Buckling and fracture characterization of pristine bundles of vertically aligned carbon nanotubes using quantitative in situ TEM axial compression

This work investigates the mechanical deformation and fracture characteristics of pristine bundles of vertically aligned multi-walled carbon nanotubes (MWCNTs) subjected to axial compression in situ transmission electron microscope (TEM). Accurate measurements of force-displacement data were collected simultaneously with real-time TEM videos of the deformation process. Two distinct regimes were observed in the force-displacement curve: (1) an initial elastic section with a linear slope, followed by (2) a transition to a force plateau at a critical buckling force. Morphological data revealed coordinated buckling of the pristine bundle, indicating strong van der Waals (VdW) forces between the nanotubes. The experimental setup measured an effective modulus of 83.9 GPa for an MWCNT bundle, which was in agreement with finite element analysis (FEA) simulations. FEA also highlighted the significant role of VdW forces in the bundle mechanical reactions. Furthermore, we identified nickel nanoparticles as key players in the fracture behavior of the bundles, acting as nucleation sites for defects. The direct mechanical measurements of MWCNT bundles provide valuable insights into their mechanical deformation and fracture behavior, while correlating it to the morphology of the bundle. Understanding these interactions at the bundle level is crucial for improving the reliability and durability of VACNTs-based components.

Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.used to create anisotropic films, useful for applications that require directional properties, such as nanocomposite materials [6][7][8], nanoelectromechanical systems [9][10][11][12], energy storage [13][14][15], and sensors [16][17][18][19].
To effectively harness VACNT's exceptional mechanical properties and to determine their lifetime and durability, it is essential to study their mechanical deformation under mechanical stress.A VACNT forest is characterized by a hierarchical structure of nanotubes at different length scales, including individual nanotubes, bundles, and the overall forest.Consequently, the mechanical response of VACNT forests is influenced by various factors, such as the dimensions, density, and orientation of the individual tubes, as well as the characteristics of the CNT bundles.For example, forests consisting of multi-walled CNTs (MWCNTs) with larger diameters have been found to be stiffer and more capable of recovering their original shape after compression [20].This is due to the good ability of MWCNTs to form new interactions between tubes via van der Waals (VDW) forces, which allows them to recover more readily.Additionally, MWCNTs buckle under compressive deformation, exhibiting localized ripples along the tubes' outer wall, rather than the irreversible kink-bends of singlewalled CNTs (SWCNTs) [21][22][23][24].This bent configuration of the MWCNT results in a strong repulsion between its walls that acts as a restoring force, straightening them back.The surface roughness of CNTs [25], the presence of defects [26,27], and the density of CNT-CNT contacts [28,29] also influence the mechanical response of CNT forests.
An effective method for assesing the nonlinear response of CNTs against compression is to conduct quantitative mechanical testing using in situ transmission electron microscope (TEM).This method combines nanoscale image analysis with quantification of the entire process through force-displacement curves.Recent studies examined buckling and fracture modes of pristine MWCNTs within the TEM, employing an AFM apparatus [21,[48][49][50] or a piezoelectrically driven nanoindenter [24,51,52].These investigations have correlated load-displacement (F-D) data to the onset of buckling and rippling, unveiling insights into the compression-induced instabilities of CNTs.Notably, experiments have highlighted that CNTs exhibit reversible deformation in response to repeated compression [24,48,53,54] and that the buckling mode is dependent on their aspect ratio [21,50].Furthermore, Young's modulus values vary widely (0.2-1.075 TPa [21,24,27,[48][49][50][53][54][55]) among MWCNT types due to factors such as geometry (e.g.diameter, length, alignment), crystalline structure, and concentration of defects [27,55], which are influenced by the growth method.For instance, the arc-discharge method yields highly crystalline CNTs with a modulus of 1 TPa, while while chemical vapor deposition (CVD) grown CNTs typically range from 0.1 to 0.4 TPa [21].Additionally, bamboo-like CNTs (bCNTs), known for their hollow compartments and bamboo knots along the axis,as well as high concetration of defects, exhibit a modulus below 0.2 TPa [24,49] and can be produced via various synthesis methods [56].
