数理解析研究所講究録
1191 巻 2001 年 1-8

1

LOCALIZED PATTERNS IN PLANAR GAS-DISCHARGE SYSTEMS

H.-G. Purwins, Yu. A. Astrov, I. Brauer and M. Bode
Institut fiir Angewandte Physik, University of Muenster,
Corrensstr.
D-48149 Muenster, Germany$\mathit{2}/\mathit{4}_{\mathrm{J}}$ABSTRACT

A summary is given of parts of the work that has been done on pattern formation in planar-and dc-gas-discharge systems with high ohmic and dielectric barrier respectively, at theInstitute of Applied Physics of the University of Muenster. In addition, a qualitative reactiondiffusion model is reviewed that takes account of many of the effects that have been observed
experimentally.
$\mathrm{a}\mathrm{c}$1. Introduction
Pattern formation in gas discharge systems is well-known since long time. It is
astonishing that these phenomena have attracted so little attention in the field of “Nonlinear
Dynamics and Pattern Formation“ so far. It is the goal of the present review to report on thework that has been performed at the Institute of Applied Physics on localized patterns andthat may serve to fill the gap in investigating gas discharge systems.

2. Experimental Set-Up and Results
The experimental investigations have been carried out with four different devices:
A quasi-l-dimensional -system operated at room temperature where the edge of a thinmetallic plate is opposite to the edge of a thin high-ohmic semiconductor wafer. The
electrodes are separated by a discharge gap with discharge length ranging from some 100
to some mm. The pressure of the gas is about 10-100. The driving voltage is up
to about 1
A quasi-2-dimensiona1-system operating at room temperature with a high-ohmic
semiconductor layer with diameter in the range of some cm parallel to a glass plate coatedwith ITO and being transparent with respect to the radiation emitted from the discharge, and the voltage up to 1
gap. Roughly the discharge length is 1 mm, the pressure 100$\mathrm{d}\mathrm{c}$$\mathrm{h}\mathrm{P}\mathrm{a}$$\mu \mathrm{m}$$\mathrm{k}\mathrm{V}.\iota$$\mathrm{d}\mathrm{c}$$\mathrm{h}\mathrm{P}\mathrm{a}$.

$\cdot 2$$\mathrm{k}\mathrm{V}$A quasi-2-dimensiona1 -system similar to the former one. However, to increase theresistivity of the semiconductor the latter can be cooled down to about 90 K. In addition,
the semiconductor resistivity can be controlled by an external -source. The discharge,
length ranges from about 100
to about 1 mm, the pressure is in the order of 100
and the voltage rises up to some
A quasi-2-dimensiona1 -system consisting of two parallel dielectric layers having a
diameter in the order of several cm and a transparent ITO-contact at the outer sides. The
dielectric plates are separated by a discharge space with a discharge length of
the amplitude of theapproximately 1 mm. The pressure is in the order of some 100, and the period is in the range of.
driving voltage is up to some$\mathrm{d}\mathrm{c}$$\mathrm{I}\mathrm{R}$$\mathrm{h}\mathrm{P}\mathrm{a}$$\mu \mathrm{m}$$\mathrm{k}\mathrm{V}.3$$\mathrm{a}\mathrm{c}$$\mathrm{h}\mathrm{P}\mathrm{a}$$\mathrm{k}\mathrm{V}$$10^{-5}\mathrm{s}^{4}$

2
Among other things, self-organized patterns in the distribution of the discharge current do

occur. These patterns can be observed optically due to the fact that excited states in the
discharge gap emit light. Therefore, locally the current density distribution is reflected by theradiation density distribution which is approximately proportional to the current. All patterns
listed below are recorded by optical means. Table 1 gives a listing of some of the observed
patterns of the current distribution in the discharge space.

