Determination of tensile strength distribution of nanotubes from testing ...

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COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 2937–2942 www.elsevier.com/locate/compscitech

Determination of tensile strength distribution of nanotubes from testing of nanotube bundles T. Xiao

a,*

, Y. Ren b, K. Liao

c,* ,

P. Wu a, F. Li d, H.M. Cheng

d

a

d

Department of Physics and Institute of Advanced Materials, Shantou University, Shantou 515063, People’s Republic of China b School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, Singapore c School of Chemical and Biomedical Engineering, Nanyang Technological University, Singapore 639798, Singapore Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang 110016, People’s Republic of China Received 19 September 2007; accepted 15 October 2007 Available online 22 October 2007

Abstract Single-walled carbon nanotube (SWNT) bundles are subject to tensile testing using a self-developed nano-mechanical testing device. The tensile strength of SWNT bundles, as well as reported data of multi-walled carbon nanotubes (MWNTs) and SWNT ropes, are studied by statistical approach. It is shown that the tensile strength distributions of carbon nanotubes (CNTs) or SWNT ropes can be adequately described by a two-parameter Weibull model. Considering the nonlinear elastic behavior of CNTs, a relation between individual CNT and CNT bundle strengths is studied, and a method for determining the tensile strength distribution of individual CNTs or CNT sub-bundles from experimental measurements on CNT bundles is proposed. Therefore the process for determining the strength distributions of individual CNTs can be greatly simplified by testing CNT bundles instead. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: A. Carbon nanotubes; B. Tensile strength; B. Stress–strain curves; B. Non-linear behavior

1. Introduction Carbon nanotubes (CNTs) are potential candidates for individual super load-bearing materials or reinforcements in advanced nanocomposites. To date, long, aligned single-walled carbon nanotube (SWNT) bundles can be synthesized by catalytic decomposition of hydrocarbons in the form of ropes or ribbons [1]. Contrasting to the belief that CNTs are made of perfect hexagonal carbon lattice structure, it has been shown experimentally that as-synthesized CNTs may contain numerous defects on their walls [2], and even after annealing by graphitization at high temperatures [3]. These defects result in local strain-energy concentration or decrease of atomic binding energy. As is true for many structural materials, it is reasonable to *

Corresponding authors. E-mail address: [email protected] (T. Xiao).

0266-3538/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compscitech.2007.10.004

assume that the tensile strength of CNTs is also controlled by defects and should be described on the basis of probability approach. By examining three nanotube data sets (two for multi-walled carbon nanotubes (MWNTs) and one for multi-walled WS2 nanotubes), Wagner and coworkers have shown that all data sets can be accurately fitted by Weibull statistics and thus the model seems to be applicable at the nanometer as well as at the micrometer and macroscopic scale [4]. In this paper, we first present results from tensile testing of SWNT bundles using a nano-mechanical testing device [5]. The data obtained was analyzed with Weibull statistics, and compared with data previously reported by Barber et al. for individual MWNTs [4], and by Yu et al. for individual MWNTs [6] and for individual SWNT ropes [7]. Having examined the tensile strength distribution of individual CNTs, a model for tensile strength of CNT bundles is developed. Chi et al. have proposed two approaches for

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T. Xiao et al. / Composites Science and Technology 68 (2008) 2937–2942

determining single fiber strength distribution from measurements on fiber bundles, based upon an analysis of the force–strain relation of fiber bundles [8]. Following the idea, the relation between the tensile strength distribution of individual CNTs and the force–strain relation of CNT bundles is studied. In the present study, however, the nonlinear stress–strain behavior of individual CNTs at large strain is incorporated, and it is shown that the nonlinear effect has rendered the force–strain relation of CNT bundles different from the linear case for fiber bundles [9]. Using the model, the tensile strength distribution of individual CNTs can be estimated from experimental measurements on CNT bundles. As strength measurement of individual CNTs is still a great challenge to date, the process for determining the strength distributions of individual CNTs can be greatly simplified by testing CNT bundles instead, and by analyzing the results using the proposed model. 2. Tensile testing of SWNT bundles All SWNT samples used in this study were synthesized by catalytic decomposition of hydrocarbon described elsewhere [10]. Long (several tens of centimeters), aligned SWNT ropes with diameter of about 100 lm were obtained. These SWNT ropes were carefully separated manually into several thinner, 3 mm-long SWNT bundles for testing. The diameters of the SWNT bundles, determined by electron microscopy, were in the range of 15– 25 lm. A multi-purpose nano-mechanical testing device (NMTD) for testing micro/nano structures, shown schematically in Fig. 1, is developed in our group. The hardware of the device consists of three main components: the motion control unit, the force acquisition unit, and the side-view image acquisition and analysis unit. The software system is an interactive software shell based on the Lab-

Fig. 1. Schematics of the nano-mechanical testing device: (A) loading tip, (B) force transducer, (C) displacement motor, (D) data acquisition and control unit, (E) lens/CCD camera, (F) rigid stage, (G) epoxy adhesive, (H) carbon nanotube bundle.

