nucleation and diffusion-controlled growth of electroactive centers

Electrochimica Acta 50 (2005) 4736–4745

Nucleation and diffusion-controlled growth of electroactive centersReduction of protons during cobalt electrodeposition

M. Palomar-Pardavea,∗, B.R. Scharifkerb, E.M. Arcec, M. Romero-Romoa

a Universidad Autónoma Metropolitana-Azcapotzalco, Departamento de Materiales, C.P. 02200 México, D.F., Mexicob Universidad Simón Bolıvar, Departamento de Qu´ımica, Apartado 89000, Caracas 1080A, Venezuela

c Instituto Politecnico Nacional-ESIQIE, Departamento de Ingenier´ıa Metalurgica, Apartado 75-876, C.P. 07300 México, D.F., Mexico

Received 3 February 2005; received in revised form 1 March 2005; accepted 2 March 2005Available online 2 April 2005

Abstract

A theory is presented describing, for the first time, the temporal evolution of the fractional surface area,S(t), of 3D non-interacting nucleigrowing at a rate limited by diffusion of electrodepositing ions onto substrates of a different nature. Likewise, an equation has been deriveddescribing the potentiostatic current–time transients arising from the formation and growth of such nuclei with redox reactions occurrings e surface ofi th of cobalta©

K

1

toTaecdrippsrbi

n ofrans-rest.cerrent

f in-ree-unt-rousulti-

th.ions

fromucle-uslyclei,ob-

ofcon-e pro-

0d

imultaneously on their surfaces. An equation is also proposed to describe the current due to redox reactions taking place on thnteracting growing nuclei. The latter is used to describe the experimental current transients recorded during nucleation and growt applied potentials where the proton reduction reaction occurs simultaneously with the electrocrystallization process.2005 Elsevier Ltd. All rights reserved.

eywords:Current–time transients; 3D nucleation; Diffusion; Proton reduction

. Introduction

Secondary reactions such as the proton reduction reac-ion often take place concurrently with the electrodepositionf metals M from aqueous solutions of their ions M(ac)

z+.hese reactions are in general not desired but in many casesre unavoidable. Consequently, they affect the properties oflectrodeposits and lower the efficiency of the deposition pro-esses. For instance, the Faradaic efficiency for chromiumeposition is frequently as low as 30%[1]. The protoneduction reaction (PR) in many cases also hinders stud-es of metal deposition processes[2]. During potentiostaticulses, the simultaneous occurrence of PR on the electrode-ositing new phase gives rise to complex current–time tran-ients that are not appropriately described by existing theo-ies [3–5]. Other examples of electrode reactions catalysedy emerging deposits include the reduction of nitrate dur-

ng the electrodeposition of cobalt[6] or thallium [7]. The

∗ Corresponding author. Tel.: +52 5553189472; fax: +52 5553189087.E-mail address:[email protected] (M. Palomar-Pardave).

description of processes where the electrocrystallizatiometals occurs simultaneously with concurrent charge tfer reactions, particularly PR, is therefore of great inteAbyaneh and Fleischmann[8] have described the influenof secondary redox processes on the potentiostatic cutransients when phase growth is limited by the rate ocorporation of adatoms to two-dimensional (2D) or thdimensional (3D) growth centers. However there is moing experimental evidence that indicates that in numecases, the electrodeposition of metals occurs through mple nucleation followed by diffusion-controlled 3D growThus the aim of the present work is to develop expresscapable of describing current–time transients arisingmetal electrocrystallization processes where multiple nation and diffusion-controlled growth occurs simultaneowith further reactions on the surface of the growing nuparticularly the reduction of protons. The expressionstained are based on current formulations[9–11]of the theoryfor multiple nucleation with diffusion-controlled growththree-dimensional centers. It is shown that the kineticstants characterising both the nucleation process and th

013-4686/$ – see front matter © 2005 Elsevier Ltd. All rights reserved.oi:10.1016/j.electacta.2005.03.004

M. Palomar-Pardav´e et al. / Electrochimica Acta 50 (2005) 4736–4745 4737

ton reduction reaction can be successfully obtained from po-tentiostatic current transients recorded during the electrode-position of cobalt onto vitreous carbon electrodes.

