Electrolytic nucleation of silver on a glassy carbon electrode

J. Electroanal. Chem., 107 (1980) 323--336

323

© Elsevier Sequoia S.A., Laussane -- Printed in The Netherlands

ELECTROLYTIC NUCLEATION OF SILVER ON A GLASSY CARBON ELECTRODE PART I. MECHANISM OF CRITICAL NUCLEUS FORMATION *

A. MILCHEV, E. VASSILEVA and V. KERTOV

Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1040 Sofia (Bulgaria) (Received 14th August 1978; in revised form 1st September 1979)

ABSTRACT The initial stage of electrodeposition of silver on a glassy carbon electrode is investigated using a double-pulse potentiostatic method. Data are obtained for the concentration dependence of the steady-state nucleation rate I and conclusions are drawn for the probable mechanism of critical nucleus formation.

INTRODUCTION

In a series of papers recently published [ 1--3] the atomistic approach to the nucleation process [4,5] was applied to the initial stage of metal electrodeposition. Theoretical expressions for the steady-state nucleation rate I were derived considering two different mechanisms of critical nucleus formation: (1) through direct attachment of ions from the volume of the electrolyte, and (2) through surface diffusion of adatoms on the substrate. These investigations were an attempt to remove the contradiction between the macroscopic classical theory and the experiment arising from the very small size of the critical nucleus usually obtained in the electrolytic case. Unfortunately, the detailed analysis of the experimental data for the steady-state nucleation rate [6] did not allow a finite conclusion to be drawn on the applicability of the two theoretical models. In all cases the data for I were obtained in a very narrow overvoltage interval and their interpretation on the basis of both classical and atomistic theoretical expressions lead to approximately the same qualitative and quantitative results. Therefore, it was necessary to carry out new experiments to obtain data for the steady-state nucleation rate in a sufficiently wide supersaturation interval. In a series of two papers we present the results from such an investigation. The kinetics of silver electrodeposition was studied and information on the nucleation rate was obtained by investigating the number N of silver nuclei as a function of time t at a constant supersaturation. The aim of the first paper is: (1) to describe the experimental method; * This paper is dedicated to Professor R. Kaischev on the occassion of his 70th birthday.

324

(2) to clarify the probable mechanism of critical nucleus formation -- direct attachment of ions or surface diffusion of adatoms; (3) to provide information on the distribution of clusters on the electrode surface. The second paper presents the basic results from the investigation of the number of nuclei as a function of time. The experimental data for the steadystate nucleation rate are discussed on the basis of both classical and atomistic models. EXPERIMENTAL

Electrodes and electrolyte The cathodes used were glassy carbon rods of 2 mm diameter, polished with diamond paste and placed in a teflon capillary or sealed in glass. The anode was a silver ring with an area of a b o u t 1.5 cm 2. The electrolyte was an aqueous solution of silver nitrate with a concentration varying from 0.5 X 10 -3 tool cm -3 to 12 × 10 -3 mol cm -3 in different experiments. The glassy carbon electrodes were always polarized with reference to the equilibrium potential of the bulk silver in the corresponding silver nitrate solution. The temperature of the electrolyte was kept at 36°C.

Experimental method The principle of the double-pulse potentiostatic method used for investigating the " n u m b e r of nuclei -- t i m e " dependence is as follows. A double rectangular pulse (Fig. 1) is applied to the electrodes of the electrolytic cell. The overvoltage ~ of the first pulse is selected sufficiently high to initiate a nucleation process on the cathode. During the second pulse the nuclei grow to visible sizes and can be easily registered in an optical microscope. After counting, the crystallites can be dissolved by applying a positive voltage to the electrodes. The block diagram of the pulse generator constructed for the present investigation is shown in Fig. 2. The operational amplifier A and the resistances R and

T t

--

Fig. 1. F o r m of the double potentiostatic pulse.

325 "t'Y I1 P

T

I

R

~ R1

R

O

Fig. 2. Block diagram of the pulse generator.

