INSTITUTE OF PHYSICS PUBLISHING
NANOTECHNOLOGY
Nanotechnology 17 (2006) 634–639
doi:10.1088/0957-4484/17/3/003
Statistical characterization of single-wall carbon nanotube length distribution Shiren Wang, Zhiyong Liang, Ben Wang and Chuck Zhang Department of Industrial and Manufacturing Engineering, Florida A&M University, Florida State University College of Engineering, Tallahassee, FL 32310-6046, USA E-mail:
[email protected]
Received 10 October 2005 Published 6 January 2006 Online at stacks.iop.org/Nano/17/634 Abstract This paper describes an effective method for quantifying the length and length distribution of large populations of single-wall carbon nanotubes using atomic force microscopy and SIMAGIS software. The results of the measurements were modelled with the Weibull distribution, resulting in a statistically confirmed fit. The fitted Weibull distribution was used to predict the length effect factor and elastic modulus as functions of nanotube properties in composite materials. The prediction shows that the length factor for the elastic modulus tends to increase with enhanced loading but decrease with rising rope diameter. The statistical characterization presented indicates a pathway for the future theoretical modelling and related experimental investigation of carbon nanotube application. (Some figures in this article are in colour only in the electronic version)
1. Introduction Carbon nanotubes are molecular-scale tubes of graphitic carbon with outstanding properties, such as exceptional elastic modulus, tensile strength, unique thermal conductivity, and electronic conductivity [1, 2]. Carbon nanotubes have been regarded as the most promising next generation reinforcement and are actively being investigated in polymer composite application due to the high strength-to-weight ratio and modulus-to-weight ratio. However, many issues need to be resolved for future industrial application. Although carbon nanotubes are relatively easy to synthesize, it is difficult at the synthesis level to control the geometric configurations, including length, chirality, number of walls. Modelling the effective properties for carbon nanotube reinforced polymer is made complicated due to the complex micromechanical characteristics and nanoscale length distribution. A key step to understanding carbon nanotube synthesis and industrial application is the accurate measurement and characterization of nanotube length. While some progress has been made with in situ techniques, these approaches typically only give an average length and little information about the distribution [3, 4]. Direct imaging techniques are more tedious but typically give more information. Among these imaging approaches, atomic force microscope (AFM) appears to be 0957-4484/06/030634+06$30.00 © 2006 IOP Publishing Ltd
the most useful technique. Many prior reports have utilized AFM as a means of measuring the length distribution but generally there are a few ( 0.
(1)
Both scale and shape parameters can be estimated by maximum likelihood estimation (MLE); however, there is no closedform solution and the estimates must be made iteratively. One alternative method is to estimate the shape and scale parameters through the cumulative distribution function using linear regression. This method counts the cumulated frequency of SWNT lengths from the sample to create an empirical distribution: l l b F(l) = f (x) dx = abx b−1 e−ax dx = 1 − exp(−al b ), 0
636
0
(2)
(4)
The A, B parameters in equation (4) can be easily estimated with the least squares method as follows: ˆ B] ˆ = arg min yi − ( Axi + B)22 [ A,
(5)
resulting in the following estimates for intercept and slope; n n n n 2 i=1 yi i=1 x i − i=1 x i i=1 x i yi ˆ (6) A= n 2 n 2 n i=1 x i − i=1 x i Bˆ =
n
n n xi yi − i=1 x i i=1 yi . n n 2 2 n i=1 x i − i=1 x i
n
i=1
(7)
Back-transforming yields the Weibull parameter estimates n n n n i=1 xi yi − i=1 x i i=1 yi ˆ aˆ = exp( B) = exp n n 2 2 n i=1 x i − i=1 x i n n n n (8) yi xi2 − xi xi yi i=1 i=1 i=1 i=1 , bˆ = Aˆ = n 2 n 2 n i=1 x i − i=1 x i (9) where xi = ln(li ), yi = ln(−ln(1 − Fi )). After SWNT lengths were accurately quantified using AFM and SIMAGIS, the empirical cumulative distribution was computed so that the scale and shape parameters could be estimated, as shown in equations (8) and (9). The hypothesis that the observed length distribution of SWNTs is a member of a certain parametric family of Weibull distributions can be
Statistical characterization of single-wall carbon nanotube length distribution
statistically tested. Pearson and Fisher [11, 12] introduced a chi-square goodness-of-fit test: χ2 =
k (Oi − E i )2 Ei i=1
(10)
where Oi and E i are the observed and expected frequencies for bin i , respectively. The expected frequency is calculated from (11) E i = n (F(lu ) − F(ll )) where F is the cumulative distribution function for the theoretical Weibull distribution, lu is the upper limit for class i , ll is the lower limit for class i , and n is the sample size. The computed chi-square test statistic did not result in a statistically significant value for a type I error rate α = 0.05, so the null hypothesis that the observed data follow a Weibull distribution was accepted: χ2 = χ < 2
k (Oi − E i )2 = 39.2 Ei i=1
2 χ(a,k−c)
=
2 χ(0.05,27)
(12)
= 40.11.
