 a.u. is simulated, which corresponds approximately to the T(32:0) tube shown in Ref.
[7] A linear (eigenvalue) buckling analysis is performed and the results are compared to the standard textbook solutions as well as to the atomistic model results.
2. Method A three-dimensional FE shell analysis was performed using the standard software package ANSYS, Rev. 5.7 [21]. The 4-noded shell element, Shell181, was used to develop the FE model for the carbon nanotube. This element has six degrees of freedom at each node: translations in the x-, y-, and z-directions, and rotations about the x-, y-, and z-axes, and is suitable for analyzing thin to moderately thick shell structures. The boundary conditions used at one end of the tube involved radial, axial, and tangential constraints, while at the loading end only radial and tangential constraints were applied. All nodes are coupled in the axial direction. A typical discretized FE model with external loading and boundary conditions indicated is shown in Fig. 1. A com-
Fig. 1. Three-dimensional FE model.
P.S. Das, L.T. Wille / Computational Materials Science 24 (2002) 159–162
161
pressive unit load (1 MPa) was applied as the external loading on the tube. The materials properties used as input were: Young’s modulus, E, of 1 TPa (consistent with atomistic simulations [6]) and Poisson’s ratio, m, of 0.3. The origin of the global coordinate system is at the midpoint of the central axis of the system. In cylindrical coordinates, the convention used was: z-axial, x-radial, y-h. Prior to the eigenvalue solution, a prestressed structure (PSTRES, ON) was created with a unit pressure load. This structure was submitted to the eigenvalue solver to determine the critical pressure. Since the eigenvalues are simply the load multipliers, and the prestressed structure had a unit load, the lowest eigenvalue corresponds to the theoretical critical buckling pressure. Fig. 2. First mode shape.
3. Results and discussion The instability modes for an axially compressed cylinder are overall column buckling and local wall buckling, either of which can be elastic or inelastic. The type of buckling to which a particular cylinder is susceptible is dependent both on the ratio of length to radius of gyration, L=r, and the ratio of cylinder diameter to wall thickness, D=t. Generally, column buckling is controlled by the L=r ratio, while shell buckling is dependent on the D=t ratio [22]. Since L=r is much smaller than D=t ¼ 24 in the present FE model, shell buckling dominates the simulations. This is confirmed by the results shown in Figs. 2 and 3. In addition, the shape of the buckling appears to consist predominantly of diamond shaped bulges. This is in agreement with theoretical results [22] which state that this mode will dominate when the parameter: 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi L D Z¼2 1 m2 ð1Þ D t
Fig. 3. Eighth mode shape.
2
is within the bounds 2.85 and 1.2ðD=tÞ C, where ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ 1= 3ð1 m2 Þ, as is the case here. Therefore, the theoretical buckling stress rxc can be represented by: rxc ¼
2CE : D=t
ð2Þ
Table 1 shows the first 10 modes and the corresponding buckling pressures, as obtained from the FE calculations. Although the first mode (lowest eigenvalue) is found to be 48 044 MPa, the predicted diamond buckling occurs at the eighth mode, i.e. at 49 456 MPa. The corresponding mode
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Table 1 Collapse pressures from three-dimensional FE model for different mode shapes Shape no.
Buckling pressure (MPa)
1 2 3 4 5 6 7 8 9 10
48 044 48 044 49 050 49 134 49 134 49 416 49 416 49 456 49 456 49 626
shapes 1 and 8 are shown in Figs. 2 and 3. We note that the eighth mode deformation is remarkably similar to the atomistic simulations [7]. The numerically obtained FE value for the critical buckling pressure is within 4% of the theoretical value, given by Eq. (2). It is to be noticed that the boundary conditions have some influence on Eq. (2). For a simply supported carbon nanotube under axial compressive loading, with the tangential constraints removed, the critical buckling pressure from the present FE model is found to be 24 022 MPa, i.e. close to half of the classical value. This variation has also been found by previous authors in studies of cylindrical shells [23,24]. Summarizing, we have shown that the FE method can be reliably applied to the problem of elastic deformation of carbon nanotubes. The results can be related to findings from continuum mechanics and are compatible with atomistic simulations, at considerably less computing time. However, the FE method cannot take into account atomistic details such as chirality and needs experimentally determined input, such as Poisson ratios. A full understanding of elastic instabilities in systems such as Carbon nanotubes can only be obtained by a combined FE/atomistic study.