3. Qualitative Reaction-Diffusion Model for$\mathrm{d}\mathrm{c}$-Discharge and Theoretical Results

Most of the work has been performed using the three-component reaction diffusion
systems similar to,$\mathrm{u}_{\mathrm{t}}=\mathrm{D}_{\mathrm{u}}\Delta \mathrm{u}+\mathrm{f}(\mathrm{u})-\mathrm{v}-\kappa_{3}\mathrm{w}+\kappa_{1}-\kappa_{2}\int_{\Omega}\mathrm{u}\mathrm{d}\Omega$$\tau \mathrm{v}_{\mathrm{t}}=\mathrm{D}_{\mathrm{v}}\Delta \mathrm{v}+\mathrm{u}-\mathrm{v}$,$\Theta \mathrm{w}_{\mathrm{t}}=\mathrm{D}_{\mathrm{w}}\Delta \mathrm{w}+\mathrm{u}-\mathrm{v}$

,
(1)

with
$\mathrm{f}(\mathrm{u})=\lambda \mathrm{u}-\mathrm{u}^{3}$,$\mathrm{D}_{\mathrm{u}},\mathrm{D}_{\mathrm{v}},\mathrm{D}_{\mathrm{w}},\tau,\Theta,\lambda\geq 0$,$(\mathrm{u},\mathrm{v},\mathrm{w})=(\mathrm{u}(\mathrm{x};\mathrm{t}),\mathrm{v}(\mathrm{x};\mathrm{t});\mathrm{w}(\mathrm{x};\mathrm{t}))$,$\mathrm{x}\in\Re^{1},\Re^{2},\Re^{3}$

The system of equations has been treated analytically and numerically with Neumann
and periodic boundary conditions or with no boundary restrictions on infinite domain.
The equation can be derived from an equivalent circuit for a layer system that consists of
two high ohmic electrodes with linear behaviour having laterally homogeneous extension
and being separated by a homogeneous material with -shaped current-voltage
characteristic. The high ohmic layer may represent high ohmic semiconductor electrodes
or regions of anode and cathode fall. The material with -shaped characteristic is a gas inour case. Dynamical behaviour is introduced to the system by considering a distributed
capacity going along with the high ohmic layers and taking account of dielectricrelaxation and a distributed inductivity describing charge carrier relaxation (Ref. 5-8, 12,34, 39, 41-52). Table 1 contains also a review of some of the patterns resulting fromanalytical and numerical observations.
$\mathrm{S}$$\mathrm{S}$4. Acknowledgements
We are grateful to the German BMBF and the Deutsche Forschungsgemeinschaft for
supporting this work.

3$\Phi\forall\infty\triangleleft.$$\underline{\sigma\backslash \alpha \mathrm{r}}^{\mathrm{r}}$

$.$

$-$

$rightarrow$$rightarrow$$\underline{\mathrm{r}^{1}\circ}.0$$\underline{\mathrm{o}}.$$\iota_{\cap}\mathrm{N}$$-\wedge\iota_{\cap}$$rightarrow\infty \mathrm{O}\wedge$$\backslash O^{\wedge}\triangleright.\alpha_{\forall^{\wedge}}^{\infty}\infty.\forall.$$,\overline{\underline{=\mathrm{Q})0}}$$\dot{\mathrm{o}}^{\mathrm{e}}\iota\cap$$-\cdot\triangleright 0^{\wedge}$$(\underline{\overline{.\omega}.}$$0rightarrow$$-\wedge\triangleright\wedge\hat{\infty}$$rightarrow$$..\triangleleft\infty.\cdot.$$\forall 1\cap$$.. .. ..$$\mathrm{v}\iota_{\cap}$$|\cap\forall\triangleleft.$$\Xi\Xi\Xi$$\Xi\Xi$$\Xi\Xi\Xi$$\Xi\Xi$$.rightarrow.-\cdotrightarrow$$.-.rightarrow$$\prime 0.0\varpi$$rightarrow\circ 1\propto)$$.. .. ..$$.. ..$$.-\cdot-\cdotrightarrow$$.rightarrow.-$$\varpi_{\mathrm{I}}r\circ$$\varpi_{\mathrm{I}}\tau\varpi$$\varpi_{1}\varpi_{1}$$rightarrow \mathrm{N}$$rightarrow \mathrm{c}\triangleleft \mathrm{c}^{r})$$rightarrow \mathrm{o}\mathrm{t}$$\mathrm{N}^{1}\iota_{\cap}.\frac{\dot{\mathrm{c}}}{\dot{\varpi}}.$$-\cdot$$\circ \mathrm{l}[mathring]_{\dot{\mathrm{o}}_{\wedge}^{\mathrm{i}}\propto})$$\overline{\Xi-\circ}$$rightarrow \underline{\triangleleft\triangleright\circ\tau_{\wedge}.}$$o\backslash \triangleright$$rightarrowrightarrow$$\frac{\propto)}{\mathrm{I}}\triangleleft.

