Surface diffusion and galvanostatic transients. I

ELECTROANALYTICAL CHEMISTRY AND INTERFACIAL ELECTROCHEMISTRY E l s e v i e r S e q u o i a S.A., L a u s a n n e - P r i n t e d in T h e N e t h e r l a n d s

485

SURFACE D I F F U S I O N AND GALVANOSTATIC T R A N S I E N T S . I

S. K. RANGARAJAN Central Electrochemical Research Institute, Karaikudi 3 (India)

(Received April 3rd, 1967)

i. Two observations which have led to the postulation of surface-diffusion as a possible slow step in galvanostatic metal deposition (and dissolution) experiments are that (i) the rise-times of ~-t curves and (ii) the "steady-state" ~-values are appreciably greater than those predicted b y the charge-transfer step alone. Various authors 1-1o have shown that a "surface-diffusion" model is adequate, in m a n y cases, for the interpretation of galvanostatic transients. Attempts have also been made to evaluate the relevant parameters characterising the model. Under conditions not involving diffusion in the solution phase, the basic steps constituting the overall transfer M ~+ (solution) ~,~ M (lattice) are: (A) charge transfer at the planar site and (13) adatom transport to the latticebuilding sites. Assuming edges to be "ideal sinks" (or sources) and an average "intergrowth-line distance", l, we can evaluate the following constants: (i) io, exchange current density and Cado, adatom concentration defined in (A) above; (ii) Da/l 2 (time -i) where D~ is the surface-diffusion coefficient introduced

by (B). 2. We give below the conclusions drawn b y BOCKRIS et al. and the procedure adopted b y them to evaluate i0, cad ° and D~/l 2. (a) ~]t-~o = a + bt where a = R T i / z F i o and b ---ERT/z ~( F2)] (i/c ad°)

(I)

d In (~, -- ~ o~)/dt = - vo/c aa0

(2)

n'-~ oo = R T / z F ( i / i o + i / z F v o )

(3)

and a study of ~,_~o~-log i curves can be used to find i0, ac and ~ . (b) vo, defined in eqns. (2) and (3), is termed the "surface-diffusion velocity" and is identified 2-10 with D~C~dO/l 2 and hence is supposedly independent of i0. v0 = D ~ c ~oo/l 2

(4)

(c} The rise-times for the galvanostatic and the potentiostatic transients, zg and zp, respectively, are given 9 as ~g = 2 c a ~ ° / v o

(5)

Zp = 2CadO/(2Vo + p c aa °)

(6)

and

j . Electroanal. Chem., 16 (1968) 485-492

486 so

s . K . RANGARAJAN that

(~g/~) = 2 +io/zFvo,

2;-+o

(7)

It has been suggested 9 t h a t (io/vo) can also be deduced from eqn. (7). (d) A sort of "coverage effect" for the cathodLc rate is introduced4,6, 8 in the following w a y allowing for a "finiteness" of m e t a l a t o m s on the surface.

i = io[(C~d/c~do) exp (~F~I/RT) --

(G -- c ad)/(G -- Cadd) exp ( -aoF~/RT)]

(8)

In eqn. (8), G is the total n u m b e r of m e t a l a t o m s on the surface. W i t h

r = io/zFvo

(9)

and

O' = Cad°/(G-cad°) ,

(!o)

the i-~/relationship was proved to be:

i/io = [exp (c~~F~/RT) - exp ( - ~ e F~I/RT)] • [I + r exp (a a F~/RT) + 0' r exp ( - a c F~/RT)]-I

(II)

E q u a t i o n (II) predicts, in particular, surface-diffusion controlled limiting currents and experimental evidence is adduced6,S to support this claim. (e) T h e theory of galvanostatic transients t h a t results in eqns. ( I ) - ( I I ) is worked out2, s assuming t h a t the instantaneous adion concentration is g o v e r n e d b y the equation

3c aolat = i / z F - [(D a/l 2)(c ~d/c ad d) - I]

(12)

with Cad=Cad o at t=o. 3. We now show t h a t m a n y of the equations cited need modifications to avoid erroneous conclusions.

