[Comprehensive Chemical Kinetics] Electrode Kinetics: Principles and Methodology Volume 26 || Chapter 5 Hydrodynamic Electrodes

Chapter 5

Hydrodynamic Electrodes

CHRISTOPHER M.A. BRETT and ANA MARIA C.F. OLIVEIRA BRETT

1. Introduction

Many voltammetric techniques involve measuring the system response to a perturbation in the current or potential applied to the electrode; this measurement has to be done usually over a limited time scale in order to avoid natural convection and other bulk solution effects which lead to irreproducibility. Considerable simplicity is introduced experimentally if measurements can be made under steady-state or quasi-steady-state conditions without random contributions from natural convection. This is possible with hydrodynamic electrodes which, whilst not solving all an electrochemist’s problems, go a long way towards doing so. Material transport is by forced convection due to movement of the electrode or movement of the solution; since the rate of transport is higher than in a stagnant system, so is the sensitivity. Once a mathematical description of the mass transfer has been obtained, evaluation of kinetic and mechanistic parameters can be carried out. By changing the potential of any electrode, we can alter the rate of electron transfer; for hydrodynamic electrodes we can also do so by altering the rate of convection. This latter control variable makes possible the determination of rate constants over a wide range of orders of magnitude.

The first hydrodynamic electrode to be invented was the dropping mercury electrode [l] . It has a cyclic operation and can thus be considered only as quasi-steady-state; its hydrodynamic character derives from drop growth. The principal advantage of a dropping electrode is that a fresh electrode surface is constantly exposed to the solution; however, there are few electrode materials available and mathematical solution of the mass transport to the drop surface is complicated by the fact that the surface is expanding.

The development of solid hydrodynamic electrodes, which have the advantage of fixed area and a wide range of available electrode materials, occurred rather later. This was due mainly to the lack of a theoretical description of the mass transport. Levich’s work on mass transfer to electrodes, which was largely unknown to the non-Russian-speaking world except for the occasional indication, e.g. ref. 2, only became widely available with the publication in 1962 of the English translation of Physicochemical Hydrodynamics [ 31. This dealt with many

References p p . 434-441

356

electrochemical and engineering problems and catalysed the development of hydrodynamic electrodes. An important contribution was also made by boundary layer theory [4].

The most well-known and widely used solid hydrodynamic electrode is the rotating disc electrode (RDE) and its double electrode analogue, the rotating ring-disc electrode (RRDE). Other commonly used electrodes rely on movement of the solution: the tubular (channel), wall-jet (impinging jet) electrodes, and other configurations. A number of mono- graphs and reviews have appeared on hydrodynamic electrodes, of which the most representative are given in refs. 5-14.

The goal of this chapter is to describe the application of hydrodynamic electrodes to. the study of electrode kinetics and the kinetics of electrode and coupled homogeneous reactions. In order to do this, it is important to describe first the mass transport and how to fulfil experimentally the conditions described by the mass transport equations, i.e. electrode construction and operation.

Combination of hydrodynamic electrodes and non-steady-state techniques, though more complex to analyse theoretically, is very powerful in its application with increased sensitivity. These more recent developments and their applications to electrochemical kinetics will be discussed.

2. Mass transport

2.1 INTRODUCTION

In order to be able to describe quantitatively the flux of electrons at an electrode, we first have to know the flux of material reaching the electrode which can result from convection, diffusion, and migration processes. In general, we may write for a species j

j = cj v - D,Vcj + u ~ c ~ V @ convection diffusion migration

where u j is the mobility of ion j , cj its concentration and @ represents the electric field. For our purposes, we will assume the presence of an excess of inert supporting electrolyte such that the migration term can be neglected (see Sect. 2.3.8).

The convection term is obtained by analysis of the velocity profile of the system, which is derived from two equations. The first of these is the equation of continuity

- ap = - V(pv) at

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which, assuming constant density in time and space, reduces to

v v = 0 (3)

The second is the law of conservation of momentum which, for a fluid of constant density and viscosity, is the Navier-Stokes equation

av 1 - + v v v = - V P + v P v + g a t P

(4)

where P is the pressure difference across the system. In nearly all the cases we consider, &/at is equal to zero (however, see Sect. 6). Solution of eqns. (2) and (4) thus enables us to solve the general convective- diffusion equation

ac a t - = DV2c - VVC

Clearly, the solution of this equation at forced-convection electrodes will depend on whether the fluid flow is laminar, in the transition regime, or turbulent. Since virtually all kinetic investigations have been performed in the laminar flow region, no further mention will be made of turbulent flow. The reader interested in mass transport under turbulent flow is recommended to consult refs. 14 and 15.

Before looking at specific systems, it is helpful to consider some of the terms and dimensionless groups which are commonly used to describe mass transport.

I t should also be pointed out that the problem of mass transfer is analogous to that of heat transfer. Therefore, with the appropriate transformations, results for one can be used for the other and vice versa.

2.2 USEFUL CONCEPTS IN THE SOLUTION OF MASS TRANSPORT EQUATIONS

2.2.1 Diffusion layer

Let us suppose that there is a layer of solution close to the electrode within which all concentration changes due to electrode reaction occur and that transport within this layer is entirely by diffusion. For hydrodynamic electrodes, this approximation is reasonable since the diffusion layer is very thin owing to the effects of forced convection. Following Nernst, we assume that the concentration varies linearly within the diffusion layer such that the flux, j , at the electrode is

where 6, is the thickness of the Nernst diffusion layer. In practice, we use the calculated value of ac/az at the electrode surface. For the


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diffusion-limited current we have

hf, is known as the mass transfer coefficient and has the dimensions of a rate constant. Whilst this expression is not strictly correct, it is very useful for the visualisation of an approximate diffusion-layer thickness.

2.2.2 Hydrodynamic boundary layer

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, I j H . The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance; this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 6H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7 ] takes a figure of 10% (see below). I t has been shown that

Thus, the assumption of no convection within the diffusion layer is not unreasonable for normal values of D and v.

2.2.3 Dimensionless groups

In the solution of mass transport problems, several dimensionless groups are used in order to reduce the number of variables. Diffusion layer thicknesses etc. are expressed in much of the literature in terms of these dimensionless variables.

(i) The Reynolds number, Re

V l Re = - V

(9)

where u is a characteristic velocity, 1 a characteristic length, and Y the kinematic viscosity. Re therefore describes the fluid flow. Below a certain value, Recrit, the flow is laminar and above it the flow is turbulent with a transition regime around Recrit.

(ii) The Schmidt number, Sc

V s c = - D

is the ratio between the kinematic viscosity and the diffusion coefficient. Since v is primarily a characteristic of the solvent and D of

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the electroactive species, Sc is thus determined purely by the physical properties of the solution. For example, Sc x 1000 for aqueous solutions.

(iii) The Peclet number, Pe

(11) vl D

Pe = - = R e - S c

represents the relative contributions of convective and diffusion transport. Its use is thus particularly pertinent for a treatment of hydrodynamic electrodes. In aqueous solution and outside the diffusion layer, Pe is much larger than unity, even for low flow velocities.

(iv) The Sherwood number, Sh

or

Comparison with eq. (6) shows that

1 Sh = - 6,

and comparison with eq. (7) shows that Sh is proportional to the mass transfer coefficient, k .

2.3 APPLICATION OF THE MASS TRANSPORT EQUATIONS TO SPECIFIC SYSTEMS

2.3.1 General

Table 1 shows the particular forms of the convective diffusion equation for different geometries. It is fortunate that, due to the symmetrical nature of hydrodynamic electrodes, some of these terms may be neglected. Also, the major part of investigations conducted are under conditions of steady-state flow where &/at = 0. The exception to this is, of course, the cyclic operation of the DME.

We first consider the case of a rotating disc electrode, where mass transfer is particularly simple, and then go on to consider other hydrodynamic electrodes where the situation is more complex. A summary of limiting currents calculated for various electrode geometries will be found in Table 3 (p. 384).

Double electrodes are particularly useful in kinetic studies. Inter- mediates produced on the generator (upstream) electrode are transported to the downstream electrode where they react further. This is useful for the study of short-lived species, the quantity reaching the second


TABLE 1

The convective-diffusion equationa

- = D v 2 c - vvc

in various coordinate systems;

ac a t

w (5, 0

Diffusion Convection

Cartesian

Cylindrical polar

Spherical polar

aSee Fig. 1 for explanation of coordinates.

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electrode depending on any homogeneous reactions or decomposition occurring in solution. The most widely used of these is the RRDE. It is a prerequisite to know what fraction of species which reacts at the upstream electrode reaches the downstream electrode without kinetic complications. This, for &/at = 0, is the steady-state collection efficiency, N o , and will be calculated for common electrode geometries.

It will be seen that the form of the expression for N o is quite general for double electrodes.

2.3.2 Rotating electrodes

In this section, we consider mass transport-controlled currents to disc and concentric ring electrodes on a planar spinning disc surface. For other less common rotating electrodes, e.g. rotating hemisphere, see Table 3.

(a ) Rotating disc and rotating ring electrodes

The problem of laminar fluid flow to a rotating disc has been amply discussed in the literature [7, 101. We may describe the velocity components as [17,18] (see Fig. 1)

u6 = r w G ( y )

U, = - ( c J u ) ” ~ H(y)

u, = r w F ( y )

where y is a dimensionless distance from the electrode surface

with CJ the rotation speed in rad s-l . The variation of the flow functions F, G , and H with distance and resultant streamlines are shown in Fig. 2. The particular form of the velocity components satisfies the Navier - Stokes equation and the equation of continuity. Note that u, is independent of the radial coordinate.

We are interested primarily in the convection pattern close to the electrode surface in order to calculate the flux of electrons. Following Levich [19] , we say

uz,z-*Q - cz2

U & , O 0 (16)

u , , , , ~ Crz

where C = 0.510 w3l2 u - ~ ’ ~ . The convective-diffusion equation in cylindrical polar coordinates in


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Fig. 1. Coordinate systems for common electrode geometries. (a) Cylindrica symmetry: (i) ring-disc electrodes, (ii) tubular electrodes; (b ) Cartesian symmetry channel electrodes; (c) spherical symmetry: dropping mercury electrode.

the steadv state is

where we neglect radial diffusion as being negligible compared with radia convection. We are now in a position to calculate the diffusion-limitec current at a rotating disc or rotating ring electrode. The appropriatl boundary conditions are

z + m C’C, bulk concentration r = O c = o

z = o c = o

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r

Fig. 2. The rotating disc. (a) Flow functions; (b) schematic streamlines.

If we define the dimensionless variables

c - c, y = -

C,

then it can be easily shown that eqn. (17) is transformed into

ay a2r

with boundary conditions

x = o y = l

where p = 1 for a disc electrode, p = 3 for a ring electrode and r l , r2 and r 3 are defined as in Fig. 1.

The limiting current at a disc or ring electrode is given by rP

iL = 2nnFD r(g),, dr rp-1

The integral (22), after undergoing Laplace transformation [ 201 with respect to E p , becomes


364

which is the Airy equation [21] whose solution with the boundary conditions (21) is, after the inversion

A i ' ( 0 ) A i ( 0 ) r ( 3 ) ti l I3

The Nernst diffusion layer thickness as defined in eqn. (6) is given by c , /( ac/az), and is

When p = 1 (disc electrode), 6, is independent of r . This shows us immediately that the rotating disc electrode is a uniformly accessible surface, whereas the rotating ring electrode is not. Substitution of eqn. (24) into eqn. (22) and evaluation gives the well-known

iL = 0.62048 .rrnFD213 u112 v-'16 cm (r: - r:-' )2/3 (26) For a rotating disc electrode

iD,L = 0.62048 . r r r~FD~/~ u1I2 c , r : (26a)

and for a rotating ring electrode

i R V L = 0.62048 . r r r~FD~/~ w1/2 v-lI6 c, ( r ; - r3 2 ) 2/3 (26b) In the case of a disc electrode, because of the uniform accessibility, we

is directly proportional to the electrode area. The ratio of limiting see currents at disc and ring electrodes is

with

which is purely a function of geometric parameters, an important simplifying factor in many applications.

There are other equivalent ways of solving eqn. (17): for example, by double integration [3] . However, as will be shown in following sections, the approach outlined above is particularly valuable as it may be applied directly to other hydrodynamic electrodes.

This derivation makes a number of assumptions. Firstly, we assume that there is no disruption to the laminar flow pattern due to a finite disc surface, finite cell size, or eccentricity in disc rotation. To what extent design factors affect measured currents will be discussed further in the section on electrode construction. It is sufficient at this point to say that the criteria for negligible disruption can be met.

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A second assumption made is that radial diffusion is unimportant relative t o radial convection. This problem has been investigated by Smyrl and Newman [22] who show that inclusion of the radial diffusion term only increases the limiting current by 0.12%. It can thus be neglected in practice.

More serious, when it is desired to extract accurate information, are the approximations made in the velocity components [eqn. (16)] . These approximations are valid only at high Schmidt numbers, corresponding to the diffusion layer being much thinner than the hydrodynamic boundary layer (about 5% at Sc = lo3) . In reality, u, and u, are a polynomial series where we have taken just the first terms. I t was shown by Gregory and Riddiford [23] that, even at Sc = l o 3 , this gives a significant error with correspondingIy higher errors at lower Schmidt numbers. They proposed replacing the numerical coefficient of 0.62048 in eqn. (26) by the empirical equation

0.554 0.8934 + 0.316(D/~)O.’~

This deviates by less than 0.5% from the “exact” numerical solution at Sc = 100 [ 231 . Newman [ 241 proposed another expression

0.62048 1 + 0.2980 S C - ” ~ + 0.14514 S C - ~ ’ ~

which for Sc > 100, gives a maximum deviation of only 0.1% compared with the numerical solution. In the accurate measurement of diffusion currents by means of transport-limited currents, the Newman formula is generally used. Using eqn. (30), we may write [25]

D = v [(J-’ - 0.122939)”2 - 0.14901 -3 (31) where

0.62048 nF w ‘ I2 v ’ ’ ~ c ,

iL

J - 1 =

Alternatively, D may be evaluated graphically [ 131 .

( b ) Rotating ring-disc electrode

The rotating ring-disc electrode (RRDE) is probably the most well- known and widely used double electrode. It was invented by Frumkin and Nekrasov [26] in 1959. The ring is concentric with the disc with an insulating gap between them. An approximate solution for the steady- state collection efficiency N o was derived by Ivanov and Levich [ 271 . An exact analytical solution, making the assumption that radial diffusion can be neglected with respect to radial convection, was obtained by Albery and Bruckenstein [28, 291. We follow a similar, but simplified, argument below.


366

We consider the reactions

Disc A -+ B

Ring B + A

such that

JR

ID No =- -

The evaluation of N o involves solving the equation

ay a2r x a g = i Q

(33)

for the zones of the disc electrode (p = l), gap (p = 2), and ring electrode @ = 3). The definitions of t p and x are as for the RDE [eqns. 19(a) and (b)l but

(34) C

y = - CO

where c,, is the concentration of B at the electrode surface. The boundary conditions are

x = o y = l

X + - y = o for all zones. Also

Disc 0 5 r 2 r ,

Gapr, 2 r 2 r,

Ringr, 5 r 5 r3

electrode wall

bulk solution (35)

y, = 1 uniform surface concentration

(36)

(37)

where each region of tp (r) is associated with its corresponding yp. The solution is via single Laplace transformation in all three zones and

Thus with

we obtain the result for the steady-state collection efficiency

which has been described as a rather messy equation [30] . Here we have

The equation possibly looks more elegant if we use the fact that

(43)

and write

)] (44) N o = G(&j a + p2’3 G(L) a - (1 + a + p)2’3 [ G( a( l + a + p ) F(8) has been tabulated, but is readily worked out on a programmable desktop calculator. Table 2 shows values of N , for common radius ratios.

The important point about eqn. (41) is that it is dependent solely on the electrode geometry. This simplifies kinetic studies with double electrodes considerably. There have been numerous experimental verifi- cations of N o for different systems. Digital simulation also shows very good agreement with the analytical solution [31].

For very thin-gap, thin-ring electrodes, we may write an approximate expression for N o .

[ 1 [ f (aY3 1 (45) 2 3

Nb’ = - (a’ + 6’) F 7 - 1 + ( p y 3 1 --

where


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TABLE 2

Values of the steady-state diffusion-controlled collection efficiency, N o

A. Rotating ring-disc electrode. Wall-tube electrode.

~

1.02 1.04 1.06 1.08 1.10

1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.10 1.12 1.14 1.16 1.20 1.30 1.40

0.1013 0.1293 0.1529 0.1737 0.1923 0.2092 0.2247 0.2526 0.2772 0.2992 0.3192 0.3544 0.4237 0.4762

0.0947 0.1215 0.1444 0.1647 0.1829 0.1996 0.2149 0.2426 0.2670 0.2890 0.3090 0.3443 0.4141 0.4673

0.0902 0.1162 0.1385 0.1582 0.1761 0.1925 0.2076 0.2350 0.2593 0.2812 0.3011 0.3364 0.4065 0.4600

0.0869 0.1121 0.1339 0.1533 0.1708 0.1869 0.2019 0.2289 0.2530 0.2748 0.2947 0.3299 0.4001 0.4539

0.0843 0.1089 0.1302 0.1493 0.1665 0.1824 0.1972 0.2240 0.2479 0.2695 0.2893 0.3245 0.3947 0.4487

B. Wall-jet ring-disc electrode.

1.02 1.04 1.06 1.08 1.10

1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.10 1.12 1.14 1.16 1.20 1.30 1.40

0.0591 0.0759 0.0902 0.1030 0.1145 0.1251 0.1348 0.1526 0.1683 0.1826 0.1957 0.2189 0.2657 0.3020

0.0561 0.0723 0.0863 0.0988 0.1101 0.1205 0.1301 0.1476 0.1632 0.1774 0.1904 0.2136 0.2603 0.2966

0.0540 0.0698 0.0834 0.0956 0.1067 0.1169 0.1264 0.1437 0.1592 0.1733 0.1862 0.2092 0.2558 0.2922

0.0524 0.0678 0.0811 0.0931 0.1040 0.1141 0.1234 0.1405 0.1558 0.1698 0.1826 0.2055 0.2519 0.2883

0.0510 0.0661 0.0792 0.0910 0.1017 0.1116 0.1209 0.1378 0.1529 0.1668 0.1795 0.2023 0.2485 0.2848

The usefulness of this will become apparent in the discussion of kinetic collection efficiencies (Sect. 5.6).

Another question we can ask is: what will be the ring current measured if the disc is passing its limiting current? We know that

= P2’3 iq,L i D , L

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C. Double tube electrode. Double channel electrode,

1.05 1.10 1.15 1.20 1.30 1.40 1.50

1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00

0.1368 0.1'991 0.2433 0.2799 0.3063 0.3304 0.3512 0.3696 0.3859 0.4006

0.1297 0.1907 0.2346 0.2691 0.2975 0.3216 0.3425 0.3610 0.3774 0.3922

0.1247 0.1846 0.2280 0.2623 0.2907 0.3148 0.3357 0.3542 0.3707 0.3855

0.1209 0.1798 0.2228 0.2569 0.2851 0.3092 0.3301 0.3486 0.3651 0.3800

0.1154 0.1726 0.2148 0.2485 0.2765 0.3004 0.3213 0.3398 0.3563 0.3713

0.1115 0.1675 0.2090 0.2423 0.2700 0.2939 0.3147 0.3331 0.3496 0.3646

0.1085 0.1635 0.2045 0.2374 0.2650 0.2887 0.3094 0.3278 0.3443 0.3592

and the collection effect can be expressed by

where i k , L is the limiting current when the disc is passing current. Thus

The factor No/p2'3 is called the shielding factor: it will be smallest for thin-gap, thin-ring electrodes.

