ARTICLES

Space-Charge-Dominated Mass SpectrometryIon Sources: Modeling and Sensitivity

Mark Busman and [an SunnerDepartment of Chemistry

Cur tis R. VogelDepartment of Mathematical Sciences, Montana State Univers ity, Bozeman, Mont ana, USA

The factors determining the sensitivity of space-charge-dominated (SCD) unipolar ionsources, such as electrospray (ESP) and corona atmospheric pressure ionization (API) havebeen stu died theoretically. The most important parameters are the ion density and ion drifttime in the vicinity of the sampling orifice. These are obtained by solving a system ofdifferential equations, "the space-charge problem." For some simple geometries, analyticalsolutions are known. For a more realistic "needle-in-can" geometry, a solution to thespace-charge problem was obtained using a fmite-element method . The results illustratesome general characteristics of SCD ion sources. It is shown that for typical operatingconditions the minimum voltage required to overcome the space-charge effect in coronaAPI or ESP ion sources constitutes a dominant or significant fraction of total appliedvoltage . Further, the electric field and the ion density in the region of the ion-samplingorifice as well as the ion residence time in the source are determined mainly by the spacecharge . Finally, absolute sensitivities of corona API ion sources were calculated by using ageometry-independent treatment of space charge. (J Am Soc Mass Spectrol1l 1991, 2, 1-10)

M ass spectrometric techniques using high­pressure ion sources are becoming increas­ingly important in analytical chemistry . At­

mospheric pressure ionization mass spectrometry(API/MS) is used for analysis of sub-parts per billionlevels of trace gases in ambient air [1-4]. Equippedwith a nebulizer, such a source becomes the interfacein a liquid chromatograph/mass spectrometer [5]. Inthese methods, the atmospheric gas is usually ionizedby a corona discharge, but 63Ni is also used . In elec­trospray (ESP) [6, 7), the ions instead originate from acharged spray. The electron capture detector (ECD)also operates at atmospheric pressure [8-10]. In ionmobility spectrometry (IMS), ions are separated onthe basis of their mobility in atmospheric gas [11, 12].

In the API devices mentioned above, except the ionmobility spectrometer and the ECD, the ions passthrough a sampling orifice into the vacuum system fordetection by mass spectrometry. The analytical signalthus depends on the efficiency of analyte ion forma­tion (and destruction), of analyte ion transport to thevicinity of the orifice, of anaIyte ion extraction through

Address repri nt requests to Jan Sunner, Department of Chemistry,Montana State Universi ty. Bozeman. MT 59717 .

@ 1991 American Society for Mass Spectrometry1044-0305/91 /$3.50

the orifice, and of ion extraction from the supersonicjet, as well as on the transmission through the massspectrometer. We will discuss the processes up to theorifice.

For the understanding of ion formation and trans­port it is useful to categorize ion sources with respectto the major ion loss mechanism. At low ion densiti­ies, ions are lost mainly by diffusion (diffusion­dominated loss) or by drift in the Laplacian held, thatis, the electric held created by applying different po­tentials to the walls or electrodes in the ion source(Laplacian field dominated loss) . At increasing ion den­sities in the bipolar (ions of both polarities present)ion source , diffusion first becomes ambipolar [13])andis then replaced by ion-ion (or ion-electron) recombi­nation as the main ion loss mechanism (recomb­ination-dominated loss). This is the situation when 63Nicauses the ionization, as in the ECD [14] and in someAPI [15] and IMS sources [11]. Models of recombina­tion-dominated ion sources have been published [16,17]. The unipolar ion source (ions of only one polaritypresent) becomes space-charge-dominated (SCD) at highion density. The electric field within the source is thendetermined mainly by the space charge, and ion driftto the walls is the dominating ion loss mechanism.

Received April 30, 1990Accepted August 9. 1990

2 HUSMAN ET AL. J Am Soc Mass Spectrom 1991,2, 1-10

Figure 1. Simple geometries for SCD ion sources discussed inthe text. (a) Planar geometry, infinite parallel planes; (b) cylin­drical geometry, concentric cylinders of infinite length; (c)spherical geometry, concentric spheres; (d) point-to-plane ge­ometry; (e) " needle-in-can" geometry. The ion supply surfacesare given a positive polarity in the figures.

This is the case, for example, in the ECD directly afterthe electron-extraction pulse has been applied to theanode [14].

Of the different types of ion sources, the SCD oneis the least understood. For ESP, corona API, andlMS, the problem is to what extent ion transport isdetermined by the space charge as opposed to thekilovolt potential applied to the various electrodes. Inmass spectrometry, .the ECD model of Gobby et al .[14J is perhaps the only quantitative treatment ofspace-charge effects at high pressure. However, re­sults in discharge physics (lSI are often applicable tomass spectrometry ion sources. Thus, space-chargeeffects in systems with simple geometries were treatedby Townsend [19]. A related low-pressure treatmentis due to Langmuir [20). A discussion of a sphericalsystem can be found, for example, in ref . 21. A modelof a point-to-plane corona that considers space-chargeeffects has been presented by Sigmond (22). How­ever, the implications of these results for mass spec­trometric ion sources, in particular with regard tosensitivity, seem never to have been investigated.

