19 Experimental techniques for electrode kinetics – …...19 Experimental techniques for electrode kinetics – non-stationary methods 19.1 Overview In electrochemical kinetics,

19

Experimental techniques for electrode kinetics– non-stationary methods

19.1 Overview

In electrochemical kinetics, the electrode potential is the most important vari-able that is controlled by the experimentalist, and the current is usually mea-sured as the response. Ideally, one would like to measure current and potentialat constant, well-determined concentrations of the reactants. However, gen-erally the concentrations of the reacting species at the interface are differentfrom those in the bulk, since they are depleted or accumulated in the courseof the reaction. So one must determine the interfacial concentrations. Thereare two principal ways of doing this. In the first class of methods one experi-mental variable, typically either the potential or the current, is kept constantor varied in a simple manner, the other observables are measured, and thesurface concentrations are calculated by solving the transport equations un-der the conditions applied. In the simplest variant the overpotential or thecurrent is stepped from zero to a constant value; the transient of the othervariable is recorded and extrapolated back to the time at which the step wasapplied, when the interfacial concentrations were not yet depleted. This is theclass of techniques that we cover in this chapter.

In the other class of method the transport of the reacting species is en-hanced by convection. If the geometry of the system is sufficiently simple,the mass transport equations can be solved, and the surface concentrationscalculated. They will be treated in the following chapter.

Besides the potential and current steps mentioned above, there are severalother methods by which the system can be perturbed; the more importantones are listed in Tab. 19.1. Usually, at the starting point of the perturbationthe system is in equilibrium. Alternatively, it can be in a stationary state,in which all the fluxes, in particular the current, are constant. If the systemreturns to a stationary or equilibrium state after the perturbation, one speaksof a transient technique; the first four methods in the table are of this kind. Inthis case, we can obtain information about different processes occurring withdifferent velocities by analyzing the response at different time scales.

228 19 Experimental techniques for electrode kinetics – non-stationary methods

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Table 19.1. Overview over the various perturbation methods; V (t) denotes thepotential, I(t) the current, Q(t) the charge, and CA(t) the concentration of anadsorbing species.

Alternatively, the perturbation can be periodic, and after an initial, tran-sitory period the response will be periodic as well. This is true for the last twomethods listed. In this case, the variation of the frequency of the perturbationis the key to studying processes occurring at different velocities. If the am-plitude of the perturbation is small enough, the Butler-Volmer equation canbe linearized and the current is proportional to the potential at the interface.Also in the case of periodically perturbation, the frequency of the response isthe same as that of the perturbation. When the perturbation is large, we speakabout a nonlinear response. In summary, in order to investigate the kineticsof different processes occurring at an electrochemical interface we first haveto determine the potential region at which the process of interest occurs, andsecondly we have to tune the time scale or frequency with the time constantsof the process.

In simple cases the measured current is proportional to the rate of anelectrochemical reaction. The interpretation becomes complicated if severalreactions take place simultaneously. Since the measured current gives onlythe sum of the rate of all charge-transfer reactions, the elucidation of the

19.2 Effect of mass transport and charge transfer on the current 229

reaction mechanism and the measurement of several rate constants becomesan art. A number of tricks can be used, such as complicated potential orcurrent programs, auxiliary electrodes etc., which work for special cases.

There are several good books on the classical electrochemical techniques[136–141]. Here we give a brief outline of the most important methods. Wemostly restrict ourselves to the study of simple reactions, but will considerone example in which the charge-transfer reaction is preceded by a chemicalreaction.

19.2 Effect of mass transport and charge transfer on thecurrent

Generally the current density j that is measured is determined both by therate of the electrochemical reaction and by the transport of the reacting speciesto the interface. Since both processes are in series, the slower of them deter-mines the overall current. From an electrochemist’s point of view there is littleinterest in transport processes as such, and we would like to eliminate theireffect on the data. For this purpose it is convenient to define a few quantities.

If transport were infinitely fast, the concentrations csox and c

sred of nonad-

sorbing reacting species would be the same at the interface as in the bulk.The measured current density would solely be determined by the reaction,and would equal the kinetic current density:

jk = nF (koxc0red − kredc

0ox) (19.1)

where c0 denotes a bulk concentration, and n is the number of electrons trans-

ferred (n = 1 for outer-sphere electron-transfer reactions). In the case of asimple reaction obeying the Butler-Volmer equation jk is given by Eqs. (9.10)and (9.13) with c

s = c0.

