Probing electron transport and thermodynamic properties in expanded warm dense matter

Abstract

The expanded warm dense matter regime, an intermediate state between condensed matter and high-energy-density plasma at low density, remains poorly constrained experimentally, particularly regarding transport properties across the metal-to-nonmetal transition. Here, we explore expanded warm dense aluminum using a newly developed pulsed Joule heating platform combined with first-principles simulations. Using a largely model-independent approach with time-resolved diagnostics, we measure thermodynamic and transport properties from ambient density down to 0.6 g ⋅ cm⁻³ and temperatures exceeding 37 kK, providing benchmark data for electrical conductivity and equation-of-state models. Our results reveal electronic state localization associated with the onset of the metal-to-nonmetal transition and show distinct conductivity behaviors depending on the thermodynamic path followed during expansion. Crucially, electronic relaxation time – rather than ionization – dominates conduction in this supercritical regime. These observations advance our understanding of expanded warm dense matter and open perspectives for future studies on alloys and high-Z elements relevant to giant impact modeling and fusion energy.

Despite the ubiquity of warm dense matter states, there is lack of well-established models describing the expanded regime. To address this gap, the authors provide theoretical and experimental analysis of warm dense aluminium by means of pulse Joule heating. This provides constraints to the equation of state and insight into the electrical conductivity behaviour.

Introduction

Understanding matter under extreme conditions is a major challenge across multiple fields, ranging from astrophysics and geophysics – where it contributes to modeling stellar, planetary, and terrestrial interiors^1–4–, to condensed matter physics, where it guides the search for novel materials exhibiting exotic properties such as room-temperature superconductivity^5,6. At the intersection of these different physics, for warm and dense conditions, prevails a complex state of matter characterized by strong ion coupling, partial ionization and electron degeneracy⁷. Conventional models fail to describe this warm dense matter (WDM) state, and advanced numerical simulations, usually based on density-functional theory (DFT)⁸ or path integral Monte-Carlo⁹, are often required. Over the past decade, WDM has been shown to play a predominant role underpinning major advancements in inertial fusion experiments¹⁰ or in modeling deep planetary interiors^11,12. The accurate description of such a complex state of matter requires a detailed and reliable knowledge of key physical properties, including equation-of-state, electrical conductivity, thermal conductivity, and opacity across a wide range of conditions spanning from the solid to plasma states.

Generating and probing WDM in the laboratory has driven experimental innovations¹³, through high-energy and x-ray free-electron lasers (XFELs)¹⁴, improving drastically diagnostic precision and reliability, and continuously challenging theoretical predictions required for interpreting measurements¹⁵. Despite these achievements, no unified model yet fully describes the conditions met in warm and dense planetary interiors from the inner core to the outer envelopes, nor the various stages leading to the production of energy through inertial confinement fusion. Planetary interiors feature intermediate zones, at pressures (~ 0.1 − 10 GPa), temperatures (~ 1 − 100 kK) and densities (~ 0.1 − 1 g ⋅ cm⁻³) corresponding to expanded warm dense states^16,17, which remain difficult to accurately model. This regime of low density, high temperature and moderate pressure is also typical of the after-shock state of proto-planetary cores occurring in the aftermath of catastrophic events such as giant impacts^18,19, and encountered during inertial confinement fusion experiments^20,21. As a matter of fact, expanded WDM is generated during the laser-hohlraum interaction, in the form of gold bubbles impairing the laser-matter conversion process, or in the capsule implosion process^22,23.

While compressed warm dense states have been mainly explored, the expanded regime remains understudied. Among the scarce experimental data available, the vast majority refers either to particle beams^24–26 or resistive heating experiments^27–30. These experiments provided valuable thermodynamic and transport property data, but they suffer from uncertainties and model dependency. Particularly, inferring temperature with good accuracy still limits the ability of such experiments to improve the modeling of equations of state or transport properties in the expanded regime. To overcome these limitations, an alternative platform based on pulsed Joule heating³¹ was developed at CEA to generate and probe multiple thermodynamic quantities of expanded WDM, enabling direct evaluation of the equation-of-state and electrical conductivity without relying on theoretical modeling.

In this work, we present a model-independent experimental approach revealing that electronic relaxation time dominates conduction in the supercritical expanded warm dense matter regime. This phenomenon is accompanied by signals of electronic state localization, indicating the onset of the metal-to-nonmetal transition. These results not only advance our understanding of the solid-to-plasma transition but also provide benchmark data for improving models used in hydrodynamic and magnetohydrodynamic simulations involved in the modeling of inertial confinement fusion experiments or planetary magnetic fields.

