STRESS TRANSFER EFFICIENCY IN CARBON NANOTUBE BASED ROPES

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6 TH INTRNATIONAL CONFRNC ON COMPOSIT MATRIALS STRSS TRANSFR FFICINCY IN CARBON NANOTUB BASD ROPS Luis Zalamea* and R. Byron Pipes** [R. Byron Pipes]: bpipes@purdue.

1 6 TH INTRNATIONAL CONFRNC ON COMPOSIT MATRIALS STRSS TRANSFR FFICINCY IN CARBON NANOTUB BASD ROPS Luis Zalamea* and R. Byron Pipes** [R. Byron Pipes]: **Schools o Materials ngineering, Aeronautics and Astronautics, *Chemical ngineering and Birck Nanotechnology Center, Purdue University, West Laayette, IN , USA Keywords: Carbon Nanotubes, Helical Ropes, Shear Lag, Stress Transer Introduction Semi-empirical models derived or pseudo homogeneous ropes [] have been employed to calculate the eective stiness o ropes made out o MWNT []. Those approximate models predict a signiicant reduction with respect to the Young s modulus o an individual ilament by considering the eect o miss-orientation o the twisted ilament with respect to the applied load, migration o ilaments within the iber and radial compression. Such models do not explicitly take into account the load transer eiciency rom the rope to an individual iber, but have provided satisactory approximate results, and a convenient way to understand and compare experimental data reported recently on macroscopic array. Several variations o the standard shear lag model can be employed as an alternative to estimate load transer rom a matrix into a iber o inite length embedded in a homogeneous media composed by the other ibers [3]. The main objective o the present work is to employ previously calculated values o shear stiness o single walled carbon nanotube arrays [4] to estimate the magnitude o overlap length necessary to attain eicient load transer between ilaments in a rope. On a irst approximation the model neglects the eects o helical wrapping which has been demonstrated to be o second order or ropes less than approx. 40nm dia.. This poses a problem because it invalidates one possible mechanism or enhancement o load transer; the radial stress generated due to rictional orces. The results obtained indicate that due to the low interacial shear modulus, the lengths necessary to attain a signiicant raction o the stiness o the constituent ilament are on the order o several microns. Fabrication techniques capable o conerring such degree o perection at that scale, in a rope do not exist at the present time. Model Development. Geometry Conversion The eective modulus o an individual carbon nanotube under tension or lexure is not equivalent to that o a graphene sheet, since there is an area at the center that must be taken into account and does not contribute to the stiening o the structure. The assumed wall thickness is equivalent to the separation distance between two consecutive graphene layers, which can vary rom 0.3 to 0.39 according to experimental reports [5]. This ramework insures that no structure based on olded and rolled graphene sheets can have stiness larger than that o graphite. For SWNT the assumed wall thickness is equivalent to the separation distance between two SWNT orming an array. Due to this approximation, in the converted geometry the individual carbon nanotubes are in contact with their neighbors.. Shear Lag Models As explained in detail elsewhere [4], in order to keep solutions or beam bending analytical, the hexagonal symmetry o the carbon nanotube array can be replaced by a rectangular equivalent, or it can be replaced by a concentric cylinder array to consider one nanotube immersed in a homogeneous matrix, as depicted in Fig.. Fig.. Approximate Geometries or SWNT Array

2 A shear lag model developed to study stress transer in concentric shells as those in MWNT [6], can be readily extended to the situation mentioned above. The input parameters are the shear stiness o the interacial regions and the approximations or the diameters o each layer, both o which have been calculated and reported elsewhere [4]. Typical values or the shear stiness o the interacial region o SWNT are as ollows: 7-lem. 6-lem. 7-lem. Interacial Area, [m *0-5 ] Interacial nergy, [J/m *0-3 ] Interacial Modulus, [MPa] Table. ective Properties o the Interacial Regions in Shear The bold segment o the interacial region indicates the area over which shear stress transer takes place, or the hypothetical situation in which the center ilament is pulled by the array. Note however that the present work neglects the capillary energy component that will arise as a result o creation o new surace area. Instead, the center ilament is considered to be immersed in a matrix composed o the other tubes. This assumption allows a very simple orm o the shear transer parameter, as deined by Cox [3] in his seminal 95 paper The well-known analytical solution o a shear lag model with this parameter results in the ollowing expression or the eective Young s modulus [3]: e ( L ) tanh = L () The previous equation provides a irst order approximation at the eiciency to transer load applied only to the external layer o a seven element rope into the centermost element. The calculation o the shear transer parameter, has received considerable attention in the literature, especially or applications to iber pullout tests, where the outer diameter o the structure is not clearly deined [7]. For the problem at hand it has been assumed that the external radius o the matrix is equivalent to the distance rom the center o the rope to the center o the outermost layer o elements, ollowing the work o pan [8]. This approximation results in the ollowing expression or the shear lag parameter: = r na ln G int () Alternatively a shear lag analysis based on cylindrical solid bodies in contact yields the ollowing expression or the same parameter [6]: = R A A + h G int (3) In the previous expression, h is the distance between the average radii o both cylinders, and R is the average radius o the interacial region. The shear modulus was converted rom rectangular to circular geometry in Fig., by considering the dierences in cross sectional areas and the perimeter over which the load transer takes place. These variations o the shear lag parameter lead to dierences in the actual value but not on the trend, as shown: Fig.. ective Young s modulus as a unction o Length and Shear Stiness ay The dierences between the two estimations o the shear lag parameter are not relevant compared to those due to the shear modulus o the interacial region (or pseudo homogeneous matrix). A irst order estimate or the stiness o the interacial regions in an array o carbon nanotubes, can be obtained rom the corresponding value or graphite: 4.5 GPa [5], however, recent reports have employed hybrid atomistic continuum models to estimate the bounds or such elastic constant [9], and the results suggest slightly lower values, in accordance with the calculations based on energetics under lexure [4]. In principle the shear modulus o the interacial region

