the child–langmuir law and analytical

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IOP PUBLISHING PLASMA SOURCES SCIENCE AND TECHNOLOGY

Plasma Sources Sci. Technol. 18 (2009) 014005 (14pp) doi:10.1088/0963-0252/18/1/014005

The ChildLangmuir law and analyticaltheory of collisionless to collision-dominated sheathsM S Benilov

Departamento de F sica, Universidade da Madeira, 9000 Funchal, Portugal

Received 6 July 2008, in nal form 11 September 2008Published 14 November 2008Online at stacks.iop.org/PSST/18/014005

AbstractThis paper is concerned with summarizing simple analytical models of space-charge sheathsand tracing their relation to the ChildLangmuir model of an ion sheath. The topics discussedinclude the ChildLangmuir law and model of a collisionless ion sheath, the MottGurney lawand model of a collision-dominated ion sheath, the Bohm model of a collisionless ionelectronsheath, the SuLamCohen model of a collision-dominated ionelectron sheath, ion sheathswith arbitrary collisionality, high-accuracy boundary conditions for the ChildLangmuir andMottGurney models of an ion sheath and the mathematical sense of ChildLangmuir typemodels of an ion sheath from the point of view of modern theoretical physics.

(Some gures in this article are in colour only in the electronic version)

1. Introduction

In his 1911 paper [ 1], Child treated what he called theelectrostatic effect produced by ions. He considered positiveions moving without collisions between two innite parallelplates, i.e. in a plane-parallel vacuum diode. There is aninnitely large number of ions at the positive plate and theirvelocity there is zero. The electric eld between the platesis steady state and distorted by the space charge of theions. Solving the equation of motion of an ion jointly withthe Poisson equation, Child found an analytical solution fordistributions of the electrostatic potential, electric eld and theion density between the plates. One of the results was thatthe current which is possible to be carried by positive ionswith a given distance and a given potential difference betweenthe plates is limited, however big the capability of ion supplyfrom the positive plate. This limitation is caused by the ionspace charge, i.e. by the mutual repulsion of the ions, and thelimiting current occurs when the electric eld at the positiveplate is zero. The limiting current varies directly as the three-halves power of the applied voltage and inversely as the squareof the distance separating the plates. In 1913, Langmuir [ 2]applied similar equations to the case where the conductiontakes places by electrons rather than ions, as a part of his studyof the effect of space charge on thermionic emission currentsin high vacuum.

The works [ 1, 2] represent a basis of the theory of space-charge-limited currents, and the formula for the space-charge-limited current in a plane-parallel diode is known as theChildLangmuir law, or Childs law, or the three-halves powerlaw. The theory has different applications, and the one whichis of particular interest for gas discharge and plasma physicsemergedinthe1920s:inthepapers[ 3, 4] Langmuir suggestedamodel of a high-voltage near-cathode sheath which is virtuallyelectron-free and formed by positive ions that enter the sheathwith energies negligible compared with those they acquirein the sheath itself, and these ion sheaths are in a naturalway described by Childs theory of space-charge-limited ion

current. Thismodel has initiateda hugenumber of publicationsconcerned with the analytical theory of space-charge sheaths.By now, this theory is well developed (e.g. review [ 5] andreferences therein) and includes both simple and sophisticatedmodels, thesimplest andstillmostwell knownbeingtheChildLangmuir model.

Nowadays, simulation methods are available that allowunied modeling of the whole volume occupied by the ionizedgas, without a priori dividingit into a quasi-neutral plasmaandspace-charge sheaths. However, analytical models of space-charge sheaths, and in the rst place simple ones, still retaintheresignicance. One of thereasonsfor this is,of course, theirmethodical value. Another reason is as follows: the above-mentioned unied simulation methods are available for coldplasmas but are more difcult to develop for plasmas with

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Plasma Sources Sci. Technol. 18 (2009) 014005 M S Benilov

a high density of charged particles and, consequently, a highdegree of quasi-neutrality in thebulk. Theexampleof the latterare plasmas of high-pressure arc discharges, where methods of unied simulationof near-cathode layers are justappearingandtheChildLangmuir sheathmodel or modelsnot very differentfrom it still remain a workhorse; e.g. review [ 6] and references

therein.This paper is concerned with summarizing simpleanalytical models of space-charge sheaths and tracing theirrelation to the ChildLangmuir model of an ion sheath. Bothcollision-freeandcollision-dominatedsheaths are treated. Theoutline of the paper is as follows. The ChildLangmuir modelof a collisionless ion sheath and its analog for a collision-dominated sheath, the MottGurney model, are describedin section 2. Section 3 is concerned with sheath modelsaccounting for the presence of ions and electrons, which arecapable of providing descriptions of a sheath with a moderatevoltage and of a transition from an ion sheath to a quasi-neutral plasma. The models treated in this section includethe Bohm model of a collisionless sheath and its analog fora collision-dominated sheath, the SuLamCohen model. Amodelofan ionsheath with anarbitrarydegree ofcollisionality,which describes a smooth transition from the ChildLangmuirmodel for a collisionless sheath to the MottGurney modelfor a collision-dominated sheath, is treated in section 4. Alsogiven in section 4 are high-accuracy boundary conditions forthe ChildLangmuir and MottGurney models. Another topicdiscussed in section 4 is the mathematical sense of modelsof ion sheaths from the point of view of modern theoreticalphysics.

2. Ion sheaths2.1. Collisionless ion sheath: the ChildLangmuir law and sheath model

Following [ 1], we consider a vacuum diode consisting of twoinnite parallel plates, a positive one (anode) positioned atx =0 and a negative one (cathode) at x =d . Singly chargedpositive ions are emitted into the gap by the anode with a zerovelocity. There is a voltage U applied to thediode. The systemof governing equations includes the equation of motion of anion, the equation of conservation of the ion ux across the gapand the Poisson equation:

m ivi dvi

dx = e d

dx, j i = en ivi , 0 d

2

dx 2 = en i.(1)

Here mi, ni and vi are the particle mass, number densityand velocity of the ions, j i is the density of electric currenttransported by the ions (a constant positive quantity) and isthe electrostatic potential. While writing the rst equation, theacceleration of an ion, d vi/ dt , was represented as vidvi/ dx .

The unknown functions vi(x) and (x) satisfy boundaryconditions

vi(0) =0, ( 0) =0, (d) = U. (2)The rst equation in ( 1) may be integrated to give the

equation of conservation of the total energy of an ion. Finding

the integration constant with the use of the rst two boundaryconditions ( 2), one nds vi = 2e /m i. Using this resultand the second equation in ( 1) in order to eliminate n i fromthe Poisson equation in terms of , one arrives at an equationinvolving only one unknown function, (x) . Since it involvesneither the independent variable x nor the rst derivative

d/ dx , it can be solved analytically. To this end, one canmultiply this equation by d / dx and integrate over x. Theequation obtained may be written as

e 208m ij 2i

1/ 2

ddx2

= + C1. (3)

Here and further C 1 , C2 , ... are integration constants. Notethat the constant C1 is related to the electric eld at the anode

surface: C1 = e 208m ij 2i

1/ 2

ddx x=02. Solving equation ( 3)

for dx/ d and integrating over , one arrives at the followingsolution for the function (x) :

43

+ C11/ 2

2C1 =8m ij 2i

e 20

1/ 4(x + C2) .

