Equivalent-Continuum Modeling of Nano-Structured Materials

Transcription

1 Coposites Science and Technology, Vol. 6, no. 14, pp (00) Equivalent-Continuu Modeling of Nano-Structured Materials Gregory M. Odegard a,*, Thoas S. Gates b, Lee M. Nicholson c, and Kristopher E. Wise d a National Research Council, NASA Langley Research Center, MS 188E, Hapton, VA 3681 b NASA Langley Research Center, MS 188E, Hapton, VA 3681 c ICASE, NASA Langley Research Center, MS 188E, Hapton, VA 3681 d National Research Council, NASA Langley Research Center, MS 6, Hapton, VA 3681 Abstract A ethod has been proposed for developing structure-property relationships of nano-structured aterials. This ethod serves as a link between coputational cheistry and solid echanics by substituting discrete olecular structures with equivalent-continuu odels. It has been shown that this substitution ay be accoplished by equating the olecular potential energy of a nano-structured aterial with the strain energy of representative truss and continuu odels. As iportant exaples with direct application to the developent and characterization of singlewalled carbon nanotubes and the design of nanotube-based structural devices, the odeling technique has been applied to two independent exaples: the deterination of the effectivecontinuu geoetry and bending rigidity of a graphene sheet. A representative volue eleent of the cheical structure of graphene has been substituted with equivalent-truss and equivalentcontinuu odels. As a result, an effective thickness of the continuu odel has been deterined. The deterined effective thickness is significantly larger than the inter-planar spacing of graphite. The effective bending rigidity of the equivalent-continuu odel of a graphene sheet was deterined by equating the olecular potential energy of the olecular odel of a graphene sheet subjected to cylindrical bending (to for a nanotube) with the strain energy of an equivalent-continuu plate subjected to cylindrical bending. KEY WORDS: nanotechnology, nanotubes, continuu echanics, olecular echanics, flexural rigidity, graphene sheet Noenclature Molecular odel and force field E el E g E ns E vdw E θ - Non-bonded electrostatic potential energy - Molecular potential energy of graphene sheet - Potential energy of nano-structured aterial - Non-bonded van der Waals potential energy - Bond-angle variation potential energy * Corresponding author. Tel.: E-ail address: g..odegard@larc.nasa.gov (G.M. Odegard).

2 E ρ E τ E ω K ρ - Bond stretching potential energy - Bond torsion potential energy - Bond inversion potential energy - Force constant associated with the stretching of bond K θ - Force constant associated with angle variation of bond angle K ω K ω L N p α r nt φ p - Force constant associated with bond inversion - Modified force constant associated with bond inversion - Length of carbon nanotube - Nuber of atos per carbon nanotube - Total nuber of carbon atos in carbon structure - Interatoic spacing of carbon atos in graphite - Radius of carbon nanotube - Average inversion angle (radians) φ σπ - Angle between σ and π bonds θ - Defored bond-angle Θ - Equilibriu bond-angle ρ Ρ - Defored bond length of bond - Equilibriu bond length of bond Truss and continuu odels a n A - Elastic rod type of outer portion of truss representative volue eleent - Cross-sectional area of rod of truss eber type n A nt b D I n r - Cross-sectional area of carbon nanotube - Elastic rod type of inner portion of truss representative volue eleent - Bending rigidity of a continuu plate - Moent of inertia of a continuu plate - Defored distance between joints of rod of truss eber type n

3 n R t Y g n Y - Undefored distance between joints of rod of truss eber type n - Thickness of a continuu plate (wall thickness of continuu tube) - Young s odulus of graphene sheet - Young s odulus of rod of truss eber type n Y nt r c - Young s odulus of carbon nanotube - Mid-plane radius of continuu nanotube wall r c i - Inner radius of continuu nanotube wall r c o Λ c Λ t ν - Outer radius of continuu nanotube wall - Mechanical strain energy of the continuu odel - Mechanical strain energy of the truss odel - Poisson s ratio of graphene sheet Boundary conditions u i x i ε ij ε γ - Displaceent coponents of equivalent-continuu representative volue eleent - Cartesian coordinate syste of the representative volue eleent - Strain coponents in the equivalent-continuu representative volue eleent - Applied strain - Applied shear strain 1. Introduction Nano-structured aterials have generated considerable interest in the aterials research counity in the last few years partly due to their potentially rearkable echanical properties [1]. In particular, aterials such as carbon nanotubes, nanotube and nanoparticle-reinforced polyers and etals, and nano-layered aterials have shown considerable proise. For exaple, carbon nanotubes could potentially have a Young s odulus as high as 1 TPa and a tensile strength approaching 100 GPa. The design and fabrication of these aterials are perfored on the nanoeter scale with the ultiate goal to obtain highly desirable acroscopic properties. One of the fundaental issues that needs to be addressed in odeling acroscopic echanical behavior of nano-structured aterials based on olecular structure is the large difference in 3

