Shear and Friction between Carbon Nanotubes in Bundles and Yarns

Letter pubs.acs.org/NanoLett

Shear and Friction between Carbon Nanotubes in Bundles and Yarns Jeffrey T. Paci,*,†,§ Al’ona Furmanchuk,† Horacio D. Espinosa,‡ and George C. Schatz*,† †

Department of Chemistry and ‡Department of Mechanical Engineering, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, United States § Department of Chemistry, University of Victoria, P.O. Box 3065, Victoria, British Columbia, Canada V8W 3V6 S Supporting Information *

ABSTRACT: We perform a detailed density functional theory assessment of the factors that determine shear interactions between carbon nanotubes (CNTs) within bundles and in related CNT and graphene structures including yarns, providing an explanation for the shear force measured in recent experiments (Filleter, T.et al. Nano Lett. 2012, 12, 732). The potential energy barriers separating AB stacked structures are found to be irrelevant to the shear analysis for bundles and yarns due to turbostratic stacking, and as a result, the tube−tube shear strength for pristine CNTs is estimated to be 300 MPa, which is in agreement with the analogous calculations of ref 38). However, kBT is 26 meV/ atom at room temperature, so the effects of thermal energy need to be made before making even qualitative estimates of shear strength based on these results. The average thermal energy is also higher than the largest barrier to shear (∼20 meV/atom). This suggests the possibility of easy shear motion of sheets relative to each other at room temperature. This suggestion is the same as that made in ref 39 based on the LJ potential. We base ours on much higher barriers. Also, tube curvature and chirality considerations mean that neighboring tubes in bundles and yarns will have at most limited access to the registry. Therefore, it is the turbostratic case that appears to be the most important to consider for shear between tubes. The authors of ref 15 examined the corrugation associated with graphite in detail, using a vdWs-corrected LDA method. They predict that turbostratic stacking should result in a C44 (shear modulus) approximately equal to zero, in the limit of infinite sheet size. Such a shear modulus implies a shear strength of zero. Kolmogorov and Crespi also examined the turbostratic case using a LDA functional.13 They found a very small corrugation against sliding, and an energy for the turbostratic system that was only approximately 4 meV/atom higher than AB stacking. This is consistent with thermal energy at room temperature facilitating easy shear motion of sheets. The experimental shear strengths for turbostratic graphite of 0.02 to 0.04 MPa40 and 0.06 to 0.13 MPa18 are also consistent with very small corrugation. Motivated by these studies, we did simulations to investigate the turbostratic case using a pair of graphene sheets. The unit cell for the simulations is illustrated in Figure 1a. The 14 atoms in one layer were rotated by cos−1(11/14) relative to the atoms in the other as described in refs 13 and 41, an angle that results in accidental angular commenseration. Shearing one sheet relative to the other (separation = 3.25 Å), along the length and in the direction of the unit cell vector (6.15 Å, 2.13 Å), results in an energy versus displacement profile (Figure 1c) with a minimum to maximum energy difference of 0.006 meV/atom. The inflection point adjacent to the energy maximum in the curve suggests a shear strength of ∼0.24 MPa, roughly consistent with the experimental shear strengths of the previous paragraph. The predicted shear strength is negligible for this small unit cell, and larger cells have been predicted to be associated with even smaller shear strengths.15 Ours are computationally demanding simulations (PBE, plane wavenumber and k-space sampling to convergence). Our intention with these results is to provide a confirmation of the conclusions of previous work and is not to perform an exhaustive examination of the shear between pristine graphitic sheets, as this has already be reported in, for example, ref 15. The postannealing shear strength of 2 MPa reported in ref 19 is large compared to 0.24 MPa, suggesting that there is a contribution to the strength in the experiments of this reference from a source or sources other than corrugation. These results have implications for the interpretation of the pullout experiments of ref 1. (27,0) tubes, which are achiral, were used in the MM3 simulations on which the magnitudes of the forces due to the vdWs registry corrugation were estimated. Because the chemical vapor deposition (CVD) process used to grow the tubes results in tubes of various chiralities, it is the C

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Figure 2. (a) A particle of mass m connected to a block by a spring with a force constant k, encountering a corrugated substrate. (b) Graphical representation of the solutions to eq 4. The solid line corresponds to the force −U′(q) from the substrate. Case (I) corresponds to a stiff spring with force kq (dashed line) in which case q = 0 is the only solution to eq 4. In case (II) the spring is soft, and q = ±q1 are the stable solutions to this equation.

