Defense Applications of Nanomaterials - American Chemical Society

Chapter 19

Properties of Single Wall Carbon Nanotubes: A Density Functional Theory Study 1

2

1

3

1,*

B. Akdim , X. Duan , Z. Wang , W. W. Adams , and R. Pachter Downloaded by UNIV MASSACHUSETTS AMHERST on August 4, 2012 | http://pubs.acs.org Publication Date: January 21, 2005 | doi: 10.1021/bk-2005-0891.ch019

1

Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, O H 45433 Aeronautical Systems Center, Major Shared Resource Center for High Performance Computing, 3005 Ρ Street, Wright-Patterson Air Force Base, O H 45433 Center for Nanoscale Science and Technology, Rice University, Houston, TX 77005 Corresponding author: [email protected] 2

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*

We present a study of single-wall carbon nanotubes (SWCNTs) for C(n,n) and C(n,0), n=(4,6,8,10), modeled as isolated tubes or crystalline-ropes, using a full-potential linear combination of atomic orbitals (FP-LCAO) density functional theory (DFT) approach. Structural and mechanical properties agree well with previous theoretical and experimental work. The study provides further insight, for example, into structural aspects of SWCNTs, where we note a different trend for bond lengths in armchair and zigzag tubes. An up-shift of the Raman radial breathing mode (RBM) due to intertube interactions was calculated and compared to experiment and previous theoretical studies. Our calculations provide an example for a computational approach to be used in studying SWCNT materials for defense related applications.

© 2005 American Chemical Society

In Defense Applications of Nanomaterials; Miziolek, A., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2005.

265

Downloaded by UNIV MASSACHUSETTS AMHERST on August 4, 2012 | http://pubs.acs.org Publication Date: January 21, 2005 | doi: 10.1021/bk-2005-0891.ch019

266 Studies of carbon nanotubes, recently reviewed (/), suggest a wide range of applications (2). Moreover, with the advent of the synthesis, controllable growth, manipulation and characterization of these materials (3,4,5,6), it has become possible to gain significant insight from the calculation of properties, due to the possibility of comparison with experiment. Indeed, resonance Raman spectroscopy is well known for measuring die RBMs of SWCNTs and characterizing tube radii (7,8), in comparison with theory (9,10). The rolled structure of a graphene sheet gives rise to unprecedented electronic properties of carbon nanotubes, where the semiconducting or metallic characteristics depend on chirality and radii. These properties were first predicted by theory (11,12), and then confirmed by experiment (13,14). Indeed, electronic nanodevices such as field-effect transistors were proposed (15,16,17), holding promise also for defense related applications. In addition, carbon nanotubes also attracted considerable attention for field emission, and could possibly be utilized for defense applications such as field emission cathodes in high-power microwave tubes (18), also studied theoretically (19). Furthermore, the exceptional mechanical properties of nanotubes, as demonstrated by their tensile strength and elastic moduli (20), render them appropriate for composites applications (21). A Young's modulus (YM) measurement for multi-wall carbon nanotubes (MWCNTs) was carried out by the thermal vibrational amplitudes technique (22), reporting a value of 1.8TPa. Krishnam et al. (23) performed a similar experiment for SWCNTs, and obtained a value of 1.3±0.5TPa. An experiment by Selvetat and co-workers (24) for single-wall crystalline-ropes using an A F M with a special substrate to allow for a direct measurement of Young's modulus, resulted in an average value of 1.28±0.59TPa. The recently reported experiment by Demczyk et al. (25), estimated the tensile strength and Y M to be O.lSTPa, and 0.9TPa, respectively. In this study, we report an evaluation of structural, mechanical, and vibrational properties of SWCNTs [C(n,n), and C(n,0), n=4,6,8,10], applying a FP-LCAO method (26,27), as it has previously been pointed out that all-electron full-potential local density approximation (LDA)-based calculations are appropriate for treating van der Waals interactions in systems such as fullerenes, graphene and graphite (28,29). Although the full-potential all-electron scheme is computationally intensive, our work contributes to the further validation of previous theoretical studies. Large-scale pseudopotential DFT calculations were also recently performed by Yoon et al. (30), in order to study the effects of intertube interactions on the transport properties of SWCNT junctions. We also studied crystalline-ropes, optimizing the intertube distance, in comparison with other theoretical studies using the pseudopotential LCAO approach (31), tight-binding methods (32,33), or an empirical force-constant model (34), as well as with experimental data whenever available.


