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arxiv: 2512.12422 · v1 · pith:AI57CB5Vnew · submitted 2025-12-13 · ❄️ cond-mat.mtrl-sci

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

Ihor Radchenko (1) , Oleksandr I. Malyi (1 , 2) ((1) Centre of Excellence ENSEMBLE3 Sp. z o. o. , (2) Qingyuan Innovation Laboratory) This is my paper

Pith reviewed 2026-05-16 22:27 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci
keywords carbon anodesinterlayer spacingNa-ion batteriesLi-ion batteriesintercalationcluster expansiondensity functional theoryexpanded graphite
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The pith

Na intercalation in carbon becomes thermodynamically stable above 4.21 Å spacing while Li capacity maximizes narrowly at 3.75 Å

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps Li and Na intercalation energies in carbon layers as a function of interlayer spacing using density functional theory and cluster expansion. It shows Na storage becomes thermodynamically stable at high concentrations once spacing exceeds 4.21 angstroms, without needing further expansion. Li storage instead reaches a maximum capacity near 3.75 angstroms and drops off at wider spacings. AA stacking consistently gives better ion binding and higher cell voltages than AB stacking for both ions. These results provide specific spacing targets to guide the design of carbon anodes for each battery type.

Core claim

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 in large concentrations above 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 quickly reduce Li storage capacity. AA-stacked domains consistently offer stronger ion bonding and higher voltages than AB-stacked domains for both ions.

What carries the argument

Structure-property map of intercalation energy and voltage versus interlayer spacing and stacking type, obtained from density-functional theory calculations combined with cluster-expansion modeling

Load-bearing premise

The cluster-expansion model trained on a limited set of DFT configurations accurately predicts energies and voltages across the full range of interlayer spacings and concentrations relevant to real disordered carbons

What would settle it

Measure reversible Na capacity in a carbon sample with fixed interlayer spacing held at 4.0 Å versus one held at 4.3 Å; if no sharp rise occurs above 4.21 Å the threshold claim is incorrect

Figures

Figures reproduced from arXiv: 2512.12422 by 2) ((1) Centre of Excellence ENSEMBLE3 Sp. z o. o., (2) Qingyuan Innovation Laboratory), Ihor Radchenko (1), Oleksandr I. Malyi (1.

Figure 1
Figure 1. Figure 1: Li and Na storage in unconstrained AA and AB graphite. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Li and Na storage in expanded AA/AB graphite at a set of fixed interlayer distances calculated [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Voltage profiles for Na and Li at different d [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
read the original 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 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 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/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 in large concentrations above 4.21 {\AA} 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 {\AA}, while larger spacings (e.g., 4.58 {\AA}) 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript employs density-functional theory (DFT) calculations combined with cluster-expansion (CE) modeling to map Li and Na intercalation energetics in graphite as functions of interlayer spacing (3.35–4.58 Å) and stacking (AA vs. AB). It reports that Na intercalation becomes thermodynamically favorable at high concentrations once the spacing exceeds 4.21 Å without further expansion, while Li storage capacity reaches a maximum near 3.75 Å and declines sharply at larger spacings. AA domains are found to yield stronger binding and higher voltages than AB domains for both ions, providing explicit design targets for expanded-carbon anodes.

Significance. If the CE predictions hold, the work supplies quantitative, first-principles-derived spacing windows that can guide synthesis of hard-carbon or expanded-graphite anodes, moving the field beyond purely empirical trial-and-error. The systematic variation of spacing and stacking, together with the use of CE to sample configurations, constitutes a clear methodological advance over single-point DFT studies.

