Two-dimensional progressive and instantaneous nucleation with overlap: The case of multi-step electrochemical reactions

Electrochimica Acta 56 (2011) 2399–2403

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Two-dimensional progressive and instantaneous nucleation with overlap: The case of multi-step electrochemical reactions Alexander Milchev ∗ , Ivan Krastev Rostislaw Kaischew Institute of Physical Chemistry, Bulgarian Academy of Sciences, Acad. G.Bonchev Str. Bl. 11, 1113 Sofia, Bulgaria

a r t i c l e

i n f o

Article history: Received 26 July 2010 Received in revised form 9 November 2010 Accepted 9 November 2010 Available online 18 November 2010

a b s t r a c t Two-dimensional nucleation and growth phenomena are examined in case of multi-step electrochemical reactions accounting for the cluster overlap. Theoretical expressions are derived for the current of progressive and instantaneous nucleation at a multinuclear-monolayer, direct attachment mechanism of growth at a constant overpotential. © 2010 Elsevier Ltd. All rights reserved.

Dedicated to the memory of our teachers, Professor Rostislaw Kaischew and Professor Evgeni Budevski, who contributed significantly to the theoretical and experimental studies of the two- and three-dimensional electrochemical nucleation and growth phenomena. Keywords: Two-dimensional nucleation Progressive Instantaneous Multi-step reactions

1. Introduction The process of two-dimensional nucleation was first considered by Gibbs [1] and Volmer and Weber [2] and Volmer [3]. Later, significant results on this subject were reported also by Brandes [4], Erdey-Gruz and Volmer [5], Kaischew and Stranski [6], Kaischew [7,8], Kaischew et al. [9] and Budevski et al. [10–14] (see also the books of Volmer [15], Thirsk and Harrison [16], Budevski et al. [17], Milchev [18] and the references cited therein). The authors of the above mentioned articles, who have examined the twodimensional electrochemical nucleation and growth phenomena considered one-step ions’ discharge. It is our purpose to comment upon the more general case of multi-step reactions, thus supplementing the existing theory of electrochemical two-dimensional progressive and instantaneous nucleation. Parsons [19] was among the first who made significant contribution to the theory of multi-step electrochemical reactions. Later his results were commented and included in the books of Bockris and

∗ Corresponding author. Tel.: +359 2 979 2557/2558/8727550; fax: +359 2 971 26 88. E-mail address: [email protected] (A. Milchev). 0013-4686/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.electacta.2010.11.025

Reddy [20], Damaskin and Petrii [21] and Damaskin et al. [22] (see also the recent review article of Fletcher [23] and the references cited therein). Recently we examined the nucleation and crystal growth phenomena in case of multi-step electrochemical reactions [24–30], deriving theoretical expressions both for the linear size and the growth current of single two- and three-dimensional (2D and 3D) clusters and for the currents of progressive and instantaneous nucleation. The more general case of growth under combined charge transfer and diffusion limitations was considered for single 3D clusters and experimental data were obtained for the exchange current density. For, neither the cluster overlap nor the overlap of nucleation exclusion zones was taken into account in [24–30] the present article extends the theoretical considerations of two dimensional nucleation accounting for the 2D clusters overlap at a multinuclear-monolayer, direct attachment mechanism of growth. 2. Theory If a single circular or polygonized 2D cluster grows on a flat surface at a constant overpotential  and ions’ transfer control, the theoretical formulae for the time dependence of the cluster’s edge length l2D (t), the growth rate 2D = dl2D (t)/dt, the growth cur-

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A. Milchev, I. Krastev / Electrochimica Acta 56 (2011) 2399–2403

