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Maximum nanotube volume fraction and its effect on overall elastic properties of nanotube-reinforced composites
Composites: Part B 40 (2009) 212–217
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Maximum nanotube volume fraction and its effect on overall elastic properties of nanotube-reinforced composites Bing Jiang, Charlie Liu, Chuck Zhang *, Richard Liang, Ben Wang Department of Industrial and Manufacturing Engineering, High Performance Materials Institute, Florida A&M University-Florida State University College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310-6064, USA
a r t i c l e
i n f o
Article history: Received 9 November 2008 Accepted 9 November 2008 Available online 17 November 2008 Keywords: A. Nano-structures B. Mechanical properties C. Micromechanics
a b s t r a c t The potential capability of improving overall elastic modulus of nanotube-reinforced composites is a fundamental concern in nanotechnology applications. Based on geometric analysis and micromechanics estimation, this study reports that the ratio of surface-to-surface distance of adjacent carbon nanotubes (CNTs) to the CNT diameter plays a key role in improving the overall elastic modulus of the CNT-reinforced composites when the tubes are perfectly aligned, completely separated from other tubes, and ideally bonded with the composite matrix. With the decrease of this ratio, that is, decrease of the surface-tosurface distance of adjacent CNTs and/or increase CNT diameter, the improvement capability increases. However, theoretical and experimental results show that an increase of the CNT diameters degrades the elastic moduli of CNTs. This paper discusses the criterion of choosing CNTs with larger diameter and addresses the factors influencing the surface-to-surface distance of adjacent CNTs. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Since the discovery of carbon nanotubes (CNTs) by Iijima [1], interest in CNTs has grown rapidly because of their unique superior properties. Both experimental and theoretical studies suggest that CNTs have extraordinary mechanical and electrical properties [2– 4]. For example, CNTs have a Young’s modulus about 1 TPa and a very high tensile strength due to their defect-free structure. As the mechanical properties of composites depend directly upon the embedded fiber mechanical behavior, replacing conventional microsized fibers with CNTs can potentially improve composite properties, such as tensile strength and elastic modulus. Consequently, composites of CNTs dispersed in metallic or polymeric matrices have attracted considerable attention in recent years [5,6]. However, how the elastic properties of CNT-reinforced composites can be improved relative to those of the composite matrix, even under ideal conditions, is not clearly understood. The objective of this paper is to clarify this problem. In other words, we want to answer the questions: what factors affect the potential capability to improve the overall elastic properties of CNT-reinforced composites, and how the factors do actually influence the capability? The potential improvement of elastic properties of CNT-reinforced composites will be estimated based on geometry and micromechanics. The elastic properties of a CNT-reinforced composite are determined by factors including composite constituent elastic proper-
* Corresponding author. Tel.: +1 850 410 6355. E-mail address: [email protected] (C. Zhang). 1359-8368/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2008.11.003
ties and microstructures, such as CNT elastic modulus [7,8], bundles [9], waviness [10], aspect ratio and orientation distribution [11], as well as interface between the CNTs and the polymer matrix [12–15]. It is generally believed that if the full advantage of the extraordinary properties of CNTs for reinforcement of composites can be achieved, the following conditions must be obeyed: CNTs must be (1) in a straight line; (2) perfectly aligned; (3) completely separated from other CNTs by the composite matrix; (4) infinitely long; and (5) ideally bonded with the composite matrix. However, the capability of elastic property enhancement of a CNTreinforced composite by CNTs is also related to another key factor – the maximum volume fraction or loading of CNTs. The maximum volume fraction achievable depends on CNT geometry, surfaceto-surface distance between adjacent CNTs and their packing pattern. Packing problems, i.e. how densely objects can fill a volume, or equivalently, how objects can be tightly packed in a space, are among the most ancient and persistent problems in mathematics and engineering. This was the 18th problem of Hilbert’s 23 famous problems. The maximum packing volume fraction is sensitive to the shape of particles packed. For example, the highest packing volume fraction of sphere in three-dimensional Euclidean space is believed as 0.7405, as realized by stacking variants of the face-centered cubic (fcc) lattice packing. This is the famous Kepler conjecture, which was only proven [16]. For two-dimensional congruent circles, it is well known that hexagonal packing is the optimal packing and its maximum packing volume fraction is 0.9069 [17]. The infinitely long circular cylinder with parallel in hexagonal packing has the same
B. Jiang et al. / Composites: Part B 40 (2009) 212–217
maximal packing volume fraction as that of the two-dimensional optimal packing of congruent circles, that is, 0.9069 [18,19]. This maximum packing volume fraction is much higher than that of sphere in three-dimensional space mentioned above, which suggests that reducing aspect ratio of particles decreases the maximum packing volume fraction. It is generally believed that the highest packing volume fraction of random packing of a given congruent particles is smaller than that of regular packing of the particles, although it is difficult to prove this conjecture [20]. The conventional hexagonal close packing, which yields maximum volume fraction of infinitely long cylinders, is shown in Fig. 1a (top view), in which each cylinder touches other six neighbor cylinders. In a real composite material system, however, cylinders or CNTs cannot touch each other even if there is no other phase in between two adjacent CNTs. The reasons can be illustrated as follows: the surfaces of CNTs are atomic smooth and the CNTs are subject to van der Waals interaction. Due to the balance of the van der Waals repulsive force and attractive force, the CNTs remain apart from each other at a certain distance that forms a surface-to-surface distance. Fig. 1b shows an ideal packing pattern in which all have the same diameter, length and the surface-to-surface distance, and the CNTs are hexagonally close packed. The volume fraction of the infinitely long CNTs, /inf, is
p 1 /inf ¼ pffiffiffi ; 2 3 ðS=d þ 1Þ2
ð1Þ
where d is the CNT diameter; and S the surface-to-surface distance between two adjacent CNTs. If we keep d as a constant, /inf increases with the decrease of S. When S ? 0, /inf reduces to that of the conventional hexagonal close packing and tends to its maximum value, 0.9069. 2. Determination of the maximum loading The determination of the surface-to-surface distance is complicated. Here we examine two special cases and outline a general case. If there are no other phases or materials in between CNTs, it is generally believed that the minimum surface-to-surface distance of two adjacent CNTs is the equilibrium van der Waals distance, which is about 0.34 nm. However, for a polymer CNTreinforced composite, in order to obtain best mechanical properties, polymer molecules must be in between CNTs to bond them together. Accordingly, the surface-to-surface distance must be larger than the equilibrium van der Waals distance. To illustrate this, suppose there is only one atom in the middle of two adjacent CNTs, which can also be imagined as a pi-stacking polymer chain in between CNTs. The surface-to-surface distance between the two CNTs should be doubled to 0.68 nm, since the atom has the equilibrium van der Waals distance to both of the CNTs. In an actual CNT-
a
D
b
D
S
d
Fig. 1. Hexagonal close packing: (a) without interface and (b) with interface.
