Vol. 117, No. 3

OXYGEN

i oc po~[H+]3/2 exp (

FV

-- --~-)

REDUCTION

[20]

This equation has a Tafel slope of RT/F, and has the correct o x y g e n p a r t i a l p r e s s u r e a n d p H - d e p e n d e n c e (11).

Acknowledgment The a u t h o r wishes to t h a n k P r a t t & W h i t n e y A i r craft Division of U n i t e d A i r c r a f t Corporation, and t h e sponsors of this w o r k u n d e r the T A R G E T fuel cell p r o g r a m , for permission to publish this paper. The help of t h e various m e m b e r s of IGT who assisted in p r e p a r a t i o n is also g r e a t l y a p p r e c i a t e d . Manuscript s u b m i t t e d Aug. 1, 1969; revised m a n u script received ca. Nov. 12, 1969. A n y discussion of this p a p e r will a p p e a r in a Discussion Section to be published in the D e c e m b e r 1970 JOURNAL.

REFERENCES 1. D. T. S a w y e r and L. V. I n t e r r a n t e , J. Electroanal. Chem., Z, 310 (1961). 2. M. W. Breiter, Electrochim. Acta, 9, 441 (1964). 3. C. C. Liang and A. L. J u l i a r d , J. Electroanal. Chem., 9, 390 (1965). 4. E. I. K h r u s h e h e v a , N. A. Shumilova, and M. R. Tarasevieh, Elektrokhim., 1, 730 (1965). 5. L. Mfiller a n d L. N. Nekrasov, Dokl. Akad. Nauk. S.S.S.R., 157, 416 (1964). 6. L. Mfiller a n d L. N. Nekrasov, J. Electroanal. Chem., 9, 282 (1965). 7. J. P. Hoare, This Journal, 112, 602 (1965). 8. G. Bianchi and T. Mussini, Electrochim. Acta, 10, 445 (1965). 9. J . J . Lingane, J. Electroanal. Chem., 2, 296 (1961). 10. A. D a m j a n o v i e a n d J. O'M. Bockris, Electrochim. Acta, 11, 376 (1966). 11. A. D a m j a n o v i c a n d V. Brusic, ibid., 12, 615 (1967). 12. A. Damjanovic, A. Dey, and J. O'M. Bockris, ibid., 11, 791 (1966). 13. A. Damjanovic, M. A. Genshaw, and J. O'M. Bockris, This Journal, 114, 466 (1967). 14. A. Damjanovic, M. A. Genshaw, and J. O'M. B o c k ris, J. Chem. Phys., 45, 4057 (1966). 15. A. J. A p p l e b y , J. Electroanal. Chem., 24, 97 (1970). 16. H. A. L a i t e n e n and C. G. Enke, This Journal, 107, 773 (1960).

ON OXIDE-FREE

Pt

335

17. A. K. N. R e d d y , M. Genshaw, and J. O'M. Bockris, J. Electroanal. Chem., 8, 406 (1964). 18. S. D. James, This Journal, 114, 1113 (1967). 19. S. S c h u l d i n e r and T. B. W a r n e r , ibid., 112, 212 (1966). 20. J. P. Hoare, ibid,, 109, 858 (1962). 21. H, W r o b l o w a , M. L. B, Rao, A. Damjanovic, and J. O'M. Bockris, J. Electroanal. Chem., 15, 139 (1967). 22. G. Lewis and M. Randell, " T h e r m o d y n a m i c s and F r e e E n e r g y of Chemical Substances," M c G r a w Hill Book Co., N e w Y o r k (1923). 23. Monsanto Corporation, Technical B u l l e t i n 1-239. 24. J. Bravacos, M. Bonnemay, E. Levart, a n d A. A. Pilla, Compt. Rend. Acad. Sci. Paris, 265, 337 (1967). 25. A. J. A p p l e b y and A. Borucka, This Journal, 116, 1212 (1969). 26. V. S. Bagotskii and I. E. Yablokova, Z. Fiz. Khim., 27, 1663 (1953). 27. L. Miiller and L. N. Nekrasov, Dokl. Atcad. Nauk. SSSR, 154, 437 (1964). 28. L. Miiller and V. V. Sobol, Elektrokhim., 1, 111 (1965). 29. L. N. Nekrasov, ibid., 2,438 (1966). 30. H. P. Stout, Discussions Faraday Soc., 1, 246 (1947). 31. A. C. Riddiford, Electrochim. Acta, 4, 170 (1961). 32. T. P. Hoar, Proc. 8th Meeting CITCE, Madrid 1956, p. 439, B u t t e r w o r t h s , London (1958). 33. H. Mauser, Z. Elektrochem., 62, 419 (1958). 34. D. S. G n a n a m u t h u and J. V. Petrocelli, This Journal, 114, 1036 (1967). 35. J. O'M. Bockris, J. Chem. Phys., 24, 817 (1956). 36. B. E. C o n w a y and P. L. Bourgault, Can. J. Chem., 40, 1690 (1962). 37. D. J. I r e s and G. J. Janz, "Reference E l e c t r o d e s - T h e o r y and Practice," p. 365, A c a d e m i c Press, New Y o r k (1961). 38. J. P. Hoare, This Journal, 112, 849 (1965). 39. B. E. C o n w a y and E. Gileadi, Trans. Faraday Soc., 58, 2493 (1962). 40. W. BSld and M. Breiter, Electrochim. Acta, 5, 145 (1961). 41. O. A. Petrii, R. V. Marvet, and Zh. N. Malysheva, Elektrokhim., 3, 962 (1967). 42. R. Parsons, Trans. Faraday Soc., 54, 1053 (1958). 43. T. Biegler and R. Woods, J. ELectroanal. Chem., 20, 347 (1969). 44. A. Damjanovic, P r i v a t e communication (1969).

