The semitransparent rotating disc electrode

J. Electroanal. Chem., 82 (1977) 199--208

199

© Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

THE SEMITRANSPARENT ROTATING DISC ELECTRODE *

W. JOHN ALBERY, MARY D. ARCHER *1 and RUSSELL G. EGDELL *2

Physical Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ (England) (Received 17th January 1977 )

ABSTRACT At a semitransparent rotating disc electrode, species are generated photochemically close to the electrode and they then react on the electrode. The theory is developed for a species being generated, transported by convection and diffusion and decaying by first order kinetics. The theory is tested with the ferrioxalate system. The shape of the photocurrent voltage curves show that photogenerated trisoxalatoiron(II) reacts on the electrode. The limiting photocurrent decreases with increasing rotation speed. Reasonable agreement with theory is found.

INTRODUCTION

This paper describes both theory and experiments for the semitransparent rotating disc electrode. In order to combine photochemistry and electrochemistry there are great advantages in using a semitransparent electrode. If one shines light through the solution at the electrode then the solution has to be transparent enough to allow the passage of the light to the electrode. This means that only a small fraction of the light can be absorbed close to the electrode. When one shines the light through the electrode, then the solution can contain much higher concentrations of absorbing species so that a large part of the light can be absorbed in the diffusion layer of the electrode. The reason for rotating the electrode is that one controls the supply of reactant to the electrode; a genuine steady state can be established and the rotation speed is a us.eful experimental variable. The purpose of the experiments is two fold. Firstly, one can use electrochemical detection to m o n i t o r photochemical intermediates and products and secondly, one can use photochemical generation to make unstable species with interesting electrochemistry, for instance intermediates in electrochemical reactions [ 1 ]. The electrode is particularly useful for the investigation of photogalvanic cells. This paper describes the theory of the electrode and its application to the trisoxalatoiron(III) system.

* In honour of the 60th birthday of Benjamin G. Levich. *1 Now at Department of Physical Chemistry, Cambridge. ,2 Now at Inorganic Chemistry Laboratory, Oxford.

200

THE BASIC DIFFERENTIAL EQUATION

We assume that the rotating disc electrode is uniformly illuminated and that the intensity of the light is not so large that the solution becomes bleached. Then the basic convective diffusion equation for the semitransparent rotating disc electrode for the system, Aa-~B ~ products is D ( a 2 b / D x 2) + V x ( ~ b / ~ x ) + ¢Io ea e x p ( - - e a x ) - - k b = 0

(1)

where a is the concentration of the absorbing species, b is the concentration of the photochemical product or intermediate, e is the extinction coefficient (on the natural logarithmic scale) for A at the wavelength of light used, I0 is the flux of light at the electrode surface in Einstein cm - 2 s- 1 , k describes a first order process which removes B, ¢ is the q u a n t u m efficiency, and Vx is the velocity of flow towards the electrode. This equation is the same as the usual equation for a rotating disc electrode, except that it contains the I0 term which describes the photochemical generation of B. The intensity of the light decreases according to the Beer-Lambert law, and the concentration of A is assumed to be large enough for a to be taken as constant. The system has four characteristic lengths: the thickness of the h y d r o d y n a m i c layer XH = (V/CO)1/2 the thickness of the diffusion layer XD = A 1 ( D / C ) 113

(2)

the thickness of the reaction layer X k -'- ( D / h ) 112 and thickness of the absorbance layer X~ = (ea) - 1

(3)

where [2] C = 0.510

LO3/2

p--1/2

(4)

co is the rotation speed in radians s- i , v is the kinematic viscosity, and A 1 = 1.288. The length XD describes the stagnant layer on the electrode. For x < XD transport is mainly by diffusion while for x > X D transport is mainly by convection. The length X k describes the distance over which B can diffuse before it is destroyed. When X k is smaller than X D then B must be generated within a distance of about three times Xk of the electrode or else it will not have much chance of reaching the electrode. When X k is greater than XD the situation is more complicated since convection is a much more efficient transport mechanism than diffusion and Xk compares diffusion to kinetics. The length X e describes the distance over which the light is absorbed according to the Beer-Lambert law. We now introduce three further parameters which describe the ratio of the thickness of the diffusion layer to that of the other layers: Ho = X D / X H = A I ( D / 0 . 5 1 v) 113 ~ 10 -1 K, ----X D / X

k

(5)

201

(6)

= XD/Xe

We normalise the concentration b with b = u¢fl X D I o / D

and the distance in the direction normal to the electrode with × = X/XD

Then from eqn. (1) a2u/ax 2 + ( v x X n / D )

a u / a x + exp(--flX) -- ~:2u = 0

(7)

