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

This paper presents a robust modeling framework that couples pulsed-power electrical circuit simulations with one-dimensional hydrodynamic codes to accurately design and optimize experiments for generating expanded warm dense matter in thin metallic foils confined within sapphire cells.

The Gist

The Big Picture: What is "Warm Dense Matter"?

Imagine a material that isn't quite a solid, isn't quite a liquid, and isn't quite a gas. It's in a weird, messy middle ground called Warm Dense Matter (WDM). Think of it like a crowded dance floor where everyone is moving fast (hot) but still bumping into each other (dense).

Scientists need to understand this state of matter to study things like the inside of giant planets or to build better fusion energy reactors. But it's hard to study because it's tricky to create in a lab and even harder to predict with math.

The Experiment: The "Electrical Pancake"

The researchers set up an experiment to create this state.

• The Setup: They took a very thin sheet of metal (like a microscopic aluminum foil) and sandwiched it between two thick, hard plates made of sapphire (like the glass in a watch face).
• The Action: They zapped this metal sandwich with a massive, super-fast burst of electricity (pulsed power).
• The Result: The electricity heats the metal so quickly (in less than a millionth of a second) that it melts, boils, and turns into a hot, expanding plasma. Because the sapphire plates hold it in, the metal can only expand in one direction, like a pancake puffing up.

The Problem: The "Black Box"

The challenge is that when you zap the metal, two things happen at the same time:

1. The Electrical Circuit: The electricity flows through the wires, the switch, and the metal. As the metal heats up and changes shape, its ability to conduct electricity changes, which changes the flow of current.
2. The Physical Movement: The metal gets hot, expands, and moves. As it moves, it changes the shape of the circuit, which changes the electricity again.

It's a feedback loop. If you try to calculate the electricity without knowing how the metal moves, you get it wrong. If you try to calculate the metal's movement without knowing the electricity, you also get it wrong.

The Solution: A "Tandem Bicycle" Model

The authors built a computer program that acts like a tandem bicycle.

• Rider 1 (The Electrical Model): This part simulates the power generator, the switch, and the wires. It calculates how much current is flowing.
• Rider 2 (The Hydrodynamic Model): This part simulates the metal foil. It calculates how the metal heats up, expands, and changes density.

These two riders are locked together. Every tiny fraction of a second, they talk to each other:

• "Hey, the metal just got hotter and thinner," says Rider 2.
• "Okay, I'll adjust the current flow because the metal is now a worse conductor," says Rider 1.
• "Okay, I'll update the heat and pressure based on that new current," says Rider 2.

How They Tested It

To make sure their "tandem bicycle" works, they tested it in three different ways, like checking a car's engine at different levels:

1. The "Known Power" Test: They fed the computer the actual electricity measurements from the real experiment and asked, "Can you predict how the metal moves?"

   • Result: Yes, very well. The computer predicted the speed and expansion of the metal almost perfectly. This told them which mathematical "rules" (Equations of State) best describe how the metal behaves.

2. The "Known Conductivity" Test: They fed the computer the actual electrical conductivity of the metal (how well it conducts) and asked, "Can you predict the electricity and the movement?"

   • Result: Yes. The computer successfully predicted the voltage and current, matching the real experiment. This proved the two parts of the model were talking to each other correctly.

3. The "Pure Prediction" Test: This was the hardest. They gave the computer no data from the real experiment. They just gave it the laws of physics and asked, "Can you predict the whole experiment from scratch?"

   • Result: It was very close. The computer predicted the speed, current, and voltage with good accuracy. There were small differences (like a 10% error in voltage at the very end), but the overall picture was correct.

Why This Matters

The paper concludes that this computer model is a robust and efficient tool.

Instead of just guessing how to set up future experiments, scientists can now use this "tandem bicycle" model to design them. They can simulate different scenarios on the computer to see what will happen before they ever turn on the real machine. It helps them understand the physics of warm dense matter without needing to rely solely on expensive and difficult experiments.

In short: They built a digital twin of a high-speed electrical explosion. They proved it works by comparing it to real explosions, and now they can use it to plan future experiments with confidence.

Technical Summary

Technical Summary: Hydrodynamic Simulations of Expanded Warm Dense Foil Heated by Pulsed-Power

Problem Statement
Warm Dense Matter (WDM), the intermediate regime bridging condensed matter and plasma, is critical for fields ranging from planetary science to inertial confinement fusion (ICF). However, modeling this regime is challenging due to the coexistence of ionic correlation effects and electron degeneracy, which neither classical plasma physics nor conventional condensed matter theories describe accurately. While various experimental techniques exist to produce WDM, data regarding the expanded (lower-than-nominal density) regime is particularly valuable for validating equations of state (EOS) and conductivity models relevant to ICF hohlraum configurations.