(4)

A relationship between integration constants C1 and C2 isfound by means of the second boundary condition ( 2):

C2 = 83/ 4

3 e 20m ij 2i

1/ 4

C 3/ 21 . (5)

The constant C1 is found by means of the third boundarycondition ( 2), which can be written in a dimensionless form as

(1 + b1)1/ 2(1 2b1) + 2b3/ 21 =a 1, (6)where a1 = (81m id 4j 2i / 32e 20 U 3)1/ 4 may be interpreted asthe square root of the dimensionless ion current density andb1 =C1/ U as the squared dimensionless electric eld at theanode surface.

Equation ( 6) represents an algebraic equation relating the(positive) unknown b1 to a (positive) parameter a 1. It admitsno real positive roots at a 1 > 1. There isone root in the interval0 < a 1 1, which is shown in gure 1. It should be stressedthat b1 =0 at a 1 =1. Note that a simple approximate formulafor this root may be obtained by means of a rational-fractioninterpolation (Pad e approximant) over a

21 between the two-term asymptotic expansion of the function b1(a 1) at a 1 0and the value b1 =0 at a1 =1:

b1 =24316a 21

1 a 2127 + 5a 21

. (7)

One can see from gure 1 that this formula is quite accurate.The solution is complete now. One can conclude that the

problem being considered is solvable provided that the ioncurrent density does not exceed a value that corresponds toa 1 =1, namely,

j i = 32e20U 3

81m id 4

1/ 2

. (8)

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0.2 0.4 0.6 0.8 1

0

1

2

3

4

b 1 ,

b 2

a 1 , a 2

b1b 2

Figure 1. The square root of the dimensionless current densityversus the squared dimensionless electric eld at the emittingelectrode in the ChildLangmuir and MottGurney models. Solid:solutions to equations ( 6) and (15). Dashed: approximate solutions,equations ( 7) and (16).

If the density of the ion current emitted by the anode is belowthis value, then every ion emitted will reach the cathode, j ithe current density in the gap equals the ion emission currentdensity and is treated as a known quantity, the electric eldat the anode is non-zero and its value is given by formula ( 7)(or may be determined by numerically solving equation ( 6)).As the emission capability of the anode increases and thecurrent in thegapapproaches the limiting value( 8), theelectriceld at the anode surface vanishes and the current in the gapstops increasing. This limitation is due to the space chargewhich is created in the gap by the charge carriers (ions).

Obviously, the above treatment is equally applicable tothe case where the charge carriers are electrons emitted by thecathode rather than ions emitted by the anode. In particular,the limiting electron current is given by equation ( 8) with m ireplaced by the electron mass [ 2].

The above treatment represents a basis of the theory of space-charge-limited currents, and equation ( 8) is the ChildLangmuir law.

Let us apply the above treatment to a physical object otherthan a vacuum diode, namely, a high-voltage near-cathodesheath in a gas discharge [ 3, 4]. Citing Langmuir [ 4]: If theelectrodeis 50 voltsor more negative with respect to theplasmathe sheath will contain no appreciable number of electrons andthe positive ions will enter the sheath with energies negligiblecompared to those they acquire in the sheath itself. Thepositive ions are assumed to cross the sheath and reach thecathode without collisions with neutral particles. The valuex =0, which in the case of a vacuum diode was attributed tothe anode, is now attributed to the outer edge of the sheath,i.e. a plasmasheath boundary, and x = d to the cathodesurface, so d designatesthe thicknessof thesheath. Thezero of potential is attributed to the sheath edge and the sheath voltageis designated by U .

Under the above assumptions, the near-cathode sheathis described by equations ( 1) and boundary conditions ( 2).Since the electric eld in the plasma is much smaller thana characteristic eld inside the sheath, the electric eld at theouter edge of the sheath may be set equal to zero while treatingthe sheath:

ddx x=0 =0. (9)

A solution to the stated problem is given by equation ( 4) inwhich C1 is set equal to zero in accordance with the boundarycondition ( 9). (This also implies C2 = 0 in accordancewith equation ( 5).) In other words, a collisionless ion sheathis described by the same solution that describes a vacuumdiode with a space charge-limited ion current. In particular,the ion current density, the sheath voltage and thickness arerelated by the ChildLangmuir law ( 8). However, the physicalinterpretation of the latter equation is now different. CitingLangmuir [ 4]: In the usual applications of [equation ( 8)] thevoltage and the distance between the electrodes are known andwe wish to calculate the current density that can ow whenthe current is limited by space charge. In the present case,however, the current density [ j i] is xed by conditions withinthe plasma and since the applied voltage is usually known theequation can be used to calculate only the thickness [ d ] of thesheath.

The ChildLangmuir law ( 8) was applied to thedescription of a near-cathode layer of a gas discharge in the1923 papers by Ryde [ 7] and Langmuir [ 3]. The conceptof a high-voltage electron-free near-cathode sheath, whichconstitutes the basis of such approach, was formulated in thelatter work.

Let us introduce a characteristic value of the iondensity in the sheath: n (s)i = j i/e eU/m i. Eliminating fromequation ( 8) j i in terms of n

(s)i , one can obtain

d =25/ 4

3 0U n (s)i e . (10)Thus, the thickness ofthe ion sheathis of the order of the Debyelength in which the electron thermal energy kT e is replaced bythe electrostatic energy eU .

Equation ( 3) with C 1 = 0 allows one to nd the electriceld at the cathode surface in terms of the ion current densityj i and the sheath voltage U (but not the sheath thickness):

ddx x=d

=8m ij 2i U

e 20

1/ 4

. (11)

This equation is used in the theory of arc discharges forevaluation of the effect of the electric eld at the cathodesurface over the electron emission current [ 8] and is usuallycalled the Mackeown equation.

The above-described ion sheath model is based on theassumption that the electron contribution to the space chargein the sheath is negligible. The electrons can enter the sheathboth from the adjacent plasma and on being emitted from thecathode. Theabove-cited conditionof the sheath voltagebeinghighenough ensures thatthe plasma electronscannotovercome

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theretardingelectric eld andenter thesheath, but thequestionof the contribution of theemitted electrons remains. If needed,this contribution can be taken into account [ 7, 8]. On the otherhand, a simple estimate follows from the fact that the electriccurrent in the sheath transported by the emitted electrons iscomparable toormuch smaller than theion current (in thecases

of an arc or, respectively, glow discharge): since the velocityof the electrons is much higher than the ion velocity, it followsimmediately that the space charge of the emitted electrons isnegligible [ 7].

2.2. Collision-dominated ion sheath: the MottGurney lawand sheath model

The above theory of space charge-limited currents in a plane-parallel vacuum diode can be readily modied for the casewhere the interelectrode gap is lled with a gas and motionof charge carriers in the gap is collision-dominated ratherthan collision-free. In this case, the rst equation in ( 1) is

replaced byvi = i

ddx

, (12)

where i is the mobility of the charge carriers (ions).Equation ( 12) is written under the assumption that diffusion of the ions is negligible compared with drift in the electric eld.(This assumption is justied provided that the ion gas is coldenough: the thermal energy of an ion should be much smallerthan the electrostatic energy eU . The thermal energy of an ionequals the thermal energyof a neutral particle plus a differenceof the order of work of the electric eld over the ion mean freepath. The thermal energy of a neutral particle is always muchsmaller than eU , work of the electric eld over the ion meanfree path in the collision-dominated case is also much smallerthan eU . Hence, the above-mentioned assumption is alwayssatised in the collision-dominated case.) For brevity, let usrestrict consideration to the case where the frequency of ionatom collisions does not depend on velocity, then i may betreated as constant. The rst boundary condition in ( 2) mustbe discarded.

