Two floating sheath solutions with strong electron emission in planar geometry have been proposed, a “space-charge limited” (SCL) sheath and an “inverse” sheath. SCL and inverse models contain different assumptions about conditions outside the sheath (e.g., the velocity of ions entering the sheath). So it is not yet clear whether both sheaths are possible in practice, or only one. Here we treat the global presheath-sheath problem for a plasma produced volumetrically between two planar walls. We show that all equilibrium requirements (a) floating condition, (b) plasma shielding, *and* (c) presheath force balance, can indeed be satisfied in two different ways when the emission coefficient γ > 1. There is one solution with SCL sheaths and one with inverse sheaths, each with sharply different presheath distributions. As we show for the first time in 1D-1V simulations, a SCL and inverse equilibrium are both possible in plasmas with the same upstream properties (e.g., same N and T_{e}). However, maintaining a true SCL equilibrium requires no ionization or charge exchange collisions in the sheath, or else cold ion accumulation in the SCL's “dip” forces a transition to the inverse. This suggests that only a monotonic inverse type sheath potential should exist at any plasma-facing surface with strong emission, whether be a divertor plate, emissive probe, dust grain, Hall thruster channel wall, sunlit object in space, *etc*. Nevertheless, SCL sheaths might still be possible if the ions in the dip can escape. Our simulations demonstrate ways in which SCL and inverse regimes might be distinguished experimentally based on large-scale presheath effects, without having to probe inside the sheath.

## I. INTRODUCTION

A floating sheath solution under strong electron emission was first theorized in the seminal 1967 paper “*Heat Flow Through a Langmuir Sheath in the Presence of Electron Emission*” by Hobbs and Wesson.^{1} The authors added an emitted electron population to Langmuir's sheath equations and showed that when the emission coefficient γ approaches unity, the negative sheath potential develops a “potential well”. They predicted that if γ continues increasing beyond unity, the well should suppress the additional emission, saturating the sheath potential at a negative value. In a more recent paper,^{2} a fundamentally different positive valued “inverse sheath” potential was predicted to be possible at any surface with γ > 1.

It is important to determine whether space-charge limited (SCL) and inverse sheaths are both possible at strongly emitting surfaces, or whether just one type is physically realizable. Both analytical solutions appear valid on paper, but they rely on initial assumptions. The Hobbs-Wesson model and other SCL sheath derivations^{1,3–8} postulate that the sheath potential is negative, and then derive a self-consistent set of charge density and potential distributions in the sheath. The inverse sheath model^{2} postulates that the sheath potential is positive and derives another self-consistent set of density and potential distributions. SCL and inverse sheath solutions also contain contradictory assumptions about the ion flow velocity at the sheath edge. For either kind of sheath to exist, a compatible presheath structure must be present. The presheath side of the strong emission problem has not been studied as much.

Further theoretical analysis is needed to determine whether one, two, or perhaps more equilibria are possible when plasmas contact strongly emitting surfaces. We could try to answer the question by looking at past experiments on sheaths at emitting surfaces.^{7,9–13} Although there is some evidence consistent with the SCL sheath theory and some consistent with inverse sheath theory, depending on interpretation, there is not enough information to conclusively identify the equilibrium states for reasons we discuss later. Plasma-surface interaction with emission has also been analyzed fundamentally by 1D simulations.^{3,6,8,14–18} But most of the past studies used source boundary plasma injection simulations that do not model a bounded plasma in a realistic way; natural presheath gradients associated with charge production and collisions are absent. Careful consideration of the presheath problem is crucial not only for ensuring a realistic sheath formation but also for predicting the large-scale (easier to measure) effects of emitting surfaces in experiments.

This paper will present a unifying analysis of the strong emission problem and address open questions. In Sec. II, we show on theoretical grounds that whenever the emission coefficient exceeds unity, two distinct equilibria are possible regardless of the upstream bulk plasma properties. One has SCL sheaths and Bohm presheaths, while the other has inverse sheaths and inverted presheaths. Both are demonstrated and compared in direct kinetic simulations with volumetric sources and collision operators. Time-dependent simulations in Sec. III show that the initial conditions and collisions in the sheath determine whether the system evolves to a SCL or inverse equilibrium. In Sec. IV, we discuss plasma applications with emitting surfaces where the differences between SCL and inverse sheaths have key practical consequences. In Sec. V, we offer new suggestions for distinguishing SCL and inverse equilibria in experiments, and revisit previous measurements in the literature. Concluding remarks are given in Sec. VI.

## II. SCL AND INVERSE REGIMES – DEMONSTRATION AND COMPARISON

In this section, we will determine from first principles what steady states are possible when a plasma interacts with emitting boundaries. We will support the theoretical aspects of our discussion with 1D-1V simulations. Our direct kinetic code produces noise-free data, giving a crisp illustration of the sheath and presheath structures. Including volumetric sources and collisions enables us to demonstrate realistic aspects of the equilibrium states that are not observable in simulation models such as Refs. 3 and 8 with source boundary plasma injection.

### A. Simulation model

The simulations here are configured in such a way that the upstream plasma properties are the control variables, sustained irrespective of wall losses and varying emission intensity. We start with a uniform Maxwellian plasma state with length L = 10 cm, temperatures T_{e} = 4 eV, and T_{i} = 0.05 eV, and density N_{0} = 10^{15 }m^{−3}. These parameters resemble a typical low temperature laboratory plasma where the electrons are a few eV, and the ions originate from a colder neutral gas. The ion mass m_{i} is 1 amu. The ratio λ_{D}/L exceeds 200.

When the simulation starts, the plasma evolves and interacts with the walls. Time advancement for the electron and ion species is governed by the kinetic equations solved over a uniform 1D-1V grid.

The number of spatial grid points 3001 is sufficient for good resolution of the resulting sheaths. A grid of 350 velocity points for each species, spread over their relevant velocity ranges, enables good resolution of the VDFs (velocity distribution function). Advancement of (2) is carried out in four substeps: *x* advection, *v* advection (governed by the calculated electric field E), charge source, and collisions. Charge source i-e pairs with the same initial state temperatures in f_{0(e,i)} (1) are produced uniformly in space except for the regions 1 cm from the walls. The total charge produced each time step is adjusted proportional to the ion flux Γ_{i} in order to balance the wall losses, thereby keeping the mean density ⟨N⟩ equal to the chosen initial value N_{0}.

Electrons undergo BGK (Bhatnagar-Gross-Krook) “thermalization collisions” that relax the EVDF (electron VDF) at each x towards a 4 eV Maxwellian. For this study we use a coefficient e_{therm} = 7 just large enough to keep the “upstream” (near the midplane) plasma electrons at a 4 eV Maxwellian. If much lower values of e_{therm} are used, the equilibrium midplane EVDF begins to show a loss tail and peaks from emitted electrons reaching there.

For ion collisions, we use a realistic charge exchange (CX) operator (5) that removes fast ions from f_{i} at each x and replaces them with an equal number of cold ions. The collision frequency should be proportional to velocity, such that the mean free path is independent of velocity. For simulations here, we set i_{mfp} to half the plasma width (5 cm), which is large compared to the sheath widths (≤5 mm).

Emitted electrons have a Maxwellian distribution with a typical thermionic temperature T_{emit} = 0.3 eV. Emission is controlled by specifying the plasma-facing side of the EVDF at the boundaries. It is updated at each time step such that the total emitted flux equals a chosen coefficient γ times the plasma electron influx Γ_{ep} from the previous step. For the left wall

Γ_{ep} is calculated at the wall if φ(x) is classical monotonic. If an emission barrier is present in φ(x), Γ_{ep} is evaluated from the wall-directed flux at the potential minimum, to make sure that the emitted electrons returned to the wall are not counted in Γ_{ep}.

