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Mechanical behavior of functionalized nanotubes
Chemical Physics Letters 387 (2004) 247–252 www.elsevier.com/locate/cplett
Mechanical behavior of functionalized nanotubes S. Namilae *, N. Chandra, C. Shet Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA Received 17 July 2003; in final form 17 July 2003 Published online: 4 March 2004
Abstract The potential use of carbon nanotubes as reinforcements in composite materials is greatly enhanced by improving the fibermatrix interface bonding. A proposed method to increase the fiber-matrix bonding is by functionalizing the nanotubes. We use molecular dynamics and statics simulations to study the tensile mechanical response of functionalized carbon nanotubes. It is found that the there is a marginal increase in stiffness in functionalized nanotubes, and the stiffness values are found to increase with the increase in number of attachments. When deformed at high temperatures, formation of topological and failure are found to occur at lower strains. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction The exceptional mechanical properties of carbon nanotubes (CNT) and their potential use in structural components and devices have stimulated great interest and extensive research ever since their discovery by Ijjima [1] in 1991. Experimental estimates [2–4] of Young’s modulus of CNTs are in the range of 0.32–1.47 TPa, while the theoretical estimates [5–7] vary from 0.5 to 5.5 TPa. The failure strength of these materials is experimentally estimated to be in the order of 150 GPa [4]. These excellent mechanical properties combined with their extremely high strength to weight ratio make them potential candidates as reinforcing fibers in super strong composites. In spite of outstanding properties of individual nanotubes, the elastic modulus of nanotube composite is much lesser, for e.g. Shaffer and Windle [8] report Young’s modulus value of 150 MPa in CNT– PVA composite films. The main obstacles in improving the stiffness and strength of nanotube are dispersion, alignment and interfacial load transfer. In order to optimize the properties of nanotube-based composites a *
Corresponding author. Fax: +1-850-410-6337/6420. E-mail addresses: [email protected] (S. Namilae), chandra@ eng.fsu.edu (N. Chandra). 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.01.104
clear understanding of the load transfer and interface mechanics at nanoscale is required. Chemical attachment or cross-linking of nanotube walls and polymeric matrix has been proposed as one of the techniques to improve the interfacial bonding. Based on molecular dynamics simulation Frankland and coworkers [9] report that CNT matrix shear strength and critical length for load transfer improve considerably by chemically cross-linking the CNT and matrix. Gong and coworkers [10] obtained improved mechanical properties in CNT epoxy composite by using surfactant coating on nanotube. They infer that this is due to increased interfacial strength with surfactant forming a weak bond between CNT and epoxy. Lordi and Yao [11] based on molecular mechanics simulation suggest that molecular level entanglement as a possible method to strengthen the interface. Garg and Sinnott [12] have recently studied the compressive properties of functionalized tubes, and reported that the force required for buckling reduces marginally by chemical attachment of hydrocarbon chains. Chandra and Ghonem [13] have noted that thermomechanical load transfer between fibers and matrix in conventional composites is generally affected by both chemical and mechanical bonding. While chemical bonding arises from the formation of new phases,
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mechanical bonding occurs by interlocking of asperities. Use of functionalized nanotubes as fibers in composites could enhance the cross-linking (chemical bond) a nd interfacial strength between matrix and fiber. These chemical attachments are also expected to act as ‘tethers’ (mechanical bond) and help in load transfer when the composite is deformed either slowly or rapidly [13,14]. Though these concepts are well established in conventional composites at micrometer scale interfaces, it is not clear if the same concepts could be transferred to nanoscale interfaces. Researchers have attached different functional groups to the walls of nanotubes using various experimental techniques. For example Michelson and coworkers [15] have fluorinated CNTs using alcohol solvents, while Pekker and coworkers [16] attached hydrogen using ammonia. Khare and coworkers [17] have hydrogenated nanotubes using electric discharge. Chen and coworkers [18] have attached alkyl chains using amidization, whereas Sun and coworkers used amidization and esterification to attach carboxylic acids [19,20]. The latter also processed PVA thin films embedded with functionalized nanotubes, which could be used for preparation of aligned nanocomposites. Apart from possible applications in nanocomposites, functionalization of CNTs is proposed as a useful technique for altering electronic properties and fabrication of sensors [21]. In order to use chemical attachments as a means to achieve increased interfacial strength, it is necessary to have an understanding of mechanical properties of functionalized nanotubes. The objective of this Letter is to study the effect of functionalization on the mechanical response CNTs under tension, and explore its origin. We also examine the effect of different number and types of (hydrocarbon) attachments primarily on the elastic response and also inelastic response and fracture.
