Mechanical behavior of carbon nanotube yarns with stochastic microstructure obtained by stretching buckypaper

Composites Science and Technology xxx (2018) 1e12

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Mechanical behavior of carbon nanotube yarns with stochastic microstructure obtained by stretching buckypaper A. Sengab, R.C. Picu* Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, 12180, USA

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 November 2017 Received in revised form 8 February 2018 Accepted 9 February 2018 Available online xxx

The development of yarns composed primarily from carbon nanotubes (CNTs) has been pursued recently with the intent of transferring to the yarn the exceptional mechanical and transport properties of individual nanotubes. In this work we study the process of yarn formation by dry stretching buckypaper, and the mechanical behavior of the resulting yarns, function of the CNT length and of the state of the CNT assembly before stretching. The analysis is performed using a coarse grained, bead-spring representation for individual CNTs. It begins with a random buckypaper structure composed from CNTs of diameter 13.5 Å. This structure is stretched to form a yarn. This occurs once the stretch ratio becomes larger than a threshold which depends on the CNT length. At the threshold, adhesion stabilizes a highly aligned packing of CNT bundles. Packing defects and pores, reminiscent of the initial structure of the buckypaper, are incorporated in the yarn. The yarn is further tested in uniaxial tension. The defects have little effect on the mechanical behavior of the resulting yarns. However, the behavior depends sensitively on the degree of packing of the CNTs in the sub-bundles forming the yarn. Therefore, the initial structure of the buckypaper has little effect on the performance of the yarn. Increasing the CNT length increases the yarn flow stress and this is associated with the residual tortuosity of the CNTs in the yarn. Decreasing the temperature or increasing the strain rate lead to a small increase of the flow stress. These results have implications for yarn design, which are discussed in the article. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Significant efforts have been devoted over the last decade to the development of materials composed primarily from carbon nanotubes (CNT). The objective is to convey the exceptional nanoscale properties of individual CNTs [1], to their macroscopic assemblies. To this end, strongly aligned structures have been considered, with CNTs forming two dimensional buckypaper [2], or one-dimensional yarns [3,4]. Such materials are expected to have high strength and high thermal and electrical conductivities, and to find applications in various fields such as structural composites, special textiles, and electrical power transmission lines. CNT yarns have been produced by a number of techniques, of which the most notable are in-situ direct spinning [5e7], drawing from aligned CNT forests [3,8,9], and drawing or/and spinning of CNT aerogels and buckypaper mats [10]. Drawing leads to CNT alignment, while spinning increases the density of the yarn and

* Corresponding author. E-mail address: [email protected] (R.C. Picu).

improves its mechanical properties. The strength of CNT yarns produced to date is much lower (10e500 times lower) than that of individual CNTs which is approximately 50 GPa [11]. This situation is attributed to the poor load transfer between nanotubes or, in presence of a matrix, between nanotubes and the matrix. A broad range of values have been reported for the strength of such fibers. Yarns which have not been densified or twisted have strength ranging from ~100 MPa [4,7,12] to ~1 GPa [13]. The buckypaper of initially randomly oriented CNTs yields at small stresses of several tenths of MPa and then strain hardens and fails at a nominal stress between 100 and 200 MPa [14,15]. If such samples are unloaded before failure and re-tested after a wait period, the response is much stiffer and the strength increases significantly, function of the level of pre-strain applied [14,15]. For example, Cheng et al. [15] report that preconditioning buckypaper by uniaxial deformation up to 30% strain followed by unloading, leads to a much stiffer response upon reloading in the same direction, with a failure stress of 400 MPa. If the pre-strain applied is of 40%, the strength of the resulting structure increases to above 600 MPa. This history dependence is associated with the