A variant of the RRDE in which the ring is split into two parts (the rotating split ring-disk electrode) was invented to enable the detection of different intermediates produced at the disc electrode [ 321 . It was shown that the current a t the ring segments is proportional to the length and that there is n o interference between the segments [ 9, 33 J .

( c ) The rotating double ring electrode

In reactions involving gas evolution, the RRDE can be problematic in that bubbles may become trapped at the centre of the disc electrode. To obviate this, a rotating double ring electrode was suggested [34] . The collection efficiency, N o , is given by eqn. (41) if we define

ri - r 3 r 3 - 3

1 ro a=-

where ro is the internal radius of the smaller ring. When ro is zero (RRDE), we obtain the rotating ring-disc expressions for 01 and p. Digital

References pp. 434-441

Fig. 3. Establishment of Poiseuille flow and diffusion layer under laminar flow in a tube.

simulation gives the same results as the analytical solution [ 3 5 ] . These electrodes have also been used for electrogenerated chemiluminescence studies [ 3 6 ] (see Sect. 7.1).

2.3.3 Tubular and channel electrodes

In these electrode configurations, the solution moves past the electrodes embedded in the wall of a tube or channel. It turns out, as is to be expected, that for high Schmidt numbers (thin diffusion layer) the mass transport in the appropriate dimensionless variables is virtually iden tical for both electrodes.

(a ) Tubular electrodes

Laminar fluid flow in tubes has been described by Levich [ 3 ] . An entry length, 1, , is necessary to establish Poiseuille flow, given approximately by

1, - 0.1R - Re ( 5 2 )

where R is the radius of the tube: this is derived from the boundary layer thickness a t a flat plate. The velocity profile after this point (see Fig. 3 ) is

there being no radial or angular convection. uo is the fluid velocity in the centre of the tube and coordinates are defined in Fig. 1. The time- independent convective diffusion equation is

Making the assumption of a thin diffusion layer compared with the tube radius, we approximate

(55) 2 ( R - r )

R = 2p u, = uo

3 7 1

where. as before

c - c , CCC

Y = -_

and

Analogously to rotating electrodes, we take p = 1 for the upstream of two electrodes (generator) and p = 3 for the downstream (detector). Since, in electrochemical experiments, radial diffusion will be much less than axial convection, we can say that

and eqn. (54) becomes

a7 a27 X g = g

This equation is of exactly the same form as the dimensionless convective- diffusion equation at the RDE (p. ). Furthermore, in dimensionless variables, the boundary conditions are exactly the same as for the rotating disc [eqn. (19)]. We thus arrive, following the same procedure, at the expression for the limiting current

i , = 5.43 nF c , D213 Vj’3 (x, - xp-l )2 ’3

where Vf is the volume flow rate defined by

u,rR2 2

Vf = -

(59)

It is interesting that iL is independent of the viscosity of the solution; note, however, that this is only true for high Schmidt numbers (thin diffusion layer). Also, iL is dependent only on the cube root of the fluid flow (cf. square root at the RDE). This lower sensitivity can result in larger errors in the determination of mass transfer and kinetic parameters.

The ratio of limiting currents at generator and detector is


372

where

(62) x 3 x 2

x1 x1 P = - - -

Note the similar expression at the rotating electrode [eqn. (27)]. A Nernst diffusion-layer thickness is defined as

( 6 3 4

(63b)

& N = 1 . 5 ~ 1 / 3 u;1/3 ~ 1 / 3 x1/3

- 1-74 D 113 v - 1 / 3 ~ ~ 1 1 3 f

-

In this case, h N is dependent on the axial coordinate: the tubular electrode is not uniformly accessible. This complicates the mathematical description of partially kinetically controlled reactions at the TE. How- ever, for total kinetic control (irreversible reaction at the foot of the wave), the flux is uniform as radial convection is uniformly zero along the tube.

Digital simulation [37] has shown that the approximations made by Levich are valid under most conditions: limits to his assumptions are given, particularly with regard to potential scan rates.

( b ) Channel electrodes

The velocity profile at a channel electrode is

u, = uo (l+)

where v o is the maximum fluid velocity in the x direction and h is the half-height of the channel as shown in Fig. 1. Poiseuille flow will be established after an inlet length I, which, according to Schlichting [4] is given by 1, = 0.11 (uo h / v ) . Neglecting the contribution of axial diffusion as being small compared with axial convection, we have

ac a z C ax ay2

u , - - - - -

Assuming a thin diffusion layer

Then, by putting

we can follow exactly the same argument as at the tubular electrode. The reason for the similarity is that the approximations we have made

373

regarding diffusion and convection have removed any curvature effects. Thus, the diffusion-limited current is

i , = 0.925 nF c , D213 Vf1i3 (h2 d) - l i3 W ( X , - 3cp-1 ) 2 i 3 (68)

where, in this case, V, = (4/3)hdu0 and w is the width of the electrode.

( c ) Double tube and double channel electrodes

Since the form of the dimensionless convective-diffusion equation for tube and channel electrodes is exactly the same as for rotating electrodes, we can immediately conclude that the steady-state collection efficiency, N o , under conditions of uniform surface concentration at the generator electrode (which corresponds to the limiting current at the generator or to any point on a reversible wave) is, once again

where

3c3 x2

X 1 x1 P = ---

Values for typical electrode lengths are shown in Table 2. Double channel electrodes were first used by Gerischer et al. [38] and

this expression for No was obtained by Braun [39] and subsequently experimentally verified [ 401 .

2.3.4 Electrodes based on impinging jets

When a jet of fluid submerged in a medium of that fluid strikes a surface perpendicularly, it spreads out radially over that surface. Original interest in these systems was due to mass transfer investigations of down- ward directed jets of vertical-take-off aircraft [ 4 1 ] , though other applications such as electrochemical machining are important.

We may identify several regions as shown in Fig. 4 . In particular, there is the stagnation region (Region 111) and the wall-jet region (Region IV) which give rise to the wall-tube and wall-jet electrodes, respectively. The relative sizes of electrode and impinging jet are thus most important.


374

Fig. 4. Mass transfer from an impinging jet electrode. (From ref. 48 by permission of the publisher, The Electrochemical Society, Inc.). I, Potential core region; 11, established flow region; 111, stagnation region; IV, wall-jet region.

(a ) Wall-jet electrodes

The laminar flow hydrodynamics for a radial wall-jet were first considered by Glauert [41] and subsequently by Scholtz and Trass [42] . A more complete evaluation for electrochemical purposes has recently appeared [ 4 3 ] . In Fig. 5 are shown the velocity profiles and schematic streamlines. Close to the wall we find

( 7 0 4

(70b)

u, = cz2r -15 /4

u , = Czr-11’4

which satisfy the equation of continuity and where 3 114

c = with M the flux of exterior momentum flux given by

Here, a is the jet diameter and h is a constant approximately equal to unity. If we rewrite the velocity components as

3 7 5

Fig. 5 . The wall jet. (a) Flow functions. (b) schematic streamlines. Here u = (15M/2ur3)”’ f‘(q); uz = 0.75 ( 4 0 M ~ / 3 r ’ ) ’ / ~ h ( q ) where q = ( 1 3 5 M / 3 2 v 3 r ’ ) I f 4 z . (From ref. 43.)

CZ

rm I u, =

where m = l l j 4 , we seen that they are identical to the Levich approximations for the velocity components a t the RDE if we put m = - 1, and use C = 0.5100~’~v-~’~.

We define dimensionless variables 1/3 r-4’3

1


376

t P = (;I c - c,

Y = - C ,

where

q = 4 (5-rn)

ac a 2 c

to arrive a t the convective-diffusion equation

*c=s with p = 1 for a disc electrode and p = 3 for a ring electrode. The value of q is 9/8 whereas, at the rotating disc electrode, it is 3. For the limiting current, identical boundary conditions [eqn. (19)] apply. We arrive directly at

(75) ’ - 1.59k nFc a - 1 / 2 ~ 2 / 3 v - 5 / 1 2 V 3 / 4 ,.9/8 - 918 2 i 3 1L - m f ( p r p - 1 )

The ratio of limiting currents at ring and disc is once again ,62’3. In order for eqn. (75) to be correct, and as can be seen from the streamlines in Fig. 5, a minimum nozzle exit/electrode separation is necessary [44, 451.

The wall-jet disc electrode is clearly not uniformly accessible (current density a r-5i4 ). Another important point is that i, depends on the three- quarter power of the flow rate: it is more sensitive in this sense than rotating or tube/channel electrodes.

Yamada and Matsuda [46] arrived at the same result for the limiting current by another method. They determined the constant k experimentally to be 0.86, a value which was confirmed by later observations [44].

We can conclude that the steady-state collection efficiency, when we can assume uniform surface concentration on the disc, will be given by formula (41) owing to the form of the convective-diffusion equation with

Values of N o are approximately 60% of the values for the same radius ratio at the RRDE (see Table 2). Experimental verification has been provided [ 441 .

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( b ) Wall-tube electrodes

In the case of the wall-tube electrode, the electrode is smaller than the jet diameter. The original treatment was due to Frossling [47] who demonstrated that, close to the wall

( 7 7 4

(77b)

u, = - 1.31 a3/2 ,,-1/2 z 2

u, = 1.31 a3'2 v-ll2 rz

Figure 6 shows the schematic streamlines and relevant coordinates. A disc electrode concentric with the nozzle is thus uniformly accessible since u, is independent of r. a is a function of flow rate and has been shown experimentally by Chin and Tsang [48] to be equal to

Using these formulae, Albery and Bruckenstein have derived the limiting current a t a wall-tube electrode [ 49 3 .

We put q = 3 and m = - 1 in eqns. (73) and (74) and the only differences from the rotating disc are that the constant term is slightly different and w is replaced by Vf/r$.

Thus

Note that iL depends on V:" whereas, for the wall-jet electrode, it depends on V:'4. This equation only holds for 0.1 <rT/zT <2.5. Mass transfer is more efficient than at an RDE; however, the electrode has to be smaller. Nevertheless, in applications where it is difficult to fabricate a moving electrode (i.e. photoelectrochemical and semiconductor), it could be very valuable. From the theoretical point of view all that has to be done is replace w by 0.98 Vf/r: in all the equations for a rotating disc or r ingdisc electrode to obtain the wall-tube analogue. In particular, the steady-state collection efficiency, No [eqn. (41)] , is the same not only in form but also in numerical value for the same radius ratios [ 501 (Table 2).

Experimental work has been published on a ring-disc electrode which is intermediate in geometry between the wall-jet and wall-tube configurations. Consequently, and as expected, intermediate collection efficiency values were measured [ 5 11 .

2.3.5 The dropping mercury electrode

Although it was the first hydrodynamic electrode to be invented, the mathematical solution of mass transport at the DME is complex owing to the fact that no genuine steady-state can be attained such that &/at # 0


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r = o

Fig, 6 . Wall tube electrode geometryschematic streamlines. (From ref. 49 . )

in the convective-diffusion equation. Additionally, due to the increasing surface area of the electrode, there are contributions from, amongst others, charging currents and solution depletion. However, there are few problems with electrode poisoning compared with solid electrodes and a large cathodic window so that methods have been developed to circumvent the difficulties. These will be discussed further below.

(a ) The diffusion-limited current

Detailed expositions of limiting current deviations may be found in standard polarography texts, e.g. refs. 5 and 6. Here, we try to pick out the more salient points.

In the calculation, our time scale is the lifetime, 7, of a mercury drop. Its radius a t time t is given by

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ro = (F) where m l is the mass flow rate and p the density of mercury. At time 7, the drop falls and another begins to form.

Firstly, owing to the importance of the IlkoviC equation, we follow Ilkovic’s argument [52] in the calculation of the limiting current, which although not completely rigorous, was later shown by MacGillavry and Rided [ 5 3 ] to be correct under Ilkovic’s assumptions. There are three steps to the argument.

(a) Assume the electrode is a plane surface and obtain the current by solution of Fick’s second law

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with appropriate boundary conditions

x+m, t 2 o c = c, bulk solution

x = o t = O c = c,

t > O c = o limiting current

to lead to

which is the Cottrell relation. (b) Take account of increasing drop area using eqn. (80).

(c) The growth of the drop causes an increase in the concentration gradient resulting in a multiplication factor of 4 7 / 3 for it. Thus we obtain, finally, the instantaneous limiting current

itIL = 709 nc, D1/2m:/3 t 1 I 6 ( 8 5 4

iL = 607 nc, D”’ mf13 r1I6 (85b)

and averaging over the lifetime of the drop -

These are both forms of the Ilkovic equation. The first has been shown experimentally to be only reasonably correct in terms of time dependence; however, the expression for the average current is substantially correct. The principal defect of this derivation is that it takes no account of electrode curvature.

Following Levich [3] , we present below the formalism for a rigorous deviation based on the spherical natilre of the DME. We do, however, neglect tangential (stretching) forces on the surface of the mercury drop, i.e. the u@ velocity component. In spherical coordinates

The radial velocity component, ur, is [from eqn. (SO)]

where r,, = yt1I3. Introducing the dimensionless variables


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we obtain

The left-hand side of the above equation gives the IlkoviC expression. It is thus possible to consider the right-hand side as correction terms to the IlkoviC equation. These corrections must take into account spherical diffusion, solution depletion in the neighbourhood of the drop due to previous drops, contact area with and shielding due to the capillary, and solution stirring.

Various approaches have been used to calculate the correction terms [ 54, 551. They normally take the form

m

z = iIkovic 1 + 1 An(D1/2rn;1 /3 t1 I6 ) (90) n = l

First attempts [56] began by considering the limiting current at a stationary sphere and then taking into account the increase in area. This led to a first-order correction term which represents spherical diffusion (proportional to t ). In this and subsequent treatments, depending on approximations made in the derivation with regard to area expansion and stretching effects, A has been evaluated between 17 and 39.

The most rigorous solutions [ 57, 581 include a second-order correction

Koutecky [57] evaluated A l = 34.7 and A 2 = 100, while Matsuda [58] found A l = 31.1 and A 2 = 294. However, for typical values of D and m , these second-order terms can be neglected. Nevertheless, early in the drop life, the observed currents are lower than those predicted and later on they are higher (possibly due to convection effects).

The former observation is concerned with the effective electrode area. In the early part of drop life, its size is similar to that of the capillary orifice. A significant part of the drop is thus not in contact with the solution, a fact which qualitatively explains the lower observed currents. Also, close to the capillary surface, the diffusion process will be restricted, the so-called shielding effect. This is particularly pertinent with modern polarographic equipment where mechanical drop timers are often used in conjunction with short drop times. These problems have been discussed recently [ 591 . The following modification was proposed

i = iIkovic (1 + A , D 1 / 2 t 1 / 6 r n i 1 ’ 3 ) [ 1 - 1 1 7 . 8 B / ( r n 1 t ) 2 ’ 3 ] (92)

where B is the contact area between the drop and capillary. This equation has been shown to hold down to about T = 0.3 s.

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Another approach is to treat the shielding effect in terms of loss of diffusion layer, which automatically takes into account the contact area, as proposed by Matsuda [ 5 8 ] .

It has been observed that the current obtained from the first drop at a DME is 10--20% higher than that at subsequent drops, an effect which is ascribed to depletion of electroactive species close to the electrode. How- ever, this has not been fully explained quantitatively.

(6) Effect o f the mercury column height

For natural dropping, the flow rate of Hg will be dependent on the column height. I t is also dependent on the depth of the tip's immersion in the solution and on the work needed to expand against the surface tension of mercury. I t is thus possible t o express the effective pressure as

where the second term represents the solution back pressure and the third is the pressure due to the surface tension of mercury 2y/F0, with To the average drop radius; h, is the corrected column height. In practice, the second term can be neglected but the third term cannot. We thus arrive, after evaluation of the constants, a t

If we put K = m / h c , then this may be rewritten as

3.1K ( m l 7 ) l j 3

m = KhHg- ( 9 5 )

Since m , is proportional and 7 inversely proportional to the column height ( m , 7g = 2nry), then the product m i 7 should be virtually constant for a given capillary and independent of h, . Thus

iIkovic = const. x m:/371/6

iIkovic a ( 9 7 )

( 9 6 )

is then

which is a characteristic of diffusion-limited currents a t the DME. Pure kinetic currents will, of course, be independent of column height.

( c ) Charging current

Possibly the biggest disadvantage of the conventional DME is the capacitative current contribution due to drop growth. This phenomenon usually accounts for the major part of the so-called residual current. We


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may describe i, as

. dQ dA - = Ci (E , - E ) - dt d t 1, =

where Ci is the integral double-layer capacitance and E , the point of zero charge. From eqn. (80) , we may write

Thus

i, = 5.67 x Ci (E , - E ) m?l3 t-'13 (100a)

(100b) - i, = 8.5 x Ci(E, - E ) my3 r-'13

where i, is measured in pA. We are therefore in a position to compare Faradaic and non-Faradaic

currents and at M they are of comparable magnitude (using Ci - 20 pF cm-2 ). However, i, decays with t ' I 3 whereas i, increases with t ' I 6 .

This is, of course, an important reason for the sampling of current towards the end of drop life, which is used, for example, in the pulse polarography techniques. Pulse techniques are also useful when the integral capacitance, Ci, varies with potential; in these cases graphical elimination of i, can be difficult .

I t is clear that, since the charging current is proportional to (m: /t)'13, i t is therefore directly proportional to the column height. Comparison with eqn. (97) shows that increasing column height will therefore dis- criminate against the Faradaic current.

Recent investigations [ 601 have shown that shielding effects for Faradaic and non-Faradaic processes are the same.

( d ) Polarographic maxima

There is, as yet, no complete theory to explain the phenomenon of polarogaphic maxima which is manifested by the observed current over- shooting the limiting current. The causes are convective mass transport in the solution and adsorption. Three types of maximum have been identified [61] :

(i) variations of surface tension in the mercury initiate streaming. The velocity distribution in solution at the top of the drop near the capillary is rather different from that at the bottom of the drop due to shielding [62, 631. There is thus an uneven current distribution. Maxima tend to be high and sharp.

(ii) due to high mercury flow rates and gives small, rounded maxima. (iii) arises during the adsorption of sparingly soluble surfactants, but

only those with a compact molecular structure [64] which leads to

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two-dimensional condensed layers of surfactant on the mercury surface. The result is turbulent motion of the mercury drop surface. Examples of these surfactants are camphor and adamantanol.