Results

Experimental platform

Pulsed Joule heating experiments were performed on 12 μm thick aluminum foils confined in a sapphire cell (see Fig. 1), using a setup similar to  our previous work³¹ (see “Methods”). The resistive heating process is achieved on a timescale of hundreds of nanoseconds, triggering a one-dimensional expansion and uniform thermodynamic conditions across the foil length and cross-section (see Supplementary sections I and II). Experiments were conducted on two different generators delivering hundreds of kA electrical currents. Each generator is made of a 4 μF capacitor but displays distinct circuit resistance and inductance so that the pulse duration and intensity allow achieving various thermodynamic paths.

Fig. 1. Experimental platform achieving expanded warm dense states.

A uniform heating process is achieved by means of a pulsed power generator, equivalent to an RLC circuit, delivering a multi-kA and sub-μs current pulse. Experiments were performed on 12 μm thick aluminum foils confined in a sapphire cell. The foil thermodynamic changes are recorded by various electrical and optical diagnostics: current and voltage probes, Photonic Doppler Velocimetry (PDV), Pressure by Ruby Luminescence (PRL). Implemented diagnostics provide direct measurements of pressure, density, internal energy and DC conductivity.

The combination of a sub-microsecond electrical drive with the inertial confinement provided by the sapphire plates is sufficient to ensure a homogeneous heating process. Such homogeneity is ideal for measuring directly with good accuracy the foil thermodynamic properties at the foil-confinement interface, continuously from the solid to plasma phase³¹.

Given the uniform heating process, the variation in internal energy within the foil is directly linked to the energy deposited by resistive heating. As demonstrated in our previous study³¹, contributions from fluid kinetic energy, mechanical work, and heat dissipation can be neglected in most cases, since measurements are performed over a timescale (0.1–0.8 μs) shorter than the characteristic heat diffusion time. Nevertheless, these contributions are systematically evaluated and incorporated into the internal energy calculation to verify the energy balance (see Supplementary section V). Ultimately, the change in internal energy results primarily from integration of the electrical power deposited, calculated from the current and voltage measurements.

The density decrease over the heating process is obtained from a Photonic Doppler Velocimetry (PDV) diagnostic³² while the pressure increase inside the foil is obtained by the luminescence wavelength shift of a ruby plate^30,33, measured for a uniaxial strain state^31,34. This measurement is possible because the pressure across the foil section is uniformly distributed³¹. Similarly, we use the displacement of the foil-ruby interface, obtained by time integration of the velocity signal, to calculate the variation of the foil volume over time and obtain the time evolution of density. Using the characteristic wave formalism described by Landau and Lifchitz³⁵ in combination with the adiabatic equation-of-state of sapphire^30,34, the consistency of the density and pressure profiles can be verified for each experiment, thereby increasing the accuracy of the collected data (see Supplementary section III).

The static electrical conductivity σ is also directly determined from the time evolution of the density, current, and voltage, which are obtained simultaneously in each experiment. Having access to three independent thermodynamic variables (here, internal energy, pressure, and density) is sufficient to fully characterize the thermodynamic states explored during the experiment, allowing the entire data analysis to be carried out without relying on any external EOS or conductivity model. This provides a highly controlled and well-constrained description of the thermodynamic transformation induced by each electrical discharge, from the solid-liquid transition up to the onset of the plasma phase.

Thermodynamic paths across expanded warm dense matter

Multiple thermodynamic paths can be explored with our experimental platform by varying the dimensions of the foil and the drive input energy. This high flexibility is illustrated in Fig. 2 by the measurement of three thermodynamic quantities along two distinct paths, which gives access to a wide range of conditions without compromising the quality of the results.

Fig. 2. Thermodynamic paths for different heating conditions.

Thermodynamic paths obtained after heating aluminum foils with two generators charged at 35 kV and 75 kV, respectively. The internal energy variation ΔE_i is presented as a function of the density ρ with pressure indicated by the colorbar ranging from 0.25 to 17 GPa. Our experimental data (circles) are compared to : (A) the experimental works of Gathers²⁷ (dark blue area in the inset - at 0.3 GPa), Clérouin et al.³⁴ (cross - density inferred from pressure measurements) and the recent data of Ofori-Okai et al.³⁶ (up triangle - pressure not measured) ; (B) DFT-MD calculations (square) and three equation-of-state models: BLF³⁹ (star), SESAME³⁸ (plus), and from Hébert et al.³⁷ (left triangle). Error bars originate from the uncertainties on current, voltage, velocity and pressure direct measurements and propagated using a Monte-Carlo algorithm. The gray shaded area refers to the range where melting is observed on current, voltage, pressure and velocity signals (see Supplementary sections III and IV) and on the radial distribution functions g(r) obtained by DFT-MD (see Supplementary Fig. 15).

