Discussion

Extracting nucleation rates from current�/time transients. Concludingremarks

Stephen Fletcher *

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

Received 2 December 2001; accepted 24 May 2002

Abstract

Some time ago, Abyaneh and Fleischmann submitted several papers to this journal in which they modelled the heterogeneous

nucleation of crystals as a first order kinetic process. In an invited response, I argued that such an approach was seriously flawed,

because it ignored nucleation rate dispersion. More recently, instead of responding directly to this criticism, Abyaneh and

Fleischmann have challenged the validity of the Deutscher�/Fletcher experiments that first established the physical reality of

nucleation rate dispersion. This note, therefore, serves two purposes. Firstly, it confirms the validity and high rigour of the original

Deutscher�/Fletcher experiments. Secondly, it identifies the errors in the Abyaneh�/Fleischmann papers. In conclusion, it is

emphasised that nucleation rate dispersion is real, universal, and must always be taken into account in experiments involving

heterogeneous nucleation. # 2002 Elsevier Science B.V. All rights reserved.

Keywords: Heterogeneous nucleation; Nucleation rate dispersion; Active sites; Current�/time transients; Alamethicin

I am grateful to Abyaneh and Fleischmann [1] for

their comments.On one matter, at least, we are in full agreement. . .

they surmise, and I can confirm, that I am indeed

dismayed by their lack of appreciation for the works of

Deutscher and Fletcher! I had hoped that Professor

Fleischmann in particular would have appreciated the

subtlety and beauty of the modern theory, given his

early efforts to establish nucleation as an electrochemi-

cal phenomenon. He was instrumental in proving the

widespread existence of nucleation phenomena in elec-

trochemical systems, an achievement still widely under-

rated in the electrochemical literature. For example, you

will search in vain for the word ‘nucleation’ in the index

to Bard and Faulkner’s textbook [2]. Yet many systems

of interest in electrochemistry, ranging from metal

deposition to corrosion, exhibit a phase change con-

trolled by nucleation kinetics. It is an important

problem.

Given the importance of nucleation, it is desirable

that researchers in the field should reach a common

understanding of the underlying physics, and a common

understanding of the mathematical tools for describing

them. Inevitably, the only way to attain this common

understanding is to embark on a focussed dialogue

concerning the existing theory.

Unfortunately, what Abyaneh and Fleischmann have

done in their latest comments is to expand the discussion

beyond purely theoretical questions, and criticised the

experimental work of Deutscher and Fletcher [3�/6].

This is unfortunate, because the experimental work of

Deutscher and Fletcher contains some of the best data

that we have. Widening the discussion in this way also

makes it difficult to provide a brief response, though I

shall try. Essentially, I shall assert that the works of

Deutscher and Fletcher stand unblemished, whereas the

experimental and theoretical works of Abyaneh and

Fleischmann stand in need of correction. Before em-

barking on this course, however, let me first present to

the general reader, who is perhaps unfamiliar with some

of the arcana of nucleation theory, a simple thought

experiment that distinguishes the approaches of the two

groups.Consider a ‘perfect’ electrode, i.e. one without defects,

impurities, grain boundaries, inclusions, etc. which

terminates in a ‘perfect’ surface without reconstruction.* Tel.: �/44-1509-222-561; fax: �/44-1509-223-925

E-mail address: stephen.fletcher@lboro.ac.uk (S. Fletcher).

Journal of Electroanalytical Chemistry 530 (2002) 119�/122

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 7 - 4

Every location on the surface of such an electrode is

defined to be identical to its neighbours; ipso facto

nucleation takes place at the same rate everywhere, and

nucleation rate dispersion is absent. Now consider whathappens if I pick up a piece of emery paper, or

sandpaper, or indeed any rough object, and scratch

the surface of the electrode. Does the new surface now

present only one type of active site towards nucleation

of a second phase, or more than one type?

If you are absolutely certain that only one type of site

exists on the scratched surface, you need never consult

the papers of Deutscher and Fletcher. They would beirrelevant. If, however, you even suspect that there may

be more than one kind of nucleation site, with each site

having its own activation energy towards nucleation,

then you will need a different formulation of nucleation

theory than that provided by Fleischmann. This differ-

ent formulation is the one provided by Deutscher and

Fletcher, and it has been justified both experimentally

[3] and theoretically [4]. Unfortunately, in the matter ofreal-world data, no compromise is possible between

these formulations. You have to choose either the

general model of Deutscher and Fletcher, or the

particular model of Fleischmann.