213
reinforced composite system, the surface-to-surface distance between CNTs is larger than 0.68 nm because the pi-stacking polymer chain conformations do not exist at all locations. The complete discussion of the surface-to-surface distance is quite complex because many factors affecting are it, such as CNT geometry, polymer chain conformation, interaction between CNTs and polymer chains, and processing conditions of CNT-reinforced composites. We shall here provide a sketchy, albeit representative, overview based on the polymer chain conformation. In the processing of CNT-reinforced composites, CNTs need to be separated from bundles, and meanwhile, dispersed uniformly in a polymer matrix for maximizing their contact surface area with the matrix. This implies polymer chains should be put in between CNTs. This processing is related to the polymer chain conformation. If a flexible linear polymer chain is in a good solvent and the concentration of polymer chains is so low that there is no entanglement with each other, the size of polymer chains can be estimated by its square-root-mean end-to-end distance or its radius of gyration [21]. Based on the self-avoiding walk analysis, the end-to-end distance of a flexible polymer chain, R, can be obtained by R = lN3/5, where l is the monomer size and N is the number of monomers per polymer chain, which is equivalent to the molecular weight of the polymer chain. Suppose N is 100 and l is 0.154 nm or C–C bond distance. In fact, the monomer length of a typical polymer chain is larger than 0.154. We can find the polymer chain size is about 2.44 nm. If this polymer chain bonds two adjacent CNTs, the surface-to-surface distance is about 2.44 nm. If we want to reduce the surface-to surface distance between two adjacent CNTs, we must squeeze the polymer chain into a smaller size along the surface-to-surface distance direction. Because of this, the processing of a CNT-reinforced composite becomes difficult; therefore, CNT separation and dispersion are hard to obtain. Table 1 indicates the changes of the maximum volume fraction with CNT diameter and the surface-to-surface distance, which is obtained from Eq. (1). Examining four typical surface-to-surface distances, that is, 0.34 nm, 0.68 nm, 1.0 nm and 2.44 nm, with the CNT diameter varying from 1.0 nm to 30 nm, we can see that if CNT diameter is 1.0 nm, the maximum CNT volume fraction is only 50.51 vol.%, even though there is no material in between CNTs (S = 0.34 nm). If there is one atom in between CNTs, the maximum CNT volume fraction drops to 32.13 vol.% with the CNT diameter being 1.0 nm, which is only about one-third of maximum volume fraction of traditional fiber-reinforced composites. We believe that this is the maximum CNT volume fraction if CNTs have 1.0 nm diameter. If the surface-to-surface distance increases further, the maximum volume fraction reduces to 22.67 vol.% for S = 1.0 nm and only 7.66 vol.% for S = 2.44 nm at d = 1.0 nm. With the increase of CNT diameter, the surface-to-surface distance influence on the maximum CNT volume fraction becomes smaller. We can see that if CNT diameter is 30 nm, all four cases have the maximum CNT volume fraction more than 77 vol.%. An actual CNT cannot have infinite length, and therefore, the aspect ratio of a CNT, that is length-to-diameter ratio of the CNT, is finite. Fig. 2a shows a CNT-reinforced composite with CNTs as cylinders of the same diameter and length and CNTs have the same surface-to-surface distance and the same end-to-end distance. The model to estimate the maximum packing volume fraction of CNTs with finite aspect ratio is shown in Fig. 2b in which the surface-to-surface distance is, again, S; the end-to-end distance between two CNTs is H; and the length of CNTs is L. Accordingly, the representative volume element (RVE) has the length of L + H. The maximum packing volume fraction of CNTs with a finite aspect ratio, /a, can be estimated by
/a ¼
/inf ; 1 þ S=ðd aÞ ðH=SÞ
ð2Þ
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Table 1 Maximum volume fraction of infinitely long CNT with different diameters and surface-to-surface distances. CNT diameter (nm)
S = 0.34 (nm)
S = 0.68 (nm)
S = 1.0 (nm)
S = 2.44 (nm)
1 2 3 4 5 6 7 8 9 10 15 20 25 30
0.5051 0.6625 0.7316 0.7704 0.7951 0.8122 0.8248 0.8344 0.8420 0.8482 0.8671 0.8768 0.8827 0.8867
0.3213 0.5051 0.6027 0.6625 0.7027 0.7316 0.7534 0.7703 0.7840 0.7951 0.8299 0.8482 0.8595 0.8671
0.2267 0.4031 0.5101 0.5804 0.6298 0.6663 0.6943 0.7166 0.7346 0.7495 0.7971 0.8226 0.8385 0.8493
0.0766 0.1840 0.2758 0.3499 0.4096 0.4583 0.4987 0.5325 0.5613 0.5860 0.6709 0.7204 0.7527 0.7756
a
4.0
b S
Fiigure 4
L
H
Logarithm of the aspect ratio
3.5
0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.150 0.100 0.0750 0.0500 0.0375 0.0313 0.0250 0.0188 0.0125 0.00625 0
0.0313 0.0375
3.0
0.500 2.5
0.0500
0.400 0.0750
2.0
0.300 0.100
1.5
0.150 1.0
0.600
0.0250
0.200
0.0188
0.5
0.0125 0.0 0
1
2
3
4
5
S/d Fig. 3. Relationship among the aspect ratio of the nanotubes, S/d and the maximum volume fraction of the nanotubes.
where a = L/d is the CNT aspect ratio. If a ? 1, that is equivalent to the CNTs having infinite length, /a ? /inf. /a is smaller than /inf when a is finite. The change of the maximum CNT volume fraction with S/d and a can be obtained from Eq. (2) and is illustrated at H/S = 1 in Fig. 3. Generally, fiber diameters in conventional fiber-reinforced composites are in micrometers even in millimeter ranges. If the surface-to-surface distance between fibers is on the order of equilibrium van der Waals distance, the ratio S/d for the conventional composites is very small. It can be seen from Fig. 3 that when S/d is smaller than 0.01, the maximum CNT volume fraction is about 90 vol.%. In this range, the maximum volume fraction is insensitive to the CNT aspect ratio. However, in CNT-reinforced composites, the CNT diameters have the same order or even smaller than the surface-to-surface distance, which results in large S/d ratio. If S/d is unity and a is larger than 100, the maximum CNT volume fraction approach to 20 vol.%. When S/d further increases, say, to S/d = 3, the maximum volume fraction reduces to 5 vol.%. When S/d increases, the maximum volume fraction becomes more sensitive to the CNT aspect ratio. When S/d is larger than 2, decreasing the CNT aspect ratio from 100 to 10 significantly reduces the maximum volume fraction. To reveal how the reduction affects the overall elastic properties of CNT-reinforced composites, we use the micromechanics to estimate the effect of S/d and a on the overall elastic modulus.
3. Micromechanics analysis Micromechanics analysis is a very powerful tool to predict overall properties of a composite from the constituents and microstructures, and has been used successfully in various materials such as
4.0
160 200 120 240
3.5
Logarithm of the aspect ratio
Fig. 2. (a) Scheme of packing of an actual aligned nanotubes; (b) scheme of packing to estimate the volume fraction of the nanotubes.
80.0
50.0 35.0
25.0 20.0 17.5 15.0
40.0
3.0
350 320 280 240 200 160 120 80.0 60.0 50.0 40.0 35.0 30.0 25.0 20.0 17.5 15.0 12.5 10.0 7.50 5.00 2.50 0
30.0
60.0
2.5
12.5 10.0
2.0
7.50 5.00
1.5
2.50 1.0 0.5 0.0 0
1
2
3
4
5
S/d Fig. 4. The changes of the maximum overall Young’s modulus of the composite relative to that of polymer matrix with CNT aspect ratio and the S/d.