A. Steady-state potential Whereas the complications inherent in deriving an exact p o t e n t i a l - t i m e profile under galvanostatic conditions are o b v i o u s n , 12, the problem of obtaining the i - ~ (t-->oo) relationship is e x t r e m e l y simple. As long as the mass transfer problems in the solution phase are ignored and t >>~g, one need only solve Daa~c~d/ax~+(io/zF)[exp(c~cF~loo/RT)-(ca~(x)/cad°) exp(-oc~F~oo/RT)] = o

(13)

The b o u n d a r y conditions are

(acad/ax)~=z = o

(14)

and Cad=Cad °

at

x=o;

(15)

and

i/io = [exp (de F~ oolRT) - (c avlc ad d) exp ( - ~ ~F~ oolRT)] where eav =

(i#)

f

~ 0 Ca~ (x) d x

j. Electroanal. Chem., I6 (1968) 485-492

(I6)

487

SURFACE DIFFUSION AND GALVANOSTATIC TRANSIENTS

and ~ = n ( t ~> r~).

(i7)

In eqns. (i3)-(i7), no restrictions are placed on the magnitude of i and the model is the same as assumed by BOCKRIS et al. The solution of (I3) with (14)-(17) gives i -- i0 [exp (ae F ~ oo/RT) - exp ( -oc a F ~ oo/RT)] •tanh El2 exp ( - ~ ~ F~1 oo/2RT)]

=

• [I/f21 -exp (~ a F ~ o~/2RT) .

(18)

[/io/zFc~doD~ • l

(I9)

(In eqn. (18) and in the following equations, we use ~/and ~ ~oin an interchangeable sense.) Equation (18) shows that the i-~ relationships under potentiostatie and galvanostatic (steady) conditions are identical. Let i~/io ~ I. Then (~c +O~a)F~/RT = (i/io)~ coth f~ (20) Equation (3) differs radically from eqn. (20) if vo =Dacacl°/l 2, as assumed by MEHL AND BOCKRIS (eqn. (14)). That it is wrong to identify Vo with (Dacaa°/l 2) is shown in Appendix I. B. Galvanostatic rise-times Although the i vs. ~1 behaviour (eqn. (18)) is derived easily---without any restriction on the magnitude of i (or ~)--the same cannot be said of galvanostatic rise-times. Two limiting cases 14, however, offer themselves for easy analysis. (i) ~ = o

(21)

Writing (io/zF) (exp (ae F~I/RT) - I) = ¢ (t)

(22)

~b(p) = p f o e x p ( - p t ) ¢(t) dt

(23)

and

it can be proved that q)(p) = (i/zV) EP/(P + k2) + k2/(p + k2) "tanh ( ~ / ~ . / ) .

(!/(W(p + k2)/o ~. l) ) ]-1 (24)

(ii) z F ~ / R T < I

Let ~(P) = P f o exp (-pt)-~/(t) dt

(25)

Then (ae + o~,) F ~1(p)IRT = (i/io) [p/(p + ks) + k2/(p + ks). tanh (V(P + k2)ID a" l)/(V(p + k2)/D ~./)] -1

(26)

In eqns. (24) and (25), ks = io/ (zF c a~ 0)

(27) j. Electroanal. Chem., 16 (I968) 485-492

488

s.K. RANGARAJAN

Whereas eqn. (24) is derived without assuming that ~ is small, (25) is true for small zF~/RT. The nature of the poles of functions in the right-hand side of eqns. (24) or (25) must be studied to evaluate the rise-time. Such a study suggests that, when surface diffusion is indeed important,

(28)

7:g ~_ (2Xo/k2) coth Xo, where Xo is a non-zero real root of tanh X + X ( X ~ / ~ -

1) = o

(29)

(Note: There is no ambiguity in eqn. (28) because eqn. (29) has only three real roots"

-Xo, o and Xo). ]Xol < ~ and for large f2,

~g~ 2(~- ~)/k~ Illustration

f~ X0

(30) io 2,44

50 6.5

~oo 9.5

A comparison of eqn. (30) and eqns. (2) or (5) shows that Tg# 2CadO/Vo=212/D~. Under conditions favourable to surface diffusion, z g - 2l ~zFc ~d°/ioD ~ and Tg/Tp ~-~ (~-~__1)

(31)

I t is interesting that such a change in the order of rise-times in the potentiostatic and galvanostatic cases is possible. Both zg and vp depend upon io and C~d° (cf. ref. 9)' We have shown above that the steady-state overpotential and the galvanostatic rise-times as given b y eqns. (3)-(5) are incorrect. An alternative expression for z~/vp (cf. eqns. (7) and (31)) was also presented, but, under limiting conditions (~ >>I), it is possible to retain eqns. (3) and (5) b y abandoning (4) and writing instead,

vo/c~a° "" k2/f~

(32)

The significance of eqn. (32) is explained in Appendix I.

