Journal of Electroanalytical Chemistry 453 (1998) 25–28

Non-steady-state electrochemical nucleation under potentiostaticconditions

V.A. Isaev

Institute of High Temperature Electrochemistry, Ural Di6ision, Russian Academy of Sciences, 20 S. Ko6ale6sky Str., 620219 Ekaterinburg, Russia

Received 28 July 1997; received in revised form 9 October 1997

Abstract

The kinetics of electrochemical nucleation at constant overpotential have been analysed. The influence of the concentrationchange of single adatoms (monomers) on the process of nucleation is considered. The Fokker–Planck kinetic equation was solvedaccounting for this effect. Expressions were derived for the number of nuclei, for the nucleation rate and for the nucleationinduction time. © 1998 Elsevier Science S.A. All rights reserved.

Keywords: Electrodeposition; Phase transition; Nucleation; Theory

1. Introduction

Under potentiostatic conditions electrochemical nu-cleation proceeds at constant supersaturation Dm=zeh, where z is the valency of the depositing ions, e isthe elementary electric charge and h is the overpoten-tial. The value of Dm determines the rate of nucleation.However, under these conditions it is possible that theconcentration of single adatoms (monomers) changeswith time. This process can essentially affect the nucle-ation kinetics.

The problem of the kinetics of nucleation at constantsupersaturation and at changing monomer concentra-tion has been considered previously [1–4].

The concentration of the monomers can change withtime for different reasons. For example, when a con-stant overpotential is applied to an electrochemicalsystem, the concentration of single adatoms(monomers) changes with time from an initial value toa stationary one of the given overpotential. In theprocess of electrodeposition, active surface electrodesites may appear at which accumulation of adatomsand nucleation takes place. During the electrodeposi-tion process the adatoms can also diffuse deep into theelectrode and participate in other heterogeneous elec-

trode processes. All the reasons mentioned have aninfluence on the change of the concentration of singleadatoms (monomers) with time.

In this work the Fokker–Planck equation describingthe nucleation process has been solved for an arbitrarytime variation of the monomer concentration at con-stant supersaturation. Further, the analytical expres-sions for the number of nuclei formed on the electrodesurface, for the non-steady state nucleation rate, andfor the induction time of nucleation, have been ob-tained accounting for the concentration change ofadatoms during the process of potentiostaticelectrodeposition.

2. Theory

The process of electrochemical nucleation is usuallymodelled as a series of bimolecular reactions betweendepositing ions and clusters of the new phase. Then thenucleation process can be described by the Fokker–Planck equation [5,6]:

(Z(t

= −(a(g

(1)

0022-0728/98/$19.00 © 1998 Elsevier Science S.A. All rights reserved.PII S0022-0728(98)00010-2

V.A. Isae6 / Journal of Electroanalytical Chemistry 453 (1998) 25–2826

a=AZ−(

(g(BZ) (2)

where Z(g,t) is the concentration of clusters consistingof g atoms, a(g,t) is the flow of clusters. Coefficients Aand B are given by:

A=n1−n2 (3)

B= (n1+n2)/2 (4)

where n1(g,t) and n2(g,t) are the frequencies at whichions join or leave a cluster of size g, respectively. Theflow a can be written in the form:

a= −BZe

(

(g�Z

Ze

�(5)

where Ze(g,t) is the equilibrium cluster size distributionat which a=0.

The initial and boundary conditions for our problemare:

Z(g,0)=0, g\1 (6)

Z(1,t)=x(t)Z(1) (7)

Z(m,t)=0, m�gk (8)

where Z(1,t) is the concentration of single adatoms(monomers), Z(1) is the steady-state concentration ofmonomers at a given supersaturation, gk is the size ofthe critical cluster. The nucleation rate is J(t)�a(gk,t).Inserting Ze(g,t), Eq. (7) must be added by conditionZ(1,t)=Ze(1,t). The equilibrium cluster size distribu-tion at constant supersaturation and changingmonomer concentration is given by the relation:

Ze(g,t)=Z(1,t) exp[−G(g)/kT ] (9)

where G(g) is the reversible work required for theformation of a cluster of size g, k is Boltzmann’sconstant and T is the temperature.

At constant supersaturation the functions n1, n2, Aand B depend only upon the cluster size and areindependent of time. The critical size of clusters is alsoindependent of time.

Let us suppose that Z1(g,t) is the cluster size distribu-tion at x=1, i.e. at the constant monomer concentra-tion. Next we apply a Laplace transform to Eq. (1), Eq.(2) or Eq. (5). We find that the Laplace transforms of Zand Z1 are related by Z0 (p)=px̃(p)Z0 1(p). Hence:

Z(g,t)=ddt

& t

0

Z1(g,t %)x(t− t %) dt (10)

For the nucleation rate we obtain:

J(t)=ddt

& t

0

J1(t %)x(t− t %) dt % (11)

where J1(t) is the non-steady-state nucleation rate atx=1, i.e. at a constant monomer concentration Z(1).

