Geometry and Self-stress of Single-Wall Carbon Nanotubes and Graphene via a Discrete Model Based on a 2nd-Generation REBO Potential

Abstract

The main purpose of this paper is to evaluate the self-stress state of single-walled Carbon NanoTubes (CNTs) and Flat Graphene Strips (FGSs) in their natural equilibrium state, that is, the state prior to the application of external loads. We model CNTs as discrete elastic structures, whose shape and volume changes are governed by a Reactive Empirical Bond-Order (REBO) interatomic potential of second generation. The kinematical variables we consider are bond lengths, bond angles, and dihedral angles; to changes of each of these variables we associate a work-conjugate nanostress. To determine the self-stress state in a given CNT, we formulate the load-free equilibrium problem as a minimum problem for the interatomic potential, whose solution yields the equilibrium nanostresses; next, by exploiting the nonlinear constitutive dependence we derive for nanostresses in terms of a list of kinematical variables, we determine the equilibrium values of the latter; finally, from the equilibrium values of the kinematical variables we deduce the natural geometry and, in particular, the natural radius.

Our theoretical framework accommodates CNTs of whatever chirality. In the achiral case, the stationarity conditions implied by energy minimization are relatively easy to derive and solve numerically, because we can count on maximal intrinsic symmetries and hence the number of independent unknowns is reduced to a minimum; for chiral CNTs, we prefer to solve the minimum problem directly.

The natural-radius predictions we achieve within our discrete-mechanics framework are in good agreement with the results of calculations based on Density Functional Theory (DFT) and Tight Binding (TB) theory; the same is true for our predictions of the self-energy, that is, the energy associated with self-stress (called cohesive energy in the literature); we surmise that our discrete mechanical model may serve as a source of benchmarks for Molecular-Dynamics (MD) simulation algorithms.

We find that self-stress depends on changes in both bond and dihedral angles in achiral CNTs and, in addition, on changes in bond length in chiral CNTs. Our analysis applies also to FGSs, whose self-stress and self-energy we evaluate; we find that in FGSs self-stress is associated exclusively with changes in bond angle.

Appendices

Appendix A: Details about the Computation of Dihedral Angles

1.1 A.1 Armchair CNTs

With reference to Fig. 4, and for $\mathbf {c}_{1},\mathbf {c}_{2},\mathbf {c}_{3}$ the basis vectors of the Cartesian frame there shown, let us introduce the following unit vectors:

$$ \begin{aligned}[c] &\mathbf{a}:=\mathrm{vers}\, \overrightarrow{\mathit{HB}_{2}}=- \cos\varphi ^{A}\mathbf {c}_{1}+\sin\varphi^{A} \mathbf {c}_{3}, \\ &\mathbf{a}_{1}:=\mathrm{vers}\, \overrightarrow{H'A_{2}}= \cos\varphi ^{A}\mathbf {c}_{1}+\sin\varphi^{A} \mathbf {c}_{3}, \\ &\mathbf{a}_{2}:=\mathrm{vers}\, \overrightarrow{B_{2}A_{2}''}=- \cos 2\varphi^{A}\mathbf {c}_{1}+\sin2\varphi^{A} \mathbf {c}_{3}, \\ & \mathbf{b}:=\mathrm{vers}\, \overrightarrow{A_{1}B_{1}}= \mathbf {c}_{1}, \\ & \mathbf {c}:=\mathrm{vers}\, \overrightarrow{A_{1}B_{2}}=\cos \frac{\alpha }{2}\mathbf{a}+\sin\frac{\alpha}{2}\mathbf {c}_{2}, \\ & \mathbf {d}:=\mathrm{vers}\, \overrightarrow{A_{1}B_{2}'}= \cos\frac{\alpha }{2}\mathbf{a}-\sin\frac{\alpha}{2}\mathbf {c}_{2}, \\ & \mathbf {d}_{1}:=\mathrm{vers}\, \overrightarrow{B_{1}A_{2}'}= \cos\frac{\alpha }{2}\mathbf{a}_{1}-\sin\frac{\alpha}{2} \mathbf {c}_{2}. \end{aligned} $$

