Spontaneous Branching of Anode-Directed Streamers between Planar Electrodes

Abstract

Non-ionized media subject to strong fields can become locally ionized by penetration of fingershaped streamers. We study negative streamers between planar electrodes in a simple deterministic continuum approximation. We observe that for sufficiently large fields, the streamer tip can split. This happens close to the limit of "ideal conductivity". Qualitatively the tip splitting is due to a Laplacian instability quite like in viscous fingering. For future quantitative analytical progress, our stability analysis of planar fronts identifies the screening length as a regularization mechanism.Streamers commonly appear in dielectric breakdown when a sufficiently high voltage is suddenly applied to a medium with low or vanishing conductivity. They consist of extending fingers of ionized matter and are ubiquitous in nature and technology [1,2]. The degree of ionization inside a streamer is low, hence thermal or convection effects are negligible. However, streamers are nonlinear phenomena due to the space charges inside the ionized body that modify the externally applied electric field. While in many applications, streamers by a strongly nonuniform background electric field are forced to propagate towards the cathode through complex mixtures of gases [2-4], we here investigate the basic phenomenon of the primary anode-directed streamer in a simple nonattaching and non-ionized gas and in a uniform background field as in the pioneering experiments of Raether [5]. In previous theoretical work, it is implicitly assumed that streamers in a uniform background field propagate in a stationary manner [6][7][8]. This view seems to be supported by previous simulations [9,10].In this paper we present the first numerical evidence that anode directed (or negative) streamers do branch even in a uniform background field and without initial background ionization in the minimal fully deterministic "fluid model" [1,[6][7][8][9][10], if the field is sufficiently strong. We argue that this happens when the streamer approaches what we suggest to call the Lozansky-Firsov limit of "ideal conductivity" [6]. The streamer then can be understood as an interfacial pattern with a Laplacian instability [11], qualitatively similar to other Laplacian growth problems [12]. For future quantitative analytical progress, we identify the electric screening length as a relevant regularization mechanism. Our finding casts doubts on the existence of a stationary mode of streamer propagation with a fixed head radius. FIG. 1. Evolution of spontaneous branching of anode directed streamers in a strong homogeneous background field at times t = 300, 365, 420 and 450. Model, initial and boundary conditions are discussed in the text. The planar cathode is located at z = 0 and the planar anode at z = 2000 (shown is 0 ≤ z ≤ 1400). The radial coordinate extends from the origin up to r = 2000 (shown is 0 ≤ r ≤ 600). The thin lines denote levels of equal electron density σ with increments of 0.1 or 0.2 as indicated by the labels. The thick lines denote the higher electron de...

An adaptive grid refinement strategy for the simulation of negative streamers

“…After the first claim [13] that a streamer filament within this deterministic continuum model in a sufficiently high field can evolve into an unstable state where it branches spontaneously, streamer branching was observed in more simulations [14,15,16]. While the physical nature of the Laplacian instability was elaborated in simplified analytical models [13,17,18,19,20], other authors challenged the accuracy of the numerical results [21,22,23,24].…”

“…While the physical nature of the Laplacian instability was elaborated in simplified analytical models [13,17,18,19,20], other authors challenged the accuracy of the numerical results [21,22,23,24]. Based on purely numerical evidence, their questions were justified, as all simulations within the minimal model up to now were carried out on uniform grids, and the numerical convergence on finer grids could hardly be tested.…”

“…Up to now, all simulations that have been carried out on the minimal streamer model were performed on a uniform grid [26,28,13,14]. However the use of a uniform grid on such a large system has several limitations.…”

“…In [13,14] the simulations were performed in a radially symmetric geometry with 2000 × 2000 = 4 · 10 6 grid points. Since the number of variables is of at least 10 per grid point (the electron and ion densities both at old and new time step, the electric potential, the electric field components and strength, and the terms containing the temporal derivatives of both electrons and ions), the total number of variables will be at least 40 · 10 6 .…”

“…This procedure allows us to test the numerical convergence of branching, and also to calculate streamers efficiently in longer systems without introducing too many grid points. Moreover, the simulations in [13,14] were performed on the same uniform grid for both the particle densities and the electric potential. As the Poisson equation for the electric potential has to be solved on the complete large non-ionized outer space, this grid had to be much larger than the actual domain in which the particles evolve.…”