🔬 materials science

A Transferable Machine-Learned Potential Enabling Large-Scale Copper-Carbon Simulations from Equilibrium to Extreme Conditions

This paper introduces NEP-CuC, a transferable machine-learned potential that accurately simulates copper-carbon systems across diverse conditions—from equilibrium to extreme non-equilibrium scenarios—enabling detailed atomistic insights into thermal management, metallization, and interfacial processing behaviors.

Original authors: Feiyu Zhou, Wei Qiu, Yan Chen, Jiahui Liu, Yilun Liu, Shengying Yue, Penghua Ying

Published 2026-10-14

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CC0 1.0

Original authors: Feiyu Zhou, Wei Qiu, Yan Chen, Jiahui Liu, Yilun Liu, Shengying Yue, Penghua Ying

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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

Copper and carbon are two of the most industrially important materials on Earth, yet they behave in ways that are difficult to predict when they meet. Copper is a soft, flexible metal that conducts heat and electricity with ease, while carbon is a versatile element that can form hard diamonds, slippery graphite, or the ultra-strong sheets of graphene. When engineers try to join them—such as placing a copper layer on a diamond surface to manage heat in high-performance electronics—the resulting interface is a complex zone where the orderly atoms of the metal and the rigid bonds of the carbon struggle to find a stable arrangement. Understanding exactly how these atoms interact, how they bond, and how they fail under stress is crucial for building better electronics and stronger materials, but watching this happen at the atomic level is notoriously difficult. Traditional computer models often have to choose between describing the metal correctly or the carbon correctly, rarely managing both at once, which leaves a gap in our ability to simulate real-world manufacturing processes.

To bridge this gap, a team of researchers has developed a new, highly flexible computer model capable of simulating copper and carbon together with remarkable accuracy. This model, known as NEP-CuC, acts as a universal translator for the atomic world, allowing scientists to watch how copper and carbon behave from calm, stable conditions all the way to extreme, violent collisions. The researchers trained this model using a massive library of atomic snapshots, teaching it to recognize the subtle differences between copper atoms bonded to other copper atoms, carbon atoms bonded in various shapes, and the messy, chaotic interactions that occur when the two materials crash into each other. By testing this model against a wide range of scenarios, from the gentle growth of carbon layers on copper to the violent shock of a high-speed impact, the team demonstrated that a single set of rules can now describe these diverse environments without needing to switch between different, incompatible descriptions.

The power of this new tool lies in its ability to handle the full spectrum of atomic behavior. In the quiet, stable world of equilibrium, the model accurately predicts how copper and carbon arrange themselves in perfect crystals and how they vibrate as they warm up. It correctly calculates the stiffness of diamond and the flexibility of copper, matching the results of the most precise quantum mechanical calculations available. But the true test of a model like this is not just how well it handles calm conditions, but how it performs when things get chaotic. The researchers put the model through its paces by simulating the formation of amorphous carbon, a disordered form of the element that appears in many industrial coatings, and by subjecting copper to shock compression, mimicking the conditions of a high-speed impact. In these extreme tests, the model successfully captured how the atomic structure breaks down and reforms, showing that it can follow the material through processes that are far too fast and complex for older methods to track.

One of the most significant findings emerged when the team simulated the process of depositing copper onto a diamond surface, a common step in creating thermal management systems. They watched how the initial energy of the copper atoms as they landed on the diamond shaped the final structure of the interface. When the copper atoms arrived with low energy, they settled into a neat, orderly layer that bonded strongly to the diamond. However, when the atoms arrived with higher energy, they created a rougher, more disordered boundary zone. The researchers found that this initial roughness had a lasting effect: the stronger the initial impact, the weaker the final bond became. Specifically, as the energy of the incoming copper atoms increased from one to fifty electronvolts, the maximum stress the interface could withstand before breaking dropped significantly, falling from over twelve gigapascals to roughly six and a half gigapascals. This suggests that in manufacturing, the speed and force with which materials are joined are just as important as the materials themselves, as a violent start can leave a permanent weakness in the final product.

