Abstract
Understanding the structure and dynamics of molten Carbon extends beyond the study of carbon-rich planetary interiors, with perspective applications in nuclear fusion, material science and industrial processes. While the recent X-ray observation of liquid C rekindled the interest in its microscopic structure, it also reminds of the challenges in rationalising the complex dynamics extending beyond the hydrodynamic regime. Primary among them is the theoretical description of non-hydrodynamic processes in one-component liquids. Here, we report collective longitudinal and transverse propagating modes in molten Carbon at T = 5500 K and pressure range 10-40 GPa from ab initio simulations and machine learned molecular dynamics. We observe an unusual two-peak shape of the longitudinal current spectral functions pointing at a branch of non-acoustic propagating modes in the wave number range k > 1 Å−1. By applying a generalized collective modes framework to recover the time dependence of time correlations in the system, we identify the peak at lower energy as a non-hydrodynamic mode, and ascribe it to out-of-phase motion of particles encapsulated in cages of their nearest neighbors.
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References
Hull, C. J., Raj, S. L. & Saykally, R. J. The liquid state of carbon. Chem. Phys. Lett. 749, 137341 (2020).
Reitze, D. H., Ahn, H. & Downer, M. C. Optical properties of liquid carbon measured by femtosecond spectroscopy. Phys. Rev. B 45, 2677 (1992).
Knudson, M. D., Desjarlais, M. P. & Dolan, D. H. Shock-wave exploration of the high-pressure phases of carbon. Science 322, 1822 (2008).
Kraus, D. et al. Probing the complex ion structure in liquid carbon at 100 GPa. Phys. Rev. Lett. 111, 255501 (2013).
Principi, E. et al. Atomic and electronic structure of solid-density liquid carbon. Phys. Rev. Lett. 125, 155703 (2020).
Kraus, D. et al. The structure of liquid carbon elucidated by in situ X-ray diffraction. Nature 642, 351 (2025).
Galli, G., Martin, R. M., Car, R. & Parrinello, M. Structural and electronic properties of amorphous carbon. Phys. Rev. Lett. 63, 988 (1989).
Galli, G., Martin, R. M., Car, R. & Parrinello, M. Ab initio calculation of properties of carbon in the amorphous and liquid states. Phys. Rev. B 42, 7470 (1990).
Dharma-wardana, M. W. C. & Perro, F. Density-functional study of C, Si, and Ge metallic liquids. Phys. Rev. Lett. 65, 76 (1990).
Bethkenhagen, M. et al. Carbon ionization at gigabar pressures: An ab initio perspective on astrophysical high-density plasmas. Phys. Rev. Res. 2, 023260 (2020).
Hansen, J.-P. & McDonald, I. R.Theory of Simple Liquids (London: Academic) (1986).
de Schepper, I. M. et al. Hydrodynamic time correlation functions for a Lennard-Jones fluid. Phys. Rev. A 38, 271 (1988).
Bryk, T. Non-hydrodynamic collective modes in liquid metals and alloys. Eur. Phys. J. Spec. Top. 196, 65 (2011).
Inui, M. et al. Anomalous dispersion of the acoustic mode in liquid Bi. Phys. Rev. B 92, 054206 (2015).
Inui, M. et al. Low energy excitation in liquid Sb and liquid Bi observed in inelastic x-ray scattering spectra. J. Phys.: Condens. Matt. 33, 475101 (2021).
Hosokawa, S. et al. Transverse acoustic excitations in liquid Ga. Phys. Rev. Lett. 102, 105502 (2009).
Hosokawa, S. et al. Transverse excitations in liquid Sn. J. Phys. Condens. Matt. 25, 112101 (2013).
Bryk, T., Ruocco, G., Scopigno, T. & Seitsonen, A. P. Pressure-induced emergence of unusually high-frequency transverse excitations in a liquid alkali metal: Evidence of two types of collective excitations contributing to the transverse dynamics at high pressures. J. Chem. Phys. 143, 104502 (2015).
Bryk, T., Demchuk, T., Wax, J.-F. & Jakse, N. Pressure-induced effects in the spectra of collective excitations in pure liquid metals. J. Phys.: Condens. Matt. 32, 184002 (2020).
Bryk, T. & Mryglod, I. Optic-like excitations in binary liquids: Transverse dynamics. J. Phys. Condens. Matt. 12, 6063 (2000).
