relaxation of electrode processes with simultaneous consideration of double-layer charging and...

ELECTROA.NALYTICAL CHEMISTRY AND INTERFACIAL ELECTROCHEMISTRY 277 Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands

R E L A X A T I O N OF E L E C T R O D E PROCESSES W I T H SIMULTANEOUS CONSIDERATION OF DOUBLE-LAYER CHARGING AND FARADAIC CURRENT. P A R T I

K A R E L HOLUB*

Department of Chemistry, New York University, New York, N. Y. zooo 3 (U.S.A.)

(Received October I7th, 1967)

I t was recently shown by DELAHAY 1-8 that for some non-steady-state and periodic electrode processes it is necessary to take into account simultaneously the charging of the double layer and the faradaic process in the solution of the mass transfer problem. I t was also pointed out that the current for non-steady-state or periodic electrode processes generally cannot be separated into two independent parts corresponding to the charging of a supposedly ideal polarized electrode a n d t o the pure faradaic process, respectively. The combination of these ideas, which were originally developed on the basis of the model of charge separation and recombination, was formulated in the form of three general conditions for transport (diffusion) equations1, ~. An explicit form of the time-derivatives appearing in the general equations was suggested 4 for small perturbations and was applied to the derivation of the electrode admittance with either any exchange current s or with an infinite exchange current s . The same approach is followed here for non-steady-state electrode processes, and the differences between this t reatment and previous classical treatments are noted. A general formulation is given for any galvanostatic or potentiostatic perturbation of small amplitude and is then applied to specific cases.

FORMULATION AND SOLUTION

We consider a charge transfer reaction,

e e

A ~ B , - - z e

which takes place on a plane electrode. The oxidized (A) and reduced (B) species correspond, for example, to a metal ion in solution and the metal in the corresponding amalgam-electrode. The formulation will be given for this case but the results are also valid when both A and B are soluble in solution. Furthermore, we assume that the changes of concentrations of constituents other than A and B can be completely neglected in the analysis of the mass transfer process.

The time-derivatives of the surface excesses (F) of A and B, which appear in the first two general equations 1,2 are expressed as linear functions of the potential E and the concentrations of A and B for a finite exchange current (irreversible process). The time-derivative of the charge density of the electrode (q), which appears

* Present address: J. Heyrovsky Polarographic Inst i tute , Opletalova 25, Prague i, Czechoslovakia.

j . Electroanal. Chem., 17 (1968) 277-287

27~ K. HOLUB

in the third general equation, is expressed in the same fashion. This procedure is identical with that already applied in the derivation of the electrode admittanceS.6 and is justified for perturbations of sufficiently small amplitude. Only two of the above three variables are needed when the exchange current is infinite (reversible processes). The expression for the faradaic current is linearized, as usual, and so is the Nernst equation for small perturbation. The model of the double layer used here is only approximate. Thus, surface excesses are reduced to surface concentrations; the surface concentrations are assumed to be in equilibrium with the potential and the volume concentrations at the electrode.

The problem is formulated for galvanostatic or potentiostatic pulse of any shape (for which the linear approximation is adequate) by the system of diffusion equations

OCa/at=D~OZCa/OX 2, ~Cb/~t=DbO2Cb/~X 2 (I)

with the following initial and boundary conditions

t = o : i = o , E = E e q (2)

t=o, ___x_>o/. t > o , __x--~oo) ca=c~ (sign+), c~=c~ (s ign-) , (3)

(DaOc~/Ox- DbOc~/Ox =d( Pa + P~)/dt, (4) t >o, x =o: i i=zF(-D~Oca/Ox+dFa/dt) +dq/dt, (5)

[iy = zF( -- D aaC ~/Ox + d_P~/dt). (6)

Notations are quite conventional and need not be described except that to note that i is the experimental current and i~ represents the faradaic current. Equation (5) is not a boundary condition for potentiostatic perturbation; this equation allows the calculation of i once the diffusion problem is solved. Note also that eqn. (6) holds for a finite exchange current; it should be replaced by the linearized Nernst equation for an infinite exchange current.