Rotating Ring-Disk Electrodes II. Digital Simulation of First and Second-Order Following Chemical Reactions Keith B. Prated and Allen J. Bard* Department of Chemistry, The University of Texas at Austin, Austin, Texas ABSTRACT A digital simulation technique has been e m p l o y e d to t r e a t t h e s t e a d y - s t a t e and t r a n s i e n t ring c u r r e n t b e h a v i o r at t h e r o t a t i n g r i n g - d i s k electrode (RRDE) for cases w h e r e the i n t e r m e d i a t e g e n e r a t e d at the disk electrode undergoes a first- or s e c o n d - o r d e r homogeneous chemical reaction leading to a nonelectroactive species. W h e r e comparisons w e r e possible, t h e results w e r e found to b e in good a g r e e m e n t w i t h p r e v i o u s a p p r o x i m a t e t h e o r e t i c a l t r e a t ments. W o r k i n g curves a r e p r o v i d e d w h i c h allow d e t e r m i n a t i o n of r a t e constants of the homogeneous reactions from r i n g - c u r r e n t - r o t a t i o n r a t e data. The r o t a t i n g r i n g - d i s k electrode (RRDE) was i n troduced b y F r u m k i n and N e k r a s o v (1) as a means of detecting i n t e r m e d i a t e s of electrode reactions. A n i n t e r m e d i a t e , B, is g e n e r a t e d at t h e d i s k electrode b y the reaction A • ne ~ B [1] and B is d e t e c t e d at t h e r i n g electrode by a p p l y i n g a 1 P r e s e n t a d d r e s s : U n i v e r s i t y of T e x a s a t E1 Paso, E1 Paso, Texas. * Electrochemical Society Active Member.

sufficient p o t e n t i a l such t h a t all B reaching t h e ring is t r a n s f o r m e d b a c k to A b y t h e reaction

B • ne ~ A

[2]

If B is a stable species, t h e n t h e ratio of the c u r r e n t at t h e ring electrode to t h e c u r r e n t at t h e disk elect r o d e is a function only of the g e o m e t r y of t h e elect r o d e (2). This ratio is called t h e collection efficiency, N, and is given b y t h e expression

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336

J. Electrochem. Soc.: E L E C T R O C H E M I C A L N ------{r/id

k, B .~ X

[4]

in which B reacts to give a n electro-inactive species X. This t r e a t m e n t is valid only for e x t r e m e l y t h i n - r i n g t h i n - g a p electrodes and t h e n only for certain values of kl/~. I n a later paper, Albery, Hitchman, a n d Ulst r u p (4) modified the previous t r e a t m e n t to make it applicable to a w i d e r r a n g e of electrode geometries. A l b e r y (5) has also treated the r i n g c u r r e n t t r a n s i e n t s for this m e c h a n i s m in the case of a constant c u r r e n t at the disk electrode. /~lbery and B r u c k e n s t e i n (6) have also give*~ an approximate t r e a t m e n t of the second-order EC m e c h anism

k~

B+C-*

Y+Z

[5]