The stated problem is rather similar to the problem of avacuum diode considered in the preceding section and maybe solved in a similar way. The Poisson equation may berewritten as

0 id

dx

d2

dx 2 =j i . (13)

Integrating equation ( 13) over x , solving theobtained equationfor d/ dx and integrating over x once again, one arrives at thefollowing solution for the function (x) :

= 8j i90 i

1/ 2

(x + C3)3/ 2 + C4. (14)

The integration constant C3 is related to the electric eld at the

anode surface, C 3 = 0 i

2j i ddx x=0

2, and is governed by the

equation(1 + b2)3/ 2 b

3/ 22 =a 12 , (15)

where a 2 = (8d 3j i/ 90 iU 2)1/ 2 and b2 =C3/d . Again, thisis an algebraic equation for the (positive) unknown b2 , which

admits no real positive roots at a2 > 1 and has one root in theinterval 0 < a 2 1. This root is shown in gure 1. One cansee from gure 1 that this root is accurately approximated bythe simple formula

b2

=

32

9a22

1 a 228 + a

22

. (16)

Again, this formula was obtained by means of a rational-fraction interpolation between the two-term asymptoticexpansion of the function b2(a 2) at a2 0 and the valueb2 =0 at a2 =1.The physical meaning of the obtained solution is quitesimilar to that in the problem of vacuum diode. The space-charge-limited ion current is

j i =90 iU 2

8d 3 . (17)

This formula, which is usually called the MottGurney law,represents an analog of the ChildLangmuir law, equation ( 8),for the collision-dominated case. It was derived in [ 9] inconnection with conduction current in semiconductors andinsulators and in [ 10] as an analog of the ChildLangmuir lawfor a plane-parallel diode lled with a high-pressure gas.

The above treatment, including solution ( 14) with C 3 =C4 = 0 and the MottGurney law, can be in a natural wayapplied to a collision-dominated near-cathode ion sheath, justin the same way as the ChildLangmuir law applies to acollision-free near-cathode ion sheath. Let us introduce acharacteristic value of the ion density in the sheath: n(s)i =j id/e iU . Eliminating j i fromequation ( 17), oneagainarrives

at equation ( 10), except for the numerical coefcient (3 / 23/ 2rather than 2 5/ 4/ 3). Hence, the thickness of the ion sheathagain is of the order of the Debye length evaluated in terms of the electrostatic energy.

An analogue of the MottGurney law for the case of constant mean free path, where the cross section of ionatomcollisions does not depend on velocity, was given in [ 11].

3. Sheaths formed by ions and electrons

The applicability of the model of an electron-free sheathintroduced by Langmuir [ 3, 4] is limited to the case of high

sheath voltages. A question arises regarding the modelsapplicable at moderate sheath voltages. Obviously, suchmodels must take into account not only positive ions butalso electrons entering the sheath from the plasma against theretarding sheath electric eld.

The model of an electron-free sheath represents areasonable approximation in the bulk of a high-voltage near-cathode sheath, where the plasma electrons cannot penetrateand their density ne is much smaller than ni the positiveion density. On the other hand, the model clearly loses itsvalidity in an outer section of the near-cathode sheath: if thetwo densities are virtually equal in the (quasi-neutral) plasmaoutside the sheath, then n e in an outer section of the sheath,while being smaller than ni, is still comparable to it. Thisouter section of a high-voltage near-cathode sheath can be

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called an ionelectron layer. A question arises of nding asolution describing this layer and matching it with the solutiondescribing the electron-free bulk of the sheath (which can becalled the ion layer). This can be done by applying the limitof high sheath voltages to a model of a moderately negativesheath mentioned in the previous paragraph. Results of such

analysis must in a natural way describe both the electron-freeion layer and the ionelectron layer.Both above-mentioned questions are addressed in this

section, rst for a collisionless sheath and then for a collision-dominated one.

3.1. Collisionless ionelectron sheath: the Bohm model

In order to describe a collisionless negative sheath formed bycold (monoenergetic) ions and Boltzmann-distributed plasmaelectrons, it is sufcient to supplement the Poisson equation inthe ChildLangmuir ion sheath model with a term accountingfor space charge contributed by the electrons: the third

equation ( 1) is replaced by

0d2dx 2 = e(n i n e), n e =n0 exp

ekT e

, (18)

where T e is the electron temperature and n0 is the electronnumber density at a point where = 0. This equationwas introduced by Langmuir; cf equations (68) and (70) of [4]. Note that the assumption of cold ions is justied inthis case provided that the ion temperature is much smallerthan the electron temperature. The assumption of Boltzmanndistribution of the density of plasma electrons is justied if theux of the electrons from the plasma to the electrode surface ismuch smaller than the chaotic electron ux inside the plasma.The latter is the case if the potential of the surface at which thesheath is formed is around or below the oating potential.

This is the onlyamendment thathas tobe introduced in theequations of the electron-free collision-free ion sheath modelin order to render it applicable to a collision-free ionelectronsheath; the rst and second equations ( 1) remain unchanged.There is, however, a fundamental difference between the twomodels as far as boundary conditions on the plasma side areconcerned. The ion sheath model breaks down at a nitedistance from the cathode: a solution cannot be extendedbeyond a point at which the electric eld vanishes, and this

point represents an outer edge of the ion sheath. There isno such breakdown in the model of an ionelectron sheath:a solution can be extended to innitely large distances fromthe cathode, with the ion and electron densities tending tothe same constant value and the potential also tending to aconstant value. In modern terms, one can say that the ionelectron sheath solution can be asymptotically matched with asolution describing the quasi-neutral plasma, in contrast to theion sheath solution which neglects the presence of electronsand therefore cannot be matched with a plasma solution. Thisfeature of theionelectron sheathmodel anditsmost importantconsequence, the Bohm criterion, were described in 1949 byBohm [12]. Indications in the same direction can be foundalready in the 1929 paper by Langmuir [ 4]; e.g. the rstparagraph on p 976 and fourth paragraph on p 980.

Obviously, there is no sense in talking of a sheath edgein the Bohm model. Therefore, the denition of the axis xused up to now needs to be modied: in the current sectionand in the next one we will assume that the origin is positionedat the electrode (or wall) surface and the axis is directed fromthe electrode into the plasma. (Note that the ion velocity v i

and the ion current density j i become negative.) The boundarycondition at the electrode surface is the same as in ( 2):

= U. (19)Boundary conditions on the plasma side of the sheath, i.e. atinnitely large x , read

n i n s , n e n s, v i vs, 0,(20)

where n s is the density of the charged particles on the plasmaside of the sheath, vs = (j i)/en s is the speed with whichthe ions leave the plasma and enter the sheath, the potentialon the plasma side of the sheath is set equal to zero. (Thelatter means that n0 =n s.) The words innitely large x havehere the conventional theoretical-physics meaning: boundaryconditions ( 20) apply at distances from the electrode muchlargerthan thescaleof thicknessof thesheath but much smallerthan a characteristic length scale in the quasi-neutral plasma.

Note that some authors, including Bohm himself, use theterm sheath edge in connection with theboundary conditions(20). This is shorter to write than region on distances fromthe electrode much larger than the scale of thickness of thesheath but much smaller than a characteristic length scale inthe quasi-neutral plasma, but has proved confusing. We stress

once again that an ionelectron sheath has no denite edge; of course, one can dene a sheathedgeas a point where the chargeseparation equals 1% (or 3%, 5%, etc), but any such denitionwill be arbitrary.