To calculate the limiting current due to B, we assume that the electrode destroys all the B that reaches it giving the boundary condition a t x = 0,

u=O

and we have as well a s X --> oo

u ---> 0

We are then interested in the flux of material reaching the electrode given by j = D ( a b / a x ) x = O = ¢flIou'o

where u~ (a b/0X)x=O. We define a photoelectrochemical collection efficiency to describe the recovery of the photogenerated species: (8)

g h v = j / I o = ¢bflU'o

THREE ZONES FOR CONVECTIVE

TERM

To solve eqn. (7) we consider three zones. First inside the diffusion layer (× < 1), we ignore the convective term and simplify the equation to ~2U/~X2 + exp(--fl×)

-- K2u = 0

(9)

Outside the diffusion layer (× > 1) we ignore the diffusion term (a2u/ax2). The velocity towards the disc is given by [3] Vx = (coy) 1/2 H ( x / X H ) = (~v) x/2 H(Ho× ) where the function H is shown in Fig. 1. As shown in Fig. 1 for x <

~X H

or

X < XH

Vx ~- Cx 2

(10)

and f o r X > X H O r × > X H Vx ~- 0.88 (COY)1/2

(11)

where XH = 1.3 H~01

(12)

Substitution of eqn. (10) or (11) in eqn. (8) then gives us t w o approximate dif-

202

..... ~,/i

--

I~

v ori

li

ogi

' I

H I

I

It /i

i ~

/

/

I

i i/

~

/

/

Ii

/

z

0.0~i 5

n 15

I0

n 20

Z5

Fig. 1. V a r i a t i o n o f H, the f u n c t i o n describing c o n v e c t i v e f l o w t o w a r d s the e l e c t r o d e w i t h distance n o r m a l t o the e l e c t r o d e . F o r X < I the c o n v e c t i v e f l o w is ignored. F o r 1 < X <~ XH

the parabolic approximation shown by the broken line is used. For X > XH the limiting value is used.

ferential e q u a t i o n s : f o r I < X < XH 2.1 × 2 ( a u / a × ) + e x p ( - / 3 × ) - ~2u = 0

(13)

while f o r × > ×H 3.7 H~0 2 ( a u / a x ) + exp(--fl×) - - ~2u = 0

(14)

H e n c e we m u s t solve eqn. (14) w i t h the b o u n d a r y c o n d i t i o n t h a t u -* 0 as × -~ oo t o p r o v i d e a value o f u, UH, at × = ×H. This in t u r n p r o v i d e s t h e b o u n d a r y c o n d i t i o n f o r the s o l u t i o n o f eqn. (13) t o give t h e value o f u, u l , a t × = I n e e d e d to solve eqn. (9). THE SOLUTION OF THE EQUATION F O R X > XH

T h e s o l u t i o n o f eqn. (14) gives UH = e x p ( - - ~ X H ) / ( g 2 + 3.7 H o 2 fl)

(15)

This e q u a t i o n describes the p h o t o s t a t i o n a r y s t a t e set u p in t h e s t r e a m o f solution f l o w i n g t o w a r d s t h e e l e c t r o d e . T h e e x p r e s s i o n differs f r o m t h e n o r m a l h o m o g e n e o u s case b e c a u s e o f the c o n v e c t i v e dilution. T h e r o t a t i n g e l e c t r o d e c o n t i n u a l l y p u m p s fresh s o l u t i o n i n t o the b e a m o f light. This e f f e c t is d e s c r i b e d b y the/TOO 2 t e r m w h e r e H~0 2 ~ 1 0 2. THE SOLUTION OF THE EQUATION F O R 1 <~ X < XH

T h e s o l u t i o n o f eqn. (13) gives Ul = UH + e x p

i_ 2( 2~

1 --

+

2.1

1

exp

J-'~X +

L"

a:2 -j dx

2 -xJ

203

Approximations for the integral can be found for either/3 < 1 or ~ < 1. For < 1 we find 1

1

while for ~ < 1 we find ul -~- UH + exp(--~)

E2([3)/2.1

(17)

where E2(~) is the exponential integral [4]. When K > 1, any B made outside the diffusion layer decomposes before reaching the electrode. When ~ > 1, very little light penetrates through the diffusion layer and the photochemical generation takes place within the diffusion layer. Hence a significant contribution to the current from material generated outside the diffusion layer will only be found when both ~ and ~ are smaller than unity. Under these conditions the light penetrates outside the diffusion layer and B is stable enough to survive the journey to the electrode. With these conditions both eqns. (16) and (17) reduce to Ul ~- UH + 0.48