Space-charge effects are important not only in dis­charge physics but also for many devices in scienceand technology. Examples are aerosol precipitators[231, membrane ion exchangers [24], and semiconduc­tor devices [25]. The modeling of the se, as well as ofspace-charge effects in gases, is part of the subject ofnonlinear transport theory [24].

The relationship between the charge density p andthe electrostatic potential U is given by Gauss's law(using 51 units),

A

c

B

- V. VU = V. E = p / E (1)

where f is the electric field vector and E is the electricpermittivity. Assuming no ionization, the continuityequation is

ap _ _-+V'J =OiJt

(2)

Equations 1, 3, and 5 form a nonlinear system ofpartial differential equations, the " space-charge prob­lem ." The boundary conditions may be specified bydetermining some combination of the component of Enormal to the surface or U on the surface, &0, of aregion 0 , for example,

At high pressure, the main ion transport processesare ion drift, diffusion, and convection. The ion cur­rent density is given by

where j is the ion current density vector. Here we areinterested in steady-state solutions, and eq 2 simpli­fies to

(6)

(7)

XEIiO

(2J )1/2

E(X)= -x+ESKE .

and similarly for Vp or o, The solution to these equa­tions should give the electric field distributionthroughout the ion source as well as the potentialdistribution, the ion density distribution, and ion drifttimes .

Neglecting diffusion, the space-charge problem, eqs1, 3, and 5, has analytical solutions for some simplegeometries: (1) infmite parallel plates, (2) two concen­tric cylinders, and (3) two concentric spheres; seeFigure 1 (19, 21, 26]. The electric field between twoparallel plates (Figure Ia), is given by

(4)

(3)V· J= 0

where K is the coefficient of ion mobility, D is thediffusion constant, and F is the gas flow vector. In atypical SeD ion source, diffusion is important onlyvery close to the walls, and convection mainly affectsthe efficiency of ion sampling. Thus, eq 4 simplifies to

(5) where J is the ion current density, K is the ion

J Am Soc Mass Spectrom 1991, 2, 1-10 SI' ACE-CHARGE DOMINATED ION SOURCES 3

Results and Discussion

where T is the radius, TO is the radius of the innersphere, Eo is the electric held, and Jo is the ioncurrent density at the inner sphere surface. Jo isrelated to the discharge current 10 by

(12)

orifice at the point where the ions become entrainedin the viscous flow through the orifice, p(t res). In thiscase, 1 is given by

In most SCD ion sources, the primary ionization oc­curs in a (small) localized region. The remainder ofthe ion source constitutes the ion drift region, throughwhich the ions are transported to the detector orvacuum orifice and where analytically useful ion­molecule reactions may occur. The modeling of theionization region is difficult for a 6JNi source and isvery complicated for the corona API and likely alsofor the ESP source . Therefore, only the drift regionwill be modeled here. However, it will be shown thatit is mainly in this region that the sensitiv ity of SCDion sources is determined. As a consequence thetreatment will be equally applicable to different ion­ization methods . It is also assumed that all ions havethe same charge and the same mobility.

Excluding the ionization region in the modelingcauses a problem with the boundary condition at theion supply surface, eq 6. Usually we know the totalapplied voltage, VI'

Mathematical Modeling ofSpace-Charge­Dominated Ion Sources

where 5 is the gas flow rate . This would seem to be avery reasonable assumption, and we will see later thateq 12 is in good agreement with experimental obser­vations.

Here we will consider solutions to the space-chargeproblem with particular emphasis on p and Ires' Ini­tially, the boundary value problem in the modeling ofSCD ion sources and the analytical solution for thespherical geometry, eq 8, is discussed. A more diffi­cult geometry, the "needle-in-can," Figure 'le, is nu­merically approximated with a Unite-element method.Finally, a simple geometry-independent treatment forion transport and kinetics is presented and is used tocalculate analytical sensitivities of a corona API ionsource .

(9)

(10)

Jo = 10 I(h76)

In previous articles, with Kebarle, we studied thefactors that determine the sensitivities for analytes inambient air corona API/MS [27, 28). We were able toexplain the relaiiue sensitivities in terms of the ther­modynamics and kinetics for protonation of the ana­lytes by the main reagent ions, H 30 +(H20h . How­ever, we were not able to clarify what factors deter­mine the absolule sensitivities in API. In the presentarticle it is shown that the variation of absolute ionsignals with important ion source parameters can beexplained as a consequence of the ion source beingspace-charge-dominated.

It is noteworthy that in both the planar and sphericalcases, eqs 7 and 8, the space-charge solution is afunction of the boundary conditions at the ion supplysurface only, that is, Eo, ro, and Jo.

Equation 8 is easily integrated to give the potentialdistribution U(r) and the overall drift voltage betweenthe ion supply surface and the counter electrode, Yd 'The charge density is obtained from eq 5, and the iondrift time from the ion supply surface from

mobility, Eo is the electric held at the "current source"plate or "ion supply surface," and x is the distancefrom this surface.

