.. 111.C FILE Coe4

oOFFICE OF NAVAL RESEARCH

Contract N00014-83-K-0470-P0O003

I R&T Code NR 33359-718

Technical Report No. 110

The Behavior of Microdisk and Microring Electrodes.Mass Transport to the Disk in the Unsteady State:

Chronopotentiometry

by

M. Fleischmann and S. Pons

Prepared for publication in J. Electroanal. Chem.

Department of ChemistryUniversity of UtahSalt Lake City, LT 84112

July 15, 1988 4

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Arlinaton. VA 22217 I11. TITLE (IncIUO S*Cunyafic=AtV* ON),me Ieh'vsor o0Mcrds and Microring Electroies. Mass Transport to the Disk in the Unsteady State:Chronopotentiometry

1j~ALj 1 ~R~S. Pons

13a. TYPE OF REPORT FRO TME O~E

Technical - RO 7Z___r TO 7/88~~' SP~fOi

16 SUPPLEMENTARY NOTATION

19 AaSTRACT Conrinue on reverse f nelOlsary and idenltify by block number)

Attached.

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I. Eklia=0L Chfwi. 00 f 1988) JECO9789ELsirvier Sequoia S.A.. Lauxanne - Printed in The Netherlands

The behavior of microdisk and microring electrodes.Mass transport to the disk in the unsteady state

Chronopotentiometry

Dq-pwof Ch-Wy 7UT UpuvavuWsa S-~A-WOMu H.-a S09 SNVH (GraN BfkdX)

stead" PownDV-aun of Choluhb UnPIEOW Of UVt S& akf CRY. UT 8411P (.SA)

(RaItsov 9th Februy 1968; in revised farn 3rd Maey 1968

ABST'RACT

- We duather the um dgssdet r~esp of a fidw dis diecuode under ainihbo of Wotangapplied flux() hSmi oita nrnm by solvnthe differIeuataan, the criwrlrcidr~al aarilsm y= with the use of suitable discowtuixa intogpaLa In anonb.

we; ,set th e tsult for the m.of Unm map anspesisery whichwo uti may beinri use"u for

There has been much recent work on tealyiofmass transport to finteelectrode geometries. One reason for this activity tieS in the increased interest in newapplications of microelectrodes [1]. The applicability of the preferred sphericalmicroelectrode geometry has so far been somewhat restricted e.g., to the elec-trodeposition of ensembles [Z,31 or single mercury droplets [4-61, the clectrolys~sCfdispersions (7,81, and the dropping mercury inicroelectrode (9]. Disk, band, and thei recently introduced ring microelectrodes [5,10-121, are in general more easil.constructed, but the necessary mathematical analysis has so far proved to be ratherintractable. The mathematical difficulties are due to the discontinuities at the edzcNof the electrodes (e.g. constant concentration Or flux over the surface of tr-electrode, zero flux over the adjacent insulating surface). The diffusion lirruted il.becomes infinite at these discontinuities ( the combined effects of the finite ratesthe surface reactions and of the distribution of potential and concentration act,

0022-072 8/88/S3.50 0 1988 Elsevier Sequoia S.A.

,aP QtVcilObI'- to DTIC dOo" iao%

t~r~ ttd1y legjble 1 eptoduciiOfl

the surface i.e.. the "tertiary current distribution." however, will limit the rates atthe edges for real systems). Disk and ring 'mcroelectrodes have the advantage thatquasi-sphcrical diffusion fields are established at relatively short times; in contrast,diffusion to line or band electrodes does not reach a steady state, the flux varying asI/In t. Spherical diffusion fields at small electrodes give rise to high rates of masstransport to the surface so that the kinetics of fast heterogeneous reactions and offast reactions in solution can be studied under steady state conditions (e.g., see theprevious analysis, [13)). While it is clear that fast reactions can always be studied bydecreasing the size of the electrode, there are some applications that are amenableto study at "larger" microelectrodes. A variety of analytical and simulation proce-dures have been used in attempts to develop adequate descriptions of the chro-noamperometric and chronopotentioeric responses at disk and ring electrodese.g., see [11,3.In this series of papers, we develop a general approach to the - -analyas of the non-steady state, and we apply the method here to the chronopo. ,.. L 0.

tentiometric case for constant and uniform flux over the srftace. The result is anexact expression that is valid at all tiams and for any size disk, and is therefore e+applicable to conventional electroanalyical experiments, as well as thoas at micro- c.r' -electrodeL The approach to the problem is basd on the propertie of diacoitinuousintegrals (see e.S., refs. 13421nd 2whih we have extended from the previousanalysis (the prediction of i r -fer coefficients fee constant concentrationand constant flux conditions in the steady sate) to include time dependent masstransfer. In addition, we include the reults for a linearly swept current experiment.