To date, studies of VACNT networks under compressive deformation were primarily conducted on macroscopic forests [34,39,[42][43][44][45][46][47] or by performing mesoscopic simulations [57][58][59].However, mechanical and morphological analyses of isolated bundles have not yet been successfully performed due to the experimental challenges in sample preperation and measurement, and thus the failure mechanisms of VACNT bundles under compression remain unclear.Understanding the relationship between the bundle's microstructure and its mechanical properties is notably intricated, due to the structural variations of the nanotubes between different samples.
Herein, we present the first attempt to use in situ TEM to study VACNT bundles, focusing on sequential compression loadings for the analysis of the buckling and fracture processes in vertically aligned bundles of MWCNTs.As buckling is the predominant mode of deformation of VACNTs under compression, it significantly affects their structural stability.Therefore, we analyse the collective buckling of VACNT bundles under compression, correlating the morphological and structural distortions that lead to failure, including the role of the metal catalyst and the CNT-CNT interactions.Our findings could help predict the behavior of VACNT bundles under compression, thus guiding the design and optimization of CNT structures to yield improved mechanical performance and reliability for structural and electromechanical applications.

Materials and methods
Vertically aligned MWCNTs were grown by plasma enhanced chemical vapor deposition (PECVD) on a silicon substrate with a long and tall wedge geometry, specially designed for TEM viewing [60].The VACNTs were grown by DC/RF PECVD (Black Magic 2, Aixtron, Germany), at 700 °C using a nickel catalyst (3 nm) and C 2 H 2 :NH 3 20:80 sccm feedstock, for 1 h [24].In this reaction, acetylene is utilized as a carbon source, and ammonia as a hydrogen-rich reducing agent [61].The PECVD growth method produces highly straight and aligned nanotubes, which is crucial for measuring accurately the mechanical properties.The fabrication process does not require additional micromachining processes such as focus ion beam milling, which causes gallium ion irradiation damage [62,63].The substrate was then mounted onto the TEM holder, perpendicular to the electron beam (figure 1(a)).
The in situ nanomechanical setup involves compressing bundles of MWCNTs with a flat nanoindenter while simultaneously viewing the deformation process in a TEM (figure 1).The experiments were conducted using an FEI Tecnai 20 TEM (Hillsboro, OR, USA) equipped with a Bruker (Hysitron, Minneapolis, MN, USA) PicoIndenter 95 (PI-95) TEM holder.The TEM was operated in bright-field mode at 200 keV with a field-emission electron source.The bundles were viewed at medium magnification to ensure complete visibility and to protect against radiation and knockout damage [64,65].The PicoIndenter utilizes a 3D piezoelectric drive to precisely position the nanoindenter perpendicular to the bundles inside the TEM (figure 1(b)).The bundles were not completely straight due to the intrinsic waviness of the tubes, but they occupied the same plane as the loading axis as evidenced by them both occupying the same focal plane.
The mechanical properties of two bundles of MWCNTs were studied by applying force along their axis using a displacement-controlled mode at a rate of 5 nm s −1 .The deformation process was recorded using 200 data points per second of force and normal displacement, and digital capture of the Gatan One View camera (Gatan Inc., Pleasanton, CA, USA) at 10 frames per second and 4K resolution videos.The TriboScan software's (Hysitron, Minneapolis, MN, USA) frame grabber feature was used to record and synchronize the load-displacement data and the real-time video.The synchronized imaging was used to validate the measured displacement.Seven sequential deformation cycles were conducted, compressing the two bundles simultaneously, each one was carried out until one of the bundles was fractured.Then, post-fracture TEM images were taken at higher magnifications for failure analysis.The collective buckling characteristics of the bundles were assessed, and the critical force for buckling (P cr ) was determined from the F-D curves by calculating the crossing point between two different slopes of the pre-and post-buckling regions linear fits, using OriginLab software (Origin Pro 2016, OriginLab Corporation, Northampton, MA, USA).The pre-buckling region was identified based on the linear rise in force and the postbuckling region was identified by a decrease in the slope, indicating a softening transition.A stress-strain curve of cycle 7 was calculated to extract the effective modulus of bundle 2, through a linear fit of the elastic regime.
