Why is there no generally accepted theory of the plasma-sheath transition and the Bohm criterion? Open Access

The theory of the transition from quasi-neutral plasma to space-charge sheaths and related topics, in particular, the Bohm criterion, has been an active research field in plasma physics for several decades. Search of Web of Science with “Bohm criterion” in the field “Topic” returned a list of 161 papers published during the last 10 years, from 2015 to 18 September 2024. Gas discharge-related aspects are reviewed in Refs. 1–7. The 2020 review⁷ includes 357 references, and there is a significant number of subsequent publications, which are referenced as follows. Additionally, one should mention works in which the formation of space-charge sheaths and the Bohm criterion were studied from the point of view of pure mathematics, e.g., Refs. 8–21.

Broadly, works on the theory of plasma-sheath transition and the Bohm criterion may be divided into three groups. The above-mentioned pure-mathematics publications, exemplified by Refs. 8–21, represent one of the groups. Brief reviews of these works can be found in Refs. 20 and 21. There are many mathematical results on the sheath formation based on using the ion motion equation written in the fluid approximation, coupled with the Poisson equation with the electron density expressed in terms of the electrostatic potential, typically by means of the Boltzmann distribution. (In mathematical works, this system of equations is usually referred to as the Euler–Poisson system.) The existence and stability of stationary solutions assuming the fluid Bohm criterion were established, which validated mathematically the criterion. There are not many mathematical results with rigorous proofs concerning the sheath formation based on the Vlasov–Poisson system, which is relevant to the kinetic Bohm criterion (and is similar to the Euler–Poisson system except that the ion motion is described by means of the Vlasov equation instead of the fluid equation of motion). The solvability of boundary-value problems for the Vlasov–Poisson system under the kinetic Bohm criterion was investigated in Ref. 20 and it was shown that the kinetic Bohm criterion is a necessary condition for the solvability.

The pure-mathematics works on the theory of plasma-sheath transition appear to be virtually unknown to physicists working on the gas discharges. Referring to the above-mentioned reviews,^2–7 one can see that relevant references are given in the 2003 work² but appear to be missing in the subsequent works.

The other two groups of works on the theory of plasma-sheath transition and the Bohm criterion are physics-oriented. One of these groups is comprised of publications in which the plasma-sheath transition is studied by means of the perturbation theory, the small parameter being the ratio of a characteristic Debye length to a length scale characterizing the adjacent quasi-neutral plasma region (presheath). The method of matched asymptotic expansions is used, which is a singular perturbation technique adequate for problems describing regions with significantly different scales. Examples of this approach are works.^1,5,22–31

In Refs. 9, 11, and 19, the treatment of plasma-sheath transition by the method of matched asymptotic expansions was recast in pure-mathematics terms. In Refs. 15 and 16, the same was done for the method of composite asymptotic expansions, which is a singular perturbation technique close to the method of matched asymptotic expansions. On the other hand, the method of matched asymptotic expansions is nowadays virtually out of use in gas discharge physics and most of the modern literature on the physics of plasma-sheath transition contains no trace of the results obtained by this method.

The third group of works on the theory of plasma-sheath transition and the Bohm criterion, which is physics-oriented as well and is by far much more numerous than the second group, includes publications that use approximate approaches based on assumptions, as opposed to more formal methods, primarily the method of matched asymptotic expansions. A representative example of this approach is the 1982 paper,³² in which the sheath edge is identified by a certain value of the electric field. As another example, one can mention works^33–46 based on the recently popular assumption that the Sagdeev potential attains a maximum at the sheath edge. Many other assumptions have been put forward; however, none of the assumptions and corresponding approximate approaches have gained widespread acceptance.

One should mention a heated debate that took place in the literature about basic concepts of the approximate approaches such as the sheath edge and its location, modifications, and/or generalizations of the fluid Bohm criterion with account of collisions and/or ionization and/or geometrical effects in the sheath, disproportionate contribution of the low-speed part of the ion distribution function to the kinetic Bohm criterion, divergence of the kinetic Bohm criterion if any part of ion distribution functions corresponds to ions leaving the sheath, and the necessity of modification/generalization of the kinetic Bohm criterion. The debate started in the 1980s and peaked in the 2000s and 2010s, e.g., Refs. 47–69 and references therein with regard to the fluid Bohm criterion and^31,70–78 with regard to the kinetic Bohm criterion.

A significant number of papers on the plasma-sheath transition and the Bohm criterion have been published in recent years. The effect of ion-neutral collisions on the plasma-sheath transition was studied theoretically in Ref. 79 and numerically in Refs. 80 and 81. A sheath edge criterion was proposed, and approaches to a kinetic generalization of the Bohm criterion were discussed in the review.⁷ The assumption of maximum of the Sagdeev potential at the sheath edge was used to derive generalized Bohm criteria for collisional sheaths with ionization,³⁹ collisional sheaths with negative ions,⁴⁰ magnetized collisional sheaths with positive and negative ions,⁴¹ magnetized collisional sheaths with ionization,^43,45 sheaths in warm electronegative collisional plasma in contact with electron emitting surfaces,⁴⁴ and magnetized sheaths with the ion stress.⁴⁶ In Ref. 42, PIC modeling results were found to be in close agreement with a collisional Bohm criterion obtained in Ref. 33 with the use of the Sagdeev potential, and an alternative derivation of this criterion was given from a generalized Bohm criterion,⁷⁰ obtained with the use of positive-exponent velocity moments of the kinetic equation.

In Refs. 82–84, the Bohm criterion was analyzed over a spatially extended presheath–sheath transition region, as opposed to a sharp boundary, and the conclusion was drawn that the Bohm speed should vary inside this transition region. The Bohm speed was computed with account of detailed transport physics, including heat flux, collisional isotropization, and thermal force for both electron and ion transport and for a high recycling regime of a divertor. By comparison with kinetic simulation results, these expressions were shown to be accurate in the transition region rather than a single point at the sheath entrance, over a broad range of collisionality.

Different forms of the modified Bohm criterion were proposed in Refs. 85 and 86. In Ref. 87, an approximation of the truncated electron velocity distribution function in an oblique magnetic field was used to find the ion velocity at the sheath edge as a function of the magnetic field angle. In Ref. 88, spatially resolved laser Thomson scattering measurements in a cathode presheath are reported and compared with a newly derived approximate expression for the electric field at the sheath edge that accounts for collisional and ionization effects. In Ref. 89, values of the electric field at the sheath edge are determined theoretically with the use of experimental values of radius of the sheath on a hairpin probe. The time evolution of the sheath and presheath in a collisionless plasma with cold ions and Boltzmann-distributed electrons is analyzed analytically and numerically in Ref. 90 in one dimension in planar, cylindrical, and spherical geometries.

In summary, there is no universally accepted theoretical description of the plasma-sheath transition and the Bohm criterion in gas discharge science, in spite of the efforts invested by able researchers over decades and in spite of the existence of results of a mathematical nature, including pure-mathematics results obtained in the course of the past two decades. Active research in this area continues.

The concept of the space-charge sheath and quasi-neutral plasma goes back to Langmuir^91–93 and is transparent and understandable. However, given the above-described state-of-the-art, it is clear that the adequate theoretical description of the plasma-sheath transition is nontrivial and requires some sophistication.

The use of relatively complex mathematical methods is not by itself an obstacle for a work being accepted by the gas discharge community; the theory of high-voltage collisionless capacitive RF sheaths^94–96 is an excellent example. The question then arises: what is missing in the published works that prevented the emergence of a generally accepted theoretical description of the plasma-sheath transition and the Bohm criterion? In more general terms, what kind of mathematical results would have a better chance of being accepted in gas discharge science?