R1 play the role of a summing amplifier. The first and the second potentiostatic pulses are applied to the electrolytic cell by turning on the electronic switches K3 and K4. The amplitudes of the pulses depend on the current of the precise sources I1 and I2. The duration of the first pulse is determined b y the monostable multivibrator M, and the duration of the second pulse depends on the time during which K1 is on. The characteristics of the pulse generator are as follows: amplitudes of the first and second pulses 0--0.5 V; duration of the first pulse 5--5 X l 0 s ps; rise and fall time of the pulses < 0.8 ps; temperature drift < 10 #V/°C; voltage of etching 0 - - 3 V. Undoubtedly the double-pulse method is very useful, especially for investigating the initial stage of the electrodeposition processes. However, there are two questions concerning the principle of this method which must be clarified in advance. As is known, a direct voltage applied to an electrolytic cell attains its constant value ~7 after a time interval necessary for charging of the electrical double layer. Therefore, at the very beginning of the potentiostatic pulse the overvoltage is a function of time and changes according to the formula 77(t) = ~ [1 -- exp(--t/RCea)] where R is the ohmic resistance of the electrolyte and C~ is the capacity of the double layer. As can be seen from the formula the overvoltage attains 99% of its nominal value after a time of about 5 RCd~. Therefore, it is necessary first to measure the quantities R and Ccu and to determine the lower limit of the pulse durations which could be used in the concrete experimental investigation. In our case, the quantities R and Cd~ were determined by measuring the impedance of the electrolytic cell with an RC bridge. The values obtained were R = 12~2 and C~ = 22 X 10 -8 F, and consequently 13 ps resulted for the product 5 RCed.

326

A similar value (~ 30 ps) was obtained by analysing the galvanostatic ~7/t relationship. So, the shortest pulses used in our experiment were selected to be. 300 ps. The second question which has to be clarified concerns the overvoltage of growth. Obviously, the nuclei formed during the first pulse will grow during the second only if they have exceeded the critical size corresponding to the overvoltage of growth ~ . This means that part of the nuclei formed at the end of the first pulse will be unavoidably lost. Therefore, it is important to estimate the percentage of the lost nuclei. The problem could be theoretically considered on the basis of the classical nucleation theory [7], but here only an empirically obtained estimate will be presented. Figure 3 shows data for the number of silver nuclei formed at a constant overvoltage and duration of the first pulse plotted as a function of the overvoltage of growth. As seen, the dependence N/7?g does not rise continuously but reaches a plateau at an overvoltage of about 10 mV. This means that the number of nuclei dissolved because of their thermodynamic instability at the overvoltage of growth is very small and probably commensurable with the specific dispersion of the experimental data. (Analogous results were obtained by Toschev and Markov for the case of mercury deposition on a platinum single-crystal electrode [8] .) Therefore, in the present investigation the overvoltage of growth was selected to be 20 mV. As mentioned above, the potentiostatic pulse technique allows the crystallites formed on the cathode to be dissolved by applying a positive voltage pulse. In the case under consideration the silver crystallites could be fully dissolved by polarizing the glassy carbon electrode to a positive overvoltage of 0.9 V for about 60 s. Unfortunately, we could not use this method for cleaning the electrode surface because it was established that the anodic etching activates the glassy carbon substrate. In order to demonstrate the influence of anodic etching, the electrode surface was isolated by a varnish so that only a small part in the centre of the electrode could come in contact with the electrolyte. Then the electrode was polarized to a positive overvoltage of 0.9 V for 30 min. Finally, the varnish was dissolved and silver nuclei were deposited on the whole elec-

60

1,0

o

o

0

o

o

2o

IQ

20

30

~0

~glmY

P

Fig. 3. D e p e n d e n c e o f the n u m b e r of nuclei N on the overvoltage o f g r o w t h ~Tg 07 = 0.10 V, t-- 2 ms).

327

trode surface by a cathodic pulse. The uniform distribution of the crystallites formed on the freshly polished glassy carbon substrate can be seen in Fig. 4a. Figure 4b shows the structure of the anodically treated part of the electrode and Fig. 4c shows the higher density of the silver crystallites deposited. This result indicates that a periodically repeated process of anodic etching may lead to a continuous increase of the electrode surface activity. Therefore, data for the number of nuclei obtained in this way could not be simply interpreted. For this reason, after each cathodic pulse the silver crystallites were dissolved in nitric acid. It was, of course, firmly established beforehand that this procedure does not change the state of the electrode surface. MECHANISM OF CRITICAL NUCLEUS FORMATION

At present there is reliable information that the direct attachment of ions plays the determining role during the electrodeposition of silver from aqueous silver nitrate solutions. This conclusion has been drawn from detailed experimental studies of the current of growth of both silver crystals formed on a foreign substrate [9,10] and isolated faces of single silver crystals grown in a capillary [11,12]. Unfortunately, data for the mechanism of critical nucleus formation could not be obtained in this way because it is difficult to measure the current owing to the random behaviour of single ion attachment and detachment. A possible way to clarify this point is by studying the kinetics of the nucleation process. Classical model

