Angular Dependence of Four Mechanical Properties of Graphynes

In 1987, Baughman, Eckhardt and Kertesz proposed the existence of 7 sets (or “families” as we call them) of novel two-dimensional carbon allotropes called “graphynes”. These allotropes are composed of acetylene chains of length <i>n</i>, [–C≡C–]<i>n</i>, and connected by aromatic rings or sp² carbon-carbon bonds [1]. Although graphdiynes (GYs with <i>n</i> = 2) [2] and graphtetraynes (GYs with <i>n</i> = 4) [3] have been synthesized several years ago, the densest form of GY with <i>n</i> = 1 was only recently synthesized by several groups [4-6]. This result is significant given that graphyne has been considered a potential candidate to replace graphene in certain applications [7]. In 2022, we conducted [8] a comprehensive computational study to predict the mechanical properties of 70 GYs, 10 GYs (for 1 ≤ <i>n</i> ≤ 10) of each original family proposed by Baughman, Eckhardt and Kertesz. The elastic compliances <i>C_ij</i> of all structures were obtained from molecular dynamics calculations, and the following mechanical quantities were obtained from them along two main directions: the Young’s modulus, shear modulus, Poisson’s ratio and linear compressibility. In this work, we present the angular dependence of the four mechanical quantities on the direction of application of the tensile stress. First, the rules for transforming the elements <i>C_ij</i> of the elastic compliance tensor under rotation of an angle θ in the GY structure plane were revised and applied. The resulted equation <i>C’_ij</i>(θ) = R(θ) ● <i>C_ij</i> ● R^-1(θ), with R(θ) being the rotation matrix by angle θ [9], was used to generate the polar plots of all mechanical quantities of the 7 GY families (for 1 ≤ <i>n</i> ≤ 10). As demonstrated by Huntington [10], the shear modulus is independent of θ, and thus the corresponding polar plots exhibit only circles. Additionally, for the symmetric GY structures, or the families known in the literature by the Greek letters γ-, β- and α-GYs (which correspond to our designated GY families 1, 4 and 7, respectively), also exhibit circular results, as expected. However, the most intriguing results are observed for the non-symmetric GY families. We show that the Young’s modulus can vary by up to a factor of ten depending on the direction of the applied tensile stress. Furthermore, the Poisson’s ratio can be altered from values less than one to values greater than two for some of the asymmetric GY families. The linear compressibility of asymmetric GYs presents even more intriguing results. For instance, the known 14,14,14- and 14,14,18-GYs present a change in sign of their linear compressibility, from positive to negative as the angle is altered. This indicates the existence of directions along which the linear compressibility of these structures is null. The nonlinear dependence of the elastic moduli and Poisson’s ratio on <i>n</i> contrasts with the linear dependence on <i>n</i> of the linear compressibility in both symmetric and asymmetric GYs. We discuss potential applications of these distinctive properties of different GYs. The authors would like to acknowledge the support and grants provided by the São Paulo Research Foundation (FAPESP) (#2020/02044-9 and #2023/02651-0), the Brazilian Agency CNPq (#303284/2021-8) and by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior-Brasil (CAPES)-Finance Code 001.<br/><br/><b>References</b><br/><br/>[1] 1987, DOI: 10.1063/1.453405<br/>[2] 2010, DOI: 10.1039/B922733D<br/>[3] 2018, DOI: 10.1016/j.nanoen.2017.11.005<br/>[4] 2022, DOI: 10.1038/s44160-022-00068-7<br/>[5] 2022, DOI: 10.1021/jacs.2c06583<br/>[6] 2022, DOI: 10.1016/j.carbon.2022.08.061<br/>[7] 2022, https://bigthink.com/the-future/graphyne/<br/>[8] 2022, DOI: 10.1016/j.cartre.2022.100152<br/>[9] 2006, Nye, J. F. <i>Physical properties of crystals</i>. Oxford University Press.<br/>[10] 1958, Huntington, H. B. “The Elastic Constants of Crystals” in <i>SOLID STATE PHYSICS, Advances in Research and Applications</i>. Volume 7. Edited by F. Seitz and D. Turnbull. Academic Press Inc.