3 can be close to that o in plane graphite (80 GPa), although it would require a very high density o crosslinks. In this case the length necessary to obtain the same eiciency is reduced drastically, as presented graphically in Fig.. The main conclusion that can be drawn rom Figure 4 is that there is an enormous incentive or eorts aimed at strengthening the interace, because they have very relevant eects on the overlap length necessary to insure proper load transer. Several experimental [0,] as well as theoretical [,3] works have reported on this issue..3 Conventional Semi-mpirical Rope Models The book by Hearle et al.[] was ollowed or the calculation o the eective Young s modulus o a helical rope (yarn). Although some parameters are empirical it can provide a irst order approximation to be compared against the corresponding value rom shear lag analyses presented beore and experimental data []. There are three actors modiying the eective stiness o a helical rope o any material: the missorientation o the ibers with respect to the yarn due to twisting, the radial compression due as a result o axial deormation (poisson s eect) o the yarn as well as the ibers, and inally the inite length o constituent ibers, which results in the necessity o migration as a mechanism to insure cohesion in the rope, as hypothesized by Hearle et al. in their book. The ollowing equation combines the eects o mis-orientation, radial contraction and inite length: e cos = k = 3L ( k csc ) F D μ Q (4) The actor is the helical angle to which the individual ibers are twisted in order to orm a cohesive yarn. The Poisson s or radial contraction eect can be calculated by means o the ollowing equation: ( ) ( + y ) ln( cos)+ + cos + + ( ) F = ( + )sin 3+ y ( ) 4+ ( ) cos cos (5) This eect might be especially important, as indicated by the experimental indings o Baughman and coworkers, who observed unusually large Poisson s ratios or yarns made out o multiwalled carbon nanotubes. However such large contractions must be analyzed careully to understand their relationship with changes in packing and volume raction Finally the eect o inite length o the ibers was treated by Hearle et al. assuming that all ibers slowly migrate radially across the yarn, providing a mechanism to accommodate inite length ibers without loosing cohesion. In equation (5) μ is the riction coeicient between ibers (F=μN) and Q is the migration length, i.e., the length necessary or a iber to migrate all the way rom the outer surace to the center and back to the surace o the yarn. D is the diameter o a iber and L is its length. The migration length should be smaller than the iber length in order or the stress transer mechanism to be ully eective. However it would be possible to have migration lengths larger than the iber length and still have some degree o load transer. In the limit, the migration length becomes ininite and the stiness reduces to zero. The values used or the calculation o eective modulus as a unction o iber length are presented in Table, and the results are plotted in Figs Parameter Value Helical Angle 5 deg Yarn diameter μm Fiber diameter 0 nm Fiber Length 00 μm Yarn Poisson s R MWNT Poisson s R Migration Length μm Friction Coeicient (?) Table. Yarn and Fiber Properties or Hearle s Models The values o shear modulus estimated rom lexural experiments cannot be used directly to estimate a riction coeicient because there is no unique relationship between both. The radial compression eect was calculated or a yarn Poisson s ratio o.35 and a iber Poisson s ratio o 0.. The shaded region indicates the range o lengths measured by Baughman and coworkers in their report. The ollowing igure was generated using a value o 0. or the riction coeicient. Recent experiments o graphite [5] suggest that it could eventually deviate appreciably rom the assumed value i the two graphene suraces are not 3

