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Streamer Propagation in a Point-to-Plane Geometry - Comsol

Excerpt from the Proceed**in**gs of the COMSOL Conference 2009 Milan**Streamer** **Propagation** **in** a Po**in**t-**to**-**Plane** **Geometry**M. Quast *,1 and N. R. Lalic 11 Gunytronic Gasflow Senoric Systems*Correspond**in**g author: Gunytronic Gasflow Senoric Systems, Langenharter Straße 20, A-4300 St. Valent**in**;m.quast@gunytronic.comAbstract: Corona discharge is used **in** severalapplications such as surface treatment ofpolymers, pho**to**copy**in**g or dust removal **in** aircondition**in**g. **Streamer** formation is undesirablefor most of these applications. Therefore, severalstudies have been dedicated **to** **in**vestigate theformation and propagation of streamers, whichare still not fully unders**to**od.The most suitable models **to** describe streamersare hydrodynamic models, which are usuallysolved us**in**g the f**in**ite difference method.In this paper we present the successfulsimulation of the formation and propagation of astreamer. This simulation is based on a twodimensionalhydrodynamic plasma model whichhas been solved by the f**in**ite element methodus**in**g COMSOL Multiphysics.Keywords: Hydrodynamic model, PlasmaPhysics, Corona discharge, **Streamer** formation.1. IntroductionWhen apply**in**g a high positive voltage **to** asharp tip **in** atmospheric pressure gas, a positivecorona sets **in**. In this corona a high number ofelectrons and positive ions are created due **to**impact ionization. Because of these collisions,pho**to**ns are emitted and the region of highestionization (few millimeters from the tip) caneven be observed by the human eye. This regionis called the ionization zone. The positive ionsdrift **to**wards the cathode (i.e. away from the tip).S**in**ce these ions do not have enough energy forfurther ionization, no additional ions are created**in** the region between ionization zone andcathode, which is called drift zone.At sufficiently high electric fields, th**in**plasma channels (diameter **in** the order of 1 mm)of several millimeter length, called streamer, canoccur. Such a plasma channel is formed by asmall region of high electron concentration (i.e.:the streamer head) which travels **to**wards theanode. In its trace the streamer head leaves achannel of positive ions, which form the actualplasma channel.Gunytronic Gasflow Sensoric Systemsdevelops a gasflow sensor based on ionization ofthe **in**vestigated gas by means of positive corona.A constant ion production rate is crucial for thesensor pr**in**ciple and therefore streamers areundesirable. Understand**in**g the physics ofstreamer formation is therefore important for thedevelopment of this new sensor technology.Another paper on the physics of this sensor willbe presented as oral presentation at theCOMSOL Conference 2009.Several attempts **to** simulate streamerformation and propagation have been made **in**the past [1]. Kulikovsky used a hydrodynamicplasma model which he solved numerically us**in**gthe f**in**ite difference method [2]. Here we presenta way **to** solve a hydrodynamic plasma model bythe f**in**ite element method us**in**g COMSOLMultiphysics.2 Govern**in**g equationsTo model the formation and propagation of astreamer **in** air a hydrodynamic plasma model isused. This model describes the generation,annihilation and movement of three species(electrons, positive ions and negative ions),described by equations (1) **to** (3) [2]:(1)Here: n e , n p and n n denote the concentration ofelectrons, positive ions and negative ions,respectively. D e is the diffusion coefficient ofelectrons and µ e is the mobility of electrons. isthe electrostatic field. S ph and S i describe the rateof generation of ions and electrons due **to** pho**to**ionization and impact ionization, respectively.More details on S ph can be found **in** appendix A.