The CNT bundle was modelled using finite element analysis (FEA, Abaqus 2020, Johnston, RI, USA) to validate the experimental results and study the evolution of stress in the sample.The model was based on the dimensions of bundle 2 at cycle 7, where the dimensions were more uniform and therefore better suited to computer analysis.The bundle was modelled as a single column, four columns, and 16 columns, to investigate the effect of the attachment of the nanotubes to each other due to VDW forces.The finite element mesh consisted of 16 720 linear hexahedral elements of type C3D8R and 21 220 nodal points.The columns were attached via Tie connections, plane stress conditions were assumed, and buckling analysis was performed to extract the P cr .The Young's modulus of the bundle was set according to a calculated effective modulus of cycle 7 (83.9GPa) and the Poisson's ratio was set to 0.3.The methodology was validated by extracting the P cr from the simulations and comparing it to the experimental results.

Results and discussion
Compression experiments were performed on two bundles as shown in figure 1(b).The lengths of bundles 1 and 2 were 2640 and 2760 nm, respectively.The number of tubes in each bundle was estimated by counting the catalyst nanoparticles (NPs), which appear as small black dots in the TEM image of figure 1(b).Accordingly, bundle 1 contained ∼37 tubes, and bundle 2 contained ∼61 tubes.It is important to note that the catalyst NPs are not of uniform size, resulting in tubes with a wide range of diameters.The tubes' lengths were not uniform either, making the bundles wider at the base and narrower at the top; bundle 1 ranged from 330 nm at the base to 150 nm at the tip, and bundle 2 ranged from 490 nm at the base to 170 nm at the tip.This non-uniformity in the tubes' structure is typical for the CVD growth method.Figures 1(c) and (d) reveal a tubular structure with bamboo-like compartments and nickel NPs at the top, encapsulated by the graphitic multilayer.The interwall spacing is 0.34 nm confirming the CNT structure [1] and the presence of the catalyst NP at the top of the tube suggests tip growth [56].The nanotubes in each bundle are strongly attached by VDW interactions, which influence their mechanical behavior.
Through the in situ TEM experiments, we observed the axial buckling process of two separate bundles of MWCNTs compressed simultaneously, and investigated the fracture process throughout seven cycles of compression.The load versus displacement (F-D) curve of a representative loading cycle (cycle number 2) is presented in figure 2(a), highlighting key points in the mechanical response.Figure 2(b) shows their corresponding TEM images.Since the bundles were not of the same length, the diamond tip of the indenter first approached the longer bundle 1, and only later did it touch bundle 2 as well.At the beginning of loading, a single longer tube pointing out of the bundle was compressed, before the indenter reached the bulk of the bundle at 180 nm displacement (figure 2(a-I)).We then see a linear increase of the force with the displacement, indicating the elastic region.When the critical load for buckling (P cr ) of the bundle is reached, the linear rise in force transitions to a force plateau (figure 2(a), buckling onset in the F-D curve), yet does not drop sharply, as often seen in the buckling behavior of VACNT forests [31,34,38,40,66].At that point, the bundle fails by buckling and starts to deflect laterally (bend) towards its right side, as can be seen in the TEM stills (figure 2(b-II)).The P cr of the left bundle is estimated from the graph to be 6.6 μN.Based on a circular cross-section of 210 nm diameter at the contact plane, the critical stress is 190 MPa.In the postbuckling regime, the F-D curve exhibits a reduced stiffness (decreased slope) because of lower rigidity in bending [21,67,68], and the system deviates from its' ideal linearly elastic response of the pre-buckling regime.At point II in the F-D curve (figure 2(a)), the indenter reaches the second bundle (bundle 2) and starts to compress both bundles simultaneously (it should be noted that bundle 2 does not react before that point, hence we consider these two bundles separated).As a result, a large increase in the force and stiffness is observed, until bundle 1 fractures at point III (figure 2(a)).The TEM image in figure 2(a)-III shows bundle 1, a few frames prior to the