2. Experimental

Cobalt was electrodeposited onto vitreous carbon from a3.5 mM solution of CoCl2 in a 1 M NH4Cl (pH 4.5) solu-tion [12]. All solutions were prepared from analytical gradereagents, using ultrapure water from a Millipore-Q system.A thermostated, jacketed three-electrode glass cell, was usedthroughout this work. The working electrode was built from a3 mm diameter vitreous carbon rod. In order to expose only aportion of the electrode surface (0.0707 cm2) to the solution,the rod was sealed in a Teflon-filled glass tube. The exposedsurface was polished with Buehler Microcloth to a mirrorfinish with alumina powders down to 0.05 mm and treated inan ultrasonic bath in pure water for 10 min. The counter elec-trode was a graphite rod with a surface area much larger thanthat of the working electrode. A saturated calomel electrode(SCE) was used as the reference electrode. This electrode wasconnected to the main cell through a Luggin capillary posi-tioned 2 mm from the vitreous carbon disc. Electrode poten-tials were controlled with an EG&G PARC 273A potentiostatconnected to a microcomputer with EG&G M270 softwaref timet ega-t lr era-t terf g aL y of±

3

car-b balt.S atives ak at− e tot uponr ross-i ep at− po-s alt,o etalo eousc yclicv d

an-s ten-t am-m ition

Fig. 1. Voltammogram recorded at a vitreous carbon electrode in 3.5 mMCoCl2 + 1 M aqueous NH4Cl (pH 4.5) at 100 mV s−1. Potential scan startedat 400 mV in the negative direction.

(Ep >−1.220 V), the curves fit closely to the behaviourdescribed by models considering multiple nucleation anddiffusion-controlled growth of three dimensional cobalt crys-tallites[5]. However, these transients change shape as the steppotentials are made more negative.

Fig. 3 shows a diagnostic test frequently applied for thecharacterisation of current–time transients involving 3D nu-cleation processes with diffusion-controlled growth. Experi-mental transients are compared with theoretical expressions[13] in normalised coordinates with respect to the current den-sity maximum,jm, and the corresponding time of this maxi-mum,tm, with instantaneous and progressive nucleation (seebelow) as limiting cases. Good agreement of experimentaldata with the progressive nucleation limiting case was ob-tained for potentials down to−1.220 V. However, at increas-ingly negative potential values, the experimental data failed

F g thee lutionc lsi .

or experimental control and data acquisition. Current–ransient measurements were performed with single nive potential steps within the−1.100 to−1.300 V potentiaange. All experiments were done at a controlled tempure of 25± 0.01◦C by the circulation of thermostated warom a water bath through the jacket of the cell, by usinauda Heating Circulator Model C6-CP with an accurac0.01◦C.

. Results

Fig. 1shows a voltammogram recorded at a vitreouson electrode during the deposition and dissolution of comall currents close to zero were detected in the negweep at potentials positive to the cobalt reduction pe1220 mV, followed by a further increase in the current du

he reduction of protons. The current remains cathodiceversion of the sweep towards negative potentials, cng over the current recorded during the negative swe

700 mV. This current loop is due to the fact that the deition of metals onto its own, in this case cobalt on cobccurs at lower overpotentials that the deposition of mnto different nature substrates, in this case cobalt on vitrarbon. This feature has been frequently observed in coltammograms when nucleation processes are involve[3].

Fig. 2shows a family of potentiostatic current–time trients following negative potential steps. For step poials more positive than those observed for the voltetric cathodic peak corresponding to cobalt depos

ig. 2. Family of potentiostatic current–time transients obtained durinlectrodeposition of cobalt onto vitreous carbon from an aqueous soontaining 3.5 mM CoCl2 + 1 M aqueous NH4Cl (pH 4.5) at the potentia

ndicated. The initial potential prior to each potential step was 400 mV

4738 M. Palomar-Pardav´e et al. / Electrochimica Acta 50 (2005) 4736–4745

Fig. 3. Reduced-variable plots of the experimental current–time transientsshown in Fig. 2. Comparison of transients obtained at−1.275 V (�),−1.225 V (�), and−1.175 V (�) with theoretical dimensionless plots forinstantaneous (– – –) and progressive (—) nucleation (see Scharifker andHills [13]).

to fit the 3D progressive nucleation model, with particularlylarge deviations seen whent/tm > 1.

Fig. 4 presentsj versust−1/2 plots for two experimen-tal transients shown inFig. 2, one at a low overpotential(E=−1.150 V) and expected to be free from the influenceof PR, and the other recorded atE=−1.275 V, where the in-fluence of PR is evident. It can be observed that whent> tmthe transient free from PR satisfies the Cottrell equation (bro-ken line inFig. 4), whereas the transient deviates from such

F ft

behaviour in the presence of the concurrent PR. This is dueto the reduction of protons at the more negative potential,which in this case is a process occurring simultaneously withCo deposition. The PR reaction supplies a current densitybeyond that provided by the flux of Co2+ ions. The diffu-sion coefficient of Co2+ in solution,D= 1× 10−5 cm2 s−1,was found from the slope of thej versust−1/2 plot of thecurrent–time transient recorded atE=−1.150 V. Detailedanalysis of transients (from analysis of experimental currentdensity maxima according to Scharifker and Hills[13]) ob-tained during cobalt electrodeposition at low overpotentials,apparently free from the influence of the PR, are given inPalomar-Pardave et al.[12]. However it is clear, as shownabove, that analysis of current transients obtained at higheroverpotentials (E<−1.175 V) requires that the PR reactionbe taken into account in addition to the nucleation process. Itis important to stress that under our experimental conditions,convection problems due to the formation of H2 bubbles arenegligible because no H2 bubbles were noticed to form dur-ing the recording of the experimental current–time transientsshown inFig. 2. Water electrolysis that could be provokedduring these experiments cannot produce enough hydrogenmolecules to saturate the solution around the electrode sur-face.