According to the classical theory of nucleation the steady-state nucleation rate I could be expressed by the equation: I = ZoFDk exp[--Ak/kT]

(1)

Here, A~ is the work for critical nucleus formation, P the Zeldovich factor, Z0/cm -~ the number of sites on the substrate where nucleation can proceed and D~/s -~ the flux of the ambient phase particles to the critical nucleus. As is well known, it is Dk that depends on the mechanism of critical nucleus formation.

"~

•

o

I

• .

Fig. 4. Distribution of silver nuclei before (a) and after (c) the anodic treatment of the glassy carbon surface. (b) Structure of the anodically treated part of the electrode. Magnification 60x.

328 In the case of direct attachment of ions D ~ is usually expressed as a product of the surface area Sk of the critical nucleus and the density ic of the cathodic current: D da = Skic/ze

=

K'~7-~C1-~ exp[aze~7/kT]

(2)

In this equation * z is the valency of the depositing ion, e the elementary electric charge, c the bulk concentration of the electrolyte, a the transition coefficient and K' is a constant n o t depending on c and r~. In the case of surface diffusion the following expression for Dk holds good: D~d = Z1LkDs/a

(3)

Here, Lk is the length of the critical nucleus periphery, D s the surface diffusion coefficient, a the jump distance at surface diffusion and Z1 the number of single adsorbed atoms per unit area. Atomistic model At high supersaturations the critical nucleus consists of a few atoms only and the steady-state nucleation rate is given by the expression [ 1--3 ] : I = Z0co +,~ exp [-- cb(nk)/kT] exp [nkze~?/kT]

(4)

In this equation nk is the number of atoms in the critical nucleus and O(nk) is a quantity depending on the energetic state of the nk atomic cluster. Here, CO+,k gives the frequency of attachment of an ion to the cluster consisting of nk atoms and corresponds to the flux of the ambient phase particles, but referred to one atomic site. In the case of direct attachment of ions w .da , k is given as follows [1,14] : dak O~+n

= K~kC exp[_(U~k + azeE)/kT]

(5)

In eqn. (5) U~k is the energy barrier for the transfer of an ion from the electrolyte to the n k atomic cluster at a potential difference between the electrode and the electrolyte E = 0 and Kdak is a frequency factor also accounting for the possible ways of single-ion attachment. If an overvoltage 77 is applied to the system, the potential difference E can be presented as E = E0 -- 7, E0 being the equilibrium potential difference between the bulk metal and the electrolyte. Taking into account the dependence of Eo on the concentration c of the electrolyte, according to the Nernst equation, an expression similar to eqn. (2) results for wdak e,ak LO+n

= Kd~cl-C, exp[azeT?/kT]

(6)

sd is In the case of surface diffusion CO÷,k sd k =. LO+n

gS+dk(Zl/Zo) e x p [ _ ( E s d

+

given by

U'nk)/kT ]

(7)

In eqn. (7) the ratio (Z1/Zo) gives the probability to find an adatom in a given adsorption site, E~d is the activation energy for surface diffusion, U',k the * As pointed out by Bindra et al. [13 ] the exact expression for D k must depend on the configuration of the activated state. Therefore eqn. (2) is an approximation.

329 energy barrier for adatom attachment to the cluster and K÷nk sd is again a frequency factor. Obviously, in order to present D~d and ¢0~k in an explicit form it is necessary to derive a theoretical expression for the adatom concentration Z1. The thermodynamic method which could be used for this purpose is as follows. Let us consider an electrochemical system consisting of an inert, ideally polarizable working electrode, an electrolyte with ionic activity aMe+ and an ideally non-polarizable, bulk counter electrode made of the depositing metal. The equilibrium state of such a system can be described through the equality of the electrochemical potentials of species in the coexisting phases:

Here, fie1 is the electrochemical potential of the metal ions in the electrolyte, / ~ the electrochemical potential of the atoms in the bulk metal and ~ad is the electrochemical potential of the adsorbed atoms on the inert substrate. As is well known (see ref. 15) the quantities/ael,/~ and fiad are defined by the equations: ~ 1 = P~l + z e ~ l

