Exploring the Bounds on the Young’s Modulus and Gravimetric Young’s Modulus

The continuous discovery of ultrahigh-modulus materials has increased the record for the Young’s modulus and the gravimetric Young’s modulus. However, the theoretical bounds on these moduli are still unknown. The upper bounds depend on the limits of the stiffness, alignment, and density of chemical bonds. From these limits, we here develop theoretical expressions for predicting the Young’s modulus, $Y_{\max }$ = [${\hslash }^2$/($m_e^2{a}_B^2$)]${\rho }_e$, and the gravimetric Young’s modulus, $Y_{\rho ,\max }=[{\hslash }^2$/($m_e{m}_p{a}_B^2$)]($N_e/A$), for ideal extreme-modulus solids, where ħ, $m_e$, $m_p$, $a_B$, ${\rho }_e$, and $N_e/A$ are the reduced Planck constant, electron mass, proton mass, Bohr radius, mass density of valence electrons, and the ratio of valence-electron number to atomic mass. By substituting the values of the nonconstant parameters (${\rho }_e$ and $N_e/A$) for all elements into these expressions, the upper bounds on the Young’s modulus and gravimetric Young’s modulus are predicted to be 3074 GPa and 1036 GPa g${}^{-1}$ cm${}^3$. These predictions are supported by the fact that the Young’s modulus and gravimetric Young’s modulus from a large set of experiments and first-principles calculations fall within these bounds. Moreover, by applying lateral pressure to linear carbyne crystals, the first-principles-calculated maximum Young’s modulus and gravimetric Young’s modulus are 2973 GPa and 968 GPa g${}^{-1}$ cm${}^3$, respectively, which are near the predicted bounds. These carbyne crystals are predicted to have space group R-3m.

Figure 1

Brief history of the Young’s modulus (red bars) and gravimetric Young’s modulus (blue bars) in the highest-modulus direction.

Figure 2

Illustration of ideal extreme-high-modulus solids, in which all chemical bonds are stiff, aligned in the tensile direction, and densely packed.

Figure 3

(a) Mass density of valence electrons (ρ_e), and (b) ratio of the number of valence electrons to the atomic mass (N_e/A) for different elemental solids.

Figure 4

Predicted bounds on the Young’s modulus and gravimetric Young’s modulus (dotted red lines), compared with the Young’s modulus and gravimetric Young’s modulus calculated from the MPDB [23], the database of experimental elastic tensors [24], data in Table S1 within the Supplemental Material [1, 8, 9, 10, 13, 16, 17, 18, 19, 20, 21], and our calculated data for compressed carbyne chains.

Figure 5

(a) Top and side views of the carbyne crystal structure. Computational cell is denoted by the dashed box. (b) Atomic and electronic structures of compressed carbyne chains. (c) Young’s modulus (Y) for compressed carbyne crystals as a function of atomic volume (V_a). Binding energy is defined as the sum of the energy of noninteracting carbyne chains minus the total energy of compressed carbyne chains (normalized per atom). (d) Predicted crystal structure that minimizes the total energy for both cumulene and polyyne chain crystals, in which the primitive cell is denoted by the red box. In this structure, there are three types of carbon chains, which have axial symmetries of 3 (gray chains), 3₁ (green chains), and 3₂ (brown chains), where the 3₁ and 3₂ chains are related by mirror planes. Predicted equilibrium unit-cell parameters are a = b = 6.409 Å and c = 1.281 Å for the cumulene crystal and a = b = 6.436 Å and c = 2.565 Å for the polyyne crystal. Atomic fractional coordinates for the primitive cells are (0,0,0) for the cumulene crystal and (0,0, 0.2546) for the polyyne crystal.