The purpose of this paper is to offer a personal opinion on this issue and to review in this context theoretical results concerning the plasma-sheath transition, with a particular focus on the Bohm criterion, in the hope of contributing to the achievement of a more or less universally accepted description of the plasma-sheath transition in gas discharges.

The paper is organized as follows. Criteria of mathematics suitable for the gas discharge theory (“good” math), as these authors see them, are formulated in Sec. II. In Sec. III, the classic model of collisionless positive space-charge sheaths and conventional derivations of the fluid and kinetic Bohm criteria are recapitulated. A concise description of the method of matched asymptotic expansions is given in Sec. IV, with the aim to show how the method works and thus dispel misconceptions about it. A brief review is given of works dedicated to the asymptotic theory of plasma-sheath transition. In Sec. V, asymptotic results on the collisionless sheath-plasma transition and the fluid and kinetic Bohm criteria are interpreted without resorting to asymptotic arguments, in the hope to make them more readily understandable and convincing. Conclusions are summarized in Sec. VI. In the  Appendix, the collisional Bohm criterion, obtained by assuming that the Sagdeev potential attains a maximum at the sheath edge, is revisited, and a significant issue with this approach has been identified. Also discussed in the  Appendix is an alternative form of the Bohm criterion involving derivatives of the charged-particle densities with respect to the electrostatic potential, employed, in particular, in Refs. 82–84 and 97.

Many excellent works employing mathematical methods have been published in the theory of gas discharges. In these authors' humble opinion, a good example are works,^94–96 which were already mentioned in the Introduction. The presentation in the paper⁹⁵ is especially impressive: the analytical theory of a complex phenomenon is presented transparently and understandably in just seven journal pages! Another example is the work,⁹⁸ where an analytical solution was found for the classic Tonks–Langmuir model.

Intuitively, “good mathematics” refers to a mathematical treatment that goes beyond the elementary, produces important new results for a real physical situation or a useful idealization (e.g., for a reasonable limiting case), and provides sufficient detail to be reproducible but not much more (trivialities are skipped). There are also less obvious criteria, which are different for each researcher. In view of the authors, the criteria most important for the plasma-sheath transition theory are as follows.

• Features of the solution stemming from the approximations being used (e.g., asymptotic approximations) should not be misinterpreted as real physical features.

• The mathematical sense of the treatment should be clearly identified, and only standard mathematical methods should be used.

• The essence of the work and its results should be understandable to physicists who have little interest in mathematical details.

In what follows, these criteria are applied to the problem of plasma-sheath transition, including the fluid and kinetic Bohm criteria.

Collisionless space-charge sheaths formed by the ions and the electrons were first considered in the seminal paper of Bohm of 1949.⁹⁹ Bohm's treatment, with some changes, may be summarized as follows.

In most discharge conditions, the ionized gas in the bulk of the discharge is quasi-neutral. The space charge is localized in thin layers near solid surfaces bounding the discharge; the so-called space-charge sheaths. Positive sheaths, which are formed on negative electrodes (cathodes) and insulating walls, are schematized in Fig. 1. Here, x is the distance from the surface, $φ$ is the electrostatic potential, and $ni$ and $ne$ are the number densities of the ions and the electrons. (It is assumed that the plasma contains one neutral species, one species of singly charged positive ions, and electrons.) The ions entering a positive sheath from the quasi-neutral plasma are accelerated by the sheath electric field in the direction of the surface. The plasma electrons are repelled by the sheath electric field, and their density is related to the local electrostatic potential by the Boltzmann distribution. The density of electrons emitted by the cathode is usually small, and the presence of such electrons may be disregarded.

The scale of thickness of the sheath is represented by the characteristic Debye length $λD$⁠. Let L be the length scale characterizing the adjacent quasi-neutral plasma region (presheath). L must necessarily be much larger than $λD$⁠, otherwise the ionized gas would not be quasi-neutral on the scale L. We consider the case where the ion-neutral collisions, ionization, and geometrical effects are negligible inside the sheath. This implies that $λD≪$ $λi,d,l$⁠, where $λi$ is the mean free path for the ion-neutral collisions, d is the ionization length, and l is the characteristic geometrical dimension, and L may be set equal to the smallest of these three lengths scales.

Now, the boundary-value problem (9), (8), (10) is closed. Note that the rhs (right-hand side) of Eq. (9) involves only the unknown function $Φ$ and does not involve the independent variable $ξ$ nor the first derivative $dΦ/dξ$⁠. Such equations are readily solved in quadratures for the dependence $ξ(Φ)$⁠; however, this is unnecessary for the purpose of the subsequent analysis.

If $B<1$⁠, the second term on the rhs of Eq. (11) must be dropped. The resulting equation is compatible with the boundary condition (8), similar to the case $B=1$⁠, except that the solutions decay for $ξ→∞$ exponentially rather than algebraically.

The importance of the Bohm criterion stems from the fact that it shows the way to match solutions describing quasi-neutral plasmas and positive collisionless space-charge sheaths. To use the Bohm criterion, one needs to interpret the words “enter the sheath.”

Bohm⁹⁹ explicitly speaks of a sheath “edge” while interpreting the mathematical results, and although he admits that it cannot be given a precise definition, he suggests that the sheath begins roughly when the ions reach the speed $uB$⁠. However, the concept of a sheath edge is inconsistent with the mathematical derivation: the expansion of the sheath Eq. (9) in powers of electrostatic potential, measured from the plasma potential, amounts to investigation of asymptotic behavior of solutions to this equation at large distances from the surface and is therefore of asymptotic nature and does not involve any particular point separating the sheath and the plasma. Note that Bohm himself indicates that the electrostatic potential remains close to the plasma potential over distances of many sheath thicknesses in extent; cf. Eq. (3) above.

The term “edge of the sheath” appears also in another classic work.⁹¹ However, the term “sheath” in Ref. 91 refers to the inner, electron-free, section of a high-voltage space-charge sheath rather than to the space-charge sheath on the whole, as in Bohm's work⁹⁹ and most of the modern literature. Langmuir's analysis⁹¹ is based on the Child–Langmuir law and is valid only in this inner section (which may be termed ion sheath or ion layer) rather than in the space-charge sheath on the whole. However, the transition to the quasi-neutral plasma occurs in the outer section of the space-charge sheath, where the densities of the ions and the electrons are comparable, which may be termed ion-electron layer and is described by the Bohm⁹⁹ analysis. In other words, the edge that bounds the ion sheath and allows a meaningful and unambiguous definition with the use of the Child–Langmuir law should not be confused with “sheath edge” in the context of the Bohm criterion, which means a “boundary” that limits a sheath containing both the ions and the electrons, i.e., a “boundary” between the space-charge region and the quasi-neutral plasma. [In mathematical terms, this can be explained as follows. The ion sheath equation, which leads to the Child–Langmuir law, is obtained by neglecting the first term on the rhs of Eq. (7), accounting for the electron density. Such a model breaks down at a finite distance from the surface: a solution to the ion sheath equation cannot be extended beyond a point at which the electric field vanishes, and this point represents the edge of the ion sheath. There is no such breakdown in the Bohm model: a solution to the full Eq. (7) can be extended to infinitely large distances from the surface, with the ion and electron densities tending to the same constant value and the potential also tending to a constant value.] This topic has been discussed in the literature, e.g., Ref. 4 and references therein, and is touched upon in Sec. IV C.

