The nucleation and growth of two-dimensional anodic films under galvanostatic conditions

J. Electroanal. Chem., 124 (1981) 247--262 Elsevier Sequoia S.A., Lausanne -- Printed in The Netherlands

247

THE NUC LEA T I O N AND GROWTH OF TWO-DIMENSIONAL ANODIC FILMS UNDE R G A L V A N O S T A T I C CONDITIONS

G.J. HILLS, L.M. PETER, B.R. SCHARIFKER * and M.I. DA SILVA PEREIRA Department of Chemistry, The University, Southampton S09 5NH (England) (Received 28th July 1980; in revised form 1st December 1980) ABSTRACT The electrochemical nucleation and growth of two-dimensional anodic films on liquid electrodes has been studied by the galvanostatic method. Overpotential--time relationships have been derived from a model of the electrocrystallisation process which assumes that the rate-determining step involves incorporation of ad-molecules into the lattice of the expanding growth centres. Comparison of these theoretical curves with experimental transients has shown that non-steady state effects are important in nucleation under galvanostatic conditions. The experimental examples considered are the formation of Cd(OH)2 on Cd(Hg), Hg2Cl2 on Hg and HgS on Hg.

INTRODUCTION Insoluble passivating films are f o r m e d on a variety of metals and semiconductors u n d er electrochemical conditions, and t h e y are i m p o r t a n t in m a n y contexts including, for example, corrosion and s e m i c o n d u c t o r technology. The films are generally f o r m e d by oxidation of the parent material, and in m a n y cases th ey interact so strongly with the substrate t hat t hey spread o u t as monomolecular layers. The crystallographic and t h e r m o d y n a m i c properties of such thin deposits may differ considerably from those of the corresponding bulk phase, although direct characterisation o f these properties is n o t easy. Reliable in f o r matio n a b o u t these films can only be obtained from in situ measurements, and here electrochemical m e t h o d s are powerful since t h e y are able to det ect small changes in the electrical variables of the system. Thus the m ost obvious evidence o f the t h e r m o d y n a m i c differences between the thin layer and bulk phases is th at their equilibrium electrode potentials can differ by several millivolts, although the role played by two-dimensional phases in the establishment o f equilibrium at electrodes of the second kind is n o t well understood. Crystallographic changes in orientation and packing, on the o t h e r hand, can o f t e n be deduced f r om the charge required to com pl et e the m onol ayer. Many o f the properties o f anodic films are reflected in the way which t h e y grow u n d e r controlled electrochemical conditions. Some of the earliest studies o f the electrocrystallisation of calomel films on m e r c u r y [1--3] made use of the galvanostatic m e t h o d , but at the time no satisfactory m odel of the nucle* Present address: Universidad Simon Bolivar, Departamento de Quimica, Caracas, Venezuela. 0022-0728/81/0000--0000/$02.50, © 1981, Elsevier Sequoia S.A.

248

ation and growth processes existed. The advent of the fast potentiostat led to a re-examination of the system under constant potential conditions, and at the same time the first adequate theory was developed [4,5]. The potentiostatic m e t h o d corresponds in principle to a measurement at constant supersaturation, but unfortunately the experimental realisation of this condition is made difficult by the errors which arise from the potential drop in the electrolyte solution. These errors are particularly serious when the rate of an electrode process depends strongly on potential, and as a result potentiostatic transients for the nucleation and growth of t w o dimensional films are often distorted, making a correct analysis difficult or even impossible [6]. The ohmic distortion is important at short times if nucleation is preceded by transient adsorption [7--9], as well as at high overpotentials where the current densities are large. Analysis of such distorted transients is difficult since the two sources of error appear in a non-linear combination [6]. The galvanostatic m e t h o d offers the advantage that the macroscopic ohmic overpotential is constant throughout the experiment, so that correction of the ohmic drop, iRa, is possible, in principle at least, to any degree of precision (provided that the ohmic resistance due to the film itself is negligible). The potentiostatic m e t h o d has been preferred in most previous studies because the analysis of the i--t transients does n o t depend on any assumptions a b o u t the potential-dependence of the rate constants for nucleation and growth; the results are interpreted with the help of an essentially geometrical description of the growth process [4]. Here we show h o w the geometrical model of the nucleation and growth of two dimensional layers can be transposed using the boundary conditions appropriate to the galvanostatic experiment. The resulting expressions have been used to analyse the nucleation and growth reactions associated with the deposition of several anodic films on liquid substrates. At the same time the experimental work has established the limitations of the galvanostatic m e t h o d so that a realistic comparison of the merits of the potentiostatic and galvanostatic methods for the study of electrocrystallisation is now possible * THE GALVANOSTATIC TRANSIENT