(53)

In terms of these unit vectors, the cosines of the dihedral angles $\varTheta_{1}^{A},\varTheta_{2}^{A},\varTheta_{3}^{A}$ read:

$$ \begin{aligned}[c] &\cos\varTheta_{1}^{A}= \frac{\mathbf{b}\times \mathbf {c}}{|\mathbf{b}\times \mathbf {c}|}\cdot\frac{\mathbf {d}_{1}\times\mathbf{b}}{|\mathbf {d}_{1}\times\mathbf {b}|}=\frac{1}{\sin^{2}\beta^{A}} \biggl( \sin^{2}\frac{\alpha}{2}-\cos ^{2}\frac{\alpha}{2} \sin^{2}\varphi^{A} \biggr), \\ & \cos\varTheta_{2}^{A}=\frac{\mathbf {c}\times\mathbf{a}_{2}}{|\mathbf {c}\times\mathbf {a}_{2}|}\cdot \frac{\mathbf{b}\times \mathbf {c}}{|\mathbf{b}\times \mathbf {c}|}=\frac{1}{\sin^{2}\beta^{A}} \biggl( \cos^{2} \varphi^{A}\sin^{2}\frac {\alpha}{2}-\sin^{2} \varphi^{A} \biggr), \\ & \cos\varTheta_{3}^{A}=\frac{\mathbf {c}\times\mathbf{a}_{2}}{|\mathbf {c}\times\mathbf {a}_{2}|}\cdot \frac{\mathbf {d}\times \mathbf {c}}{|\mathbf {d}\times \mathbf {c}|}=-\frac{1}{\sin \beta^{A}} \sin\frac{\alpha}{2}\cos{ \varphi^{A}}. \end{aligned} $$

(54)

Together with (18), these relations, which are equivalent to the first three of (20), permit to compute the first derivatives of $\beta^{A}$ and $\varTheta_{i}^{A}$ ($i=1,2,3$) with respect to $\alpha$:

$$ \begin{aligned}[c] \beta^{A},_{\alpha}&=- \frac{\sin\frac{\alpha}{2}\cos\varphi^{A}}{2 \sin\beta^{A}}, \\ \varTheta_{1}^{A},_{\alpha}&= -\frac{\sin\frac{\alpha}{2} \sin\varphi ^{A}+2 \beta^{A},_{\alpha}\cos\beta^{A} \sin \frac{\varTheta_{1}^{A}}{2}}{\sin\beta^{A}\cos\frac{\varTheta_{1}^{A}}{2} }, \\ \varTheta_{2}^{A},_{\alpha}&=-\frac{\beta^{A},_{\alpha}\cos\beta^{A} \sin \varTheta_{2}^{A}}{\sin\beta^{A}\cos\varTheta_{2}^{A}}, \\ \varTheta_{3}^{A},_{\alpha}&=2 \varTheta_{2}^{A},_{\alpha}. \end{aligned} $$

(55)

1.2 A.2 Zigzag CNTs

With reference to Fig. 5, we introduce the unit vectors

$$ \begin{aligned}[c] &\mathbf{a}:=\mathrm{vers}\,\overrightarrow{A_{1}H}= \cos\varphi^{Z}\mathbf {c}_{2}+\sin\varphi^{Z} \mathbf {c}_{3}, \\ &\mathbf{a}_{1}:=\mathrm{vers}\,\overrightarrow{\mathit{HA}_{3}}= \cos2\varphi ^{Z}\mathbf {c}_{2}+\sin2\varphi^{Z} \mathbf {c}_{3}, \\ &\mathbf{b}:=\mathrm{vers}\,\overrightarrow{A_{1}B_{2}'}=- \sin\beta ^{Z}\mathbf {c}_{2}+\cos\beta^{Z} \mathbf {c}_{1}, \\ &\mathbf{b}_{1}:=\mathrm{vers}\,\overrightarrow{B_{1}A_{2}'}=- \sin\beta ^{Z}\mathbf {c}_{2}-\cos\beta^{Z} \mathbf {c}_{1}, \\ &\mathbf {d}:=\mathrm{vers}\,\overrightarrow{A_{1}B_{2}}=\sin \beta^{Z}\mathbf {a}+\cos\beta^{Z}\mathbf {c}_{1}, \\ &\mathbf {d}_{1}:=\mathrm{vers}\,\overrightarrow{B_{1}A_{2}}= \sin\beta^{Z}\mathbf {a}-\cos\beta^{Z}\mathbf {c}_{1}, \\ &\mathbf {d}_{2}:=\mathrm{vers}\,\overrightarrow{A_{2}B_{3}}= \sin\beta^{Z}\mathbf {a}_{1}+\cos\beta^{Z} \mathbf {c}_{1}. \end{aligned} $$