The other limiting case is that of an infinitely fast reaction, when the cur-rent is determined by transport only. It is customary to call such a reactionreversible, and denote the corresponding current density, which is determinedby transport alone, as the reversible current density jrev. It is determinedby the transport, usually by diffusion, because right at the electrode surfacetransport of the reacting species is by diffusion alone – convection cannotcarry a species right to the surface because the component of the solutionflow perpendicular to the surface must vanish. One also speaks of a diffusioncurrent density jd in this case. It is obtained from the following considera-tions: If the reaction is infinitely fast, the electrode is in equilibrium with thereacting species at the interface; hence the concentrations c

sox and c

sred are de-

termined solely by Nernst’s equation. The current is obtained by solving thediffusion equation with these surface concentrations as boundary conditions.The diffusion current density is then obtained from:

jd = −ziFDi

dci

dx

x=0

(19.2)


where x is the coordinate in the direction perpendicular to the surface, whichis situated at x = 0, i denotes the reacting species, Di its diffusion coefficient,and zi its charge number. In the special case when the surface concentrationof the reacting species is negligibly small compared with c

0, we speak of adiffusion limited current density jlim; under these conditions every moleculeof species i arriving at the surface is immediately consumed.

Since transport and electrochemical reactions are in series, the slower pro-cess determines the overall current. Hence we can obtain the rate constantsof the reaction only, if the reversible current jrev is not much slower than thekinetic current. This limits the magnitude of the reaction rates that can bemeasured with any given method.

19.3 Potential step

The principle of this method is quite simple: The electrode is kept at theequilibrium potential at times t < 0; at t = 0 a potential step of magnitudeη is applied with the aid of a potentiostat (a device that keeps the potentialconstant at a preset value), and the current transient is recorded. Since thesurface concentrations of the reactants change as the reaction proceeds, thecurrent varies with time, and will generally decrease. Transport to and fromthe electrode is by diffusion. In the case of an infinitely fast reaction (reversiblereaction), i.e. when the potential is stepped to a region that is far removedfrom the equilibrium value, the concentration of the reacting species at theinterface is reduced to zero immediately upon application of the potential step.The gradient of the concentration at the surface decreases with the inverse ofthe square root of time (see Fig. 19.1). Thus, the current is only determinedby transport, it does not depend on the applied potential and its variationwith time is given by the Cottrell equation:

jrev =zFD

1/2c

π1/2t1/2(19.3)

In the case of a simple redox reaction obeying the Butler-Volmer law, thediffusion equation can be solved explicitly, and the transient of the currentdensity j(t) is (see Fig. 19.2, upper panel):

j(t) = jk(η) expλ2t erfc λt

1/2 ≡ jk(η)A(t) (19.4)

where λ is a constant given by:

λ =

j0

F

1

coredD

1/2red

expαe0η

kT+

j0

F

1

cooxD

1/2ox

exp− (1− α)e0η

kT

(19.5)

j0 and α are the exchange current density and the transfer coefficient of theredox reaction, and jk(η) is the kinetic current density defined above.

19.3 Potential step 231

At long times, for λt1/2 1, we can use the asymptotic expansion of

the error function, and j → jrev (see Problem 2). This does not contain anyinformation about the rate constant.

At short times, for λt1/2 1, the function A(t) can be expanded:

A(t) ≈ 1− 2λ

t/π (19.6)

Under these conditions a plot of j versus t1/2 gives a straight line, and the ki-

netic current can be obtained from the intercept (see Fig. 19.2, middle panel).Furthermore, the rate constant may be obtained for different overpotentialsand in consequence the transfer coefficient can be also calculated. If the reac-tion is fast the straight portion can be too short for a reliable determinationof jk; in this case one should obtain estimates for jk and λ from this plot, anduse them in fitting the whole curve to Eq. (19.4).

A more elegant method consists in using the Laplace transform, whichmany mathematics packages contain as a standard option. In general, theLaplace transform f(s) of a function f(t) is defined as:

f(s) = ∞

0e−st

f(t) dt (19.7)

where the variable s has the meaning of a frequency. In the potentiostaticmethod, the Laplace transform of the current density takes on the simpleform:

j(s) =zFc

√D

√s(k +

√sD)

(19.8)

It is convenient to introduce an auxiliary function:

Fig. 19.1. Concentration profile for a reversible, infinitely fast, reaction. Time in-creases from the back towards the front, the distance from the electrode from rightto left.


t

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Fig. 19.2. Current transients after application of a potential step for various re-action rates (upper panel). Current transient plotted vs. t

1/2 (middle panel); thestraight lines give the short-time limit according to Eq. (19.6). Plot of Y (s) ob-tained by Laplace transform (lower panel).