The two thermodynamic paths (circles), referred to as low- and high-ΔE_i, were measured on 12 μm thick aluminum foils heated by discharges obtained with the generators charged at 35 kV and 75 kV respectively (see Supplementary section IV). The high-ΔE_i path shows the largest range of expanded states currently accessible with our generators, spanning densities from the equilibrium value at ambient conditions of 2.7 g ⋅ cm⁻³ down to ~ 0.6 g ⋅ cm⁻³ for pressure levels ranging from 0.5 − 17 GPa and internal energy reaching up to ~ 75 MJ ⋅ kg⁻¹ (see Supplementary sections V and VI). These thermodynamic paths differ significantly from standard reference curves such as Hugoniot or quasi-isentropic trajectories, as they correspond to a specific transformation in which the density decreases while the pressure increases under the combined effects of energy deposition and inertial confinement by the sapphire plates.

Only a limited number of experimental platforms are currently capable of probing expanded warm dense matter over such a broad thermodynamic range. Comparison with previous experimental studies (see Fig. 2a), including the pulsed-power measurements of Gathers²⁷ (dark blue area in the inset) and Clérouin et al.³⁴ (cross), as well as the recent XFEL isochoric-heating data of Ofori-Okai et al.³⁶ (up triangle), shows that our measurements both reproduce and unify these state-of-the-art results while substantially extending the explored expanded-WDM regime. Moreover, whereas some of these earlier studies relied on only two independent thermodynamic variables, our approach provides a complete thermodynamic characterization of the investigated states.

Indeed, with this approach, one can probe extensively the thermodynamic landscape of the system in very few experiments, and construct isochoric, isobaric or iso-energy curves often used for constraining equation-of-state models (see Supplementary section IV). To further validate our approach, we performed DFT-MD simulations (data can be found in Supplementary section VII) for a range of densities consistent with the probed density levels and temperatures going up to 37 kK (see Fig. 2b). Along the low-ΔE_i path, our experimental data (circles) are in very good agreement with the DFT-MD calculations (squares), validating our experimental approach in a temperature-density range favorable to DFT-MD calculations. By comparing our results to three equation-of-state models available in the literature (see Supplementary section VIII), it emerges that the model from Hébert et al.³⁷ is the most suited to accurately reproduce the low-ΔE_i path, when the SESAME³⁸ and BLF³⁹ models display more than 50% discrepancy in the range 1.0 to 2.0 g ⋅ cm⁻³. This observation is consistent with the fact that DFT-MD data were used to parametrize the model of Hébert et al. up to 15 kK.

This observation does not hold for the high-ΔE_i path, however. Once the density drops below ~ 2.0 g ⋅ cm⁻³, none of the three EOS models is capable of accurately reproducing the complete set of our experimental data over the probed range. Down to 1.3 g ⋅ cm⁻³, the model of Hébert et al. is clearly the most appropriate, while the BLF model shows the best agreement between 1.3 g ⋅ cm⁻³ and 0.9 g ⋅ cm⁻³. Surprisingly, it is the SESAME model that stands out for the lowest densities (ρ < 0.9 g ⋅ cm⁻³) and highest energies (ΔE_i > 40 MJ ⋅ kg⁻¹), whereas it showed a significant deviation of up to more than 30% between 2.5 g ⋅ cm⁻³ and 1.0 g ⋅ cm⁻³. In contrast to these models, the DFT-MD calculations exhibit again a very good agreement with our measurements, even at the highest energy and lowest density levels.

Based on DFT-MD data, the temperature at 1.3 g ⋅ cm⁻³ reaches 16 kK, exceeding the upper limit of validity reported by Hébert et al. for their model (~ 15 kK), explaining why such a deviation is observed at a density lower than 1.0 g ⋅ cm⁻³. Below 1.3 g ⋅ cm⁻³, the deposited energy by the Joule effect in the aluminum ribbon is three times larger, causing a drastic increase in temperature while the pressure variation decreases during the expansion process. In this density range, the SESAME and BLF models predict temperatures in the range of 18–36 kK and 12–25 kK, respectively (see Supplementary Figs. 18, 19). In contrast, DFT-MD calculations indicate that temperatures higher than 37 kK are reached at densities lower than 0.8 g ⋅ cm⁻³. In the final moments of the experiment, the velocity measurements reach a plateau (see Supplementary Fig. 4), suggesting a quasi-steady state of confinement pressure comparable to an isobaric state, although expansion and Joule heating continue.