where C, Y, and Z are electro-inactive species. Unfortunately, the treatment of this mechanism is valid only for very small values of the parameter k2C~ where C~ is the bulk concentration of species A. Albery, Hitchman, and Ulstrup (4) have presented experimental w o r k to support the modified treatment of the first-order case, and Johnson and Bruckenstein (7) have investigated the second-order case. In both of these studies, however, the systems chosen did not strictly conform to the above mechanisms in that in all cases one product of the homogeneous following reaction was the starting species, A. Thui, the experimental studies were more related to the catalytic reaction mechanism; a theoretical t r e a t m e n t of that case will be g i v e n in a l a t e r paper. In this paper, we present the results o i the applica~ tion of a digital simulation t e c h n i q u e (8) to the firstand second-order EC mechanisms. In addition to steady-state kinetic collection efficiencies, ring c u r r e n t t r a n s i e n t behavior is presented. This simulation techn i q u e can be applied to a n y electrode and the results are valid for a n y value of the rate parameters, k l / ~ and k~C~A/~.

Digital Simulation Method The basic method for the digital simulation of the RRDE i n c l u d i n g the simulation of the first-order EC m e c h a n i s m has been presented in a previous paper (8). Only two modifications of the simulation techn i q u e previously presented are necessary in order to simulate the second-order EC mechanism. First, the collection efficiency for such a process will be determ i n e d not only b y the value of the rate constant, k~, the rotation rate, oJ, and the b u l k concentration of species C, C~ For the second-order case, then, a n additional parameter, m, given b y r~ = C~176

Fc (J,K) = m

[3]

If species B undergoes a chemical reaction which depletes i t s concentration as it passes from the disk to the ring, t h e n the observed collection efficiency u n d e r these conditions, Nk, (the kinetic collection efficiency) will be smaller t h a n that found i n the absence of these reactions. I n this case, the collection efficiency is a function, not only of electrode geometry, b u t also of the rate constant of the reaction, the rotation rate, ~, and other solution parameters. A l b e r y a n d B r u c k e n s t e i n (3) have given a n approximate t r e a t m e n t of steady-state kinetic collection efficiencies for the first-order EC mechanism (where EC denotes an electron t r a n s f e r followed by a chemicaI reaction)

[6]

must be specified. Thus i n setting up the initial conditions, the fractional concentration of species A in a n y box will be FA(J,K) ----1.0 [7] a n d the initial fractional concentration of species C in a n y box will be

March 1970

SCIENCE

[8]

A second difference between the treatment of the first- and second-order cases concerns the form of the dimensionless rate parameter. The pertinent rate law for the second-order mechanism is --dCc/dt =

--dCB/dt = k2CBCc

[9]

In terms of the simulation this becomes --~Cc = --~CB = k~CBCc• Dividing through by C~

[I0]

and letting

Cc/C~

----Fc

[Ii]

CB/C~

= FB

[12]

and then Eq. [10] becomes --~Fc = --~FB = k2FBCcAt Replacing Cc by FcC~ [12] and [25], ref. (8))

[13]

and using the equation (Eq.

At ----tk/L = vi/aDA-i/s~-IL -i (0.51) -2/3

[14]

one obtains --AFc • --AFB ----kutkC~

[15]

The product k2tkC~ is the dimensionless rate parameter applicable to this mechanism and is called X K T C . Thus XKTC = k2tkC~ ----k2COA~-lvl/3DA -1/3 (0.51)-2/3

[16]

where as before v is the kinematic viscosity and DA is the diffusion coefficient of species A. The effects of kinetics on t h e concentration in any box m a y be t a k e n into account b y replacing the existing concentrations of B and C, FB and Fc by F'B (J,K) ----FB (J,K) -- AFB (J,K)

[17]

F'c (J,K) ----Fc (J,K) -- hFc (J,K)

[I8]

Other t h a n these two modifications, the simulation is identical to that previously presented. A digital computer program based on the digital simulation model with the appropriate modifications was used to obtain the results w h i c h will be presented. These results were obtained by first selecting a value for the appropriate dimensionless rate p a r a m e t e r X K T = (0.51)-2/3kw-l~-l/SDA -1/3

[19]

for the first-order case or XKTC for the second-order case. I n the second-order case, a value for the p a r a m eter m was also specified. The computation was then carried out u n t i l a steady state was attained. This computation yields both steady-state kinetic collection efficiencies, Nk, a n d the ring c u r r e n t transients as functions of the dimensionless rate p a r a m e t e r s and m. I n all cases unless otherwise stated, L = 50 and D,~ ---- 0.45 and the appropriate correction factors were used.

