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

Using density-functional theory and cluster expansion, this study establishes that while Na-ion intercalation becomes thermodynamically favorable at interlayer spacings above 4.21 Å, Li-ion capacity is maximized at approximately 3.75 Å, thereby providing fundamental design principles for optimizing carbon anodes for both battery chemistries.

The Gist

The Big Picture: The "Shoebox" Problem

Imagine you are trying to store people (ions) inside a stack of flat, rigid sheets (graphite layers).

• Lithium (Li) is like a small child. They fit perfectly into the standard gaps between the sheets.
• Sodium (Na) is like a large adult. The standard gaps are too narrow; the adult simply cannot squeeze in without breaking the stack or getting stuck.

For years, scientists knew that standard graphite works great for Lithium batteries but fails for Sodium batteries. To fix this, researchers started making "expanded" graphite—sheets that are pulled slightly further apart. They hoped this would let the "adults" (Sodium) fit in.

However, there was a big debate: Does the Sodium actually fit inside the crystal layers, or does it just hide in the messy cracks and holes between them? Also, no one knew exactly how far apart to pull the sheets to get the best performance for both types of batteries.

This paper uses powerful computer simulations to act like a "molecular architect," testing different distances between the sheets to find the perfect design rules.

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The Key Findings

1. The "Goldilocks" Zone for Lithium

For the small Lithium ions, the paper found that there is a very specific, narrow "sweet spot" for the distance between the sheets.

• The Analogy: Think of a sandwich. If the bread is too close together, the filling (Lithium) gets squished and can't get in. If the bread is too far apart, the filling falls out or doesn't stick to the bread.
• The Result: Lithium performs best when the gap is about 3.75 Å (a tiny unit of measurement).
  • If the gap is smaller, the sheets push back too hard.
  • If the gap is larger (like 4.58 Å), the Lithium loses its grip and the battery capacity drops drastically.
• Takeaway: If you want a high-capacity Lithium battery, you need to keep the sheets relatively close together.

2. The "Wide Open Door" for Sodium

For the larger Sodium ions, the rules are completely different.

• The Analogy: Imagine a large adult trying to enter a room. If the door is slightly ajar, they can't get in. But if you open the door wide, they can walk right in.
• The Result: Sodium cannot enter standard graphite at all. However, once the gap between the sheets is widened to about 4.21 Å or more, Sodium can enter and store itself effectively without needing to push the sheets apart further.
• Takeaway: For Sodium batteries, the bigger the gap (up to a point), the better. The paper confirms that Sodium does store inside the crystal layers if they are expanded enough, settling the debate that it only hides in the cracks.

3. The "Stacking" Secret (AA vs. AB)

The paper also looked at how the sheets are stacked on top of each other.

• The Analogy: Imagine stacking plates.
  • AB Stacking: The plates are offset (like a staircase).
  • AA Stacking: The plates are perfectly aligned (like a tower).
• The Result: The "perfectly aligned" (AA) stack is actually better for holding both Lithium and Sodium. It creates a stronger bond and higher voltage than the offset (AB) stack. It's like a perfectly aligned tower holding weight better than a leaning one.

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Why This Matters (The Trade-Off)

The most important discovery in this paper is a design trade-off.

• What works for Sodium hurts Lithium: If you make the sheets very far apart to help the big Sodium ions, you ruin the battery for the small Lithium ions.
• What works for Lithium hurts Sodium: If you keep the sheets close together for Lithium, the big Sodium ions can't get in at all.

The Conclusion:
You cannot use the exact same "expanded graphite" recipe for both batteries.

• To build a great Sodium battery, you need to engineer the material with wide gaps (around 4.58 Å).
• To build a great Lithium battery, you need narrower, specific gaps (around 3.75 Å).

This research gives engineers a clear "instruction manual" on how to tune the spacing of carbon sheets to create the next generation of batteries, ensuring they know exactly how far apart to pull the layers depending on which metal ion they want to store.

Technical Summary

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

Problem Statement
Graphite is the standard commercial anode for Li-ion batteries (LIBs) but is thermodynamically incompatible with Na-ion batteries (SIBs) due to the unfavorable intercalation of larger Na ions into tightly spaced graphite layers. Consequently, researchers have turned to disordered carbon structures, such as hard carbon and expanded graphite, which feature increased interlayer distances, defects, and nanopores. However, the specific contribution of interlayer spacing versus other structural motifs (e.g., defects, pores) to electrochemical performance remains debated. Specifically, it is unclear whether Na intercalation occurs within the crystalline, graphite-like domains of these materials or is restricted to disordered regions. Furthermore, the distinct design requirements for optimizing Li versus Na storage in expanded carbon architectures are not fully understood, leading to a reliance on trial-and-error experimentation rather than rational design.