Moreover, one can observe that the velocity, current, voltage, and pressure signals exhibit changes in slope during the initial moments of the discharge (see Supplementary Figs. 3, 4 and 5). These inflection points correspond to the onset of melting, occurring between 2.4 g ⋅ cm⁻³ and 2.6 g ⋅ cm⁻³ (see Fig. 2). The onset of the liquid state around 2.4 g ⋅ cm⁻³ is also confirmed here by the radial distribution functions g(r) obtained by DFT-MD (see Supplementary Fig. 15) and consistent with the previous work of Hébert et al.³⁷.

Ultimately, the very good agreement between our experimental results and DFT-MD calculations enables us to constrain the thermodynamic properties of aluminum with high confidence over the conditions explored in this work, spanning densities from ρ₀ down to ρ₀/5 and temperatures reaching at least 37 kK. The ability of our platform to both reproduce previous state-of-the-art measurements and extend them into a previously unexplored high-energy regime further reinforces the reliability of the thermodynamic characterization achieved here.

The robustness of this thermodynamic characterization is particularly important for the investigation of transport properties, whose interpretation is often limited by uncertainties in the underlying thermodynamic state. Here, the simultaneous and independent determination of density, pressure, and internal energy provides a well-constrained framework for analyzing the evolution of the electronic properties of aluminum throughout the expanded warm dense matter regime.

Electrical conductivity

One of the key elements driving advances in modeling the evolution of planetary properties, such as cooling rates and the dynamo effect, or predicting the growth of magnetic instabilities in inertial confinement fusion, involves developing a solid understanding of regime changes in transport properties, such as the metal-to-nonmetal transition⁴⁰. However, in the vast majority of dedicated experiments, either transport properties are inaccessible^4,26,41 or thermodynamic conditions are not sufficiently constrained^{28,36,42–45}. In our approach, the good knowledge of the encountered thermodynamic conditions provides a unique opportunity to measure the DC conductivity in a consistent way.

Figure 3 presents the measured conductivity data, compared with previous experimental results in Fig. 3a and with DFT-MD calculations and conductivity models in Fig. 3b. Our measurements span conductivity levels from 0.28 to 5 MS ⋅ m⁻¹ for densities decreasing down to 0.6 g ⋅ cm⁻³, and reveal two distinct behaviors depending on the thermodynamic path.

Fig. 3. Electrical conductivity for different heating conditions.

Experimental conductivity data (circles) measured along the low- and high-ΔE_i paths are compared in (A) with previous results from Gathers²⁷ (dark blue shaded region - at 0.3 GPa), Clérouin et al.³⁴ (cross - densities inferred from pressure measurements), and THz spectroscopy measurements by Ofori-Okai et al.³⁶ (triangle - pressure not measured). In (B), DC conductivity calculations are shown from DFT-MD (square) and from the model of Desjarlais⁵⁰, using the equation-of-state models of Hébert et al.³⁷ and BLF³⁹ for the low- and high-ΔE_i paths, respectively (for more details, see Supplementary Fig. 14). Error bars originate from the uncertainties on current, voltage, velocity and pressure direct measurements and propagated using a Monte-Carlo algorithm. The gray shaded area refers to the range where melting is observed on current, voltage, pressure and velocity signals (see Supplementary sections III and IV) and on the radial distribution functions g(r) obtained by DFT-MD (see Supplementary Fig. 15).

Along the low-ΔE_i trajectory, the measured conductivity data are consistent with previous experimental results^27,34,36, while providing a continuous connection between these independent data-sets, and highlighting a very good agreement with the DFT-MD calculations. In contrast, the conductivity measured along the high-ΔE_i path exhibits a pronounced deviation as the density decreases, with differences reaching up to ~ 100% at 1.7 g ⋅ cm⁻³. In addition, the conductivity along the low-ΔE_i is consistently reducing with decreasing density as expected in a metallic state, whereas a minimum is observed along the high-ΔE_i path below 0.9 g ⋅ cm⁻³, suggesting the onset of a different conduction regime and/or thermodynamic state. This minimum in the conductivity is well-known and has previously been observed in more dilute regimes^28,30,40,42. Here, however, it occurs at significantly higher density and temperature conditions than previously reported. This change in transport behavior is usually associated with the so-called metal-to-nonmetal transition.