4 length scales. On the opposite ends of the length scale spectru are coputational cheistry and solid echanics, each of which consists of highly developed and reliable odeling ethods. Coputational cheistry odels predict olecular properties based on known quantu interactions, and coputational solid echanics odels predict the acroscopic echanical behavior of aterials idealized as continuous edia based on known bulk aterial properties. However, a corresponding odel does not exist in the interediate length scale range. If a hierarchical approach is used to odel the acroscopic behavior of nano-structured aterials, then a ethodology ust be developed to link the olecular structure and acroscopic properties. In this paper, a ethodology for linking coputational cheistry and solid echanics odels has been developed. This tool allows olecular properties of nano-structured aterials obtained through olecular echanics odels to be used directly in deterining the corresponding bulk properties of the aterial at the acroscopic scale. The advantages of the proposed ethod are its siplicity and direct connection with coputational cheistry and solid echanics. In addition, the proposed ethod has been deonstrated with two independent exaples that have direct application to the developent and characterization of single-walled carbon nanotubes (SWNT). First, the effective geoetry of a graphene sheet has been deterined. A representative volue eleent (RVE) of the graphene layer has been odeled as a continuous plate with an effective thickness that has been deterined fro the bulk in-plane properties of graphite []. Second, the effective bending rigidity of a graphene sheet has been deterined. The bending rigidity of the elastic plate has been described in ters of the atoic interactions that doinate the overall bending behavior.. Carbon nanotubes In 1991 Iijia [3] obtained transission electron icrographs of elongated, nano-sized carbon particles that consisted of cylindrical graphitic layers, known today as carbon nanotubes (Figure 1). Since then, carbon nanotubes have becoe a priary focus in nanotechnology research due to their apparent exceptionally high stiffness and strength [1]. One of the fundaental issues that scientists and engineers are confronting is the characterization of the echanical behavior of individual carbon nanotubes. Two independent coponents of this issue, the wall thickness and bending rigidity of carbon nanotubes, are discussed below..1 Wall thickness Many experiental [4-8] and theoretical [9-14] studies have been perfored on single- and ulti-walled carbon nanotubes. In particular, deforation odes and nanotube stiffnesses have been closely exained. Physical properties, such as effective cross-sectional area and oent of inertia, and echanical properties, such as Young s odulus and Poisson s ratio, are traditionally associated with the acroscopic-length scale, where the characteristic diensions of a continuu solid are well defined. The deterination of these properties has been attepted in any of the studies cited above without proper regard to an acceptable definition of the nanotube geoetry. Accurate 4