L ep (L pn ) describes the recomb**in**ation rate ofelectrons and positive ions (positive and negativeions). S att is the rate of attachment of electrons **to**neutral molecules.The mobility of ions is much lower than that ofelectrons, therefore ions are assumed **to** beimmobile.Expressions **to** evaluate the generation andannihilation rates **in** equations (1) **to** (3) can befound **in** [3]The electric field is determ**in**ed us**in**g thePoission equation:where e is the magnitude of the charge of anelectron.3 Solv**in**g the model3.1 **Geometry** of the modelThe modeled geometry is a so called po**in**t-**to**planegeometry. In our case a metal needle witha sharp tip (radius of curvature = 2.5 x 10 -5 m) is**in** front of a grounded metal plate (20 mm x 1mm). The distance between needle and metalplate is 20 mm. Figure 1 shows the **in**vestigatedgeometry.Figure 1. Model geometry of a 20 mm long needle(red) **in** front of a 20 mm broad metal plate (blackbordered rectangle at the right). The grey borderedrectangle **in** front of the needle is a subdoma**in** usedfor the simulation.entered **in** the Subdoma**in** Sett**in**gs of thisapplication mode.Only S ph , the rate of pho**to**ionization, has **to** betreated **in** a different way, s**in**ce it requires the**in**tegral of a function of n e over the completecomputational doma**in** (see Appendix A). Theevaluation of the orig**in**al formulation of S ph isvery time consum**in**g, therefore a simplified way**to** evaluate S ph has been applied: thecomputational doma**in** has been divided **in****to**several smaller doma**in**s and S ph has beenevaluated for a s**in**gle po**in**t with**in** such asubdoma**in** us**in**g “Subdoma**in** IntegrationVariables”.Equation (4) is def**in**ed **in** the “Electrostatics”Application Mode (es-mode).The complete model was then solved step bystep us**in**g the “Solver Manager”. For the firsttime step (t = 0) an **in**itial high electronconcentration (n e = 10 7 cm -3 ) has been assumed**in** a small square (50 µm x 50 µm) 3.75 mm **in**front of the tip. This **in**creased electronconcentration is necessary **to** **in**itiate a streamer.Such an **in**creased electron concentration couldbe caused by e.g.: ionization by cosmic rays.Throughout the computational doma**in** an **in**itialconcentration of n p , n e and n n (10 3 cm -3 ) isassumed. The needle has been set **to** a voltage of10 kV. Then the es-mode is solved us**in**g these**in**itial sett**in**gs. This solution is s**to**red and thefirst time step (1 ns) of the chcd-mode is solvedus**in**g the s**to**red electric field. The solution of thechcd-mode is now used as **in**itial sett**in**gs **to**calculate the electric field for the first time step(us**in**g es-mode). The solutions of the first timestep (chcd-mode and es-mode) are now taken **to**solve the second time step (stepsize = 1 ns) ofthe chcd-mode. Figure 2 shows a block diagramof the solv**in**g procedure.3.2 Numerical modelEquations (1) **to** (3) can be solved us**in**g the timedependent “Convection and Diffusion”Application Mode (chcd-mode) available **in** theChemical Eng**in**eer**in**g Module, where n p , n e andn n are the dependent variables. The generationand annihilation rates (e.g. S i , L ep ,…) can beFigure 2. The solutions of the “Convection andDiffusion” Application Mode (chcd) and “Electrostatic”Application Mode (es) of one time step areused **to** calculate the solution of chcd of the next time

step. To solve the es-mode, the solution of chcd of thesame time step is used.4. Results of the simulationThe procedure described **in** section 3.2 is applied**to** simulate the first 18 ns of streamer formation.Simulat**in**g the streamer formation is very timeconsum**in**g, therefore, the chcd-mode is onlysolved **in** a small doma**in** **in** front of the needle.This doma**in** is shown as gray bordered rectangle**in** figure 1.Figure 3 shows the **in**itial situation of the model.One can see the **in**itial high electronconcentration **in** a small area **in** front of theneedle (red rectangle). This additional charge hasa small **in**fluence on the electric field, which canbe seen on the shape of the electric field l**in**es(red l**in**es).Figure 3. Illustration of the electron concentration att = 0 ns. An **in**creased electron concentration of 1 x10 7 cm -3 is assumed **in** a small square **in** front of theneedle (red square). Red l**in**es represent electric fieldl**in**es.The formation and propagation of the streamercan be divided **in****to** three different regimes.In the first 8 nanoseconds the electron cloudtravels **to**wards the needle (due **to** the electricfield of the needle). Additionally, the electroncloud expands **in** radial direction due **to** diffusionand mutual electric repulsion of the electrons.This behavior can be seen **in** figure 4 a, b and c,which show the electron concentration at t = 3ns, 5 ns and 8 ns, respectively.Figure 4. a, b, and c show the electron concentrationat t = 3 ns, 5 ns and 8 ns, respectively. Red l**in**esrepresent electric field l**in**es.In the second regime, which occurs between 8 nsand 15 ns, one can only observe the radialexpansion of the electrons caused by mutualrepulsion of electrons. The movement of theelectron cloud **to**wards the tip cannot beobserved because the electric field caused by theelectron cloud itself is greater than the electricfield of the needle. Figures 5 a,b and c show theelectron concentration at t = 8 ns, 10 ns and 15ns, respectively.