A solution to the stated problem may be found inquadratures in the same way as the solution to the ChildLangmuir ion sheath model was found in section 2.1. Theion velocity may be expressed as

vi = v2s 2em i . (21)Eliminating from the rst equation in ( 18) (the Poissonequation) n i and n e in terms of with the use of the secondequation in ( 1), equation (21), and the second equation in(18), one arrives at an equation involving only one unknownfunction, (x) , and not involving the independent variablenor the rst derivative. Multiplying this equation by d / dx ,integratingover x andnding theintegrationconstantby takinginto account that d / dx 0 as 0 in accordance withthe last boundary condition ( 20), one obtains02

ddx

2

=n sm iv2s 1 2em iv2s

1/ 2

1

+ n skT e exp ekT e 1 . (22)

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Solving this equation for d x/ d and integrating over , onecan obtain a solution in quadratures. Without elaborating thispoint, we note the following. At large x, where is small,the right-hand side of equation ( 22) may be expanded in andassumes the form

n se2

2 1kT e

1m iv2s

2

+ . (23)The left-hand side of equation ( 22) is positive, so the rst termin (23) must be non-negative. One comes to the inequality

vs kT em i . (24)Thus, the ions leave the quasi-neutral plasma and enter thesheath with a velocity equal to or exceeding u B = kT e/m i;the famous Bohm criterion [ 12].

There is a huge number of works dedicated to the Bohm

criterion and, in more general terms, different aspects of thetransition from a quasi-neutral plasmato a collisionless sheath;see reviews [ 5, 13] and references therein. The most reliableresults have been obtained by means of the method of matchedasymptotic expansions, which is a standard tool for solvingmulti-scale problems (e.g. [ 1419]) and represents a powerfulalternative to intuitive approaches. Except in a few articialcases, the Bohm criterion is satised with the equality sign.To a rst approximation, the plasma-sheath transition is rathersimple: the above-described Bohm solution for the sheath canbedirectlymatched witha solutiondescribingthe quasi-neutralplasma and the Bohm criterion with the equality sign plays therole of a boundary condition for both the sheath and plasmasolutions. The situation turns more complex in a secondapproximation: an intermediate region separating the sheathand the quasi-neutral plasma must be introduced [ 2022].The necessity of such a region is clear from the fact that theion velocities in the sheath and in the quasi-neutral plasmaapproach the Bohm velocity from above and, respectively,below; hence a direct matching is impossible beyond the rstapproximation and an intermediate region must be introducedwhere the ions moving to the electrode pass the Bohm velocityor, as one can say (e.g. [ 13] and references therein), the ionacoustic sound barrier is broken.

We consider at the moment the case of a moderately

negative sheath on an electrode or an insulating wall, wherethe sheath voltage U is of the order of kT e /e the electrontemperature measured in electronvolts. Variations of potentialin the sheath are also of the order of kT e /e . The charged-particle densities are of the order of n s . Equation ( 22) with vsreplaced by the Bohm velocity uB may be written as

2D2

ekT e

2

ddx2

= 1 2ekT e

1/ 2

2 + exp ekT e

,

(25)

where D = ( 0kT e /n 0e2)1/ 2 is the Debye length evaluated onthe plasma side of the sheath. Assuming that in the sheath theterm on the left-hand side of this equation is of the same orderof magnitude as the terms on the right-hand side, one nds thatthe thickness (length scale) of the sheath is of the order of D .

The above-described Bohm model is applicable not onlyto moderately negative near-electrode and near-wall sheathsbut also to high-voltage near-cathode sheaths, where U kT e /e . Hence, the ChildLangmuir model can be derivedfrom the Bohm model by means of the limiting transition , where =eU/kT e . An investigationof this limitingtransition is of interest in order to better understand errorsintroducedby differentapproximationsof the ChildLangmuirmodel and to eventually improve the overall accuracy. Suchan investigation, performed by means of the method of matched asymptotic expansions in [ 23], revealed two sub-layers: the ionelectron layer, which is an outer section of the space-charge sheath where the electron and ion densitiesare comparable, and the ion layer, which occupies the bulk of the space-charge sheath and in which the electron density isexponentially small compared with the ion density.

Scalings in the ionelectron layer are the same as those ina moderately negative sheath, i.e. the length scale, thecharged-particle densities and variations of potential are of the ordersof D , ns, and kT e /e , respectively. Let us estimate orders of magnitude of parameters in the ion layer. In the ion layer, is of the order of U and considerably exceeds kT e /e ; hencethe right-hand side of equation ( 25) equals (2e/kT e)1/ 2 toa rst approximation and is of the order of 1/ 2 . Equating theorder of magnitude of the left-hand side of equation ( 25) to 1/ 2, one nds that the thickness of the ion layer is of the orderof D 3/ 4.

It follows from equation ( 21) that the ion speed in the ionlayer is of the order of uB 1/ 2 . Consequently, the ion densityis of the order of ns 1/ 2. The electron density in the ionlayer, being governed by the Boltzmann distribution ( 18), is

exponentially small with respect to the large parameter .It follows from the above asymptotic estimates that the

thickness (length scale) of the ion layer substantially exceedsthe thickness of the ionelectron layer. Hence, the ion layerhas a more or less distinct edge. On the other hand, nounambiguousdenitionof this edge canbe given toanaccuracybetter than O( 3/ 4) , which is the order of the ratio of thethickness of the ionelectron layer to the thickness of the ionlayer. The voltage drop in the ion layer is much higher thanthat in the ionelectron layer. The ion speed and density in theion layer are much higher and, respectively, lower than in theionelectron layer. The scale of ion density in the ion layer,

n(il)

i = ns 1/ 2

, and the length scale = D3/ 4

are relatedby the equation

= 0U n (il)i e , (26)which is similar to equation ( 10) and has the same physicalmeaning (the thickness of the ion layer is of the order of theDebye length evaluated in terms of the local ion density andthe electrostatic energy).

The above-described asymptotic structure is shown ingure 2.

It can be shown on the basis of the above asymptoticestimates that the ion layer to a rst approximation is describedby the ChildLangmuir model, and the (relative) error of this approximation is of the order of 1/ 2 . In other words,

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x

n i ,

n e

ni

ne

Quasi-neutralplasma

vi=-2u

B

=-1.5kT e /e

Ion-electronlayer

Ion layer

vi=-u

B

= 0

Figure 2. Asymptotic structure of the collision-free high-voltagecathode sheath.

terms which are missing from the solution, or are calculatedincorrectly, are of the order of 1/ 2 relative to the leadingterms. It should be stressed that this estimate refers to theaccuracy of the ChildLangmuir model compared with theBohm model. In other words, this is just one component of the overall error, other components being errors inherent to theBohm model, e.g. errors originating in the neglect of ionatomcollisions in the sheath.

3.2. Collision-dominated ionelectron sheath: theSuLamCohen model

A theory of collision-dominated space-charge sheaths in aweakly ionized plasma was developed in 1963 by Su and Lam[24]andCohen[ 25] inconnection with a problem ofa sphericalelectrostatic probe in a quiescent weakly ionized plasma withconstant electron and heavy-particle temperatures and withoutionization and recombination. The work [ 24] was concernedwith the case of high negative probe potentials and the work [25] with the case of moderate potentials. These were therst works on the sheath theory in which matched asymptoticexpansions were applied. Rened asymptotic treatments weregiven in [26, 27]. In this section, a summary is given of asimplied version of the theory which was developed in thework [28] and is based on the assumptions of cold ions andBoltzmann-distributed electrons.

