A Turing model of pattern formation in atmospheric pressure gas discharges

0.72 MB·English

by  Xi Chen

A Turing model of pattern formation in atmospheric pressure gas discharges
Xi Chen,1,∗ Yao Zhou,1,∗ Xi-Ming Zhu,1 Yi-Kang Pu,1 F. Iza,2 and M. A. Lieberman3
1Department of Engineering Physics, Tsinghua University, Beijing 100084, China
2Department of Electronic and Electrical Engineering,
Loughborough University, Leicestershire, LE11 3TU, UK
3Department of Electrical Engineering and Computer Science,
University of California, Berkeley, California 94720, USA
(Dated: January 19, 2011)
In this letter we propose a Turing model of the formation of patterns of visible light emission
intensityinatmosphericpressuregasdischarges. Theelectrondensityandtheelectrontemperature
1 take the roles of activator and inhibitor respectively in a two-reactant Turing model, with the
1 activator diffusion coefficient being much smaller than that of the inhibitor, and ionization and
0 excitation from excited state atoms considered as the dominant reaction processes. The model
2 predictsstriationsina1Dsystem,whichquantitativelyagreewithexperimentalresultsintermsof
the spatial variation scale lengths. Additionally, the model also predicts changes in the discharge
n
structure observed experimentally when input power and gas pressure are varied.
a
J

8
Plasmas, especially those at atmospheric pressure, of- electron temperature as the activator and the inhibitor
1
ten exhibit non-uniform and/or non-stationary emis- respectively into a Turing model to explain pattern for-