The reversible potential Erev,mono of a monolayer may well differ from the normal reversible potential Erev of the electrode concerned, b u t as a general rule, the nucleation of a new phase requires an overpotential 7;

=E -- Erev.mono

(1)

The rate at which the film covers the surface depends on the rate of appearance of nuclei of the new phase and on the rate of their lateral growth. Both of these processes depend on the overpotential [10,11]. During the initial stages of film formation, the simplest geometry of the new phase which we can consider is a circular patch, originating from a nucleus at * Since this report was submitted, Barradas and Porter [ 37 ] have discussed the derivation of the potential--time relationships for 2-]:} electrocrystallisation under galvanostatic conditions. Their final equations are similar to those given here.

249 its centre and growing uniformly in all directions along the plane of the surface. If the rate at which the patch grows is controlled by the incorporation of material at its periphery, the current due to the growth of an isolated patch at constant potential is given by [12]

I = 27rzFMhk2t/p

(2)

where M/p is the molar volume of the deposit, h is its height, k is the rate of lattice incorporation, z F is the molar charge and t the age of the patch. The growth and subsequent interaction of a number of centres on an electrode surface has been treated [4] for the case where their distribution is random. The interrmption of growth which occurs when the edges of adjacent centres come into contact is dealt with by a statistical argument due to Avrami [ 13]. (A similar and independent treatment has been given by Evans [ 14] .) The results of these arguments are best summarised in the Avrami theorem, which relates the actual advance of the transformation to the transformation that would occur if we imagined that the interaction between centres did n o t occur. In terms of the coverage 0 0 = 1 -- exp(--0ex )

(3)

where 0ex is the extended coverage which corresponds to the imaginary situation where the growth is n o t influenced by interactions between adjacent centres. The essentially statistical nature of the arguments leading to eqn. (3) is n o t entirely clear in the original discussion [ 13 ], whereas the equivalent approach given by Evans [14] is clearly based on the Poisson formula. The Avrami theorem can be generalised for cases in which the rate constants of nucleation and growth are n o t constant with time, provided that at any time the rates which describe the progress of the transformation are uniform in all the space in which the transformation occurs. The equations which describe the current at constant potential (when the rate constants for nucleation and growth are constant) are [12]

I = (27rzFMhNck:t/p) exp(--wM2Nck:t:/p:)

(4)

for instantaneous nucleation, and

I = (lrzFMhANok2t2/p) exp(--v:M:ANok:t3/3p 2)

(5)

for progressive nucleation. Instantaneous nucleation occurs when the centres grow from N c randomly distributed nuclei formed at once, whereas in progressive nucleation the probability that new nuclei appear is uniform with time, so that N = NoAt, where No is the number density of active sites and A is a 2-D nucleation rate constant. We s h a l l n o w make use of the Avrami theorem to obtain an expression which relates the rates of nucleation and growth to the faradaic current flowing through the electrode and to its coverage with the deposit. In general, the current I s flowing through the external circuit can be formally deconvoluted as [151 Ig = It + Idl + lads

(6)

