Twisting Carbon Nanotube Ropes with the Mesoscopic Distinct

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Twisting carbon nanotube ropes with the mesoscopic distinct element method: geometry, packing, and nanomechanics Yuezhou Wang, Igor Ostanin, Cristian Gaidau, and Traian Dumitrica Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.5b03208 • Publication Date (Web): 27 Sep 2015 Downloaded from http://pubs.acs.org on September 29, 2015

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Twisting carbon nanotube ropes with the mesoscopic distinct element method: geometry, packing, and nanomechanics ˘ ¶ and Traian Dumitrica∗,§,† Yuezhou Wang,† Igor Ostanin,‡ Cristian Gaidau, Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA, Skolkovo Institute of Science and Technology, Moscow, Russia, Department of Physics, University of Illinois at Urbana-Champaign, IL 61801, USA, and Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA E-mail: [email protected]

Abstract The geometry and internal packing of twisted ropes composed of carbon nanotubes (CNTs) are considered, and a numerical solution in the context of mesoscopic distinct element method (MDEM) is proposed. Compared to the state of the art, MDEM accounts in a computationally tractable manner for both the deformation of the fiber and the distributed van der Waals cohesive energy between fibers. These features enable us to investigate the torsional response in a new regime where the twisted rope develops packing rearrangements and aspect-ratio dependent geometric nonlinearities. MDEM ∗

To whom correspondence should be addressed Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, MN 55455, USA ‡ Skolkovo Institute of Science and Technology, Moscow, Russia ¶ Department of Physics, University of Illinois at Urbana-Champaign, IL 61801, USA § Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA †

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emerges as a robust simulation method for studying twisted agglomerates comprising semi-flexible nanofibers.

Introduction Modern technology offers a variety of synthetic fibers for rope fabrication. In addition to nylon and polyester fibers, macro-scale yarns composed of carbon nanotubes (CNTs) 1 have begun to be manufactured by twist-spinning. 2–4 It is well known that the emergent mechanical behavior of laid ropes depends on both the strength of the fibers and on the twisted geometry. 7 CNTs are the strongest and stiffest synthesized nanofibers, with a Young’s modulus (E) of 1030 GPa. 5 Their shear modulus (G) of about 0.5 TPa 6 positions them as the stiffest torsional nano-fibers. To manufacture advanced rope structures 8,9 with such outstanding nanofibers, it is crucial to understand the relationship among geometry, internal packing, and the individual nanofiber mechanics. The ability to quantify total energy as a function of twist is an important step in this direction. All-atom simulations uncovered remarkable insights into the torsional mechanics of CNT systems. 6,10–12 Unfortunately, these methods are impractical for simulating twisted CNT assemblies approaching realistic dimensions. To reduce the computational complexity and hence the computational cost, mesoscopic modeling simulations are needed. However, the application of the current models to the problem at hand encounters significant challenges. Drawing from the bead-and-spring modeling approaches developed for semi-flexible polymers, 13,14 discretization schemes of CNTs into a sequence of point masses have been pursued. 15 Due to the limited sliding friction, 16–18 there is a poor load transfer between CNTs. When a crystalline close-packed aligned CNT rope is twisted, it is expected that the nanofibre constituents will acquire a variety of deformations, including geometric and physical twist. (Physical twist is the relative rotation of elements along the axis of the rope; geometric twist is bending in the direction of the binormal vector.) Unfortunately, the coarse graining of a


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group of atoms into a point mass cannot be trained to describe the torsional deformations of the individual CNTs. In addition, theories and simulations of defects in twisted ropes 19–21 evidenced the relationship between packing defects and rope twist. The development of such defects cannot be described with large diameter bead-and-spring models of the CNTs. These models introduce interfacial corrugated interactions, which prohibit smooth sliding 15,22 and the CNTs’ relaxation into the lowest energy state. We propose to simulate CNT rope twisting with the recently developed mesoscopic distinct element method (MDEM), 22,23 which is the mesoscopic version of a genuine solid mechanics method called the distinct element method. 24 MDEM (i) simulates efficiently large ensembles of CNTs, (ii) captures the CNT linear elasticity in elongation, bending, and twisting deformation modes, and (iii) accounts for the distributed van der Waals (vdW) interactions in a manner that eliminates corrugation artifacts entirely. These features are essential for the problem at hand. In our prior work, we have informed the method from atomistic data and verified it against molecular dynamics (MD) simulations and continuum models for ring aggregates formed by CNTs. 22,25 Here we use MDEM to access the deformation regime needed to investigate packing in twisted CNT ropes. We achieve packing rearrangement conditions, where the vdW adhesion between CNTs decreases and even vanishes, and rope geometries dictated by the CNT mechanics. By contrast, previous simulations focused on the load transfer regime and considered twisted ropes comprising only a few CNTs. 10,15,26

