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, ${Y}_{\mathrm{max}}$ = [${\ensuremath{\hbar}}^{2}$/(${m}_{e}^{2}{a}_{B}^{2}$)]${\ensuremath{\rho}}_{e}$, and the gravimetric Young's modulus, ${Y}_{\ensuremath{\rho},\mathrm{max}}=\phantom{\rule{0.1em}{0ex}}[{\ensuremath{\hbar}}^{2}$/(${m}_{e}{m}_{p}{a}_{B}^{2}$)](${N}_{e}/A$), for ideal extreme-modulus solids, where \ensuremath{\Elzxh}, ${m}_{e}$, ${m}_{p}$, ${a}_{B}$, ${\ensuremath{\rho}}_{e}$, 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 (${\ensuremath{\rho}}_{e}$ 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${}^{\ensuremath{-}1}$ 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${}^{\ensuremath{-}1}$ cm${}^{3}$, respectively, which are near the predicted bounds. These carbyne crystals are predicted to have space group R-3m.