250

where the subscripts refer to the faradaic, double-layer and adsorption components. At any potential we can rewrite eqn. (6) explicitly as Ig=I~+Cdl

d~

+ 1--0

dt

(7)

where F is the surface concentration of adsorbed species. For typical values of Cdl, the charge CdlA~? is negligibly small compared with the faradaic charge for the formation of a monolayer phase. Similarly the contribution from ionic adsorption will only be significant if the adsorption is highly potential dependent. Neglecting these two contributions to Ig for the moment, we assume that I~ ~- Ig, and it follows that (8)

dO/dt = Ig/qmo n

where qmon is the charge required to complete a monolayer of product. Thus, in the galvanostatic experiment, we assume that the total rate of transformation is kept constant, although the rate constants for nucleation and growth vary throughout the transient. It is convenient to eliminate time as an independent variable, and use instead 0 = I g t / q m o n . For progressive nucleation, the equation 0 = 1 -- exp(---~M2NoAk2t3/3p

(9)

2)

follows from eqns. (3) and (4). Solving for t: t = [--fl~' ln(1 - - 0 ) ] '/3

(10)

where tip = 7 r M 2 N o A k 2 / 3 p 2

(11)

It follows by elimination of t that I = 3qmon(l

-

-

O) (fl~/2[--In(l -- 0)]} 2/3

(12)

from which •p can be obtained as a function of the current and of the coverage: ~p = [ I / 3 q m o n ] 3

1 / ( ( 1 -- 0)3[--In(1 -- 0)] 2}

(13)

In the galvanostatic experiment I = Ig, and tip tends to infinity as 0 -* 0 and as 0 -* 1. tip passes through a minimum at a value of 0 found from the derivative of eqn. (13); p0mi n = 1 --

e x p ( - ~ ) ~-- 0.49

(14)

Similarly for instantaneous nucleation we define [3i = 7 r M 2 N c k 2 / p 2

(15)

and it follows that fli = [ I / 2 q m o , ] 2 1 / ( ( 1 -- 0)2[--ln(1 -- 0)]}

(16)

Again in a galvanostatic experiment (I = Ig), fli tends to infinity at the limits 0 -~ 0 and 0 -~ 1, but now its minimum is at iOmin

=

1 -- exp(--~) ~ 0.39

(17)

251

f(O)

10'

10c

e

o

o15

Fig. 1. The variation of/~i (curve 1)and ~p (curve 2) with 0 at constant current, predicted for instantaneous progressive nucleation by eqns. (16) and (13) respectively.

Armstrong and Harrison [16] have obtained an expression which is equivalent to eqn. (16) (although the m e t h o d of derivation is not given explicitly in their paper). The variation of tip and fli with 0 at constant current is shown in Fig. 1. The experimentally accessible quantity is, however, not fl, but ~?. Since fl = riO?, 0) and r~ = rl(O), dO

~

0 dO

a-0 n

(18)

If we assume that the rate constants for nucleation and growth depend only on r~, and n o t on 0, we can eliminate the second term in eqn. (18) and obtain ( a f t ) d~ d/~= a-~0

(19)

Therefore the functional relationship between fl and rl can be obtained along a galvanostatic transient, comparing fl with 77 at constant 0. If the assumption that lattice incorporation is the rate-determining step is justified, the same fl-~ relationship should be obtained regardless of the applied current density. As a first approximation, however, we can make use of the empirical relation obtained from potentiostatic studies [17]: In/max = a + bT?

(20)

252

where/max is the current at the maximum of the potentiostatic transient. By taking the first derivatives of eqns. (4) and (5) to find the current maxima and equating their logarithms to eqn. (20), we can then use eqn. (12) and its equivalent for instantaneous nucleation to obtain expressions for the W--0 curves for the two cases. Thus for progressive nucleation 2 2 77 = (--a + In I + ~[ln(~)-1] - - l n [ ( 1 -- 0)[--ln(1 - - 0 ) ] 2 n ] } / b

(21)

and for instantaneous nucleation 1 77 = (--a + In I + i1 [ln(~) -- 1] -- ln[(1 -- 0) [--ln(1 - -

0)]1/2]}/b

(22)

By substituting the appropriate values of 0mi n into eqns. (21) and (22) we recover our original assumption given by eqn. (20). EXPERIMENTAL SECTION