Bryk, T. & Mryglod, I. Longitudinal optical-like excitations in binary liquid mixtures. J. Phys. Condens. Matt. 14, L445 (2002).
Mryglod, I. M., Omelyan, I. P. & Tokarchuk, M. V. Generalized collective modes for the Lennard-Jones fluid. Mol. Phys. 84, 235 (1995).
Scopigno, T., Balucani, U., Ruocco, G. & Sette, F. Density fluctuationsin molten lithium: inelastic x-ray scattering study. J. Phys. Condens. Matt. 12, 8009 (2000).
Scopigno, T., Ruocco, G. & Sette, F. Microscopic dynamics in liquid metals: The experimental point of view. Rev. Mod. Phys. 77, 881 (2005).
Bryk, T. & Ruocco, G. Generalised hydrodynamic description of the time correlation functions of liquid metals: Ab initio molecular dynamics study. Mol. Phys. 111, 3457 (2013).
Kresse, G. & Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B 47, 558 (1993).
Kresse, G. & Hafner, J. Ab initio molecular-dynamics simulation of the liquid-metal-amorphous-semiconductor transition in germanium. Phys. Rev. B 49, 14251 (1994).
Kresse, G. & Furthmüller, J. Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set. Comput. Mat. Sci. 6, 15 (1996).
Kresse, G. & Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169 (1996).
Blöchl, P. E. Projector augmented-wave method. Phys. Rev. B 50, 17953 (1994).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B 59, 1758 (1999).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 77, 3865 (1996).
Sun, J., Ruzsinszky, A. & Perdew, J. P. Strongly constrained and appropriately normed semilocal density functional. Phys. Rev. Lett. 115, 036402 (2015).
Perdew, J. P. SCAN meta-GGA, strong correlation, symmetry breaking, self-interaction correction, and semi-classical limit in density functional theory: Hidden connections and beneficial synergies? APL Comput. Phys. 1, 010903 (2025).
Behler, J. & Parrinello, M. Generalized neural-network representation of high-dimensional potential-energy surfaces. Phys. Rev. Lett. 98, 146401 (2007).
Behler, J. Four generations of high-dimensional neural network potentials. Chem. Rev. 121, 10037 (2021).
Thompson, A. P. et al. LAMMPS - a flexible simulation tool for particle-based materials modeling at the atomic, meso, and continuum scales. Comput. Phys. Commun. 271, 108171 (2022).
Singraber, A., Behler, J. & Dellago, C. Library-based LAMMPS implementation of high-dimensional neural network potentials. J. Chem. Theory Comput. 15, 1827 (2019).
Jakse, N. et al. Machine learning interatomic potentials for aluminium: application to solidification phenomena. J. Phys. Condens. Matter 35, 035402 (2022).
Demmel, F. & Jakse, N. Diffusion in liquid metals is directed by competing collective modes. Phys. Rev. B 111, L081104 (2025).
Sandberg, J., Voigtmann, T., Devijver, E. & Jakse, N. Feature selection for high-dimensional neural network potentials with the adaptive group lasso. Mach. Learn: Sci. Technol. 5, 025043 (2024).
Bryk, T. et al. Collective excitations in supercritical fluids: Analytical and molecular dynamics study of “positive” and “negative” dispersion. J. Chem. Phys. 133, 024502 (2010).
Acknowledgements
T.B. was supported by the National Research Foundation of Ukraine under Project 2023.05/0019. N.J. acknowledges the CINES, TGCC and IDRIS under Project No. INP2227/72914/gen5054, as well as CIMENT/GRICAD for computational resources. This work has been partially supported by MIAI Cluster AI (ANR-19-P3IA-0003). Discussions within the French collaborative network in artificial intelligence in materials science GDR CNRS 2123 (IAMAT) are also acknowledged. The calculations have been performed using the ab initio total-energy and molecular dynamics program VASP (Vienna ab initio simulation program) developed at the Institute für Materialphysik of the Universität Wien26,27,28,29.
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T.B. and N.J. designed the research. T.B, N.J. and J.-F.W. performed and analyzed the analytical and numerical calculations. T.B., N.J. and G.R. discussed the results and wrote the paper.
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Bryk, T., Ruocco, G., Wax, JF. et al. Direct evidence of non-acoustic collective modes in dynamics of molten Carbon. Commun Phys (2026). https://doi.org/10.1038/s42005-026-02602-x
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DOI: https://doi.org/10.1038/s42005-026-02602-x