fracture, and points to the location where it was initiated.The force then increases sharply as a result of the compression of bundle 2, which consists of almost twice as many tubes as bundle 1.At point IV in the F-D curve (figure 2(a)), the indenter reaches its predetermined maximum displacement and begins to unload.The TEM image in figure 2(b)-IV shows the resulting fracture of bundle 1 and the buckling-bend of bundle 2. The deformation mechanism of CNT bundles is defined by their structural characteristics.The bundles presented herein consist of slightly bent CNTs, equilibrated by contact VDW forces between the tubes [69] and exhibiting good overall axial orientation.Consequently, each bundle tends to buckle near the indenter head in a coordinated fashion [33].Moreover, the nanotubes are constrained to the substrate at the bottom, which limits their rearrangement under compression and activates the following deformation mode of bundle 1.As the indenter presses down on the bundle, the nanotubes reorganize themselves to fit into a smaller volume.In response to increasing loads, the CNTs near the indenter reorient uniformly in one lateral direction, as seen in figure 2(b)-II.The bundle bends outwards and the localized bending stress leads to the collective buckling of the nanotubes.As the compression process progresses, CNTs within the bundle become increasingly bent and packed, preventing them from rearranging themselves to reduce stress.Finally, under a load of 15.2 μN (figure 2(a)-III), bending moments produced tensile stresses at the right surface of the bundle.This initiated a single crack, which then propagated swiftly across the bundle.The brittle, rapid failure was accompanied by an abrupt drop in force preceding the crack (figure 2(a)-III).It is worth emphasizing that since the tubes were not perfectly straight in the beginning of the experiment, the mechanical testing was not entirely uniaxial, but rather included beneding moments which diminished the tubes' load-carrying capacity, leading to a reduction in the critical force.
Furthermore, the PECVD synthesis method produces bCNTs that exhibit a high density of defects.These defects not only diminishes the Young's modulus of the pristine tube (measured at 0.2 TPa [24,49]) but also serve as stress concentration points, making the bundles more susceptible to deformation and reducing their overall mechanical strength.It is expacted that MWCNT with a greater degree of crystallinity would be less compliant to compression while demonstarting superior mechanical strength.It is noteworthy that the bundles in this study did not exhibit periodic buckling, a phenomenon that has been observed in other studies that have examined the compression of macroscopic VACNT forests [30-32, 34, 35].Periodic buckling is characterized as a series of buckling instabilities that extend across the entire thickness of the sample and produce an accordion-like wavy pattern.However, since the CNTs in this study are significantly shorter than the macroscopic forests previously examined, repeated coordinated buckling cannot occur.
The direct observation of the fracture process through the TEM provides insight into the role of the catalyst in the fracture process as well.High-magnification TEM images of the post-test fractured bundles were used for post-failure analysis (figure 3). Figure 3(a) presents the two bundles prior to the deformation, with the location of the fractures and the trajectory of crack propagation indicated by blue dashed arrows.Post-fracture TEM images of the seven cycles are presented in the insets in figures 3(b)-(g).Even though the failure occurred too quickly to be captured in real-time by the camera, a frame-by-frame analysis provided clues for the crack opening.The deformation process video (see video SV1) and the images in figure 3(a) demonstrate crack initiation and opening at the location of the maximum tensile stress, which corresponds to mode I fracture.The crack then swiftly propagates across the bundle towards the compressive side, as indicated by the blue arrows.The direction of the arrows indicates the direction of fracture propagation and depends on the existing curvature of the bundle at each cycle.For example, in the experiment of cycle 2 (figure 2), the fracture evolved at the right side of the bundle where tensile stresses were concentrated and progressed towards the compressed side on the left.
