A semi-analytical approach for calculating the equilibrium structure and radial breathing mode frequency of single-walled carbon nanotubes

Abstract

A semi-analytical model for determining the equilibrium configuration and the radial breathing mode (RBM) frequency of single-wall carbon nanotubes (CNTs) is presented. By taking advantage of the symmetry characteristics, a CNT structure is represented by five independent variables. A line search optimization procedure is employed to determine the equilibrium values of these variables by minimizing the potential energy. With the equilibrium configuration obtained, the semi-analytical model enables an efficient calculation of the RBM frequency of the CNTs. The radius and radial breathing mode frequency results obtained from the semi-analytical approach are compared with those from molecular dynamics (MD) and ab initio calculations. The results demonstrate that the semi-analytical approach offers an efficient and accurate way to determine the equilibrium structure and radial breathing mode frequency of CNTs.

Appendices

Appendix 1: Bond angle calculation

A Cartesian coordinate system with the origin at the center of the circular cross-section is defined as shown in Fig. A1. Since the points A, $B_j$, and $B_k$ are all on the same cylindrical surface, $\overrightarrow{\varvec{OA}}$, $\overrightarrow{\varvec{OB}_j}$, and $\overrightarrow{\varvec{OB}_k}$ can be expressed as: $\overrightarrow{\varvec{OA}} = \{R,0,0\}$, $\overrightarrow{\varvec{OB}_j} = \{R \cos \beta _j, R \sin \beta _j, h_j\}$, $\overrightarrow{\varvec{OB}_k} = \{R \cos \beta _k, R \sin \beta _k, h_k\}$ with $j \ne k$ and $j,k = 1,2,3$. Therefore,

$$\begin{aligned} \begin{aligned}&\overrightarrow{\varvec{AB}_j} = \{R(\cos \beta _j - 1), R \sin \beta _j, h_j\}, \\&\overrightarrow{\varvec{AB}_k} = \{R(\cos \beta _k - 1), R \sin \beta _k, h_k\}. \\ \end{aligned} \end{aligned}$$

(A.1)

We obtain

$$\begin{aligned} \begin{aligned}&\cos \angle B_jAB_k = \frac{\overrightarrow{\varvec{AB}_j} \cdot \overrightarrow{\varvec{AB}_k}}{|\overrightarrow{\varvec{AB}_j}||\overrightarrow{\varvec{AB}_k}|} \\&\quad = \frac{R^2(\cos \beta _j{-} 1)(\cos \beta _k{-} 1){+}R^2 \sin \beta _j \sin \beta _k{+}h_j h_k}{r_j r_k}. \end{aligned}\nonumber \\ \end{aligned}$$

(A.2)

Recalling Eq. (2), we have

$$\begin{aligned}&1 - \cos \beta _j = 2\sin ^2\frac{\beta _j}{2} = 2\frac{l_j^2}{4R^2},\nonumber \\&\sin \beta _j = 2\sin \frac{\beta _j}{2}\cos \frac{\beta _j}{2} = 2\frac{l_j}{2R}\sqrt{1-\frac{l_j^2}{4R^2}}. \end{aligned}$$

(A.3)

Substituting Eq. (A.3) with $h_j = r_j \sin \alpha _j$ and $l_j = r_j \cos \alpha _j$ into Eq. (A.2), we have

$$\begin{aligned}&\cos \angle B_jAB_k \nonumber \\&\quad = \frac{1}{r_j r_k} \left[ 4R^2\frac{l_j^2}{4R^2}\frac{l_k^2}{4R^2} +4R^2 \left( \frac{l_j}{2R}\sqrt{1-\frac{l_j^2}{4R^2}} \right) \right. \nonumber \\&\qquad \left. \left( \frac{l_k}{2R}\sqrt{1-\frac{l_k^2}{4R^2}} \right) +h_j h_k \right] \nonumber \\&\quad =\frac{r_j r_k }{4R^2}\cos ^2\alpha _j \cos ^2\alpha _k \nonumber \\&\qquad +\, \cos \alpha _j \cos \alpha _k \sqrt{1-\frac{r_j \cos ^2\alpha _j}{4R^2}} \sqrt{1-\frac{r_k \cos ^2\alpha _k}{4R^2}} \nonumber \\&\qquad +\, \sin \alpha _j \sin \alpha _k. \end{aligned}$$

(A.4)

