Corrosion Science, 1970, Vol. 10, pp. 679 to 686. Pergamon Press. Printed in Great Britain

REVIEW ARTICLE

RECENT DEVELOPMENTS IN MODELS FOR THE ELECTROCRYSTALLISATION OF ANODIC FILMS*

R. D. ARMSTRONG, J. A. HARRISON and H. R. THIRSK Electrochemistry Research Laboratories, Department of Physical Chemistry, University of Newcastle upon Tyne, Newcastle upon "lyne, NEI 7RU, U.K.

THE PROBLEM of electrochemical phase formation (of both anodic films and metals) has been much discussed. In metal deposition early work emphasized the importance of two-dimensional nucleation? Subsequent to the work of Frank and others 2 which showed that screw dislocations can contribute self-perpetuating growth sites it has been realized that two-dimensional nucleation and growth will only be observable on perfect surfaces as shown by the recent work of the Bulgarian school. 3

The emphasis in anodic film formation has been somewhat different, though the problem is in many ways the same. The work of Mueller 4 suggested that the initial stages of anodic film formation involve a dissolution-precipitation process and interest in this mechanism has recently been reviewed. 5 The alternative view, that anodic electrocrystallization involves a solid state process with nucleation and growth as the primary mechanism, however, has much experimental support, particularly on perfect surfaces. 6 In this review we shall briefly describe the theoretical models appropriate for the description of these processes.

1. Processes with the slow step nucleation and growth The major theoretical problem is the calculation of the i-t and n-t relations for

potentiostatic and galvanostatic impulses. (a) Centres which interact solely with themselves. We shall consider centres which

grow at a velocity v (cms -1) which is time independent at constant potential.* These may be two-dimensional or three-dimensional centres. For the latter the possibility exists that the rate of growth orthogonal to the surface (v~) is different from that parallel to the surface 0'1). Nucleation can frequently be assumed to occur at a constant rate A (nuclei cm-2s-1) or at a fixed number of sites, Ns (nuclei cm-2) at t = 0. When centres interact only with themselves the use of the AvramF theorem is appropriate. For three-dimensional centres t nucleated instantaneously and where there is no necessity for mass transport through the centres, under potentiostatic conditions s

*Manuscript received 3 December 1969. "l'For centres comparable in size with the critical nucleus v is expected to be a function of the size

of the centre. 679

680 R . D . ARMSTRONG, J. A. HARRISON and H. R. THIRSK

i nFv2 = - p [1 - - exp ( - - nNov2tS)]. (I) M

At short t imes (1) gives 9

whereas at long times

nF i = - - pv2v~Nor~t 2 (2) M

nFvo i = " P. (3)

M

For three-dimensional centres with a nucleation rate constant in t ime the analogous relation 8 is

nFv2 [1 e x p ( ~v3At!)] i - - ~/- p - - (4)

giving at short t imes '~

nFpP2 i -- v~Aru 3. (5) 3M

When the three-dimensional centres require mass t ranspor t through the newly formed layer so that i(t -+ oo) ~ 0 the exact solution of the problem is difficult. An approximate solution s has been given by assuming that the rate of or thogonal advance of a centre is proport ional to the uncovered surface, i.e.

v2 = v2oexp (--_nv~;ts). (6)

Then for progressive nucleation

i = n F V 2 ° p [ 1 - - e x p ( ~v3AtS)]exp( n v ~ t ~ M - . (7)

This predicts a max imum current

I m - -

at a t ime

/'/FP2o 4 M

- - - p ( 8 )

= { 31n2~ 1/3 t,, \ -v2A ! . (9)

For two-dimensional centres with instantaneous nucleation x°

*The centres are assumed to be right circular cones. The relationships for other shapes of centres are similar?

Recent developments in models for the electrocrystallization of anodic films 681

i = q.27tv2Not exp ( - - 7zv2t2No) (lO)

whilst for progressive nucleation ~°

i = q . nv2At~'exp ( n v ~ - (11)

where q is the total charge passed in the formation of the layer. These relations can be applied either to macroscopic cylinders growing on the

electrode surface or to the formation of a monomolecular layer. For the latter case we shall write q as qMON-

Galvanostatic conditions are much more difficult to deal with. However, a solution has been given s for the growth of a two-dimensional layer with instantaneous nucleation

v (t) = 2(~N0) 1/2 (q - it) In (12) \ q - - i t / d

which predicts that v will have a minimum value at a time given by

/min : q ( e l / ~ - - 1) (13) iel12

To derive the rl-t relationship the functional dependence of v on 11 must be known. Since a monotonic relationship is expected 11 should also have a minimum value at this time.