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1 NT Lijkl ¼ ½/M LM ; ijpq þ /NT hLijmn Amnpq iNT /M Ipqkl þ /NT hApqkl iNT
ð3Þ
where /NT and /M are the volume fraction of the CNTs and the maM trix, respectively; LNT ijkl and Lijkl the stiffness tensors of CNTs and the matrix; Iijkl the fourth-order unit tensor; Aijkl the strain concentration tensor which can be determined by Eshelby tensor Sijkl as follows: 1 NT 1 M Aijkl ¼ ½Iijkl þ Sijmn ðLM mnpq Þ ðLpqkl Lpqkl Þ :
ð4Þ
In Eq. (3), hi is the average of ‘’ over the volume of the composite, which is related to the CNT shape, orientation and volume fraction. The Eshelby tensor, however, is only dependent on the CNT shape and Poisson ratio of the matrix. Theoretical analysis and experimental results show that for a fiber-reinforced composite, if fibers are perfectly aligned, the maximum and the minimum overall elastic modulus of the fiber-reinforced composite are along the directions that are parallel and perpendicular to the fiber direction, respectively [22,25]. The overall elastic modulus of a randomly-orientated-fiber-reinforced composite is in between the maximum and the minimum overall elastic modulus of the perfectly-aligned-fiber-reinforced composite. Since the case with perfectly-aligned-fibers gives the upper and lower bound of the overall elastic modulus and is easy to analyze, we will just focus on this special case. The enhancement of elastic modulus of the CNT-reinforced composite relative to that of the polymer matrix can be characterized by E33 =Ematrix , where Ematrix is Young’s modulus of the polymer matrix and E33 is Young’s modulus of the CNT-reinforced composite along the 3-direction, or in the CNT axial direction. A larger E33 =Ematrix implies higher enhancement of elastic modulus, and, if E33 =Ematrix is unity, no enhancement. E33 can be determined from Eq. (3) by E33 ¼ 1=M 3333 , where M ijkl ¼ ðLijkl Þ1 . If the CNTs are perfectly aligned, completely separated and ideally bonded with the polymer matrix, the maximum elastic modulus along the CNT axis direction can be obtained from Eq. (3). Table 2 shows the enhancement of Young’s modulus of CNT-reinforced composites with infinitely long CNTs. In the calculation, CNTs are assumed as isotropic and their Young’s moduli are 1 TPa and Poisson ratios are 0.2. The polymer matrix has 2.5 GPa Young’s modulus and 0.3 Poisson ratio. We can see that if the CNT diameter is 30 nm, E33 =Ematrix is about 340, that is, E33 =ENT 85%. This implies that the 85% of elastic modulus of the CNT is taken. However, if the CNT diameter is
Table 2 The enhancement of Young’s modulus of infinitely long CNT-reinforced composite, E33 =Ematrix , at different CNT diameters and surface-to-surface distances. (Young’s modulus of CNT does not change with its diameter.) CNT Diameter (nm)
S = 0.68 (nm)
S = 1.0 (nm)
S = 2.44 (nm)
1 2 3 4 5 6 7 8 9 10 15 20 25 30
129.19 202.51 241.34 265.33 281.39 293.10 301.68 308.37 313.53 318.23 332.35 339.44 344.07 346.76
91.45 161.81 204.54 232.57 252.28 266.87 278.06 286.90 294.09 300.04 319.03 329.21 335.55 339.90
31.57 74.41 111.03 140.58 164.41 183.86 199.96 213.46 224.95 234.81 268.67 288.43 301.35 310.46
1.0 nm, the enhancement of Young’s modulus drops to 129.19 ðE33 =ENT 32:29%Þ for S = 0.68 nm and 91.45 ðE33 =ENT 22:86%Þ for S = 1.0 nm. If the S further increases to 2.44 nm, the enhancement of Young’s modulus reduces to 31.57 and E33 =ENT 7:89%. This suggests that CNT elastic modulus of CNTs can be taken only 7.89% for S = 2.44 nm. The relationship between E33 =Ematrix with S/d and a is shown in Fig. 4. If we keep S/d as a constant, E33 =Ematrix reduces dramatically with a decrease of the CNT aspect ratio, when the CNT aspect ratio is smaller than 100. When S/d is small, say smaller than 0.01, and a is large, say larger than 1000, we can see that E33 =Ematrix is about 350, that is, E33 =ENT 87:5%, which is very close to Young’s modulus of the CNTs, ENT. However, if we keep a as 1000 and increase S/d, we can see that E33 =Ematrix becomes very small. For example, if S/d = 1, E33 =Ematrix is about 80, which is about one-fourth of that at S/d = 0.01. If S/d increases to 4, E33 =Ematrix decreases further to 15, that is E33 =ENT 3:75%. The above discussion strongly suggests that to take full advantage of the extraordinary properties of the CNTs, we must decrease S/d. In other words, we should decrease the surface-tosurface distance and/or increase the CNT diameter. If the elastic modulus of CNTs or CNT bundles do not change with their diameters, increasing the diameter of the CNTs or CNT bundles would improve the overall elastic modulus due to increase of the maximum CNT volume fraction. Unfortunately, many theoretical and experimental results indicate that the elastic modulus of CNTs or CNT bundles becomes lower with the increase of the diameter of the CNTs or the CNT bundles [7–9]. For instance, increasing a SWNT diameter from 0.35 nm ((5, 5) SWNT) to 1.35 nm ((20, 20) SWNT) decreases Young’s modulus from 1.12 TPa to 0.97 TPa, which declines 13.3% [7]. With the increase of diameter, Young’s modulus of CNT bundles drops much more than that of individual SWNTs. Experimental results show that if the diameter increases from 3 nm to 13.5 nm, Young’s modulus of the CNT bundles drops from 899 GPa to 298 GPa, which loses 66.9% [9]. All the experimental and theoretical results (e.g. [7– 9]) show that the relationship between Young’s modulus and diameter of CNTs or CNT ropes has the following form:
E ¼ m þ n ðd=d0 Þc ;
ð5Þ
where m, n and c are material parameters CNTs or CNT ropes, respectively; d0 is a reference CNT diameter; E is Young’s modulus of CNTs or CNT ropes and d is, again, the diameter of CNTs or CNT ropes and d P d0. For SWNTs, the theoretical prediction shows that c = 2 [7]. However, for a CNT rope, c is about 1 based on the exper-
Elastic modulus of the composite
metals, ceramics and composites [22]. If the interaction among the CNTs is estimated by Mori–Tanaka’s method [23], the overall elastic moduli of the CNT-reinforced composite, Lijkl , can be written as [24]
EA
EBCritical EC
EMatrix 0
φA
φB
1
Volume Fraction Fig. 5. Conceptual scheme for choosing CNTs or CNT bundles with larger diameter.