Figure 3. (a) Particle on the stiff spring closely follows the motion of the block (κ < 1). (b) When the spring is sufficiently soft, slow motion of the block causes discontinuous flips from one stable position to another (κ > 1).

electron−hole pairs), where γ is the damping due to this coupling. For simplicity, we assume the potential

interaction between turbostratic-type sheets that is the better guide to the magnitude of the corrugation between them than those that can access the registry. Thus, whereas the corrugation potential associated with the tubes in ref 1 was estimated to contribute a small amount (∼0.1 of the 1.7 nN/ CNT−CNT interaction), the current work suggests that the corrugation contribution is negligible. Errors in the MM3 potential used in ref 1 are therefore irrelevant to the final conclusion. Elastic Instability Transitions. If a sliding system is sufficiently flexible, the kinetic friction force is nonzero in the limit as sliding velocity goes to zero.22,42 This is due to the occurrence of elastic instability transitions. In such cases, the kinetic friction is approximately equal to the static friction. Otherwise, the time-averaged kinetic friction force is zero. Such considerations are important for the interpretation of the pullout experiments of ref 1 because the pullout velocity is ∼1 × 10−13 Å/fs that is slow (typical sliding velocity in macroscopic friction experiments = 1 cm/s = 10−7 Å/fs). These considerations also lead to more general insight into the mechanisms of friction that operate between CNTs in bundles and yarns. Here we closely but briefly follow the presentation of Persson in ref 22. Consider the sliding system shown in Figure 2a. A particle of mass m is connected to a block by a spring with a force constant k. The particle experiences the corrugated potential U(q), and the particle equation of motion is mq ̈ = −U ′(q) − mγq ̇ + k(x − q)

⎛ 2πq ⎞ ⎟ U (q) = U0 cos⎜ ⎝ a ⎠

(2)

At equilibrium q̇ = 0 so eq 1 gives ⎛ 2π ⎞ ⎛ 2πq ⎞ ⎟ + k(x − q) = 0 U0⎜ ⎟sin⎜ ⎝ a ⎠ ⎝ a ⎠

(3)

When the spring is centered over the local maximum in the potential, eq 1 reduces to ⎛ 2π ⎞ ⎛ 2πq ⎞ ⎟ − kq = 0 U0⎜ ⎟sin⎜ ⎝ a ⎠ ⎝ a ⎠

(4)

Solutions to this equation fall into two cases, which are illustrated graphically in Figure 2b. One case corresponds to a stiff spring (I) and the other to a soft spring (II). In case (I), q = 0 is the only solution to eq 4. The particle closely follows the motion of the block, so if the block moves slowly, so does the particle (see Figure 3a). For case (II), three solutions exist, q = 0 and q = ±q1. Three solutions exist only when the initial slope of the function U0(2π/a)sin(2πq/a) is greater than the slope of kq (see Figure 2b). This occurs when κ ≡ U0/ε > 1, where ε = ka2/4π2 is the elastic energy. The q = 0 solution is unstable in this case, so the particle will occupy a displaced position (q = q1 or −q1). If κ > 1, sliding at slow velocities causes discontinuous flipping from one stable position to another (see Figure 3b). This flipping is called an elastic instability transition. Thus, if κ > 1, a finite friction force results even in the limit of zero pulling velocity of the block.

