Atomic structure and mechanical properties of carbyne

The atomic structure and mechanical properties of the carbyne (monatomic linear chains), containing from 2 to 21 carbon atoms, are theoretically investigated by ab-initio methods. We demonstrate the existence of a stable cumulene structure in the inner part of chains with the number of atoms $N\ge 10$. We present a general stress-strain diagram of chains until the moment when they break, which enables to determine their strength, elasticity, and fragility. These diagrams can be utilized to calibrate empirical potentials, especially for large deviations of the atoms from the equilibrium positions. For chains with $N\ge 4$, the relationship between the strength of the chain and the binding energy of the edge atom in the chain is established. The existence of scale-effect and “even-odd” effect for such properties as strength, elasticity, and fragility is observed. We demonstrate that the five-atom carbon chains show the maximum strength value.

Figure 1

Supercell for calculation of atomic and electronic structure of finite chain containing five atoms, as an example (scheme).

Figure 2

The dependence of force $F$ on strain $e$ for the whole chain with odd (a) and even (b) number of atoms, and on the value of strain $\varepsilon$ at the edge atomic bond in “odd” (c) and “even” (d) chains: $F_c$ is critical stress for instability of atomic bond (bond strength); ${\varepsilon }_c$ and $e_c$ are critical strain of instability of atomic bond and the chain as a whole, respectively; ${\varepsilon }_f$ and $e_f$ are fracture strain for atomic bond and chain, respectively; $k_Y$ is coefficient of elasticity of the chain; $N$ is number of atoms.

Figure 3

The distribution of the electron density in the bulk of the chain and its cross section passing through the middle of the edge atomic bond in the unloaded state (dashed line) and at the moment of instability of the chain (solid line).

Figure 4

The dependence of the length of interatomic bond $a_{i,j}$ on the number of atoms in the chain $N$.

Figure 5

The dependence of the value of bond length alternation (BLA) on the number of atoms in the chain $N$: ${\text{BLA}}_{1,3}$ and ${\text{BLA}}_{2,4}$ are the values of $\text{BLA}$ for atomic bonds $a_{1,2}$ and $a_{2,3}$; $a_{2,3}$ and $a_{3,4}$, respectively.

Figure 6

The dependence of the binding energy on the number of atoms: $E_1^b\left(N\right)$ and $E_2^b\left(N\right)$ for the first and second atoms from chain edge, $E_{\mathrm{cum}}^b$ for inner atoms, $E^b\left(N\right)$ for energy of separation, calculated in the Eq. (8), $E_0^b\left(N\right)$ for the average binding energy per atom [solid line—calculated in Eq. (11)].

Figure 7

The effect of the number of atoms in carbyne on its strength $F_c$, fragility $e_c$, and hardness $k_Y$.

Figure 8

The relationship between the strength of the chain $F_c$ and the binding energy $E_1^b$ of the edge atom: $•$—odd chains; $\circ$—even chains.