The test statistic value was very close to being statistically significant, so further testing was conducted for validation purposes. Similar work was repeated more than ten times; the results support the above conclusion that the Weibull distribution is a sufficient model. Since the above SWNTs were randomly selected for making suspensions and randomly taken for the AFM measurement, and the sample size is large enough (sample size n = 651), it can be concluded that the observed length distribution is typical for all the SWNTs. Hence it can be safely assumed that the length distribution of SWNTs is a member of the Weibull family.
3. Theoretical application of length distribution Due to the extraordinary properties of carbon nanotubes, they are actively being investigated for use in polymer composite applications. However, modelling the effective properties of an SWNT-reinforced polymer is difficult because of complexities related to: (1) the structure and properties of the SWNTs; (2) the orientation and dispersion of the SWNTs within the polymer; (3) the characteristics of the interface and load transfer between the SWNTs and the polymer; (4) an understanding of the impact of the SWNTs on the molecular mobility of the polymer chains. Accurate models of how these issues influence the effective properties of the SWNT-reinforced polymer will be necessary to optimize the fabrication and effective properties of SWNT polymer systems. In particular, modelling of effective behaviour for SWNT reinforced composite is complicated by the range of length scales characteristic of these materials. It will be necessary for models developed at these different length scales, from atomistic simulations to continuum theories, to work in concert to accurately model the nanocomposite mechanical response.
Since the above analysis indicates that the SWNT length distribution can be represented by a specific Weibull distribution, this result would help to accurately model the SWNT reinforced polymer composite. Cox [13] proposed a shear-lag model for theoretically estimating the short glassfibre composite modulus. Fukuda and Kawaka [14] expanded it by considering the variations in both fibre length and orientation. Subsequently, some corrections were made to this expanded model [15] that has been extensively cited by the composite research community [16–18]. This model, expressed in equation (13), was applied here to compute SWNT/epoxy composite stiffness on the basis of above SWNT length distribution and random alignment. According to the experimental measurement of the modulus for SWNTs, 640 GPa has been suggested as a reasonable value (E f = 640 GPa) [19–22]. For epoxy resin, Yamini and Young [23] reported the elastic modulus E m = 2.5 GPa and Poisson ratio νm = 0.3. The Weibull distribution parameters were estimated using the method described in this paper, resulting in a = 5 × 10−6 , b = 2.4. The Young’s modulus of nanocomposite can then be estimated using the following rule of mixture [13–15]: E com = χ1 χ2 E f Vf + E m (1 − Vf ) lmax 1 tanh(βl/2) χ1 = f (l)l dl 1− lmean lmin βl/2 lmax tanh(βl/2) 1 abl b exp(−al b ) dl 1− = lmean lmin βl/2 1/2 1/2
Em 2 × 2(1+ν 2G m m) = β = E f r 2 ln (R/r ) E f r 2 ln (R/r ) 1/2 Em = E f (1 + νm )r 2 ln (R/r ) π/2
χ2 = (cos θ )2 − ν12 (sin θ )2 (cos θ )2 g(θ ) dθ
(13)
0