5 values of acroscopic physical and echanical properties are crucial in establishing a eaningful link between nanotube properties and the properties of larger structures, such as nanotube-reinforced polyer coposites. Therefore, caution should be used when applying continuu-type properties to nano-structured aterials. In any studies, it has been assued that the nanotube wall thickness is erely the interplanar spacing of two or ore graphene sheets [5, 8-13], which is about 0.34 n in single-crystal graphite. While this siple idealization appears to have intuitive erit, it does not necessarily reflect the effective thickness that is representative of continuu properties. In order to avoid this proble, Hernandez et al. [10] proposed the use of a specific Young s odulus, i.e., Young s odulus per unit thickness. Even though this approach is convenient for studies concerned with the relative stiffnesses of nanotubes, it is of little use when odeling a nanotube as a continuu structure. Another proposed solution to this dilea is to assue that the nanotube is a solid cylinder [15, 16]. This ethod is certainly convenient, however, significant inconsistencies arise when coparing oduli data of single wall nanotubes (SWNT) and ultiwalled nanotubes (MWNT) when both are assued to be solid cylinders. It follows that in order to properly odel the echanical behavior of a SWNT using continuu echanics, the effective geoetry ust be known. If the nanotube is odeled as a continuous hollow cylinder with an effective wall thickness, then a siple first step is to odel the flat graphene sheet in order to deterine the effective wall thickness. In the current study, the effective thickness will be calculated as an exaple of the proposed odeling approach.. Bending rigidity Due to the aspect ratio and tube-like geoetry of SWNT, any studies have been conducted concerning the buckling and bending response of nanotubes using both theoretical [1, 15, 17-0] and experiental [1, ] approaches. In particular, Overney et al. [17] conducted a coputational study and calculated a bending paraeter of a graphene sheet based on the vibrational odes of a nanotube. Yakobson et al. [18] used coputational ethods to study the buckling of carbon nanotubes. They odeled the nanotubes as shells with a bending rigidity proportional to the Young s odulus and shell-wall thickness. Using the coputationally obtained bending paraeters, they calculated the Young s odulus and wall thickness of the shell. Govindjee and Sackan [19] theoretically investigated the validity of continuu echanics at the nano-scale by exaining the bending of ulti-walled carbon nanotubes. They also assued that the bending rigidity of each layer is proportional to the Young s odulus and oent of inertia. In each of these studies it is clear that the bending and buckling behavior of carbon nanotubes is highly dependent on the bending properties of the graphene sheet. Therefore, it follows that the bending behavior of a graphene sheet ust be well-understood, and should be described in ters of the atoic properties of graphene. According to the classical elasticity theory, it is assued that the bending rigidity, D, of an isotropic solid is related to the cross-sectional geoetry and the in-plane odulus [3]: 5

6 D = Y t g 3 1(1 ν ) (1) and D g = Y I () which are for the bending of plates and beas, respectively. In equations (1) and (), Y g, ν, I, and t are the Young s odulus, Poisson s ratio, oent of inertia, and plate thickness of a graphene sheet, respectively. It sees logical that these equations could describe the bending properties of graphene sheets and nanotubes due to geoetric siilarities to solid plates and/or beas. However, Ru [0] has pointed out that a discrepancy exists in the use of equation (1) when describing bending properties of carbon nanotubes. If the bending rigidity described by this equation is used in the bending analysis of carbon nanotubes, then an equivalent wall thickness ust be used that is very sall copared to the inter-planar spacing of graphene sheets. For exaple, Yakobson et al. [18] showed that the bending rigidity of carbon nanotubes is uch saller than that described by equation (1) if the typically assued effective thickness of 0.34 n (the inter-plane spacing of layers of graphite) of graphene sheets is used. They derived an effective thickness and Young s odulus of n and 5.5 Tpa, respectively, for a SWNT. The arguent of Ru [0] is further supported by exaining the cheical structure of a graphene sheet. At first glance, it appears that a graphene sheet would have very little bending rigidity since the atoic C-C bonds lie very close to the neutral axis during cylindrical bending, unlike continuous elastic plates in which there is aterial that is not on the neutral axis that contributes a resistance in bending that is proportional to the in-plane Young s odulus (equations (1) and ()). Therefore, the priary atoic bonds, which are the ain contribution to the in-plane elastic properties of graphene, should provide little, perhaps negligible, contribution to the bending rigidity of graphene sheets. The resistance to bending ust be due to a different atoic interaction, which is discussed in section Modeling procedure The proposed ethod of odeling nano-structured aterials with an equivalent-continuu is outlined below. Since the approach uses the energy ters that are associated with olecular echanics odeling, a brief description of olecular echanics is given first followed by an outline of the equivalent-truss and equivalent-continuu odel developent. 3.1 Molecular echanics An iportant coponent in olecular echanics calculations of the nano-structure of a aterial is the description of the forces between individual atos. This description is characterized by a force field. In the ost general for, the total olecular potential energy, E ns, for a nanostructured aterial is described by the su of any individual energy contributions: (3) E = E + E + E + E + E + E ns ρ θ τ ω vdw el 6