(18)

THE SOLUTION OF THE EQUATION F O R X ~ 1

Inside the diffusion layer we solve eqn. (9) with the boundary conditions at × = 0 and × = 1 to obtain for the gradient at the surface of the electrode: , ~ Iu coshK--(~/~)sinh~--exp(--~)] U0-sinh~ z+ ~2_~2 "

(19)

~g k

p

f

~.=1

1 log A'

A

,e l//¢2 /9 UH

C p = H___o 2 :3"7

P.,

Fig. 2. Approximate solutions for the photoelectrochemical collection efficiency Nhv assuming that ¢ = 1. The different approximations hold for different values of ~ and/~ where (=XD/Xe) compares the thickness of the diffusion layer to the distance over which the light is absorbed and v. (= XD/Xk) compares the thickness of the diffusion layer to the distance a photogenerated species diffuses before decomposing. The parameter H 0 (=XD/XH) compares the thickness of the diffusion layer to the thickness of the hydrodynamic layer.

204

When e i t h e r K > 1 or fi > 1, then, as discussed above, the term in u 1 is negligible and we need only consider the second term in the square bracket. Under these conditions eqn. (19) reduces to

On the other hand when b o t h ~ < 1 a n d ~ < 1 then, expanding the hyperbolic and exponential functions in eqn. (19) and using eqn. (18), t

u0 ~- UH + 0.98

(20)

From eqns. (15) and (18) the term in uH is only significant when ~2 + 3.7 H o 2 fl < exp(--1.3/-IF0-O1

~ ) ~"

1

(21)

The approximation holds when the term is significant. In Fig. 2 we show approximate solutions for the collection efficiency (Nh, = eft u~) and the values of fl and K for which the approximations hold; we have assumed that ¢ = 1. DISCUSSION OF THE APPROXIMATE SOLUTIONS

It can be seen that in Fig. 2 the different approximate solutions pass smoothly over one into the other at the boundaries in the "case diagram". We now discuss each of the solutions. In case A the collection efficiency is unity. Here the light is absorbed close to the electrode (fl > 1) and all the photogenerated material reaches the electrode and is there destroyed. In case B the kinetics of the decomposition are sufficiently rapid to destroy the material in its passage across the diffusion layer. Hence material that reaches the electrode has to be generated close to the electrode. In cases C, D and E on the other hand the kinetics are sufficiently slow to allow material generated outside the diffusion layer to reach the electrode. In case C the light does n o t penetrate far outside the diffusion layer; of the species that reach the electrode, roughly equal amounts are generated inside and outside the diffusion layer. In case D the light penetrates outside the h y d r o d y n a m i c layer. The collection efficiency again becomes constant (~ 10-2). Most of the material generated in the stream flowing to the electrode is lost by radial convection; only a small constant fraction reaches the electrode. In case E the kinetics for the decomposition are sufficiently rapid for the concentration in the stream to be determined n o t by the convective dilution but by the decomposition. The variation of the current with rotation speed for the different cases is as follows. For cases A, B and D there is no variation. This is because in cases A and B everything happens inside the diffusion layer. For case D increasing the speed increases the convective dilution but also increases the rate of transport to the electrode. For case C the current d e c r e a s e s as rotation speed is increased; it varies with co-1/2. This is because more rapid rotation reduces the stagnant diffusion layer and therefore there is less space where the species can be generated w i t h o u t being swept away by radial convection. For case E the current increases with rotation speed. Here the more rapid transport carries the species to the electrode before it decomposes.

205 EXPERIMENTAL

The apparatus and the construction of the semitransparent platinum rotating disc electrode have been described previously [1]. The flux of light through the electrode was measured using the "Ferrioxalate" actinometer described by Hatchard and Parker [5]. Electrochemical measurements were made on neutral aqueous oxalate (K2C204) and varying concentrations of Fe(C2Oa)~-. All potentials were measured with respect to the saturated calomel electrode. Solutions containing oxalate ion exhibit p h o t o e l e c t r o c h e m i c a l effects in the absence of Fe(III) [6] b u t these currents were negligible compared to those measured in the presence of Fe(III). RESULTS AND DISCUSSION

