Thermodynamics of Linear Carbon Chains

Linear carbon chains (LCCs) are one-dimensional materials with unique properties, including high Debye temperatures and restricted selection rules for phonon interactions. Consequently, their Raman C-band frequency’s temperature dependence is a probe to their thermal properties, which are well described within the Debye formalism even at room temperatures. Therefore, with the basis on a semiempirical approach we show how to use the C band to evaluate the LCCs’ internal energy, heat capacity, coefficient of thermal expansion, thermal strain, and Grüneisen parameter, providing universal relations for these quantities in terms of the number of carbons atoms and the temperature.

Figure 1

Representative experimental results for LCCs ($N=35$, 39, 45, and 86): (a) experimental ${\mathrm{\Delta }\omega }_{\mathrm{LCC}}(T)$ evolution with $T$. (b) The energy per $N$, $u(T)$, presents a quadratic, universal, and unified behavior with $T$; (c) $\alpha (T)$ shows a linear universal behavior with $T$, where LCCs with $N$ from 35 to 45 atoms present very similar $\alpha (T)$. For $N=86$, however, $\alpha (T)$ increases at a slightly higher rate with increasing $T$. (d) The heat capacity per $N$, $c_v(T)$, presents a linear, universal, and unified behavior with $T$. (e) A $T^2$ universal dependence is observed for the thermal strain ${ϵ}_T(T)$ with $N$ from 35 to 45 atoms presenting very similar ${ϵ}_T(T)$, while for $N=86$ ${ϵ}_T(T)$ increases at a slightly higher rate with increasing $T$; (f) Every LCC presents a distinct linear dependence of ${ϵ}_T$ with ${\mathrm{\Delta }\omega }_{\mathrm{LCC}}(T)$. The curves show the predictions of the pure Debye model (solid lines) compared with the model proposed (dashed lines) and the insets in (a)–(f) suggest that for temperatures $300\mathrm{K}\le T\le 700\mathrm{K}$ the observables evolve more slowly with increasing $T$. Figures S3 and S4 in the Supplemental Material [56] show the results including all the 10 LCCs measured in this work.

Figure 2

(a) The distribution function $x/(e^x-1)$ with $x=T_{\mathrm{ph}}/T$ leads us to identify the high-temperature limit ($T_{\max }$) of the model. The distribution is most relevant for $0\le x\le 4$, which implies that $T_{\max }=T_{\mathrm{LCC}}/4\approx 700\mathrm{K}$ (see inset). (b) Universal dependences with $N$ for both ${\gamma }_P(N)=\ln {(N-20)}^{0.18}$ (red solid lines) and $d^2{\omega }_{\mathrm{LCC}}/d{T}^2=A+B/N$ (blue solid line with $A=-1.63\times {10}^{-4}{\mathrm{cm}}^{-1}{\mathrm{K}}^{-2}$ and $B=35\times {10}^{-4}{\mathrm{cm}}^{-1}{\mathrm{K}}^{-2}$). The symbols (red circles and blue diamonds for ${\gamma }_P$ and $d^2{\omega }_{\mathrm{LCC}}/d{T}^2$, respectively) represent the experimental data. (c) ${\omega }_{\mathrm{LCC}}^0(N,0)=(1757+3980/N){\mathrm{cm}}^{-1}$, where ${\omega }_{\mathrm{LCC}}^0(N,0)$ is the C-band frequency at $T=0\mathrm{K}$. The open circles represent the experimental data. The inset shows a representative case at $T=700\mathrm{K}$ for $\mathrm{\Delta }(N,T)={\mathrm{\Delta }\omega }_{\mathrm{LCC}}^{\mathrm{Corr}}-{\mathrm{\Delta }\omega }_{\mathrm{LCC}}^{\mathrm{Debye}}$, which shows how the difference between ${\mathrm{\Delta }\omega }_{\mathrm{LCC}}^{\mathrm{Corr}}$ proposed here [dashed lines in Fig. 1] and ${\mathrm{\Delta }\omega }_{\mathrm{LCC}}^{\mathrm{Debye}}$ [solid lines in Fig. 1] progresses with $N$.