- Physical Review Journals
All JournalsPhysics Magazine

PRX Energy

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Rational Design Principles for Na- and Li-Ion Carbon Anodes from Interlayer Spacing Control

Ihor Radchenko2 and Oleksandr I. Malyi1,2,*

  • 1Qingyuan Innovation Laboratory, 1 Xueyuan Road, Quanzhou 362801, People’s Republic of China
  • 2Centre of Excellence ENSEMBLE3 Sp. z o. o., Wolczynska Str. 133, Warsaw 01-919, Poland

  • *Contact author: oleksandrmalyi@gmail.com
PDF

PRX Energy 5, 023010 – Published 11 June, 2026

DOI: https://doi.org/10.1103/g52s-znhz

Abstract

Graphite, the standard commercial anode for Li-ion batteries, is thermodynamically incompatible with Na-ion batteries, leading researchers to search for alternative C-based structures (e.g., hard carbon, expanded graphite). In a simplified picture, the main idea of such a search relies on identifying disordered C structures with a large interlayer spacing and distribution of local structural motifs (e.g., pores) with target electrochemical properties. Such an exploration is typically done via trial-and-error experimentation, which often does not allow precise understanding of the role of interlayer distance and even Na−or Li-ion intercalation in the electrochemical performance. Motivated by this, using density-functional theory and cluster expansion, we establish a structure-property relationship for Li and Na intercalation across a range of graphite interlayer spacings and stacking arrangements. We show that Na intercalation becomes thermodynamically possible at high concentrations once the interlayer spacing exceeds 4.21 Å even without a change in interlayer spacing. Conversely, Li intercalation has a narrow optimal window, with maximum capacity (close to the commercial limit) at approximately 3.75 Å, while larger spacings (e.g., 4.58 Å) quickly reduce Li storage capacity. We also find that AA-stacked domains consistently offer stronger ion bonding and higher voltages than AB-stacked domains for both ions. Our results, thus, not only explain the role of metal-ion intercalation in the electrochemical performance, but also clarify the fundamental design trade-offs for expanded C anodes, offering practical targets and interlayer distance ranges for the independent optimization of the next generation of negative electrodes for metal-ion batteries.

View figure in article

Physics Subject Headings (PhySH)

Popular Summary

Sodium-ion batteries are a cheaper alternative to their traditional lithium-ion counterparts. However, they cannot work with well-established graphite electrodes and instead require carbon-based materials with larger internal spacings, which must be designed through a slow trial-and-error process. Here, the authors investigated how internal carbon layer spacing and alignment affect Na and Li storage via intercalation. They found that “AA stacking,” where the carbon hexagons align perfectly, is better both for Na and Li intercalation; however, Li and Na have different optimal spacing. Sodium requires layers expanded over 25% compared to graphite to give useful capacity. Conversely, lithium storage peaks within a narrow carbon spacing window (around 12% relative to graphite) and falls if carbon layers are too far apart or too close together. This reveals a fundamental conflict: carbon electrodes that work for sodium will not work well with lithium. These results can serve as a blueprint to aid the design carbon-based electrodes for next-generation batteries.

Article Text

Supplemental Material

References (55)