C. Illustration (i) Surface-diffusion process in the electrodeposition of silver from molten halidesl0: T = 4 5 o°, i0 = I9O A c m -2, C~d0= 9 "10 -10 mol cm -2, F v o " 3 . 3 A cm -2, k2~2 .to 6 sec -1. The correct values for f2 and Da/l 2 are f~ =54, D~/12''6.o x lO 2 sec -1 from eqns. (20) and (3o), whereas eqn. (4) would give f~"~7.4 and D~/12,,,3.66 x lO 4. (ii) Deposition of gallium~ T = + o . 5 °, i o = I . 4 X l o - 4 A cm -2, Cad0=5"IO -11 m cm -2, zFVo=5.5 XlO -5 A cm -2, k2~IO sec -1, f~"~3, D~/12"I sec-1- Equation (4) gives f~< !.6 and D~/12,~4 sec -1.

D. i-~ Curves under "coverage-effects" If we assume, with BOCKRIS, the expression for i as given b y eqn. (8), it is easy to set up and solve steady-state diffusion equations appropriate to the model used. The result is i/io = ~exp (ae F~/RT) - exp ( - ~a F~/RT)~ • tanh ~Q x

f exp~-- aa F ~ / R T + 0 ' e x p (o~oF~/RT)J " [[2 Vexp-( - aa F~ /RT) + 0 ' e x p (ao F~ [RT)l-1

(33) J. Electroanal. Chem., 16 (I968) 485-492

SURFACE DIFFUSION AND GALYANOSTATICTRANSIENTS

489

(~, 0' are given b y eqns. (I9) and (IO), respectively). T h a t eqn. (33) is essentially different from (II) is obvious. The error c o m m i t t e d b y using (II) (in evaluating various parameters) can be serious indeed. E q u a t i o n (33) reduces to (I8) as 0'-+o, as it should b e - - w h e r e a s (ii) reduces to an erroneous i-~; relationship. Again, eqn. (II) predicts limiting currents4, 6,8 while (33) does not. On the other hand, (33) implies that, if 0' < ~, a change in cathodic Tafel slope from its true value of 2.303 RT/oce F to one t h a t is twice this, is possible. On the (cathodic) ~-log i plot, eqn. (33) m a r k s three t y p e s of behaviour: (i) i >>(ac +aa)F~/RT linear ~ - i as given b y eqn. (20) ; d~/di ~_~/io (if) 0' exp (~o + ~ a) F~/RT ~ I, exp ( ~ F~/2RT) >>Y2; linear 27-1og i slope 2.303 RT/ac F intercept in i0. (iii) 0' exp (~c +o~)F~/RT >>I, ~V0' exp (oceF~/RT) > 5 linear ~-log i slope intercept

2 (2.3o3 RT/~xoF) in (io/~0')

A change-over in the cathodic Tafel slopes as indicated above b y (if) and (iii) is a direct consequence of " t h e coverage-effect" introduced in the model (eqn. (18) predicts only a transfer* to (if) from (i) and no transition to (iii) is suggested). E. To sum up, the theory of the surface-diffusion model predicts t h a t : (a) The steady-state overpotential, B, is related to i through (~o + ~ ) F ~ / R T = i ~ coth f~/io if [zF~I/RT ]~ I. (b) The galvanostatic rise-time, zg, for small current densities is given approxim a t e l y by: zg = 2(tanh Xo/Xo)(I/k2) where Xo is the real root of t a n h X + X ( X ~ / f ~ - i) = o (c) The "velocity of surface-diffusion", vo of BOCKRIS, wrongly identified with

(Dac~a°/l~), can at best be formally e q u a t e d with k2°/f~ to obtain b e t t e r results under limiting conditions (~: large). (d) The i - ~ curves deduced from eqn. (33) u n d e r "coverage-effects" are erroneous and a corrected equation for the same model gets rid of the prediction of "limitingcurrents". On the other hand, a doubling of the slopes is predicted. (e) The theoretical analysis (eqns. (18) and (33)) shows t h a t if surface-diffusion ever controls at near-equilibrium conditions, it subsequently continues to d o m i n a t e in the anodic side, until, of course, the theoretical model itself fails to represent the phenomenon. And, for large anodic overpotentials (or currents), even though a Tafel * If Y~is large enough, an anomalous intermediate behaviour is possible, viz., a Tafel behaviour with the slope (RT/(cXe+ ~a/2)F) over the region: i ,~ exp (aaF~/RT)< (~/5) 2. Such an effect, if observed, can be important evidence in favour of the surface-diffusion model.