### B. Overview of the equilibrium states

Using the uniform plasma initial condition (1) and following the advancement process (2–6), the plasma will evolve to an equilibrium with sheaths and presheaths. Setting γ = 0 leads to a “classical” 1D bounded plasma equilibrium as described in Chapters 1 and 2 of Stangeby's^{19} book. In the following analysis, we will compare the (a) classical equilibrium as a control case, (b) SCL equilibrium with γ = 10 *and* (c) inverse equilibrium with γ = 10. As designed, the simulated plasmas in all three cases have the same “upstream” properties, e.g., same ⟨N⟩ and same T_{e} at the midplane, to facilitate a direct comparison among the cases. The only difference between the setup of the SCL and inverse cases is that the CX collisions were turned off in a thin region 3.33 mm from the walls in the SCL in order to prevent cold ions from becoming trapped in the virtual cathode. The charge source cutoff in Eq. (3) was used in all runs for the same reason. In Sec. III we will show what happens if ions accumulate in a SCL sheath.

To compare the equilibria, we start from the potential distributions φ(x) shown in Figs. 1(a) and 1(b). It is important to distinguish the sheath potential drop Ф_{sh} ≡ φ_{wall} - φ_{se} from the presheath drop Ф_{ps} ≡ φ_{se} – φ_{mid} (“se” denotes the sheath edge). The SCL case has the same presheath drop as the classical γ = 0 case, but the sheath potential is weaker, and is nonmonotonic. The inverse case carries a much weaker presheath drop, and has a *positive* sheath potential (φ_{wall} > φ_{se}).

Figs. 1(c) and 1(d) show representative trajectories of particles that enter the sheaths. The lines were calculated by tracing contours of equal f_{e,i}. The contours are not exact representations of trajectories due to collisions, but are a decent approximation over the sheath length. Electrons entering the classical and SCL sheaths are decelerated. Those entering with small velocities are reflected back into the plasma, while those entering with sufficiently high velocities manage to reach the wall. The SCL's potential dip returns some emitted electrons to the wall, while the rest get accelerated into the plasma. Because the inverse sheath potential is positive, all electrons entering it flow into the wall. The inverse sheath pulls some emitted electrons back to the wall, allowing the rest to enter the quasineutral plasma. Ion trajectories show that all ions entering the sheath are lost in the classical and SCL cases. They get accelerated by the sheath to velocities far above the ion thermal velocity. In the inverse case, thermal velocity ions entering the sheath are decelerated and return to the plasma.

The color plots in Fig. 2 show the equilibrium distribution functions f_{e,i}(x,v_{e,i}) from the left wall to the midplane. The classical and SCL f_{e} have closed contours, while the inverse f_{e} does not. Another unique aspect of the inverse case is that the maximum f_{e} in the quasineutral plasma is located near the edge rather than the midplane. The classical and SCL f_{i} show that ions everywhere are moving preferentially towards the nearest wall. Conversely, the inverse f_{i} shows that ions move in both directions.

Overall, the structures of the color contours in Fig. 2 suggest that the structure of a plasma facing SCL sheaths is qualitatively similar to the well-understood classical case. On the other hand, the inverse regime is a unique mode of plasma-surface interaction. We will find that the distributions of all important quantities including, density, pressure, and flow velocity not just in the sheath but throughout the plasma are very different in the inverse equilibrium compared to the classical and SCL.

### C. Why are two strong emission equilibria possible?

Some properties of SCL and inverse sheaths have been predicted in previous theoretical papers. But the theoretical derivations depend on the initial assumptions about sheath edge quantities. How can these assumptions be correct in one equilibrium situation, but break down in another? Here we will explain from first principles why SCL and inverse sheaths are both possible, and elucidate the presheath side of the problem.

For a plasma to be in equilibrium with a wall that draws no current, three interconnected conditions must be met, (a) floating condition, (b) wall shielding, *and* (c) presheath force balance. Below, we will review how the conditions are satisfied without emission, and then explain how two solutions appear under strong emission.

#### 1. Floating condition

Normally at a floating surface, a classical negative sheath potential forms to limit the electron flux to the ion flux.^{19} In Eq. (7), the left hand side gives the electron flux from a Maxwellian electron source of temperature T_{e} in terms of the sheath potential Ф_{sh}. The factor (1−γ) accounts for emission, where γ is defined as the average number of electrons emitted per incident plasma electron. The right hand side gives the ion flux assuming that ions enter a source-free sheath at the sound speed. The equilibrium Ф_{sh} is then calculable in terms of known plasma properties. For the T_{e} and m_{i} used here, Eq. (7) predicts Ф_{sh} = −11.3 V when γ = 0, close to the observed value −11.6 V.

Emission weakens the sheath potential in a manner predictable from Eq. (7). But it is not obvious what happens when γ > 1 as the left hand side is negative for any Ф_{sh}, and the ion flux cannot be negative. When Hobbs and Wesson added emission to the conventional sheath equations, their calculation showed that when γ reaches a critical value γ_{cr} below unity and |q_{e}|Ф_{sh,cr} ≈ −1T_{e}, the electric field at the wall reaches zero.^{1} They predicted that if γ rises further, a potential dip would form to suppress the extra emission, saturating the “effective γ” near γ_{cr} and saturating the sheath potential near Ф_{sh,cr}. (Later in Sec. III Fig. 7 we will show the dip formation and saturation effect.) Sheaths with a finite dip are more complex analytically, so most papers treat the marginal case. Marginal SCL sheath models can be used to estimate φ_{min} − φ_{se}, where φ_{min} is the potential at the dip minimum. The SCL sheath shown in Fig. 1(b) has φ_{min} − φ_{se} = −2.8 V = −0.7T_{e}/|q_{e}|, consistent with Sheehan's model^{6,7} which for T_{e}/T_{emit} = 4.0/0.3 predicts φ_{min} − φ_{se} ≈ −0.65T_{e}/|q_{e}|.

An inverse sheath potential can also maintain the floating condition when γ > 1. The fundamental reason is that whenever γ > 1, the wall no longer needs to attract ions.^{2} In an inverse regime, the ion flux can be as low as zero. The wall draws the full thermal flux of electrons from the plasma, as the trajectories show in Fig. 1(c). The inverse sheath allows the same flux of emitted electrons to enter the plasma, and pulls back the “extras,” so that the *net* electron flux to the wall is also zero. The inverse sheath potential that maintains current balance^{2} is Ф_{sh} = T_{emit}ln(γ)/|q_{e}|. It does not depend explicitly on m_{i}, in contrast to the negative Ф_{sh} in Eq. (7). The Ф_{sh} = 0.7 V in the inverse simulation agrees with the predicted 0.3 eV × ln(10)/|q_{e}|.

SCL and inverse sheaths lead to sharply different particle and energy influxes from the plasma, as compared in Table I. Due to the lack of electron confinement, the influx from plasma electrons Γ_{ep} in the inverse state is 4 times larger than the SCL, and 33 times larger than the classical γ = 0 case. The emitted flux is larger in the inverse case than the SCL because the emission coefficient was used as a control variable, Γ_{emit} = γΓ_{ep}. In both cases, most emitted electrons are returned to the wall, and any further increase of emission intensity would increase the returned flux without significantly changing the influx of plasma electrons or ions. The electron *heat flux* is largest in the inverse case due to the high Γ_{ep}. The mean impact energy per electron is roughly 1T_{e} in all cases as is expected^{19} for a 1D-1V Maxwellian source, but is slightly higher in the SCL and inverse cases because of the extra energy gained accelerating from the potential minimum to the wall. In the classical and SCL states, the impact energies of ions are high relative to T_{i} due to the accelerating potentials. In the inverse case, both the flux of ions and their impact energies are very low. The ion flux in realistic inverse equilibrium situations will never be exactly zero because some ions will be able to overcome the positive sheath potential due to their thermal energies, or due to energy gain in a presheath.