The rapid propagation of the crack and the flat fractures' surface (figures 3(b)-(g)) indicate a brittle fracture.Additionally, catalyst NPs are present in each fracture surface, marked by yellow arrows in figures 3(b)-(g).We suggest that the nickel NPs act as a site for the initiation of defects or promote weakness at the interfaces between individual CNTs in the bundle.It seems that the bundle deforms under compression as a uniform body, due to the strong VDW forces between the tubes and the consequent coordinated buckling.Under this assumption (later examined through simulations), the catalyst NPs can be perceived as inclusions or holes inside the bundle, with stress accumulating around them.The increased stress at the interface between CNTs initiates defects, and thus facilitates the fracture of the bundle.Additionally, it can be observed that catalyst NPs on the surface of fracture 2 (figure 3(d), marked by red arrows) were detached from the tube structure.The nickel NP is initially encapsulated by the CNT, as shown in figure 1(d).The Ni atoms tend to form covalent bonds with C atoms, which are weaker than the regular C-C bonds of the nanotubes and result in longer bond lengths [70][71][72].At the proximity of the Ni NP, the C-C bond strength is weakened due to stretching (longer bond length) and bending (the slant between the two bonds is altered) of the covalent bonds between the catalyst and the nanotube [71].It has been shown as well that the formation of Ni-C bonds around the NP reduces the pi-bonds ratio in the CNT [70].This reduction can be attributed to the rehybridization of C atoms caused by defects in the sp 2 system and possibly by the formation of new bonds with nickel.Consequently, the tube is more prone to fail under lower stress at the location of the NP.
To validate the experimental methodology, we performed FEA simulations.To ensure high accuracy, only bundle 2 from cycle 7 was considered for modeling, possessing a uniform diameter of 400 nm (figure 4(b)) and loading of only one bundle, in the linear elastic regime.We calculated the stress-strain curve (figure S1) and determined the effective modulus (E eff ) of the bundle by fitting a linear regression to the elastic region.The E eff value was calculated to be 83.9GPa, which is in the upper range of previously reported values [34,39,[42][43][44][45][46][47], and was used in the FEA simulations.It is worth noting that due to the low aspect ratio of the bundle, column failure through pure compression is more likely than column buckling.Nevertheless, the bundle is composed of slender CNTs that do undergo coordinated buckling under compression, which is why buckling is observed for the bundle in both the graph (figure 4(a)) and TEM imaging (figure 4(c)).Additionally, calculating the intrinsic Young's modulus of the CNTs is not feasible due to the complexity of the structure.
The bundle can be modeled via two approaches.The first one estimates that the bundle behaves as a uniform body and is therefore modeled as a single column (figure 4(d)).The second approach divides the single column into smaller units -4 and 9 columns (figures 4(e) and (f), respectively) while maintaining the cross-sectional area.To evaluate the contribution of VDW forces to the overall stiffness, we compared the two conditions-with column attachments (accounting for VDW) and without (no VDW).
The simulation applied compressive force and mapped the deformations throughout the column and calculated the P cr .The simulation results showed that the maximum surface stress concentrates around the indenter head, which is consistent with our experimental observations of buckling and subsequent fractures.The P cr values were compared to the experimental results (figure 4(a)), to validate the experimental methodology and confirm the calculation of the E eff .Figure 4  connected and un-connected conditions are shown in figure 4(d) (1 column), 4e (4 columns), and 4f (9 columns).The simulated P cr values of the single column, as well as the connected 4 and 9 columns, yielded a good fit to the experimental results, with 187, 218 μN, and 197 μN, respectively.It should be noted that due to the columns' attachments, which add spatial constraints and produce more force, the 4 and 9 connected columns yielded higher force than the single column and closer to the measured value.
