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5 0 0 2 Structures, Symmetries, Mechanics and Motors of n a carbon nanotubes J 5 Z. C. Tu and Z. C. Ou-Yang ] l l Institute of Theoretical Physics, The Chinese Academy of Sciences a h P.O.Box 2735 Beijing 100080, China - s e m ABSTRACT . t a Thestructuresandsymmetriesofsingle-walledcarbonnanotubes(SWNTs)are m introduced in detail. The physical properties of SWNTs induced by their sym- - metries can be described by tensors in mathematical point of view. It is found d that there are 2, 4, and 5 different parameters in the second, third, and fourth n o rank tensors representing electronic conductivity (or static polarizability), the c second order nonlinear polarizability, and elastic constants of SWNTs, respec- [ tively. Thevaluesofelasticconstantsobtainedfromtight-bindingmethodimply 3 that SWNTs might be very weakly anisotropic in mechanical properties. The v further study on the mechanical properties shows that the elastic shell theory 4 in the macroscopic scale can be applied to carbon nanotubes (CNTs) in the 4 mesoscopic scale, as a result, SWNTs can be regarded as an isotropic material 5 with Poisson ratio, effective thickness, and Young’s modulus being ν = 0.34, 5 0 h = 0.75˚A, Y = 4.70TPa, respectively, while the Young’s moduli of multi- 2 walled carbon nanotubes (MWNTs) are apparent functions of the number of 0 layers, N, varying from 4.70TPa to 1.04TPa for N = 1 to . Based on the / ∞ t chirality of CNTs, it is predicted that a new kind of molecular motor driven a byalternatingvoltagecanbeconstructedfromdoublewalledcarbonnanotubes m (DWNTs). - d n INTRODUCTION o c : Carbon is the core element to construction of organic matters and has always v i attracted much attention up to now. Many decades ago, people only knew X two kinds of crystals consisting of carbon: graphite with layer structure and r diamond with tetrahedral shape. The situation was changed in 1985 when a Kroto et al. synthesized bucky ball—a football-like molecule consisting of 60 carbonatoms[1]whichmarkedthebeginningofcarbontimes. Afterthat,Iijima synthesized MWNTs in 1991 [2] and SWNTs in 1993 [3]. Simply speaking, a SWNT can be regarded as a graphitic sheet with hexagonal lattices that was wrapped up into a seamless cylinder with diameter in nanometer scale and 1 length from tens of nanometers even to several micrometers if we ignore its two end caps, while a MWNT consists of a series of coaxial SWNTs with layer distance about 3.4˚A [4]. SWNTs have many unique properties. Viewed from the chirality, some of them are chiral but others are achiral. Viewed from the electronic properties, some of them are metallic but others are semiconductive. Moreover,their elec- tronicpropertiesdependsensitivelyontheirchirality[5–8]. Theconductivityof metallic SWNT does notsatisfy Ohm’s law because the electrontransportin it is ballistic [9–11]. Otherwise, theoretical [12–15] and experimental [16] studies have suggested that SWNTs also possess many novel mechanical properties, in particular high stiffness and axial strength, which are insensitive to the tube diameters and chirality. MWNTs have the similar mechanical properties to SWNTs [17–19]. In purely theoretical point of view, we should consider the following two facts: (i) As quasi-one-dimensional structures with periodic boundary conditions, SWNTs might show anisotropic physical properties which may depend on the tube diameters and chirality. Generally speaking, the physical properties of crystalscanbe representedbytensors[20]. Forexample,the electronicconduc- tivity