VIEWÒ platform. Equipped with a video enhanced sideview image monitoring system, the NMTD is able to continuously capture, display, and store images during testing. The force and displacement resolution are 0.2 lN and 1 nm, respectively. Each SWNT bundle was mounted vertically between a rigid stage surface and a replaceable lead rod that was firmly attached to the force transducer of the NMTD (Fig. 2). Epoxy adhesive was used to fix the SWNT bundle at both ends, between the rigid stage and the tip of the lead rod. Displacement, force, as well as images of a SWNT bundle were recorded simultaneously during loading. A pull test was performed on the lead rod prior to testing of SWNT bundle to ensure that it was firmly attached to the transducer. No slipping of the lead rod was found upon loading to full-scale force. The strain rate was controlled at 0.001 s1. No slipping of SWNT bundle from the mounting grip was observed from the video images recorded. A total of 12 SWNT bundles were tested. All of the samples were broken in the region between the two mounting ends, typical images of a SWNT bundle before and after tensile testing are shown in Fig. 2. An electron microscopy image of the fracture surface is shown in Fig. 3. It is seen that fracture occurred within a region of about 30 lm, without pullout of long threads of SWNT sub-bundles, which rules out the possibility of separation due to slipping and pullout. Taking into account the 65% volume fraction of SWNT in the bundle [5], the tensile strengths of SWNT bundles were estimated in the range of 10–52 GPa (mean 23 GPa). The stress–strain curves of four SWNT bundles tested are shown in Fig. 4, here the stress on a bundle is calculated based on the original cross-sectional areas. Worth noticing are the numerous kinks, big and small, that appear on the stress–strain curves. These kinks or load drops, presented in all of the

Fig. 2. A mounted SWNT bundle, in which the upper end is attached to a lead rod and the lower end is fixed on a rigid surface, is subjected to tensile loading: before breaking (left), and moments after broken into two parts (right).


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3. Strength distributions of individual CNTs and CNT ropes As CNTs are essentially one-dimensional macromolecules with identical periodic structures that are sensitive to structural defects, presumably, the strength distribution of individual CNTs follows the Weibull distribution. The probability of failure for a CNT at an applied stress, r, is m r ð1Þ P f ¼ 1 exp r0

Fig. 4. Stress–strain curves of four SWNT bundles. Each kink on these curves represents one or more sub-bundles in the bundle have failed while the survived sub-bundles bear the remaining load until final rupture of the sample. Smaller kinks may not be resolvable in the figure.

samples tested, are indicative of sub-bundle failures. Small kinks are not resolvable at the scale of Fig. 4 presented here. Only the big kinks, obvious from the figure, that span from low to high load (1.92–54.6 GPa), were analyzed. The as-prepared SWNT ropes consist primarily of many bundles of SWNT with diameter of 10–40 nm [10], and it is estimated that each of the SWNT bundles prepared for the tensile test contain from a few hundred to about a thousand sub-bundles. We assume that there is no interaction between these SWNT sub-bundles, and they are stressed equally during loading. This assumption is supported by the evidence that large gaps exist between these sub-bundles, from electron microscopy examinations. The applied tensile load on a SWNT bundle is evenly redistributed among the remaining intact sub-bundles after failure of one or more sub-bundles at a specific load, until final failure of the entire SWNT bundle.

2

n = 114

LN(-LN(1-Pf ))

Fig. 3. A scanning electron microscopy image of the fracture surface of a SWNT bundle.

where r0 the characteristic strength, and m the Weibull modulus which controls the dispersion of the distribution. The tensile failure of the SWNT sub-bundles is represented by kink points (load drop) on the stress–strain curve, such as those seen on Fig. 4. Based on the kink points the failure stress of the sub-bundle, rf, is calculated. The probability of failure, Pf, at a applied stress, rf, is then estimated by the median rank method, Pf = (i 0.3)/ (n + 0.4), where i is the rank of tensile strength value in ascending order and n is the total number of samples. In our case, n is 114, the total number of kinks collected from stress–strain curves of 12 SWNT bundles tested. Taking double logarithm on both sides of Eq. (1), m is obtained from linear regression of the strength data. It is shown in Fig. 5 that our strength data fit the regression line fairly well, demonstrating that Weibull model is applicable for SWNT sub-bundles. Results of Weibull statistics for our case are tabulated in Table 1. Furthermore, three reported experimental data sets, including Barber’s data for 26 individual MWNTs produced by chemical vapor deposition (CVD) method [4], Yu’s data for 19 individual MWNTs and 15 individual SWNT ropes produced by arc-discharge (AD) method [6,7], were statistically analyzed in the same way and their Weibull statistics results are also tabulated in Table 1 for comparison. Note that the characteristic strength for Barber’s data is considerably higher than ours (and also Yu’s data) because the length of their samples is