We then consider that proton reduction occurs simultane-ously with the diffusion-limited 3D growth of cobalt centers,a

H

T tion[ :

j

w ett andS c-t

S

w ulkomA n ont tively.T obaltr

ig. 4. Cottrell analysis of current transients fromFig. 2. j–t−1/2 curves o
ransients obtained at−1.275 V (�) and−1.150 V (©). j
s indicated by the following reaction:

+interface+ e− → Hadsorbed (1)

his reaction is the first step of the proton reduction reac14] and the current densityjPR associated to it is given by

PR(t) = P1S(t) (2)

ith P1 =zPRFkPR, where zPRF is the molar chargransferred during the proton reduction process,kPRhe rate constant of the proton reduction reaction(t) = (2c0M/πρ)1/2θ(t) the fractional surface area of elerodeposited cobalt,

(t) =(

2c0M

πρ

)1/2

×{

1 − exp

{−P2

[t − 1 − exp(−P3t)

P3

]}}(3)

herec0 is the concentration of the metal ion in the bf the solution,ρ the density of the deposit,M its molarass,P2 =N0πkD, k= (8πc0/ρ)1/2 andP3 =A, with N0 andbeing the number density of active sites for nucleatio

he electrode surface and the rate of nucleation, respeche current associated with the contribution due to the ceduction process (j3D-dc), on the other hand, is given by[10]:

3D-dc(t) = P4t−1/2θ (t) (4)


Fig. 5. Comparison between experimental transient (©) obtained duringdeposition of cobalt atE=−1.275 V and the theoretical current transient(—) after non-linear fitting of Eq.(5) to the experimental data. The val-ues of the parameters obtained from the best-fit wereP∗

1 = 5.9 mA cm−2,P2 = 73.53 s−1, P3 = 0.28 s−1, and P4 = 1.25 A cm−2 s1/2. The individualcontributions to the overall current due to the nucleation process (j3D-dc)and to hydrogen reduction (jPR) are also shown.

where

P4 = 2FD1/2c0

π1/2

and

θ(t) ={

1 − exp

{−P2

[t − 1 − exp(−P3t)

P3

]}}

The current due to the overall process (jtotal) is the sum ofboth contributionsjPR+ j3D-dc and is hence given by

jtotal(t) = (P∗1 + P4t

−1/2)

×(

1 − exp

{−P2

[t − 1 − exp(−P3t)

P3

]})

(5)

where

P∗1 = P1

(2c0M

πρ

)1/2

Fig. 5 shows a comparison of an experimental transient ob-tained atE=−1.275 V during electrodeposition of cobaltonto vitreous carbon and the current–time transient gener-

ated by non-linear fitting (using the Marquardt–Levenbergalgorithm) of Eq.(5) to the experimental data. During the it-erative fitting process, the parametersP∗

1 ,P2,P3 andP4 wereallowed to vary freely.

From the value ofP4 obtained from the best-fit of Eq.(5)to the data, an average value for the diffusion coefficient ofCo2+ of 1.31× 10−5 cm2 s−1 was found, seeTable 1, whichis in good agreement with that determined from the Cottrellequation for transients free from effects due to PR (proton re-duction), and also with values found in the literature for ionsof this kind. Thus,D can also be estimated even when the PRreaction dominates, from deconvolution of the contributionsto the total current due to the electrocrystallization and redoxprocesses on the surface of the growing crystallites, usingEq.(5). FromP∗

1 , the value for the proton reduction rate con-stant found at−1.275 V was 6.1× 10−8 mol cm−2 s−1. Cor-reia and Machado[15] studied the proton reduction processon nickel surfaces electrodeposited onto Pt using steady-statepolarization measurements and found values forkPR withinthe range (0.11–1.9)× 10−8 mol cm−2 s−1, while Hitz andLasia[16], using the galvanostatic step technique, found val-ues forkPR around (2.88–4.05)× 10−8 mol cm−2 s−1 duringPR on polycrystalline nickel in 1 M NaOH. From the fittingparameters, information regarding the electrodeposition pro-cess may be obtained too, such asD,N0 orA, correspondingto the physical characteristics of the present system (for ex-a

re-l sur-fc . Thefia thisp eterss oft get n.T

n

w r-at r

Table 1Best-fit parameters and kinetic data resulting from analysis of the experimen

