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Explicit pore pre-conditioning and inverse design of amorphous nanoporous carbon

This paper presents an inverse design framework for amorphous nanoporous carbon that treats pore geometry as a design input rather than a synthesis output, successfully reconstructing target pore size distributions and structural properties by growing carbon around predefined pores and optimizing construction parameters via particle swarm optimization.

Original author: Matteo Alberghini, Giacomo Pierotti, Maryam Amini, Mauro Sgroi, Alessio Tommasi

Published 2026-11-05

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CC BY 4.0

Original author: Matteo Alberghini, Giacomo Pierotti, Maryam Amini, Mauro Sgroi, Alessio Tommasi

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 author. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a material so full of tiny holes that it can hold a charge like a battery, filter the air we breathe, or clean our water. This material is nanoporous carbon, a sponge-like substance made of pure carbon atoms arranged in a chaotic, disordered web. Its power does not come from the carbon itself, but from the shape and size of the empty spaces trapped inside it. If the holes are too big, the material cannot hold a charge efficiently; if they are too small, ions cannot get inside. For decades, scientists have tried to design these materials by simulating how they are made in a lab, heating and cooling carbon until it settles into a shape. However, this approach is like trying to sculpt a statue by throwing clay at a wall and hoping it lands in the right form. The final shape of the holes is a result of the process, not a choice the scientist can make. They can set the temperature or the speed of cooling, but they cannot directly specify that the material must have a specific pattern of holes to work perfectly with a particular battery fluid or gas.

A team of researchers has flipped this process on its head. Instead of hoping the holes form correctly, they start with the holes they want and build the carbon around them. In a new study, they demonstrated a method to design amorphous nanoporous carbon by first placing empty spheres of specific sizes into a virtual box, mimicking the exact distribution of hole sizes needed for a specific job. They then grow carbon atoms ring by ring around these empty spheres, carefully arranging the atoms so they do not crush the holes. Because the computer cannot predict exactly how the carbon will settle just by looking at the hole sizes, the team used a smart search algorithm to tweak the rules of the construction until the resulting material matched a desired blueprint. Once the structure was built, they gave it a brief, high-temperature shake to smooth out the rough edges and fix the chemical bonds, leaving the carefully designed holes intact. The result is a material where the geometry of the void is a specification, not an accident.

The researchers tested this method against two very different challenges. First, they tried to recreate a real-world carbon structure that had been made in a previous simulation using standard heating and cooling methods. They fed the computer the target size of the holes and asked it to build a matching structure. The search algorithm adjusted the construction rules, such as how far the carbon atoms stay from the hole walls and how stiff the carbon rings are, until the new structure looked almost identical to the original. The final material matched the density, surface area, and hole size distribution of the reference within about five percent. This proved that the method could reverse-engineer a complex, disordered structure just by looking at its final shape.

To push the method even further, the team tried to build a structure with a "bimodal" distribution, meaning it needed two distinct groups of hole sizes at the same time: some very small and some much larger. This is a difficult shape that standard simulation methods have struggled to create because they usually produce only one average size. Using a specific type of carbon growth rule, the team successfully built a structure with both small and large holes, matching the target pattern. However, they found that not all construction rules worked for every target. When they tried to build the same bimodal shape using a different set of rules, the computer failed to place the holes correctly, revealing that the choice of how carbon atoms connect matters deeply depending on the desired outcome. By analyzing the history of the search, the researchers could tell which rules were essential for the design and which ones were irrelevant, allowing them to spot a failed attempt before even finishing the build.

The final step involved a short period of heating, known as annealing, to fix the local chemistry. The initial construction created a network that had the right holes but was chemically incomplete, with many carbon atoms missing the bonds they needed to be stable. The heating process allowed the atoms to move and reconnect, fixing these weak spots without destroying the carefully placed holes. After this brief treatment, the material's internal structure became much more like a real, stable carbon network, with the right number of chemical bonds and a density that matched the reference. The holes remained in their designed positions, though they widened slightly as the carbon skeleton tightened around them. The study shows that while the local chemistry still requires the laws of physics to settle, the overall shape of the pores can now be a design choice. This shifts the focus from asking "what will this process make?" to "what do we need this material to do?" and building it to fit.

Technical Summary: Explicit Pore Pre-Conditioning and Inverse Design of Amorphous Nanoporous Carbon

Problem Statement
Disordered nanoporous carbons are critical for energy storage, gas separation, and water purification, where performance is dictated not just by surface area but by the specific geometry of the void network, particularly the pore size distribution (PSD). Existing modeling approaches face significant limitations:

Idealized Geometric Models: Use rigid slit pores or simple shapes that lack the curvature, non-hexagonal rings, and connectivity of real disordered carbon, leading to inaccurate predictions of properties like capacitance.
Reconstruction Methods (e.g., Reverse Monte Carlo): Invert measured data to fit atomic positions but require a pre-existing sample and suffer from non-uniqueness; they constrain local order more than the global pore network.
Mimetic Molecular Dynamics (MD): Simulate physical synthesis processes (e.g., quenching or annealing) to produce relaxed structures. However, the pore geometry is an output of the simulation, determined by density and thermal schedules. Users cannot specify a target PSD as an input; matching a specific distribution requires a difficult search over protocol parameters, and no mimetic protocol has been shown to produce arbitrary shapes (e.g., bimodal distributions).