The ion velocity $vi$ in the sheath is higher than $uB$ and $vi$ in the presheath is lower than $uB$⁠, and there must certainly be a point somewhere between the sheath and the presheath at which $vi=uB$⁠. Calling this point the sheath edge is not illegitimate, but it creates the impression that the Bohm criterion is a physical feature that applies at a certain point in space (the sheath edge) rather than an asymptotic feature that emerges in the course of an approximate analysis. Such interpretation is incorrect and inconsistent with Bohm⁹⁹ derivation, which is asymptotic in nature. However, it is this interpretation that is used in the vast majority of physics-oriented works on the theory of plasma-sheath transition, the aim being to modify (generalize) Bohm's criterion to account for ion-atom collisions and/or ionization and/or geometric effects in the sheath. In the framework of such an approach, the sheath edge is defined as a point in space where some or other specific condition is met, and some or other form of the modified Bohm criterion is postulated at this point.

There is no unique, i.e., non-arbitrary, way to implement this approach. Many researchers worked on this issue, and there was a heated debate that peaked in the 2000s, e.g., Refs. 47–69. However, no general understanding has emerged. In particular, there is no widely accepted definition of the sheath edge. Sometimes, it is defined by a specified degree of separation of charges, e.g., values of $1.9%$ or $2.3%$ were assumed in the recent paper.⁴² However, it was shown by numerical integration of the fluid equations¹⁰⁰ that the position of the edge of the positive sheath on a floating surface differs significantly depending on whether it is defined using the equality $vi=uB$ or in terms of the charge separation and depends on the discharge parameters in different ways.

In fact, definitions of the sheath edge used in most works are more sophisticated than a specified degree of separation of charges. Some of these definitions are based on the physical intuition of the authors, some are of a more mathematical nature. A representative example of intuition-based works is the 1982 paper by Godyak,³² in which the sheath edge is defined as a point where the electric field has reached the value of $kTe/eλD$⁠. This is an intuitively clear definition: the sheath “starts” where the electric field is strong enough, and $kTe/eλD$ is a natural scale of the sheath electric field. Still, this definition is arbitrary, e.g., in Ref. 42, the electric field at the sheath edge is set to be 10 or 20 times lower. An example of a more mathematical nature is the work,⁶⁵ where the sheath edge (“collisionally modified Bohm point”) was identified with a removable singularity in the sheath equation derived in that work.

Similarly, no widely accepted formulation has emerged of the Bohm criterion, modified to account for collisions and/or ionization and/or geometric effects in the sheath, as can be seen from reviews^1–7 and references therein. As a further example, one can mention work,³³ in which a collisional Bohm criterion was obtained by assuming that the Sagdeev potential attains a maximum at the sheath edge. There is a significant number of subsequent works,^34–46 in which this approach was used for the investigation and derivation of different forms of the Bohm criterion. However, the collisional Bohm criterion, derived in this way, can be obtained without invoking the Sagdeev potential and, when written with the equality sign, represents a corollary of the transport equations written under the assumption of quasi-neutrality; see the  Appendix. As a consequence, this criterion is trivially fulfilled, with the equality sign, at each point of the quasi-neutral region and thus does not provide any specific condition concerning the ions leaving the quasi-neutral plasma and entering the sheath. It is difficult to see how this criterion could be useful. The same is true for the expressions for the Bohm speed.^82–84,97

In summary, no self-consistent and widely accepted formulation of the Bohm criterion modified with account of collisions and/or ionization and/or geometrical effects in the sheath has emerged, in spite of several decades of active research.

There is an aspect in this criterion which is not intuitively clear: the integral on the lhs of (16) diverges at $v=0$ unless $fs(v)$ vanishes at $v=0$ and, additionally, tends to zero for $v→0$ fast enough. There seem to be no obvious reasons why the distribution of the ions entering the sheath should satisfy these conditions.

There was a heated controversy about this point with a surge in the 2010s, e.g., Refs. 31 and 70–78. Some authors insisted that the kinetic Bohm criterion in the classical form (15) or (16) is valid but critically dependent on the contribution from slow ions entering the sheath. Others argued that the classical kinetic Bohm criterion is unphysical and must be remedied, for example, by using the positive-exponent velocity moments of the plasma kinetic equation, by a careful treatment of the full Boltzmann equation, or by a fluid-moment hierarchy approach.

Three problems in the theory of near-wall space-charge sheaths have been discussed in the preceding sections.

• No widely accepted definition of the sheath edge;

• No widely accepted formulation of the Bohm criterion with account of ion-atom collisions and/or ionization and/or geometrical effects in the sheath;

• Controversy about the kinetic Bohm criterion,

It has already been indicated that the derivation of the Bohm criterion is of asymptotic nature and does not involve any boundary separating the quasi-neutral plasma from the space-charge sheath; hence, all definitions of the sheath edge, i.e., a particular point where the charge separation begins, are inevitably arbitrary. In other words, the confusion around the sheath edge, i.e., the first of the problems listed above, arises from the failure to recognize the asymptotic nature of the concepts of quasi-neutral plasma and space–charge sheath. The same failure originates from the second and third problems, as will be discussed as follows.

This illustrates the importance of criterion (A) of good math proposed in Sec. II: features of the solution stemming from the approximations being used (e.g., asymptotic approximations) should not be misinterpreted as real physical features.

Let us now discuss the theory of plasma-sheath transition in light of criterion (B) of good math proposed in Sec. II: the mathematical sense of the treatment should be clearly identified, and only standard mathematical methods should be used.

The problem of plasma-sheath transition involves two strongly different length scales: the Debye length $λD$⁠, which characterizes the sheath thickness, and the length scale L characterizing the adjacent quasi-neutral plasma region (presheath). The concept of a quasi-neutral plasma and a space-charge sheath is only meaningful provided that $λD≪L$⁠. Hence, the problem of the plasma-sheath transition is by its nature a problem of perturbation theory with the small parameter being the ratio $λD/L$²² and must be treated as such. In other words, the basis of the theory of the plasma-sheath transition must be perturbation methods; any other methods, for example, methods of kinetic theory, may be useful as auxiliary ones, but using them as a replacement for perturbation methods is methodologically incorrect and such a procedure cannot but be arbitrary. The adequate tool is the method of matched asymptotic expansions, which is a singular perturbation technique for problems describing regions with significantly different scales.

The method of matched asymptotic expansions goes back to classic Prandtl's 1904 viscous boundary layer theory and was formalized in the 1950s. It is a powerful technique that automatically exploits all the simplifications justified by the presence of different scales and distinctly reveals regions governed by different physical mechanisms along with these mechanisms. In contrast to approximate approaches widely used in the theory of plasma-sheath transition, the method of matched asymptotic expansions, being a perturbation technique, does not rely on arbitrary assumptions; its accuracy is controlled by the smallness of the ratio $λD/L$⁠, which can be readily estimated for given conditions.

The method of matched asymptotic expansions has been applied to the problem of plasma-sheath transition starting from the early 1960s.^22–24 However, the method is virtually out of use in the modern literature on the theory of plasma-sheath transition. Labels such as a mathematical trick, a mathematical formality, and a purely mathematical construct are encountered, which sounds quite strange considering that this is a classic singular perturbation technique which, according to the prominent scientist and historian of science, has profoundly influenced science, engineering, and mathematics.¹⁰¹ Even when the validity of results obtained by the method of matched asymptotic expansions is recognized, the derivation of these results is plainly ignored and they are viewed as postulated ad hoc, with the conclusion that the good physical intuition of the author had fortuitously led him to the correct result(!)