7 where E ρ, E θ, E τ, and E ω are the energies associated with bond stretching, angle variation, torsion, and inversion, respectively (the reader should refer to a olecular echanics text, e.g. [4], for a detailed description of these energy ters). The nonbonded interaction energies consist of van der Waals, E vdw, and electrostatic, E el, ters. The suation occurs over all of the corresponding interactions in the considered volue of the nano-structured aterial. Various functional fors ay be used for these energy ters depending on the particular aterial and loading conditions considered [4]. Obtaining accurate paraeters for a force field aounts to fitting a set of experiental or calculated data to the assued functional for, specifically, the force constants and equilibriu structure. In situations where experiental data are either unavailable or very difficult to easure, quantu echanical calculations can be a source of inforation for defining the force field. 3. Truss odel In order to siplify the calculation of the total olecular potential energy of olecular odels with coplex olecular structures and loading conditions, an interediate odel ay be used to substitute for the olecular odel. Due to the nature of olecular force fields, a pin-jointed truss odel ay be used to represent the energies given by equation (3), where each truss eber represents the forces between two atos. Therefore, a truss odel allows the echanical behavior of the nano-structured syste to be accurately odeled in ters of displaceents of the atos. This echanical representation of the lattice behavior serves as an interediate step in linking the olecular potential with an equivalent-continuu odel. In the truss odel, each truss eleent corresponds to a cheical bond or a significant non-bonded interaction. The stretching potential of each bond corresponds with the stretching of the corresponding truss eleent. Traditionally, atos in a lattice have been viewed as asses that are held in place with atoic forces that reseble elastic springs [5]. Therefore, bending of truss eleents is not needed to siulate the cheical bonds, and it is assued that each truss joint is pinned, not fixed. The echanical strain energy, Λ t, of the truss odel is expressed in the for: t Λ = n AY R n n n ( r n R) (4) n where n n A and Y are the cross-sectional area and Young s odulus, respectively, of rod of truss eber type n. The ter ( r n n R) is the stretching of rod of truss eber type n, where n R and n r are the undefored and defored lengths of the truss eleents, respectively. In order to represent the cheical behavior with the truss odel, equation (4) ust be equated with equation (3) in a physically eaningful anner. Each of the two equations are sus of energies for particular degrees of freedo. The ain difficulty in the substitution is specifying equation (4), which has stretching ters only, for equation (3), which also has bond-angle variance and torsion ters. No generalization can be ade for overcoing this difficulty for every nano-structured syste. A feasible solution ust be deterined for a specific nano- 7

8 structured aterial depending on the geoetry, loading conditions, and degree of accuracy sought in the odel. 3.3 Equivalent-continuu odel For any years, researchers have developed ethods of odeling large-area truss structures with equivalent-continuu odels [6-31]. These studies indicate that various ethods and assuptions have been eployed in which equivalent-continuu odels have been developed that adequately represent truss structures. In general, the equivalent-continuu odel is defined as a continuu that has the following characteristics: 1. Truss lattices with pinned joints are odeled as classical continua where icropolar [3] continuu assuptions are not necessary.. Local deforations are accounted for. 3. The teperature distribution, loading and boundary conditions of the continuu odel siulate those of the truss odel. 4. The sae aount of theroelastic strain energy is stored in the two odels when defored by identical static loading conditions. The paraeters of the equivalent-continuu odel, such as the elastic properties and geoetry, are deterined based on the above characteristics. In soe cases the strain energy of the continuu, Λ c, can be easily forulated analytically and copared directly with equation (4) to obtain the equivalent-continuu properties. In other cases, especially with coplex geoetries and deforations, nuerical tools need to be used to deterine the continuu paraeters. Once the properties of the equivalent-continuu odel have been deterined, the echanical behavior of larger structures consisting of the nano-structured aterial ay be predicted using the standard tools of continuu echanics. 4. Exaple 1: Effective geoetry of a graphene sheet In this section, a graphene sheet is odeled as a continuous plate with a finite thickness that represents the effective thickness for the deterination of continuu-type echanical and physical properties. By using the ethodology described above, the olecular echanics odel is substituted with a truss odel and subsequently an equivalent-plate odel. The continuu odel ay then be used in further solid echanics-based analyses of SWNT. 4.1 Representative volue eleent To reduce the coputational tie associated with odeling the graphene sheet, a representative volue eleent (RVE) for graphene was used in this study (Figure ). The selected RVE allows each degree of freedo of the carbon ato associated with bond stretching and bond-angle variation in the hexagonal ring to be copletely odeled by truss and continuu finite eleent odel nodal-displaceent degrees of freedo. Also, this RVE allows the displaceents on the boundary of the proposed cheical, truss, and continuu odels to correspond exactly. Furtherore, acroscopic loading conditions applied to a continuous graphene plate can be easily reduced to periodic boundary conditions that are applied to the RVE. 8