(1)

where the displacement coordinates x and q are defined in the figure. The friction force mγq̇ originates from the coupling of the particle to the substrate excitations (i.e., phonon modes and D

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For CNTs, k = EA0/L0, where E is Young’s modulus, A0 is the cross sectional area of a tube, and L0 is the length of the tube between sites at which significant intertube interactions affecting shear occur. This gives the expression κ=

root-mean-square force of interaction of an individual defect, and Nd is the number of pinning centers in Ac. The force at a defect is correlated with the force at another defect up to a distance of ξ away. The authors of ref 45 also describe shear calculations for a vacancy defect. They used the classical force field AMBER, which may not accurately reproduce the forces associated with moving a defect relative to a pristine sheet, particularly at separations largely dictated by vdWs forces. Our higher-level theory result suggests reasonable agreement with their conclusions regarding the insensitivity of resistance to shear to the presence of such a defect; 0.039 eV is a small barrier. When the length L0 exceeds the length of the bundle remaining within the sheath in the pullout experiments of ref 1, the relevant elasticity becomes that of the cantilever and not the CNTs in the bundle. In such cases, the defects are in elastic contact with the cantilever. The expression for κ becomes

4π 2U0L0 a 2EA 0

(5)

For the tubes contained in the bundles of ref 1., we use E ∼ 1 TPa (523 GPa homogenized), the modulus of nondefective tubes, and A0 ∼ π*(1.1 × 10−9m)2. As suggested by the discussion below, E may be somewhat smaller, but this does not affect the argument. The case of neighboring pristine chiral tubes that do not achieve any significant level of AB stacking is straightforward. U0 is effectively zero (see Figure 1), so no significant static or kinetic friction in the limit as velocity goes to zero is expected. In the case of defective tubes, a detailed analysis is needed. The CVD reaction conditions are hot, and they may be too hot to produce functional groups such as carbonyl and ether pairs at single-atom vacancy defects, a common type of defect in CNTs. For the case of such a defect in a chiral tube, the potential wells are deeper, because the atoms at such defects will tend to have covalent interactions with atoms in neighboring graphitic layers. Shearing a single-atom vacancy defect at fixed separation from a pristine sheet (separation = 3.25 Å), using PBE and enough plane waves and special kpoints for convergence, gives a pinning barrier of 0.039 eV and a well minimum to maximum distance of 1.3 Å. This shear was performed along the unit cell vector (6.15 Å, 2.13 Å) with one atom removed from the cell shown in Figure 1a. The potential associated with the vacancy is not the simple sinusoid on which eq 5 is based (defects are expected to be randomly located), so we set a = 1.3 Å to develop a semiquantitative idea of what such a barrier implies in terms of instability transitions. Using the formula and setting κ = 1 suggests that such a defect would need to be at least L0 ∼ 1400 Å from anything that significantly interfered with the smooth sliding of the tubes (e.g., another defect) for the defect to result in energy dissipation due to tube elasticity. The 1400 Å is the elastic coherence length for a CNT interacting with this pinning barrier. For example, as the cantilever pulls on bundles as in the experiments of ref 1, energy will be dissipated by such a defect due to the elasticity of the tubes provided that the defect is at least 1400 Å from another defect that is closer to the cantilever. Smaller defect− defect separations mean there will be no instability, so only the q = 0 solution applies; there will be no fast motion at the first defect in the limit of zero pulling velocity of the second defect, so there will be almost no friction. Defect pairs at less than 1400 Å separations will have a roughly equal probability of net pushing in the direction of applied force or net pulling resisting the applied force. Such considerations apply equally well to defects that are large and that have potentially serious deleterious effects on tube and bundle strengths. Still, a finite concentration of defects that are randomly distributed always gives rise to a finite pinning barrier.43,44 For interactions between sheets, such as the one associated with the 0.039 eV barrier just described, there is an area Ac = ξ2 over which there is short-range order. Inside the area, the lattice is almost regular and the defects are randomly located, so the pinning forces acting on the lattice nearly compensate for each other. Random walk arguments indicate that the total pinning force acting on area Ac is of the order f rmsN1/2 d , where f rms is the

