IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 1, FEBRUARY 1998 7

Numerical Simulation of Axisymmetric Anode SpotFormation in Glow Discharge at Elevated Pressure

Raphael Sh. Islamov and Echtiram N. Gulamov

Abstract—A two-dimensional nonstationary fluid model is usedto simulate dc elevated pressure glow discharges to describe thegrowth of the instability near an anode inside a cylindricallysymmetric parallel plane surface geometry. This model is basedon a fluid description of electron and ion transport coupledwith Poisson’s equation. Numerical results for the case of aglow discharge in nitrogen gas are presented. The transition ofa radially uniform glow discharge into a state with spots orring structures is obtained from the two-dimensional simulationswithout any special initiating perturbation. The influence ofdiffusion, full current through spot, and volume recombinationon the normal current density effect in the near-anode regionof the glow discharge has been investigated. The applicability ofthe similarity principle to the anode spot development processesis analyzed. Prerequisites to the existence of the complex shapecurrent structure in the form of a ring with a central spot arediscussed.

Index Terms—Current structures, glow discharges, numericalsimulation, self-organization.

I. INTRODUCTION

T HE uniform glow discharge desired in a gas laser canbe affected by a variety of instabilities with varying time

scales and growth rates. The most harmful instability leadsto the electrical breakdown (arcing) of the lasing gas whichresults when the excitation energy exceeds a critical value. Inmost cases, the contraction of self-sustained quasi-stationaryglow discharges in elevated pressure gases is preceded by theformation of anode spots [1]. Anode spots, often symmetricallyarranged and very sharply defined, were first described byLehmann [2] who found their occurrence in an air glowdischarge, and were reported by other workers for several othergases and conditions (see, e.g., [3]–[7]).

The most credible cause of the plasma inhomogeneitiesformation near anode is the negativity of volt-ampere char-acteristics (VAC) of the anode layer [8], which is in con-formity with the contraction dynamic and the weak sen-sitivity of instability evolution to discharge conditions andgas composition. A phenomenological model based on anS-shaped VAC of the electrode layer is developed in [7]to explain single spots or regular patterns of spots at theelectrode of gas discharges. A more physical model of sta-tionary current-density perturbations in near-electrode plasmaregion is proposed in [9]. Numerically within the frame

Manuscript received April 1, 1996; revised October 1, 1997.The authors are with the Research Center for Technological Lasers of RAS,

Shature, Moscow Region, 140700, Russia.Publisher Item Identifier S 0093-3813(98)01471-4.

of pure electrodynamic phenomena, the possibility of aninstability growth in anode region has been shown in [10].The formation of stationary anode spots in elevated pressurenitrogen glow has been demonstrated by numerical simulationsfor a wide range of pressure and currents [11], [12]. Severalphenomenological aspects of uniform glows and spots are stilluncertain, however, as are many details of their theory. Amongthese phenomena are the prerequisites to the formation ofcurrent spots, generally, and complex sharp current structures,specifically. In addition, the class of equations which wehave considered is of substantial theoretical interest as one ofthe physically clear models giving rise to solitary dissipativestructures.

We present here results from two-dimensional numericalmodeling of the glow discharge in nitrogen gas based onthe method described in [11]. The set of fluid equations forelectrons and ions coupled to Poisson’s equation is solvedinside a cylindrically symmetric parallel plane surface geome-try. In Section II, the basic equations and boundary conditionsare described and the applicability of the model is discussed.Simulation results and discussions are presented in Section III.Section IV contains the concluding remarks.

II. BASIC EQUATIONS

Most of the two-dimensional simulations of direct currentglow discharges reported so far are based fully [10]–[17]or partly [18], [19] on the continuum or fluid drift-diffusionapproximation. Recent comparisons of fluid with kinetic par-ticle [20] have shown that the fluid approximation providessurprisingly reasonable results for a wide range of appli-cations and regimes even down to pressures at which thisapproximation would be considered highly suspect. Under thenitrogen pressure of several tens torr, collision mean free pathis about a 0.001 cm. The mean free path is much less than thecharacteristic discharge dimensions of this work. Therefore,the continuum modeling can be adopted to describe the plasmabehaviors in a modeling of glow discharge.

The simplest set of equations containing the basic physicsnecessary for spot formation are the continuity equations forelectrons and positive ions coupled with Poisson’s equationfor the electric field, along with various constituent relationsdischarge plasma coefficients

div

(1)

0093–3813/98$10.00 1998 IEEE

8 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 1, FEBRUARY 1998

Fig. 1. Model geometry. 1: anode, 2: cathode, 3: computation volume.

div

(2)

(3)

where and are the particle density,mobility, and diffusivity for the electrons and positive ions,respectively, is the frequency of direct ionization, isthe electron-ion recombination coefficient, is the electricpotential, is the current density, is the (unsigned) electroniccharge, and the permittivity of free space.