A micrometer driven hanging mercury drop (HMDE) filled with mercury or 1 At% Cd(Hg) was used in the experiments. The importance of electrode geometry and preparation in studies of this kind has been demonstrated elsewhere [18]. The HMDE was fitted into a glass cell equipped with a platinum counter electrode and an adjustable Luggin capillary connected to the reference electrode. HgO, Hg2C12 and red HgS on mercury were used as reference electrodes. All measurements were carried out in a light-tight Faraday box at room temperature, using nitrogen purged solutions prepared from triply distilled water and AnalaR reagents. Galvanostatic transients were obtained with a 'PGP' control device which switched rapidly from potentiostatic (P) to galvanostatic (G) control. The instrument was developed from a circuit published by Bruckenstein and Miller [ 19], and it incorporates an operational amplifier bridge circuit which subtracted the ohmic contribution from the galvanostatic transient. Rapid, essentially transient-free switching was accomplished by a solid-state VMOS device, the control sequence being dictated by the TTL logic status of an external function generator. The ohmic potential drop between the tip of the Luggin capillary and the working electrode was measured as the response to a short current pulse which produced a small perturbation of the electrode potential in the double layer region. The bridge circuit was then adjusted carefully to compensate exactly for the iR drop. Transients were then recorded on a digital storage oscilloscope and transferred to an X--Y recorder for analysis. R E S U L T S F O R T H E G R O W T H O F Cd(OH)2, Hg2C12 A N D HgS

A schematic potential transient for the galvanostatic deposition of an anodic film on a liquid substrate is shown in Fig. 2. The instantaneous potential step (a--b) at the beginning of the transient is due to the ohmic drop across the electrolyte solution, and in the experiments reported here, it was eliminated by the bridge network described above. The subsequent rise in potential (b--c) is followed by a number of potential maxima (c, d, e ...). Each maximum corresponds to the onset of a nucleation and growth process which leads to the formation of a monolayer of product [16,20].

253 @

E

~

a't 60

40

b. 20

-i;

t,

p

G

O

t P'

t/ms

io

'

~'o

'

6'o

'

8'o

'

~6o

Fig. 2. Potential (a) and current (b) profiles for a typical PGP experiment. The control sequence is potential (P), current (G), open circuit (O) and finally potentiostatic again (P'). Fig. 3. Potential transient for the galvanostatic deposition of cadmium hydroxide onto hanging amalgam drop electrode from 1 M NaOH at 15 mA cm -2.

An experimental transient for the deposition of Cd(OH)2 on cadmium amalgam is shown in Fig. 3. The initial double layer charging is almost linear, indicating that there is no significant ionic adsorption prior to the nucleation of the Cd(OH)2 phase. This conclusion is in accord with the evidence obtained from capacitance measurements in the same system [21,22]. Three potential maxima can be observed on the transient; the charge passed between the first and second peaks is 340 pC cm -2, whereas that between the next two peaks is 300 pC cm-'. These charges agree reasonably well with those needed to complete a monolayer in the [0001] plane of/3-Cd(OH)2 [23] (303 pC cm-2), and also coincide with the experimental results of potentiostatic measurements (320 + 20 pC cm -2) [22]. The coverage 0rain at the potential minima for both monolayers corresponds very closely to the value of 0.39 predicted by eqn. (17) for the instantaneous nucleation and growth of two dimensional centres. The fl--~ relationship was then examined by calculating values of fli from eqn. (16) and plotting them on a logarithmic scale against overpotential as shown in Fig. 4 for the first layer of Cd(OH)2. Evidently for low current densities, a logarithmic relationship between fli and r~ is obeyed and as expected, the descending (©) and ascending portions (o) of the galvanostatic transient give data which fall on a c o m m o n line. As the galvanostatic current is increased, however, the data from the two parts of the transient no longer coincide; the overpotential rises more steeply beyond 0min than we should expect if lattice incorporation is the rate determining step. In addition, transients obtained at different current densities do not give data which fall on a straight line. The line which has been drawn through the data has a slope of (0.9 mV)-', but clearly it is steeper than the lines which can be drawn through the data from individual transients. We therefore compared the transients at a c o m m o n value of 0 = 0min, and Fig. 5 shows that this treatment gives a value of (a log fli/ar~) = (0.75 mV)-'. Both