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fact that the pre-strain allows adhesion-driven network reorganization leading to denser and more stable structures with higher adhesion energy. Increasing the density of yarns is known to greatly enhance their strength. A simple pressing with a spatula of a CNT yarn leading to the formation of a denser ribbon has been recently shown [7] to increase the strength from ~150 MPa to 3 GPa [16]. Increasing the density of un-twisted yarns has been achieved in Ref. [9] by passing the yarn through a die. The maximum strength reported was 1 GPa for a yarn of 35 mm diameter. Yarns compacted using this method to a diameter of 75 mm had a strength of only 200 MPa. Densification can be also produced by twisting the yarn and this leads to a significant increase of the strength relative to the yarn drawn without spinning. Naraghi et al. [10] report a strength just below 1 GPa for yarns obtained by dry spinning from mats, while a strength of 3 GPa is reported in Ref. [17] for twisted yarns of few microns in diameter. A higher strength of 9.6 GPa is reported in Ref. [18] for sub-micron thick films grown from pre-aligned CNTs and tested upon densification by rolling. Enhanced CNT alignment and enhanced strength are observed when using a polymeric embedding medium during drawing [19e24]. A quite different approach to increasing the yarn strength involves the formation of chemical bonds between nanotubes, which can be achieved either by CNT functionalization or by irradiation. This improves the load transfer between CNTs, but introduces defects in the nanotubes, therefore reducing their intrinsic strength [25]. Irradiation has been shown to increase the strength of very small bundles of mostly parallel CNTs to more than 10 GPa [26], but the effect on larger (micron-scale) yarns is less spectacular. For example [7], reports a strength enhancement of less than 20% upon cross-linking densified CNT ribbons, while the densification step alone increases the strength ~20 times over the undensified, but aligned, CNT yarn. A review of works dedicated to the effect of irradiation on the stiffness and strength of CNT yarns is presented in Ref. [27]. Given the large variability of experimental results and the complexity of the problem, modeling has been used to identify the relationship between yarn structure and its strength. Models usually consider perfect CNT bundles, in which nanotubes are parallel to the bundle axis and are either spanning the model [28] or are shorter than the bundle length and randomly staggered in the axial direction [29e32]. Here we refer to such arrangements of parallel CNTs as ‘perfect bundles,’ and to the more random arrangements resulting from spinning or drawing buckypaper, as ‘yarns.’ Perfect bundle twisting was simulated using a fully atomistic model in Ref. [33], and with a coarse grained representation in Ref. [34]. These models generally indicate that if the axial deformation mode of fibers is not engaged due to poor inter-tube load transfer, the overall strength is rather small. Introducing inter-tube friction or cross-links leads to more significant load transfer and an increase of the yarn strength to 1 GPa and higher. These models fall short of investigating realistic yarn structures and focus on studying the inter-tube load transfer in idealized configurations. The effect of the waviness of nanotubes was investigated in Ref. [35] using a coarse grained model of CNTs. However, the geometry was defined by introducing waviness arbitrarily, with no relation to the nanostructure of realistic CNT yarns. The simulations indicate that the yarn strength depends on the formation and failure of CNT contacts and, in general, on the yarn nanoscale geometry. In this work we study the mechanics of yarns with stochastic nanoscale structure. We avoid working with perfect bundles and generate yarns by directly stretching buckypaper mats produced by random deposition of nanotubes. The resulting structures are imperfectly packed. The effects of the CNT length, of the initial structure of the mat and of the test temperature on yarn strength

are studied. 2. Models and methods A coarse grained model is used in this work to represent CNTs. This model is similar to the bead-spring representation frequently used in polymer physics [e.g. [36]], and was used for CNTs in a number of works [34,37]. Each filament is represented as a chain of spherical beads which interact along the CNT axis through axial and angular potentials that mimic the axial and bending stiffness of the filament, and with beads not belonging to the same CNT, via a Lennard-Jones (LJ) potential tailored to represent the adhesion energy per unit length of two parallel and relaxed CNTs in contact, g. The axial and angular potentials are harmonic and the respective elastic constants are computed in terms of the axial and bending rigidity of the CNT (E0 A0 and E0 I0 , respectively) by requiring the equivalence of strain energies stored in the actual CNT and in the bead-spring model. The stiffness of the harmonic spring representing axial interactions results ka ¼ E0 A0 =sa , where sa is the distance between consecutive beads along the filament, while that of the bending potential is kb ¼ E0 I0 =sa . E0 ; A0 and I0 are the effective modulus, cross-sectional area and moment of inertia of the CNT. The description applies equally to single wall (SWCNT) and multiwall (MWCNT) nanotubes. The models are advanced in time with molecular dynamics,
-Hoover thermostat. All using the Verlet algorithm and the Nose simulations are performed on a massively parallel computer at the Rensselaer Polytechnic Institute Center for Computational Innovation (CCI), using LAMMPS from Sandia National Labs. We use parameters relevant for (10,10) CNTs of diameter d0 ¼ 13.5  A calibrated based on atomistic simulations in Ref. [38]: ka ¼ 2 270:4 eV= A , kb ¼ 6499 eV and g ¼ 0:23 eV= A. This value of g corresponds to the equilibrium distance between the parallel axes of two relaxed CNTs in contact, d0 , and to the minimum of their interaction potential. CNTs of large diameters collapse to form flattened tubes resembling bi-layer graphene ribbons [19]. For d0 < 1…2 nm, the circular shape prevails, while for d0 > 4…5 nm CNTs tend to flatten [39e41]. In this work we focus on structures of CNTs of small diameter which remain approximately circular. Bending buckling of SWCNT, which happens when the bending angle is larger than ~40 [42], is not represented in the model. MWCNT do not buckle in the bending mode. The discrete representation of the CNT may lead to artificial surface roughness. In order to minimize this effect, the density of beads along the filament is increased to sa ¼ d0 =4, which means that 4 spherical beads represent a cylindrical CNT segment of aspect ratio 1. To demonstrate the behavior of this representation and to clarify certain aspects of the mechanics of adhesive interactions between one-dimensional objects, we consider a close packed perfect bundle of 7 chains arranged in a hexagonal pattern, with axes parallel to each other, and overlapping over their entire length L0 , Fig. 1(a). The outer 6 chains have their left end fixed and the middle chain is pulled out [33,43]. The system is allowed to fully relax in the radial direction. As the central CNT is pulled out, both axial and adhesion energies vary. The CNTs are loaded by forces of magnitude g acting as shown in the sketch of Fig. 1(a). Note that only segments AB and CD are loaded axially, while segment BC of both outer and inner CNTs is not loaded. Fig. 1(b) shows the variation of the total adhesive energy of the system with the relative imposed displacement, L0  x, at 20 K and 300 K. The slope of the line is 1.4, which is equal to 6g, as predicted for the model in Fig. 1(a). The curves corresponding to the two temperatures overlap, which indicates that adhesive interactions are temperature-independent. This implies that the stress component associated with the