First-Order Results Steady-state behavior.--The simulated collection efficiency vs. X K T curve for a n electrode a p p r o x i m a t ing the t h i n - r i n g t h i n - g a p electrode t r e a t e d by A l b e r y and B r u c k e n s t e i n is shown in Fig. 1 a n d 2. I n Fig. 1, the simulation is compared with A l b e r y and B r u c k e n stein's Eq. [6.4] (3) in the region of small XKT. The entire simulated curve is shown i n Fig. 2 as is the comparison with their Eq. [5.8] (3). These graphs show clearly the regions in which the approximate t r e a t m e n t s o f A l b e r y and B r u c k e n s t e i n apply for a n electrode of this geometry. These regions of applicability are quite close to those suggested by A l b e r y and Bruckenstein. I n a later paper (4), A l b e r y pointed out that the t h i n - r i n g t h i n - g a p t r e a t m e n t is not valid for most

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Vol. 117, No. 3

ROTATING

RING-DISK

ELECTRODES

337

0.12 o4

O.IC

O.OE

Nk

02t

Nk O.OE

Ol

o

02

0.4

06

i o

08

X K T '/z

002

o

o:,

o12

Fig. 3. Collection efficiency vs. (XKT)'/2 for a first-order following reaction for two electrodes of different geometries

o13

XKT

Fig. 1. Collection efficiency vs. XKT for a first-order following reaction at a thin-ring thin-gap electrode (IR1 = 2000, IR2 2040, IR3 = 2080). (~) Comparison with Albery's Eq. [6.4] (2). XKT = k l ~ - Z D - 1 / % z / 3 ( 0 . 5 1 ) - 2 / 3 .

a. rl = 0.3635

r2 = 0.3777

r3 = 0.4835

IR1 = 200

IR2 = 208

IR3 ---- 266

b. rl = 0.4770

r2 = 0.4869

r3 -= 0.5222

IR1 = 820

IR2 =- 837

IK3 = 898

(~) Comparison with Albery's Fig. 2 (4). XKT (0.51) -2/3.

--~

kw-1D-1/3v

1/3

010[

N•kLO[

0.08 Nk

0.8

O06

06

004

0.02

o

~

~

6

s

~0 -

~2

,"4

,%

XKT

Fig. 2. Collection efficiency vs. XKT for a first-order following reaction at thin-ring thin-gap electrode (IR1 = 2000, IR2 = 2040, IR3 = 2080). (~) Comparison with Albery's Eq. [5.8] (2). XKT kl~-

ZD- 1/3~P/3(0.51)-2/3.

I0

20

30

0

80

TO

80

XKT

Fig. 4. Collection efficiency vs. XKT for a first-order following reaction for three electrodes of different geometries. XKT ~

klm-lD-1/3u1/3(0.51) -:2/3. a, r J r l practical electrodes, a n d he presented a modified theory which should be applicable to a wider range of electrode geometries. The results of the simulation of the two electrodes for which A l b e r y presented his calculations are shown in Fig. 3 as well as Albery's calculated values for the collection efficiency as a function of (XKT) 1/g. The a g r e e m e n t over this range of X K T values is excellent. It is interesting to consider the effect of electrode geometry on the range of applicability of different electrodes to studies of first-order following reactions. The results of the simulation of t h r e e different electrodes are given in Fig. 4. Electrode A is the electrode previously presented which approximates the t h i n - r i n g t h i n - g a p electrode. Electrode B is one of Albery's electrodes. It has a fairly t h i n gap a n d a moderately thin ring. Electrode C which has been used in these laboratories has a comparatively wide gap and ring. The simulated working curves show, as expected, that the t h i n - r i n g t h i n - g a p electrode (A) is the most applicable of the three for studying e x t r e m e l y fast reactions, while electrode C, with a wider gap is more useful for slow reactions. The point to be made here is that while electrodes similar to electrode A should be used for studying fast reactions, they are not useful for slow reactions (k ~ 0.1) because even at the lowest usable rotation rates such a reaction does not proceed to any detectable extent i n the time required for species B to reach the ring electrode. I n these cases, electrodes with gaps as wide as that in electrode C or even wider should be used

50

r2/rl

=

| . 0 4 , r3/r2 =

| . 2 8 ; c, r 2 / r l =

= 1.02, r.3/r2 = 1.02; 1.14, r3/r2 = 1.70.