(56)

For the dihedral angles, we find:

$$ \begin{aligned}[c] &\cos\varTheta_{1}^{Z}= \frac{\mathbf {d}_{1}\times \mathbf {c}_{1}}{\sin\beta^{Z}}\cdot\frac {\mathbf {c}\times\mathbf{b}}{\sin\beta^{Z}}=\cos\varphi^{Z}, \\ &\cos\varTheta_{2}^{Z}=\frac{\mathbf {d}_{1}\times \mathbf {d}_{2}}{\sin\alpha}\cdot \frac {\mathbf{b}_{1}\times \mathbf {d}_{1}}{\sin\alpha}=\frac{1}{\sin^{2}\alpha} \bigl(\sin^{2}\alpha-2 \sin^{2}\beta^{Z}\sin^{2}\varphi^{Z}\bigr) , \\ & \cos\varTheta_{4}^{Z}=\frac{\mathbf {d}_{1}\times \mathbf {d}_{2}}{\sin\beta^{Z}}\cdot \frac{\mathbf {c}_{1}\times \mathbf {d}_{1}}{\sin\beta^{Z}}=\bigl(1+\cos\varphi^{Z}\bigr)\cos \beta^{Z} , \end{aligned} $$

(57)

a set of relations equivalent to the first, second, and fourth, of (25). With these and (23), we find:

$$ \begin{aligned}[c] &\beta^{Z},_{\alpha}= \frac{\cos\frac{\alpha}{2}}{2 \cos\beta^{Z}\cos \frac{\varphi^{Z}}{2}}, \\ &\varTheta_{1,\alpha}^{Z} =0, \\ &\varTheta_{2}^{Z},_{\alpha}= \frac{\beta ^{Z},_{\alpha}\cos\beta^{Z} \sin \varphi^{Z}- \cos\alpha\sin\varTheta_{2}^{Z}}{\sin\alpha\cos\varTheta _{2}^{Z}}, \\ &\varTheta_{4}^{Z},_{\alpha}=2 \varTheta_{2}^{Z},_{\alpha}. \end{aligned} $$

(58)

Appendix B: 2nd-Generation REBO Potentials for Hexagonal Lattices

2.1 B.1 General Form

As anticipated in Sect. 2, the 2nd-generation REBO potentials developed for hydrocarbons by Brenner et al. in [3] accommodate up third-nearest-neighbor interactions, through a bond-order function depending also on dihedral angles. In general, given a substance or a group of substances in the hydrocarbon family, the appropriate potential is tailored by fitting the parameters to the available experimental data and ab initio calculations; the behavior of electron clouds is not accounted for explicitly, and quantum effects are ignored. In spite of these limitations, the predictions obtained with the use of REBO potentials, when compared with those obtained by ab initio or TB methods, have been always found accurate qualitatively, and sometimes even quantitatively. In fact, REBO potentials do incorporate much of the physics and chemistry involved in covalent bonding, as well as Coulomb interactions and many-body effects; if necessary, they also can accommodate bond-breaking and bond-formation.

