Anharmonicity and Universal Response of Linear Carbon Chain Mechanical Properties under Hydrostatic Pressure

Isolated linear carbon chains (LCCs) encapsulated by multiwalled carbon nanotubes are studied under hydrostatic pressure ($P$) via resonance Raman scattering. The LCCs’ spectroscopic signature C band around $1850{\mathrm{cm}}^{-1}$ softens linearly with increasing $P$. A simple anharmonic force-constant model not only describes such softening but also shows that the LCCs’ Young’s modulus ($E$), Grüneisen parameter ($\gamma$), and strain ($ϵ$) follow universal $P^{-1}$ and $P^2$ laws, respectively. In particular, $\gamma$ also presents a unified behavior for all LCCs. To the best of our knowledge, these are the first results reported on such isolated systems and the first work to explore universal $P$-dependent responses for LCCs’ $E$, $ϵ$, and $\gamma$.

Figure 1

(a) Resonance Raman profile of an isolated LCC@MWCNT. The $G$ band is located around $1575{\mathrm{cm}}^{-1}$ and the C band is located around $1850{\mathrm{cm}}^{-1}$. Inset: Representative spectra showing the evolution of the LCCs’ C bands with $P$. (b) The C band Raman frequencies evolution ${\omega }_{\mathrm{LCC}{n}_1}$, ${\omega }_{\mathrm{LCC}{n}_2}$, ${\omega }_{\mathrm{LCC}{n}_3}$, and ${\omega }_{\mathrm{LCC}{n}_4}$ with $P$. The frequencies linearly decrease with distinct $d{\omega }_{\mathrm{LCC}}/dP$. (c) $\mathrm{\Delta }\omega ={\omega }_{\mathrm{LCC}}-{\omega }_{\mathrm{LCC}}^0$ as a function of $P$: the loading cycles show a universal but nonunified $d{\omega }_{\mathrm{LCC}}/dP$.

Figure 2

(a) Young’s modulus $E$, (b) strain $ϵ$, and (c) Grüneisen parameter $\gamma$ as a function of $P$ for each LCC. Both $E$ and $\gamma$ follow a $P^{-1}$ universal law, while $ϵ$ follows a $P^2$ universal law. (d) $(\mathrm{\Delta }\omega /\omega )(E/\gamma )=-P$, an important parameter for nanometrology, is universal and unified. (a)–(d) the solid lines are experimental data fittings. Insets: (a) shows the $E(P)$ vs $P$ graphic enlarged to highlight the $P\to \infty$ limit. The horizontal dashed lines correspond to the values $E$ (4.60 GPa) for each LCC. (b) shows the evolution of $ϵ$ with relation to the relative changes of ${\mathrm{C}}_1$ with increasing $P$. (c) shows the $\gamma (P)$ vs $P$ graphic zoomed to highlight the $P\to \infty$ limit. The horizontal dashed lines correspond to the values of $\gamma$ calculated with $E=0.3\mathrm{TPa}$ for each LCC [50]. The vertical dashed lines in (a) and (c) stand for $P_c=0.1\mathrm{GPa}$.