A model of a collision-dominated sheath formed bycold ions and Boltzmann-distributed plasma electrons canbe obtained from the MottGurney model in the same wayas the Bohm model for the collisionless case is obtainedfrom the ChildLangmuir model. First, the Poisson equationis supplemented with a term accounting for space chargecontributed by the electrons. Thus, the system of equationsincludes the second equation ( 1) and equations ( 12) and (18).Second, the boundary conditions at the edge of the ion sheathmust be replaced by conditions of matching with the quasi-neutral plasma. These boundary conditions apply at innitelylarge values of x, which again means distances from theelectrode much larger than the scale of thickness of the sheathbut much smaller than a characteristic length scale in thequasi-neutral plasma. In order to derive these conditions, let us

multiply the rst equation ( 18) by d/ dx , remove n i by meansof the second equation ( 1) and equation ( 12) and ne by meansof the second equation ( 18) and integrate over x . The resultingequation may be written as

2D

2

e

kT e

2

d

dx

2

=

x+ exp

e

kT e+ C5 , (27)

where D = ( 0kT e /n 0e2)1/ 2 as before and = n0kT e i / (j i). At large x where the plasma is quasi-neutral, theleft-hand side of this equation is negligible and one nds thefollowing asymptotic behavior:

n e =n0x

+ , n i =n0x

+ , (28)

=kT e

eln

x+ o (1) . (29)

As usual, dots designate terms of higher orders of smallness inthe asymptotic parameter; in this case, in the large parameterx and o( 1) designate terms that tend to zero at large x .

Expressions ( 28) and (29) are substantially differentfrom the corresponding expressions in the Bohm model,equation ( 20): while the (rst-approximation) solution ina collisionless sheath tends at large x to constant values,in a collision-dominated sheath it manifests an algebraicasymptotic behavior for charged-particle densities and alogarithmic behavior for the potential. In other words,parameters of a collisionless plasma at distances from theelectrode surface much larger than the scale of thickness of thesheathbut much smaller than a characteristic lengthscale in thequasi-neutral plasma are constant to a rst approximation, and just these values are implied when one speaks of the densityof the charged particles and potential on the plasma side of thesheath or the velocity with which the ions leave the plasmaand enter the sheath. Nothing of the latter makes any sense inthe case of a collision-dominated sheath. On the other hand,an algebraic or logarithmic intermediate asymptotic behavioris quite common in multi-scale problems and the method of matched asymptotic expansions is very well suited for suchcases, soa theoryof collision-dominated sheaths canbe readilydeveloped.

Since there is no way to unambiguously identify apotential on the plasma side of the sheath, a voltage drop in the

sheathcannotbe unambiguously dened. It is appropriate thento dene a combined voltage drop across the sheath and theadjacentplasmaregionwhere theion uxto thecathodesurfaceis generated. In other words, let us choose a reference pointfor the potential at the external boundary of the plasma regionadjacent to the sheath where the ion ux to the cathode surfaceis generated. Then n0 signies the charged-particle densityat this boundary and U appearing in the boundary condition(19) signies a combined voltage drop across the sheath andthe adjacent plasma region. As an example, let us considera spherical electrostatic probe in a quiescent weakly ionizedplasma with constant electron and heavy-particle temperaturesand without ionization and recombination, treated in [ 24, 25].In this example, the ion current to the probe is generatedin the whole volume of the quasi-neutral plasma up to the

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undisturbed plasma at distances from the probe much largerthan the probe radius. Then n0 represents the charged-particledensity in the undisturbed plasma and = 0 correspondsto the plasma potential. The near-cathode plasma region inarc discharges can be considered as another example. In thisexample, the ion current to the cathode is generated in the

so-called ionization layer (e.g. [ 6] and references therein),which is a region of quasi-neutral plasma where deviationsfrom ionization equilibrium are localized. It is the ionizationlayer that plays a role of the plasma region adjacent to thesheath in this example, so n 0 represents the charged-particledensity on the plasma side of the ionization layer and U is thecombined voltage drop in the sheathandin the ionization layer.

The ion current to a negative electrode generatedin a collision-dominated quasi-neutral plasma is governedby ambipolar diffusion and may be evaluated as j i =eD adn c/ dx , where D a is the coefcient of ambipolardiffusion which in the case of cold ions is related to theion mobility by the formula D a

= kT e i/e and dn c/ dx is

the derivative of the charged-particle density evaluated in thequasi-neutral approximation at the electrode surface. Nowone can understand the meaning of the length appearingin equations ( 27)(29): =n0(dn c/ dx) 1 , i.e. it is a scale of thickness of the plasma region adjacent to the sheath wherethe ion ux to the cathode surface is generated. (In theabove-mentioned examples, equals the probe radius or,respectively, the ionization length [ 6].) Note that D =( 0kT e/n 0e2)1/ 2 has in this section the meaning of Debyelength evaluated at the external boundary of the plasma regionadjacent to the sheath where the ion ux to the cathode surfaceis generated. The applicability of the present analysis is

obviously subject to the inequality D , which justiesa division of the near-electrode region into a quasi-neutralplasma and a space-chargesheath. On the other hand, D mustbe not too small so that the sheath thickness is much largerthan the mean free paths of the charged particles.

We consider at the moment the case of a moderatelynegative near-electrode or near-wall sheath, where the sheathvoltage is of the order of kT e/e . Variations of potential in thesheath are also of the order of kT e /e . Assuming that in thesheath the term on the left-hand side of equation ( 27) andthe rst term on the right-hand side are of the same order of magnitude, one nds that the scale of thickness of the sheathis 2/ 3D

1/ 3. Assuming that the second term on the right-handside of equation ( 27) is in the sheath of the same order as therst term and the term on the left-hand side, one nds that thepotential distribution in the sheath may be represented as

= kT e

e23

ln D

+ , (30)

where = (X) is a dimensionless function of the stretchedcoordinate X =x/2/ 3D

1/ 3.The electric eld in the sheath is of the order of

kT e /e2/ 3D

1/ 3. It follows from the second equation ( 1)and equation ( 12) that n i in the sheath is of the order of n0( D / ) 2/ 3. It follows from equation ( 30) that exp ekT e is of the order of ( D / ) 2/ 3; hence n e in the sheath is of the orderof n0( D / ) 2/ 3, i.e. comparable to n i as it should be.

The function (X) is governed by equation ( 27) writtenin the dimensionless form

12

ddX

2

= X + e + C6, (31)

where C6

=C5( / D)2/ 3. Boundary conditions are obtained

from (19) and (29):

X =0 : = w , X : = ln X + o(1),(32)

where w is related to U by the formula

U =23

kT ee

ln D

+ kT e

ew (33)

and should be treated as a given positive parameter.The boundary-value problem ( 31), (32) may be

convenientlysolved as follows. The constant C6 is removed byintroducing the new independent variable

X

=X

C6. After

this, equation ( 31) is solved for d / dX and the obtained rst-order differential equation is integrated numerically with thesecond boundary condition ( 32). Note that the function ( X)is universal, i.e. does not depend on any control parameters.A graph of this function can be found in [ 28] and a number of subsequent works (e.g. [ 29, 30]). After the function ( X) hasbeen determined, the constant C6 is found: C6 = X w , whereX w is the value of the argument at which this function attainsthe value w , i.e. the root of the equation ( X) = w .As discussed above, U represents a combined voltagedrop across the sheath and the adjacent plasma region andthere is no way to unambiguously separate contributions of the

sheath and the adjacent plasma region. On the other hand,the rst term on the right-hand side of equation ( 33) involvesparameters and n0 , characterizing the plasma region, whilethe second term involves parameter w , characterizing thepotential distribution in the sheath. Therefore, it is natural toassume, by convention, that theseterms representcontributionsof, respectively, the plasma region and the sheath.