It is natural to wonder if a ‘reasonably uniform’

electrode surface might perhaps be prepared, upon

which the number of scratches, steps, or point defects

was decreased to an insignificant level. Alas, no. Thepoint is, with �/1013 atoms cm�2, even a few hundred

misplaced atoms would be sufficient to destroy the

energetic uniformity. This is because, by definition,

nucleation always occurs most rapidly on the active

sites that constitute the upper outliers of the rate

distribution, and this distribution is phenomenally

wide due to the high sensitivity of nucleation rates to

the interfacial free energies of the active sites. If therewere 100 highly energetic sites, for example, then that is

where nucleation would occur first. In short, nucleation

dispersion is unavoidable. This is the reason why

nucleation is so irreproducible in practice, and why

crystals decorate scratches. In the final analysis, any

satisfactory theory of heterogeneous nucleation MUST

take into account the fact that the interface is energe-

tically non-uniform.One can also reach the same conclusion from a

different angle. Imagine I have a collection of fine wires

made of different metals (such as Ag, Pt, Cu, etc.), and

suppose that I collect these together in a tightly packed

bundle and use their cross-sections as an electrode

surface. Thus a patchwork of different metals is exposed

to the solution, creating a micro-heterogeneous surface.

Suppose further that I deposit nuclei of a new phase onthis micro-heterogeneous surface. Do all regions of the

surface exhibit the same nucleation rate? Of course not.

The rate will vary from region to region, depending on

the local composition and structure. In this extreme case

it would be absurd to dispute the very existence of

nucleation rate dispersion. It follows, therefore, that we

are not arguing about the existence of nucleation rate

dispersion per se*/it clearly exists*/but rather, whetherit exists to a significant extent on an ‘ordinary’ electrode

surface prepared in an ‘ordinary’ way. To decide this

question, we merely need to know how uniform the

interfacial free energy must be before nucleation rate

dispersion can be ignored. The answer, as shown in ref.

[4], is that the interfacial free energy must be very, very

uniform indeed. Indeed, it needs to be uniform beyond

what has been achieved by anyone anywhere in theworld today, and possibly uniform beyond what is

thermodynamically possible! So once again we are

forced to conclude that any satisfactory theory of

heterogeneous nucleation MUST take into account the

fact that the interface is energetically non-uniform.

This is as clear as I can explain the existence of

nucleation rate dispersion without using maths. The rest

of this reply to the Abyaneh�/Fleischmann commentsmust necessarily be mathematical and historical.

Turning now to the specific comments of Abyaneh

and Fleischmann [1], they basically contain three types

of error. First, misrepresentations of the original papers

of Deutscher and Fletcher. Second, a hubristic insistence

that the assumptions of Fleischmann et al. are ‘well

established principles’. Third, terminological inexacti-

tudes. Let us consider each of these in turn.First, concerning the misrepresentations of the origi-

nal papers of Deutscher and Fletcher, we make the

following corrective remarks. (1) No errors of experi-

mental design were made. (2) Deconvolution does not

depend on the probabilistic character of nucleation. It

merely requires that nucleation events be independent.

Independence can readily be established by a series of

simple tests as discussed in the original papers. (3)Deconvolution does not depend on the reversibility of

the growth process, because the shapes of the growth

curves simply cancel. (4) Nucleation rate dispersion is

predicted by both the classical and atomistic models of

nucleation. We chose the classical model only as an

example; we could just as easily have chosen the

atomistic model. In any case, the existence of nucleation

rate dispersion is an empirical fact that is completelyindependent of any assumed model. (5) Deutscher and

Fletcher have nowhere claimed to have observed a

‘marked time dispersion’ as erroneously stated several

times by Abyaneh and Fleischmann in their reply. In

fact, what we have claimed is a marked rate dispersion.

(6) Deutscher and Fletcher have never described the

work of Abyaneh and Fleischmann as ‘heretical’.

Metaphors drawn from religion seem to us whollyinappropriate in a scientific discussion.