B. Jiang et al. / Composites: Part B 40 (2009) 212–217
iments by Salvetat et al. [9]. Table 3 displays the enhancement of Young’s modulus of the infinitely long CNT-reinforced composite considering the degeneration of elastic properties of CNTs or CNT ropes with the increase of their diameters. The simulation assumes that m = 0, n = 1 TPa, d0 = 1 nm, and c = 1. We can see from Table 3 that E33 =Ematrix decreases dramatically with diameter of CNTs or CNT ropes when S = 0.68 nm and 1.0 nm. For example, if CNT diameter is about 30 nm, E33 =Ematrix is about 11.70 for S = 0.68 nm, which is only one-tenth of that at d = 1 nm. However, for S = 2.44 nm, when CNT diameter is small (d 6 2 nm), the enhancement of Young’s modulus increases with CNT diameter increase. After that (d > 2 nm), the enhancement of Young’s modulus decreases with CNT diameter increases. Based on the discussion above, whether the overall elastic properties of CNT-reinforced composite could be improved or not through increasing diameter is determined by the competition between increasing maximum volume fraction and decreasing of elastic properties due to the increase of CNT diameter. Fig. 5 illustrates the conceptual scheme of how to consider the competition. Here we use the term of inclusion, which is widely used in micromechanics, to denote reinforcement phase. The inclusion could be CNTs or CNT bundles. Suppose the inclusions considered here are perfectly aligned and infinitely long as well as ideally bonded with the composite matrix based on micromechanics. The overall elastic properties of the composite would be almost linear with the inclusion volume fraction. The overall Young’s modulus of the composite must be equal to that of the polymer matrix at inclusion volume fraction vanishing and should be the same as that of inclusion at CNT volume fraction being unity. Here, we consider two types of composites: one has inclusion A (A-composite) whose diameter is dA and Young’s modulus is EA and the other composite has inclusion B (B-composite) whose diameter is dB and dB > dA. If the surface-to-surface distance between inclusions, say S, keeps constant and inclusion A is replaced by inclusion B, what criterion of elastic modulus of inclusion B should be satisfied so that the overall Young’s modulus of the composite can keep the same as or be higher than that before being replaced? From Eq. (1), we can obtain the maximum volume fraction of A- and B-composites as /A and /B, respectively. According to the schemes shown in Fig. 5, we can determine the critical Young’s modulus of the inclu, as follows: sion B, ECritical B
ECritical Ematrix /A Ematrix B ¼ þ 1 : EA EA /B EA
ð6Þ
From Eq. (1), we have
1.0 1.0 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.050 0
0.85 0.80 0.8
0.65 0.60 0.55
0.6
dSWNT / dI
216
0.50
0.40 0.35
0.45
0.30
0.4
0.25 0.20
0.2
0.15 0.10
0.0 0
1
2
3
4
5
S / dSWNT Fig. 6. Changes of the critical Young’s modulus with dSWNT/dI and S/dSWNT.
2 ECritical Ematrix Ematrix 1 þ ðdA =dB Þ B ¼ þ 1 : 1 þ ðS=dA Þ EA EA EA
ð7Þ
As a result, the criterion to choose a larger diameter inclusion B to replace inclusion A is
EB P ECritical ; B
ð8Þ
where Ematrix are Young’s modulus of the matrix. If the inclusion is SWNT, we can use Eq. (7) to estimate the critical Young’s modulus of inclusion B, which could be CNTs or CNT bundles. In the calculation, we assume ESWNT/Ematrix = 500. This value is chosen because for a polymer CNT-reinforced composite, the polymer matrix has, in general, about 2 GPa Young’s modulus and the SWNT has about 1 TPa Young’s modulus. Fig. 6 exhibits the results obtained from Eq. (7). We can see that when S/dSWNT is very small, the critical Young’s modulus of the nanotube with larger diameter is insensitive to dSWNT/dI because the maximum volume fraction is close to 90 vol.%. However, if S/dSWNT is large, the critical Young’s modulus is very sensitive to dSWNT/dI. For example, if S/dSWNT is unity, when CNT diameter is doubled, the critical Young’s modulus should be at least about 55% of the SWNT. If the diameter is 10 times of that of the SWNT, the critical Young’s modulus is about 22% of that of the SWNT. 4. Conclusions
Table 3 The enhancement of Young’s modulus of infinitely long CNT-reinforced composite, E33 =Ematrix , at different CNT diameters and surface-to-surface distances. (Young’s modulus of CNT change with its diameter by the relation given in Eq. (5), and m = 0 TPa, n = 1.0 TPa, d0 = 1.0 nm and c = 1.0.) CNT diameter (nm)
ENT (GPa)
S = 0.68 (nm)
S = 1.0 (nm)
S = 2.44 (nm)
1 2 3 4 5 6 7 8 9 10 15 20 25 30
1000.00 500.00 333.33 250.00 200.00 166.67 142.85 125.00 111.11 100.00 66.67 50.00 40.00 33.33
129.19 101.51 80.76 66.59 56.51 49.05 43.30 38.75 35.06 32.01 22.30 17.18 13.89 11.70
91.45 81.21 68.51 58.46 50.76 44.76 39.98 36.11 32.92 30.23 21.46 16.63 13.58 11.47
31.56 37.61 37.49 35.63 33.36 31.10 29.00 27.10 25.39 23.86 18.22 14.69 12.29 10.57
The main findings from this study are summarized below: (1) The maximum CNT volume fraction depends on CNT geometry, surface-to-surface distance between adjacent CNTs and CNT packing pattern. Since CNT diameter is on the same order of the equilibrium van der Waals distance, increase of the surface-to-surface distance and/or decrease of CNT diameter will reduce the maximum CNT volume fraction dramatically. (2) The maximum enhancement of elastic properties of CNTreinforced composite is related to the maximum CNT volume fraction. Decreasing the maximum CNT volume fraction significantly degrades the enhancement capability of CNTreinforced composites. (3) Under the criterion given in Eq. (8), the enhancement capability could be improved by means of increasing CNT diameter and/or decreasing the surface-to-surface distance between adjacent CNTs.
B. Jiang et al. / Composites: Part B 40 (2009) 212–217
Acknowledgement This work has been sponsored by the Air Force Research Laboratory, the NSF IUCRC Program, IUCRC members, and the Florida State University Cornerstone Research Program. References [1] Iijima S. Nature (London) 1991;354:56. [2] Saito R, Dresselhaus G, Dresselhaus MS. Physical properties of carbon nanotubes. London: Imperial College Press; 1998. [3] Dresselhaus MS, Dresselhaus G, Avouris Ph. Carbon nanotubes: synthesis, structure, properties, and application. Springer; 2001. [4] Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS. Appl Mech Rev 2002;55:495. [5] Calvert P. Nature (London) 1999;399:210. [6] Thostenson ET, Ren Z, Chou T-W. Compos Sci Technol 2001;61:1899. [7] Yao N, Lordi VJ. Appl Phys 1998;84:1939. [8] Popov VN, Doren VE, Balkanski M. Solid State Commun 2000;114:395. [9] Salvetat J-P, Briggs GAD, Bonard J-M, Bacsa RR, Kulik A. Phys Rev Lett 1999;82:944. [10] Fisher FT, Bradshaw RD, Brinson LC. Appl Phys Lett 2002;80:4647.
[11] [12] [13] [14] [15] [16] [17] [18]
[19] [20] [21] [22] [23] [24] [25]
217
Jiang B, Liu C, Wang Z, Zhang C, Wang B. Compos Part B – Eng 2007;38:24–34. Schadler LS, Giannaris SC, Ajayan PM. Appl Phys Lett 1998;73:3842. Qian D, Dickey EC, Andrews R, Rantell T. Appl Phys Lett 2000;76:2868. Ajayan PM, Schadler LS, Giannaris C, Rubio A. Adv Mater 2000;12:1671. Zhu J, Peng H, Rodriguez-Macias F, Margrave JL, Khabashesku VN, Imam AM, et al. Adv Funct Mater 2004;14:643. Hales TC. LANL e-print archive; 1998.. Donev A, Stillinger FH, Chaikin PM, Torquato S. Phys Rev Lett 2004;92:2555061–6-4. Bezdek A, Kuperberg W. Applied geometry and discrete mathematics: DIMACS series in discrete mathematics and theoretical computer science, vol. 4. Providence (RI): American Mathematics Society; 1991. Pach J, Agarwal PK. Combinatorial geometry. New York: Wiley-Interscience; 1995. Donev A, Cisse I, Sachs D, Variano EA, Stillinger FH, Connelly R, et al. Science 2004;303:990. Flory PJ. Principles of polymer chemistry. Ithaca: Cornell University; 1953. Nemat-Nasser S. Micromechanics: overall properties of heterogeneous materials. Amsterdam, New York: North-Holland; 1993. Mori T, Tanaka K. Acta Metall 1973;21:571. Weng GJ. Int J Eng Sci 1990;28:1111. Thostenson ET, Chou TW. J Phys D: Appl Phys 2002;35:L77.