254

7

O

PO

oo

105 ol

10o

I/Acm-a

J

10 4

o" 10 ~

3

J

10-'

o o I O

10 2

f

¢D

/

io

(E-823)/mY vs Hg/HgO 2'0

'

a'0

6'0

'

,'0

2'0 (E-823)/mV

' 30 vs Hg/HgO

Fig. 4. Semilogarithmic plot of ~i vs. potential for the deposition of the first monolayer of on cadmium amalgam. The plot shows data obtained at four different current densities. Cd(OH)2

Fig. 5. Semilogarithmie plot of the applied current density vs. the potential minimum for the deposition of the first monolayer of Cd(OH)2 on cadmium amalgam.

values of (~ log ~ i / ~ ) are much higher than the equivalent quantities derived from potentiostatic data (Armstrong [23] reports that (a log Ima,,/~7?) is (4.5 + 2 mV) -1 for the same system, which corresponds to (~ log 13i/~7) = (2.3 -+ 1 m y ) -1. A galvanostatie transient for the deposition of calomel on mercury is shown in Fig. 6. Comparison with Fig. 3 shows that in the present case the initial charging of the double layer is no longer linear and that the nucleation and growth process clearly sets in at a much higher coverage of the electrode with adsorbed ions. In the ease of Cd(OH)2 deposition, the nucleation spike occurs at very low surface coverages (0 < 0 < 0.02), but in the ease of calomel, 0 rises to 0.2--0.5 before the relaxation due to the phase formation is evident. The adsorption processes which precede phase formation give rise to large charging currents in potentiostatie measurements [4] as well as to high double layer capacities of the mercury/chloride system near the reversible potential of calomel [7,24]. The second layer of calomel, on the other hand, gives rise to a well developed transient with a minimum at 0 = 0.39, indicating again that instantaneous nucleation occurs. A plot of log/3 i vs. t7 for the second layer of calomel is shown in Fig. 7, and like similar plots for Cd(OH)2 deposition, it is clear that

255

o• I • ,o,,0

#.]_i

30

ElmV

v s HglHg2C[ 2

]

20

l° J

s

t/ms ,'o

o

I'o

E/mV

y~ Hg/H%CI 2

2'0

30

Fig. 6. Potential transient for the galvanostatic deposition of calomel onto hanging mercury drop electrode from 0.1 M HC1 at 23 m A c m -2. Fig. 7. Semilogarithmic plot of Hi vs. potential for the deposition of the second monolayer of calomel on mercury.

the data from the descending and ascending branches of the transients diverge as the current density is increased. A similar analysis for the first layer of calomel was not made because the current due to adsorption/desorption processes cannot be neglected. However, at the potential minimum dT?/dt = O, and so the effects of double layer charging should vanish, provided that the adsorption processes are at equilibrium. A plot of log Ig vs. ~?min is therefore shown in Fig. 8 for the deposition of the first layer of calomel; the slope (a log Ig/a~7) is (0.7 mV) -1, and if the nucleation process is assumed to be instantaneous, (a log flila~) would be (0.3 mV) -1. A similar plot of log I~ vs. ~Tmin for the deposition of the second layer of calomel is shown in Fig. 9, and its slope corresponds to (a log ~i/a~) = (10 mV) -1. These extraordinarily low values demonstrate the extreme potential dependence of the nucleation and growth processes which are involved in the electrocystallisation of these monomolecular anodic films. As a final example of the application of the galvanostatic m e t h o d to twodimensional phase transformation studies Fig. 10 shows a transient for the for"mation of an anodic film of HgS on mercury. The immediately striking feature of the transient is that the first monolayer appears to be deposited reversibly; there is no nucleation spike until the second layer begins to deposit. We have shown elsewhere [25] that the first monolayer of HgS on mercury is formed by

256

|/A cm-2

l/Acre -2

10c

10°

aD/'~~ ' / ~

o ooo O(D

0

o co

,0-'

0

~o

010(30 co

10-'

/OO

E/mV v.s Hg/Hg2C[2 ,'0

0

;5

2'o

E/mV vs Hg/Hg2C[ ~ 0

'

~0

'

2'0

'

3'o

Fig. 8. S e m i l o g a r i t h m i c p l o t o f t h e applied c u r r e n t d e n s i t y vs. t h e p o t e n t i a l m i n i m u m for t h e d e p o s i t i o n o f t h e first m o n o l a y e r o f c a l o m e l o n m e r c u r y .

