Pattern Formation in Electric Discharges

Electric discharges with low current densities can be described by a minimal model of reaction, drift and diffusion of two charged species coupled nonlinearly to an externally imposed electric field. The intrinsic ionization–field–coupling of the model leads to pattern formation both in stationary or periodically driven barrier discharges and in transient streamer discharges. In particular, we discuss negative streamer discharge fingers extending into a non–ionized area. They belong to the class of Laplacian growth phenomena. For quantitative predictions, a systematic derivation of the corrections to the velocity of a moving boundary due to curvature or relaxation effects is still missing. This is because negative streamer fronts are so–called pulled fronts, for which standard approximation schemes fail. We present new approaches and results. 1 Modelling Electric Discharges 1.1 Phenomenology, Spatio-temporal Patterns Electric discharges are commonly known from natural phenomena like sparks whose lengths can vary from cm’s to km’s, or St. Elmo’s fire (a corona discharge at the tops of ship masts). Neon tubes are the best known example from a whole list of technical applications that are under continued use and investigation. For an overview of phenomena in gas discharges, we refer to [1]. Discharges can occur not only in gases, but also in fluids or solids – in just any matter that can turn from a state of low or vanishing conductivity to a state of high conductivity, when a sufficiently strong field is applied. The name “discharge” is a historical one like many other ones in this field, since it refers to the discharge of the voltage stored in a capacitor, when the matter between the capacitor plates becomes conducting. However, a sufficiently strong voltage source also might sustain the voltage difference between the outer electrodes despite the “discharge” in the bridging matter. Electric discharges can create a low temperature plasma locally. In contrast to high temperature plasmas as in fusion reactors or stars that exist due to heating and confinement, low temperature plasmas exist under nonequilibrium conditions due to external excitations like electromagnetic fields. Hence they generically are inhomogeneous in space and time and form spatio-temporal patterns. The homogeneous glow of the neon tube is an exception and a result of the art of the engineer. The onset of spatio-temporal instabilities limits the range of technical applicability in quite a number of cases. D. Reguera, L.L. Bonilla, and J.M. Rub́ı (Eds.): LNP 567, pp. 270–282, 2001. c © Springer-Verlag Berlin Heidelberg 2001 Pattern Formation in Electric Discharges 271 Below, we will focus on streamer and barrier discharges in simple matter like pure nitrogen under normal conditions. Streamers are the initial breakdown mode of insulating matter on length scales from cm’s to m’s after a voltage shock has been applied. They are extending finger–like patterns of ionized matter, and also the precursors of a later short–circuit, if the conducting channel eventually contacts both electrodes. Initially during the streamer phase, the degree of ionization stays relatively low. Barrier discharges are operated in a stationary or periodically changing electric field. They consist of a layered structure of a discharge with at least one resistive layer in dc fields or at least one dielectric layer in ac fields. High currents and degrees of ionization are then prevented by the dielectric or resistive layers, just as in the case of a large external load resistance in the electric circuit. 1.2 The Minimal Model of Low Current Discharges and the Ionization–Field–Coupling Pattern formation occurs in nonlinear spatially extended systems under nonequilibrium conditions. The nonequilibrium here is due to an externally applied electric field. The nonlinear mechanism to be explored is an ionization–field– coupling: a sufficiently high electric field leads to a multiplication of charge carriers by a local impact ionization reaction. The generated charges drift in the electric field. If their density is sufficiently high, they will modify the field and hence change the local reaction rates and drift velocities. Accordingly, the evolution of the local degree of ionization is determined by a fully dynamic process. The situation is captured by the following model: In general, the continuity equations for particles of species i are ∂tρi +∇ · ji = fi ({ρj},E) , ji = qiμiρiE−Di∇ρi , (1) where ρi is the particle density, qi = ± the sign of the charge, μi the mobility and Di the diffusion constant of species i. If the degree of ionization is sufficiently small, dissipative heating can be neglected and the neutral background stays unchanged. The current ji then can be approximated as in (1) by particle diffusion and Ohmic friction. This is true in gases, liquids or solids. In a semiconductor, the current ji can saturate for large E. fi is the source term describing the reactions of the particles in the local field. If the currents stay sufficiently small, also magnetic and relativistic effects can be neglected, and we only need the Poisson equation of electrostatics