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Composites: Part B journal homepage: www.elsevier.com/locate/compositesb
Maximum nanotube volume fraction and its effect on overall elastic properties of nanotube-reinforced composites Bing Jiang, Charlie Liu, Chuck Zhang *, Richard Liang, Ben Wang Department of Industrial and Manufacturing Engineering, High Performance Materials Institute, Florida A&M University-Florida State University College of Engineering, 2525 Pottsdamer Street, Tallahassee, FL 32310-6064, USA
a r t i c l e
i n f o
Article history: Received 9 November 2008 Accepted 9 November 2008 Available online 17 November 2008 Keywords: A. Nano-structures B. Mechanical properties C. Micromechanics
a b s t r a c t The potential capability of improving overall elastic modulus of nanotube-reinforced composites is a fundamental concern in nanotechnology applications. Based on geometric analysis and micromechanics estimation, this study reports that the ratio of surface-to-surface distance of adjacent carbon nanotubes (CNTs) to the CNT diameter plays a key role in improving the overall elastic modulus of the CNT-reinforced composites when the tubes are perfectly aligned, completely separated from other tubes, and ideally bonded with the composite matrix. With the decrease of this ratio, that is, decrease of the surface-tosurface distance of adjacent CNTs and/or increase CNT diameter, the improvement capability increases. However, theoretical and experimental results show that an increase of the CNT diameters degrades the elastic moduli of CNTs. This paper discusses the criterion of choosing CNTs with larger diameter and addresses the factors influencing the surface-to-surface distance of adjacent CNTs. Ó 2009 Elsevier Ltd. All rights reserved.
1. Introduction Since the discovery of carbon nanotubes (CNTs) by Iijima [1], interest in CNTs has grown rapidly because of their unique superior properties. Both experimental and theoretical studies suggest that CNTs have extraordinary mechanical and electrical properties [2– 4]. For example, CNTs have a Young’s modulus about 1 TPa and a very high tensile strength due to their defect-free structure. As the mechanical properties of composites depend directly upon the embedded fiber mechanical behavior, replacing conventional microsized fibers with CNTs can potentially improve composite properties, such as tensile strength and elastic modulus. Consequently, composites of CNTs dispersed in metallic or polymeric matrices have attracted considerable attention in recent years [5,6]. However, how the elastic properties of CNT-reinforced composites can be improved relative to those of the composite matrix, even under ideal conditions, is not clearly understood. The objective of this paper is to clarify this problem. In other words, we want to answer the questions: what factors affect the potential capability to improve the overall elastic properties of CNT-reinforced composites, and how the factors do actually influence the capability? The potential improvement of elastic properties of CNT-reinforced composites will be estimated based on geometry and micromechanics. The elastic properties of a CNT-reinforced composite are determined by factors including composite constituent elastic proper-
* Corresponding author. Tel.: +1 850 410 6355. E-mail address: [email protected] (C. Zhang). 1359-8368/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2008.11.003
ties and microstructures, such as CNT elastic modulus [7,8], bundles [9], waviness [10], aspect ratio and orientation distribution [11], as well as interface between the CNTs and the polymer matrix [12–15]. It is generally believed that if the full advantage of the extraordinary properties of CNTs for reinforcement of composites can be achieved, the following conditions must be obeyed: CNTs must be (1) in a straight line; (2) perfectly aligned; (3) completely separated from other CNTs by the composite matrix; (4) infinitely long; and (5) ideally bonded with the composite matrix. However, the capability of elastic property enhancement of a CNTreinforced composite by CNTs is also related to another key factor – the maximum volume fraction or loading of CNTs. The maximum volume fraction achievable depends on CNT geometry, surfaceto-surface distance between adjacent CNTs and their packing pattern. Packing problems, i.e. how densely objects can fill a volume, or equivalently, how objects can be tightly packed in a space, are among the most ancient and persistent problems in mathematics and engineering. This was the 18th problem of Hilbert’s 23 famous problems. The maximum packing volume fraction is sensitive to the shape of particles packed. For example, the highest packing volume fraction of sphere in three-dimensional Euclidean space is believed as 0.7405, as realized by stacking variants of the face-centered cubic (fcc) lattice packing. This is the famous Kepler conjecture, which was only proven [16]. For two-dimensional congruent circles, it is well known that hexagonal packing is the optimal packing and its maximum packing volume fraction is 0.9069 [17]. The infinitely long circular cylinder with parallel in hexagonal packing has the same
B. Jiang et al. / Composites: Part B 40 (2009) 212–217
maximal packing volume fraction as that of the two-dimensional optimal packing of congruent circles, that is, 0.9069 [18,19]. This maximum packing volume fraction is much higher than that of sphere in three-dimensional space mentioned above, which suggests that reducing aspect ratio of particles decreases the maximum packing volume fraction. It is generally believed that the highest packing volume fraction of random packing of a given congruent particles is smaller than that of regular packing of the particles, although it is difficult to prove this conjecture [20]. The conventional hexagonal close packing, which yields maximum volume fraction of infinitely long cylinders, is shown in Fig. 1a (top view), in which each cylinder touches other six neighbor cylinders. In a real composite material system, however, cylinders or CNTs cannot touch each other even if there is no other phase in between two adjacent CNTs. The reasons can be illustrated as follows: the surfaces of CNTs are atomic smooth and the CNTs are subject to van der Waals interaction. Due to the balance of the van der Waals repulsive force and attractive force, the CNTs remain apart from each other at a certain distance that forms a surface-to-surface distance. Fig. 1b shows an ideal packing pattern in which all have the same diameter, length and the surface-to-surface distance, and the CNTs are hexagonally close packed. The volume fraction of the infinitely long CNTs, /inf, is
p 1 /inf ¼ pffiffiffi ; 2 3 ðS=d þ 1Þ2
ð1Þ
where d is the CNT diameter; and S the surface-to-surface distance between two adjacent CNTs. If we keep d as a constant, /inf increases with the decrease of S. When S ? 0, /inf reduces to that of the conventional hexagonal close packing and tends to its maximum value, 0.9069. 2. Determination of the maximum loading The determination of the surface-to-surface distance is complicated. Here we examine two special cases and outline a general case. If there are no other phases or materials in between CNTs, it is generally believed that the minimum surface-to-surface distance of two adjacent CNTs is the equilibrium van der Waals distance, which is about 0.34 nm. However, for a polymer CNTreinforced composite, in order to obtain best mechanical properties, polymer molecules must be in between CNTs to bond them together. Accordingly, the surface-to-surface distance must be larger than the equilibrium van der Waals distance. To illustrate this, suppose there is only one atom in the middle of two adjacent CNTs, which can also be imagined as a pi-stacking polymer chain in between CNTs. The surface-to-surface distance between the two CNTs should be doubled to 0.68 nm, since the atom has the equilibrium van der Waals distance to both of the CNTs. In an actual CNT-
a
D
b
D
S
d
Fig. 1. Hexagonal close packing: (a) without interface and (b) with interface.