The model also revealed that the way heat moves across the boundary between copper and diamond depends heavily on the orientation of the crystal surfaces. When the copper and diamond are aligned in a specific way, heat flows across the interface much more efficiently than when they are aligned differently. This difference is so consistent that the model predicts one orientation will always conduct heat about twice as well as the other, regardless of the temperature. This kind of detailed insight is vital for engineers designing heat sinks for computers or lasers, where even a small improvement in heat flow can prevent a device from overheating. By providing a single, reliable framework that works for both metals and carbon, this new model allows scientists to simulate entire manufacturing processes, from the initial deposition of atoms to the final mechanical testing of the product, without having to stop and switch to a different type of calculation.

The researchers did not stop at static simulations; they also used the model to watch how these materials respond to rapid changes, such as the shock waves generated by a high-velocity impact. In these simulations, the model showed how the shock wave travels through the copper, splitting into different fronts depending on the direction of the impact. It revealed that the internal structure of the metal, including the size of its tiny crystal grains, influences how the shock wave spreads and how the material deforms. While the overall speed of the shock wave remained consistent with what is seen in experiments, the model showed that the local stress and the way the metal bends and twists varied significantly based on the microscopic arrangement of its grains. This level of detail helps explain why materials sometimes fail in unexpected ways under extreme stress, offering a clearer picture of the hidden mechanics that govern material failure.

Ultimately, this work provides a new way to link the history of how a material is made to its final performance. The study shows that the conditions present during the initial stages of processing, such as the energy of the atoms as they land on a surface, leave a permanent signature on the atomic structure. These signatures, in turn, dictate how the material will behave when it is later subjected to stress or heat. By capturing these connections, the new model offers a powerful tool for designing better materials, allowing engineers to predict how changes in the manufacturing process will affect the final product. It moves the field beyond simple guesses and isolated calculations, providing a comprehensive view of how copper and carbon interact across the full range of conditions they might encounter in the real world. This capability opens the door to designing more efficient thermal management systems, stronger composite materials, and more durable electronic components, all by understanding the fundamental dance of atoms that holds them together.

Technical Summary: NEP-CuC for Large-Scale Copper–Carbon Simulations

Problem Statement
Copper–carbon (Cu–C) systems are critical for applications ranging from diamond thermal management and metallization to graphene growth and carbon coating. However, atomistic simulations of these systems face a significant modeling challenge: a single interatomic potential must accurately describe metallic bonding in crystalline, defective, liquid, and compressed copper; covalent bonding across diverse carbon environments ($sp$, $sp^2$ , $sp^3$ ); and heterogeneous interactions at interfaces, including weakly bound regions and short-range repulsive encounters during energetic collisions. Existing classical potentials (e.g., EAM, MEAM, reactive force fields) typically have defined domains of applicability and struggle to maintain accuracy across these distinct regimes simultaneously. While machine-learned potentials (MLPs) offer flexibility, previous Cu–C models have been restricted to specific structural or chemical environments (e.g., carbon growth on solid Cu or interfacial heat transport), lacking a unified framework capable of handling both equilibrium properties and strongly non-equilibrium processes.

Methodology
The authors developed NEP-CuC, a transferable neuroevolution potential (NEP) designed to operate within a single framework for Cu–C systems.

Dataset Construction: A candidate pool of 15,979 geometries was assembled from public datasets and purpose-built configurations, covering crystals, defects, surfaces, interfaces, thermal disorder, deformation, and close atomic encounters. From this pool, 2,727 unique configurations were selected using structural-family stratification and farthest-point sampling based on NEP89 descriptors. These were uniformly relabeled with energies, forces, and virials using a consistent density functional theory (DFT) protocol (PBE functional).
Model Training: Five independently trained NEP models (M1–M5) were generated using stratified train/validation splits. The ensemble disagreement serves as an internal indicator of model sensitivity.
Physical Corrections: To address limitations of the local potential, two corrections were incorporated:
1. A Ziegler–Biersack–Littmark (ZBL) correction to describe short-range nuclear repulsion during close atomic encounters.
2. A runtime D3(BJ) dispersion correction to supplement calculations for dispersion-sensitive systems (e.g., graphene on Cu), applied without modifying the trained parameters.
Validation Strategy: The potential was rigorously tested against source-derived held-out data and independent external datasets. Benchmarks included equilibrium properties (lattice constants, elastic constants, defect energies), energy profiles, lattice dynamics (phonon dispersions), finite-temperature phonon transport, and non-equilibrium processes (shock compression, amorphous carbon formation, and Cu deposition on diamond).

