A Van der Waals-like theory of plasma double layers

Abstract

A new theory is presented that describes plasma double layers in terms of multiple roots of the charge density expression. Analogous to Van der Waals’ equation for simple fluids, the system is described using simple analytical expressions that contain the essential nonlinearity of the physics. Both theories predict multiple states and transitions between the states. Van der Waals’ theory is for the liquid–gas phase transition; the theory presented here is for double layers between two plasmas. Except within the double layer, the plasma is assumed to be quasineutral, that is, the charge density is almost zero. The expression used for charge density includes linear shielding at low potentials and current continuity at high potentials. The theory is independent of the details of the expression used for the charge density; it only requires that the charge density be a nonmonotonic function of potential. Multiple roots exist because of this nonlinearity; linear theories such as Debye shielding allow for only a single root. For two semi-infinite plasmas in planar geometry, the charge density has a single root when the separation of plasma potentials is less than a critical value. Above that, the root bifurcates into two symmetric branches that asymptotically approach the two boundary plasma potentials. The transition between the two roots is a double layer. For a plasma expanding spherically from a source into a uniform background plasma, the charge density equation roots have two branches that never cross. One branch corresponds to the source plasma and disappears abruptly at some radius away from the source. The other branch corresponds to the background plasma and disappears close to the source. The two branches coexist for a limited range of radii. A double layer provides the transition between the two branches. An ancillary condition, similar to the Maxwell construction, is used to locate this transition. The location of the double layers calculated using this theory is consistent with laboratory measurements.