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Fig. 1. (a) Schematic representation of a bundle of 7 parallel chains used for the pullout test. The axial forces acting on the CNTs and associated with the variation of the adhesion energy are shown. Variation of the total adhesion energy of the system (b) and the pull-out force (c) versus the overlap length at 20 K and 300 K for d ¼ d0 (curves overlap), and at 300 K for d ¼ 0:95d0 (dashed yellow lines). (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

variation of the adhesion energy during deformation, sad , is athermal. Fig. 1(c) shows the pull-out force, F, versus the imposed displacement at 20 K and 300 K. If one considers that a frictional shear force per unit length of the contact between two CNTs, ff exists, the pull-out force is given by:

F ¼ 6g þ 6ff x

(1)

The curves in Fig. 1(c) are horizontal and do not exhibit any periodic pattern; this indicates that ff ¼ 0 at both temperatures and the effective roughness is negligible. In order to test that roughness effects are minimal even under radial compression (which may

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occur naturally in yarns with stochastic microstructure), we reduce the distance d between the CNT axes and repeat the pull-out test. Fig. 1(b) and (c) show the variation of the adhesion energy and of the axial force during the test performed at 300 K, with d ¼ 0:95d0 . The frictional force vanishes under this level of compression as well and the effect of roughness is still too weak to be observed. The axial energy is negligible relative to the adhesion energy. This is due to the fact that the axial energy is inversely proportional to the axial rigidity of the CNT, which is large, and to the fact that CNTs are free to relax by sliding along their contour, which drastically limits axial deformation. We begin by creating buckypaper mats which are then stretched uniaxially to produce yarns. The mats are produced by depositing on a plane CNTs of same length, L0 , with random positions of their centers of mass and random orientations. The mats have a 3D structure and are allowed to relax in the direction perpendicular to the deposition plane. The CNTs interact with the deposition plane through purely repulsive non-bonded interactions, and the plane is removed after the mat is created and relaxed. The deposition continues until the specified density, r, is achieved. The density is defined here, for convenience, in the projection of the network on the supporting plane, as the total length of CNT per unit area. Filaments are deposited in an Ls  Ls domain of the deposition plane, with Ls ¼ 1084 nm. We work with models of density 1 r ¼ 0:0183  A and vary the CNT length L0 : CNTs of length L0 ¼ 135:5 nm, 271.2 nm, 542.2 nm and 800 nm are considered in separate models. These models are generated, relaxed and then deformed using periodic boundary conditions. This represents a system of infinite extent and of constant density. Vacuum padding (traction free boundary conditions) is applied in the direction perpendicular to the mat. After the mats are relaxed, the resulting structure is subjected to tensile uniaxial in-plane loading at 300 K. We denote this deformation as ‘preconditioning.’ Traction free boundary conditions are imposed in the other in-plane direction. This deformation is performed by moving two in-plane parallel boundaries of the model away from each other with a velocity of 5  A/ps, up to a total stretch of l ¼ 2:4. Subsequently, the model is unloaded and re-equilibrated at 300 K. As discussed below, compact, adhesion-stabilized yarns are obtained upon unloading provided the mat was stretched to a sufficient degree. The mechanical behavior of these yarns is tested in uniaxial tension, which is applied in the same direction with the preconditioning stretch. The yarns are deformed by applying a velocity of 5  A/ps to one of their free ends, while the other end is kept fixed. This corresponds to an effective strain rate of 0.2 ns1. Traction free boundary conditions are used in both directions perpendicular to the stretch direction. We consider also perfect bundles, which provides a reference for the yarns obtained by stretching mats. The perfect bundles are composed form CNTs with parallel axes, staggered randomly in the axial direction, and fully relaxed under traction free boundary conditions in the radial direction. Similar models of CNT bundles have been considered in many other works [29,31e33,43,44]. We also consider perfect yarns in which some degree of randomness is introduced by perturbing the parallel arrangement of CNTs. In these models, nanotube axes are randomly misaligned in the initial state   by angles in the range 0  3 and the CNTs are shuffled in the radial direction. The structure is relaxed at 300 K and further tested in uniaxial tension. 3. Self-organized CNT buckypaper As-deposited buckypaper mats may evolve under the action of adhesion forces. The self-organization of filamentary structures

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Fig. 2. Map of configurations of stable mats in the J  rL0 plane. Three types of structure exist: ‘locked,’ which remain in the as-deposited state upon relaxation, ‘disintegrating’, which evolve into isolated bundles that lose connectivity and do not form a network, and ‘cellular’, which self-organize due to adhesion into cellular networks of bundles. The symbols indicate the 4 types of mats used to produce yarns by preconditioning stretch. These have the same density and L0 ¼ 135.5, 271.2, 542.4 and 800 nm, increasing from left to right, for A, B, C and D, respectively.

