Hydrodynamic simulations of expanded warm dense foil heated by pulsed-power

Warm Dense Matter lies at the frontier between condensed matter and plasma, and plays a central role in various fields ranging from planetary science to inertial confinement fusion. Improving our understanding of this regime requires experimental data that can be directly compared with theoretical and numerical models over a broad range of conditions. In this work, a pulsed-power experiment is described in which thin metallic foils, confined within a sapphire cell, are Joule-heated to achieve the expanded warm dense matter regime. Designing such an experiment is challenging, as it requires simultaneously predicting the electrical response of the pulsed-power driver and the hydrodynamic evolution of the heated material. To tackle this challenge, a modeling framework has been developed that couples an electrical description of the pulsed-power system, including the driver, the switching stages and the load with a one-dimensional hydrodynamic code. This coupling allows the electrical energy deposition and the load thermodynamic evolution to be consistently linked through the material electrical conductivity. This approach takes advantage of the simplicity of a 1D geometry while retaining the essential physics and allowing to reproduce various measurements with good accuracy, such as expansion velocity, current and voltage. This numerical approach therefore constitutes a robust and efficient method for designing and optimizing future Warm Dense Matter experiments using pulsed-power facilities.

💡 Research Summary

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This paper presents a comprehensive modeling framework that couples an electrical description of a pulsed‑power driver with a one‑dimensional hydrodynamic code (ESTHER) to simulate the formation and evolution of expanded warm‑dense matter (WDM) generated by Joule heating of thin metallic foils confined in a sapphire cell. The authors begin by motivating the need for accurate experimental data in the intermediate regime between condensed matter and plasma, where ion‑ion correlations and electron degeneracy make conventional models inadequate. They describe the experimental configuration: foils of 5–20 µm thickness, 10 mm length, and 6 mm height are sandwiched between sapphire anvils and driven by two pulsed‑power generators (EPP1: up to 140 kA, 1.5 µs rise; EPP2: up to 530 kA, 0.83 µs rise). Diagnostics include Rogowski coils for current, voltage probes, B‑dot/D‑dot probes, photon‑Doppler velocimetry (PDV) for expansion velocity, and ruby photoluminescence for pressure.

The electrical circuit is modeled as a series R‑L‑C network whose resistance R(t) and inductance L(t) are time‑dependent. Three contributions are identified: fixed driver parameters (R_f, L_f), a spark‑gap switch (R_SG(t), L_SG(t)), and the load (R_W(t), L_W(t)). The current equation is discretized with a second‑order centered finite‑difference scheme, yielding an explicit recurrence that naturally incorporates the evolving R and L. The spark‑gap dynamics are described using a modified Braginskii model: the arc is treated as a cylinder of constant conductivity σ and a radius a(t) that grows according to a power‑law integral of the current. This provides analytical expressions for R_SG and L_SG, which include a β parameter to avoid divergence at the first time steps and a factor N representing the number of parallel spark‑gaps. The authors show that early‑time resistance can be orders of magnitude larger than the fixed circuit resistance, while inductance initially contributes a substantial fraction of the total.

Model validation is performed using short‑circuit experiments on four generators (EPP1, EPP2, EOLE, GEPI2) covering a current range from 0.1 MA to 5 MA. Since direct measurement of R_f and L_f is difficult, the authors infer these values by fitting the late‑time portion of the measured current waveforms (where the circuit behaves linearly) and then refine β using the early‑time region where the spark‑gap dominates. The resulting simulated current waveforms agree with measurements within 5 % across all generators, demonstrating the robustness of the electrical model.

The hydrodynamic part couples the circuit output to ESTHER, which solves the 1‑D planar expansion of the foil. Energy deposition is linked to the material’s electrical conductivity σ(ρ,T), creating a feedback loop: the current heats the foil, raising its temperature and reducing its resistivity, which in turn modifies the current distribution. Three energy‑deposition strategies are explored: (1) current‑based Joule heating (I²R), (2) voltage‑based power input (V·I), and (3) a more detailed electron‑ion energy exchange model. The authors find that the current‑based approach provides the most direct and accurate representation for the present geometry.

Simulation results reproduce key experimental observables: the expansion velocity measured by PDV, the current waveform, and the load voltage. Discrepancies are typically below 5 %, confirming that a 1‑D approximation, when combined with a realistic, time‑dependent electrical circuit, captures the essential physics of the experiment. The paper concludes that this coupled framework offers a fast, reliable tool for designing and optimizing future pulsed‑power WDM experiments, reducing computational cost while preserving fidelity. The authors suggest that extending the approach to multi‑dimensional geometries would allow investigation of asymmetries, magneto‑hydrodynamic instabilities, and more complex material behavior, paving the way for deeper insight into the warm‑dense regime.