Pulsed-Power Experiment Unlocks New Insights into Extreme States of Matter

Warm Dense Matter, a state between solid and plasma, underpins phenomena across planetary science and inertial confinement fusion. Luc Revello, Laurent Videau, and Frédéric Zucchini, alongside their colleagues at CEA and Université Paris-Saclay, present hydrodynamic simulations addressing the complex challenge of creating and characterising this regime experimentally. Their research details a novel modelling framework which couples electrical descriptions of pulsed-power systems with one-dimensional hydrodynamic codes, allowing for consistent prediction of energy deposition and material evolution. This approach represents a significant advance in designing and optimising Warm Dense Matter experiments, offering a robust and efficient method for interpreting data and validating theoretical models.

Recent work details a novel modeling framework designed to accurately simulate experiments exploring this challenging regime.

Researchers have developed a coupled system linking electrical descriptions of pulsed-power facilities with one-dimensional hydrodynamic codes, enabling consistent prediction of both electrical response and material evolution during Joule-heating experiments. This approach addresses a key difficulty in warm dense matter research: the simultaneous need to model complex electrical drivers and the hydrodynamic behaviour of heated materials.
The study focuses on experiments employing pulsed-power to Joule-heat thin metallic foils confined within sapphire cells, achieving the expanded warm dense matter state. Designing these experiments presents significant challenges, demanding accurate prediction of the pulsed-power driver’s electrical characteristics alongside the hydrodynamic evolution of the heated material.

To overcome this, a modeling framework was created that couples the electrical description of the pulsed-power system, including the driver, switching stages, and load, with a one-dimensional hydrodynamic code. This coupling consistently links electrical energy deposition with the thermodynamic evolution of the material via its electrical conductivity.

This numerical approach leverages the simplicity of a 1D geometry while retaining essential physics, successfully reproducing measurements such as expansion velocity, current, and voltage with good accuracy. The framework models pulsed-power drivers as equivalent electrical circuits, capable of simulating generators delivering currents from hundreds of kiloamperes up to several megaamperes.

Validation against short-circuit measurements across a wide range of operating conditions confirms the model’s robustness. These foils are subjected to high currents to achieve the expanded warm dense regime, a state between condensed matter and plasma. Designing these experiments requires simultaneous prediction of both the electrical response of the pulsed-power driver and the hydrodynamic evolution of the heated material.

To address this complexity, a modelling framework was developed that couples an electrical description of the pulsed-power system with a one-dimensional hydrodynamic code named ESTHER. This coupling links electrical energy deposition and the thermodynamic evolution of the load through the material’s electrical conductivity.

The electrical model represents any pulsed-power driver as an equivalent electrical circuit, capable of delivering pulsed currents ranging from hundreds of kiloamperes up to several megaamperes. Validation of this electrical model was performed against short-circuit measurements taken across a wide range of operating conditions, ensuring accurate representation of the driver’s behaviour.

The hydrodynamic code, ESTHER, simulates the one-dimensional expansion of the heated foil, accounting for the material’s response to the deposited energy. This approach leverages the simplicity of a 1D geometry while retaining essential physics, allowing for accurate reproduction of measurements such as expansion velocity, current, and voltage.

The numerical method constitutes a robust and efficient tool for designing and optimising future warm dense matter experiments utilising pulsed-power facilities. This simplification considerably reduces computational cost while preserving the relevant physics of energy deposition and material response.

Reproduced discharge profiles validate hydrodynamic modelling of Joule heating

Discharge currents ranging from 50 kA to 5 MA were accurately reproduced through modeling and experimentation. Rising edges of these currents were comprised between 0.8 and 1.5 microseconds. This agreement validates the numerical scheme used to solve the current equation and the spark-gap model employed in the research.

The optimization procedure used to determine intrinsic circuit properties was also supported by these results. The ESTHER code, a one-dimensional Lagrangian hydrodynamic solver, was applied to model the warm dense regime. This code originally designed for laser, matter interactions and shock wave propagation, was adapted to simulate Joule heating over a microsecond timescale.

The fluid motion was described by conservation equations of mass, momentum, and energy, solved using a finite volume scheme ensuring conservation across cell interfaces. To accurately model material behaviour, an equation of state relating pressure and internal energy was implemented within ESTHER. Thermal conduction was modeled using Fourier’s law, incorporating tabulated data from sources such as the Y.

S. Touloukian tables and average atom calculations. Radiative effects were found to be negligible under the experimental conditions and were therefore omitted from the simulations.

Simulations were performed on an aluminium foil load measuring 1.04cm in length, 5.9mm in height, and with an initial thickness of 18.3μm, confined between two 3mm thick sapphire plates. This load was connected to the EPP2 generator charged to 35kV. The assumption of uniform heating and one-dimensional expansion remained valid up to 0.47 microseconds, allowing access to derived quantities including pressure and temperature profiles.

Pulsed-power driven experiments and hydrodynamic modelling validate warm dense matter creation

Warm dense matter, existing between solid and plasma states, is crucial to understanding phenomena in planetary science and inertial confinement fusion. Achieving a detailed understanding necessitates comparing experimental data with theoretical and numerical models across a wide range of conditions. Recent work has described a pulsed-power experiment utilising thin metallic foils confined within a sapphire cell, Joule-heated to create an expanded warm dense matter regime.

A novel modelling framework was developed to simultaneously predict the electrical response of the pulsed-power driver and the hydrodynamic evolution of the heated material. This framework couples an electrical description of the pulsed-power system with a one-dimensional hydrodynamic code, linking energy deposition and material thermodynamic evolution through electrical conductivity.

The resulting simulations demonstrate good agreement with experimental measurements of expansion velocity, current, and voltage, establishing a robust and efficient method for designing and optimising future warm dense matter experiments. Slight discrepancies between simulation and experiment likely stem from a combination of the equation-of-state and conductivity models, highlighting the interconnectedness of thermodynamics and electrical transport in these simulations.

The developed numerical framework successfully simulates a confined exploding foil experiment driven by pulsed-power, incorporating a current solver and switch dynamics validated against short-circuit current measurements. By initially using experimentally measured electrical power to impose energy deposition, and subsequently coupling the electrical model to a hydrodynamic solver via an experimentally derived conductivity, the simulations closely matched experimental results.

This self-consistent approach allows for complete description of the experiment, encompassing both the electrical driver response and the coupled electrical and hydrodynamic evolution of the load. Beyond reproducing observed quantities, the simulations provide access to properties inaccessible through direct measurement and offer a means to test different equation-of-state and electrical conductivity models. This methodology therefore represents a valuable tool for interpreting existing experiments, guiding future designs, and improving the modelling of warm dense matter properties.