Method We start by outlining the basic components our mesoscopic model. A CNT is represented by a linear sequence of rigid cylindrical elements, Fig. 1(a). This constitutes a coarsegrained model of the CNT in question, as each element represents a segment of the CNT. Specifically, we have discretized a (10,10) CNT such that each cylindrical element with length T = 0.35 nm, radius rCNT = 0.68 nm, and thickness t = 0.335 nm, represents 220 carbon


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atoms. 22 These mesoscopic elements are traked in the Lagrangian frame. In the PFC3D 27 implementation, they are evolved deterministically in time as rigid bodies with a damped scheme based on a velocity Verlet algorithm for translations and a 4-th order Runge-Kutta for rotations. 28

Figure 1: (a) Adjacent distinct elements on a CNT connected by a parallel-bond contact. Restoring forces Fn and Fs and moments Mn and Ms in response to the normal and shear relative displacements Un and Us , and rotations θn and θs , respectively. (b) Parallel distinct elements on two CNTs connected by a vdW contact. (left) Restoring forces in response to changes in center-to-center distance R and alignment angle θ. (right) Restoring moment at a nonzero crossing angle γ. n1 and n2 indicate the axes of the elements.

The cylindrical elements are coupled by parallel-bonds contact, which physically reproduce the behavior of an elastic glue between adjacent elements. 22 The relative displacements of the two elements are resolved with the help of versors n and s, which are located into and


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Table 1: Parameters for the mDEM model of a (10,10) CNT used in our simulations. 22 A (nm2 ) I (nm4 ) 1.427 0.348

J (nm4 ) kn (eV/nm4 ) 0.696 4,740

ks (eV/nm4 ) 2,110

perpendicular to the contact plane, Fig. 1(a). On one hand, the element normal and shear relative displacements Un and Us are resisted by forces Fn and Fs , respectively. On the other hand, the relative rotations θn and θs are resisted by moments Mn and Ms , respectively. These forces and moments are created by linear-elastic normal kn and shear ks springs that are uniformly distributed over a finite-sized section lying on the contact plane. The explicit laws are Fn = −kn AUn , Fs = −ks AUs , Mn = −ks Iθs , Ms = −kn Jθn , where A = 2πrCNT t, 2 I = πrCNT t(rCNT + 0.25t2 ), and J = 2I are area, moment of inertia and polar moment of

inertia of the parallel-bond ring, kn = E/T , and ks = G/T , Table 1. 22 In the above laws, we highlight the need for a rigid body treatment of the coarse grained elements in order to resolve the resistance to a torsional deformation. The chain of parallel-bonded distinct elements describes the CNT as an Euler-Bernoulli beam. Fig. 1(b) shows two mesoscopic elements interacting via a vdW contact. This contact has been developed by us via a procedure that involves integration of the atomistic vdW interactions between two CNTs. 22 Ref. 22 gives the explicit form for the vdW contact energy U (R, θ, γ). Here we only note that in order to avoid the corrugation artifacts and to describe the vdW energy lowering when CNTs are crossing, U is anisotropic, i.e. it depends not only on the center-to-center distances R but also on the θ and γ alignment angles. Additionally, U is truncated in a smooth manner at distances four times larger than the CNT radius. The vdW contact can act simultaneously with the parallel-bonds and create additional forces and moments according to FR = −∂U/∂R, RFθ = −∂U/∂θ, and Mγ = −∂U/∂γ. As in MD, 16 our vdW contact describes a smooth sliding of two parallel (10,10) CNTs against each other at an interspacing of 1.72 nm. We note that the elements in vdW contact will also experience forces resulted from a viscous damping contact representing the microscopic energy loss during sliding. 23 5 ACS Paragon Plus Environment