Since J=dN/dt, where N(t) is the number of nucleiforming, so Eq. (11) is followed by the generalexpression:

N=& t

0

J1(t %)x(t− t %) dt % (12)

Eq. (10) and Eq. (11) were found in other forms inRef. [1] and Eq. (12) was obtained in Ref. [4]. Theseequations are valid for any type of nucleation at con-stant supersaturation and for changing monomer con-centrations with time.

A number of approximate expressions for the func-tion J1(t) exist [7]. We use the approximate solution[8,9]:

J1(t)=Jst�

1+2 %�

n=1

(−1)n exp(−n2t/t)n

(13)

where Jst is the steady-state nucleation rate at monomerconcentration equal to Z(1), t is the nucleation induc-tion time, which is needed to establish a steady-statesize distribution of subcritical clusters.

The steady-state nucleation rate is equal to [10,11]:

Jst=ZnB(gk)Ze(gk) (14)

where Zn is the non-equilibrium factor (Zeldovichfactor):

Zn= [g/2pkT ]1/2 (15)

g= − [(2G/(g2]g=g k(16)

According to Ref. [8] the quantity t is given by:

t=2kT/pgB(gk) (17)

Thus for the calculation of t it is necessary to determinethe values of g and B(gk). For the complete solution ofour problem it is also necessary to take into account theconcentration changes of monomer particles x(t).

In the case of electrochemical nucleation the re-versible work to form a cluster of size g is given by:

G(g)=bg2/3−zehg (18)

where b is a constant depending upon the geometricalform of the cluster. For the hemispherical cluster b=(18pn2)1/3s, where n is the volume of one atom of thenew phase and s is the surface tension of the elec-trolyte�cluster interface. In this case the radius of thecritical cluster is equal to:

rk=2sn/zeh (19)

From Eq. (15), Eq. (16) and Eq. (18) it follows that:

g=zeh/3gk (20)

Zn= [fh/6pgk]1/2 (21)

where f=ze/kT.Let us write the coefficients A and B in the Fokker–

Planck equation for electrochemical nucleation. It fol-

V.A. Isae6 / Journal of Electroanalytical Chemistry 453 (1998) 25–28 27

lows from Eq. (3) that the coefficient A can be writtenas:

A= j(g)s(g)/ze (22)

where j(g) is the average current density at the elec-trolyte�cluster interface, s(g) is the area of the elec-trolyte�cluster interface. The averaging j(g) is fulfilledamong all the clusters of size g. Outside the criticalregion this quantity can be applied to a single nucleusand then A=dg/dt.

From the conditions of equality of frequencies n1 andn2, at the overpotential equal to 2sn/zer, the coefficientA can be expressed in the form [12]:

A=j0s(g)

ze{exp[af(h−hp)]−exp[(1−a)f(hp−h)]}

(23)

where J0 is the exchange current density at the elec-trolyte�cluster interface, a is the transfer coefficient, andhp=2sn/zer.

For gBgk, AB0 and for g\gk, A\0.The ‘diffusion coefficient’ in space of sizes B can

accordingly be written as:

B=j0s(g)2ze

{exp[af(h−hp)]+exp[(1−a)f(hp−h ]}

(24)

For critical size we obtain:

B(gk)= j0s(gk)/ze (25)

The analogous frequencies of joining and leaving ofions to/from clusters were used in Refs. [13,14]. It isusually supposed that the values J0 and a are indepen-dent of cluster size.

The substitution of Eq. (20) and Eq. (25) into Eq.(17) gives the following expression for the inductiontime t determined by redistribution of nuclei by sizes:

t=4s/pfj0h2 (26)

By analogy with Ref. [3] one can find the concentrationchange of adatoms (monomers) during the process ofpotentiostatic electrodeposition. The following equationcan be written for monomer concentration Z(1,t):

dZ(1,t)dt

=j01

ze!

exp[afh ]−Z(1,t)Z(1,0)

exp[− (1−a)fh ]"

(27)

where j01 is the exchange current density at the elec-trolyte�electrode interface. The concentration ofadatoms (monomers) is changed from the initial Z(1,0)to the stationary Z(1) Z(1,�)=Z(1,0) exp( fh).

The solution of Eq. (27) gives:

x(t)=1−d exp[− t/t1] (28)

t1= (zeZ(1,0))/j01 exp[(1−a)fh ] (29)

d=1−exp(− fh) (30)

where t1, is the induction time determined by theaccumulation of adatoms on the electrode surface.

Substitution of Eq. (13) and Eq. (28) into Eq. (12)gives the following expression for the number of nucleiforming on the electrode:

N(t)

=Jst�

t−p2

6t−dt1+

dp(tt1)1/2

sin p(t/t1)1/2 exp(− t/t1)

−2t %�

n=1

(−1)n un2−t/t1

n2(n2−t/t1)exp(−n2t/t)

n(31)

where m=1−d.After differentiation the nucleation rate is given by:

J(t)

=Jst�

1−dp(t/t1)1/2

sin p(t/t1)1/2 exp(− t/t1)

+2 %�

n=1

(−1)n un2−t/t1

n2−t/t1

exp(−n2t/t)n

(32)

The analogous equations for d=1 were obtained ear-lier [1] by means of a more complicated method with-out use of the general expression (Eq. (12)).