− m−2 s1/

1111111

E (mV) P∗1 (mA cm−2) P2 (s−1) P3 (s−1) P4 (mA c

100 0.16 3.54 0.02 1.35125 0.79 12.13 0.03 1.45150 1.18 19.34 0.06 1.42175 2.16 39.39 0.12 1.38225 3.37 73.10 0.13 1.41275 5.89 73.53 0.28 1.25300 7.05 125.58 0.59 1.48

mple seeTable 1).FromTable 1it becomes clear that the rate constants

ated to cobalt nucleation and to proton reduction on theace of these nuclei,A andkPR, respectively, as well,N0, in-rease as the applied potential becomes more negativegures corresponding to the potential dependence ofA, N0nd lnkPR can be found as supplementary material toaper. From the potential dependence of some paramhown inTable 1it is possible to estimate both the sizehe critical nucleus (nc) [17–19]and the value of the charransfer coefficient,αPR, of the hydrogen adsorption reactiohe former can be estimated in accordance with Eq.(6)

c =(

kT

ze0

) (d lnA

dE

)− αCo (6)

herek is the Boltzmann constant,T the absolute tempeture,e0 the elementary charge of the electron, andαCo

he transfer coefficient for cobalt reduction[18]. The latte

tal current–time transient inFig. 2according to the model described by Eq.(5)2) kPR (×108)

(mol cm−2 s−1)A (s−1) N0 (×106)

(cm−2)D (×105)(cm2 s−1)

0.16 0.02 0.02 1.250.82 0.03 0.06 1.451.22 0.06 0.10 1.352.24 0.12 0.20 1.203.49 0.13 0.36 1.366.10 0.30 0.36 1.087.31 0.59 0.66 1.51


can be assessed, assuming thatkPR can be modelled by aButler–Volmer type relationship[20], by using Eq.(7)

kPR = k0PRexp

[−αPRzFE

RT

](7)

As can be noticed from the slope of the lnkPR versusE plot,it is possible to estimate the value ofαPR. In our case wefoundαPR= 0.32, considering thatz= 1 andT= 298 K. It isimportant to mention that a similar value (0.34) was foundby Rodrigues et al.[21] for hydrogen adsorption on a Nibase alloy from KOH solutions with different concentrations,using both Tafel plots and impedance measurements. More-over, Elumalai et al.[22] found the following values ofαPR,0.42 (Co), 0.34 (Pd), 0.30 (Ni) 0.34 (Co0.8Ni0.2) and 0.31(Co0.4Pd0.6) from ac impedance measurements, using differ-ent kinds of electrodes consisting of either pure metals orpowdered alloys or mixtures, as indicated in brackets, all im-mersed in 6 M KOH aqueous solution. Losiewicz et al.[23]also found a similar value (0.35) forαPR using a compos-ite Ni P + TiO2 + Ti electrode, immersed in 5 M KOH, fromimpedance experiments.

On the other hand, the value that we found for the sizeof the critical nucleus,nc, estimated from the slope of thelnA versusEplot and Eq.(6) was zero. This means that eachcobalt atom adsorbed on an active site is a stable “cluster”w tial.So lec-t enceo trodef

N

w ale yA si-t 0 0)s

ur-r hep timet y freef

4

ionsi clud-i fromt edb witht eim two-d rly

forming three-dimensional (3D) deposits[19]. Phase growthmay involve either the direct incorporation of ions reducedat the deposit–solution interface or the formation of adatomsdiffusing along the surface to lattice incorporation sites. Un-der various conditions or at different stages, the rate of theelectrocrystallization process may be limited by mass trans-fer or by the incorporation of atoms to growth centers. Butin any case, the growth of the new phase involves changes ofthe surface area,S, of the deposit, i.e., of the area availablefor reduction of metal ions as well as other species that maybe reduced on the surface of the growing nuclei.Svaries be-cause of the time-dependent size of the growing nuclei, char-acterised by, e.g., their radiiri(t−u), wheret−udenotes theage at timet of nuclei born at timeu, and possibly becausethe number density of nuclei on the electrode surface,N, in-creases with time too. Moreover the amount of M depositedduring the reaction is related to the electric chargeq passed.Hence the temporal variation of the growing phase surfaceS(t) may be expressed in general as a compound functionS(t) =S{ri [q(t−u)],N(t)}, the current–time transient arisingfrom all electrochemical processes occurring at the growingsurface are clearly related toS(t).

4.1. Non-interacting growing centers

on-t starto

r

w theo

of andg facea

S

Fp as[

N

T ‘in-s e nu-c s ont ver

N

T on’,w argen

hich can grow irreversibly at a given electrode potenuch a value ofnc was found by Arbib et al.[24] in the casef rhodium electrodeposition onto a polycrystalline gold e

rode. Moreover, we also found that the potential dependf the active nucleation sites on the vitreous carbon elec

ollows the empirical equation

0 = mexp(−nE) (8)

ith m= 2.27 cm−2 andn= 1.22 V−1. The same empiricquation, with different values ofmandn, was observed brbib et al. [24] to be valid in the case of rhodium depo

ion onto both polycrystalline and single crystal gold (1urfaces.