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The MDEM simulations proceed in two stages. First, torsional moments M along the rope axis are applied in incremental steps on grip distinct elements located at the two ends. At each incremental step, all elements are evolved in time by holding only the grip elements under the corresponding twisting moment in order to prevent the CNTs from twisting back. With this protocol we relax the structure and avoid high distortions localized at the grips. The constituent CNTs will form helices so the parallel-bonds representing the helical state will store a superposition of bending and twist due to both physical and geometric twist experienced by individual CNTs. Second, at the end of the loading process the applied constraints are only maintained and the system is evolved under a longer time interval. The applied loading rates and final relaxation time were decided after making sure that convergence was achieved. In both stages, the grip elements are allowed to change their axial positions.

Results and discussion In Fig. 2 we describe the twisting of a rope with hexagonal packing, containing 91 CNTs, each with length L = 67.8 nm. The rope has an initial radius of 8.6 nm. The simulated morphologies, Fig. 2(a), display an increase in the twist angle β measured between the outermost CNTs and the rope axis. (The twist angle is lower for the internal CNTs.) Additionally, there is a dwindling down in profile to form a “spindle" morphology. This progression is possible due to the smooth vdW sliding and the employed free boundaries, which allow for the relaxation of the tensile strain accumulated in the individual CNTs. As the rope twists, the individual helical shape of the constituent CNTs with length L implies that the length of the bundle shortens with the radial distance from the bundle center. Associated with this process there is an increase in bending energy, Fig. 2(b)-left. If sliding is prohibited by the applied boundary conditions and/or by the artificial corrugation effects, the individual CNTs would experience elongation. The energy relaxation


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Figure 2: (a) Rope morphologies at different twist angles β. Cross-sections and Q values are shown below. CNT length is L = 67.8 nm. (b) Time evolution of the energy stored in the parallel and vdW contacts, and of Q. Light gray area is the loading stage (steady loading with 1.0 nm·nN/ps). Dark gray is the stabilization stage under M = 64 nm·nN. A-D mark the rope morphologies shown in (a).


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mechanism would involve hydrostatic pressures acquired in the interior of the rope and increased contact between CNTs. 15,26 Instead, under free boundary conditions and no corrugation sliding, here we obtain local CNT rearrangements which interrupt the original 6-fold coordination. In Fig 2(b)-right, structural rearrangements occurring in the core (dark gray nanotubes) are visible in the changes in vdW energy and topological charge Q. 21,29 When Q = 0 the rope is packed defect-free. Initially, there is a steady increase of the vdW cohesion energy with twist, due to changes in R, θ, and γ between the distinct elements in vdW contact. When Q 6= 0 there is a change in packing that interrupts the decrease in cohesive energy. The positive values of Q indicate an excess in 5-fold defects. For example, the Q = 2 charge is due to two 5-fold defects. Negatively-charged 7-fold packing defects are also encountered. The jump to Q = 3 is due to the emergence of a 7-fold and two 5-fold defects. Finally, when the applied twist is only maintained (the dark gray areas in Fig. 2(b)), there are still significant packing rearrangements as indicated by the increase in Q. These changes are effective at lowering the vdW energy, in spite of the surface introduced with the creation of the two “spindle” ends. The obtained non-monotonic dependence of cohesive energy on twist, showing improvement in cohesion when Q 6= 0, is consistent with the predictions of previous simulations. 20,21 It confirms that packing defects are caused by a packing frustration effect. Nevertheless, responsible for the Q > 6 in the final stages of twisting (dash line in Fig. 2(b)) are not only these packing defects but also the formation of local voids, which originate instead in the relaxation of the torsional energy into bending energy. This explanation transpires more clearly from the similar simulations presented in Fig. 3(a), which considered a shorter CNT rope comprising 91 CNTs with L = 41 nm. Due to the smaller CNT length, this rope presents a smaller vdW cohesive energy. Upon twist, it develops a large void in its core. Void formation represents an effective way to relax the twist energy stored in CNTs: As it can be seen in Fig. 3(a)-bottom, there is 41% of twist energy release into bending associated with the formation of the “bird cage”. We have performed complementary simulations on the


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Figure 3: (a) “Bird caging" of a twisted CNT rope with L = 41 nm is visible in its crosssection (top). The strain energy during the “bird cage" formation. Light gray area is the loading stage. Dark gray area is the stabilization stage under M = 64 nm·nN (bottom). (b) Alternative “spindle" morphology arising when the rope is only under geometric twist. In the cross-section, red and blue elements represent 5- and 7-fold packing defects, respectively.