267


Computational Details In all SWCNTs calculations we applied DMOL3 (26), using a doublenumeric polarized basis set, with an atomic cutoff radius of 10.4 (a.u.). The exchange-correlation term follows the Perdew and Wang parameterization (35). Calculations were carried out with full geometry optimization, assuming a hexagonal symmetry to reduce computation time. Different k-point samplings of the irreducible Brillouin zone were tested, adopting IS k-point for crystallineropes (bundles) and 5 k-point for an isolated nanotube. Periodic boundary conditions were used in 3-D to simulate a crystalline-rope in the hexagonal structure, with a triangular unit cell in the plane perpendicular to the tube axis. A wide lateral separation, larger than 45 A, was used to model isolated tubes. In addition, for the C(6,6) SWCNT, we performed calculations of a super-cell of five unit-cells and found insignificant changes in the structural properties. Thus, all reported results were carried out using only one unit-cell with no defects.

Results and Discussion

Structural and Mechanical Properties Some of the early theoretical work on carbon nanotubes includes that of Lu (34), using an empirical force-constant model to calculate elastic moduli for SWCNTs and MWCNTs, assuming that the interwall distances are those of the graphite interlayer distance (3.4A) with no further optimization. In addition, Hernandez et al. (32) performed a comparative tight-binding study of isolated carbon nanotubes and other compositions (B N C ), using 3.4A to predict the properties of a crystalline-rope with no further optimization, whereas SanchezPortal and co-workers (31) used a pseudopotential LCAO method to calculate properties of isolated carbon nanotubes. Notably, interwall carbon nanotubes interactions (36), were also most recently studied (37,38,39,40). Intertube distances for the SWCNTs considered were optimized by calculating the energy as a function of intertube separation, allowing for a foil relaxation of atom coordinates at each intertube distance. The results were taken as the intertube distance evaluated from the center of one tube to die center of an adjacent one and listed for selected tubes (Figure 1, Table I). With the exception of C(4,0), where we obtained a value of 12% lower than 2R enige+3.4A, our results differ by about 2-4.5% when compared to other theoretical work. Interestingly, a previous DFT study of carbon nanotube packings has shown that hexagonal packing does result in intertube distances smaller than 3.4 A (41). x

y

z

av



268

The nanotube crystalline-rope interaction energies were found to be, on average, 0.2 (eV/A) and 0.17 (eV/A), for armchair and zigzag tubes, respectively. These results agree well with a previous theoretical study (42). The average radii (R rage)> and the (C-C) and (C-C) bond lengths, derived from the optimized geometry of the equilibrium structure, are summarized in Table I. The subscripts h and v designate horizontal and vertical bonds, respectively; the vertical bonds in zigzag tubes correspond to the double bonds, and the horizontal bonds to the single ones, whereas in the armchair tubes the correspondence of the bonds is reversed. Radii were shown to be within 1.5% of tight-binding (32) and pseudopotential LCAO (31) calculations. In Figure 2, we plot the carbon-carbon bond lengths of various nanotubes. For zigzag tubes, the horizontal bond lengths, in the direction of the circumference of the tube, were shown to be larger for small-diameter tubes, whereas the vertical bonds, in the direction of the tube axis, have a smaller bond length, increasing with tube radius. Both the horizontal and vertical bond lengths converge to the graphite carbon-carbon bond length experimental value (43) of 1.419A. In the case of armchair tubes, the difference is insignificant, and their values are shown to be within the DFT graphite bond length value. Young's modulus was calculated by carrying out a full self-consistent field calculation for each strain imposed along die tube axis, allowing for a full relaxation of atomic positions. The tubes were then compressed and stretched to about -4% and +4%, respectively. A smaller range of strains [-1%, +1%], as previously suggested (31), resulted in incorrect predictions of Poisson's ratio. ave

h

v


269 The total energy was then fitted to a third order polynomial as a function of the strain (e), from which Young's modulus was derived (31) (Table II).

« i i i i • I- i

i *i

• i i i

\ ( C - C ) of'an.0) h


r

:

(C-C) ofC(n,n) v

r

(C-C) ofC(n,n) h

~

"

(C-C) ofQn,0) v

, i , , , , i , , , . t • , i

1

2

3

4

• i « . i < l . . i . '

5

6

7

Radius, A

Figure 2: Bond lengths as a function of the tube radius (A). (See page 7 of color insert.)