In the spherical geometry, where the inner concen­tric sphere is the ion supply surface (Figure 1c), theelectric held is given by

E(r) = (TO)2{E5_ 2Jo2 (r~ _ r3))112

(8)r 3Kera

First, it is important to consider what parameters maybe important for the sensitivity of SCD ion sources.Clearly, when analyte ions are formed by chemicalionization, the ion residence time, Ires' or the drift timeof the reagent ions from the point of formation to thesampling orifice, will be important. Further, the totalion current 1, passing through the orifice could con­ceivably be determined by the ion current densityJ, atthe orifice wall,

(11)

where A is the orifIce surface area . Alternatively, 1could be determined by the ion density close to the

(13)

where Vi is the voltage over the ionization region andVd is the total ion drift voltage. Since we do not knowVi' there will be a range of possible potentials for theion supply surface. This problem will be dealt withhere by obtaining solutions for the whole range ofphysically reasonable boundary conditions.

Ions exit the drift region as they hit the ion sourcewalls or pass thr ough a vacuum orifice. The boundarycondition here may be defined by specifying the po­tential or electric field at this surface .

It was mentioned in the Introduction that solutionsto the space-charge problem are known for some

4 BUSMAN ET AL. J Am Soc Mass Spectrom 1991, 2, 1-10

(14)(8wai )1/2

3dot ""res

potential,:> ' , , ,

,,~

, .- .~ Ion Density

the ion density increases correspondingly. A furtherdecrease in Vd leads to an unphysical situation with anegative Eo value. Therefore, Vo = 5224 V and Eo = 0VIcm represents the space-charge limit. Thus, Vd =

5224 V is the minimum voltage required to overcomethe space charge in this particular case . Stated differ­ently, at Vd = 5224 V the space-charge-limited (maxi­mum) current (29) is 6 /LA. This concept of a space­charge limit is central to the understanding of 8mion sources. The dashed line in Figure 2a shows theelectric field distribution that Vd = 5224 V would in­duce in the absence of space-charge effects-theLaplacian field. The difference relative to the SeDcase is dramatic. With space charge, the electric fieldsare strongly suppressed dose to the ion supply sur­face, whereas they are enhanced close to the counterelectrode. Clearly, any analysis of these ion sourcesbased upon Laplacian fields would likely be in error .

Figure 2b shows the ion density, electric held,potential, and ion drift time as a function of radius forthe special case of Vd = 7693 V (£0 =40,000 V Icrn)and other parameters as in Figure 2a. Though notshown in the figure, the effect of changes in Vd (or Eo)on ion drift time and ion density is very small . Thedecrease in ion density and the concurrent increase inion drift time with radius are worth noting. That thisbehavior is fundamental to all 8m ion sources irre­spective of geometry is shown later.

Of particular interest for practical applications isthe dependence of the various source parameters on10 (total current) and on ion source size '1' The depen­dence on 10 is shown in Figure 3 with other parame­ters as in Figure 2. It is seen that Vd and P""mp areproportional and Ires inversely proportional to 1Jf2.This also follows from eqs 14-16, which are easilyderived for the space-charge limit and for r0 <c r t -

10" 10.1 10° 101 102 103

Current(lJI\ )

Figure 3. Variation of ion residence time, drift voltage, and iondensity with discharge current; spherical ion source geometry asin Figure 3. The drift time scale is from 10- 5 to 10-1 5•

o::> .:0

0.9 ~.5<-

I

.. . . .' 1.8

.'.'

2.5

Ion Density

2.0

'--"Drifl.,.' Time

, .

Potent ial

\ .. .- '..'.'

_~t:'placi~n Fieh.!

1.5 2.0 2.5Rad iustcm )

A

1.0

1.0

0.5

0.5

II1IIII ;'I' 1: \: ,, '

~::::,J~ _

"---~-~----~-~---' 8

oo

oo

8

simple geometries. It is instructive to first briefly dis­cuss the spherical geometry, Figure 1c. Figure 2 showscalculated results for the spherical geometry with in­ner radius TO = 0.1 em, outer radius r1 = 3 ern, dis­charge current 10 = 6 /LA, and mobility K = 10-4

m 2 /(v . s) typical for ions at atmospheric pressure[13]. The electric field was obtained from eq 8, thepotential by integration of eq 8, the drift time td fromeq 10, and the ion density from eq 5. Figure 2a showsthe electric field and potential distributions for sixdifferent choices of Vd from 5224 to 7600 V. Thecorresponding Eo values range from 0 to 40 kV[cxa.At 1 arm, substantial gas-phase ionization sets in at 30kVIcm [29], and for the drift region only fields lowerthan this would have to be considered. It is seen inFigure 2 that in the larger outer region of the ionsource the electric field is insensitive to Eo!

