Sodium from Melting to the Critical Point: One Potential Instead of a Thousand Calculations ✛
Context
Liquid sodium serves as a coolant in fast reactors, so its properties are needed up to the critical region, where direct experiments are difficult. Around this topic, there are already works on individual fragments: self-diffusion calculations in the normal liquid, simulations of highly expanded states, studies of the liquid–vapor interface, and the melting curve at high pressure. This work stands somewhat apart because it unites all these directions into a single model covering the entire region from solid to critical point.
Methods
The authors construct temperature-dependent deep potentials for sodium, training them on quantum molecular dynamics data with PBE, AM05, and r2SCAN functionals. A key feature of the approach: in addition to the free energy, a model of electronic entropy is trained separately, allowing the internal energy to be recovered and caloric properties to be calculated. The reference dataset for r2SCAN contains 759 configurations in the density range 0.01–1.00 g/cm³ and temperatures 0.2–5.0 kK. Molecular dynamics simulations are performed in cells up to 2000 atoms, and for direct sound velocity calculation, a cell of 8000 atoms is used with excitation of a longitudinal acoustic mode.
Results
The r2SCAN-based model yields critical parameters for sodium: temperature 2508(8) K, density 0.203(4) g/cm³, pressure 0.249(6) kbar, consistent with recommended values. The density of BCC sodium at normal pressure differs from experiment by only 0.06%. Direct solid–liquid coexistence simulations give a melting temperature of 346(2) K, which is 25 K below the recommended value but falls within the expected sensitivity to free energy errors of a few meV/atom. Surface tension decreases from 177.8 mN/m at 0.5 kK to 1.02 mN/m at 2.48 kK, approaching zero near the critical point. The self-diffusion coefficient along the critical isobar increases from 0.351 Ų/ps at 380 K to about 30 Ų/ps in the critical region, while shear viscosity drops from 0.653 mPa·s at 0.38 kK to 0.06 mPa·s near the critical point.
Implications
The work shows that a single neural network potential trained on r2SCAN data can simultaneously reproduce equilibrium thermodynamics, phase boundaries, acoustic and transport properties from room temperature to the critical region. This differs from neighboring works, which typically focus on one regime or one class of properties. Here, for the first time, end-to-end consistency is achieved between the equation of state, its derivatives, interfacial characteristics, and transport coefficients for an alkali metal.
Future development
The approach can be transferred to other liquid metals, especially those where electronic excitations and meta-GGA functionals are important. Further development will allow refining critical parameters and transport properties in regions inaccessible to direct experiment, as well as constructing consistent equations of state for multicomponent coolants.
Impact
The results are directly useful for thermophysical modeling of fast reactors and other systems with liquid metal coolants, where reliable data are needed up to accident conditions with overheating.
Next steps
The next step could be expanding the training set to improve accuracy near the critical point and applying a similar scheme to other alkali metals or sodium-based alloys.
Key open problems
For thermophysics of liquid metals, the question of reliable determination of critical parameters and transport properties in the rarefied liquid remains open, where experimental data are extremely scarce. The work shows that machine-learned potentials based on meta-GGA functionals can fill this gap, but a systematic uncertainty remains associated with the choice of exchange-correlation functional.
The knowledge machine's view
This section is addressed to the author of the paper and their colleagues. Neighbouring works were found by meaning-based search across our archive, not by word overlap. We do not review: we show the field around the work.
What the work does
This work combines finite-temperature quantum calculations with deep neural network potentials to describe the thermodynamics, phase boundaries, surface tension, and transport properties of liquid sodium from melting to the critical point. The r2SCAN-based model reproduces critical parameters, melting temperature, surface tension, self-diffusion, and viscosity, demonstrating end-to-end consistency.
Where its strength is
For the first time, a single potential simultaneously describes equilibrium, acoustic, interfacial, and transport properties of a metal over such a wide range of conditions, including a separate model of electronic entropy for caloric properties.
Where it could go next
- It might be interesting to see how the approach with temperature-dependent potentials and a separate electronic entropy model transfers to describing phase transitions in other materials where electronic excitations play a role, such as metallic nitrogen at high pressures.based on: Metallic Nitrogen: Wires from Thin Air arXiv:2603.02716
- Closely related is work on neural network self-attention for analyzing quantum systems; perhaps architectural ideas from there could help improve the representation of electronic entropy or accelerate training of potentials for sodium.based on: Electron Jazz: How Neural Network Self-Attention Unveils a Quantum Symphony arXiv:2607.08616
Worth trying
- Test incorporating dynamic structure factor data to further constrain transport properties.
- Check whether joint training of the potential and the electronic entropy model improves predictions of caloric properties near the critical point.
- Use an active learning approach to expand the dataset in the low-density, high-temperature region.
Why this may matter
An end-to-end description of liquid sodium properties from melting to the critical point is important for applications in fast reactors, where direct experimentation is difficult, and demonstrates the feasibility of a unified neural network potential for metals.
Nearby works in our archive
- The Mystery of Two Gaps in a New Superconductorsimilarity 0.51
- Metallic Nitrogen: Wires from Thin Airsimilarity 0.50
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- Electron Jazz: How Neural Network Self-Attention Unveils a Quantum Symphonysimilarity 0.49
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The archive is sparse around this work: the closest review is only 0.51, against a usual 0.64. It looks like few have drilled nearby.
The melting temperature of sodium, calculated by direct coexistence simulation, turned out to be 25 K below the experimental value, although the free energy error of the potential is only 3.5 meV/atom. Such sensitivity explains why even small model inaccuracies can noticeably shift phase transitions.
Key numbers
- critical temperature: 2508 K
- critical density: 0.203 g/cm³
- critical pressure: 0.249 kbar
- BCC density deviation from experiment: 0.06%
- sound velocity near critical point: about 600 m/s