Second, regarding the theories of Fleischmann et al.,

we emphasise the following points. (1) The number of

crystals is not, in general, a Poisson random variable,

S. Fletcher / Journal of Electroanalytical Chemistry 530 (2002) 119�/122120

though it may appear so under some special conditions

(infinite N0). (2) The first order nucleation law is

physically unrealistic and there is no credible evidence

for it anywhere in the electrochemical literature. (3)‘Instantaneous nucleation’ is a contradiction in terms,

and simply does not exist. (4) The double potential step

method does not work as advertised. You cannot switch

nucleation on and off instantaneously. If you could, it

would violate the whole spirit of nucleation theory,

because in the real world the population of critical nuclei

responds slowly to rapid changes in driving force, due to

the fact that the nuclei must assemble molecule bymolecule via a series of step-wise size fluctuations.

Third, concerning terminological inexactitudes, there

are a number of quite bizarre examples in the Abyaneh�/

Fleischmann comments that, at times, make them

completely incomprehensible. Take, for example, the

word ergodic. As is well known, a random process is

stationary if all of its statistical moments are time

invariant, and in addition it is ergodic if its expectationvalues correspond to its time average values. By defini-

tion, therefore, all ergodic processes are stationary

processes. Now, since nucleation is not even stationary,

having both an acceleratory period (induction time) and

a deceleratory period (due to the exhaustion of active

sites) it is trivially obvious that nucleation is not ergodic.

Indeed, no transient process is ergodic. So why do the

authors insist that ‘We believed (and we still believe)that such a discussion (of the work of Deutscher and

Fletcher) should have been delayed until proper tests of

the ergodicity of the systems have been carried out’?

With respect, this is nonsense. The non-ergodicity of

nucleation has no bearing whatever on the reality of

nucleation rate dispersion.

The strange references to ‘time dispersion’ have

already been commented upon: that phrase too ismeaningless. In addition, I have no idea what is meant

by their phrase ‘compound stochastic system.’ Conven-

tional compound stochastic processes are well defined;

they are processes in which one time-dependent random

process triggers another; but Abyaneh and Fleischmann

seem to mean something quite different. My guess is

that they are simply distinguishing multi-nuclear ob-

servations from single nucleus observations, but if so Icannot see what point is being made. If my interpreta-

tion is correct, ‘compound nucleation’ is just the same

nucleation that we have all been dealing with for the

past several decades. There are also scattered references

to ‘nucleation rate constants’ when what are meant are

nucleation rates, an error which I have pointed out on

many occasions and which Professor Fleischmann

unaccountably refuses to correct. My guess is that‘ensemble average’ refers to the sample mean. There is

also a reference to ‘death processes’, without any further

explication. How are we to understand this? The only

death process normally considered in the theory of

stochastic processes arises in the description of reversible

processes (Kolmogoroff’s equations) rather than in

irreversible processes such as the nucleation of metals.

Are the authors seriously suggesting that macroscopi-cally growing metal crystals disappear again at over-

potential? If not, what is the point of this remark?

Examples of woolly thinking of this type abound, and

could be discussed almost indefinitely. However, this

small sample of errors concerning stochastic processes

should be enough to convince even the most sceptical

reader that something is seriously wrong with the

Abyaneh�/Fleischmann approach. Indeed, the wholetenor of the reply of Abyaneh and Fleischmann suggests

that they have completely missed the point of the

Deutscher�/Fletcher papers. Could it be that they have

confused nucleation rate dispersion with some sort of

induction time dispersion? I am beginning to think so.

Given the terminological confusions that evidently

exist regarding the probabilistic formulation of nuclea-

tion, let me briefly try to summarise present knowledgeusing standard notation. The observation of one active

site chosen at random constitutes a Bernoulli trial (i.e. a

simple yes/no decision depending on whether there is a

growing crystal or not), and so the number of crystals

N (t ) observed during a series of experiments on a fixed

number of N0 active sites of equal activity (Fleisch-

mann’s own model) is in fact a binomial random

variable not a Poisson random variable [7�/9]. Thevariance is zero when all the sites are empty and zero

when all the sites are full, as we would expect. The paper

of Bindra et al. [10], mentioned in the comments [1],

completely misses this latter point. The defects of the

Bindra approach were pointed out 17 years ago [8].