can be expressed by second-rank tensor and the elastic constants can be described as fourth-rank tensor. These tensors can be derived from the structures and symmetries of SWNTs. Therefore fully discussing structures and symmetries of SWNTs is one of the topic in this chapter. (ii) The SWNT is a single layer of carbon atoms. What is the thickness of the layer? It is a widely controversial question. Some researchers take 3.4 ˚A, the layer distance of bulk graphite, as the thickness of SWNT [15,16]. Others define an effective thickness (about 0.7˚A) by admitting the validity of elastic shell theory in nanometer scale [12,13]. The present authors have proved that the elastic shell theory can indeed be applied to SWNTs [21] which supports the latter standpoint. Because the two standpoints have little effect on the mechanicalresultsinmanycases,thecontroversyisstillbeingdiscussed[22–28]. Thereforeitisnecessarytopointoutwhenthetwostandpointswillgivedifferent results. Inthe appliedpointofview, the presentauthorshavepredictedamolecular motor constructed from a DWNT driven by temperature variation [29] which induces to moleculardynamics simulationsby Dendzik et al. [30]. Using molecular dynamics simulations, Kang et al. recently predicted a carbon-nanotube motor driven by fluidic gas [31]. In this chapter, a conceptual motor of DWNT driven by alternating voltage will be proposed based on previous work [29]. 2 STRUCTURES, SYMMETRIES AND THEIR IN- DUCING PHYSICAL PROPERTIES OF SWNTS To describe the SWNT, some characteristic vectors require introducing. As shown in Fig.1, the chiral vector C , which defines the relative location of h two sites, is specified by a pair of integers (n,m) which is called the index of the SWNT and relates C to two unit vectors a and a of graphite (C = h 1 2 h na +ma ). Thechiralangleθ definestheanglebetweena andC . For(n,m) 1 2 0 1 h nanotube, θ =arccos 2n+m . The translational vector T corresponds 0 2√n2+m2+nm h i to the first lattice point of 2D graphitic sheet through which the line normalto the chiralvectorC passes. The unit cellofthe SWNT is the rectangledefined h by vectorsC and T, while vectorsa and a define the area of the unit cell of h 1 2 2D graphite. The number N of hexagons per unit cell of SWNT is obtained as a function of n and m as N = 2(n2+m2+nm)/d which is larger than 8 for R SWNTs in practice, where d is the greatest common divisor of (2m+n) and R (2n+m). Thereare2N carbonatomsineachunitcellofSWNT becauseevery hexagon contains two atoms. To denote the 2N atoms, we use a symmetry vector R to generate coordinates of carbon atoms in the nanotube and it is defined as the site vector having the smallest component in the direction of C . From a geometric standpoint, vector R consists of a rotation around the h nanotube axis by an angle Ψ = 2π/N combined with a translation τ in the directionofT; therefore,RcanbedenotedbyR=(Ψτ). Usingthe symmetry vector R, we can divide the 2N carbon atoms in the|unit cell of SWNT into two classes [32]: one includes N atoms whose site vectors satisfy A =lR [lR T/T2]T (l =0,1,2, ,N 1), (1) l − · ··· − another includes the remainder N atoms whose site vectors satisfy B =lR+B [(lR+B ) T/T2]T l 0 0 − · [(lR+B ) C /C2]C (l=0,1, ,N 1), (2) − 0 · h h h ··· − wnehaerreestBn0eig≡hb(oΨr0a|τt0o)m=s to(cid:16)A2π0r0ac|noCsd(hθ|0r−0π6is)(cid:12)(cid:12)trh0eccoasr(bθ0on−-cπ6ar)b(cid:17)ornepbroensdenltesngotnhe. of the (cid:12) Ifweintroducecylindricalcoordinatesystem(r,θ,z)whosez-axisisthetube axis parallel to vector T. Its rθ-plane is perpendicular to z-axis and contains atom A in the nanotube. r the distance from some point to z-axis, and θ the 0 angle rotating around z-axisfrom an axis which is verticalto z-axis and passes through atom A in the tube to the point. In this coordinate system, we can 0 express Eqs.