theory of square wave try

8/3/2019 Theory of Square Wave try

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Theory of Square Wave VoltammetryLouis R a m a l e y a n d M a t t h e w S . Krause , Jr.1Department of Chemistry, University of Ari zon a , Tucson, A r i z . 85721When the sum of a synchronized square wave and stair-case potential i s applied to a stationary electrode, thedouble layer charging current may be mad e negligibleby measurement a t a suitable time afte r the pulses oc-cur. The staircase allows the electrode potential tobe swept over the useful potential range. The theoryfor a rev ersible system pre dicts that the resulting cur-rent-time behavior should be symme trical and bell-shaped with a peak at El/*. The peak current is a linearfunction of concentration and the square root of thesquare wave frequency. The peak curren t is a mo recomplicated function of square wave amplitude, mea-surem ent time, and staircase amplitude. The sim i-larity between this technique and s quare wave polarog-raphy is discussed.T H EUSEFULNESS of square wave polarography as developed byBarker ( I ) is well established in trace analysis. Th e abilityto present the current output in the form of a symmetricalpeak, rather th an the waveforms obtained in dc polarographyor linear sweep voltammetry, and the ability to measureFaradaic current at a t ime when the double layer chargingcurrent is negligible, are primarily responsible for its success.The sensitivity is limited by capillary noise (2) and the neces-sary resolution is obtained only at sweep rates of 1 V in 10t o 60minutes.The use of stationary electrodes should eliminate capillarynoise and allow rapid scan rates while maintaining highresolution. Mann (3, 4 ) has applied a staircase potentialwaveform t o stationary electrodes a nd significantly reducedthe effect of doub le layer charging current. Th e currentwaveforms obtained were similar to those of linear sweepvoltammetry. The superimposition of a square wave on thestaircase potential should result in higher sensitivity than inthe case of the staircase alone, and a derivative or peakcurrent waveform similar to that of square wave polarographyshould be observed.The waveforms for this technique, which the authors pro-pose be called square wave voltammetry (SWV), are shown,somewhat exaggerated, in Figure 1. Since the potentialsweep is discontinuous rather than linear, the tops of thepulses remain flat, even at rapid sw eep rates. Since the doublelayer charging current is proportional to e-IIRC,where t istime, R is the solution resistance, and C s the double layercapacitance, and the Faradaic current is approximately p ro-portional to t - l / * , the charging current decays much morerapidly than the Faradaic current, allowing measurementsto be made at a t ime when the charging current can be con-sidered negligible. Th e final current waveform is the differ-ential sum of the current flowing at an in stant, selected to re-duce the effect of the charging current, along the cathodichalf cycle of the square wave and that flowing at the sameinstant alo ng the preceeding or following anodic half cycle.

1 Present address, E. I. du Pont d e Nemours & Co., Inc.,MarshallResearch Laboratory, 3500 Grays Ferry Ave., Philadelphia, Pa.19146(1) G. C. Barker and I. L. Jenkins, Analyst, 77,68 5 (1952).( 2 ) G. C. Barker, Anal. Chim.Acta, 18, 118 (1958).(3) C. K. M ~ ~ ~ , A N A L .H E M . , ~ ~ ,484(1961).(4) C. K. Mann, ibid., 7, 326 (1965).

M E A S U R E DC U R R E N T

D I F F E R E N T I A LC U R R E N T

E l / *nT O T A L

D I F F E R E N T I A LC U R R E N T

E [ - )Figure 1. Waveforms em ployed and obtained in square wavevoltammetry

THEORYBarker et al . (9 ,Matsuda (6), and Kambara (7 ) have dis-cussed the theory of square wave polarography. Christieand Lingane (8) have derived an equation fo r staircase voltam-metry. Th e theory presented below for square wave voltam-metry is approached in a manner similar to that of Barkeret al . and Kambara .The current resulting from the application of the potentialwaveform of Figure 1 to an electrode at which the reversiblereaction

Ox + ne - +R edcan occur may be easily determined if semi-infinite inear dif-fusion to the electrode is assumed. It has been demonstrated(5, 7) that fo r a planar electrode and a reversible redox couple,the concentrations at the electrode surface are functions onlyof the electrode potential and the bulk concentrations and no tof any previous potential established at the electrode.