AJ:, the voltage Vd is reduced toward 5224 V, theelectric field at the ion supply surface, Eo, rapidlydecreases. In order to maintain a constant ion current,

1.5Radios(cm)

BFigure 2. Space-charge effects in spherical geometry. Ion sup­ply surface radius 0.1 em; outer electrode radius 3.0 em; dis­charge current 6 I'A . (a) Field strengths and potential distribu­tions versus radius for six values of applied potential Vd as seenat radius = 3 em. At the space-charge limit, Vel = 5224 V andEo= 0 Vfcm . Dashed line shows expected electric held with outspace charge for V<l = 5224 V. (b) Electric held , potential, iondensity, and ion drift time for Vel = 7693 V. Scales for drift timeand ion density are shown. Full scale for electric field is 40,000Vfcm and for potential 7693 V.

7.5x 10'

J Am Soc Mass Spectrom 1991, 2, 1-10 SPACE-CHARGE DOMINATED ION SOURCES 5

(15)

(16)

_ (210 ' 1 )1/2Vd - --

31rKE

(3dO ) 1/2

Psamp '" -8311: K'1

Versions of eqs 15 and 16 can be found in refs. 21 and22. The agreement between eqs 14-16 and the exactresults in Figure 3 are excellent for tres and Psamp '

whereas, as expected, there is a discrepancy for Vd •

For a constant 10' eqs 14-16 show that tres is propor­tional and Psamp inversely proportional to r{l2,whereas Vd is proportional to ,}/2.

Equations 14-16 can be used to make order-of­magnitude estimates of ion residence times, drift volt­ages, and ion sampling densities in any SeD ionsource. Real ion sources, of course, do not have aspherical geometry but are usually much better ap­proximated by the point-to-plane or "needle-in-can"geometry (Figure 1). Numerical solutions for suchgeometries [30], see also below, have shown that thespace-charge-limited drift voltage for a point-to-planegeometry is approximately given by eq 15, with r1being the point-to-plane distance, provided that theactual discharge current is multiplied by a factor of 2.

The absolute sensitivity will behave as Psamp if eqU is correct as anticipated. Thus, we would expectthat in a corona discharge API or ESP ion source, theion residence time, the ion sampling density, andthus the sensitivity are much more sensitive to mov­ing the corona needle or ESP capillary in the sourceaway from the sampling orifice than is the dischargevoltage.

Because of its recent success [6], the extent towhich the ESP ion source is space-charge-dominatedis of particular interest. Because of the lateral exten­sion of the spray, the geometry of this source wouldseem to be intermediate between the point-to-planeand planar geometries. The capillary-to-orifice dis­tance is typically 3-5 em, and the applied voltage, Vt ,

is in the 1000-10,000 V range [31]. These values arevery similar to corona API. However, the current inESP is lower,S x 10- 8 to 10- 6 A [31]. Using eq 15,with a correction factor of 2 for the current as men­tioned above, we calculate Vd to be one-fourth toone-half of the total applied voltage as reported byYamashita and Fenn [31]. However, in this calculationa mobility value of 1 X 10-4 m2 / (V ' s) was used, andmost of the charge may be carried by droplets ofunknown mobility. Even if the drift voltage Vd is afraction of the total ESP voltage, it can be seen fromFigure 2 that the electric field in the outer part of theESP ion source is most certainly dominated by space­charge effects.

In our treatment so far, we have assumed that allions in the drift space have the same mobility. In ESP,the situation is much more complicated. The effect onthe treatment presented here is solely through themobility values. In ESP, the mobilities of ejected

charged droplets are unknown. In addition, thedroplets undergo largely unknown changes in radiusand charge as they drift toward the orifice or capillaryinlet. Finally, many of the formed ions are multiplycharged, and again the mobility is unknown. To com­plicate matters further, it is not unlikely that thepotential distribution caused by space charge of thedroplets and the ions in turn influences the dropletevaporation and ion formation processes. Thus, thegeneral conclusion that the ESP is space-charge­dominated, with all the consequences that entails, iseasy to make, but a detailed solution is likely verydifficult. An interesting possibility is to use experi­mentally measured potential distributions in the ESPsource to make deductions about the ionization pro­cess.

The point-to-plane geometry has been extensivelystudied for corona discharges. In Sigmond's model[22], it is assumed (1) that the (elliptical) Laplacianelectric field lines are not affected by space charge and(2) that the electric field strength is constant along afield line from the point to the plane. Both turn out tobe good approximations. Sigmond's model can beused to calculate both drift times and ion densities atthe plane if the total voltage and current are known.

Finite-Element Calculations for "Needle-in-Can"Geometry

Because analytical solutions for the point-to-plane andneedle-in-can geometries are not available, we havestudied these using a fInite-element method. A gen­eral description of the fmite-element method forboundary value problems can be found in ref. 32.Computational details related to the following discus­sion will appear elsewhere [30]. By substituting eq 4with F = 0 into eq 3 and substituting - P/ E for V. VUvia eq 1, one obtains a convection-diffusion equation,

To simulate a needle-in-can geometry, axial symmetryis assumed and the system of eqs 1 and 17 is con­verted to cylindrical coordinates on a rectangular com­putational domain 0 < r <'v r = (x 2 + y2)1/2.Boundary values for both U and p must be specifiedon all sides of the rectangle except along the z axis,which corresponds to the axis of symmetry of the can.A uniform grid is set up on the rectangle, and apiecewise linear approximation is taken for both Uand p. A standard variational formulation of eqs 1and 17 yields, respectively,