Fig. 6

Bond angle calculation

Appendix 2: REBO potential and its derivatives

The general form of the REBO potential energy is given by [32]

$$\begin{aligned} w = \frac{1}{2} \sum _{a,b} V(r_{ab}) = \frac{1}{2}\sum _{a,b} \left[ V_R(r_{ab}) - \bar{b}_{ab}V_A(r_{ab})\right] , \end{aligned}$$

(A.5)

where $V_R(r_{ab})$ and $V_A(r_{ab})$ are repulsive and attractive energy between atom a and atom b, respectively. $\bar{b}_{ab}$ is the multi-body coupling modification term: $\bar{b}_{ab} = \frac{1}{2} \left( b_{ab} + b_{ba} \right)$. In this work, due to the $C_2$-axis rotational symmetry, $b_{ab} = b_{ba}$. Then the potential energy of a unit cell containing only one atom can be expressed using the bond indexes as

$$\begin{aligned} w = \frac{1}{2} \sum _{j=1}^3 V(r_{j}) = \frac{1}{2}\sum _{j=1}^3 (V_R(r_{j}) - b_{j}V_A(r_{j})), \end{aligned}$$

(A.6)

where $j = 1,2,3$ are the indexes of three bonds surrounding an atom. The terms in Eq. (A.6) are given by [32]

$$\begin{aligned} V_R(r_{j})= & {} \left( 1+\frac{Q}{r_{j}} \right) A\mathrm{e}^{-\alpha r_{j}} f_\mathrm{c}(r_{j}), \nonumber \\ V_A(r_{j})= & {} \sum _{n=1}^3 B_n \mathrm{e}^{-\beta _nr_{j}} f_\mathrm{c}(r_{j}), \nonumber \\ b_{j}= & {} \left[ 1+\sum _{k\ne j}G(\cos \theta _{jk})f_\mathrm{c}(r_{k})\right] ^{-\frac{1}{2}},\nonumber \\ \qquad j,k= & {} 1,2,3, \end{aligned}$$

(A.7)

where $\theta _{jk}$ is the angle between bond j and bond k. The cutoff function $f_\mathrm{c}(r)$ is defined as

$$\begin{aligned} f_\mathrm{c}(r)= & {} \left\{ \begin{array}{ll} 1, &{}\quad \,\, r< R_1, \\ \frac{1}{2} \left\{ 1+\cos \left[ \frac{\uppi (r-R_1)}{R_2 - R_1} \right] \right\} , &{}\quad R_1 \leqslant r \leqslant R_2,\\ 0,&{}\quad \,\, r> R_2, \end{array} \right. \end{aligned}$$

(A.8)

and $G(\cos \theta _{jk})$ is an empirical 6-order spline function. For C–C bonds, $G(\cos \theta )$ is given by [37]

$$\begin{aligned} G(\cos \theta ) = \left\{ \begin{aligned}&0.27186 +0.48922 \cos \theta \\&-0.43286 \cos ^2\theta -0.56140 \cos ^3\theta \\&+1.2711 \cos ^4\theta -0.037931 \cos ^5 \theta , \\&\,\mathrm{for} \ \theta < 109.47^\circ ,\\&0.69669 +5.5444 \cos \theta \\&+23.432 \cos ^2\theta +55.948 \cos ^3\theta \\&+69.876 \cos ^4\theta +35.312 \cos ^5\theta , \\&\,\mathrm{for} \ 109.47^\circ \leqslant \theta \leqslant 120^\circ ,\\&0.00260 -1.0980 \cos \theta \\&-4.3460 \cos ^2\theta -6.8300 \cos ^3\theta \\&-4.9280 \cos ^4\theta -1.3424 \cos ^5\theta , \\&\,\mathrm{for} \ \theta > 120^\circ .\end{aligned} \right. \end{aligned}$$

(A.9)

The constant parameters Q, A, $\alpha$, $B_n$, $\beta _n$, $R_1$, and $R_2$ are listed in Table A1 [23].

