1995 - magno - ultramicroelectrodes in kinetic investigations supported by simulation-review

8/12/2019 1995 - MAGNO - Ultramicroelectrodes in Kinetic Investigations Supported by Simulation-Review

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ELSEVIER Analytica Chimica Acta 30.5 199.5)96-105

ANALYTICACHIMICCT

Ultramicroelectrodes in kinetic investigations supportedsimulation. A review with some additional examples

Franc0 Magno *, Irma Lavagnini

b

Diparti mento di Chimica Inorganica, Metallor ganica ed Anali tica, Universitir di Padova, Via Marzolo 1, 35131 Padova, I taly

Received 7 June 1994; revised manuscript received 30 August 1994

AbstractThe potentialities of coupling ultramicroelectrodes and digital simulation techniques in the evaluation of the charge trans-

fer constant, k, of the half-wave potentials, E,,,, in multistep electrochemical reactions, of kinetic rate constants in com-plexation reactions, and of the responses of a ultramicroelectrode-based channel detector, are presented. The advantages ofcarrying out experiments under intermediate mass-transfer conditions are assessed. In the lack of theoretical studies, the digi-tal simulation can offer quite accurate solutions to these complicated questions, which can be faced introducing proper modi-fications into a unique algorithm as long as the geometry of the electrode and the boundary conditions of the diffusion equa-tion describing the electrochemical processes remain unchanged. The characteristic equations of the finite difference meth-ods used in the simulations of the mass-transfer at ultramicroelectrodes are also described.Keywords: Ultramicroelectrodes; Digital simulation; Mass transport; Kinetic hindering; Review

1. IntroductionThe advent of ultramicroelectrodes as tools in

voltammetric investigations marked a noticeable stepin the development of electroanalytical research.Electrodes, characterized by one dimension of the or-der of micrometers, namely less than 25 pm [1,2],offer unique properties [3,4] which can be summa-rized as follows: high mass flux, steady-state current,low ohmic drop, low cell time constant, influence ofthe electrode size on the kinetics of the electrochemi-cal reaction, expansion of the working range of po-tentials and possibility of performing voltammetry inhighly resistive media. Following these advantageousproperties, ultramicroelectrodes have found applica-tion in many fields such as voltammetry under dif-

* Corresponding author.

ferent mass transfer conditions (from planar tosteady-state or pseudo steady-state), liquid chro-matography [5-81, in vivo monitoring [9] capillaryelectrophoresis [lo] and electrochemiluminescence[ill. In particular, since the size of an electrode dic-tates the spatial as well as the time domain that isprobed [12], a specific application of ultramicroelec-trodes is found in the area of kinetic studies on theheterogeneous charge-transfer step [13,14] and on thecoupled homogeneous reactions [l&16]. The digitalsimulation technique [17], on the other hand, is aneffective and sometimes necessary approach in eluci-dating reaction mechanisms so that the coupling ofthe digital simulation to voltammetric data obtainedat ultramicroelectrodes becomes obvious. This paperis intended to show the effectiveness of this couplingby considering equilibrium and non-equilibrium het-erogeneous and homogeneous reactions. In particu-

0003-2670/95/$09.50 0 199.5Elsevier Science B.V. All rights reservedSSDI 0003-2670(94)00431-5


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F. Magno, I. Lavagnini / Analytica Chimica Acta 305 (1995) 96-105 91

lar, the determination of the heterogeneous constantfrom fast voltammetric measurements and from dataobtained for intermediate mass transfer conditions isdiscussed, together with how to extract the thermo-dynamic parameters from a two electron system un-der steady-state conditions, how to rationalize ahost-guest complexation reaction considered as asecond order CE mechanisms, and the response of achannel flow-through amperometric detector. Fi-nally, the basic concepts of the digital simulationmethods used are briefly illustrated.

Among the different geometries proposed in theliterature the disk and the band will be consideredhere since these electrodes are the most common be-cause of their ease of preparation.

2. Basic concepts and applicationsOn the basis of the stepwise nature of the electro-

chemical processes [18], the determination of the het-erogeneous rate constant, k, requires that the rate ofmass transfer of the electroactive material be largerthan that of the heterogeneous charge transfer step.This condition is fulfilled for m > k, where m is themass-transfer coefficient appropriate for a particularelectrochemical technique. The use of an ultramicro-electrode is convenient to achieve a very large masstransfer both in transient and in steady-state mea-surements, that is when the controlling parameter inthe voltammetric experiment is the time or the ra-dius, respectively. In the first case large values of ma (D/t) I [19 201 (D is the diffusion coefficient)are obtained at short times where perturbations due tothe RC characteristics of the experimental devicerepresent a severe drawback.