b,

For this first-order case, identical steady-state collection efficiencies are predicted w h e n the b o u n d a r y condition at the disk electrode corresponds to a constant c u r r e n t step instead of a potential step. The constant c u r r e n t b o u n d a r y condition at the disk affects only the t r a n s i e n t and not the steady-state behavior of the collection efficiency. Transient behavior.--The simulated ring c u r r e n t t r a n s i e n t s (RCT) for this m e c h a n i s m due to a potential step at the disk are shown in Fig. 5. Each curve represents the RCT for the specified value of the d i m e n sionless rate parameter, XKT. As m u s t be the case, the steady-state currents are seen to decrease with increasing values of XKT. A n interesting feature of these curves is that, u n l i k e the RCT in the absence of a following reaction, the ring c u r r e n t passes t h r o u g h a m a x i m u m before attaining steady state. This is b e cause u n d e r the potential step condition, a very large instantaneous flux of species B is generated at the initiation of electrolysis [see Fig. 2, ref. (8)]. The short time i n t e r v a l b e t w e e n the generation of this large flux of B and the detection of B at the ring does not permit the homogeneous reaction to reduce the concentration of B to its eventual, lower steady-state value. Hence a m a x i m u m i n the ring c u r r e n t is observed. Identical simulation results are obtained if the simulation is carried out with L = 1000 and no correction factors. This effect is more clearly shown in

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338

J. FAectrochem. Soc.: E L E C T R O C H E M I C A L SCIENCE

March I970

0.5. o

04

/

I0 i,

0.3 ZR

b

02 05 c

Oi d e o

o

2:o

3:o 0

t/t.

f

21o

,io

30

t/t k

Fig. S. Ring current transients for a first-order following reaction for different values of XKT. a, 0.0; b, 0.5; c, 1.0; d, 2.0; e, 4.0. Zr =

i r I ( O . 5 1 ) l / 3 n F A D C ~ ADB2/3ml/2V - 1 / 6 . X K T =

(0.51) -213. Simulated with IR1 =

k l ~ - l D - 1 / 3 ~ r 1/3

83, IR2 = 94, ~IR3 =

159.

Fig. 6 in w h i c h the RCT's h a v e each been normalized b y t h e e v e n t u a l s t e a d y - s t a t e c u r r e n t for each v a l u e of XKT. This plot also p o i n t s out that t h e v a l u e of ~t D1/3~-1/3(0.51) 2/3 at w h i c h the ring c u r r e n t is onehalf (or any o t h e r fraction) of the s t e a d y - s t a t e v al u e is a function of X K T and could in principle be used to d e t e r m i n e t h e rate constant, as has been suggested by A l b e r y (5). These results can be c o m p a r e d w i t h those obtained using a constant c u r r e n t b o u n d a r y condition at t h e disk. The results of tw o simulations w h i c h differed only in t h e b o u n d a r y condition at the disk are shown in Fig. 7. Note that no m a x i m u m is observed w h e n a constant c u r r e n t is applied to the disk. The effect of t h e r a t e p a r a m e t e r on the n o r m a l i z e d RCT's w i t h a constant c u r r e n t step at th e disk is shown in Fig. 8. As before, t h e v a l u e of th e t i m e parameter, ~t D1/3v-1/3(0.51)2/3, at a specified fraction of t h e s t e a d y - s t a t e c u r r e n t is a function of the r a t e p a r a m eter, XKT. It should also be pointed out that these n o r malized curves are i n d e p e n d e n t of t h e v a l u e of the applied constant current. Second-Order

Fig. 7. Ring current transients for a first-order following reaction for XKT = 2.0, IR1 ~ 83, IR2 z 94, IR3 z 159. tk ~-lD-Z/3vl/3(0.51)-2/3. a, Potential step at the disk electrode; b, constant current at the disk electrode.

ID

iJ~s,)05

o

0

20 tit

3.0

4.0

k

Fig. 8. Ring current transients far a first-order following reaction with constant current at the disk for different values of XKT. tk ~-ZD-1/3ul/3(O.51)-2/3.

a, 2.0; b, 1.0; c, 0.5; d, 0.0. IR1 =

83,

IR2 ~ 94, IR3 = 159. 0 , 6 84

Results

Steady-state behavior.--A series of curves of Nk Vs. ( X K T C ) (m) for different values of m, for a following s e c o n d - o r d e r r e a c t i o n and for t h e disk electrode held at a p o t en t i al w h e r e t h e l i m i t i n g c u r r e n t for the conver s i o n of A to B occurs, is shown in Fig. 9. Also shown in Fig. 9 is t h e s im u la te d collection efficiency

t0

05

0.4 o

03 Nk 0.2

15 O.i

O

0

i r(S.~;

0.5

3.0

40

50

60

7o

k2a~-ICO AD-113vll3(0.51 )-213.