] sion intensity of visible light. These phenomena, called mation in gas discharges. These two reactants are fun-
h
patterns, are common in non-equilibrium systems [1– damentalparameterscontrollingthevariousprocessesin
p
3]. Patterns including striations, constrictions, zigzags, plasmas. Whilethismodelcanbeappliedtovariousdis-
-
m hexagons, clusters, etc. have been observed in gas dis- charge systems, we use a particular one at atmospheric
s charges since the 1830s [4–10]. Models based on skin ef- pressureasanexampleheretoillustratethatthepredic-
a fects[5,6],non-uniformgasheating[11,12]andelectron- tionsofthemodelcanindeedbequantitativelyconsistent
l
p electron collisions [13] were proposed to explain these with experimental results.
s. phenomena, but all these interpretations could not ex- Theexperimentalobservationsconsideredherearethe
c plain some of the important characteristics of the pat- striations of the visible light emission reported in atmo-
i
s terns such as the discharge instabilities and the spatial spheric pressure argon discharges generated with a mi-
y
periodicity. Simulations based on the kinetic theory suc- crowave induced plasma source based on a microstrip
h
cessfully reproduced striations [4] and constrictions [14], split-ring resonator [18], where the striations are dis-
p
[ but the basic mechanism of the pattern formation is not tributed periodically along the gap between the elec-
revealedduetothemanycomplicatedprocessesincluded trodes. Some experimental images are shown in FIG.
1
in the simulations. 2, (a)-(c). This periodic distribution of the visible light
v
emission intensity suggests a periodic distribution of the
1 In 1952, A. M. Turing adopted bifurcation analysis
7 of coupled nonlinear reaction-diffusion equations to ex- excitedstateatomdensity,whichdependsontheelectron
4 densitynandtheelectrontemperatureT. Soitisimplied
plainpatternformationinreaction-diffusionsystems[15],
3 that their distributions should also be non-uniform. To
which is now referred to as the Turing model. A typ-
.
1 ical two-reactant Turing model includes an activator- describe such non-uniformity, we first consider the elec-
0 tron continuity equation:
inhibitor system, where the diffusion coefficient of the
1
inhibitor should be much larger than that of the activa-
1 ∂n ∂2n
v: tor [3]. The model was later applied to explain patterns ∂t =Da∂x2 +Gn−Ln (1)
inDCdischarges[7–10,16,17],withcurrentdensityand
i
X electric potential chosen as reactants. However, the the- Herexdenotesthepositionalongthegap, andDa isthe
r ory is limited to the two-layer discharge structure of the ambipolar diffusion coefficient, while Gn and Ln are the
a system,andthuscouldnotexplainthepatternsreported production and loss rates of the electron density respec-
recently in other gas discharge systems, including mi- tively. At atmospheric pressure, the ionization from the
crowave plasmas [18], dielectric barrier discharges [19], excited state atoms dominate the production of the elec-
boundary layer discharges [20] and breakdown processes trondensity, andthecollisional-radiativemodelsuggests
[21]. These patterns share similar characteristics, for in- that the production rate is [22]:
stance, when the pressure is decreased or the power is
G =r n2e−E1/T (2)
raised,thedischargebecomesuniform[18–20]. Suchcon- n 1
sistencysuggeststhatthereshouldbeacommonphysical
Here r is a constant and E is an effective ioniza-
1 1
mechanism for these patterns.
tion energy for excited state atoms. In atmospheric
Inthisletterweintroducetheelectrondensityandthe pressure argon discharges, the electron loss is mainly
2
due to electron recombination with argon dimer ions:
Ar+ + Ar + Ar → Ar + Ar+, e + Ar+ → Ar + Ar
2 2
[22]. The rates of these reactions can be expressed as
R = r n(Ar+) and R = L respectively, with r be-
1 2 2 n 2
ing a constant. These two processes are also the dom-
inant production and loss processes of Ar+, which are
2
locally balanced: R = R . In addition, according to
1 2
thecollisional-radiativemodel[22], thedensityofAr+ is
muchhigherthanthatofAr+,thusapproximatelyequal
2
to the electron density: n(Ar+)=n. Then we have:
L =R =R =r n(Ar+)=r n (3)
n 2 1 2 2
The patterns in gas discharges often come with a non-
uniform distribution of the electron temperature [4], the
existence of which has been proved possible in the dis-
chargesystemdiscussedhere[23,24]. Todescribesucha
no-uniformity, we start with the electron energy conser-
vation equation:
(cid:18) (cid:19) (cid:18) (cid:19)
∂ 3 ∂ 3 ∂u ∂ ∂T
p + pu +p − (κ )=G −L
∂t 2 ∂x 2 ∂x ∂x T ∂x E E
(4)
Here p is the pressure, and we assume the energy dis-
tribution of the electrons to be Maxwellian: p = nT.
u is the flow velocity, and κ = (3/2)nD is the ther-
T e
mal conductivity, where D is the diffusion coefficient
e
of the electrons. G and L denote the gain and loss
E E
rateoftheelectronenergyrespectively. Intheambipolar
regime, D is 2∼3 orders of magnitude larger than D .
e a
So, compared with the heat conduction term, the energy
flux caused by the flow, that is, terms containing u, can
FIG.1. Spatio-temporalevolutionsoftheelectrondensity(a)
be neglected. In the bulk plasma, G is mainly caused
E andtheelectrontemperature(b)distributions. Thedischarge
by the Ohmic heating [25]. When filaments form, more condition is 1 atm, 1 W.
power is coupled to these channels of high conductivity
(high electron density) than to the neighboring regions
of lower conductivity (lower eletron density). In a first The time derivative of the electron density in the first
approximation, this non-uniform power deposition can termoftheLHSisgivenby(1)andcanbeneglectedbe-
be modeled by assuming that the power density is pro- cause De (cid:29) Da, r3 (cid:29) Tr2, and r4e−E2/T (cid:29) Tr1e−E1/T
portional to the plasma conductivity, and thereby, the [22]. We also neglect the third term on the LHS because
electron density: a bifurcation analysis [3, 15] shows that this term does
not affect the periodicity of the striations or the phase
G =(3/2)r n (5) diagramofdischargetransitions(whichwillbediscussed
E 3
later). Neglectingthetwotermsin(7),dividingtheequa-
Here r3 is a constant that depends on the input power. tionbyn,andsubstituting(2)and(3)intoequation(1),