The binding energy $V$ of an atomic aggregate is written as a sum over nearest neighbors:

$$ V=\sum_{I}\sum _{J< I} V_{\mathit{IJ}}; $$

(59)

the interatomic potential $V_{\mathit{IJ}}$ is given by the construct

$$ V_{\mathit{IJ}}=V_{R}(r_{\mathit{IJ}})+b_{\mathit{IJ}}V_{A}(r_{\mathit{IJ}}), $$

(60)

where the individual effects of the repulsion and attraction functions $V_{R}(r_{\mathit{IJ}})$ and $V_{A}(r_{\mathit{IJ}})$, which model pair-wise interactions of atoms $I$ and $J$ depending on their distance $r_{\mathit{IJ}}$, are modulated by the bond-order function $b_{\mathit{IJ}}$. The repulsion and attraction functions have the following forms:

$$ \begin{aligned}[c] &V_{A}(r)=-f^{C}(r) \sum_{n=1}^{3}B_{n} e^{-\beta_{n} r}, \\ &V_{R}(p)=f^{C}(r) \biggl( 1 + \frac{Q}{r} \biggr) A e^{-\alpha r}, \end{aligned} $$

(61)

where $f^{C}(r)$ is a cutoff function limiting the range of covalent interactions, and where $Q$, $A$, $B_{n}$, $\alpha$, and $\beta$, are parameters to be chosen fit to a material-specific dataset. The remaining ingredient in (60) is the bond-order function:

$$ b_{\mathit{IJ}}=\frac{1}{2}\bigl(b_{\mathit{IJ}}^{\sigma-\pi}+b_{\mathit{JI}}^{\sigma-\pi } \bigr)+b_{\mathit{IJ}}^{\pi}, $$

(62)

where apexes $\sigma$ and $\pi$ refer to two types of bonds: the strong covalent $\sigma$-bonds between atoms in one and the same given plane, and the $\pi$-bonds responsible for interlayer interactions, which are perpendicular to the plane of $\sigma$-bonds. We now describe functions $b_{\mathit{IJ}}^{\sigma-\pi}$ and $b_{\mathit{IJ}}^{\pi}$.

The role of function $b_{\mathit{IJ}}^{\sigma-\pi}$ is to account for the local coordination of, and the bond angles relative to, atoms $I$ and $J$; its form is:

$$ b_{\mathit{IJ}}^{\sigma-\pi}= \biggl(1+\sum _{K\neq I,J} f_{\mathit{IK}}^{C}(r_{\mathit{IK}})G(\cos \theta_{\mathit{IJK}}) e^{\lambda_{\mathit{IJK}}}+P_{\mathit{IJ}}\bigl(N_{I}^{C},N_{I}^{H} \bigr) \biggr)^{-1/2}. $$

(63)

Here, for each fixed pair of indices $(I,J)$, (a) the cutoff function $f_{\mathit{IK}}^{C}$ limits the interactions of atom $I$ to those with its nearest neighbors; (b) $\lambda_{\mathit{IJK}}$ is a string of parameters designed to prevent attraction in some specific situations; (c) function $P_{\mathit{IJ}}$ depends on $N_{I}^{C}$ and $N_{I}^{H}$, the numbers of $C$ and $H$ atoms that are nearest neighbors of atom $I$; it is meant to adjust the bond-order function according to the environment of the C atoms in one or another molecule; (d) for solid-state carbon, the values of both the string $\lambda_{\mathit{IJK}}$ and the function $P_{\mathit{IJ}}$ are taken null; (e) function $G$ modulates the contribution of each nearest neighbor of atom $I$ in terms of the cosine of the angle between the $\mathit{IJ}$ and $\mathit{IK}$ bonds; its analytic form is given by three sixth-order polynomial splines. Function $b_{\mathit{IJ}}^{\pi}$ is given a split representation:

$$ b_{\mathit{IJ}}^{\pi}=\varPi_{\mathit{IJ}}^{\mathit{RC}}+b_{\mathit{IJ}}^{\mathit{DH}}, $$

(64)

where the first addendum $\varPi_{\mathit{IJ}}^{\mathit{RC}}$ depends on whether the bond between atoms $I$ and $J$ has a radical character and on whether it is part of a conjugated system, while the second addendum $b_{\mathit{IJ}}^{\mathit{DH}}$ depends on dihedral angles and has the following form:

$$ b_{\mathit{IJ}}^{\mathit{DH}}=T_{\mathit{IJ}}\bigl(N_{I}^{t},N_{J}^{t},N_{\mathit{IJ}}^{\mathrm{conj}} \bigr) \biggl(\sum_{K(\neq I,J)}\sum _{K(\neq I,J)} \bigl( 1-\cos^{2}\varTheta_{\mathit{IJKL}} \bigr)f_{\mathit{IK}}^{C}(r_{\mathit{IK}})f_{\mathit{JL}}^{C}(r_{\mathit{JL}}) \biggr), $$

