Elastic Properties of Single-Walled Carbon Nanotubes

Transcription

1 Elastic Properties of Single-Walled Carbon Nanotubes Emily Schiavone Department of Physics, Carthage College, Kenosha, WI May 19, 2010 Abstract A new and very promising development in the field of materials science is the discovery of carbon nanotubes. Research into these tiny structures is motivated by their unusual and potentially useful properties. Computational values for the Poisson s ratio seem to agree with experimental data and other computational predictions. However, both experimental and computational research into the Young s modulus of SWCNT s has yielded conflicting results. The disagreement over the value of the Young s modulus stems from the difficulty in defining an area over which the force is applied. 1 Introduction All forms of carbon exhibit unique characteristics. Two familiar macroscopic forms of carbon include diamond and graphite. In these two forms of carbon alone, we see the uniqueness of carbon. Diamond is the hardest naturally occurring substance, while graphite is one of the softest[1]. Diamond is completely transparent to visible light, while graphite is opaque and black. These disparities result from the different lattice structures of each substance(see Figure 1). Lattice structures are characteristic of crystalline solids, as opposed to amorphous solids that lack definitive geometric structure[2]. Diamond can follow an interpenetrating cubic or hexagonal lattice structure, while graphite consists of sheets of hexagonal lattices[3]. Clearly, the differences in geometry of the carbon structure at the atomic level drastically affect the macroscopic properties of the substance[4]. Carbon atoms can form a variety of structures. Diamond and graphite are well known forms of carbon. Other forms, some of which are pictured in Figure 2, include single sheets of graphite called graphene, fullerenes, also known as bucky balls, single-walled carbon nanotubes (SWCNT), multi-walled carbon nanotubes (MWCNT), and carbon nanohorns. Multiwalled carbon nanotubes are concentric cylinders made of carbon. The multi-walled carbon 1

2 Figure 1: The crystal structure of diamond and graphite[4]. nanotube was actually discovered before the single-walled variety. Single-walled carbon nanotubes can also be grouped into bundles that exhibit exceptional mechanical strength[5]. Each form of carbon possesses drastically different characteristics. For example, graphene resists tensile forces effectively; however, when these layers are stacked to form graphite, the layers of graphene easily separate when a shear force is applied along the direction of the graphene sheets. These structures can also be combined to form new arrangements. For example, so-called peapods consist of fullerenes inside of a single-walled carbon nanotube(figure 3). Under certain conditions, the fullerenes inside of these structures that resemble peapods will merge into a second nanotube within the SWCNT. Both peapods and SWCNT bundles are studied for potential application in hydrogen storage elements[6]. The properties of SWCNT s reflect the versatility of carbon. Research into SWCNT s electronic and elastic properties has demonstrated the wealth of their applications, but we can gain a more detailed understanding of their characteristics[7]. 2 Mathematical Description Single-walled carbon nanotubes are essentially sheets of graphene that are rolled up into hollow cylinders. The lattice structure of graphene, displayed in Figure 4, is essentially a tessellation of hexagons. The nuclear centers of the carbon atoms as well as the inner shell electrons are located at the vertices of the lattice. The line segments connecting these vertices represent the position of the cloud containing the valence electrons. The length of these line segments, or the bond length is 1.42Å[10].This lattice when rolled into a cylinder demonstrates the structure of a SWCNT. The ends of a carbon nanotube are naturally closed 2

3 Figure 2: Crystal lattice of different allotropes of carbon: a) diamond, b)graphite, c)graphene, d-f)fullerenes, g)swcnt[8].! Figure 3: SWCNT filled with fullerenes[9]. 3!