The above analysis applies to the case of a moderatelynegative near-electrode or near-wall sheath, where the sheathvoltage kT e w /e is comparable to kT e/e , i.e. w is of orderunity. The asymptotic nature of the results, including thescalings, remains the same also if a nite ion temperatureand deviations of the electron density from the Boltzmanndistribution (i.e. transport of the electrons) are taken intoaccount [ 25, 27].

In the case of a high-voltage near-cathode sheath, wherethesheathvoltage is considerablyhigher than kT e /e , the sheathincludes two sub-layers: the ionelectron layer and the ionlayer. Scalings in the ionelectron layer are the same asthose in a moderately negative sheath, i.e. the length scale,the charged-particle densities and variations of potential areof the orders of, respectively, 2/ 3D , n0( D / )

2/ 3 and kT e /e .Let us estimate orders of magnitude of parameters in the ionlayer. For brevity, the consideration is restricted to the casewhere the sheath voltage substantially exceeds not only kT eebut also kT ee ln D . In this case, the voltage drop in the adjacentplasma region may be neglected compared with the sheath

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voltage and the latter to a rst approximation equals U . In theion layer, is of the order of U . Assuming that the termon the left-hand side of equation ( 27) and the rst term on theright-hand side are of the same order of magnitude in the ionlayer, one nds that the thickness of the ion layer is of theorder of 2/ 3D

1/ 3 2/ 3. The electric eld in the ion layer is

of the order of U /2/ 3D

1/ 3

2/ 3. It follows from the secondequation ( 1) and equation ( 12) that n i in the ion layer is of the

order of n 0( D / ) 2/ 3 1/ 3. The electron density in the ionlayer, being governed by the Boltzmann distribution ( 18), isexponentially small.

It follows from the above asymptotic estimates that thethickness (length scale) of the ion layer substantially exceedsthe thickness of the ionelectron layer. Hence, the ion layerhas a more or less distinct edge; however no unambiguousdenition of this edge can be given to an accuracy betterthan O( 2/ 3). The voltage drop in the ion layer is muchhigher than that in the ionelectron layer. The electric eldand the ion density in the ion layer are much stronger and,respectively, much lower than those in the ionelectron layer.The scale of ion density and the length scale in the ionlayer are related by equation ( 26), the difference being thatn (il)i =n0( D / ) 2/ 3 1/ 3 and =

2/ 3D

1/ 3 2/ 3 in thepresentcase.

It can be shown on the basis of the above asymptoticestimates that the ion layer to a rst approximation is describedby the MottGurney model, the error of this approximationbeing of the order of 1 . Except for different scalings and theMottGurney modelappearing in place of theChildLangmuirmodel, all above are similar to what happens in the case of acollision-freehigh-voltage sheath andareschematically shownin gure 2.

The above-described ionelectron and ion layers alsoappear in the treatments [ 24, 26], in which a theory of high-voltage near-cathode sheaths with account of a nite iontemperature and transport of the electrons was developed inconnection to spherical electrostatic probes. The ion drift ina strong electric eld occurring in the ion layer signicantlyexceeds the ion diffusion caused by the ion pressure gradient;therefore the ion layer in the analysis [ 24, 26] is described bythe MottGurney model (with the curvature effect taken intoaccount). Additionally, an ion diffusion layer adjacent to thecathode surface appears at the bottom of the ion layer. In theion diffusion layer, ion diffusion comes into play and the ion

density rapidly falls in the direction of the cathode surface.

4. Advanced models of ion sheaths

Without trying to review all the works concerned withadvanced models of ion sheath, we will consider three topicshere. The rst one is uid modeling of ion sheaths withan arbitrary degree of collisionality, which spans the wholerange of conditions from a collisionless sheath described bythe ChildLangmuir model to a collision-dominated sheathdescribed by the MottGurney model. The second topic isa derivation of high-accuracy boundary conditions for theChildLangmuir and MottGurney models. The third topicis a mathematical interpretation of ion sheath models from thepoint of view of modern theoretical physics.

4.1. Ion sheath with arbitrary collisionality

A theory of an electron-free ion sheath with an arbitrarydegree of collisionality (e.g. [ 3136]) can be developed in theframework of the so-calleduid model. A uid model is basedon treating the ions and the atoms as separate uids coexistingwith, rather than diffusing in, each other; e.g. [ 3739]. Theuid model is understood to ensure a sufcient accuracy inapplications and is widely used in modeling of low-pressuregas discharges; e.g. [ 39] and references therein. In the case of cold ions, the uid model of an electron-free ion sheath relieson equations ( 1) with the difference that the rst equation ( 1)is written in the form of an equation of motion of the ion uid:

m ividvidx = e

ddx

ev i i

. (34)

The term on the left-hand side of this equation describesthe inertia force. If the sheath is dominated by collisions,this term is minor and equation ( 34) becomes identicalto the corresponding equation of the MottGurney model,equation ( 12). The terms on the right-hand side describethe electric-eld force and a friction force resulting fromelastic collisions of the ions with neutral particles. If thesheath is collision-free, the second term on the right-handside is minor and equation ( 34) becomes identical to thecorresponding equation of the ChildLangmuir model, rstequation ( 1). Thus, the uid model contains the ChildLangmuir and MottGurney models as limiting cases for thecollision-free and, respectively, collision-dominated sheathsand describes a smooth transition from one to the other at nitecollisionalities.

Let us assume that the origin x = 0 is positioned at thesheath edge and the x-axis is directed to the cathode as insection 2. Then the boundary conditions ( 2) and (9) remainapplicable.

As above, i is treated as constant. In this case, theabove-described problem admits a simple analytical solution [ 36],which can be conveniently found by treating the electric eldas a new independent variable and vi as an unknown function.This solution may be written in a universal form:

V =E + eE 1, (35)

=E 3

3 E 2

2 + 1

E eE

eE , (36)

=E 2

2 E + 1 eE , (37)

where the dimensionless variables are dened as

=ex

m i iv(0)i

, V =vi

v (0)i, E =

i

v (0)i

ddx

,

= e

m iv(0)2i

, (38)

with the characteristic ion velocity v (0)i = mi 2i j i/e 0 . Thesolution( 35)(37) isparametric, theroleof theparameter beingplayed by the dimensionless electric eld E .

The dimensionless coordinate may be interpreted as acharacteristic number of collisions that an ion suffers while

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0.01 0.1 1 10 100

0.1

1

10

V

E

E ,

V ,

Figure 3. Solid: solution for the ion sheath with arbitrary degree of collisionality, equations ( 35)(37). Dashed: the ChildLangmuirsolution, equations ( 39). Dotted: the MottGurney solution,equations ( 40).

crossing the distance x . Therefore, solutions that correspondto the limiting cases of collision-free and collision-dominatedsheaths are obtained from equations ( 35)(37) by applyingthe limits of small and, respectively, large (or E , which isequivalent):

V =(3) 2/ 3

21/ 3 , =

(3 ) 4/ 3

25/ 3 , E = (6) 1/ 3. (39)

V = 2 , =(2 ) 3/ 2

3 , E = 2. (40)The second equations in ( 39) and (40) represent dimensionless

forms of equations ( 4) and, respectively, ( 14), i.e. the ChildLangmuir and MottGurney solutions, as they should.

The solution ( 35)(37) is shown in gure 3. Also shownaresolutions that correspond to theChildLangmuir andMottGurney models. As expected, the uid model describes asmooth transition between the ChildLangmuir and MottGurney models.