I

20 t/T k

Fig. 6. Ring current transients for a first-order following reaction for different values of XKT. a, 4.0; b, 2.0; c, 1.0; d, 0.5; e, 0.0. Curves are normalized by the steady-state current at each value of XKT. Simulated for IR1 = 83, IR2 = 94, IR3 = 159. XKT =

kl~-ZD-1/3u1/3(0.51)-2/3.

20

Fig. 9. Collection efficiency vs. (XKTC)(m) for a second-order following reaction for different values of m = C~176 a, 0.10; b, 0.20; c, 0.50; d, 1.0; e, 10.0. (~) Simulated first-order curve. IR1 ~ 83, IR2 ~ 94, IR3 ~ 159. XKTC

i,

0

i.o

(XKTC)(m)

I.O

c u r v e for a first-order following reaction. Note that w h e n C is in tenfold excess, t h e s e c o n d - o r d e r c u r v e is almost indistinguishable f r o m the first-order curve; thus w h e n m = 10.0, the s e c o n d - o r d e r process can be t r e a t e d as a p s e u d o - f i r s t - o r d e r one. In the case of this s e c o n d - o r d e r mechanism, unlike that of a following first-order reaction, the si mul a t e d collection efficiencies p r esen t ed h e r e are v al i d only for the case id -- ilim. If a constant c u r r e n t less t h a n the limiting c u r r e n t w e r e applied to t h e disk, differ-

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Vol. 117, No. 3

ROTATING

RING-DISK

ent collection efficiencies would be observed as pointed out by A l b e r y and B r u c k e n s t e i n (6). The simulation of the effects of a constant c u r r e n t less t h a n the limiting c u r r e n t can easily be done, but there seems to be little reason to do so. I n the t r e a t m e n t of this case b y A l b e r y and B r u c k enstein (6), the c u r r e n t at the disk electrode is adjusted to a value where the concentrations of B and C are equal at the i n n e r edge of the ring electrode. U n d e r these conditions, the c u r r e n t at the ring due to species B will be d e p e n d e n t on the rate of the reaction b e t w e e n B and C and is given b y the following equation ir

BnF~r2~DA~/2v- I/~k2-1

=

[20]

The constant, B, was originally calculated to be 590 (6) but more recently has been given as 210 (7). The disk c u r r e n t necessary to produce the appropriate conditions is given b y

id = 620nFnr12DA2/3el/2~-l/6C~ (1 -- F ( a ) ) - 1 [21] where F ( a ) is a geometrical p a r a m e t e r of the electrode which has been tabulated b y A l b e r y and B r u c k enstein (2). Note that for the p a r t i c u l a r value of m, given b y m = C~176 = 1 -- F ( ~ ) [22] Eq. [21] becomes

id

620nF~r12DA213~l/2v-1/6C~

=

[23]

This is precisely the limiting c u r r e n t due to species A as predicted by the Levich equation. Thus from Eq. [20] and [23], Nk i n this case is given b y Nk = (B/620) (r2/rl)2 ~k2-1COA-1D1/~v-1/3

[24]

Nk = (B/620) (r2/r1)2(0.51)2/3(XKTC)-I

[25]

or Note that Eq. [24] m u s t at best be valid only as ,~/k2C~ approaches zero where it accurately predicts that, u n d e r these conditions, Nk approaches zero. At the other limit, as ~/k2C~ approaches infinity, Eq. [24] predicts that Nk should increase w i t h o u t bound, while in reality the actual u p p e r limit is N, the collection efficiency in the absence of kinetic complications. To compare the digital simulation results with those of A l b e r y and B r u c k e n s t e i n (6), an electrode of a given geometry was chosen, and m was adjusted according to Eq. [22]. Values of Nk at various values of XKTC with id at its limiting value can t h e n be used to investigate the useful range of Eq. [24]. Simulated collection efficiencies vs. 1/XKTC are shown i n Fig. 10 for one of A l b e r y ' s electrodes u n d e r these conditions. Also shown is the line predicted by Eq. [25] using B = 210. The s i m u l a t i o n yields curves which approach the proper limits (zero or N) for the e x t r e m e values of ~/k2C~ I n the region of small ~/keC~ the simulated curve is a p p r o x i m a t e l y linear, but the a g r e e m e n t with Eq. [25] depends on the p a r -