References [1] D. Tomanek, R.J. Enbody (Eds.), Science and Applications of Nanotubes, Kluwer/Plenum, New York, 2000. [2] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Phys. Rev. B 58 (1998) 14013. [3] M.M.J. Treacy, T.W. Ebbesen, J.M. Gilson, Nature 381 (1996) 678. [4] J.P. Salvetat, G.A.D. Briggs, J.M. Bonard, R.B. Basca, A.J. Kulik, T. St ockli, N.A. Burnham, L. Forr o, Phys. Rev. Lett. 82 (1999) 944. [5] B.I. Yakobson, C.J. Brabec, J. Bernholc, Phys. Rev. Lett. 76 (1996) 2511. [6] C.F. Cornwell, L.T. Wille, Sol. St. Comm. 101 (1997) 555. [7] C.F. Cornwell, L.T. Wille, J. Chem. Phys. 109 (1998) 763. [8] T. Ozaki, Y. Iwasa, T. Mitani, Phys. Rev. Lett. 84 (2000) 1712. [9] D. Sanchez-Portal, E. Artacho, J.M. Soler, A. Rubio, P. Ordejon, Phys. Rev. B 59 (1999) 12678. [10] G. Van Lier, C. Van Alsenoy, V. Van Doren, P. Geerlings, Chem. Phys. Lett. 326 (2000) 181. [11] L.D. Landau, E.M. Lifshitz, Elasticity Theory, Pergamon, Oxford, 1986. [12] B.I. Yakobson, C.J. Brabec, J. Bernholc, J. Computer-Aid. Mat. Design 3 (1996) 173. [13] S. Govindjee, J.L. Sackman, Sol. St. Comm. 110 (1999) 227. [14] C.Q. Ru, Phys. Rev. B 62 (2000) 9973. [15] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Phys. Rev. B 58 (1998) 14013. [16] P. Zhang, P.E. Lammert, V.H. Crespi, Phys. Rev. Lett. 81 (1998) 5346. [17] G. Friesecke, R.D. James, J. Mech. Phys. Solids 48 (2000) 1519. [18] R.E. Rudd, J.Q. Broughton, Phys. Rev. B 58 (1998) R5893. [19] E.B. Tadmor, G.S. Smith, N. Bernstein, E. Kaxiras, Phys. Rev. B 59 (1999) 235. [20] V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips, M. Ortiz, J. Mech. Phys. Solids 47 (1999) 611. [21] ANSYS Users Guide, ANSYS, Inc., Canonsburg, Pennsylvania, 2001. [22] T.V. Galambos (Ed.), Guide to Stability Design Criteria for Metal Structures, fifth ed., Wiley, New York, 1998. [23] S.B. Batdorf, M. Schildcrout, M. Stein, NACA Technical Note, no. 1343, Available on-line at: http://naca.larc.nasa. gov/new.html. [24] B.O. Almroth, AIAA J. 4 (1966) 134.
Atomistic and continuum studies of carbon nanotubes under pressure P.S. Das a, L.T. Wille b
b,*
a Col Tec Inc., 1250 Washington St., Columbus, IN 47201, USA Department of Physics, Florida Atlantic University, 777, Glades Road, 33431 Boca Raton, FL, USA
Abstract The results of finite element simulations of carbon nanotubes under pressure are presented and compared with atomistic studies. The dominant failure mode is shell buckling, characterized by diamond shaped bulges. Critical pressures are calculated for two types of boundary conditions. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction Since their discovery in 1991, carbon nanotubes have continued to receive a great deal of attention, both theoretical and experimental [1]. Their remarkable physical and chemical properties have led to suggestions of numerous applications. Although problems remain with controlled growth and high volume production, it is generally believed that of all the fullerenes, nanotubes will have the largest technological impact. One area that has received a great deal of scrutiny involves the elastic properties of these materials. On the experimental side, single-walled nanotubes (SWNTs) [2] multi-walled tubes [3] and SWNT ropes [4] have all been studied intensively with regards to their behavior under stress. Theoretically, various parameters, such as Young’s moduli and Poisson ratios, as well as deformation modes,
*
Corresponding author. Tel.: +1-561-297-3379; fax: +1-561297-2662. E-mail address: [email protected] (L.T. Wille).