κ=

4π 2U0 ka 2

(6)

where k is the force constant of the cantilever in the experiments. The force constants of these cantilevers were typically ∼15 N/m. Using k = 15 N/m, a barrier that is 0.039 eV high, and a well minimum to maximum distance of 1.3 Å gives κ = 1.0. Thus, elastic instability transitions may occur in such a system. They will occur for larger barriers that may be associated with multiple and/or larger defects. Note that the fact that κ is equal to one, to two significant figures in this example, is purely coincidental. How large does an energy barrier need to be to make the ∼1.2 nN/CNT−CNT interaction that was missing in the interpretation of the experiments of ref 1? If the energy change takes place over 1 Å, ΔE = 1.2 × 10−9 N/CNT−CNT interaction times 1 × 10−10 Å = 1.2 × 10−19 J/CNT−CNT interaction or 0.75 eV/CNT−CNT interaction. Less abrupt energy changes need to be higher in energy. The carbonyl groups at the tube ends (see Figure 6 B of ref 1.), formed when tubes break in the weak vacuum of the SEM (pressure ∼10−5 Torr), encounter energy barriers that are approximately this large; ΔE ∼ 0.55 eV/CNT−CNT interaction, assuming a CNT−CNT contact width of 1.3 nm. The modeling of carbonyl termination for ref 1 represented tubes that had broken completely cleanly and that were terminated in an ideal fashion. Ragged breaks and incomplete termination46 (i.e., dangling bonds) tend to be associated with higher energy barriers. The 1.3 nm contact width is larger than that assumed in the original interpretation of the experiments of ref 1. This larger value is meant to account for the possibility of ragged breaks. In the case examined in this reference, the energy change took place over a distance of approximately 3 Å, but as the analysis here shows this distance can be ∼1 Å. It is plausible that it is these carbonyl groups and defects, such as vacancies, that lead to the majority of the 1.2 nN/CNT−CNT interaction of force measured at the cantilever, by way of elastic instability transitions, the interactions between carbonyl groups and adjacent tubes perhaps supplemented by molecules such as oxygen that interact strongly when they bridge between the broken tube ends and adjacent tubes.47 The fact that the pullout force is approximately constant suggests that most of the force is due to interactions near the broken tube ends. This is because were defects along the lengths of the tubes to be the dominant contributors to the force, one would expect the force E

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Figure 4. (a) A turbostratic graphene bilayer with a hole in one of the layers. Carbon atoms are shaded gold and hydrogen atoms are white. (b) The layer containing the hole, shown in the absence of the other layer. (c) Energy versus displacement for the layer with the hole sheared relative to the other layer in the direction perpendicular to the long-axis of the hole.

instabilities can occur. The breaking of a hydrogen bond followed by the formation of another can result in abrupt motion, regardless of the relative sliding velocity of the materials supporting the hydroxyl groups. Therefore, dissipation is expected in the limit of zero sliding velocity. Hydroxyl groups can form at the edges of graphitic materials, by way of the cleavage of water molecules.49 Achiral tubes that establish a significant level of AB stacking constitute a special case for elastic instabilities. Details of this case are provided in the Supporting Information. A Closer Analysis of Bundles. The results of bundle pulling experiments are described in ref 50. Some experiments were performed on pristine bundles, while other bundles were heattreated. Some of the heat-treated bundles failed by fracture of the outer sheath of tubes and subsequent pullout of the inner tubes.1 The strengths of bundles for which the outer sheath of tubes (i.e., bundles that undergo pullout) and for which the entire bundle breaks are low compared to the strengths of defect-free tubes (∼30 GPa versus ∼100 GPa, assuming loading of just the outermost walls of the tubes at the bundle peripheries). This suggests that large defects are present, as is often the case for CVD-grown tubes; a Griffith formula suggests that holes with diameters >20 Å exist.51 Such hole sizes are consistent with recent experiments in which, for example, CVD-grown tubes were shown to be readily filled with water.52 Holes in CNTs grown using CVD such as those in the bundles of ref 1 tend to be functionalized with hydrogen atoms.53 The Young’s modulus of the bundles, which assumes loading of the outer walls of the tubes at the bundle peripheries, is significantly lower than 1 TPa. The modulus of CNTs can be