  1. J.-M. Tarascon and M. Armand, Issues and challenges facing rechargeable lithium batteries, Nature 414, 359 (2001).
  2. D. Guerard and A. Herold, Intercalation of lithium into graphite and other carbons, Carbon 13, 337 (1975).
  3. R. Yazami, Ph. Touzain, A reversible graphite-lithium negative electrode for electrochemical generators, J. Power Sources 9, 365 (1983).
  4. C. Grosu, C. Panosetti, S. Merz, P. Jakes, S. Seidlmayer, S. Matera, R.-A. Eichel, J. Granwehr, and C. Scheurer, Revisiting the storage capacity limit of graphite battery anodes: Spontaneous lithium overintercalation at ambient pressure, PRX Energy 2, 013003 (2023).
  5. Y. Liu, B. V. Merinov, and W. A. Goddard, Origin of low sodium capacity in graphite and generally weak substrate binding of Na and Mg among alkali and alkaline earth metals, Proc. Natl. Acad. Sci. 113, 3735 (2016).
  6. Y. Wen, K. He, Y. Zhu, F. Han, Y. Xu, I. Matsuda, Y. Ishii, J. Cumings, and C. Wang, Expanded graphite as superior anode for sodium-ion batteries, Nat. Commun. 5, 4033 (2014).
  7. D. A. Stevens and J. R. Dahn, High capacity anode materials for rechargeable sodium-ion batteries, J. Electrochem. Soc. 147, 1271 (2000).
  8. F. Wang, L. Chen, J. Wei, C. Diao, F. Li, C. Du, Z. Bai, Y. Zhang, O. I. Malyi, X. Chen, Y. Tang, and X. Bao, Pushing slope- to plateau-type behavior in hard carbon for sodium-ion batteries via local structure rearrangement, Energy Environ. Sci. 18, 4312 (2025).
  9. Z. Sun, H. Liu, W. Li, N. Zhang, S. Zhu, B. Chen, F. He, N. Zhao, and C. He, Advanced hard carbon materials for practical applications of sodium-ion batteries developed by combined experimental, computational, and data analysis approaches, Prog. Mater. Sci. 149, 101401 (2025).
  10. C. Matei Ghimbeu, A. Beda, B. Réty, H. El Marouazi, A. Vizintin, B. Tratnik, L. Simonin, J. Michel, J. Abou-Rjeily, and R. Dominko, Review: Insights on hard carbon materials for sodium-ion batteries (SIBs): Synthesis—properties—performance relationships, Adv. Energy Mater. 14, 2303833 (2024).
  11. Y. Yang, Z. Liu, Q. Zhang, J. Song, W. Li, S. Jiang, C. Zhang, J. Han, H. Yang, X. Han, and S. He, Recent advances of high-rate hard carbon anodes for sodium-ion batteries: Correlations between performance and microstructure, Adv. Funct. Mater. 36, e14132 (2025).
  12. D. Saurel, B. Orayech, B. Xiao, D. Carriazo, X. Li, and T. Rojo, From charge storage mechanism to performance: A roadmap toward high specific energy sodium-ion batteries through carbon anode optimization, Adv. Energy Mater. 8, 1703268 (2018).
  13. C.-W. Tai, W.-Y. Jao, L.-C. Tseng, P.-C. Wang, A.-P. Tu, and C.-C. Hu, Lithium-ion storage mechanism in closed pore-rich hard carbon with ultrahigh extra plateau capacity, J. Mater. Chem. A 11, 19669 (2023).
  14. L. Xie, C. Tang, Z. Bi, M. Song, Y. Fan, C. Yan, X. Li, F. Su, Q. Zhang, and C. Chen, Hard carbon anodes for next-generation Li-ion batteries: Review and perspective, Adv. Energy Mater. 11, 2101650 (2021).
  15. Y. Sun, J. Tang, K. Zhang, J. Yuan, J. Li, D.-M. Zhu, K. Ozawa, and L.-C. Qin, Comparison of reduction products from graphite oxide and graphene oxide for anode applications in lithium-ion batteries and sodium-ion batteries, Nanoscale 9, 2585 (2017).
  16. S. Yang, G. Yang, M. Lan, J. Zou, X. Zhang, F. Lai, D. Xiang, H. Wang, K. Liu, and Q. Li, Green synergy conversion of waste graphite in spent lithium-ion batteries to GO and high-performance EG anode material, Small 20, 2305785 (2024).