under the action of adhesion was studied in detail in Ref. [45]. Here we present only the main results for completeness. The important non-dimensional parameter in this process is J ¼ gL20 =E0 I0 , which represents the relative importance of adhesion and bending, with bending being the dominant energy storing CNT deformation mode in this problem. The system is also characterized by its density, r, and the CNT length, L0 , with rL0 being the relevant nondimensional parameter. The stable structures resulting from the evolution of the asdeposited mat are defined in the space J  rL0 in Ref. [45], Fig. 2. For rL0 < 5:71, the as-deposited structure is too sparse to form a percolated network and hence this case is not of interest. At higher values of rL0 and low values of J, adhesion is too weak and the asdeposited structure cannot evolve. These structures are denoted as ‘locked.’ At large J, the CNT structure self-organizes under the action of adhesion. The boundary between locked and evolving structures is defined by the equation J  ð rL0 Þ2 and shown in Fig. 2. Two types of self-organized structures are observed for J > ð rL0 Þ2 [45]. At low rL0, CNTs self-organize into isolated bundles which lose inter-bundle connectivity and hence do not percolate to form a space-filling network. At high rL0 , the as-deposited network

self-organizes into a network of bundles denoted here as ‘cellular structure.’ The systems considered in this study are shown in Fig. 2 and labeled (A) to (D). These correspond to the same projected density, r ¼ 0:0183 A1 , and to the CNT lengths L0 indicated in the previous section. System (A) is of ‘disintegrating’ type and hence it would not form a continuous network if allowed to relax. Therefore, we subject the as-deposited, unrelaxed (UR) network to preconditioning stretch and label the case (A)-UR. The characteristic time of relaxation is much longer than the time of preconditioning stretch and hence a yarn is obtained despite the small rL0 of the initial buckypaper. Systems (B), (C) and (D) are of ‘cellular’ type. We consider both the relaxed (cellular) (R) and unrelaxed (as-deposited) (UR) states of systems (B) and (C) and subject both to preconditioning stretch. System (D) is subjected to preconditioning deformation in the unrelaxed state. We consider both (R) and (UR) states of the same structure in order to investigate the importance of the starting configuration in the mechanics of the resulting yarns. Fig. 3 shows 2D (projected) views of models (C)-R and (C)UR. 4. Generation of yarns from buckypaper mats 4.1. Mat preconditioning Systems (A) to (D) of Fig. 2 are subjected to preconditioning stretch to produce yarns. Both relaxed and unrelaxed states are considered for systems (B) and (C), while for systems (A) and (D) only unrelaxed states of the mat are considered. The initial state of the mat is transversely isotropic, with CNTs oriented randomly in the plane of the mat. These structures are stretched uniaixally at 300 K up to a stretch ratio of l ¼ 2:4, with periodic boundary conditions in the plane of the mat and zero tractions in both directions perpendicular to the stretch direction. Fig. 4 shows the variation of the Cauchy stress during deformation. Due to the large Poisson contraction of the network, the cross-sectional area decreases rapidly during stretch. The area is evaluated at all stages of the deformation. The deformation takes place for l(1:5 at an approximately constant stress of about 10 MPa for systems (A) to (C), and about 30 MPa for system (D). This observation is in agreement with experimental data presented in Ref. [14] for mats of ~1 mm long MWCNTs which deform at a stress of ~75 MPa up to the maximum strain reported of 30%. Rapid strain hardening is observed in our simulations for stretches larger than a * * threshold l which depends on L0 . This threshold is l z1:8; 1:75

Fig. 3. Realizations of system (C) in the unrelaxed, as-deposited state (C)-UR, and in the relaxed, cellular state (C)-R, before preconditioning stretch. Models are three-dimensional and are shown here in projection on the plane of the mat.

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Fig. 4. Cauchy stress vs. stretch ratio curves (preconditioning deformation) for all mats considered in this study. The rapid stiffening corresponds to the formation of yarns, which are stabilized by adhesion between merging CNT bundles. Once the yarns form, the stress reaches a plateau whose value is a function of the CNT length, L0 .

and 1.65 for systems (B) (the average of (B)-R and (B)-UR is reported), (C) and (D), respectively. The threshold is evaluated as the intersection of two lines fitted to the strain hardening regime and to the initial, small stretch regime. The stress-stretch curve eventually plateaus at stress values that increase with increasing L0 . The essential feature of this evolution is the jump-in-contact occurring during the rapid strain hardening regime. This process is either rapid, occurring at a specific stretch ratio, or more gradual, in different systems and realizations. If the system is unloaded after the jump-in-contact, the structure does not relax and retains its current yarn state. Unloading before the jump-in-contact leads to significant relaxation and the final structure resembles the initial network. The stress reaches a plateau after the yarn structure forms. The models are fully unloaded at this stage. The preferential orientation of the CNTs during stretch is evaluated using the Herman orientation factor P2 ¼ 12 ð3cos2 q  1Þ. We evaluate the orientation of chain end-to-end vectors as well as the orientation of segments of length 2d0 . In the two cases, q represents

5

Fig. 6. Energy partition in systems (C)-R and (C)-UR during preconditioning stretch. fad ; fb andfax represent the ratios of the adhesion, bending and axial energies to the total energy, respectively.