rLutsko ij
1 X ¼ Avg X a¼1;n
! 1 a a a X j i m mi mj þ rab fab lab : 2 b¼1;n
ð1Þ
Here XAvg is the averaging volume, m, r, f are velocity force and radial vectors, a and b are the atomic indices, i and j are the indices of the stress tensor. lab denotes the fraction of a–b bond lying inside the averaging volume. Strains are calculated as derivative of displacement field, heavily borrowing from the concepts of finite elements. The displacement field is expressed in terms of known atomic displacements and interpolation functions. Graphitic structure can be considered as a mesh of hexagons. Each of these hexagons is further divided into triangular facets and average strain in these facets is calculated based on the displacement field given by the atoms defining the corners of the triangular facets. Strains at each atomic location are obtained as area average of the strains of immediate hexagons surrounding it. To validate the applicability of the stress and strain measure, stress–strain curves are plotted for a perfect (9,0) CNT under uniaxial tension. Displacements are applied incrementally to all the atoms of periodic (9,0) CNT in longitudinal direction. The nanotube is stabilized using conjugate gradients energy minimization scheme. Fig. 1 shows the stress–strain curves based on the three stress measures considered, virial, Lutsko and BDT stress, measured in longitudinal direction and longitudinal strain evaluated using the above described scheme. The Young’s modulus is evaluated as the slope of stress–strain curve at zero strain (initial tangent). We obtain a Young’s modulus value of 1.002 TPa for (9,0) CNT. This compares favorably with values published in the literature [2–7] which center around 1 TPa.
60
Lutsko Stress (E=0.997 TPa) Bulk stress (E=1.002 TPa)
2. Elastic deformation
Stress (GPa)
We use molecular dynamics and statics simulations based on Tersoff–Brenner bond-order potential [22] to evaluate the tensile stiffness and stress–strain response of various zigzag and armchair CNT. Stiffness can be evaluated as the second derivative of energy with respect to strain, or as the ratio of stress and strain. The later method is preferable as it involves fewer assumptions and further enables the calculation of local properties. We have used stress measures such as virial stress [23], BDT stress [24] and Lutsko stress [25,26] and developed a methodology for evaluating strain. These concepts are discussed in detail elsewhere [27], however they are mentioned here briefly for the sake of completeness. We primarily use Lutsko stress rLutsko to characterize stress. ij It is given by
BDT stress (E=1.002 TPa)
50
40
30
20
10
0 0
0.02
0.04
0.06
strain Fig. 1. Stress–strain curves for (9,0) CNT subject to uniaxial tension.
S. Namilae et al. / Chemical Physics Letters 387 (2004) 247–252
30
25
Stress (GPa)
Now we present the molecular dynamics results of nanotubes with chemically attached hydrocarbon chains. Zigzag and armchair nanotubes with varying diameter were considered for this study. All the na in length, tensile stresses were notubes are about 120 A applied by fixing and displacing both the ends of na on either side) followed by stabilization notubes (15 A for 1500 time steps with a step size of 0.2 fs. The temperature is maintained at 77 K. Stresses and strains are evaluated as average over 15 time steps. Hydrocarbon chains are attached to the nanotubes at randomly selected positions in the central region of nanotube. Lutsko stress enables us to select any averaging volume in which stresses are calculated, and the selected volume is . Stresses shown in Fig. 2, with a wall thickness of 3.4 A and strains are computed in the central region where the hydrocarbon chains are attached. In order to make consistent observations the stresses and strains were calculated within the same averaging volume for CNTs with and without hydrocarbon chain attachments. Since we are primarily interested in elastic properties most of the simulations were carried out for a total strain up to 5%. Fig. 3 shows the Lutsko stress vs. strain plots for (10,10) nanotube without any chemical attachments, and with 21 vinyl attachments in the central region. There is a marginal increase in local stiffness when the carbon chains are attached (for e.g. 0.84–0.92 TPa for (10,10) CNT). The stiffness here is evaluated as initial tangent of the stress–strain curve. Caution should be exerted in comparing this stiffness value with overall Young’s modulus because the CNTs here are of finite length and the local stiffness is calculated only for the region in which hydrocarbon chains are attached. The increase in local stiffness is further confirmed by corresponding increase in the slope of force–displacement curve. Table 1 provides the computed stiffness values for various cases of diameters, chirality, number and type of hydrocarbon attachments studied here. In all the cases, the local stiffness increases with any type of chemical attachment. This effect is more pronounced in nanotu-
249
20
15
10 (10,10) CNT with vinyl attachments (10,10) CNT no attachments
5
0.01
0.02
0.03
0.04
Strain Fig. 3. Stress–strain plot for (10,10) nanotube with and without vinyl attachments.
bes with smaller diameter. For e.g. there is an increase in local stiffness from 0.86 to 1.05 TPa in (8,0) CNT of while, in the case (15,0) nanotube with radius 3.13 A radius 5.87 A stiffness increases to 0.95 TPa (see rows 1–4 and 5–7 of the Table 1). In order to observe the effect of number of chemical attachments, different numbers of chains (21, 31 and 50) are attached to (10,10) nanotube. Stiffness increases from 0.83 to 0.93 TPa with 21 vinyl attachments; it increases to 1.02 and 1.11 TPa with 31 and 50 attachments, respectively (rows 11 and 12 of Table 1). This is understandable based on concentration of chemical attachments. Evidently, the concentration of chemical attachments is higher in CNTs with lower radius; also increasing the number of hydrocarbon chains increases the concentration of chemical attachments. Longer hydrocarbon chains with three, four and five carbon atoms were attached to (10,10) CNT and the elastic behavior was examined. For these attachments, there are two energetically stable structures i.e. cis and
Fig. 2. Volume for Lutsko stress computation.