Maxima may be removed by the addition of small amounts of certain surface-active substances, e.g. Triton-X-100 and gelatin (see Sect. 3.3.3), whose action is ascribed to their effect on the mercury surface tension. When addition of such substances is not possible, the placement of a shroud around the capillary tip has been suggested to minimise convection effects [65]. An alternative is to arrange for short drop times by mechanical means.

( e ) Short drop times. The vibrating dropping mercury electrode

Since the use of mechanical drop knockers in much polarographic equipment allows the use of a wide range of drop times, some further consideration of short drop lifetimes is convenient. Advantages of short times are that analysis of agitated solutions is facilitated 1661 (indeed the problems of concentration depletion should be lessened), polarographic maxima and kinetic waves are minimised [67-691 and faster potential scans can be applied 1701. The disadvantage is the high capacitative current contribution. Additionally we have to take account of the fact that instantaneous polarographic currents measured at short drop times do not conform to the IlkoviC equation [Sect. 2.3.6(a)] and that the mercury flow rate varies with mechanically induced drop times for small values of 7 at a constant column height 171, 721.

DMEs with short drop times are commonly known as vibrating dropping mercury electrodes (VDME). Although first described more than thirty years ago 1661, it is only recently that a VDME of relatively simple construction and with a regular drop time was described, allowing a minimum drop time of 5ms 1731. It has been shown that the limiting current is proportional to m:’3 t’’6 as in the IlkoviC equation but that the proportionality factor is different [ 691 . Measurements at small I-, where homogeneous reaction effects are minimised, in combination with measuring at long 7, holds promise in the elucidation of kinetic processes. It has been proposed that the disadvantage of the high charging current could be eliminated by synchronisation of the drop time and recording instrument, enabling the use of pulse and differential pulse-recording modes. This would improve the sensitivity and allow a wider range of application.

( f ) Other DME variants

A natural extension of the VDME is the streaming mercury electrode [74] where a fine jet of mercury issues from the capillary and is limited by a glass plate. Although some theoretical treatments are available, the poor definition of the length, radius and surface velocity of the mercury


TABLE 3

Diffusion-limited currents at hydrodynamic electrodes under laminar flow conditionsa

Flow parameter, f k Comments Ref.

b A. Uniformly accessible electrodes . (iL = knFc, D2’3~-1/6 nr: f ’ / * )

Rotating disc Rotating hemisphere

Rotating cone

w w

Wall-tube Vf/r Stationary disc in laminar

uniform flow U/2 r I

B. Dropping mercury electrodec. (iL = hnc,D”2 m;l3 t ’ ’6) .

m l

0.62 Sect. 2.3.2(a) 0.474 0.451 0.433 0.62 (sin8)-’I2

k value of 0.451 shown to be best experimentally [ 811

rotation axis and 8 is the angle between

electrode surface

i 0.61 Sect. 2.3.4(b)

0.780 Experimental deviations 0.753 of * 7% found

709 Instantaneous current 607 Average current ( t = 7)

Sect. 2.3.5

78 79 80 82

83 84

Ref. Flow parameter, f Limiting current expression Comments

C. Non-uniformly accessible electrodes.

Tube Vf Channel Vf Wall-jetb Vf

rotating so1utionbvd w ’ Stationary disc electrode in uniformly

5.43 nFc,D2I3 x2I3 V:I3 0.925 nFc, D2I3 ( h 2 d)- ’ I3 wx2I3 V;/3

Sect. 2.3.3(a) Sect. 2.3.3(b) Sect. 2.3.4(a) 1.59 n ~ c , ~ 2 / 3 V-S/12 314 ~ 3 1 4

r1 f

0.761 nFc, D2I3 v-Il6 7~1“:w‘~’~ 85

Stationary disc electrode in rotating 72 flow caused by rotating w" 0.422 ~ F c , D ~ ' ~ v-Ir6 r r ; w1'1i2 86 2

g *c1 'p

cu

I

3 0 agesides the method described in the text, most of these expressions may be derived from general equations describing axial laminar flow past bodies of revolution [ 7 6 , 7 7 1 . bThe i, expression for the ring-electrode analogue is obtained by replacing r; by ( r P i 2 - $I2 )2'3.

CThis is the Ilkovic equation, which assumes the DME is uniformly accessible. dThese electrodes are not uniformly accessible, despite appearances, as the convective flow is in towards, and not outwards from, the centre of the disc. w = rotation speed (rad s - l ) ; Vf = volume flow rate (em3 s - l ) ; rT = radius of wall-tube; U = linear velocity of solution (ems-' ); 0 = angle between cone surface and rota,tion axis ;x = length of tubular/channel e1;ctrode; w = width of channel electrode; d = width of channel; h = half-height of channel; w = rotation speed of solution (rad s - l ) ; w = rotation speed of rotating disc (rad s-l) .

IP1

A

w ul m

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stream have meant that it has fallen out of use. Other DME variants to enhance the current have been described based on rotating the capillary [74, 751 : i t is probably easier nowadays to coat an RDE with a film of mercury.

2.3.6 Other hydrodynamic electrodes

Besides the RDE and DME, other uniformly accessible electrodes under laminar flow have been described. They include the rotating hemispherical and rotating cone electrodes, which were developed to obviate the problem of trapped gas bubbles a t the centre of an RDE: their use is not widespread.

A summary of limiting current expressions at many hydrodynamic electrodes is given in Table 3.

2.3.7 Current and potent ial distributions

Current and potential distributions are affected by the geometry of the system and by mass transfer, both of which have been discussed. They are also affected by the electrode kinetics, which will tend to make the current distribution uniform, if it is not so already. Finally, in solutions with a finite resistance, there is an ohmic potential drop (the iR drop) which we minimise by addition of an excess of inert electrolyte. The electrolyte also concentrates the potential difference between the electrode and the solution in the Helmholtz layer, which is important for electrode kinetic studies. Nevertheless, it is not always possible t o increase the solution conductivity sufficiently, for example in corrosion studies. It is therefore useful to know how much electrolyte is necessary to be “excess” and how the double layer affects the electrode kinetics. Additionally, in non-steady-state techniques, the instantaneous current can be large, causing the iR term to be significant. An excellent overview of the problem may be found in Newman’s monograph [87].

A commonly employed method t o minimise ohmic potential drop effects is to place the reference electrode very close to the working electrode by means of a Luggin capillary. The disadvantage of very close placement, which may be unacceptable, is disturbance of the fluid flow. To avoid this, other methods are sometimes used. For example, a rotating disc electrode has been described in which the reference electrode is placed in a tiny compartment within the rotating electrode assembly and linked to the solution via a tiny orifice (0.7 mm) drilled in the centre of the disc [ 8 8 ] .

One of the principal reasons for the extensive use of the rotating disc electrode is its uniform accessibility. However, if the solution conductivity is not sufficiently high, it does not have a uniform current distribution. Experimental investigations have provided criteria for the concentration of inert electrolyte necessary to add to ensure uniformity [89]. Current

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distributions at ring [go] and ring-disc electrodes [91] have also been investigated. An additional problem with the latter, and indeed any double electrode, is coupling effects due to the superposition of potential fields [92] ; this can be compensated for by an electronic equivalent circuit [ 931 .

Studies have also been conducted into the current distribution at tubular electrodes [94]. At the DME, it is non-uniformity of current and potential distributions, during drop growth, which is one of the causes of polarographic maxima [Sect. 2.3.5(d)].

3. Experimental

Detailed expositions of the practical use of hydrodynamic electrodes can be found in various references, e.g. refs. 95-97. A commonly heard criticism of hydrodynamic electrodes is that they, and often the associated flow systems, are difficult to construct. We do not believe the difficulty to be so great and if judicious care is taken, it is perfectly possible to make the electrodes in most research laboratories.

3.1 ELECTRODE MATERIALS FOR HYDRODYNAMIC ELECTRODES

Choice of electrode materials depends usually on potential range in the solvent being studied and available purity. In the case of hydrodynamic electrodes, we also have to consider carefully the ease of machining in order to conform to the shapes and forms required by the theoretical equations.

Solid electrode performance can be affected by the electrode’s previous history. A freshly polished electrode surface is virtually free of functional groups. To what extent its electrochemical behaviour changes in use depends very much on the electrode material and electrochemical pretreatment procedures [ 981 .

3.1.1 Metals

It is most common to use solid metals for hydrodynamic electrodes and it is usually possible to choose a metal which is correct for the particular application. Inert electrodes are usually of Pt, Au or Ag. In the study of electrodeposition or electrodissolution, the respective metal is employed.

The advantage of using metallic electrodes is that they are relatively easy to shape and to polish.

3.1.2 Carbon

Carbon exists in various forms which are conducting. In general, reactions tend to be less reversible at carbon electrodes than at metallic electrodes. There is also a problem of reproducibility. Reactivity is due


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not only to the bulk structure of carbon but also to the surface groups [99] ; it has been shown that links with hydrogen, hydroxyl, and carboxylic groups may be formed on the surface. The electrode behaviour can be very pH-sensitive.

Several types of carbon are in common use as electrodes. The most used of these is glassy carbon (GC) [ loo] : it is the most reproducible but very difficult to machine as it is hard and brittle. One thus tends to be confined to the dimensions and shapes which come from the manufacturer. Each manufacturer has his own fabrication method (or more than one). A common problem is that small holes can appear in the middle of the piece of GC; if this occurs, there is no option but to machine more away. Additionally, the GC is not always homogeneous. In the authors’ opinion, the best glassy carbon is Tokai. While it is not always possible to use this because of geometrical considerations, better reproducibility will be obtained with it, especially after electrochemical pretreatment [ 1011 .

The response of pyrolytic graphite electrodes depends on whether the orientation is edge plane or basal plane [102]. Spectral grade graphite impregnated with Ceresin wax or paraffin wax under vacuum can also be used.

Finally, it is possible to fill a small shallow hole the size of the electrode required with carbon paste. These pastes are made from particles of graphite and a suitable hydrophobic diluent such as Nujol [103], silicone rubber [ 1041 , paraffin [ 1051 , epoxy resin [ 1061, Teflon [ 1071, or Kel-F [108]. A comparison of seven different carbon pastes has been made [ log] .

3.1.3 Other solid electrode materials

Rotating optically semii-transparent electrodes for spectroelectro- chemical or photoelectrochemical studies can be fabricated by vapour deposition techniques on a quartz substrate. In this way, tin oxide, platinum and gold electrodes, amongst others, can be made. Electrical contact is with silver paint.

In the study of the electrochemistry of single crystals, e.g. semi- conductors, at rotating electrodes, an RDE can be fabricated with a shallow hole and the crystal, after appropriate machining, cemented in place with silver epoxy resin.

3.1.4 Mercury

Mercury is a very widely used electrode material for studying cathodic processes owing to its very high hydrogen over-potential; however, its anodic range is small. For use in dropping electrodes, mercury purity is most important. Its purification has been described extensively and is in four parts.

(i) Removal of surface scum and oxides by filtering through perforated filter paper (Whatman No. 40).

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(ii) Removal of dissolved base metals (Zn, Cd) by agitating under 2 M HNO, for 1-3 days by means of a vacuum aspirator. The appearance of hollow bubbles of mercury on the surface of the solution signifies that this has been achieved.

(iii) Vacuum distillation to remove noble metals (Pt, Au, Hg, etc.). There are various types of distillation apparatus available. Care must be taken that the still does not become a reservoir of impurities!

(iv) Washing with water, drying and filtering once more. There are two useful tests for mercury purity which are easy to carry

out. If base metals are present, the mercury will leave a thin film on a glass vessel. Secondly, if a small quantity of mercury is shaken in a stoppered flask with about three times its volume of pure distilled water, foaming will occur lasting for 5-15 s if it is pure.

Mercury used for electrochemical purposes should be recycled : distillation, once performed, should not be necessary again.

Great care should be taken with the use of mercury on account of its toxicity. Spills should be dealt with immediately and mishaps avoided by the use of trays placed under the apparatus. Additionally, the laboratory should be well ventilated.

Mercury can be electrodeposited on solid electrodes if its use is desired in conjunction with solid hydrodynamic electrodes to increase their cathodic range. There are various procedures for carrying this out, but a convenient way is by using a dilute solution (- lo-’ M) of Hg(N0,)2 in 0 .1MHN03. There is some tendency to form mercury droplets rather than a homogeneous film on the electrode surface [110].

3.2 CONSTRUCTION OF SOLID HYDRODYNAMIC ELECTRODES

3.2.1 General considerations

Certain criteria have to be met in the construction of hydrodynamic electrodes, such that the laminar flow pattern, which is used in the derivation of the theoretical equations, is conformed to. Thus edge effects, which are due to the fact that electrode and surrounding mantle are not of infinite size and which are also dependent on cell dimensions, must be minimised. The shape of the electrode and mantle is important; the surfaces must be smooth and there must be no discontinuities or electrolyte penetration a t the electrode/mantle junction.

Materials used for insulating sheaths should be inert and easy to machine; they are generally plastics or casting epoxy resins. Epoxy resins are easier to handle because of their moulding ability but they are not chemically inert to certain species, including many non-aqueous solvents. Additionally, care must be taken when preparing the epoxy resins that air bubbles do not appear in the mixture: prior treatment of the adhesive and hardener mixture under vacuum for a short while reduces the problem.


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This is important as holes in the surface derived from air bubbles can give rise to turbulence, as well as not giving a leak-free electrode/mantle junction.

Plastic insulation sheaths are best fitted by machining to just less than the correct size for the electrode. Heating the plastic causes it to expand and allows it to be slipped over the electrode. They have varying degrees of chemical inertness: the most inert are Teflon and Kel-F. The second of these is probably the most convenient since its thermal expansion coefficient is closer to that of metals than is that of Teflon. The problem of matching thermal expansion coefficients between electrode and insulating sheath is a very real one. When experiments are all to be conducted at one temperature (usually 298K), the difficulty can be cir- cumvented by storing the electrodes in a glass tube, for example, immersed in a thermostatted bath when not in use. If this is not done, then after some time it is probable that the insulation will stand proud of the electrode or vice versa and further machining is then necessary.

Other electrode insulation materials have been employed, such as various types of glass and ceramic for rotating electrodes. They will be specified below in more detail.

Connection between the electrode material and lead or shaft is made in one of four ways: soldering, with silver-loaded epoxy resin, spot welding, or by spring-loaded contacts.

Once the electrode and sheath have been machined to shape, it is necessary to polish the electrode. Metallographic paper is usually used first, followed by abrasive of progressively finer particles size down to 0.1pm. This can be alumina, dry or made into a slurry with water or glycerol, or diamond lapping compound. Polishing can be done manually or preferably on a motor-driven polishing table. Afterwards, the electrode should be thoroughly rinsed with distilled water. Before each experiment, a very light polish should be given manually to the electrode; if filming has occurred, then stronger polishing will be necessary. In any polishing procedure, it is important that the hardness of the abrasive be not much greater than that of the material being polished. Also, steps should be taken to ensure that particles of abrasive do not become embedded in the electrode or surrounding insulation.

3.2.2 Disc electrodes

The simplest way to construct these electrodes is to solder or glue with silver epoxy resin a cylindrical specimen of the electrode material to a stainless steel shaft. A mantle of epoxy resin or plastic is then made around the disc to the shape required.

Edge effects at rotating disc electrodes caused by using different electrode shapes have been discussed by Blurton and Riddiford [lll]. Their conclusion was that a bell-shaped mantle was best, although cylindrical electrodes, which are widely used, are perfectly sufficient in

391

most cases so long as the insulation sheath is large compared with the disc electrode radius. This edge effect results from the assumption in the theory that there is no interaction between the flow above and below the plane of the disc.

An alternative manufacturing method, used particularly by the Soviet researchers, is based on fitting cones. These are introduced internally or externally (in which case there has to be a screw fitting) [112]. The electrode shaft is either solid or a hollow tube with electrical connection being provided by a spring-loaded contact which links the electrode to the contact pin at the top of the shaft.

An RDE for use at high temperatures nad been described [113]. For application to RDEs made of semiconductor crystals, some advantages are to be gained by having two electrode contacts, one for current and the other potentiometric [ 1141 .

3.2.3 Ring-disc electrodes

It is more difficult to manufacture these electrodes than the simple disc electrode since the ring must be exactly concentric with the disc. Additionally, in many applications the insulation gap must be thin (0.25mm or less) as must the ring. For rotating ring-disc electrodes, typical dimensions are a disc radius of 3-4mm and an outer ring radius of 4-5mm. For wall-jet ring-disc electrodes, these dimensions can be approximately halved.

Some sort of construction jig is indispensable in order to ensure con- centricity of ring and disc. Various methods of assembling parts have been described based on press-fitting and cone-type arrangements. Whichever construction method is employed, the most difficult part is making the insulation gap between disc and ring, whether it is of epoxy or plastic. In the simplest method [49], the disc electrode is cemented to a steel or brass shaft and covered with insulator which is machined to slightly larger than the ring’s inner radius. The ring electrode is cemented to a brass cylinder of exactly the same cross-section and press-fitted over the assembly; all parts must be very smooth to avoid gouging the insulation gap which would lead to electrolyte penetration. Finally, the outer sheath is press-fitted. More specific details of this and other construction methods may be found in the literature [115,116].

It is often convenient to make a number of studies with the same ring electrode but different disc materials for comparison purposes. Such ring- disc electrodes with demountable discs have been described based on press-fitting [ 105, 117, 1181, cone-fitting [ 119, 1201, and screw-thread [ 1211 principles.

In order to increase the collection efficiency or to conform to theoretical equations for kinetic collection efficiencies only valid for very thin gaps, plastic or epoxy insulation is not very useful since with these it


392

is difficult to make gaps less than about 0.2 mm in width. Construction of an RRDE with a gap of 0.01 mm made by a corundum stylus has been described [ 1221.

For use over a wide temperature range, it is necessary to match the thermal expansion coefficients of electrode and insulation sheath. RRDEs of glassy carbon embedded in borosilicate glass for use up to 450°C [ 1231 and gold sputtered on to a chromium or titanium substrate on a Macor ceramic cylinder for use up to at least 125OC [ 1241 are examples.

3.2.4 Tubular and channel electrodes

The construction of tubular electrodes may be divided into two basic types: integral and demountable. Channel electrodes are only of the latter type. Final dimensions must satisfy the entry length criterion for Poiseuille flow (pp. 370 and 372).

The principle of the integral method of construction involves placing the electrode itself round a tightly fitting forming rod together with connection tubes (glass or plastic) and cementing [125, 1261. Alter- natively, epoxy resin may be cast round the forming rod. When the resin is cured, the former is removed and the electrode polished with alumina slurry on a steel rod [127]. This is not only to give a mirror finish, but also to remove any ridges inside the tube. The disadvantage of the integral method is that one cannot see whether the electrode surface is smooth and clean; however, experimental results will give the answer!

Demountable electrodes are of two kinds depending on whether tube or channel electrodes are employed. In the first case, the individually machined sections are bolted together round a former [128, 1291 but, here, turbulence problems may arise owing to ridges at the joints between sections. In the second case, one face of the cell is made separately from the rest and the channel electrode(s) embedded in it or insulation cast round them. After machining and polishing, the cell is bolted together [ 38,391.

3.3 OPERATION

This section is concerned with cell design and arrangement, and fluid flow control.