However owing to the fact that the ohmic drop fora disk ultramicroelectrode is given by [4] AE, = R,i= p(i, + i,)/(4a) a (a* + a*>/~ a a where p isthe resistivity of the solution and a the radius of theelectrode, the use of an ultramicroelectrode allowsvery fast voltammetric experiments limiting the oc-currence of prohibitive perturbations. Under diffu-sive steady-state conditions, m a D/a [19], highmass-transfer rates can be obtained as long as theelectrode radius, a, is sufficiently small. Also forelectrochemical processes with coupled chemical re-actions the two approaches, use of the time or of thedimension of the electrode as controlling parameters,

are possible. In the former case where the experi-mental time defines a time window, t, in which thechemical reaction occurs to a definite extent depend-ing on its characteristic lifetime, t,, (for a first orderreaction the usual dimensionless kinetic parameter ish = kt = t/t, [18]) the advantageous properties of ul-tramicroelectrodes in reducing the noise remainand the treatment of data is that typical of the planardiffusion electrodes. In the latter case the extent of thekinetic effect is dependent on the ratio of the concen-tration change due to chemical reaction and therestoring effect of the mass transfer, that is by thevalue of the dimensionless parameter k/CD/u*> at/t, [15]. The variety of the systems studied, of theexperimental conditions adopted and of the objectspursued points out that a digital simulation approachis in many cases an obvious choice.2.1. Evaluation of k

In the determination of the parameters of thecharge transfer reaction, different experimental con-ditions can be chosen and the digital simulation ap-proach represents a valid support. Data obtained atfast scan rates are distorted by ohmic drops so thatdifferent strategies, on-line [20] or post factum [21-251, must be followed for extracting the correctFaradaic component from the total current, which foran anodic scan is given byi dE( t)tuf=iF+iC.=iF+Cd--- dt (1)where i, and i, indicate the Faradaic and the charg-ing components, respectively, and E(t) is the effec-tive potential applied to the working electrode withE(t) =E(t) -R,i,,,=E,+vt-RR,it,, (2)

Other symbols in Eqs. 1 and 2 have their usualmeaning. Substituting Eq. 2 into Eq. 1 yieldsi Go,tot= i, + CC, - R,C,xIt appears that the Faradaic current is not obtained bythe simple subtraction of the capacitive current UC,of the blank, recorded in the absence of electroac-tive species, from the total current because thevoltammograms at fast scan rates are distorted to avarying degree depending on the change of the totalcurrent and the time constant of the cell. The calcula-tion of the term R,C,di,,,/dt from the experimental


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98 F. Magna, I. Lavagnini /Analytica Chimica Acta 305 (1995) 96-105voltammogram allows the Faradaic current to beproperly evaluated. However, for the evaluation of kthe corrected voltammogram representing i, againstthe effective potential E(t) cannot be treated follow-ing the well known theory reported in the literature,since the potential waveform is no longer triangularin shape [2,26]. Fig. 1A shows the modification of thereal applied potential waveform and Fig. 1B provesthat the Faradaic current, properly extracted, nevercoincides perfectly with the voltammetric curve ob-tained in the absence of an uncompensated resis-tance. Correct procedures can be the convolution ap-proach [21], the global analysis [27] and the digitalsimulation [2,28]. In the context of the digital simula-tion approach, the term Z?,Z,,, in Eq. 2 can be ac-counted for by calculating numerically the real po-tential applied to the working electrode, E(t), for anypotential step in which the potential range is divided.Fig. 2 shows quasi-reversible voltammetric curves(A) free of significant ohmic drop, (B) affected onlyby uncompensated solution resistance R,, (C) bydouble-layer capacitance C,, and (D) by a finite R,C,time constant. The ohmic drop effects in the abovefigures were calculated introducing in a well estab-lished algorithm [29] an iterative procedure 1301which searches for the effective potential E(t) as aroot of the Eq. 4E(t) -E(t) +R;i,(E(t))

+ RCd RCdI+-dtE(r-Ar)=O (4)This equation was derived inserting Eq. 1 in Eq. 2

and approximating the first derivative d E(t)/d twith (E(t - At) - E(t))/At.The Faradaic current i,(E(t)) is calculated by theButler-Volmer equation. Consequently the k valuecan be calculated by fitting simulated curves on ex-perimental responses even if they are affected by RCdistortion.

On decreasing the scan rate the effects due to R,and C, become negligible so that it is worthwhile toexplore the possibility of extracting k values fromdata obtained under mixed spherical/semi-infinitelinear diffusion conditions. At the same time theshape of the voltammograms changes from that char-acterized by potential separated anodic and cathodicpeaks to a sigmoidal one (Fig. 3).