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same rope in which the twist stored in individual CNTs was only geometric. (We insured the lack of physical twist by carrying out additional relaxations of all angular degrees of freedom, including those of the grip elements, at each incremental step.) We now obtained “spindles” with small Q values, such as the one shown in Fig. 3(b) exhibiting 5- and 7-fold packing defects. Such packings with Q ≤ 6 are expected when deformation energy of the filament is small. 21 These alternative simulations confirm the role of large physical twist in the “bird cage” formation.

Figure 4: (a) A straight-to-helix transformation in a rope with L = 163 nm. The color represents the modulus of the bending moment (in eV) acting on the corresponding distinct element. (b) Time evolution of the strain energy stored in the parallel-bonds. Light gray is loading (steady rate with 0.68 nm·nN/ps). Dark gray is the stabilization stage where M = 64 nm·nN. A-C mark the morphologies shown in (a). For long and thin twisted CNT ropes, “spindle” is not necessarily the lowest energy state. 10 ACS Paragon Plus Environment

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Figure 5: Phase diagram of straight-to-helix transition, as obtained by the MDEM. The simulation showcased in Fig. 4(a) uncovers the formation of a helical geometry when twisting an initially straight rope containing 91 CNTs with L = 163 nm. The evolution of the twist and bending energy components during this deformation are presented in Fig. 4(b). The first shown “spindle” in Fig. 4(a) corresponds to the end of the loading process. When M = 64 nm·nN on the grip elements is maintained, we initially observe a sizable increase in bending energy, which is associated with the progression of the “spindle" profile. Next, a small lateral instability triggers a dramatic decrease (increase) in twist (bending) energy. This energy redistribution is associated with the formation a two-windings helix, which at the end of the equilibration stage becomes compact, see last geometry in Fig. 4(a). For a broader view, we have investigated helix formation in ideally-crystalline ropes 17.2– 24.1 nm in diameters comprising CNTs up to 326 nm in length. When the transformation occurred, we obtained fully-converted helical geometries with similar pitch and radii, but larger number of windings. Fig. 5 shows that twisted ropes with aspect ratio ξ > 15 tend to be unstable against helical deformation when the applied load exceeds a critical value, which decreases with ξ. The obtain critical load scaling as ξ −0.36 is nonstandard. It differs from the ξ −1 straight-to-helical transition scaling specific to thin elastic rods. 30


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Conclusion The presented simulations of twisted CNT ropes demonstrate that MDEM can capture a rich non-linear behavior. The initial stages of twisting are dominated by the creation of “spindle” morphologies and packing rearrangements. There is no contradiction with the thoroughly investigated conclusion that twisting represents as a way to enhance the frictional contact between CNTs. 8,15 We are targeting the different regime enabled by the ability to let the individual CNTs slide freely while they are being deformed. After the “spindle” formation, the rope geometry is dominated by the amount of physical twist stored in CNTs. The strain may relax locally, with forming defects similar with the “bird-cage” defects encountered in wire cables. 7 High aspect ratio ropes exhibit a robust straight-to-helix transition. Twist insertion to produce helical CNT ropes has been accomplished in experiments 8,9 but to our knowledge was never simulated. Our simulated helical ropes resemble the experiment ones 8,9 in compactness (the formed loops are in contact) and “liveliness” (they can untwist) characteristics. Due to the computational restrictions, the MDEM simulated ropes are still much shorter and thiner than the experimental ones. 8,9 Simulating larger ropes would require a more computationally efficient MDEM model, with a larger number of atoms captured into one distinct element. Based on these results we believe the method is able to offer further important insights into the structure and mechanics of helical and superhelical CNT ropes and, more broadly, into the complex and widespread problem of finding the relationship among the geometry, internal packing, and fiber mechanics in twisted agglomerates comprising semi-flexible nanofibers. The MDEM scheme emerges as a complementary tool to theory, 19–21 where the regime of fiber deformation energy dominating the inter-fiber cohesion remains a challenge.