Y (s) =1

j(s)√

s=

1zF√

Dc+

√s

zFkc(19.9)

19.4 Current step 233

Thus, a plot of Y (s) versus the square root of the frequency gives a straightline, and the rate constant can be obtained from the slope (see Fig. 19.2, lowerpanel). The intercept contains only parameters pertaining to mass transport.Note that the slope decreases with the rate k, and the uncertainty becomeslarge for fast reactions.

There are two difficulties with this method. The first one is due to the factthat in reality the potentiostat keeps the potential between the working andthe reference electrode constant; there is an ohmic resistance RΩ between thetip of the Luggin capillary (see Chapter 4) and the working electrode, givingrise to a potential drop IRΩ (I is the current). Since I varies in time, so doesthe potential drop by which η is in error. However, modern potentiostats cancorrect for this to some extent. The second difficulty is more serious. Imme-diately after the potential step the double layer, which acts as a capacitor,is charged, and double layer-charging and the Faradaic current due to thereaction cannot be separated. If the reaction is fast, the surface concentra-tions change appreciably while the double layer is charged, and Eqs. (19.4)and (19.5) no longer hold. This limits the range of rate constants that can bedetermined with this method to k0 ≤ 1 cm s−1.

19.4 Current step

A related technique is the current-step method: The current is zero for t < 0,and then a constant current density j is applied for a certain time, and thetransient of the overpotential η(t) is recorded. The correction for the IRΩ dropis trivial, since I is constant, but the charging of the double layer takes longerthan in the potential step method, and is never complete because η increasescontinuously. The superposition of the charge-transfer reaction and double-layer charging creates rather complex boundary conditions for the diffusionequation; only for the case of a simple redox reaction and the range of smalloverpotentials | η | kT/e0 is the transient fairly simple:

η(t) =kT

e0

1j0

+2B

F

t

π

1/2

−RTC

B

F

2

j (19.10)

with:B =

1

c0oxD

1/2ox

+1

c0redD

1/2red

(19.11)

where C is the double-layer capacity at the equilibrium potential. A plot of η

versus t1/2 does not give the exchange current density directly by extrapola-

tion; the double-layer capacity must be determined separately.These equations cannot be used at higher overpotentials | η |≥ kT/e0. If

the reaction is not too fast, a simple extrapolation by eye can be used. Thepotential transient then shows a steeply rising portion dominated by double-layer charging followed by a linear region where practically all the current is


t / ms1 2 3

50

100

!V

m /

Fig. 19.3. Potential transient for a current step.

due to the reaction (see Fig. 19.4). Extrapolation of the linear part to t = 0gives a good estimate for the corresponding overpotential.

If the reaction is too fast for this procedure, a double-pulse method canbe used: The current pulse is preceded by a short but high pulse which isdesigned to charge the double layer. The height of the pulse is adjusted insuch a way that the transient η(t) is horizontal at the beginning of the secondpulse, and this portion is then extrapolated to t = 0. This method is onlyapproximate, and adjusting the height of the first pulse is tedious, but it doesextend the range of application to faster reactions. Even so the current pulsemethod is limited to reactions with k0 ≤ 1 cm s−1 just like the potential stepmethod.

19.5 Coulostatic pulses

In the sixties Delahay [142] and Reinmuth [143] developed the idea to measurethe rates of electrochemical reactions by charging the double layer with a veryshort pulse. The rate constant is determined by analyzing the subsequentrelaxation of the potential to the equilibrium conditions. Although this is anexcellent method to measure fast reactions, it is underused.

The experimental setup is very simple and inexpensive – see Fig. 19.4. Acoulostat can be home-made and consists of a condenser, a current supplyand a fast relay. A condenser with a capacity Cc much smaller than that ofthe double layer Cd injects an amount of charge Q0 into the cell during a veryshort time. If the time constant for discharging the condenser is much shorterthan that for the double layer, τc = RΩCc τk = RkCd, no leaking of chargethrough the charge transfer resistance Rk occurs during the pulse. Thus, atthe end of the pulse the system is at open circuit and the charge accumulatedat the double layer is Q0. This is an advantage in comparison with othermethods since no compensation for the electrolyte resistance is necessary.