THEORETICAL CONSIDERATIONS

General tim depnde soiutoFor any simple electrochemical experiment involving a single reactant, we must

solve the time dependent diffusion equation in circular cylindrical coordinates:

c D 2c +D 8c 82cat 3,?2 F ar 3z2

where c is the concentration of the reactant, and r is the radial distance coordinatemeasured from the center of the disk electrode which is imbedded in the insulatingplane at z - 0. The general initial condition is, at

r>O, z>O, t-0 C-c, (2)

where c' is the bulk concentration. Laplace transformation of eqn. (2) gives

al iae ale cm+r + r + T - : qcE+- -0 (3)

where

q- . s/D (41

and s is the Laplace transformation variable. The last term on the LHS of eqn. (3) is

a constant and hence determines the parucular integral of eqn. (1). We then seek thesolution for the complementary function from

LCF :cF Ic'-' - - +. 4- - q'c 0 (5)

a-2 r ar 3:2I

Separations of variables of the form

CCF - V eXp[ -f(X. q)z] (6)

simplify the differenual equation (5) to the familiar Bessel form

d 20 1 dD '& 0 (7)-" + r-r w - 0 (7

dr12 r dr

where

a2, [/(X, q)]Z q' (8)

For ist. ., we can choose the simple formf (A, q).(, + q 2)1/2 (9)

so that

wit - andepen( + o q. (10)

and eqa. (7) is recoveed from eqn. (5) with a independent of q. (The factorsinvolved when considering a choice for this function will be discusmd elsewhere.)Theefore, the solution to Bessel's differential equation (3) bcomes

- - f :(, q) exp(-/(A, q):).(ar) da (i)

where A0 is the Bsael function of the first kind, order 0, and we choose g(A, q) tosatisfy the boundary conditions.

ChronopotentiometryWe consider here the solution for eqn. (11) for the chronopotentiometry problem

for a constant a constant uniform flux - Q(mol cm - 2 s - ) over the surface at allt > 0. The boundary conditions at the surface of the electrode become

O4r<a, z-0, t>O DLel - - Q " L

r >a, -, t>O DI[ 1.0- "';

We take the Laplace transform of n. 12), and substitute in eqn. (11) afterdifferentiation under theintegral and obtan, for t > 0.

D[ E - - -- fg(.,q)/(X.q)Jo(ar)d- O<r<a.z-0 (13)

_.- .. .,m .- _ . 1 i n m a z S mm mm mm m m Il m~fi

4

and

D[ j ,0 r>a, :-O (14)

The boundary conditions suggest that we can apply the discontinuous integrals

10 r>a (15)

fJo(ar)J (aa) da 1/2a r'a (16)Il/a r < a (17)

in the solution of the problem. We determine the conditions for which g(X, q) fitsthe boundary conditions (13) and (14). From eqn. (13), we see that

og(,. q)f(X, q)Jo(ar) da-- D 0<<a.- (18)

or

' o g(,i. q)f(X, q)Jo(ar) da - (19)

Therefore if we let

Aq)- J(a" (20)f~x q-(X, q)

the complete solution to eqn. (3) having boundary conditions (13) and (14) is

Z a da (21)s Ds f [- f(X, q)

and at z - 0,

-. _ ss f 0o J ° (aw ) Jj(aa) f(daq) (22)

We point out here that the interpretation of the arguments of the Bessel functionsdepends on the nature of the assumption of the form of f(X, q) and, indeed, on thenature of Ae experiment (e.g. compare with the discussion of the chronoamperomet-"ic case( The use of the simple form (9) gives

E: _ 0 Jo(ar)J(aa da (23)S= s =(') J() +. +q,)'/

for galvanostatic conditions with a now independent of q; we can therefore invertirmmediately to the t-domain:

Qa 00 1/"/ dac"c - "f J,(ar)J,(aa) erf(DL/2atl/2)"! (24)

where erf( y) denotes the error function.