The problem of the growth of successive monomolecular layers potentiostatically on an electrode was solved earlier by direct simulation. 1° A more exact solution has recently been obtained for the metal deposition case and the i - t curve which was calculated is shown in Fig. 1. For this case e

i (t --> oo) z qMos (nv2A) l/s22/s exp [-2/s]. (14)

(b) Centres which interact with the boundaries o f the electrode. When only one centre is formed on an electrode the i - t , t I - t transient is determined by the interaction of this centre with the boundary of the electrode and therefore depends on the position of the nucleus with respect to the boundaries. For a circular nucleus growing on a circular electrode of radius R, nucleated at a distance a, from the centre of the electrode, a under potentiostatic conditions

i = qMON 2nvZt (15)

forO ~<vt < ~ R - - a

682 R.D. ARMSTRONG, J. A. HAaalSON and H. R. Tnms~:

Flo. 1.

\

, / ,,',. ,;', ; ' , [ # I $ % s S •

-I / \ I ",<'" I / 2~,i 3?,,." 4 . - " "- . s , , t ~ .~" I , ~ " " ~'.. I,- ~ " " ~ ' ~ -

t

Potentiostaiie transient for metal deposition by two-dimensional nucleation.

[ i = qMON 2 r~ -- Cos -x 2aR- v2t2 (16)

f o r R - - a ~ vt ~ R + a . Corresponding relations have been derived for a square nucleus on a circular

electrode, s Another problem which may arise in the case o f a real electrode is that o f two-

dimensional nucleation on a macroscopic electrode with a large number of small bounded areas. The solution o f this problem is rather more difficult, but we can approximate it by assuming that after growing to a certain size (radius r ' ) the centre ceases to grow. The i - t transient for progressive nucleation is xl

i, s t, = qMoy ~v"-t°'A exp ( - - n~ 2Ata)-

it ~ t' = qMoy ~v2t'2A exp [-- nv2A(]t '2 - - t'2t)] (17)

where t ' = r'/i,. (c) Centres which interact with bulk diffusion. The above type o f calculation can

be extended to cope with diffusion. Only overlap o f the nuclei themselves ignoring overlap of the diffusion layer can be simply treated. For the nucleation and growth of a single two-dimensional layer two situations have been calculated:

(a) The case when a large number o f nuclei are formed initially so that the individual diffusion zones have overlapped and diffusion is linear after t ---- 0. The potentiostatic i - t transient is 12

i ---- n F A ' C X~ D f nFA'aC2D (exp - -

_ ~ n F ]

Recent developments in models for the electrocrystallization of anodic films 683

RT 7 Jm'~J

"er fc[{CD'A' ( exp-nF1]]\ttRT/jm' ]" (18)

(b) The situation when progressive nucleation or few nuclei form at t = 0. In this case analogues of equations (10) and (11) can be calculated. Using the Avrami theorem and the rate of growth of a single nucleus by cylindrical diffusion 18 gives

i = q~OZNoD exp (--rcOZDNot) (19)

i = qnO2DAt exp (-- nOZDAt~) (20)

for instantaneous and progressive nucleation respectively 0 is a constant. 13 Equation (19) is a failing transient different from equation (18) or the well-known equation

i = nFA'C ~/_o (21) ~t

expected for completely uniform linear diffusion. Equation (20) has a similar form to equation (10).

2. Processes involving surface diffusion The complete mathematical solution for the nucleation and growth under potentio-

static conditions of. a single layer including overlap under control by diffusion in solution---electrochemical reaction--surface diffusion would be very difficult. An attempt has been made to predict the i-t transients by a graphical simulation. 14 Experiment and theory have been compared for nickel deposition on to mercury.

