Discussion

Extracting nucleation rates from current�/time transients. Commentson three papers by Abyaneh and Fleischmann published in this issue

Stephen Fletcher

Department of Chemistry, Loughborough University, Ashby Road, Loughborough, Leicestershire LE11 3TU, UK

Received 6 March 2001; accepted 24 May 2002

Abstract

Electrochemical nucleation presents the theorist with a number of problems. Perhaps the most difficult of these is how to model

the appearance rate of crystals as a function of time. Recently, Abyaneh and Fleischmann published several papers in which they

modelled the appearance rate of crystals as a first-order kinetic process. Though conceptually simple, I argue in this note that such

an approach is seriously flawed. Indeed, it violates the basic physics of heterogeneous nucleation, because it omits consideration of

spatial non-uniformities in the specific interfacial free energies of crystal j solid interfaces. As a result, the first-order kinetic model

fails to predict nucleation rate dispersion, which is known to occur in nature. A better model is needed. # 2002 Elsevier Science

B.V. All rights reserved.

Keywords: Heterogeneous nucleation; Nucleation rate dispersion; Active sites; Current-time transients; Crystal growth

1. Discussion

In Refs. [1�/3] Abyaneh and Fleischmann explore two

of the classical problems of electrochemical nucleationtheory. The first is the problem of deriving current�/time

transients from nucleation rates, and the second is the

inverse problem of deriving nucleation rates from

current�/time transients. Specifically, they explore the

ramifications of choosing different growth geometries

for the case of bulk deposition of nuclei under interfacial

control; they compare a computer-fit method with a pre-

pulse method for extracting nucleation rates from realdata; and they modify a first-order kinetic model of

nucleation to include the effects of potential pre-pulses.

Throughout all three papers the authors assume that

the number of nuclei as a function of time is given by an

equation of the type

N(t)�N(�)[1�exp(�At)] (1)

where N (�) is the total number of active sites and A is a

constant activation rate of sites. Indeed this equation

provides the starting point for their entire theoretical

approach; it provides the basis for extracting the limits

of ‘instantaneous’ and ‘progressive’ nucleation in Part I;

it provides the null hypothesis for Part II (namely that

all available active sites are converted into nuclei at

sufficiently high potential); and, after differentiation, it

supplies the differential equation used in Part III to

describe the effects of potential pre-pulses.

By contrast, I assert that Eq. (1) is always unphysical

when applied to real systems.

Why do I say this? The first reason is that hetero-

geneous nucleation events (including all electrochemical

nucleation events) take place at very different rates at

different locations on solid surfaces, so that overall rates

are actually the sums of many different rates. Typically,

these different rates have a vast span of values, from

more than 1000 nuclei per second on the most active

sites, to less than one nucleus per day on the least active

sites. Such a span of values, which was first described in

1988 [4], is called ‘nucleation rate dispersion’ and its

physical origin is now well understood [5]. Essentially, it

arises from the extraordinary sensitivity of the nuclea-

tion rate to the specific interfacial free energy of the

crystal j solid interface. In real systems there are always

differences in specific interfacial free energies between

different locations on solid surfaces, and consequently

these locations exhibit vastly different nucleation rates.E-mail address: stephen.fletcher@lboro.ac.uk (S. Fletcher).

Journal of Electroanalytical Chemistry 530 (2002) 105�/107

www.elsevier.com/locate/jelechem

0022-0728/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved.

PII: S 0 0 2 2 - 0 7 2 8 ( 0 2 ) 0 0 9 7 5 - 0

The second reason why Eq. (1) is unphysical is that it

neglects non-steady state effects caused by rapid changes

in driving force. The theory of how potential pulses

cause nucleation rates to become time-varying waspublished nearly a decade ago [6], and since then time-

varying nucleation rates have been observed many times

by direct deconvolution. By 1998, enough evidence had

accumulated to suggest that the nucleation rate never

switched on and off at the same time that the electrode

potential switched on and off. Accordingly, it could

safely be concluded that ‘instantaneous’ nucleation did

not exist [7].Alas, in Refs. [1�/3] the authors have chosen to ignore

this extensive literature, and have therefore been led into

duplicating some of its more notable conclusions, but

without shedding any new light on the physical origins

of the phenomena. For example, in Part II they

conclude, ‘‘In the case of the pre-pulse method. . . we

can even go so far as to challenge the fundamental

precepts on which this method is based’’. As explainedabove, this same point was reached 3 years ago, in a

major journal, on the basis of more and better data [7].