9 4. Molecular echanics odel The specific fors of the energy ters in equation (3) used in this exaple were taken fro the AMBER force field of Kollan and coworkers [33, 34]. Due to the nature of the aterial and loading conditions in the present study, only the bond stretching and bond-angle variation energies were included. Torsion, inversion, and non-bonded interactions were assued to be negligible for the case of a graphene lattice subjected to sall deforations. For this exaple, the olecular potential energy of a graphene sheet with carbon-to-carbon bonds is expressed as a su of siple haronic functions: ρ θ ( ) ( ) g E = K ρ Ρ + K θ Θ (5) where the ters Ρ and Θ refer to the undefored interatoic distance of bond and the undefored bond-angle, respectively. The quantities ρ and θ are the bond length and angle after stretching, respectively (see Figure 3). K ρ and K θ are the force constants associated with the stretching and angle variance of bond and bond-angle, respectively. Using the paraeters for the AMBER force field [33], the force constants used in this exaple are: K K ρ θ kcal nj = = ole n bond n kcal 10 nj = 63 = ole rad angle rad 7 (6) The equilibriu bond length, Ρ, is n, and the undefored bond-angle, Θ, is 10.0 degrees. 4.3 Truss odel In order to express the echanical strain energy, Λ t, of the truss odel in ters of the variable truss joint angles that are specified in olecular echanics (θ -Θ ), the RVE has been odeled with additional rods between nearly adjacent joints to represent the interaction between the corresponding carbon atos (Figure 3). In order to represent the cheical odel, which has bond stretching and variable angles as degrees of freedo, with a truss odel that has stretching degrees of freedo only, two types of elastic rods, a and b, are incorporated into the truss RVE. Alternatively, a torsional spring eleent ay be used to odel the bond-angle variation, however, the nuber of degrees of freedo of every truss joint (and the coplexity of the odel) is significantly increased. 9

10 The echanical strain energy, Λ t, of the discrete truss syste shown in Figure 3 is expressed in the for of equation (4) as: AY AY R R a a b b Λ t = + a a b b ( r R a ) b ( r R) (7) where the superscripts correspond to rod types a and b, respectively. Coparing equations (5) and (7), it is clear that the bond stretching ter in the equation (5) can be related to the first ter of equation (7) for the rods of type a: a a AY K ρ = (8) a R where it is assued that ρ = r a a and Ρ = R. However, the second ters in equations (5) and (7) cannot be related directly. In order to equate the constants, the cheical bond-angle variation ust be expressed in ters of the elastic stretching of the truss eleents of type b. For siplicity, it ay be assued that the prescribed loading conditions consist of sall, elastic deforations only, even though, in general, the proposed odeling approach could be applied to larger deforations. This assuption is not an over-siplification for the graphene sheet since the deforations for highly stiff linear-elastic aterials subjected to any practical loading conditions are quite sall. In order to express the Young s odulus of the rods of type b in ters of the bond-angle force constant, a relationship between the change in the bond angle and the corresponding change in length of the type b truss eleent is required. If it is assued that the changes in bond angle are sall, then it can be easily shown that (see Figure 4): ( r R) b b θ Θ (9) a R The right-hand side of equation (9) is four ties larger than the right hand side of equation (8) given by Odegard et al. [35]. This discrepancy is due to an iproveent in the assuptions used to derive equation (9) in the current study. Substitution of equation (9) into equations (5) and (7) results in the following approxiation: Therefore, the Young s oduli of the two rod types are: b b b RAY K θ = (10) 4 10