(65)

where function $T_{\mathit{IJ}}$ is a tricubic spline depending on $N_{I}^{t}=N_{I}^{C}+N_{I}^{H}$, $N_{J}^{t}$, and $N_{\mathit{IJ}}^{\mathrm{conj}}$, a function of local conjugation.

2.2 B.2 The Form Used in This Paper

We write here the expressions we use for the functions $V_{A}$, $V_{R}$ and $b_{\mathit{IJ}}$ defined in (61) and (62); we also record the form of their first derivatives, because they enter Eqs. (46) and (51).

The attractive and repulsive part of the potential, and their first derivatives are:

$$ \begin{aligned}[c] & V_{A}(r)=-\sum _{n=1}^{3}B_{n} e^{-\beta_{n} r},\qquad V'_{A}(r)=\sum_{n=1}^{3} \beta_{n} B_{n} e^{-\beta_{n} r}, \\ & V_{R}(r)= \biggl( 1 + \frac{Q}{r} \biggr) A e^{-\alpha r} ,\qquad V'_{R}(r)=-\frac{Q}{r^{2}} A e^{-\alpha r} - \alpha \biggl( 1 + \frac {Q}{r} \biggr) A e^{-\alpha r}. \end{aligned} $$

(66)

The bond order function $b_{\mathit{IJ}}$ (62) specializes to $b_{a}$ and $b_{b}$, respectively, for $a$- and $b$-type bonds in achiral CNTs (cf. Eqs. (40)), and it specializes to $b_{i}$ for the typical bond in chiral CNTs (cf. the second of (50)):

$$ \begin{aligned}[c] & b_{a}= \bigl( 1+ 2 G( \beta) \bigr)^{-\frac{1}{2}} + 2 T \bigl( 1 - \cos ^{2} \varTheta_{1} \bigr), \\ & b_{b}= \bigl( 1+ G(\alpha)+G(\beta) \bigr)^{-\frac{1}{2}} + T \bigl( 2\bigl( 1 - \cos^{2} \varTheta_{2} \bigr)+\bigl( 1 - \cos^{2} \varTheta_{3} \bigr)+\bigl( 1 - \cos ^{2} \varTheta_{4} \bigr) \bigr), \\ & b_{i}= \bigl( 1+ G(\theta_{i+1})+G(\theta_{i+2}) \bigr)^{-\frac {1}{2}} + T \bigl( 2\bigl( 1 - \cos^{2} \varTheta_{i1} \bigr)+\bigl( 1 - \cos^{2} \varTheta_{i2} \bigr)+\bigl( 1 - \cos^{2} \varTheta_{i3} \bigr) \bigr) . \end{aligned} $$

(67)

The following derivatives of bond-order functions are found in Eq. (46):

$$\begin{aligned} & b_{a},_{\beta}= - \bigl( 1+ 2 G(\beta) \bigr)^{-\frac{3}{2}} G'(\beta ), \end{aligned}$$

(68)

$$\begin{aligned} & b_{a},_{\varTheta_{1}} = 4 T \cos\varTheta_{1}\sin \varTheta_{1}, \end{aligned}$$

(69)

$$\begin{aligned} & b_{b},_{\alpha}= -\frac{1}{2} \bigl( 1+ G(\alpha)+ G( \beta) \bigr)^{-\frac{3}{2}} G'(\alpha), \end{aligned}$$

(70)

$$\begin{aligned} & b_{b},_{\beta}= -\frac{1}{2} \bigl( 1+ G(\alpha)+ G( \beta) \bigr)^{-\frac{3}{2}} G'(\beta), \end{aligned}$$

(71)

$$\begin{aligned} & b_{b},_{\varTheta_{2}} = 4 T \cos\varTheta_{2}\sin \varTheta_{2}, \end{aligned}$$

