Numerical validation of the d/R scaling law based on the extended paschen's law via 2D fluid and particle-in-cell simulations

To understand plasma discharge characteristics and to design and operate high-performance plasma equipment effectively, an accurate understanding of discharge scaling laws is essential. Traditionally, Paschen's law describes breakdown voltage using the product of pressure and electrode gap distance (pd). Recently, the extended Paschen law has been proposed, incorporating the frequency-to-pressure ratio (f/p) and the electrode gap-to-electrode radius ratio (d/R), demonstrating improved predictive capabilities across a wider range of pressures, frequencies, and geometries. While the f/p scaling has been validated using both fluid and particle-in-cell (PIC) simulations, the d/R scaling has remained unverified in two-dimensional geometries due to high computational costs and limited geometric flexibility in experimental setups.

In this study, we numerically validate the d/R scaling law derived from the extended Paschen formulation through both 2D fluid and PIC simulations. Notably, both approaches independently exhibit consistent scaling trends, supporting the validity of the f/p and d/R scaling across fundamentally different physical models. This cross-consistency strengthens the theoretical foundation of the extended Paschen law.

Our findings provide a reliable and predictive framework for analyzing RF-driven plasma behavior and offer practical guidance for optimizing discharge parameters, refining electrode geometries, and improving high-frequency plasma source design.