19.5 Coulostatic pulses 235

Cell

relay

C.E.

W.E.

Cd

Rk

Cc Vc

i2

i1 R

Fig. 19.4. Basic setup for the coulostatic method..

Thus at t = 0 the overpotential at the interface is η0 = Q0/Cd. When aredox couple is present (Ox/Red), the double layer subsequently dischargesthrough the Faradaic resistance Rk (see Fig. 19.5) and the potential decay isrecorded. In the absence of mass transport limitations, and if the perturbationis sufficiently small such that the Butler-Volmer equation can be linearized(see Eq. (9.15)) the transient η(t) decays exponentially:

η(t) = η0 exp(−t/τk) (19.12)

The exchange current, and hence the rate constant, can be calculated fromRk according to:

j0 =RT

zFRk(19.13)

Since the injected charge is known, the value of the double-layer capacity canbe obtained from the initial overpotential η0.

Equation (19.12) is only valid when mass transport plays no role, which isalways true at very short times. Taking into account the boundary conditionsand solving the Ficks laws the concentration profiles at the interface can beobtained. Figure 19.6 shows the results for different times after the applicationof the coulostatic pulse. We consider that a charge pulse Q0 has been applied.The system attempts to recover the equilibrium conditions and an oxidationor reduction process starts. When the double layer begins to discharge theconcentrations cox of the oxidized or cred of the reduced species at the interfacegradually change. However, the profile attains a maximum (or minimum) asa consequence of the depletion of the charge. This extremum shifts to longertimes and becomes broader at larger distances from the interface. Then theperturbation propagates to the bulk and becomes attenuated. The more exactexpression for the transient η(t) considering the diffusion processes is now:


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Fig. 19.5. Potential transients after application of a coulostatic pulse for variousreaction rates (upper panel). Logarithmic plot (middle panel); the straight linesgive the short-time limit according to Eq. (19.12). Plot of Y (s) obtained by Laplacetransform (lower panel).

η(t) = η01

b− a

b exp(a2

t)erfc(at1/2)− a exp(b2

t)erfc(bt1/2)

(19.14)

where:

a =τ

1/2d + (τd − 4τk)1/2

2τk, b =

τ1/2d − (τd − 4τk)1/2

2τk(19.15)

19.5 Coulostatic pulses 237

Fig. 19.6. Concentration profile after application of a coulostatic pulse.

and:

τ1/2d =

RTCd

n2F 2

1

D1/2ox Cox

+1

D1/2red Cred

(19.16)

Figure 19.5a shows the normalized transients η(t)/η0 for three different rateconstants. For comparison, the response for an infinitly fast reaction controlledby mass transport is also shown. The limitations of the analysis using Eq.(19.12) are shown in Fig. 19.5b. A deviation from linearity is observed evenfor the slowest reaction at times as short as 10 µs.

Just as in the potential step method, a more convenient analysis can beperformed in the frequency domain. The Laplace transform η(s) of the over-potential obeys a much simpler equation. We define a function Y (s) of thefrequency s which correlates with the kinetic and mass transport parametersthrough:

Y (s) =1

(Q0/η(s)− Cd) s1/2= Rks

1/2 +τ

1/2d

Cd(19.17)

The evaluation of the rate constant can be done by plotting the functionY (s) against s

1/2, (see Figure 19.5c), which results in a straight line withRk as slope and the diffusion parameter τ

1/2d divided by the double layer

capacity as intercept. In this way the evaluation of the kinetic parametersis independent of the knowledge of the diffusion parameters. This would nothave been the case if the data had been fitted directly with Eq. (19.14). Thecoulostatic method has been successfully used to determine rate constant offast reactions and values of the same order of magnitude as those obtained byturbulent pipe flow (see next chapter) have been obtained [144].


19.6 Impedance spectroscopy

An alternative strategy to investigate electrochemical reactions is directly towork directly in the frequency domain. In impedance spectroscopy a sinu-soidally varying potential with a small amplitude is applied to the interface,and the resulting response of the current measured. It is convenient to use acomplex notation, and write the applied signal in the form:

V (t) = V0eiωt (19.18)

where it is understood that the real part of this equation describes the physicalprocess. When the amplitude V0 is sufficiently small, V0 kT/e0, the responseof the interface is linear, and the current I takes the form:

I(t) = I0eiωt (19.19)

where the amplitude I0 of the current is generally complex (i.e., the currentresponse has a phase shift denoted by −ϕ):

I0 =| I0 | e−iϕ (19.20)

The impedance of the system is the ratio:

Z = V0/I0 =| Z | eiϕ (19.21)

Typically, the frequency ω of the modulation is varied over a considerablerange, and an impedance spectrum Z(ω) recorded. Various electrode processesmake different contributions to the total impedance. In many cases it is use-ful to draw an equivalent circuit consisting of a number of simple elementslike resistors and capacitors, arranged in parallel and in series. However, incomplicated systems more than one equivalent circuit with the same overallimpedance may exist, and the interpretation becomes difficult.