Quantity . | Classical . | SCL . | Inverse . |
---|---|---|---|

γ | 0 | 10 | 10 |

Γ_{i} (10^{19 }m^{−2} s^{−1}) | 1.03 | 1.04 | 0.09 |

Γ_{ep} (10^{19 }m^{−2} s^{−1}) | 1.03 | 8.95 | 33.89 |

Φ_{sh} (V) | −11.6 | −2.1^{a} | 0.7 |

Φ_{ps} (V) | −3.2 | −3.2 | −0.7 |

Mean ion impact energy | 14.15 | 4.50 | 0.28 |

Mean electron impact energy | 3.90 | 4.61 | 4.38 |

n_{edge}/n_{mid} | 0.41 | 0.42 | 2.23 |

Quantity . | Classical . | SCL . | Inverse . |
---|---|---|---|

γ | 0 | 10 | 10 |

Γ_{i} (10^{19 }m^{−2} s^{−1}) | 1.03 | 1.04 | 0.09 |

Γ_{ep} (10^{19 }m^{−2} s^{−1}) | 1.03 | 8.95 | 33.89 |

Φ_{sh} (V) | −11.6 | −2.1^{a} | 0.7 |

Φ_{ps} (V) | −3.2 | −3.2 | −0.7 |

Mean ion impact energy | 14.15 | 4.50 | 0.28 |

Mean electron impact energy | 3.90 | 4.61 | 4.38 |

n_{edge}/n_{mid} | 0.41 | 0.42 | 2.23 |

^{a}

Note that in the SCL state, the sheath potential is −2.1 V when defined as φ_{wall} − φ_{se}, but the barrier seen by plasma electrons entering the sheath is φ_{min} − φ_{se} = −2.8 V. Also note the outcome that the sheath and presheath potentials have the same amplitude in the inverse case, is a coincidence for this simulation, not a universal property of inverse equilibria.

We note the charge source production is also lower in the inverse case by design, to keep ⟨N⟩ equal in each case. That way the influx variations in Table I are attributable to the different sheath and presheath potentials rather than density. If the charge production rate in a plasma was the controlled quantity, then the equilibrium ⟨N⟩ would be higher when an inverse sheath is present compared to a classical or SCL sheath.

#### 2. Wall shielding

For a plasma equilibrium to be possible, the sheath not only has to maintain the floating condition but must also shield the plasma from the charge on the wall. Shielding is ensured in analytical derivations by requiring that the electric field decays towards zero far from the wall.^{20} In a bounded plasma, the electric field amplitude should reach zero at a single upstream point, and not oscillate in space. The shielding condition puts some hidden restrictions on the properties of the possible classical, SCL, and inverse equilibrium states.

Shielding requires that the total charge in the sheath be opposite to the wall charge. When γ = 0, a negatively charged wall is shielded by a classical sheath with a single layer of net positive charge, see Fig. 3(a). Ions need to enter the sheath pre-accelerated to a “Bohm speed” v_{Bohm} to enable n_{i} to drop more slowly than n_{e} in the sheath.^{20} The mean flow speed seen at the classical sheath edge in Fig. 3(d) is close to the typical sonic Bohm speed estimate^{19} v_{Bohm} = c_{s} = (T_{e}/m_{i})^{1/2}. Calculating the Bohm speed is more intricate when kinetic effects^{21} and multiple ion species^{22} are considered. A key general consequence of the Bohm criterion is that it essentially determines the ion flux, $\Gamma i$ = n_{edge}v_{Bohm} and allows just one classical sheath potential equilibrium for each γ < 1. There could be infinitely many sets {$\Gamma i$, Ф_{sh}} that satisfy the floating condition (7). But ion fluxes below the Bohm flux are incompatible with the shielding requirement, while fluxes exceeding the Bohm flux are usually incompatible with presheath requirements (a singularity would develop).

How does the shielding work when the surface emits a strong electron flux? We can start with the fact that to maintain the floating condition when γ > 1, there must be an emission barrier, which requires there to be a positively charged wall and a negatively charged sheath. The SCL and inverse sheath in Fig. 3 are both physically possible because both have a net negative charge dominated by emitted electrons. The key difference is that the inverse sheath is a single layer of net negative charge, while the SCL is a double layer whose total charge is negative. The positive charge layer in the SCL sheath in Fig. 3(b) looks subtle in magnitude but has a profound influence on the solution because it requires the same Bohm criterion to be satisfied as for a non-emitting surface. SCL sheath models predict that emitted electrons reaching the sheath edge only cause a small modification to v_{Bohm}.^{1,4,6} It follows that a SCL equilibrium should have a similar sheath edge flow velocity and similar Γ_{i} as when γ = 0. This is observed in our simulations, see Table I and Fig. 3(d).

The inverse sheath model^{2} drops the Bohm condition and makes a different initial assumption about ion flow; the ions are cold and have zero velocity at the sheath edge. This ensures that n_{i} drops to zero in the sheath, and the required negative shielding charge layer easily forms. In experiments or simulations, n_{i} will drop more slowly in the sheath due to finite ion energy/temperature, as in Fig. 3(c). But one can be confident that an inverse solution should always exist. Because n_{e} increases from the inverse sheath edge towards the wall, the charge in this region is still everywhere negative no matter how slowly n_{i} decreases. That self-consistently enables a solution where φ(x) monotonically increases from the sheath edge to the wall. It is interesting to observe that from the potential minimum to the wall, n_{i} decreases in the inverse case because most ions entering the sheath get turned around, but n_{i} *increases* from the potential minimum to the wall in the SCL case because the ions decelerate and still reach the wall.

It is useful to predict the width of sheaths in terms of plasma properties. If we put control parameters <N> = 10^{15 }m^{−3} and T_{e} = 4 eV into the typical estimate^{19} of classical sheath width, ∼10λ_{D} = 10(ε_{0}T_{e}/q_{e}^{2}N)^{1/2}, we get 4.7 mm, in good agreement with where quasineutrality begins to break down around x = 5 mm in Fig. 3(a). In theory, a SCL sheath should be similar in width to the classical, but the edge location may be more ambiguous because the positive charge layer is weaker. By inspection n_{e} and n_{i} in the SCL sheath also begin to separate around x = 5 mm (the slight drop of n_{i} at x = 3.33 mm is due to the CX cutoff, unrelated to sheath formation). Inverse sheaths have a sharper edge because n_{e} and n_{i} separate in opposite directions. The width of an inverse sheath was predicted in Ref. 2 to be (2ε_{0}T_{emit}lnγ/q_{e}^{2}N)^{1/2} which will be much smaller than a classical sheath width for the same N and T_{e}. The prediction gives 0.28 mm, in good agreement with the observed 0.3 mm inverse sheath width in Fig. 3(c).