A mathematical model can be devised for a simpler situation. The familiar adatom model is shown in Fig. 2. The step line is the edge of a two-dimensional nucleus or part of a screw dislocation. The mathematical analysis of this model including bulk diffusion for the potentiostatic case has been carried out by Rangarajan x5 for two cases:

®

KI~ I K2 Solution ( ~ Da , ~ [ _ . .

" Me ta I

FIG. 2. The adatom model including bulk diffusion.

684 R . D . ARMSTRONG, J. A. HARRISON a n d H . R . THIRSK

(a) Edge incorporation fast; (b) edge incorporation slow. The solution is available only as a Laplace transform. The more general case (b) gives

k 2 ~ p D -] n D r , (p) = nF (k I C - - k2 C,°,) V'L---/ l + nl) l

Zoo L Xoo (S] + (22)

where the symbols are defined in the appendix and

[ (p) = f exp (-- p t ) . i(t) dr. 0

(23)

A discussion of the steady state solution assuming that bulk diffusion le, 17 has been subtracted experimentally and 1 al/rl will not be given here. A galvanostatic theory not including bulk diffusion has been suggested but this has difficulties associated with it.is

Equation (22) has been calculated numerically for the a.c. 1'~ and potentiostatic pulse ~° methods. The conclusions in the a.c. impedance case are:

(i) a detailed fit of experimental data to theoretical calibration curves must be made before slow diffusion or slow lattice incorporation can be distinguished as the rate determining step.

(ii) a Randles plot (Rs vs. to-J) may still be observed even if surface diffusion or lattice incorporation plays a role.

In the potentiostatic case equation (22) has been inverted numerically 1~ (for last lattice incorporation), as shown in Fig. 3. The two limits represent D',/12k2 large or small. The limits correspond qualitatively to capture of adatoms by the edge or

q~

E 0,L I

I I I I / I 0 2 4 6 8 10

FIG. 3. B e h a v i o u r of the cu r r en t w h e n a p o t e n t i o s t a t i c pu l se s t a r t i n g f r o m 1"1 = 0 is a p p l i e d to the a d a t o m m o d e l o f F ig . 2.

© D J P = 10e; + 17./12 = 10s; A D . / I ~ = l 0 s . • Do~I* = 10s; A D./I* = 10 -a .

q = 5"13 mV, k, : 3 '16 × 10-*" c m s - ' , ks = 2.59 × I0 a s -1, D = 10 -~ cm2s - l .

Recent developments in models for the electrocrystaUization of anodic films 685

deposition and dissolution without capture. Figure 3 gives some guide to the values D~, l, k2, for surface diffusion to be significant. It is our feeling, as only a narrow range of the parameters put the current in between the limits, that surface diffusion will rarely be observable in practice.

3. Processes involving dissolution-precipitation

In this model for the formation of an anodic film 4, 5.z1.~2 it is assumed that when the concentration of the metal cation or complexed cation exceeds a certain value C* at the electrode surface, nuclei of the anodic film are formed in the solution. This model seems to imply that monolayer anodic films will not be formed (since all nuclei formed in the bulk of the solution will be three-dimensional).

Under galvanostatic conditions with

M ~ M "+ + ne

and C (x, o) = 0

the concentration C* is achieved at the electrode surface at a time given by zz,'a4

nZ F2 D n C .2

4i ~

This assumes that the anion concentration is sufficiently high that its concentration is not disturbed. Likewise for the concentration of M in a metal amalgam.

For a reversible metal-dissolution reaction the dependence on potential is given by

E : E ° + - - R T I n 2 i t ~ n F n F D ~ t

while for a completely irreversible reaction E is invariant with time. Where the mechanism is reversible metal dissolution M M n + + ne followed by a chemical reaction

Mn+ ~ k f M ,, K _ k f k b kb

the appropriate relationship is ~

iv ~ = C*nFDI(1 q-K)n ~" _ n lK i erf [(k f +kb)~v ~]

2 2 ( k f + k b ) ~

and not that given elsewhere. *5,zl,2z The dissolution-precipitation model also predicts that under conditions of a

constant diffusion layer thickness 5, the current at which anodic film formation occurs will be given by

c* i = n F D - -

so that e.g. in a rotating disc experiment this current will be a sensitive function of

*It is difficult to understand why experimental results should fit an erroneous relationship.