Table 1also shows that upon deconvolution of the cent using Eq.(5), D remains essentially invariant with totential, as required. The presence of PR in current–

ransients that, based on their shape, were apparentlrom this parallel reaction is especially noteworthy.

. Discussion

The electrodeposition of metals from aqueous soluts a heterogeneous process involving various stages, inng the transport of solvated electrodepositing specieshe bulk of the solution to the electrode surface, followy their reduction at the electrode–solution interface

he formation of nuclei[17]. Further growth of the nuclay occur along the surface of the substrate, leading toimensional (2D) phases[18], or extend also perpendicula

The radius of isolated nuclei growing under diffusion crol on the electrode surface, a condition that holds at thef the nucleation process, is given by[5]:

0(t − u) = [2Dc0M/ρ]1/2(t − u)1/2 (9)

hereu is the time of birth of the nucleus, referred tonset of the electrode potential perturbation.

For instantaneous multiple nucleation,u is equal to zeror all nuclei formed. Assuming they are hemisphericalrow in isolation from all others, then each will have a surrea given by

i = 2πr2i (10)

or nuclei appearing at a constant rateA onN0 active siteser unit area of electrode surface,N(t) may be expressed

5]:

(t) = N0[1 − exp(−At)] (11)

his expression has two limiting cases. One is termedtantaneous nucleation’ and holds for an extremely largleation rate on a small number density of active sitehe surface. In this case,N0/A→ 0 and the equation aboeduces to

(t) = N0 (12)

he opposite condition is that of ‘progressive nucleatihich takes place at very low nucleation rates on a lumber density of active sites. In this case,N0/A→ ∞ and


(11)becomes

N(t) = N0At (13)

In the instantaneous nucleation limit, the temporal variationof the total surface areaS(t) is given by the product of theN0 nuclei by their corresponding areas given by(10). Thetime-dependent radii of nuclei is given by Eq.(9):

S(t) = 2π(2DcM/ρ)N0t (14)

Therefore, the surface area of the deposit increases linearlywith time. In the general case, nuclei appear at different timesu after the onset of the potential step and at a given timet they have different agest−u, then the surface area gen-erated by non-interacting progressively nucleated growinghemispheres is:

S(t) = 2π

(2Dc0M

ρ

) ∫ t

0(t − u)

dN

dudu (15)

where dN/du is obtained from the time derivative of Eq.(11),which substituted in Eq.(15)yields

S(t) = 2π

(2Dc0M

ρ

)AN0

∫ t

0(t − u) exp(−Au) du (16)

whereA andN0 were taken out of the integral by assum-ing that they are independent of time. The surface area ofn eng

S

Si

S

F oftl ted atl ationa

ntd itedb on-i

j

w

j

Fig. 6. Dimensionless variation of deposit area as a function ofAt for mul-tiple nucleation of non-interacting centers growing in 3D under diffusioncontrol. The broken lines represent the quadratic and linear behaviours atvery low and highAt values, respectively.

where

Φ = 1 − e−At

(At)1/2

∫ (At)1/2

0eλ

2dλ (21)

is the Dawson integral[25]. For large values ofAt, Φ tendsto 1 and Eq.(20) is simplified to

jex(t) = zFDc0(πD)1/2(8πMc0/ρ)1/2N0t1/2 (22)

which corresponds to instantaneous nucleation. For small val-ues ofAt, Φ tends to (2/3)At, and Eq.(20) reduces to

jex(t) = zFDc0(πD)1/2(8πMc0/ρ)1/2(2/3)AN0t3/2 (23)

which represents progressive nucleation. Eqs.(22) and (23)are asymptotic solutions of Eq.(19) and represent limitingcases.

As already stated in Section3, the hydrogen reductioncurrent on the surface of the electrodepositing nuclei is givenby:

jPR(t) = zPRFkPRS(t) (24)

wherezPRF is the charge transferred during the proton re-duction process,kPR the rate constant of the proton reductionreaction, andSthe surface area of the electrodeposited nuclei,i w-i beg h oft

j

t

j

on-interacting nuclei growing under diffusion control is thiven by

(t) = 2π

(2Dc0M

ρ

)N0

A[At + exp(−At) − 1] (17)

(t) may be expressed non-dimensionally asSnd(t) by divid-ng Eq.(17)by 2π[2Dc0M/ρ]N0/A,

nd(t) =(

S(t)

2π(2Dc0M/ρ)(N0/A)

)= At + exp(−At) − 1

(18)

ig. 6shows a plot ofSnd as a function of time. The areahe deposit increases with (At)2 for small values ofAt andinearly at higher values, as nucleation sites are exhausonger times and the limiting case of instantaneous nucles given by Eq.(14) is approached.

