Transient convective diffusion to a disk electrode

Electroanalytical Chemistry and Interfacial Electrochemistry, 50 (1974) 2359

23

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

T R A N S I E N T C O N V E C T I V E D I F F U S I O N TO A DISK E L E C T R O D E

KEMAL NISANCIOGLU and JOHN NEWMAN

Inorganic Materials Research Division, Lawrence Berkeley Laboratory, and Department of Chemical Engineering, University of CaliJornia, Berkeley, Calif. 94720 (U.S.A.) (Received 20th August 1973)

INTRODUCTION

Nonsteady-state methods are commonly employed in electrochemistry for the study of electrode kinetics and mass transfer in electrolytic solutions. The transient behavior of an electrochemical system is determined by a number of processes of varying complexity, which may occur concurrently at the electrode surface and in the bulk solution. To name a few, these processes include convective diffusion I and the capacitive discharge of the electric double layer due to a faradaic reaction and/or transfer of charge through the bulk solution in the presence of a nonuniform electric field 2"3. The fundamental treatment of transient diffusion at electrode surfaces has been of interest since the classical study of the problem early in the century by Roseburgh and Lash-Miller 4. The rotating disk electrode has become popular for convective-diffusion problems because previous applications of this geometry have been quite fruitful in the assessment of its hydrodynamic conditions 5 6 mass transfer characteristics 7"8 and the current distribution in the presence of an electric field and complex electrode kinetics 9"10. Levich 8.11 has solved for the first time the tjcansient diffusion equation for the disk geometry. Since convection is ignored in that treatment, the results are valid only for very short times. Subsequent analytic efforts12,13 with the consideration of axial convection are also limited to small time intervals due to approximate methods of analysis. Fairly accurate numerical solutions are available for response to flux step 1 and concentration step 14"15 at the surface. However, analytic results are always more desirable for design calculations, determination of relaxation times, and investigation of complex boundary conditions involving electrode kinetics and capacitive effects. Krylov and Babak ~6 have recently attempted an exact solution of the axial convective-diffusion equation by a classical perturbation expansion technique. They have reported results for the concentration step and flux step conditions. Selman I s derived independently the same solution for the concentration step. These results provide considerable improvements over the previous analytic work but are still confined to relatively short times if a reasonable number of terms are to be retained in the series expansion. The purpose of this paper is to contribute to the past effort by presenting an alternative treatment for large times, so that the results can be employed interchangeably with the short-time series of Krylov and Babak within their ranges of applicability.

24

K. NI~ANCIO~LU, J, NEWMAN

In treating problems of this kind, it is commonly assumed that the concentration variations in the solution are confined to a thin region near the electrode surface (high P6clet numbers). Consequently, the axial velocity component can be approximated by 5-9 (1)

vr = - a y 2 Q , ~ / ~ / v

where a=0.51023, y is the axial distance from the electrode surface, f2 is the angular rotation speed, and r is the kinematic viscosity. It is further assumed that the disk is uniformly accessible, and thus radial convection can be ignored. One should realize, however, that the latter assumption is introduced merely as a mathematical convenience. It is well accepted by now that radial convection becomes significant below the limiting current 9. The experimental data of Nanis and Klein 17 seem to indicate that this assumption may lead to appreciable error especially during the transient build-up of overpotential after a step increase in the current. THEORETICAL FORMULATION

Without the radial terms, the transient equation for convective diffusion reads

OC

~C

a5 +

~2C = o -~y2

t2)

We introduce the dimensionless variables 0 = f2(D/v)~(a/3) ~ t

(3)

= y(av/3D)~x/-~/v

(4)

(concentration step)

0 = (co-

c)/c~

0 = (c~-

c)/(0c/0()¢ = o

(5)

or

(flux step)

(6)

Equation (2) thus becomes

~0 aO

~0 ~20 = 3~2~7-~ +

(7)

Consider the build-up case after a step increase in the concentration or flux. The boundary conditions are

O=Oas ~ - ~ / 0

Oat 0=0

(8)

f

0 = 1 at ( = O, 0 > 0

(concentration step)

(9)

or t?O/~( = - 1 at ( = O, 0 > 0

(flux step)

(10)

It is possible to express 0 in terms of a steady-state and a transient part, 0 = O"s-O t

(build-up)

(11)

TRANSIENT CONVECTIVE D I F F U S I O N TO A DISK E L E C T R O D E

25

so that each part satisfies eqn. (7) separately. The boundary conditions for 6~ S are O ~ = 0 as ~ o o

(12)

O ~ = 1 at ~ = 0

(concentration step)

or

~Oss/~( = - 1 at ( = 0

(flux step)

(13)

l

These yield the solutions Os s _ -

or

1

[

oo

r(4/3) j ~

O ss =

e - ~ dx (concentration step)

e - ~ dx (flux step)

(14) (15)

where the integral ca,n be found as a tabulated ~8 function of ~. The transient part of concentration satisfies the conditions O ~ = 0 as ~--,oo Ot