(72)

$$\begin{aligned} & b_{b},_{\varTheta_{3}} = 2 T \cos\varTheta_{3}\sin \varTheta_{3}, \end{aligned}$$

(73)

$$\begin{aligned} & b_{b},_{\varTheta_{4}} = 2 T \cos\varTheta_{4}\sin \varTheta_{4}. ; \end{aligned}$$

(74)

the derivatives found in Eq. (51) are:

$$\begin{aligned} & (b_{i+1}),_{\theta i}=-\frac{1}{2} \bigl( 1+ G( \theta_{i})+ G(\theta _{i+2}) \bigr)^{-\frac{3}{2}} G'(\theta_{i}), \end{aligned}$$

(75)

$$\begin{aligned} & b_{i},_{\varTheta_{i1}} = 4 T \cos\varTheta_{i1}\sin \varTheta_{i1} , \end{aligned}$$

(76)

$$\begin{aligned} &b_{i},_{\varTheta_{ij}} = 2 T \cos \varTheta_{ij}\sin\varTheta_{ij},\qquad i=1,2,3,\quad j=2,3 \end{aligned}$$

(77)

(subscripts should be taken modulo 3).

In Eq. (63), the angular-contribution function $G$ is:

$$ G(\theta)= \textstyle\begin{cases} G_{1}(\theta), & 0\leq\theta< 0.6082\pi\\ G_{2}(\theta), & 0.6082\pi\leq\theta< \frac{2\pi}{3} \\ G_{3}(\theta), & \frac{2\pi}{3}\leq\theta\leq\pi \end{cases}\displaystyle , \quad G_{j}(\theta)=\sum_{i=0}^{5}d_{ji} \ (\cos\theta)^{i} ,\ j=1,2,3, $$

(78)

whence

$$ G'_{j}(\theta)=-\sin\theta \Biggl(\sum _{i=1}^{5} i d_{ji}(\cos \theta)^{i-1} \Biggr),\quad j=1,2,3. $$

(79)

The polynomial coefficients $d_{ji}$ are computed following [3]; they are reported in the following table:

$d_{ji}$		i
		0	1	2	3	4	5
j	1	0.37545	1.40678	2.25438	2.03128	1.42971	0.50240
	2	0.70728	5.67744	24.09702	57.59183	71.88287	36.27886
	3	−0.64440	−6.20800	−20.05900	−30.22800	−21.72400	−5.99040

The parameters of the binding energy $V$, the same as in [3, 47], are:

$$\textstyle\begin{array}{@{}l@{\quad}l@{\quad}l@{}} B_{1} = 12388.79197798\ \mbox{eV}, & \beta_{1} = 47.204523127\ \mbox{nm}^{-1}, & Q = 0.03134602960833\ \mbox{nm}, \\ B_{2} = 17.56740646509\ \mbox{eV}, & \beta_{2} = 14.332132499\ \mbox{nm}^{-1}, & A = 10953.544162170\ \mbox{eV}, \\ B_{3} = 30.71493208065\ \mbox{eV}, & \beta_{3} = 13.826912506\ \mbox{nm}^{-1}, & \alpha = 47.465390606595\ \mbox{nm}^{-1}, \\ T = -0.004048375. \\ \end{array} $$

Appendix C: The Traction Problem

We here derive the explicit form of the balance equations for the case when a pure-traction load is applied to an achiral CNT. We made use of the solution to this problem in Sect. 5.2, when we compared the folding energy of CNTs with the energy stored in such a traction problem.