\frac{\triangleright}{\dot{\mathrm{o}}}$$.\omega=$$rightarrow \mathrm{C}\triangleleft \mathrm{C}’)rightarrow\iota_{\cap^{\wedge}}$$\underline{\infty}$$\underline{\triangleleft\cdot}\underline{\backslash _{Q}}\wedge\cdot$$\underline{\Psi\iota_{\cap \mathrm{N}}}\triangleleft^{:}.$$\iota_{\cap.-\wedge\iota_{\cap}}-$$rightarrow(\triangleleftrightarrow O\backslash \cdot$$\omega\triangleright<$$-\wedge\circ \mathrm{l}\wedgerightarrowrightarrow$N$\aleph.

$\ddot{\Xi}\ddot{\Xi}\ddot{\Xi}\ddot{\mathrm{g}}$\triangleleft.$$o\backslash \mathrm{c}\triangleleft \mathit{0}\backslash \cdot\triangleleft.\wedge$$\ddot{\Xi}\ddot{\Xi}$$\ddot{\Xi}\ddot{\mathrm{E}}\ddot{\Xi}\ddot{\Xi}$$.rightarrow.rightarrow$$.-\cdot-.rightarrow.-$$*\underline{\overline{\omega}.}$$.rightarrow.-.-\cdot-$$\varpi r\circ$$\varpi_{1}\varpi \mathrm{r}_{1}\varpi$$\frac{\ddot{\mathrm{S}}}{\dot{\varpi}}$$.\mathrm{o}_{1}r\mathrm{o}\prime 0\varpi_{\mathrm{I}}$$. \cdot.\circ \mathrm{J}\mathrm{o}\mathrm{o}^{1}\frac{\ddot{\in}}{\mathrm{o}}1$$rightarrow \mathrm{N}rightarrow\circ \mathrm{l}$$\mathrm{o}\mathrm{t}\circ\uparrow$$\dot{\varpi}().\dot{\circ}\zeta)\alpha 0\alpha 0$$\circ \mathrm{o}\mathrm{c}\overline{3}()$$-\circ \mathrm{l}-(\triangleleft$$\mathrm{N}$$.\mathrm{o}^{1}\mathrm{o}\mathrm{Q}\cup \mathrm{c}\dot{\mathrm{d}}()\alpha 0^{1}$$\alpha 0$$\frac{\infty}{\omega \mathrm{z},\Xi}$$.\cdot\infty\omega$$\underline{\omega}$$.>\approx\underline{\mathrm{o}\supset\infty}.1$$\frac{\infty}{\not\subset,\omega}$$\mathrm{s}_{\mathrm{c}3}\in-$$\mathrm{q}.\frac{\mathrm{o}}{\mathrm{o}}\Xi.

\underline{\varpi}\frac{\overline\circ \mathrm{Q}\supset}{\mathrm{O}}$$(.-\underline{\underline{\mathrm{c}3}}\mathrm{q}\mathrm{G}\mathrm{c}3$$\frac{\varpi}{\mathrm{c}3}\frac{\infty}{\mathrm{g}}.\underline{\frac{\underline{\mathrm{O}\mathrm{E}}}{\alpha}}.