(1) and (2) as [33]: A = ρ,lΨ,lτ [lτ/T]T (l=0,1,2, ,N 1), (3) l { − } ··· − and lΨ+Ψ lτ +τ B = ρ,lΨ+Ψ 2π 0 ,lτ +τ 0 T l 0 0 (cid:26) − (cid:20) 2π (cid:21) −(cid:20) T (cid:21) (cid:27) (l =0,1,2, ,N 1), (4) ··· − 3 F T a 1 a Al Bl 2 R C h A B 0 0 Figure 1: The unrolled honeycomb lattice of a SWNT. By rolling up the sheet along the chiral vector C , that is, such that the point A coincides with the h 0 point corresponding to vector C , a nanotube is formed. The vectors a and h 1 a are the real space unit vectors of the hexagonal lattice. The translational 2 vectorTis perpendicular to C andruns inthe directionofthe tube axis. The h vector R is the symmetry vector. A , B and A,B (l = 1,2, ,N) are used 0 0 l l ··· to denote the sites of carbon atoms. z(z ) l y l 0 l A x l l 0 y A x 0 Figure 2: The coordinates of SWNT. 4 C where ρ = | h|. In Eqs.(1)-(4), the symbol [ ] denotes the largest integer 2π ··· smaller than , e.g., [7.3]=7. ··· In the following contents of this section, we will derive the general forms of thesecond,third,andfourthranktensorsbyfullyconsideringthesymmetriesof SWNTs. As shown in Fig.2, oxyz is the initial coordinate system whose z-axis is the tube axis and x-axis passes through atom A . If we use R to act l times 0 on the initial system, we can get the coordinate system O x y z . Obviously, l l l l the transformation from the bases xˆ,yˆ,zôf the initial system to the new bases xˆ ,yˆ,zˆ can be expressed as, l l l xˆ xˆ l  yˆl =(aij) yˆ , (5) zˆ zˆ l     where (a ) is the matrix with elements a = a = coslψ, a = sinlψ, ij 11 22 12 a = sinlψ, a =1, a =a =a =a =0. 21 33 13 31 23 32 − Above all, let us consider the second-rank tensor S. s and s(l) (i,j = ij ij 1,2,3;l=1,2, ,N)denoteitscomponentsinthecoordinatesystemOxyzand ··· (l) O xy z . The transformation law of the components is s = a a s [20], l l l l ij im jk mk wheretheEinsteinsummationconventionisused. ButthesymmetryofSWNTs requires s(l) = s . From this condition and the transformation law we derive ij ij out s = s , s = s , s = s = 0, s = s = 0. Especially, there 11 22 12 21 13 31 23 32 − are only 2 different nonzero parameters in symmetric 2nd-rank tensors which represent the electronic conductivity or static polarizabilities: s = s = 0, 11 22 6 s =0. 33 N6 ext,letusdealwiththethird-ranktensorDwhosecomponentsaredenoted byd andd(l) (i,j,k =1,2,3)inthecoordinatesystemOxyz andOx y z ,re- ijk ijk l l l l spectively. Fromthetransformationlawofthecomponentsd(l) =a a a d ijk iq ju kv quv [20]andthesymmetryofSWNTs,wecanobtainthenon-vanishingcomponents of D: d = d , d = d , d = d , d = d , d = d , 113 223 123 213 131 232 132 231 311 322 − − d = d , d . If some physical property requires d =d (e.g. the sec- 312 321 333 ijk ikj − ondorderpolarizationeffect),thenthereareonly4differentnonzeroparameters in its components: d = d = d = d d /2, d = d = d = 123 213 132 231 1 113 223 131 − − ≡ d d /2, d = d d , d d . Thus the relation PNL = d E E 232 ≡ 2 311 322 ≡ 3 333 ≡ 4 i ijk j k between the nonlinear polarization and field can be expressed in matrix form, PNL =d E E +d E E x 1 y z 2 x z  PPyNNLL ==dd2E(Ey2E+z −Ed2)1E+xdEzE2 . (6) z 3 x y 4 z  Similarly,wecanobtainthenonzerocomponentsofthefourth-ranktensorC fromthe transformationlawc(l) =a a a a c (i,j,k,m=1,2,3)[20] ijkm iq ju kv mw quvw and the