We consider a simple redox reaction obeying the Butler-Volmer equation.At small overpotentials, the charge-transfer impedance is:

Zk =RT

Fj0(19.22)

Double-layer charging gives rise to an impedance:

ZC =−i

ωCd(19.23)

These two impedances are in parallel. The resistance RΩ between the workingand the reference electrode is purely ohmic, and is in series with the other two.

At high frequencies diffusion of the reactants to and from the electrode isnot so important, because the currents are small and change sign continuously.Diffusion does, however, contribute significantly at lower frequencies; solving

19.6 Impedance spectroscopy 239

R!

ZC

Zk

ZW

Fig. 19.7. Equivalent circuit for a simple redox reaction.

the diffusion equation with appropriate boundary conditions shows that theresulting impedance takes the form of the Warburg impedance:

ZW =RT

n2F 2

1

credD1/2red

+1

coxD1/2ox

1− i

(2ω)1/2(19.24)

which is in series with Zk, but parallel to ZC . The resulting equivalent circuitis shown in Fig. 19.7, and in this simple case there is no ambiguity about thearrangement of the various elements.

There are several ways to plot the impedance spectrum Z(ω) or Z(ν). Acommon procedure is to plot the absolute value |Z| of the impedance and thephase angle ϕ as a function of the frequency (see Fig. 19.8). In the exampleshown we chose values of: RΩ = 1 Ω, C = 0.2 F m−2, j0 = 10−2 A cm−2,diffusion coefficients of Dox = Dred = 5 × 10−6 cm2 s−1, and concentrationsof 10−2 M for both species. We assumed the presence of a supporting elec-trolyte with a higher concentration so that transport is by diffusion alone.At high frequencies the double-layer impedance ZC is low and short circuitsthe charge-transfer branch. The impedance is then determined by the ohmicresistance RΩ , and the phase angle is almost zero. At frequencies in the rangeof 103 − 104 Hz, most of the current flows through the capacitive branch.Therefore the phase angle is higher in this region. At lower frequencies ZC

is large, and the current flows mostly through the charge-transfer branch.The exchange current density can be evaluated from the data in the rangeof 10 − 103 Hz. At lower frequencies transport is dominant, the current isdetermined by ZW , and the phase angle rises towards 45.

The form of such an impedance spectrum is readily understood if onerealizes that it can be obtained from the current transient for a small potentialstep by Fourier transform. High frequencies correspond to short times, andlow frequencies to long times. Thus double-layer charging dominates at shorttimes and high frequencies, diffusion at long times and low frequencies.

For diagnostic purposes a plot of −Im(Z) versus Re(Z), a Nyquist plot,is useful, since certain processes give characteristic shapes. For example, the


100 101 102 103 104

10

100

lg10 ( / Hz)

lg10

( |Z|

/ )

100 101 102 103 1040

10

20

30

40

/ de

gree

s

lg10 ( / Hz)

Rk+R

R

kinetic diffusion

Fig. 19.8. Absolute value of the impedance and phase angle as a function of thefrequency.

Warburg impedance shows up as a straight line with a slope of 450, a capacitorin parallel with a resistor gives a semicircle (see Problem 1). A simple charge-transfer reaction results in the beginning of a semicircle at high frequencies,which goes over into the Warburg line at low frequencies (see Fig. 19.9). Whenthe charge transfer is fast, only a vestige of the semicircle can be seen.

Impedance spectroscopy is a good all-around method, giving both qualita-tive and quantitative information. It is easier to use than the pulse methods,but is limited to small deviations from equilibrium. Again, the upper limit ofrate constants that can be measured is limited by double-layer charging, andis about the same as for the potential and current pulse methods.