A general point to be mentioned here is that of deoxygenation of solutions [130]. The reduction of oxygen interferes with many experiments: the solution is deoxygenated with an inert gas (usually nitrogen or argon) by bubbling a stream of the gas through the solution. This reduces the partial pressure of oxygen in the solution and provides a blanket of inert gas. Nitrogen is most commonly used, but should be purified before use, even if it is bought as “Oxy-free” or grade “U”. Purification can be achieved by passing through Dreschel bottles filled with a solution of a reducing agent such as Na-9,1O-anthraquinone-2-sulphonate

393

in NaOH solution in the presence of zinc amalgam followed by drying over silica gel if a non-aqueous solvent is to be used. A more efficient and less messy procedure is to pass the gas over a combination of molecular sieves and finely divided copper filings (BTS catalyst). Finally, whichever purification method is used, the inert gas should be bubbled through a flask containing the solvent before introduction into the solution.

3.3.1 Rotating assemblies

The cell for rotating electrodes, Fig. 7, is usually cylindrical and surrounded by a water jacket for thermostatting purposes, but as long as the cell walls are more than 1 cm or so from the rotating assembly, there are usually no cell edge effects. The auxiliary electrode is very often con- tained in a separate compartment behind a glass frit in order to avoid contamination problems. A Luggin capillary, where required, can be positioned in various ways: unless it is more than 0.5cm from the electrode, it must be placed under the centre of the disc in order to avoid a non-equipotential surface; this can cause some problems with disturbance of the fluid flow.

For good accuracy and reproducibility, it is most important that the rotation speed of the electrode be precisely controlled. This signifies that the motor should be of good quality and have a reasonable power reserve; the motor speed is usually controlled by some kind of tachometer feedback mechanism. In the early days, the speed was usually measured by using the stroboscopic effect, which limited the possible rotation speeds. However, by use of a photoelectric sensor and a slotted disc [131], the exact speed can be displayed digitally and a full range of rotation speeds achieved. Since, when not operated near their power limits, motor speeds are linearly proportional to the applied voltage, it is a simple matter to arrange an external voltage input to modulate the rotation speed (see Sect. 6.5).

The actual drive can be of the belt type (indirect) or axial (direct). While the former puts less stress on the rotating parts, the second is simpler and is very often used nowadays. Eccentricity of rotation must be minimised: nevertheless, a slight eccentricity will not affect the electrochemical measurements so long as solution which has reached the electrode does not then pass over it again [132,133]. Electrical contacts with the external electronics can be of the silvercarbon brush type or enclosed mercury. A problem with the former is that particles from the brushes can contaminate the solution if proper care is not taken and, additionally, they give quite a lot of electrical noise at low rotation speeds: mercury is to be preferred in these instances (up to about 50 Hz). At higher rotation speeds, noise from mercury contacts increases considerably and silver-carbon brushes have to be used.


3 94

Iner t gas 7

1

Fig. 7 . Typical rotating ring-disc electrode cell. A, rotating ring-disc electrode; B, reference electrode with Luggin capillary; C, counter electrode; D, teflon lid; E, porous frit; F, thermostatted water jacket.

3.3.2 Flow systems

Flow systems operate by gravity feed or pumping. Although the DME belongs to the former class, it will be discussed separately below. Gravity feed gives a regular flow, but the range of possible flow rates is limited compared with pumping; with a good pumping system, flow rates in the range from to 0.5cm3 s-' are possible. However, for any motor- driven pump, whether via a worm-drive with coupled syringe or a peristaltic pump, there is a problem of flow pulsation, particularly in the latter case. Pulsation may be damped by using a long length of thin tubing (preferably Teflon) and/or a hollow glass ball which is semi-filled, acting as a solution capacitor between the pump and the cell.

Whichever system is used, connections to the electrochemical cell must usually be made with plastic tubing. All plastics are, to some extent, permeable to oxygen [ 1341. This can be obviated by surrounding the tubing with a second, larger tube through which inert gas is passed, as well as being bubbled through the solution in the solution reservoir.

Unfortunately, if a peristaltic pump is employed, then sheathing the tubes in the pump is not possible: these are normally of silicone rubber, which is exceptionally permeable to oxygen. Where deoxygenation is necessary, the peristaltic pump is arranged to suck the solution and the cell is placed as close to the reservoir as possible. A note of caution is that, in this conformation, air can enter the solution relatively easily so that junctions must be well sealed. The arrangement of the flow systems is very similar to that employed in flow-injection analysis [ 135,1361.

For use with tubular and channel electrodes (Fig. 8), no extra special

395

E E

\ ( b )

/ /

I - +

1 1 A B

Fig. 8. Typical tubular electrode assemblies. (a) Integral construction. A, Generator electrode; B, detector electrode; C, reference electrode; D, counter electrode; E, porous frits; F, ball and socket joints; G, epoxy resin. (b) Demountable type. A, Generator electrode; B, counter electrode; C, Teflon spacers; D, reference electrode; E, Teflon cell body; F , brass thread. (From ref. 128.)

In

/ C

* out

D

Fig. 9. Typical wall-jet cell, A, Disc electrode contact; B, ring electrode contact; C, Ag/AgCl reference electrode; D, Pt tube counter electrode; E, cell inlet; F, Kel-F cell body. (From ref. 44.)


396

Inert gas A

Fig. 10. Typical dropping mercury electrode assembly. A, Mercury reservoir; B, Tygon tubing; C, Pt wire connection to mercury; D, capillary; E, reference electrode; F, counter electrode.

difficulties arise so Iong as, if they are of the sandwich type, there are no leaks, especially round 0 ring joints. The same problem may occur with wall-jet cells (Fig. 9). If there are any small leaks caused by incorrectly fitting 0 rings, then this can usually be solved by a piece of Teflon tape on the screw thread. Additionally, for wall-jet cells, care needs to be exercised in filling the cell with solution so that trapping of air bubbles is avoided.

3.3.3 The dropping mercury electrode assembly

The basic construction of a DME is as shown in Fig. 10. Mercury flows from a reservoir at height h above the top of a glass capillary, through the capillary and into the test solution. The connecting tubing would ideally be glass, but this is impractical and unnecessary: Tygon, Teflon, and various other plastics have been employed. Before use, the tubing should be thoroughly cleaned and dried to avoid introduction of impurities. Electrical connection is made via a Pt wire in the reservoir or just above

3 97

the capillary itself. The inner bore of the capillary is - 0.05 mm. Home- made capillaries can be made from thermometer or barometer tubing.

There are various factors which affect the performance of the DME: (i) The mass flow rate of mercury. This is usually proportional to the

square root of the corrected mercury column height, h, [see Sect. 2.3.6(b)]. Measurement of rn, is via weight change (weighing the drops) or by measuring how long is needed for the mercury to descend from one level to another in the column. This is conveniently achieved by making/ breaking contact with an external electrical circuit [ 1371 .

(ii) The drop time, T , depends on the bore of the capillary. Natural drop times for a column of 50 cm height and 0.05 mm capillary bore are of the order of 3-5s. With mechanical drop knockers, it is possible to alter T significantly [see Sect. 2.3.6(c)].

(iii) The capillary tip can cause shielding as described on p. 380. This can be minimised by using a tapered tip capillary.

(iv) The purity of the mercury. If there are impurities in the mercury, they tend to clog the capillary, thus affecting the drop time as well as the electrochemical response. Mercury purification is described on p. 388. Clogging only tends to happen with impure mercury and infrequent use of the DME. It can be avoided [138] by removing the cell from the capillary after experimentation, rinsing with distilled water or other solvent while the mercury is still dropping, and only stopping the flow after the solvent has dried. In this way, no electrolyte can creep into the capillary. By the same token, the flow of Hg should be started before immersion in the test solution. If clogging does occur, it can often be dealt with by sucking distilled water through with a vacuum pump; if this fails, dilute nitric acid should be tried. However, it is essential in this case that no mercury is left in the capillary as it will tend to form insoluble mercury salts. When this happens, usually the only remedy is a new capillary, but sometimes cutting 1 cm or so from the end of the capillary will suffice.

3.4 INSTRUMENTATION

Three-electrode control systems are widely available in the market and there are also four-electrode systems for double working electrodes. The construction is either integral or modular. It is perfectly possible to construct the necessary electronics in-house and, in this case, modular construction is suggested as being more flexible. Operational amplifiers and other components of high quality should be used, particularly for kinetic applications. The elements of a bipotentiostat (independent control of two working electrodes) and a galvanostat are described in ref. 139.

Three or four-electrode systems together with the use, when appropriate, of a Luggin capillary solve most of the problems of uncompensated resistance in solution. However, a t times positive feedback

References pp . 434-441

398

compensation is necessary [93] as in the case of the DME to compensate for charging currents.

Various circuits have been described to measure collection efficiencies based on galvanostatic control of the upstream electrode with the downstream electrode being held a t the limiting current for the reaction taking place there. It is also possible to measure No by a potentiostatic shielding experiment. For, an irreversible electrode reaction, measurement of N o in these two different ways will, in principle, give different results if the upstream electrode is not uniformly accessible.

Analogue control of the electrode potential or current is achieved by a function generator: there are now very expensive waveform generators on the market which enable any potential on current waveform to be applied.

The use of digital instrumentation to control the experiments is becoming more widespread. A microcomputer can generate any waveform and apply it to the potentiostat or galvanostat. It can also analyse the response from the electrode to the applied perturbation. Various possibilities are clear when one considers that the flow rate or rotation speed can also be programmed [140]. The application of a variety of pulse waveforms with current sampling and a.c. with phase-sensitive detection is available: a good account of the possibilities may be found in ref. 141. However, whereas analogue instruments exhibit limitations in the complexity of waveform generation, with digital waveform generators there may be poor voltage resolution.

Electrical noise should generally be avoided in order to keep high signal- to-noise ratios. In principle, much information can be extracted from noise 11421, but most people prefer to analyse the signal! Interference may result from the line or other machines nearby as well as from moving electrical contacts in the apparatus. A high quality power supply is important and all electronics should be kept in an earthed box with screened cables to the electrode connections. It is often advisable to pass the signal through a band-pass filter. In certain experiments, involving a.c. measurements for example, it may be necessary to enclose the whole electrochemical apparatus within a Faraday cage.

4. Application of hydrodynamic electrodes to electrode kinetics

4.1 INTRODUCTION

In the absence of coupled homogeneous reactions, the current observed at an electrode is controlled by mass transport, electrode kinetics, or a mixture of the two. Control is wholly by mass transport at all points of a current-voltage curve for a reversible reaction and at the limiting current for quasi-reversible and irreversible reactions.

399

For the simple redox reaction 0 + n e + R , we may write

k D , O k -1 k D , R

where kf , are the mass transfer coefficients [eqn. (7)] and the asterisks represent concentrations just outside the double layer. Control of k', and k is via the potential of the electrode; control of k f, is by alteration of the rotation speed or of the flow or drop time. Thus, a reaction which is reversible for low kf , may not continue to be so when we increase the convection rate. The effects of this will become clearer in the following sections.

Unless the electrode is uniformly accessible, the mass transfer coefficient will vary over the electrode surface. This introduces certain complications in the quantitative analysis of experimental results.

We first consider the case of reversible reactions before going on to discuss the general case of mixed kinetic and transport control.

4.2 REVERSIBLE REACTIONS

In this case, the rate constant for the electron transfer is much faster than mass transport. This means that thermodynamic equilibrium is readily established and, for 0 + n e + R

Using the fact that, for diffusion control i" - .

iE 1

[O]* = [OI - and

where the superscripts c and a represent reduction and oxidation currents, respectively, we arrive at

RT i t - i D R s

nF z - i L ( D o ) E = E - t - l n y -

where s = 2/3 for all hydrodynamic electrodes except the DME, where s = l j Z . This may be rewritten as

RT i i - i E = E',,, -I- - In-

nF i - i a L


400

where E: /2 , the half-wave potential for a reversible reaction, is given by

Ei,z is independent of the flow parameters and of [O], and [RJ , . Very often, it can be assumed that D, = D o , in which case E i I 2 = E". If D, = 1.5Do, n = 1 and s = 2/3, Ei12 is shifted from E" by only 7 mV.

There are a number of ways to test for the reversibility of an electrode reaction. Any deviation is an indication of partial (or total) kinetic control.

(i) E I l 2 is independent of [O] (ii) The wave shape is independent of the fluid flow velocity, i.e.

rotation speed or flow rate. Thus, since the limiting current varies with the fluid flow velocity, so does the current at all points of the current-voltage curve. A plot of i vs. rotation speed or flow rate raised to the appropriate power will be linear for any potential and will have a zero intercept. (This is also true for diffusion-limited currents of any current-voltage curve.)

(iii) A plot of log [(iL - i)/(i - i t ) ] vs. E gives a straight line of slope 0.0591/nV at 298 K.

(iv) Another criterion which is sometimes useful is that E 5 4 - E;,4 = 0.0565/nV at 298 K.

An alternative way of describing the reversible current-voltage curve, which is sometimes found in the literature, is obtained by putting

and [R] - .

Substitution of this relation in eqn. (106) gives

4.3 CURRENT-VOLTAGE CURVES. THE GENERAL CASE

Pure kinetic control can only be evidenced at the foot of an irreversible wave. In all other cases and at all other parts of the irreversible wave, mass transfer has to be taken into account.

For irreversible processes, we can neglect the contribution made by the back reaction. In the Butler-Volmer formulation of electrode kinetics, k ; is given as

h = hb exp [- a(nF/RT) ( E - E")] (110)

h', = k b exp [(l - a)(nF/RT) ( E - E*)] (111)

and similarly

401

where a is the transfer coefficient. Clearly, we can also neglect the back reaction if only one of the species 0 or R is present in bulk solution.

Mixed kinetic control is defined by

where p and v are the orders of the electrochemical reduction and oxidation, respectively. In general, p and v are unity, but if there is adsorption blocking the electrode surface, the values will be fractional. To simplify the discussion we will consider only the reduction process, i.e. assume that [R] is zero. The flux due to the electrode kinetics is

j = h ' , [ O ] i (113)

j = h L ( [ O ] , - [ O I * ) (114)

(115)

We write the mass transport flux as

= j L - k b [o] , Equating (113) and (115)

Using the fact that

we obtain

M

= j , I+] or

where

ik = nFAj, = nFAk', [O]; (121)

Equation (120) is true at any particular point on any electrode: since the diffusion layer thickness is constant for a uniformly accessible electrode, in this case we simply replace j by i. Otherwise, the equation must be integrated over the area of the electrode.


402

A

.

0

w ’/2

w - ’h

Fig. 11. Electrode kinetics a t the rotating disc electrode [eqn. (122) ] . (a) Curve A, i, vs. CJ’” ; curve B, i vs. C J ~ ’ ~ at a lower overpotential showing the effect of electrode kinetics where curve B’ is the line obtained if the reaction were reversible. ( b ) Analysis of curve B i n (a) by plotting i-’ vs. ~ 3 - l ’ ~ . hi is obtained from the intercept and D from the slope.

Analysis for uniformly accessible electrodes is by plotting i ‘’IJ vs. ilk which leads to ik and hence k ’1. If p is unknown, then plot log i vs. log i /kb to give y. Other plots are also possible [143]. In the particular case where p = 1 (no adsorption)

For the rotating disc electrode, we plot i-’ vs. w-’’’ : we can determine D from the slope and ik from the intercept. Application of this relation has been thoroughly discussed [144] (Fig. 11).

403

p = o

Fig. 12. Nomogram for the graphical evaluation of electrode reactions of fractional order p according to eqn. (125). K = 0 corresponds to total mass transport control and K = m to pure kinetic control. y and K are given by eqn. (124) . (From ref. 145.)

An alternative method is to eliminatej (or i) from eqn. (115) and write

k m l : = G W I , - [OI*) (123) If we define

then eqn. (123) reduces to

1-7, TP = - K

Like eqn. (121), this is only valid for the whole electrode when the surface is uniformly accessible. I t is most commonly evaluated graphically by using a nomogram such as that shown in Fig. 1 2 [145].

In general, because of non-uniform accessibility of the electrode, equations for current-voltage curves have to be solved either by using approximations or numerically. For historical reasons, the dropping mercury electrode was the first t o be treated theoretically [ 146- 1491 and the equations obtained depend on whether the steady-state, expanding-plane, or expanding-sphere models are utilised [ 1501 .

Matsuda and co-workers have provided generic approximate expressions for current-voltage curves for first-order reactions at a variety of hydrodynamic electrodes. They take the form


TABLE 4 rp 0 rp

Value of 0 and A for common electrode geometries in the equations for the current-voltage curve [eqn. (126)la

Electrode U A Ref.

y 1 i 6 w - 1 i 2

Rotating ring y 1 / 6 w - 1 / 2

y 1 / 6 w - l ' 2

Rotating disc

Ring of rotating ring-disc electrode

Stationary disc in uniformly (reactant produced at disc)

rotating fluid y 1 / 6 w-1/2

Dropping mercury (EP model)b +2

Tubular v3 R x i i 3 Detector of double channel electrode

(reactant produced at generator) ~ ~ - 1 1 3 (h2 d 1 1 i 3 ~ : / 3

0.620 0.620B(r)

151 152

0.799C(r, E K ) 153 (see also 154)

0.761 B ' ( r ) 1.13 0.839

85 149 155 (see also 156)

0.616 C'(x, Ed,+) 157

aThe values of A are approximate; B and €3' are functions of r; C is a function of radius and ring potential; C' is a function of electrode length and detector potential.

eqn. (126) for the DME, D is raised to the power 0.5.

405

where

kh is the standard rate constant and u is a mass transport dependent expression. A is a number which is a constant for a uniformly accessible electrode: in other cases, it depends on the electrode geometry and has been numerically evaluated. I t is clear that when

A < ( k b / D 2 l 3 ) u(e-"r + e('-&)f) (1 29) expression (126) reduces t o the reversible case [eqn. ( l o g ) ] . Values of u and A are given for a number of electrode geometries in Table 4. Since all the expressions are of the same form as the RDE, the type of analysis described by Jahn and Vielstich [144] to extract rate constants may be directly applied.

In general, the half-wave potential is dependent on the rate of mass transfer (drop time at the DME [ 1581 ) and is only not so for reversible reactions. If we consider that just the species Ox exists in bulk solution, then it is fairly easy to show that eqn. (126) may be rewritten as

Logarithmic analysis is therefore possible and can be utilised in the determination of ET12 and (an) from quasi-reversible voltammetric waves [159]. For the totally irreversible case, when there is a significant overpotential, we obtain, for all electrodes for a cathodic process

RT i: - i E = EiYi + - In 7

LWlF 1

and for an anodic process

(131

(132

Another important factor which can significantly affect the shape and position of the voltammetric wave, and hence rate constants, is adsorption of product or reactant on the electrode surface. For a linear adsorption isotherm, if the reactant is adsorbed, a reduction wave will be shifted towards more negative potentials and if the product is adsorbed, towards more positive potentials [ 1601 . Non-linear adsorption isotherms give rise to pre-waves (product adsorption) and post-waves (reactant adsorption), a phenomenon which was first discussed by BrdiEka at the DME and since then by many authors [161, 1621. At the RDE,


406

adsorption of reactants and products following a Langmuir isotherm has recently been theoretically treated [163].