4 by half fullerenes, which eliminate any dangling bonds. The resulting structure resembles a cylinder closed at both ends by hemispheres. Figure 4: Graphene lattice[11].! There are three conformations of SWCNTs. These are zigzag, armchair, and general chirality or helical nanotubes [3, 5]. These names correspond to different configurations and describe the pattern that traces the circumference of the nanotube(figure 5). Therefore, the type of nanotube depends on which axis the sheet of graphene is rolled along. The direction that the lattice is rolled is called the chiral vector, Ch. The chiral vector is a linear combination of the two primitive translation, or basis vectors, of the graphene lattice, a and b. In other words, C h = n a + m b, where n, m ɛ Z. Typically, carbon nanotubes are classified by an ordered pair of these two integers: (n, m). The zig-zag variety of carbon nanotube is formed by wrapping the graphene lattice along the primitive translation vector a, thus forming a zig-zag pattern around the circumference of the nanotube. This direction corresponds to n 0 and m = 0, so a zig-zag SWCNT is denoted by (n, 0). Wrapping the graphene lattice along a vector π/6 radians above a results in an armchair SWCNT. In an armchair structure, n = m, making the representative ordered pair (n, n). Rolling the lattice along a vector that lies in between a and the chiral vector of an armchair carbon nanotube will result in a general chirality 4

5 nanotube. Due to the rotational symmetry of the graphene lattice, these three structural classes are the only possible variations of the SWCNT[3]. Previous research has found that armchair nanotubes are strictly metallic, while zig-zag and helical nanotubes can be either metallic or semiconducting[5].! Figure 5: Structural classes of SWCNT s[12]. 3 Applications There are many significant proposed applications of carbon nanotubes, some of which are currently in use. Depending on their morphology and diameter size, carbon nanotubes can be either semiconducting or metallic[5]. They easily conduct thermal energy and can be polarized. In addition, researchers can produce carbon nanotubes with a small amount of imperfections [7]. Based on these desirable characteristics, carbon nanotubes have been hypothesized to be used for conducive films, nanoprobes, transistors, infrared emitters, mechanical reinforcements, catalytic supports, logic gates, interconnects, biosensors, and hydrogen storage elements [5, 13]. Already carbon nanotubes have been incorporated into a device that measures the mass of small atoms[14]. Researchers also use carbon nanotubes to create new materials, such as thin, transparent and flexible films to be used for solar cells, touch screens, or flat panel displays, among other ideas[13]. Carbon nanotubes may provide quantum wires and heterojunction devices for nanoscale circuits. Carbon nanotubes may 5

6 also be used as nanoprobes for high-resolution imaging, nanolithography, nanoelectrodes, drug delivery, sensors, and field emitters[5]. Even though the potential applications of carbon nanotubes are known, the ability to reliably produce these structures on large scales remains elusive[7]. In addition, disagreement over the definition of the Young s modulus stands in the way of properly characterizing carbon nanotubes. Researchers use both experimental and computational approaches to learn about the elastic properties of carbon nanotubes. 4 Elastic Properties Elastic properties include Young s modulus, the shear modulus, the Lamé modulus, and Poisson s ratio[3]. Here we will focus on Young s modulus and Poisson s ratio of SWCNT s. 4.1 Young s Modulus Knowing the Young s modulus of a SWCNT is important because it can be used to predict the elongation or compression of the nanotubes under a certain stress. SWCNT s are interesting because they are simultaneously very stiff and light-weight[19]. We can think of a carbon nanotube as similar to a spring. If we apply a tensile force on one end of the spring, the spring stretches. This stretch depends on the force applied as well as the stiffness of the spring[15]. For a Hooke s law spring in one-dimension, we know that F = k( L), where F is the applied force, k is the stiffness of the spring, and L is the stretch, or change in length of the spring. In the case of a carbon nanotube, the stretch depends on the force applied and the strength of the interatomic bonds. Given a material of a certain length L, and cross-sectional area A, to which we apply a force F (See Figure 6), the strain, ɛ, describes the fractional change in length, ɛ = L L. The force per unit area defines the stress, σ, applied to a material[15]. In other words, σ = F A. 6