Not uncommon are claims that the Bohm criterion is based on the asymptotic limit $λD/L=0$ as opposite to $λD/L≪1$⁠; or that the asymptotic sheath edge may equivalently be characterized by a field singularity in the quasi-neutral plasma solution and is a necessary reference point to construct the intermediate layer connecting plasma and sheath; or that in the asymptotic limit $λD/L→0$ the Bohm criterion applies only at a sharp sheath entrance, where the ion speed reaches the Bohm speed; or that while the sheath width scales with the Debye length, the presheath has no natural characteristic length scale. This attests to the confusion existing in the sheath physics-related literature about important aspects of the singular perturbation theory.

Thus, there are profound misconceptions about the method of matched asymptotic expansions in the modern gas discharge literature. Let us consider a concise description of the method, with the aim to try to dispel these misconceptions and demonstrate that the method is in fact simple and physically transparent, automatically exploits all the simplifications that are justifiable, and distinctly reveals governing physical mechanisms in different regions.

Here, $ε=λD/L$ and $λD=ε0kTe0/n0e2$⁠. Other relevant equations and boundary conditions should be transformed into dimensionless variables as well.

The concept of a quasi-neutral plasma and a space-charge sheath is only meaningful provided that $λD≪L$⁠; hence, the coefficient $ε2$ on the lhs of Eq. (17) must be considered as a small parameter. Since this coefficient is multiplied by the only derivative in this equation, setting $ε=0$ will result in a reduction in the differential order of the problem; therefore, the solutions for $ε=0$ and small but non-zero $ε$ are substantially different. Such-type problems are called singular perturbation problems, to distinguish them from problems where perturbations introduced into the solution by the presence of a small parameter are small as well.

The solution to the problem is sought in the form of expansion in the small parameter $ε$⁠. Since physical mechanisms governing different regions (the quasi-neutral plasma and the near-surface space-charge sheaths, in this case) are different, the form of the expansion in different regions is not the same. Just one term of the expansion in each region is sufficient in virtually all cases to identify governing mechanisms and the underlying physics. The expansions describing different regions are asymptotically matched, in contrast to “gluing” solutions at one point as is done in a popular approximate approach sometimes called patching.

In the next section, the main ideas of the method are illustrated by a simple example. The readers are referred to textbooks (e.g., Refs. 102–107) for more detail and further examples, including much more complex ones.

Equation (21) provides an explicit expression for $y1(0)$⁠. This solution satisfies the second boundary condition (22) but not the first one.

On the other hand, the second boundary condition (19) cannot be applied to the expansion (23) since the range of $x≫1$⁠, to which this boundary condition refers, is beyond the range of x of order $ε$⁠, where expansion (23) is valid.

One needs to somehow connect expansions (20) and (23) to formulate the missing boundary condition for $y1(i)$⁠. As shown above, expansion (20) loses validity for x of order $ε,$ i.e., its applicability is limited by inequality $x≫ε$⁠. Similarly, applicability of expansion (23) is limited by inequality $x≪1$⁠, since for x of order unity the rhs of Eq. (18) cannot be approximated by a constant and Eq. (24) becomes invalid.

This is the condition of asymptotic matching of van Dyke.¹⁰² 

The functions $y1(o)$ and $y1(i)$⁠, given by Eqs. (21) and (30), respectively, represent the full first-approximation solution to the problem considered. The fact that the asymptotic matching was possible confirms the assumptions made in the course of the derivation concerning the form of the first terms of the outer and inner expansions.

It is seen that the inner and outer solutions represent good approximations to the exact solution in the inner and outer regions, respectively, as they should.

The difference between this approximation and the exact solution (31) is not seen on the graph and is of the order $ε3$ in the inner region and $ε2$ in the outer region, i.e., the first-approximation uniformly valid solution is very accurate in this example.

The above derivation has naturally revealed the existence of two regions governed by different mechanisms, where different approximations are appropriate: the outer region, where the spatial scale is unity and the second-derivative term on the lhs of Eq. (18) is minor, and the inner region, where the spatial scale is $ε$⁠, and the second-derivative term is comparable to other terms but the rhs of Eq. (18) is virtually constant. This example clearly shows how the method of matched asymptotic expansions automatically exploits all the simplifications that are justified by the presence of different scales and distinctly reveals regions governed by different mechanisms. It should be stressed that results given by the method are unique: no other researcher will obtain different results in the same problem with the same accuracy, in contrast to intuitive methods that lead different researchers to different results.

The mathematical example considered in this section was chosen to be quite simple, with the aim of illustrating the main ideas of the method of matched asymptotic expansions. Textbooks provide innumerous further examples, including much more complex ones from various areas of science, engineering, and applied mathematics, e.g., Refs. 102–107.

The method of matched asymptotic expansions was first applied to the plasma-sheath transition for the case of collision-dominated sheath.^23,24 Refined treatments for this case were given in Refs. 28 and 29. The case of collisionless sheath, which is the focus of this work, was considered in Ref. 25 and a refined treatment was given in Ref. 27.

Works^24,25,27,29 were concerned with sheaths on moderately negative surfaces. Asymptotic solution for the collision-dominated sheath^24,29 involves two asymptotic regions: the quasi-neutral plasma and the space-charge sheath. Nothing similar to the Bohm criterion arises in the course of the asymptotic analysis. The same two asymptotic regions also appear in the case of collisionless sheath,^25,27 but only in the first approximation in $ε$⁠. The fluid version of the Bohm criterion, with the equality sign, arises in the course of the first-approximation analysis; cf. Eq. (21) in Ref. 27. Beyond the first approximation, a direct matching of the plasma and sheath expansions is impossible, an intermediate asymptotic region, with a thickness on the order $λD4/5L1/5$⁠, needs to be introduced, and the Bohm criterion does not hold. It should be stressed that the intermediate region, while being asymptotically thin compared to the quasi-neutral plasma, is asymptotically thick compared to the space-charge sheath. Note that while this region has appeared in the course of asymptotic analyses^25,27 as a purely mathematical concept on the grounds that a direct matching of the plasma and sheath expansions is impossible, this region has a clear physical interpretation⁶⁸ and the most adequate term for it is “transonic layer,” as originally suggested in Ref. 26.

The works^23,28 were concerned with the case of collision-dominated sheath on a strongly negative surface. Asymptotic analysis in this case involves, in addition to the small parameter $ε$⁠, a large parameter characterizing the sheath voltage U: $χ=eU/kTe0$⁠. The space-charge sheath includes three different asymptotic regions. One of them is the ion layer (or the ion sheath), which occupies the main part of the sheath and in which the ion density significantly exceeds the electron density. The ion layer is governed by the Mott-Gurney law, which was derived in Ref. 108 in connection with conduction current in semiconductors and insulators and in Ref. 109 as an analog of the Child–Langmuir law for a plane-parallel diode filled with a high-pressure gas. The second asymptotic region is the diffusion layer, which is much thinner than the ion layer and is positioned “at the bottom” of the ion layer, i.e., separates the ion layer from the surface. The third asymptotic region is the ion-electron layer, where the densities of the ions and the electrons are comparable and the transition to quasi-neutrality occurs. The ion-electron layer is much thinner than the ion layer and in this sense may be considered as its boundary (edge). On the other hand, the ion-electron layer is much thinner than the quasi-neutral plasma; therefore, the ion-electron layer cannot be assigned any meaningful edge. Consequently, there is no meaningful edge for the space-charge sheath as a whole.