symmetry of SWNTs: c = c , c = c , c = c , 1111 2222 1112 2221 1121 2212 − − c =c ,c =c ,c = c ,c =c ,c =c =c 1122 2211 1133 2233 1211 2122 1212 2121 1221 2112 2222 − − c c ,c = c =c +c +c ,c = c ,c =c , 2121 2211 2111 1222 2122 2212 2221 1233 2133 1313 2323 − − − c = c , c = c , c = c , c = c , c = c , 1323 2313 1331 2332 1332 2331 3113 3223 3123 3213 − − − 5 c = c , c = c , c = c , c = c , c . If we 3131 3232 3132 3321 3311 3322 3312 3321 3333 − − consider the elastic property, the elastic constants can be expressed by the fourth-rank tensor whose components satisfy c = c = c = c ijkm ijmk jikm kmij [20]. Thus there are only 5 different non-vanishing parameters (c ,c ,c ,c ,c ) 1 2 3 4 5 in its components and c = c c , c = c = c = c c , 1111 2222 1 1133 3311 2233 3322 2 ≡ ≡ c = c , c = c = c = c = c = c = c = c c , 3333 3 1313 2323 1331 2332 3113 3223 3131 3232 4 ≡ c =c c ,c =c =c =c =(c c )/2. Thestress-strain 1122 2211 5 1212 2121 1221 2112 1 5 relation σ =C≡ε can be expressed by the matrix nota−tions, σxx c1 c5 c2 εxx  σyy   c5 c1 c2  εyy   σzz = c2 c2 c3  εzz , (7)  σyz   c4  γyz   σxz   c4  γxz        σxy   (c1−2c5)  γxy  where γ = 2ε , γ = 2ε ,γ = 2ε [34]. The axial Young’s modulus Y xy xy xz xz yz yz z defined as the stress/strain ratio when the tube is axially strained and Poisson ratio ν defined as the ratio of the reduction in radial dimension to the axial z elongation can be expressed as Y = c 2c2/(c +c ) and ν = c /(c +c ), z 3− 2 1 5 z 2 1 5 respectively. Obviously, the numbers of different parameters in the expressions S, D, C of SWNTs are more than that in isotropic materials (see also Table 1) if the differentparametersareindependent, whichimpliesthatSWNTs mightpossess anisotropic physical properties. All parameters are functions of n, m, which revealsthephysicalpropertiesdependonthechiralityanddiametersofSWNTs to some extent. Table 1: The non-zero parameter numbers of different rank physical property tensors for isotropic materials and single-walled carbon nanotubes. Tensors S(2nd-rank) D(3rd-rank) C(4th-rank) Isotropic materials 1 0 2 carbon nanotubes 2 4 5 Asexamples,wewillgivetheformsofsecondandfourth-rankordertensors— the static polarizabilities and elastic constants of SWNTs, respectively. It is well known that the relation between the polarization P and external electric field E is P = αE [35], where α is the static polarizability, a second- rank tensor. From above discussions, we known α can be expressed as matrix form, α xx α= αyy , (8) α zz   6 with α =α . Benedict et al. [36] have studied the polarizabilities and their xx yy resultsareshowninTable 2, wherewehavechangedtheir valuesto polarizabilities per atom. From Table 2 we find that the polarizabilities of SWNTs are sensitive to the tube indexes (n,m), particularly, the value of α is extremely zz large when n m is a multiple of three, which corresponds metallic tubes. − Table 2: Static polarizabilities per atom of various tube indexes and radii [36]. When n m is multiple of three, α is extremely large and is not given. zz − (n,m) ρ(˚A) α (˚A3/atom) α (˚A3/atom) zz xx (9,0) 3.57 1.05 (10,0) 3.94 18.62 1.10 (11,0) 4.33 17.04 1.20 (12,0) 4.73 1.23 (13,0) 5.12 23.98 1.29 (4,4) 2.73 0.92 (5,5) 3.41 1.02 (6,6) 4.10 1.13 (4,2) 2.09 9.87 0.84 (5,2) 2.46 0.89 Otherwise, we calculate the elastic constants of SWNTs through the tight binding method [13] with considering the curvature and bond-length change effects. The results are listed in Table 3 which suggests that the elastic properties of SWNTs slightly depend on the tube indexes (n,m). We also