We can evaluate the average concentration at - 0 by integrating eqn. (24) overthe surface of the disk:

- f = 12 25)

With the substitutions

I2 - Dt (26)

0 - a/ (27)

eqn. (25) can be written in terms of dimensionless variables and parameters

CA, C - a

c 2Qa Va 21 (28)

The function 01 is tabulated in Table I as a function of the dimensionlessparameter (Dt/a2 ).

At long times and sufficiently small values of the flux, we do not observe atransition time and always reach the steady state value [13]

4- " (29)3s'D

If Q is sufficiently large we will get a sharp transition time as the surfaceconcentration of the reactant approaches zero; for this condition eqn. (28) can bewritten2 Qa-.Jo [ .-(Ia) erf(p) d# . 2Qa 1 (30)

from which the transition time can be obtained. The problems to be discussed inthis series of papers ae always cast in the form of such definite integrals, and thesefrequently converge slowly. Accurate evaluation is readily obtained, however, throughthe use of standard Bulirsch-Stoer numerical integration methods. In addition.convergence can always be speeded by suitable rearrangements. For instance, eqn.(30) can be rewritten in the equivalent form

2Qa I [ (La~l~dO ro[r(La 112F_2 _J ( hf )~ -$J ~J~ erf(R )- 1) - (31)

The value of the first integral is known to be 4a/31rI .M. The second integralconverges rapidly since the error function approaches I with increasing B. Figure 1illustrates the square root of the dimensionless transition times Dr/a 2 as a functionof the inverse of the dimensionless flux 2Qa/Dc'. The values are close to thosepredicted by Aoki and Osteryoung (181 even though these authors based theiranalysis on a uniform surface concentration boundary condition; the expressions

6

TABLE 1

Value of the funcuon Ot(Dh/aZ)

Dt/a- 0 ( Dr/a" )Dr/a 0(Dt/a2

)

4.0000X I0 Z.1217x 10- 4

2.7778 2.0533 x 10-2.7778 x 10 :.5460x10-4 2.0408 2.3064x 10-'2.0408 x 106 Z.9702 x 10-' 1.5625 2.5372 x 101.5625 x 106 3.3944 x 10-4 1.2346 2.7473 x 10-1.2346 x 10' 3.8186x 10- ' 1.0000 2.9381 x 10-l.O000x 10' 4.2427 X 10 2.5000 X 10- 4.10D45 x 10 -Z.000x 10' .4826 x 10- 1.1111 X 10- 1 4.5961 x 10-'1.1111 X 0' 1.2720 x 10 - 6.2O x 10- 2 4.g524 x 10- '6.2500 x 104 1.6954 x 10- 4.0000x10- 5.0085x 10-4.OO O x 10 2.1185 X 10-' 2.7778 X 10- 2

5.1132 x 10- '2.7771 x 104 2.5414x10' 2.0"X10" 5.13x 10 42.04M x 104 2.9600x 10- 3

1.5625x 10- ' .248x10't

1.3625 x 10' 3.3863 x 10' .2346 X10- 5 XI0'-1.2346 x 104 3.03 x 10-' 1.00 x0- 5.32dx 10-1.000x10' 4.230x 10- ' 2.000X10 - 1 5,48X10'-

2.5=O X 10' 8.4318 X 10- 3 1.1111 X 10- 3 5.5358x 10-1.lllxlO 1.265 x 10- ' 6Z5xlO

-' 5.5623x10'-

6.250x 10 1.6751 X 10-' 4.OOOO X 10 - 4 5.5783 x 10-'

4.0000x 10' 2.0868x 10 2.7771x10-4 5.5889x 10-2.7778 x 10' 2.495"7x 10' 2.040xO

-' 5.5964x10'-

2.0,08 x 10' 2.9018 x 10- 1 1.562 x I4 5.6021 x 10-'1.5625 x 102 3.3051 X 10- 1.234X10- 4 5.6065 X10-t1.2346 x 102 3.7055 X 10-2 1.OOOX 10 4 5.6101 X 10"'lO000X 101 4.1032X 102 2.5000XlO- 5.6260X 10-'2.5M x 101 7.9259x 10- 1.1111 x 10-