Similarly, the authors deduce that an unacceptably small

value for the size of the critical nucleus is obtained

despite an apparently reasonable estimate of the inter-

facial energy. This has also been discussed previously in

the literature (Ref. [5], final page) and the explanation

given in terms of nucleation rate dispersion. Finally,they also state: ‘‘The most striking feature of the results

was our inability to attain a pre-pulse potential beyond

which the slope of the rising transient remained con-

stant’’. But if we take into account the reality of

nucleation rate dispersion*/as we must*/then this is

not striking at all. Instead, it is merely the expected

response in which increasing numbers of crystals appear

as the potential is increased.Neglect of the literature is also evident in Part III. For

example, it is stated that ‘‘The current belief is that, at

an appropriate pre-pulse potential, permanent conver-

sion of all available sites into nuclei can be induced, such

that coverage at a lower potential. . . is achieved by

growth only’’. However, it would have been fairer to say

that this belief is not universally shared. The effects of

double potential steps on nucleation have been verycarefully studied, the results have been reported in the

literature [6,7], and the conclusions disagree with the

statement above.

Also in Part III, the first-order kinetic model of

nucleation represented by Eq. (1), which is valid at one

potential (E2, say), is extended to include a pre-pulse to

an earlier potential (E1, say), where E1�/E2. Unfortu-

nately, this extension of the model only compounds thedifficulties. First, the activation rates should be time

varying to take into account the baleful influence of

non-steady-state effects. Second, if the classic pre-pulse

method is going to be modelled properly, the value of

the second potential E2 should be so low that the

activation rate is negligible when the pre-pulse is

omitted. This strict condition on E2 is necessary because

a background activation rate at E2 would needlessly

complicate the interpretation of data.

We are now in a position to confront the most

complex question of all: if E2 is set so low that the

activation rate is negligible when E2 is reached in one

step, why is the activation rate sometimes non-negligible

when E2 is reached in two steps?This is simply not answerable in terms of the

Abyaneh�/Fleischmann model. But it is answerable in

terms of the theory of non-steady state nucleation [6]. As

Abyaneh correctly notes in Part III, during a pre-pulse

some nuclei that are formed at the first (high) potential

dissolve at the second (low) potential, thus causing a

number of sites to become available for nucleation

again. But it is important to understand that these sites

are not responsible for the nucleation at E2, because E2

is deliberately chosen to be far too low for nucleation to

occur afresh on empty sites. What is happening, in fact,

is that a small but significant fraction of the nuclei

formed at the first potential are continuing to grow at

the second potential, even though they have sub-critical

size there (see Ref. [6], and references therein). The sole

reason they are able to do this is that they are not in

their steady state distribution.

To clarify this point, it is helpful to distinguish

between the number of nuclei in the steady-state

population of nuclei, and the number of nuclei that

grow to macroscopic size in a given interval of time

(‘crystals’). Only the latter ‘count’ toward the nucleation

rate. Although the number of nuclei in the steady-state

population certainly decreases at the lower potential,

this does not cause the total number of macroscopic

crystals to decrease. On the contrary, the number of

macroscopic crystals continues to increase for a short

while after the driving force has decreased, an effect

which Deutscher and I have called ‘nucleation persis-

tence’ [6]. This effect is readily observed experimentally,

and is probably what is being observed in Part III of the

current papers. So, contrary to the conclusion of Part

III, the number of macroscopic crystals at the lower

potential neither decreases due to dissolution, nor

increases due to nucleation on empty sites. In fact, it

increases because a small fraction of nuclei left over

from the pre-step disappear from the now-unstable

distribution by growing, even though the vast majority

of their unstable siblings disappear by dissolving. The

effect is transient, probabilistic, and can be modelled

successfully only by considering the full non-steady state

of nucleation [6,7].

In summary: Eq. (1) is an unphysical choice for

modelling electrochemical nucleation processes because

it fails to predict nucleation rate dispersion and time-

S. Fletcher / Journal of Electroanalytical Chemistry 530 (2002) 105�/107106

varying nucleation rates. When better models are used,

such as those in Refs. [4�/7], all of the apparently

surprising results in Refs. [1�/3] are readily explained.

Acknowledgements

I am grateful to Professor L.M. Peter for inviting me

to submit this note.

References

[1] M.Y. Abyaneh, J. Electroanal. Chem. 530 (2002) 82.

[2] M.Y. Abyaneh, M. Fleischmann, J. Electroanal. Chem. 530 (2002)

89.

[3] M.Y. Abyaneh, J. Electroanal. Chem. 530 (2002) 96.

[4] R.L. Deutscher, S. Fletcher, J. Electroanal. Chem. 239 (1988) 17.

[5] R.L. Deutscher, S. Fletcher, J. Electroanal. Chem. 277 (1990) 1.

[6] S. Fletcher, in: M.I. Montenegro, M.A. Queiros, J.L. Daschbach

(Eds.), Microelectrodes: Theory and Applications, Kluwer, Dor-

drecht, 1991, pp. 341�/355.

[7] R.L. Deutscher, S. Fletcher, J. Chem. Soc. Faraday Trans. 94

(1998) 3527.

S. Fletcher / Journal of Electroanalytical Chemistry 530 (2002) 105�/107 107