The development of the instability on the cathode occursover typical times of less than 1s and, in the calculations inthe axisymmetric geometry, results in a disruption of the radialuniformity of the entire discharge (including the anode region).That does not make it possible to study the developmentof the slower anode instability (typical times on the orderof 1 ms [5]) from the initially radially uniform state of thedischarge. That is why the reference region was chosen tobe between the anode and the positive column (see Fig. 1).Excluding the cathode region from consideration makes itpossible to obviate the influence of a rapidly forming cathodespot and examine an array of problems that are not directlyrelated to the conditions on the cathode, but are importantin a number of technical applications characterized by anelongated, volumetrically uniform, positive column (such as ingas-discharge chambers with a thoroughly sectioned cathodeand a flat, solid anode [1]). That makes it possible to alsoavoid a number of difficulties related to the applicability ofthe model (1)–(3) for describing regions of cathodic drop inpotential, negative glow, and Faraday dark space [18]. Wenote that such a formulation of the problem, i.e., betweenthe anode and the positive column, was used earlier in thework reported in [10], whereas an alternative formulation, i.e.,between the positive column and the cathode, was used in thework reported in [16].

The local field approximation for electrons is used toobtain the transport coefficients (ionization rate, mobilities,diffusivities). It is assumed that the electron energy distributionis in equilibrium with the electric field, and the transportcoefficients which are involved in these equations are supposedto depend on the local coordinates only through the localelectric field Due to the moderate field andrelatively small field variations (of order 2) in the regionunder consideration, the energy gained by the electrons fromthe field is rather locally balanced by the energy lost throughcollisions. For these conditions (small anisotropy, diffusion-

like electron motion), the energy relaxation length can beestimated as where and are the mean freepaths for elastic and inelastic collisions, respectively. Sincethe characteristic quasi-neutrality breakdown dimension in theanode region significantly exceeds the electron energyrelaxation length the anode region and positive columnare long enough for electrons to come in an equilibriumstate. Besides, the relaxation time for achieving a steadyelectron energy distribution function is shot compared to thecharacteristic time of spot development.

The boundary conditions have been discussed in more detailpreviously [11]; thus only the mathematical problem will beconsidered now. The following conditions have been taken:

for the anode constfor the positive column

for the axis of symmetry, and outer surface In this case, is a unit

vector with the external normal for the discussed surfaces,is the average thermal velocity of

electrons, and is the probability of the electron reflectionfrom the electrode surface. The boundary conditions on theouter surface maintain the conditions for the electronand ion impermeability. Thus the calculation region may beinterpreted as a periodic pattern cell. The boundary conditionfor the electron flux in the positive column region is retainedtypically for the cathode form (with the effective secondaryemission coefficient but it satisfies the condition

whenDue to the severe nonlinearity and strong coupling of

(1)–(3), the treatment of these equations is a difficult numericalproblem. In the present work, the numerical solution of theset of equations (1)–(3) uses the time-implicit finite differ-ence method of second-order spatial coordinate approximationand first-order time approximation. Taking into account theabsence of any smoothing of calculations, the method ofcalculation essentially eliminates the effect of the numericaldiffusion. Using the integro-interpolation method results in theconservatism property of the net current. The discussed resultsconform to axially symmetrical geometry with uniform gridspacing. Considering (3) for charge density determination

the potential is estimated from the nonstationaryequation of the current continuity, being the result of (1) and(2)

div (4)

At each time step, the implicit difference equations for thesystem of differential equations (2) and (4) can be written asmatrix operators

(5)

(6)

The nonlinear and strong coupled equations (5) and (6) aresolved for and respectively, one after another (which isrepeated if required), with a unified implicit two-level iterationmethod with the optimal choice of the iteration parametersequence [21, ch. XIII]. The Laplacian operator option as a

ISLAMOV AND GULAMOV: NUMERICAL SIMULATION OF AXISYMMETRIC ANODE SPOT FORMATION 9

resolving process enables us to use the total cyclic reductionmethod [21, ch. III] (with similar properties as a fast Fouriertransform). The cyclic reduction method requires a uniformgrid spacing in the -direction. Therefore, the gap distance wasdivided into uniformly spaced grid points (typically

or 64, but or 128 was also examined). Thedirection was divided up into uniformly spaced

grid points. Consequently, this unified solver for the implicitequations costs arithmetic operation countper iteration. The solver performs well and allows us to solveeither (5) or (6) with a minimal (one or more) amount ofiteration to achieve a desired low convergence rate. At eachtime step, it may be that a few extra consecutive iterations arerequired because of strong coupling of (5) and (6). Except ina few particular time points such as the initiating moment orsome moments at which a solution is reorganized, the timestep was limited by the equation