O~

at

0 = 0

/

(16)

/

O ~ = 0 at ~ = 0, 0 > 0

(concentration step)

c~O'/O~ = 0 at ~ = 0, 0 > 0

(flux step)

(17) (18)

The calculation of transient overpotential decay after a step decrease in the surface concentration or flux may also be of interest 17. However, it is straightforward to show that O = Ot

(decay)

(19)

and therefore a separate formulation is not necessary, contrary to the analysis given by Nanis and Klein 17. The solution to O t can be derived conveniently in terms of a boundaryvalue problem since eqn. (7) is separable, and the conditions (16) to (18) are homogeneous in the ~-coordinate. Let us express O ~ in the form 0t

= ~ B.Z.(~) e-4"° n=0

(20)

.

where Z. is an eigenfunction, and 2. is the eigenvalue associated with it. Substitution into eqn. (7) and conditions (16) to (18) yields the Sturm-Liouville system Z/ ....

2,7,

]

z.( oo ) = o

[

n t :~

Ln"~- ~nZn -~- O

Z,(0) = 0, Z~(0)= 1 (concentration step) [ or

Z',(O) = O, Z , ( O ) = 1 (flux step)

(21)

I

This system has been solved here numerically by a method commonly employed in this laboratory 19-21. The coefficients B, are given by

26

K. NISANCIO(~LU, J. N E W M A N

B. =

6 )ss e~3 Z.(C)d

e{3 Z2(C)dC

(22)

and were evaluated by numerical integration. Table 1 lists the eigenvalues and the coefficients B, after they have been extrapolated to zero mesh size. The first three eigenfunctions are plotted in Figs. 1 and 2 for concentration step and flux step, respectively. The results are compared with the short time series of Krylov and Babak in Figs. 3 and 4. The two series match quite well over a certain range of 0 for each case even though only three terms of each series were used to plot these figures. Agreement with previous numerical calculations ~ ~4. is is also quite satisfactory. Application of the superposition integral makes possible the treatment of TABLE 1 T H E FIRST T E N E I G E N V A L U E S A N D THE R E L A T E D C O E F F I C I E N T S B, O F T H E EIGENFUNCTIONS n

Concentration

0 l 2 3 4 5 6 7 8 9

0.4

step

Flux step

2,

B,

2,

B,

7.21644439 18.1596045 31.1962389 45.7926549 61.6691473 78.6461928 96.5966836 115.424957 135.05591 155.42872

1.12818046 0.90505798 0.7907692 0.718387 0.666834 0.627481 0.596032 0.570071 0.548117 0.52920

2.58078493 12.3099728 24.4331401 38.3054830 53.5740271 70.0220380 87.5010784 105.902059 125.140833 145.15016

0.663516066 0.081564022 0.034457046 0.01962199 0.0128965 0.0092267 0.0069829 0.0O55048 0.0044645 0.0037089

I

I

I

I

Concentration step

Flux step

0.3

O.2 Zn

0.1

0

Z n

o

-o.I -I

-0.2 o

I

I

0

I I

I

C Fig. 1. The first three eigenfunctions for the concentration-step case. Fig. 2. The first three eigenfunctions for the flux-step case.

27

TRANSIENT CONVECTIVE D I F F U S I O N TO A DISK ELECTRODE J

6 "

~

T

I

;

Concentrotion step -- -- Short-time series 15, 16 - Long-time series

5 '\

1.0-

o

~4

I

I

i

I lO-i

I

I

I --

I

1 IO°

i

Flux slep - - . - - S h o r t - time series 16 0 . 8 Long-tlme series ~ . ~ ' ~

\ '~

\.\ I

"\,

3

--

0.6

@O0A

2 j

0.2i

0

I

] lO-i

I0-2

I

0

~ I0°

_

I

0 lOi

IO-Z

0

I0~

Fig. 3. C o m p a r i s o n of the short-time and the long-time series for a concentration step. ( - . - ) Short-time series of refs. 15 and 16; ( - - ) long-time series. Fig. 4. C o m p a r i s o n o f the short-time and long-time series for a flux step. ( . - )

of ref. 16: (

Short-time series

) long time series.

more complex boundary conditions:

c- ca = [Co(0)- c®] oo(0,0 +

fo dco 0

Oc(O-O',~)dO'

(23)

0=0'

or equivalently, C--Coo~

--

oF

= ~;[Of(0-0" 0]d0'

(24)

o

where the subscript 0 denotes conditions at the electrode surface, and Oc and Or represent the concentration-step and the flux-step solutions, respectively. Differentiation of eqn. (23) yields an explicit expression for the flux at the surface, ~O=

co dco f

_o+Jo 0-1_

80= I 0=0-0,

d0"

(25)

k -o_0,

which may prove more useful for certain calculations. DISCUSSION AND C O N C L U S I O N S