19.7 Cyclic voltammetry 241

19.7 Cyclic voltammetry

When faced with an unknown electrochemical system, or setting out on anew project, one generally starts with cyclic voltammetry. The electrode po-tential is varied cyclically and with a constant rate between two turning points(i.e., the applied potential varies in sawtooth-like fashion), and the current isrecorded. Often the decomposition potentials of the solvent – for water, theonset potentials of hydrogen evolution and oxygen evolution – are chosen asturning points, but others may be chosen for special purposes. Sweep ratesvary between a few mV s−1 up to 103− 104 V s−1, depending on the purposeof the investigation. The resulting current-potential plot, the cyclic voltam-mogram, gives a survey over the processes occurring in the range studied.

As an example, Fig. 19.10 shows a cyclic voltammogram of a polycrys-talline platinum electrode in 1 M H2SO4; it was recorded with a scan rateof 100 mV s−1, a typical rate for the investigation of adsorption processes.Starting from 0 V vs. SHE, we see in the upper part of the curve, the posi-

0 20 40 60 80 1000

20

40

60 ko=0.01 cm s-1

Re (Z) / cm-2

- Im

(Z) /

cm

2

0 20 40 60 80 1000

20

40

60

10 12 14 16 18 2002468

10

- Im

(Z) /

cm

2

Re (Z) / cm-2

ko=0.1 cm s-1

1 Hz

10 kHz

1 Hz

10 kHz

Fig. 19.9. Nyquist plot for a simple redox reaction for two different rate constants.


0

0.5

0.5

0.5 1 1.5

mc Am / j

-2

! / V

Fig. 19.10. Cyclic voltammogram of polycrystalline Pt in 1 M H2SO4 on SHE scale.

tive direction, first the desorption of adsorbed hydrogen; the different peakscorrespond to different facets of single crystal surfaces on the polycrystallinematerial. At about 350 mV all hydrogen is desorbed, and the small residualcurrent is due to double-layer charging. At about 850 mV PtO is formed atthe surface, and oxygen evolution begins only at about 1.6 V, even thoughits thermodynamic equilibrium potential is at 1.23 V; as discussed in Section13.3, its kinetics are slow and complicated. In the reverse sweep the PtO layeris desorbed; there is only a small double-layer region, and the adsorption ofhydrogen begins again at 350 mV.

Polycrystalline metals are a badly defined superposition of various crystalfaces. Actually, the response depends strongly on the surface structure and onthe ions of the electrolyte. Figure 19.11 shows cyclic voltammograms of thethree low index planes of Pt single crystals at a scan rate of 50 mV/s. Theinterpretation of the voltammogram of Pt(111) in sulfuric acid solution hasbeen extensively discussed in the literature. The potential region of hydrogenunderpotential adsorption, 0.07 < φ < 0.3 V, is clearly separated from thepotential region for adsorption/desorption of bisulfate anions, 0.3 < φ <

0.5 V. At a more positive potentials, the OHads formation starts, which ishindered in the presence of adsorbing bisulfate anions. Cycling the electrodepotential into the region where oxygen adsorption and desorption take place,leads to a successive disordering of the single crystal.

The characteristic features of the voltammogram of an ordered Pt(100)surface in sulfuric acid solution are two distinct peaks at 0.3 and 0.4 V, whichmainly correspond to the coupled processes of hydrogen adsorption and bisul-fate anion desorption on the (100) terrace sites and the (100) and (111) stepsites, respectively. The potential region of Hupd is followed, first by the re-versible adsorption of OHads in the potential range 0.7 < φ < 0.85 V, andthen by the irreversible formation of platinum oxide at potentials more posi-tive than 0.9 V.


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Fig. 19.11. Cyclic voltammogram of the three principle single-crystal surface of Pton RHE scale. Data by courtesy of G. Beltramo, Julich, and J. Feliu, Alicante [145]

In the case of Pt(110) surfaces, depending on the heat preparation treat-ment, it is possible to produce two different surface Freconstructions. The 1x1reconstruction can be produced by rapid gas-phase quenching (in argon with3 % hydrogen), and the 1x2 or missing row reconstruction can be produced byslow cooling of the flame annealed crystal. The voltammograms of these twomodifications differ significantly. The voltammetric features include reversiblehydrogen adsorption/desorption peaks in the potential range of 0.05−0.35 V,probably overlapping with bisulfate adsorption/desorption). Two peaks ap-pear in the Pt(110)-(1x2) and are broader than the sharp peak observed inthe Pt(110)-(1x1). These differences are attributed to the openness of themissing row structure.