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Supporting Information Available Video shows the straight-to-helix transformation in a rope comprising 91 CNTs with L = 326 nm under M = 52 nm·nN. This material is available free of charge via the Internet at http://pubs.acs.org/.

References (1) Iijima, S. Helical Microtubules of Graphitic Carbon. Nature 1991, 354, 56-58. (2) Jiang, K.; Li. Q.; Fan, S. Nanotechnology: Spinning Continuous Carbon Nanotube Yarns. Nature 2002, 419, 801. (3) Li, Y.-L.Kinloch, I. A.; Windle, A.H. Direct Spinning of Carbon Nanotube Fibers from Chemical Vapor Deposition Synthesis. Science 2004, 304, 276-278. (4) Ericson, L. M.; Fan, H.; Peng, H.; Davis, V. A.; Zhou, W.; Sulpizio, J.; Wang, Y. H.; Booker,R.; Vavro, J.; Guthy, C.; Parra-Vasquez, A. N. G. M.; Kim, J.; Ramesh, S.; Saini, R. K.; Kittrell, C.; Lavin, G.; Schmidt, H.; Adams, W. W.; Billups, W. A.; Pasquali, M.; Hwang, W. F.; Hauge, R. H.; Fischer, J. E.; Smalley, R. E. Macroscopic, Neat, Single-Walled Carbon Nanotube Fibers. Science 2004, 305, 1447. (5) Dumitrică, T.; Hua, M.; Yakobson, B. I. Symmetry-, Time-, and TemperatureDependent Strength of Carbon Nanotubes. Proc. Natl. Acad. Sci. U.S.A. 2006, 103, 6105-6109. (6) Zhang, D.-B.; James, R.D.; Dumitrică, T. Electromechanical Characterization of Carbon Nanotubes in Torsion via Symmetry Adapted Tight-Binding Objective Molecular Dynamics. Phys. Rev. B 2009, 80, 115418. (7) Costello, G. A. Theory of Wire Rope. 2007 (Springer Verlag, New York).


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(8) Lima, M. D.; Li, N.; Jung de Andrade, M.; Fang, S.; Oh, J.; Spinks, G. M.; Kozlov, M. E.; Haines, C. S.; Suh, D.; Foroughi, J.; Kim, S. J.; Chen, Y.; Ware, T.; Shin, M. K.; Machado, L. D.; Fonseca, A. F.; Madden, J.D. W.; Voit, W. E.; Galvao, D. S.; Baughman, R.H. Electrically, Chemically, and Photonically Powered Torsional and Tensile Actuation of Hybrid Carbon Nanotube Yarn Muscles. Science 2012, 338, 928932. (9) Shang, Y.; Li, Y.; He, X.; Du, S.; Zhang, L.; Shi, E.; Wu, S.; Li, Z.; Li, P.; Wei, J.; Wang, K.; Zhu, H.; Wu, D.; Cao, A. Highly Twisted Double-Helix Carbon Nanotube Yarns. ACS Nano 2013, 7, 1446-1453. (10) Qian, D.; Liu, W. K.; Ruoff, R. S. Load Transfer Mechanism in Carbon Nanotube Ropes. Compos. Sci. Technol. 2003, 63, 1561-1569. (11) Liew, K. M.; Wong, C. H.; Tan, M. J. Twisting Effects of Carbon Nanotube Bundles Subjected to Axial Compression and Tension. J. Appl. Phys. 2006, 99, 114312. (12) Jeong, B.-W.; Lim, J.-K.; Sinnott, S. B. Elastic Torsional Responses of Carbon Nanotube Systems. J. Appl. Phys. 2007, 101, 084309. (13) Underhill, P.T.; Doyle, P.S. On the Coarse-Graining of Polymers into Bead-Spring Chains. J. Non-Newtonian Fluid Mech. 2003, 122, 3-31. (14) Koslover, E. F.; Spakowitz, A. J. Discretizing Elastic Chains for Coarse-Grained Polymer Models. Soft Matter 2013, 9, 7016-7027. (15) Mirzaeifar, R.; Qin, Z.; Buehler, M. J. Mesoscale Mechanics of Twisting Carbon Nanotube Yarns. Nanoscale 2015, 7, 5435-5445. (16) Carlson, A. T. Dumitrică, T. Extended Tight-Binding Potential for Modelling Intertube Interactions in Carbon Nanotubes. Nanotechnol. 2007, 18, 065706-065710.