to decrease with a decrease in the remaining tube overlap length. Push−pull effects along the lengths of the tubes are probably small. An analogous argument would seem to apply to the straightening of any kinks and bends that might be present in the tubes in the bundles. They would be expected to affect the pullout force but in a way that leads to the force being dependent on overlap length. According to the argument, the pullout force should vary linearly with the radius of the bundle. Larger defects than the single-atom vacancy may lead to higher energy barriers upon sliding. For a barrier of 1 eV, a value typical for weak covalent bonds, setting κ = 1 in eq 5 gives L0 = 53 Å for a = 1.3 Å. One would expect elastic instability transitions to occur as a result of tube elasticity in a system with defects separated by distances greater than L0, for systems to which such a parameter set applies. Of course, eq 6 suggests that such defects would also cause energy dissipation due to the elasticity of the cantilever, in the limit of zero pulling velocity. Covalent cross-linking of tubes resulting from the bonding of atoms at a vacancy defect in one tube to a similar defect in an adjacent tube might also lead to barriers ∼1 eV.3 However, it seems improbable that such cross-links would lead to the constant pullout force as the tubes slide relative to one another as observed in the experiments of ref 1, as there could not be many such cross-links or the pullout force would be larger (force ∼ f rmsN1/2 d ). Certain types of functional group interactions lend themselves more readily than others to elastic instabilities. The hydroxyl groups of ref 48 provide one example. When such groups are involved in the formation and breakage of hydrogen bonds with analogous groups on neighboring surfaces, F

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low when they contain holes.54 However, Raman spectroscopy suggests that the defect density in the tubes in our bundles is very low.1 The homogenized strengths of the bundles, taking into account their full cross-sectional areas (engineering fracture stresses), are ∼2−3 GPa.50,55 Carbon nanotube misalignment, twists, kinks, and so forth could be influencing parameters during bundle pullout. However, transmission electron microscope (TEM) images such as those shown in ref 55 and others examined by us (not shown) show some evidence of misalignment, although it appears to be minor. We used TEM measurements of the intertube spacing to argue that functionalization must be minor.1 Modeling of a geometry more similar to the tested bundles with, for example, the Kolmogorov-Crespi potential would be beneficial to explore misalignment, twists, and kinks.13 Twists and kinks pose a challenge as their density could be low yet they would have big effects. If the kinks and twists lead to nanotube breakage when pulled, then that leads to functional group formation and/or incomplete termination. The results presented above enable a reevaluation of the factors which determine the shear interactions between nanotubes in a bundle based on the pullout experiments presented in ref 1. The results (see the Supporting Information for details) show that of the 1.7 nN per CNT/CNT interaction that was measured, 1.2 nN comes from carbonyl group and defect interactions, and 0.4 nN from the energy needed to create new surface (derived from the adhesion energy). Polygonization of the CNTs leads to less than a 0.1 nN force and other contributions are negligible. Large Holes and Shear. To investigate the shear interactions between a tube containing a large hole (diameter ∼20 Å) and an adjacent pristine graphitic layer, we ran simulations using PBE-D2/DZP. For the model described in relation to Figure 1a, we find that PBE-D2/DZP predicts a negligible shear strength (maximum energy difference over the displacement 20 Å) such as those that seem to be present in the tubes of the bundles used in the pullout experiments of ref 1 also contribute. However, if these holes are terminated with hydrogen atoms as is expected for the reaction conditions used in these experiments, their contribution to shear resistance is not large; a fully hydrogen-terminated hole with an 11 Å long-axis is associated with a pinning barrier of 0.06 eV. Pinning can result in friction when graphitic materials are slid relative to one another. Graphitic materials are stiff, so they can have large coherence lengths. This means force cancellation can occur as defects or functional groups push and pull on one another. However, random walk arguments indicate that cancellation is never complete, with the force resisting sliding growing with the number of defects or functional groups according to f rmsN1/2 d , where f rms is the root-mean-square force associated with the defects or functional groups and Nd is their number. When sliding occurs in the limit of zero velocity, as in the experiments of ref 1, friction is due to elastic instability transitions. The tubes are stiff, so their elasticity does not contribute much to instabilities. However, the cantilevers used for the pullout are soft, so friction occurs, that is, energy is dissipated during the pullout. The occurrence of these instabilities and the pinning barriers predicted using DFT account for the dissipation that was missing from the analysis of ref 1. The majority of the dissipation is due to interactions at or near the broken tube ends with push−pull effects along the lengths of the tubes playing a minor role. It seems that differences in the amount of end group functionalization combined with less important instability transitions lead to the smaller shear interactions that were found in the sliding of CNT tube walls reported by Zettl et al.66 This occurs because the end group functionalization chemistry in these types of systems is sensitive to differences in concentrations of species