213
reinforced composite system, the surface-to-surface distance between CNTs is larger than 0.68 nm because the pi-stacking polymer chain conformations do not exist at all locations. The complete discussion of the surface-to-surface distance is quite complex because many factors affecting are it, such as CNT geometry, polymer chain conformation, interaction between CNTs and polymer chains, and processing conditions of CNT-reinforced composites. We shall here provide a sketchy, albeit representative, overview based on the polymer chain conformation. In the processing of CNT-reinforced composites, CNTs need to be separated from bundles, and meanwhile, dispersed uniformly in a polymer matrix for maximizing their contact surface area with the matrix. This implies polymer chains should be put in between CNTs. This processing is related to the polymer chain conformation. If a flexible linear polymer chain is in a good solvent and the concentration of polymer chains is so low that there is no entanglement with each other, the size of polymer chains can be estimated by its square-root-mean end-to-end distance or its radius of gyration [21]. Based on the self-avoiding walk analysis, the end-to-end distance of a flexible polymer chain, R, can be obtained by R = lN3/5, where l is the monomer size and N is the number of monomers per polymer chain, which is equivalent to the molecular weight of the polymer chain. Suppose N is 100 and l is 0.154 nm or C–C bond distance. In fact, the monomer length of a typical polymer chain is larger than 0.154. We can find the polymer chain size is about 2.44 nm. If this polymer chain bonds two adjacent CNTs, the surface-to-surface distance is about 2.44 nm. If we want to reduce the surface-to surface distance between two adjacent CNTs, we must squeeze the polymer chain into a smaller size along the surface-to-surface distance direction. Because of this, the processing of a CNT-reinforced composite becomes difficult; therefore, CNT separation and dispersion are hard to obtain. Table 1 indicates the changes of the maximum volume fraction with CNT diameter and the surface-to-surface distance, which is obtained from Eq. (1). Examining four typical surface-to-surface distances, that is, 0.34 nm, 0.68 nm, 1.0 nm and 2.44 nm, with the CNT diameter varying from 1.0 nm to 30 nm, we can see that if CNT diameter is 1.0 nm, the maximum CNT volume fraction is only 50.51 vol.%, even though there is no material in between CNTs (S = 0.34 nm). If there is one atom in between CNTs, the maximum CNT volume fraction drops to 32.13 vol.% with the CNT diameter being 1.0 nm, which is only about one-third of maximum volume fraction of traditional fiber-reinforced composites. We believe that this is the maximum CNT volume fraction if CNTs have 1.0 nm diameter. If the surface-to-surface distance increases further, the maximum volume fraction reduces to 22.67 vol.% for S = 1.0 nm and only 7.66 vol.% for S = 2.44 nm at d = 1.0 nm. With the increase of CNT diameter, the surface-to-surface distance influence on the maximum CNT volume fraction becomes smaller. We can see that if CNT diameter is 30 nm, all four cases have the maximum CNT volume fraction more than 77 vol.%. An actual CNT cannot have infinite length, and therefore, the aspect ratio of a CNT, that is length-to-diameter ratio of the CNT, is finite. Fig. 2a shows a CNT-reinforced composite with CNTs as cylinders of the same diameter and length and CNTs have the same surface-to-surface distance and the same end-to-end distance. The model to estimate the maximum packing volume fraction of CNTs with finite aspect ratio is shown in Fig. 2b in which the surface-to-surface distance is, again, S; the end-to-end distance between two CNTs is H; and the length of CNTs is L. Accordingly, the representative volume element (RVE) has the length of L + H. The maximum packing volume fraction of CNTs with a finite aspect ratio, /a, can be estimated by
/a ¼
/inf ; 1 þ S=ðd aÞ ðH=SÞ
ð2Þ
214
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Table 1 Maximum volume fraction of infinitely long CNT with different diameters and surface-to-surface distances. CNT diameter (nm)
S = 0.34 (nm)
S = 0.68 (nm)
S = 1.0 (nm)
S = 2.44 (nm)
1 2 3 4 5 6 7 8 9 10 15 20 25 30
0.5051 0.6625 0.7316 0.7704 0.7951 0.8122 0.8248 0.8344 0.8420 0.8482 0.8671 0.8768 0.8827 0.8867
0.3213 0.5051 0.6027 0.6625 0.7027 0.7316 0.7534 0.7703 0.7840 0.7951 0.8299 0.8482 0.8595 0.8671
0.2267 0.4031 0.5101 0.5804 0.6298 0.6663 0.6943 0.7166 0.7346 0.7495 0.7971 0.8226 0.8385 0.8493
0.0766 0.1840 0.2758 0.3499 0.4096 0.4583 0.4987 0.5325 0.5613 0.5860 0.6709 0.7204 0.7527 0.7756
a
4.0
b S
Fiigure 4
L
H
Logarithm of the aspect ratio
3.5
0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.150 0.100 0.0750 0.0500 0.0375 0.0313 0.0250 0.0188 0.0125 0.00625 0
0.0313 0.0375
3.0
0.500 2.5
0.0500
0.400 0.0750
2.0
0.300 0.100
1.5
0.150 1.0
0.600
0.0250
0.200
0.0188
0.5
0.0125 0.0 0
1
2
3
4
5
S/d Fig. 3. Relationship among the aspect ratio of the nanotubes, S/d and the maximum volume fraction of the nanotubes.
where a = L/d is the CNT aspect ratio. If a ? 1, that is equivalent to the CNTs having infinite length, /a ? /inf. /a is smaller than /inf when a is finite. The change of the maximum CNT volume fraction with S/d and a can be obtained from Eq. (2) and is illustrated at H/S = 1 in Fig. 3. Generally, fiber diameters in conventional fiber-reinforced composites are in micrometers even in millimeter ranges. If the surface-to-surface distance between fibers is on the order of equilibrium van der Waals distance, the ratio S/d for the conventional composites is very small. It can be seen from Fig. 3 that when S/d is smaller than 0.01, the maximum CNT volume fraction is about 90 vol.%. In this range, the maximum volume fraction is insensitive to the CNT aspect ratio. However, in CNT-reinforced composites, the CNT diameters have the same order or even smaller than the surface-to-surface distance, which results in large S/d ratio. If S/d is unity and a is larger than 100, the maximum CNT volume fraction approach to 20 vol.%. When S/d further increases, say, to S/d = 3, the maximum volume fraction reduces to 5 vol.%. When S/d increases, the maximum volume fraction becomes more sensitive to the CNT aspect ratio. When S/d is larger than 2, decreasing the CNT aspect ratio from 100 to 10 significantly reduces the maximum volume fraction. To reveal how the reduction affects the overall elastic properties of CNT-reinforced composites, we use the micromechanics to estimate the effect of S/d and a on the overall elastic modulus.
3. Micromechanics analysis Micromechanics analysis is a very powerful tool to predict overall properties of a composite from the constituents and microstructures, and has been used successfully in various materials such as
4.0
160 200 120 240
3.5
Logarithm of the aspect ratio
Fig. 2. (a) Scheme of packing of an actual aligned nanotubes; (b) scheme of packing to estimate the volume fraction of the nanotubes.
80.0
50.0 35.0
25.0 20.0 17.5 15.0
40.0
3.0
350 320 280 240 200 160 120 80.0 60.0 50.0 40.0 35.0 30.0 25.0 20.0 17.5 15.0 12.5 10.0 7.50 5.00 2.50 0
30.0
60.0
2.5
12.5 10.0
2.0
7.50 5.00
1.5
2.50 1.0 0.5 0.0 0
1
2
3
4
5
S/d Fig. 4. The changes of the maximum overall Young’s modulus of the composite relative to that of polymer matrix with CNT aspect ratio and the S/d.