Figure 19.12 shows voltammograms for gold single crystal electrodes.There is no detectable hydrogen adsorption region; the hydrogen evolutionreaction is kinetically hindered, and sets in with a measurable rate only atpotentials well below the thermodynamic value. There is a much wider double-layer region in which other reactions can be studied without interference. Athigher positive potential we observe the formation of an oxide film, and itsreduction in the negative sweep.

On both Au(111) and Au(100) the behavior is complicated by surface re-construction, which has already be treated in Chapter 16. In particular thereconstruction of Au(100) entails a fairly large change in energy. In weaklyadsorbing electrolytes it is lifted at potentials positive of the pzc, which is ev-idenced by a distinct peak in a slow cyclic voltammogram (see bottom panel).When the potential is scanned back towards negative potentials, the recon-struction is slow, and the corresponding peak is broader and not so high.Though the Au(111) surface is already densely packed, it exhibits a hexago-nal reconstruction in the vacuum. Similarly to Au(100), this reconstruction islifted at sufficiently positive potentials. Since the change in the surface struc-ture is small, it only gives rise to small features in the voltammogram, which

Fig. 19.12. Cyclic voltammograms of Au(111) and Au(100)


-0.8 -0.6 -0.4 -0.2 0.0

-0.8

-0.6

-0.4

-0.2

0.0

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2-4

-2

0

2

4

/ V vs. SHE

j /

A c

m-2

0.1 M H2SO4

Ag(111) Ag(100) Ag(110)

j /

mA

cm

-2

/ V vs. SHE

0.05 M KClO4

Fig. 19.13. Cyclic voltammograms of silver single-crystal surfaces. Data from [81].

fade between the peaks caused by the oxidation and reduction of the surfaceat potentials above 1 V.

Silver electrodes are interesting because of their wide double layer potentialregion and, contrary to gold, they do not show reconstruction processes. Thecyclic voltammograms in Fig. 19.13 illustrate the electrochemical behavior ofthe different low index surface orientations. Similar to gold, the hydrogen evo-lution reaction is shifted to much more negative potentials than on platinum.There is a noticeable variation of the catalytic activity of the different sur-faces. The inset of the Figure shows the details of the double layer regions for adilute, non-adsorbing electrolyte. The minima corresponding to the potentialof zero charge can be easily distinguished.

0 200 400 600-200

2

-2

0

1

-1

! / mV

mc Am / j

-2

equlibrium potential

Fig. 19.14. Cyclic voltammogram of a simple electron transfer reaction.


A simple redox reaction shows a characteristic cyclic voltammogram exhib-ited in Fig. 19.14, which shows the situation after several cycles have alreadybeen performed, so that the original starting point has become irrelevant. Inthis example both the oxidized and the reduced species have the same concen-trations in the bulk. We explain the shape of the curve for the positive sweep.At the lower left corner the potential is negative of the equilibrium potential,and a cathodic current is observed. Since this current has been flowing forsome time, ever since the current became negative in this sweep, the concen-tration of the oxidized species at the surface is considerably lower than in thebulk. In the positive sweep the absolute magnitude of the overpotential, andhence also the cathodic current, become smaller, and the oxidized species isfurther depleted, while the reduced species is enriched. Therefore the currentbecomes zero at a potential below the equilibrium potential, and an anodiccurrent starts to flow. With increasing potential, the rate of the anodic reac-tion becomes faster, and the current increases. However, simultaneously thereduced species is depleted at the surface, so that the current passes througha maximum, and becomes smaller as the surface concentration of the reducedspecies tends to zero. Usually the sweep direction is reversed soon after themaximum has been passed. Mutatis mutandis the same arguments can beused for the negative sweep.

This type of cyclic voltammogram is formed by the interplay of diffu-sion and the charge-transfer reaction; if the sweep rate is fast, double-layercharging also makes a significant contribution to the current. If the exchangecurrent density and the transfer coefficient of the redox reaction, and fur-thermore the double-layer capacity, are known, the shape of the curve canbe calculated numerically by solving the diffusion equation with appropriateboundary conditions. Conversely, these parameters can be determined froman experimental curve by a numerical fitting procedure. However, the curvesare sensitive to the rate of the redox reaction only if the sweep rate is so fastthat the reaction is not transport controlled throughout. For fast reactionsthis typically involves sweep rates of the order of 103 V s−1. The whole pro-cedure is useful only if the required computer programs are readily available.For slow reactions, as they often occur on organic electrochemistry, this is asuitable method, but not for fast reactions.