Sluyters-Rehbach et al.[9] have shown that the curreensity due to 3D nucleation and growth processes limy diffusion of the metal ions, in the general case of n

nteracting nuclei is given by

ex(t) = zFc0π(2D)3/2(Mc0

ρ

)1/2

×N0t1/2

[1 − exp(−At)

At1/2

∫ (At)1/2

0exp(λ2) dλ

]

(19)

hich may be expressed as

ex(t) = zFDc0(πD)1/2(8πMc0/ρ)1/2N0t1/2Φ (20)

ndicated by Eq.(17) for the case of non-interacting grong centers. The overall current–time transient will theniven by the sum of the contributions due to the growt

he deposit and proton reduction:

(t) = j3D(t) + jPR(t) (25)

hat is,

(t) = zPRFkPR2π(2Dc0M/ρ)N0/A[At + exp(−At) − 1]

+ zFDc0(πD)1/2(8πc0M/ρ)1/2N0t1/2Φ (26)


or

j(t) = P5

[(P6t

(1 + exp(−At)

At− 1

At

))]+ P6P7t

1/2Φ

(27)

with

P5 = 2πzPRFkPR(2Dc0M/ρ) (28)

P6 = N0 (29)

and

P7 = 2πzFDc0(2Dc0M/ρ)1/2 (30)

Fig. 7 shows the result of plotting Eq.(26). It is importantto note that even from the very early stages of the depositionprocess, the reduction of protons may alter the shape of cur-rent transients from that resulting from nucleation and growthprocesses alone.

4.2. Interacting growing centers

The three-dimensional growth of hemispheres from theelectrode surface plane towards the bulk solution involvesa ‘2.5’ dimensional problem that cannot be directly solvedwith the Avrami theorem[26] relating the area (or volume)o enceoi thet-i ontot h thatt sus-t ring

F f 3Dn , Eq.( ode-p l-c ositionak

electrodeposition. The overlap of these 2D diffusion zonescan be readily accounted for with the Avrami theorem,

θ(t) = 1 − exp{−θex(t)} (31)

whereθex(t) is the ‘extended’ coverage that would occur inthe absence of overlap. It then follows that the fractionθ(t) ofthe electrode surface covered by such zones is given by[10]

θ(t) = 1 − exp{−α[At − 1 + exp(−At)]}= 1 − exp{−αAtΘ} (32)

with α=πD(8πc0M/ρ)1/2N0/A. The functionΘ(At) related tothe ‘extended’ coverage is given by

Θ = 1 − (1 − e−At)/At (33)

As with non-interacting nuclei, two limiting cases may bedistinguished when interactions among nuclei are considered.If N0 is very small and/orA is very large, thenα→ 0, and therate of nucleation is limited by the rate of depletion of activesites. In this case, Eq.(32) reduces to

θ(t) = 1 − exp(−αAt) (34)

On the contrary, ifα→ ∞, then nucleation is controlled bythe extension of the exclusion zones, and Eq.(32)transformsinto

θ

T eous,E ev n.F hd

ineda me ofi

beenf sing

Fi

ccupied by overlapping entities to that taken in the absf overlap. To solve this problem, Scharifker and Hills[13]

ntroduced the so-called diffusion zones. These are hypocal circular entities around growing centers, projectedhe electrode surface and with time-dependent radii suche mass flowing to them equates with that required toain the growth of hemispherical 3D nuclei generated du

ig. 7. Temporal variation of the current due to multiple nucleation oon-interacting centers growing under diffusion control (broken line)20), and with parallel proton reduction on the surface of the electrosit (continuous line), according to Eq.(27). Both equations were caulated by using parameters with values corresponding to cobalt depnd the following data.A= 5 s−1,N0 = 107 cm−2,D= 5× 10−6 cm2 s−1, and

PR= 0.1× 10−8 mol cm−2 s−1.

(t) = 1 − exp[−(2/3)α(At)2] (35)

hese two limiting cases correspond to the instantanq. (34), and progressive, Eq.(35), nucleation limits. Thalue for parameterα is an indicator for their characterisatioig. 8shows graphs corresponding to Eq.(32)generated witifferent values ofα.

Complete coverage of the surface is more readily attasα increases, i.e., as the system passes from a regi

nstantaneous nucleation to progressive nucleation.From the above considerations, various routes have

ollowed to derive the potentiostatic current transient ari

ig. 8. θ–At curves generated with Eq.(32) with different values toα, asndicated in the figure.


from multiple nucleation with diffusion-controlled growth ofthree-dimensional centers relating the locally hemisphericalflux to the planar flux to the overall ensemble of growingcenters[5,10,11,27–29]. The current density given by thediffusive flow to an electrode of fractional areaθ is given by

j(t) = zFDc0θ(t)/δ(t) (36)

whereδ(t) is the width of the diffusion layer at timet. Ac-cording to Heerman and Tarallo[27], evaluation ofδ(t) toaccount for the fact that in the general case nuclei arise at dif-ferent times on the surface, may be carried out inserting thecurrent from Eq.(20) for non-interacting nuclei in Eq.(36),with θ(t) =α[At− 1 + exp(−At)] for non-overlapping diffu-sion zones. The result is:

zFDc0N0πkDtΦ/(πDt)1/2

= zFDc0N0πkD/A{[At − 1 + exp(−At)]/δ(t)} (37)

where

δ(t) = (πDt)1/2[At − 1 + exp(−At)]/[AtΦ]

