Excerpt from the Proceedings of the COMSOL Conference 2009 MilanStreamer Propagation in a Point-to-Plane GeometryM. Quast *,1 and N. R. Lalic 11 Gunytronic Gasflow Senoric Systems*Corresponding author: Gunytronic Gasflow Senoric Systems, Langenharter Straße 20, A-4300 St. Valentin;email@example.comAbstract: Corona discharge is used in severalapplications such as surface treatment ofpolymers, photocopying or dust removal in airconditioning. Streamer formation is undesirablefor most of these applications. Therefore, severalstudies have been dedicated to investigate theformation and propagation of streamers, whichare still not fully understood.The most suitable models to describe streamersare hydrodynamic models, which are usuallysolved using the finite 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 finite element methodusing COMSOL Multiphysics.Keywords: Hydrodynamic model, PlasmaPhysics, Corona discharge, Streamer formation.1. IntroductionWhen applying 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 toimpact ionization. Because of these collisions,photons 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 towards the cathode (i.e. away from the tip).Since these ions do not have enough energy forfurther ionization, no additional ions are createdin the region between ionization zone andcathode, which is called drift zone.At sufficiently high electric fields, thinplasma 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 towards 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 investigated gas by means of positive corona.A constant ion production rate is crucial for thesensor principle and therefore streamers areundesirable. Understanding 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 inthe past . Kulikovsky used a hydrodynamicplasma model which he solved numerically usingthe finite difference method . Here we presenta way to solve a hydrodynamic plasma model bythe finite element method using COMSOLMultiphysics.2 Governing 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) :(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 photoionization and impact ionization, respectively.More details on S ph can be found in appendix A.
L ep (L pn ) describes the recombination rate ofelectrons and positive ions (positive and negativeions). S att is the rate of attachment of electrons toneutral 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 The electric field is determined using thePoission equation:where e is the magnitude of the charge of anelectron.3 Solving the model3.1 Geometry of the modelThe modeled geometry is a so called point-toplanegeometry. In our case a metal needle witha sharp tip (radius of curvature = 2.5 x 10 -5 m) isin front of a grounded metal plate (20 mm x 1mm). The distance between needle and metalplate is 20 mm. Figure 1 shows the investigatedgeometry.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 subdomain usedfor the simulation.entered in the Subdomain Settings of thisapplication mode.Only S ph , the rate of photoionization, has to betreated in a different way, since it requires theintegral of a function of n e over the completecomputational domain (see Appendix A). Theevaluation of the original formulation of S ph isvery time consuming, therefore a simplified wayto evaluate S ph has been applied: thecomputational domain has been divided intoseveral smaller domains and S ph has beenevaluated for a single point within such asubdomain using “Subdomain IntegrationVariables”.Equation (4) is defined in the “Electrostatics”Application Mode (es-mode).The complete model was then solved step bystep using the “Solver Manager”. For the firsttime step (t = 0) an initial high electronconcentration (n e = 10 7 cm -3 ) has been assumedin a small square (50 µm x 50 µm) 3.75 mm infront of the tip. This increased electronconcentration is necessary to initiate a streamer.Such an increased electron concentration couldbe caused by e.g.: ionization by cosmic rays.Throughout the computational domain an initialconcentration 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 using theseinitial settings. This solution is stored and thefirst time step (1 ns) of the chcd-mode is solvedusing the stored electric field. The solution of thechcd-mode is now used as initial settings tocalculate the electric field for the first time step(using es-mode). The solutions of the first timestep (chcd-mode and es-mode) are now taken tosolve the second time step (stepsize = 1 ns) ofthe chcd-mode. Figure 2 shows a block diagramof the solving procedure.3.2 Numerical modelEquations (1) to (3) can be solved using the timedependent “Convection and Diffusion”Application Mode (chcd-mode) available in theChemical Engineering 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 appliedto simulate the first 18 ns of streamer formation.Simulating the streamer formation is very timeconsuming, therefore, the chcd-mode is onlysolved in a small domain in front of the needle.This domain is shown as gray bordered rectanglein figure 1.Figure 3 shows the initial situation of the model.One can see the initial high electronconcentration in a small area in front of theneedle (red rectangle). This additional charge hasa small influence on the electric field, which canbe seen on the shape of the electric field lines(red lines).Figure 3. Illustration of the electron concentration att = 0 ns. An increased electron concentration of 1 x10 7 cm -3 is assumed in a small square in front of theneedle (red square). Red lines represent electric fieldlines.The formation and propagation of the streamercan be divided into three different regimes.In the first 8 nanoseconds the electron cloudtravels towards 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 linesrepresent electric field lines.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 towards 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 linesrepresent electric field lines.After 15 ns the electron concentration showsanother characteristic behavior: the electronconcentration rapidly increases within a sickleshaped region at the side of the electron cloudfacing the needle. This increase of n n is causedby photoionization. As described in Appendix A,the rate of photoionization (S ph ) is a function ofthe total electron concentration and increaseswith increasing electron number. Between 0 and15 ns the number of electrons is too little tocause a considerable amount of photoionization.After 15 ns the number of electrons is highenough and photoionization leads to an increaseof the electron concentration. Since S ph alsodepends on the magnitude of the electric field,photoionization is stronger in the vicinity of thetip.Figure 6. a, b, and c show the electron concentrationat t = 15 ns, 17 ns and 18 ns, respectively. Red linesrepresent electric field lines.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 towards the highvoltage tip. A similar shape of the streamer headhas been reported by others .5. ConclusionsA hydrodynamic plasma model has been appliedto simulate the generation and annihilation ofthree species (electrons, positive ions andnegative ions) in an atmospheric pressure gas.Starting from an initial high electronconcentration within a small area, the formationof a streamer head could be shown.
The model was fully implemented and solvedwith COMSOL Multiphysics using predefinedapplication 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 moving 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 ofphotons as a function of the distance.In order to get S ph at the point (x 1 ,y 1 ) one has tointegrate over the complete computationaldomain:7. Appendix A: Calculation of the rate ofphotoionization S phWhen electrons with high energy collide withneutral molecules, these molecules can beionized (impact ionization), but also photons areemitted. These photons are the main source ofphotoionization described by S ph .Let’s assume a point (x 1 , y 1 ) in the computationaldomain. To evaluate S ph one has to calculate thenumber of photons at (x 1 , y 1 ).The intensity of photon emission of a source areais :where: A ≈ 0,0038 is the ratio between emittedphotons 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 photon intensity of a point source decreaseswith the square of the distance. Considering this,and the fact that photons are also absorbed, thenumber of photons at (x 1 , y 1 ) emitted from anarea at (x 2 , y 2 ) is:P(x 1 , y 1 ,x 2 , y 2 )= (6)