Au = b(p)/E (18)

[KB(u) + DA]p = (K/e)c(p) (19)

where A and B are matrices resulting from the dis­cretization of the Laplacian operator and E' V, re-

6 BUSMAN ET AL. J Am So c Mas s Spectrom 1991, 2, 1-10

where W is the ion drift velocity vector related to fand o,

This is the unipolar charge drift formula (22). This is avery powerful formula for the analysis of SCD ionsources. A main reason is that the formula is totallygeometry-independent; it is equally valid for planar,

It is worth noting that the axial electric fteld for Case 1(Figure Sa) behaves similarly to that in the sphericalcase (Figure 2a) when close to the space-charge limit.Near the bottom of the can, however, the electric fieldagain increases slightly. This is the qualitative behav­ior expected for the planar case, see eq 7.

Applications of the Unipolar Formula to SeDIon Sources

In the analytical and fmite-element solutions above, itwas seen that as the ions moved through the driftspace the charge density decreased monotonically.That this should be a general phenomenon for SCDsources can be seen from the following argument.Imagine a small unipolar "cloud" of ions movingalong the electric field lines through a gas . Each ion isdrifting in the electric field induced by the chargesand potentials in the surrounding space . For a smallcloud, this field is the same for all ions. However,because the ions in the cloud also mutually repel eachother, it follows that the cloud will expand. Mathe­matically, this expansion or space-charge "blow-out"[14] is expressed by the so-called unipolar charge driftfonnJlla [26]. Because of the importance of this for­mula, a short derivation will be given here.

The continuity equation, ignoring diffusion andgas convection, was given by eq 2. In that equation,the derivative iJ()/ at refers to a fixed point in space.The time derivative along the path of a charged cloudis given by

(21)

(25)

(23)

(20)

(22)

(24)

(dP ) iJp __di path = at + W . vp

!=pW

iJp _ _ _ _ __-= -V·J= -pV'W -WVpiJt

(dP ) _ - _ _ --- = - p V . W = - pI( V . E = pK V • VUdt path

(do ) K p2

di path = - -f-

Integration of eq 24 from t = 0 to t = t yields

11K-- '" + -tp( t) p( t = 0) e d

Using eqs 2, 20, and 1, we can write

spectively. The components of the vectors u and pare, respectively, the approximate values of U and pat the grid points. The discrete nonlinear system ofeqs 18, 19 is then solved by using Newton's method.

Since typical values of KE are much larger than thediffusion coefficient D in eq 17, a boundary layerforms at the "outflow boundary, " where the electricheld vector E points out of the computational do­main . This means that the density o makes an ex­tremely rapid transition from interior behavior toboundary behavior in a very narrow region near theoutflow boundary. Failure to adequately resolve thisboundary layer leads to (nonphysical) oscillations inthe approximate solution throughout the computa­tional domain. If a uniform mesh is used, the meshspacing must be at least as small as the width of theboundary layer to prevent these oscillations. This leadsto systems of equations with an intractably large num­ber of unknown coefficients. To remedy this diffi­culty, we make use of a concept from computationalfluid dynamics known as "artificial viscosity." Thus,the diffusion coefficient D in eq 17 is made artificiallylarge, about 1000 times larger than is typical at 1 atm(131. This has the effect of producing a much widerboundary layer, which can be resolved with a muchcoarser grid with many fewer unknowns. The result­ing approximation is poor near the outflow boundarybut is quite accurate away from the boundary, wherevp is small.

Figure 4 shows the result from finite-element calcu­lations on an 11 x 11 grid for a needle-in-can geome­try with I = 5 em and r1 = 5 em (Figure Ie), The ionsupply surface, in front of the needle tip, has a radiusof 1.0 em, a potential of 5 kV, and a charge density of10-4 C/m3 (6.2 X 108 ions /cm''). The charge densityis zero at all other walls.

The only difference between Case 1 and Case 2 isin the boundary conditions. In Case 1, the potential ofthe can wall is 0 V, whereas in Case 2 it is increased to5 kV in an effort to "focus" the ions down the centeraxis. The arrows in Figure 4a and b indicate theelectric field, The divergence toward the can wall inCase 1 distinguishes this field from the nearly ellipti­cal held in the point-to-plane geometry. The solidlines are equipotentials (in steps of 500 V). Figure 4cand d show the charge density contours. The chargedensity is seen to decrease monotonically downfieldfrom the ion supply surface . In Case 1, the side wallof the can has a noticeable effect of "attracting" ionsaway from the center axis. The electric field strengthalong the axis at r = 0 is shown in Figure Sa, andFigure 5b shows the charge density along the sameaxis. Finally, Figure 5c shows the current densitydistribution along the bottom of the can . It is interest­ing to note the radical difference in the axial helddistribution, the much larger decrea se in ion densityfor Case 2, and the rather moderate focusing of theion current distribution achieved. It should be observedalso that the total current is different in the two cases .