give the correspondingaxialYoung’s moduli andPoissonratios ofsingle-walledcar- bon nanotubes with different indexes. Moreover, we find (c +c ) c c , 1 5 4 2 − ≈ c 2(c +c )and(c +c ) 4c ,i,e.,theremightbeonlytwoindependentpa- 3 1 5 1 5 2 ≈ ≈ rameters in the elastic constants, which implies that the mechanicalanisotropy ofSWNTsissoweaklythatwecanregardthemasapproximatelyisotropicma- terials(Remark: theisotropicmaterialshavetwoindependentelasticconstants, see also Table 1). It is necessary to discuss the meanings of strains and stresses in nanometer scale. Strains are geometric quantities so that their definitions in macroscopic theoryofelasticitystillholdforSWNTs. Butwemustredefinestressesbecause they are not well-defined quantities for SWNTs. Given strains, we can calculate the energy variation of the SWNTs due to the strains through quantum mechanics in principle. The stresses are defined as the partial derivatives of energy variation with respect to the strains. In fact, stresses are not necessary concepts. We can directly determine the elastic constants by strains and the corresponding energy variation. Otherwise, we do not separate c and c in Table 3. Up to now, we do not 1 5 know how to separate them. A possibility is that we need not do that when we discuss the mechanical properties of SWNTs. 7 Table3: Theelasticconstants(unit: eV/atom)andcorrespondingaxialYoung’s moduli (unit: eV/atom), Poissonratios of single-walled carbon nanotubes. (n,m) c +c c c c Y ν 1 5 2 3 4 z z (6,0) 29.04 7.03 56.97 22.72 53.56 0.24 (8,0) 29.34 7.05 57.68 23.43 54.29 0.24 (10,0) 29.50 7.07 57.92 23.79 54.53 0.24 (50,0) 29.51 7.08 58.97 24.36 55.57 0.24 (6,6) 29.99 7.08 56.97 24.09 53.63 0.24 (8,8) 29.76 7.08 57.92 24.22 54.55 0.24 (10,10) 29.66 7.08 58.33 24.28 54.96 0.24 (50,50) 29.51 7.08 59.00 24.38 55.60 0.24 (6,4) 30.21 7.07 56.15 23.13 52.83 0.23 (7,3) 30.00 7.07 56.60 22.53 53.27 0.24 (8,2) 29.64 7.07 57.44 22.62 54.07 0.24 MECHANICAL PROPERTIES OF CNTS In this section, we will continue to discuss the mechanical properties of CNTs in detail. We start from the concise formula proposed by Lenosky et al. in 1992 to describe the deformation energy of a single layer of curved graphite [37] Eg = (ǫ /2) (r r )2+ǫ ( u )2 0 ij 0 1 ij − X(ij) Xi X(j) + ǫ (1 n n )+ǫ (n u )(n u ). (9) 2 i j 3 i ij j ji − · · · X X (ij) (ij) Thefirsttwotermsarethecontributionsofbondlengthandbondanglechanges to the energy. The last two terms are the contributions of the π-electron res- onance. In the first term, r = 1.42 ˚A is the initial bond length of planar 0 graphite, and r is the bond length between atoms i and j after the deforma- ij tions. In the remaining terms, u is a unit vector pointing from atom i to its ij neighbor j, and n is the unit vector normal to the plane determined by the i three neighbors of atom i. The summation is taken over the three near- (j) est neighbor j atoms to i atom, and tPaken over all the nearest neighbor (ij) atoms. The parameters (ǫ ,ǫ ,ǫ ) =P(0.96,1.29,0.05)eV were determined by 1 2 3 Lenosky et al. [37] through local density approximation. The value of ǫ was 0 not given by Lenosky et al., but given by Zhou et al. [38] ǫ = 57eV/˚A2 from 0 the force-constantmethod. In 1997, Ou-Yang et al. [39] reduced Eq.(9) into a continuum limit form withouttakingthebondlengthchangeintoaccountandobtainedthecurvature 8 elastic energy of a SWNT 1 E(s) = k (2H)2+k¯ K dA, (10) c 1 Z (cid:20)2 (cid:21) where the bending elastic constant k =(18ǫ +24ǫ +9ǫ )r2/(32Ω)=1.17eV (11) c 1 2 3 0 with Ω=2.62 ˚A2 being the occupied area per atom, and k¯ /k = (8ǫ +3ǫ )/(6ǫ +8ǫ +3ǫ )= 0.645. (12) 1 c 2 3 1 2 3 − − In Eq.