#### 3. Presheath force balance

Although SCL and inverse sheaths can both self-consistently satisfy the floating condition and shielding requirements, we saw that their existence depends on how ions enter the sheath. To prove that both sheaths are realistic, it is necessary to show that each has a self-consistent, realistic presheath solution. There would actually be an infinite number of inverse-like and SCL-like sheath solutions on paper if the ion velocity at the sheath edge were arbitrary. Most solutions are not compatible with any presheath that could form. The allowed presheaths must have spatial gradients that maintain a force balance everywhere and enable a smooth transition to the corresponding sheath.^{19} Presheath gradients are not captured in simulations that inject plasma at a source boundary into a collisionless domain^{3} (where the quasineutral plasma is flat and often has a supersonic flow velocity). Our simulations offer a realistic demonstration of the presheath-sheath matching, even though the source and collision terms are not configured to model a real device.

The γ = 0 control simulation reproduces the familiar “Bohm presheath.” Its main purpose is to accelerate ions from zero velocity at the midplane to the Bohm speed ∼c_{s} at the sheath edge, as was seen in Fig. 3(d). The plasma density generally drops along a Bohm presheath as ions accelerate (although the density can rise if T_{e} drops at a sufficient rate^{19}). Because electrons get Maxwellized towards 4 eV by BGK collisions, and wall losses only deplete a small tail of the EVDF, the electrons in the classical simulation are roughly isothermal with T_{e} = 4 eV. Therefore, the Boltzmann approximation accurately relates the plasma density and potential, see Fig. 4(a). Under isothermal T_{e}, presheath models predict the density at the edge relative to the midplane n_{edge}/n_{mid} to be about 0.5, or less if significant ion-neutral friction is present.^{19} Here the ion friction is weak (CX mean free path is half the system size) and n_{edge}/n_{mid} = 0.41. The observed presheath drop Ф_{ps}= −3.2 V = −0.8T_{e}/|q_{e}| is consistent with typical theoretical estimates.^{19}

As discussed earlier, SCL sheath existence also requires the Bohm criterion to be satisfied at the sheath edge. But does that mean the entire presheath in a SCL equilibrium is similar to when γ = 0? To address this question, one must consider what emitted electrons do to the presheath. In a SCL equilibrium, emitted electrons enter the presheath with a flux much larger than the Bohm ion flux and thermalize over a collisional length, so their effect on the presheath gradients cannot be assumed negligible. Ahedo studied the collisional presheath-sheath matching problem with emission thermalization and predicted that for given plasma properties, γ weakly affects the presheath solution.^{4} This has not yet been tested in an empirical way to our knowledge. Source injection models^{3,8} do not capture the collisional thermalization effects (emitted electrons transit to the source boundary and disappear).

The BGK thermalization operator in Eq. (4) thermalizes the emitted beam in a gradual manner. The SCL f_{e} in Fig. 2 shows that the beam enters the quasineutral region and dissipates over a few cm. The SCL equilibrium data corroborates Ahedo's^{4} predictions in that the distributions of potential, density, and ion flow velocity overlap with the γ = 0 run from the midplane to the sheath edge. Although we do not include intermediate data here, the presheath is almost identical for all γ when the sheath is classical or SCL. Only Ф_{sh} varies with γ according to the floating equation (7). One interpretation for why emitted electrons do not alter the presheath is that the negative sheath potential accelerates them to a low spatial density beam. As shown in the SCL sheath edge EVDF in Fig. 4(b), the emitted electrons entering the presheath form a peak with low density compared to the bulk electrons, so the presheath physics is dominated by the latter and the Boltzmann relation is still a good approximation, Fig. 4(a). The presheath should remain unchanged even if γ increases beyond 10, as the extra emission gets suppressed by a larger dip and does not reach the presheath. Overall, this is the final piece of the puzzle needed to confirm that a SCL sheath with Bohm presheath is a possible equilibrium state for any γ > 1.

The inverse sheath also has a compatible presheath, with numerous differences compared to Bohm presheaths. Some properties of inverted presheaths were first shown in Hall discharge particle-in-cell (PIC) simulations in Ref. 17. Here we make the uniqueness of the inverse regime more clear by directly comparing it to the classical and SCL equilibrium for a plasma with the same ⟨N⟩ and T_{e}. First, what determines an inverted presheath potential distribution? In the idealized limit T_{ion} → 0, φ(x) must be flat in the inverted presheath. Otherwise, there would be a local potential minimum somewhere, and all trapped ions would settle there. We can see from the trajectories in Fig. 1(d) that thermal ions are trapped and the presheath drop is quite weak in Fig. 1(a), but Ф_{ps} is still nonzero due to finite ion energy effects discussed soon.

Another general property of inverted presheaths is that the plasma density gradient is opposite in sign to the typical Bohm presheaths, as shown in Fig. 4(a). To see why, first consider the EVDF at the inverse sheath edge. Because all plasma electrons entering an inverse sheath get lost, the only electrons entering the plasma from the sheath edge originate from the wall, as evident by the trajectories in Fig. 1(c). Making reasonable assumptions that electrons are emitted with a Maxwellian velocity distribution with some temperature T_{emit}, and neglecting electron collisions in the sheath, those reaching the inverse sheath edge have a (half) Maxwellian distribution with temperature T_{emit} (as they do not get accelerated). The plasma electrons entering the sheath will have a velocity distribution which we assume for discussion purposes is (half) Maxwellian with some temperature T_{e,in}. So the EVDF at the inverse presheath-sheath edge takes the form of two half-Maxwellians with temperatures T_{e,in} and T_{emit}, see Fig. 4(b). Zero current at the *edge* requires that the partial densities of the incident and outgoing electron components satisfy

Even when the ion flux is nonzero, its contribution to current balance will not significantly modify the relationship in (8). In general, T_{emit} will be less than T_{e,in}, meaning that the cold electrons will dominate the plasma density at the presheath edge. The high density of cold electrons entering the presheath is evident by the white cloud in the inverse f_{e} in Fig. 2 and is responsible for the breakdown of the Boltzmann relation for the inverse case in Fig. 4(a).

In an ideal inverse regime, plasma ions are all trapped and no net charge is lost to the walls. Correspondingly, no net charge production takes place in the presheath. (In a device, any ions created above the surface would have to be lost by recombination or by escape along other dimensions.) There can also be no net flow velocity of ions (or electrons) in the presheath in the direction normal to the surface. Therefore the source-dependent and velocity-dependent terms that normally appear in the 1D moment equations will vanish in inverted plasmas.^{17} It follows that the total pressure gradient must vanish.

For now let us assume dp_{i}/dx ≪ dp_{e}/dx. (In the ideal T_{i} → 0 limit, p_{i} = 0 anyway.) Neglecting ion pressure is a valuable approximation because we can deduce that p_{e} at the edge must equal p_{e} at the midplane, irrespective of the complex collisional processes that mix the hot and cold electrons in between.

Solving for n_{e,in} and n_{e,out} by combining (8) and (10) leads to a simple relationship between the plasma density at the sheath edge (n_{edge} = n_{e,in} + n_{e,out}) and the midplane.

T_{e,mid} and T_{emit} correspond to the properties of the upstream bulk plasma and wall material that would be known or controllable in an experiment. T_{e,in} at the sheath edge may depend on the system under consideration. One could envision realistic situations with T_{e,in} anywhere from T_{e,mid} to T_{emit}. For example if an emitting plate is placed in a large plasma chamber with weak e-e collisionality, the bulk electrons will flow freely into the plate, so T_{e,in} equals the bulk electron temperature. In the other extreme, if e-e collisionality is strong enough to ensure Maxwellization throughout the presheath, then an equilibrium is possible where the EVDF is approximately Maxwellian everywhere, and T_{e} decreases from the source at T_{e,mid} to a (thermionically) emitting plate at temperature T_{emit}. In that case, the EVDF at the inverse sheath edge would be approximately full Maxwellian, such that T_{e,in} = T_{emit}. The plasma density would be inversely proportional to the temperature, keeping p_{e} uniform, n(x)T_{e}(x) = n_{e,mid},T_{e,mid}. There would be a heat flow driven by dT_{e}/dx. Or if there is no heating of electrons in the plasma, T_{e,mid} can equal T_{emit}, and dT_{e}/dx = 0. In other words, a near zero energy loss inverted plasma state is possible where ions of uniform density are trapped between thermionically emitting boundaries that emit and collect electrons at equal rates. A somewhat related gradient-free plasma between inverse sheaths has been demonstrated in PIC simulations, see Fig. 5 of Ref. 2.