= (πDt)1/2Θ/Φ (38)

From Eqs.(36) and (38)it follows that the current density forthe ensemble of interacting growing nuclei is given by

j

w

b

E tly[ tedw i,

q

a bleo e in af nare

j

q

w dif-f itht ,E ded

coverage and the thickness of the diffusion layer

〈δ(t)〉 = (πDt)1/2[1 − 3Φ/(2At)]1/2/Φ1/2 = (πDt)1/2Ψ

(43)

θex(t) = πD(8πMc0/ρ)1/2N0tΦ[1 − 3Φ/(2At)]1/2/Φ1/2

= btΦ[1 − 3Φ/(2At)]1/2/Φ1/2 = btΦΨ (44)

with

Ψ = [1 − 3Φ/(2At)]1/2/Φ1/2 (45)

Substituting〈δ(t)〉 andθ(t) = 1− exp[−θex(t)] from Eqs.(43)and (44)into Eq. (36) yields the current density within themean-field approximation[11,27,28].

j3D(t) = [zFDc0/(πDt)1/2]1/Ψ{1 − exp[−btΦΨ ]} (46)

Fig. 9shows the currents given by Eqs.(39) and (46)accord-ing to Heerman and Tarallo[11,27]and Heerman et al.[28],respectively.

4.2.1. Concurrent reactionsIf the surface of the electrodeposited nuclei turns out to

be more active for the reduction of protons or other speciesin solution than the surface of the substrate, then the currentd d byt

4 foc nz

r

F froma( aneNNt

3D(t) = [zFDc0/(πDt)1/2]Φ/Θ{1 − exp[−btΘ]} (39)

here

= (2π)3/2D(Mc0/ρ)1/2N0

valuation ofδ(t) andΘ may be carried out self-consisten28] by integration of Eq.(20)to obtain the charge associaith the growth of the ensemble of non-interacting nucle

ex(t) = (2/3)zFDc0(πD)1/2(8πMc0/ρ)1/2N0t3/2

× [1 − 3Φ/(2At)] (40)

nd then writing both the current density for the ensemf independent growing nuclei and its associated charg

orm that presents an analogy with a diffusion flux to a plalectrode:

ex(t) = zFDc0{[πD(8πMc0/ρ)1/2N0t]/(πDt)1/2}Φ= zFDc0{bt/(πDt)1/2}Φ = zFDc0θex(t)/〈δ(t)〉

(41)

ex(t) = (2/3)(zFc0/π)(πDt)1/2πD(8πMc0/ρ)1/2

×N0t[1 − 3Φ/(2At)]

= (2/3)(zFc0/π)〈δ(t)〉θex(t) (42)

here〈δ(t)〉 represents the mean-field thickness of theusion layer att. Solution of the set of two equations wwo unknowns given by the flux, Eq.(41), and the chargeq. (42), yields self-consistent expressions for the exten

ensityjPR(t) of the associated process may be describehe general expression (Eq.(24)).

.2.1.1. The relation betweenS(t) andθ(t). In the absence overlap between diffusion zones, the radius,r0, of a growingenter att born atu is given by(9). The radius of the diffusioone around it,rd, at the same timet is given by

d = 23/4π1/4D1/2(c0M/ρ)1/4(t − u)1/2 (47)

ig. 9. Current–time transients corresponding to cobalt deposition3.5 mM solution of Co2+ in 1 M NH4Cl according to Eqs.(39) and

46) from Heerman and Tarallo[11,27] (continuous lines) and Heermt al.[28] (symbols), withA= 1.0 s−1,N0 = 1.0× 106 cm−2 (♦); A= 0.8 s−1,

0 = 0.5× 106 cm−2 (×);A= 0.7 s−1,N0 = 0.2× 106 cm−2 (�);A= 0.6 s−1,

0 = 0.1× 106 cm−2 (�); A= 0.4 s−1, N0 = 0.05× 106 cm−2 (©). The Cot-rell response is also shown with a dotted line.


whereas the fractionS(t) of electrode surface covered by de-posit is given by

S(t) = (2c0M/πρ)1/2θ(t) (48)

In general this is an approximation because diffusion zonesoverlap much sooner that the overlap of nuclei sets in, thusdiminishing the mass flow feeding the hemispherical growthof the deposit. Thus, Eq.(48) is only strictly satisfied at shorttimes, but the approximation still holds good at longer timesin view of the fact thatθ(t) approaches unity well beforeS(t) and that in the transition from hemispherical to planardiffusion, the nuclei growth law changes fromr∼ t1/2 to t1/6.Taking this approximation as sufficiently good even whenoverlap is taking place at longer times and substituting Eq.(48) into Eq.(24), then

jPR(t) = zPRFkPR(2c0M/πρ)1/2θ(t) (49)

Thus, consistent with Eq.(49) the current density due tothe associated electrochemical process occurring on the elec-trodeposited surface will vary asθ(t). Fig. 10shows a plot ofjPR(t) with α= 1, zPR= 1 andkPR= 1× 10−7 mol cm−2 s−1

and the data related to cobalt deposition.