Table 5 Parameters for $V_R$ and $V_A$

Next, the derivatives of the REBO potential are obtained. Denoting $\frac{1}{2}\sum _{j=1}^3 V(r_{j})$ as W, the first order partial derivatives are

$$\begin{aligned}&\frac{\partial W}{\partial r_j} = \frac{1}{2}\left[ \frac{\mathrm{d}V_R}{\mathrm{d}r_j}(r_j) - b_{j}\frac{\mathrm{d}V_A}{\mathrm{d}r_j}(r_j) - \sum _{k \ne j}\frac{\partial b_{k}}{\partial r_j}V_A(r_k)\right] ,\nonumber \\\end{aligned}$$

(A.10)

$$\begin{aligned}&\frac{\partial W}{\partial \cos \theta _{jk}} = -\frac{1}{2}\left[ \frac{\partial b_{j}}{\partial \cos \theta _{jk}}V_A(r_j)+\frac{\partial b_{k}}{\partial \cos \theta _{jk}}V_A(r_k)\right] \nonumber \\ \end{aligned}$$

(A.11)

$\frac{\mathrm{d}V_R}{\mathrm{d}r_j}(r_j)$, $\frac{\mathrm{d}V_A}{\mathrm{d}r_j}(r_j)$, $\frac{\partial B_{k}}{\partial r_j}$ and $\frac{\partial B_{j}}{\partial \cos \theta _{jk}}$ can be obtained from Eqs. (A.7) and (A.8) as

$$\begin{aligned}&\frac{\mathrm{d}V_R}{\mathrm{d}r_{j}}(r_j) = -\left[ \alpha \left( 1+\frac{Q}{r_{j}} \right) + \frac{Q}{r_{j}^2} \right] A\mathrm{e}^{-\alpha r_{j}} f_\mathrm{c}(r_{j}) \nonumber \\&\qquad +\left( 1+\frac{Q}{r_{j}} \right) A\mathrm{e}^{-\alpha r_{j}} \frac{\mathrm{d}f_\mathrm{c}}{\mathrm{d}r}(r_{j}), \end{aligned}$$

(A.12)

$$\begin{aligned}&\frac{\mathrm{d}V_A}{\mathrm{d}r_{j}}(r_j) = \sum _{n=1}^3 B_n \mathrm{e}^{-\beta _nr_{j}} \left( \frac{\mathrm{d}f_\mathrm{c}}{\mathrm{d}r}(r_{j}) - \beta _n f_\mathrm{c}(r_{j}) \right) , \end{aligned}$$

(A.13)

$$\begin{aligned}&\frac{\mathrm{d} b_{k}}{\mathrm{d} r_j} = -\frac{1}{2}\left[ 1+\sum _{l\ne k}G(\cos \theta _{kl}) f_\mathrm{c}(r_{l})\right] ^{-\frac{3}{2}} G(\cos \theta _{kj}) \frac{\mathrm{d}f_\mathrm{c}}{\mathrm{d}r} (r_{j}), \end{aligned}$$

(A.14)

$$\begin{aligned}&\frac{\mathrm{d} b_{k}}{\mathrm{d} \cos _{jk}} = -\frac{1}{2}\left[ 1+\sum _{l\ne k}G(\cos \theta _{kl})f_\mathrm{c}(r_{l})\right] ^{-\frac{3}{2}} \frac{\mathrm{d} G(\cos \theta _{kj})}{\mathrm{d} \cos \theta _{jk}} f_\mathrm{c}(r_{k}),\nonumber \\ \end{aligned}$$

(A.15)

where

$$\begin{aligned} \frac{\mathrm{d} f_\mathrm{c}}{\mathrm{d} r} = \left\{ \begin{array}{ll} 0 , &{}\quad r< R^{(1)}, \\ \frac{-\uppi }{2(R^{(2)} - R^{(1)})} \sin \frac{\uppi (r-R^{(1)})}{R^{(2)} - R^{(1)}}, &{}\quad R^{(1)} \leqslant r \leqslant R^{(2)},\\ 0, &{}\quad r> R^{(2)}, \end{array} \right. \nonumber \\ \end{aligned}$$

(A.16)

and $\frac{\mathrm{d} G(\cos \theta )}{\mathrm{d} \cos \theta }$ can be easily obtained from Eq. (A.9). Then, $\nabla W$ can be computed by combining Eq. (A.10) with Eqs. (10) and (11). With $\nabla W$ computed, the unit cell position vector ${\varvec{x}}=\left\{ r_1, r_2, \alpha _1, \alpha _2, \alpha _3\right\}$ is optimized by following the energy minimization procedure given in Algorithm 1. The equilibrium configuration and the radius of the CNT can then be obtained.