The time-derivatives of t h e / " s and q are introduced as indicated before, and the following notation is introduced: one of the superscripts, a, b, 7, on the symbols q, /'~, /'~ indicates the first derivatives of q, /'~ o r / ' ~ with respect to Ca, co or E. The symbol / ' z represents /'~ +/ '~ , and the superscript notation is also used for / 'z . Moreover, we have one of the two conditions

t >o : i=i( t ) for galvanostatic process (7)

t >0 : E =E(t) for potentiostatic process. (8)

We introduce

a=(Ca--Ca~)/CaL b=(c~--cb')/c~L I=iFza/(zFO~ca~), (9 a)

v-=v~, v=(acb /~E) / cd=-zF / (RT) , ~ I=E-E ,q , (9 b)

T = D . t / ( F z . ) 2, X = x / F z ~ (IO)

and obtain (see list of symbols in Appendix)

~a/OT=~Za/OXZ, Ob/OT=DO2b/OX z :(Ii)

j. Electroanal. Chem., 17 (1968) 277-287

R E L A X A T I O N OF E L E C T R O D E P R O C E S S E S . I 2 7 9

T = o : v=o, I = o , (12)

T = o , + X > o t- T > o , +X-+oo] a = o (sign+), b = o (s ign-) , (13)

(aa/BX-ROBb/aX = da/dT +7db/dT + edv/dT, (14)

T > o, X =o: ~I-- - ~a/~X +hada/dT +hbdb/dT +hndv/dT, (15) -a~a/BX +aada/dT +a bdb/dT +andv/dT =iaa + i~b +inv. (16)

The following relationships hold in eqn. (16) for irreversible processes

a = I , aj=Tj(j=a,b,~), ij(j=a,b,~) as defined in the Appendix. (17)

For reversible processes one has in eqn. (i6)

a = o , aj=o(j=a,b,~7), ia=-t* , i b = I , i n = - I . (18)

If we choose for q, / 'a, /'b, the independent variables Ca(X=O) and E, we have ya=! , y~=o, kb=o, hb=0.

Finally we have

I =I(T) for galvanostatic, or (19)

v-----v(T) for potentiostatic perturbation. (20)

The system for eqns. (II)-(2o), as solved by Laplace transform, yields:

I +~/s r+~ Vs

= l__ S+S ra--ia (21) Vs hn/s l+haVs h Vs'

~Ts 1 + Vs r + yl/s sa~- i n a~s+saa-ia Sab--ib

where ~ and I are Laplace transform of v and I, respectively. Equation (21) can also be written in the form:

3 4 = is-½ ~ bjsJ/2/~ a~sm (22)

j=0 J=0

where the coefficients aj and bj can be easily calculated from eqn. (21). For galvanostatic perturbations we write on the basis of eqn. (22) the ex-

pression for ~ in a more convenient form 4

~ = i X [Aj/(¢s(rj+#s))] (23) j = l

where A~ can be calculated from eqn. (22) and from the r / s which are the roots of the equation:

4

a4 -1 2~ aj#=o. (24) j = 0

The expression for i for potentiostatic perturbations, can be written on the basis of eqn. (22) in the.form:

3 l=~[B~s+ B~/s+ Bo + Y~ Cj/(i/s+ej)], (25)

j = l

j. Electroanal. Chem., 17 (1968) 277-287

280 K. HOLUB

where the coefficients B , (k =o,I,2) can be found from eqn. (22) and ~j are the roots of the equation: :,

3 ba -1 ~ bjxJ=o. (26)

j=o

Formulas (23) and (25) can be used directly for obtaining v and I for any admissible (within the linear approximation) kind of perturbation. The inverse transform is in general

± f v = Aj I ( r - r)exp(rj2~)erfc(rA/~)d~, (27) f=1 0

dv d ~" v('r)

{28) C , f : v ( T - 7 : ) [ - - ~ - o,exp({}j2z)erfc(o,Uz)] dT

Jffil

REVERSIBLE ELECTRODE PROCESSES

To treat reversible electrode processes we use eqn. (21) in which the coefficients are those of eqnl (18). We also introduce the following notation