( 5 ) G. C. Barker, R. L. Faircloth, and A. W. Gardner, AtomicEnergy Research Establ. (Gt. Brit.) C/R-1786.(6) H. Matsuda, 2. lektrochem.,60,489 (1957).(7) T. Kambara, Bull. Chem. SOC.apan, 27 527 (1954).(8) J. H. Christie and P. J. Lingane, J. Electroanal. Chem., 10,176 (1965).1362 ANALYTICAL CHEMISTRY


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A E ( m ~ )Figure 2. Effect of LIE on measured current (1)

The init ial and boundary conditions may be stated as fol-lows:a t t = 0, C ox = C * , CII~~I 0, 0 5 x 5 03

fo r r > 0, Cox- * , CRcd- , x-a n d f o r 0 < I < 7 (C,,),,, = c1

7 < r < 27 (Co,)z_o= C?( j - 1 ) ~ t < r (C O ~) , -~~C,

where t is t ime? x is distance from th e electrode surfacc, andCj is the concentration of the oxidized species established byelectrode potential E , during thejtb half period, T , of the squarewave.Ficks seco nd law of linear diffusion( 1 )?c% Do d?COx-at ax

may be solved using the method of the Laplace transform (9 ,10, 11) in conjunction with the superposit ion principle (Z2)to give the following expression for the conce ntration of oxi-dized species as a function of distance and time

-tX(C , - C, - J erfc __-2D01[t - j - ~ ) T I ,

(9) P. Delahay, New Instrumental Methods in Electrochemistry,(10) R. V. Churchill, Operational Mathematics, 2nd ed., Mc-(1 1 ) H. S . Carslaw and J. C . Jaeger, Conduction of Heat in Solids,(12) J. Crank, The Mathematics of Diffusion, Oxford University

Interscience, New Yo rk, 1954,p 46.Graw-Hill Book Co.. Inc ., New York, 1958.2nd ed., Oxford University Press, London, 1959,p 62.Press, London, 1956, p 9.

where D O s the diffusion coefficient of the oxidized species.The current may be obtained in the usual manner by dif-ferentiating Equ ation 2 with respect to .r and sett ing x = 0in th e resulting expression. This produ ces

The Cs may be obtained from the following expressions= C * Q , (4)( exp (E , - El .) nFiRT- ~ _ _ _c, = C 1 - x p ( E - El .) nF RT

In the above expressions, .4 is the electrode area, Dn the dif-fusion coeffiiiznt of the rzduced species, and the other symbolshave their usual meaning. If it is further specified that thecurrent will only be measured at a certain t ime during eachhalf period of the s quare wave, thent = ( j - 1)r T 87 0 < / 3 i l (6 )

where p is the fraction of th e squ are wave half period at whichthe current is measured.Substituting Equations 6 an d 4 into Equation 3 results in

To obtain the differential sum of the currents flowing duringthe anod ic an d cathodic half cycles, the current flowing duringan even value o f j is subtracted from that flowing during thepreceeding or following odd value of j , ori,,,,= i, - ,-.:i c l i f = i,-] - ,

f o r i = 1, 3, 5 , 7 . .f o r J = 2 , 4, 6 , 8

The ditkrzntial current thus becomesI,,,f = nFADo/C* (-l)J[ QI - 1

?,I 2 Ti-1T-p I-Q , - 281-~.1 $. , , , $. ~.. ----~-1-1-.-Q + Q j - z( j - 2 + p ? ) l 2 (!3)1 ?