Nomenclature l2D h

2D cluster edge length (cm) 2D cluster’s height equal to the atomic diameter (cm) non-dimensional numerical constants depending on the 2D cluster geometry growth current of a single 2D cluster (A) current density of stationary 2D progressive nucleation without overlap (A cm−2 ) stationary nucleation rate (cm−2 s−1 ) non-dimensional quantity accounting for the overpotential dependence of the growth current in case of single- and multi-step electrochemical reactions current density of progressive 2D nucleation with overlap (A cm−2 ) current density of instantaneous 2D nucleation with overlap (A cm−2 ) number of instantaneously formed 2D nuclei (cm−2 )

b, b I1,2D j2D Ist,2D Q2D

j2D,p j2D,ins N0

rent I1,2D (t) and the current density j2D (t) of stationary progressive nucleation without cluster overlap read [27,29]: j0,2D b VM Q2D t 2bzF

l2D (t) = 2D =

(1)

j0,2D b VM Q2D 2bzF

I1,2D (t) =

(2)

2 2 b hVM j0,2D

2bzF

2 Q2D t

(3)

actual surface fraction covered by growing two-dimensional crystals at t = u. According to the Kolmogorov–Avrami theory [31–34], the latter quantity is given by:  = 1 − exp(−ext ) where

2 b hVM j0,2D

4bzF

 ˛ zF  1 R T

 z

f

 (1 − ˛ )zF  1

− exp −

+ ˛2

v

R T

 F  R T

 z − z f

− exp −

v

− ˛2

 F  R T

(6)

Suppose now that 2D circular, disk-shaped clusters (b = , b = 2) appear progressively on a plain surface and may overlap during the growth. In this case, the current j2D,p (t) of progressive two-dimensional nucleation is given by the general formula (see e.g. [27] and the references cited therein): j2D,p (t) = Ist,2D



(8 )

du

0

is the fraction of the total surface area, which would be covered by the 2D clusters if none overlap. Substituting Eqs. (3), (8) and (8 ) into Eq. (7) and solving the integrals, for the current j2D,p (t) of progressive two-dimensional nucleation accounting for the cluster overlap one obtains: j2D,p (t) =



  2 2 2 2 2 2 3 Ist,2D Q2D t exp − VM j0,2D Ist,2D Q2D t hVM j0,2D zF 3(zF)2

 (9)

The current–time relationship given by Eq. (9) displays a maximum at time t = tp,m ,

tp,m =

2/3

21/2 zF

(10)

1/2

1/2 VM j0,2D Ist,2D Q2D

and the value of the maximal current j2D,p (tmax ) is given by

 j2D,p,m =

2 2 4zFh3 j0,2D Ist,2D Q2D

1/3

VM

j2D,p,m tp,m =



2zFh 2 exp − VM 3

 2

exp −

(11)

3



ln

j2D,p (t) t2



= ln −

with

a

slope

 zF

2 2 Ist,2D Q2D hVM j0,2D

 3(zF)2

[1 − (u)]I1,2D (t − u) du

(12)



(9 )

2 2 2 3 VM j0,2D Ist,2D Q2D t

2 j2 I Q2 [/3(zF)2 ]VM 0,2D st,2D 2D

and

an

intercept

2 ln[(/(zF)hVM j0,2D Ist,2D Q2D ], which allows a straightforward interpretation of experimental results. Fig. 1 (circles) shows a theoretical current transient calculated by means of Eq. (9) using the following values of the constants involved1 : z = 2, zf = 0,  = 1, VM = 7.1 cm3 /mol, h = 2.4 × 10−8 cm, ˛2 = 0.5,  = 0.030 V, j0,2D = 0.2 A cm−2 , Ist,2D = 1 × 105 cm−2 s−1 ,  T = 298 K and Q2D = Q2D,2 = 1.62, which correspond to a two-step electrochemical reaction of type (a) (Table 1). In order to evaluate the significance of the multi-step electrochemical reactions’ effect let us now assume that the circles in Fig. 1 represent an experimental current transient and fit it making use of a theoretical equation containing two free parameters, P1 and P2 :

j2D,p (t) = P1 t 2 exp(−P2 t 3 )

t



contains only material constants characterizing the deposited species and does not depend on the nucleation and growth kinetics. It is readily seen that Eq. (9) can also be presented as a linear relationship, ln[j2D,p (t)/t2 ] vs. t3 ,