0= r / ( I+ # r ) , P = ( I +#r)/(I +#~), N=r/ (y+e) , M=!/h~, L = h n M N / P O (29)

and deduce a result similar to eqn. (22), namely

M N I P + ]/s = I p~ • V-- s • L(P + Us) U s - (N + Us) (M + Us) " (30)

We obtain explicit forms of this result for potentiostatic or galvanostatic perturbations by introducing any particular perturbation and deducing the resulting inverse Laplace transform. Thus, for a galvanostatic perturbation

~= I.K~ Us{x~ + Us) + Us(x~ + Us) ' (3i) where

K ~ = M N ( x I - P ) / [Pe (L - ! ) ( x l - x2)], K 2 = K I ( P - x 2 ) / ( x I - P) (32)

with

xl=p+~p~+Q, x2=p-Vp2+Q, p = [PL - ( M +N)]/[2(L - I)], Q =MN/(L - I).

Similarly we have for potentiostatic perturbation

1 = ~ (Po/MN) {s (L - I) - LIUs - L2Us/(P + Us) }

where

LI = M + N - P , L ~ = M N - P L .

(33)

(34)

(35)

(36)

APPROXIMATE SOLUTION

We now solve the system of eqns. (11)-(2o) by changing it into a system of integral equations. This approach yields an approximate solution which allows direct comparison with results obtained in classical treatments for T-+o. This approach

J. Electroanal. Chem., 17 (1968) 277-287

RELAXATION OF ELECTRODE PROCESSES. I 281

also enables one to estimate an error affecting the approximate solution. We first introduce some notations. Let A be a matrix with n rows and n columns of elements ag~ and x a column vector of n components x~. The quantities IIAll o~ and Ilx[[ are defined by the following

]]Allo~--Max ~ ]aM, Ilxll=Maxlx~l (37)

If we have 7 another vector y given by the equation y =A.x then

IlYlI- < IIAII ® [[xl[.

Let f(T) and g(T) be two functions, and let us denote by f*g the convolution integral

In the same way we denote by A , x a vector y whose components y~ are defined by the relation:

y~ = ~ a~z,x~ (39) l=l

If ak, or xz is a function of t, we define [[A[[ and ][x[[ for a given interval, o<t<_T, by the relations:

[Ix[l= Max Ix, I, [[AI[= Max [a~,x[. (4 ° ) *, O~ t~ T R,O< t~ T

Similarly we have for y = A * x [[y[]_< ][A][-]]x[[. By noting that we have at x = o (see eqns. (I i) and (13))

(3a/OX),I = --(I/]/~--T),a, D(Ob/OX),I = ~ ) , b (4 I)

we deduce from eqn. (41) and eqns. (11)-(2o) the following system of integral equations :

hnv + h~a + h ~b = I , 1 - ( x /V~-T) , a ,

+ a + ),b = - (llV ),a -

Vnv + y ,a + y bb = -- (I/V~-T),a + G,a + i b,b + in,v.

(42a)

(42b)

(42c)

We now write this system of integral equations in matrix formulation in a way which is different for galvanostatic and potentiostatie perturbations. In the first case, I * I is a known function and we must find v (and perhaps also a and b). Conversely, v is a known function for potentiostatic perturbation and we have to obtain I (and perhaps also a and b). I t is seen immediately from eqn. (42) that we can directly calculate just I , I but this does not mat ter so much because we can measure the integral of the current during the perturbation just as well as the current. We denote for a galvanostatic perturbation

( i ) / " I ' x = ' Y = t o ) , (43)

j . Electroanal. Chem., 17 (1968) 277-~87

282 K. HOLUB

= h°) (i , o) (o o o) (44> ~Yn Y~ ~b ' I r , K2 = i n i~ ib '

KI = (I/]/~T). H~. (45)

The system of integral equat ions can be now writ ten in the simple form:

Gx = y - (K~ - K~) ,x (46)

where x is the vector to be found. If we assume tha t det(G) # o , we get from eqn. (42)

x =z - (M1 -M2)*x (47)

where z=G=ly, Mj=G-1Kj ( j = ! , 2 ) . (48)