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S. Namilae et al. / Chemical Physics Letters 387 (2004) 247–252
Table 1 Stiffness values for various nanotubes with and without chemical attachments S. No
Nanotube
˚ Radius (A)
Chemical group
Number of Attachments
Stiffness (TPa) (w/o attachments)
1 2 3 4 5 6 7 8 9 10 11 12
(8,0) (10,0) (12,0) (15,0) (8,8) (10,10) (12,12) (10,10) (10,10) (10,10) (10,10) (10,10)
3.13 3.91 4.69 5.87 5.42 6.78 8.13 6.78 6.78 6.78 6.78 6.78
–C2 H3 a –C2 H3 –C2 H3 –C2 H3 –C2 H3 –C2 H3 –C2 H3 –C3 H5 b –C4 H7 c –C5 H9 d –C2 H3 –C2 H3
21 21 21 21 21 21 21 21 21 21 31 50
0.862 0.854 0.859 0.849 0.721 0.837 0.784 0.837 0.837 0.837 0.837 0.837
a
–C2 H3 –C3 H5 c –C4 H7 d –C5 H9 b
Stiffness (TPa) (with attachments) 1.05 1.04 0.977 0.951 0.889 0.932 0.906 0.94 1.03 0.95 1.02 1.11
is –CH@CH2 . is –CH2 –CH@CH2 . is –C H@CH–CH@CH2 . is –C H2 –CH@CH–CH@CH2 .
trans isomers. We have restricted this study to extended configurations (trans) as they have higher possibility of interacting with polymer matrix in composite applications. It is observed that there is a stiffness increase in all the cases though the magnitude of increase does not show a clear trend. The stiffness values increase as the chain length is increased from two carbon atoms (0.93 TPa) to four carbon atoms (1.03 TPa) but there is a small drop with pentene attachments (0.95 TPa) (rows 6 and 8–10, respectively). In order to observe the effect of functionalization more precisely, we have plotted contour plots based on BDT stress for (10,10) nanotube with a single vinyl group attached to it. The plots are shown at 0%, 2%, 3%
and 4.5% strain (Fig. 4). BDT or atomic stress can be defined as: ! X j 1 1 i rAtomic ma mai maj þ ¼ Atom rab fab ; ð2Þ ij 2 X b¼1;n where XAvg is the averaging volume, m, r, f are velocity force and radial vectors, a and b are the atomic indices, i and j are the indices of the stress tensor. Based on the definition of BDT stress, stresses are observed at zero strain; these are similar to residual stresses at macroscopic scale. Contour plots show that there is considerable stress concentration at the location of chemical attachment. From the definition of atomic stress above (Eq. (2)), value
Fig. 4. Contour plots of stress for (10,10) CNT with one vinyl attachment at 0% strain (a), 1% strain (b), 2% strain (d), 3% strain and 4.5% strain (e). Schematic is shown in (c).
S. Namilae et al. / Chemical Physics Letters 387 (2004) 247–252
3. Plastic deformation and fracture Yakobson and coworkers [28] have observed formation of topological defects such as 5–7–7–5 Stone–Wales defect in nanotubes subject to tensile deformation at high temperature. They envisage formation of these defects as a possible mechanism of strain release during plastic deformation. Belytchsko and coworkers [29] have observed that formation of these topological defects precedes fracture in nanotubes. In order to study the effect of functionalization on plastic deformation and fracture, the functionalized nanotubes were subjected to tensile deformation at 3000 K. Preliminary results indicate that topological defects are formed at lower strains in functionalized nanotubes as compared to CNT without chemical attachments. For example in (10,10) CNT with 21 vinyl chains attached at the center, first defects originate at an overall strain of 6.5% as compared to 13% in a similar tube without chemical attachments. It is interesting to note that the topological defects are first observed in the region away from the center. At higher strains more topological defects are formed at the location of functional attachments. This results in fracture of functionalized CNT at lower strain of 14% where as the CNT without chemical attachments does not fail completely at this strain. The failure in the functionalized CNT occurs near the functional attachments. Garg and Sinnot have noted that the chemical attachments are unstable and detach from the nanotubes nanotubes are when small diameter (less than 5 A) compressively deformed. We observe similar behavior at large tensile strains (greater than 6%) and high temperatures, although the attachments remain stable at lower strains and temperatures.