4.2.1.2. Simultaneous processes.The overall current den-sity, j(t), due to simultaneous occurrence of nucleation with3 em-i lei isg

j

Sf

j

F s tak-iz toc

Fig. 11. Current transients obtained with Eq.(51)(—) considering the phys-ical constants for Co, asα= 1, zPRFkPR= 146. The individual contributionsto the overall current due to the 3D diffusion-limited nucleation process(j3D-dc) and to the reduction of substances over these nuclei (jPR) are alsoshown.

It is important to mention that during derivation of Eq.(51), we took advantage of the fact that previously,

Sluyters-Rehbach et al.[9] have shown that2FD1/2c0

π1/2 t−1/2 =1/α(At)1/2 (see page 11 in Ref.[9]). The current transientas given by Eq.(51) is plotted inFig. 11. The dotted linesindicate the partial current densities due to the growth ofthe deposit and the associated electrochemical reaction on itssurface. The shapes of these transients are similar to thoseobtained in the absence of parallel reactions (seeFig. 9).However, the associated reduction process introduces signif-icant distortions on the shape of the overall transient, withthe effects being more obvious at longer times. An equationsimilar to(51)may be obtained substituting Eq.(39) insteadof Eq.(4) into (50), the resulting equation is:

j(t) = {P∗1 + (P4t

−1/2)Φ/Θ)}[1 − exp(−btΘ)] (52)

Eqs.(51) and (52)differ in the model used to describe the 3Dnucleation mechanism in either the Scharifker et al. model[10] or the Heerman and Tarallo expression[27]. Notwith-standing, for the case of cobalt deposition and proton re-duction discussed in this work, interpretation of the experi-mental data using either Eq.(51) or (52) yielded results thatwere practically indistinguishable (the figures correspondingto experimental current transient-fitting using Eq.(52)can befound as supplementary material to this paper).

5

le ofd elec-t ledg tiont pro-t ribes

D diffusion-controlled growth and associated electrochcal reaction on the surface of the electrodeposited nuciven by

(t) = j3D-dc(t) + jPR(t) (50)

ubstituting Eqs.(4), (32) and (49)into Eq.(50) we get theollowing equation:

(t) = [zPRFkPR(2c0M/πρ)1/2 + 1/α(At)1/2]

× (1 − exp{−α[At − 1 + exp(−At)]}) (51)

ig. 10. Current density associated with an electrochemical procesng place on the electrodeposited surface according to Eq.(49), with α= 1,

PR= 1 andkPR= 1× 10−7 mol cm−2 s−1. Surface growth correspondsobalt deposition, see text.

. Conclusions

A theoretical model has been proposed that is capabescribing potentiostatic current–time transients due to

rochemical phase formation involving diffusion-controlrowth of 3D nuclei, with a simultaneous reduction reac

aking place on the growing surface of the nuclei e.g.,on reduction. We have shown that such a model desc


adequately the experimentalj–t curves obtained during elec-trodeposition of cobalt onto vitreous carbon at high overpo-tentials, and that proton reduction occurs simultaneously withthe nucleation and growth of cobalt on the electrode. It hasbeen shown also that due to the PR, current–time transientsare distorted when compared to those arising from phase for-mation in the absence of PR, thus impeding determination ofkinetic parameters pertaining to the phase formation processwith current theory. The model presented here allows deter-mining the kinetic parameters describing the phase formationprocess (N0 andA) from experimental data and also yieldsthe heterogeneous rate constant of the PR reaction.

Acknowledgements

This work was supported by CONACyT-NSF (projects42437/A-1 and 47178-F), DEPI-I.P.N. and by UAM-A-DCBI(projects 2260220 and 2260225). EMA, MPP and MRR wishto thank SNI (Mexico), and BRS FVPI (Venezuela), for thestipends received. We also like to express our gratitude to theanonymous reviewers of this paper for their criticisms andsuggestions that contributed to improve our work.

Appendix A. Supplementary data

canb cta.2

R

8)

ental, p.

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chemistry, Aspect, London, 1990 (Chapter 7).[ . 23

[ ar, J.

[ J.

[ ss-

[ ns,

[[[ 001)

[

Supplementary data associated with this articlee found, in the online version, at 10.1016/j.electa005.03.004.

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nucleation and diffusion-controlled growth of electroactive centers

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