Polygonization and anomalous graphene interlayer spacing of multi-walled carbon nanofibers

The graphene interlayer spacing in pure graphite is known to have a minimum value of $d_{\mathit{min}}=0.3354\mathrm{nm}$, while defective graphites typically have larger interlayer spacings. Using x-ray diffraction, we find that the graphene interlayer spacing in multi-walled carbon nanofibers heat treated above $\approx 2800\mathrm{K}$ is distinctly smaller than $d_{\mathit{min}}$. To explain this unusual observation, we investigate the structural properties of carbon nanotubes using a multiscale approach rooted in extensive first-principles calculations, specifically allowing the nanotube cross sections to polygonize. We show that, whereas normal nanotubes are favored energetically at low temperatures, the configuration entropy associated with Stone-Wales defect creation at high temperatures makes the polygonal shape of large nanotubes or nanofibers thermodynamically stable, accompanied by a reduction in the graphene interlayer spacing. These unique predictions are confirmed in further experimental tests.

Figure 1

SEM (a) or TEM (b) image showing that as-grown fibers have circular cross sections (a) and heat-treated fibers are well graphitized (b). (c) Experimentally measured interlayer spacing of the fibers as a function of the heating temperature (Ref. 28). The dotted line marks the spacing of natural graphite. The error bar is within $4\times {10}^{-5}\mathrm{nm}$, smaller than the size of the data points.

Figure 2

The buckled-graphite model considered to investigate the role of the local curvature $\left(K_l\right)$ in determining interlayer interactions as shown on top of (a). The structures are stacked in $AA$ or $AB$ fashion and extend in three dimensions. (a) The optimum interlayer spacing $\left(d_{\mathit{opt}}\right)$ as a function of the local curvature. (b) The energy difference $\left(\Delta E_{\mathit{int}}\right)$ between the structure shown on top of (a) and that of the same two layers but with an infinite separation. (c) Formation energy [Eq. (2)] of single-walled polygonal nanotubes as a function of the local curvature. The solid lines are fits using Eq. (2) to the data points from LDA-DFT calculations. The inset illustrates a polygonal nanotube consisting of $N_p$ piecewise flat sides, with $N_p=6$. (d) Energy cost $\left(\Delta E\right)$ per unit area to make 30-walled polygonal nanotubes relative to the energy of a circular nanotube as a function of the innermost tube width $\left(W_{\mathit{in}}\right)$ and the polygon order $\left(N_p\right)$.

Figure 3

(Color online) (a) Total free energy of single-wall polygonal nanotubes per unit length relative to that of the conventional (or circular) carbon nanotubes as a function of the local curvature at the tube circumference $W=2000\mathrm{nm}$. (b) Dependence of the two critical temperatures on $W$. TEM images [(c) and (d)] and SEM image (e) confirm the polygonization of the nanotubes upon high-temperature heat treatment $(T⩾2800\mathrm{K}$).