with: E com : Young’s modulus of nanocomposite; χ1 : length effect factor for discontinuous fibres; χ2 : orientation effect factor, equal to 1/5 for 3D plane randomness, 3/8 for 2D randomness; E f : Young’s modulus of reinforced fibre; E m : Young’s modulus of resin matrix; Vf : reinforced fibre volume fraction; R: average separation for the reinforced fibre norm and the length; r : reinforced fibre radius. The length effect factor was calculated with the change of SWNT loading (figure 6). Obviously, the length effect factor increases with rising tube loading which would keep improving the mechanical properties of the nanocomposite. As observed in equation (13), the larger the length effect factor, the larger the modulus value. As long as the length effect factor approximates to one, it is equivalent to the continuous fibre reinforced composite. The gain of the length effect factor from rising tube loading originates from the decreasing of the separator space between SWNTs, which helps to distribute the applied stress. In contrast, the length effect factor reduces quickly when the SWNT rope diameter is increasing. In particular, if the diameter is too large, the nanotube reinforcement effect would be close to zero since the length effect factor would be around zero. Due to the gain of the length effect factor with volume fraction increase, the elastic modulus of the SWNT composite 637
S Wang et al 0.85
1
(a)
0.84 d=4.5nm
0.8 Length Effect Factor
Length Effect Factor
0.83 0.82 0.81 0.8 0.79 0.78
0.7 0.6 0.5 0.4
0.77
0.3
0.76
0.2
0.75
(b)
0.9
0
0.05
0.1
0.15
0.2
0.25
0.3
0.1
0.35
Vf=30%
0
10
Vf
20 30 Rope diameter (nm)
40
50
Figure 6. (a) Length effect factor versus volume fraction and (b) length effect factor versus nanotube rope diameter. d: diameter of nanotube ropes.
70
Theoretic data(d=4.5nm) Theoretic data(d=16nm) Theoretic data(d=30nm) Experiment data
50 d=4.5 40 d=16
30 20
50
40
30 Vf=30%
d=30 20
10 0
(b)
60 Elastic Modulus(GPa)
Elastic Modulus(GPa)
60
70
(a)
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
10
0
Vf
10
20 30 Rope diameter(nm)
40
50
Figure 7. (a) Elastic modulus of composite versus volume fraction and (b) elastic modulus of composite versus nanotube rope diameter. d: diameter of nanotube ropes.
continuously goes up (figure 7(a)). Obviously the gain of the length effect factor for the short fibre reinforced composite, such as SWNT composite, is very important. Even though individual SWNTs have exceptional mechanical properties, the experimental results are far from the theoretical values. The best experimental value for the modulus of SWNT composite was approximately 17 GPa for 30% tube loading [24, 25]. This is not comparable to those for traditional glass fibre or carbon fibre reinforced composites. Figure 7(b) indicates that modulus should decrease dramatically as SWNT rope diameter increases. As shown in figure 7(a), the experimental results are close to the prediction at large diameters, for instance, about 30 nm. The negative effect of the diameter indicates that the SWNTs should be dispersed as well as possible to get isolated SWNTs so that a maximal contribution of the SWNTs to composite properties would be acquired.
4. Conclusions AFM and SIMAGIS can be used to accurately quantify the length of SWNTs. The Weibull distribution is reasonable 638
for describing the SWNT length distribution. The statistical distribution of the SWNT length helps to reveal the micromechanical characteristics of SWNTs and accurately predict the performance of the final product, especially in the SWNT reinforced polymer composite. Statistical analysis was combined with the Cox shear-lag model to estimate the elastic modulus of nanocomposite. The theoretical prediction shows that the length effect factor for modulus tends to decrease with rise of the rope diameter and increase with enhanced SWNT loading, which leads to change in the final material properties. Poor performance of the nanocomposite possibly arises from heterogeneous dispersion, which results in large diameter of the SWNT ropes.