Writ ing Ax instead of (M1-M~)*x, we can rewrite the system of integral equations in eqn. (42) in the simplest form:

x = z - A x . (49)

The solution can be wri t ten simply in the form

x = ( E - A + A 2 - A 3 +... + ( - i)nA n + . . . ) z (50)

where E is the unit operator, i.e.,

Ez =z (51)

If we take a finite number of terms in eqn. (50) we obviously obtain only an approximate solutionT, s. I f we take for example, as n ' t h approximate solution

x(n) = ( E - A +A2-T-... + ( - I)nAn)z (52)

we obtain the estimate of error

[[M1-M2[ [n+l I[x-x(~>ll <_ ~ - [ IM~-M211 Ilzl[- (53)

This estimate is only useful for I IM1-M21[ < 1 but, in a given case, the solution of eqn. (50) is valid for any ][M1-M2[[.

For the zero-th approximation, x(0) = z , we deduce an estimate of the error from eqn. (53)

1 + ¢~" [[Melloo . / A T 4 []G-1HI[[~ V ~ - - I,z[[ • (54)

l lx-z l l _< " [IG-1HII[ I - [¢~ffI[G-IHII[~o+TIIM2I[o~]

I t is immediately seen from eqn. (54) tha t for T---> o we have x--->z with an error of order 1/T.

For potentiostat ic per turbat ion we write

(/:1 t " = ' Y = l - ~ v ! (55) \b / in,v--Tnv~



oOOi) = o G - - ( 5 6 )

ia 0 7a

and we use the same K1 and HI as in eqns. (44) and (45). We can now apply eqns. (45)-(54), keeping in mind that x, y, K2 and G have

different meaning than for a galvanostatic perturbation. In particular, we have for the zero-th approximation, x(°) = z =G-~y, and we again find that the exact solution x-+x(o) with an error of order tIT.

COMPARISON WITH CLASSICAL FORMULATION

We compare the present results with those derived by the classical approach, i.e., without consideration of the double-layer charging in the analysis of the diffusion problem. We start with galvanostatic perturbations, and let the current, in our notations, be given by

I = / ° ' T < o (57) ( I1, T >o .

The dimensionless potential is given in classical t reatment (denoted by the subscript c) by:

vc-->IxT'(I/kn) for T-+o. (58)

We have in our t reatment

v2v-+I1T. (b3/a4) for T-+o (59)

where the subscript N indicates our values, and b3 and a4 are coefficients from eqn. (22). The limit given by eqn. (59) can easily be obtained from eqn. (27) after integration and by noting that

4 A~ =b3/a4.

j=l

The limit can be deduced directly from x-~G-ly where x and y are defined by eqn. (43) and G by eqn. (44). We deduce from eqns. (58) and (59)

I kn ~ ~b

vN : v, = (knbs/a @ : I = : I. (60) ksnkaCbI

The potential in both treatments is a linear function of time, but the proportionality constant is different. This coefficient, in the classical formulation, is proportional to Ilk m i.e., to the reciprocal of dq/dE. In the present treatment, the coefficient is proportional to a more involved expression containing dF/dc's, dq/dc's and dq/dE.

Results for large T's are obtained as follows. We deduce from eqns. (23) or (27) and (57)

2rjVT exp(rjeT)erfc(rjUT)] (61) v=I i ~ Aj [ - I + + J =1 ~,j2 7

j. Electroanal. Chem., 17 (I968) 277-287

284 K. HOLUB

The last term approaches zero when min~[rj[~/T >>r, and the asymtotic expression of v i~

2 G-i~r V1. blao- boa11 (62) V Ii t ~ i~r ao i J

This formula gives the same relation for the proportionality coefficient between and t~ as the classical theory 9. The difference between the two treatments appears in the constant term which is now more involved than in the classical treatment. I t should be kept in mind that eqn. (61) is valid only when v is so small to be in the region of the linear approximation. Thus, there is a lower and upper value of T for which eqn. (62) can be applied. Finally, we note that eqn. (61) is formally the sum of four terms whereas the corresponding classical formula contains only two terms.