Since T_{e,in} should be between T_{e,mid} and T_{emit}, it follows from Eq. (11) that the quasineutral plasma density in any inverted plasma should increase from the bulk to the sheath edge by a factor ranging from (T_{e,mid}/T_{emit})^{1/2} to T_{e,mid}/T_{emit}. This factor can be high because generally the bulk plasma temperature exceeds T_{emit}, often by more than an order of magnitude. Returning to the simulations here, recall that the charge source and BGK operator create 4 eV Maxwellian electrons throughout the volume. Hence in all simulations, the plasma electrons approaching the walls from the sheath edge are roughly 4 eV half Maxwellian. Based on Eq. (11), we can predict a density ratio for the inverse case of n_{edge}/n_{mid} of 4/(4 × 0.3)^{1/2 }= 3.7. The observed ratio is only 2.2. The discrepancy probably originates because the ion pressure is not negligible. We found that if the CX mean free path was reduced to 2 mm to help suppress ion pressure, n_{edge}/n_{mid} increases to 4.

Let us now address the effects of nonzero ion pressure. Recall that when the ions are cold, the presheath electric field must vanish, and no ion acceleration occurs. The only role of ions is to neutralize whatever density distribution the electrons take to maintain a uniform p_{e}. That requires n_{i} to increase with n_{e} towards the boundaries. So if the ions instead have a finite temperature (suppose uniform T_{i} for simplicity) there is an ion pressure force –m_{i}T_{i}(dn_{i}/dx) directed towards the midplane. A presheath electric field must then form to sustain the force balance on ions. Although the inverted presheath E field may carry the same sign as the Bohm presheath E field, see Fig. 1(a), its purpose is quite different. Fig. 5(a) shows that the E field offsets dp_{i}/dx in the inverse equilibrium. Meanwhile in the classical and SCL cases, the electric field's fundamental purpose is to *accelerate* ions, so the field force is unrelated to, and stronger than, dp_{i}/dx.

The presheath electric field in an inverse regime can accelerate ions and electrons “incidentally,” depending on its strength relative to thermal energies and collisional mean free paths. By inspection of the inverse f_{i} in Fig. 2, the ions appear to acquire higher speeds closer to the left wall, consistent with acceleration, but no ions reach the sound speed because the inverted presheath drop −0.7 V is much weaker than the −3.2 V Bohm presheath drops. The few ions that escape to the wall are replaced by the volumetric charge source. Most ions entering the inverse sheath reflect and return to the plasma. This is evident by the presence of positive velocity ions to the right of the sheath in the inverse f_{i}. The mean (flow) velocity of ions moving left and right is near zero. As long as the flow velocity is small, <v_{i}> ≪ c_{s}, Eq. (9) is still accurate, and the *total pressure* must be constant in the presheath. Figure 5(b) confirms that p_{i} + p_{e} is conserved in the quasineutral region in the inverse equilibrium, but not in the classical or SCL. Bohm presheaths intrinsically need a pressure drop.^{19}

## III. HOW ARE THE TWO STRONG EMISSION EQUILIBRIA REACHED?

We have confirmed that SCL and inverse equilibria are possible whenever the emission coefficient exceeds unity. This result is irrespective of the bulk plasma parameters such as temperature, density, collisionality, and ion species. An important question to consider now is, which equilibrium will occur in a given laboratory situation? We will show here that collisions in the sheath play a decisive role.

### A. Classical sheaths with and without CX collisions

Collisions are an important component of presheath modelling but are often neglected in models of the sheath. It is expected that collisions should not have a substantial effect when the mean free paths of electrons and ions are small compared to the sheath width. The ion mean free path in the previous simulations i_{mfp} = 50 mm is indeed much larger than the sheath widths (5 mm or less). To test the significance of CX collisions in the sheath, we compare the classical γ = 0 equilibriums in simulations with and without a S_{coll(i)} cutoff inside the sheath. Fig. 6 shows the equilibrium density profiles in the classical simulation analyzed in Sec. II, where the CX collisions were omitted within 3.33 mm of walls (as in the SCL case). Superimposed in Fig. 6 is the outcome of an equivalent classical simulation without the CX cutoff. Note that both cases show a noticeable charge distortion at x = 10 mm due to the *charge source* cutoff (3). Without the CX cutoff, some slow-moving ions are created near the wall, leading to a slightly higher n_{i} near the wall. The n_{e} there also increases slightly in order to keep the net shielding charge similar. The charge everywhere outside the sheath is virtually identical in both runs.

Table II confirms that the CX cutoff in the sheath does not significantly affect any important transmission factors. For example, the wall potential, ion flux, and the mean electron impact energy are equal to within a percent. The mean ion impact energy without the cutoff is less by 3%. This slight reduction is consistent with the expectation that about 3.33 mm/i_{mfp} = 6.6% of incoming ions will lose their energy in the cutoff region and only gain back about half of the sheath potential energy on average.

Quantity . | with CX cutoff . | without CX cutoff . |
---|---|---|

Γ_{i} (10^{19 }m^{−2} s^{−1}) | 1.034 | 1.029 |

φ_{wall} − φ_{mid} | −14.83 | −14.85 |

Mean ion impact energy | 14.15 | 13.73 |

Mean electron impact energy | 3.898 | 3.896 |

Quantity . | with CX cutoff . | without CX cutoff . |
---|---|---|

Γ_{i} (10^{19 }m^{−2} s^{−1}) | 1.034 | 1.029 |

φ_{wall} − φ_{mid} | −14.83 | −14.85 |

Mean ion impact energy | 14.15 | 13.73 |

Mean electron impact energy | 3.898 | 3.896 |

### B. Strongly emitting sheaths with and without CX collisions

The above comparison was a control case confirming that it is reasonable to neglect a small amount of CX collisions (i_{mfp} ≫ ∼λ_{D}) in planar sheath modelling. But this only holds when the sheath is classical monotonic. Next, we begin new simulations starting from the classical equilibrium states in Sec. III A and gradually increasing γ from zero to 10 in time. A reasonable experimental analogy would be raising the temperature of a plasma-facing metal to raise its thermionic flux. Sunrise can cause a similar increase of photoemission coefficient for rotating objects in space. A secondary emission coefficient at a surface can increase in time if T_{e} rises. (Here the source temperatures are kept the same as γ is raised.)

We observe in Fig. 7(a) in both simulations that the potential minimum becomes less negative as γ rises above zero, consistent with the sheath potential weakening expected by zero current considerations, Eq. (7). With and without the CX cutoff, the potential minima follow the same path until they begin to diverge at a time near when γ = 1, which is the time when the electric field sign reverses at the wall, as shown in Fig. 7(b).

