Particle-in-cell study of the ion-to-electron sheath transition
  The form of a sheath near a small electrode, with bias changing from below to above the plasma potential is studied using 2D particle-in-cell (PIC) simulations. Five cases are studied: (A) an electrode biased more than the electron temperature ($T_e/e$) below the plasma potential, (B) an electrode biased less than $T_e/2e$ below the plasma potential, (C) an electrode biased nearly at the plasma potential, (D) an electrode biased more than $T_i/2e$ but less than $T_e/2e$ above the plasma potential, and (E) an electrode biased much greater than $T_e/2e$ above the plasma potential. In case (A), the electron velocity distribution function (EVDF) is observed to be Maxwellian with a Boltzmann-type exponential density decay through the ion sheath and presheath. In cases (B) and (C), the EVDFs exhibit a loss-cone type truncation due to fast electrons overcoming the small potential difference between the electrode and plasma. No sheath is present in this regime, and the plasma remains quasineutral up to the electrode. The EVDF truncation leads to a presheath-like density and flow velocity gradient. In case (D) an electron sheath is present, and essentially all ions are repelled. Here the truncation driven behavior persists, but is accompanied by a shift in the maximum value of the EVDF that is not present in the negative bias cases. In case (E), the flow shift becomes greater and the loss-cone moves further into the tail of the EVDF. In this case the flow moment has significant contributions from both the flow shift of the EVDF maximum, and the loss-cone truncation. 
Brett Scheiner, Scott D. Baalrud, Matthew M. Hopkins, Benjamin T. Yee,   Edward V. Barnat
physics.plasm-ph