Overall, the simulations provided reliable coarse estimation for the critical buckling force, but there are a few limitations that introduce some degree of error.Firstly, the simulations employ the effective modulus of the bundle rather than the actual modulus of the CNTs, which can lead to some deviations when modeling the bundle as separate columns.Secondly, the modeled geometry is not hollow, as in the case of CNTs.This affects the calculation of the moment of inertia since the cross-section is that of a full circle rather than a hollow one.As a result, geometry factors, such as aspect ratio, have a greater impact on the computed critical force.This is clearly seen in figures 4(e) and (f), where the 4 columns simulations, with lower aspect ratio, result in a higher computed force.Lastly, more error is introduced through the boundary conditions of 'tie connections' since they only simulate the VdW forces between the nanotubes, making the results only partially accurate.However, despite this limitation, the simulation still yields a good understanding of the VdW role in the bundle behavior, as the estimated force significantly undershoots when the 'tie' boundary conditions is omitted.
Furthermore, the attachment of the columns to each other has a significant effect on P cr , as the values drop significantly without attachment (figure 4(a), dotted lines).Similar to VDW forces, the attachments in the simulation add spatial constraints to the bundle during compression, increasing the force needed to initiate buckling.Additionally, the connections between the nanotubes improve the stress distribution under load.Based on this, we can conclude that there is good agreement between the experimental results and the FEA simulations, proving that VDW forces play a significant role in determining the strength of a bundle.The good fit of the P cr values confirms that the effective modulus used in the simulations is correct.Noteworthy, the E eff value of 83.9 GPa is relatively high in comparison to values reported in the literature for VACNTs forests [34,39,[42][43][44][45][46][47].However, this outcome is expected since we measured the forces of a single bundle directly, while the studies in the literature conducted measurements on microscopic forests or foams, which mainly contain air between nanotubes.In these studies, the actual cross-area between the indenter and nanotubes is therefore significantly smaller than the one used in the calculations, resulting in lower values of modulus.Our direct measurement is thus probably closer to the actual value.

Conclusions
To summarize, the buckling behavior and the fracture evolution of pristine MWCNT bundles were investigated utilizing in situ TEM compression test.The in situ TEM demonstrated the deformation mechanism of the bundles.The F-D curves and morphological images were used to characterize the buckling behavior.VDW forces cause the nanotubes in the bundles to buckle in a coordinated manner.The deformation process of the bundle demonstrates a brittle fracture and crack initiation at the location of the maximum tensile stress, which spreads quickly across the bundle toward the compressive side.Furthermore, the fracture evolution was revealed through morphological analysis.According to our findings, Ni NPs act as a site for the initiation of defects or promote weakness at the interfaces between individual CNTs.Catalyst NPs resemble inclusions or holes inside the bundle, causing an increase in stress around them, and facilitating bundle fracture.The P cr results of FEA simulations were in accordance with the experimental results for a single column, four columns, and nine columns (ranging between 187 and 238 μN), validating the experimental approach.The accuracy of the simulation also supports the assumption that a CNT bundle behaves as a uniform body under axial compression.The effective modulus of the bundle at cycle 7 was estimated to be 83.9GPa.Through in situ TEM, structure-property relationships can be studied to effectively control CNT properties.The acquired knowledge and control over the synthesis process enable the production of certain types of CNTs with desirable properties.Our study exemplifies that large catalyst particles reduce the mechanical durability of MWCNT bundles and may facilitate fracture.In this case, it may be advantageous to reduce the catalyst diameter or to implement another method of removing the NP from the tubes' ends.
Future work may include comparative studies between different types of CNT structures, controlled by the synthesis method.The synthesis method should also be adjusted to yield more uniform tubes and hence bundles (e.g.tubes' diameter, length, width, and also number of tubes in each bundle), wich will improve the accuracy of the mechanical measurement.It would be also intersting to investigate how the mechanical behavior in the bundle level affects the mechanical behavior at the macroscopic level (forest).Such study may be possible through a combination of in situ deformation experiemnts of isolated VACNT bundles within TEM and VACNT forests within scanning electron microscope (SEM).