The specific challenge addressed in this work is the design and simulation of a pulsed-power experiment where thin metallic foils, confined within a sapphire cell, are Joule-heated to achieve the expanded WDM regime. Designing such experiments is difficult because it requires simultaneously predicting the electrical response of the pulsed-power driver and the hydrodynamic evolution of the heated material. Furthermore, accurate interpretation of the transition from solid to partially ionized plasma requires a fully coupled numerical framework that links electrical energy deposition with thermodynamic evolution.

Methodology
The authors developed a modeling framework that couples an electrical description of a pulsed-power system with a one-dimensional (1D) hydrodynamic code, ESTHER. The approach relies on a 1D planar geometry, which simplifies the problem by assuming expansion occurs in a single preferential direction due to high acoustic impedance confinement (sapphire), allowing the neglect of magnetic field effects and reducing computational cost while retaining essential physics.

The methodology proceeds through three distinct modeling stages:

1. Electrical Circuit Modeling: The pulsed-power driver, spark-gap switch, and load are modeled as an equivalent RLC circuit. The current evolution is solved numerically using a second-order centered finite difference scheme to handle time-dependent resistance and inductance.

   • Driver: Characterized by fixed intrinsic parameters ($R_f, L_f, C$).
   • Switch: Modeled using a modified Braginskii arc model, where the spark-gap resistance and inductance evolve based on the current integral and the number of parallel channels ($N$).
   • Validation: This electrical model was validated against short-circuit measurements from four different generators (EPP1, EPP2, EOLE, GEPI2) covering currents from ~140 kA to ~5 MA.

2. Hydrodynamic Simulation: The ESTHER code, a 1D Lagrangian hydrodynamic solver, was used to simulate the foil's thermodynamic evolution. The code solves conservation equations for mass, momentum, and energy, incorporating an EOS and thermal conduction. The Joule heating source term is integrated directly into the energy equation.

3. Coupling Strategies: Three approaches were tested to link the electrical and hydrodynamic domains:

   • Approach A (Experimental Power): The source term is the experimentally measured power ($P = I_{exp} \times U_{exp}$). This isolates the performance of different EOS models (SESAME, BLF, H´ebert et al.).
   • Approach B (Experimental Conductivity): The source term is derived from the experimental electrical conductivity ($\sigma_{exp}$) combined with the circuit solver. This validates the coupling of the current solver with the hydrodynamics.
   • Approach C (Fully Numerical): A self-consistent approach where the electrical conductivity is calculated from a tabulated model (modified Lee-More/Desjarlais) based on the local thermodynamic state (density and temperature) of each hydrodynamic cell. This removes dependence on experimental input signals.

Key Results

• Electrical Validation: The proposed circuit model, including the spark-gap dynamics, showed excellent agreement with short-circuit current measurements across a wide range of generators and operating conditions (50 kA to 5 MA).
• EOS Selection: When using experimental power as a source, the BLF and H´ebert et al. EOS models reproduced experimental expansion velocities, densities, and internal energies significantly better than the SESAME model, particularly regarding the solid-to-liquid transition and the subsequent velocity plateau.
• Coupled Validation: Using experimental conductivity as a source term, the coupled simulation accurately reproduced experimental current and voltage signals. The BLF and H´ebert et al. models provided the best agreement for voltage evolution, though slight discrepancies in foil thickness calculations were noted.
• Fully Numerical Simulation: The fully self-consistent approach (Approach C) successfully reproduced the general trends of expansion velocity, current, and voltage. While minor temporal shifts were observed (attributed to the interplay between EOS-predicted temperatures and the conductivity model), the simulations effectively bracketed the experimental behavior. The study highlighted that discrepancies in the fully numerical approach likely arise from the combined influence of the EOS (affecting temperature predictions) and the conductivity model, rather than a single deficiency.

Significance and Claims
The paper claims that this numerical approach constitutes a "robust and efficient method" for designing and optimizing future Warm Dense Matter experiments using pulsed-power facilities. By successfully coupling the electrical response of the driver with the hydrodynamic evolution of the load, the framework allows for:

1. Design Optimization: Predicting the thermodynamic trajectory of targets before experimentation.
2. Model Testing: Providing a straightforward methodology to test different EOS and electrical conductivity models under relevant conditions.
3. Access to Unmeasurable Quantities: Granting access to properties like temperature and internal energy that are difficult or impossible to measure directly in these experiments.

The authors emphasize that while the fully numerical simulations show slight discrepancies compared to experimental data, the overall agreement validates the framework as a powerful tool for interpreting existing experiments and guiding future designs, without needing to invent new applications beyond the scope of the described pulsed-power foil experiments.