🔬 condensed matter

The origin of complex behavior of liquid carbon: an insight from computer simulation

This study utilizes molecular dynamics simulations with a machine-learning potential (GAP-20) to demonstrate that liquid carbon exhibits a relatively low critical temperature, a finding that may influence the interpretation of experimental graphite melting point measurements.

Original authors: Yu. D. Fomin

Published 2026-03-02

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CC BY 4.0

Original authors: Yu. D. Fomin

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 Mystery of the "Shy" Carbon: Why We Can't Agree on When Graphite Melts

Imagine you are trying to find the exact moment a block of ice turns into water. You'd expect a clear line: solid here, liquid there. But with graphite (the stuff in your pencil lead), scientists have been arguing for over a century about exactly when it melts. Some say it happens at 4,000 degrees; others swear it's 6,700 degrees. That's a huge difference!

This paper is like a detective story where the author, Yu. D. Fomin, uses a super-powerful computer simulation to solve the mystery. Here is the simple breakdown of what they found.

1. The Problem: The "Fuzzy" Boundary

Think of graphite melting like a crowd of people at a party. Usually, you can clearly see when the crowd turns from a standing group (solid) to a dancing group (liquid). But with carbon, the "dance floor" seems to disappear.

The authors realized that previous experiments might be missing a crucial detail: The Critical Point.

In physics, the "critical point" is like a magical threshold where a substance stops being clearly a liquid or a gas and becomes a Supercritical Fluid. It's a weird state where the substance is so hot and chaotic that it's like a "ghost" between liquid and gas. It's hard to measure because its properties (like how much it expands or how much heat it holds) go crazy near this point.

2. The Tool: A Machine-Learning Crystal Ball

To investigate this, the author didn't use a furnace (which is hard to control at these temperatures). Instead, they used a computer simulation powered by a "Machine Learning Potential" called GAP-20.

Think of GAP-20 as a highly trained virtual assistant that knows the rules of how carbon atoms interact. The author fed 4,000 carbon atoms into a virtual box and watched them dance at different temperatures and pressures, essentially running a "virtual experiment" millions of times faster than real life.

3. The Big Discovery: The "Low" Critical Point

The simulation revealed something surprising. The authors found that carbon's critical point (the moment it turns into that chaotic supercritical fluid) happens at a much lower temperature than many scientists thought—around 4,235 K.

Here is the "Aha!" moment:
This temperature is suspiciously close to the temperature where graphite melts.

Imagine trying to measure the temperature of a cup of coffee, but the cup is sitting right next to a roaring bonfire. The heat from the fire messes up your thermometer.

The Old View: Scientists thought graphite melted, turned into a nice, calm liquid, and then eventually turned into gas much later.
The New View: The paper suggests that as soon as graphite melts, it immediately hits the "bonfire" of the critical point. It doesn't become a calm liquid; it instantly becomes a Supercritical Fluid.

4. Why the Experiments Disagree

If you are trying to measure the melting point, but the substance immediately turns into a chaotic, fluctuating supercritical fluid, your measurements will be all over the place.

The paper uses a great analogy for this: The "Jittery" Thermometer.
Near the critical point, the substance is so sensitive that a tiny change in pressure or temperature causes massive changes in its behavior.

Heat Capacity: The amount of heat the carbon can hold spikes wildly.
Expansion: The carbon expands or shrinks unpredictably.

Because of this "jitteriness," different experiments get different results. One experiment might catch the carbon just as it's spiking in heat, making it look like it's at a higher temperature. Another might catch it in a quiet moment. It's like trying to take a clear photo of a hummingbird's wings with a slow camera; you get a blur, and everyone argues about what the wings actually look like.

5. The Conclusion: It's Not a Bug, It's a Feature

The author admits that their computer model might not be 100% perfect (it was trained mostly on solid carbon, not super-hot gas). However, the idea is likely correct.

The Takeaway:
The reason we can't agree on the melting point of graphite isn't because our thermometers are broken. It's because graphite doesn't just melt; it explodes into a chaotic supercritical state immediately.

The "melting point" isn't a single, clean line. It's a messy, fluctuating zone where the rules of physics get a little fuzzy. To get the right answer, future scientists need to stop treating liquid carbon like a calm pool of water and start treating it like a stormy sea right next to a hurricane.

In short: Carbon is a drama queen. It melts, but then it immediately throws a wild party (the critical point) that makes it impossible to measure exactly when the party started.

1. Problem Statement

The melting point of graphite remains one of the most controversial topics in high-temperature physics. Experimental measurements over the last century have yielded a vast dispersion of results, estimating the melting temperature ( $T_m$ ) anywhere from 4000 K to 6700 K.

