Location of the Liquid-Vapor Critical Point in Aluminum
In May 2026, the group of Associate Professor Kai Luo from the Department of Applied Physics at Nanjing University of Science and Technology published a paper in Physical Review B titled Location of the liquid-vapor critical point in aluminum (https://doi.org/10.1103/nq4w-nbbw). This work systematically studies the long-debated liquid-vapor critical point of pure aluminum using deep potential molecular dynamics with first-principles accuracy. It provides a reliable range for the critical temperature, density, and pressure of aluminum. The proposed computational framework based on Gaussian Mixture Model identification offers a new tool for studying critical phenomena. The results have direct reference value for determining whether the unloading path after ultrafast laser ablation or shock loading passes through the vapor–liquid two-phase region, and for modeling matter under extreme conditions.
2. Research Background(1) Review of Gas–Liquid Critical Phenomena
At a given temperature and pressure, many substances can coexist in liquid and vapor phases (e.g., water boiling continuously in an open pot: bubbles form in the liquid with a clear interface). If one raises the temperature (or changes the pressure) along the gas–liquid equilibrium line, the liquid becomes less dense, the vapor becomes denser, and the difference between the two phases gradually decreases. The gas–liquid critical phenomenon is the endpoint reached by this equilibrium: at the critical point, the densities of the two phases become equal, the interface disappears, and both surface tension and latent heat of vaporization tend to zero. Above this point, the substance enters a supercritical fluid region where “liquid” and “gas” can no longer be distinguished. Near the critical point, molecular arrangements are highly inhomogeneous and fluctuation scales become large, so compressibility increases anomalously; critical opalescence can sometimes be observed. The critical point is commonly described by three numbers: critical temperature , critical pressure
, and critical density
.
Boiling water and evaporation are processes of liquid turning into vapor; if pressure changes, the boiling point changes. Similarly, aluminum at high temperatures has both liquid and vapor phases, but unlike water, it cannot be directly observed in the laboratory. One must rely on theory and computation to infer the critical point – the endpoint of the gas–liquid phase boundary on a pressure–temperature diagram: below the critical point one can distinguish a dense liquid from a rarefied vapor; at the critical point the interface disappears, and it becomes a supercritical fluid.
The ideal gas model treats molecules as point particles with no volume and no mutual attraction, obeying . This equation of state can only roughly describe dilute gases, not real liquids, and therefore has no critical point. The van der Waals equation considers that molecules occupy a finite volume
and attract each other (parameter
); in molar volume form:
Below a certain temperature, it can produce a “plateau” of gas–liquid coexistence on a –
diagram; the cusp of the coexistence region is the critical point. This simplified model yields a critical exponent
, but real liquids, due to their complex molecular interactions, typically deviate from this universal constant.
(2) Why study the critical parameters of aluminum?
Aluminum is a paradigmatic simple metal and is a common target material and model system in high-energy-density physics. The liquid–vapor critical point marks the top of the coexistence region: once ,
, and
are determined, one can judge whether matter at a given temperature and pressure lies in the two-phase region or in a single-phase (including supercritical) region. This is crucial for understanding the states and transport under extreme conditions.
- Ultrafast laser ablation and micromachining: Material vaporization and plume formation often involve behavior close to the critical region; if the adopted critical temperature is significantly too low or too high, simulations may incorrectly judge whether the system has entered a supercritical state or still maintains a droplet–vapor two-phase structure.
- Shock loading and pressure release: If the unloading path after a strong shock crosses into the gas–liquid two-phase region on the phase diagram, boiling and nucleation may occur; properties such as electrical conductivity and opacity often vary sharply near the two-phase region, and the critical point location affects whether the path “grazes” or crosses the coexistence region.
- Theory and equations of state: Literature estimates of
for aluminum have differed by thousands of kelvin, hindering validation of multiphase equations of state and liquid metal models. Narrowing the critical point to a smaller uncertainty provides a referenceable benchmark for subsequent experimental design and simulation.
- Broader significance: A similar methodology can be extended to the gas–liquid behavior of other metals and alloys under extreme conditions, serving modeling for materials processing, inertial confinement fusion, and discussions of matter states inside planets.
Aluminum is a typical simple metal, but literature estimates of its liquid–vapor critical point have long been scattered: reported values for the critical temperature range from about 5000 K to 9500 K, and the critical density
also spans a wide interval; some works did not even provide
.