7 Figure 6: The Young s modulus[16].! When considering a three-dimensional object oriented within rectangular coordinates with a strain along the z-axis, the definition of the Young s modulus, E or sometimes Y, is the magnitude of the applied stress in the z-direction, σ z, divided by the strain, ɛ z, also along the z-axis[17]. Thus E = σz ɛ z = F/A L/L. Young s modulus measures the stiffness of a material[17]. The value of the Young s modulus of carbon nanotubes was expected to match the Young s modulus of a single graphene layer along the surface( 1.02TPa) or even the Young s modulus of diamond ( 1.063TPa)[18]. The difficulty in determining Young s modulus for SWCNT s is that a force cannot be applied to the full cross sectional area. Instead, the force only acts on discrete points along a perimeter. However, the Young s modulus of SWCNT s cannot be equated to the Young s modulus of graphene either, because something unique happens when the sheet is curved into a tube. This difficulty leads to an ambiguity of the definition of Young s modulus Computational Research There are three methods for defining Young s modulus for computational research. Some researchers define the volume of the nanotube to be a hollow cylinder with a certain thickness. The most commonly used thickness is 3.4Å(equal to the separation between layers of graphene in graphite and also equal to the separation between walls of MWCNT s)[19, 20]. 7

8 Alternatively, other scientists refer to this concept as the surface Young s modulus. The surface Young s modulus has dimensions of force per unit length, which allows the quantity to be more easily defined. The surface Young s modulus takes the form E S = F ɛ z C, where C is the perimeter of the SWCNT. An final method of determining Young s modulus is to take the second derivative of the strain energy with respect to the axial strain per atom of the nanotube[19]. The method of calculating Young s modulus in this approach is less obvious than in other strategies. To start, we can manipulate the definition of the Young s modulus to get σ z = Eɛ z. Euler s theorem gives us the relation V = 1 2 σ ikɛ ik, where V is the free energy, σ ik is the stress tensor, and ɛ ik is the strain tensor[24]. For the case of uniform tension along the z-axis, the elastic free energy reduces to V = 1 2 σ zɛ z We can then substitute Eɛ z for σ z, V = 1 2 (Eɛ z)ɛ z = 1 2 E(ɛ z) 2. The elastic free energy in the uniaxial load case is analogous to the elastic potential energy, U, of a linear spring, which we know to be U = 1 2 k( L)2. If we take the second derivative of the free energy per atom with respect to the strain in the z-direction, we do in fact find Young s modulus, d 2 dɛ 2 z V = E. 8

9 Computational models fall into two categories based on the type of calculation involved. These categories are classical and quantum. Quantum methods can be divided further into ab intio, or simulations that calculate quantities based on first principles using the Schrödinger wave equation, and simulations that use varying levels of approximation of the potential energy in the wave equation. The other approach uses classical mechanics. The model that use classical mechanics are useful in determining mechanical properties but are insufficient for investingating any electronic properties. When simulating nanoscale materials, the number of atoms in a single simulation may run from a few hundred or even a few thousand to several billion atoms. However, the number of atoms that even a highperformance platform using the most elegant ab intio methods can manage is limited to a few hundred atoms. Consequently these simulation programs use approximations and parameterizations to reduce the total number of atom-atom interactions[3]. In computational research of carbon nanotube structures, there exists disagreement over both the value of the Young s modulus and its definition. In addition, some studies found correlation between Young s modulus and the radius and chirality of SWCNT s and others did not. WenXing et al. conducted extensive studies using molecular dynamics simulation finding the Young s modulus of SWCNT to be in the range of ± 11.5 GPa. The study concluded that the Young s moduli of SWCNT s are weakly affected by changes in the nantoube chirality and radius, and the results are listed in Table 1[21]. Lu simulated SWCNT s and MWCNT s using an empirical force-constant model, but found no correlation between elastic moduli and helicity, radius or the number of walls (See Table 2)[18]. H.F Ye et al. use three different theoretical models, two of which are improved versions of the first[22]. These improved models, the eigenvalues modified method and the eigenvalues and eigen vectors modified method, closely resemble the commonly used finite element method and confirm that there is a dependence of Young s modulus on both the radius and the chirality of the nanotube. For small radii, the results showed the surface Young s modulus to be relatively small( 0.345TPa nm), and as the radius of the nanotube increased, the surface Young s modulus asymptotically approached the surface Young s modulus of graphene( 0.360TPa nm). Popov reached a similar conclusion for the relationship of the nanotubes radius and its Young s modulus. These results are shown in Figure 7. 9