0.6

0.4

02

12 01-.42 -0 2sa-0.4

-06

-0.4 I-I:.:I -0 75 -0.5 -0.25 0 0.25 0.5 0.75

E /voltFig. 1. (A) Potential-time curves relative to the cyclic voltammo-grams in Fig. 1B. (solid line) applied potential; (broken line) ef-fective potential in the presence of an uncompensated solution re-sistance of 55,000 0, and of a double-layer capacitance of 5~lo-* F. (B) Faradaic currents against the effective potential in theabsence (solid line) and presence (broken lined) of ohmic drop ef-fects. Calculations based on: transfer coefficient, 0.5; charge trans-fer constant, 3.3 cm/s; scan rate, 128,700 V/s; electrode radius,2.5 pm; electroactive species concentration, 10 mM, diffusion co-efficient, 8.4 X 10m6 cm*/s.

Since no analytical description of the voltammet-ric signal for intermediate situations has been re-ported until now, the resort to the digital simulation


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F. Magna, I. Lavagnini /Analytica Chimica Acta 305 (1995) 96-105 99

E /Volt E /VoltFig. 2. Simulated cyclic voltammograms versus the applied potential (A) in the absence of ohmic drop, in the presence (B) of only an un-compensated solution resistance of 55,000 0, (C) of only a double layer capacitance of 5 X IO-t2 F, and (D) of a combination of theseproperties. Other simulation parameters as in Fig. 1.for this particular problem is mandatory [14,28,31].Fig. 4 shows calculated dimensionless peak separa-tion values A&, (the dimensionless potential 5 isdefined by &= nF(E - E)/RT) against the dimen-sionless kinetic parameter A = ka/D for a fixedvalue of the dimensionless diffusion parameter p =a( nFu/RTD) I * . On a particular p curve determinedby the experimental values of the parameters a andu, a measured value of A , gives one A value whichin its turn furnishes k via A = ka/D. The k val-

ues calculated by this procedure are larger than thosederived applying the Nicholson method 1321 to dataobtained at scan rates of the order of magnitude ofhundreds V s- r, but agree with those obtained bymore sophisticated techniques [ 191.2.2. Determination of the half -wave potenti als of amultistep process

The digital simulation approach holds its validityalso when the steady-state conditions are operative.


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100 F. Magno, I. Lavagnini /Analytica Chimica Acta 305 (1995) 96-105

nF(E - E)/RTFig. 3. Simulated cyclic voltammograms for a disk electrode un-der different diffusion conditions. Dimensionless diffusion param-eter, (a) 12; (b) 2; (c) 0.01. Other simulation parameters as in Fig.1.

Typically this is true when one considers a multistepcharge transfer process since the analytical relation-ships developed until now [12] treat only a one step

,L 10 20 30 40 50 60li

Fig. 4. Calculated dimensionless peak separation values A ver-sus the dimensionless kinetic parameters A for a dimensionlessdiffusion parameter p of 12.

charge transfer which involves n 2 1 electrons. Pol-cyn and Shain [33] and later Ryan [34] have demon-strated that, in cyclic voltammetric experiments, a re-versible two-step charge transfer process can be re-duced to a reversible two-electron one-step processwhen the difference between the standard potentialsassumes appropriate values. Otherwise the shapechanges from reversible to apparently quasi-reversi-ble and in the presence of a kinetic hindering the dis-proportionation-comproportionation reaction mayplay a relevant role. Fig. 5 compares linear sweep (A>and steady-state (B) voltammetric curves relative tothree different electrochemical systems. It features aquite lower resolution of the steady-state voltamme-try but its superiority in defining the overall numberof electrons transferred since the height of the curvedoes not change for different values of the thermody-namic and kinetic parameters approaching in any casethe limiting value ifisk = 8FDcba. It can be re-marked that the curves shown in Fig. 5A and B arecalculated with the same algorithm, inserting onlydifferent values of the dimensionless p parameter.2.3. Study of coupled chemical reactions.