CNT is altered to diamond like SP3 hybridization. Shenderova and coworkers [30] define nanostructural stiffness as the force required to cause unit elongation in a nano-scale structure. They report that the nanostructural stiffness is more for SP3 hybridized diamond nano rods than that for multi walled or single walled nanotubes. This alteration in chemical structure of functionalized nanotube could be one of the reasons for increase in stiffness. Further, to enable the corresponding changes in bond lengths and bond angles, the radius of curvature of the CNT is increased at the location of hydrocarbon attachments. Fig. 5 shows the radius of curvature variation in (10,10) CNT with and without chemical attachments. The peaks in Fig. 5 correspond to the locations of chemical attachment; it can be observed as that the radius at these locations is about 7.3 A compared to 6.78 A. In addition, there is a decrease in radius of curvature in regions adjoining the location of attachment. Because of this, there is a reasonable deviation from the ideal cylindrical structure of nanotubes resulting in sort of ‘serrations’ in the region of chemical attachments. The contour plots of stress show that surrounding the high stress region at the location of chemical attachment there are regions of low stress. We can infer from this that the peaks and valleys in structure lead to stress fluctuations, which in turn result in the increased local stiffness in functionalized region. In the case of tensile deformation at high temperature, topological defects are formed at lower strains in regions away from chemical attachments, in comparison to nanotubes without chemical attachments. This can be rationalized based on the observation of stress
with vinyl attachments without attachments
7.3 7.2 7.1
Radius
of stress depends upon the number of atoms in the neighborhood of the concerned atom and the forces of interactions between them. The neighborhood of the atom to which vinyl group is attached has higher number of neighbors than rest of the atoms. This results in expected higher stress at this location. The plots, however show that surrounding regions exhibit lower stresses than corresponding regions of defect-free CNT. In contrast to this behavior, uniform stress distribution is observed in CNT without chemical attachments. These stress fluctuations show that the effect of chemical attachment is not limited to the location where bond changes occur, instead a diffuse behavior is observed.
251
7 6.9 6.8 6.7 6.6 100
4. Discussion There is an alteration in chemical bonding at the site of hydrocarbon attachment; the SP2 hybridization in
200
300
400
Atom Number Fig. 5. Radius variations in (10,10) CNT with and without chemical attachments. The peaks correspond to the locations of hydrocarbon attachments.
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fluctuations and lower stiffness in the region where hydrocarbon chains are attached. We can conjecture that this increased local stiffness leads to higher strains in the regions away from region of chemical attachments resulting in defects formed at lower overall strains. Combined with the fact there is fracture at lower strains in functionalized nanotubes we can conclude that in the regions where chemical attachments are made nanotubes are stiffer and more brittle than nanotubes without chemical attachments.
5. Summary The effect of functionalization on mechanical behavior of carbon nanotubes has been investigated. We find that functionalized CNT in general have higher local stiffness than nanotubes without chemical attachments. When deformed at high temperature, formation of topological defects and fracture are observed to occur at lower strains.
Acknowledgements Authors are thankful to Leon VanDommelen, Ashok Srinivasan and Vijayraghavan Rangachari for useful discussions. Funding provided by FSU foundation is gratefully acknowledged
References [1] S. Iijima, Nature 354 (1991) 56. [2] M.F. Yu, B.S. Files, S. Arepalli, R.S. Ruoff, Phys. Rev. Lett. 84 (2000) 5552. [3] A. Krishnan, E. Dujardin, T.W. Ebbesen, P.N. Yianilos, M.M.J. Treacy, Phys. Rev. B 58 (2001) 14013. [4] B.G. Demczyk, Y.M. Wang, J. Cumings, M. Hetman, W. Han, A. Zettl, R.O. Ritchie, Mater. Sci. Eng. A 334 (2002) 173.