AstoLE,theexcitationofexcitedstateatomsdominates we obtain a set of coupled nonlinear reaction-diffusion
the loss of the electron energy [22]: equations for the electron density and the electron tem-
perature:
L =(3/2)r n2e−E2/T (6)
E 4
∂n ∂2n
=D +r n2e−E1/T −r n (8)
Herer4 isanotherconstant,andE2 isaneffectiveexcita- ∂t a∂x2 1 2
tion energy, with E2 < E1. Based on the considerations ∂T =D ∂2T +r −r ne−E2/T (9)
above, (4) becomes: ∂t e∂x2 3 4
∂n ∂T ∂n∂T ∂2T In equations (8) and (9), which describe the gas dis-
T ∂t +n∂t −De∂x ∂x −nDe∂x2 =r3n−r4n2e−E2/T chargesystem,theelectrondensitypromotesitsownpro-
(7) duction via stepwise ionization, while the electron tem-
3
perature self-inhibits: a perturbation that increases the
electron temperature results in increased energy losses
due to the increase in excitation processes (third term
on the RHS of equation (9)), driving the electron tem-
perature down and “inhibiting” the initial perturbation.
In addition, D (cid:29)D . So, the two reactants satisfy the
e a
conditionsofatypicalTuringactivator-inhibitorsystem,
and thus patterns can result from the dynamics of the
system.
Withtheparametersofanatmosphericpressureargon
discharge [22], including reaction rates, ionization and
excitation threshold energies, the equations (8) and (9)
are solved numerically under uniform initial conditions
(1014 cm−3, 1 eV) and fixed boundary conditions (1013
cm−3,0.1eV)(infacttheboundaryandinitialconditions
havelittleinfluenceonmajorquantitativecharacteristics
ofthefinalstriations,includingperiodsandamplitudes).
The spatio-temporal evolutions of the electron density
andtheelectrontemperatureobtainedareshowninFIG.
1. The profiles evolve from uniform at the beginning to
steadyperiodicstructuresattheend,whichcanresultin
striationsofvisiblelightemissionintensityshowninFIG.
2, (B). The opposite phase distributions of the electron
densityandtheelectrontemperatureshowninFIG.1are
consistent with simulation results based on the kinetic
theory [4]. The magnitude of the spatial period of the
striations, which has an order of ∼ 0.1 mm, agrees with
the experimental results (FIG. 2, (b)).
It has been observed in experiments that the number
FIG. 2. Phase diagram: discharge structure transition as
of striations increases with the input power (FIG. 2, (b)
predicted by the proposed model. Images (a)-(c) show ex-
and (c)). This behavior is also predicted by the model:
perimentallyobserveddistributionsofvisiblelightemittedin
with bifurcation analysis [2, 3], the spatial period of the argonmicrodischarges(scalebar=250um)sustainedbetween
striations λ can be determined with the parameters in coplanarelectrodesat 825MHzatthreepressure/powercom-
(8) and (9): binations. (A)-(C) are the theoretical results of the emission
densitydistributionswiththesamepressure/powercombina-
(cid:26) r r [log(r r /r r )]2(cid:27)−1/4 tion as (a)-(c) respectively.
λ=2π 2 3 · 1 3 2 4 (10)
D D E −E
e a 1 2
Note that the spatial period of the striations decreases arefoundtobestable, andthusthestriationsoccur. On
when r , which represents the Ohmic heating term (5), the other hand, (12) is required for the discharge to be
3
increases. In other words, the higher the input power, stable: when it is not satisfied, either because the input
the more striations exist. power is too small (small r in (5)) or there is a large
3
In experiments, it has also been observed that when recombination loss rate (large r in (3)), the discharge is
2
the pressure decreases, the discharge structure changes difficult to ignite and is not stable.
from periodic to uniform, i.e. striations disappear (FIG. Based on the physical mechanisms governing the gain
2, (a)). Such phenomena can also be explained with the and loss terms (2), (3), (5) and (6), we can assume the
model: bifurcation analysis [2, 3] also suggests that the parameters in the model scale with pressure p and input
formation of the striations demands the following two power P as follows:
conditions:
r ∼p1P0,r ∼p2P0,r ∼p0P1,r ∼p1P0,
(cid:115) 1 2 3 4
DDer2− ET22r3 >2 DDe · E1T−2E2 ·r2r3 (11) De ∼p−1P0,Da ∼p−1P0, (13)
a 0 a 0
Again, with the parameters of the atmospheric pressure
E
r − 2r <0 (12) argon discharge [22], the criteria (11) and (12) can be
2 T2 3
0 presented in a phase diagram, FIG. 2. In the phase on
HereT =(E −E )/[log(r r /r r )]. When(11)issatis- the lower right, where the power is low and the pressure
0 1 2 1 3 2 4
fied,periodicdistributionsinsteadoftheuniformprofiles is high, inequality (12) is not satisfied and therefore it is
4
not possible to generate stable plasmas under such con- temperature act as activator and inhibitor in a classical
ditions. The phase in the middle is where the discharge two-reactant Turing model. Therefore similar patterns
is spatially periodic, and atmospheric pressure gas dis- should be expected in systems where the stepwise ion-
charge (∼ 1 atm, ∼ 1 W) is usually in that phase. If ization/excitation dominate the production of the elec-
we lower the pressure (to ∼ 0.6 atm) or raise the power trondensity/lossoftheelectronenergy. Extensionofthe
(to ∼3 W), we enter the phase on the upper left, where model is expected to be able to reproduce the selection
inequality (11) does not hold and the discharge is uni- among different patterns observed in higher dimensional
form. Despite the simplicity of the proposed Turing systems, or explain more complicated pattern behaviors,
model, the phase diagram shown in FIG. 2 is in good such as the traveling waves.
agreement with experimental observations and therefore
suggests that phase transitions between unstable, peri-
The authors would like to thank Y. Lan, Q. Ouyang
odic and uniform discharges can be explained in terms
and D. Economou for helpful discussions. This work is
oftheactivatorandinhibitorrolesoftheelectronenergy
supportedbytheNSFCGrantNo. 11075093andtheNa-
and the electron temperature.
tional Innovation Research Project for Undergraduates.
Even though (5) implies a uniform electric field dis-
tribution, the existence of periodic structure solutions
obtained from the model is rather insensitive to this as-
sumption. Infact,numericalsimulationresultsshowthat
the model can still reproduce striations, when (5) is re- ∗ AlsoattheDepartmentofPhysics,TsinghuaUniversity.
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