contained defects or functional groups that led to significant shear strength between tubes, the resulting transverse pressure would contribute to the prevention of CNT slippage. For most staple-fiber yarns, the primary function of transverse pressure is to increase the area of real contact with the flexibility of the fibers in compression being significantly less important.28 CNTs with one or two walls are stringlike, so they will be flexible to bending under transverse forces of this size.61,62 Tubes with a sufficient number of walls are stiffer than steel under bending deformations, so they are expected to be inflexible under such transverse forces.63 For yarns, the tension in an individual fiber increases with its radius squared. For incompressible fibers, the friction between them increases with radius. This means the finer the fiber, the stronger the yarn. Amorphous carbon is often associated with the surfaces of CNTs. Its size, density, distribution, surface chemistry, and whether or not it is strongly adhered to the surfaces of the tubes, will affect its impact on yarn performance. Holes have a significant impact on the strengths of graphitic materials,51 while chemical functionalization with groups such as epoxides has a lesser impact on strength.64 This suggests the use of functionalization (or cross-linking) to increase the forces that oppose sliding in staple-fiber yarns composed of otherwisepristine CNTs. The yarns described in refs 27 and 30 may derive a significant amount of their strength from cross-links, in these cases by way of the polymerization of 1,5-hexadiene and hexa(methoxymethyl) melamine, respectively. Covalent crosslinking of epoxide-functionalized tubes using epoxy resin curing as has been done for mats composed of CNTs,65 resulting in materials with high tensile strengths. As suggested in ref 65, tubes with small numbers of walls are best, as inner walls add weight without significantly improving the load bearing characteristics of the tubes. The bundling of pristine tubes is not beneficial, as load transfer is minimal due to the nearly absent shear strength. F, the shear strength integrated over the area of real contact, is largest for covalent bonds. This means that fewer covalent cross-links than defects should be necessary to prevent tube slip within staple-fiber yarns. In the absence of cross-linking chemistries, twist and fiber migration are of central importance in staple-fiber yarns. Fiber migration is necessary for yarns in which fiber−fiber contacts are associated with small shear strengths (noncrystalline, noncovalent interactions) to facilitate the generation of significant tension. This means the tubes need to be long enough to have paths within the yarn that can be reasonably described by twist angle and migration. For a yarn with a 10 μm diameter, this would seem to be a tube length of ∼100 μm, that is, at least several multiples of the yarn diameter. Groups are manufacturing such materials by spinning CNT arrays and forests (tube length ∼1 mm), and the materials they produce have engineering fracture stresses of ∼2−3 GPa.23−27,29 An (untwisted) yarn made of pristine 2.2 nm tubes as long as the yarn (continuous fiber) would have an engineering fracture stress of ∼50 GPa, a Young’s modulus (homogenized) of ∼500 GPa, and a failure strain identical to pristine tubes. Using analogous tubes of smaller diameters would result in yarns with higher fracture stresses and Young’s moduli. Staple fiber yarns require a mechanism to transfer load between tubes. Were this to be accomplished using covalent cross-linking between otherwise pristine 2.2 nm tubes, the resulting yarns could have a fracture stress greater than 50% × 50 GPa = ∼25 GPa, as graphitic material that is nearly 100% functionalized is ∼50% as strong as its pristine analogue.64 Similarly, one would expect the H