(5)

2.1. Progressive nucleation





2D (u ) du 0

(4)

In the more complex case when z electrons are exchanged in z one-electron successive steps, zf of which are fast and precede the rate determining one, that is repeated  times and is followed by z − zf −  fast steps the quantity Q2D in Eqs. (1)–(4) reads [27]: Q2D,2 = exp

2

t

Note that the product j2D,p,m tp,m , 2 2 Ist,2D Q2D t

In Eqs. (1)–(4) j0,2D is the exchange current density referred to unit edge area, VM is the molar volume of the depositing metal, Ist,2D is the stationary nucleation rate, h is the cluster’s height, which equals the atomic diameter and b and b are constants depending on the√cluster geometry. Thus b = 1, b = 4 for a quadratic cluster, b = (3/2) 3, b = 6 for a hexagon, b = , b = 2 for a disk with a radius l(t), etc. The quantity Q2D in Eqs. (1)–(4) determines the overpotential dependence of the growth process and for a z-step, one-electron reaction is given by [27]: Q2D,1 = exp

 t 

ext = Ist,2D

2

j2D (t) =

(8)

(13)

(7)

0

where I1,2D (t − u) is the growth current, at the time moment t, of a single 2D cluster formed at the time moment u < t and (u) is the

1 The values of z, zf , , VM and h correspond to copper electrodeposition. The values of ˛2 , , j0,2D and Ist,2D are arbitrarily chosen.

A. Milchev, I. Krastev / Electrochimica Acta 56 (2011) 2399–2403

2401

-16 2,0 -18

5

1,2 0,8 0,4

0

10

20

30

40

t/s

50

60

70

80

1

2

3 -5 3

3.99 × 103 A2 cm−6 s−1 respectively, i.e. numbers, practically equal to 4 × 103 A2 cm−6 s−1 , as expected. If, however, the same val2 ues of P1 and P2 are used to evaluate the product j0,2D Ist,2D but assuming that the “experimental” current transient in Fig. 1 corresponds to a two-step electrochemical reaction of type (b),  2 = 5.22 (Table 1) one obtains j0,2D Ist,2D = 3.83 × i.e. Q2D = Q2D,2 2 102 A2 cm−6 s−1 and j0,2D Ist,2D = 3.85 × 102 A2 cm−6 s−1 , respectively, i.e. values approximately one order of magnitude lower than the real one. Finally, assuming that the “experimental” current transient corresponds to a single-step, two-electron reaction of type (c) 2 i.e. Q2D = Q2D,1 = 2.91 (Table 1), for j0,2D Ist,2D from P1 and P2 it results

1.23 × 103 A2 cm−6 s−1 and 1.24 × 103 A2 cm−6 s−1 , i.e. more than three times lower than 4 × 103 A2 cm−6 s−1 . Practically the same values are obtained for the product 2 Ist,2D if the data for the current j2D,p (t) are plotted in semij0,2D logarithmic coordinates ln[j2D,p (t)/t2 ] vs. t3 in accordance with Eq. (9 ) (circles in Fig. 2) and the free parameters P1 and P2 are determined from a simple linear fit (line in Fig. 2) based on the formula:



t2

0

4

5

6

3

10 t /s

The obtained result is demonstrated by the line in Fig. 1 and the values obtained for the two free parameters are P1 = 2.93 × 10−8 A cm−2 s−2 and P2 = 1.50 × 10−5 s−3 . Bearing in mind the constants involved in Eq. (9), from P1 and P2 for 2 Ist,2D one obtains 3.97 × 103 A2 cm−6 s−1 and the product j0,2D

j2D,p (t)

-26

90

Fig. 1. Theoretical “progressive” current transient (circles), calculated by means of Eq. (9) and non-linear fit (line) based on Eq. (13).

ln

-22

-24

0,0 -10



-20

2

10 j2D,p /Acm

-2

-2 -2

ln[(j2D,p/t )/Acm s ]