The application of ultramicroelectrodes to studiesof homogeneous reactions coupled to the chargetransfer step has been thoroughly discussed. In par-ticular, the voltammetric responses relative to CE,EC, ECE, EC mechanisms under steady-state condi-tions for the spherical and the hemispherical geome-tries have been mathematically described [15,35,36].Apart from the problem of the equivalence amongsphere, hemisphere and disk, which is usually ac-counted for by using a simple transformation coeffi-cient [37], the digital simulation is particularly effec-tive when treating higher order kinetics or intermedi-ate mass transfer conditions. However the complex-ity of an accurate description of the edge effect in thedigital simulation model results in only few papersdealing with complicated electrochemical processes[16,24,31,38,39].In the context of a research on the application ofdisk ultramicroelectrodes in studies of the complexa-tion between electroinactive-host ( @cyclodextrin)electroactive-guest (ferrocene carboxylic acid)species, we simulated the voltammetric responsesrelative to a second order CE mechanism under dif-


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F. M agno, I, Lauagnini/A nalyt ica Chimi ca Acta 305 1995) 96-105 101

0I,,,,L ,,,,,,,,,,,,,,,,I,,,I

10 -5 0 5 ,o 15 23 25 JonF E - E)/ RT

Fig. 5. (A) Linear sweep and (B) steady-state voltammetric curvesfor a reversible one-electron two-step charge transfer. (a) Ef = 200mV and Ei = 0 mV, (b) EF = 0 mV and Ef = 0 mV, (c) Ef = 0mV and Et = 200 mV.ferent mass-transfer and kinetic conditions [40]. Thehost (L&guest (A) inclusion system considered re-duces to the following reaction sequenceA+L-----ALK,, =

IA


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102 F. M agna, I. Lavagnini /Anal yt ica Chimi ca Acta 305 1995) 96-105

Table 2Voltammetric data relative to acid dissociation equilibriaAcid Limiting currents normalized to the value of the

reduction of free hydrogen ionsa bCH,COOH 0.240 d 0.231 e 0.235

H,PO; 0.094 d 0.093 f 0.120 fa Experimental values recorded with a disk electrode of radius 12.5pm, and an acidic analytical concentration of 1X 10m3 M.b Values simulated with a kinetic model. Values calculated via Eq. 5.d From Ref. 41. Input parameters: k, = 7.56 X lo5 s-; k_ 1= 4.2 X 10 M-s-; D, = 0.125.f Input parameters: k, = 2.5 X lo4 s-; k_ 1 = 6.2X 10 M- s-;D, = 0.102.

the ratios of the limiting currents in the presence, i,,nd absence, z,,~, of the ligand species on the bulkligand concentration over the entire range of the con-centrations explored:

11I++ lLlb

-=I,,,,. (Lib (5)0

A related second example, currently in progress, isthe study of the reduction of the hydrogen ions com-ing from weak acids. Considering a weak acid as acomplex between the hydrogen ion and the conju-gated anion, the limiting currents recorded at ultra-microelectrodes must depend on the ratio between thediffusion coefficients of hydrogen ions and the weakacid, the value of the equilibrium constant, the valueof the rate constants and the ratio of the analytical HAand A- concentrations. The data reported in Table 2show that at a concentration level of 1 X 10m3 M,acetic acid gives an equilibrium current whilstH,PO, produces a kinetically controlled current. Infact for the acetic acid the experimental limiting cur-rent ratio practically coincides with the value calcu-lated simulating the kinetic model inserting the re-ported kinetic constants [35] and with that calculatedby Eq. 5. On the contrary, H,POi gives an experi-mental current ratio quite inferior to the equilibriumresponse and coincident with that derived simulatingthe kinetic model [42]. Therefore the coupling of thedigital simulation with steady-state current measure-ments can furnish, in principle, the parameters of dif-ferent acids in different media. The onset of a kinetic

control of the complex dissociation can be experi-mentally tested by noting that the limiting currentdepends not only on the ratio of the host to the guestspecies but also on the radius of the microelectrodeused [15].2.4. Response of a channel f low-throughamperometric detector

Passing from quiescent solutions to hydrodynami-cally forced mass-transfer conditions, the resort to anumerical approach is even more advisable. Recentstudies have dealt with ultramicroelectrode-basedchannel detectors to support theoretically the use ofthese tools in flow injection and chromatographicanalysis. The main objects of these investigations arethe prediction of the effect of the size of the elec-trode, disk and band, and of the solution flow rate onthe mass-transport limited current [5-71, the achieve-ment of a steady-state response during scanning ex-periments and the extension of the use to normalphase chromatography [8]. Following the scheme re-ported in Fig. 6, the general two-dimensional diffu-sion-convection equation reduces toac r a c a cl ac-= Dat I - -ax2 az2 1 - K-g

assuming V, depending only on z via a parabolic re-A

Fig. 6. (A) Real three-dimensional model of the flow throughchannel and (B) two-dimensional model used with the parabolicvelocity profile of the solution.