In order to test the theory, experiments were carried o u t on trisoxalatoiron(III). The advantage of this system is that first there are no kinetic complications since the Fe(II) product is stable and secondly the effective light intensity through the electrode can be measured directly by actinometry. Under the conditions used both Fe(II) and Fe(III) form stable trisoxalato complexes [7]. Figure 3 shows a set of typical current--voltage curves. The photocurrents are the difference between the current at an illuminated and at an unilluminated electrode. In the dark one finds a moderately irreversible wave for the reduction of the trisoxalatoiron(III). The limiting currents for this wave obey the normal w 1/2 dependence of the Levich equation [8]. On illumination the overall photochemical reaction is: !2 C2042- + Fe(C204)33- -~ Fe(C204)~- + CO2 (g) Hence in Fig. 3 at anodic potentials there is no current in the dark because there is no Fe(II) in the bulk of the solution, but, on illumination, the current from the photogenerated Fe(II) is observed. The faster the rotation speed the smaller is the limiting current. At cathodic potentials the photogenerated Fe(II) does not react on the electrode but the current from Fe(III) is smaller because of its photochemical decomposition in the diffusion layer; this difference gives an anodic photocurrent. At intermediate potentials both Fe(II) and Fe(III) react slowly on the electrode and so the photocurrent passes through a minimum. The shape of the dark current-voltage curve is given by iD/iD = ~'A/( 1 + ~'A)

(22)

where i D is the dark current, iD is the limiting value of the dark current, h A = k ' A X D / D , and k~ is the rate constant for the reaction of Fe(III) on the electrode.

As shown in the appendix, the photocurrent-voltage curve is given (for iD < < iD) by i * / i ~ = [~B + (iD/iD - - 1)--1]/( 1 + ~e)

(23)

where i* is the photocurrent, i* is the limiting value of the photocurrent, ~s = k ' B X D / D , and k~ is the rate constant for the reaction of Fe(II) on the electrode.

In eqn. (23) at anodic potentials the ~B term dominates the numerator and eqns. (22) and (23) have similar forms. The photocurrent is caused by the reac-

206

500

x/

250

/20 az ×

0

\

1"5

__X m X m X ~ X m X - -

X~X mXmX&X#~

i/#A

-20 :OHz°--°"'~" °"'°'~ °t

o.0"°'-% 2 F"o.'~o

"// .o/

'\

IC

X/

05 Y

0"0

/

7

\

-0"5 ~/~

s.z

3--0--0~0~0~ -40

o..OI°/°/~// 0//

-I0

/

-I 5

9/

o?~H__Zo__o_oo.~~.o I

0.4 013 0!2 o!J 0.0-0!(-0~2-0!3

0!20'-I

E/V

]

I

00-01-0!2-0!3 E/v

Fig. 3. Typical current voltage curves for [Fe(C204) 3 - ] = 50 mM. Rotation speeds are shown by each curve: (X) Dark currents, (o) photocurrents. Fig. 4. Tafel plots from eqns. (24) and (25) for dark currents (X) and photocurrents (o) at different rotation speeds. [Fe(C204) 3 - ] = 50 raM.

tion of Fe(II). At cathodic potentials XB is small and eqn. (23) reduces to: i* /i* "" iv /iDL

The photocurrent is caused by the removal of Fe(III) in the diffusion layer. Differentiation of eqn. (23) shows that at intermediate potentials the photocurrent passes through a minimum at the potential where kA ~- k~. Equations (22) and (23) can be rearranged to give respectively YA = log(iD/iD -- 1 ) - - ½ log CO = --log(kA ~ l l 2 X D / D )

(24)

and YB = log

-- 1) -- ~ log co -- log(1

iD/i D -- 1

D

Figure 4 shows typical plots of eqns. (24) and (25) for a series of rotation speeds. Linear Tafel plots are found for k A and k~. Allowing for the ohmic resistance of the electrode the values of the transfer coefficients when added together equal unity. The two lines intersect at a b o u t --0.25 V which is the value for E for the F e ( C 2 0 4 ) ~ - / F e ( C 2 0 4 ) ] - redox couple [9]. Hence we can conclude that the species reacting on the electrode in the photoelectrochemical experiments is indeed Fe(C204) 2-. We n o w turn to the size of the limiting photocurrent at anodic potentials, i*. The limiting photocurrents decrease with increasing rotation speed. The gra-

