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Submitted to Nanotechnology, 10/15/97
A. Garg and S. B. Sinnott*
Department of Chemical and Materials Engineering
The University of Kentucky
Lexington, Kentucky 40506-0046
This is a draft paper for a talk at the Fifth Foresight Conference on Molecular Nanotechnology. The final version has been submitted for publication in the special Conference issue of Nanotechnology.
This paper is available on the web at http://www.foresight.org/Conferences/MNT05/Papers/Sinnott/index.html
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Proximal probe technology has provided researchers with new ways to investigate and manipulate matter on the nanometer scale. We have studied, through molecular dynamics simulations, using a many-body empirical potential, the indentation of a hydrogen-terminated, diamond (111) surface, with a proximal probe tip that consists of an open, hydrogen-terminated, (10,10) carbon nanotubule. The simulations showed that upon indenting 1.8 Å, the tubule deforms but returns to its original shape upon retraction. The Youngís modulus of the tubule was determined using the predicted Euler buckling force and was found to be comparable to measured and calculated values. In a second series of simulations, an open (10, 10) nanotubule was heated to 4500 K and allowed to close. We find that at this temperature the resulting cap contains numerous imperfections, including some not mentioned previously in the literature.
Advances in proximal probe technology over the last decade have given researchers new abilities to investigate, process and manipulate matter at the nanometer scale, where material properties can be significantly different from the bulk. As this fascinating new realm of nanometer-scale materials engineering continues to be explored and understood, our ability to manufacture powerful new devices will increase. Current experimental efforts have focused on using proximal probes such as the Atomic Force Microscope (AFM) to study the nanometer-scale properties of thin films and substrates1 and manipulate individual atoms in an effort to create nanometer-scale structures and switches.2,3,4,5,6 Theoretical efforts have concentrated on providing information about the mechanisms involved in microscope tip-surface interactions7,8,9,10,11 and exploring new applications and materials not yet available for experimental study.12,13
During much of the same time period, there has also been intense interest in a new class of materials known as carbon nanotubules. Since their discovery by Iijima,14 they have been under concentrated study by numerous research groups.15 Carbon nanotubules, which may be thought of as rolled up sheets of graphite, can have different helical structures and can be single-walled (SWT) or multi-walled (MWT).14,16 In the latter case, 2-50 tubules can be positioned concentrically within one another. Changes in the helical structure and diameter can affect the overall electronic properties of nanotubules which range from semiconducting to metallic.17,18,19 There has also been considerable interest in the mechanical properties of carbon nanotubules because of their predicted20,21,22,23,24,25 and measured26,27 high modulus in the direction of the tubule axis.
Recently, Smalley and co-workers have attached individual carbon nanotubules to silicon cantilevers in conventional AFM instruments.28 This is an important advancement in proximal probe microscopy because such tips can not only conduct, they should also be resistant to mechanical damage while at the same time providing superior imaging capabilities due to their relative sharpness. Although only briefly mentioned by Smalley and coworkers, such tips, will be important in nanometer-scale AFM measurements because of the well-defined contact area between the tip and the sample.
Over the last few years there has been much discussion in the literature about the mechanisms by which nanotubules form under various conditions, such as in the presence of metal catalysts like iron to produce SWTs.29 For example, Charlier et al. have performed first principles molecular dynamics (MD) simulations of the closure of (10, 0) and (5, 5) tubules at about 3000 K and have shown that the closed (10, 0) tubule is more stable than the open tube by about 18 eV.30 (see Ref. 15 or 21 for a discussion of these reference numbers). In addition, these simulations provided researchers with a glimpse into the atomic-scale mechanisms by which tubules might close.
This paper examines, through MD simulations using a well-known, classical, empirical, many-body potential, the indentation of a hydrogen-terminated, diamond (111) surface, C(111):H, with an AFM tip which has as its tip an open, hydrogen-terminated, (10,10) carbon nanotubule. We also replicate the simulations of Charlier et al. with this classical potential using a larger tubule system.
The MD simulations integrate Newtonís equations of motion using a third-order Nordsieck predictor-corrector method31 with a time step of 0.15 fs. In this approach the forces on the individual carbon atoms are determined from an improved reactive-empirical bond order (REBO) hydrocarbon potential that realistically characterizes the properties of small hydrocarbon molecules, graphite and diamond.32 This potential includes improved analytic functions for intramolecular interactions and an expanded fitting database, which includes radicals, small hydrocarbon molecules, graphite and diamond. Because the atoms are treated as hard spheres, forces from electronic effects like orbital resonance and symmetry are neglected. In addition, since this potential is comparatively short ranged, long-ranged forces, like H-bonding, are not included.
The potential predicts elastic constants for diamond of c11=10.78 x 1011 N/m2, c12=1.31 x 1011 N/m2 and c44=6.8 x 1011 N/m2 which compare favorably with the respective experimental values of c11=10.76 x 1011 N/m2, c12=1.25 x 1011 N/m2 and c44=5.8 x 1011 N/m2.33 It has been successfully used to simulate numerous processes including the deformation of carbon nanotubules,21,24,25 the indentation of diamond surfaces7,10 and amorphous carbon thin films,12,16 surface patterning,12,13 nanofriction and tribochemistry,34 ion-bombardment of polymers,35 the formation of fullerenes from graphitic ribbons,36 and surface chemical reactions.37,38,39,40
The carbon nanotubule used for the indentation simulation is of type (10,10) and consists of 20 unit cells, terminated with 10 hydrogen atoms at each end for a total of 820 atoms in the tip. The C(111):H surface is composed of 20 layers of carbon with each layer containing 370 atoms. Periodic boundary conditions are applied in the plane of the surface and both end layers of the substrate are hydrogen terminated (370 atoms). Thus the substrate contains 8140 atoms and the entire system consists of 8960 atoms.