686 R . D . ARMSTRONG, J. A. HARRISON and H. R. THIRSK

r o t a t i o n speed , whereas fo r a sol id s ta te r e ac t i on a d i f fe ren t d e p e n d e n c e on r o t a t i o n

speed w o u l d be an t i c ipa t ed . In pa r t i cu l a r fo r a to ta l ly i r revers ib le me ta l d i s so lu t i on

r eac t ion no r o t a t i o n speed d e p e n d e n c e is an t i c ipa t ed .

N O M E N C L A T U R E A = rate of nucleation (cm-2s-0

A' = area of electrode C = bulk concentration

C ° = adatom concentration at q = 0 D = diffusion coefficient E = electrode potential k = rate constant (s-0

k~ = forward electrochemical rate k~ = back electrochemical rate M = molecular weight m" = number of moles in a monolayer No = number of nuclei

n = number of electrons q = charge in the formation of a two-dimensional layer

q~tos = charge in the formation of a two-dimensional monomolecular layer t = time (s) v = rate of advance of a centre (cms -t) p = density (gm/cm -a) r = transition time

rra = l~/m27tD ~ r,~ = 12/m2nD ~ Zoo = (k~ + x /pD) p + kz x /pD

X.~o = (k~ + ~/(p + 1/rm) D ) ( p + l/rm) 4- kz vZ(P -t- l/rm) D Sx = Z e,,, (k~ + ~/(p + I/vm) D) p/Xmo era = 1, °hi ~ 0 e,, = 2, m = 1 ,2 . . .

R E F E R E N C E S

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Status SolM 13, 577 (1966); Electrochhn. Acta 11, 1697 (1966). 4. W. J. MUELLER, Z. Electrochem. 33, 401 (1927). 5. J. O'M. BOCKRIS, M. A. V. DEVANATHAN and A. K. N. REDOY, Proc. R. Soc. A297, 327 (1964). 6. R. D. ARMSTRONG and J. A. HARRISON, J. electrochem. Soc. 116, 328 (1969). 7. M. AVRAMI, J. chem. Phys. 9, 177 (1941). 8. R. D. ARMSTRONG, M. FLEmCHMANN and H. R. THmSK, J. electroanalyt. Chem. 11,208 (1966). 9. M. FLEISCHMANN and H. R. TmRSK, Trans. Faraday Soc. 51, 71 (1955).

10. A. BEWlCK, M. FLEISCHMANN and H. R. TmRSK, Trans. Faraday Soc. 58, 2200 (1962). 11. R. D. ARMSTRONG and J. W. OLDFIELD, unpublished work. 12. D. J. ASTLEY, J. A. HARRISON and H. R. TmRSK, J. electroanalyt. Chem. 19, 325 (1968). 13. F. C. FRANK, Proc. R. Soc. A201, 586 (1950). 14. M. FLEISC~ANN, J. A. HAkRISON and H. R. TmRSK, Trans. Faraday Soc. 61, 2742 (1965). 15. M. FLEISC~ANN, S. K. RANGARAJAN and H. R. TmRSK, Trans. Faraday Soc. 63, 1240 (1967). 16. M. FLEISCHMANN and J. A. HARmSON, J. electroanalyt. Chem. 12, 183 (1966). 17. M. FLEISCHMANN and J. A. HARRISON, Electrochim. Acta 11,749 (1966). 18. S. K. RANGARAJAN, J. electroanalyt. Chem. 16, 485 (1968). 19. S. K. RANGARAJAN, J. electroanalyt. Chem. 17, 61 (1968). 20. J. A. HARRISON, J. electroanalyt. Chem. 18, 337 (1968). 21. A. K. N. REDOY, M. A. V. DEVANATHAN and J. O'M. BOCKmS, d. electroanalyt. Chem. 6, 61

(1963). 22. M. A. V. DEVANATHAN and S. LAKSHMANAN, Electrochim. Acta 13, 667 (1968). 23. H. J. S. SAND, Phil. Mag. 1, 45 (1901). 24. Z. KARAOGLANOFF, Z. Elektrochem. 12, 5 (1906). 25. B. Br-HR and J. TARASZEWSKA, J. electroanalyt. Chem. 19, 373 (1968).