j. Electroanal. Chem., 16 (1968) 485-492

490

s.K. RANGA,RAJAN

behaviour m a y be observed, the slope is equal to 2(2.3o3 R T / ~ F) (twice its " t r u e " slope) and "io" calculated in the usual way from this Tafel-intercept =i0 app =io/~. No experimental confirmation, however, is reported* t. Lastly, a comment on the model. In the foregoing discussions, a criticism of the theory of BOCKRIS at al. (assuming the same model as theirs) was presented. We point out that "a slow rise-time" and "a large overpotential" are not unique to a "slow surface-diffusion" model. A model that assumes a fast diffusion on the surface followed b y a slow incorporation at the lattice-building sites exhibits similar characteristics 13-15. If the condition Da(~C~o/Sx) =ks(cad--Cad°) at the edge (x =o) is used instead of (I5), it is easy to show t h a t a rise-time, zg~-I/ks, results for the latter model. More exactly, zg~_k~/E~+k2l/k3] and vp---k2 (see Appendix I) for large ~. This amounts to identifying v0 with (k, cadO/l). Although a similar discussion of i - ~ curves for this model could be useful, we omit this, for the sake of brevity. Results concerning the ~-t profiles for the generalised galvanostatic case will be communicated later. ACKNOWLEDGEMENT

The author thanks Professor K. S. G. Doss, Director, Central Electrochemical Research Institute, Karaikudi, India, for very useful discussions. SUMMARY

The theoretical results concerning the galvanostatic transients in the problem of electrodeposition, with surface-diffusion as a slow-step, are derived. The explanations offered and the theory presented b y BOCKRIS at al. are critically discussed, and modifications are proposed. APPENDIX I

The error involved in using eqn. (12) instead of the more accurate eqn. (13) is reflected through the approximation implicit** in

Daa2cad/~x 2 ~--Da/l~ (cad--Cad °)

(A.I)

(A.I) helps to get rid of the dependence of Cad on x and therefore Cad-----C~dav where cad av

= I/l f caa'dx. o

(A.2)

The solution of eqn. (12) is a function of t only and with * I t h a s b e e n n o t e d d u r i n g g a l l i u m d e p o s i t i o n a n d dissolution, t h a t d ~ / d l o g i a ~ 4 5 - 5 5 m V a n d t h u s , if surface-diffusion control as d i s c u s s e d above, exists, d~/d(logi~)~___(2:3o3)2RT/a~,F, m a k i n g a v a l u e of 2. 5 for a a plausible. Also d~/di = RT/F(o~e + as) (I/i0 + I/ZFvo) (eqn. (23) of ref. 6), s h o w s a r e a s o n a b l e a g r e e m e n t w i t h e x p e r i m e n t s if ac + a s = 3. Again, t h e r e l a t i o n s h i p , d In fold In (H2GaO3-) ~ I --ill3, is t r u e for m e c h a n i s m (A) (ref. 6) as well as (D). T h e a b o v e r e a s o n i n g s h o w s t h a t m e c h a n i s m (A) need n o t be r u l e d o u t as i m p r o b a b l e . t Also see ref. (7) w h e r e i0,a ( T a f e l ) ~ i 0 , c (Tafel). ** (A.I) does n o t i m p l y a linear c o n c e n t r a t i o n g r a d i e n t (cf. refs. 2-8).

j . Electroanal. Chem., 16 (1968) 485-492

SURFACE DIFFUSION AND G&LVANOSTATIC TRANSIENTS

i = i0 [exp (ac FB/RT) - (c aaaV/c ad°) exp ( - - a aF*I/RT)]

491

(A.3)

or eqn. (8), the derivation of the ~-t profile as given b y (11)-(3) is immediate. B u t even for ]zFB/RT[~ I, (A.I) is v e r y a p p r o x i m a t e , if not arbitrary. A more realistic, though heuristic, procedure is given below: I n t e g r a t e (13) from x = o to x = l with respect to x. Then using (14) and (A.2)-(A.3),