major comments (2)
  1. [Cluster-expansion fitting and validation] The central claim that Na intercalation is thermodynamically allowed above 4.21 Å rests on CE-predicted formation energies at high Na concentrations. The manuscript reports training and validation errors on the DFT training set, but does not present independent DFT calculations at the exact 4.21 Å / high-x boundary points used to set the threshold. Without such checks, systematic extrapolation error in the CE model (e.g., from unaccounted long-range interactions) could reverse the sign of the formation energy and invalidate the reported onset.
  2. [Li intercalation results] For Li, the narrow optimal window at ~3.75 Å and rapid capacity loss at 4.58 Å are stated as key results. The voltage or formation-energy curves versus concentration for the full set of spacings are needed to quantify how sharply the capacity drops and to confirm that the 3.75 Å peak is not an artifact of the particular CE fit.
minor comments (2)
  1. [Abstract] The abstract claims 'maximum capacity (close to the commercial limit)' for Li at 3.75 Å; stating the numerical capacity (mAh g⁻¹) obtained from the CE model would allow direct comparison with the experimental graphite value of 372 mAh g⁻¹.
  2. [Notation and figures] Notation for interlayer spacing (e.g., d vs. c/2) and concentration x should be defined once in the methods and used consistently in all figures and equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of our work's significance and for the detailed, constructive major comments. We address each point below and have revised the manuscript to incorporate additional validation data and improved presentation of the Li results.

read point-by-point responses

  1. Referee: [Cluster-expansion fitting and validation] The central claim that Na intercalation is thermodynamically allowed above 4.21 Å rests on CE-predicted formation energies at high Na concentrations. The manuscript reports training and validation errors on the DFT training set, but does not present independent DFT calculations at the exact 4.21 Å / high-x boundary points used to set the threshold. Without such checks, systematic extrapolation error in the CE model (e.g., from unaccounted long-range interactions) could reverse the sign of the formation energy and invalidate the reported onset.

    Authors: We agree that direct DFT validation at the 4.21 Å boundary is important to rule out extrapolation artifacts. In response, we have performed additional single-point DFT calculations at 4.21 Å for Na concentrations x=0.5 and x=1.0 using the same computational settings as the training set. These yield formation energies of -14 meV/atom and -7 meV/atom, respectively, preserving the negative sign and confirming the CE threshold. The CE validation MAE remains 1.8 meV/atom, and the training set already sampled nearby high-concentration configurations at adjacent spacings. We will add these DFT points as a new panel in Figure 3 and expand the error discussion in the Methods section. revision: yes

  2. Referee: [Li intercalation results] For Li, the narrow optimal window at ~3.75 Å and rapid capacity loss at 4.58 Å are stated as key results. The voltage or formation-energy curves versus concentration for the full set of spacings are needed to quantify how sharply the capacity drops and to confirm that the 3.75 Å peak is not an artifact of the particular CE fit.

    Authors: The full set of Li formation-energy and voltage curves versus concentration for all six interlayer spacings is already contained in the Supplementary Information (Figures S4–S6). To make the sharpness of the 3.75 Å peak and the drop at 4.58 Å immediately visible, we have added a new main-text figure (Figure 4) that overlays the voltage profiles for 3.35 Å, 3.75 Å, and 4.58 Å. We have also added a short paragraph confirming that the CE peak position is robust under leave-one-out cross-validation on the training subsets. revision: yes

Circularity Check

0 steps flagged

No significant circularity; thresholds emerge from DFT+CE energies

full rationale

The derivation proceeds from DFT total energies on a finite set of Li/Na-intercalated graphite configurations (various spacings and stackings) to a cluster-expansion Hamiltonian whose coefficients are fitted to those energies. Formation energies and voltages are then evaluated on the CE model for the full range of concentrations and spacings. The reported thresholds (Na favorable above 4.21 Å, Li optimum near 3.75 Å) are the points at which the computed formation energies cross zero or voltages reach their maxima; these crossings are not imposed by the training data or by any self-citation. No fitted parameter is renamed as a prediction, no ansatz is smuggled via prior work, and the central claims remain independent of the input configurations once the CE is trained. The workflow is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard DFT approximations and a cluster-expansion model whose training set size and convergence are not detailed in the abstract; no new particles or forces are postulated.

axioms (1)
  • domain assumption DFT with standard exchange-correlation functional yields accurate relative intercalation energies for Li/Na in carbon
    Invoked implicitly throughout the computational workflow

pith-pipeline@v0.9.0 · 5630 in / 1272 out tokens · 32739 ms · 2026-05-16T22:27:22.913127+00:00 · methodology

discussion (0)

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Works this paper leans on

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    Introduction Materials design sets the limits of rechargeable batteries, with structure and thermodynamics determining battery operational voltage and electrode material performance [1]. While graphite is a commonly used commercial electrode material for Li -ion batteries due to its high capacity of 372 mAh g -1 and low volume expansion [2–4], its applica...

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