Non-linear diffusion and inhomogeneity of the magnetic field in single-turn coils: Insights from 3D multiphysics modeling

This paper utilizes fully 3D multiphysics finite element modeling to demonstrate that the highly inhomogeneous magnetic fields generated in destructive single-turn coils are caused by the nonlinear diffusion of electric current, temperature, and magnetic fields driven by the skin effect, rapid heating, and coil deformation.

The Gist

Imagine trying to create a magnetic field so incredibly strong (over 100 times stronger than a MRI machine) that it would crush a normal magnet. To do this, scientists use a "single-turn coil." Think of this coil not as a sturdy spring, but as a single, thick copper ring. When you blast a massive amount of electricity through it for a tiny fraction of a second (microseconds), it creates a super-powerful magnetic field. But there's a catch: the force is so intense that the copper ring literally explodes. It's a "one-shot" experiment where the machine destroys itself to create the field.

The problem is that inside this exploding ring, things are chaotic. The electricity, heat, and magnetic field don't spread out evenly. They are messy and uneven, which makes it hard to know exactly what the magnetic field looks like at any specific spot inside the ring.

The "Traffic Jam" of Electricity
The researchers used a powerful 3D computer simulation to watch what happens inside this ring in slow motion. They discovered that the electricity behaves like a crowd of people rushing through a hallway, but with a twist:

1. The Skin Effect (The Edge Rush): At the very beginning (0.3 microseconds), the electricity doesn't want to go through the middle of the copper. It's like a crowd that only wants to hug the walls. Because of a physics rule called the "skin effect," the current rushes to the very edges of the copper ring's inner surface.
2. The Heat Trap: Because all that electricity is crammed into the edges, those edges get incredibly hot, very fast. It's like friction heating up a brake pad.
3. The Migration (Moving to the Middle): As the edges get hotter, the copper there becomes "grumpier" to electricity (its resistance goes up). The electricity, looking for an easier path, starts to drift away from the hot edges and moves toward the cooler middle of the copper ring.
4. The Explosion: Eventually, the magnetic pressure becomes so strong (like a giant invisible hand squeezing the ring) that the copper starts to deform and the ring explodes. However, the simulation showed that the electricity had already moved to the middle before the ring actually started to physically blow apart.

Why the Magnetic Field is "Lumpy"
Because the electricity is constantly moving around—first hugging the edges, then drifting to the middle, then spreading deep into the copper—the magnetic field it creates is also constantly changing shape.

• Early on: The field is relatively smooth, kind of like a calm pond, because the electricity is neatly hugging the edges (similar to how a perfect ring of magnets creates a smooth field).
• Later: As the electricity gets messy and moves around, the magnetic field becomes "lumpy" and uneven. Some spots have a stronger field, some have a weaker one, and the peak of the field might even shift slightly away from the exact center.

The Big Takeaway
The paper claims that by using a full 3D computer model (instead of assuming the ring is perfectly symmetrical), they finally saw this "non-linear diffusion." They proved that the magnetic field isn't static; it's a dynamic, shifting landscape caused by the electricity running away from the heat it creates.

This is crucial because scientists need to know exactly how "lumpy" the field is to interpret their experiments correctly. If they think the field is smooth but it's actually bumpy, they might misread the data about the materials they are studying. The simulation acts like a high-speed camera, showing us the invisible dance of electricity and heat that happens right before the coil blows up.

Technical Summary

Technical Summary: Non-linear Diffusion and Inhomogeneity in Single-Turn Coils via 3D Multiphysics Modeling

Problem Statement
The single-turn coil (STC) method is a primary technique for generating destructive pulsed magnetic fields exceeding 100 T, with pulse durations in the microsecond range. While this method has enabled the discovery of novel phase transitions in condensed matter (e.g., solid O₂, CdCr₂O₄, VO₂) at fields up to 1000 T, it is inherently limited by the coil's explosion due to immense magnetic pressure (4 GPa at 100 T). A critical challenge in utilizing these fields is the highly inhomogeneous distribution of the magnetic field in both time and space. This inhomogeneity arises from complex dynamic phenomena, including the skin effect, rapid temperature rise, and coil deformation. Previous numerical studies have largely relied on models assuming cylindrical symmetry, which fails to account for the "feed gap" (neck part) of the coil—a geometric feature essential for understanding the true inhomogeneity of the field.