4.2. High-accuracy boundary conditions for theChildLangmuir and MottGurney models

There are two sources of errors in the ChildLangmuirand MottGurney models of electron-free sheath comparedwith the corresponding models taking into account thepresence of the electrons (the Bohm model and, respectively,the SuLamCohen model with cold ions and Boltzmann-distributed plasma electrons): the disregard of the electrondensity term of the Poisson equation and the trivial boundaryconditions at the sheath edge. The electron density term of the Poisson equation in the bulk of the sheath is exponentiallysmall in the large parameter

=eU/kT e , so the error caused

by dropping this term also is exponentially small. The errorcaused by the trivial boundary conditions is of algebraic order

( 1/ 2 for the ChildLangmuir model and 1 for the MottGurney model), i.e. essentially higher, and it is this error thatlimits the accuracy of the model on the whole. It is naturalin such a situation to try to derive more accurate boundaryconditions and thus improve theoverall accuracy of themodel.

There has been a signicant number of works concerned

with this task; see, e.g. [ 23, 30, 32, 33, 4048]. Boundaryconditions were found that indeed ensure the exponentialaccuracy of the ChildLangmuir and MottGurney models[23, 43, 46]. Let us start with the ChildLangmuir model. Letus transform equation ( 25) to variables normalized in such away that they are of order unity in the bulk of the sheath:

12

ddX

2

= ( 1 + 2 ) 1/ 2 2 1/ 2 + 1/ 2e , (41)where X = x/ D 3/ 4, = /U . The right-hand side of equation ( 41) contains three terms which are small at large , all of them of different orders of smallness. The simplest

model of an ion layer is obtained by dropping all the threeterms, i.e. by retaining only the term (2 ) 1/ 2 . One can readilysee that this approximation corresponds to the original ChildLangmuir model, in which the ion velocity and electric eldat the edge of the ion layer are neglected as well as the voltagedrop in the ionelectron layer. The most accurate model isobtained by dropping only the last term on the right-handside of equation ( 41), which is exponentially small, whileretaining the other two terms, which are small algebraically.Theresultingequation readsafter transformation to theoriginalvariables

2D

2 ekT e

2

ddx2

= 1 2ekT e1/ 2

2. (42)

This equation, which coincides with equation ( 25) withoutthe last term on the right-hand side, describing the electrondensity, represents the desired exponential-accuracy model of an ion layer. The procedure of its derivation is equivalent toasymptotic matching of a solution describing the region where

kT e/e , i.e. the ion layer, with a solution describing theregion where e/kT e is of order unity, i.e. the ionelectronlayer.Let us recast the exponential-accuracy model ( 42) into

a form of the ChildLangmuir model with new (non-trivial)boundary conditions at the edge of the ion layer (electron-freesheath). First, one must dene what is the edge of the ion layerin this new model; forexample, it canbe associated with a givenvalue of the electric eld, or the potential, or the ion velocity.A natural denition which is usual in multi-scale asymptoticmethods is obtained by assuming that the model is used in thewhole region where its solution exists; thus the edge of the ionlayer is a point at which the ion-layer solution breaks down.A solution to equation ( 42) exists provided that the right-handside of this equation is non-negative, i.e. at 3kT e/ 2e,and breaks down at a point where described by this solutionreaches the value

3kT e / 2e or, in other words, where the

electric eld described by this solution vanishes. It followsfrom equation ( 21) that vi = 2uB at this point. Thus, the

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above denition results in the following boundary conditionsfor an ion-layer solution at the edge of the ion layer:

ddx =0, v i = 2u B , =

32

kT ee

. (43)

Boundary conditions ( 43) admit a simple interpretationshown in gure 2: the ions are accelerated in the ionelectronlayer from theBohm velocity, vi = u B , on the plasma side of the ionelectron layer, to twice theBohm velocity, vi = 2u B ,at the edge of the ion layer; the voltage drop in the ionelectronlayer is 32

kT ee . We stress that these boundary conditions have a

merely illustrative purpose; they have been obtained with theuse of an extrapolation (the ion-layer solution loses its validitybefore the electric eld described by this solution vanishes)and do not amount to assuming that there is a real physicalpoint at which d/ dx =0 and vi = 2uB (of course, no suchpoint exists: the electric eld at a point where vi = 2u B isnot exactly zero; however it is much smaller than the electriceld inside the ion layer).

One can readily check that the ChildLangmuirequations ( 1) jointly with boundary conditions ( 43) are exactlyequivalent to the exponential-accuracy model ( 42). In otherwords, boundary conditions( 43) indeed ensure theexponentialaccuracy of the ChildLangmuir model. A further discussionof the exponential-accuracy model of a collision-free ion layerand a comparison of its results with exact results as well asresults given by various approximate models can be foundin [46].

An exponential-accuracy model for a collision-dominatedion layer formed by cold ions in a weakly ionized plasmamay be developed in a similar way [ 43]. If the frequencyof collisions ionatom does not depend on velocity and theion mobility may be treated as constant, then the exponential-accuracy model is equivalent to the MottGurney model withthe following boundary conditions at the edge of the ion layer:

ddx =0, =

23

kT ee

ln D 1.0082

kT ee

. (44)

The rst boundary condition represents a denition of an edgeof the ion layer: it is once again dened as a point wherethe extrapolation of the ion-layer electric eld vanishes. Therst and second terms on the right hand-side of the secondboundary condition may be interpreted as the voltage drop inthe plasma region adjacent to the sheath where the ion ux tothe cathode surface is generated and, respectively, the voltagedrop in the ionelectron layer. The numerical coefcientin the second term was determined by means of numericalcalculations [ 43, 47] and is quite close to unity, meaning thatthe voltage drop in the collision-dominated ionelectron layeris very close to kT e /e .

The above-described models ensure the accuracy of several per cent for sheath voltages exceeding kT e/e three orfour times or more. Since the voltage drop across a sheath on aoating surface is no smaller than approximately 4 kT e /e , thesemodels are accurate enough for all near-cathode and near-wallsheaths.

4.3. Mathematical sense of ChildLangmuir type models from the point of view of modern theoretical physics

The mathematical sense of Langmuirs model of an electron-free ionsheathwith the ion velocity, electric eld andpotentialvanishing at the sheath edge became clear after the methodof matched asymptotic expansions was developed in the1950s1960s and applied to the problem of plasma-electrodetransition: the model perfectly ts into the formalism of matched asymptotic expansions in the large parameter equal totheratio of thesheathvoltage to theelectron temperature, andasolution to this model represents therst term of an asymptoticexpansion valid in the bulk of the space-charge sheath. Notethat a similar statementcan be made concerning Bohms modelof a collisionless ionelectron sheath [ 12]: it perfectly ts intothe formalism of matched asymptotic expansions in the smallparameter equal to the ratio of the Debye length to the ion meanfree path, and Bohms solution represents the rst term of anasymptotic expansion describing a collisionless space-charge

sheathwith cold ions on a moderately negative surface. Such at is a manifestation, on the one hand, of the power of physicalintuition of Langmuir and Bohm, and on the other hand, of thecapability of the method of matched asymptotic expansionsto reveal and exploit the underlying physics of layers of fastvariations, such as viscous boundary layers in uid mechanics,shock waves in gas dynamics, skin layers in electromagnetictheory or, in this case, different layers constituting a space-charge sheath or separating a sheath from the plasma.

The method of matched asymptotic expansions has beenwidely used for treating problems with layers of fast variationin different areas of theoretical physics, in particular, in uidmechanics. One of the very important advantages of thismethod with respect to intuitive methods is that the methodof matched asymptotic expansions provides an estimate of accuracy of a solution. Solutions obtained by means of thismethod are smooth, in contrast to solutions given by intuitivemethods, which inevitably suffer discontinuities, or havediscontinuous derivatives, at boundaries separating differentregions. Another very important advantage is that this is aregular procedure in the sense that the results do not dependon the researchers taste, in contrast to intuitive methods whichusually lead different researchers to different solutions of thesame problem.

