Laboratoire de Physique des Plasmas Global Model of Micro-Hollow Cathode Discharges Pascal CHABERT Laboratoire de Physique des Plasmas, Ecole Polytechnique, Palaiseau FRANCE Claudia LAZZARONI Laboratoire des Sciences des Procédés et des Matériaux, Université Paris 13, Villetaneuse FRANCE

Acknowledgments Micro-Hollow Cathode Discharges (MHCD’s): -Antoine Rousseau -Nader Sadeghi

100µm Microdischarges History of microplasmas : plasma displays (since 60s..) –Microhollow cathode type (Schoenbach 1996) DC source –Microarray (Eden 2001) AC source –Microneedle (Stoffels 2002) rf source –Micro APPJ (Schulz-von der Gathen 2008) rf source Applications in various fields: Surface treatment/ Light sources (excimer)/ Biomedicine

Global models Volume-averaged (0D) model: densities and temperatures are uniform in space Particle balance: Electron power balance:

Particle balance: low pressure Particle losses at the walls. The term P  contains volume and wall losses. For the electron particle balance at low pressure, wall losses dominate and in one dimension: A plasma transport theory is required to relate the space-averaged electron density to the flux at the wall

Particle balance: issues in µdischarges In microdischarges, ionization is often non uniform. Classical low-pressure transport theories do not apply Fortunately in some instances volume losses dominate (recombination) However, evaluation of wall losses is a critical point for high-pressure discharges modeling

Other issues Properly evaluate the reaction rates: ―Electron energy distribution function is unknown ― In microdischarges, T e is often strongly non uniform in space, and may vary in time… Properly evaluate the electron power absorption: ―Distinguish between electron and ion power (and other power losses in the system) ―Equivalent circuit analysis is often useful. I-V characteristic of a device is a good starting point

So why one would use global models? They are easily solved, sometimes analytically. Therefore they provide scaling laws and general understanding of the physics involved They can be coupled to electrical engineering models to give a full description of a specific design They allow very complex chemistries to be studied extensively. Numerical solutions of global models with complex chemistries only take seconds

Laboratoire de Physique des Plasmas Micro Hollow cathode Discharges (MHCD’s) DC Gas : Argon Pressure : Torr Excitation : DC

I-V Characteristics: mode jumps self-pulsing Aubert et al., PSST 16 (2007) 23–32 confined inside the micro-hole normal: cathode expansion

Equivalent electrical circuit P. Chabert, C. Lazzaroni and A. Rousseau, J. Appl. Phys. 108 (2010) C. Lazzaroni and P. Chabert, Plasma Sources Sci. Technol. (2011) Hsu and Graves, J. Phys.D 36 (2003) 2898 Aubert et al., PSST 16 (2007) 23

Non-linear plasma resistance normal: cathode expansion abnormal: confined in the hole R p controls the physics of the transition between the two stable regimes A 3 =0 R p = switch

Stable and unstable regimes Stable regimes Self-pulsing regime Normal regime Abnormal regime 2 equilibrium dI d /dt=0 dV/dt=0 equilibrium

Electrical signals/ Phase-space diagram

Electrical model of the MHCD Total absorbed power simply: What is the physical origin of R p ? What fraction of the total power goes into electrons?

Decomposition in two main regions The resistance of the positive column is small Makasheva et al, 28th ICPIG(2007) 10

Structure in the cathodic region

Calculation of sheath thickness (d) One-dimensional cylindrical geometry: Ions/electrons fluxes: r 0 R d sheath bulk Cathode Sheath

Sheath thickness vs pressure -decrease of the sheath size with the increase of pressure -maxima of both emission lines located after the sheath edge -same trends for the evolution of d than that of the maxima of the emission line -the sheath edge coincide with the maxima of the ionic line C. Lazzaroni, P. Chabert, A. Rousseau and N. Sadeghi, J. Phys. D: Appl. Phys. 43 (2010)

Abnormal and self-pulsing regime P and V high enough

Electrons: Same for ions with: Power absorbed in the cathode sheath

Power absorbed by electrons Fraction of power absorbed by electrons: Power absorbed in the volume under consideration:

Power balance Power loss at the wall is negligible

Particle balance

Results in the self-pulsing regime Experiments Global model - with excited species/2 - without excited species*3 P=150 T -n e (with exc states) ≈ 6 n e (without exc states)  Importance of multi-step ionization -n e (without exc states) = n e (exp)/3.7 -n e (with exc states) = 1.7 n e (exp) Model with excited states in better agreement with experiment

Results in the self-pulsing regime This has been observed experimentally: Aubert et al., Proceedings of ESCAMPIG, Lecce, Ed. M. Cacciatore, S. D. Benedictis, P. F. Ambrico, M. Rutigliano, ISBN X, Vol. 30G (2006). P=150 Torr Strong and short T e peak  e-impact Multi-step ionization Recombination

Electron temperature dynamics C. Lazzaroni and P. Chabert J. Appl. Phys. 111 (2012)

Conclusions (1) Global models are very useful to understand the general behavior of plasma discharges They provide analytic solutions and scaling laws and allow fast numerical solutions of complex chemistry However, they rely on many assumptions and therefore need to be benchmarked by experiments or more sophisticated numerical simulations

Conclusions (2) The need for many simplifications and assumptions forces one to propose physical explanations, sometimes at the expense of being “rigorous” Global models do not explain the detailed and microscopic phenomena taking place in plasma discharges. they offer a representation of reality that becomes more accurate when more realistic fundamental assumptions are made