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Kinetic simulation of plasmas in which equilibrium occurs over ion time scales poses a computational challenge due to disparity with electron time scales. Hybrid electrostatic particle-in-cell (PIC) algorithms are presented in which most of the electrons reach thermodynamic equilibrium [Maxwell–Boltzmann (MB) distribution function] each time step. Conservation of charge enables convergence of the nonlinear Poisson equation. Energy conservation is used to determine the temperature of the Boltzmann species. This article first develops an algorithm where all the electrons have a MB energy distribution, either with a full MB distribution or with a truncation of the high energy tail. Second, high energy PIC electrons are added to the truncated distribution so that high energy electrons are modeled kinetically by PIC and low energy electrons (the majority) are modeled by the MB distribution. Collisions for PIC electrons are included via a Monte Carlo model, while for the MB electrons, the distributions are integrated with energy dependent cross sections. The MB model is not constrained by the electron time scale which decreases the required computer time by about the square root of the mass ratio of ion to electron. However, the hybrid boundary conditions are more complex and the simulation is not quite self-consistent. Comparison between full PIC and the PIC–MB hybrid is made for simulations of photo-ionized sustained discharges and current-driven dc discharges.

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The electric field of two semi-infinitely wide knife-edge cathodes with arbitrary separation is calculated by using a Schwarz–Christoffel transformation. This geometry could also represent a trench (or scratch) on a flat surface. It is found that the magnitude of the electric field on the knife-edge cathodes depends strongly on the ratio h/a, where h is the height of the knife-edge cathodes and 2a is the distance between the cathodes. When h/a increases, the magnitude of the electric field on the cathode’s surface decreases. This shows the screening of one cathode by another cathode; for example, keeping the height fixed and decreasing the distance between the cathodes, the field enhancement on the corner decreases. Analytic approximations for the divergent electric field in the immediate vicinity of the sharp edge are derived for the cases where h/a>>1, and h/a≪1. These results lead to insight on the relationship of the density of field emitter in field emitting arrays and field emission from rough surfaces.

Abstract.A new approach to the kinetic simulation of plasmas in complex geometries, based on the Particle-in-Cell (PIC) simulation method, is explored. In the two dimensional (2d) electrostatic version of our method, called the Arbitrary Curvilinear Coordinate PIC (ACC-PIC) method, all essential PIC operations are carried out in 2d on a uniform grid on the unit square logical domain, and mapped to a nonuniform boundary-fitted grid on the physical domain. As the resulting logical grid equations of motion are not separable, we have developed an extension of the semi-implicit Modified Leapfrog (ML) integration technique to preserve the symplectic nature of the logical grid particle mover. A generalized, curvilinear coordinate formulation of Poisson's equations to solve for the electrostatic fields on the uniform logical grid is also developed. By our formulation, we compute the plasma charge density on the logical grid based on the particles' positions on the logical domain. That is, the plasma particles are weighted to the uniform logical grid and the selfconsistent mean electrostatic fields obtained from the solution of the logical grid Poisson equation are interpolated to the particle positions on the logical grid. This process eliminates the complexity associated with the weighting and interpolation processes on the nonuniform physical grid and allows us to run the PIC method on arbitrary boundary-fitted meshes.

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