  17. Y. Li, Y. Hu, M. Titirici, L. Chen, and X. Huang, Hard carbon microtubes made from renewable cotton as high-performance anode material for sodium-ion batteries, Adv. Energy Mater. 6, 1600659 (2016).
  18. B. Zhang, C. M. Ghimbeu, C. Laberty, C. Vix-Guterl, and J. Tarascon, Correlation between microstructure and Na storage behavior in hard carbon, Adv. Energy Mater. 6, 1501588 (2016).
  19. Y. Li, A. Vasileiadis, Q. Zhou, Y. Lu, Q. Meng, Y. Li, P. Ombrini, J. Zhao, Z. Chen, Y. Niu, X. Qi, F. Xie, R. Van Der Jagt, S. Ganapathy, M.-M. Titirici, H. Li, L. Chen, M. Wagemaker, and Y.-S. Hu, Origin of fast charging in hard carbon anodes, Nat. Energy 9, 134 (2024).
  20. G. Kresse and J. Hafner, Ab initio molecular dynamics for liquid metals, Phys. Rev. B 47, 558 (1993).
  21. G. Kresse and J. Furthmüller, Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci. 6, 15 (1996).
  22. G. Kresse and J. Furthmüller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54, 11169 (1996).
  23. P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994).
  24. See Supplemental Material at http://link.aps.org/supplemental/10.1103/g52s-znhz for all the details of the calculations.
  25. A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, Commentary: The Materials Project: A materials genome approach to accelerating materials innovation, APL Mater. 1, 011002 (2013).
  26. P. Trucano and R. Chen, Structure of graphite by neutron diffraction, Nature 258, 136 (1975).
  27. D. T. Larson, S. Carr, G. A. Tritsaris, and E. Kaxiras, Effects of lithium intercalation in twisted bilayer graphene, Phys. Rev. B 101, 075407 (2020).
  28. P.-O. Löwdin, On the quantum-mechanical calculation of the cohesive energy of molecules and crystals. Part II. Treatment of the alkali metals with numerical applications to sodium, J. Chem. Phys. 19, 1579 (1951).
  29. S. Grimme, Semiempirical GGA-type density functional constructed with a long-range dispersion correction, J. Comput. Chem. 27, 1787 (2006).
  30. J. Klimeš, D. R. Bowler, and A. Michaelides, Chemical accuracy for the van der Waals density functional, J. Phys.: Condens. Matter 22, 022201 (2010).
  31. J. Klimeš, D. R. Bowler, and A. Michaelides, Van der Waals density functionals applied to solids, Phys. Rev. B 83, 195131 (2011).
  32. M. Dion, H. Rydberg, E. Schröder, D. C. Langreth, and B. I. Lundqvist, Van der Waals density functional for general geometries, Phys. Rev. Lett. 92, 246401 (2004).
  33. K. Lee, ÉD Murray, L. Kong, B. I. Lundqvist, and D. C. Langreth, Higher-accuracy van der Waals density functional, Phys. Rev. B 82, 081101 (2010).
  34. A. Tkatchenko and M. Scheffler, Accurate molecular van der Waals interactions from ground-state electron density and free-atom reference data, Phys. Rev. Lett. 102, 073005 (2009).
  35. A. Van De Walle, M. Asta, and G. Ceder, The alloy theoretic automated toolkit: A user guide, Calphad 26, 539 (2002).
  36. A. Van De Walle, Multicomponent multisublattice alloys, nonconfigurational entropy and other additions to the Alloy Theoretic Automated Toolkit, Calphad 33, 266 (2009).
  37. Add-ons and helpers for pymatgen tailored for Inverse Materials Design group, (2025). https://github.com/yantar92/IMDgroup-pymatgen/ (accessed December 8, 2025).