Nowadays, the field has moved on and contemporary

interest centres on N0 active sites of different activity, i.e.

nucleation rate dispersion. Unfortunately this area iscomparatively unexplored from a stochastic point of

view and only one analytical solution is known. This is

for the special case where the distribution of nucleation

rates can be approximated as a gamma random process

and N0 is asymptotically large. The probability density

function of the number of observed crystals N (t) turns

out to be that of the negative binomial distribution, and

the corresponding mean and variance have been pub-lished [7]. A unique and interesting feature of nucleation

rate dispersion is that the standard deviation of N (t) can

exceed its mean. (Unfortunately, this signature effect

will probably be hard to see for the case of finite N0.)

This is the current state of the art: further studies are

desirable.

Concerning the experimental systems mentioned by

Abyaneh and Fleischmann, there is little to add, exceptto comment on their unrepresentative character. The

nucleation of PbO2 on platinum, over most of the

overpotential range, is a rare example of nucleation at

super-high nuclear density (I know of only nickel that

S. Fletcher / Journal of Electroanalytical Chemistry 530 (2002) 119�/122 121

behaves similarly), and this would exclude it from

analysis by the Deutscher�/Fletcher experimental

method (though not the Deutscher�/Fletcher theory)

because the nuclei are too close together to remainindependent for long enough for decent measurements

to be made. From the point of view of stochastic

measurements there are other serious problems with

this system. Once crystals of PbO2 have been deposited,

it is almost impossible to remove vestigial fragments of

material left behind by the re-reduction process. These

are readily seen by microscopy and tend to act as new

nucleation sites in future measurements. As a conse-quence, the results become time-dependent.

The formation of ion channels in lipid bilayers by

alamethicin is an interesting example of a reversible

stochastic process, but the ‘quantized’ currents observed

should have raised serious doubts in the authors’ minds

about whether the process really is one of nucleation. I

strongly doubt it. The fluctuating sizes of nuclei inherent

in all nucleation models should manifest as fluctuatingcurrent responses, whereas constant current responses

are seen. In other words, the ion channels behave more

like stochastically triggered trapdoors. This suggests a

rate-determining step connected with a conformational

change rather than a nucleation process. In any case

such a system is hardly relevant to metal deposition

processes, which, to the best of my knowledge, have

never generated even one example of reversible nuclea-tion. So the death of nuclei is irrelevant in the present

context. Finally, the extraordinary claim at item (ix) of

ref. [1] cannot be allowed to pass without comment. This

states that ‘. . . in the development of work on the

compound stochastic systems (sic), it will not be

necessary to use arrays of microelectrodes in order to

ensure adequate separation of the diffusion fields; the

same objective can be achieved using vitreous carbonsubstrates.’ Well, if Abyaneh and Fleischmann were

actually to try this experiment, they would immediately

find that the large and slow capacitive charging currents

of vitreous carbon obscured the small currents needed to

ensure the diffusional independence of the nuclei.

In summary, all the criticisms of Abyaneh and

Fleischmann regarding nucleation rate dispersion are

ill founded. In a short note such as this, it is not possible

to track every error in their comments to its source, but

the principal defects are clear enough. The reader who

wishes to follow the correct mathematical arguments in

their full rigour is strongly recommended to consult refs.

[3�/9].

The first-order nucleation law for N (t) was a reason-

able hypothesis 40 years ago, before microelectrode

arrays were developed, but today it is unsustainable. It is

not a ‘well established’ principle. It is a discredited

principle.

References

[1] M.Y. Abyaneh, M. Fleischmann, J. Electroanal. Chem. 530

(2002) 108.

[2] A.J. Bard, L.R. Faulkner, Electrochemical Methods, Fundamen-

tals and Applications, 2nd ed., Wiley, New York, 2001.

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

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

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

(Eds.), Microelectrodes: Theory and Applications NATO-ASI,

vol. 197, Kluwer Academic Publishers, Dordrecht, 1991, pp. 341�/

355.

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

(1998) 3527.

[7] R.L. Deutscher, S. Fletcher, J. Electroanal. Chem. 164 (1984) 1.

[8] S. Fletcher, J. Electroanal. Chem. 164 (1984) 11.

[9] S. Fletcher, J. Electroanal. Chem. 215 (1986) 1.

[10] P. Bindra, M. Fleischmann, J.W. Oldfield, D. Singleton, Faraday

Disc. Chem. Soc. 56 (1973) 180.

S. Fletcher / Journal of Electroanalytical Chemistry 530 (2002) 119�/122122