Jerusalem, Israel, and by the United States National Science Foundation (NSF).

Figure 1 .
Figure 1.(a) Diagram illustrating the experimental setup for in situ TEM nanoindentation; (b) a TEM image showing the initial position of the diamond tip and two MWCNTs bundles in a compression experiment; (c) a TEM image showing a magnified view of the end of the bundle, showing nanotubes of different diameters and large nickel nanoparticles; (d) CNT walls encapsulating a nickel nanoparticle with a diameter of 46 nm at higher magnification.These nickel nanoparticles play an important role in the failure mechanism of the nanotubes.

Figure 2 .
Figure 2. (a) Load-displacement curve of a representative compression cycle 2 under axial loading and relaxation.The graph presents four stages in the compression of the two bundles: (I) the beginning of compression of bundle 1, characterized by a linear rise in force; (II) the initial touch of bundle 2, which results in a steep rise in force as the two bundles are compressed simultaneously; (III) failure of bundle 1, seen as a large drop in force; (IV) compression of only bundle 2 up to a maximum displacement of 500 nm; (b) the corresponding TEM images of the four stages marked in the F-D curve: (I) the two bundles facing the indenter at the start of axial compression; (II) the bucklingbend of bundle 1.Note that the bundle bends towards the direction of the initial curvature of the bundle (right) (III) the two bundles are compressed simultaneously.The point of fracture of bundle 1 is marked by the red arrow.(IV).The resulting fracture is marked by a red circle.The buckling-bend of bundle 2 is marked by an arrow.TEM images were gathered in bright-field mode at 200 kV.
(a) shows the F-D curve of cycle 7, with point (b) corresponding to the starting position (figure 4(b)) and point (c) corresponding to a few frames before fracture (figure 4(c)).The F-D curve (figure 4(a)) exhibits a linear region up to a displacement of ∼35 nm, transitions to a force plateau, and finally fails at point (c).The calculation of the experimental P cr provided an estimation of 238 ± 12 μN (as marked with an 'X' in figure 4(a)).The modeled P cr of the three FEA simulations, under column attachment conditions and without, are marked by the dashed lines for comparison (figure 4(a)).The P cr values of the FEA simulations for

Figure 3 .
Figure 3. Fracture evolution of the two bundles (a) a TEM image of the two MWCNT bundles prior to deformation.The location of the fractures and the trajectory of crack propagation are indicated by the blue dashed arrows; (b)-(g) high-magnification TEM images of the posttest fractured bundles: (b) corresponds to fracture 6, (c) corresponds to fracture 4, (d) corresponds to fracture 2. Red arrows indicate Ni NP vacancy or detachment.(e) corresponds to fracture 5, (f) corresponds to fracture 3, and (g) corresponds to fracture 1. Yellow arrows indicate Ni NP at the fracture surface.

Figure 4 .
Figure 4. (a) F-D curve of cycle 7.Point (b) corresponds to the starting position of the experiment, as shown in image (b).Point (c) corresponds to a few frames before failure, as shown in image (c).The experimental P cr value is marked in 'X'.The calculated P cr values from the FEA simulations are marked by five lines-dashed lines denote the connected columns conditions and the dotted lines denote the unconnected columns conditions.The blue lines correspond to 4 columns, green lines correspond to 9 connected columns, and the red line corresponds to 1 column; (b) a TEM image of the bundle at the start of the experiment, showing its initial dimensions; (c) a TEM image of the bundle, few frames before failure.The tensile and compressive forces are illustrated by the red arrows; FEA simulations of columns under compression, showing deformation contours and P cr : (d) a single column; (e) 4 columns; (e) 9 columns.('con' stands for connected-i.e.constrained by VDW forces, and 'un-con' stands for un-connected).