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B. Jiang et al. / Composites: Part B 40 (2009) 212–217
1 NT Lijkl ¼ ½/M LM ; ijpq þ /NT hLijmn Amnpq iNT /M Ipqkl þ /NT hApqkl iNT
ð3Þ
where /NT and /M are the volume fraction of the CNTs and the maM trix, respectively; LNT ijkl and Lijkl the stiffness tensors of CNTs and the matrix; Iijkl the fourth-order unit tensor; Aijkl the strain concentration tensor which can be determined by Eshelby tensor Sijkl as follows: 1 NT 1 M Aijkl ¼ ½Iijkl þ Sijmn ðLM mnpq Þ ðLpqkl Lpqkl Þ :
ð4Þ
In Eq. (3), hi is the average of ‘’ over the volume of the composite, which is related to the CNT shape, orientation and volume fraction. The Eshelby tensor, however, is only dependent on the CNT shape and Poisson ratio of the matrix. Theoretical analysis and experimental results show that for a fiber-reinforced composite, if fibers are perfectly aligned, the maximum and the minimum overall elastic modulus of the fiber-reinforced composite are along the directions that are parallel and perpendicular to the fiber direction, respectively [22,25]. The overall elastic modulus of a randomly-orientated-fiber-reinforced composite is in between the maximum and the minimum overall elastic modulus of the perfectly-aligned-fiber-reinforced composite. Since the case with perfectly-aligned-fibers gives the upper and lower bound of the overall elastic modulus and is easy to analyze, we will just focus on this special case. The enhancement of elastic modulus of the CNT-reinforced composite relative to that of the polymer matrix can be characterized by E33 =Ematrix , where Ematrix is Young’s modulus of the polymer matrix and E33 is Young’s modulus of the CNT-reinforced composite along the 3-direction, or in the CNT axial direction. A larger E33 =Ematrix implies higher enhancement of elastic modulus, and, if E33 =Ematrix is unity, no enhancement. E33 can be determined from Eq. (3) by E33 ¼ 1=M 3333 , where M ijkl ¼ ðLijkl Þ1 . If the CNTs are perfectly aligned, completely separated and ideally bonded with the polymer matrix, the maximum elastic modulus along the CNT axis direction can be obtained from Eq. (3). Table 2 shows the enhancement of Young’s modulus of CNT-reinforced composites with infinitely long CNTs. In the calculation, CNTs are assumed as isotropic and their Young’s moduli are 1 TPa and Poisson ratios are 0.2. The polymer matrix has 2.5 GPa Young’s modulus and 0.3 Poisson ratio. We can see that if the CNT diameter is 30 nm, E33 =Ematrix is about 340, that is, E33 =ENT 85%. This implies that the 85% of elastic modulus of the CNT is taken. However, if the CNT diameter is
Table 2 The enhancement of Young’s modulus of infinitely long CNT-reinforced composite, E33 =Ematrix , at different CNT diameters and surface-to-surface distances. (Young’s modulus of CNT does not change with its diameter.) CNT Diameter (nm)
S = 0.68 (nm)
S = 1.0 (nm)
S = 2.44 (nm)
1 2 3 4 5 6 7 8 9 10 15 20 25 30
129.19 202.51 241.34 265.33 281.39 293.10 301.68 308.37 313.53 318.23 332.35 339.44 344.07 346.76
91.45 161.81 204.54 232.57 252.28 266.87 278.06 286.90 294.09 300.04 319.03 329.21 335.55 339.90
31.57 74.41 111.03 140.58 164.41 183.86 199.96 213.46 224.95 234.81 268.67 288.43 301.35 310.46
1.0 nm, the enhancement of Young’s modulus drops to 129.19 ðE33 =ENT 32:29%Þ for S = 0.68 nm and 91.45 ðE33 =ENT 22:86%Þ for S = 1.0 nm. If the S further increases to 2.44 nm, the enhancement of Young’s modulus reduces to 31.57 and E33 =ENT 7:89%. This suggests that CNT elastic modulus of CNTs can be taken only 7.89% for S = 2.44 nm. The relationship between E33 =Ematrix with S/d and a is shown in Fig. 4. If we keep S/d as a constant, E33 =Ematrix reduces dramatically with a decrease of the CNT aspect ratio, when the CNT aspect ratio is smaller than 100. When S/d is small, say smaller than 0.01, and a is large, say larger than 1000, we can see that E33 =Ematrix is about 350, that is, E33 =ENT 87:5%, which is very close to Young’s modulus of the CNTs, ENT. However, if we keep a as 1000 and increase S/d, we can see that E33 =Ematrix becomes very small. For example, if S/d = 1, E33 =Ematrix is about 80, which is about one-fourth of that at S/d = 0.01. If S/d increases to 4, E33 =Ematrix decreases further to 15, that is E33 =ENT 3:75%. The above discussion strongly suggests that to take full advantage of the extraordinary properties of the CNTs, we must decrease S/d. In other words, we should decrease the surface-tosurface distance and/or increase the CNT diameter. If the elastic modulus of CNTs or CNT bundles do not change with their diameters, increasing the diameter of the CNTs or CNT bundles would improve the overall elastic modulus due to increase of the maximum CNT volume fraction. Unfortunately, many theoretical and experimental results indicate that the elastic modulus of CNTs or CNT bundles becomes lower with the increase of the diameter of the CNTs or the CNT bundles [7–9]. For instance, increasing a SWNT diameter from 0.35 nm ((5, 5) SWNT) to 1.35 nm ((20, 20) SWNT) decreases Young’s modulus from 1.12 TPa to 0.97 TPa, which declines 13.3% [7]. With the increase of diameter, Young’s modulus of CNT bundles drops much more than that of individual SWNTs. Experimental results show that if the diameter increases from 3 nm to 13.5 nm, Young’s modulus of the CNT bundles drops from 899 GPa to 298 GPa, which loses 66.9% [9]. All the experimental and theoretical results (e.g. [7– 9]) show that the relationship between Young’s modulus and diameter of CNTs or CNT ropes has the following form:
E ¼ m þ n ðd=d0 Þc ;
ð5Þ
where m, n and c are material parameters CNTs or CNT ropes, respectively; d0 is a reference CNT diameter; E is Young’s modulus of CNTs or CNT ropes and d is, again, the diameter of CNTs or CNT ropes and d P d0. For SWNTs, the theoretical prediction shows that c = 2 [7]. However, for a CNT rope, c is about 1 based on the exper-
Elastic modulus of the composite
metals, ceramics and composites [22]. If the interaction among the CNTs is estimated by Mori–Tanaka’s method [23], the overall elastic moduli of the CNT-reinforced composite, Lijkl , can be written as [24]
EA
EBCritical EC
EMatrix 0
φA
φB
1
Volume Fraction Fig. 5. Conceptual scheme for choosing CNTs or CNT bundles with larger diameter.