or

where Qo an d Q-I are both unity. The above equations arestrictly valid only for planar electrodes and for liquid elec-trodes of large volume if the reduced species is soluble in theelectrode. At the measurement t imes encountered in suchwork, the diffusion layer thickness at a spherical electrode isusually small compared to the radius of curvature of the elec-trode and the above equations will be valid. However, forreactions in which the reduced species is soluble in the elec-trode, th e sem i-infinite diffusion as sum ption w ill be invalidand the above equations will only be approximations for elec-trodes of small volume.It can iinmediately be seen th at the difTerentia1 current is alinear function of the bulk depolarizer concentration and isdirectly proportional t o r - I t 2 or t o f 1 !2 where ,fis h e squarewave frequency.In order to determine the other characteristics of the dif-ferential current, the summation in Equation %-hereafterreferred to simply as 2-must be carried ou t. Calcula tions

VOL. 41, O. 11, SEPTEMBER 1969 1363


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I0 0.1 0 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0PFigure 3.constant 7 Effect of /3 on measured current (2) at

demonstrate that 2 , and therefore i d i f , are maximum at E I l 2and tha t the peak is bell-shaped and symmetrical as show n inFigure 1. Th e half-width of the peak is a function of n andA E , the peak-to-peak square wave amplitude. As long asA E < O.SRT/nF, the half-width is 90.5 mV for n = 1, 45.3mV for n = 2, and 30.2 mV for n = 3. The half-width in-creases slightly as A E becomes larger than RT/nF-for ex-ample, a one-electron reaction and a square wave amplitudeof 50 mV produ ce a half-width of 98.8 mV .The dependence of Z: on A E is shown in Figure 2 . T hecalculations for this figure were performed with p = 0.60an d E,,, = -0.20 mV. Again, as long as A E < O.SRT/nF,Z is linear in A E and also l inear in n. This has the result thatthe differential current is proportion al to n 2at small A E .Th e effect of p o n Z: is demonstrated in Figure 3 , where A E= 10 mV , EjtOP-0.20 mV, and n = 1. This figure indi-cates that the highest sensitivity should be obtained with thesmallest 0 t which the d ouble layer charging curren t is neg-ligible. The usual practice in square wave polarography ist o make T as small as possible and set /3 a t 1.0, thus makingT - ~ ~ ~arge and Z small. A slight gain in sensitivity can beobtained by making T large and p as small as instrumentalfactors will allow. This is shown in Figure 4, where againA E = 10 mV , Es t , , = - 0 . 2 0 mV, and n = 1. The points inthis figure were calculated for a constant measure instant-the time after the square wave polarity change at which thecurrent i s measured-of 500 psec by suitable adjustment ofp and T .

Sweep rate may be altered in tw o ways, by changing T or bychanging E.,,,, the amplitude of the staircase waveform. Theeffect of T on differential current has already been discussed.The effect of E,,, is shown in Figure 5 , where n = 1, (3 =0.60, and Z0 s the value of Z fo r E,,, , = 0.0. This effect maybe looked upon as a distortion of the square wave by the stair-case wave. Fo r example, if A E is 2.5 mV and E s t e ps -0 .5mV, the cathodic pulse will be -2 .5 mV while the anodicpulse will be only + 2 . 0 mV , a difference of 2 0 x . The resultmay be looked upon as a squ are wave with a n average ampli-tude of 2.25 mV , a distortion of 10% as indicated in Figure 5 .If ELtepas summed with the cathodic half cycle rather thanthe anodic half cycle, the current would tend to increase asEs t , ,was made larger, rather th an decrease.

'c

a0 -

65 -6.0

4.55' I 2 3 4 5 6 7 8 9 IO7 ( msec:

Figure 4. Combined effect of T and p (measure in-stant) on measured current ( T - '12 2 )

COMPARISON WITH SQUARE WAVE POLAROGRAPHYThe abo ve results are similar to those predicted by the theoryof square wave polarograp hy as originally presented by Barker

et al . ( 5 ) provided that A E < 0.5 RTlnF. The equation de-rived by Barke r et al. is

where P = e x p ( E- 1 i 2 ) RT/ nF nd the other symbols have thesam e significance as above. Equ ation 9 was derived assuming

A E5 0 m V2 0 m V

5 m V

0 . 8 80 -0.05-0.1 -0.2 - 0. 5

Figure 5. Effect of square wave distortion ( E s t e p )n measuredcurrent (2)1364 ANALYTICAL CHEMIST RY


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theory of square wave try

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