At long times Eq. (31) transforms to a linear one:

N(t)=Jst�

t−p2

6t−dt1

n(33)

This linear part gives the steady-state nucleation rate Jst

and the total induction time:

t0=p2

6t+dt1 (34)

As one can see as t1�0 Eq. (31) and Eq. (32) trans-form to the expressions for J1(t) and N(t) in case ofnucleation at constant supersaturation and constantmonomer concentration [1]. Applying Eq. (26), Eq.(29), Eq. (30) and Eq. (31) it is possible to calculate thenumber of nuclei and the nucleation induction times.For typical parameters of the electrocrystallization: h=0.05 V, s=10−5 J cm−2, j0= j01=1 A cm−2, n=2×10−23 cm3, Z(1,0)=1013 cm−2, a=0.5 and T=300 Kwe obtain d=0.86, t=1.32×10−4 s and t1=4.20×10−6 s. The N(t) relationship obtained from Eq. (31)for d=0.86 and various t1/t values is shown in Fig. 1.

3. Conclusions

The induction time t determined by redistribution ofnuclei by sizes and the induction time t1 determined byaccumulation of adatoms have different dependences

V.A. Isae6 / Journal of Electroanalytical Chemistry 453 (1998) 25–2828

Fig. 1. Belationship between number of nuclei and time according toEq. (31) for d=0.86 and various t1/t values: (1) 0, (2) 2, (3) 5 and (4)10.

ance of new active sites at which adatoms were accumu-lated and the nucleation process occurs [4].

But this method cannot be applied if the work re-quired for the formation of a cluster changes with time.

The model presented here is valid for the early stageof electrochemical phase formation, i.e. for a smallcoverage of the electrode by a new phase; at largecoverage of the electrode by the new phase it is neces-sary to take into account the interactions of neighbour-ing nuclei [16,17].

References

[1] D. Kashchiev, Cryst. Res. Technol. 20 (1985) 723.[2] V.I. Roldugin, A.I. Danilov, Yu.M. Polukarov, Elektrokhimiya

21 (1985) 661.[3] A.I. Danilov, Yu.M. Polukarov, Usp. Khim. 56 (1987) 1082.[4] V.A. Isaev, A.V. Volegov, A.N. Baraboshkin, Rasplavi 3 (1989)

114.[5] N.N. Tunizki, V.A. Kaminski, C.V. Timachev, Metodi fisico-

khimicheskoi kinetiki, Khimiya, Moscow, 1972.[6] B.Ya. Lubov, Teoriya cristallizatsii v bolshikh objemakh,

Nauka, Moscow, 1975.[7] J.W. Christian, The Theory of Transformations in Metals and

Alloys, 2nd ed., Pergamon, Oxford, 1975.[8] F.C. Collins, Z. Elektrochem. 59 (1955) 404.[9] D. Kashchiev, Surf. Sci. 14 (1969) 209.

[10] Ya.I. Frenkel, The Kinetic Theory of Liquids, Oxford Univer-sity, Oxford, 1946.

[11] Ya.B. Zeldovich, Acta Phisicochim. USSR 18 (1943) 1.[12] V.A. Isaev, V.N. Chebotin, Trudi Inst. Electrokhim. Uralsk

Filial Akad. Nauk USSR 27 (1978) 46.[13] G.J. Hills, D.J. Schiffrin, J. Thompson, Electrochim. Acta 19

(1974) 671.[14] A.I. Danilov, Yu.M. Polukarov, Electrokhimiya 17 (1981) 1883.[15] A. Milchev, Electrochim. Acta 30 (1985) 125.[16] V.A. Isaev, S.Ya. Bayankin, A.N. Baraboshkin, Elektrokhimiya

23 (1987) 1645.[17] V.A. Isaev, A.N. Baraboshkin, J. Electroanal. Chem. 377 (1994)

33.

upon the overpotential: t decreases when the overpo-tential increases; on the contrary t1 increases when theoverpotential increases.

The relationship between induction times t and t1,essentially depends upon the relationship between ex-change current density at the electrolyte�cluster inter-face ( j0) and the exchange current density at theelectrolyte�electrode interface ( j01). It is possible forj01B j0 that the total nucleation induction time t0 willbe defined by t1, i.e. by the delay of the process ofadatom accumulation.

The value of the nucleation induction time can essen-tially change if the properties of the electrode surfacechange with time, for example, new active sites appearduring the electrodeposition [15]. This kinetic methodcan be applied also in this case, but it is necessary todefine the function x(t) taking into account the appear-

.