10 Figure 7: Graphs of Young s Modulus(top), shear modulus(bottom) and Poisson s ratio(inset) according to results found by Povpov[19]. Most studies model SWCNT s as a hollow cylinder of thickness 3.4A, though at least one paper used the value of 0.66A [19]. Sears uses this model but does not assume the wall thickness[20]; instead, he determines the thickness of the equivalent continuum tube wall as a precursor to predicting its elastic properties. He utilizes two potentials: MM3 and TersoffBrenner, each of which yield different values for the wall s thickness and different elastic properties of a SWCNT. Table 3 shows this disparity and lists the results of additional simulations using the MM3 potential. Sears concludes that the radius and chirality of the nanotube has little effect on Young s modulus. However, his values for the Young s modulus are much larger than those of most other studies. 10

11 (n, m) Number of atoms Radius (nm) Length(nm) E (GPa) Armchair (8,8) (10,10) (12,12) (14,14) (16,16) (18,18) (20,20) Average ±0.618 Zigzag (14,0) (17,0) (21,0) (24,0) (28,0) (31,0) (35,0) Average ±2.887 General Chirality (12,6) (14,6) (16,8) (18,9) (20,12) (24,11) (30,8) Average ± Overall Average 929.8±11.5 Table 1: Data from WenXing et al. showing Young s modulus to be ± TPa. This study shows a weak dependence of Young s modulus on the chirality and radius of the nanotube[21]. 11

12 (n, m) Radius (nm) E (TPa) ν(tpa) Armchair (5,5) (10,10) (50,50) (100,100) (200,200) Zigzag (10,0) General Chirality (6,4) (7,3) (8,2) (9,1) Table 2: Data from Lu showing no correlation between elastic moduli and helicity or radius of the nanotube[18]. Potential (n,m) Thickness (Å) Radius (nm) E (TPa) ν(tpa) MM3 (16,0) Tersoff-Brenner (16,0) (Thickness not set by potential) (16,0) (Thickness not set by potential) (16,0) MM3 (8,0) MM3 (25,0) MM3 (48,0) MM3 (12,6) MM3 (10,10) Table 3: Data from Sears and Barta showing strong dependence of Young s modulus on wall thickness[20] Experimental Research The parameters used to create computational simulations are set by experimental values, or in some cases by calculations made by more sophisticated programs. If the experimental 12

13 values agreed with one another, then the computational models would fall into a smaller range of reasonable values for the Young s modulus. However, there also exists disagreement over experimental values. This problem arises because the Young s modulus is typically applied to continuous macroscopic materials. A SWCNT s walls have virtually no thickness, on the order of a few angstroms, and its total diameter typically ranges from about 1-40nm[18, 20, 21]. Wu uses optical spectroscopy to determine the chirality of the carbon nanotube and magnetic actuation to measure the Young s modulus of the nanotube[23]. To make the measurement of the Young s modulus, SWCNT s were suspended across a slit and placed into a magnetic field. Researchers then ran current through the nanotubes which caused the nanotubes to bend due to a Lorentz force, F = q( E + v B). The strain is then calculated from the lateral displacement of the nanotube. This study defined the nanotube to be a hollow cylinder with a wall thickness of 3.4Å and found the Young s modulus to be 0.99±0.13TPa. Many experimental results showed that the Young s modulus for SWCNT s nearly matched or exceeded that of diamond. Experimental work by Liu et al. found that the Young s modulus of SWCNT s varied inversely with the diameter of the nanotube, specifically as the diameter ranged from 8-40nm the Young s modulus of the nanotubes decreased from 1.0TPa to 0.1TPa[25]. Other experimental research by Krishnan et al., Tombler et al., and Yu et al. found values of the Young s modulus to be TPa, 1.2TPa, and TPa, respectively using methods of observing SWCNT vibrations at room temperature, bending SWCNT s with atomic force microscopy(afm), and pulling SWCNT ropes with AFM tips under a scanning electron microscope[26, 27, 28]. Other experimental approaches involve embedding the carbon nanotubes in a composite and deforming the composite with thermomechanical stresses. In one particular case, a carbon nanotube composite was cooled, ultimately shrinking the composite and compressing the carbon nanotubes. In this study, Lourie calculated the Young s modulus of SWCNT s and MWCNT s from shifts in the Raman spectra[29]. These shifts are related to the shortening of the interatomic bonds as the material is cooled and provide information about the strain of the nanotubes. The stress was determined using the concentric cylinder model described by Wagner[30]. Using the strain found by Raman spectroscopy and the concentric cylinder model, Lourie found the value to be around TPa, a value much larger than that of other studies. 4.2 Poisson s Ratio The Poisson s Ratio, denoted by ν, is defined as the transverse strain, ɛ x or ɛ y, divided by the longitudinal strain, ɛ z. More concisely, ν = ɛx ɛ z 13 = ɛy ɛ z.