This condition is a prerequisite to the positive definitenessof operators and , and thus it is sufficientfor convergence of the iteration process for either equationsindividually. The order of magnitude of the time step was

s at the pressure of 40 torr inAdditionally, the stability and computational efficiency are

somewhat improved by using implicit stabilizing correction[22, ch. V], [23] for the potential on account of the term

in (4). In the case being considered, the same solveris usable and this problem has an exact solution (i.e., a singleiteration is a necessary) for the corrected

The simulations begin with an initial condition requiring aradially uniform distribution of both electric potential and iondensity. The gap voltage is preassigned to meet a givenvalue of the net current , and it may then be adjusted ifnecessary.

Each 1000 s of the process time required a CPU time of0.5 PC AT-486DX2 hour at 40 torr gas pressure.

III. RESULTS AND DISCUSSION

Anode spots have a complex, poorly understood morphol-ogy, even for the simple plane-parallel discharge geometry. Wetherefore restrict ourselves to a limited set of conditions, ig-noring heating and plasma chemistry. The discharge geometryin all cases is plane and parallel, and cylindrical symmetry isimposed (Fig. 1). In the results presented below,has beenused as a working gas. The same transport parameters wereset in computations for a nitrogen plasma [11]. The ionizationterm is a very strong function of the value of , and thishas played a dominant role in the case under consideration.The dependencies of and on are weak andhave little effect on the principal simulation results in theconsidered conditions ( varies relatively small). Therefore,the electron mobility and diffusion coefficient are assumedto be field independent and their ratio is chosen as aparameter. This work is similar to the modeling done in ourearlier work [11], but any special initiating perturbation forthe spot development is not in use.

Fig. 2. The spatial distribution of the anode current density at various timesfor the electron characteristic energyDe=�e = 2 eV, and the electron-ionrecombination coefficient� = 2 � 10

�7 cm3/s under nitrogen pressurep = 40 torr and full currentI = 1 A.

We begin our study by considering spot development from aspontaneous perturbation which occurs because the computercannot give answers to an infinite number of decimal places,and then freely grow within a tolerance of the iterationprocedures.

A. Anode Spot Development

The development of an anode spot is a very nonlinearphenomenon. We first discuss the overall evolution. A moredetailed investigation of the current structures then follows.For the conditions considered, a current spot is developed fromthe near surface region of the anode layer on a millisecond timescale. For most of the data presented here, the net current andgas pressure were held fixed at 1 A and 40 torr, respectively.

We carried out several simulations in which differing initi-ating perturbations can occur. Fig. 2 shows the spatial dis-tribution of the current density in the glow discharge atvarious times. The ionization-recombination balance (e.g.,

) is achieved at the positive column axis. To ourknowledge, no spatially resolved measurements of dynamicsof spot development have been performed in nitrogen glow.The dynamics of the change in the current density at the anodehave been experimentally investigated in dry air [5]. It wasfound that the following phases are characteristic in dry air:postbreakdown stationary uniform distribution, contraction,and incomplete decontraction. The first and second phasesin 40 torr dry air glow last for about 3000 and 1200srespectively. These values can be correlated to Fig. 2 for

10 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 1, FEBRUARY 1998

Fig. 3. The spatial distribution of the anode current density at various timesfor the electron-ion recombination coefficient� = 0:75�10

�7 cm3/s (sameother conditions as Fig. 2).

nitrogen glow: in simulation also there is an initial delay beforethe spot starts to development, and a rapid growth of theperturbation in the amplitude.

The initiating perturbation is rather accidental. As was tobe expected, the other long-wavelength initiating perturbationcan proceed as easily (Fig. 3). In theory [9], ring domains arealso forming which are positioned either inside the circle orat its outer boundary. According to our simulations, the glowdischarge is stable against any local (shot-wavelength) pertur-bance. But the presence of some long-wavelength initiatingperturbations (as, e.g., nonuniformity of the anode potentialin [11]) drastically reduces the spot formation time. It isinteresting to note that the ring current structure (formed at

s in Fig. 3) is unstable. Nevertheless, we haverevealed that a complex shape current structure in the formof a circular spot and a ring spot concentric with it is stableunder the same conditions as for Fig. 2 [Fig. 4(a)]. The contourplot of the charge density of this current structure is shown inFig. 4(b). The influence of the grid refinement on distributionsof the current density and the anode potential dropis shownin Fig. 5. There, the is given by the formula

where The characteristic quasi-neutralitybreakdown dimension in the anode region

(a)