Inconsistencies: Existing theoretical models and experimental data often rely on unverified assumptions, such as arbitrary volume jumps upon melting (e.g., assuming a 20% jump) and constant thermodynamic properties (heat capacity, thermal expansion) across wide temperature ranges.
Critical Point Uncertainty: The location of the liquid-gas critical point ( $T_c$ ) is poorly defined. Previous estimates (e.g., Leider et al., 1973) suggest $T_c \approx 6810$ K, but these rely on assumptions that contradict newer experimental findings regarding density jumps and specific heat.
The Core Issue: It is hypothesized that the large discrepancy in experimental melting data arises because the melting process occurs in a regime where the system is close to the critical point or in a supercritical fluid (SCF) state, leading to strong thermodynamic fluctuations that are often ignored in data analysis.

2. Methodology

The authors employed Molecular Dynamics (MD) simulations to investigate the thermodynamic properties of liquid carbon and locate its critical point.

System: A cubic box containing 4096 carbon atoms with periodic boundary conditions.
Potential: The GAP-20 (Gaussian Approximation Potential) machine-learning potential was used. This model is trained on Density Functional Theory (DFT) data to accurately reproduce condensed phases of carbon.
Ensemble & Parameters:
- Simulations were performed in the Canonical Ensemble (NVT).
- Temperature Range: 4000 K to 7000 K (with specific isotherms at 4100, 4400, 5000, 6000, and 7000 K).
- Density Range: 0.05 g/ml to 2.0 g/ml.
- Time Steps: 2.0 fs; Equilibration: 10 ps; Production: 20 ps.
Analysis:
- Thermodynamic quantities (pressure, internal energy) were sampled.
- Data was fitted to a polynomial equation of state ( $X = \sum a_{ij} T^i \rho^j$ ) to calculate response functions.
- Response Functions Calculated: Isobaric heat capacity ( $c_P$ ), thermal expansion coefficient ( $\alpha_P$ ), and isothermal compressibility ( $\beta_T$ ).
- Critical Point Identification: Located via inflection points on isotherms and the maxima of response functions (Widom lines).

3. Key Results

A. Critical Point Location

Using the GAP-20 model, the authors identified the critical point parameters as:

Critical Temperature ( $T_c$ ): 4235 K
Critical Density ( $\rho_c$ ): 0.1887 g/ml
Critical Pressure ( $P_c$ ): 106.61 bar

Note: This $T_c$ is significantly lower than the historical estimate of 6810 K but is consistent with some experimental triple-point estimates ( $T_t \approx 4600-4800$ K).

B. Thermodynamic Anomalies (Widom Lines)

The simulations revealed that the melting line of graphite lies very close to the critical point. Consequently, the system exhibits strong anomalies near the melting temperature:

Heat Capacity ( $c_P$ ): Near the critical point, $c_P$ reaches values as high as $25 k_B$ per particle, far exceeding the constant values assumed in many experimental analyses.
Thermal Expansion ( $\alpha_P$ ) and Compressibility ( $\beta_T$ ): These quantities show sharp maxima (Widom lines) near $T_c$ .
Crystallization Traces: At $T=4100$ K and $\rho=1.55$ g/ml, traces of crystallinity were observed, indicating the critical point is dangerously close to the melting line in the density-temperature plane.

C. Implications for Experimental Data

The study explains the "complex behavior" of liquid carbon:

Supercritical State: Upon melting, the system may immediately enter a supercritical fluid (SCF) state rather than a distinct liquid phase.
Fluctuations: In this regime, thermodynamic quantities fluctuate wildly. Small experimental errors in pressure or temperature lead to massive errors in derived properties like density and heat capacity.
Failure of Constant Assumptions: The common assumption that liquid carbon has constant heat capacity or thermal expansion is invalid near the critical point, leading to erroneous melting point calculations.

4. Limitations and Caveats

The authors explicitly acknowledge limitations regarding the GAP-20 model:

Training Bias: GAP-20 was fitted to DFT data for condensed phases (solids/liquids) and may not be fully accurate for the dilute gas phase or the critical region.
DFT Limitations: Since the potential is derived from DFT, it inherits DFT's known difficulties with van der Waals forces at high temperatures and low densities.
Quantitative Shift: The authors suggest that while the qualitative behavior (existence of a low $T_c$ near melting) is likely correct, the absolute temperature values might be shifted. The real critical point could be higher (closer to 6000 K), but the physical mechanism (proximity to criticality) remains the same.

5. Significance and Conclusion

Resolution of Controversy: The paper proposes that the wide dispersion in experimental melting points is not due to measurement error alone, but because experiments are probing a region of strong thermodynamic fluctuations near the critical point.
Re-evaluation of Data: It suggests that previous analyses ignoring the gas phase influence and assuming constant thermodynamic properties are fundamentally flawed.
Future Directions: The authors call for new models fitted simultaneously to gas, liquid, and solid phases to accurately map the phase diagram. They also highlight the need for experimental techniques that can account for the non-ideal behavior of carbon near the critical point.

Summary: The study uses machine-learning potentials to demonstrate that liquid carbon likely exhibits a low critical temperature close to the graphite melting point. This proximity causes extreme thermodynamic fluctuations (Widom lines), which explains the historical inconsistencies in experimental melting data and challenges the validity of standard thermodynamic assumptions used in high-temperature carbon physics.