19.8 Microelectrodes

Spherical diffusion has peculiar properties, which can be utilized to measurefast reaction rates. The diffusion current density of a species i to a sphericalelectrode of radius r0 is given by:

jd = nFDic0i

1

(πDit)1/2+

1r0

(19.25)

19.9 Complementary methods 247

The first term in the large parentheses is the same as that for a planar elec-trode, and it vanishes for t → ∞. The second term is independent of time,so that a steady diffusion current is obtained after an initial period. Eventhough the region near the electrode gets more and more depleted as the re-action proceeds, material is drawn in from an ever-increasing region of space,and these two effects combine to give a constant gradient at the electrodesurface. By making the radius of the electrode sufficiently small, the diffusioncurrent density can be made arbitrarily large, as large as the kinetic currentof any electrochemical reaction, so that any rate constant could, in principle,be measured!

There are, however, obvious limitations. It is not possible to make a verysmall spherical electrode, because the leads that connect it to the circuit mustbe even much smaller lest they disturb the spherical geometry. Small discor ring electrodes are more practicable, and have similar properties, but themathematics becomes involved. Still, numerical and approximate explicit solu-tions for the current due to an electrochemical reaction at such electrodes havebeen obtained, and can be used for the evaluation of experimental data. Inpractice, ring electrodes with a radius of a fraction of a µm can be fabricated,and rate constants of the order of a few cm s−1 be measured by recordingcurrents in the steady state. The rate constants are obtained numerically bycomparing the actual current with the diffusion-limited current.

Even though their fabrication is difficult, microelectrodes have a numberof advantages over other methods:

1. Since measurements can be performed in the steady state, double-layercharging plays no role.

2. Only small amounts of solutions and reactants are required.3. Currents are small, and so is the IR drop between the working and the

reference electrode, so that microelectrodes are particularly useful in so-lutions with a low conductivity.

4. Because of their small size, they can be used in biological systems.

19.9 Complementary methods

The methods described above rely on the measurements of current and po-tential, and provide no direct information about the microscopic structure ofthe interface, though a clever experimentalist may make some inferences. Dur-ing the past 30 years a number of new techniques have been developed thatallow a direct study of the interface. This has led to substantial progress inour understanding of electrochemical systems, and much more is expected inthe future. Thus we have the possibility of applying additional perturbationsto the interface, which provide complementary information to the classicalelectrochemical variables such as potential, current and charge.


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Fig. 19.15. Interaction of light with an electrode surface.

A particular interesting perturbation is light, which implies the presenceof oscillating electromagnetic fields at the interface. Thus, besides the double-layer field we have an additional electrical field, whose direction we can changein a simple way by changing the polarisation angle γ of the light as shownin Fig. 19.15. Changing the intensity of the light, we can investigate bothlinear and non linear phenomena. By changing the wavelength of the light wechange the frequency of the oscillating fields, but in contrast to impedancespectroscopy the range is now within 1014 − 1015 Hz. So, we can follow muchfaster processes with time constants of the order of 1 to 10 femtoseconds. Res-onance phenomena corresponding to processes such as electronic and vibronictransitions can be easily identified. Many of these methods are variants ofspectroscopies familiar from other fields.

All methods in which the electrode surface is investigated as it is, in con-tact with the solution, are called in situ methods. In ex situ methods theelectrode is pulled out of the solution, transferred to a vacuum chamber, andstudied with surface science techniques, in the hope that the structure underinvestigation, such as an adsorbate layer, has remained intact. Ex situ meth-ods should only be trusted if there is independent evidence that the transferinto the vacuum has not changed the electrode surface. They belong to therealm of surface science, and will not be considered here.

19.9 Complementary methods 249

Problems

1. Consider the impedance circuit of Fig. 19.7. Show that for ZW = 0 a Nyquistplot gives a semicircle. If ZW = 0 calculate the frequency region in whichthe semicircle merges into a straight line of unit slope.

2. From Eq. (19.4) derive an asymptotic expression for the current densitywhich is valid in the region λt

1/2 1.3. Consider the generation of a species at a spherical electrode. In polar coor-

dinates the diffusion equation is:

∂c

∂t=

D

r2

∂

∂r

„r2∂c

∂r

«(19.26)

Show that this equation has a steady-state solution, and derive a generalexpression for the concentration and the diffusion current.

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19 Experimental techniques for electrode kinetics – …...19 Experimental techniques for electrode kinetics – non-stationary methods 19.1 Overview In electrochemical kinetics,