[5] D.H. Robertson, D.W. Brenner, J.W. Mintmire, Phys. Rev. B 45 (1992) 12592. [6] E. Hernandez, C. Goze, P. Bernier, A. Rubio, Appl. Phys. A 68 (1999) 287. [7] M.F. Yu, B.I. Yakobson, R.S. Ruoff, J. Phys. Chem. B 104 (2000) 8764. [8] M.S.P. Shafer, A.H. Windle, Adv. Mater. 11 (1999) 937. [9] S.J.V. Frankland, A. Caglar, D.W. Brenner, M. Griebel, J. Phys. Chem. B 106 (2002) 3046. [10] X. Gong, J. Liu, S. Baskaran, R.D. Voise, J.S. Young, Chem. Mater. 12 (2000) 1049. [11] V. Lordi, N. Yao, J. Mater. Res. 15 (2000) 2770. [12] A. Garg, S.B. Sinnott, Chem. Phys. Lett. 295 (1998) 273. [13] N. Chandra, H. Ghonem, Composites: Part A 32 (2001) 575. [14] D.R. Askeland, The Science and Engineering of Materials, third ed., PWS Publishing, Boston, MA, 1994. [15] E.T. Michelson, C.B. Huffman, A.G. Rinzler, R.E. Smalley, R.H. Hauge, J.L. Margrave, Chem. Phys. Lett. 296 (1998) 188. [16] S. Pekker, J.P. Salvetat, E. Jakab, J.M. Bonard, L. Forro, J. Phys. Chem. B 105 (2001) 7938. [17] B.N. Khare, M. Meyyappan, A.M. Cassell, C.V. Nguyen, Jie Han, Nano letters 2 (2002) 73. [18] J. Chen, M.A. Hamon, H. Hu, Y. Chen, A.M. Rao, P.C. Eklund, R.C. Haddon, Science 282 (1998) 95. [19] Y.P. Sun, K. Fu, Y. Lin, W. Huang, Acc. Chem. Res. 35 (2002) 1096. [20] Y.P. Sun, B. Zhou, K. Henbest, K. Fu, W. Huang, Y. Lin, S. Taylor, D.L. Carroll, Chem. Phys. Lett. 351 (2002) 349. [21] J. Kong, N.R. Franklin, C. Zhou, M.G. Chapline, S. Peng, K. Cho, H. Dai, Science 287 (2000) 622. [22] D. Brenner, Phys. Rev. B 42 (1990) 9458. [23] M.P. Allen, W.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, 1989. [24] Z.S. Basinski, M.S. Duesberry, R. Taylor, Can. J. Phys. 49 (1971) 2160. [25] J.F. Lutsko, J. Appl. Phys. 64 (1988) 1152. [26] J. Cormier, J.M. Rickman, T.J. Delph, J. Appl. Phys. 89 (2001) 99. [27] N. Chandra, S. Namilae, C. Shet, Phy. Rev. B, 2004, in press. [28] B.I. Yakobson, C.J. Brabec, J. Bernholc, Phys. Rev. Lett. 76 (1996) 2511. [29] T. Belytschko, S.P. Xiao, G.C. Schatz, R.S. Ruoff, Phys. Rev. B 65 (2002) 235430. [30] O. Shendorova, V.V. Zhirnov, D.W. Brenner, Crit. Rev. Solid State Mater. Sci. 27 (2002) 227.
Mechanical behavior of functionalized nanotubes S. Namilae *, N. Chandra, C. Shet Department of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University, 2525 Pottsdamer Street, Tallahassee, FL 32310, USA Received 17 July 2003; in final form 17 July 2003 Published online: 4 March 2004
Abstract The potential use of carbon nanotubes as reinforcements in composite materials is greatly enhanced by improving the fibermatrix interface bonding. A proposed method to increase the fiber-matrix bonding is by functionalizing the nanotubes. We use molecular dynamics and statics simulations to study the tensile mechanical response of functionalized carbon nanotubes. It is found that the there is a marginal increase in stiffness in functionalized nanotubes, and the stiffness values are found to increase with the increase in number of attachments. When deformed at high temperatures, formation of topological and failure are found to occur at lower strains. Ó 2004 Elsevier B.V. All rights reserved.
1. Introduction The exceptional mechanical properties of carbon nanotubes (CNT) and their potential use in structural components and devices have stimulated great interest and extensive research ever since their discovery by Ijjima [1] in 1991. Experimental estimates [2–4] of Young’s modulus of CNTs are in the range of 0.32–1.47 TPa, while the theoretical estimates [5–7] vary from 0.5 to 5.5 TPa. The failure strength of these materials is experimentally estimated to be in the order of 150 GPa [4]. These excellent mechanical properties combined with their extremely high strength to weight ratio make them potential candidates as reinforcing fibers in super strong composites. In spite of outstanding properties of individual nanotubes, the elastic modulus of nanotube composite is much lesser, for e.g. Shaffer and Windle [8] report Young’s modulus value of 150 MPa in CNT– PVA composite films. The main obstacles in improving the stiffness and strength of nanotube are dispersion, alignment and interfacial load transfer. In order to optimize the properties of nanotube-based composites a *
Corresponding author. Fax: +1-850-410-6337/6420. E-mail addresses: [email protected] (S. Namilae), chandra@ eng.fsu.edu (N. Chandra). 0009-2614/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2004.01.104
clear understanding of the load transfer and interface mechanics at nanoscale is required. Chemical attachment or cross-linking of nanotube walls and polymeric matrix has been proposed as one of the techniques to improve the interfacial bonding. Based on molecular dynamics simulation Frankland and coworkers [9] report that CNT matrix shear strength and critical length for load transfer improve considerably by chemically cross-linking the CNT and matrix. Gong and coworkers [10] obtained improved mechanical properties in CNT epoxy composite by using surfactant coating on nanotube. They infer that this is due to increased interfacial strength with surfactant forming a weak bond between CNT and epoxy. Lordi and Yao [11] based on molecular mechanics simulation suggest that molecular level entanglement as a possible method to strengthen the interface. Garg and Sinnott [12] have recently studied the compressive properties of functionalized tubes, and reported that the force required for buckling reduces marginally by chemical attachment of hydrocarbon chains. Chandra and Ghonem [13] have noted that thermomechanical load transfer between fibers and matrix in conventional composites is generally affected by both chemical and mechanical bonding. While chemical bonding arises from the formation of new phases,
