Anisotropic and isotropic elasticity and thermal transport in monolayer C$_{24}$ networks from machine-learning molecular dynamics

Two-dimensional fullerene networks have recently attracted increasing interest due to their diverse bonding topologies and mechanically robust architectures. In this work, we develop an accurate machine-learned potential NEP-C${24}$ for both the quasi-hexagonal phase (qHP) and the quasi-tetragonal phase (qTP) C${24}$ monolayers, based on the neuroevolution potential (NEP) framework. Using this NEP-C${24}$ model, we systematically investigate the elastic and thermal transport properties. Compared with C${60}$ monolayers, both C${24}$ phases exhibit markedly enhanced stiffness, arising from the combination of reduced molecular size and increased density of covalent bonds. The qTP C${24}$ monolayer shows nearly isotropic elastic properties and thermal conductivities along its two principal axes owing to its four-fold symmetry, whereas the chain-like, misaligned bonding topology of the qHP C${24}$ monolayer leads to pronounced in-plane anisotropy. Homogeneous nonequilibrium molecular dynamics and spectral decomposition analyses reveal that low-frequency ($<5$ THz) acoustic phonons dominate heat transport, with directional variations in phonon group velocity and mean free path governing the anisotropic response in qHP C${24}$. Real-space heat flow visualizations further show that, in these fullerene networks, phonon transport is dominated by strong inter-fullerene covalent bonds rather than weak van der Waals interactions. These findings establish a direct link between intermolecular bonding topology and phonon-mediated heat transport, providing guidance for the rational design of fullerene-based two-dimensional materials with tunable mechanical and thermal properties.

💡 Research Summary

In this work the authors develop a dedicated machine‑learned interatomic potential, NEP‑C24, for two-dimensional fullerene networks composed of the smallest stable fullerene molecule, C24. The potential is built within the neuroevolution potential (NEP) framework, which combines an evolutionary strategy with a feed‑forward neural network. A training set of 500 structures (400 for training, 100 for testing) was generated by performing density‑functional theory (DFT) calculations (PBE‑D3) on strained configurations of both the quasi‑hexagonal (qHP) and quasi‑tetragonal (qTP) phases. The resulting root‑mean‑square errors are 3.1 meV/atom for energy, 29.9 meV/atom for virial, and 184 meV/Å for forces, markedly better than the generic NEP‑Carbon model (520 meV/Å) and the traditional Tersoff potential (≈2000 meV/Å). Validation through radial and angular distribution functions, as well as phonon dispersion curves, shows excellent agreement with DFT, confirming that NEP‑C24 faithfully reproduces both short‑range order and harmonic inter‑atomic forces.

Elastic properties were obtained by finite‑strain DFT calculations and reproduced with NEP‑C24. For the qHP phase the elastic constants are C11 ≈ 236 N m⁻¹, C22 ≈ 295 N m⁻¹, C12 ≈ 24 N m⁻¹, C66 ≈ 110 N m⁻¹, indicating a pronounced anisotropy between the chain direction (x) and the inter‑chain direction (y). In contrast, the qTP phase exhibits C11 ≈ C22 ≈ 249 N m⁻¹, C12 ≈ −16 N m⁻¹, C66 ≈ 82 N m⁻¹, reflecting its near‑square lattice and four‑fold symmetry. From these constants the orientation‑dependent Young’s modulus E(θ), shear modulus G(θ), and Poisson’s ratio ν(θ) were derived. The qTP monolayer shows almost angle‑independent values (isotropic elasticity), whereas the qHP monolayer displays up to 30 % variation with θ, confirming strong mechanical anisotropy caused by its chain‑like bonding topology.

Thermal transport was investigated using homogeneous nonequilibrium molecular dynamics (HNEMD). By applying a zero‑net external driving force, the authors measured the steady‑state heat current and extracted the diagonal components of the thermal conductivity tensor. At 300 K the qTP phase has κₓ ≈ κ_y ≈ 12 W m⁻¹ K⁻¹, essentially isotropic. The qHP phase shows κₓ ≈ 9 W m⁻¹ K⁻¹ and κ_y ≈ 13 W m⁻¹ K⁻¹, a clear anisotropy that mirrors the elastic behavior. Spectral decomposition of the heat current reveals that low‑frequency acoustic phonons below 5 THz dominate heat conduction, contributing more than 70 % of the total conductivity. The two in‑plane acoustic branches (longitudinal and transverse) are responsible for most of the transport, while out‑of‑plane flexural modes and higher‑frequency optical modes play a minor role. Directional differences in phonon group velocities and mean free paths—particularly a reduced group velocity and shorter mean free path along the chain direction—explain the lower κₓ in qHP. Real‑space visualizations of heat flow further demonstrate that heat is carried primarily through the strong covalent bonds linking neighboring C24 molecules, rather than through weak van der Waals interactions.

Overall, the study establishes a clear structure‑property relationship: the reduced molecular size of C24 relative to C60, combined with a higher density of covalent inter‑fullerene bonds, yields significantly higher stiffness. Moreover, the specific bonding topology (chain‑like in qHP versus square‑like in qTP) directly governs both elastic anisotropy and phonon‑mediated thermal conductivity anisotropy. The newly developed NEP‑C24 potential proves capable of accurately describing mixed sp²/sp³ carbon systems at a computational cost suitable for large‑scale molecular dynamics, opening the door to systematic design of 2D fullerene‑based materials with tailored mechanical and thermal performance for applications such as flexible electronics, thermal management, and nano‑mechanical devices.