An interesting way of evaluating rate constants and charge transfer coefficients is the technique of iso-surface concentration voltammetry (ISCVA) [164] where the surface concentration of reactant is held constant over the electrode surface. A uniformly accessible electrode such as the RDE is therefore a prerequisite. At the RDE, the value of i/w"2 is kept constant and disc potential plotted against current for different ratios of This yields the kinetic parameters as well as E* and the number of electrons transferred.

4.4 HETEROGENEOUS CATALYTIC REACTIONS

In certain cases and together with the electrode reaction, in particular that of oxygen reduction at metal surfaces, a non-electrochemical regeneration mechanism operates which is heterogeneous in nature and involves the adsorbed product [165] : obviously, it is very dependent on the electrode material and the available surface states. The reaction scheme is thus of the type

Electrode A , * n e - + A 2

kh Metal surface A2-A1

For a first-order regeneration mechanism, the effect will be to shift the Levich-type plot at the limiting current upwards by a constant amount. At the RDE, the intercept is given by

i = nFAh,[A2]* (133)

and rate constants in the range 5 hh 5 lo-' cms-' can be determined.

For a second-order regeneration mechanism, there is a more complex dependence on the mass transport; however, this type of reaction is readily distinguishable from the homogeneous regeneration mechanism (Sect. 5 . 3 ) . The range of second-order rate constants amenable to measurement at the RDE is l o 4 5 hh 5 l o 6 cm4 mol-' s-l .

4.5 MULTISTEP ELECTRON TRANSFERS

Many electrode reactions, particularly those of organic compounds, take place in two or more steps, exemplified by the EE mechanism

A+e-B+e-C

At one extreme we will observe two separate one-electron waves and at the other, a two-electron wave [166]. The intermediate cases where the waves are overlapping is more difficult to analyse. It is, however, amenable to graphical logarithmic analysis, which has been treated by Ruzie e t al. at

407

the DME [ 1671 but which is possible, in principle, to apply to any hydrodynamic electrode owing t o the fact that the current-voltage curves can all be expressed in the same form. The analysis enables evaluation of limiting current, half-wave potential and (y11 for each wave.

The theoretical description of any mechanism is most complex at the DME. Solutions using the stationary plane, expanding plane and the expanding sphere models have been obtained and exhibit some differences, the last being, of course, the most rigorous [168--1701. The EEE mechanism at stationary and expanding planes has also been discussed [171, 1721. Immobilization or adsorption of the intermediate in the EE mechanism is a realistic possibility [173] .

A double electrode is clearly a convenient tool for the investigation of multistep processes. Its application to the study of multistep electron transfer is the subject of the next section.

4.6 MULTISTEP ELECTRON TRANSFER AT THE RRDE

Double electrodes are invaluable in the elucidation of multistep electron transfers. Applications have been almost exclusively at the RRDE.

4.6.1 Consecutive electron transfers

[ 1741 , viz. We consider the simple scheme first discussed by Ivanov and Levich

Ring

where the mass transfer coefficients are given by

and

kL,B - - 0.62D23 v-1 /6 112 (134b)

When the second electron transfer step at the disc is rapid (kk > k;),B, i.e. B does not have time to diffuse away) then the ring current is zero and an ( n l + n2 ) electron wave is seen at the disc. Conversely, when it is slow (kk < k;,, ), we obtain the usual

in the steady state.


408

To calculate the general behaviour, we use the steady-state approximation and the fact that

and

- n R N O j D J R -

n1

or (137)

The disc current is thus given by

(139) I D - FA

- n l k i [ A 1 * + n 2 k ; [ B l .

We are now in a position to construct expressions for the graphical evaluation of n n 2 , k '1 and k . In particular, we may write

Thus, the slope of the plot of iD/iR vs. o-1/2 gives the value of k ; and the intercept n . In a similar fashion

2 /3

(iD,L - i D ) = + ( n , + n 2 ) 3 ] + ( n 1 + n 2 ) 1R

nR N O

X k',

(145)

409

so that a plot of (iD,L - i D ) / i R vs. ol” yields n 2 from the intercept and k’, from the slope.

4.6.2 Parallel electron transfers

A similar approach may be used to evaluate kinetic parameters for parallel electron transfer of the kind

Disc A------ A*

Ring t nRe B-C

where only B is electroactive at the ring. k ;,A and k b.B are given by eqns. (134a) and (134b). The expressions obtained in the particular case are

n 2 ) + n l k b , A

k ; (147)

4.6.3 Branching mechanisms

Branching mechanisms involve both consecutive and parallel electron transfers. The most important application of the RRDE in this context has been to the electrochemical reduction of oxygen [175], on which a large amount of research has been done. Different mechanistic models give rise to different expressions linking the rate constants, which can be compared with experimental data: as in previous sections, the most important is the variation of ( iD / i R ) with rotation speed. A summary of different models has recently appeared [176] the conclusion of which is that, at platinum, the model of Damjanovic et al. [177] is correct; diagnostic criteria to test the model have been developed.

The model may be represented by (all electron transfers are 2-electron)

k :

Ring


410

and the resulting expressions allowing determination of k :, k \ and k \ are

(149) k’, + k\

. = I +

The plot of iD /i, vs. will, in general, give lines of different positive slopes and different intercepts, depending on the applied potential (which directly affects the ratio h: / k ; and h ; ) .

Nevertheless, the general case reduces to simpler expressions under certain conditions as shown in Table 5. We thus have a diagnostic for the relative contributions of the three disc processes as well as the possibility of determining the rate constants from expressions (148) and (149). Other mechanisms, and indeed any multistep electron transfer process, may be treated in the same way at the RRDE, which can be used as a diagnostic tool, and for rate constant evaluation. Note, however, that sometimes different mechanisms can give the same qualitative rotation speed dependence in limiting cases. I t is also necessary that the ring potential can be chosen such that only one of the products formed at the disc electrode is electroactive at the ring.

Diagnostic plots for heterogeneous catalytic electrode reactions at the RRDE have many features in common with those for simple parallel reactions [ 1781. This type of analysis is important in the investigation of the oxygen electrode reaction where non-electrochemical surface processes can occur.

1D.L

1D,L ID k b,A -

4.6.4 Non-first-order mechanisms

order. The general case has been treated by Filinovskii [ 1791. We have assumed above that the electron transfer steps are all first

The reaction scheme may be written as

k i h i Disc p l A +n,e-B p2B+rz2e-C I

Ring t B + n , e + E

Making measurements a t two rotation speeds enables the reaction orders p I and p2 to be found. If we let

where x = 1 or 2 for the two different rotation speeds w and w 2 , then

411

TABLE 5

Diagnostic criteria for the electroreduction of oxygen a t the RRDE [from eqn. (14811 : values of N o i D / i R

“Intercept” “Slope”

General case

k ; = 0

k ; = 0

1

k ; 1 + 2 ,

k2

+

+

2k; ~

k h B

0

k ; = 0 and lz ; = 0 1 + 0

and

Equation (151) may also be used for determining p1 when the two disc reactions are in parallel; however, in this case there is no simple expression for p 2 .

Applications to disproportionation reactions, e.g. U(V) [ 1801, and to determining the order of the electroreduction of oxygen at platinum [ 1811 have been described.

5. The application of hydrodynamic electrodes to the study of electrode processes with coupled homogeneous reactions

5.1 INTRODUCTION

Many electrochemical reactions are coupled with homogeneous chemical reactions, particularly those of organic compounds. The effect of the homogeneous reaction will be stronger the faster it is and will alter the characteristics of the voltammetric waves. Hydrodynamic electrodes offer a particularly useful way of studying these reactions as the control of mass transport, and hence the diffusion-layer thickness, 6, , allows rate constants of varying orders of magnitude to be investigated.

I t is convenient to think of a reaction layer, 6,, within which the concentrations of the products and reagents of the homogeneous reaction are perturbed from their equilibrium values. When the homogeneous rate constants are very small, the reaction layer will effectively include the


412

whole cell: there will be no effect on the electrode reaction. At the other extreme, the reaction layer will be much smaller than the diffusion layer and the i-E behaviour of the electrode reaction will be highly affected by the homogeneous reaction: concentration gradients due to the electroactive species can be neglected, which considerably simplifies the mathematics. In the limit of very large homogeneous rate constants, assuming that equilibrium is not far on the side of the electroinactive species, the reaction layer has zero thickness, and the behaviour can be described by diffusion alone, where the effective diffusion coefficient is a function of the diffusion coefficients of the electroactive and electroinactive species and the equilibrium constant of the homogeneous reaction. Of course, in practice, this limit is hardly encountered. For intermediate kinetics, when 6, 6,, the mathematical treatment cannot be simplified and numerical solutions are necessary. In these cases, analysis usually proceeds by fitting experimental data to theoretical working curves.

I t is for these reasons that most of the theoretical treatments have, until recently, considered uniformly accessible electrodes, i.e. the DME and RDE, where 6, and 6, are independent of the electrode coordinate, and also fast homogeneous reactions (6, -4

Solution proceeds via modification of the convective-diffusion equation to include the relevant kinetic terms and new boundary conditions. Accounts of the general methods for obtaining kinetic currents, first developed for the DME, can be found in, for example, refs. 182-184.

).

5.2 PRECEDING CHEMICAL (CE) REACTIONS

The simplest reaction scheme may be written as

kl k - Solution A2 F====== A, K='

k-1 k l

Electrode A, * n e + A,

The preceding chemical reaction causes the concentration of the electroactive Al to be lower than otherwise; the effect of this will be largest when K is large, when a purely kinetic current dependent on the value of k will be observed.

Modification of the convective-diffusion equation involves addition of terms to take account of the preceding reaction. Thus, for the rotating disc electrode, we obtain

413

with the boundary conditions

z = o (%lo = 0 (155b)

We also consider first the case where k l i s large, such that the reaction layer concept is useful. For the transport limited current, imax, we have the further boundary condition

z = 0 c1.0 = 0 (155c)

Solution proceeds via addition of the two convective-diffusion equations and leads to the result [ 1511, assuming a rapid chemical reaction

- D [ C p + CZ,= I K { D / [ k , (1 + K)] + 6, Jmax -

We thus conclude immediately that the term

K ( k l (:+ K ) r 2 (157)

has the dimension of length and represents the effect of the preceding chemical reaction: it is the reaction layer thickness.

There are two limiting cases. When K is small

and the electrode reaction is unaffected by the homogeneous reaction. When K S 1, % cl,= and (1 + K ) x K so that

showing that the observed current is lower than that which would be observed for A2 if it were electroactive. The equation may be rearranged to give

A plot of imax rapid ( k , large), the slope of the plot will be zero [see Fig. 13(a)].

vs. imax yields h , . If the chemical reaction is very


414

L

l m a x

Fig. 13. Diagnostic plots for electrode reactions with coupled homogeneous reactions, illustrated for the RDE. (a) CE mechanism. Curve A, no effect from chemical reaction (6k = 0); curve B, effect of preceding chemical reaction ( 6 k > O ) . (b ) Catalytic mechanism. Curve A, in the absence of parallel chemical reaction; curve B, experimental dependence predicted from eqn. (1 75).

Hale calculated the effect of a preceding monomolecular reaction when the disc electrode is galvanostatically controlled [ 1851 . His results can be applied to a wide range of homogeneous rate constants.

At the DME, the result obtained is [ 148, 186, 1871 - I,,, - - _ _ _ ~ 0 .886[ (h1 /K)7] ' l2h 7rD1/2m-1/371'6 1 (h , 7 / K ) ' / 2 P - - - 1L 1 + 0.886[(h1/~)7 j1 /2h- 1 4 - 0.14(h17/K)1/2h2

where

D2 + KD3 1 + K

D =

and

415

The second term represents a correction for spherical diffusion. This result is approximate and assumes that there is a steady state in the reaction layer. Numerical solution using the expanding-plane model leads to the approximate equation

- 1,,,- 0.886 [ k , ( l + K)7] 1 '2K-1

tanh (0.886 [ k l ( l + K ) T ] ' I 2 ) + 0.886 [ k l ( l + K ) T ] '12K-l - -

1L

(164)

which holds to within 2% error over all k , and K [188] and where we assume equality of diffusion Coefficients, but with the condition rmax 2 0.1 FL. This was extended by use of Koutecky's function F(x) [ 1891 to include small values of rmax [ 1901 . Current-potential curves have been evaluated numerically [191, 1921. The effects of sphericity and non- Nernstian electrode behaviour have also been taken into account [ 193, 1941.

Matsuda has considered CE reactions at a number of different hydrodynamic electrodes, making use of the reaction layer concept for fast chemical reactions. Assuming diffusion coefficients are equal, he arrives at the expression

( k h / 0 2 1 3 )a - e-er'(1 + e*') A + ( k $ / D 2 / 3 ) ~ ( i C L / i ~ a x ) e - a ~ ' ( 1 + e")]

(165)

i =(-+ i ; l + e

where

+ l L- A D ~ / ~ K -

i c

iLaX u [ k l (1 + K)] ' I 2

RT nF

(E:12)' = E" - - In (1 + K )

Values of A and u are the same as in Table 4. This immediately shows us the effect of the preceding chemical reaction on the half-wave potential. We can also, by analogy with Sect. 4.3, write

RT ik RT i i - i E = (E: /2) ' f - In 7 + __ In

nF imax nF L


416

for a reversible reaction. For an irreversible electrode reaction, illustrated below by a cathodic process

RT it - i E = ( E z ) ’ + -In 7

anF 1

where (E& ) ’ is a complicated function of rate constants, equilibrium constants, and solution and electrode parameters. Note that eqns. (170) and (171) are approximate equations and that logarithmic analysis of experimental data will not necessarily give a good straight line [ 1951 .

Channel and tubular electrodes have been studied in detail without making the assumption of a fast homogeneous reaction [196]. Solution was obtained numerically, but the approximate equation

i, tanh(1.290AK) + - h a x 1.290A

holds to within 1%. When the tanh term is approximately unity, then eqn. (166) is obtained since

A = o{ h (1 + K ) } K-’ D-’l6 0.923 (173)

where u and A are as in Table 4. This corresponds, for common electrode geometries, to k , (1 + K ) 2 270 s-l .

Other CE reaction schemes have been described for the DME and RDE. The reader is referred to the original references [183, 1841.

5 . 3 PARALLEL/CATALYTIC REACTIONS

These are reactions of the type

k l

k-1 Solution A2 C A1

Electrode A , k n e --f A2

The convective-diffusion equations and boundary conditions for z = 00 will be exactly the same as for the CE mechanism. However, at z = 0 we have

At the rotating disc electrode, if k , is large, we obtain in the limiting current region [151]

which is independent of rotation speed. As the rotation speed is increased, the reaction layer thickness can often no longer be assumed to be much

417

smaller than the diffusion layer thickness and a rotation rate dependence arises [see Fig. 13(b)].

If the homogeneous reaction is not very rapid, then we may write

where x = 1.61 (v/D)’16 (h, /w ) ” ’ , and can obtain h , from the plot of x vs. wV1l2 [197].

Another approach at the RDE uses the method of moments, which assumes a linear concentration gradient in the kinetic and diffusion layers [ 1981 . The expression obtained is a cubic equation which may be written as

and leads to evaluation of h l (it is assumed here that k - , is zero). Koutecky obtained the numerical solution for the catalytic current at

a DME by rigorous solution of the convective-diffusion equations by the expanding-plane model with h-, = 0 [199]. Subsequently, an approximate analytical solution was obtained which holds over the whole range of h l with an error of not more than 1% [ 1881 . The equations are

= {(1.16[klt] ’/’)’ + 1 -0 .2 [h,t] 1/2}1’2 I,ax

1,

for the instantaneous current and - - - - ((0.81 [ h , T] ‘ 1 2 ) ’ + 1 - 0.08 [ h , T ] 1/2}1/2 imax

iL -

for the average current, where iL and FL are given by the IlkoviC equation. Catalytic reactions at tubular electrodes have also been studied [ 2001 .

Expressions were obtained for a very rapid homogeneous reaction, in which case

imaX = 277nFRc x , (Dh ,) ‘ I2 (180)

imax = i, + 2.50 x lo5 nD1/3 R 2 ~ : / 3 ~ , , , k , V;’/3 (181)

and for a slow homogeneous reaction

which was solved numerically. This last analytical expression has a maximum error of about 6% as k , approaches unity. More recently, the problem has been studied further and the following expressions obtained [ 2011.

- - 1, ax - 1 + 0.55274A’ - 0.06969A4 + 0.01217A6 1 ,


418

for A 5 1.25 and

1,,,- - 1.23832A + 0.30958AP2 iL

for A > 1.25 where, for a'channel electrode

and, for a tubular electrode

These equations, if used in the correct regions, are very good approximations to the exact curve. Experimental verification was provided using the Fe2+/H202 systems [201].

Another important type of parallel reaction, which has been treated chiefly at the RDE, is where the homogeneous reaction is

We arrive at [202]

where 6 , = ( D / k - 1 ) ' / 2 , which is of similar form to the expression at the DME [ 2031 , and so at high disproportionation rates the maximum current of 2iL is observed. An analytical solution was obtained for low rates of disproportionation [ 2041 ; at higher rates, numerical solution is necessary and has been tabulated [205].

5.4 FOLLOWING CHEMICAL REACTIONS. THE EC MECHANISM

The EC mechanism is of the type

Electrode A, f n e + A,

k l k - Solution A, A2 K = -

k-1 k l where we represent the simplest possible homogeneous reaction. Many of the examples of this reaction type involve the anodic oxidation of organic compounds such as aromatic amines or diamines and aminophenols in acidic solution. The latter has been used as a test system for theoretical equations since the product of the electrode reaction is an unstable intermediate.

Qualitatively, we see that when K is small, the following homogeneous

419

reaction has little effect and its effect is larger the larger is K . Anodic waves are shifted to more negative potentials and cathodic waves to more positive potentials as a direct result of the reduced concentration of the product of the electrode reaction. Often, the kinetic parameters for the homogeneous reaction may be obtained directly from shifts in the half- wave potential. Unlike CE and parallel reactions, however, here the maximum current will be equal to the diffusion-controlled limiting current.

The convective-diffusion equation for species Al and A2 is the same as in the case of the CE reaction [eqns. (153) and (154)]. At the RDE for species A,, we have

which does not contain any kinetic terms. Boundary conditions are

(186a) - - Z + W c 1 = C l , r n , C 2 - C2,-, c 3 - c j m

(186b)

(186c)

where eqn. (186c) expresses the fact that we are considering a reversible electrode process. Mathematical solution is more complex than in the case of the CE reaction. Rigorous solution, making no assumption about the rate of the homogeneous reaction, leads to the expression

at the RDE for small K [206]. If we use the reaction layer concept with

we write

Then, by combination with eqn. (186c) and the Nernst equation, we


420

obtain [207]

RT i L - i RT k :I2

n F 2 + - In E = E e + - l n T n F 0 . 6 2 ~ - ” ~

and so

Fortunately for k l > 1, eqns. (187) and (190) give the same result. Thus we can write the following diagnostic for an EC reaction at an RDE.

vs. log,, w will have a slope of - (0.059/2n) V

(i) When eqn. (190) is valid, a plot of

(ii) A plot of iL vs. wl/’ will be a straight line with a zero intercept. Clearly, it is a simple matter using the reaction layer concept t o write

For Nernstian behaviour at the DME, the equation for the half-wave expressions for EC reactions with other stoichiometries.

potential is

where

I tanh (0.886 [ k (1 + K)T] ‘ I2 ) 0.886 [ k , ( l + K)7I1I2

* = - derived on the expanding-plane model [ 1911 . Using the expanding-sphere model, it has been shown that the effect of sphericity becomes smaller as k l decreases and as K increases but, in general, leads to higher current values at a given potential than the expanding-plane model [193]. Non- Nernstian behaviour has also been discussed with the expanding-plane model [194].