Methodology
The authors propose a generative framework that inverts the standard workflow: the pore network is defined as an input, and the carbon structure is grown around it. The process consists of five steps:

Explicit Pore Pre-Conditioning (Construction):
- Pore Placement: Spheres representing voids are drawn from a target PSD (arbitrary shape) and inserted into a periodic cell until a target pore volume fraction ( $\phi$ ) is reached.
- Ring-by-Ring Growth: A carbon network is grown stochastically around these pores. Rings are accepted based on geometric criteria (no atom enters a pore) and energetic criteria (energy below a threshold).
- Placement Potentials: Two interaction models are tested:
  - Harmonic Bonded Model: Sets ring geometry relative to the parent sheet using bond, angle, and dihedral terms.
  - Tersoff-Contact Model: Uses the Tersoff potential but excludes local bonding terms (anchors and nearby atoms), acting only on contacts between the new ring and distant parts of the network.
- Parameters: The construction is controlled by search parameters including pore volume fraction ( $\phi$ ), pore wall depth ( $A$ ) and range ( $B$ ), dihedral stiffness ( $k_\phi$ ), and a merge coefficient ( $m$ ).
Characterization:
- Generated "proxy structures" are characterized using the same routines as the targets (PSD, accessible surface area, radial distribution function, etc.).
- A custom Python library, nanocarbon, performs these calculations efficiently.
Scoring:
- A cost function is calculated as the sum of dimensionless deviations between the proxy structure's properties and the target properties.
Inverse Design Search (Optimization):
- A Particle Swarm Optimizer (PSO) searches the construction parameter space to minimize the cost function.
- The swarm proposes parameter sets, and a quadratic surrogate model assists in proposing candidates. The history of the search is analyzed to determine which parameters are actually constrained by the target (identifiability analysis).
Annealing:
- The optimal proxy structure undergoes a short molecular dynamics anneal (using the EDIP potential at 2500 K) with the pores held as repulsive walls.
- This step cures the local topology (fixing coordination defects) and compacts the network while preserving the imposed global pore geometry.

Key Results

Recovery of Global Geometry: The search successfully recovers the PSD, density, and surface area of a reference structure generated by Quenched Molecular Dynamics (QMD) within approximately 5%.
Bimodal Distribution Stress Test: Using the harmonic placement potential, the method successfully reconstructs a synthetic bimodal PSD (modes at 7 Å and 16 Å), a shape that mimetic MD has not previously achieved. The Tersoff-contact construction failed to place pores for this specific target, highlighting the dependence of the method on the choice of placement potential.
Identifiability Analysis: Analysis of the search history reveals that the pore volume fraction ( $\phi$ ) is the primary parameter constrained by the targets in successful runs. The pore wall range ( $B$ ) is constrained in some runs, while other parameters (dihedral stiffness, wall depth, merge coefficient) are "insensitive" (the cost is indifferent to them). This analysis can also flag failed searches (e.g., the Tersoff run on the bimodal target) before full construction.
Local Order and Annealing:
- Proxy structures initially have incomplete topology (high fraction of 2-coordinated atoms, low 4-coordinated atoms).
- The short anneal (approx. 15 ps at 2500 K) restores the coordination to match the reference (approx. 80% 3-coordinated, 12% 2-coordinated) and increases the skeletal density.
- The anneal causes a slight shift in the mean pore diameter toward larger values (0.8–1.5 Å) due to network compaction, but the overall shape of the PSD is preserved.
- The final structures do not rely on the repulsive walls to maintain pore openness; the walls hold negligible energy after annealing.

Significance and Claims
The paper claims to establish a method where pore geometry becomes a design specification rather than a simulation outcome.

Design Variable: By treating the PSD as an input, the method allows for the design of carbon structures for specific applications (e.g., specific electrolytes or transport requirements) rather than being limited to structures producible by known synthesis routes.
Efficiency: The cost of characterizing the pore size distribution grows linearly with structure size, making the inverse design search computationally affordable compared to full mimetic MD trajectories.
Modest Scope: The authors note limitations: the method is currently limited to elemental carbon, and the ring statistics (specifically the ratio of 5/7-membered rings to 6-membered rings) in the final annealed structures remain lower than in reference QMD structures. The local chemistry is still supplied by molecular dynamics, and the construction parameters are not necessarily physical values but tuning knobs for the generator.
Future Outlook: The work suggests that future applications could involve targets derived directly from adsorption measurements and the design of electrodes for specific electrolytes, moving the design process "in silico."

Technical Summary: Explicit Pore Pre-Conditioning and Inverse Design of Amorphous Nanoporous Carbon

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