JAm See Mass Spectrom 1991 , 2,1-10 SPACE·CH ARGE DOMINATED ION SOURCES 7

0.04

0.02 0.03

z axis (melus)

CASE 2: charge density p

0.01

0.045

p • 0, 5000 Vnos CASE 2: po!entia! U 2r.d f,'(:!d E

B

z axis (meter s)

CASE 1: charge densuy p

I 0 0.05

J

0.045

0.04

0.03

~J

\ jz axis (mcte ru z axis(meler s)

Figure 4. Results of fmite-element calculations for the needl e-in-can geometry. (a) Equipotentiallines (500-V step) and electric field distribution and (c) ion density distribution (10- S_Cjm 3 step)for Case 1. (b) and (d) show the same results for Case 2. Boundary conditions for Case 1: ionsupply surface. p = 10- 4 Cjm 3, in lower left corner; zero ion density at all other surfaces; canbottom, Z = 0, and wall, r =5 em, at zero potential; ion supply surf ace at 5000 V. Between the ionsupply surface and the can wall. the potential decreases linearly. Case 2 differs from Case 1 in thatthe can v....,rall and can top, z = 5 em, are at 5000 V. Along the can bottom, the potential isproportional to r 2

. The appr oximate position of a corona needle or ESP capillary is shown in (a).

o v, P ':' a CASE I: potential U and field E0.05 IIA

c

p • a

l1Ib-

tNeedle

cylindrical , spherical, and any othe r ion source geom­etry . However, the problem of calculating the resi­dence time remains.

An interesting consequence of eq 25 is that inLaplacian field dominated ion sources the charge den­sity will not change along the electric field lines . Thismay at first seem surprising. For example , in a spheri­cal ion source, the ions obviously must spread out asthey drift toward the outer electrode. However, theapparent conflict is resolved wh en it is realized thatthe ion cloud will compress in the radi al direction asthe electric field is decreasing.

Figure 6 shows ion densities calculated from the

unipolar formula as a function of drift time for initialion concentrations of (a) 1012, (b) 1011, and (c) 1010

ions/ern". A mobility value of 1 x 10- 4 m2 ICY . s)was used. It is seen that at long times the ion drnsity isinversely proportional to ion drift time and independent ofinitial ion drnsity. Thus, for a constant ion drift time, itis not possible to increase the ion signal by increasingthe initial charge density as might be accomplished,for example, by increasing a discharge current.

An important consequence of the unipolar formulais that it is not possible to "concentrate" ions in anseD ion source by applying any electric fields or bysurrounding a low-charge-density region by high-

8 IlUSMAN ET AL . J Am Soc Ma ss Spectrorn 1991, 2, 1-10

z component of eleeele fieldXI()!2..5r----------~--...A

10"10-6

c

ij--------- ..

10'10"

10"

10"Time(s)

Figure 6. Ion density versus drift time in a un ipolar ion sourcefor three values of Pa, the ion density at t = 0, calculated fromthe ion drift formula , eq 21.

ca•• 2

'-:/ /

///

z axis (meten)

Ca.. 1Jr~ 1.S

E

(26)

where Np is the number density of analyte molecules.For an analyte who se sensitivity is thermodynami-

charge density at the can bottom, where the orifice isusually situated, is considerably lower in Case 2 thanin Case 1. Thus , assuming that eq 12 is correct, theeffort to concentrate the ions will result in a lower ionsignal! The fact that such efforts have not been suc­cessful, as attested by the lack of such focusing incommercial instruments, also supports eq 12 over eq11. For a constant distance and potential difference,the best strategy to maximize sensitivity is to keep aconstant electric field throughout to the sampling ori­fice. This minimizes the ion residence time and , ac­cording to eq 25, results in a maximum ion density atthe orifice. This effect probably accounts for somemoderate sens itivity effects observed experimentally[33]. However, it must also be remembered that wedo not know the accuracy or limits of applicability ofeq 12.

The unipolar charge drift formula will now be usedto calculate absolute sensitivities in a corona API ionsource for trace impurities. As the reagent ions, R+,pass through the drift space, they ionize trace amountsof analyte, P,

where ku, is the (second-order) rate constant. It isassumed that the reverse reaction does not occur, thatis, we are dealing with an analyte whose sensitivity iskinetically as opposed to thermodynamically con­trolled [27]. In ambient air, the reagent ions areH 30 +(H20)h,

H :P+(H:Ph + B --+ BH +(H20) b+ (h - b + 1)H20

(27)From eq 12, the current of product ions through theorifice is given by

C..... 1/

Current clemity 1"Ill"3r--~-~--.,.:....-_-__,

"lil" charge clellIityp1.------~--~--~--__,

0.1

z axis (meters )

0.2

0.9 \ \

0.8

\-,"".,<,

./ ............ ......

C.. se 2 .~.-.~ ......_._._ ..._

N~ 0.7

~~ 0.6

8

c

charge-d ensity regions! Case 2 in Figures 4 and 5illustrates this well. Comparing it with Case 1, it isseen that giving the can wall a high potential of5000V dramatically changes the electric held distribu­tion and the ion current density, and charge density issomewhat concentrated toward the center axis of thesource. However, it is seen in Figure 5b that the

radius r (meters)

Figure 5. (a) Electric field and (b) ion dens ity distributionalong the z axis (r = 0) and (c) ion current distribution alongcan bottom for Cases 1 and 2; see text and Figure 4.