= P°l + k T In

aMe+ +

zet~e 1

(8) (9)

p~ = p ~ + ze~b~ Pad = Pad + z e U s = po d + k T l n a a d + z e U s

(10)

In eqns. (8--10) p~, p~ and Pad are the chemical potentials of the particles in the three phases; ~el, ~ and ~ are correspondingly the Galvani potentials of the electrolyte, the bulk metal and the inert substrate; a~÷ is the activity of the metal ions in the electrolyte and aad the activity of the adsorbed atoms on the inert substrate. The last two quantities depend correspondingly on the bulk concentration c of the electrolyte and on the surface concentration Z~ of the adatoms; p° l and pod express the chemical potentials of the metal ions and metal adatoms at aMe+ = 1 and aad = 1 respectively, and depend only on the temperature T. (Obviously, for the bulk metal p~ - go since its activity is always considered to be equal to unity.) Putting /~el and / ~ equal, one obtains the condition for equilibrium between the electrolyte and the bulk metal phase: zeEo= po! _ p~ + k T In aMe+ (11) This is the well-known Nernst equation in which E0 = ~b~ -- ~/el expresses the equilibrium potential difference between the bulk metal and the electrolyte. Putting ~el and Pad equal, one obtains the condition for equilibrium between the electrolyte and the adsorbed phase z e E = p° 1 -- P°ad + leT In

(aMe+/aad)

(12)

Here, E = ~ -- ~ is the actual potential difference between the inert substrate and the electrolyte. Consequently, the difference Eo - - E gives exactly the electrochemical overvoltage ~ involved in the theory of electrolytic phase formaation. Thus, combining eqns. (11) and (12) for aad , one obtains aad = exp[(p~ -- P a d ) / k T ] e x p [ z e ~ / k T ]

(13)

330 The lasl~ equation is quite general and gives the overvoltage dependence of the adatom activity aad in all cases when the surface coverage of the ideally polarizable inert electrode is determined only from the adsorption and desorption of the metal ions. (In case of, say, co-adsorption processes the simple thermodynamic treatment cannot be applied and one should theoretically consider the process by introducing the so-called "charge coverage coefficient" (or "electrosorption valency") ZE which could in principle depend on both the electrode potential and the surface concentration of the adsorbed species (see e.g. ref. [16] and the references cited therein). In order to find the connection between aad and Z, it is necessary to calculate the free energy F of the adsorbed phase and then to determine the chemical potential Pad (Pad = P°d + k T In aad ) using the general definition Pad =

T

Here Z'I = Z1S is the total number of adatoms on the substrate, S being its surface area. Following this approach (see e.g. ref. [17] ) one obtains different expressions for the activity aad depending on the approximations made in calculating the free energy F. For example, assuming the number of the adsorption sites on the substrate to be much greater than the number of the adsorbed atoms (Z0 > > Z,) for aad it follows that

(14)

aa d = Z1/Zo Combining eqs. (13) and (14) for Z~ results in Z, = Z0 exp[(p~ -- p°d)/kT] exp[zel?/kT]

(15)

This is Henri's isotherm. Assuming Z', to be commensurable with Zo for aad it follows that (16)

aad = Z,/Zo -- Z,

and Langmuir's isotherm is obtained from eqns. (13) and (16): Zl = (Z0 -- Z,) exp[(p~ -- P°ad)/kT] exp [zer~/kT]

(17)

Taking into account the possibility for a lateral interaction between the absorbed atoms for aad results in aad = (Z1/Zo -- Z l ) exp[--gZt/Zo]

(18)

and Frumkin's isotherm follows from eqs. (13) and (18): ( Z 1 / Z o - ZI) exp[--gZ1/Zo] = exp[(p~

--p°ad)/kT] exp [ze~?/kT]

(19)

In eq. (19) g is Frumkin's dimensionless attraction constant which can be obtained in an explicit form in the framework of the mean field Bragg--Williams approximation [ 18]. Equations 15, 17 and 19 unambiguously show that independently of the type of isotherm the adatom concentration Z, depends only on the overvoltage and not on the bulk concentration c of the electrolyte. This result provides a