5.6313 x 10'1.1111 x 1O 1.1472x10

- 6.200x10- 6 5.63d0x10'-

6.2500 1.4749 x 10-' 4.0 x 10 6 5.6356x10'-

they derived could not be inverted exactly over the entire time range whereas eqn.(30) is exact. It should be noted that the constant flux condition is more likely toapply for most of the duration of the experiment rather than the constant surface

concentration condition, the actual behavior lying between these two limiting

conditions. A more correct formulation of the boundary condition on the disk for

an irreversible reaction, say, would be

ac/az-kc 0<r<a, z-0 (32)

For a cathodic reaction, say, we would write

k-koexp[ aEF/RTI-constant 0<r <a (33)

provided we neglect the effects of the distribution of potential in the solutio4

;k. ie.

we neglect the primary or secondary current distribution. Equation (32) shows thatwe cannot stctly speaking assume cither the flux or the -crtr=t4"-,' "C --constant over the surface of the disk. However. the close agreement of the transition

3 33

159

0.37

0.87

0.03 0o12 0.2 0.29 O. ci

DC"/ 2Qa

Fig. 1. Plot of the squmro ot of i dimmAoulu umianot ur. (Dr/az) / 2 as a tuncum of th.LnVWSU of the dMutmuoisho fu O2Q/Dc-)

times derived using there two approaches shows that the ine tion is not verysensitive to the nature of the assumptions; subsequent p:pe= 31)v.i show that thisis also the case for other types of experimigni We note i th constant flux Scondition will hold at low current densities (as in relaxation experiments) while theConstant oncetrat condition may be approached in the limting current region(see, however, below) so that the actual behavio must lie between these two limits.Thus, while it would be possible to develop the conditions (31) and (32), it isunlikely that the accuracy of the experimental-data would allow an assessment ofthe range of validity of the various assumptions. Furthermore, the application ofeqn. (31) would also require the consideration of the distribution of the potential inthe solution, i.e. we have to consider the tertiary current distribution. We canpredict. to some extent, the probable outcome of such analyses: it is unlikely thatthe "throwing power" of any practicable system would ever be sufficiently low thatthe current density could deviate appreciably from uniformity over the surface of amicrodisk (note that an infinite flux to the edge of the disk is clearly impossibleeven on the limiting current plateau!). We therefore consider that the constant fluxboundary condition will hold under most experimental conditions.

Figure I shows the expected linear dependence at high values of 2$Qa/Dc" ,where we observe essentially linear diffusion to the electrode followed by a rapidrise at low values of 2Qa/Dc' as this parameter approaches the condition requiredfor the observation of a steady state

2 Q a z " .T _<42,44T,2 (34)

• ~ ~ ~ ~ ~ ~~c 3- n- mm~alu nlln .. . ..

The ,.alculated values of c.., may be used to derive the potential-time curves for_rpropriate models of the electrode reactions. For instance, m the case of simpleButler-Volmer kinetics, we obtain

FQ _ Qa 0 D " -- ato Dc' a- RT RT

ep- aF 0- a) 1F)

exp~T-) - expi T(i (35)

which shows that the transients are a function of a. FQ/i o and 2Qa/Dc',

Linear sweep amperometryIt is clear that since there is rapid attainment of steady state diffusion to

microdisk electrodes, constant current experiments may not be generally useful. It tsdifficult to determine the appropriate galvanostatic condition that will give atransition tme within a convenient experimental time scale. It is therefore morestraightforward to apply time dependent fluxes to the surfac It we consider, forexample, the simplest case of a linear currevt ramp

Qt) -YtI (36)

we can immediately rewrite eqn. (23) in the general form [1,13.I , l.3 -- a f ojo(a ) J, ("a do (37)"s Lis' (0312 + q 2)1/2

and by the same methods obtain the transition time

D- a I J[ a)] [jy 0erf(y) dyj =-- (38)

or

4Tyra Lit 1 (39)Dc ~2

If now the current is swept from zero, a very sharp transition is observed in thepotenual-time plot. For simple Butler-Vomer kinetics, the shape of the responsecan be determined from