Methodology
The authors employed a computational approach combining Density Functional Theory (DFT) with the Cluster Expansion (CE) method to establish structure-property relationships for Li and Na intercalation.

• Systems Modeled: The study investigated both AA-stacked and AB-stacked graphite configurations.
• Interlayer Spacing (d-spacing): Calculations were performed across a range of fixed interlayer distances (3.52, 3.75, 3.99, 4.25, and 4.58 Å) to simulate expanded graphite and hard carbon domains. Unconstrained relaxation calculations were also conducted to determine equilibrium states.
• Functionals: Multiple van der Waals (vdW) exchange-correlation functionals were screened (PBE-D2, optB88-vdW, optB86b-vdW, vdW-DF, vdW-DF2, PBE-TS). The optB88-vdW functional was selected as the primary basis due to its agreement with experimental graphite d-spacing and Na cohesive energy, with key trends validated against other functionals.
• Analysis: The study utilized convex hull analysis to identify thermodynamically stable compounds and calculated formation energies, voltage profiles, and theoretical capacities. The CE method was used to map ground-state structures for various ion concentrations.

Key Results

1. Thermodynamic Viability of Na Intercalation:

   • In pristine graphite (equilibrium d-spacing ~3.33–3.52 Å), Na intercalation is thermodynamically unfavorable.
   • However, as the interlayer spacing increases, Na intercalation becomes thermodynamically possible. The study identifies a critical threshold where Na-C compounds become stable. Specifically, at an interlayer spacing of 4.58 Å, stable Na-C compounds (e.g., NaC28, NaC24, NaC20, NaC16) form without requiring further expansion of the lattice.
   • This suggests that Na storage can occur within the crystalline domains of hard carbon/expanded graphite if those domains possess sufficiently large interlayer spacings stabilized by defects or amorphous networks.

2. Optimal Window for Li Intercalation:

   • In contrast to Na, Li intercalation exhibits a narrow optimal interlayer spacing window.
   • The maximum storage capacity for Li (approaching the commercial limit of 372 mAh/g) occurs at a d-spacing of approximately 3.75 Å.
   • At spacings larger than ~4.0 Å, the Li-C interaction weakens rapidly, and the formation energy becomes positive, rendering intercalation unfavorable. At 4.58 Å, the theoretical Li capacity drops significantly to ~80 mAh/g.

3. Stacking Effects (AA vs. AB):

   • AA-stacked domains consistently offer stronger ion bonding and higher voltages than AB-stacked domains for both Li and Na.
   • In AA stacking, ions bind symmetrically to hollow sites in adjacent layers. In AB stacking, the site is asymmetric (bonding to a top site on one layer and a hollow on the other), leading to greater repulsion, particularly for the larger Na ion at smaller spacings.

4. Voltage and Capacity Trends:

   • Li: Capacity is maximized at ~3.75 Å and declines sharply with increased spacing. Voltage profiles show a slight increase from 3.52 to 3.75 Å, followed by a reduction at larger spacings.
   • Na: Capacity shows a direct positive correlation with d-spacing, with the maximum theoretical capacity (139 mAh/g for AA-stacked) observed at 4.58 Å. Voltage profiles for Na generally increase monotonically with d-spacing in the studied range.

Significance and Claims
The paper claims to resolve long-standing debates regarding the mechanism of Na storage in carbon anodes. By demonstrating that Na intercalation is thermodynamically viable in crystalline graphite-like domains provided the interlayer spacing is sufficiently large (e.g., >4.21 Å, with optimal stability at 4.58 Å), the authors argue that Na storage in hard carbon is not solely a result of pore filling or disordered regions but can occur via intercalation in expanded crystalline domains.

Furthermore, the study establishes a fundamental design trade-off: structural motifs that favor high-capacity Na storage (very large interlayer spacings) are inherently detrimental to Li storage. This finding clarifies why materials optimized for SIBs may underperform in LIBs and vice versa. The authors propose that rational design of next-generation anodes should target specific interlayer distance windows: ~3.75 Å for maximizing Li capacity and >4.2 Å (ideally ~4.58 Å) for enabling Na intercalation in crystalline domains. These findings provide practical targets for the independent optimization of carbon anodes for metal-ion batteries.