331

possibility of drawing a conclusion about the actual mechanism of the critical nucleus formation by studying the concentration dependence of the steady-state nucleation rate I at a constant overvoltage *. Indeed, the nucleation rate could essentially depend on the bulk concentration of the electrolyte only through the kinetic coefficients Dk and ¢O.nk, and both these quantities depend on c only when the critical nuclei are formed through direct attachment of ions from the volume of the electrolyte. In case of surface diffusion mechanism D ds and sd k do not involve concentration-dependent quantities. Of course, one should Og+n bear in mind that eqns. (15), (17) and (19) are derived in a quasi-equilibrium approximation. Therefore, they can be used for expressing D~d, CO+nkSdand I only at the beginning of the nucleation process when the growth of the comparatively small number of supercritical clusters does not essentially change the conditions for the formation of new nuclei on the substrate. In the advanced stage of the phase transition one should express Z 1 by taking into account the continuous exhaustion of adatoms owing to the intensive growth of stable clusters. It must be noted in conclusion that eqns. (1) and (4) give the rate of formation of stable clusters in the supersaturated system. Thus, on investigating the I vs. c dependence we obtain information on the mechanism of transformation of the critical nuclei into stable clusters. However, this process has a random character and its mechanism probably coincides with the mechanism of critical nucleus formation. EXPERIMENTAL RESULTS AND DISCUSSION

Data for the "number of nuclei--time" dependence obtained at constant overvoltages of 0.090 V and 0.100 V and concentrations of the electrolyte in the interval 0.5 × 10 .3 mol cm -3 -- 12 × 10 -3 mol cm -3 are shown in Figs. 5 and 6. respectively. It can be seen that in both cases the slopes of the straight lines, giving directly the steady-state nucleation rate, increase when the electrolyte concentration increases. This is the first indication that the direct attachment of ions plays a determining role in the formation of the critical nucleus. Plotting the data for I in logarithmic coordinates In I vs. In c yields more convincing evidence in this respect. In this case a straight line having a slope between 0 and 1 must be obtained since a is expected to lie in this interval. Figure 7 illustrates the results of such an interpretation of the experimental data. The slopes of the straight lines are equal to 0.19 and therefore a attains the value of 0.81. The last result is in good agreement with the findings of Gerisher and Tischer [19] (a = 0.74) and Vitanov and Popov [20] (a = 0.70) who have studied the concentration dependence of the exchange current density of a bulk silver electrode in HC104 and AgNO3 aqueous solutions. This slight concentration dependence of the nucleation rate may be the reason w h y Hills et al. [7] did not find a marked concentration effect while studying the nucleation * When t h e bulk c o n c e n t r a t i o n c o f t h e metal ions changes, t h e e q u i l i b r i u m p o t e n t i a l E 0 also changes. Therefore, in o r d e r to keep t h e overvoltage 77 = E o - - E c o n s t a n t it is necessary t o polarize t h e w o r k i n g electrode to o n e a n d t h e same E, b u t always in reference to t h e equilibr i u m p o t e n t i a l E 0 of the bulk m e t a l in t h e s o l u t i o n w i t h the c o r r e s p o n d i n g c o n c e n t r a t i o n . This can be simply realized by using a Me/Me + reference electrode, d i p p e d in t h e w o r k i n g electrolyte.

332

6

4

5

3

0

1

2

3

Fig. 5. Dependence of the n u m b e r of nuclei N on time t at a c o n s t a n t overvoltage ~?= 0 . 0 9 0 V and various c o n c e n t r a t i o n s CAg+/mol cm -3. (1) 0.5 X 10 -3, (2) 1.0 X 10 -3, (3) 1.5 X 10 -3, (4) 3.0 X 10 -3, (5) 6.0 X 10 -3, (6) 12 X 10 -3.

kinetics of silver. In their experiment the concentration of silver is changed only 5.5 times. Additional information on the mechanism of the nucleation process could be obtained from the intercepts to of the straight lines shown in Figs. 5 and 6. In

Z3o I

6O

5O

6

5

4O

:oo i

i

i

3

4

5 t/ms

Fig. 6. D e p e n d e n c e o f t h e n u m b e r o f nuclei N o n t i m e t at a c o n s t a n t overvoltage 77 = 0 . 1 0 0 V and various c o n c e n t r a t i o n s CAg+/mol c m -3. (1) 0.5 X 10 -3, ( 2 ) 1.0 × 10 -3, ( 3 ) 1.5 X 10 -3, ( 4 ) 3.0 X 10 -3 , (5) 6.0 X 10 -3 , ( 6 ) 12 × 10 -3 .