The downward shift in conductivity observed between the low and high-ΔE_i paths is confirmed by DFT calculations, but its magnitude depends on the exchange-correlation functional employed. We investigated the effect of the exchange-correlation functional on the electrical conductivity of expanded warm dense aluminum in the Supplementary section VII. The treatment of the exchange-correlation seems to play a non-negligible role. As previously shown by Witte et al.⁴⁶, the conductivity can be overestimated by the standard exchange-correlation functional (GGA-PBE)⁴⁷ used here for the molecular dynamics simulations. Consistently, the meta-GGA functional (R2SCAN01⁴⁸) yields conductivity values in closer agreement with our experimental data (see Fig. 3), demonstrating that the choice of exchange-correlation functional has a non-negligible impact on the predicted conductivity in this regime. Given the high temperatures reached, especially along the high-ΔE_i path, we also investigated the impact of temperature on the exchange-correlation functional by computing the electrical conductivity using the KDT16 thermal exchange-correlation GGA functional⁴⁹ (see Supplementary section VII and Supplementary Fig. 16). These tests show that the impact of finite temperature on the exchange correlation remains relatively small for the range of thermodynamic conditions explored here and can thus be safely neglected.

Beyond the DFT-MD calculations, we also benchmark our measurements against the conductivity model of Desjarlais⁵⁰, which remains a widely used reference for the development of transport models in warm dense matter. Following the thermodynamic analysis presented in Section 1.2, we employ the Hébert et al. equation-of-state for the low-ΔE_i path and the BLF model for the high-ΔE_i path to determine the temperature required as input for the conductivity evaluation. Overall, the Desjarlais model reproduces the general trend of the conductivity along both thermodynamic trajectories, with the best agreement observed in the intermediate density range between 1.5 and 2.5 g ⋅ cm⁻³, where deviations remain below ~ 30%. However, for more expanded states (ρ≤1.5 g ⋅ cm⁻³), the model increasingly departs from both the experimental and DFT-MD data, most notably along the high-ΔE_i path, reaching discrepancies of up to ~ 170% at 1 g ⋅ cm⁻³.

To further assess the origin of these deviations, we extended the comparison to two additional equation-of-state models (see Supplementary Fig. 14). Despite substantial differences in the temperature predictions provided by the various EOS models, none of them satisfactorily reproduces the experimental and DFT-MD conductivity data below 1.5 g ⋅ cm⁻³. In particular, although the different EOS descriptions lead to temperature variations of approximately 11–75% at 0.8 g ⋅ cm⁻³ (see Supplementary Figs. 18, 19), the corresponding conductivity variations predicted by the Desjarlais model remain insufficient to account for the observed discrepancies.

Taken together, the experimental and DFT-MD results reveal a pronounced dependence of the electrical conductivity on the thermodynamic path explored in expanded warm dense matter. While the low-ΔE_i trajectory remains consistent with previous measurements^27,34,36 and standard conductivity models^50,51, the high-ΔE_i path shows a strong reduction in conductivity (downward shift), including the emergence of a minimum at low density that is absent along the low-ΔE_i trajectory. The robustness of these features across different equation-of-state models, despite substantial variations in the inferred temperatures, indicates that the observed behavior cannot be explained solely by uncertainties in the underlying thermodynamic state.

More generally, these observations suggest that the conductivity evolution in this regime may not be governed exclusively by thermal ionization effects, but could also involve substantial modifications of the underlying electronic transport mechanisms and collision processes. In particular, the appearance of a conductivity minimum along the high-ΔE_i path could indicate the onset of a distinct conduction regime related to the metal-to-nonmetal transition.

Relaxation time

In the context of the Drude model, the DC conductivity σ is expressed as a function of the effective conducting electron density $n_e^{*}$, the free electron mass m_e and the relaxation time τ_e referring to the average time interval between two consecutive collisions, $\sigma =\frac{e^2{n}_e^{*}{\tau }_e}{m_e}$, with e the electron charge. The effective electron density takes into account the renormalization of the free conducting electron density by the effective mass, which for aluminum in the thermodynamic conditions explored here is expected to be rather small. Hence, by comparing the evolution of the DC conductivity σ across the low- and high-ΔE_i path relatively to the conductivity σ₀ = 40 MS ⋅ m⁻¹ at ambient density ρ₀ = 2.7 g ⋅ cm⁻³, it becomes possible to infer the relative evolution of the ${\tau }_e.n_e^{*}$ term.

