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Atomistic Structure Generation and Neural-Network Screening of Hard Carbons to Identify High-Capacity Sodium Storage

This paper presents a scalable computational framework combining machine-learned interatomic potentials, the RAFFLE structure-generation tool, and a neural-network surrogate to model over 13,000 realistic hard carbon structures, successfully linking their microstructural features to sodium storage performance and identifying high-capacity anode candidates exceeding 800 mAh g $^{-1}$ .

Original authors: Harry Mclean, Aiden Daniel Emery, Theodore Thomas Walton, Ned Thaddeus Taylor, Steven Paul Hepplestone

Published 2026-08-19

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CC BY 4.0

Original authors: Harry Mclean, Aiden Daniel Emery, Theodore Thomas Walton, Ned Thaddeus Taylor, Steven Paul Hepplestone

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The world is moving toward electricity that comes from the sun and the wind, sources that are clean but unpredictable. To make this energy reliable, we need batteries that can store vast amounts of power and release it when the clouds cover the sun or the wind dies down. While lithium-ion batteries currently power our phones and cars, the lithium they rely on is becoming expensive and scarce. Scientists are now looking for a more abundant alternative: sodium, the common element found in table salt. Sodium is everywhere and cheap, but it is a larger, clumsier ion than lithium, making it difficult to store efficiently in the materials we currently use. The most promising candidate for holding sodium is a type of carbon called hard carbon, a disordered, sponge-like material that can trap sodium ions in its tiny internal pockets. However, this material is so complex and irregular that scientists have struggled to understand exactly how its internal structure affects its ability to store energy, leaving the design of better batteries largely a game of trial and error.

A team of researchers at the University of Exeter has taken a different approach, using powerful computers to build thousands of virtual models of hard carbon to see what makes them work best. Instead of trying to guess the perfect structure, they used a machine-learning system to generate over thirteen thousand unique, realistic models of hard carbon. These models were not simple drawings; they were detailed atomic maps containing thousands of carbon atoms arranged in disordered patterns, complete with the tiny pores and defects that exist in real materials. The researchers then used a specialized computer program, trained on the laws of physics, to simulate how sodium ions would move into these virtual structures. By testing each model, they could measure exactly how much sodium it could hold and how the voltage changed as the battery charged and discharged. This massive digital experiment allowed them to see patterns that would be impossible to spot in a physical lab, where creating and testing even a single new material can take weeks.

The simulations revealed a clear relationship between the shape of the carbon and its performance. The researchers found that the best hard carbon for storing sodium is not the dense, solid kind, but rather a lighter, more porous version. Specifically, the models showed that structures with a density around 1.25 grams per cubic centimeter and a porosity between 12 and 15 percent performed the best. In these optimal structures, the internal pores were not just empty holes; they were well-connected tunnels that allowed sodium to flow freely throughout the material. When the carbon was too dense, the sodium could not get inside, and the storage capacity was low. When the carbon was too light or the pores were isolated, the capacity also suffered. The most successful virtual model, which the researchers called Structure E, was able to store sodium at a rate of 857 milliamp hours per gram, a figure that surpasses both current experimental records and the capacity of standard lithium-graphite batteries.

To make sense of such a huge number of models without spending years on calculations, the team trained a lightweight artificial intelligence to predict the storage capacity of any given structure just by looking at its shape. This AI acted as a fast filter, scanning the entire library of thirteen thousand models in seconds to identify the most promising candidates. The AI correctly identified that high capacity was linked to low density and high porosity, and it successfully pointed the researchers toward the top-performing structures. When the researchers then ran detailed, time-consuming simulations on these top candidates to confirm the AI's predictions, the results held up. The detailed analysis showed that in the best structures, the sodium ions filled a vast, connected network of pores, spreading out evenly rather than getting stuck in isolated spots. This confirmed that the geometry of the internal voids is just as important as the chemical composition of the carbon itself.

The study suggests that the path to better sodium batteries lies in engineering the microscopic architecture of the carbon, ensuring it has the right amount of empty space and that this space is connected in a way that allows ions to move freely. The researchers identified a specific "sweet spot" for these properties, offering a clear target for chemists and engineers who manufacture these materials. By moving from guesswork to a data-driven understanding of how atomic structure dictates performance, this work provides a roadmap for designing high-capacity anodes. The approach used here, which combines realistic structure generation with machine learning, could also be applied to other materials, potentially accelerating the discovery of new energy storage solutions that are essential for a stable, renewable energy future.

Technical Summary: Atomistic Structure Generation and Neural-Network Screening of Hard Carbons

Problem Statement
Hard carbons are established anodes for lithium-ion batteries and leading candidates for sodium-ion batteries (SIBs). However, their electrochemical performance is governed by a complex, heterogeneous network of graphitic domains, defects, and nanopores. Conventional atomistic methods, such as density functional theory (DFT), are computationally prohibitive for modeling these systems at the required length scales (thousands of atoms) and time scales. While molecular dynamics (MD) simulations have bridged some gaps, they often rely on simplified structural representations that systematically underestimate topological disorder, microporosity, and structural heterogeneity. Consequently, the optimization of hard carbon anodes remains largely empirical, with precursor chemistry, pyrolysis temperature, and heating rates creating a vast, coupled configuration space that is difficult to isolate or understand atomistically.