0

-2

σ0 = 17.64 GPa m = 1.71

-4

R2 = 0.96

-6 1

2

3

4

LN(σf , GPa) Fig. 5. Weibull plot for the tensile strength of SWNT ropes from 12 SWNT bundles tested. The open circles are experimental results, while the straight line is the regression line.

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Table 1 Results of Weibull statistics for SWNT ropes in SWNT bundles, and for individual MWNTs and SWNT ropes n

r0(GPa)

m

R2

SWNTa bundles [5] Barber’s MWNTsb [4] Yu’s MWNTsc [6] Yu’s SWNTc ropes [7]

114 26 19 15

17.64 109.68 31.40 33.99

1.71 1.69 2.56 2.49

0.96 0.98 0.94 0.95

n denotes the number of specimens, r0 the characteristic strength,m the Weibull modulus, and R2 the coefficient of the correlation. a SWNTs synthesized by catalytic decomposition of hydrocarbons. b MWNTs synthesized by CVD method. c SWNTs synthesized by AD method.

substantially shorter (10 lm versus 3 mm). The low Weibull moduli shown in Table 1, ranged between 1.71 and 2.56, indicate a wide scattering of CNT tensile strength. It is seen that Yu’s data has a relatively higher Weibull modulus, which can probably be attributed to CNTs with less defects synthesized by the AD method. In all cases, the coefficient of correlation is close to unity, suggesting that the strength distributions of CNTs or CNT sub-bundles can adequately be described by a Weibull model. 4. Correlations between individual CNT and CNT bundle strengths Having established the evidences that tensile strength of SWNT and SWNT bundles are Weibull distributed, we further explore a correlation between Weibull modulus, m, for individual SWNTs and measurable parameters from tensile testing of a SWNT bundle. The force–strain relation of CNT bundles, which was studied in our previous work [11], is briefly summarized here. In a strain controlled tensile test, the probability of failure of a CNT of length, L, under an applied strain, e, is m e ð2Þ P f ¼ 1 RðeÞ ¼ 1 exp L e0 where R(e) is the probability of survival, and e0 the characteristic strain. The axial stiffness of CNTs is nonlinearly related to the applied strain at large strain [9], thus the CNT stress, rt, is E0 (1 ae)e, where E0 is the initial elastic modulus and a is a strain-dependent coefficient. Consider a bundle of N parallel, frictionless CNTs loaded in the longitudinal direction. The total tensile force, F, sustained by the remaining unbroken CNTs at strain, e, is F ¼ E0 ð1 aeÞeANRðeÞ

ð3Þ

where A is the cross-sectional area of a CNT. A typical force–strain curve from tensile testing of SWNT bundle (data in filled circles), along with theoretical predictions (more will be said about this later) are shown in Fig. 6. As shown in the figure, the slope of the force–strain curve, S0, for Eq. (3) at the origin is E0AN. From Eq. (3), the nominal stress, r, of the bundle is E0(1 ae)eR(e), and

S0

Fmax 1400 1200

Force F (mN)

Specimens

1600

1000 800

Linear α = 0

600 400

Nonlinear α=6

200

εb

0 0

1

2

3

4

5

6

7

Strain ε (%) Fig. 6. Force–strain curves of SWNT bundles. The solid circles (in blue) are experimental results, while the solid line (in red) is the curve of Eq. (9) for the nonlinear case and the dashed line is the curve of Eq. (10) for the linear case. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

the tensile strength is the maximum stress sustained by the bundle or ¼ 0 : em0 mLem ¼ aeð2em0 mLem Þ ð4Þ oe Setting a = 0 in Eq. (4), we get the linear solution of the failure strain, e1 ¼ e0 ðmLÞ