We now turn to the potentiostatic perturbation for which we have

= /vo, T > o ; vo is a constant (63) v [o, T_< o.

The classical t reatment 10-1s yields, with the present notations,

I=voi~ exp (is~T) erfc (-is#T) (64) where

is =G-G/r (65)

The present t reatment yields on the basis of eqns. (63) and (25) or (29)

+ Bo + ~ Cj [1-exp(oj~r)erfc(0jl/r)? . (66) j=l ~j

II[ =oo for T = o , both for eqns. (64) and (66). The basic difference between eqns. (64) and (66) lies in the first term on the right-hand side of eqn. (66). We have for T-+o

IN-+vo BdV~T (67)

whereas the classical solution gives

Ic-+voi~ (68)

We pursue the comparison between present and classical t reatments by considering reversible processes. The classical t reatment yields, with the present notations,

l= ~(- V~Q + skn) (69)

The limit according to classical theory

vc-+IIT/kn, for T-+o (70)

holds for a galvanostatic pulse [eqn. (57)] whereas the present t reatment leads to

v2v-+II(MN/PQ) {T/(L-I)}, for T-+o. (71)

Both results coincide in the functional dependence on T at T-+o, but the proportionality constants are different. Both treatments coincide completely for n~inlxjll/T >> I.

The formulas for potentiostatic perturbations and reversible processes are:



Ie= --Voo/V~T, (72)

IN = -- voe(PL1/MN){I /V~T + (L2/L1)exp(P2T)eric(P]/T)}. (73)

Both Ie and IN depend on T-½ at times T~. L1/(V~L2) but with different proportionality coefficients. Ic and IN coincide for P[/T >> I.

Comparison between this and the previous approaches can also be extended to coulostatic perturbations 14. We first substitute in eqn. (27) the coulostatic perturbation QI8(T) for I (T) and integrate from - o . Thus

4 v = • A~Qlexp(rj~T)erfc(r~T). (74)

1=1

The classical formula is, with present notations,

Q 1 2 Zjexp(zj2T)erfc(z~f T) (75)

v = k , (Zl-Z2) j=l

where

Z l = - z 2 , Z2=21, (76)

Zl = - - i~/2 + #(½ia)Lin/k n, 22 = -- i~/2 -- V(i~/2)2-in/k n (77)

We again find the same behavior for T->o but the constants to which the solutions converge are not the same. They are in the ratio (b3/a4) : (I/kn), for this and classical treatments, respectively. The formulas coincide in the limits for minlrall/T >>I and min[z~[]/T >> I. J

J

CONCLUSION

The classical formulation of non-steady-state electrode processes and the present approach compare as follows:

The same kind of functional dependence of the response on time prevails for short time in some cases but the proportionality constants are not the same, e.g., linear overvoltage-time dependence for a single-step galvanostatic perturbation. Hence, previous analyses in which dq/dE was deduced by extrapolation to t =0 for a galvanostatic step-perturbation are in error whenever conditions are such that the classical analysis is not applicable. In other cases, e.g., potentiostatic single-step perturbation, the functional time-dependence is not the same in both treatments. Thus, the present treatment yields a current which varies inversely with time whereas the classical formulation gives a current approaching a constant for time tending to zero.

Both treatments coincide for long times, i.e., whenever charging of the double layer becomes of little significance. Conditions for coincidence in the solutions are for irreversible processes:

min[rj[Tt >> I for a single-step galvanostatic perturbation (see r~ in eqn. (24)); J

mi.nlojlT½ >> I for a single-step potentiostatic perturbation (see Oj in eqn. (26)).

In the time-domain between the above two extreme regions the functional time-dependence is different in the two treatments.


286 K. HOLUB

ACKNOWLEDGEMENT

This work was supported by the National Science Foundation. The author wishes to express his thanks to Professor DELAHAY for suggesting this work and for his kind interest in it.