In the run with the CX cutoff, no major changes to φ_{min} occur as γ continues increasing beyond unity. This is consistent with the conventional theories of space charge saturation (φ_{min} is the parameter that saturates rather than φ_{wall}). Fig. 7(c) shows that the dip is larger when γ = 10 than when γ = 2, and φ_{wall} is higher, but φ_{min} is similar. Gyergyek and Kovačič also observed the saturation of φ_{min} with increasing γ, see Fig. 4(d) of Ref. 15. But this outcome can only happen in simulations where no cold ions are created in the SCL sheath's dip.

In the run without the CX cutoff, we see that φ_{min} continues rising after γ passes unity in Figs. 7(a) and 7(c). The key difference in this run is that when the dip forms, CX ions created there get trapped and start accumulating. At t = 0.75 *μ*s, the IVDF at the dip minimum in Fig. 7(d) shows a peak of zero velocity ions. Ion accumulation is what drove φ_{min} upwards. To understand why it happens, we first note that when cold ions collect in a potential well, their space charge will weaken the well until the ions can escape. In fact it was suggested long ago by Intrator *et al*. that the dip of any SCL-like sheath should get destroyed.^{9} The sheath would then look “classical” and all ions created in the sheath would be able to escape to the wall. To our knowledge, it was not previously recognized that a SCL sheath with a flattened dip cannot exist because the floating condition could not be sustained. Whenever γ > 1, there must be an emission barrier potential at the wall of at least T_{emit}ln(γ)/|q_{e}|. For this reason, as CX ions accumulate in the SCL dip, the sheath potential is forced to transition to a monotonic inverse structure so that (a) the floating condition is still maintained *and* (b) ions created in the sheath can escape to the *plasma* side. A temporal transition from a SCL-like sheath to inverse sheath was illustrated in more detail in our previous paper.^{18}

The SCL and inverse equilibrium states analyzed in this paper in Sec. II were reached by continuing the simulations in this section to a true equilibrium with γ = 10. The CX cutoff in the SCL equilibrium simulation ensures that all ions created by S_{coll} are born outside the dip with enough potential energy to reach the wall. It is interesting that just a small rate of CX collisions inside the sheath led to a drastically different equilibrium with a different presheath distribution and different wall fluxes.

It is natural to ask what happens if the CX collisions are turned back off near the wall after the inverse equilibrium is reached. Does the simulation remain in a similar inverse state, or transition back to a SCL? We would expect the former because collisions should not be necessary to *sustain* an inverse sheath that already exists. But simulations of the inverse regime become problematic if there are electrostatically trapped ions that never suffer collisions; they cannot reach a reasonable equilibrium. Any ions confined in a collisionless region where φ < φ_{wall} will oscillate forever (although in simulations they eventually diffuse out due to numerical energy conservation errors). In practice, the ion collisionality will always be nonzero. The structure of any inverted presheath will depend on the collisions of ions and electrons with each other and with neutrals.

Another crucial issue to emphasize is that φ(x) in an inverse equilibrium can be nonmonotonic due to the patching of a monotonically decreasing presheath potential and the monotonically increasing sheath potential, e.g., Fig. 1(a) here and Fig. 2(d) of Ref. 17. One could question why these “potential wells” can exist. They can exist because ions do not get trapped in the sheath. As long as there is a collisional thermalization of ions over presheath length scales, the trapped ions will not build up indefinitely in the potential well. Similarly, low energy plasma electrons created in the classical equilibrium do not build up forever because they eventually gain enough energy via Maxwellization to escape.

## IV. WHICH EQUILIBRIUM SHOULD OCCUR IN PRACTICE?

### A. General considerations

The strong emission problem is ubiquitous in plasma physics. Many floating plasma-facing surfaces in the laboratory and space can have intense enough emission such that the emission coefficient exceeds unity.^{23} Hot cathodes in low temperature plasma devices often emit fluxes tens or hundreds of times as large as the plasma electron saturation current.^{9,10,24} Photoemission from sunlit objects in space often exceeds the electron saturation current provided by the solar wind plasma.^{25} Secondary emission coefficients for many dielectric and metal materials exceed unity when T_{e} is tens of eV or more.^{26} It would be valuable to know from theoretical grounds whether a SCL or inverse equilibrium will occur in a given γ > 1 situation.

It can be argued that a SCL equilibrium is more natural as long as γ was below unity at some point in the past because the Bohm presheath and negative Ф_{sh} needed for a SCL state were pre-existing. Our time-dependent simulations confirm that when γ rises above unity for the first time, the system does indeed transition to a SCL-like state. However, over a longer time scale, ion accumulation inside the SCL dip forces a transition to an inverse equilibrium. The only way we were able to simulate a true SCL equilibrium here was to exclude all ion production mechanisms inside the dip. This also explains why the only previous simulation studies observing SCL-like sheaths with a stable dip were using source boundary plasma injection codes without ionization in the domain.^{3,6,8,14,15} By comparison, in the EDIPIC code Hall thruster simulation studies,^{2} where inverse sheaths were first observed, it was noted curiously that SCL equilibria were never observed when γ > 1. This is now explainable by the fact that volumetric e-n ionization occurred throughout the domain in the EDIPIC model.

In all laboratory sheaths, there must be some cold neutrals either from background gas or, even if the upstream plasma is fully ionized, from recycling at the surface. If a SCL sheath existed, we would expect some CX collisions between sheath-accelerated ions and cold neutrals to create trapped ions, ensuring a transition to the inverse equilibrium. The collision rate need not be large. Recalling that the ion mean free path in our time-dependent simulations was set to 50 cm, and the width of the dip was 1.5 mm, only about 3% of the ions passing through the dip became trapped. The full transition took only microseconds. Although the transition time will vary with collision frequencies, we expect it to be fast enough in typical laboratory plasmas operating for seconds to ensure an inverse equilibrium.

It might be possible for an equilibrium SCL sheath to exist if there is an offsetting loss mechanism that removes whatever ions are born in the dip. The most promising mechanism is ion escape in the direction parallel to the surface. Experiments with a biased anode by Forest and Hershkowitz showed that equilibria with and without a potential well are possible, depending on whether the ions have a multidimensional escape path.^{27} Recent papers with measurements of ion velocities near anodes also demonstrate the importance of multidimensional effects.^{28,29} Experiments or 2D simulations would be helpful to study whether ions can escape easily from a potential well in front of a finite length strongly emitting surface. We could argue based on the structure of the 1D SCL sheath, see Fig. 3(b), that because the dip minimum is in a region of intense net negative charge, the fringe field at the ends of a finite length surface with a SCL sheath might serve to keep ions trapped in the parallel direction too.

Ion escape in multiple dimensions could be ruled out for some surfaces like emissive probes or dust grains where any SCL sheath dip would surround the entire object, leaving no escape path. For surfaces contacting magnetized plasmas such as divertor plates, the magnetic field would suppress the ion escape parallel to the surface. Trapped ions in a SCL sheath at a divertor plate may gain enough energy to overcome the dip after Coulomb collisions with higher energy charged particles. But the energetic electrons and ions entering the sheath would also create new cold ions in the dip by e-n ionization or i-n charge exchange, likely at a faster rate than trapped ions are kicked out. All things considered, we believe the inverse equilibrium will be heavily favored in all practical situations.

### B. Practical differences between SCL and inverse regimes

Our conclusion that inverse sheaths are more stable could apply to a variety of practical situations. Since the Hobbs-Wesson article^{1} originated, most researchers have assumed that the sheaths at strongly emitting surfaces are SCL. Inverse sheaths and presheaths would change the plasma-surface interaction in several ways. In general terms, the loss rate of ions, and hence the loss rate of plasma to wall recombination is much less in the inverse regime. Any ions that reach the surface will have much lower impact energies, thereby reducing sputtering and erosion. The lack of electron confinement leads to higher electron influx and higher heat flux in the inverse than the SCL (for given T_{e}). Depending on the size of the emitting object, the different fluxes could feedback affect T_{e} and other bulk plasma properties.