In Fig. 4.A, the ratio σρ₀/σ₀ρ is calculated along the two paths and exhibits very different decays. At low-ΔE_i, this ratio is decreasing almost linearly as a function of density down to 1.5 g ⋅ cm⁻³. On the contrary, the high-ΔE_i path follows a parabolic curve with a minimum located around 1.0 g ⋅ cm⁻³ with a value of ~ 0.02, confirming a change in conduction related to the onset of the metal-to-nonmetal transition. Besides, a large gap between the two paths is also observed, and confirmed by DFT-MD calculations, from 2.4 g ⋅ cm⁻³, narrowing as density drops over the expansion down to 1.0 g ⋅ cm⁻³. At first glance, one may consider that this gap is due to the pressure difference between the two paths. However, in this regime, ionization-driven pressure changes cannot occur since the aluminum is expanding and the density is decreasing down to ρ₀/5. Hence, the pressure increase, by almost a factor of 10, at a given density is the consequence of the temperature elevation only.

Fig. 4. Relaxation time in the expanded regime.

Relative variation of σ. ρ₀/σ₀. ρ (A) calculated using experimental data and results obtained from DFT calculations for the low-ΔE_i (triangles) and high-ΔE_i (squares) paths. Error bars originate from the uncertainties on current, voltage, velocity and pressure direct measurements and propagated using a Monte-Carlo algorithm. B Variation of the effective number of conducting electrons $n_e^{*}/n_i$ and (C) the relaxation time τ_e obtained from a Drude fit of the DFT calculations. The color reflects the maximum of the relative error in the Drude fitting range. D Density of states (DOS) obtained at various densities showing the transition from a Drude-to-non-Drude behavior below 1.5 g ⋅ cm⁻³.

To explore further this behavior, conductivity spectra calculated by DFT, which predict similar decay, were fitted with the Drude formula from which we deduced $n_e^{*}$ and τ_e. As shown on Fig. 4.B, the ab initio effective number of conducting electrons per ion $n_e^{*}/n_i$ is increasing with decreasing density by a factor of ~ 1.05 and ~ 1.25 along the low- and high-ΔE_i paths, respectively. In contrast, the ab initio relaxation time τ_e shown in Fig. 4C is, like the ratio σρ₀/σ₀ρ, dramatically dropping as density decreases, by a factor of ~ 2.5 along the low-ΔE_i path, which also appears more pronounced as the temperature is higher, reaching around 5 along the high-ΔE_i path. This decrease of the relaxation time could be explained by the increasing temperature, which increases both the ionic disorder and the average electronic velocity, consistent with the earlier analysis of Sasaki et al.⁵² on warm dense copper using an analogous Drude decomposition, corroborating and extending their conclusion to aluminum. We also note that below densities of approximately ~ 1.3 g ⋅ cm⁻³, the electronic structure undergoes a pronounced modification, as evidenced by the electronic density of states shown in Fig. 4D. These changes reveal an electron localization mechanism similar to that reported by Levy et al.⁵³, associated with the onset of the metal-to-nonmetal transition. In this regime, the conductivity spectra exhibit a clear non-Drude behavior (see Supplementary Fig. 17). Consequently, at densities below ~ 1.5 g ⋅ cm⁻³, the Drude model begins to fail due to strong electron localization, as evidenced by the increasing fit errors (Fig. 4B, C).

Therefore, our experimental evidence, supported by ab initio calculations, indicates that the ionization process does not drive the conductivity in a supercritical density-temperature range of the metal-to-nonmetal transition edge. Instead, electron scattering seems to predominate in the density and temperature conditions explored in this work.

Discussion

Thanks to an innovative setup based on confined exploding metallic foils, we provide benchmark quality data on the equation of state and the dependence of conductivity on density and temperature in the rarely explored expanded warm dense matter regime. The ability of our platform to both reproduce previous state-of-the-art measurements and extend them into a previously unexplored high-energy regime further demonstrates the robustness and broad applicability of our approach. Our results highlight the limitations of standard EOS and conductivity models while demonstrating the adequacy of DFT-based molecular dynamics to provide reliable data in this regime.

A key finding regarding electrical conductivity is that it is not primarily governed by ionization in the range of conditions explored by our experiments. As suggested by our ab initio simulations, consistent with our experimental data, most of the conductivity evolution can be explained by changes in the relaxation time and electronic scattering.

A simple conductivity model solely based on ionization degrees would thus fail at capturing the physics at play for numerous systems lying in the expanded warm dense matter regime. The strong influence of electronic relaxation times could have significant implications for the description of planetary envelopes and post-impact states. The electrical conductivity is indeed a crucial ingredient in the magnetic field generation and interaction, shaping the evolution or even the formation of planetary bodies. This is also true for the dynamics of inertial confinement fusion capsule implosions, as they are sensitive to magnetic instabilities.