SUMMARY

Non-steady-state electrode processes are analyzed -with simultaneous consideration of the double-layer charging and the faradaic process in the t reatment of the diffusion problem. A treatment is given from a unified point of view for any galvanostatic or potentiostatic perturbation. The general formulas thus derived are applied to galvanostatic and potentiostatic step-perturbations and to one type of coulostatic perturbation. Differences from the results previously derived by the classical approach were found in the region where the double-layer charging influences the overall process.

APPENDIX

Lis t of symbols All parameters given below are dimensionless with the exception of quanti ty v

and the parameters fl, f2, f~. The superscript, a, b or ~, on the quantities q, / 'a, F~, / 'E, denotes the partial derivative of these quantities with respect to ca, c~, or E. The subscript E denotes the s u m / ' z = / ' a + F~.

D =D~[Da

f l = (ai1/OCa)/(zF)

f2 = (Oif/Ocb)/(zF)

f3 = (~i~/OE)/(zF)

hj = 7,j+k~ ( j=a ,b ,~ )

ia = f l P z a / D ~

ib = R f2Fza /Da

in = f~ Fza/(vc aSD a)

I = iF2a/ ( zFDaca s)

ka = q a / ( z F F z a)

kb = R q ~ / ( z F F z ~)

k n = qn/(zFvFzaCa ,)

Q = Aq/(zFca"Fza)

= RV D

R = ebb/ca ~

= F z , / ( ~ c a , F z a )

7' = R F z b / F f f '

j . EZectroanal. Chem., 17 (1968) 277-287

RELAXATION OF ELECTRODE PROCESSES• I 287

Yb = R F a b [ F z a

~77 = I ~ a 7 f ( i ) c a s F z a )

hj = y j+ka (j=a,b,rl)

= ( a c ~ l ~ c o ) l R = ~

v = (acb/dE)lcb s = - z F / ( R T )

Constants aj and bj in eqn. (22) The constants aj, b~ appearing in eqn. (22) can be derived from eqn. (21).

These constants are listed here.

ao = - iTr

~ n h~ al = -- I

~7 ~a

~ 7 h. a2 = -- I,

"~ 7 "ha

a3 .~ I

77 I

h7 h~ a4 ~ ~-

7'n 7~ bo = i a r - i o

hb

Y. + e i ~ - 7 i 7 *b

)b -r (hT-~7) ib

~b -r(hnya-ha~7)

;° 7b

bl = 7 i ~ - ( r +ib)

b2 = 9,~- (r7 ~ + 9,)

b3 = 7 ~ - Y T a

R E F E R E N C E S

I P. DELAHAX r, J . Phys . Chem., 7 ° (1966) 2067. 2 P. DELAHAY', ibid., 70 (1966) 2373. 3 P" DELAHA'V At'riD G. G. SUSBIELLES, ibid., 7 ° (1966) 315 o. 4 P. DELAHAY, K. HOLUB, G. G. SUSBI•LLES AND G. TESSARI, ibid., 71 (1967) 779. 5 K. HOLUB, G. TESS&RI 2kinD P. DELAHAY, ibid. , 71 (1967) 2612. 6 P. DELAHAY AICD K. HOLUB, J . Electroanal. Chem., 16 (I968) 131. 7 L. COLLATZ, Funct ional Ana ly s i s and Numer ica l Mathematics, Academic Press, New York, 1966. 8 B. VULIKH, Introduction to Funct ional Ana ly s i s for Scientists and Technologists, Addison-Wesley,

Reading, Mass., 1963 . 9 T. BERZIN'S AlffD P. DELAHAY, J . A m . Chem. Soc., 77 (1955) 6448.

IO M. SMUTEI~, Proc. Polarograph. Congr., I (1952) 677. i i T. KAMBARA AND I. TACHI, Bul l . Chem. Soc. ( Japan) , 25 (1952) 135. 12 P. DELAHAY, J . A m . Chem. Soc., 75 (1953) 143o. 13 H. GERISCHER AND W. VIELSTICH, Z. Phys ik . Chem., N . F . , 3 (1955) 16. 14 P• DELAHAY, J . Phys . Chem., 66 (1962) 2204.

j . Electroanal. Chem., 17 (1968) 277-287

relaxation of electrode processes with simultaneous consideration of double-layer charging and...

Documents