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S. Namilae et al. / Chemical Physics Letters 387 (2004) 247–252
mechanical bonding occurs by interlocking of asperities. Use of functionalized nanotubes as fibers in composites could enhance the cross-linking (chemical bond) a nd interfacial strength between matrix and fiber. These chemical attachments are also expected to act as ‘tethers’ (mechanical bond) and help in load transfer when the composite is deformed either slowly or rapidly [13,14]. Though these concepts are well established in conventional composites at micrometer scale interfaces, it is not clear if the same concepts could be transferred to nanoscale interfaces. Researchers have attached different functional groups to the walls of nanotubes using various experimental techniques. For example Michelson and coworkers [15] have fluorinated CNTs using alcohol solvents, while Pekker and coworkers [16] attached hydrogen using ammonia. Khare and coworkers [17] have hydrogenated nanotubes using electric discharge. Chen and coworkers [18] have attached alkyl chains using amidization, whereas Sun and coworkers used amidization and esterification to attach carboxylic acids [19,20]. The latter also processed PVA thin films embedded with functionalized nanotubes, which could be used for preparation of aligned nanocomposites. Apart from possible applications in nanocomposites, functionalization of CNTs is proposed as a useful technique for altering electronic properties and fabrication of sensors [21]. In order to use chemical attachments as a means to achieve increased interfacial strength, it is necessary to have an understanding of mechanical properties of functionalized nanotubes. The objective of this Letter is to study the effect of functionalization on the mechanical response CNTs under tension, and explore its origin. We also examine the effect of different number and types of (hydrocarbon) attachments primarily on the elastic response and also inelastic response and fracture.
rLutsko ij
1 X ¼ Avg X a¼1;n
! 1 a a a X j i m mi mj þ rab fab lab : 2 b¼1;n
ð1Þ
Here XAvg is the averaging volume, m, r, f are velocity force and radial vectors, a and b are the atomic indices, i and j are the indices of the stress tensor. lab denotes the fraction of a–b bond lying inside the averaging volume. Strains are calculated as derivative of displacement field, heavily borrowing from the concepts of finite elements. The displacement field is expressed in terms of known atomic displacements and interpolation functions. Graphitic structure can be considered as a mesh of hexagons. Each of these hexagons is further divided into triangular facets and average strain in these facets is calculated based on the displacement field given by the atoms defining the corners of the triangular facets. Strains at each atomic location are obtained as area average of the strains of immediate hexagons surrounding it. To validate the applicability of the stress and strain measure, stress–strain curves are plotted for a perfect (9,0) CNT under uniaxial tension. Displacements are applied incrementally to all the atoms of periodic (9,0) CNT in longitudinal direction. The nanotube is stabilized using conjugate gradients energy minimization scheme. Fig. 1 shows the stress–strain curves based on the three stress measures considered, virial, Lutsko and BDT stress, measured in longitudinal direction and longitudinal strain evaluated using the above described scheme. The Young’s modulus is evaluated as the slope of stress–strain curve at zero strain (initial tangent). We obtain a Young’s modulus value of 1.002 TPa for (9,0) CNT. This compares favorably with values published in the literature [2–7] which center around 1 TPa.
60
Lutsko Stress (E=0.997 TPa) Bulk stress (E=1.002 TPa)
2. Elastic deformation
Stress (GPa)
We use molecular dynamics and statics simulations based on Tersoff–Brenner bond-order potential [22] to evaluate the tensile stiffness and stress–strain response of various zigzag and armchair CNT. Stiffness can be evaluated as the second derivative of energy with respect to strain, or as the ratio of stress and strain. The later method is preferable as it involves fewer assumptions and further enables the calculation of local properties. We have used stress measures such as virial stress [23], BDT stress [24] and Lutsko stress [25,26] and developed a methodology for evaluating strain. These concepts are discussed in detail elsewhere [27], however they are mentioned here briefly for the sake of completeness. We primarily use Lutsko stress rLutsko to characterize stress. ij It is given by
BDT stress (E=1.002 TPa)
50
40
30
20
10
0 0
0.02
0.04
0.06
strain Fig. 1. Stress–strain curves for (9,0) CNT subject to uniaxial tension.