207 TABLE 1 D e p e n d e n c e of limiting p h o t o c u r r e n t on r o t a t i o n speed log i~ = c o n s t a n t -- m log co [Fe(e204)3--]/mM

m

#

0.5 1.5 5.0 15.0 50 125

0.28 0.49 0.42 0.47 0.39 0.31

0.002 0.006 0.02 0.06 0.2 0.5

dients of log iL* versus log fi plots are summarized in Table 1; values of/3 for the median rotation speed, ~, are also included. It can be seen that in the middle of the concentration range the values of m and ~ place the system in case C of Fig. 2, where the theory predicts that m ~ 0.5 for the range o f ~ values given by H 2 / 4 ~ 0.005 < ~ < 1. At low concentration the system is almost in case D and at high concentration in case A; hence m is less than 0.5. Now using the results from the actinometry experiments we can calculate the observed photoelectrochemical collection efficiency (gh~/¢) = i * / F J

(26)

where F is the Faraday constant and J is the rate ~)f decomposition (in mol s- 1 ) of Fe(III) caused by shining the light through the electrode into the actinometer. In Fig. 5 we plot the experimental points from eqn. (26) against ~. It is gratifying that all the points lie on a c o m m o n curve even though the points are taken at different concentrations and rotation speeds.

0"0 o~V~

-0'5 -I'0 =~ -15 -2.o x

-2"5

A +O9 . ° x

-3"0

I

I

-3"0 -25

t

t

t

-2"0 -1"5 -I'0 log /3

I

-0'5

I

0"0

Fig. 5. Comparison of experimental points for Nhv/dpcalculated f r o m eqn. (26) and the theoretical curve for Nh~,/¢. R o t a t i o n speeds/Hz are as follows: (X) 20; (A) 10; (+) 6.7; ([]) 5; (~7) 3.3; (*) 2.5; (©) 1.7.

208

In Fig. 5 the line is the theoretical collection efficiency (Nh,/¢) calculated from eqns. (2), (3), (4), (6), (8), (15), (18) and {19). Reasonable agreement is found, and these results confirm the theory of the semitransparent rotating disc electrode for the photochemical generation of a stable species. APPENDIX

This appendix describes the shape of the photocurrent voltage curve for cases C and D where a stable species, B, is generated outside the diffusion layer. The system is: Solution A ~ B

kk

Electrode e + A -~ B B_~B A + e T h e bulk concentration of B in the dark is zero. In the dark the flux of A is jD _-- kAao, D = ( D / X D ) ( a D _ a D)

(27)

while on illumination j * = k ' h a * -- ( D / X D ) ( a * l - - a * )

The difference, j a , is therefore (28)

j~ = kk(ao* - ao')

From eqn. (27) the dark current voltage curve is given by

(29)

"D/ j "D = iD/i D = 1 + D / k 'AX D JL

But, assuming equal diffusion coefficients, at all distances b*=a D --a*

(30)

and the flux of B is given by j. , .

(31)

= kBb o

The observed photocurrent j* is (32)

J* =j* --ja and from eqns. (28), (30), (31) and (32) we obtain •D "D

J*/J*L = i*/i*L = [ k S + ( j L / j

-- 1)--1]/(1 + kB)

where kB = k ' B X D / D . REFERENCES 1 2 3 4 5 6 7 8 9

W.J. A l b e r y , M.D. A r c h e r , N.J. F i e l d a n d A . D . T u r n e r , Discuss. F a r a d a y Soc., 5 6 ( 1 9 7 3 ) 28. W.J. A l b e r y a n d M.L. H i t c h m a n , R i n g - D i s c E l e c t r o d e s , C l a r e n d o n Press, O x f o r d , 1 9 7 1 , p. 1 2 . W.G. C o c h r a n , P r o c . C a m b r i d g e Phil. S o c . , 3 0 ( 1 9 3 4 ) 3 6 5 . M. A b r a m o v i t z a n d I.A. S t e g u n (Eds.), H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s , D o v e r , N e w Y o r k , 1 9 6 5 , p. 2 2 8 . C.G. H a t c h a r d a n d C . A . P a r k e r , P r o c . R o y . S o c . A, 2 3 5 ( 1 9 5 6 ) 5 1 8 . M. H e y r o v s k y , P r o c . R o y . S o c . A, 3 0 1 ( 1 9 6 7 ) 4 1 1 . J. L l o p i s a n d R . P a s a d o , E l e c t r o c h i m . A c t a , 1 3 ( 1 9 6 8 ) 1 1 3 1 . V . G . L e v i c h , P h y s i c o c h e m i c a l H y d r o d y n a m i c s , P r e n t i c e Hall, E n g l e w o o d Cliffs, N . J . , 1 9 6 2 , p. 6 9 . J . J . L i n g a n e , J. A m e r . C h e m . S o c . , 6 8 ( 1 9 4 6 ) 2 4 4 8 .

(33)