In the course of the procedure of the method of matched

asymptoticexpansions, all mathematical simplicationswhichare justied by the underlying physics will be duly revealedand exploited. In other words, the method of matchedasymptotic expansions produces relevant physical informationat a minimal effort; thus the idea that this method isunnecessarily heavy mathematically, which is encountered insome works on the theory of collisionless and moderatelycollisional space-charge sheaths, is unjustied.

Another sometimes encountered idea is that the methodof matched asymptotic expansions requires a great deal of mathematical expertise. This idea is also unjustied: one canreadily verify by inspecting textbooks (e.g. [ 1419]) that theformalism of this method is not complex and does not requirespecic mathematical skills. Equally unjustiedis the idea thatthe method of matched asymptoticexpansions produces results

11


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which are not robust enough for engineering purposes: suchresults always have a distinct physical sense and, if reducedto the same level of description, are no more difcult to usethan results provided by any minimally reasonable intuitiveapproach. In fact, asymptotic results are usually easier to use,since they account foronly effects that arepertinent at the level

of accuracybeing adopted, while intuitive results in most casestake into account some effects which are below the adoptedlevel of accuracy and therefore unnecessarily complicate theresults.

As far as space-charge sheaths are concerned, the methodof matched asymptotic expansions plays a dominant role inthe theory of collision-dominated sheaths. The situation inthe theory of collisionless and moderately collisional sheathsfor some reason is different: the intuitive approach basedon patching (i.e. gluing solutions at one point rather thanasymptotically matching them) is still in use, and a heateddiscussion around the two methods has unfolded during thelast decade, e.g. [ 4952] and references therein.

Without trying to follow the above-mentioned discussion,we note that the most instructive way of comparing differentmethods is toapply them toseveral clear-cut examples. In [ 46],results given by the ChildLangmuir model, patching and themethod of matched asymptotic expansions were compared inthree examples. One of the examples was a collision-free dcsheath, treated in sections 3.1 and 4.2. The second examplewas a matrix sheath. The third example was a high-voltagecollisionless capacitive RF sheath, driven by a sinusoidalcurrent source, with the ions responding to the time-averagedelectric eld and the inertialess electrons responding to theinstantaneous electric eld. (An elegant analytical solution

to this problem was given in [ 5355]; note that an analyticalsolution for the case of a collision-dominated RF sheath wasgiven in [56].)

In all three examples, the ChildLangmuir model andpatching provide results which are accurate to the rstapproximation in the sheath voltage but not to the second,irrespective of details of patching. In order to illustrate thisstatement, let us turn to the example of an RF sheath. In [ 46],this problem was treated by means of the method of matchedasymptotic expansions with an asymptotic large parameter equal to thefourthpowerof thedimensionless amplitudeof thedriving current. A two-term expansion of the dimensionless

(normalized by kT e /e ) sheath voltage was found, with the rstandsecond termsbeing of theorders of and 1/ 2, respectively.The rst term represents an analogue for an RF sheath of the ChildLangmuir solution for a dc sheath. The solutions[53, 55]and[ 54] correctlydescribethe rst term, i.e. theChildLangmuir type solution, but not the second. Furthermore, theorder of the second term in the solution [ 53, 55] is incorrect: 3/ 4 instead of 1/ 2. It follows that the ChildLangmuirtype solution and the solution [ 54] are more accurate than thesolution [ 53, 55]: termsmissing from theChildLangmuir typesolution and calculated incorrectly in the solution [ 54] are of the order of 1/ 2 , which is lower than the order of the termscalculated incorrectly in the solution [ 53, 55] ( 3/ 4).

In all the three examples, the method of matchedasymptotic expansions ensures a higheror considerably higher

asymptotic accuracy than the ChildLangmuir model andpatching. The same is true for numerical accuracy, aswas demonstrated in [ 46] by means of a comparison withexact solutions (in the cases of dc and matrix sheaths)and the numerical simulations [ 57] (in the case of anRF sheath).

These and other examples clearly show that intuitiveconsiderationscanhardly improvethequality of results beyondthose given by simple ChildLangmuir type models. If notsatisedwitha simple ChildLangmuirtype model, oneshouldbetter resort to a standard tool, i.e. to the method of matchedasymptotic expansions, or alternatively directly computedsolutions (but one has to recognize that in the latter case thenumber of parameters is too large to obtain simple analyticapproximations). They would give more accurate and reliableresults than intuitive considerations.

One more comment on Bohms 1949 paper [ 12] seemsto be in place. Authors of the recent work [ 52] assumed thatBohm was trying to patch the plasma and sheath solutions

rather than to asymptotically match them and found on thesegroundsmisinterpretations andcontradictionsin Bohmswork.One should admit that not all aspects of Bohms work [ 12] arepresented equally clearly; in particular, the term sheath edgein the context on an ionelectron sheath model is appropriatefor patching and not matching. On the other hand, there areclear indications towards matching in the text of the paper[12], for example, on p 79: ... plasma elds, comparedwith sheath elds, are so small that they produce negligiblechanges of potential over distances many sheath thicknessesin extent. To a rst approximation, therefore, it may beassumed that the plasma potential is constant, at least in so faras the processes involved in sheath formation are concerned.However, the plasma elds cannot be completely neglected,because over the long distances that they cover they are able toaccelerate positive ions up to appreciable energies.... Thesewords reect the very essence of the asymptotic multi-scaleapproach.

The clearest indication of the intentions of a theoreticalphysicist comes from his formulae. It is mentioned abovethat the solution [ 12] without a single change represents therst term of an asymptotic expansion describing a collisionlessspace-charge sheath with cold ions on a moderately negativesurface. If Bohms solution is correct from the point of the method of matched asymptotic expansions and contains

misinterpretations and contradictions from the point of viewof patching, is it not legitimate to assume that Bohm aimedatand succeeded inwhat is called today matching ratherthan patching?

Bohms paper [ 12] was written before multi-scaleasymptotic methods were developed. The term sheathedge, while being misleading in the framework of Bohmsmodel of an ionelectron sheath, is relevant to Langmuirsmodel of an electron-free ion sheath (see discussion insection 3.1), and this may have inuenced Bohm. If this termis replaced by region on distances from the electrode muchlarger than the local Debye length but much smaller than acharacteristiclength scale in thequasi-neutralplasma, thenthecorresponding statements in [ 12] will become consistent andappropriate.

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5. Conclusions

We have tried to trace the footprint of the ChildLangmuirmodel of an electron-free ion sheath with the ion velocity,electric eld and potential vanishing at the sheath edge. It isno exaggeration to say that this model gave the origins of the

modern theory of near-electrode and near-wall space-chargesheaths and still remains the most widely used model of acollisionless near-cathode sheath.

From the point of view of modern theoretical physics, theionsheath modelperfectlyts into theformalism of themethodof matched asymptotic expansions and represents therst termof an asymptotic expansion in the large parameter equal tothe ratio of the sheath voltage to the electron temperature,describing the bulk of the space-charge sheath. The method of matched asymptotic expansions has given the most accurateand reliable results in the analytical theory of sheaths andsheathplasma transition. In particular, it hasprovided a betterunderstanding and improvement of the accuracy of the ionsheath model.

Acknowledgments

The work on this paper was supported by the projectsPPCDT/FIS/60526/2004 of FCT, POCI 2010 and FEDER andCentro de Ci encias Matem aticas of FCT, POCTI-219 andFEDER.

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the child–langmuir law and analytical

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