1,6

Fig. 2. Theoretical “progressive” current transient (circles), calculated by means of Eq. (9 ) and non-linear fit (line) based on Eq. (13 ).

the references cited therein): j2D,ins (t) = N0 [1 − (t)]I1,2D (t)

Here the actual surface fraction (t) covered by growing twodimensional crystals at the time moment t is again expressed by the Kolmogorov–Avrami formula (Eq. (8)) [31–34] but with extended surface fraction  ext (t) given by: ext (t) =

 (zF)2

2 2 2 2 VM N0 j0,2D Q2D t



2  2 2 2 2 2 2 Q2D t exp − VM N0 j0,2D Q2D t hVM N0 j0,2D i2D,ins (t) = zF (zF)2

If N0 circular, disk-shaped 2D clusters appear simultaneously on the electrode surface at the time moment t = 0, the current density j2D,ins (t) of instantaneous nucleation is defined as (see e.g. [18] and

tins,m =

zF (2)

1/2

Reactions (b)

Reaction (c)

Oxy + e → X slow X + e → Red fast z = 2; zf = 0;  = 1  Q2D = Q2D,2

Oxy + e → X fast X + e → Red slow z = 2; zf = 1;  = 1  Q2D = Q2D,2

Oxy + ze → Red z=2 Q2D = Q2D,1

Progressive nucleation,  = 0.030 V   Q2D = Q2D,2 = 1.62 Q2D = Q2D,2 = 5.22

Q2D = Q2D,1 = 2.91

Instantaneous nucleation,  = 0.050 V   = 2.59 Q2D = Q2D,2 = 18.19 Q2D = Q2D,2

Q2D = Q2D,1 = 6.87

(17)

1/2

VM N0 j0,2D Q2D

 1

1/2

j2D,ins,m = (2)1/2 hN0 j0,2D Q2D exp −

2

(18)

Note that the product i2D,ins,m tins,m ,



zFh 1 exp − VM 2



(19)

again contains only material constants characterizing the deposited species and does not depend on the total number N0 of the 2D nuclei formed. Similarly to the case of progressive nucleation, Eq. (16) can be presented as a linear relationship, ln[j2D,ins (t)/t] vs. t2 ,



Reactions (a)

(16)

and the value of the maximal current i2D,ins,max is given by

ln

Table 1 Two- and one-step electrochemical reactions.



The current–time relationship given by Eq. (16) displays a maximum at time t = tmax ,

j2D,ins,m tins,m = 2.2. Instantaneous nucleation

(15)

Thus combining Eqs. (3), (8), (14) and (15) for the current density j2D,ins (t) of instantaneous nucleation one obtains:

(13 )

= ln P1 − P2 t 3

(14)

j2D,ins (t) t





= ln

2 2 2 N0 Q2D hVM j0,2D zF



−

 (zF)

2

2 2 2 2 VM j0,2D N0 Q2D t

(16 )

allowing for a direct interpretation of experimental data. Fig. 3 (circles) shows the current transient i2D,ins (t) calculated by means of Eq. (16) with the following values of the constants involved: z = 2, zf = 0,  = 1, VM = 7.1 cm3 /mol, h = 2.4 × 10−8 cm, ˛2 = 0.5,  = 0.050 V, j0,2D = 0.2 A cm−2 , N0 = 1 × 107 cm−2 , T = 298 K  and Q2D = Q2D,2 = 2.59, which correspond to a two-step electrochemical reaction of type (a) (Table 1). The line in Fig. 3 shows the non-linear fit based on the formula, j2D,ins (t) = P3 t exp(−P4 t 2 )

(20)

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A. Milchev, I. Krastev / Electrochimica Acta 56 (2011) 2399–2403

30 6

25

Q 2D

-2

20 4

15

c

10

5

10 j2D,ins /Acm

b

5

2

0

a 0

1

2

3

2

4

5

6

10 η/V

0 0

5

10

15

t/s

20

25

Fig. 3. Theoretical “instantaneous” current transient calculated by means of Eq. (16) (circles) and non-linear fit (line) based on Eq. (20).