S. Namilae et al. / Chemical Physics Letters 387 (2004) 247–252
30
25
Stress (GPa)
Now we present the molecular dynamics results of nanotubes with chemically attached hydrocarbon chains. Zigzag and armchair nanotubes with varying diameter were considered for this study. All the na in length, tensile stresses were notubes are about 120 A applied by fixing and displacing both the ends of na on either side) followed by stabilization notubes (15 A for 1500 time steps with a step size of 0.2 fs. The temperature is maintained at 77 K. Stresses and strains are evaluated as average over 15 time steps. Hydrocarbon chains are attached to the nanotubes at randomly selected positions in the central region of nanotube. Lutsko stress enables us to select any averaging volume in which stresses are calculated, and the selected volume is . Stresses shown in Fig. 2, with a wall thickness of 3.4 A and strains are computed in the central region where the hydrocarbon chains are attached. In order to make consistent observations the stresses and strains were calculated within the same averaging volume for CNTs with and without hydrocarbon chain attachments. Since we are primarily interested in elastic properties most of the simulations were carried out for a total strain up to 5%. Fig. 3 shows the Lutsko stress vs. strain plots for (10,10) nanotube without any chemical attachments, and with 21 vinyl attachments in the central region. There is a marginal increase in local stiffness when the carbon chains are attached (for e.g. 0.84–0.92 TPa for (10,10) CNT). The stiffness here is evaluated as initial tangent of the stress–strain curve. Caution should be exerted in comparing this stiffness value with overall Young’s modulus because the CNTs here are of finite length and the local stiffness is calculated only for the region in which hydrocarbon chains are attached. The increase in local stiffness is further confirmed by corresponding increase in the slope of force–displacement curve. Table 1 provides the computed stiffness values for various cases of diameters, chirality, number and type of hydrocarbon attachments studied here. In all the cases, the local stiffness increases with any type of chemical attachment. This effect is more pronounced in nanotu-
249
20
15
10 (10,10) CNT with vinyl attachments (10,10) CNT no attachments
5
0.01
0.02
0.03
0.04
Strain Fig. 3. Stress–strain plot for (10,10) nanotube with and without vinyl attachments.
bes with smaller diameter. For e.g. there is an increase in local stiffness from 0.86 to 1.05 TPa in (8,0) CNT of while, in the case (15,0) nanotube with radius 3.13 A radius 5.87 A stiffness increases to 0.95 TPa (see rows 1–4 and 5–7 of the Table 1). In order to observe the effect of number of chemical attachments, different numbers of chains (21, 31 and 50) are attached to (10,10) nanotube. Stiffness increases from 0.83 to 0.93 TPa with 21 vinyl attachments; it increases to 1.02 and 1.11 TPa with 31 and 50 attachments, respectively (rows 11 and 12 of Table 1). This is understandable based on concentration of chemical attachments. Evidently, the concentration of chemical attachments is higher in CNTs with lower radius; also increasing the number of hydrocarbon chains increases the concentration of chemical attachments. Longer hydrocarbon chains with three, four and five carbon atoms were attached to (10,10) CNT and the elastic behavior was examined. For these attachments, there are two energetically stable structures i.e. cis and
Fig. 2. Volume for Lutsko stress computation.
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Table 1 Stiffness values for various nanotubes with and without chemical attachments S. No
Nanotube
˚ Radius (A)
Chemical group
Number of Attachments
Stiffness (TPa) (w/o attachments)
1 2 3 4 5 6 7 8 9 10 11 12
(8,0) (10,0) (12,0) (15,0) (8,8) (10,10) (12,12) (10,10) (10,10) (10,10) (10,10) (10,10)
3.13 3.91 4.69 5.87 5.42 6.78 8.13 6.78 6.78 6.78 6.78 6.78
–C2 H3 a –C2 H3 –C2 H3 –C2 H3 –C2 H3 –C2 H3 –C2 H3 –C3 H5 b –C4 H7 c –C5 H9 d –C2 H3 –C2 H3
21 21 21 21 21 21 21 21 21 21 31 50
0.862 0.854 0.859 0.849 0.721 0.837 0.784 0.837 0.837 0.837 0.837 0.837
a
–C2 H3 –C3 H5 c –C4 H7 d –C5 H9 b
Stiffness (TPa) (with attachments) 1.05 1.04 0.977 0.951 0.889 0.932 0.906 0.94 1.03 0.95 1.02 1.11
is –CH@CH2 . is –CH2 –CH@CH2 . is –C H@CH–CH@CH2 . is –C H2 –CH@CH–CH@CH2 .
trans isomers. We have restricted this study to extended configurations (trans) as they have higher possibility of interacting with polymer matrix in composite applications. It is observed that there is a stiffness increase in all the cases though the magnitude of increase does not show a clear trend. The stiffness values increase as the chain length is increased from two carbon atoms (0.93 TPa) to four carbon atoms (1.03 TPa) but there is a small drop with pentene attachments (0.95 TPa) (rows 6 and 8–10, respectively). In order to observe the effect of functionalization more precisely, we have plotted contour plots based on BDT stress for (10,10) nanotube with a single vinyl group attached to it. The plots are shown at 0%, 2%, 3%
and 4.5% strain (Fig. 4). BDT or atomic stress can be defined as: ! X j 1 1 i rAtomic ma mai maj þ ¼ Atom rab fab ; ð2Þ ij 2 X b¼1;n where XAvg is the averaging volume, m, r, f are velocity force and radial vectors, a and b are the atomic indices, i and j are the indices of the stress tensor. Based on the definition of BDT stress, stresses are observed at zero strain; these are similar to residual stresses at macroscopic scale. Contour plots show that there is considerable stress concentration at the location of chemical attachment. From the definition of atomic stress above (Eq. (2)), value
Fig. 4. Contour plots of stress for (10,10) CNT with one vinyl attachment at 0% strain (a), 1% strain (b), 2% strain (d), 3% strain and 4.5% strain (e). Schematic is shown in (c).