By providing experimentally constrained thermodynamic paths and transport data in this previously underexplored regime, our work establishes a new physical framework for understanding transport across the metal-to-nonmetal transition in strongly coupled matter.

Methods

Pulsed-power generators

The experiments were performed on 2 pulsed-power generators developed at CEA DAM³¹ that allow us to adjust the characteristic time and the energy level of the current pulse. Both generator circuits are equivalent to an RLC series circuit. The characteristics of the first generator, used for acquiring data at low-ΔE_i, are 4 μF, 35 mΩ and 220 nH with charging voltages ranging from 25 kV to 55 kV. The second generator is 4 μF, 22.5 mΩ and 84.5 nH and is operated from 35 kV to 75 kV, which enables measurements at high-ΔE_i. On a dummy load of 10 nH, maximum current delivered is 100 kA in 1.4 μs and 500 kA in 0.9 μs, respectively for the low and for the high current generators. Each circuit includes switches whose intrinsic properties are taken into account in the circuit characterization process.

Target design

The target consists of a multi-layer assembly made of a 10 mm long, 6 mm high and 12 μm thick pure aluminum foil confined by ~ 3 mm sapphire plates with a square section of 10 × 10 mm². The foil edges are connected to the generator on the other edge by means of electrodes. However, depending on the generator, two target assemblies were designed to accommodate the current intensity and diagnostics.

For the experiments conducted on the first generator, leading to the low-ΔE_i path, the targets displayed a U-shape³¹, with a 100 μm thick aluminum drive conductor and a 400 μm a-cut ruby plate inserted between the foil of study and one of the sapphire plates (see Fig. 1 and Supplementary Fig. 2A, C).

For the higher-temperature experiments, conducted on the second more energetic generator, the target was designed with a flat geometry (see Fig. 2B, D in Supplementary section II). Because the intensity of the current peak is 3 to 10 times higher than that of the first generator, it was decided not to use a U-shape target to avoid having a non-negligible magnetic pressure induced by the current flow. With planar targets, the contribution of the magnetic pressure to the total pressure stays below uncertainties, i.e., ~ 0.4 GPa at 300 kA when the total pressure measured reaches up to ~18 GPa. Therefore, no ruby plate was inserted into the assembly. Instead, the electrodes were insulated with resin to avoid exceeding the spark air gap threshold (~ 35 kV ⋅ cm⁻¹) while leaving a resin-free channel to enable PDV measurement on multiple locations (see section 3.3 and Supplementary section II Fig. 2B, D).

Voltage and current measurements

Electrical diagnostics have been fielded in the terminal coaxial line of the generator just before the load to ensure proper voltage and current measurements. This includes B-dot and E-dot probes, Rogowski coils, and two NorthStar PVM-5 voltage probes directly connected to the sample. The terminal coaxial line with its insulating material can be embedded in a calibration cell in order to determine the frequency response of the electrical diagnostics (except for the NorthStar PVM-5 probes). The calibration chain works with a network analyzer and includes an amplifier and an optical chain to be able to measure low sensitivity and low frequency components.

Velocimetry

The density profile is obtained by Photonic Doppler Velocimetry (PDV)³² measurements at the foil-confinement interface. It is composed of a multi-channel 1550 nm fibered interferometer. For each channel, 2 photodiodes FEMTO HSA-X-S and HSPR operating between 30 kHz and 2 GHz connected to a Tektronix MSO68B 2.5 GHz 50 GS/s measure the PDV interferometric signals and allow us either to resolve the velocity profile by Short Time Fourier Transform (STFT) or to resolve the displacement profile by phase retrieval by Lissajous treatment. The first method is robust to low signal to noise ratio, whereas the second method is more efficient for low displacement (i.e., low number of fringes), so both are complementary. A fiber optic array (FOA), connected to 8 PDV channels, is used as a transmitter/receiver probe. This FOA is in the entrance of a single 4F optical system, made of 2 lenses, to focus all the 8 probe beams on the sample arranged in a square pattern with a 3 × 3 mm pitch (see Supplementary sections II and III)⁵⁴.