 Fig. 5. Overpotential dependence of the quantity Q2D . Line (a), Q2D = Q2D,2 (Eq.  (Eq. (6) for z = 2, zf = 1,  = 1); line (c) (6) for z = 2, zf = 0,  = 1); line (b), Q2D = Q2D,2 Q2D = Q2D,1 , Eq. (5) for z = 2).

simple linear fit (line in Fig. 4) based on the formula:



assuming that the circles in Fig. 3 represent an experimental current transient. The values obtained for the two free parameters are P3 = 1.5 × 10−5 A cm−2 s−1 and P4 = 1.1 × 10−2 s−2 and bearing 2 in mind the constants involved for the product j0,2D N0 it results

3.98 × 105 A2 cm−6 and 3.92 × 105 A2 cm−6 , respectively, i.e. practically equal to 4 × 105 A2 cm−6 , as expected. However, if we 2 use the values of P3 and P4 to evaluate the product j0,2D Ist,2D assuming that the current transient in Fig. 1 corresponds to a two = 18.19, step electrochemical reaction of type (b) (Q2D = Q2D,2

2 Table 1) for the product j0,2D N0 it results 8.1 × 103 A2 cm−6 s−1

and 7.9 × 103 A2 cm−6 s−1 , respectively, i.e. more than one order of magnitude lower than the real value 4 × 105 A2 cm−6 . Finally, assuming that the “experimental” current transient corresponds to a single-step, two-electron reaction of type (c) (Q2D = Q2D,1 = 6.87, 2 Table 1) for j0,2D N0 from P3 and P4 it results 5.65 × 104 A2 cm6 s−1 and 5.57 × 104 A2 cm6 s−1 , i.e. again lower than 4 × 105 A2 cm−6 . 2 N0 Practically the same values are obtained for the product j0,2D if the data for the current j2D,ins (t) are plotted in semi-logarithmic coordinates ln[j2D,ins (t)/t] vs. t2 in accordance with Eq. (16 ) (circles in Fig. 4) and the free parameters P3 and P4 are determined from a

ln

j2D,ins (t) t



3. Discussion The obtained results clearly show that the specific mechanism of the electrochemical reactions affects significantly the current transients both in case of progressive and in case of instantaneous two-dimensional nucleation. The difference in the values obtained 2 2 Ist,2D and j0,2D N0 in for the nucleation and growth parameters j0,2D case of reactions (a)–(c) (Table 1) is due to the different overpo  , Q2D,2 and Q2D,1 (Fig. 5). tential dependence of the quantities Q2D,2 Therefore, the Q2D () relationship should be taken into consideration when interpreting experimental current transients, applying the general formulae (Eqs. (5) and (6)) to each particular case. Note that the quantities ˛, j0,2D and Ist,2D also depend on the specific mechanism of the electrochemical reactions [27–30]. In our opinion, the theoretical interpretation of an experimental current transient should start with checking whether coordinates of the current maximum correspond to Eq. (12) or to Eq. (19) written for the deposited species.2 Note that the ratio (j2D,p,m tp,m )/(j2D,ins,m tins,m ) is just a numerical constant,



j2D,p,m tp,m 1 = 2 exp − 6 j2D,ins,m tins,m

-2 -1

ln[(j2D,ins/t)/Acm s ]

-14

-16

-18

-20

0

1

2

3 -2 2

4

5

6

(20 )

= ln P3 − P4 t 2



≈ 1.69

(21)

depending neither on the reaction mechanism nor on the properties of the depositing species. Besides that, the constant is bigger enough than unity, which means that one can easily distinguish between progressive and instantaneous two dimensional nucleation. Depending on the obtained result, the experimental data should be interpreted using either Eqs. (9), (9 ) or Eqs. (16), (16 ) in 2 2 order to evaluate the products j0,2D Ist,2D or j0,2D N0 , respectively. Unfortunately, at its present state, the theories of progressive and instantaneous two-dimensional nucleation with overlap do not allow determining separately the values of j0,2D , Ist,2D and N0 . For the purpose it is necessary to study the time of appearance and the rate of spreading of isolated two-dimensional crystals, for example,

2

10 t /s

Fig. 4. Theoretical “instantaneous” current transient calculated by means of Eq. (16 ) (circles) and non-linear fit (line) based on Eq. (20 ).