Fig. 9. Logarithm of the applied current density vs. the potential minimum for the deposition of the second monolayer of calomel on mercury. an adsorption process. A phase condensation at 0 ~ 0.99 then precedes the nucleation and growth of the second layer of HgS, and close examination of Fig. 10 reveals the phase condensation as a shoulder in the rising part of the transient just before the nucleation spike for the second layer. The deposition of the second layer of HgS is examined in more detail in Fig. 11 which compares a family of transients obtained at different current densities. In each case 0min is close to the value of 0.39 characteristic of instantaneous nucleation. The plot of log ~i vs. 77 shown in Fig. 12 exhibits the same features dicussed above for the deposition of Cd(OH)2 and calomel, Viz, a divergence between the data points obtained from the descending portions of the transients which becomes more significant as the current density is increased. Figure 13 is a plot of log Ig VS. T/rain for the second layer of HgS, and its slope (a log/g/a~min) is (4.6 MV) -1. We have taken this value of (a log i/a~7) and used it to compare an experimental transient with the potential response predicted by eqn. (22) for instantaneous nucleation. At the lower current density chosen for the comparison, the agreement is seen from Fig. 14 to be satisfactory. A more detailed study of the phase transformation processes attending the anodisation of mercury in sulphide solutions is discussed elsewhere [25]. The galvanostatic m e t h o d also allows us to measure the reversible potential of the electrode immediately after the application of a controlled a m o u n t of charge. The current is simply interrupted when a certain surface coverage has been reached, and the relaxation of the electrode potential is then followed. Figure 15 illustrates the decay of potential which follows anodisation at constant current to different degrees of surface coverage of HgS on mercury [25]. The reversible potential of the first monolayer of HgS is appreciably more negative than either the reversible potential of the bulk (red HgS) phase or of

257

E / m Y vs

Hg/Hgs

tls

0'.s

~io

~.s

2'.o

Fig. 10. Potential transient for the galvanostatic deposition of HgS onto hanging mercury drop electrode from 0.1 M Na2S + 1.0 M NaHCO3 at 0.295 m A c m -2.

E / m V vs HglHgs

30 10 3 o

Pi

o

i.

s2

20

I0

/

!

10~

10'

•

/°

j" o

7.

10c

/:" e

0'.2

0'.4

o'.6

ole

,'.o

10~

}~ 10

E / mV v~ HglHgS 20

Fig. 11. Family of galvanostatic transients for the deposition of the second monolayer of HgS on mercury. Current densities in/~A cm-2: (1) 55, (2) 110, (3) 275, (4) 550, (5) 1100. Fig. 12. Semilogarithmic plot of ~i vs. potential for the deposition of the second monolayer of HgS on mercury.

30

258

E/mV vs Hg/HgS

30

20

I I A crn 2

~"O,~o~-~o

10 -I

/

o

/

o//°

I0

/ 10-;

E/mV vs Hg/HgS '

1'0

2'0

'

8

0 . '2

O.4 '

0 -'6

O i8

~.O '

Fig. 13. Semilogarithmic plot of the applied current density vs. the potential minimum for the deposition of the second monolayer of HgS on mercury. Fig, 14. Comparison of experimental transients for the deposition of the second monolayer of HgS on mercury with the behaviour predicted by eqn. (22) for instaneous nucleation (©). Current densities: (1) 55 #A cm-2; (2) 1100 pA cm -2. (~ log i/~) was taken as 4.6 mV - l .

L

E/mY vs Hg! HgS

-1C

tls

0

~

~0

{s

2'0

Fig. 15. Galvanostatic transients for the deposition of HgS, with interruption of the current at the points indicated by the arrows. Current density: 22//A c m -2.