The reasons are: (1) Experiments: Maintaining and diagnosing aluminum at high temperature and pressure under static conditions is extremely difficult; many results rely on empirical scaling for liquid metals or extrapolation of low-temperature thermodynamic quantities. (2) Theory and simulation: Near the critical point fluctuations dominate, and finite system size and time scales make direct first-principles sampling prohibitively expensive; classical empirical potentials often cannot simultaneously describe the liquid and dilute vapor phases well. Therefore, machine-learned potentials that can achieve first-principles accuracy while scaling to thousands of atoms and nanosecond sampling times offer a practical path to narrow the uncertainty.
3. Research Results
The research team used the first-principles software ABACUS with numerical atomic orbital basis sets, employing ONCV pseudopotentials and a DZP basis. They produced training data on representative –
grids covering liquid and vapor states, used DPGEN for sampling, screening, generation, and testing, and performed large-scale molecular dynamics calculations using Deep Potential (DP). Post-processing and analysis were done with the candela program.
(1) Calibration of exchange-correlation functional
When extrapolating liquid–vapor coexistence to the critical point, the liquid branch is often more sensitive to functional errors. The authors performed NPT simulations at 1 bar and 1100–1600 K using neural network potentials trained with different functionals, comparing the liquid density with multiple experimental and compiled data. The conclusion was that PBEsol is overall the most reliable; subsequent critical-point work is based on DP trained with PBEsol.
(2) Critical-point calculation method one: Equation of state (EOS) and spinodal
The authors conducted NVT-DPMD simulations on a 4096-atom system, obtaining –
isotherms in the range 5000–8000 K. The pressure was fitted as a fourth-order polynomial in density (with temperature-dependent coefficients), from which
Figure 2: Pressure-density isotherms from DPMD and quartic-density EOS fits; liquid and vapor spinodal lines (dashed) meet at the critical point (star). The EOS path yields K,
g/cm³, and
kbar.
the conditions and
were solved simultaneously to obtain
and
, which were then substituted back into the EOS to get
and propagate errors.
(3) Critical-point calculation method two: TQMD coexistence + GMM + finite-size scaling
The system was first equilibrated at high temperature and then quenched to a target temperature, forming a liquid–vapor interface in a long, narrow simulation box. A Gaussian Mixture Model (GMM) was used to classify liquid-phase and vapor-phase regions based on the local environment and extract coexistence densities. The coexistence densities and
were fitted to obtain the critical point according to the scaling law and the law of rectilinear diameter. Finite-size scaling was performed for systems with 2000, 3000, and 4000 atoms to obtain
in the thermodynamic limit. The main text notes that finite-size extrapolation of
was not stable enough, so the final adopted critical density is from the EOS path (
g/cm³), with the corresponding
.
Figure 2 [sic]: Liquid–vapor coexistence densities from TQMD and the critical point compared with previous estimates. Blue/green markers: liquid and vapor coexistence densities identified by GMM; black square: critical point from coexistence fit; black star: critical point from EOS; scattered points: critical point estimates from various historical models and simulations.
By combining both the EOS and coexistence paths, the critical temperature can be constrained to 6531–6576 K (corresponding to the central EOS value and the thermodynamic limit from coexistence extrapolation). The reported values are g/cm³ (EOS) and
kbar; the temperature uncertainty is significantly narrowed compared to historical estimates. The table below lists only a few representative literature values versus the converged interval from this work (see Table I in the paper for a full review).
| Source | |||
|---|---|---|---|
| Young & Alder (1971, EOS) | 7151 | 0.69 | 5.458 |
| Likalter (1996, EOS) | 8860 | 0.28 | 4.68 |
| Lomonosov (2007, EOS) | 6250 | 0.703 | 1.97 |
| Povarnitsyn (2008, EOS) | 6595 | 0.698 | — |
| Morel et al. suggested range (2009) | 0.566 | — | |
| This work | 6531–6576 | 0.637 | ~1.6 |
Furthermore, the authors analyzed the vapor pressure obtained from coexistence simulations and found a trend consistent with earlier experimental data. This agreement, which was not imposed a priori, further supports the reliability of the model and sampling workflow adopted in this study.
4. Summary and Outlook
This study combines ABACUS first-principles data, deep potentials, and two complementary paths (EOS and TQMD coexistence) to significantly narrow the uncertainty of the temperature–density–pressure range of the aluminum liquid–vapor critical point under PBEsol calibration, with consistency with experimental saturated vapor pressure trends. Compared with the earlier controversy where estimates could differ by thousands of kelvin, the methodology presented in this paper offers a new approach for computing critical phenomena and plays a constructive role in statistical physics and high-energy-density physics.
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