Radial deformation and stability of single-wall carbon nanotubes under hydrostatic pressure

In this paper we have developed a theory of energetics for isolated single-wall carbon nanotubes (SWNTs) deformed in the radial direction, and applied this theory to investigate their deformation characteristics and stability under hydrostatic pressure. The starting point of the theory is the strain energy of SWNTs predicted by ab initio calculations based on the density functional theory (DFT), which shows the same behavior as that obtained for the continuum elastic shell model. We extend this result for inflated SWNTs with circular cross section to calculate the deformation energy of a deformed SWNT without performing further DFT calculations. This extension is then complemented by a van der Waals interaction, which is not fully taken into account in the DFT approximations currently in use but becomes important in highly deformed tubes. We find that the minimum pressure, $P_1$, for the radial deformation to occur is proportional to the inverse cube of tube diameter, $D$, in agreement with the recent theoretical predictions as well as the classical theory of buckling. The radial deformation of SWNTs with $D<2.5\mathrm{nm}$ is found to be elastic up to very high pressure and they hardly collapse. On the other hand, SWNTs with $D>2.5\mathrm{nm}$ show a plastic deformation and collapse if the applied hydrostatic pressure exceeds a critical value, which is about 30–40% higher than $P_1$ and also varies as $D^{-3}$ though approximately. These SWNTs with large $D$ collapse when the cross-sectional area is about 60% reduced with respect to the circular one. It is also found that for SWNTs with $D>7.0\mathrm{nm}$, the plastically deformed (collapsed) state is more stable than the inflated one. This critical value of $D$ is somewhat larger than previously predicted.

Figure 1

Schematic view of the superimposed two circumferences of a cross section in the plane perpendicular to the tube axis. $\mathbf{t}$ and ${\mathbf{t}}^{\prime }$ are unit tangential vectors at the atomic positions on the two circumferences.

Figure 2

Geometrical factors given by Eqs. (5, 6). Here, $G_0(\mathbf{r},{\mathbf{r}}^{\prime })$ and $G(\mathbf{r},{\mathbf{r}}^{\prime })$ are written as $G_0\left(\theta \right)$ and $G\left(\theta \right)$, respectively, with $\mathbf{t}\bullet {\mathbf{t}}^{\prime }=\cos \theta$.

Figure 3

Interatomic interactions, ${\varphi }_{\mathrm{DFT}}\left(r\right)$ and ${\varphi }_{\mathrm{vdW}}\left(r\right)$, obtained by the analysis of the interlayer cohesive energy of graphite.

Figure 4

Model shapes of the cross section of a deformed SWNT: (a) oval and (b) peanut. Both shapes reduce to an ellipse as a whole when the centers of two ellipses on both sides coincide with each other. Each shape, either oval or peanut, transforms into another through a racetracklike shape, which corresponds to the infinite radius of two circles on the top and bottom.

Figure 5

(Color online) Deformation energies as functions of the fractional volume decrease, $X$, defined by Eq. (18). The full DFT (LDA) results for the armchair (8, 8), (10, 10), and (12, 12) SWNTs (squares, circles, and triangles, respectively) are compared with the DFT contributions, $\Delta E_{\mathrm{DFT}}=\Delta E_{\mathrm{DFT}}^{\left(0\right)}+\Delta E_{\mathrm{DFT}}^{\left(1\right)}$, obtained by the present continuum model for the corresponding tubes with $D=1.085\mathrm{nm}$ (broken line), $D=1.356\mathrm{nm}$ (solid line), and $D=1.627\mathrm{nm}$ (dotted line). The cross sections of the unit sells used in the full DFT (LDA) calculations are also shown for the deformed (10, 10) tubes.

Figure 6

(Color online) Deformation energies of the zigzag (12, 0) and (18, 0) SWNTs obtained by the full DFT (LDA) calculations (squares and circles, respectively) and their comparisons with the DFT contributions (broken lines), $\Delta E_{\mathrm{DFT}}=\Delta E_{\mathrm{DFT}}^{\left(0\right)}+\Delta E_{\mathrm{DFT}}^{\left(1\right)}$, obtained by the present continuum model for the corresponding tubes with $D=0.940\mathrm{nm}$ and $D=1.409\mathrm{nm}$. For the latter tube $(D=1.409\mathrm{nm})$, the deformation energies with the successive inclusions of the contributions interwall and vdW interactions are also shown: $\Delta E_{\mathrm{DFT}}^{\left(0\right)}$ (dotted line), $\Delta E_{\mathrm{DFT}}$ (broken line), and $\Delta E$ given by Eq. (9) (solid line). The cross-sectional shapes of the deformed (18, 0) tube are similar to those of the (10, 10) tube (Fig. 5).

Figure 7

Deformation energies obtained by the present method for SWNTs with $D=1–7\mathrm{nm}$ as functions of $X$. The optimized shapes at selected values of $X$ are also shown.

Figure 8

Circular-to-oval transition pressure ($P_1$, closed circles), oval-to-peanut transition pressure ($P_2$, broken line), and collapse pressure ($P_{\text{collapse}}$, open circles) as functions of tube diameter. The solid and broken lines are smooth interpolations of the calculated results.