14 The opposite signs of the transverse and longitudinal strains reflects the fact that as a material is stretched along the z-axis, the thickness along x- and y-axes diminishes[17]. Figure 8: The Poisson s ratio[31]. Research on the Poisson s ratio of SWCNT s is more scarce than for that of the Young s modulus. Experimental research into Poisson s ratio is limited or nonexistent. All the articles with computational predictions of the ratio used the Poisson s ratio for graphite to validate their accuracy. Data from Lu shows no correlation between radius or chirality and the Poisson s ratio and can be found alongside the Young s modulus values, E, in Table 2. Sears concludes there is no significant relationship between the size or orientation of the nanotube and Poisson s ratio[20]. Stiff materials such as steel have a low Poisson s ratio, while soft materials, like rubber, have a high Poisson s ratio. SWCNT s are stiff materials. This means that the change in width when the nanotube is stretched longitudinally is relatively small. Therefore, a SWCNT should have a small Poisson s ratio.! 5 Analysis For the Young s modulus, the findings of both experimental and computational research fall into two ranges. The first range is around 1TPa. The majority of research finds the Young s modulus to be in this range. The second range is from 2TPa to almost 4TPa. Studies claiming the Young s modulus to be within this range include experimental studies by Lourie and computational studies by Sears[29, 20]. The Young s modulus determined by Sears uses a much smaller wall thickness that 3.4Å, which accounts for a much larger 14

15 Young s modulus. One possible explanation for the high Young s modulus reported by Lourie is that the SWCNT s are not perfectly aligned within the composite. Therefore, the thermomechanical pressure caused by cooling the composite is not applied along the axis of all the carbon nanotubes within it. Some studies find that Young s modulus has no dependence on the radius or chirality of the nanotube. Research that reports that there is no dependence include computational studies by Lu. Hernandez claims that the empirical pair potential used by Lu to study this dependence did not reflect the effects of curvature of the carbon nanotube[32]. Experimental studies like that of Lourie cannot determine the relationship between Young s modulus and the chirality of the nanotube because the concentric cylinder model used to determine Young s modulus only applies to cylinders that are transversely uniform[30]. 6 Conclusions Research into the Young s modulus of SWCNT s has produced a wide range of values. This problem arises because the standard definition of Young s modulus applies to the mechanics of continuous materials. The pressure applied to SWCNT s does not act over a continuous area. Researchers employ various strategies for overcoming this problem. Some define the property as the surface Young s modulus; others define the carbon nanotube to be a hollow cylinder of a certain thickness. Some researchers have found the Young s modulus by taking the second derivative of the free energy with respect to axial strain. Young s modulus has been calculated for SWCNT s within composites using a concentric cylinder model and through the use of magnetic actuation. These different definitions lead to disagreement over the value of Young s modulus for SWCNT s. The studies that find correlation between Young s modulus and the radius and chirality of the nanotube admit that this correlation is weak. However, studies that do not find any relationship between Young s modulus and the geometry of the nanotube use methods that are not sensitive enough to detect a slight dependence. References [1] The Allotropes of Carbon. Interactive Nano-Visualization in Science and Engineering Education (IN-VSEE). 10 December < [2] Rodgers, Glen E. Descriptive Inorganic Coordination, and Solid-State Chemistry. Thomson Learning, Inc

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