\frac{\mathfrak{c}6\Xi}{\mathrm{q}\dot=}$$\varpi\dot{\mathrm{I}}\mathfrak{c})$$y^{\backslash }\mathrm{r}\mathrm{c}.\mathrm{q}\underline{\overline{\mathrm{o}}}$$.\dot{\subset\omega.} .\mathrm{c}\circ.\Xi\underline{\mathrm{b}-}0:\underline{\mathrm{z}}$$. \frac{>}{\overline{\varpi\infty \mathrm{c}\omega}}\mathrm{Y}.\mathrm{b}0\underline{\approx\omega\frac{\mathrm{G}}{\infty}}\frac{\infty}{\mathrm{c}}$$. \mathrm{b}0\frac{\not\subset}{\infty}.\underline{\underline{\alpha\Xi\omega}}\mathrm{E}\Phi \mathrm{b}\mathit{1})\circ .\frac{\underline{\mathrm{c}}}{\overline{\omega}}$$\frac{\omega}{\cup,\circ\Xi}\overline{\underline{\infty \mathrm{c}3\circ}}$$\overline{\omega-.}.\underline{\subset 0}\alpha\omega \mathrm{c}.$$.\underline{\underline{\mathrm{o}\not\subset}}$\={o}$\omega\alpha\dot{\mathrm{o}}\mathrm{q}-\mathrm{b}0.\underline{\not\subset}\not\subset.\mathrm{b}\dot{D-} \underline{\alpha\succ.}$$\overline{\approx}$$. \cdot=\simeq\underline{0\omega 0}^{0}0\infty .\cdot\underline{\underline{.=\underline{\underline{(..)\omega\approx\infty}}}}\frac{\frac{\infty}{\circ\not\subset\circ\Xi_{3}}}{\mathrm{q}\dot=}$$.\Xi.=-\underline{\omega}$$\Xi\infty$$.\overline{\supset}\underline{s}\overline{\dot{\overline{\mathrm{b}}}}$$. \mathrm{E} \circ rightarrow \mathrm{b}\mathrm{J})\mathrm{G}\frac{\mathrm{E}}{\triangleleft}$$(\mathit{3}.rightarrow$$\dot{\mathrm{q}\underline{\mathrm{o}}}\vee \mathrm{O}\supset(.\mathrm{D}$$\mathrm{c}\not\subset.\overline{\overline{\geq}}=\mathfrak{c}9\succ 0\alpha\infty \mathrm{o}\mathrm{G}.\cdot--\cdot.$$’.\overline{.\overline{\mathrm{o}\cup\infty\omega}}$$\mathrm{c}.3^{\backslash }\succ\geq \mathrm{r}_{\mathrm{O}}$$\geq\Xi$$\frac{y3}{\not\subset,\Xi\circ} \circ\alpha\omega\infty\geq \mathrm{q}^{\circ}\overline{\mathrm{o}\dot{\omega}}$$.-\vee-.\mathrm{c}\mathrm{Q}\tilde{)}\zeta\prime r\dot{\circ\circ}$$\underline{\mathrm{o}}.\overline{\overline{\infty}}$$. \frac{\vee\infty\omega}{\Xi}$$. \overline{\overline{-0\Xi}}\triangleleft()\alpha\dot{\infty}\omega\infty.\mathrm{q}\overline{\mathrm{c}\mathrm{o}0^{\cdot})}.\frac{\underline{\mathrm{o}\ddot{\mathrm{c}}}}{()\dot{\alpha}} .\overline{\underline{\frac{\overline \mathrm{c}3}{\omega}}}\overline{\underline{y3\alpha 0}}.\Xi 0-\overline{>}$$\overline{(\triangleleft}\underline{\omega}$$\underline{\mathrm{o}}.\overline{\infty}$$\underline{\omega\cup}$$\mathrm{b}00\mathrm{J}\Phi\not\supset 0_{\overline{\mathrm{O}}}$$(.\overline{\overline{\triangleleft}}\underline{\approx}\vee.y)0\alpha\infty 0\Xi\supset\subset$$.-\underline{\mathrm{o}\omega \mathrm{N}U^{\backslash }\mathrm{o}_{d}}\mathrm{o}\mathrm{o}\alpha\infty \mathrm{E}\approx \mathrm{b}\mathit{1})\Xi\supset \mathrm{c}$$\omega 0$$.-\mathfrak{Q}.$$\mathrm{o}_{\wedge}\Xi.$$\underline{\Xi}\Xi$a$\mathrm{t}.\underline{\underline{\mathrm{z}\omega}}\frac{\omega}{=}.\underline{\mathrm{O}\mathrm{G}}.\Xi$$. \frac{\frac{\mathrm{c}9\Xi\omega \mathrm{G}}{\mathrm{t}_{4}.\omega=}}{\mathrm{b}\mathit{1}}).\frac{\underline{\mathrm{O}\not\subset(.)\mathrm{c}3}}{\mathrm{o}a,\supset}.\cdot\alpha\omega\frac{\mathrm{c}}{\infty}\omega \mathrm{b}0-\mathrm{z}.--\Xi\overline{\circ\overline{\mathrm{c}\Xi}-}.\cdot\frac{\vee\alpha\underline{\mathrm{o}\mathrm{c}}}{\dot}\overline{\frac{s}{\not\subset}}\vee \mathrm{c}\approx \mathrm{c}\omega 0\infty\supset’.\frac{\underline{\mathrm{o}\mathrm{o}-}}{\alpha,\omega}..\frac{\frac{(.)\omega}{\omega}}{\frac{\infty}{\triangleright<}}$$0\infty$$\{\underline{\omega}0\Xi$$\underline{\mathrm{r}\circ}\frac{\omega 0}{\mathrm{o}}\underline{\overline{\alpha 0}}\underline{.\omega\alpha\infty}$$\overline{\underline{\mathrm{o}\mathrm{Q}\mathrm{c}3}\overline{\overline{\omega\geq}}}.\cdot r\frac{\underline{\mathrm{o}}}{\mathrm{q}_{\frac{\supseteq}{\mathrm{o}}}\alpha 0}.\cdot \mathrm{E}\frac{\omega}{\zeta)\dot{\circ}}03\frac{\mathrm{z}}{\infty}$$.0 \underline{\mathrm{b}}0\frac{}{\succ}\Xi$$^{\circ}.

