Numerical studies of ion focusing behind macroscopic obstacles in a supersonic plasma flow W. J. Miloch * and J. Trulsen University of Oslo, Institute of Theoretical Astrophysics, Box 1029 Blindern, N-0315 Oslo, Norway H. L. Pécseli Department of Physics, University of Oslo, Box 1048 Blindern, N-0316 Oslo, Norway Received 11 December 2007; revised manuscript received 22 March 2008; published 29 May 2008 We study the potential and plasma density variations around a solid object in a plasma flow, emphasizing supersonic flows. These objects can be dust grains, for instance. Conducting as well as insulating materials are considered. In a streaming plasma, a dust grain develops an electric dipole moment, which varies systemati- cally with the relative plasma flow. The strength and direction of this dipole moment depends critically on the material. The net charge together with the electric dipole associated with the dust grains gives rise to electric fields, which affects the trajectories of nearby charged particles. The perturbation of ion orbits in streaming plasmas can give rise to a focusing of ions in the wake region facing away from the plasma flow. We study the parameter dependence of this ion focus. Our simulations are carried out in two spatial dimensions by a particle-in-cell code, treating ions and electrons as individual particles. DOI: 10.1103/PhysRevE.77.056408 PACS numbers: 52.27.Lw, 52.65.Rr I. INTRODUCTION The potential distributions surrounding solid objects, such as dust grains, in plasmas depend critically on the plasma conditions 1. In thermal equilibrium we find the standard electrostatic Debye shielding. When the grains are exposed to a streaming plasma the conditions can change signifi- cantly. One conspicuous new feature that has been noted is manifested by a focusing of ions in the region behind the grain, in the wake facing away from the flow direction 2. The ion focusing is shown to be the most likely candidate to explain the vertical alignment of macroscopic dust grains that are levitated in the sheath region of discharges used in experiments 3–5. This nonreciprocal vertical interaction between dust grains can, under certain conditions, be the source for an oscillatory instability, which leads to a phase transition of crystalline dust structures 6. The other ap- proach, which concerns the linear response of a collisionless plasma with the ions flowing around a pointlike dust grain, shows an oscillatory wake potential behind the dust 7–11. This analysis can be extended to include dust with a finite size and anisotropic charge distribution on its surface. It is shown theoretically that the finite size becomes important for dust radii comparable to the Debye length, i.e., a   D 12. The electric dipole moment will also have a strong influence on the strength and position of potential extrema 12,13. The test charge approach used for different electron to ion temperature ratios, shows that the perturbations in the poten- tial are stronger for larger ratios 14. Vertical alignment of dust grains is generally accepted to be caused by ion focus- ing, while the ion drag force 15 only modifies this effect 16,17. The full problem is in reality far more complicated than that considered in theoretical approaches. The linear theory will not hold for highly charged dust grains, when ion trajec- tories are highly distorted and some of the ions are trapped. The standard analysis does not include the wake in the plasma density, which develops due to the plasma flow around the object of finite size. We refer to the density wake as the region in the shadow of the object, where the electron and ion densities are reduced by more than approximately 50%. Therefore, for the more complete description one could consider numerical simulations, which can address nonlinear problems in a consistent way. Previous numerical works revealed the existence of an ion focusing region behind an object or a dust grain and studied this phenomenon for some selected cases and parameters 18–21. The ion focusing was also observed in other nu- merical simulations for charging of individual dust grains or objects in plasmas, where it was attributed to orbital effects in the vicinity of the object 15,22,23. Many numerical models, which were used in previous studies of the ion fo- cusing effect, were not fully self-consistent concerning the charging of the object. The dust grain was represented either as an object of finite size that was biased at an arbitrarily chosen potential 18, or by a pointlike grain with a fixed charge 20,21. Thus, these simulations could only mimic conducting objects with some uncertainty for the floating potential. However, once the finite size effect of the object becomes important, the disturbance in the plasma induces an electric dipole moment on the conducting object. This is ob- served or mentioned already for moderate radii of the object 23,24. The charge distribution becomes even more con- spicuous for insulating objects. Here, it was shown that due to the flow, an electric dipole moment will develop, which strongly influences the surrounding plasma 23,25–27. Since insulating dust grains are often used in laboratory ex- periments of dusty plasmas, one should consider insulating dust grains also in simulations. Self-consistent simulations should also include the stochastic nature of the dust charging process in plasmas. Therefore, the challenge for numerical simulations is to mimic the entire charging process. Such a self-consistent approach was proposed in particle-in-cell PIC simulations for small conducting objects with the * w.j.miloch@astro.uio.no PHYSICAL REVIEW E 77, 056408 2008 1539-3755/2008/775/0564089 ©2008 The American Physical Society 056408-1