References [1] Reich S, Thomsen C and Maultzsch J 2004 Carbon SWNTs (New York: Wiley–VCH) [2] Wong E W, Sheehan P E and Lieber C M 1997 Nanobeam mechanics: elasticity, strength, and toughness of nanorods and SWNTs Science 277 1971–5 [3] Zhou W, Islam M F, Wang H, Ho D L, Yodh A G, Winey K I and Fischer J E 2004 Small angle neutron
Statistical characterization of single-wall carbon nanotube length distribution
[4] [5]
[6] [7] [8]
[9] [10] [11] [12] [13] [14] [15]
scattering from single-wall carbon SWNTs suspensions: evidence for isolated rigid rods and rod networks Chem. Phys. Lett. 384 185–9 Kukovecz A, Kanyo T, Konya Z and Kiricsi I 2005 Long-time low-impact ball milling of multi-wall carbon SWNTs Carbon 43 994–1000 Ziegler K J, Gu Z, Peng H, Flor E L, Hauge R H and Smalley R E 2005 Controlled oxidative cutting of single-walled carbon SWNTs J. Am. Chem. Soc. 127 1541–7 Liu J et al 1998 Fullerene pipe Science 280 1253–5 Gu Z, Peng H, Hauge R H, Smalley R E and Margrave J L 2002 Cutting single-wall carbon SWNTs through fluorination Nano Lett. 2 1009–13 Islam M F, Rojas E, Bergey D M, Johnson A T and Yodh A G 2003 High weight fraction surfactant solubilization of single-wall carbon SWNTs in water Nano Lett. 3 269–73 Paredes J I and Burghard M 2004 Dispersions of individual single-walled carbon SWNTs of high length Langmuir 20 5149–52 Gan F F and Koehler K J 1990 Goodness-of-fit tests based on P–P probability plots Technometrics 32 289–303 Li G and Doss H 1997 Generalized Pearson–Fisher Chi-square goodness-of-fit test, with application model to life history data Ann. Stat. 21 772–97 Easterling R G 1976 Goodness of fit and parameter estimation Technometrics 18 1–9 Cox H L 1952 The elasticity and strength of paper and other fibrous materials Brit. J. Appl. Phys. 3 72–9 Fukuda H and Kawata K 1974 On Young’s modulus of short fiber composites Fiber Sci. Technol. 7 207–12 Jayaraman K and Kortschot M T 1996 Correction to the Fukuda–Kawata Young’s modulus theory and the
[16]
[17]
[18] [19]
[20] [21] [22] [23] [24]
[25]
Fukuda–Chou strength theory for short fiber-reinforced composite materials J. Mater. Sci. 31 2059–64 Laura D M, Keskkula H, Barlow J W and Paul D R 2000 Effect of glass fiber and maleated ethylene–propylene rubber content on tensile and impact properties of nylon 6 Polymer 41 7165–74 Coleman J N, Cadek M, Blake R, Nicolosi V, Ryan K P, Belton C, Fonseca A, Nagy J B, Gunko Y K and Blau W J 2004 High-performance SWNTs-reinforced plastics: understanding the mechanism of strength increase Adv. Funct. Mater. 14 791–8 Chou T-W and Kelly A 1980 Mechanical properties of composites Annu. Rev. Mater. Sci. 10 229–59 Salvetat J-P, Briggs G A D, Bonard J-M, Bacsa R R, Kulik A J, St¨ockli T, Burnham N A and Forr´o L 1999 Elastic and shear moduli of single-walled carbon SWNTs ropes Phys. Rev. Lett. 82 944–7 Popov V N, Van Dorena V E and Balkansk M 2000 Elastic properties of crystals of single-walled carbon SWNTs Solid State Commun. 114 395–9 Lu J 1997 Elastic properties of carbon SWNTs and nanoropes Phys. Rev. Lett. 79 1297–9 Liu T and Kumar S 2003 Effect of orientation on the modulus of SWNT films and fibers Nano Lett. 3 647–50 Yamini S and Young R J 1980 The mechanical properties of epoxy resins J. Mater. Sci. 15 1814–22 Wang Z, Liang Z, Wang B, Zhang C and Kramer L 2004 Processing and property investigation of single-walled carbon SWNTs (SWNT) buckypaper/epoxy resin matrix nanocomposites Composites A 35 1225–32 Zhu J, Kim J, Peng H, Margrave J L, Khabashesku V N and Barrera E V 2003 Improving the dispersion and integration of single-walled carbon SWNTs in epoxy composites through functionalization Nano Lett. 3 1107–13
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