Pressure by ruby luminescence

The foil pressure is measured by ruby luminescence (PRL)^30,31,34 over the heating process. The ruby fluorescence is stimulated by a pulsed Nd:YAG laser (Agilite) operating at 532 nm. The laser pulse displays a 10 μs Gaussian-shape pulse of ~ 10 mJ. After focalization, the focal spot is of 1 mm diameter at the ruby-foil interface. The collected light is then driven by a streak camera Optronis SC-10 equipped with an S25 photocathode connected to an Andor Kymera 328i spectrometer slit equipped with a 1200 l/mm grating blazed at 750 nm. The temporal R-line shift evolution is recorded at a streak speed of 250 ns ⋅ mm⁻¹. This diagnostic is coupled to the PDV in the same optical system by means of multiple dichroic beamsplitters and focusing at the foil-ruby interface³¹. The PRL diagnostic is used along the PDV to acquire data on low-ΔE_i trajectories, while the multipoints PDV system was preferred for high-ΔE_i experiments to prevent magnetic perturbations from triggering the Zeeman effect at higher current levels.

Density functional theory simulations

Molecular Dynamics (MD) simulations based on Density Functional Theory (DFT-MD) were performed at various densities within the range probed experimentally. All the simulations were run at constant density in the isokinetic ensemble at temperatures chosen so that the obtained pressures match the experiment. The molecular dynamics timestep was set between 2.42 fs and 1.45 fs, depending on the thermodynamic conditions and the total simulation time varied between 2.42 and 5.0 ps in order to accurately converge the different thermodynamic properties of interest. The internal energy variation was obtained by taking the difference between the average internal energy obtained from the MD simulations and a reference energy. This reference internal energy was obtained from a similar DFT-MD calculation of solid aluminum in the FCC structure at equilibrium density ρ₀ = 2.7 g ⋅ cm⁻³ and temperature T = 300 K. The thermodynamic conditions of all DFT-MD calculations are summarized in Supplementary Table I of the Supplementary section VII.

Afterwards, representative snapshots of atomic positions were randomly extracted from each DFT-MD trajectory in order to compute the electrical conductivity. For each of these snapshots, the optical conductivity was calculated using the Kubo-Greenwood formula⁵⁵ and finally averaged over all snapshots. In practice, we found that averaging over five snapshots was sufficient to converge the average optical conductivity. A fit of the average optical conductivity using a Drude-Smith model⁵⁶ was finally performed in order to extract the static conductivity σ. An example of calculated optical conductivity and the associated Drude-Smith fit is given at different conditions in Fig. 14 of the Supplementary section VII.

These Density-Functional Theory (DFT) simulations were carried out within the Kohn-Sham formalism using the ABINIT code⁵⁷ in the projector augmented-wave approach (PAW)⁵⁸. The exchange-correlation functional was treated in the Generalized Gradient Approximation (GGA-PBE)⁴⁷ for the MD simulations. The electrical conductivity was obtained using the GGA-PBE functional along the low-ΔE_i path and using both the GGA-PBE and the R2SCAN01⁴⁸ meta-GGA functional along the high-temperature path. The PAW data were generated with the ATOMPAW program⁵⁹, considering three valence electrons (3s²3p¹), the other electrons being considered frozen in their atomic state. The radius of the PAW spheres was set to 0.9 Å in order to ensure only minimal overlap of the spheres for all simulations. The cutoff energy, which defines the size of the plane-wave basis, was set to 1088 eV. We considered a system of 108 aluminum atoms with periodic boundary conditions. The Brillouin zone was sampled using 4 k-points for the MD simulations and 32 k-points for the subsequent Kubo-Greenwood calculations of the conductivity.

Author contributions

Conceptualization: B.J., F.Z. and L.V. Data collection: B.J., L.R., J.A., E.B., A.L., S.B., T.G. and F.Z. Analysis and discussion: B.J., F.B., J.B., F.S., J.R., C.B., L.V. and V.R. Theory and Simulation: F.B., J.B., J.R., F.S., C.B., L.V., and V.R. Writing-original draft: B.J., F.B. and J.B. Writing-review & editing: B.J., F.B., J.B., F.S., C.B., L.V. and V.R.

Peer review

Peer review information

Nature Communications thanks Toru Sasaki and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This research received no specific grant funding from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

All relevant data supporting the findings of this study have been deposited in the Zenodo database and are accessible at https://doi.org/10.5281/zenodo.21621105 (see ref. ⁶⁰).

Code availability

The first-principles molecular dynamics simulations presented in this work were performed using the ABINIT code, an open-source software package freely available at https://www.abinit.org. The specific input files and parameters used in this study are available from the repository at https://doi.org/10.5281/zenodo.21621105 (see ref. ⁶⁰).

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at https://doi.org/10.1038/s41467-026-77435-x.