EC reactions at tubular and channel electrodes have been considered [208]. An analytical solution is not possible due to the non-uniformly accessible nature of the electrode. However, an approximate equation for the half-wave potential can be written, for a reduction, as

(193) RT n F El12 = ET,, + - (0.27 + 0.50 l n p )

for K > 4, p being given by

for a tubular electrode and

(194a)

(194b)

421

for a channel electrode. Thus, a plot of vs. log uo will have a slope of (-0.059/3n)V compared with (--- 0.059/2n)V at the RDE. The range of homogeneous rate constants amenable at the tube electrode is to l o 2 s - l , whereas a t the RDE it is 10 to l o 3 s-l . The two techniques together allow coverage of a wider range of rate constants.

5.5 FOLLOWING CHEMICAL REACTIONS. THE ECE MECHANISM

Whilst i t is clearly the case that double electrodes offer a unique way of studying these reactions, in many cases information can be extracted from single electrodes: much of the theoretical basis was derived before the use of double electrodes became widespread. The reaction scheme may be written as

Electrode A, * n , e -+ A l k

Solution A1- A2 Electrode A2 * n2 e -+ A,

When k is very small, the voltammetric response corresponds to an n l - electron transfer and when k is very large to an ( n l + n2)-electron transfer in a single wave. For intermediate k, we can obtain kinetic information.

The problem was considered at the rotating disc electrode, making the assumptions of equal diffusion coefficients, that n 1 = n2 = n and that the second electron transfer is easier than the first to give [209, 2101

where napp is the apparent value of n. As can be seen, this has the correct limiting behaviour for large and small k values. Digital simulation showed reasonably good agreement [ 2111 and considered the extra reaction

k l A1 +A2 ZF=+ A, + A4

k-1

In this case, analysis has t o proceed by fitting experimental data from a wide range of rotation speeds to theoretical working curves.

At the DME, a rigorous numerical solution was obtained by Nicholson et al. using the model of the expanding plane electrode [212]. This has been followed by a number of studies on expanding-plane and expanding- sphere models together with derivation of current-potential curves [213-2151. The so-called parallel ECE mechanism, where one of the electrode reactions is a reduction and the other oxidation, has been investigated and a numerical solution obtained [ 2161 . Numerical solutions have been found to other ECE-type mechanisms at the DME [217--2191.


422

5.6 DOUBLE ELECTRODES AND HOMOGENOUS KINETICS

For complex mechanisms such as ECE or other schemes involving at least two electron transfer steps with interposed chemical reactions, double electrodes offer a unique probe for the determination of kinetic parameters. Convection from upstream to downstream electrodes allows the study of fast homogeneous processes. The general reaction scheme for an ECE mechanism can be written

A, f nD e + A,

A, (+ X) h products

Upstream electrode

Solution

Downstream electrode A, f n , e + A 3 ( o r A 1 )

The homogeneous reaction may be of first or second order; in the latter case, X is a non-electroactive species. Both these cases have been studied a t the rotating ring-disc electrode and the first-order case at a double channel electrode.

5.6.1 Second-order collection efficiencies at the RRDE

If the solution reaction is “infinitely” fast, we can measure the concentration of X from a plot of ring current vs. disc current. This is the diffusion layer titration technique where, for example, A, and A, are Br-, A, is Br, , and X is As(II1) [ZZO]. The quantity of A, measured at the ring will be less than the steady-state collection efficiency value, N o . At small disc currents, no A, will leave the disc as i t is immediately consumed by X. As the disc current increases, an A,-dominated zone moves out from the disc and the ring current begins to increase when this zone boundary reaches the inside edge of the ring. When it reaches the outside edge of the ring, the ratio iR/iD increases linearly. The characteristics of the i R vs. iD plot are shown in Fig. 14. Equations which define the titration curve are

where a, /3 and G(8) have already been defined [eqns. (28), (40), (42), and

3 (43)l and

PJ =(?r--(‘) r , 5 rJ 5 r3 (198)

M represents the diffusion-limited current of X that would be observed if i t were electroactive. Thus, the ring current begins to increase from zero

423

Fig. 14. Diffusion layer titration curve at a double electrode.

when

and the linear portion of the plot is governed by the equation

iR = NoiD - Mp2’3 (200) Analysis of the experimental curves allows the determination of the concentration of the non-electroactive species from the value of M .

It is interesting that mathematical solution proceeds in a very similar fashion to that for N o ; any double electrode system where the convective- diffusion equation can be reduced to the form

will give the same type of iR vs. i, plot with appropriate definitions of 01 and [43]. In particular, this is true for double tube/double channel and wall-jet ring-disc electrodes.

We have assumed above that the rate constant of the homogeneous reaction, h , , is infinitely large so that the boundary between the A, and X-dominated zones is clearly defined. When h , is smaller, this boundary will be less clear. The modified convective-diffusion equations are difficult to solve and numerical solution is necessary. This has been performed at the RRDE making the assumptions that [22l , 2221

(i) convection normal to the electrode can be neglected (h2 sufficiently large);


424

(ii) the ring collects all the intermediate (potential applied in the

(iii) the interpenetration of A2 and X either side of the reaction limiting-current region and ring sufficiently wide); and

boundary is equal. The results obtained are [ 2221

iR.k = 0.21nr: nFDw3l2 v 2 -'12k-' (201)

0.339r$D113 (1 - F(LY)} - o NL =

r; v113 k 2 C ,

The most convenient analysis involves plotting NL vs. w/c, . In the region OP high k, , a straight line is obtained allowing determination of the value of k , ; this corresponds to low rotation speeds. At high rotation speeds, curvature may become evident. Prater and Bard have performed digital simulation of this problem and reasonable agreement was found in the region corresponding to the assumptions made above [ 2231. Bard and co- workers have also stimulated variants of the ECE mechanism involving dimerisation of the electrogenerated species and regeneration of the disc reactant [ 224, 2251 .

5.6.2 First-order collection efficiencies a t the RRDE

In the limiting case where we have a thin-gap thin-ring electrode and consequently normal convection may be neglected, this problem can, in principle, be solved analytically. However, inversion from Laplace space is difficult and polynomial expansion is necessary except for small k , . If we define

where C = 0.510 v-l12 03/' , we obtain (i) for 4.5 > K > 3.5 (extremely thin-gap and thin-ring) [ 2261

(ii) for 4 > K > 1

where values of a, b and c are tabulated for common radius ratios and give the result of numerical inversion [ 2271

(iii) for K <1 [228]

N; = N: - ( p ' ) 2 / 3 ( 1 - A 1 1 C * ) f ~ A 1 1 A $ ~ 2 C * ( p ' ) 4 ' 3 - 2 A 2 ~ 2 T 2

(206)

425

where N: is the approximate collection efficiency for a thin-gap thin-ring electrode [eqn. (45)] and

A l = 1.288

A2 = 0 . 6 4 3 ~ " ~ D " ~ (207) 0' = 3 In ( r 3 / r , )

and for K > 0.3, C , = K-' tanh (A K )

K < 0.3, c, = A l (1 - 0.372(A K ) 2 4- 0.146 (A K ) 4 . . .} (208) which has been tabulated by Hale [ 1851. T2 is a function of geometric parameters. Since N x N" is not met in practice, a correction is included

Digital simulation by Bard and Prater agrees well with the numerical solution provided by Albery and the analytical solution [229]. Experi- mental verification using the bromination of anisole and Fe(II)/V(V) reaction was given [230]. The range of measurable rate constants is 0.2 < K < 5 which corresponds to

3 x s-l < k l < l o3 s-' if we take D as cm2 s-' . If the homogeneous reaction is pseudo-first order, being, in fact, dependent on the concentration of a second species such that

cm2 s-' and v as

kl = k2c,

then the range of k , is

3 x M-' s-' < k , < l o 7 M-' s-' Another possibility which avoids algebraic manipulation is to calibrate

an RRDE by using a reaction of known rate constant and from this determine the rate constant of the unknown reaction [231].

5.6.3 First-order collection efficiencies at the double channel electrode

Braun [39] derived the following equation which has been shown to be approximately true only for very thin gap electrodes by using the same approach as Albery and Bruckenstein for the RRDE [cf. eqn. (204)]

n 1 Nk = A - A-3 exp - 4aA3

ItD

where X a =- - 1 XI


426

and

A = ($r2 D h x , 1'3

Since then, numerical solution by Matsuda and co-workers [ 2321 has allowed accurate values of the collection efficiency to be calculated over a wider range of electrode geometries. The following approximate equations hold to within a few per cent.

(i) For 1.0 2 Nk/No 2 0.08

Nk = No exp (--A2 4- bA4 - -A6) (212)

(ii) For 0.1 2 Nk/No 2 0.002

Nk = No exp (- a' + b'A2 - c'A4 )

where a, b, c, a' , b', c' are given by summations of the type 3 3

a = 1 1 amn ampn (214) m=O n = O

The numerical coefficients have been tabulated [ 2321 . Rate constants are extracted from working curves constructed using these equations and there is reasonable agreement between experiment and theory [ 2331 . The range of measurable rate constants is very similar to that at the RRDE.

6. Transient techniques at hydrodynamic electrodes

6.1 INTRODUCTION

All the electrode kinetic methodology described until now has assumed a steady state (or quasi-steady state in the case of the DME). Many techniques at stationary electrodes involve perturbation of the potential or current: in combination with forced convection, this offers new possibilities in the evaluation of a wider range of kinetic parameters. Additionally, we have the possibility of modulating the material flux, the technique of hydrodynamic modulation which has been applied at rotating electrodes. Unfortunately, the mathematical solution of the convective-diffusion equation is considerably more complex and usually has to be performed numerically.

It has been shown that, under conditions of mass-transfer control a t the RDE, the various modulation possibilities are linked via an implicit functional relationship [ 2341

F(i,E,w) = 0 (215)

427

Thus

where (dw/di), is the electrohydrodynamical impedance under potentiostatic control, ( d o / a E ) , is the electrohydrodynamical impedance under galvanostatic control, and ( a E / a i ) , is the electrochemical impedance. Similar relationships are to be expected at other hydrodynamic electrodes. In general we can say that, since impedances contain a charge-transfer component, if this is sufficiently large then kinetic information may be obtained. Although solution of the convective-diffusion equation is usually numerical, it has been shown that approximate analytical solutions are available for any axisymmetric electrode [ 2351 .

The electrochemical impedance may be obtained from potentiostatic or galvanostatic experiments. Alternating current voltammetric techniques are well documented at the DME, as are various kinds of pulse techniques. The former has also been developed a t rotating and tubular/channel electrodes.

Hydrodynamic modulation has been performed almost exclusively at the rotating disc electrode. It has found use for analytical purposes at rotating and tubular electrodes owing to the fact that non-convectively dependent electrode processes are unaffected by the modulation [ 2361 .

We will first briefly consider some of the treatments of potential step and current step techniques at hydrodynamic electrodes. One should bear in mind, however, that this division is somewhat artificial owing to the implicit dependence of one on the other. We then treat a.c. voltammetric techniques, LSV, and finally consider hydrodynamic modulation.

6.2 POTENTIAL STEP AND PULSE TECHNIQUES

The potential step provides the theoretical background for any potentiostatic regulation experiment and a basic understanding is necessary for the mathematical solution of any controlled potential, non- steady-state voltammetric response, such as LSV, pulse or a.c. experiments. At a stationary electrode, the current response to a potential step is described by the Cottrell equation [eqn. (83)] but at hydrodynamic electrodes, it needs to be modified to take account of forced convection.

At the RDE, various approximate analytical treatments have been presented by dropping the highest order convective term [ 2371 , neglecting convection completely [ 2381 , and by assuming a linear concentration profile within a time-dependent mass transfer boundary layer [ 2391 . The last of these gives


428

where r = Dt/6$ and R = ( idc )L/ i , which shows best agreement with the numerical solution [ 2401 . Successive potential steps (pulsed potential) have been theoretically investigated at the RDE as higher quality electrodeposition is obtained under such conditions [ 2401 .

At a double electrode, such as the rotating ring-disc electrode, a potential step at the disc will produce a ring current transient, the form of which is affected only by Faradaic current components at the disc. This fact can be very useful in separating Faradaic and non-Faradaic processes.

Square-wave and pulse polarography were first invented by Barker et al. [ 241, 2421 and variants, including differential mode, have been treated by many authors. Separation of Faradaic and non-Faradaic electrode processes is thus facilitated. Heterogeneous and homogeneous rate constants have been evaluated by the expanding-plane model (including shielding effect and spherical correction) and by the expanding-sphere model by Los and co-workers [ 2431 and Galvez et al. [ 2441 . Measurements made at widely varying pulse times, together with d.c. data, reveal the kinetic and mechanistic parameters and enable discrimination between different mechanisms. More details will be found in Chap. 4.

6 . 3 CURRENT STEP, CHRONOPOTENTIOMETRIC AND PULSED-CURRENT TECHNIQUES

Just as the fundamental Cottrell equation, for the Sand equation [ 2451

equation for a potential step experiment is the current step it is, in quiescent solution, the

where ttr, the transition time, is the time after which the concentration, c, falls to zero at the electrode surface: its potential then changes so that another electrode process can occur. By using hydrodynamic electrodes, we increase the transition time enabling more accurate measurements to be made. A complication of the technique is that double layer charging can significantly affect the system response.

Until recently, there were no chronopotentiometric treatments at the DME, except those assuming the steady-state. The main reason for this is that, if it is desired to apply a current density which is constant or with known time variation, then the total current has to be programmed with the drop growth. A solution to this involving applying a total current proportional t o t 2’3 was proposed and reasonable agreement found between theory and experiment [ 2461 . More recently, the problem has been evaluated numerically, considering a general current function i(t) = i, t 4 , and applied t o simple electrode reactions [247], and coupled homogeneous reactions: CE, EC and catalytic mechanisms [ 248-2501. Equations for the determination of kinetic parameters are given. Application to the evaluation of experimental data is now awaited.

4 29

The potential response of the RDE to current steps has been treated analytically [3, 237, 2511 and accurately by Hale using numerical integration [ 2521 : this enables the elucidation of kinetic parameters [185, 2531. A current density-transition time relationship at the RDE has been established which accounts for observed differences from the Sand equation [eqn. (218)] and which has been applied to EC reactions [ 2541 . Other hydrodynamic solid electrodes have not been considered in detail, although reversible reactions a t channel electrodes have been discussed [ 255, 2561 .

In the case of current steps a t the RRDE, analysis of the resulting ring transient can give information about double layer and adsorption processes occurring at the disc electrode, since the ring current will only depend on Faradaic processes occurring a t the disc [ 2571. The response of the ring to any periodic current-forcing function at the disc has been evaluated by expressing the forcing function as a Fourier series [258]. Recently, a procedure has been described for deducing the Faradaic component of the disc current from the ring current transient [ 2591.

Pulsed-current techniques can furnish electrochemical kinetic information and have been used at the RDE. With a pulse duration of

s, good agreement was found with steady- state results [ 1441 for the kinetic determination of the ferri-ferrocyanide system [260, 2611. Reduction of the pulse duration and cycle time would allow the measurement of larger rate constants. Kinetic parameter extraction has also been discussed for first-order irreversible reactions with two-step cathodic current pulses [ 2621 . A generalised theory describing the effect of pulsed current electrolysis on current-potential relations has appeared [ 263 J .

s and a cycle time of

6.4 ALTERNATING CURRENT VOLTAMMETRY

Alternating current polarography is now common practice (see Chap. 4). It was developed primarily to enhance sensitivity, but is also very useful in kinetic studies. Only much more recently have a.c. techniques been applied to other hydrodynamic electrodes. We can, however, make the following general observations. At high a.c. frequencies, the thickness of the diffusion layer associated with the a.c. perturbation will be much smaller than the thickness of the hydrodynamic boundary layer: in this case, convection is unimportant and we measure the classical Warburg impedance. At lower frequencies, this approximation cannot be made and convection needs t o be taken into account. At the rotating disc electrode, coupling is unimportant under normal conditions in aqueous solution if the a.c. frequency is greater than 40 Hz [264]. The theory of the a.c. voltammetric response for reversible reactions at tubular and channel electrodes gives this frequency as 10 Hz for a flow rate of 0.1 cm3 s-l and 0.1 Hz for a flow rate of cm3 s-l with normal electrode dimensions [ 2651.


430

Reversible, quasi-reversible and irreversible electrode processes have been studied at the RDE [266] as have coupled homogeneous reactions without [267] and with the effect of electrode kinetics [268]. The theoretical results are very similar to those of a.c. polarography, being very phase-angle sensitive to coupled chemical reactions: in the rotation speed range where convection can be neglected, the polarographic results may be directly applied 12691.

Alternating current voltammetry at a double electrode, such as the RRDE, allows one to distinguish between the total flux of electrons and the flux of electroactive material produced at the disc electrode, owing to the phase shifting of the ring current. Transport t o the ring electrode has been calculated over the complete frequency range and values of the first- order kinetic collection efficiency calculated [ 270-2721 . Adsorption of species on the disc electrode has also been studied at the RRDE using low frequencies [273].

6.5 LINEAR SWEEP VOLTAMMETRY AT THE RDE

In LSV experiments at stationary electrodes, there can be unwanted effects due to natural convection: forced convection and a uniformly accessible electrode obviate this problem. The minimum voltage scan rate at which LSV effects appear (i.e. steady-state assumptions fail) will depend on the electrode kinetics and flow parameters. We can immediately identify two extreme situations.

(i) No peak. Convection makes a much larger contribution than the sweep rate.

(ii) Well-formed peak. Sweep-rate is much more important than convection and the equations for stagnant systems apply.

The mathematical solution was first studied by Girina et al. [274] based on the same approximation as that for potential step studies, by dropping the highest-order convective term 12371, and by Fried and Elving by using the Nernst diffusion-layer concept [275] .

Cheh and co-workers [ 276-2781 also investigated LSV at the RDE for first-order reversible, quasi-reversible, and irreversible systems. Whilst for the quasi-reversible case numerical solution cannot be approximated by any analytical expression, for the other cases this is possible.