333

c

~3~),.~y~4~

q.:o.~ov

10

•- - - - - ' o " - -

-B

-7

~

-6

qoo.ngv

-5

-4

Inc

Fig. 7. Concentration dependence of the steady-state nucleation rate I.

principle, these intercepts could be interpreted according to the non-steady-state Zeldovich theory [21]. However, it is not firmly established at present whether in the electrolytic case the induction periods observed are those predicted by the Zeldovich theory. Moreover, in the range of high supersaturations in which the present experiments were carried out, an analytical expression for the time lag has not yet been theoretically derived. Nevertheless, it could be expected that the time needed to reach a steady-state regime in the supersaturated system will decrease when the flux of the ambient phase particles increases. If so, the In t0/ln c relationship must be a straight line having a slope equal to ~ -- 1. Figure 8 shows the straight lines obtained in this way. The values of ~ calculated are 0.68 and 0.78 respectively. Two basic conclusions can be drawn from the experimental results presented so far: (1) It is very likely that the direct attachment of silver ions determines the

T

J c

-68

-6.5

-7.0

-7.5

-&0

-8

-7

F i g . 8. C o n c e n t r a t i o n

-6

dependence

-5

In¢

•

of the induction

period to .

334 rate of transformation of the critical nuclei into stable clusters. At overvoltages of 0.090 and 0.100 V the critical nucleus was found to consist of one atom only (see Part II of this paper, this volume) and therefore the conclusion on the direct attachment mechanism has to be related to the transformation of the single adsorbed atoms into two-atomic stable clusters. Obviously, there is no point in discussing "possible mechanisms" of critical nucleus formation in this case -- the one-atomic critical nuclei can be formed only through direct attachment of silver ions from the volume of the electrolyte to the glassy carbon substrate. (2) Although the physical nature of the induction periods connected with the electrolytic nucleation is not completely clarified, the experimental data for to do not contradict the theoretical predictions, at least with respect to the concentration dependence of this quantity. SATURATION NUCLEUS DENSITY The detailed experimental investigations of the "number of nuclei--time" dependence [8,22--24] show that the nucleation rate while initially increasing from 0 to a steady-state value, falls again to 0 after a sufficiently long period of time, i.e. the number of nuclei remains constant (N = Ns). (See e.g. the Nit curves obtained at overvoltages 0.090 and 0.100 V, fig. 2 in Part II of this paper, this issue, pp. 337--352.) The saturation nucleus density in the electrolytic case is usually explained in two ways: (1) The constant number Ns of nuclei is considered as being determined by the direct exhaustion of the active sites on the substrate [22]. In this case the distribution and the number of crystallites reflect the distribution and the number of active sites on the electrode surface. (2) It has been experimentally established that zones of reduced overvoltage ("nucleation exclusion zones") arise around each growing crystallite [25]. The overlapping of these zones is considered to interrupt the nucleation process and to determine the saturation nucleus density. The problem has been theoretically considered by Markov and Kashchiev [26] and expressions have been derived for the Nit dependence in b o t h limiting cases: exhaustion of active sites and overlapping of nucleation exclusion zones. However, the m e t h o d of interpretation of the experimental data proposed b y Markov and Kashchiev is rather complicated and requires a very detailed investigation of the Nit relationship. Therefore, we have used another m e t h o d for clarifying the nature of the plateaus of the N/t curves. Let us assume that the direct exhaustion of the randomly distributed active sites of the glassy carbon surface determines the constant number Ns of the silver crystallites in the plateau region of the Nit curves. In such a case, the distribution function of the distances between the nearest-neighbour crystallites can be derived as follows [27--29]. The probability P for a crystallite to have its nearest neighbour at a distance r is equal to the product of the probability P0 not to find any crystallites in a circle with a radius r, and the probability P, to find just one crystallite in a circle with a radius r + dr. According to Poisson statistics the probability Pm, a random quantity X to assume the exactly deter-

335

mined value of m, is given by the expression (20)

P,, = X m e x p [ - X ] / m !

where X is t h e expected mean value of X. Putting Y. = vr2No, for m = 0, P0 becomes Po = exp [--~r2No]

(21)

In this equation N0/cm -2 is the mean nucleus density on the electrode surface when the plateau of the N i t curve is reached. In the same way, for X = 27rr drNo and m = 1 for P,, one obtains P1 : 2~r drNo exp[--2~r drN0] ~ 2~r drNo

(22)

Hence, the following expression is valid for P: P(r) dr = 2~rr drNo exp[--~r2N0]

(23)

Figure 9 shows the theoretical curves P(r) Ar/r drawn on the basis of eqn. (23) (fluent lines) and the experimental values of P determined by measuring the distances between nearest crystallites (histograms). Here, N s denotes the total number of crystallites obtained in the plateau region of the corresponding Nit curve. It is seen from Fig. 9 that the smallest distances predicted by the theory are not observed in the experimental histograms -- both of them are shifted to the larger values of r. This means that some nucleation exclusion zones actually arise around the growing crystallites. Of course, this experiment does not allow the estimation of the role of the direct exhaustion of active sites. However, it is clear that the distribution of the crystallites obtained in the plateau of the Nit curves does not reflect the random distribution of the active sites on the glassy carbon substrate.