A theory of collisionless sheath on a strongly negative surface was developed in Ref. 110. The space-charge sheath comprises an electron-free ion layer and an ion-electron layer. A simple analytical model with exponential accuracy with respect to the large parameter $χ$ was derived for the case of cold ions. Equations describing the ion layer in this model are identical to those of the Child–Langmuir model; however, boundary conditions are different: they imply that the ions are accelerated in the ion-electron layer from the Bohm speed $uB$ to $2uB$ and the voltage drop in the ion-electron layer equals $(3/2)kTe0/e$⁠. [The same values appeared in Ref. 111; however, the final results were based on an arbitrary definition of the edge of the ion sheath; cf. Eqs. (17) and (18) of Ref. 111]. The model predicts the electric field and the ion speed at the surface and the thickness of the ion layer with an accuracy of several percent for $χ≥3$⁠. A simple analytical model of the ion layer in a collision-dominated weakly ionized plasma, which has exponential accuracy with respect to $χ$⁠, was derived in Ref. 112. The model is equivalent to the Mott–Gurney model; however, boundary conditions are different. Further comments on high-accuracy models of the ion sheath can be found in Ref. 4.

The kinetic Bohm criterion (16) may be derived by means of investigation of asymptotic behavior, at large distances from the surface, of solutions of the Poisson equation in the sheath, i.e., in the same way as the original Bohm criterion for cold ions was derived in Ref. 99 and in Sec. III B. Such derivation was given in Sec. III B of Ref. 1 and, in a more straightforward form, in the Appendix in Ref. 78. On the other hand, the kinetic Bohm criterion (16) may be derived by means of investigation of asymptotic behavior of the first-approximation solution describing the quasi-neutral plasma, e.g., the Appendix in Ref. 31. Similarly to the original Bohm's derivation, the kinetic derivations^1,31,78 refer to the case where ion-neutral collisions, ionization, and geometrical effects are negligible in the sheath.

The heated controversy about the alleged divergence of the kinetic Bohm criterion, mentioned in Sec. III D, stems from the failure to recognize the asymptotic nature of the criterion and from attributing to it a meaning that it does not have. This is a clear example of problems that arise when the criterion (A) of good math formulated in Sec. II is violated.

The asymptotic (singular-perturbation) theory briefly described in Sec. IV C provides an adequate mathematical description of the plasma-sheath transition, including the fluid and kinetic Bohm criteria. In particular, it proves that the sheath edge and modified and/or generalized fluid or kinetic Bohm criteria with account of collisions and/or ionization and/or geometrical effects in the sheath cannot be introduced but arbitrarily and that there is no divergence of the kinetic Bohm criterion for low ion energies and hence no need to remedy it.

Nevertheless, new formulations of the modified or generalized Bohm criterion, inconsistent with the asymptotic theory, have continued to appear. There are no publications which would dispute the new works from the point of view of the asymptotic theory, simply because it is clear by now that asymptotic arguments as such have little effect. One of the last papers of Raoul Franklin, a staunch defender of the asymptotic theory of the plasma-sheath transition, who passed away in 2021, has an expressive title “The quest to find the plasma edge and discover a collisionally modified Bohm criterion” and likens the enterprise to a “chimera of Homeric proportions.”⁶⁹ 

The fact that the perturbation theory by itself has had no visible effect is a good illustration of the importance of the criterion (C) of good math, proposed in Sec. II: the essence of the work and its results should be understandable to physicists who have little interest in mathematical details. This criterion is especially important in cases where the formalism is not easy to grasp for researchers unfamiliar with the method, as is the case with the method of matched asymptotic expansions.

Finding a simple and convincing interpretation of asymptotic results on the plasma-sheath transition and various forms of the Bohm criterion without resorting to asymptotic arguments is obviously a challenging task. In particular, the following questions should be answered.

• Imagine that an ideal experiment, physical or numerical, provided complete information on the variation of the speed of cold ions, $vi(x)$⁠, in the vicinity of a negative surface. Is there an objective way to identify the Bohm criterion for cold ions? In terms of Ref. 65: are there any peculiarities at the Bohm speed $uB$?

• If there is a unique Bohm criterion for collisionless sheaths, then why cannot be there a unique Bohm criterion for collisional sheaths?

• Imagine that an ideal experiment, physical or numerical, provided complete information on the ion distribution function, $f(v,x)$⁠, in the vicinity of a negative surface. Is there an objective way to identify the kinetic Bohm criterion? In other words: are there any peculiarities consistent with the kinetic Bohm criterion?

We will try to answer these questions in two steps. First, we need to figure out what kind of answers we should be looking for, in particular, what peculiarities are to be expected in the spatial distributions of the ion velocity or the distribution function. Second, results from straightforward numerical modeling will be analyzed to verify these answers, particularly to confirm that the peculiarities are indeed present.

The outer and inner approximations must coincide in the intermediate region $ε≪x≪1$⁠. Indeed, setting $x≪1$ in the first expression in Eq. (36) and $ξ≫1$ in the second expression, one finds $y1(o)≈y1(i)≈1$⁠. This is clearly seen in Fig. 3(a): $y(x,ε)$ is approximately equal to unity in the range $ε≪x≪1$⁠, and this feature is expressed the better, the smaller $ε$ is. In other words, the function $y(x,ε)$⁠, given by Eq. (35), reveals a plateau in the range $ε≪x≪1$ provided that $ε$ is sufficiently small. The latter is a natural limitation for an asymptotic feature.

One can say that the asymptotic matching of the inner and outer approximations occurs on a constant in this example. This is the same scenario that occurs in the theory of plasma-sheath transition in the case of collisionless sheath,^25,27 and the value $y=1$ may be viewed as an analog of the Bohm speed. Note that this scenario occurs also in the example considered in Sec. IV B, however, no plateau is seen in Fig. 2, in contrast to Fig. 3(a). This difference is due, in the first place, to the fact that Fig. 2 is plotted on a linear scale rather than on a semi-log one which is appropriate in this context and is used in Fig. 3(a).

In the intermediate region $ε≪x≪1$⁠, the outer and inner approximations give $y1(o)≈1/x$ and $y1(i)≈1/(εξ)$ and coincide as they should, although they are not constant, in contrast to the example (35).

A major difference between these two examples is as follows. Values of function (35) in the outer and inner regions, given by Eqs. (36), are of the same order of magnitude (unity). In contrast, values of function (37), given by Eqs. (38), are of different orders: of the order unity in the outer region and of the order $1/ε$⁠, i.e., much larger, in the inner region. As a consequence, asymptotic matching occurs in essentially different ways: on a constant (equal to 1) in the first example and on an algebraic function (⁠ $1/x$⁠), which describes the increase in y from order unity in the outer region to order $1/ε$ in the inner region, in the second example. This difference is clearly seen in Figs. 3(a) and 3(b): while $y(x,ε)$ is virtually constant (and equal to 1) in the range $ε≪x≪1$ for small $ε$ in Fig. 3(a), in Fig. 3(b)  $y (x,ε)$ is variable in the range $ε≪x≪1$ and cannot be characterized by a number.

The scenario of asymptotic matching on an algebraic function, exemplified by Fig. 3(b), occurs in the theory of plasma-sheath transition for the case of collision-dominated sheath,^23,24,28,29 and also for the case of moderately collisional sheath.

Thus, function (35) together with Fig. 3(a) and function (37) together with Fig. 3(b) illustrate general scenarios relevant to the plasma-sheath transition: matching on a constant in the case where ion-atom collisions in the sheath are rare, and matching on an algebraic function in the case of collisional sheath. It follows that plasma parameter distributions for the case of collisionless sheath, plotted on a semi-log scale, reveal a plateau in the intermediate region between the inner and outer regions, i.e., between the sheath and the quasi-neutral region, provided that the Debye length is small enough. This plateau represents the Bohm criterion. There is no such plateau for collisional sheaths and hence no sense in talking about any form of the Bohm criterion.

