Plasma–solid interaction represents a major concern in many applications such as power-interruption and plasma–metal processing. Characterized by high-current density and voltage drop, the arc roots dissipate intensive heat to electrode vaporization, which participates in the ionization and, thereby, significantly alters the plasma properties and gas dynamics. Most of the arc root models feature approaches based on surface temperature or (temperature dependent) current density. Due to the complexity of conjugated heat transfer across arc roots involving three-phase interactions of plasma with liquid spots and solid electrodes, accurately determining the surface temperature distribution is extremely computationally demanding. Hence, models hitherto fail to quantitatively estimate neither the molten spot size nor the total amount of vaporization. In this work, we propose an arc root model featuring a hemispherical structure that correlates the molten spot size with the heat partition between conduction and vaporization to estimate the energy dissipation at arc roots and, thus, to trace the vaporization rate. Following local partial pressure adjusted Langmuir vaporization, we deduce an analytical solution of molten spot size for quasi-steady-state, which compares favorably with experiments. Specifically, the vaporization dominates over conduction for large molten spots as in the case of high-current arcs. However, for low-current arcs, the vaporization heat is trivial compared with conduction. Furthermore, we integrate this arc root model into a study case of arc plasma based on the magnetohydrodynamics method. The simulated arc voltage and arc displacement match with the experiment. This model is expected to find broad applications in power interruption and plasma etching.

Pattern formation in plasma-solid interaction represents a great research challenge in many applications from plasma etching to surface treatment, whereby plasma attachments on electrodes (arc roots) are constricted to self-organized spots. Gliding arc discharge in a Jacob's Ladder, exhibiting hopping dynamics, provides a unique window to probe the nature of pattern formation in plasma-surface interactions. In this work, we find the existence of negative differential resistance (NDR) across the sheath is responsible for the observed hopping pattern. Due to NDR, the current density and potential drop behave as activator and inhibitor, the dynamic interactions of which govern the surface current density re-distribution and the formation of structured spots. In gliding arc discharges, new arc-roots can form separately in front of the existing root(s) which happens periodically to constitute the stepwise hopping. From the instability phase-diagram analysis, the phenomenon that arc attachments tend to constrict itself spontaneously in the NDR regime is well explained. Furthermore, we demonstrate via a comprehensive magnetohydrodynamics (MHD) computation that the existence of a sheath NDR can successfully reproduce the arc hopping as observed in experiments. Therefore, this work uncovers the essential role of sheath NDR in the plasma-solid surface pattern formation and opens up a hitherto unexplored area of research for manipulating the plasma-solid interactions.

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