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(12) Ponder, J. W. TINKER Molecular Modeling Package version 6.3. Available from: http://dasher.wustl.edu/tinker/ (Web site location verified on June 29, 2014). (13) Kolmogorov, A. N.; Crespi, V. H. Phys. Rev. B 2005, 71, 235415. (14) Lebedeva, I. V.; Knizhnik, A. A.; Popov, A. M.; Lozovik, Y. E.; Potapkin, B. V. Phys. Chem. Chem. Phys. 2011, 13, 5687. (15) Savini, G.; Dappe, Y. J.; ad J. C. Charlier, S. Ö .; Katsnelson, M. I.; Fasolino, A. Carbon 2011, 49, 62. (16) Persson, B. N. J. Phys. Rev. B 1991, 44, 3277. (17) Perdew, J. P.; Burke, K.; Ernzerhof, M. Phys. Rev. Lett. 1996, 77, 3865. (18) Yang, J.; Liu, Z.; Grey, F.; Xu, Z.; Li, X.; Liu, Y.; Urbakh, M.; Cheng, Y.; Zheng, Q. Phys. Rev. Lett. 2013, 110, 255504. (19) Suekane, O.; Nagataki, A.; Mori, H.; Nakayama, Y. Appl. Phys. Express 2008, 1, 064001. (20) Parsegian, V. A. Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists; Cambridge University Press: Cambridge, 2006. (21) Steele, W. A. The International Encyclopedia of Physical Chemistry and Chemical Physics; Pergamon Press Ltd.: Oxford, 1974; Topic 14, Vol. 3. (22) Persson, B. N. J. Sliding Friction, 2nd ed.; Springer-Verlag: Germany, 2000. (23) Zhang, X.; Li, Q.; Holesinger, T. G.; Arendt, P. N.; Huang, J.; Kirven, P. D.; Clapp, T. G.; DePaula, R. F.; Liao, X.; Zhao, Y.; Zheng, L.; Peterson, D. E.; Zhu, Y. Adv. Mater. 2007, 19, 4198. (24) Zhang, X.; Li, Q.; Tu, Y.; Li, Y.; Coulter, J. Y.; Zheng, L.; Zhao, Y.; Jia, Q.; Peterson, D. E.; Zhu, Y. Small 2007, 3, 244. (25) Tran, C. D.; Humphries, W.; Smith, S. M.; Huynh, C.; Lucas, S. Carbon 2009, 47, 2662. (26) Ryu, S.; Lee, Y.; Hwang, J. W.; Hong, S.; Kim, C.; Park, T. G.; Lee, H.; Hong, S. H. Adv. Mater. 2011, 23, 1971. (27) Min, J.; Cai, J. Y.; Sridhar, M.; Easton, C. D.; Gengenbach, T. R.; McDonnell, J.; Humphries, W.; Lucas, S. Carbon 2013, 52, 520. (28) Hearle, J. W. S.; Grosberg, P.; Backer, S. Structural Mechanics of Fibers, Yarns, and Fabrics; Wiley-Interscience: New York, 1969. (29) Beese, A. M.; Wei, X.; Sarkar, S.; Ramachandramoorthy, R.; Roenbeck, M. R.; Moravsky, A.; Ford, M.; Yavari, F.; Keane, D. T.; Loutfy, R. O.; Nguyen, S. T.; Espinosa, H. D. What Limits the Strength of Carbon Nanotube Yarns? Exploring Processing-StructureProperty Relationships. ACS Nano; Submitted for publication. 2014. (30) Boncel, S.; Sundaram, R. M.; Windle, A. H.; Koziol, K. K. K. ACS Nano 2011, 5, 9339. (31) Gonze, X.; et al. Comput. Phys. Commun. 2009, 180, 2582. (32) Kresse, G.; Joubert. Phys. Rev. B 1999, 59, 1758. (33) Torrent, M.; Jollet, F.; Bottin, F.; Zerah, G.; Gonze, X. Comput. Mater. Sci. 2008, 42, 337. (34) Grimme, S. J. Comput. Chem. 2006, 27, 1787. (35) Sánchez-Portal, D.; Ordejón, P.; Artacho, E.; Soler, J. M. Int. J. Quantum Chem. 1997, 65, 453. (36) Troullier, N.; Martins, J. L. Phys. Rev. B 1991, 43, 1993. (37) Troullier, N.; Martins, J. L. Phys. Rev. B 1991, 43, 8861. (38) Wang, L. F.; Ma, T. B.; Hu, Y. Z.; Wang, H. Phys. Rev. B 2012, 86, 125436. (39) Shibuta, Y.; Elliott, J. A. Chem. Phys. Lett. 2011, 512, 146. (40) Liu, Z.; Yang, J.; Grey, F.; Liu, J. Z.; Liu, Y.; Wang, Y.; Yang, Y.; Cheng, Y.; Zheng, Q. Phys. Rev. Lett. 2012, 108, 205503. (41) Beyer, H.; Müller, M.; Schimmel, T. Appl. Phys. A: Mater. Sci. Process. 1999, 68, 163. (42) Shinjo, K.; Hirano, M. Surf. Sci. 1993, 283, 473. (43) Larkin, A. I.; Ovchinnikov, Y. N. J. Low. Temp. Phys. 1979, 34, 409. (44) Blatter, G.; Feigelman, M. V.; Geshkenbein, V. B.; Larkin, A. I.; Vinokur, V. M. Rev. Mod. Phys. 1994, 66, 1125. (45) Guo, Y.; Guo, W.; Chen, C. Phys. Rev. B 2007, 76, 155429. (46) Xia, Z.; Curtin, W. A. Phys. Rev. B 2004, 69, 233408. (47) Chung, J.; Lee, K. H.; Lee, J.; Troya, D.; Schatz, G. C. Nanotechnology. 2004, 15, 1596. (48) Liu, Y.; Szlufarska, I. Appl. Phys. Lett. 2010, 96, 101902.