JAm SocMass Spectrom 1991,2.1 -1 0 SPACE-CHARGE DOMINATED IO N SOURCES 9

r1(cm)

0.01 0 .1 1 100.3m s

3ms

A RH' B I H,O' (H,OI

,.

:1.-!- .1~AH'.

:1~ , BW r pyrtr..........

r 31 31I I , Izo 1 5 10 20 15 10

r 1 / - Nee dl e o riCi c:e <b at anc e / -.

Figure 9. Reagent and analyte (pyridine) ion signals versusneedle-to-orifice dist ance in an API ion source with 0.1 ppbpyridine . (al Calculated results, source condi tions as in Figure6e; RHo+- = reagent ion ; BH+ = aiialyte ion. (b) Experimentalresults from ref. 27. AH+ is prolon ated acetone, the intensity ofwhich is thermodynamically con trolled .

~ 106

iio' //

a::~ I10-3 10-2 10-' 10° J01 10' 103 10'

Concemrationtppb)

Figure 8. Analyte ion signal versus analyte concentration in anAPI ion source for an analyte with kinetically controlled sens itiv­ity . Upper lines sho w calculated results for ion residence timesof 0.3 and 3 IDS, res pectively; experimental points from ref . 27.

ions is nearly complete, and the ion signal decreasesas p(t,es ) decreases. The most noteworthy feature inFigure 7 is the intermediate plateau region for low­concentration analytes, where the analyte ion signaldoes not change appreciably. The reason for this istha t the increase in t res is nearl y canceled by a corre ­sponding decrease in p( tres) . The plateau was experi­mentally observed in the author's previous article (seeref 27), We observe that with eq 11 such a plateauwould not be predicted; instead lp + would be pro­portional to 71(1,'2). The upper axis in Figure 7 showsthe needle-to-orihce distance calculated from the iondrift time and eq 13 with 10 = 3 /-lAoIt is seen that inorder to obtain a drift time of less than 10- 6 s, theneedle-to-orifice distance must be made very smalland comparable to the orifice radius. Because theioniz ation processes as well as the initial ion-moleculechemis try become verI important at short distances,and also because one can expect severe disturbancesin the ion-sampling mechanism, a more detailedmodel would be required for these short times.

In some instruments. a "curtain gas," for exampledry N2 • is used in front of the orffice. In the curtaingas, the unipolar formula still holds, but reactions 26or 27 are quenched. A comparison of curves c (no

10"

10'

10'

where [p+I/O]H30+(H20h) + [P+) is the equilib­rium ion concentration ratio,

10' '--__'--__""-__""-__.J-_ _ .J-_---'

Time(s)Figure 7. Analyte ion inten sities in a COrona API source versusion drift time calculated with eq 25. The uppe r x axis showsneedle-to-orifice distance for a discharge current of 3 p.A; seetext. (a) Po = 1012 ions /cm'' , c. = 100 ppb; (b) Po = 1010

ions/em 3, c. = 100 ppb : (c) Po = 1012 ions /cm3, c. = 1 ppb ; (d)Po = 1010 tons / em", c. = 1 ppb; (e) as (c) bu t 4-mm curtain gasin front of orifice.

cally controlled [27]. eq 25 should be substituted sim­ply by

The calculation of the sensitivity of a typical coronaAPI source by use of eq 28 will now be illustrated.The following conditions are assumed . The geometryis point-to-plane with 10 -= 3 IJA. The rate constant forformation of analyte ions is k26 = 1 X 10- 9

cm3/ (molecule' s). The gas flow through the O.l -mmion-sampling orifice is calculated to be about 1.0 cm3 jsat 1 atm [34). The fraction of ions ent ering the orificethat reaches the mass spectrometer detector was esti­mated to be 1%. This represents a reasonable guessonly . For these conditions. Figure 7 shows calculatedanalyte ion signals as a function of ion residence timefor three different combinations of analyte concentra­tions and initial (reagent) ion density. At short tres'the analyte ion signal increa ses with t res. The reasonis that conversion from reagent to analyte ions in­creases while p(t,.,,;) remains relatively constant; seethe initial flat portion of the graphs in Figure 6. Atlong t res • on the other hand, the conversion to analyte

10 BUSMANET AL . JAm Soc Mass Spectrom 1991, 2,1-10

curtain gas) and e (4 mm of curtain gas) in Figure 7illustrates the effect of the curtain gas. The ion resi­dence time in the curtain gas was calculated by usingeq 14, and it varied in the 10-4-10- 3-s range. Aresidence time in the curtain gas of 0.2 ms was esti­mated previously from experimental data (27).

Figure 8 shows a calculation of analyte ion currentsas a function of concentration for two different resi­dence times . The same source conditions as for Figure7 were used. The circles show corresponding experi­mental results (27). The agreement is good consider­ing that the ion intensities were artificially suppressedby about a factor of 10 in the experiments. The resi­dence time was estimated to be 0.3 ms (27), also ingood agreement with the present calculations (Figure8).