259 E/mY vs Hg/Hg2CI2 20 --o

c, e x p ( - A G . / R T )

0

2

4

6

8

t/s

-10

n

Fig. 16. G a l v a n o s t a t i c t r a n s i e n t s for t h e d e p o s i t i o n o f calomel, w i t h i n t e r r u p t i o n o f t h e curr e n t at t h e p o i n t s i n d i c a t e d b y t h e arrows. C u r r e n t d e n s i t y : 180 ~ A c m -2. Fig. 17. S c h e m a t i c s t e a d y - s t a t e d i s t r i b u t i o n o f cluster sizes at a p o t e n t i a l 171, at w h i c h t h e critical size is n 1. As t h e o v e r p o t e n t i a l increases t o 772, t h e critical size decreases t o n~ a n d t h e clusters inside t h e s h a d e d area b e c o m e supercritical, c l is t h e surface c o n c e n t r a t i o n o f m o n o m e r s at t h e p o t e n t i a l 71 a n d AG o is t h e s t a n d a r d free e n e r g y of f o r m a t i o n o f clusters o f size n.

the subsequent monolayers. In fact, here we have clear evidence for the underpotential deposition of an anodic film by an adsorption process. The absence of any decay in potential when the current is interrupted is proof of the reversibility of the adsorption phenomenon. By contrast, the subsequent monolayers of HgS are clearly deposited by a nucleative mechanism, since the characteristic relaxation of the overpotential associated with nucleation and growth can be seen in Fig. 15. Figure 16 displays a set of open circuit decay curves obtained during the deposition of calomel on mercury. The difference between the reversible potentials of the monolayers is evident once more, although in this case the reversible potential of the first monolayer is more positive than that of the second and subsequent layers. The reversible potential of the calomel electrode is approached only after the second monolayer has been completed and after a period at open circuit, during which a recrystallisation phenomenon appears to be involved in the establishment of the reversible potential of the system. DISCUSSION

Although the study of anodic electrocrystallisation by potentiostatic techniques is simplified in principle by the constant electrochemical rates maintained throughout the experiment, the galvanostatic method has some practical advantage, since it circumvents the experimental errors which arise in the control of potential in potential step measurements. The results of this study have shown that the experimental transients obtained at low current densities agree fairly well with the curves predicted for the instantaneous nucleation mech-

260

anism. This is a surprising result, since we have shown elsewhere [18] that the potentiostatic transients for the Cd(OH)2 system are characteristic of the progressive nucleation mechanism. This is evidently not an instrumental artifact; we were never able to obtain galvanostatic transients which had the shape expected for progressive nucleation. Evidently the formal equivalence between the t w o methods which have been established theoretically does n o t correspond to the experimental situation. Theoretical treatments of two-dimensional electrocrystallisation at constant potential generally start from the assumption that the rate constant for nucleation has its equilibrium value A which corresponds to the steady state distribution of clusters [26,27]. Transient effects in nucleation have been considered by a number of authors [28--32], and in the potentiostatic case the experimental evidence for such effects is that an induction time is needed before the steady-state cluster distribution is reached and A rises to its equilibrium value. Although the induction time is clearly seen in many experiments involving the electrochemical nucleation of three-dimensional centres [ 33--35], it has n o t been characterised in the case of the potentiostatic two-dimensional electrocrystallisation of anodic films, and experimental transients show little evidence for non-equilibrium nucleation. In the galvanostatic case, and indeed in any situation where the supersaturation varies with time, transient effects could conceivably change the mechanism of nucleation entirely, and for this reason we discuss the validity of the steady-state approximation below. Nucleation occurs through fluctuations of the cluster size distribution [26,27], and at any given supersaturation, an equilibrium size distribution can be defined in terms of the Boltzmann distribution function and the free energy of formation of each cluster at that supersaturation. Under electrochemical conditions, the supersaturation is uniquely defined by the potential. In terms of classical nucleation theory [ 25], the critical cluster radius r c is given for centres of spherical geometry by r c = 2 o M / ( p z F [ rll )