the angle between the end-to-end vector and the loading direction, and the angle between CNT segments of specified length and the loading direction. The operator < > represents ensemble averaging. P2 is zero for random orientations and 1 for perfect alignment. Fig. 5 shows the variation of P2 during the preconditioning deformation for system (C). P2 increases from close to zero in the initial state (P2 is not exactly zero due to the preferential arrangement of the CNTs in 2D in the initial state) to ~0.9 at l ¼ 2.4, indicating almost perfect alignment at these large stretches. P2 measured on the chain and segment scales are statistically identical and there is little difference between the alignment of the (R) and (UR) systems. It is interesting to compare this prediction with experimental results. In Ref. [46] it is reported that millimeter long MWCNT mats of diameter 3e8 nm were dry stretched and the degree of alignment was evaluated by X-Ray scattering and Raman spectroscopy. The Herman orientation factor evaluated based on their data is 0.37 and 0.54 at 20% and 40% strain, respectively. An equivalent orientation factor of 0.45 is reported in Ref. [14] for a similar mat stretched by 30%. These data points are added to Fig. 5 for reference. Fig. 6 shows the energy partition during the preconditioning stretch for systems (C)-R and (C)-UR. The adhesion, bending and axial energies are computed separately and their ratios to the total energy, fad ; fb ; fax , are shown in Fig. 6. The dominant contribution is due to adhesion. The relaxed cellular structure (C)-R has finite adhesion energy in the initial stage, while the unrelaxed structure (C)-UR starts with a smaller value of the adhesion energy, as expected. The axial energy makes a negligible contribution to the total energy in all cases. Nevertheless, axial interactions make a significant contribution to stress, as discussed below. The fraction of bending energy is almost constant as the mat transforms into a yarn during stretch. As the adhesion energy increases in absolute value during this process, the bending energy increases too. 4.2. Stabilization of defects in the yarn

Fig. 5. Evolution of the orientation factor, P2 , during preconditioning stretch for systems (C)-R and (C)-UR. P2 is evaluated based on the end-to-end vector of CNTs, and based on CNT segments of aspect ratio 2. Experimental data from Ref. [46] (red circles) and [14] (black triangle) are added for reference. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

The yarn formation process outlined above suggests that a large number of packing defects may be incorporated in the yarn. These originate from the random structure of the mat and can be only partially eliminated by stretching. Defects are stabilized against collapse under the action of adhesive forces by folded CNT bundles, as shown in Fig. 7. The figure shows the evolution of a cell of the cellular network of (C)-R during preconditioning. The initial cell of

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It is of interest to inquire under what conditions such defective yarns form. It is apparent that, for given strength of adhesion, the yarn cannot be stabilized above a certain number density of such defects. To address this issue, we use the periodic model shown in Fig. 8. In this schematic, the yarn is composed from bundles with a wavy pattern forming pores of size 2L  2h (Fig. 8(a)). In realistic yarns, these bundles form a 3D web and the pores are irregular. The crosssectional view is shown in Fig. 8(c) which represents schematically apparently isolated bundles; this configuration is commonly observed experimentally, e.g. Refs. [9,14,17,19,47]. The pores are kept open by bent bundles such as that shown in red in Fig. 8(b). In the schematic of Fig. 8(b), a repeat unit of the structure in Fig. 8(a) is composed from segments AB, BC and CD. Segments AB and CD are straight and represent bundles containing 2n CNTs. Segment BC is bent and is composed from n CNTs. The number density of such packing defects is controlled by parameter L. The structure stores strain energy in segment BC and in the folded bundle of r CNTs shows in red. We aim to identify the key parameter describing this structure and to evaluate its range for which the yarn is stable. The total energy of a quarter of the structure shown in Fig. 8(b) can be written as:

Fig. 7. Three stages of the evolution of a cell of the cellular mat structure (C)-R during preconditioning. (a) initial mat structure (l ¼ 1), (b) before jump-in-contact (l ¼ 1.8), and (c) after jump-in-contact and yarn formation (l ¼ 2.2). These images suggest that highly bent components of the initial cellular structure are incorporated in the yarn. A similar evolution is observed in unrelaxed yarns (e.g. (C)-UR).

CNT bundles in the mat is shown in Fig. 7(a). As the stretch ratio, l, increases, the cell elongates and the bundles re-organize (Fig. 7(b)). Jump-in-contact occurs when the opposite faces of this collapsing structure touch (Fig. 7(c)). At this stage, the distorted regions at the right and left end of the structure are locked in the bent configuration.

1 ðEIÞBC h2 1 p ðEIÞr ; U ¼  2agn þ 6 þ 2 ðL  2aÞ3 2 2 h

(2)

where the first term represents the adhesion energy associated with merging two sub-bundles of size n into a larger bundle of size 2n along segments AB and CD, the second is the bending energy of segment BC, and the last term is the bending energy stored in the folded element. The adhesion energy per unit length of a bundle of n CNTs is given by Ref. [45]:




pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g0n ðnÞ ¼ g 3n  12n  3 ;

(3)

with g being the adhesion energy per unit length of two CNTs with

Fig. 8. Schematic representation of a yarn composed from CNT bundles forming pores. (a) represents the longitudinal projection, while (c) represents the cross-section in which bundles appear isolated. (b) shows a schematic of one of the pores and surrounding bundles in the longitudinal projection and the parameters used in the model. n and r represent the number of CNTs in the respective bundles.

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A. Sengab, R.C. Picu / Composites Science and Technology xxx (2018) 1e12 Table 1 Parameters characterizing the various yarns considered in this work: mean density of the yarn, the mean number of CNTs per cross-section ; N, the mean adhesion bead energy per bead, Ead , and the coefficient of variation of the distribution of adhesion energy per bead, CVEbead . ad

bead

System

Density (g/cm3)

N

Ead

(A) (B)-UR (B)-R (C)-UR (C)-R (D)

0.608 0.708 0.737 0.81 0.895 0.914

70 70 68 67 72 66

1.6463 1.6192 1.5481 1.5912 1.7063 1.5928

(eV)