11 Y ρ K R = a A a a Y 4K = R A θ b b b (11) The strain energy of the truss odel ay then be expressed in ters of the force constants: θ t a a 1K b Λ = + ( R ) ρ b K ( r R) b ( r R) (1) 4.4 Equivalent-plate odel Working with the assuptions discussed herein, the next step in linking the olecular and continuu odels is to replace the equivalent-truss odel with an equivalent-continuous plate with a finite thickness (Figure 3). For this exaple, it is assued that the truss and continuu odels are equivalent when the elastic strain energy stored in the two odels are equal under identical displaceent boundary conditions. The value of the plate thickness that results in equal strain energies is the assued effective thickness of the graphene sheet. While the echanical properties of the truss eleents have been deterined as described above, those of the graphene sheet were taken fro the literature. Values for the in-plane echanical properties of graphite have been easured acroscopically, i.e., without any assuptions regarding the graphene sheet thickness. For this exaple, the values of the Young s odulus and Poisson s ratio of bulk graphite are Y g = 1008 GPa and ν = 0.145, respectively []. For siplicity, it is assued that graphite is isotropic since no out-of-plane deforations are considered here. Given that there are no consistent data available on the properties of a single graphene sheet, these properties of the graphene sheet are based on the in-plane properties of bulk graphite. 4.5 Boundary conditions In order to deterine an effective plate thickness, both the truss and continuu odels were subjected to three sets of loading conditions. For each set of loading conditions, a corresponding effective thickness was deterined. The loading conditions correspond to the three fundaental in-plane deforations of a plate, that is, unifor axial tension along x 1 and x and pure shear loading in the x 1 and x plane. For a uniaxial deforation along the x direction (load case I), the RVE ay be subjected to the following boundary conditions (Figure 5) [35]: 11

12 u1 = νεx u = εx u 1 = νεx 3 3 (13) where the displaceent coponents are parallel to the corresponding RVE coordinates. The total strain energy can be calculated using [3]: g c VY ν Λ = ε kk +εijεij 1+ν 1 ν ( ) (14) where V is the volue of the RVE, i,j,k = 1,,3 (using suation notation), and the coponents of the strain tensor, ε ij, are given by: 1 u j u i ε ij = + xi x j (15) For the RVE under the conditions given in equation (13): c 3 3 Λ = R ty 4 a ( ) g ε (16) where t is the thickness of the continuu plate. For a uniaxial deforation along the x 1 direction (load case II), the boundary conditions are (Figure 6) [35]: u1 = εx u = νε u 1 x = νεx 3 3 (17) The total strain energy of an equivalent-continuu RVE under this condition is given by equation (16). For a pure shear strain in the x 1 -x plane (Load case III), the RVE ay be subjected to the following boundary conditions (Figure 7) [35]: u = 0 u u 1 3 = γx = 0 1 (18) The total strain energy of an equivalent-continuu RVE under this condition is: c Λ = g 3 3 Y a ( R ) t 8 1+ ( ν) γ (19) 1

13 4.6 Results and discussion The strain energy of the truss odel was calculated using a finite eleent analysis (ANSYS 5.7 ) for all three boundary conditions. The strain energies of each truss eleent were sued to obtain the total strain energy for the RVE. The resulting strain energy was equated with equations (16) or (19) for Load cases I and II or III, respectively, and the corresponding effective thickness of the equivalent-continuu plate, t, was deterined. Each extensible rod was odeled using a finite truss eleent (LINK1) with two degrees of freedo at each node (displaceents parallel to x 1 and x ). The cross-sectional areas of the type a rods were divided by a factor of, since these rods are sharing their total area with adjacent RVEs. For Load cases I and II, the resulting effective thickness was calculated to be 0.69 n. For Load case III, the effective thickness was 0.57 n. These values are significantly larger than the widely accepted value for the graphitic inter-planar spacing, 0.34 n, and uch larger than the value suggested by Yakobson et al. [18] of n. It ay be assued that during uniaxial loading of a carbon nanotube, with the force and strain known, the Young s odulus can be calculated using: Y ( A ) nt nt 1 (0) where A nt is the cross-sectional area of the hollow continuu cylinder with a constant id-plane radius, r c. The inner radius, r c c, and outer radius, r, of the tube are (Figure 8): i o r c i r c o c t = r c t = r + (1) where c r t /. The cross-sectional area of the hollow continuu cylinder is: A nt c = π tr () The calculated cross-sectional areas using the effective thickness values obtained above are shown in Table 1. The ratios of the calculated Young s odulus based on the effective thickness obtained for the three load cases, their average, and fro Yakobson et al. [18], to the Young s odulus based on inter-planar spacing, are also provided in Table 1. It was assued that the echanical properties of the SWNT wall are equal to those of the equivalent-plate odel. The results shown in Table 1 suggest that easured and calculated values of echanical properties of carbon nanotubes that are dependent on the diensions of the continuu tube ay differ significantly based on the assued geoetry. 13