(23)

and the free energy of nucleation by A G = 167ro3M2 / 3 p 2 z 2 F 2 ~ 2

(24)

where a is the interfacial free energy between the new phase and the electrolytic solution. N o w in the galvanostatic experiment the overpotential increases initially as most of the imposed current is used to charge the double layer. The critical radius therefore decreases rapidly as the electrode potential rises b e y o n d the reversible potential. At equilibrium, i.e. at constant supersaturation, nucleation occurs when a subcritical cluster accepts an additional atom or molecule and is p r o m o t e d to the other side of the free energy maximum. This t y p e of nucleation is a stochastic process which has been appropriately termed thermal nucleation [36]. However, in the galvanostatic or potentiodynamic methods, the location of the free energy maximum shifts to smaller cluster sizes as the overpotential increases. If the relaxation of the cluster distribution is t o o slow to follow the charge in potential, clusters may be simply p r o m o t e d to become supercritical as the critical cluster size moves past them to smaller values. This is an essentially deterministic process, and it has been given the

261 name athermal nucleation, albeit in a non-electrochemical c o n t e x t [36]. If the rate of change of potential is sufficiently rapid, a significant fraction of the subcritical nuclei will be p r o m o t e d in this way, and most of the nuclei formed during the rising portion of the galvanostatic transient may owe their origin to this effect. Figure 17 illustrates h o w these clusters may become supercritical as the critical size moves past them under potentiodynamic conditions. The galvanostatic transient passes through a sharp maximum, after which the overpotential begins to fall again as the growing centres expand and the current density at the growing edge decreases. If the number of nuclei formed by the thermal mechanism during this relaxation is very much smaller than the number formed athermally at short times, the nucleation will appear to be instantaneous. A similar p h e n o m e n o n would be expected to occur under potentiostatic conditions, particularly at high overpotentials, b u t the experimental evidence is not convincing, although the effects may well be obscured by the potentiostat rise-time and ohmic errors. We have, however, observed that potentiostatic transients at higher overpotentials show the linear i--t dependence at short times which is characteristic of instantaneous nucleation. Clearly further experimental studies are needed in order to clarify some of the questions raised by our galvanostatic results. Leaving aside the problem of w h y the galvanostatic transients correspond to instantaneous nucleation, we turn n o w to the deviations from the predicted behaviour which b e c o m e increasingly evident at higher current densities. Our theoretical treatment of the galvanostatic transients supposes that the ratedetermining step is lattice incorporation, regardless of the degree of surface coverage reached. However, this assumption is likely to be suspect in the limits of very low or very high surface coverage, when the free edge area of the growing film becomes small and the current density into the growing edge correspondingly high. Under these conditions we should expect that the overall rate of growth might be limited b y some other process, for instance chargetransfer or surface diffusion. The overpotential would then be expected to be higher than predicted by the simple model. An additional problem which we have touched on above is that some fraction of the total current is used to charge the double-layer. However, the error involved in the neglect of this factor is small, provided that the differential capacitance concerned is of the order of magnitude of the double layer capacitance in the absence of appreciable specific adsorption of ions. We prefer to conclude that surface diffusion can account for the deviation of the experimental transients from the predictions of the geometric model. It is significant that the overpotential rise more steeply above the theoretical transient as the surface coverage increases b e y o n d 0rain. Under these conditions the electrode m a y be thought of as a random distribution of contracting free patches on an otherwise covered surface. The geometry of the diffusion problem is particularly unfavourable since the 'reservoir' of ad-molecules decreases as the free patch shrinks in area, and the surrounding edge competes for material. By contrast in the early stages of film growth the circular patches of growing film are surrounded by circular diffusion zones which can supply much more material to the growing edges. A quantitative treatment of these effects is difficult because the geometry of the diffusion fields is very complex. It is clear from this discussion that any extension of the

262

theoretical model b e y o n d the simple geometric approach outlined above runs into serious difficulties, and we conclude that at low current densities or low overpotentials the present theoretical description is remarkably successful. ACKNOWLEDGEMENTS

Grateful thanks are offered to the following bodies for financial support for research studentships. B.S. to the Fund for Research in Hydrocarbons (FONINVES) of Venezuela and M.P. to the Instituto Nacional de Investigaq~o Cientifica of .Portugal.

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