CV Ebead ad

0.0762 0.0840 0.0960 0.0582 0.1346 0.0814

parallel axes separated by d0. The difference between the adhesion energy per unit length of a bundle of 2n CNTs and that of two bundles of n CNTs each results:


pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

pffiffiffi

gn ¼ 2g0n ðnÞ  g0n ð2nÞ ¼ 2 12n  3  24n  3 gz2 ng (4) The terms ðEIÞBC and ðEIÞr are the bending rigidities of segment BC (bundle size n) and of the folded element (bundle size r). We assume that the folded element does not buckle when severely bent. Since CNTs are free to slide axially in the bundle during bending, ðEIÞBC ¼ nE0 I0 and ðEIÞr ¼ rE0 I0 . The strain energy in the folded bundle is approximated as the strain energy in a beam bent into a half circle of diameter 2h. The variables of this problem are a and h, and the total system energy is minimized with respect to them. The condition of existence of a stable yarn emerges from the geometric inequalities:

a  L=2; h > 0

(5)

These lead to the restriction:


r 2=3 1 gn L 2 >a n E0 I0 n

(6)

where a is a numerical coefficient, a ¼ 9:3. It is interesting to observe that the left side of inequality (6) is the J parameter of the

gn L bundle of size n, i.e. Jn ¼ ðEIÞ , while the ratio r=n is of the order of 2

BC

1. With Eq. (4), one can also write Jn ¼ E2IgLpffiffinffi a9:3. This indicates 2

0 0

that the stability of the yarn depends most markedly on the density of defects. The inequality provides a lower bound for L, for given n. Yarns of thin bundles (small n) are more likely to be stable at given L and defect density.

7

4.3. Structural characterization of yarns It is of interest to characterize the structure of the yarns produced by preconditioning stretch. Table 1 shows the density of all yarns considered and the average number of CNTs per yarn crosssection, N. The density is evaluated based on the total CNT length in the yarn and the yarn length and mean cross-sectional area. The cross-section area is evaluated by projecting all beads of the model on a plane perpendicular to the yarn axis. The density of projected beads in this plane is then evaluated: it is almost constant in the central part of the projection, where the yarn axis intersects the plane of projection, and shows a gradual decay towards the boundary of the yarn. The perimeter of the yarn is identified by finding the locus of the points in the projection corresponding to the half maximum of the projected density distribution. This mean cross-sectional area is used to compute the mean density and the Cauchy stress. The density is approximately the same in relaxed (R) and unrelaxed (UR) cases, but increases with increasing CNT length. The average number of CNTs per cross-section is approximately identical in all these systems. The yarns have a rather irregular structure, as can be also seen in Fig. 7. Fig. 9 shows 3 cross-sections taken at different locations along one of the yarns. The filled circles represent the points of intersection of CNTs with the respective cross-sectional plane, while the outer circles are added for clarity and represent schematically the outer surface of the yarn. The yarns are composed from sub-bundles (see also Fig. 8(c)), as observed experimentally. The packing within one such subbundle is approximately hexagonal, but the mean distance between CNTs, d, is not always equal to that corresponding to maximum adhesion, d0 . To quantify the effect of packing, we compute the mean adhesion energy per bead (length of CNT segment equal to d0 =4) and compare this quantity with the mean adhesion energy per bead evaluated with a perfect bundle of same number of CNTs. The perfect bundle with 68 CNTs has a mean adhesion energy per bead of 1.909 eV, while the yarns have adhesion energies of about 1.6 eV (Table 1). The coefficient of variation of the adhesion energy distribution in each yarn is on the order of 0.1 (Table 1). We interpret the ~15% difference between the mean adhesion energies of the yarns and perfect bundle by comparing the situation when all n CNTs in the average cross-section are grouped in a single bundle, with the situation when these are split in k sub-bundles of n/k CNTs each. The adhesion energy for a bundle of n CNTs is given kg0 ðn=kÞ

by Eq. (3). The ratio gn0 ðnÞ , for n ¼ 68 (Table 1), decreases by 15% n when k increases from 1 to 4. This indicates that, from an energetic point of view, the simulated yarn packing is equivalent to a set of 4

Fig. 9. Three cross-sections of a yarn ((C)-R). The filled circles represent points where CNTs intersect the plane of the cross-section, while the outer circles are added for clarity and represent schematically the outer surface of the yarn.

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Fig. 10. Probability distribution function (PDF) of bending energy per CNT in yarn (C)R. The distribution is multi-modal, demonstrating the existence of a subset of highly bent CNTs.

Fig. 12. Dependence of the flow stress of yarns measured at 5% strain on the CNT length, L0 . Results for preconditioned yarns obtained by stretching CNT mats are shown together with data for perfect bundles.

5. Mechanical behavior of yarns perfect bundles of 17 CNTs each (Fig. 8(c)). The present analysis provides an indication of the causes leading to, and the degree of, yarn sub-structuring. It is of interest to quantify the density of the highly bent subbundles that stabilize packing defects in yarns (Fig. 7 and the red element shown in Fig. 8(b)). This corresponds to the density of defects discussed in Section 4.2. To this end, we evaluate the probability distribution functions (PDF) of the adhesion, axial and bending energies per CNT in the yarn. The adhesion and axial energies have Gaussian distributions. However, the PDF of the bending energy is multi-modal. This indicates that a majority of CNTs are relatively straight and carry low bending energy, while a smaller fraction are highly bent. Fig. 10 shows the PDF of the bending energy per CNT for system (C)-R. A large peak is observed at small energies (of mean 260 eV/CNT), while two broad peaks of mean ~3 and ~5 times that of the low energy peak develop. These high energy peaks correspond to approximately 20% of the CNTs in the model. It should be observed that the present model does not allow the CNTs to buckle under bending. Bending buckling is expected under these circumstances only in SWCNT. Bending buckling would reduce the stored bending energy and the moment associated with the strongly bent sub-bundles, therefore favoring yarn stabilization.