We note that, in general, calculated values are within the experimental estimation, and in particular, in good agreement with recent data (25). In addition, we examined the effects of intertube interactions on tube radii and Young's moduli, using 3.4A (Figure 3a) and optimized distances (Figure 3b). Our results show that intertube interactions have only a small effect on Young's moduli. Young's modulus increases with the tube radius for zigzag nanotubes, while only small variations are shown for armchair tubes. In both cases, the values converge to the in-plane C elastic modulus of a graphite sheet, measured as 1.02TPa. Young's moduli using the FP-LCAO optimized intertube separations are shown in Figure 3b and summarized in Table II. We obtained larger values when compared to those evaluated with the interlayer graphite distance of 3.4A. Clearly, these results are due to the lower values obtained for the intertube distances (Table I). It is interesting to note that for C(4,0), with a radius smaller than 2.4A, we predict a different trend, compared to the results using the interlayer graphite distance (Figure 3a), thus suggesting a different strength. This is in agreement with Stojkovic et al. (44), reporting a high stiffness for small radii tubes. We further investigated the effect of strain (e) on the tubes calculating Poisson's ratio (a), as listed in Table III. Intertube interactions appear to have overall a small effect on Poisson's ratio, although a higher response to an external force for small radii tubes was calculated. Our results were found to be lower than the tight-binding values (32,33). n


270

Table I. A summary of calculated structural parameters. ITD denotes the intertube distance (A).


SWCNT C(4,0)

(C-Cfc, (C-C), ^average ITD (2R +3.4) (C-C), (C-C) ^average ITD (2H-.+3.4) (C-C) (C-C) Raverage ITD averaue

(6,0)

v

(4.4)

h

v

(8.0)

(C-C)* (C-C), Raveraje ITD (2R verai» +3.4) a

(10.0)

(C-C) (C-C)v Raverage ITD (2R»«,«+3.4) (C-C),, (C-C), Raveiajjc ITD (2R +3.4) (C-C)* (C-C), Raverap ITD (2R .+3.4) (C-C)» (C-C), Raveragc ITD (2R_+3.4) h

(6,6)

lvmiK

(8,8)

WM

(10,10)

Crystalline-rope 1.460 1.400 1.660 6.03 6.72 1.435 1.405 2.402 7.980 8.200 1.421 1.418 2.746 [2.194(31)] 8.600 8.900 r8.99(J/j] 1.422 1.411 3.158 9.497 9.720 1.421 1.411 3.933 [3.979(5/), 3.955(32)] 11.04 11.27 [ll.36( J/), 11.31(52)] 1.415 1.417 4.070 [4.14(5/), 4.1(52)] 11.20 11.54 [11.68(5/), 11.60(52)] 1.413 1.416 5.420 [5.498(5/)] 13.90 14.24 [14.40(52)] 1.412 1.414 6.760 [6.864(5/), 6.8(52)] 16.60 16.92 [17.13(5/), 17.00(52)]


Isolated 1.463 1.398 1.665 — —

1.435 1.404 2.399 —

1.421 1.417 2.746

1.425 1.409 3.160

1.420 1.411 3.929

1.415 1.417 4.077

1.413 1.416 5.416

1.412 1.414 6.761


2

b

C

a

Cystalline-rope YM Y M *5R YM (TPaA) 2.77 0.82 1.06 1.03 3.23 0.95 3.27 0.96 1.03 3.54 1.09 1.01 3.53 1.04 1.12 3.34 0.98 1.08 3.40 1.00 1.11 3.40 1.00 1.09 Y M *8R (TPaA) 2.84 3.28 3.27 3.43 3.49 3.32 3.37 3.35

b

YM" 1.04 1.01 1.03 1.08 1.14 1.07 1.10 1.10

C

0.84 0.97 0.96 1.01 1.03 0.98 0.99 0.99

Isolated YM

1.24(32)0.972(34)

1.22(52)0.975 (34) 1.22, 1.09 (31)

y-

A

1

C

a

Q

d

rsl p ° , where V = 2nLR8R; SR. edge-to-edge intertube separation; L unit cell length, R radius of the V. de tube; "Young's modulus*8R (TPa*A) values are provided due to the controversy regarding the definition of

Defense Applications of Nanomaterials - American Chemical Society

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