0.4

,.~ OA

P

B.

0.3

q : 0,13 V

q = 0.1t. V

Ns=/,77

Ns=771

0.2

0.2

0.1

[1.1

50

100

150

F

50

rlyrn

100

150

m~

Fig. 9. Distribution of the distances between nearest-neighbour crystallites.

336 ACKNOWLEDGEMENTS

The authors are indebted to Professor R. Kaischev for his stimulating interest in this work and to Dr. A. Popov for measuring the impedance of the electrolytic cell. The comments of Professor E. Budevski, Dr. T. Vitanov and Dr. G. Staikov are gratefully acknowledged.

REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29

A. Milchev, S. Stoyanov and R. Kaischev, Thin Solid Films, 22 (1974) 255. A. Milchev, S. Stoyanov and R. Kaischev, Thin Solid Films, 22 (1974) 267. A. Milchev, S. Stoyanov and R. Kaischev, Electrokhimiya, 13 (1977) 855. D. Walton, J. Chem. Phys., 37 (1962) 2182. S. Stoyanov, Thin Solid Films, 18 (1973) 91. A. Milchev and S. Stoyanov, J. Electroanal. Chem., 72 (1976) 33. G.J. Hills, D.J. Schifrin and J. Thompson, Electrochim. Acta, 19 (1974) 671. S. Toschev and I. Markov, Bet. Bunsenges. Phys. Chem., 73 (1969) 184. A. Scheludko and G. Bliznakov, Bull. Bulg. Acad. Sci. (Phys.), 1 (1951) 239. S. Toschev, A. Milchev and E. Vassileva, Electrochim. Acta, 21 (1976) 1055. T. Vitanov, A. Popov and E. Budevski, J. Electrochem. Soc., 121 (1974) 207. V. Bostanov, G. Staikov and D.K. Roe, J. Electrochem. Soc., 122 (1975) 1301. P. Bindra, M. Fteischmann, J.M. Oldfield and D. Singleton, Faraday Discuss. Chem. Soc., 56 (1973) 180. R. Kaischev, Annu. Univ. Sofia, Fac. Physicomat., Livre 2 (Chem.), 42 (1945/1946) 109. B.B. Damaskin and O.A. Petrii, Vvedenie v elektrokhimicheskuyu kinetiku, Vyshaya Shkola, Moscow, 1975, p. 22 (in Russian). W.J. Lorenz, H.D. Hermann, N. Wutrich and F. Hilbert, J. Electrochem. Soc., 121 (1974) 1167. J.P. van der Eerden, G. Staikov, D. Kashchiev, W.J. Lorenz and E. Budevski, Surface Sci., 82 (1979) 364. W.L. Bragg and E.J. Williams, Proc. Roy. Soc. Lond., 145A (1934) 699. H. Gerischer and R.P. Tischer, Z. Electrochem., 61 (1957) 1159. T. Vitanov and A. Popov, personal communication, 1978. J.B. Zeldovich, Acta Physicochim. URSS, 18 (1943) 1. R. Kaischev and B. Mutaftschiev, Electrochim. Acta, 10 (1965) 643. S. Toschev, A. Milchev, K. Popova and I. Markov, C. R. Acad. Bulg. Sci., 22 (1969) 1413. I. Markov and E. Stoycheva, Thin Solid Films, 35 (1976) 21. I. Markov, A. Boynov and S. Toschev, Electrochim. Acta, 18 (1973) 377. I. Markov and D. Kashchiev, J. Cryst. Growth, 16 (1972) 170. S. Chandrasekhar, Rev. Mod. Phys., 15 (1943) 1. H. Schmeisser, Thin Solid Films, 22 (1974) 99. J.C. Zanghi, J.J. Metois and R. Kern, Phil. Mag., 29 (1974) 1213.