A transport-limited theory for the coalescence/de-coalescence plasma patterns on a plasma-liquid interface

The fundamental mechanism of self-organized anode patterns in plasma electrolysis is as puzzling as their beauty. Yet, a comprehensive understanding of pattern formation and self-organization remains elusive. In this work, we propose a transport-limited theory for an unexpected plasma pattern consisting of coalescence and de-coalescence oscillations (CDO) when operating a cathodic DC glow discharge at a moderate current of ~-26 mA. Time-resolved characterization yields a liquid conductivity- and viscosity-dependent CDO frequency of ~200 Hz, indicating a fluid transport process as the frequency is far smaller than any reaction timescales inherent to plasma processes. We therefore attribute the observed CDO dynamics to the advective transport of liquid phase cations due to capillary waves that arise from the electrostatic Maxwell pressure on the plasma-liquid interface. A Maxwell stress-modified dispersion relation of viscous capillary waves is developed. The derived analytical solution of the capillary wave frequency is comparable to the experimental data, suggesting a strong correlation between the induced capillary waves and the measured CDO frequency. Laser-assisted visualization further resolves the existence of the capillary waves only when the CDO pattern is being generated, confirming the hypothesized connection between the dynamics of the plasma and the dynamic liquid behavior. Ultimately, the finding of this work reflects the intrinsic electrostatic coupling between the liquid Debye layer and the plasma sheath at the plasma-liquid interface.