Table 6 shows the results obtained for the reversible reduction 0 t n e -+ R. For an irreversible reduction with soluble product, the three regions of peak behaviour dependent on dimensionless sweep rate, u, are the same as for a reversible reaction. In this case, for u > 10

i 5 = 0.496 a1’2a1/2 1L

and

% TABLE6 -a $ LSV a t the RDE for a reversible reaction 0 -k n e --f R in terms of dimensionless sweep-rate, u [276Ia 0 m 01 b P

No peak Transition region Peak: equations for stagnant system apply

co A ip / iL constantb ED - E1,,/mV (298 K) 9 I

2 Soluble product a < 3 3 < a < 1 0 a > l O 0.718 A

Insoluble product a < 2.7 2.7 < a < 4 a > 4 0.98 - 22/n

aDimensionless sweep rate, 0 = nFuh2 /RTD where u is sweep rate in mV s-l bip/iL = const. (nFu/RTw)’I2 ( v / D ) l# .

432

enabling calculation of a and kh . We can see from Table 6 and eqns. (219) and (220) that, besides the

suppression of natural convection, other advantages of LSV at the RDE are weak dependence on the physical properties of the electrolyte and the simultaneous determination of peak and limiting current in a single experiment.

Recently, a numerical solution has been obtained for the LSV response to a homogeneous catalytic reaction at an electroactive-monolayer-film- covered rotating disc electrode [ 2791.

6.6 HYDRODYNAMIC MODULATION AT THE RDE

This technique was proposed by Bruckenstein and co-workers [ 280, 2811 and is useful in that the current due to the modulation of the fluid flow is essentially free of any electrode surface-controlled contributions in most cases. Thus, it can be used as an analytical tool to increase sensitivity [ 2821 . Step changes were originally considered but this was later extended to sinusoidal hydrodynamic modulation (SHM) in the limiting current region and then to the region of mixed convective- diffusion/kinetic control [ 283-2871 . If the modulation frequency is a’, then the modulation, which is small, can be described by

01/2 = wg2 + Awl” cos u’t

where w o is the centre rotation speed. For small modulation, the Levich equation can be applied and deviations from it analysed directly. The current response is

i = ?+ Aicos(w,u’t-@)

Another possible modulation programme is

w = oo (1 + E cos a’t)

with

i = 7 + Ai, cos (a‘t - 41 ) + Ai2 cos (o’t -- 42 ) + . . . The problem was solved numerically for the potentiostatic case and

experimental verification provided [ 283, 2841 . Analytical expressions were derived for uniform surface concentration and uniform flux if the modulation frequency is not too high [288]. Recently, the whole frequency range, including high modulation frequencies, has been reviewed with a new theoretical solution and experimental verification [289, 2901.

Besides kinetic applications, which are still t o be fully realized, hydrodynamic modulation is useful for Schmidt number and diffusion coefficient measurements not only in Newtonian fluids but also in visco- elastic polymer solutions (Ostwald fluids) [291].

433

7. Hydrodynamic electrodes and spectroscopic techniques

Electrochemically generated radicals may be photochemically active, measurable by ESR techniques, or both. Conversely, species generated photochemically in solution may be electrochemically active. By using hydrodynamic electrodes with known flow patterns, the kinetics of these systems can be studied more easily.

7.1 ELECTROGENERATED CHEMILUMINESCENCE (ECL)

In ECL, two electrochemically produced radicals react together by homogeneous electron transfer such that one of the products is in a photochemically excited state and thus luminesces [ 2921 . The rotating ring-disc electrode is particularly suited to these studies as one radical can be generated a t the disk and the other at the ring. Digital simulation allows the evaluation of experimental data. An example is the auto- annihilation reaction from the system 9,lO-diphenylanthracene (DPA) in DMF between DPA-’ and DPA+‘ [ 2931 . The steady-state continuous production of ECL in a flow cell has been suggested for possible applications to lasers and displays [ 2941 .

7.2 OPTICALLY TRANSPARENT ELECTRODES

Rotating electrodes have been used considerably owing to the simplifi- cations introduced by the disc’s uniform accessibility.

( i ) Semi-transparent rotating disc electrode. A disc electrode is prepared by spraying a thin film of Pt, Au or SnOz on to the end of a quartz rod. Light shined through the rotating quartz rod generates species photochemically close to the electrode where they then react. Determination of the decay kinetics of the photochemically generated species, with and without homogeneous reactions before reacting on the electrode, is possible. Applications to photogalvanic cells have been described [ 295, 2961.

( i i ) Ring-disc electrode with transparent disc. Photochemically generated species from near the disc surface are detected electrochemically at the ring [297]. Clearly, the radial transport time puts an upper limit or the rate constant values that can be determined; this will be smaller than for the semi-transparent rotating disc electrode. Analysis proceeds by using theory for kinetic collection efficiencies a t the RRDE.

(iii) Ring-disc electrode with transparent ring. The transparent ring is used to record spectrophotometrically species electrochemically generated a t the disc electrode [ 298, 2991. EC mechanisms of aromatic amines have been investigated experimentally and by digital simulation and very good agreement is found [ 299,3001.


434

7.3 ESR AND TUBULAR ELECTRODES

Radicals are generated at a tubular electrode and are then transported by laminar flow into the ESR cavity which, as a downstream detector, is analogous to a second electrode. The theoretical response for the cases where the radicals are stable or decompose by first- or second-order kinetics has been derived and experimentally confirmed [ 126, 301,3021. The flow-rate dependence is different for each of the three situations which provides a diagnostic for the type of kinetics. Further information may be obtained from galvanostatic transients which allow the elucidation of electrode and radical surface processes [303]. Very recently, an in situ channel tube electrode has been described for electrochemical ESR which also allows shorter-lived species to be observed and smaller surface coverages to be analysed [ 304-3061 .

References

1 2 3

4 5 6

7 8

9

1 0

11 1 2 1 3 1 4

1 5

1 6 17 18 19. 20

21

22 23 24

J. Heyrovsk?, Chem. Listy, 16 (1922) 256. V.G. Levich, Discuss. Faraday Soc., 1 (1947) 37. V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962. H. Schliehting, Boundary Layer Theory, Pergamon Press, London, 1955. I.M. Kolthoff and J.J. Lingane, Polarography, Wiley, New York, 2nd edn., 1952. J. Heyrovsky and J. K ~ t a , Principles of Polarography, Academic Press, London, 1965. A.C. Riddiford, Adv. Electrochem. Electrochem. Eng., 4 (1966) 47. R.N. Adams, Electrochemistry at Solid Electrodes, Marcel Dekker, New York, 1969, pp. 67-114. W.J. Albery and M.L. Hitchman, Ring-Disc Electrodes, Clarendon Press, Oxford, 1971. Yu.V. Pleskov and V.Yu. Filinovskii, The Rotating Disc Electrode, Consultants Bureau, New York, 1976. F. Opekar and P. Beran, J. Electroanal. Chem., 69 (1976) 1. S. Bruckenstein and B. Miller, Ace. Chem. Res., 1 0 (1977) 54. E. Pungor, Zs. Feher and M. Viradi, Crit. Rev. Anal. Chem., 9 (1980) 97. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Wiley, New York, 1980. J.S. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973. L. Prandtl, Proc. I11 Int. Math. Congr., Heidelberg, 1904. T. von Karman, Z . Angew. Math. Mech., 1 (1921) 233. W.G. Cochran, Proc. Cambridge Philos. SOC. Math. Phys. Sci., 30 (1934) 365. V.G. Levich, Acta Physicochim. URSS, 17 (1942) 257. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, p. 1019. M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965, p. 446. W.H. Smyrl and J. Newman, J. Electrochem. Soc., 118 (1971) 1079. D.P. Gregory and A.C. Riddiford, J. Chem. Soc., (1956) 3756. J. Newman, J. Phys. Chem., 70 (1966) 1327.

43 5

25 26 27 28 29 30

31 32 33 34

35

36

37 38 39 40 41 42 43 44 45

46 47

48 49 50 51 52 53 54

55 56 57 58 59 60 61 62

63 64

65 66 67

S. Bruckenstein, J. Electrochem. Soc., 122 (1975) 1215. A.N. Frumkin and L.I. Nekrasov, Dokl. Akad. Nauk SSSR, 126 (1959) 115. Y.B. Ivanov and V.G. Levich, Dokl. Akad. Nauk SSSR, 126 (1959) 1029. W.J. Albery, Trans. Faraday SOC., 62 (1966) 1915. W.J. Albery and S. Bruckenstein, Trans. Faraday SOC., 62 (1966) 1920. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Wiley, New York, 1980, Chap. 8. K.B. Prater and A.J. Bard, J. Electrochem. SOC., 117 (1970) 207. B. Miller and R.E. Visco, J. Electrochem. SOC., 115 (1968) 251. G.K. Dikusar, Electrokhimiya, 11 (1975) 1411. V.Ju. Filinovskii, I.V. Kadija, B.Z. Nicolic and M.B. Nakic, J. Electroanal. Chem., 54 (1974) 39 and references cited therein. J. Margarit and M. Levy, J. Electroanal Chem. 49 (1974) 369; J. Margarit, G. Dabosi and M. Levy, Bull. SOC. Chim. Fr., 7-8 (1975) 1509. S. Clarenbach, E.W. Grabner and E. Brauer, Ber. Bunsenges. Phys. Chem., 77 (1973) 908. J.B. Flanagan and L. Marcoux, J. Phys. Chem., 78 (1974) 718. H. Gerischer, I. Mattes and R. Braun, J. Electroanal. Chem., 10 (1965) 553. R. Braun, J. Electroanal. Chem., 19 (1968) 23. W.J. Albery, C.M.A. Brett and P. Wood, unpublished results. M.B. Glauert, J. Fluid Mech., 1 (1956) 625. M.T. Scholtz and 0. Trass, AIChE., 9 (1963) 548. W.J. Albery and C.M.A. Brett, J. Electroanal. Chem., 148 (1983) 201. W.J. Albery and C.M.A. Brett, J. Electroanal. Chem., 148 (1983) 211. W.J. Albery, C.M.A. Brett and A.M. Oliveira Brett, in E. Pungor, I. Buzas and G.E. Veress (Eds.), Modern Trends in Analytical Chemistry, Elsevier, Amsterdam, 1984. J. Yamada and H. Matsuda, J. Electroanal. Chem., 44 (1973) 189. N. Frossling, cited in H. Schlichting, Boundary Layer Theory, Pergamon Press, London, 1955, pp. 73-75. D.-T. Chin and C.H. Tsang, J. Electrochem. SOC., 125 (1978) 1461. W.J. Albery and S. Bruckenstein, J. Electroanal. Chem., 144 (1983) 105. D.-T. Chin and R.R. Chandran, J. Electrochem. SOC., 128 (1981) 1904. H.Do Duc, J. Appl. Electrochem., 10 (1980) 385. D. Ilkovic, Collect Czech. Chem. Commun., 6 (1934) 498. D. MacGillawy and E.K. Rideal, Rec. Trav. Chim., 56 (1937) 1013. J. Koutecky and M. von Stackelberg, in P. Zuman (Ed.), Progress in Polargraphy, Vol. 1, Interscience, New York, 1962 pp. 21-42 and references cited therein. J. Newman, J. Electroanal. Chem., 15 (1967) 309. J.J. Lingane and B.A. Loveridge, J. Am. Chem. SOC., 72 (1950) 438. J. Koutecky, Czech. J. Phys., 2 (1953) 50. H. Matsuda, Bull. Chern. SOC. Jpn., 26 (1953) 342. T.E. Cummings and P.J. Elving, Anal. Chem., 50 (1978) 480. T.E. Cummings and P.J. Elving, Anal. Chem., 50 (1978) 1980. H.H. Bauer, Electroanal. Chem., 8 (1975) 170. S. Levine, H.B. Mead, R.N. O’Brien and A.C. Brett, J. Electroanal. Chem., 82 (1977) 227. R. Guidelli, J. Electroanal. Chem., 100 (1979) 711. A.N. Frumkin, N.V. Fedorovich, B.B. Damaskin, E.V. Stenina and V.S. Krylov, J. Electroanal. Chem., 50 (1974) 103. J.H. Bauer and A.K. Shallal, J. Electroanal. Chem., 7 3 (1976) 367. D.A. Berman, P.R. Saunders and R.J. Winder, Anal. Chem., 23 (1951) 1040. R.E. Cover and J.G. Connery, Anal. Chem., 41 (1969) 918,1191.

436

68

69 70 71

72 73 74

75 76 77 78 79 80 81 82 8 3 8 4 8 5 86

87

88 89 90

91

92 9 3 94

95 96

97

98 99

100 101

102 103 104 105 106 107

108

R.E. Cover and J.T. Folliard, J. Electroanal. Chem., 30 (1971) 143; 33 (1971) 463. M. Lovrii., T. Magjer and M. Branica, Croat. Chem. Acta, 57 (1984) 157. S. Wolf, Angew. Chem. , 72 (1960) 449. A.M. Bond and R.J. O’Hal.oran, J. Electroanal. Chem., 68 (1976) 257; 97 (1979) 107. D.R. Canterford, J. Electroanal. Chem., 77 (1977) 113; 97 (1979) 113. T. Magjer, M. Lowik and M. Branica, Croat. Chem. Acta, 53 (1980) 101. I.M. Kolthoff and Y. Okinaka, in P. Zuman and I.M. Kolthoff (Eds.), Progress in Polarography, Vol. 2, Wiley, New York, 1962 p. 357. H.J. Mortko and R.E. Cover, Anal. Chem., 45 (1973) 1983. H. Matsuda, J. Electroanal. Chem., 15 (1967) 109. H. Matsuda, J . Electroanal. Chem., 16 (1968) 153. D.-T. Chin, J. Electrochem. Soc., 118 (1971) 1434. J. Newman, J. Electrochem. Soc., 119 (1972) 69. D.-T. Chin, J. Electrochem. SOC., 119 (1972) 1699. J.T. Kim and J. Jorne, J. Electrochem. Soc., 126 (1979) 1937. E. Kirowa-Eisner and E. Gileadi, J. Electrochem. Soc., 123 (1976) 22. S.L. Marchiano and A.J. Arvia, Electrochim. Acta, 1 2 (1967) 801. H. Matsuda and J. Yamada, J. Electroanal. Chem., 30 (1971) 261. H. Matsuda, J. Electroanal. Chem., 38 (1972) 159. S. Hamada, M. Itoh, H. Matsuda and J. Yamada, J. Electroanal. Chem., 91 (1978) 107. J.S. Newman, Electrochemical Systems, Prentice-Hall, Englewood Cliffs, NJ, 1973, Chaps. 18-20. V.A. Belyaeva, Zh. Fiz. Khim., 36 (1962) 1385. W.J. Albery and M.L. Hitchman, Trans. Faraday Soc., 67 (1971) 2408. P. Pierini and J. Newman, J. Electrochem. Soc., 125 (1978) 79 and references cited therein. P. Pierini and J. Newman, J. Electrochem. Soc., 124 (1977) 701 and references cited therein. Ch. Dorfel, D. Rahner and W. Forker, J. Electroanal. Chem., 107 (1980) 257. M. Shabrang and S. Bruckenstein, J. Electrochem. Soc., 122 (1975) 1305. R. Alkire and A.A. Mirarefi, J. Electrochem. Soc., 120 (1973) 1507; 124 (1977) 1043,1214. L. Meites, Polarographic Techniques, Wiley, New York, 2nd edn., 1965. J. Heyrovsky and P. Zuman, Practical Polarography, Academic Press, New York, 1968. D.T. Sawyer and J.L. Roberts, Experimental Electrochemistry for Chemists, Wiley, New York, 1974. E. Bishop, IUPAC Meeting Commission V-5,1975. J.O. Besenhard and H.P. Fritz, Angew, Chem. Int. Ed. Engl., 22 (1983) 950. W. van der Linden and J.W. Dieker, Anal. Chim. Acta, 119 (1980) 1. R.C. Engstrom and V.A. Strasser, Anal. Chem., 56 (1984) 136 and references cited therein. See, for example, T. Geiger and F.C. Anson, Anal. Chem., 52 (1980) 2448. R.N. Adams, Anal. Chem., 30 (1958) 1576. E. Pungor, E. Svepesviry and J. Havas, Anal. Lett., 1 (1968) 213. C. Olson and R.N. Adams, Anal. Chim. Acta, 22 (1960) 582. J. Wang, Anal. Chem., 5 3 (1981) 2280. L.N. Klatt, D.R. Connell, R.E. Adams, I.L. Honigberg and J.C. Price, Anal. Chem., 47 (1975) 2470. J.E. Anderson, D.E. Tallman, D.J. Chesney and J.L. Anderson, Anal. Chem., 50 (1978) 1051.

437

J. Lindquist, J. Electroanal. Chem., 52 (1974) 37. M. Stulikovi, J. Electroanal. Chem., 48 (1973) 33. K.F. Blurton and A.C. Riddiford, J. Electroanal. Chem., 1 0 (1965) 457. See, for example, M.P. Belyanchikov, Yu.V. Pleskov and V.G. Pominov, Zh. Fiz. Khim., 34 (1960) 782. J. Wojtowicz and B.E. Conway, J. Electroanal. Chem., 13 (1967) 333. T. Hepel and M. Hepel, J. Electroanal. Chem., 112 (1980) 365. Yu. V. Pleskov and V.Yu. Filinovskii, The Rotating Disc Electrode, Consultants Bureau, New York, 1976, pp. 365-367. F. Opekar and P. Beran, J. Electroanal. Chem., 69 (1976) pp. 77-80. G.V. Zhutaeva and N.A. Shumilova, Elektrokhimiya, 2 (1966) 606. A.N. Doronin, Elektrokhimiya, 4 (1967) 1193. G.W. Harrington, H.A. Laitinen and V. Trendafilov, Anal. Chem., 45 (1973) 433. D.R. Reinus, M.S. Grilikhes, M. Ya. Zarzhevskii and Yu. A. Katyshev, Elektrokhimiya, 1 5 (1979) 228. S. Menezes, L.F. Schneemeyer and B. Miller, J. Electrochem. SOC., 128 (1981) 2167. M.A. Sokolov, B.K. Filanovskii, Yu. N. Shestopalov and M.S. Grilikhes, Elektrokhimiya, 15 (1979) 718. J. Phillips, R.J. Gale, R.G. Wier and R.A. Osteryoung, Anal. Chem., 48 (1976) 1266. D.K. Roe and M. Aparicio-Razo, Anal. Chem., 56 (1984) 118. W.J. Blaedel, C.L. Olson and L.R. Sharma, Anal. Chem., 35 (1963) 2100. W.J. Albery, B.A. Coles and A.M. Couper, J. Electroanal. Chem., 65 (1975) 901. W.J. Albery, C.M.A. Brett, J.F. Burke and P. Wood, unpublished work. K. Stulik and V. Hora, J. Electroanal. Chem., 70 (1976) 253. P.L. Meschi and D.C. Johnson, Anal. Chem., 52 (1980) 1304. D.T. Sawyer and J.L. Roberts, Experimental Electrochemistry for Chemists, Wiley, New York, 1974, pp. 133-143. See, for example, G.E. Troshin, A.N. Domarev, F.F. Lakomov, V.I. Kumantsov and I.M. Sosonkin, Elektrokhimiya, 17 (1981) 897. M.B. Bardin and A.I. Dikusar, Elektrokhimiya, 6 (1970) 1147. C.M. Mohr, Jr. and J. Newman, J. Electrochem. SOC., 122 (1975) 928. D.T. Sawyer, R.S. George and R.C. Rhodes, Anal. Chem., 31 (1959) 2. G. Nagy, Zs. Feh& and E. Pungor, Anal. Chim. Acta, 52 (1970) 47. C.B. Ranger, Anal. Chem., 5 3 (1981) 20A. See, for example, J.J. Lingane, Ind. Eng. Chem. Anal. Ed., 16 (1944) 329. J. Heyrovsky and P. Zuman, Practical Polarography, Academic Press, New York, 1968, p. 5. A.J. Bard and L.R. Faulkner, Electrochemical Methods, Fundamentals and Applications, Wiley, New York, 1980, Chap. 13. See, for example, S.C. Creason and R.F. Nelson, J. Electroanal. Chem., 21 (1969) 549; 27 (1970) 189. P. He, J.P. Avery and L.R. Faulkner, Anal. Chem., 54 (1982) 1313A. S.C. Creason, J.W. Hayes and D.E. Smith, J. Electroanal. Chem., 47 (1973) 9. See, for example, F. Opekar and P. Beran, J. Electroanal. Chem., 69 (1976) pp. 41-43. See, for example, D. Jahn and W. Vielstich, J. Electrochem. SOC., 109 (1962) 849. D.A. Frank-Kamenetskii, cited in V. Levich, Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 74. P. Delahay, J. Am. Chem. SOC., 75 (1953) 1430.