One can go a step further and try to devise a more specific example that would illustrate some features characteristic of the plasma-collisionless sheath transition. It follows from Eqs. (6) and (12) that the ion velocity in the sheath for $x≫λD$ or, equivalently, for large values of the normalized coordinate $ξ$ tends to the Bohm speed proportionally to $ξ−2$ (i.e., the difference $vi−uB$ decays proportionally to $ξ−2$⁠). This is different from the first above-described example: the function $y1(i)(ξ)$ given by Eq. (36) tends to unity for $ξ≫1$ exponentially. The ion velocity, given by first-approximation solutions describing the quasi-neutral region under conditions of a collisionless sheath, tends to the Bohm speed in the vicinity of the surface proportionally to $x$⁠. Again, this is different from the first above-described example: the function $y1(o)(x)$ given by Eq. (36) for small x tends to unity proportionally to x.

As desired, function $y1(i)(ξ)$ tends to unity for $ξ→∞$ proportionally to $ξ−2$ and $y1(o)(x)$ tends to unity for $x→0$ proportionally to $x$⁠. Asymptotic matching occurs on a constant as in the first example; however, the outer and inner approximations tend to unity in the intermediate region in the third example slower than in the first one. For this reason, the plateau in the range $ε≪x≪1$ in Fig. 3(c) appears for smaller values of $ε$ and is less pronounced than in Fig. 3(a).

Thus, function (39) reproduces some features characteristic of the plasma-collisionless sheath transition and therefore provides, together with Fig. 3(c), a specific example better suited to illustrating the plasma-collisionless sheath transition than the general example provided by function (35) and Fig. 3(a).

The task of this section is to find the objective meaning of the fluid Bohm criterion without invoking asymptotic arguments. To this end, let us consider a distribution of the speed of cold ions, $vi(x)$⁠, in the vicinity of a negative surface, obtained from an accurate numerical solution of a system of relevant equations. The system must include the Poisson equation and must not involve any apriori division of the computation domain into a quasi-neutral plasma and a space-charge sheath. Are there any peculiarities at the Bohm speed in the computation results, which would allow us to objectively and unambiguously identify the Bohm criterion for cold ions?

Let us set, as in Ref. 65, $Te=3eV$ and $λi=(naσ0)−1vi2/(v02+vi2)$⁠, where $σ0=10−18m2$⁠, $v0=550ms−1$⁠, and $na$ is the number density of argon atoms. A boundary condition at the surface is $ne=ji/ekTe/2πme$⁠, two boundary conditions far away from the surface are those of quasi-neutrality and the ion motion being drift-dominated. Note that the exact position of the point where the latter conditions are applied is irrelevant, provided that it is sufficiently far away from the surface. Note also that the Bohm speed under these conditions is $uB=2.7×103ms−1$⁠.

Computed distributions of ion velocity in the near-wall region for several combinations of the plasma pressure p and the ion current density $ji$ are shown in Fig. 4. Here, the characteristic ion mean free path and the Debye length are defined as $λi0=λi(uB)$ and $λD=ε0kTeuB/eji$⁠, respectively. Three circles on each curve represent points where the charge separation $(ni−ne)/ni$ decreases, in the direction from wall into the plasma, to $50%$⁠, $20%$⁠, and $10%$⁠.

The variants shown in Fig. 4(a) are chosen so that collisions of ions with atoms in the sheath are rare, $ε=λD/λi0≤0.1$⁠. One can see that this figure is qualitatively similar to Fig. 3(c) as expected. In particular, there is a plateau, in Fig. 4(a), that appears for $ε≤10−3$⁠. As expected, this plateau occurs at the Bohm speed $uB$⁠. Note that the reduction of ion speed from $uB$ by $30%$ occurs as x increases by the factor of 83 for $ε=10−4$ and 13 for $ε=10−3$⁠, i.e., the plateau is pronounced rather well.

The variants shown in Fig. 4(b) are chosen so that the sheath is moderately collisional, $λD=λi0$⁠. In this case, the quasi-neutral region is represented by the whole plasma slab; therefore, the small parameter $ε$ should be defined in terms of the slab width $Δ$⁠: $ε=λD/Δ$⁠. Note for definiteness that Fig. 4(b) refers to $Δ=4cm$⁠. The ion velocity in Fig. 4(b) is normalized by its value at $x=Δ$ so that it is of order unity in the quasi-neutral region regardless of $ε$⁠, which facilitates comparison with the various lines shown in Fig. 3(b). One can see that Fig. 4(b) is qualitatively similar to Fig. 3(b) as expected. In particular, there is no plateau in both figures.

Thus, the pattern of plasma-sheath transition revealed by numerical calculations is precisely as predicted by the asymptotic reasoning and illustrated by the simple examples shown in Figs. 3(c) and 3(b). In particular, the calculations reveal a plateau in the distribution of the ion velocity, representing the Bohm criterion, for small enough values of the ratio $λD/λi0$⁠, and “small enough” means the order of $10−3$ or smaller.

Figure 4 provides a clear answer to the first and second questions formulated at the beginning of Sec. V. Spatial distribution of the ion velocity, given by accurate numerical modeling that includes solving the Poisson equation without apriori introduction of a quasi-neutral plasma and a space-charge sheath and plotted on a semi-log scale, reveals a plateau in the intermediate region between the sheath and the quasi-neutral region, provided that ion-atom collisions in the sheath are sufficiently rare, and this plateau occurs at the Bohm speed. In other words, the ion velocity at distances from the surface much larger than the sheath thickness scale $λD$ but much smaller than L the length scale characterizing the adjacent quasi-neutral plasma region [ $L=λi0$ in the numerical calculations shown in Fig. 4(a)], is approximately constant and close to the Bohm speed $uB$⁠. This is the ultimate physical meaning of the fluid Bohm criterion and its only legitimate interpretation; any other interpretation is inevitably arbitrary.

The plateau in Fig. 4(a) is pronounced the better, the smaller the ratio $λD/λi0$ is. This clearly illustrates the fact that the Bohm criterion is an asymptotic theorem valid in the first approximation in the small parameter $λD/λi0$⁠. This theorem cannot be generalized to higher approximations, since in higher approximations the plateau does not occur.

No plateau exists in cases where the ion-atom collisions in the sheath are not rare; cf. Fig. 4(b). Similarly, no plateau exists in cases of sheaths with ionization; cf. Fig. 2 of Ref. 78. The ion velocity in the region $λD≪x≪L$⁠, i.e., between the sheath and the quasi-neutral region, is variable in these cases and cannot be characterized by a single number. Therefore, there is no way to unambiguously identify a speed with which ions enter the sheath. Hence, all versions of Bohm criterion, modified with account of collisions and/or ionization in the sheath, are inevitably arbitrary.

The task of this section is to find the objective meaning of the kinetic Bohm criterion without invoking asymptotic arguments. To this end, let us consider an ion distribution function, $f(v,x)$⁠, in the vicinity of a negative surface, obtained from an accurate numerical modeling that includes solving the Poisson equation without apriori introduction of a quasi-neutral plasma and a space-charge sheath. If the condition $λD≪L$ is satisfied, then the kinetic Bohm criterion is valid and information about it must be contained in the distribution function. Yet, how can this information be extracted? The difficulty is that the distribution function $fs(v)$ of ions leaving the quasi-neutral region and entering the sheath in the kinetic Bohm criterion (16), which is evaluated in the first approximation in the small parameter $λD/L$⁠, cannot be replaced by the “exact” distribution function $f(v,x)$ for any non-zero x, since $f(0,x)≠0$ for all x except at the surface and hence the integral on the lhs of Eq. (16) would diverge.