Finally, Figure 9a shows calculated intensities for0.1 ppb of pyridine in ambient air with k26 = 2 X 10-9

cm3/(molecule . s) as a function of the distance fromthe needle to the curtain gas orifice. The correspond­ing experimental result is shown in Figure 9b [27].The introduction of 4-mm-deep curtain gas into thecalculations has the effect of slightly curving the ana­lyte ion signal downwards with decreasing needle-to­orifice distance, in agreement with experiment.

Acknowledgment

Mike Ikonomou is thanked for he lpin g J. S. in a doomed effortto improve the sen sitivity of API /MS. This work was partlysupported by a grant from the MonlanajEPSCOR II Program ofthe Na tional Science Foundation (grant no . Rl1-8921978).

References

1. French, J. B.; Thomson, B. A.; Davidson, W. R.; Reid, N.M.; Buckley , J. A. In Mass Spectrometry in the EnvironmentalSciences; Karasek, F. W.; Hutzinger, 0.; Safe,S., Eds.;Plenum: New York, 1984; pp 101-121.

2. Mitchum, R. K.; Korfmacher, W. A. A,tal. Chem. 1983, 55,1485A.

3. Proctor, C. J.; Todd, J. F. J. Drg. Mass Spec/rom. 1983, 18,509.

4. Ketkar, S. N.; Dulak, J. G.; Fite, W. L.; Buchner, J. D.;Dheandhanoo, S. Anal. Chern. 1989, 61,260.

5. Covey, T.; Lee, E.; Bruins, A .; Henion, J. Anal. Chem. 1986.58,14S1A.

6. Fenn , J. B.; Mann. M.; Meng, C. K.; Wong . S. F.; White­hous e, C. M. Science 1989. 246. 64.

7. Wong, S. F.; Meng, C. K.; Fenn, J. B. J. Phys. Chern. 1988,92, 546.

8. Lovelock, J. E. Anal. Chem. 1963, 35, 474.9. Dressler, M. Selective Gas Chromatographic Detectors; Elsevier:

Amsterdam, 1986; p 217.10. Mock . R. S.; Grimsrud, E. P. Anal. Chern. 1988. 60, 1684.11. Dam, R. In Plasma Chro11Ultography; Carr, T. W., Ed .;

Plenum: New York, 1984; pp 177-213.12. Revercomb, H. E.; Mason, E. A. Anal. Chern. 1975 47, 970.13. McDan iel, E. W.; Mason, E. A. The Mobility and Diffusion of

Ions in Gases; Wiley : New York, 1973.14. Gobby, 1'. L.; Grimsrud, E. P.; Warden. S. W. Anal. Chem.

1980, 52, 473.15. Carroll. D. I.; Dzidic, 1.; Stillwell, R. N .; Homing, M. G.;

Homing, E. C . Anal. Chern. 1974, 46, 706.16. Siegel, M. W.; McKeown, M. C. ]. Chromatogr. 1976, 122,

397.17. Siegel, M. W. In Plasma Chro11Ultography; Carr, T., Ed.;

Plenum: New York, 1984; pp 95-113.18. Loeb, L. B. Electrical Cnronas. Their Basic Physical Mecha-

nisms; University of California Press: Berkeley, 1965.19. Townsend, J. S. Phi/. Mag. 1914, 28, 83.20. Langmuir, I. Phys. Rev. 1913, 2, 450.21. Chapman, S. ]. A/mos. Sci. 1977, 34, 1801.11. Sigmond, R. S. In Electrical B=kdown ofGases; Meek , J. M.;

Craggs, J. D. , Eds ., Wiley : Chichester, 1978; pp 319-384.23. Hinds, W. C. Aerosol Technology. Properties, Behavior, and

Measuremetll of Airborne Particles; Wiley: New York, 1982.24. Rubinstein, 1. Electro-diffusion; SIAM: Philadelphia, 1990.25. Selberherr, S. Analysis and Simulation of Semiconductor De-

vices; Springer-Verlag: Wien, 1982.26. Sigmond, R. S. t. Appl. Phys. 1982, 53, 891.zr. Sunner, J. ; Nicol, G; Kebarle, P. Anal. Chern. 1988, 60, 1300.28. Sunner, J.; Ikonomou, M. ; Kebarle, P. Anal. Chem. 1988,

60, 1308.29. Loeb, L. B. Basic Processes of Gaseous Electronics; University

of California Pre ss : Berkeley, 1955.30. Vogel, c.; Busman, M.; Sunner, J. in preparation.31. Yamashita, M.; Fenn, J. B. f. Phys. Chem. 1984, 88. 4451.32. Axelsson, 0 .; Barker, Y. A. Finite Element Solutions to Bound­

ary Value Problems; Academic: New York, 1984.33. Potjewyd, J. Focussing of Ions in Atmosphen'c Pressure Gases

UsingElectrostatic Fields, Ph.D. thesis, University of Toronto,1983.

34. Dushman, S. Scientific Foundations of Vacuum Technique; Wi.ley: New York, 1958.