4 commensurate with each other. It could be negligibly small resulting in a very large impact on the stiness, as presented graphically on Fig. 4, in which the migration length was kept constant at 0. mm (00μm, equivalent to the iber length). riction coeicients or shear modulus are responsible or the low eective stiness: Fig. 6. Model Comparisons Fig. 4. Modeling Baughman s xperiments Fig. 5. Ultralow Friction Coeicient Interestingly, or the lengths reported by Baughman et al. [] and usual macroscopic values o the riction coeicient (~0.), Hearle s models predict a rather mild decrease in stiness, unless the migration length alls bellow its critical limit. In contrast, their experiments show a drastic reduction in stiness. i.e., the eective modulus measured is only.3% o the expected value, taking into account reductions by volume raction, hexagonal packing raction and the act that the constituent nanotubes are partially hollow. The explanation or this unusually low value may lie in the surprisingly low value o the riction coeicient or graphitic structures..4 Model Comparison The ollowing plot presents a comparison between the two models, showing that very low The value o the interacial shear modulus that corresponds to a riction coeicient o ~0. is very low and close to the corresponding value reported or graphite [5]. For very low riction coeicients as those mentioned above, the corresponding shear stiness becomes negligibly small..5 Conclusions According to the experimental data analyzed, the overlap necessary to insure proper load transer among untwisted SWNT is rather large (on the order o microns). Fabrication techniques aimed at optimizing the overlap length would enhance signiicantly the load transer eiciency o ropes. Interestingly, very short overlap lengths would be enough to transer load satisactorily, provided they are chemically modiied to increase the interacial shear stiness. There are great, yet unexploited, potential beneits or surace modiied carbon nanotube systems (both SWNT and MWNT) that have not been realized on a macroscopic scale. In the case o MWNT, the capillary orces are not as eective because the strength o the interaction decreases as size increases, as calculated by Girialco et al. [6]. However experimental evidence suggests that riction between shells in MWNT can be very low acilitating sliding o one MWNT past another and making large overlap lengths theoretically possible. Frictional resistance among MWNT is still a subject o debate and it is a very relevant parameter in the design o ropes, as demonstrated by the calculations presented To summarize, the design o CNT structures should be addressed on a multiscale manner, seeking to extract advantages o the dierent load transer mechanisms that dominate each scale. i.e. commensurability and shear stiness at the

5 nanoscale, and twisting-induced stresses at the microscale Reerences [] J.Hearle, et al., Structural Mechanics o Fibers, Yarns and Fabrics, V., Wiley-Interscience, (969). [] M.Zhang, et al., Multiunctional Carbon Nanotube Yarns, Science, 306(5700), 358 (004). [3] H.Cox, lasticity and Strength o Paper and Other Fibrous Materials, Brit. J. Appl. Phys. 3, 7. (95). [4] R.B.Pipes, et al., Flexural Delection as a Measure o van der Waals Interaction, Comp.Sci&Tech., 66(9),, (006). [5] M.Dresselhaus, et al., Carbon Nanotubes, Springer- Verlag, (000). [6] Zalamea et al., Stress Transer in Multiwalled Carbon Nanotubes, in press in Comp.Sci&Tech. [7] J.A. Nairn, On the Use o Shear-Lag Methods or Analysis o Stress Transer in Unidirectional Composites, Mechanics o Materials, 6, 63, (997) [8] N. Pan, Development o a Constitutive Theory or Short Fiber Yarns: Mechanics o Staple Yarn Without Slippage ect, Textile Res. J., 6(), 749, (99) [9] J.Z. Liu, Q.S. Zheng, L.F. Wang, Q. Jiang, Mechanical properties o single-walled carbon nanotube bundles as bulk materials, Journal o Mechanics and Physics o solids, 53, 3, (005) [0] W. Guo, W. Zhong, Y. Dai, and S. Li, Coupled deect-size eects on interlayer riction in multiwalled carbon nanotubes, Phys. Rev. B, 7, (005) [] M. Huhtala, A.V. Krascheninnikov, J. Aittoniemi, S.J. Stuart, K. Nordlund, and K. Kaski, Improved mechanical load transer between shells o multiwalled carbon nanotubes, Phys. Rev. B, 70, (004) [] J.P. Salvetat, et al., lastic and Shear Moduli o Single-Walled Carbon Nanotube Ropes, Phys. Rev. Lett., 8(5), 944, (999) [3] Cs. Mikó, M. Milas, J.W. Seo, R. Gaál, A. Kulik, and L. Forró, ect o ultraviolet light irradiation on macroscopic single-walled carbon nanotube bundles, Appl. Phys. Lett., 88, 5905, (006) [4] L. Shen and J. Li, Transversely isotropic elastic properties o multiwalled carbon nanotubes, Phys. Rev. B, 7, 0354 (005) [5] M. Dienwiebel, G.S. Verhoeven, N. Pradeep, J.W.M. Frenken, J.A. Heimberg and H.W. Zandbergen, Superlubricity o Graphite, Phys. Rev. Lett., 9, 60, (004) [6] L.A. Girialco, M. Hodak, and R.S. Lee, Carbon nanotubes, buckyballs, ropes, and a universal graphitic potential, Phys. Rev. B, 6(9), 304, (000) 5

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