_ 4y t a "/ Dti, / -a'F exp(_1

10 D 1a{CP RT +e P RT fe£o • - F _exp( (I - a)7F (.40)"eiTW RT

Table 2 gives values of $3 as a function of Dria.The application of other boundary conditions representing other electrochermcal

experiments (e.g. cyclic amperometry, ac impedance measurments, and to casesinvolving reactions in solution coupled to the electrode processes) to this form of

TABLE 2

Values of the function VjDt/la)Dr.,, a , 0, Dt la;) Dtia-, '(Dt ,'a"

L.0000 lu tO 5112XI10-

5.00W0 3.6511 x 10 - : ,

3.1623 10- I 8778x 10-' 7.0711 39321 x 10-

3.5355 x 10-2

-'.3473 x 10- 1.0000 4.0842 x 10 -3.7796x 10 -: 6826x 10-' 1.1180 4,1155 x 10" '- -4,4721x10() 3"556x'10 1 1952 4.1313x10 -

5.0000x 10-, 4 6945 x 10-' 1.4142 4.1631 x10-

7.0711 x 10-2

?3831 x 10 - 1.5811 4.1792 x 10I.M~O x 10

- t 1.8728 x 10

- 2.2361 42116 x 10- "

't CA\, L.'1.1ISO x 10- 2.3385 x 10-1 3.1623 4.2277 x 10'1.1952 x 10 -' 2.6706 x 10- 3

3.7796 4.2327 x 10-1.4142 x 10-' 3.7299X 10- ' 4.4721 4.2360 X 10-1.5811 x I0- 4.6526 x 10- 3 5.0000 4.2376 x 102.2361 x 10 9.2073X1O"' 7.0711 4.2409X 10"'3.1623x 10

- 18023x10- ' 1.000X 10' 4.2425 X 10

3.5355 x 10-' 2.2W x 10- 1l.1lOx 102 4.2421 x 10"3.7796 x 10-' 2.5269 x 10- 2 1.192 X 10 4.2430 x 10-4.4721 X 10-' 3.44a6X 1 1.4142 X 10' 4.2433 x 10-'5.0000 X I0" 4.213SX10-1 1.5811 X 10 4.2435 x 10-'7.0711 x 10- 7.4851 x 10

- 2 2.2361 x 10' 4.2438 X 10" '

1.0000 1.1751 x 10-' 3.1623 $K I 4.2440xI0'-1.1180 1.3151 Xo - I

3.5355KO2

4.24d x i0 t

1.1952 1.3971 x 10-' 3.77% X 102 4.2440x I0'1.4142 1.5820 X 10- 4.4721 X 101 4.24d0x 10"'1.5811 1.7040X 10

- 5.0000X 10 4.2441 e I10-'

2.2361 2.2654x 10" 7.0711 X 10' 4.2441 x 10-'3.1623 2.9695 x 10- 1 1.0000 X 10' 4.2441 x 10-3.5355 3.1727x I0- 3.1623 X 10' 4.2441 x 10"'4.4721 3.5219 X 10- ' 1.0000 X 10' 4.2441 x 10- )

analysis, as well as extension of the analysis to include the ring geometry will bediscussed elsewhereJl.

ACKNOWLEDGEMENT

We thank the Office of Naval Research for support of this work.

GLOSSARY OF SYMBOLS USED

a Radius of disk. cmc Concentration trol cm-3c. Bulk concentration. mol cm- 3c ,, Average concentration. mol cm-

CCF Concentration complementary functioncS Surface concentration, mol cm-3

10

D Diffusion coefficient. cms t

E Electrode potential. VF Faraday constant. 96485 C mol -

10 Exchange current density. A cm --J0. J, Bessel functionsk Heterogeneous rate constant, cm -ko Heterogeneous standard rate constant. --nm -I ( DO '

Q Flux. mol cnif 2 S-IR Gas constant. 8.3141J rnor K'r Radial coordinate. cmS Laplace transform variable

T Temperature, KV Concentration ampituzdez Coordinate normal to plane of disk. cma Transfer coefficient (when in exponent)a Continuous dummy integration variable18 ad

Y Flux sweep rate. mol cm-2 s-A Continuous dummy integration variable17 Overpotential. V

* i ~ji(a)fj P 2y

Transition tim s

REFERENCES

I M. Fletschmarim S. Pons, D. Robson and P. Schmidi. Ultivmacroelectrodes. Datatech SciencnMorgatnic. NC. 1987.

