Liquid surface model for carbon nanotube energetics

In the present paper we developed a model for calculating the energy of single-wall carbon nanotubes of arbitrary chirality. This model, which we call as the liquid surface model, predicts the energy of a nanotube with relative error less than 1% once its chirality and the total number of atoms are known. The parameters of the liquid surface model and its potential applications are discussed. The model has been suggested for open end and capped nanotubes. The influence of the catalytic nanoparticle, atop which nanotubes grow, on the nanotube stability is also discussed. The suggested model gives an important insight in the energetics and stability of nanotubes of different chirality and might be important for the understanding of nanotube growth process. For the computations we use empirical Brenner and Tersoff potentials and discuss their applicability to the study of carbon nanotubes. From the calculated energies we determine the elastic properties of the single-wall carbon nanotubes (Young modulus, curvature constant) and perform a comparison with available experimental measurements and earlier theoretical predictions.

Figure 1

(Color online) Construction of a nanotube from a graphene sheet. The vectors ${\vec{a}}_1$, ${\vec{a}}_2$ are the basis vectors in the graphene sheet. The chiral index $n$ is the number of steps in the ${\vec{a}}_1$ direction and the chiral index $m$ is that in the ${\vec{a}}_2$ direction. ${\vec{C}}_h=n{\vec{a}}_1+m{\vec{a}}_2$ is the chirality vector, and $\vec{T}$ is the translation vector and is in the direction of the tube axis. The chiral vector directions for both zigzag $(m=0)$ and armchair $(m=n)$ nanotubes are indicated.

Figure 2

(Color online) Different parts of the nanotube considered in the liquid surface model are shown. (a) Open end nanotube; (b) capped nanotube. $\Delta x$ is the width of the edge region of the nanotube, $N_e$ and $S_e$ are the number of atoms in the edge region and the area of this region, respectively. $N_i$ and $S_i$ are the number of atoms in the inner region and its surface area, respectively. $L$ and $R$ are the length and the radius of the nanotube. $N_{\text{cap}}$ and $S_{\text{cap}}$ are the number of atoms in the cap and its surface area, respectively.

Figure 3

(Color online) Area per atom in the inner and edge regions of a nanotube. $⟨a⟩$ indicates the average bond length in a nanotube.

Figure 4

(Color online) Comparison of the binding energy per atom calculated using the Tersoff potential and that obtained from the liquid surface model for the open end nanotubes. The curves show the values predicted by the model and different symbols (squares, triangles, stars, dots) correspond to the calculated values. The plots are shown for $n=5–10$. Each curve in one plot corresponds to a different $m$ value.

Figure 5

(Color online) Relative deviation of the liquid surface model predictions of the binding energy per atom from that calculated using the Tersoff potential, Eq. (36), for the open end nanotubes. Plots are shown for $n=5–10$. Each curve in one plot corresponds to different $m$ values.

Figure 6

(Color online) Plots (a)–(n): Comparison of the binding energy per atom calculated using the Brenner potential and that obtained from the liquid surface model for the capped nanotubes. The curves show the values predicted by the model and dots correspond to the calculated values. Plots (o) and (p) show the binding energies per atom calculated for capped (dots) and open end (squares) nanotubes of close sizes with chiralities (28,0) and (5,0). The chirality of the nanotubes is indicated in the inset to the plots.

Figure 7

(Color online) Relative deviation of the liquid surface model predictions of the binding energy per atom from that calculated using the Brenner potential, Eq. (36), for the capped nanotubes. Each dot in the plot corresponds to a certain nanotube of a given length and chirality.

Figure 8

(Color online) Comparison of the binding energy per atom calculated using Brenner (dots) and Tersoff (squares) potentials for capped nanotubes with chirality (5,0); (5,5); (6,0); (9,0). The line shows the binding energy per atom calculated using the liquid surface model.

Figure 9

(Color online) Single-wall carbon nanotube atop a catalytic nickel nanoparticle, $z_0$ is the distance between the nanotube and the catalytic nanoparticle, $L$ is the length of the nanotube, $R$ is the nanotube radius, and $R_{\text{part}}$ is the radius of the nanoparticle.

Figure 10

(Color online) Binding energy per atom calculated using the Brenner potential (dots) for a capped nanotube of chirality (28,0). Different lines show the binding energy per atom calculated using the liquid surface model with accounting for the interaction with the catalytic nanoparticle and correspond to the different values of the parameter $\epsilon$ which describes the interatomic interaction. The corresponding values of $\epsilon$ are indicated in the inset.