  38. Scripts for supercomputer job submission for vasp and atat via slurm, (2025). https://github.com/yantar92/IMDgroup-gorun (accessed December 8, 2025).
  39. S. P. Ong, W. D. Richards, A. Jain, G. Hautier, M. Kocher, S. Cholia, D. Gunter, V. L. Chevrier, K. A. Persson, and G. Ceder, Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis, Comput. Mater. Sci. 68, 314 (2013).
  40. A. Hjorth Larsen, et al., The atomic simulation environment—a Python library for working with atoms, J. Phys.: Condens. Matter 29, 273002 (2017).
  41. A. Walle and G. Ceder, Automating first-principles phase diagram calculations, J. Phase Equilib. 23, 348 (2002).
  42. M. K. Aydinol, A. F. Kohan, G. Ceder, K. Cho, and J. Joannopoulos, Ab initio study of lithium intercalation in metal oxides and metal dichalcogenides, Phys. Rev. B 56, 1354 (1997).
  43. J.-K. Lee, S. Lee, Y.-I. Kim, J.-G. Kim, K.-I. Lee, J.-P. Ahn, B.-K. Min, C.-J. Yu, K. Hwa Chae, and P. John, Structure of multi-wall carbon nanotubes: AA′ stacked graphene helices, Appl. Phys. Lett. 102, 161911 (2013).
  44. O. I. Malyi, K. Sopiha, V. V. Kulish, T. L. Tan, S. Manzhos, and C. Persson, A computational study of Na behavior on graphene, Appl. Surf. Sci. 333, 235 (2015).
  45. K. Persson, Y. Hinuma, Y. S. Meng, A. Van Der Ven, and G. Ceder, Thermodynamic and kinetic properties of the Li-graphite system from first-principles calculations, Phys. Rev. B 82, 125416 (2010).
  46. M. P. Mercer, C. Peng, C. Soares, H. E. Hoster, and D. Kramer, Voltage hysteresis during lithiation/delithiation of graphite associated with meta-stable carbon stackings, J. Mater. Chem. A 9, 492 (2021).
  47. J. Holland, A. Bhandari, D. Kramer, V. Milman, F. Hanke, and C.-K. Skylaris, Ab initio study of lithium intercalation into a graphite nanoparticle, Mater. Adv. 3, 8469 (2022).
  48. L. Shulenburger, A. D. Baczewski, Z. Zhu, J. Guan, and D. Tománek, The nature of the interlayer interaction in bulk and few-layer phosphorus, Nano Lett. 15, 8170 (2015).
  49. M. S. Dresselhaus and G. Dresselhaus, Intercalation compounds of graphite, Adv. Phys. 51, 1 (2002).
  50. P. Ge, Electrochemical intercalation of sodium in graphite, Solid State Ion. 28–30, 1172 (1988).
  51. L. Li, W. Zhang, W. Pan, M. Wang, H. Zhang, D. Zhang, and D. Zhang, Application of expanded graphite-based materials for rechargeable batteries beyond lithium-ions, Nanoscale 13, 19291 (2021).
  52. Y. Morikawa, S. Nishimura, R. Hashimoto, M. Ohnuma, and A. Yamada, Mechanism of sodium storage in hard carbon: An x-ray scattering analysis, Adv. Energy Mater. 10, 1903176 (2020).
  53. J. Mathiesen, R. Väli, M. Härmas, E. Lust, F. Von Bülow, K. Jon, K. M. Ø. Jensen, and P. Norby, Following the in-plane disorder of sodiated hard carbon through operando total scattering, J. Mater. Chem. A 7, 11709 (2019).
  54. G. Wang, M. Yu, and X. Feng, Carbon materials for ion-intercalation involved rechargeable battery technologies, Chem. Soc. Rev. 50, 2388 (2021).
  55. Y. Tang, X. Wang, J. Chen, X. Wang, D. Wang, and Z. Mao, PVP-assisted synthesis of g–C3N4–derived N-doped graphene with tunable interplanar spacing as high-performance lithium/sodium ions battery anodes, Carbon 174, 98 (2021).

Outline

Information

PRX Energy 5, 023010– Published 11 June, 2026
  • Received 1 January 2026
  • Revised 15 March 2026
  • Accepted 2 April 2026
Crossmark - Check for updates button
Creative Commons Logo

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation