Phase diagram of single-wall carbon nanotube crystals under hydrostatic pressure

In a number of studies it has been shown that single-wall carbon nanotube crystals undergo phase transformations and shape changes as a function of nanotube radius and hydrostatic compression. When large radius nanotubes form a crystal, the circular tube cross section deforms to a hexagonal shape, while under hydrostatic pressure the cross section can transform from circular to oval or to a so-called racetrack shape. At still higher pressures cross linking of nanotubes has been inferred. A model which explains and describes these phenomena is presented and a phase diagram is computed. It is shown that the intertube interaction plays a dominant role at low pressure only, and that the curvature energy is important over the whole pressure range. The model and the general conclusions should apply also to other nanotube materials such as those based on BN and $\mathrm{Mo}{\mathrm{S}}_2$.

Figure 1

Geometry of a hexagon-shaped polygonized nanotube and of a racetrack-shaped nanotube with characteristic dimensions.

Figure 2

Maximal circumference of round nanotubes as a function of the dimensionless parameter $K$ according to Eq. (25).

Figure 3

(Color) Phase diagram of single-wall carbon nanotube crystals as a function of nanotube circumference and hydrostatic pressure. Areas with round (R), oval (O), hexagonal (H), racetrack (T), cross-linked (X) nanotubes are indicated with red, dark blue, green, yellow, light blue, respectively. The letters indicate the states in order of decreasing stability. The marking (XX) indicates that both the oval and racetrack-shaped nanotubes are cross linked. (a) for $u_c=2\mathrm{eV}{\mathrm{\AA }}^2∕\text{atom}$, $u_{ig}\left(s_0\right)=-0.02\mathrm{eV}∕\text{atom}$, $u_{ic}\left(s_0\right)=-0.0442\mathrm{eV}{\mathrm{\AA }}^{1∕2}∕\text{atom}$; (b) as (a), but neglecting the racetrack state; (c) for $u_c=1\mathrm{eV}{\mathrm{\AA }}^2∕\text{atom}$, $u_{ig}\left(s_0\right)=-0.02\mathrm{eV}∕\text{atom}$, $u_{ic}\left(s_0\right)=-0.0442\mathrm{eV}{\mathrm{\AA }}^{1∕2}∕\text{atom}$; (d) for $u_c=2\mathrm{eV}{\mathrm{\AA }}^2∕\text{atom}$, $u_{ig}\left(s_0\right)=-0.01\mathrm{eV}∕\text{atom}$, $u_{ic}\left(s_0\right)=-0.0221\mathrm{eV}{\mathrm{\AA }}^{1∕2}∕\text{atom}$. In all cases ${\rho }_X=2\mathrm{\AA }$ and $s_0=3.1\mathrm{\AA }$.

Figure 4

Interaction between curved regions in racetrack-(a) and hexagon (b)-shaped nanotubes.

Figure 5

(Color) As Fig. 3a, but with $u_{ic}\left(s_0\right)=-0.0442*1.1\mathrm{eV}{\mathrm{\AA }}^{1∕2}∕\text{atom}$ for the hexagonal state only as explained in the text.

Figure 6

Contributions to the enthalpy for racetrack-shaped nanotube relative to a round nanotube as a function of the order parameter $\eta$ for $C=42\mathrm{\AA }$ and $P=5\mathrm{GPa}$. (c) curvature energy; (i) intertube interaction energy; (pv) $PV$ term; and (t) total enthalpy.

Figure 7

Separation distance $S$ as a function of pressure between nanotubes with a circumference $C=42\mathrm{\AA }$ and round (R), hexagonal polygonized (H), oval (O), and racetrack-shaped (T) cross sections. The pressure at which the radius of curvature reaches the cross-linking radius is indicated with vertical arrows for oval- and racetrack-shaped nanotubes.

Figure 8

Order parameter $\eta$ as a function of pressure in crystals of nanotubes with a circumference $C=42\mathrm{\AA }$ and round (R), hexagonal polygonized (H), oval (O), and racetrack-shaped (T) cross sections. Arrows as in Fig. 7.