1=m

ð5Þ

which is used as a reference strain in subsequent analyses. Eq. (4) is a transcendental equation which can be solved by a fixed-point iteration method [11,12], and the solution,eb, the failure strain of a CNT bundle taking into consideration its nonlinear elastic behavior, is expressed as [11] 1 1 aeb m eb ð6Þ e1 ¼ 1 2aeb Substituting eb into Eq. (3) and using Eq. (2), the maximum tensile force of the CNT bundle is m eb ð7Þ F max ¼ S 0 ð1 aeb Þeb exp L e0 Substituting Eq. (5) into Eq. (6), and then from Eq. (7) the Weibull modulus of individual CNTs is solved: m¼ ln

h

12aeb 1aeb

S 0 ð1aeb Þeb F max

i

ð8Þ

Here S0, eb, and Fmax are measurable parameters obtainable from the force–strain curve of a tensile test of a CNT bundle (as shown in Fig. 6). From Eq. (8) the Weibull modulus m, and from Eqs. (6) and (5) the characteristic strain, e0, the two-parameters for individual CNTs, can be determined.


5. Data analysis and discussion With the analysis presented earlier, we are able to estimate Weibull modulus, m, for individual using tensile strength data of CNT bundles for both the linear and nonlinear cases. Simply fitting the experimental data in Fig. 6 with a cubic function, we obtain slope S0 = 74244, and Fmax = 1440 mN at eb = 0.028. Substituting these parameters and a = 6 (a typical value for SWNT) into Eq. (8) [9], we get the Weibull modulus m = 4.36 for the nonlinear case. Substituting these known parameters into Eq. (3) we get e 4:36 ð9Þ F ¼ 74244eð1 6eÞ exp 0:0414 When a = 0 in Eq. (8), we get the Weibull modulus m = 2.72 for the linear case, for which Eq. (3) reads e 2:72 ð10Þ F ¼ 74244e exp 0:0405 Curves of Eq. (9) for the nonlinear case and of Eq. (10) for the linear case are plotted in Fig. 6. The two curves differ little at small strains and both fit the experimental data well. The force–strain curve of the nonlinear case (solid line) skews left because of the larger m, at large strains. It shows that the SWNT nonlinear elastic behavior does have an effect on the strength distribution. The experimental data (solid circles) have a tendency of skewing left, consistent with the prediction of the nonlinear model. It is interesting that the experimental force–strain data for carbon fiber bundles (open circles in Fig. 7, taken from Ref. [13]) also skew to the left, away from theoretical prediction (so-

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lid line in Fig. 7) at large strains. The observed data trend for carbon fiber bundles could be better explained by the present nonlinear model. As mentioned in Section 2, the SWNT bundles used in our tests consist of many SWNT sub-bundles [10], and the kink points on the experimental stress–strain curves (Fig. 4) are believed to be SWNT and/or SWNT sub-bundle failures. These sub-bundles follow the weakest link model if there are strong local load sharing among the SWNTs within them. From the results, the strength distribution of sub-bundles comprising enough individual SWNTs can also be described by the Weibull model [14]. Hence the present model for individual frictionless SWNTs is also applicable to SWNT sub-bundles, containing hundreds of SWNTs [10]. Otherwise the frictionless assumption is practicable for weak local load sharing. Since only limited number of kinks is observed on a stress–strain curve, more than one SWNT sub-bundle could have failed at a kink point. This phenomenon may have something to do with rupture stability of SWNT bundles. Works by Hemmer [15] and Lee [16] on simultaneous fiber failure behavior for fiber bundles may shed light on this very issue. 6. Conclusions In sum, SWNT bundles are subject to tensile testing using a nano-mechanical testing device which was developed in our group. The tensile strength of SWNT bundles, as well as individual MWNTs and SWNT ropes, are studied by statistical approach based on the experimental data. It is shown that the tensile strength distributions of CNTs or SWNT ropes can be adequately described by a twoparameter Weibull model. Considering the nonlinear elastic behavior of CNTs, a model for a bundle constituted by parallel frictionless CNTs is proposed. Comparison with experimental results shows that the model can properly characterize the force–strain relation of SWNT bundles, and the nonlinear elastic behavior of SWNTs does have an effect on the strength distribution of SWNTs. Assuming a nonlinear elastic behavior will produce a more accurate Weibull modulus for individual CNTs, which is a critical design parameter for CNT-based nano-composites. Because testing individual CNT is still extremely time consuming and difficult, testing a CNT bundle and using the proposed model for estimating distribution parameters for individual CNTs could be a practical alternative. Acknowledgement T. X. and P. W. acknowledge the support of the National Natural Science Foundation of China (Grant No. 10572079). References

Fig. 7. Comparison of a theoretical force–strain curve (solid line) with experimental data (open circles) for carbon fiber, adapted from Ref. [13].

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