(10), H, K, and dA are mean curvature, Gaussian curvature, and area element of the SWNTs surface, respectively. In 2002, we obtained the total free energy [21] of a strained SWNT with ε ε in-plane strainε = x xy at the i-atomsite, where ε , ε , and ε are i (cid:18) εxy εy (cid:19) x y xy the axial, circumferential, and shear strains, respectively. The total free energy contains two parts: one is the curvature energy expressed as Eq.(10); another is the deformation energy [21] 1 E = k (2J)2+k¯ Q dA, (13) d d 2 Z (cid:20)2 (cid:21) where 2J = ε +ε and Q = ε ε ε2 , are respectively named “mean” and x y x y − xy “Gaussian” strains, and k =9 ǫ r2+ǫ /(16Ω)=24.88eV/˚A2, (14) d 0 0 1 (cid:0) (cid:1) k¯ = 3 ǫ r2+3ǫ /(8Ω)= 0.678k . (15) 2 − 0 0 1 − d The value ofk¯ /k isso exce(cid:0)llentlyclose(cid:1)to the value ofk¯ /k showninEq.(12) 2 d 1 c that we can regardthat they are,in fact, equal to each other. We assume both k¯ /k and k¯ /k are equal to their average value, 2 d 1 c k¯ /k =k¯ /k = 0.66. (16) 1 c 2 d − This is the key relationthatallowsto describethe deformationsofSWNT with classic elastic theory. Thus, the deformation energy of a SWNT, the sum of Eqs.(10) and (13) 1 1 E(s) = k (2H)2+k¯ K dA+ k (2J)2+k¯ Q dA (17) d Z (cid:20)2 c 1 (cid:21) Z (cid:20)2 d 2 (cid:21) can be expressed as the form of the classic shell theory [34]: 1 E = D (2H)2 2(1 ν)K dA c 2Z − − (cid:2) (cid:3) 1 C + (2J)2 2(1 ν)Q dA, (18) 2Z 1 ν2 − − − (cid:2) (cid:3) 9 where D = (1/12)Yh3/(1 ν2) and C = Yh are bending rigidity and in-plane − stiffnessofshell. νisthePoissonratioandhisthethicknessofshell. Comparing Eq.(17) with Eq.(18), we have (1/12)Yh3/(1 ν2)=k c −  Yh/(1 ν2)=k . (19) d  1 ν =− k¯ /k = k¯ /k 1 c 2 d − − −  From above equations we obtain the Poissonratio, effective wall thickness, and Young’s modulus of SWNTs are ν = 0.34, h = 0.75˚A and, Y = 4.70TPa, respectively. Our numerical results are close to those given by Yakobson et al. [12]. Through above discussion, we can declare that: (i) Eqs.(16), (17) and (18) implythatelasticshelltheoryinmacroscopicscalecanbeappliedtotheSWNT inmesoscopicscaleprovidedthatitsradiusarenottoosmall. (ii)SWNTcanbe regardas being made fromisotropicmaterialswith Y =4.70TPaandν =0.34. Its effective thickness can be well-defined as h=0.75˚A. We now turn to discuss the axial Young’s modulus of MWNT. A MWNT can be thought of as a series of coaxial SWNTs with layer distance d=3.4˚A. Due to the deformation energy of SWNT, we can write the deformation energy of MWNT as [21] N 1 E(m) = k (2J)2+k¯ Q dA d 2 Z (cid:20)2 (cid:21) Xl=1 N N 1 − + πk L/ρ + (∆E /d)πL(ρ2 ρ2), (20) c l coh l+1− l Xl=1 Xl=1 where ρ is the radius of the l-thlayerfrom inner one, N is the layernumber of l MWNT, and ∆E = 2.04eV/nm2 [40] being the interlayer cohesive energy coh − per area of planar graphite. L is the length of MWNT. The second term in Eq.(20) expresses the summation of curvature energies on all layers given in Eq.(10),andthethirdtermrepresentsthetotalinterlayercohesiveenergywhich actually arises from the relatively weaker Van der Waals’ interactions. On this account, we can reasonably believe that the axial strain ε and circumferential x strain ε still satisfy ε = νε for every layer of SWNT in the MWNT when y y x − uniform stresses apply along axial direction. Thus Eq.(20) becomes N N N 1 E(m) = kd(1 ν2)ε2 2πρ L+ πk L/ρ + − ∆E πL(ρ +ρ ). (21) 2 − x l c l coh l+1 l Xl=1 Xl=1 Xl=1 The axial Young’s modulus of the MWNT Y is defined as m 1 ∂2E(m) Y (N)= , (22) m V ∂ε2 x 10