S. Namilae et al. / Chemical Physics Letters 387 (2004) 247–252
3. Plastic deformation and fracture Yakobson and coworkers [28] have observed formation of topological defects such as 5–7–7–5 Stone–Wales defect in nanotubes subject to tensile deformation at high temperature. They envisage formation of these defects as a possible mechanism of strain release during plastic deformation. Belytchsko and coworkers [29] have observed that formation of these topological defects precedes fracture in nanotubes. In order to study the effect of functionalization on plastic deformation and fracture, the functionalized nanotubes were subjected to tensile deformation at 3000 K. Preliminary results indicate that topological defects are formed at lower strains in functionalized nanotubes as compared to CNT without chemical attachments. For example in (10,10) CNT with 21 vinyl chains attached at the center, first defects originate at an overall strain of 6.5% as compared to 13% in a similar tube without chemical attachments. It is interesting to note that the topological defects are first observed in the region away from the center. At higher strains more topological defects are formed at the location of functional attachments. This results in fracture of functionalized CNT at lower strain of 14% where as the CNT without chemical attachments does not fail completely at this strain. The failure in the functionalized CNT occurs near the functional attachments. Garg and Sinnot have noted that the chemical attachments are unstable and detach from the nanotubes nanotubes are when small diameter (less than 5 A) compressively deformed. We observe similar behavior at large tensile strains (greater than 6%) and high temperatures, although the attachments remain stable at lower strains and temperatures.
CNT is altered to diamond like SP3 hybridization. Shenderova and coworkers [30] define nanostructural stiffness as the force required to cause unit elongation in a nano-scale structure. They report that the nanostructural stiffness is more for SP3 hybridized diamond nano rods than that for multi walled or single walled nanotubes. This alteration in chemical structure of functionalized nanotube could be one of the reasons for increase in stiffness. Further, to enable the corresponding changes in bond lengths and bond angles, the radius of curvature of the CNT is increased at the location of hydrocarbon attachments. Fig. 5 shows the radius of curvature variation in (10,10) CNT with and without chemical attachments. The peaks in Fig. 5 correspond to the locations of chemical attachment; it can be observed as that the radius at these locations is about 7.3 A compared to 6.78 A. In addition, there is a decrease in radius of curvature in regions adjoining the location of attachment. Because of this, there is a reasonable deviation from the ideal cylindrical structure of nanotubes resulting in sort of ‘serrations’ in the region of chemical attachments. The contour plots of stress show that surrounding the high stress region at the location of chemical attachment there are regions of low stress. We can infer from this that the peaks and valleys in structure lead to stress fluctuations, which in turn result in the increased local stiffness in functionalized region. In the case of tensile deformation at high temperature, topological defects are formed at lower strains in regions away from chemical attachments, in comparison to nanotubes without chemical attachments. This can be rationalized based on the observation of stress
with vinyl attachments without attachments
7.3 7.2 7.1
Radius
of stress depends upon the number of atoms in the neighborhood of the concerned atom and the forces of interactions between them. The neighborhood of the atom to which vinyl group is attached has higher number of neighbors than rest of the atoms. This results in expected higher stress at this location. The plots, however show that surrounding regions exhibit lower stresses than corresponding regions of defect-free CNT. In contrast to this behavior, uniform stress distribution is observed in CNT without chemical attachments. These stress fluctuations show that the effect of chemical attachment is not limited to the location where bond changes occur, instead a diffuse behavior is observed.
251
7 6.9 6.8 6.7 6.6 100
4. Discussion There is an alteration in chemical bonding at the site of hydrocarbon attachment; the SP2 hybridization in
200
300
400
Atom Number Fig. 5. Radius variations in (10,10) CNT with and without chemical attachments. The peaks correspond to the locations of hydrocarbon attachments.
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fluctuations and lower stiffness in the region where hydrocarbon chains are attached. We can conjecture that this increased local stiffness leads to higher strains in the regions away from region of chemical attachments resulting in defects formed at lower overall strains. Combined with the fact there is fracture at lower strains in functionalized nanotubes we can conclude that in the regions where chemical attachments are made nanotubes are stiffer and more brittle than nanotubes without chemical attachments.
5. Summary The effect of functionalization on mechanical behavior of carbon nanotubes has been investigated. We find that functionalized CNT in general have higher local stiffness than nanotubes without chemical attachments. When deformed at high temperature, formation of topological defects and fracture are observed to occur at lower strains.
Acknowledgements Authors are thankful to Leon VanDommelen, Ashok Srinivasan and Vijayraghavan Rangachari for useful discussions. Funding provided by FSU foundation is gratefully acknowledged
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