\mathrm{r}_{\mathrm{I}}\frac{(-)}{\infty}.\frac{.\dot{\subset\omega}}{\mathrm{o}}\frac{\omega}{\infty}\infty$$\overline{\omega_{\mathrm{I}}\geq}\Xi_{\wedge,\wedge}\mathrm{q}_{\frac{\underline{\Xi}}{\dot{\mathrm{p}}}}.\circ\circ 0$$. \frac{-}{\infty}\dot{\overline{\mathrm{p}}}(-\dot{\varpi^{\mathrm{Q})}(.)}.\frac{\alpha}{(=.}\alpha-y30\mathfrak{Q}.

$\frac{\varpi_{\omega}}{\approx}$q\={o}\mathrm{b}0\overline{\mathrm{Q})}\cup 0_{1}\circ$$\circ\omega$$\overline{\in \mathrm{c}3\not\subset\circ}$$\overline{\mathrm{c}6}$$\frac{\frac{\circ\omega}{\mathrm{c}S}}{\mathrm{o}}\infty\frac{\infty}{\omega}.$$\underline{\infty}$$\overline{.\underline{\infty 0}}$$\overline{.\underline{\mathrm{o}\infty}}$$.\overline{\vee\alpha}$$\frac{}{\mathrm{c}3}\underline{\frac{\mathrm{E}}{\omega}}$$\infty=$$\overline{\dot{\mathrm{q}}=}$$.rightarrow-$$\mathrm{b}00$$\dot{\infty}->_{\backslash \infty}$$\dot{\mathrm{G}^{\backslash }\infty}>\underline{\infty}$$.\Xi\overline{-}$$. \overline{\overline{\overline{\Phi}}}\not\subset\frac{\infty}{\Xi\not\subset\omega}\mathrm{b}\mathrm{J})$$. \frac{\frac{\dot>}{\mathrm{o}}}{=^{\mathrm{c}3}}\overline{(-)}\backslash \frac{y)}{\alpha \mathrm{g}\Xi\omega}$$\mathrm{c}\omega$$. \underline{\mathrm{o}}\frac{\omega}{\infty}$$\overline{\overline{\omega>}}\alpha\Xi\omega$$\succ\alpha$$. \frac{}{\infty}\frac{\underline{\mathrm{o}}}{\alpha}\frac{\mathrm{c}3\Xi}{\mathrm{q}\dot=}$$\overline{\triangleleft\backslash }=$$=\dot{\mathrm{t}}\approx-\overline{\neq-}$$0\infty\overline{\dot{\mathrm{q}}=}$$\vee\infty\overline{(.)}$$-\cdot\overline{\dot{\mathrm{c}}=}\mathrm{c}6$4