The indentation sequence is shown in Figure 1, where Figure 1(a) shows the initial configuration. The top 100 atoms of the carbon nanotubule (furthest from the surface) and the 1480 atoms at the base of the substrate (furthest from the tip) are held rigid throughout the simulation. Moving toward the middle of the system, a Langevin thermostat31 is applied to the next 200 atoms of the nanotubule and 2590 atoms of the substrate. The remaining atoms of both the nanotubule and the substrate evolve in time according to Newtonís equations of motion with no constraints. The simulation is carried out at 300 K.
The rigid atoms at the end of the nanotubule are moved towards the substrate in steps of 0.05 Å and the system is equilibrated for 400 time steps in between displacement steps. This process is continued until the nanotubule indents the substrate about 1.8 Å. After indentation, the tip is retracted using identical displacement and time steps. The force on the tip atoms is calculated at each time step. To minimize fluctuations in the force caused by the relaxation of the atoms, the forces are averaged over the last 100 time steps.
As the indentation proceeds, the tip compresses along and then deforms slightly at the end that is in contact with the C(111):H surface (Figure 1(b)). After retraction (Figure 1(c)), the nanotubule and substrate both return to their starting configurations.
Figure 1. C (111):H substrate is indented by (10, 10) carbon nanotube tip. (a) Time = 0.00 ps, (b) time = 4.37 ps, (c) time = 8.57 ps.
In a second simulation, the indentation was continued until the nanotubule deformed plastically through a buckling mechanism, thus reducing the load on the tip (Figure 2). Figure 2 (inset) shows that the force on the tip atoms increases almost linearly with the displacement of the nanotubule and that the maximum force felt by the tip is approximately 100 nN, before the onset of buckling. This force is the Euler buckling force.23
where I is the stress moment over the cross section of the nanotubule radius (I » p r4/4).28 The radius (r) and the length (L) of the carbon nanotubule are 6.78 Å and 47.8 Å respectively. Using Eq. (1) we determined the Youngís modulus of the tubule tip to be 1394 GPa. This calculated Youngís modulus shows excellent agreement with previous theoretical estimates for SWTs of 1400-5500 GPa.22,23 The experimentally determined average value for MWTs was found to be 1800 GPa,27 which also shows good agreement with the calculated value.
The Euler buckling force varies with changes to the length of the tube. In other words, increasing the length of the nanotubule tip will result in a gentler probe that can be pushed against a softer surface without damaging it, though the modulus of the tube will not change. Hence, our system can be directly compared to the system examined by Smalley and co-workers28 with a tip that was approximately 6 m m long and buckled at a load of about 5 nN. They assumed a modulus of about 1000 GPa.
Figure 2. Force on the carbon nanotube versus the distance from the top of the tip to the substrate. Only the loading portion of the curve is shown. The inset shows best fit linearity to the data.
The closure of a (10,10) "armchair" carbon nanotubule that has 10 single bonded carbon atoms at the ends is considered in these simulations. The nanotubule again consists of 20 unit cells (800 atoms) as shown in Figure 3. 130 atoms at one end of the nanotubule are held rigid throughout the simulation. Moving towards the middle of the tubule, a Langevin thermostat is applied to the next 280 atoms. The remaining atoms evolve in time according to Newtonís equations of motion with no constraints.
The simulations are carried out at temperature of 4500 K. The open end of the nanotubule is allowed to relax for 72.1 ps followed by cooling to 300 K at the rate of 1 K/fs (Figure 3). As simulation progresses, the atoms at the end of the nanotubule move to come together, thus closing the end. By the end of the simulation, the closed end consists of 8 pentagons, 10 heptagons, 6 two-coordinate carbon atoms and 1 one-coordinate carbon atom (3 two-coordinate carbon atoms and 1 one-coordinate carbon atom are in the chain hanging at the closed end). The number of heptagons is high due to the presence of the 3 two-coordinate carbon atoms, which results in a closed end that is not perfectly hemispherical. The REBO potential predicts that an ideally closed (10,10) tubule is more stable than an open-ended (10,10) tubule by about 0.53 eV/atom, for the system of 150 carbon atoms. However, due to the large number of imperfections, the closed tubule from the simulation described above does not gain this much energy.
Figure 3. Initial and final views of the (10, 10) nanotubule at 4500 K. (a) Initial front view, (b) Initial top view, (c) Final front view (rotated to show chain attachment), (d) Final top view.
Allowing the nanotubule to relax for nano- or micro-second time scales might decrease the number of imperfections in the tubule. However, these time scales are not accessible in these simulations, which are restricted to relaxation times of the order of hundreds of ps.
Charlier et al. have predicted through first principles simulations that for a (10, 0) tubule the closed structure is 1.8 eV/atom more stable than the open structure. This difference is most likely due to the higher accuracy of the first principles approach. Hence, while the REBO correctly predicts qualitative trends, it does not match the quantitative accuracy of density functional theory.
We have shown through classical molecular dynamics simulations the atomic-scale mechanisms by which a tubule tip would respond during the indentation of a hydrogen-terminated diamond (111) surface. The Youngís modulus for the nanotubule was determined to be 1394 GPa, which is in good agreement with previous work. We have also studied the closing of nanotubules at 4500 K and compared our results to first principles simulation results.
The authors gratefully acknowledge the support of NASA Ames Research Center (Grant Number NAG 2-1121).