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(17) Harris, J. M.; Iyer, G. R. S.; Bernhardt, A. K.; Huh, J. Y.; Hudson, S. D.; Fagan, J. A.; Hobbie, E. K. Electronic Durability of Flexible Transparent Films from Type-Specific Single-Wall Carbon Nanotubes. ACS Nano 2012, 6, 881-887. (18) Kim, J.-W.; Siochi, E. J.; Carpena-Nunez, J. ;Wise, K. E.; Connell, J. W.; Lin, Y.; Wincheski, R. A. Polyaniline/Carbon Nanotube Sheet Nanocomposites: Fabrication and Characterization. ACS Appl. Mater. Interfaces 2013, 5, 8597-8606. (19) Grason, G.M. Topological Defects in Twisted Bundles of Two-Dimensionally Ordered Filaments. Phys. Rev. Lett. 2010, 105, 045502. (20) Bruss, I.R.; Grason, G.M. Non-Euclidean Geometry of Twisted Filament Bundle Packing. Proc. Natl. Acad. Sci. U.S.A. 2012, 109, 10781-10786. (21) Bruss, I.R.; Grason, G.M. Topological Defects, Surface Geometry and Cohesive Energy of Twisted Filament Bundles. Soft Matter 2013, 9, 8327-8345. (22) Ostanin, I.; Ballarini, R.; Potyondy, D.; Dumitrică, T. A Distinct Element Method for Large Scale Simulations of Carbon Nanotube Assemblies. J. Mech. Phys. Sol. 2013, 61, 762-782. (23) Ostanin, I.; Ballarini, R.; Dumitrică, T. Distinct Element Method Modeling of Carbon Nanotube Bundles with Intertube Sliding and Dissipation. J. Appl. Mech. 2014, 81, 061004. (24) Cundall, P. A.; Strack, O. D. L. A Discrete Numerical Model for Granular Assemblies. Geotechnique 1979, 29, 47-65. (25) Wang, Y.; Gaidău, C.; Ostanin, I.; Dumitrică, T. Ring Windings from Single-Wall Carbon Nanotubes: A Distinct Element Method Study. Appl. Phys. Lett. 2013, 103, 183902.


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(26) Vilatela, J. J.; Elliott, J. A.; Windle, A. H. A Model for the Strength of Yarn-Like Carbon Nanotube Fibers. ACS Nano 2011, 5, 1921-1927. (27) PFC3D (Particle Flow Code in 3 Dimensions), Version 5.0. Minneapolis: ICG Inc., 2008. (28) Johnson, S. M.; Williams, J. R.; Cook, B. K. Quaternion-Based Rigid Body Rotation Integration Algorithms for Use in Particle Methods. Int. J. Numer. Meth. Eng. 2008, 74, 1303-1313. (29) Q =

P

n (6 − n)V

(n), where n is the number of nearest neighbors of the CNT, and V (n)

is the total number of all internal CNTs of the bundle that have n nearest neighbors. A CNT is considered internal if it has 6 nearest neighbors in the untwisted state. A nearest√ neighbor element is situated within a 3R distance, where R = 1.76 nm, which is the equilibrium interspacing of two parallel CNTs. When identifying the nearest neighbors, we approximate the distance between two helical CNTs with the smallest R between elements located on the two tubes. We monitored Q in a cross section located in the middle of the rope, Fig. 2(a). Due to symmetry considerations, the distinct elements at this location will tilt but not displace along the rope axis. The distance between neighbors not always coincides with the separation between the corresponding elements in this cross section. 20 For this reason, we have include adjacent elements (connected by parallel bonds with the elements in the cross-section) into our calculation. (30) Love, A. E. Treatise on the Mathematical Theory of Elasticity, Dover Books (1927).


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Twisting Carbon Nanotube Ropes with the Mesoscopic Distinct

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