have been determined by a number of atomistic techniques, including classical molecular dynamics (MD) with empirical potentials [5–7] or a tightbinding description [8], as well as several types of first-principles calculations, such as density-functional based pseudopotential theory [9] and Hartree–Fock theory [10]. Attention has also been given to continuum descriptions of nanotubes, based on the theory of shells [11]. Yakobson et al. [12] wrote down analytical expressions for the energy of a shell in terms of local stresses and deformation. The parameters entering this expression were derived from atomistic simulations. The various deformation modes and critical strains were calculated analytically and found to be in good agreement with the results of the microscopic studies. Govindjee and Sackman [13] used Bernoulli–Euler beam bending theory to investigate the applicability of continuum theory to an atomic system such as a nanotube. They emphasize that the tube should be broken down in a number of parallel crosssections, rather than described as a rigid sheet or rod as had often been assumed. A similar point
0927-0256/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 0 2 5 6 ( 0 2 ) 0 0 1 9 4 - 5
160
P.S. Das, L.T. Wille / Computational Materials Science 24 (2002) 159–162
was made by Ru [14] who emphasized that the ‘‘straight normal postulate’’ of continuum shell theory (infinite subdivisibility of the shell without interlayer slips) evidently breaks down once atomic dimensions are reached. Thus, attempts to reconcile continuum theory with atomistic simulations [5] had to use parameters, such as the effective thickness of the shell, whose physical origin was doubtful at best. Similar expressions are often used to interpret experimental observations (see, e.g., Ref. [15] and references therein) and may therefore also be questioned. Finally, a clear breakdown of the continuum theory occurs when the tubes undergo plastic deformation, since the continuum model is insensitive to the chirality of the wrapping index, while tight-binding MD simulations [16] show a dependence of the elastic limit on chirality. Clearly, the continuum approximation can be very valuable and may be the only feasible approach for large and complex systems, but its applicability deserves further scrutiny. Here we propose to study the use of finite element (FE) calculations to elucidate the elastic behavior of SWNTs. The canonical problem of relating and reconciling atomic and continuum descriptions has received a great deal of attention lately, particularly in the context of multiscale computational methods. A general scheme for passing between the two limits, applicable to systems with reduced dimensionality such as films and rods, was recently described by Friesecke and James [17]. A coarsegrained MD method which provides a seamless coupling cross length scales was developed by Rudd and Broughton [18]. Alternatively, Tadmor et al. [19] describe a FE method designed to incorporate atomic energy functionals, assuming that such expressions are known for the system under study. An adaptive FE scheme, particularly well suited to study systems with grains or internal interfaces, was developed by Shenoy et al. [20] Although our eventual aim is to develop a combined FE/MD code, as an essential and non-trivial intermediate step, we present here pure FE simulations and compare them to the results of earlier MD work [6,7]. In this study, a cylindrical nanotube with wall thickness 1.0 a.u., length 30.0 a.u. and radius 12 a.u. is simulated, which corresponds approximately to the T(32:0) tube shown in Ref.
[7] A linear (eigenvalue) buckling analysis is performed and the results are compared to the standard textbook solutions as well as to the atomistic model results.
2. Method A three-dimensional FE shell analysis was performed using the standard software package ANSYS, Rev. 5.7 [21]. The 4-noded shell element, Shell181, was used to develop the FE model for the carbon nanotube. This element has six degrees of freedom at each node: translations in the x-, y-, and z-directions, and rotations about the x-, y-, and z-axes, and is suitable for analyzing thin to moderately thick shell structures. The boundary conditions used at one end of the tube involved radial, axial, and tangential constraints, while at the loading end only radial and tangential constraints were applied. All nodes are coupled in the axial direction. A typical discretized FE model with external loading and boundary conditions indicated is shown in Fig. 1. A com-
Fig. 1. Three-dimensional FE model.
P.S. Das, L.T. Wille / Computational Materials Science 24 (2002) 159–162
161
pressive unit load (1 MPa) was applied as the external loading on the tube. The materials properties used as input were: Young’s modulus, E, of 1 TPa (consistent with atomistic simulations [6]) and Poisson’s ratio, m, of 0.3. The origin of the global coordinate system is at the midpoint of the central axis of the system. In cylindrical coordinates, the convention used was: z-axial, x-radial, y-h. Prior to the eigenvalue solution, a prestressed structure (PSTRES, ON) was created with a unit pressure load. This structure was submitted to the eigenvalue solver to determine the critical pressure. Since the eigenvalues are simply the load multipliers, and the prestressed structure had a unit load, the lowest eigenvalue corresponds to the theoretical critical buckling pressure. Fig. 2. First mode shape.
3. Results and discussion The instability modes for an axially compressed cylinder are overall column buckling and local wall buckling, either of which can be elastic or inelastic. The type of buckling to which a particular cylinder is susceptible is dependent both on the ratio of length to radius of gyration, L=r, and the ratio of cylinder diameter to wall thickness, D=t. Generally, column buckling is controlled by the L=r ratio, while shell buckling is dependent on the D=t ratio [22]. Since L=r is much smaller than D=t ¼ 24 in the present FE model, shell buckling dominates the simulations. This is confirmed by the results shown in Figs. 2 and 3. In addition, the shape of the buckling appears to consist predominantly of diamond shaped bulges. This is in agreement with theoretical results [22] which state that this mode will dominate when the parameter: 2 pffiffiffiffiffiffiffiffiffiffiffiffiffi L D Z¼2 1 m2 ð1Þ D t
Fig. 3. Eighth mode shape.