The inverse regime could also open up new methods of manipulating plasma properties. For example, while SCL theories assumed that emitted electrons entering the plasma are sheath-accelerated to energies comparable to the bulk electron temperature, inverse sheaths allow cold electrons (e.g., tenths of eV thermionic electrons) to be injected deep into the quasineutral region. They can only gain energy by taking it from hotter particles (or some other heating source). Negatively charged ions and dust grains could be extracted by a strongly emitting object with an inverse sheath but not a classical or SCL sheath. The sharp distortion of plasma density above objects with inverted presheaths may trigger interesting multidimensional effects in magnetized and unmagnetized plasmas.

For a specific application, inverse theory suggests that a different interpretation of emissive probe measurements is needed when using the floating point method. To infer the plasma potential from φ_{plasma} = φ_{probe} – Ф_{sh} (neglecting presheath effects), planar SCL theory has long been used to estimate Ф_{sh} at around −1T_{e}/|q_{e}|.^{30,31} Inverse sheath theory^{2} predicts that the probe floats above plasma potential according to Ф_{sh} = T_{emit}ln(γ)/|q_{e}|. In fact, it has been reported in experiments that emissive probes float above the value of plasma potential inferred via cold probe I–V traces, see Fig. 4 of Ref. 32, Fig. 4 of Ref. 33, and Fig. 5(b) of Ref. 34. Such probe measurements are inconsistent with SCL sheath theories including the cylindrical geometry solutions derived in Ref. 35.

The potential distribution around spherical objects is also important. It is known experimentally that strongly emitting dust grains become positively charged.^{36} But the grain *potential* relative to the surrounding plasma is not easily measurable. Simulations show states with nonmonotonic (SCL-like) and monotonic (inverse-like) φ(r) distributions, see Fig. 5(b) of Ref. 37 and Fig. 1 of Ref. 38. To our knowledge, past simulations with nonmonotonic φ(r) did not contain ion CX collisions. Our study suggests that if there is any nonzero rate of CX collisions, the only possible equilibrium sheath potential around even curved strongly emitting objects including probes and dust grains is a monotonic inverse shaped φ(r).

Hall thrusters are known to have secondary emission coefficients exceeding unity at the channel walls under certain operating conditions, based on measured plasma electron temperatures and emission yields of the wall materials.^{39} The secondary emission is thought to be responsible for the saturation of the thrust field and electron temperature observed in Hall thruster experiments. Fluid models in the past used SCL sheath transmission factors to set boundary conditions along parts of the wall where γ > γ_{cr}, see Refs. 40–42. But recent kinetic (PIC) simulations found that inverse sheaths can form. The 1D simulations^{43} (radial direction only) show that inverse sheaths lead to a rapid replacement of plasma electrons with secondaries such that the near-wall conductivity (NWC) is far larger, and the net plasma electron temperature lower, compared to when the sheath potential is negative. The 2D HT PIC simulations (r,z) by Taccogna elegantly show the transition of the radial potential profiles from classical to SCL-like to inverse along the z direction that one could expect in experiments due to the increase of T_{e} (and γ) along z, see Fig. 6 of Ref. 44. The role of inverse sheaths on electron temperature saturation is also demonstrated in Fig. 7 of Ref. 44.

Secondary emission coefficients above unity at tokamak plasma-facing materials including tungsten and lithium are possible at tens of eV plasma temperatures.^{45–47} Expressions for sheath potential and transmission factors in terms of γ were always assumed to saturate at the critical SCL value γ_{cr} predicted by Hobbs and Wesson, see Sec. 25.5 of Stangeby's book.^{19} If an inverse sheath formed instead, it could significantly alter the divertor plasma. Flow velocities would no longer need to reach c_{s}. Another expected consequence would be a major reduction of erosion and impurity generation. Under typical tokamak SOL (scrape-off layer) conditions with a target T_{e} and T_{i} of tens of eV, the ion acceleration in the classical presheath and sheath potentials brings ion impact energies above the ∼120 eV sputtering threshold^{19} of tungsten. Under the same temperatures, inverse sheaths would keep impact energies of almost all ions below the threshold. One could argue that an undesirable increase of heat flux would occur in the inverse regime, but normally heat flux is fixed by the input power to the SOL, neglecting radiation losses. The heat flux could also potentially be mitigated by deliberate inducement of strong thermionic emission from tungsten plates in a conduction-limited regime. As discussed in Sec. II C 3, when e-e collisionality is strong, an equilibrium inverted plasma is possible where T_{e} drops smoothly from its upstream value to T_{emit} at the sheath edge, and the density rises to maintain a uniform p_{e}. A high density region of sub-eV plasma could help induce divertor detachment, thereby reducing the need to inject impurities.

The lunar sheath has an important influence on the motion of dust and the charging of spacecraft on the moon.^{25} Theories predicted the lunar sheath to have two possible φ(x) distributions^{48,49} during daytime when the moon emits a photoelectron flux up to four times^{50} the influx of electrons from the solar wind plasma. The theoretical solutions qualitatively resemble the SCL and inverse sheaths discussed here. (They are not equivalent because the solar wind is pre-accelerated to a supersonic flow velocity. Nonmonotonic and monotonic solutions under strong emission still exist with pre-accelerated ions, e.g., see Refs. 51 and 52, but there is no true inverse equilibrium with confined ions). Attempts to infer the φ(x) shape near the moon from Lunar Prospector data so far are inconclusive, as discussed in Ref. 53. Interestingly, it was argued in Ref. 48 based on minimum potential energy considerations that the nonmonotonic φ(x) should be the stable one. However, we propose that the ion accumulation consideration would require the equilibrium lunar sheath φ(x) to be of the monotonic type. We note that nonmonotonic and monotonic φ(x) profiles have both been observed in particle simulations of the photoemitting lunar sheath without collisions.^{50} It would be interesting to add the relevant ionization sources from CX and photoionization to simulation models. Based on our time-dependent simulations, we predict the lunar sheath to enter a SCL-like state initially at sunrise, and then should transition to an inverse-like equilibrium as ions accumulate in the potential well (unless the time required for this transition is larger than one lunation).

## V. WERE SCL OR INVERSE REGIMES OBSERVED IN PAST EXPERIMENTS?

Although we predicted inverse sheaths to be present in certain applications including Hall thrusters and around emissive probes, there is not a direct measurement of the fine details of the sheaths to compare to. It is naturally difficult to measure sheaths in complex plasma devices. There have been a variety of simpler experiments designed to study sheath physics, see Ref. 23 for a 2013 review. Some interesting studies of electron-emitting surfaces were conducted. Most took place at the time when only the SCL theory was available. A more recent experimental study reported both SCL and inverse sheath measurements.^{13} It is worthwhile to discuss some of the former experiments from the viewpoint that two different equilibria are possible.