Q

339

ELECTRODES

ticular geometry. The slope of the simulated curve as ~/k2C~ approaches zero yields a value of B of about 260, b u t it seems that the use of the entire simulated w o r k i n g curve would be preferable to an approximate linearization n e a r ~/k2C~ = 0. A l b e r y and Bruckenstein's recommended experim e n t a l procedure involves variation of id at values below the limiting c u r r e n t and at constant ~ to obtain the ir values used in fitting Eq. [24]. It appears that a more straight forward procedure w h e n w o r k i n g curves such as those in Fig. 9 are available, would involve m a i n t a i n i n g id at its limiting v a l u e and studying the variation of Nk with ~ and m. The value of k2 is t h e n obtained b y fitting the e x p e r i m e n t a l results to these w o r k i n g curves. Transient behavior.--A description of RCT's for second-order following reactions has not been given previously. The simulation shows t h a t t h e RCT's for the second-order m e c h a n i s m are similar to those for the first-order case. As shown in Fig. 11, the extent to which a m a x i m u m is predicted i n the potential step case is a function of the ratio of the b u l k concentrations of species A and C. If A is i n excess, then the m a x i m u m is small because very little B disappears on its way to the ring and the RCT approaches the RCT in the absence of a following reaction. If C is in excess, then the m a x i m a are similar to those obtained for the first-order case. Conclusion

The digital simulation technique is capable of describing both the steady-state a n d t r a n s i e n t behavior at the RRDE for systems i n v o l v i n g homogeneous reactions following the electrode reaction for a wide range of electrode geometries, ~ and R. Although the simulation technique permits the calculation of the t r a n s i e n t behavior of the ring current, there are m a n y e x p e r i m e n t a l problems involved in m e a s u r i n g these transients. The most i m p o r t a n t p r o b l e m is that adsorption of the intermediate, B, on the disk electrode will delay a n d smear out the RCT while desorption of B produced from adsorbed A will produce a m a x i m u m similar in form to the one predicted for this kinetic case, even in the absence of kinetic effects. B r u c k e n stein (9) has considered cases like this and their effects on transients. Since adsorption of either product or reactant is so f r e q u e n t l y found, the observation of the m a x i m a predicted b y the simulation will be difficult. For the same reason, the use of RRDE t r a n s i e n t m e a s u r e m e n t s to investigate the kinetics of homogeneous reactions is p r o b a b l y less profitable and more difficult, t h a n the corresponding steady-state m e a s u r e -

b

0

0.2

.

5

/

N~

o

il-

~

"o

XKTC "i Fig. 10. Collection efficiency vs. (XKTC) - 1 for a second-order following reaction, m = 0.60. IR1 ~ 200, IR2 = 208~ IR3 = 266. XKTC = k2o~-lC~ (0.51)-213. Curve a is a plot of Eq. [25] ; curve b is the simulated curve.

0

1.0

2.0

3.0

t/tk

Fig. 11. Ring current transients for a second-order following reaction with a potential step at the disk. tk = ~-lD-113vII3(0.51)-2/3. IRI = 83. IR2 = 94, IR3 = 159. The curves are all generated for (XKTC)(m) = 2.0 at different values of m; also shown is the first-order curve, a, First-order; b, 10.0; c, 5.0; d, 1.0; e, 0.5; f, 0.1.

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340

J. Electrochem. Soc.: E L E C T R O C H E M I C A L

ments. We might point out that adsorption interferes i n a similar way in other types of electrochemical t r a n s i e n t m e a s u r e m e n t s (e.g., cyclic v o l t a m m e t r y a n d chronopotentiometry). The method of o b t a i n i n g rate constants proposed here involves setting the disk c u r r e n t at its limiting value, observing the ring c u r r e n t as a function of ,~, and fitting the results to w o r k i n g curves for the p a r ticular RRDE employed. This technique should prove just as useful, and operationally somewhat easier, t h a n the technique suggested b y A l b e r y and B r u c k e n s t e i n (6). The relatively small value of m which yields essentially pseudo-first-order behavior (m > 10) suggests that a good fit to the first-order w o r k i n g curves does not necessarily indicate that a second-order r e action is not involved.

Acknowledgment The support of the Mobil Oil Corporation and the E a s t m a n Kodak Company for fellowships to one of us (K.B.P.) a n d the Robert A. Welch F o u n d a t i o n a n d National Science F o u n d a t i o n (GP 6688X) are gratefully acknowledged. The authors are indebted to

M a r c h 1970

SCIENCE

J. T. Maloy for h e l p f u l discussions d u r i n g the course of this work. Manuscript s u b m i t t e d May 29, 1969; revised m a n u script received ca. Nov. 17, 1969. A n y discussion of this paper will appear in a Discussion Section to be published in the December 1970 JOURNAL.