Fig. 11. Stress-strain curves for all yarns considered in this study.

The yarns produced from buckypaper mats by preconditioning stretch are equilibrated at 300 K under zero traction boundary conditions. Further, their mechanical behavior is tested in uniaxial loading in the direction of the preconditioning stretch, with a strain rate of 0.2 ns1. The results are discussed below in terms of CNT length, packing, test temperature and strain rate. 5.1. Effect of the CNT length Fig. 11 shows the Cauchy stress vs. engineering strain curves for all yarns considered. After a short transient, the yarns deform at essentially constant stress. The flow stress ranges from 30 MPa to 160 MPa, increasing monotonically with increasing L0 . The initial state of the mat ((R) or (UR)), has a negligible effect on the mechanical behavior of the yarns. The discussion of Section 4.3 indicates that the initial state has little effect on the structure of the yarns as well. This is one of the important results of the present study. Fig. 12 shows the flow stress measured at 5% strain versus L0 . The graph quantifies the trend observed in Fig. 11. A similar increasing trend of the flow stress with L0 was reported in Ref. [48], for yarns spun from CVD-grown CNT forests of various thicknesses. In these experiments, the trend reverses when the CNT length becomes larger than a threshold which depends on the yarn fabrication method. In order to clarify the origin of this dependence on L0 in our models, we construct and test perfect bundles of hexagonally packed CNTs, with axes parallel to each other. The CNTs are randomly staggered in the direction of the bundle axis. The bundles are allowed to relax in the radial direction in order to insure proper contacts between CNTs, and are tested under the same conditions as the yarns. The general behavior of these structures is similar to that shown in Fig. 11, in that they flow at essentially constant stress beyond an initial transient. The flow stress of perfect bundles measured at 5% strain is shown in Fig. 12 function of L0 . We observe no L0 dependence in this case. This result is expected since the stress is dominated by adhesive interactions and the adhesive energy of the system is independent of L0 . Furthermore, the flow stress of perfect bundles is more than twice that of the yarns with stochastic microstructure, which points to the importance of packing in the mechanics of these structures. This issue is discussed further in Section 5.2.

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We attribute the L0 dependence of the flow stress in the preconditioned yarns to the tortuosity of the CNTs e an effect similar to that of entanglements in polymeric systems. We observe that the increase of the flow stress is due mainly to the axial component which is negative in the case of small L0 (see also Section 5.3) and increases gradually towards positive values as L0 increases.

5.2. Effect of packing In perfect, close-packed bundles, any CNT which is not at the surface of the bundle has 6 nearest neighbors. Similarly, preconditioned yarns tend to have close-packed structure within subbundles (Fig. 9). However, the average distance between yarn CNTs in the radial direction is different from that corresponding to the perfect hexagonal packing, d0 , and the total adhesion energy is smaller in the preconditioned yarns. To investigate the effect of packing on the flow stress, we construct perfect bundles with perturbed packing. Specifically, we start with a perfect bundle arrangement and select a fraction x of the CNT to which we give random misalignments in the range   0 - 3 . These slightly misaligned CNTs are also shifted radially to arbitrary locations within the bundle. These perturbed structures are allowed to relax during a 300 K equilibration performed with zero boundary tractions. The resulting structures are tested in uniaxial tension. Fig. 13(a) shows the resulting stress-strain curves for the perfect bundle with L0 ¼ 271:2 nm and for several other bundles with increasing percentage of

9

CNTs with perturbed packing, x. We observe that the flow stress decreases monotonically with increasing the degree of packing disorder. The adhesion energy of these sub-optimally packed structures decreases with increasing disorder, x. Perturbing the initial orientation and position of only x ¼ 5% of the CNTs leads to a reduction of the flow stress by ~50% relative to the perfect bundle case. As the fraction increases, the flow stress reduction is more gradual. The reduction tends to level off once the fraction becomes larger than x ¼ 50% (Fig. 13(b)). Interestingly, the flow stress of defective bundles with large x (x > 50%)becomes almost equal to that of the preconditioned yarn of same L0 (yarn (B)-R). This suggests that, in these structures controlled by adhesion, the flow stress of the yarn is more sensitive to the packing of CNTs within sub-bundles than to the fact that the yarn is divided in sub-bundles. This downplays the importance of the packing defects and pores in the overall small-strain deformation mechanics of preconditioned yarns. This conclusion is supported by a number of published results. A similar effect is obtained in the simulations reported in Ref. [34] using perfect bundles in which the strength of adhesion was varied. In Ref. [7], it is observed that improving CNT packing by pressing the yarn into a denser ribbon, leads to a drastic increase of the strength by almost one order of magnitude. Pressing in the radial direction is unlikely to eliminate packing defects associated with heavily bent sub-bundles, but should increase the degree of close-packing within sub-bundles. The effect of densification was also discussed in Ref. [9] in the context of reducing the porosity level of the yarn by passing it through dies of smaller and smaller diameter. The effect of CNT packing on load transfer was studied in Ref. [35] using bundle models in which CNTs have an intrinsic waviness in the relaxed state. A friction force which scales linearly with the contact area was introduced at CNT-CNT contacts. This leads to a strong dependence of the flow stress on the total contact area in the bundle and hence on the degree of waviness. The effect reported here is not due to the introduction of an artificial friction force and emerges naturally from the stochastic packing of CNTs in the preconditioned yarn. 5.3. Effect of temperature and strain rate

Fig. 13. (a) Stress-strain curves for the perfect bundle of L0 ¼ 271.2 nm and for bundles with perturbed packing and of same L0 . Parameter x represents the fraction of CNTs in the perfect bundle with perturbed initial orientation and position in the bundle. The curve for yarn (B)-R is added for reference. (b) Flow stress at 5% strain in bundles with different values of x, from (a).