Methodology
The authors propose a scalable, three-stage computational workflow to link hard carbon microstructure with sodium-storage performance:

Large-Scale Structure Generation:
- The study utilizes the RAFFLE structure-generation framework, an active-learning tool that combines genetic algorithms with learned atomic environments to guide atom placement.
- Universal Machine-Learned Interatomic Potentials (MLIPs) are employed to evaluate energetics at near-DFT accuracy with MD-scale costs. Specifically, a $\Delta$ -learned CHGNet model was used for initial high-throughput screening, followed by re-evaluation and relaxation using the MACE-MPA-0 potential (with D3 correction).
- This workflow generated 13,096 realistic hard carbon models containing between 702 and 4,378 atoms (average 2,002). The structures were targeted to match experimentally accessible ranges, with generated structures spanning $sp^3$ fractions from 0.085 to 0.662 and densities from 1.246 to 3.307 g cm $^{-3}$ .
- Initial hosts included single graphene sheets and multilayer flakes, with non-interacting templates (spheres/cylinders) used to efficiently introduce artificial pores.
Explicit Intercalation and Validation:
- A subset of representative structures (Structures A–E) underwent explicit sodium intercalation calculations to validate the workflow.
- These calculations reproduced characteristic voltage profiles (sloping-to-plateau) and allowed for the analysis of sodium distribution, pore connectivity, and local bonding environments (via 2-body distribution functions).
Neural Network Surrogate Screening:
- To screen the full library of 13,096 structures without performing prohibitively expensive explicit intercalation for each, a lightweight neural network (NN) surrogate was trained.
- Input Features: The model uses frozen descriptors from the MACE-small model (pooled per-atom features) augmented by five geometric void descriptors (void points, clusters, total volume, heuristic capacity, and cell volume).
- Training Data: The NN was trained on a dataset of 64 structures where capacity was estimated using a heuristic void-filling protocol.
- Architecture: A compact residual multilayer perceptron with three output heads predicting capacity mean, log-variance, and standard deviation.
- Performance: The model achieved a mean absolute error (MAE) of 21.3 mA h g $^{-1}$ and an $R^2$ of 0.946 under stratified k-fold cross-validation.

Key Results

Structure-Property Relationships: The generated library spans a broad range of properties. Analysis reveals that capacity is strongly dependent on density and porosity.
- Density Threshold: Structures with densities greater than 2.25 g cm $^{-3}$ exhibit very low capacities (bottom 5%). Conversely, structures with densities below 1.5 g cm $^{-3}$ fall within the top 33% of capacities.
- Optimal "Sweet Zone": The highest capacities are found in structures with densities of approximately 1.25 g cm $^{-3}$ , $sp^3$ fractions of 0.15–0.18, and porosities of 12–15%.
Capacity Range: The explicit calculations for the representative set (A–E) yielded capacities ranging from 88 mA h g $^{-1}$ (dense, low porosity) to 857 mA h g $^{-1}$ (low density, high porosity). Structure E, identified as the highest capacity candidate by the surrogate, exceeded experimental hard carbon references (~580 mA h g $^{-1}$ ) and theoretical predictions.
Role of Pore Morphology: Capacity is not solely a function of total pore volume but is critically governed by pore connectivity and topology.
- Structure C (distribution of small pores) outperformed Structure B (single large pore) despite similar densities and porosities, demonstrating that pore architecture dictates capacity.
- Structure E's high capacity is attributed to its high porosity (21%) and low $sp^3$ fraction (0.11), facilitating an extensive, connected pore network.
Sodium Storage Mechanism: Planar density analysis confirms that sodium accumulates preferentially in low-density regions (pores). In dense structures, sodium is confined to sparse, isolated sites, whereas in porous structures, it fills a connected network. The intercalant remains dispersed rather than forming bulk-like metallic clusters, even at maximum loading.

Significance and Claims
The paper claims to provide a scalable framework that links hard carbon microstructure to sodium-storage performance, offering atomistic design principles for high-capacity anodes.

Methodological Advance: The primary contribution is the successful integration of universal MLIPs with the RAFFLE framework to generate thousands of realistic, large-scale hard carbon models, overcoming the size limitations of DFT.
Efficiency: The neural network surrogate enables the screening of over 13,000 structures in seconds, a task that would otherwise require months of explicit intercalation calculations.
Design Principles: The study identifies that optimizing hard carbon performance requires moving beyond simple density or $sp^3$ fraction metrics. Instead, pore geometry, connectivity, and elongation are the dominant descriptors. The authors posit that a "sweet zone" exists for specific combinations of density, $sp^3$ fraction, and porosity.
Broader Impact: The workflow provides a route toward atomistically informed design, applicable to exploring synthesis-dependent microstructures, including precursor chemistry, pyrolysis conditions, heteroatom doping, and pore engineering.

The authors maintain a modest stance, acknowledging that the surrogate model exhibits quantitative deviations for specific structures (e.g., over-predicting Structure E) due to the limited size of the training set and the heuristic nature of the initial capacity estimation. However, they assert that the model successfully captures qualitative trends and provides a robust tool for ranking disordered carbon libraries.

Technical Summary: Atomistic Structure Generation and Neural-Network Screening of Hard Carbons

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