REFERENCES 1. A. N. F r u m k i n and L. N. Nekrasov, Doklady Akad. Nauk. SSSR, 126, 115 (1959). 2. W. J. Albery and S. Bruckenstein, Trans. Faraday Soc., 62, 1920 (1966). 3. W. J. A l b e r y and S. Bruckenstein, ibid., 1946 (1966). 4. W. J. Albery, M. L. Hitchman, and J. Ulstrup, ibid., 64, 2831 (1968). 5. W. J. Albery, ibid.,63, 1771 (1967). 6. W. J. A l b e r y a n d S. Bruckenstein, ibid., 62, 2584 (1966). 7. D. C. Johnson and S. Bruckenstein, J. Am. Chem. Soc., 90, 6592 (1968). 8. K. B. P r a t e r and A. J. Bard, This Journal, 117, 207 (1970). 9. S. B r u c k e n s t e i n a n d D. T. Napp, J. Am. Chem. Soc., 90, 6303 (1968).

Technica Notes

Q

Ultrafine Porous Polymer Membranes as Battery Separators Joseph L. Weininger* and Fred F. Holub General Electric Research and Development Center, Schenectady, New York I n nonaqueous galvanic systems, based on organic solvents, it is i m p o r t a n t to keep soluble ions of the cathodic reactant from the anode. Conventional ion exchange m e m b r a n e s have been used (1,2) for this purpose, b u t have not been able to support appreciable c u r r e n t densities. I n solvent exchange only an insufficient a m o u n t of organic solvent, e.g. propylene carbonate or d i m e t h y l sulfoxide, replaces w a t e r in the ion exchange resin. To obviate this problem and obt a i n a separator for organic electrolyte cells, a n alternate method of ion exclusion is reported in this note. It is possible to restrict migration of solvated ions by reducing the pore size of microporous m e m b r a n e s below a certain limit. Different methods are k n o w n of producing microporous polymers b y the i n t r o d u c tion of additives into a thermoplastic resin and following this by the r e m o v a l of the additive to leave a porous structure. For example, Sargent and Safford (3) introduced anionic surfactants into polyethylene and processed the mixture. After the additive is leached out, there r e m a i n s a microporous structure. Similarly, in the p r e s e n t method ultrafine porous p o l y m e r m e m b r a n e s are prepared. These m e m b r a n e s are flexible, about 50% porous, and, most importantly, have extremely fine pore size. The average pore size is b e t w e e n 40 and 120A. Membranes were prepared by adding sodium b e n zoate or other salts of benzoic acid to the melted p o l y m e r in a weight ratio of 70-85 parts of benzoate to 30-15 parts of polymer. The salt does not dissolve in the polymer, b u t forms a dispersion or colloidal suspension. I n processing, the p o l y m e r was milled on differential rolls with the benzoate salt at 140~176 for polyethylene or 170~176 for polypropylene. The * Electrochemical Society Active Member,

m i x t u r e s were t h e n cooled close to the softening point of the p o l y m e r and sheeted into 0.005-in. thick films. The final leaching of the sheets occurred in w a t e r at 20~176 The salt was generally extracted i n 1-16 hr. However, leaching is almost complete w i t h i n 5-10 min. I n some cases the polymers were also irradiated with high energy electrons of 20 Mr at a dose rate of 10 M r / m i n before or after extraction of the salt. This irradiation step was employed to improve the t h e r m a l and mechanical properties of the porous p o l y m e r for possible later use at greater t h a n a m b i e n t pressures and temperatures. I r r a d i a t i o n after leaching was more effective in s t r e n g t h e n i n g mechanical and t h e r m a l properties, but b y this t r e a t m e n t the smallest pores were closed b y cross-linking. M e m b r a n e s were tested r o u t i n e l y for porosity, gas permeability, and electrolytic conductivity. The porosity was d e t e r m i n e d b y density measurements, the gas p e r m e a b i l i t y from nitrogen flow as a function of applied pressure, and the electrolytic conductivity was measured at 1 kHz with a conductance bridge after filling the m e m b r a n e with 1M KC1. The k n o w n porosity of 40-50% corresponds to the initial weight ratio of polymer and additive. The conductivity is of the same order of m a g n i t u d e as that of the solution filling the m e m b r a n e . This shows an open continuous pore structure. Finally, small gas p e r m e ability is a measure of small pore diameters. F r o m the m a g n i t u d e of the conductivity, a tortuosity factor can be calculated. It is the ratio of the actual electrolytic conductance to that expected at the given porosity a n d thickness for a structure with straight pores, n o r m a l to the surfaces. The properties of three m e m b r a n e s are listed i n Table I. They are given as examples of the magnitudes

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