A simple way to overcome this difficulty is to use the kinetic Bohm criterion in the form (34): there is no problem with evaluating the integral in this equation in terms of the “exact” distribution function. (We restrict the consideration with the usual case where the ion energy distribution is analytic at low energies, which requires that $∂f/∂v|v=0=0$⁠.) Moreover, the lhs of (34) may be evaluated not only for ions entering the sheath, but for arbitrary x.

Figure 5 illustrates the application of this reasoning to the Tonks–Langmuir model, which has served as a test case for studies of the kinetic Bohm criterion since the work.⁹⁸ (The Tonks–Langmuir model describes a limiting regime occurring at very low pressures in which ions created at different locations inside the discharge column fall without collisions to the wall. The model was proposed in a seminal paper¹¹³ and is one of the classic models of gas discharge physics, e.g., textbooks.^114–116) The quasi-neutral region in this model is represented by the whole of the discharge column except for the near-wall sheath region. Therefore, x the distance from the wall in Fig. 5 is normalized by l the half-width of the column (thus $x/l=1$ corresponds to the plane of symmetry of the discharge) and $ε$ is defined as $ε=λD/l$⁠, where the Debye length $λD$ is evaluated at the plane of symmetry. As in Fig. 4(a), the variants shown in Fig. 5 are such that $ε≤10−1$⁠, i.e., the ionization in the sheath is a weak effect. For each $ε$⁠, B was evaluated for different distances from the surface by means of numerical solution of the Tonks–Langmuir problem (including the Poisson equation). Plotted in Fig. 5 are inverse normalized values of B, to facilitate the comparison with Figs. 3(c) and 4(a). The circles on the curves in Fig. 5 represent points where the charge separation $(ni−ne)/ni$ decreases, in the direction from the wall into the plasma, to $10%$⁠, $1%$⁠, and $0.1%$⁠. For $ε=0.1$⁠, the charge separation at the plane of symmetry is about $4%$ and the only point shown is the one marking the charge separation of $10%$⁠.

As expected, Fig. 5 is qualitatively similar to Figs. 3(c) and 4(a). In particular, there is a plateau in Fig. 5 in the intermediate region between the sheath and the quasi-neutral region for $ε=10−4$ and $ε=10−3$⁠, and this plateau occurs at B⁻¹ equal to the Bohm energy $kTe/2$ as it should. Note that the reduction of $B−1$ by $30%$ from $kTe/2$ occurs as x increases by factors of 46 and 7.5, respectively.

Thus, if ionization in the sheath is negligible, which means that the ratio $λD/l$ should be of the order of $10−3$ or smaller, the weighted inverse ion kinetic energy B is approximately constant in the region $λD≪x≪L$⁠, similar to what happens to the ion speed in the case of monoenergetic ions as seen in Fig. 4(a), and is approximately equal to $(kTe/2)−1$⁠. This is the ultimate physical meaning of the kinetic Bohm criterion.

It is interesting to note that Eq. (44) is not the only definition of the weighted inverse kinetic energy, suitable for such representation. Another example, tailored for the Tonks–Langmuir model, was given in Ref. 78.

There is no universally accepted theoretical description of the plasma-sheath transition and the Bohm criterion in gas discharge science, despite efforts invested by able researchers over decades and despite the existence of results of a mathematical nature, including recently obtained pure-mathematics results. The use of relatively complex mathematical methods is not by itself an obstacle to a work being accepted by the gas discharge community, as has been seen in the past. The question then arises: what should the mathematical results be to have a better chance of being accepted in gas discharge science?

In this work, the criteria of “good mathematics” are proposed that, in the opinion of the authors, are the most important for the theory of the plasma-sheath transition. The existing theoretical results on the plasma-sheath transition, in particular, the results on the Bohm criterion, are reviewed in this light.

The conclusion is that the theoretical framework adequate for the task is the perturbation theory, the small parameter being the ratio of the Debye length $λD$ to the length scale L characterizing the presheath (the quasi-neutral plasma region adjacent to the sheath). The very concept of a quasi-neutral plasma and a space-charge sheath makes sense only under the condition that the ratio $λD/L$ is small, so the above conclusion is the only logical one. The suitable tool is the method of matched asymptotic expansions, which is a perturbation technique for problems describing regions with significantly different scales. The method automatically exploits all the simplifications that are justified by the presence of different scales and distinctly reveals regions governed by different physical mechanisms along with these mechanisms. In contrast to the approximate approaches used in the vast majority of works on the theory of plasma-sheath transition, the method of matched asymptotic expansions, being a perturbation technique, does not rely on arbitrary assumptions; its accuracy is controlled by the smallness of the ratio $λD/L$⁠, which can be readily estimated for given conditions.

The asymptotic treatment of plasma-sheath transition has been supported by pure-mathematics analysis; an important confirmation that appears to be virtually unknown to gas-discharge physicists.

The asymptotic theory provides an adequate mathematical description of the plasma-sheath transition, including the fluid and kinetic Bohm criteria. In particular, it provides a proof that the sheath edge and modified and/or generalized fluid or kinetic Bohm criteria with account of collisions and/or ionization and/or geometrical effects in the sheath cannot be introduced but arbitrarily, and that there is no divergence of the kinetic Bohm criterion for low ion energies and hence no need to remedy it. Nevertheless, new modifications of the Bohm criterion that take collisions into account continue to be published, without mentioning that all such modifications are arbitrary and each new modification is just another item in a long list of existing ones.

There are high-quality experimental works in which the measurement results are interpreted in terms of Bohm criterion at the sheath edge, e.g., Refs. 73, 88, and 117–119. Revisiting these experimental results to interpret them in light of the asymptotic theory would be an interesting and important task.

There are works in which approximate models of the plasma-sheath transition are supported by references to results of numerical calculations. Unfortunately, in many cases such validation is methodologically incorrect and not convincing. For example, a comparison of results given by a theoretical model employing an approximate description of the plasma-sheath transition, based on fluid equations, with results of PIC modeling says little about the accuracy of the approximations on the plasma-sheath transition employed by the theoretical model. (It is of course true that PIC modeling generally describes low pressure gas discharges more accurately than the fluid equations, but in this aspect this is irrelevant.) An adequate validation of an approximate model of the plasma-sheath transition is obtained by comparison with results of numerical calculations performed using the same description of the transport of the charged particles but with the solution of the Poisson equation in the entire discharge region without dividing this region into a quasi-neutral plasma and a space-charge sheath.

Examples of such comparison are seen in Figs. 4–6. Figures 4 and 5 explain the physical meaning of the fluid and kinetic versions of the classical collisionless Bohm criterion and provide its only legitimate interpretation; any other interpretation is inevitably arbitrary. On the other hand, Fig. 6 shows that the Bohm criterion for collisional sheaths derived from the assumption of a maximum of the Sagdeev potential at the sheath edge, which has become popular in the modern literature, is satisfied at each point of the quasi-neutral region and does not provide any additional information concerning the ions leaving the quasi-neutral plasma and entering the sheath. The same is true for expressions for the Bohm speed derived from an alternative form of the Bohm criterion involving derivatives of the charged-particle densities.

The work was supported by FCT—Fundação para a Ciência e a Tecnologia of Portugal under projects UIDB/50010/2020 (https://doi.org/10.54499/UIDB/50010/2020), UIDP/50010/2020 (https://doi.org/10.54499/UIDP/50010/2020), and LA/P/0061/2020 (https://doi.org/10.54499/LA/P/0061/2020).

The authors have no conflicts to disclose.

M. S. Benilov: Conceptualization (lead); Investigation (equal). N. A. Almeida: Investigation (equal).

The data that support the findings of this study are available within the article.

