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: [email protected] (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