Key Contributions and Results

Accuracy and Transferability: NEP-CuC demonstrates broad physical fidelity across elemental Cu, diverse carbon environments, and heterogeneous Cu–C configurations.
- Equilibrium Properties: The model reproduces DFT lattice parameters and elastic constants for Cu and diamond with high accuracy. It correctly orders surface energies and stacking-fault energies for Cu. For Cu–C interfaces, it accurately predicts equilibrium separation but underestimates the work of adhesion by approximately 12%.
- Carbon Environments: The potential captures the density-dependent coordination of amorphous carbon and reproduces the thermal conductivity trends of bulk diamond and amorphous carbon, though it systematically underestimates the absolute thermal conductivity of diamond compared to experimental and carbon-specific MLP references.
- External Benchmarks: On external datasets for carbon deposition on Cu(111) and nanoporous carbon, NEP-CuC achieves force errors comparable to models specifically trained for those tasks, despite being trained on a general Cu–C pool.
Computational Performance: The compact NEP representation enables large-scale simulations. On a single NVIDIA RTX 4090D GPU, the model achieves a throughput of approximately $2.4 \times 10^7$ atom steps per second for systems containing up to ~1 million atoms, comparable to established NEP89 and significantly faster than other MLPs like MACE-MPA-0 or CHGNet for similar system sizes.
Non-Equilibrium Dynamics:
- Shock Compression: The potential successfully captures orientation-dependent shock-front splitting in Cu (single front for [100] vs. elastic/plastic splitting for [110] and [111]) and reproduces the Hugoniot relation for polycrystalline Cu.
- Amorphous Carbon: It correctly predicts the transition from $sp^2$ to $sp^3$ bonding and the associated increase in thermal conductivity as density increases.
Process-Structure-Property Linkage: A major application demonstrated is the simulation of Cu deposition on diamond. The study links the initial incident energy of Cu atoms (1–50 eV) to the resulting interphase structure and mechanical response.
- Higher deposition energies lead to broader Cu-density transition regions, reduced near-interface Cu ordering (lower fcc fraction), and increased carbon coordination disorder.
- Despite an increase in geometric Cu–C contact density at higher energies, the mean peak nominal tensile stress at 500 K decreases significantly (from $12.35 \pm 0.71$ GPa at 1 eV to $6.65 \pm 0.98$ GPa at 50 eV).
- The simulations reveal that preparation history leaves persistent structural signatures that dictate mechanical failure modes, with higher-energy depositions leading to deformation localized closer to the heterogeneous interphase.

Significance and Claims
The paper claims that NEP-CuC provides a unified atomistic framework capable of linking processing conditions, interfacial structure, and physical response across Cu–C systems. Its significance lies in:

Bridging Scales: Enabling simulations of million-atom systems and non-equilibrium processes (shock, deposition) that are beyond the reach of routine first-principles molecular dynamics.
Unified Description: Successfully capturing metallic, covalent, and heterogeneous interactions within a single potential, addressing the central challenge of modeling Cu–C materials.
Physical Insight: Demonstrating that preparation history (specifically incident energy during deposition) creates persistent structural differences in the interphase that directly influence mechanical strength, even when contact density increases.
Modest Limitations: The authors acknowledge that while the potential captures trends and relative behaviors well, quantitative transferability remains property-dependent. Specifically, the model underestimates diamond thermal conductivity and carbon point-defect energetics, and absolute interfacial conductance values show model dependence. The study emphasizes the need for physical validation beyond average prediction errors and notes that the current simulations omit electron–phonon coupling, which may affect thermal resistance comparisons with experiment.

The work establishes NEP-CuC as a tool for investigating Cu/diamond thermal management, diamond metallization, and non-equilibrium Cu–C processing, offering a pathway to extend such multielement modeling strategies to other metal–carbon systems.

Technical Summary: NEP-CuC for Large-Scale Copper–Carbon Simulations

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