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F. M agna, I. Lat iagnini/A nalyt ica Chimi ca Acta 305 1995) 96-105 103

001 I I I-1 0 0.5 0.0 0.5 1 o

6

Fig. 7. Dimensionless concentration profiles over the whole space6 = tanh(2.x) at a flow rate of (1) 0.005 ml/min, (2) 0.5 ml/min,(3) 2.0 ml/min. The band electrode lies between 6 = -0.5 and6 = 0.5. Calculations based on channel thickness, 50 pm; channelwidth, 0.4 cm; band electrode length, 5 pm; electroactive speciesdiffusion coefficient, 5 X lo- cm2/s.lationship, and V, equal to zero. By the digital simu-lation technique it is possible to calculate the concen-tration profiles of the analyte over the electrode (Fig.7) and to assess edge effects also under laminar flowconditions. At low flow rate the concentration profiledevelops in the bulk of the solution from the edgesof the band, while at high flow rate the convectivemass-transfer cancels the depletion of the electroac-tive species due to the electrochemical reaction. Thepresence of the edge effects causes an enhancementof the current values in respect to those predicted bythe classical Weber and Purdy equation [43] whichrequires a dependence of the current on the volumeflow rate value raised to l/3.2.5. Simulation algorithms

The essence of any digital simulation approachconsists of a finite difference approximation of thepartial differential equation describing the spatialmass-transfer towards an electrode. The methodsproposed for simulating voltammetric responses onultramicroelectrodes are the implicit alternating di-rection (ADI) [31], the Crank-Nicholson (CN) [25],the fast quasi-explicit finite difference (FQEFD) [44]and the Hopscotch methods [45]. All these methods,unconditionally stable and potentially very accurate,

combine explicit and implicit finite difference ap-proximation and finally result in implicit (ADI andCN), quasi explicit (FQEDF) or explicit (Hopscotch)methods.

In particular, for a two-dimensional diffusionmodel, in the CN scheme the concentration of eachspecies at the time n + 1 in the i,j node of the spacegrid is given by

DAt

s2c,y l + sx2cyj+ sy2c,fl; + sy2cyjx .2

where S,Cltj = C, + ,j - 2c:, + c:\ ,.j; s;c;, =c:,+ I - 2C:f, + Czj_ i; and Ax = Ay.

The determination of the concentration values attime 12+ 1 needs the solution of coupled linear equa-tions. The mathematical problems arising in this ap-proach are simplified when a modified CN formal-ism is used (i.e., the ADI method). In this algorithmeach time step is subdivided into two equal steps inwhich a different dimension is treated implicitly. Thediffusion equation modifies into

and1 DAtC n+l =C+i + _-.I ,I 1s2c$ + s,c:,+]2 Ax2 x

The FQEFD method is an improvement of theclassical explicit finite difference (EFD) method sinceit is more stable. The finite difference equation comesfrom that of the EFD methodcn+l - fyj =1.J ~[c~ l,j-2c:lj c:~l,J]

where Clyj is substituted by the average of the previ-ous and of the new concentration values, that is C:j= cc:; + C&Y )/2.

In the Hopscotch method the finite differenceequation assumes the form


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104 F. Magna, I. Lauagnin i/Analytica Chimica Acta 305 (1995) 96-105

where 19:~= 1 or 0 depending on n + i + j is odd or [49] able to treat different mass transfer and kineticeven, respectively. problems.

Taking advantage from the definition of the func-tion Oyj it is possible to obtain the operative equa-tion Ci,j+ = 2C[f1 - Clj [46] and to calculate thenew concentrations explicitly. Acknowledgements

In the study of the ohmic drop effects we used thewell-known EFD method [29] since the very largevalue of the scan rate used in conjunction with an ul-tramicroelectrode makes the diffusion condition pla-nar in character. The employment of the two-dimen-sional model in this particular case would have beenunnecessary time-consuming. The Hopscotch algo-rithm was used in the other cases. In particular, ourstarting point was the study of Michael et al. [39]where the diffusion equation was rewritten in a suit-able conformal space. Refinements were introducedconcerning the a priori definitions of the thickness ofthe diffusion layer and of the reaction layer. The for-mer was expressed in terms of the dimensionlessmass-transfer parameter p

The financial support of the National Council ofResearch (CNR) and the Ministry of University andScientific Research (MURST) is acknowledged.

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Elb171

[8[91

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3. ConclusionsThe examples reported are a fair illustration of the

effectiveness of the coupling of a digital simulationapproach with voltammetric data recorded at ultrami-croelectrodes. The main feature of the algorithm usedby us is its quite wide applicability to describe dif-ferent mass transfer conditions only by changing aninput parameter. In our opinion this approach canrepresent a valid indication to the development of ageneralized simulator [48] and of an expert system

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1995 - magno - ultramicroelectrodes in kinetic investigations supported by simulation-review

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