Several past authors measured the potential above emitting surfaces with emissive probes. Potential data may sound ideal for distinguishing SCL and inverse equilibria, but there are some issues to consider. Potential measurements in the upstream plasma relative to an emitting surface^{7,54} are insufficient to determine the sheath potential because presheath and multidimensional bulk gradients also affect the measured potential difference. Even in low density experiments where the sheath is large enough to probe, identifying the sheath part of φ(x) is not straightforward. Some past measured potential profiles were classified as SCL sheaths because φ(x) was nonmonotonic, but the surface potential was above the upstream potential in some of those profiles, see Fig. 6 of Ref. 9 and Fig. 4 of Ref. 13. That is inconsistent with the SCL theory which *requires* upstream ions to fall into the wall. It is now known that an inverse sheath patched to an inverted presheath can combine to make a nonmonotonic φ(x), as in Fig. 1(a) of this paper and Fig. 2 of Ref. 17. The wall potential in an inverse equilibrium can be below or above the maximum upstream potential. This theoretically follows from our discussion in Sec. II C 3. We showed that the inverted presheath drop Ф_{ps} depends on the ion pressure balance, not directly on γ. So for a given presheath, there will always be ranges of γ where the sheath potential T_{emit}ln(γ)/|q_{e}| is larger or smaller than Ф_{ps}.

Overall, to identify a SCL or inverse sheath by the shape of φ(x), one would at least need to know where the sheath edge is located. One could in principle differentiate the measured φ(x) twice to analyze the distribution of net charge n_{i}-n_{e}. But as observed in Fig. 3(b), the net charge in the ion-rich layer of a SCL sheath is weak compared to the electron-rich layer, so a high degree of precision would be needed to demonstrate its presence. We should also point out that more theoretical justification is needed to know that any emissive probe measurements inside emitting sheaths are accurate enough to compare to theories. (The idea to use emissive probes in sheaths was inspired by Yamada and Murphree's experiment^{55} where they observed data *qualitatively* consistent with the potential expected in a non-emitting sheath.) Another issue is that in the low plasma density experiments with sheaths large enough to probe, the sheath may be multidimensional and comparable to the characteristic size of the object, presheath or device, such that a comparison to 1D theories that assume very thin sheaths is inappropriate. At sufficiently low plasma density, the emitting surface will behave as if in vacuum^{56,57} where the resulting φ(x) looks like an inverse sheath, but is not a plasma sheath.

Overall, it will be valuable to measure other properties besides the potential to distinguish inverse and SCL equilibria. Ion impact energies are a promising candidate. The 1993 experiment by Schwager *et al*.^{10} measured the energies of ions that passed through a hole in a floating thermionically emitting plate. They expected under strong emission that the ions would be accelerated by a SCL sheath and Bohm presheath to beyond 1T_{e}, as in the SCL equilibrium simulated here. They reported a “major discrepancy” that the measured ion energies in their Fig. 6 dropped much closer to zero. We suggest that the low ion impact energies are consistent with the lack of ion acceleration expected in the inverse regime. Table I of their experiment^{10} is also noteworthy. It indicates that the quasineutral plasma density decreased towards the plate when it was weakly emitting, and increased when it was strongly emitting, consistent with the density gradient reversal we demonstrated here in Fig. 4(a).

We must emphasize that the density gradients in any device will be sensitive to the charge sources, heating, collisions, magnetic fields, and multidimensional flows, and may not necessarily resemble profiles produced by our 1D simulations. For example, the quasineutral density can increase towards the boundary even in a Bohm presheath if there is a substantial T_{e} drop,^{19} which is common in conduction-limited divertors. Low temperature, low density weakly collisional plasmas often have nonlocal, non-Maxwellian EVDFs, complicating the prediction of sheath and presheath properties. Transit of emitted electrons from surface to surface in weakly collisional plasmas can alter the global current balance and reduce the effective emission coefficient.^{58,59} But even if the exact outcome cannot be calculated, there are compelling reasons to expect some measurable large-scale (presheath-scale) changes to the plasma density distribution around a surface when the emission intensity is raised high enough such that the sheath changes from classical to inverse. Future experiments to look for such transitions are encouraged.

Non-invasive laser-induced fluorescence (LIF) measurements of ion velocities near floating surfaces with a variable emission would also be valuable. Substantial differences between ion speeds and directions in the presheath and sheath are expected in the inverse regime compared to a classical and SCL. LIF measurements near a floating thermionically emitting plate as the effective emission coefficient is raised from zero to well above unity should produce compelling enough data to infer whether a transition to the SCL or inverse equilibrium occurred. Thermionic emission is recommended for fundamental studies of emission because the emitted flux can be varied independent of the bulk plasma properties, and a γ > 1 situation is easy to create. Experiments that induce strong secondary emission from surfaces in filament discharges also give interesting results^{11,12} but are difficult to compare to basic theories because (a) the interior distribution function in a filament discharge is complicated with high energy and low energy electron components, (b) the effective γ is coupled to densities and energies of both components, (c) γ > 1 is achieved when the filament electrons produce the dominant flux, which is at low pressure conditions with very low plasma densities, leading to large non-planar sheaths. Nevertheless, even for non-planar objects, a transition from an ion-attracting mode to an ion-repelling mode might be measurable with LIF.

## VI. CONCLUSIONS

In the end, the answer to the question posed in the title is yes, but some qualifiers are necessary. It is true that two different equilibria are possible when a plasma contacts a floating strongly emitting surface. One equilibrium is a SCL sheath patched to a Bohm presheath, similar to the presheath above a non-emitting surface with classical sheaths. The other equilibrium is an inverse sheath patched to an inverted presheath, which has unique distributions of density, pressure, and flow velocity. In this paper, we compared the characteristics of both equilibrium states in detail. Theoretical estimates of the sheath and presheath properties in terms of the bulk plasma properties {N, m_{i}, T_{e}} and emission properties {T_{emit}, γ} were discussed and found to be in good quantitative agreement with the results of our 1D direct kinetic simulations.

Our simulations showed key features of SCL and inverse equilibrium states that are not observable using conventional source injection simulation models. For example, we used a volumetric charge source that allowed the Bohm presheath-sheath transition to form self-consistently in the SCL equilibrium. Volumetric thermalization of emitted electrons with bulk electrons played an important role in the inverse equilibrium, causing the presheath density gradient to be opposite in sign to the Bohm presheath. Volumetric charge exchange collisions were shown to cause ion accumulation in a SCL sheath's dip, forcing a transition to an inverse equilibrium. The only way we were able to demonstrate a true SCL equilibrium was to omit ion production in the dip region.

Assuming that some CX collisions are always present in real sheaths, we predict that only equilibria with monotonic inverse type sheaths should be possible at any strongly emitting surface, whether a divertor plate, Hall thruster channel wall, emissive probe, dust grain, or even the sunlit side of the moon. Because the sheaths in complex devices such as Hall thrusters are coupled to plasma properties, sources, collisions of electrons and ions, and multidimensional gradients, the effects of inverse sheaths cannot be quantified without application-specific analyses. It is still safe to say that when inverse sheaths are present, the particle fluxes, energy balance, and presheath gradients will be significantly different from whatever is predicted if SCL sheaths are assumed.

The existence of SCL sheaths cannot be ruled out entirely if cold ions created in the dip could escape, perhaps by multidimensional flow, recombination, or instabilities. Now that various differences between SCL and inverse equilibria were demonstrated in this paper, diagnostic methods for identifying them in experiments can be developed. This may not require measuring the fine structure of the sheath because measurable differences of plasma density gradients and ion velocity distributions over presheath length scales are also expected. In a fundamental study, one could diagnose the surroundings of a floating thermionic plate as its emission intensity is varied. Ideally, the plate size and plasma density should be large enough to ensure a planar sheath, and the plate should be small enough relative to the device such that it does not perturb the global plasma properties much. As the effective γ is raised from zero to well above unity, the presheath structure near the plate is expected to undergo a major change if the classical sheath switches to inverse, but not if the sheath potential saturates at the SCL threshold.

## ACKNOWLEDGMENTS

This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. The authors thank Jeffrey B. Parker for help with improving the code efficiency.