(b)

Fig. 4. The complex shape anode current structure: (a) stationary spatialdistribution of the anode current density and (b) contours of constant chargedensity. The contour labels are scaled by10

10 e=cm3: The minimal values ofcharge density (not shown) occur atz = 0 on the center of central spot and onthe ring and equals to�5.43� 1010 and�4.58� 1010 e=cm3; respectively.All conditions were the same as in Fig. 2.

closely resembles in shape. Note that the influence of thegrid refinement on the discharge is not negligible; however,the complex shape current structure is stable in both cases.The net currents through the central spot and the ring spot arealso about the same in both cases.

It is noted that with the constraint const, the presentmodels obey similarity laws [24]. Therefore, it is not neces-sary to calculate for other discharge conditions if similarityparameters are the same. To check the similarity character,calculations were also made for the pressure of 25 torr, thenet current 1 A, and the ion-electron recombination coefficient3.2 10 cm s for cm. The profile of the numberdensities of ions and electrons divided by as a function ofthe spatial position multiplied by agreed exactly with thatcalculated at the pressure of 40 torr, the same net current 1 A,and the ion-electron recombination coefficient 210 cm sfor cm as expected. Other quantities and qualities,including properties of stability and equality of the net currents

ISLAMOV AND GULAMOV: NUMERICAL SIMULATION OF AXISYMMETRIC ANODE SPOT FORMATION 11

(a)

(b)

Fig. 5. Radial profiles of the anode current density (a) and the anodepotential drop (b) for two grids (solid curve:M � N = 64 � 38; brokencurve:M �N = 32� 19): All other conditions were the same as in Fig. 4.

through the circular spot and the ring spot [see Fig. 4(a)], alsoagreed.

This complex structure is stable for a sufficient narrow rangeof radii where the boundary conditions are placed. Fig. 6shows the dynamic of decay of the ring current structure afterchanging the value of from 1.65–1.82 cm. Under theseconditions, the structure is unstable, and in the end the samespot is formed as in Fig. 2.

To determine the effect of the ion-electron recombinationon discharge stability, we made simulations at the pressure of40 torr in which the and varied, but all other parameterswere held fixed. The numerical analysis has revealed theexistence of some minimal value cm sthat allows for the occurrence of ring structure with centralspot. When the complex structure decays at anyvalues of It seems likely that in the case of ,multiple stationary solutions for glow problems may beexpected on physical grounds.

B. Normal and Subnormal Anode Spot

In the next stage of the investigation, we scan the responseby altering the diffusivity coefficient. We retain the same setof parameters, but is allowed to vary.

Fig. 7 shows the radial profiles of the current densityfor various values of electron characteristic energy. As the

is reduced, the spot tends to become a more abrupt

Fig. 6. Decay of the complex shape anode current structure after changingthe place of the outer boundary condition fromR = 1:65 to 1.82 cm. Solidcurve: t = 0; dotted curve:t = 800 �s; broken curve:t � 1400 �s. Allother conditions were the same as in Fig. 4.

profile, but the magnitude remains intact. To make an estimateof the thickness of the peripheral region, we consider theradial motions of charge particles. The equation for the radialbalance of ions can be obtained by combining (1) and (2)and integrating over the volume within the narrow radial layer

from to between anode andpositive column

where is the ambipolar diffusion coefficient and is theradial component of the Laplacian. As the characteristic quasi-neutrality breakdown dimension in the anode regionwe obtain

(7)

where

Consequently, thickness of peripheral regionmay be esti-mated by the formula

(8)

where is the position of the peripheral region. The simplestestimation of by the expression

(9)

reflects the dependencies on diffusivity and recombinationcoefficients, but the value is less than expected from oursimulations and experiment [6] by a factor of 5. In the case

12 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 26, NO. 1, FEBRUARY 1998

Fig. 7. Radial profiles of the current density at the anode for the electron-ionrecombination coefficient� = 0:75 � 10�7 cm3/s under nitrogen pressurep = 25 torr and full currentI = 1 A. Broken curve:De=�e = 1 eV;solid curve:De=�e = 2 eV; dotted curve:De=�e = 4 eV. Comparisonof the computational and analytical values oflog[m(ro)=m(r)] in periphery(r � ro = 1:4 cm) is also shown (in arbitrary unit) in the case ofDe=�e = 2

eV. +: numerical data,z = L: Straight line: formula (10).

studied here, and the asymptotic solutionof (7) at has the form

const (10)

withIt seems likely that the length is a refined value of

known as an ion avalanche spreadsduring its motion along the gap (under conditions whereionization is of importance). The computational radial profileof density in periphery is in reasonably good agreementwith the analytic formula (10), that is illustrated by Fig. 7,and it testifies that the contribution of the numerical diffusionis small.