The results enable the assessment of time constants, r = K Sc~3/~2

(26)

for build-up or decay of a concentration gradient after a step change in the surface concentration or flux. For a concentration step, K=0.45142, and for a flux step, K = 1.2623. These results are accurate insofar as the radial dependence of concentration can be ignored, such as in heat-transfer studies ~4 and mass transfer in nonelectrolytes. In electrolytic mass transfer, eqn. (26) is sufficient for making estimates, but correction for nonuniform current distribution is probably

28

K. NISANCIOQLU, J. NEWMAN

necessary for more accurate calculations. A complete analysis of the convective diffusion equation with radial dependence appears to be very complicated and rather demanding in numerical effort. However, an asymptotic calculation for large times may be tractable to determine the necessary time constants 22. Double-layer effects may become important in transient electrode processes 2.3. Delahay and coworkers z3 24 have discussed how to treat the conditions at an electrode surface in the presence of mass transfer, faradaic reaction, and double-layer charging. Equations (23) to (25) can be applied conveniently to treat these effects. In m a n y cases, one can express the flux as a function of the surface concentration, so that a numerical solution of the integral eqn. (24) or (25) is necessary 22. The capacitive effect of the double layer has been investigated in a solution with a single reactant and excess supporting electrolyte 22 and was found to be negligible in situations where the transient process is mass transfer controlled av. These integral equations can also be used in a straightforward manner to calculate the potential in current-controlled applications, or, vice versa, the current in potentiostatic cases. ACKNOWLEDGMENT

This work was supported by the United States Atomic Energy Commission. SUMMARY

Mass transfer to a rotating disk electrode is calculated at large times after a concentration step or a flux step at the surface. Radial dependence of concentration is ignored. Further application of results to treat more complex boundary conditions is discussed. NOMENCLATURE

a B, c D

K Sc t

vy y Z, 4, v 0 0 O

0.51023 coefficients in series for O' concentration/mol c m - 3 diffusion coefficient/cm 2 s-1 (1/2o)(3/a) ~, constant defined in eqn. (26) v/D, Schmidt number time/s axial velocity/cm s 1 axial distance from disk/cm eigenfunctions in series for O t eigenvalues in series for O' kinematic viscosity/cm 2 s 1 angular rotation speed/radians s-1 characteristic time constant/s dimensionless time dimensionless concentration dimensionless axial distance from disk

TRANSIENT CONVECTIVE DIFFUSION TO A DISK ELECTRODE

29

Su]J'ixes c f 0 ss t

concentration step flux step evaluated at the electrode surface steady state part transient part evaluated far away from the disk

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13

J. M. Hale, J. Electroanal. Chem., 6 (1963) 187, K. Ni~ancio~gtu and J. Newman, J. Electroehem. Soe., 120 (1973) 1339. K. Ni~ancio~lu and J. Newman, J. Electroehem. Soe., 120 (1973) 1356. T. R. Roseburgh and W. Lash-Miller, J. Phys. Chem., 14 (1910) 816. Th. v. Khrmfin, Z. Angew. Math. Mech., 1 (192i)233. W. G. Cochran, Proc. Cambridge Phil. Soe., 30 (1934) 365. V. G. Levich, Aeta Physieochim. URSS, 17 (1942) 257. V. G. Levich, Physicoehemical Hydrodynamics Prentice Hall, Englewood Cliffs, 1962. J. Newman, J, Electroehem. Soc., 113 (1966) 1235. J. Newman, J, Electrochem. Soe., 114 (1967) 239. V. G. Levich, Acta Physicochim. URSS, 19(1944) 133. Yu. G. Siver, Russ. J. Phys. Chem., 34(1960)273: (Zh. Fiz. Khim., 34 (1960) 577), V. Yu. Filinovskii and V. A. Kiryanov, Dokl. Phys. Chem., 156 (1964) 650; (Dokl. Akad. Nauk SSSR, 156 (1964) t412). 14 D. R. Olander, Int. J. Heat Mass Transfer, 5 (1962) 826. 15 J. R. Selman, Ph.D. Thesis (UCRL-20557), University of California, Berkeley, June 1971. 16 V. S. Krylov and V. N. Babak, Soy. Electrochem., 7 (1971) 626; (Elektrokhimiya, 7 (1971) 649). 17 L. Nanis and I. Klein, d. Electrochem. Soc., 119 (1972) 1683. 18 M. Abramowitz and I. A. Stegun (Eds.), Handbook of Mathematical Functions, National Bureau of Standards, Washington, 1964, p. 320. 19 J. Newman, hut. EngiH. Chem. Fundam., 7 (1968) 514. 20 J. Newman in A. J. Bard (Ed.), Electroanalytie,d Chemistry, Vol. 6, Marcel Dekker, New York, 1973, p. 187. 21 J. Newman, Electrochemical Systems, Prentice Hall, Englewood Cliffs, 1973. 22 K. Ni~ancio~lu, Ph.D. Thesis (LBL-1880), University of California, Berkeley, 1973. 23 P. Delahay, J. Phys. Chem., 70 (1966) 2373. 24 K. Holub, G. Tessari and P. Delahay, J. Phys. Chem., 71 (1967) 2612.