55. References
1.

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3.

4.
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H. Willebrand, C. Radehaus, F.-J. Niedernostheide, R. Dohmen, and H.-G. Purwins,
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published
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published
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Dielectric Barrier Discharge System, Proc. Int. Conf. on Phenomena in Ionized
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size travel in direction of the smaller filament, measured in an -system, 1999, to be
published
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published
31. I. Brauer and H.-G. Purwins, rather dense molecules made of filaments may moveeratically on the active area defining a”molecular gas“ state, measured in an acsystem, 1999, to be published
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the same is observed with the ”liquid“ state, in both cases filaments are evaporated
and condensed continuously, measurements are performed with Si electrode at 90 K,
1999, to be published$\mathrm{a}\mathrm{c}$$\mathrm{a}\mathrm{c}$$\mathrm{d}\mathrm{c}$$\mathrm{a}\mathrm{c}$7
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43. J. Berkemeier, T. Dirksmeyer, G. Klempt, and H.-G. Purwins, Pattern Formation ona Nonlinear Periodic Electrical Network, Z. Phys. -Condensed Mater 65, 255-258$\mathrm{B}$(1986)

44. H.-G. Purwins, and C. Radehaus, Pattern Formation on Analogue Parallel Networks,
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Synergetics 42, 137-154 Springer 1988
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Instabilitites“,Nato ASI, Series E: Applied Sciences 183, 91-107 Kluwer 1990
46. R. Woesler, P. Sch\"utz, M. Bode, M. Or-Guil, and H.-G. Purwins, Oscillations of
Fronts and Front Pairs in Two-and Three-Component Reaction-Diffusion Systems,
Physica D91, 376-405 (1996)
47. C. Schenk, M. Or-Guil, M. Bode, H.-G. Purwins, Interactingpulses in threecomponent reaction-diffusion systems on two-dimensionaI domains, Phys. Rev. Lett.
78, 3781-3784 (1997)
48. C.P. Schenk, A.W. Liehr, M. Bode, and H.-G. Purwins, Interaction of Two-and
Threedimensional Moving Localized Solutions in a Three-Component ReactionDiffusion-Model: A Particle Approach, Physica D, resubmitted 16.10.2000

8
49. M. Or-Guil, M. Bode, C. Schenk, H.-G. Purwins, Spot Bifurcations in ThreeComponent Reaction-Diffusion Systems: The Onset ofPropagation, Phys. Rev. E57,
6432-6437 (1998)
50. C.P. Schenk, A.W. Liehr, M. Bode, and H.-G. Purwins, Quasi-Particles in a ThreeDimensional Three-Component Reaction-Diffusion System, in: E. Krause, W. J\"ager(Eds.):High Performance Computing in Science and Engineering 1999, Springer,
2000, 355-364
51. C. Schenk, P. Sch\"utz, M. Bode, and H.-G. Purwins, Interaction ofSelforganizedQuasiparticles in an Two-Dimensional $Reactionrightarrow Diffusion$ System: The Formation of
Molecules, Physical Rev. E57, 6480-6486 (1998)
52. A.W. Liehr, M. Bode, and H.-G. Purwins, The Generation ofDissipative QuasiParticles near Turing’s Bifurcation in Three-Dimensional Reaction-Diffusion
Systems, Project Dynamical Structures in Three-Component Reaction-Diffusion
Systems of the H\"ochstleistungsrechenzentrum Stuttgart (HLRS)