14 5. Exaple : Effective bending rigidity of a graphene sheet In this section, a graphene sheet is again odeled as a continuous plate. The bending rigidity of the plate is assued to be independent of the in-plane echanical properties and thickness of the plate. By using the proposed odeling ethod, the olecular echanics odel is used to deterine the effective bending rigidity of the continuu plate. 5.1 Molecular echanics odel It has been shown that the only significant change in the electronic structure of a flat graphene sheet when subjected to pure bending is the change in the π-orbital electron density on either side of the graphene sheet (i.e. inversion, see Figure 9) [36-41]. This indicates that the inversion alone contributes to the bending resistance of graphene sheets. Bakowies and Theil [36] have suggested that the increase in the total olecular potential energy per carbon ato of a carbon cluster (i.e. a structure fored by a single plane of carbon atos, such as carbon Fullerenes and SWNT) with respect to a flat graphene sheet ay be closely approxiated as: E ω ω = K φ (3) p where K ω is a force constant and φ p is the average inversion angle defined as (in radians): φ = p p π φσπ p (4) where p is the total nuber of carbon atos in the carbon structure considered and φ σπ is the angle defined in Figure 9. For siplicity in the specific case of carbon nanotubes, the π-orbital axis vector technique [4] can be used to show that the change in the olecular potential energy due to inversion can be expressed in ters of the nanotube radius, r nt : ω K E ω = (5) ( r nt ) where K ω is a odified inversion force constant given by: ( ) ω ω K = rad n K (6) Values of the force constant, K ω, have been deterined using coputational cheistry data fro several studies [36, 43-46] and equation (5). Even though the coputational cheistry data was obtained for the specific case of bending of graphene sheets to for coplete nanotubes or nanotube-like structures, the concepts of structural echanics can be ipleented 14

15 such that these values could apply to any bending ode of graphene sheets. The values are shown in Table along with the overall average, which is: The total nuber of carbon atos per nanotube is [43]: K ω ev n = (7) ato N nt 4πr L = 3α nt (8) where α is the interatoic spacing of carbon atos and L nt is the nanotube length. Therefore, the total change in the olecular potential energy of a nanotube due to inversion is: ω E N nt 4πL K = nt 3α r ω (9) 5. Equivalent-continuu plate In the case of a graphene sheet subjected to pure bending, the total olecular potential energy is easily described with a single force constant by using equation (9), unlike the exaple discussed in the previous section where an interediate truss odel was needed in order to calculate the strain energy under various loading conditions. Therefore, the truss odel is not needed in this particular exaple and an equivalent-continuu odel has been developed by equating the total olecular potential energy of the olecular odel and the strain energy of the continuu plate directly. The strain energy of a plate of length L c subjected to cylindrical curvature (Figure 8) is [47]: c c πdl Λ = (30) c r where r c is the radius of curvature, D is the bending rigidity, and the superscript c denotes the strain energy associated with the continuu plate. In this case, D is not defined by equations (1) or () for the reasons discussed above. In order to deterine the bending rigidity of the equivalent-continuu plate that represents the actual effective bending rigidity of the graphene sheet, the strain energy of the continuu plate and the increase in the total olecular potential energy of the graphene sheet ust be equivalent when subjected to pure bending. Equating equations (9) and (30) results in: ω 4K D = (31) 3 α 15