(e.g., water and oxygen molecules) remaining on the surfaces of the materials under different vacuum conditions.67 This analysis also has implications for the manufacture of yarns from CNTs. Pristine tubes exhibit too little static friction (shear strength integrated over the area of real contact) to be spun into strong yarns, for the tube lengths that are most-often currently synthesized. Defects such as holes increase intertube friction but they weaken the tubes significantly. Chemical functionalization can increase tube−tube friction, and functionalization only has a minor impact on the strengths of the tubes, but push−pull effects can lead to substantial force cancellation. Covalent bonds, that is, bonds that maximize shear strength integrated over the area of real contact, are as tube weakening per bond as functionalization but can lead to much larger load transfer.55 This suggests covalent cross-links are the most promising route to the production of strong CNT-based yarns.

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ASSOCIATED CONTENT

S Supporting Information *

Additional details for accurately modeling corrugation, pristine achiral tubes, contributions to the pullout force, and additional figure, table, and references. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: jeff[email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors gratefully acknowledge support from the ARO through MURI award No. W911NF-09-1-0541. The authors thank Professor T. Filleter for supplying information about the cantilevers used in the pullout experiments. J.T.P. thanks Matthew Ford for thoughtful discussions. H.D.E. acknowledges discussions with Alexander Moravsky and Xiaoding Wei concerning implications of the findings here reported to yarn mechanics.

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