B. Jiang et al. / Composites: Part B 40 (2009) 212–217
iments by Salvetat et al. [9]. Table 3 displays the enhancement of Young’s modulus of the infinitely long CNT-reinforced composite considering the degeneration of elastic properties of CNTs or CNT ropes with the increase of their diameters. The simulation assumes that m = 0, n = 1 TPa, d0 = 1 nm, and c = 1. We can see from Table 3 that E33 =Ematrix decreases dramatically with diameter of CNTs or CNT ropes when S = 0.68 nm and 1.0 nm. For example, if CNT diameter is about 30 nm, E33 =Ematrix is about 11.70 for S = 0.68 nm, which is only one-tenth of that at d = 1 nm. However, for S = 2.44 nm, when CNT diameter is small (d 6 2 nm), the enhancement of Young’s modulus increases with CNT diameter increase. After that (d > 2 nm), the enhancement of Young’s modulus decreases with CNT diameter increases. Based on the discussion above, whether the overall elastic properties of CNT-reinforced composite could be improved or not through increasing diameter is determined by the competition between increasing maximum volume fraction and decreasing of elastic properties due to the increase of CNT diameter. Fig. 5 illustrates the conceptual scheme of how to consider the competition. Here we use the term of inclusion, which is widely used in micromechanics, to denote reinforcement phase. The inclusion could be CNTs or CNT bundles. Suppose the inclusions considered here are perfectly aligned and infinitely long as well as ideally bonded with the composite matrix based on micromechanics. The overall elastic properties of the composite would be almost linear with the inclusion volume fraction. The overall Young’s modulus of the composite must be equal to that of the polymer matrix at inclusion volume fraction vanishing and should be the same as that of inclusion at CNT volume fraction being unity. Here, we consider two types of composites: one has inclusion A (A-composite) whose diameter is dA and Young’s modulus is EA and the other composite has inclusion B (B-composite) whose diameter is dB and dB > dA. If the surface-to-surface distance between inclusions, say S, keeps constant and inclusion A is replaced by inclusion B, what criterion of elastic modulus of inclusion B should be satisfied so that the overall Young’s modulus of the composite can keep the same as or be higher than that before being replaced? From Eq. (1), we can obtain the maximum volume fraction of A- and B-composites as /A and /B, respectively. According to the schemes shown in Fig. 5, we can determine the critical Young’s modulus of the inclu, as follows: sion B, ECritical B
ECritical Ematrix /A Ematrix B ¼ þ 1 : EA EA /B EA
ð6Þ
From Eq. (1), we have
1.0 1.0 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.050 0
0.85 0.80 0.8
0.65 0.60 0.55
0.6
dSWNT / dI
216
0.50
0.40 0.35
0.45
0.30
0.4
0.25 0.20
0.2
0.15 0.10
0.0 0
1
2
3
4
5
S / dSWNT Fig. 6. Changes of the critical Young’s modulus with dSWNT/dI and S/dSWNT.
2 ECritical Ematrix Ematrix 1 þ ðdA =dB Þ B ¼ þ 1 : 1 þ ðS=dA Þ EA EA EA
ð7Þ
As a result, the criterion to choose a larger diameter inclusion B to replace inclusion A is
EB P ECritical ; B
ð8Þ
where Ematrix are Young’s modulus of the matrix. If the inclusion is SWNT, we can use Eq. (7) to estimate the critical Young’s modulus of inclusion B, which could be CNTs or CNT bundles. In the calculation, we assume ESWNT/Ematrix = 500. This value is chosen because for a polymer CNT-reinforced composite, the polymer matrix has, in general, about 2 GPa Young’s modulus and the SWNT has about 1 TPa Young’s modulus. Fig. 6 exhibits the results obtained from Eq. (7). We can see that when S/dSWNT is very small, the critical Young’s modulus of the nanotube with larger diameter is insensitive to dSWNT/dI because the maximum volume fraction is close to 90 vol.%. However, if S/dSWNT is large, the critical Young’s modulus is very sensitive to dSWNT/dI. For example, if S/dSWNT is unity, when CNT diameter is doubled, the critical Young’s modulus should be at least about 55% of the SWNT. If the diameter is 10 times of that of the SWNT, the critical Young’s modulus is about 22% of that of the SWNT. 4. Conclusions
Table 3 The enhancement of Young’s modulus of infinitely long CNT-reinforced composite, E33 =Ematrix , at different CNT diameters and surface-to-surface distances. (Young’s modulus of CNT change with its diameter by the relation given in Eq. (5), and m = 0 TPa, n = 1.0 TPa, d0 = 1.0 nm and c = 1.0.) CNT diameter (nm)
ENT (GPa)
S = 0.68 (nm)
S = 1.0 (nm)
S = 2.44 (nm)
1 2 3 4 5 6 7 8 9 10 15 20 25 30
1000.00 500.00 333.33 250.00 200.00 166.67 142.85 125.00 111.11 100.00 66.67 50.00 40.00 33.33
129.19 101.51 80.76 66.59 56.51 49.05 43.30 38.75 35.06 32.01 22.30 17.18 13.89 11.70
91.45 81.21 68.51 58.46 50.76 44.76 39.98 36.11 32.92 30.23 21.46 16.63 13.58 11.47
31.56 37.61 37.49 35.63 33.36 31.10 29.00 27.10 25.39 23.86 18.22 14.69 12.29 10.57
The main findings from this study are summarized below: (1) The maximum CNT volume fraction depends on CNT geometry, surface-to-surface distance between adjacent CNTs and CNT packing pattern. Since CNT diameter is on the same order of the equilibrium van der Waals distance, increase of the surface-to-surface distance and/or decrease of CNT diameter will reduce the maximum CNT volume fraction dramatically. (2) The maximum enhancement of elastic properties of CNTreinforced composite is related to the maximum CNT volume fraction. Decreasing the maximum CNT volume fraction significantly degrades the enhancement capability of CNTreinforced composites. (3) Under the criterion given in Eq. (8), the enhancement capability could be improved by means of increasing CNT diameter and/or decreasing the surface-to-surface distance between adjacent CNTs.
B. Jiang et al. / Composites: Part B 40 (2009) 212–217
Acknowledgement This work has been sponsored by the Air Force Research Laboratory, the NSF IUCRC Program, IUCRC members, and the Florida State University Cornerstone Research Program. References [1] Iijima S. Nature (London) 1991;354:56. [2] Saito R, Dresselhaus G, Dresselhaus MS. Physical properties of carbon nanotubes. London: Imperial College Press; 1998. [3] Dresselhaus MS, Dresselhaus G, Avouris Ph. Carbon nanotubes: synthesis, structure, properties, and application. Springer; 2001. [4] Qian D, Wagner GJ, Liu WK, Yu MF, Ruoff RS. Appl Mech Rev 2002;55:495. [5] Calvert P. Nature (London) 1999;399:210. [6] Thostenson ET, Ren Z, Chou T-W. Compos Sci Technol 2001;61:1899. [7] Yao N, Lordi VJ. Appl Phys 1998;84:1939. [8] Popov VN, Doren VE, Balkanski M. Solid State Commun 2000;114:395. [9] Salvetat J-P, Briggs GAD, Bonard J-M, Bacsa RR, Kulik A. Phys Rev Lett 1999;82:944. [10] Fisher FT, Bradshaw RD, Brinson LC. Appl Phys Lett 2002;80:4647.
[11] [12] [13] [14] [15] [16] [17] [18]
[19] [20] [21] [22] [23] [24] [25]
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Jiang B, Liu C, Wang Z, Zhang C, Wang B. Compos Part B – Eng 2007;38:24–34. Schadler LS, Giannaris SC, Ajayan PM. Appl Phys Lett 1998;73:3842. Qian D, Dickey EC, Andrews R, Rantell T. Appl Phys Lett 2000;76:2868. Ajayan PM, Schadler LS, Giannaris C, Rubio A. Adv Mater 2000;12:1671. Zhu J, Peng H, Rodriguez-Macias F, Margrave JL, Khabashesku VN, Imam AM, et al. Adv Funct Mater 2004;14:643. Hales TC. LANL e-print archive; 1998.