Figure 5. a, b, and c show the electron concentrationat t = 8 ns, 10 ns and 15 ns, respectively. Red l**in**esrepresent electric field l**in**es.After 15 ns the electron concentration showsanother characteristic behavior: the electronconcentration rapidly **in**creases with**in** a sickleshaped region at the side of the electron cloudfac**in**g the needle. This **in**crease of n n is causedby pho**to**ionization. As described **in** Appendix A,the rate of pho**to**ionization (S ph ) is a function ofthe **to**tal electron concentration and **in**creaseswith **in**creas**in**g electron number. Between 0 and15 ns the number of electrons is **to**o little **to**cause a considerable amount of pho**to**ionization.After 15 ns the number of electrons is highenough and pho**to**ionization leads **to** an **in**creaseof the electron concentration. S**in**ce S ph alsodepends on the magnitude of the electric field,pho**to**ionization is stronger **in** the vic**in**ity of thetip.Figure 6. a, b, and c show the electron concentrationat t = 15 ns, 17 ns and 18 ns, respectively. Red l**in**esrepresent electric field l**in**es.This behavior can be seen **in** figure 6 a,b and cwhich show the electron concentration at t = 15ns, 17 ns and 18 ns, respectively.This region of high electron concentration is thestreamer head, which travels **to**wards the highvoltage tip. A similar shape of the streamer headhas been reported by others [4].5. ConclusionsA hydrodynamic plasma model has been applied**to** simulate the generation and annihilation ofthree species (electrons, positive ions andnegative ions) **in** an atmospheric pressure gas.Start**in**g from an **in**itial high electronconcentration with**in** a small area, the formationof a streamer head could be shown.

The model was fully implemented and solvedwith COMSOL Multiphysics us**in**g predef**in**edapplication modes.6. References1. E. E. Kunhardt, Proc.XVII Int. Conf. onPhenomena **in** Ionized Gases, p 345. J. S. Bakosand Z. Sorlei, Budapest (1985)2. A. A. Kulikovsky, Positive streamer **in** a weakfield **in** air: A mov**in**g avalanche-**to**-streamertransition, Physical Review E, 57, 7066 - 7074(1998)3. A. A. Kulikovsky, Two-dimensionalsimulation of the positive streamer **in** N 2 betweenparallel-plate electrodes, J. Phys. D.: Appl. Phys,28, 2483 - 2493(1995)4. E. M. van Veldhuizen, S. Nijdam, A. Luque,F. Brau and U. Ebert, Proc. Hakone XI, 3-Dproperties of pulsed corona streamers, OleronIsland (2008)where: r 12 is the distance between (x 1 ,y 1 ) and(x 2 ,y 2 ) and f(r 12 ) describes the absorption ofpho**to**ns as a function of the distance.In order **to** get S ph at the po**in**t (x 1 ,y 1 ) one has **to****in**tegrate over the complete computationaldoma**in**[3]:7. Appendix A: Calculation of the rate ofpho**to**ionization S phWhen electrons with high energy collide withneutral molecules, these molecules can beionized (impact ionization), but also pho**to**ns areemitted. These pho**to**ns are the ma**in** source ofpho**to**ionization described by S ph .Let’s assume a po**in**t (x 1 , y 1 ) **in** the computationaldoma**in**. To evaluate S ph one has **to** calculate thenumber of pho**to**ns at (x 1 , y 1 ).The **in**tensity of pho**to**n emission of a source areais [3]:where: A ≈ 0,0038 is the ratio between emittedpho**to**ns and the number of impact ionizationevents. is the Townsend ionization coefficient(cm -1 ), v d is the electron drift velocity (cm/s) andn e is the electron concentration (cm -3 ).The pho**to**n **in**tensity of a po**in**t source decreaseswith the square of the distance. Consider**in**g this,and the fact that pho**to**ns are also absorbed, thenumber of pho**to**ns at (x 1 , y 1 ) emitted from anarea at (x 2 , y 2 ) is[3]:P(x 1 , y 1 ,x 2 , y 2 )= (6)