2 P. Bindra A.P. Brown. M. Fleischmnn and D. Pletcher. 1. ElectroanaI. Chem., 58 (1975) 31.3 P. Bindra A.P. Brown. M. Fleischzann and D. Pletcher. J. Electroanal. Cheat.. 58 (1975) 39.4 G. Gunawardenh. G.J. Hills and B. Scharifker. 1. Electroanal. Chemt.. 130 (1981) 99.5A.M. Bond. M. Fleischmann. S.B. Khsoo. S. Pons and J. Robinson. Extended Abstracts 165th Meetng-

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:2 q 31 St4 Fldschin. J. Dawsbb and I. Pns. J. ElacuOMai Che. , W2,0. 32 NI. FkWcin wid S, Poes.)J. Eleavnl Chnk JE97 97Ke-. C .

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ABSTRACTS DISTRIBUTION LIST, SDIO/IST

Dr. Robert A. Osteryoung Dr. Donald M. SchleichDepartment of Chemistry Department of ChemistryState University of New York Polytechrnic Institute of New YorkBuffalo, NY 14214 333 Jay Street

Brooklyn, New York 01Dr. Douglas N. BennionDepartment of Chemical Engineering Dr. Stan SzpakBrigham Young University Code 633Provo, UT 84602 Naval Ocean Systems Center

San Diego, CA 92152-5000Dr. Stanley PonsDepartment stry Dr. George BlomgrenUni y of Utah Battery Products Division

t Lake City, UT 84112 Union Carbide Corporation25225 Detroit Rd.

Dr. H. V. Venkatasetty Westlake, OH 44145Honeywell, Inc.10701 Lyndale Avenue South Dr. Ernest YeagerBloomington, MN 55420 Case Center for Electrochemical

ScienceDr. J. Foos Case Western Reserve UniversityEIC Labs Inc. Cleveland, OH 41106111 Downey St.Norwood, MA 02062 Dr. Mel Miles

Code 3852Dr. Neill Weber Naval Weapons CenterCeramatec, Inc. China Lake, CA 93555163 West 1700 SouthSalt Lake City, UT 84115 Dr. Ashok V. Joshi

Ceramatec, Inc.Dr. Subhash C. Narang 2425 South 900 WestSRI International Salt Lake City, Utah 84119333 Ravenswood Ave.Menlo Park, CA 94025 Dr. W. Anderson

Department of Electrical &Dr. J. Paul Pemsler Computer EngineeringCastle Technology Corporation SUNY - Buffalo52 Dragon Ct. Amherst, Massachusetts 14260Woburn, MA 01801

Dr. M. L. GopikanthDr. R. David Rauh Chemtech Systems, Inc.ETC Laboratory Inc. P.O. Box 1067111 Downey Street Burlington, MA 01803Norwood, MA 02062

Dr. Joseph S. Foos Dr. H. F. GibbardETC Laboratories, Inc. Power Conversion, Inc.111 Downey Street 495 BoulevardNorwood, Massachusetts 02062 Elmwood Park, New Jersey 07407

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ABSTRACTS DISTRIBUTION LIST, SDIO/IST

Dr. V. R. Koch Dr. Gary BullardCovalent Associates Pinnacle Research Institute, Inc.52 Dragon Court 10432 N. Tantan AvenueWoburn, MA 01801 Cupertino, CA 95014

Dr. Randall B. Olsen Dr. J. O'M. BockrisChronos Research Laboratories, Inc. Ementech, Inc.4186 Sorrento Valley Blvd. Route 5, Box 946Suite H College Station, TX 77840San Diego, CA 92121

Dr. Michael BinderDr. Alan Hooper Electrochemical Research BranchApplied Electrochemistry Centre Power Sources DivisionHarwell Laboratory U.S. Army Laboratory CommandOxfordshire, OXl ORA UK Fort Monmouth, New Jersey 07703-5000

Dr. John S. Wilkes Professor, Martin FleischmannDepartment of the Air Force Department of ChemistryThe Frank J. Seiler Research Lab. University of SouthamptonUnited States Air Force Academy Southampton, Hants, S09 5NH UKColorado Springs, CO 80840-6528