With reference to (6), for $F$ be the magnitude of the axial traction, the load potential takes the following form:

$$ \mathbf {f}\cdot\widehat{\mathbf {d}}(\mathbf {q})=F\delta\widehat{\lambda}(a,b,\alpha), $$

(80)

where $\delta\widehat{\lambda}(a,b,\alpha)$ is the load-induced change in length of the CNT under study. It follows that the balance equations are:

$$ \begin{aligned}[c] &\sigma_{a}= F \frac{\widehat{\lambda},_{a}}{n_{1} n_{2}}, \\ &\sigma_{b}= F\frac{\widehat{\lambda},_{b}}{2 n_{1} n_{2}}, \\ &\tau_{\alpha}+2\beta,_{\alpha}\tau_{\beta}+ \varTheta _{1},_{\alpha} \mathcal {T}_{1}+2 \varTheta_{2},_{\alpha} \mathcal {T}_{2} +\varTheta _{3},_{\alpha} \mathcal {T}_{3} +\varTheta_{4},_{\alpha} \mathcal {T}_{4}= F\frac{\widehat {\lambda},_{\alpha}}{2 n_{1} n_{2}}. \end{aligned} $$

(81)

The mappings $(a,b,\alpha)\mapsto\widehat{\lambda}^{A}(a,b,\alpha)$ and $(a,b,\alpha)\mapsto\widehat{\lambda}^{Z}(a,b,\alpha)$ are here defined with the use of Eqs. (29) and (36)), respectively; hence, we have that:

$$ \left [ \textstyle\begin{array}{@{}c@{}} \widehat{\lambda^{A}},_{a}(a,b,\alpha) \\ \widehat{\lambda^{A}},_{b}(a,b,\alpha) \\ \widehat{\lambda^{A}},_{\alpha}(a,b,\alpha)\\ \end{array}\displaystyle \right ]= \left [ \textstyle\begin{array}{@{}c@{}} 0 \\ 2n_{2}\sin\frac{\alpha}{2} \\ n_{2}b\cos\frac{\alpha}{2}\\ \end{array}\displaystyle \right ], \qquad \left [ \textstyle\begin{array}{@{}c@{}} \widehat{\lambda^{Z}},_{a}(a,b,\alpha) \\ \widehat{\lambda^{Z}},_{b}(a,b,\alpha) \\ \widehat{\lambda^{Z}},_{\alpha}(a,b,\alpha)\\ \end{array}\displaystyle \right ]= \left [ \textstyle\begin{array}{@{}c@{}} n_{1} \\ -n_{1}\cos\beta^{Z} \\ bn_{1}\beta^{Z},_{\alpha}\sin\beta^{Z}\\ \end{array}\displaystyle \right ]. $$

(82)

Equations (81)–(82) can be so specialized as to hold in the case of a FGS subject to a uniform traction load along the armchair or the zigzag direction: it is enough to take $\varphi=0$ and, consequently, $\alpha+2\beta=2\pi$, $\varTheta_{i}=0$, $i=1,\ldots,4$.

Appendix D: Computational Results

In this Appendix we collect a number of tables summarizing the results of our computations. Numerical values for the natural geometric parameters are shown in Tables 2 and 3. Table 2 also shows: (i) the percent difference of bond lengths $a$ and $b$ with respect to $r_{0}=0.14204\ \mbox{nm}$, the C–C distance in graphene computed according to the potential chosen in this study; (ii) the percent differences between the natural and nominal values of the bond angles $\alpha$ and $\beta$ ($\alpha_{0}$ and $\beta_{0}$ have been computed by substituting $\alpha_{0}^{A}=2\pi/3$ in (19), solving for $\beta_{0}^{A}$ (A case), and by substituting $\beta_{0}^{Z}=2\pi/3$ in (23), solving for $\alpha_{0}^{Z}$ (Z case)).

Table 2 Natural geometry of achiral CNTs

Table 3 Dihedral angles (degrees).

Table 4 Geometry of CNTs using a 1st-generation REBO potential (from [26])

Table 5 Nanostresses associated to bond and dihedral angles (nN nm).

Table 6 Natural curvature, roll-up energy, percent difference of roll-up and ‘thin-plate’ energy, dihedral contribution to the roll-up energy

Table 7 $f$ is the axial traction to be applied to each rim atom in order to have an energy increment equal to the roll-up energy; $F$ is the total axial traction; $F/2\pi\rho$, the axial traction per unit rim length, should be compared with the ultimate load reported in the literature [13]

Table 8 Natural radius and nanostresses in chiral CNTs

Table 9 Roll-up energy and dihedral contribution to energy for chiral CNTs