In Ref. 33, a collisional Bohm criterion was derived by assuming that the Sagdeev potential^120,121 attains a maximum at the sheath edge. There are a number of subsequent works along these lines.^34–46 

Equation (A1) is similar to the equation of motion of a particle under the effect of a conservative force with the potential V, with x and $φ$ having the meaning of time and position of the particle. This equation was introduced in Ref. 120 [cf. Eq. (78) of Ref. 120] for studying the ion acoustic shock waves by means of analogy to an oscillator in a potential well, and the effective potential energy V is sometimes called the Sagdeev potential (Sec. 8.3.1 of Ref. 121).

The rhs of Eq. (11) represents the dimensionless form of the two-term expansion of the function $−dV(φ)dφ$ in the vicinity of the point $φ=0$⁠. It follows that $dV(φ)dφ|φ=0=0$ and $d2V(φ)dφ2|φ=0$ has the same sign as $(B−1)$⁠.

Let us return to Bohm's model of collisionless space-charge sheaths,⁹⁹ considered in Secs. III A and III B. If the Bohm criterion is satisfied with the inequality sign, $B<1$⁠, then $d2V(φ)dφ2|φ=0<0$ and the Sagdeev potential $V(φ)$ has a maximum at $φ=0$⁠. We remind that $φ=0$ corresponds in Bohm's model to the infinite distance from the surface, where the plasma is neutral.

As before, $vi$ is the velocity with which the ions move in the direction to the surface, $λi$ is the ion-atom mean free path, and $uB=kTe/mi$ is the Bohm speed; $E=dφ/dx$ is projection of the electric field along the direction from the plasma to the surface (which is positive).

For the specific cases of constant mean free path or constant collision frequency, Eq. (A5) may be readily resolved with respect to $vi$ and the solution coincides with analytical expressions for $vi$ in terms of E, deduced in Ref. 33 [cf. left-hand sides of Eqs. (12) and (13) in Ref. 33] and denominated the lower limit of Bohm's sheath criterion for collisional sheaths. Note that in Ref. 42, the lhs of Eq. (13) of Ref. 33 [i.e., the solution to Eq. (A5) for constant collision frequency] was found to closely agree with results of PIC simulations, with the conclusion that the collisional Bohm criterion³³ provides an accurate approximation.

It is natural to try to find an objective meaning of the collisional Bohm criterion (A5), similar to how it was done above for the classical fluid and kinetic Bohm criteria and is illustrated in Figs. 4 and 5. To this end, let us consider Fig. 6, which refers to a weakly collisional sheath and shows results of the same numerical calculations that are plotted in Fig. 4(a). Here, $D1$ designates the lhs of Eq. (A5); $D1−1/2$ is chosen for the graph to facilitate a comparison with Fig. 4(a) [in the limiting case of absence of collisions, when the second term on the lhs of Eq. (A5) is negligible, $D1−1/2≈vi/uB$]. The dashed lines in Fig. 6 represent the separation of charges, $D2=(ni−ne)/ni$⁠, and are added to readily identify the space-charge sheath and the quasi-neutral region (these are domains where $D2$ is of the order unity or small, respectively).

Inside the space-charge sheath, the dependence $D1−1/2(x)$⁠, shown in Fig. 6, is qualitatively similar to the dependence $vi(x)$ in Fig. 4(a). There is no similarity in the quasi-neutral region: $vi(x)$ in Fig. 4(a) is decreasing in the quasi-neutral region, with a plateau $vi≈uB$ for small $ε$ in the region immediately adjacent to the sheath, which is characteristic of the Bohm criterion; in contrast, $D1−1/2$ in Fig. 6 is very close to 1 in the whole quasi-neutral region. Thus, the collisional Bohm criterion (A5) is fulfilled at each point of the quasi-neutral region.

The latter is, of course, unsurprising. The relation (A4) with the equality sign represents a trivial consequence of the quasi-neutrality condition $ni=ne$⁠. Therefore, Eq. (A5) is a corollary of the fluid transport equations written under the approximation of quasi-neutrality and must hold asymptotically throughout the quasi-neutral region.

Thus, the inequality (A3) at the edge of a collisional sheath with the $≤$ sign may be postulated by analogy with the alternative form of the collisionless Bohm criterion, Eq. (A6), without the necessity to invoke the Sagdeev potential. The procedure based on postulating the inequality (A3), with the derivatives $dni/dφ$ and $dne/dφ$ being evaluated from the fluid transport equations, was employed in Refs. 82–84 and 97. It was stressed that the Bohm criterion applies over a spatially extended presheath–sheath transition region, as opposed to a sharp boundary, and expressions for the Bohm speed, which is spatially inhomogeneous in this approach, were derived with account of detailed transport physics. In Refs. 82–84, these expressions were compared with results of PIC simulations and a good agreement in the quasi-neutral region was found.

Postulating a maximum of the Sagdeev potential at the edge of a collisional sheath is no better justified than postulating the inequality (A3). On the other hand, replacing the former with the latter streamlines the derivation of the collisional Bohm criterion (A5) and clearly reveals its similarity with the method employed in Refs. 82–84 and 97. In fact, the analysis in the preceding section starting from Eq. (A3) can be regarded as a simple illustration of the method,^82–84,97 and the above Fig. 6 is an analog of Fig. 2 of Ref. 82, Fig. 4 of Ref. 83, and Figs. 7 and 9 of Ref. 84.

In the framework of Bohm's model of positive collisionless space-charge sheaths,⁹⁹ the inequality (A6) is a legitimate alternative form of the classical Bohm criterion. We stress that the model⁹⁹ considers only the sheath itself while the neutral plasma appears only as a boundary condition at infinity. However, when the same inequality (A6) is applied at the sheath edge in the problem of the plasma-sheath transition, i.e., at a point that is located at a finite distance from the surface and separates the positive space-charge sheath and the quasi-neutral plasma, then this inequality becomes equivalent to the condition $dni/dx≤dne/dx$⁠, which is trivially satisfied (with the equality sign) at all points within the quasi-neutral plasma, thereby providing no new information.

Therefore, the collisional Bohm criterion Eq. (A5), as well as expressions for the Bohm speed derived in Refs. 82–84 and 97, represent corollaries of the fluid transport equations written under the approximation of quasi-neutrality and hold throughout the quasi-neutral region. These corollaries may be useful, e.g., the above-mentioned agreement of the Bohm speed, computed in Refs. 82–84 on the basis of fluid equations, with results of PIC simulations attesting to a good accuracy of the fluid equations in the quasi-neutral plasma. On the other hand, the collisional Bohm criterion Eq. (A5) and the expressions for the Bohm speed,^82–84,97 being valid for any point in the quasi-neutral region, do not allow one to single out any specific condition concerning the ions leaving the quasi-neutral plasma and entering the sheath.

The latter is very clear from a comparison of Figs. 4(a) and 6. These figures represent the results of an accurate numerical solution of a system of equations that includes the Poisson equation and does not involve any apriori division of the computation domain into a quasi-neutral plasma and a space-charge sheath and refer to exactly the same conditions. There is a plateau $vi≈uB$ in the section of the quasi-neutral plasma adjacent to the sheath in Fig. 4(a) for $ε=λD/λi0$ small enough, namely, $ε=10−3$ and $10−4$⁠; a clear manifestation of the classical Bohm criterion. In contrast, there are no such-type peculiarities in Fig. 6: the collisional Bohm criterion Eq. (A5) is valid throughout the whole quasi-neutral region and thus does not provide any specific condition concerning the ions leaving the quasi-neutral plasma and entering the sheath. In this sense, Eq. (A5) is no different from the quasi-neutrality condition itself, as is clearly seen in Fig. 6. It is difficult to see how such criteria could be useful.