109 110 111 112

113 114 115

116 117 118 119

120

121

122

123

124 1 2 5 126 127 128 129 130

131

132 133 134 135 136 137 138

139

140

1 4 1 142 143

144

145

146 147 J. Koutecky, Chem. Listy, 47 (1953) 323.

438

148 149 150 151 152 153 154

155 156 157 158

159 160 161 162 163 164

165 166 167

168 169 170

171 172 173 174 175 176

177

178 179 180 181 182

183

184

185 186 187 188 189 190 191 192

J. Koutecky, Collect. Czech. Chem. Commun., 18 (1953) 597. H. Matsuda and Y. Ayabe, Bull. Chem. SOC. Jpn., 28 (1955) 422. I. Rufid, J. Electroanal. Chem., 49 (1974) 407. J. Koutecky and V.G. Levich, Dokl. Akad. Nauk SSR, 117 (1957) 441. H. Matsuda, J. Electroanal. Chem., 35 (1972) 77. K. Tokuda and H. Matsuda, J. Electroanal. Chem., 44 (1973) 199. W.J. Albery, S. Bruckenstein and D.T. Napp, Trans. Faraday Soc., 62 (1966) 1932. H. Matsuda, J. Electroanal. Chem., 15 (1967) 325. L.N. Klatt and W.J. Blaedel, Anal. Chem., 15 (1967) 1065. K. Tokuda and H. Matsuda, J. Electroanal. Chem., 52 (1974) 421. R. Pollard, W.P. Kwan and C.G. Law, Jr., J. Electroanal. Chem., 139 (1982) 275. I. Rufik, A. Barik and M. Branica, J. Electroanal. Chem., 29 (1971) 411. R. Guidelli, J. Electroanal. Chem., 18 (1968) 5. R. BrdiEka, Collect. Czech. Chem. Commun., 12 (1947) 522. E. Laviron, Electroanal. Chem. 12 (1982) 53 and references cited therein. E. Laviron, J. Electroanal. Chem., 124 (1981) 19; 140 (1982) 247. B. Miller, M.I. Bellavance and S. Bruckenstein, J. Electrochem. Soc., 118 (1971) 1082. J.D.E. McIntyre, J. Phys. Chem., 7 1 (1967) 1196. W.J. Albery, Electrode Kinetics, Clarendon Press, Oxford, 1975, pp. 132-140. J. Cipak, I. Rufic and Lj. Jefic, J. Electroanal. Chem., 75 (1977) 9 and references cited therein. I. Rufik, J. Electroanal. Chem., 52 (1974) 331. I. RuZik and D.E. Smith, J. Electroanal. Chem., 57 (1974) 129. J . Galvez, R. Saura, A. Molina and T. Fuente, J. Electroanal. Chem., 139 (1982) 15. M. Lovrih, J. Electroanal. Chem., 102 (1979) 143. S. Komorsky and M. Lovrih, J. Electroanal. Chem., 112 (1980) 169. M. Lovrih, J. Electroanal. Chem., 144 (1983) 45; 153 (1983) 1. Yu. B. Ivanov and V.G. Levich, Dokl. Akad. Nauk SSSR, 126 (1959) 1029. L.N. Nekrasov, Elektrokhimiya, 2 (1966) 438. K.-L. Hsueh, D.-T. Chin and S. Srinivasan, J. Electroanal. Chem., 153 (1983) 79 and references cited therein. A. Damjanovic, M.A. Genshaw and J.O’M. Bockris, J. Chem. Phys., 45 (1966) 4057. J.D.E. McIntyre, J. Phys. Chem., 73 (1969) 4111. V. Yu. Filinovskii, Elektrokhimiya, 5 (1969) 991. L.N. Nekrasov and E.N. Potapova, Elektrokhimiya, 6 (1970) 806. L.N. Nekrasov and T.K. Zolotova, Elektrokhimiya, 4 (1968) 864. R. BrdiEka, V. H a n d and J. Koutecky, in P. Zuman and I.M. Kolthoff (Eds.), Progress in Polarography, Vol. 1, Wiley, New York, 1962, p. 145. S.G. Mairanovskii, Catalytic and Kinetic Waves in Polarography, Plenum Press, New York, 1968. Yu.V. Pleskov and V.Yu. Filinovskii, The Rotating Disc Electrode, Consultants Bureau, New York, 1976, Chap. 6. J.M. Hale, J. Electroanal. Chem., 8 (1964) 332. J . Koutecky, Collect. Czech. Chem. Commun., 19 (1954) 857. J. Koutecky and J. Cizek, Collect. Czech. Chem. Commun., 21 (1956) 836. C. Nishihara and H. Matsuda, J. Electroanal. Chem., 51 (1974) 287. J . Weber and J. Koutecky, Collect. Czech. Chem. Commun., 20 (1955) 980. C. Nishihara and H. Matsuda, J. Electroanal. Chem., 73 (1976) 261. H. Matsuda, J. Electroanal. Chem., 56 (1974) 165. J. Galvez, A. Serna, A. Molina and D. Marin, J. Electroanal. Chem., 102 (1979) 277.

439

J. Galvez, A. Molina and T. Fuente, J. Electroanal. Chem., 107 (1980) 217. J. Galvez, R. Saura and A. Molina, J. Electroanal. Chem., 147 (1983) 53. C. Nishihara and H. Matsuda, J. Electroanal. Chem., 103 (1979) 261. K. Tokuda, K. Aoki and H. Matsuda, J. Electroanal. Chem., 80 (1977) 211. D. Haberland and R. Landsberg, Ber. Bunsenges. Phys. Chem., 70 (1966) 724. P. Beran and S. Bruckenstein, J. Phys. Chem., 72 (1968) 3630. J. Koutecky, Collect. Czech. Chem. Commun., 18 (1953) 311. L.N. Klatt and W.J. Blaedel, Anal. Chem., 40 (1968) 512. K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 76 (1977) 217. Yu.V. Pleskov and V. Yu. Filinovskii, The Rotating Disc Electrode, Consultants Bureau, New York, 1976, p. 234. R. BrdiEka, V. H a n d and J . Koutecky, in P. Zuman and I.M. Kolthoff (Eds.), Progress in Polarography, Vol. 1, Wiley, New York, 1962, p. 178. J. Ulstrup, Electrochim. Acta, 1 3 (1968) 1717. K. Holub, J. Electroanal. Chem., 30 (1971) 71. L.K.J. Tong, K. Liang and W.R. Ruby, J. Electroanal. Chem., 13 (1967) 245. Z . Galus and R.N. Adams, J. Electroanal. Chem., 4 (1962) 248. B.A. Coles and R.G. Compton, J. Electroanal. Chem., 127 (1981) 37. P.A. Malachesky, L.S. Marcoux and R.N. Adams, J . Phys. Chem., 70 (1966) 4068. S. Karp, J. Phys., Chem., 72 (1968) 1082. L.S. Marcoux, R.N. Adams and S.W. Feldberg, J. Phys. Chem., 7 3 (1969) 2611. R.S. Nicholson, J.M. Wilson and M.L. Olmstead, Anal. Chem., 38 (1966) 542. H. Sobel and D.E. Smith, J. Electroanal. Chem., 26 (1970) 271. I. RuHii, H. Sobel and D.E. Smith, J. Electroanal. Chem., 65 (1975) 21. J. Galvez and A. Molina, J. Electroanal. Chem., 110 (1980) 49. J . Galvez, A. Molina, R. Saura and F. Martinez, J. Electroanal. Chem., 127 (1981) 17. C. Nishihara and H. Matsuda, J. Electroanal. Chem., 85 (1977) 17. I. RuZib, D.E. Smith and S.W. Feldberg, J. Electroanal. Chem., 52 (1974) 157. I. RuZik and D.E. Smith, J. Electroanal. Chem., 58 (1975) 145. W.J. Albery, S. Bruckenstein and D.C. Johnson, Trans. Faraday Soc., 62 (1966) 1938. W.J. Albery and S. Bruckenstein, Trans. Faraday SOC., 62 (1966) 2584. W.J. Albery, M.L. Hitchman and J. Ulstrup, Trans. Faraday Soc., 65 (1969) 1101. K.B. Prater and A.J. Bard, J. Electrochem. SOC., 117 (1970) 335. K.B. Prater and A.J. Bard, J. Electrochem. SOC., 117 (1970) 1517. V.J. Puglisi and A.J. Bard, J. Electrochem. Soc., 119 (1972) 833. W.J. Albery and S. Bruckenstein, Trans. Faraday SOC., 62 (1966) 1946. W.J. Albery, J.S. Drury and M.L. Hitchman, Trans. Faraday SOC., 67 (1971) 2162. W.J. Albery, Trans Faraday SOC., 6 3 (1967) 1771. W.J. Albery and M.L. Hitchman, Ring-Disc Electrodes, Clarendon Press, Oxford, l,971, p. 127.

193 194 195 196 197 198 199 200 201 202

203

204 205 206 207 208 209

210 211 212 213 214 215 216

217 218 219 220

22 1 222

223 224 225 226 227

228 229

230

231

232 233 234

235

W.J. Albery, M.L. Hitchman and J. Ulstrup, Trans. Faraday SOC., 64 (1968) 2831. P.A. Malachesky, K.B. Prater, G. Petrie and R.N. Adams, J. Electroanal. Chem., 16 (1968) 41. K. Aoki, K. Tokuda and H. Matsuda, J. Electroanal. Chem., 79 (1977) 49. K. Aoki and H. Matsuda, J. Electroanal. Chem., 94 (1978) 157. C. Deslouis, I. Epelboin, C. Gabrielli, Ph. Sainte-Rose Fanchine and B. Tribollet, J. Electroanal. Chem., 107 (1980) 193. C. Deslouis, C. Gabrielli and B. Tribollet, J. Electrochem. SOC., 130 (1983) 2044.

440

236 J. Wang, Talanta, 28 (1981) 369. 237 V.Yu. Filinovskii and V.A. Kiryanov, Dokl. Akad. Nauk SSSR, 156 (1964) 1412. 238 R.P. Buck and H.E. Keller, Anal. Chem., 35 (1963) 400. 239 S. Bruckenstein and S. Prager, Anal. Chem., 39 (1967) 1161. 240 K. Viswanathan and H.Y. Cheh, J. Appl. Electrochem., 9 (1979) 537. 241 G.C. Barker and I.L. Jenkins, Analyst, 77 (1952) 685. 242 G.C. Barker and A.W. Gardner, Z. Anal. Chem., 173 (1960) 79. 243 L.F. Roeleveld, B.J.C. Wetsema and J.M. Los, J. Electroanal. Chem., 75 (1977)

839 and references cited therein. 244 J. Galvez, A. Molina and C. Serna, J. Electroanal. Chem., 124 (1981) 201 and

references cited therein. 245 H.J.S. Sand, Philos. Mag., l ( l 9 0 1 ) 45. 246 H.L. Kies, J. Electroanal. Chem., 45 (1973) 71, 76 (1977) 1. 247 J. Galvez, J. Electroanal. Chem., 132 (1982) 15. 248 J. Galvez and A. Molina, J. Electroanal. Chem., 146 (1983) 221. 249 J. Galvez and R. Saura, J. Electroanal. Chem., 146 (1983) 233. 250 J. Galvez, T. Fuente, A. Molina and R. Saura, J. Electroanal. Chem., 146 (1983)

243. 251 Yu.G. Siver, Zh. Fiz. Khim., 34 (1960) 577. 252 J.M. Hale, J. Electroanal. Chem., 6 (1963) 187. 253 V.I. Chernenko, Elektrokhimiya, 7 (1971) 820. 254 D.A. Scherson and P.N. Ross, J . Electroanal. Chem., 119 (1981) 209. 255 F.A. Posey and R.E. Meyer, J. Electroanal. Chem., 30 (1971) 359. 256 K. Aoki and H. Matsuda, J. Electroanal. Chem., 90 (1978) 333. 257 W.J. Alhery and M.L. Hitchman, Ring-Disc Electrodes, Clarendon Press,

Oxford, pp. 149-151. 258 S. Bruckenstein, A. Tokuda and W.J. Albery, J. Chem. SOC. Faraday Trans. I, 73

(1977) 823. 259 W.J. Albery, M.G. Boutelle, P.J. Colby and A.R. Hillman, J. Chem. Soc. Faraday

Trans. I, 7 8 (1982) 2757. 260 K. Viswanathan, M.A. Farrell-Epstein and H.Y. Cheh., J. Electrochem. Soc., 125

(1978) 1772. 261 K. Viswanathan and H.Y. Cheh., J. Electrochem. Soc., 125 (1978) 1616. 262 P.C. Andricacos and H.Y. Cheh, J. Electroanal. Chem., 121 (1981) 133. 263 D.-T. Chin, J. Electrochem. Soc., 130 (1983) 1657. 264 K. Tokuda and H. Matsuda, J. Electroanal. Chem., 8 2 (1977) 157. 265 R.G. Compton and G.R. Sealy, J. Electroanal. Chem., 145 (1983) 35. 266 K. Tokuda and H. Matsuda, J. Electroanal. Chem., 90 (1978) 149. 267 E. Levart and D. Schuhmann, J. Electroanal. Chem., 53 (1974) 77. 268 K. Tokuda and H. Matsuda, J. Electroanal. Chem., 95 (1979) 147. 269 D.E. Smith, Electroanal. Chem., l ( 1 9 6 6 ) 1. 270 W.J. Albery, J.S. Drury and A.P. Hutchinson, Trans. Faraday Soc., 67 (1971)

2414. 271 W.J. Albery, A.H. Davis and A.J. Mason, Discuss. Faraday Soc., 56 (1973) 317. 272 W.J. Alhery, R.G. Compton and A.R. Hillman, J. Chem. SOC. Faraday Trans. I,

74 (1978) 1007. 273 W.J. Alhery and A.R. Hillman, J. Chem. Soc. Faraday Trans. I, 75 (1979) 1623. 274 G.P. Girina, V.Yu. Filinovskii and L.G. Feoktistov, Elektrokhimiya, 3 (1967)

941. 275 I. Fried and P.J. Elving, Anal. Chem., 37 (1965) 464, 803. 276 P.C. Andricacos and H.Y. Cheh, J. Electrochem. Soc., 127 (1980) 2153, 2385. 277 P.C. Andricacos and H.Y. Cheh, J. Electroanal. Chem., 124 (1981) 95. 278 G.C. Quintana, P.C. Andricacos and H.Y. Cheh, J. Electroanal. Chem., 144

(1983) 77.

441

M. Low% and M. Branica, Electrochim. Acta, 28 (1983) 1261. B. Miller, M.J. Bellavance and S. Bruckenstein, Anal. Chem., 44 (1972) 1983. S. Bruckenstein, M.I. Bellavance and B. Miller, J . Electrochem. SOC., 120 (1973) 1351. B. Miller and S. Bruckenstein, Anal. Chem., 46 (1974) 2026, 2033. B. Miller and S. Bruckenstein, J. Electrochem. SOC., 121 (1974) 1558. K. Tokuda, S. Bruckenstein and B. Miller, J. Electrochem. Soc., 122 (1975) 1316. K. Tokuda and S. Bruckenstein, J. Electrochem. SOC., 126 (1979) 431. Y. Kanzaki and S. Bruckenstein, J. Electrochem. Soc., 126 (1979) 437. C. Deslouis, I. Epelboin, C. Gabrielli and B. Tribollet, J. Electroanal. Chem., 82 (1977) 251. W.J. Albery, A.R. Hillman and S. Bruckenstein, J. Electroanal. Chem., 100 (1979) 687. C. Deslouis, C. Gabrielli, Ph. Sainte-Rose Fanchine and B. Tribollet, J . Electrochem. Soc., 129 (1982) 107. B. Tribollet and J. Newman, J. Electrochem. Soc., 130 (1983) 2016. C. Deslouis and B. Tribollet, J. Electroanal. Chem., 142 (1982) 95. L.R. Faulkner and A.J. Bard, Electroanal. Chem., 1 0 (1977) 1. J.T. Maloy, K.B. Prater and A.J. Bard, J. Am. Chem. Soc., 93 (1971) 5959. G.H. Brilmyer and A.J. Bard, J. Electrochem. Soc., 127 (1980) 104. W.J. Albery, M.D. Archer and R.G. Egdell, J. Electroanal. Chem., 82 (1977) 199. W.J. Albery, W.R. Bowen, F.S. Fisher and A.D. Turner, J. Electroanal. Chem., 107 (1980) 1,11. D.C. Johnson and E.W. Resnick, Anal. Chem., 44 (1972) 637. J.E. McClure, Anal. Chem., 42 (1970) 551. H. Debrodt and K.E. Heusler, Ber. Bunsenges. Phys. Chem., 81 (1977) 1172. H. Debrodt, K.E. Heusler and E.W. Grabner, Ber. Bunsenges. Phys. Chem., 83 (1979) 1019. W.J. Albery, A.T. Chadwick, B.A. Coles and N.A. Hampson, J. Electroanal. Chem., 75 (1977) 229. W.J. Albery, R.G. Compton, A.T. Chadwick, B.A. Coles and J.A. Lenkait, J. Chem. SOC. Faraday Trans. I, 76 (1980) 1391. W.J. Albery and R.G. Compton, J. Chem. SOC. Faraday Trans. I, 78 (1982) 1561. B.A. Coles and R.G. Compton, J. Electroanal. Chem., 144 (1983) 87. R.G. Compton, D.J. Page and G.R. Sealy, J. Electroanal. Chem., 161 (1984) 129. R.G. Compton, D.J Page and G.R. Sealy, J. Electroanal. Chem., 163 (1984) 65.

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[Comprehensive Chemical Kinetics] Electrode Kinetics: Principles and Methodology Volume 26 || Chapter 5 Hydrodynamic Electrodes