2 For copper electrodeposition j2D,ins,m tins,m = 3.96 × 10−4 A cm−2 .

j2D,p, m tp,m = 6.70 × 10−4 A cm−2

and

A. Milchev, I. Krastev / Electrochimica Acta 56 (2011) 2399–2403

Finally, we should point out that electric current is certainly an important quantity providing valuable information on the 2D electrochemical phase formation kinetics. At the same time, direct measurements of the 2D island growth and the step propagation rate by means of precise microscopic techniques can also contribute significantly for clarifying the actual mechanism of the 2D nucleation and growth phenomena. In particular, STM (surface tunnelling microscopy) and AFM (atomic force microscopy) in- and ex situ studies could be extremely important in this respect (see e.g. [43–45] and the references cited therein). Apparently, the best would be to combine electric current measurements and direct observation of the growing crystal surface.

(j2D,p/j2D,p,m) , (j2D,ins/j2D,ins,m)

1.0 0.8 0.6

1

2

1.5

2.0

0.4 0.2 0.0 0.0

0.5

1.0

2.5

(t/tp,m) , (t/tins,m)

3.0

3.5

References

Fig. 6. Non-dimensional plots (j2D,p /j2D,p,m ) vs. t/tm and (j2D,ins /j2D,ins,m ) vs. t/tm based on Eqs. (21) and (22).

making use of the unique capillary method developed by Kaischew et al. [9] and Budevski et al. [10–14,17]. Finally, Fig. 6 shows non-dimensional plots (j2D,p /j2D,p,m ) vs. t/tm and (j2D,ins /j2D,ins,m ) vs. t/tm based on Eqs. (9) and (16), j2D,p (t) = j2D,p,m (tp,m )



j2D,ins (t) = j2D,ins,m (tins,m )

t tp,m





2

t tins,m



exp

2 3



 exp

1−

1 2

2403

t tp,m

 1−

3 

t tins,m

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10]

(22) [11] [12]

2  (23)

which are often used to verify the correspondence between theory and experimental data. However, one should not forget that analyses based on Eqs. (22) and (23) inevitably fix theory and experiment in the current maximum and sometimes this leads to ostensible agreement between theory and experimental results. Therefore, verifying whether an experimental current transient could be linearized either in coordinates ln[j2D,p (t)/t2 ] vs. t3 or in coordinates ln[j2D,ins (t)/t] vs. t2 seems to be a better qualitative criterion.

[13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

4. Conclusions The theoretical considerations performed in this study are restricted to ions direct attachment to the edges of the growing two-dimensional clusters. The reason is that while surface diffusion plays major role in case of 2D and 3D cluster growth on solid surfaces from a vapour phase, the significance of its contribution to the electrochemical nucleation and crystal growth phenomena is still an open question (see e.g. Budevski et al. [17] and the references cited therein). It is our opinion that contribution of surface diffusion depends strongly on the particular “adsorbed atom-substrate” bond energy, adatoms (or adions) partial solvation shell, eventual residual charge of adatoms, etc. and therefore may differ significantly depending on the particular experimental conditions. As for experimental systems to which the developed theoretical model could be applied we should say that 2D underpotential and/or overpotential electrodeposition of any metal ion of valence z > 1 may, under certain circumstances, proceed as a multi-step electrochemical reaction. Examples in this respect could be electrodeposition of Cd on Cd 0 0 0 1 [35], Cu on Au 1 1 1 [36], Pd on Au 1 1 1 [37], Co on Au [38–40], Ni on brass [41], Cd on Ag 1 1 1 [42], etc.

[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45]

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