Methodology
To address the limitations of symmetric models, the authors employed a fully 3D finite element method (FEM) multiphysics simulation using a broken cylindrical symmetry model of a single-turn coil.

• Simulation Setup: The study utilized Ansys LS-DYNA (Ver. R13.1.1) to perform coupled mechanical, electromagnetic, and thermal calculations. The model accounted for transient changes in material properties, including resistivity, conductivity, and mechanical deformation.
• Geometry and Mesh: A 3D model of a $\phi$4 mm copper coil was created using FUSION360. The mesh was refined near the inner surface to accurately capture the skin effect.
• Input Parameters: The simulation was driven by a 3 $\mu$s current pulse peaking at 750 kA (scaled from experimental data used to generate 120 T fields). The study analyzed current waveforms scaled to 75, 375, 750, and 1125 kA.
• Validation: The model was validated against experimental observations from the PINK-03 system at the University of Electro-Communications, specifically comparing the relationship between maximum current and peak magnetic field, as well as the time-to-peak magnetic field.

Key Results
The 3D simulation revealed a highly non-linear diffusion process of electric current, temperature, and magnetic fields that evolves dynamically during the pulse:

1. Non-linear Current Diffusion:

   • Early Stage (0.3 $\mu$s, ~50 T): Due to the skin effect, electric current density is concentrated at the two edges of the inner surface of the coil. This distribution resembles a Helmholtz coil geometry, resulting in a relatively homogeneous magnetic field.
   • Mid-to-Late Stage (1.2–1.5 $\mu$s, ~120 T): As Joule heating increases the temperature at the edges, the electrical resistivity of the copper rises sharply. This causes the current density to diffuse away from the edges and concentrate in the middle region of the inner surface.
   • Late Stage (>1.8 $\mu$s): The current density further diffuses from the surface into the interior of the conductor.

2. Thermal and Mechanical Dynamics:

   • The rise in temperature at the coil edges leads to a significant drop in electrical conductivity, driving the current redistribution.
   • Pressure exceeds 2 GPa at the edges by 0.8 $\mu$s, well before significant mechanical displacement occurs. The coil deformation (expansion) becomes significant only after 2 $\mu$s, ensuring that the 100 T field generation and associated experiments occur before the coil is destroyed.
   • The simulation confirmed the melting of copper at the inner surface and edges, indicated by discontinuous jumps in resistivity.

3. Magnetic Field Inhomogeneity:

   • The magnetic field distribution is not static; it evolves from a relatively smooth profile in the first half of the pulse (<1.5 $\mu$s) to a highly inhomogeneous profile in the second half.
   • The field exhibits a "saddle point" distribution relative to the coil center. Notably, the magnetic field strength near the inner surface of the coil can exceed that at the geometric center.
   • The time required to reach the maximum magnetic field is slower in the neck region compared to the coil center, a finding consistent with previous experimental reports but now explained by the 3D current distribution.

Significance and Claims
The paper claims that the primary source of time- and position-dependent magnetic field inhomogeneity in single-turn coils is the non-linear diffusion of electric current driven by temperature-dependent resistivity and the specific geometry of the feed gap.

• Beyond Symmetry: By breaking cylindrical symmetry, the study demonstrates that the feed gap and the resulting current redistribution are critical factors previously overlooked in 2D or symmetric 3D models.
• Mechanism of Inhomogeneity: The authors establish that the inhomogeneity is not merely a result of mechanical deformation (which occurs too late to affect the peak field generation significantly) but is primarily driven by the thermal-electrical coupling that shifts the current path from the edges to the center during the pulse.
• Utility: The calculated results provide a necessary framework for interpreting experimental data in the >100 T regime, where the magnetic field environment is highly dynamic and non-uniform. The authors suggest that these insights are essential for the further development of destructive pulsed magnetic field methods and the design of experiments requiring precise knowledge of the local magnetic field environment.