Given the potential applications of such materials in space structures, it is of interest to investigate the effect of temperature on the stress-strain curve of preconditioned yarns. Fig. 14 shows stress-strain curves for system (B)-R deformed at 100 K and 200 K, with the corresponding curve for 300 K reproduced from Fig. 11. The flow stress increases as the temperature decreases. A similar result was observed experimentally in Ref. [49], where tests performed with ropes of 300 mm in diameter, made from twisted CNT yarns composed from millimeter-long CNTs are reported. The ultimate tensile strength increases from 209 MPa to 252 MPa as the temperature decreases from 298 K to 123 K. The flow stress at 123 K is also about 20% higher than that at room temperature. The increase reported in Fig. 14(a) in the same range of temperatures is ~35%. Fig. 14(b) shows the stress components associated with the variation of the adhesion, axial and bending energies with strain, corresponding to the total stress-strain curves obtained at 100 K and 300 K (Fig. 14(a)). The temperature dependence seen in Fig. 14(a) is mainly associated with the axial contribution. The contributions from adhesion and bending vary slightly with temperature, most likely due to changes in the packing structure of the yarns. This conclusion requires further discussion. In the initial, unloaded state, the stress components balance to zero. The

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entropic and hence thermal in nature [50,51]. The present result belongs to the same class. In yarns stabilized by other types of interactions, such as in presence of cross-links between CNTs or with strong frictional or viscous interactions, the role of excluded volume interactions is expected to be reduced. In these cases, the contracting trend of the yarn caused by the adhesive interactions is balanced by engaging the axial deformation mode via inter-CNT load transfer. In the case of chemically cross-linked yarns, or if the structure is stabilized by Coulomb friction, the effect of temperature on mechanical behavior is expected to be minimal. The temperature effect observed in this model suggests that the behavior is also strain rate dependent. This effect is shown in Fig. 15 for yarn (B)-R which was tested at strain rates equal to half and twice the nominal strain rate of ε_ 0 ¼ 0:2 ns1 considered in the study. We note that no strain rate dependence is observed in the case of the perfect bundle, which implies that the effect observed here is associated with the disordered structure of the yarns produced from buckypaper by stretching. 6. Implications for yarn design The present results have implications for designing yarns of increased strength:

Fig. 14. (a) Stress-strain curves for yarn (B)-R at three temperatures. (b) Axial, bending and adhesion stress components for the same yarn at 100 K and 300 K.

(1) For yarns produced by dry spinning from mats, it is desirable that the mat is stabilized only by adhesion, while CNT-CNT friction is kept to a minimum. The mat should be of small density and composed from thin bundles. This allows the CNTs and the CNT bundles to align, which improves packing before the jump-in-contact that stabilizes the yarn. Once the yarn is formed, inter-CNT interactions (friction, cross-links) are desirable since they increase the flow stress and the strength of the structure. (2) Porosity is detrimental only in that it increases the crosssectional area of the yarn, hence reducing the specific strength. (3) The maximum stress in yarns controlled by adhesive interactions is limited. Substantial increases can be obtained by cross-linking (e.g. via irradiation), which should be applied after yarn formation.

adhesive contribution is always positive as adhesion tends to collapse the yarn into a perfect bundle of length L0 . Hence, in order to fulfill the boundary conditions, the axial contribution has to be negative. This effect was also observed in Ref. [29] for the case of a perfect CNT bundle. The axial contribution is of excluded volume type. Such axial compressive stress components have been reported in a number of works in polymer physics, where it was also shown that the effect is

7. Conclusions

Fig. 15. (a) Stress-strain curves for yarn (B)-R tested at 200 K and at strain rates twice and half of the nominal strain rate of ε_ 0 ¼ 0:2 ns1 considered in the study.

The mechanical behavior of yarns produced by dry stretching buckypaper mats is studied in this work. These structures are stabilized by adhesive interactions between CNTs and contain packing defects reminiscent of the structure of the initial mat. The yarns deform at approximately constant stress, beyond an initial stress ramp-up. The flow stress of yarns produced from relaxed and unrelaxed buckypaper is approximately identical. The flow stress of yarns is about half of the flow stress of the perfect bundle of same number of closely packed and perfectly aligned CNTs. This indicates that packing is an essential aspect of the mechanics of these adhesion-stabilized structures. Increasing the CNT length leads to an increase of the flow stress of the yarns, but does not affect the flow stress of the perfect bundles without packing defects. Therefore, the dependence of the flow stress on the CNT length is attributed to CNT tortuosity in the yarns, similar to the effect of entanglements in polymers. We also observe that decreasing the temperature or increasing the strain rate lead to an increase of the flow stress. These observations have implications for the design of adhesion-stabilized yarns.

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