(A.4)

ac aaaVlat = --(O Jl) (ac,olax)x-o + ilzF . At this stage, we resort to an a p p r o x i m a t i o n

(ac aolax) x=0

= (c a ~ a v - - C a~ (X =

0))I8

(A.5)

= (CadaV--Cao0)/6 if Cad (x=O)=Cad °

(A.6)

(A.5) assumes a linear concentration gradient in the interval (o, 6), a n d for x > 6, Cad =Cad av (t). Such "cut-off" models can give interesting results under limiting conditions. Thus, an " e s t i m a t e " for 8, viz., V ~ , where z is the "life-time" for an adatom, suggests itself, z,,~I/k2 and thus 6~#D/k2 ( 4 l). An essential condition for this linearisation to hold is (6/l)~ I (note t h a t

& ~o/&)x=t = o # &ad/x)~o) i.e., ~ >>I (A.4) and (A.5) then give

~CaaaV/at = -- (Da/6l) (Cada~- Cad°) + i/zF

(A.7)

Hence, under limiting conditions, one could possibly identify VO/Cado with Da/6l =k2/f~. This explains how an a g r e e m e n t can be found at all, under the condition >>i, if we take Vo/Cad° =k2/f~ instead of equating Vo with DaCad°/l 2. This heuristic reasoning m a y be e x t e n d e d to the case when the lattice incorporation is slow. We abandon, then, the a s s u m p t i o n of an equilibrium at the growth-edges, i.e., caa(o)#Cad °. Instead, we have

Daacaa/Ox = ka(Cad--Cad °)

at x = o .

(A.8)

(A.5) is still true b u t (A.6) is not.

(D a ac aa/3x) ~-o = D a (c a a - cad (X = 0))/6 = k~(c ~

(A. 9)

( x = o ) - c ao °)

Eliminating Cad (X =O), (A.4) can be rewritten as

acaaav/t

=

-

{D a/[/6(i +Da/ka 8)] }. (Cada v - Caa o) + i / z F

(A.IO)

(A.Io) reduces to (A.7) as (ka6/D~)->oo. Thus

VO/Cad° "~ k2/Ef~ + k~l/k3? (~: large) -+ka/l, if k21/ka >>f~ >>I. REFERENCES I 2 3 4

H. GERISCHER, Z. Elektrochem., 62 (1958) 256. W. MEHL AND J'. O'M. BOCKRIS, Can. J. Chem., 37 (1958) 19o; J. Chem. Phys., 27 (1957) 817. A. R. DEsPic AND J. O'M. BOCKRIS, J. Chem. Phys., 32 (196o) 389. J. O'M. BOCKRIS AI,rD M. ENYO, Trans. Faraday Soc., 58 (1962) 1187.

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S. K. RANGARAJAN

492 5 6 7 8 9 io ii 12 13 14 15

H. KITA, M. ENYO AND J. 0'3/[. ]3OCKRIS, Can. J. Chem., 39 (1961) I67o. J. O'M. BOCKRIS AND M. ENYO, dr. Electroehem. Sot., lO9 (1962) 48. J. O'M. BOCKRIS AND H. KITA, J: Electrochem. Soe., lO9 (1962) 928. J. O'M. BOCKRIS AND A. DANJANOVIC, Modern Aspects of Electrochemistry, No. 3, edited b y J. O'M. BOCKRIS AND B. E. CONWAY, B u t t e r w o r t h s , L o n d o n , ch. IV. A. DAMJANOVIC AND J. O'M. BOCKRIS, J. EIectrochem. Soc., i i o (1963) lO35. T. B. REDDY, dr. Electrochem. Soc., 113 (1966) 117. M. FLEISCHMANN AND H. •. THIRSK, Advan. Electrochem. Eleetrochem. Eng., edited b y P. DELAHAY, Interscience, N e w York, 1963, ch. I I I . S. K. ~RANGARAJAN, Can. J. Chem., 43 (1965) lO52. M. FLEISCHMANN AND J. A. HARRISON, J. Electroanal. Chem., in press. M. FLEISCHMANN, S. Z . RANGARAJAN AND H. ]~.. TrllRSK, Trans. Faraday Soc., 63 (1967) 124o. J. A. HARRISON, S. K. RANGARAJAN AND H. ~R. THIRSK, dr. Eleetrochem. Soc., 113 (1966) 112o.

j . Electroanal. Chem., 16 (1968) 485-492