The origin of the critical value that allows for the occur-rence of ring structure with central spot may be qualitativelyexplained as follows. Since with decreasing recombination thethickness of the peripheral region increasesin according with (9)] and compares with the annular regionbetween central spot and ring, one could expect in such asituation that the complex current structure is unstable. Theion density spreading during its motion along the gap isevaluated at a distance cm and can be neglected.It was assumed in the above analyzes that the spot diameter ismuch larger than the thickness of the near-anode plasma layer.For comparison purposes, the characteristic quasi-neutralitybreakdown dimension is of the order of 0.01 cm in theanode region.

Fig. 8 shows the current dependence of the stationary cur-rent density in the spot center for two values of bothand The current dependence is understood by reference toFig. 7 and (8). As long as the spot radius is smaller in compari-son to the current density increases logarithmically with anincrease in the discharge current. Once a nearly flat current

Fig. 8. Current densityJ=p2 in the spot center as a function of thefull current I through the spot at the nitrogen pressure of 25 torr. Solidcurve: De=�e = 2 eV, � = 0:75 � 10�7 cm3/s; broken curve:De=�e = 1 eV, � = 0:75 � 10�7 cm3/s; dotted curve:De=�e = 2

eV, � = 1:5 � 10�7 cm3/s.

density profile is formed (as in Fig. 7), a phenomenon ofthe normal-current-density effect is observed. The differencebetween the phenomena of normal and subnormal spots is dueto different kinds of lateral losses. The shape of the subnormalspot is governed by centrally located ionization processes anddiffusion charge particles, while the normal anode spot isformed under the influence of the ionization processes andthe transport of charge particles in complicated electric field[see, e.g., the distributions ofand on Figs. 4(b) and 5(b),respectively].

The stationary current density in the spot center for lowvalues of (i.e., cm s torr) is close tothe experimental value of the normal anode current density in

[4].

IV. CONCLUSION

The near-anode region of direct current glow discharges in acylindrical geometry has been investigated by means of two-dimensional numeric modeling. Radial profiles of the anodecurrent density and anode potential drop are presented alongwith the two-dimensional charge distributions for conditionschosen to illustrate the transition from a radially uniform toa contracted discharge in elevated pressure nitrogen. Thesecalculations have been made possible by the use of efficientnumerical methods. It has been demonstrated that a develop-ment of the anode spot exhibits a nonthreshold nature andowes its origin to the purely electrodynamical phenomenonlong before the emergence of the bulk thermal effects. Also,it is shown that a complex shape current structure in theform of a circular spot and a ring spot concentric with it isnot stable against redistribution of a current from the ringspot into the circular spot except in certain conditions. Thenumerical results obtained should stimulate the experimentaland analytical investigations of the near-anode glow regionusing modern methods.

ISLAMOV AND GULAMOV: NUMERICAL SIMULATION OF AXISYMMETRIC ANODE SPOT FORMATION 13

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Raphael Sh. Islamovwas born in Bashkortostanin 1954. He received the Diploma and the C.Sc.(Ph.D.) degrees in physics from Moscow Institutefor Physics and Technology, Moscow, Russia, in1977 and 1983, respectively.

Since 1982, he has been with the Research Centerfor Technological Lasers of the Russian Academyof Sciences, Shature, Russia, where he is engagedin research on high-power lasers and their applica-tions. His research interests include the theoreticalinvestigations and modeling of kinetic and nonlinear

processes in physics of low-temperature plasma (particularly, near-electrodephenomena), physics of gas lasers, and interactions between laser beams andmaterials.

Echtiram N. Gulamov was born in Kurdamir,Azerbaijan, in 1961. He received the Diploma inquantum electronic engineering from the MoscowInstitute for Engineering and Physics in 1984 andthe C.Sc. (Ph.D.) degrees in physics from the Re-search Center for Technological Lasers of the Rus-sian Academy of Sciences, Shature, Russia, in 1995.

Since 1984, he has been with the Research Centerfor Technological Lasers of the Russian Academy ofSciences, where he has been engaged in research onhigh-power CO2 lasers and their applications. His

research interests include gas discharges, gas lasers and their applications.