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

TL;DR

This work develops a coupled 1D framework that integrates a pulsed-power current solver with a one-dimensional hydrodynamic code (ESTHER) to simulate Joule heating and expansion of a confined foil in the Warm Dense Matter regime. It validates the electrical model against short-circuit data across multiple pulsed-power facilities and then demonstrates two heating strategies: using experimentally measured power as a source term and using a conductivity-based self-consistent deposition. The hydrodynamic validation favors BLF and Hébert et al. EOS over SESAME for the solid–liquid transition, while the fully self-consistent conductivity-coupled simulations reproduce current, voltage, and velocity trends and provide access to unmeasured quantities such as temperature. The framework offers a robust, efficient tool for designing WDM experiments and for testing EOS and transport models under relevant microsecond timescales and mega-ampere currents.

Abstract

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.

Figures (9)

• Figure 1: Schematic of the confined exploding foil experiment.
• Figure 2: Equivalent electrical circuit used to model a generic pulsed-power system, including the driver stage, the switch and the exploding-wire load.
• Figure 3: Influence of $\beta$ and $N$ on the resistance and inductance of a generic spark-gap in a generic pulsed-power system (for, $C = 4 ~\rm{\mu F}$, $R_f = 30 ~\rm{m \Omega}$, $L_f = 100 ~\rm{nH}$, charged at $U_0 = 50~\rm{kV}$, with an inter electrode gap $d = 4~\rm{cm}$). Where the line style indicates the value of $\beta$, dotted for $\beta = 10^{-10}$, dashed-dotted for $\beta = 10^{-8}$ and dashed for $\beta = 10^{-6}$, while the line color denotes the number of channels, blue for $N = 1$, purple for $N=4$ and red for $N=16$.
• Figure 4: Short-circuit current measurements from four pulsed-power generators, compared with simulations using our discharge model (with parameters in Table.\ref{['tab:facility_specs']}). Solid lines represent the measured currents, with line thickness indicating the experimental uncertainty, while dashed lines correspond to the simulated currents. Colors identify the different facilities and their respective charging voltages.
• Figure 5: Different coupling paths used for the energy-deposition term in the hydrodynamic simulations. The purple path relies directly on the experimental current and voltage measurements; the blue path uses the experimental electrical conductivity together with the discharge-current model; and the red path is fully numerical, combining the discharge-current model with a tabulated conductivity model.