2
is within the bounds 2.85 and 1.2ðD=tÞ C, where ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ¼ 1= 3ð1 m2 Þ, as is the case here. Therefore, the theoretical buckling stress rxc can be represented by: rxc ¼
2CE : D=t
ð2Þ
Table 1 shows the first 10 modes and the corresponding buckling pressures, as obtained from the FE calculations. Although the first mode (lowest eigenvalue) is found to be 48 044 MPa, the predicted diamond buckling occurs at the eighth mode, i.e. at 49 456 MPa. The corresponding mode
162
P.S. Das, L.T. Wille / Computational Materials Science 24 (2002) 159–162
Table 1 Collapse pressures from three-dimensional FE model for different mode shapes Shape no.
Buckling pressure (MPa)
1 2 3 4 5 6 7 8 9 10
48 044 48 044 49 050 49 134 49 134 49 416 49 416 49 456 49 456 49 626
shapes 1 and 8 are shown in Figs. 2 and 3. We note that the eighth mode deformation is remarkably similar to the atomistic simulations [7]. The numerically obtained FE value for the critical buckling pressure is within 4% of the theoretical value, given by Eq. (2). It is to be noticed that the boundary conditions have some influence on Eq. (2). For a simply supported carbon nanotube under axial compressive loading, with the tangential constraints removed, the critical buckling pressure from the present FE model is found to be 24 022 MPa, i.e. close to half of the classical value. This variation has also been found by previous authors in studies of cylindrical shells [23,24]. Summarizing, we have shown that the FE method can be reliably applied to the problem of elastic deformation of carbon nanotubes. The results can be related to findings from continuum mechanics and are compatible with atomistic simulations, at considerably less computing time. However, the FE method cannot take into account atomistic details such as chirality and needs experimentally determined input, such as Poisson ratios. A full understanding of elastic instabilities in systems such as Carbon nanotubes can only be obtained by a combined FE/atomistic study.
References [1] D. Tomanek, R.J. Enbody (Eds.), Science and Applications of Nanotubes, Kluwer/Plenum, New York, 2000. [2] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Phys. Rev. B 58 (1998) 14013. [3] M.M.J. Treacy, T.W. Ebbesen, J.M. Gilson, Nature 381 (1996) 678. [4] J.P. Salvetat, G.A.D. Briggs, J.M. Bonard, R.B. Basca, A.J. Kulik, T. St ockli, N.A. Burnham, L. Forr o, Phys. Rev. Lett. 82 (1999) 944. [5] B.I. Yakobson, C.J. Brabec, J. Bernholc, Phys. Rev. Lett. 76 (1996) 2511. [6] C.F. Cornwell, L.T. Wille, Sol. St. Comm. 101 (1997) 555. [7] C.F. Cornwell, L.T. Wille, J. Chem. Phys. 109 (1998) 763. [8] T. Ozaki, Y. Iwasa, T. Mitani, Phys. Rev. Lett. 84 (2000) 1712. [9] D. Sanchez-Portal, E. Artacho, J.M. Soler, A. Rubio, P. Ordejon, Phys. Rev. B 59 (1999) 12678. [10] G. Van Lier, C. Van Alsenoy, V. Van Doren, P. Geerlings, Chem. Phys. Lett. 326 (2000) 181. [11] L.D. Landau, E.M. Lifshitz, Elasticity Theory, Pergamon, Oxford, 1986. [12] B.I. Yakobson, C.J. Brabec, J. Bernholc, J. Computer-Aid. Mat. Design 3 (1996) 173. [13] S. Govindjee, J.L. Sackman, Sol. St. Comm. 110 (1999) 227. [14] C.Q. Ru, Phys. Rev. B 62 (2000) 9973. [15] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Phys. Rev. B 58 (1998) 14013. [16] P. Zhang, P.E. Lammert, V.H. Crespi, Phys. Rev. Lett. 81 (1998) 5346. [17] G. Friesecke, R.D. James, J. Mech. Phys. Solids 48 (2000) 1519. [18] R.E. Rudd, J.Q. Broughton, Phys. Rev. B 58 (1998) R5893. [19] E.B. Tadmor, G.S. Smith, N. Bernstein, E. Kaxiras, Phys. Rev. B 59 (1999) 235. [20] V.B. Shenoy, R. Miller, E.B. Tadmor, D. Rodney, R. Phillips, M. Ortiz, J. Mech. Phys. Solids 47 (1999) 611. [21] ANSYS Users Guide, ANSYS, Inc., Canonsburg, Pennsylvania, 2001. [22] T.V. Galambos (Ed.), Guide to Stability Design Criteria for Metal Structures, fifth ed., Wiley, New York, 1998. [23] S.B. Batdorf, M. Schildcrout, M. Stein, NACA Technical Note, no. 1343, Available on-line at: http://naca.larc.nasa. gov/new.html. [24] B.O. Almroth, AIAA J. 4 (1966) 134.