16 Therefore, the effective bending rigidity is directly proportional to the inversion force constant and the inverse square of the interatoic spacing of graphite. The equivalent bending rigidity for the equivalent-continuu plate is calculated using equation (31) and the average value of the force constant K ω, given in Table, and with α = n [] to give D = 1. ev. This value is 44% higher than that used by Yakobson et al. [18] and Robertson et al. [43] (D = 0.85 ev). The strain energy of a graphene sheet is presented in Figure 10 for different nanotube radii for all of the coputational cheistry data and the equivalent continuu plate odel. It is assued that the graphene sheet behaves elastically in bending for all nanotube radii. Clearly, the trends of the coputational cheistry data and the equivalent-continuu odel are in agreeent. 6. Suary A ethod has been presented for odeling structure-property relationships of nano-structured aterials. This ethod serves to link coputational cheistry, which is used to predict olecular properties, and solid echanics, which describes acroscopic echanical behavior based on bulk aterial properties. This link is established by replacing discrete olecular structures with equivalent-continuu odels. It has been shown that this replaceent ay be accoplished by equating the olecular potential energy of nano-structured aterials with the echanical strain energy of a representative continuu odel. The developent of an equivalent-truss odel ay be used as an interediate step in establishing the equivalentcontinuu odel. The proposed odeling ethod has been applied to deterine the effective geoetry and effective bending rigidity of a graphene sheet. A representative volue eleent (RVE) of the cheical structure of a graphene sheet has been substituted with RVEs of equivalent-truss and equivalent-continuu odels. As a result, an effective thickness of the continuu odel has been deterined. This effective thickness has been shown to be significantly larger than the inter-planar spacing of graphite. The effective bending rigidity of an equivalent-continuu plate odel of a graphene sheet was also deterined using the proposed ethod. The olecular potential energy of the olecular odel of a graphene sheet subjected to cylindrical bending (to for a nanotube) was equated with the strain energy of an equivalent-continuu plate subjected to cylindrical bending. Acknowledgeents This work was perfored while Dr. Odegard and Dr. Wise held National Research Council Research Associateship Awards at NASA Langley Research Center. 16

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21 Table 1 - Ratio of calculated Young's odulus for different wall thicknesses of SWNT with respect to the Young's odulus calculated with the inter-planar spacing Continuu Continuu Ratio of Young's wall thickness cross-sectional area oduli [n] [n ] [%] Load cases I and II r c 0.49 Load case III r c 0.60 Average of I, II, and III r c 0.5 Yakobson et al r c 4.86 Inter-planar spacing r c 1.00 Table - Values of the force constant associated with bond inversion of a graphene sheet Study K ω [ev n /ato] Bakowies and Thiel [36] Robertson et al. - EP1 [43] Robertson et al. - EP [43] Robertson et al. - LDF [43] 0.01 Sawada and Haada [44] Miyaoto et al. [45] 0.00 Hernandez et al. (n,n) [46] 0.01 Hernandez et al. (n,0) [46] 0.0 Average

22 Figure 1 End section of a single-walled carbon nanotube. Figure Representative volue eleent of a graphene sheet.

23 Figure 3 Representative volue eleents for the cheical, truss, and continuu odels. Figure 4 Scheatic of the defored geoetry of the representative volue eleent. 3

24 u x u 1 x 1 x 3 Figure 5 Load case I: Extension along x. u x u 1 x 1 x 3 Figure 6 Load case II: Extension along x 1. 4

25 u x x 1 x 3 Figure 7 - Load case III: Pure shear. Carbon nanotube r nt nt L Equivalentcontinuu tube r c o r c t r c i c L Figure 8 Geoetry of a carbon nanotube and an equivalent-continuu tube. 5

26 φ σπ π bond σ bond Figure 9 Bond inversion of carbon ato in a graphene sheet. Strain Energy [ev/ato] Bakowies and Thiel (1991) Robertson et al. (199) EP1 Robertson et al. (199) EP Robertson et al. (199) LDF Sawada and Haada (199) Miyaoto et al. (1994) Hernandez et al. (1999) (n,n) Hernandez et al. (1999) (n,0) Equivalent Plate with D = 1. ev Nanotube Radius [n] Figure 10 - Strain energy of a graphene sheet subjected to cylindrical bending. 6