Electrocrystallization. Fundamentals of Nucleation and Growth

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ELECTROCRYSTALLIZATION Fundamentals of Nucleation And Growth

This Page Intentionally Left Blank

ELECTROCRYSTALLIZATION Fundamentals of Nucleation And Growth

by

Alexander Milchev Institute of Physical Chemistry Bulgarian Academy of Sciences Sofia, Bulgaria

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

0-306-47552-9 1-4020-7090-X

©2002 Kluwer Academic Publishers New York, Boston, Dordrecht, London, Moscow Print ©2002 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Kluwer Online at: and Kluwer's eBookstore at:

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For my family, for my mother and my sister and for my teacher Academician R.Kaischew


Contents

Preface 1. THERMODYNAMICS OF ELECTROCHEMICAL NUCLEATION 1.1 1.2 1.3 1.4

General concepts Nucleation work Theory of mean sepration works Atomistic considerations

2. KINETICS OF ELECTROCHEMICAL NUCLEATION 2.1 General formulation 2.2 Classical theory of stationary nucleation 2.3 Atomistic theory of stationary nucleation 2.4 Non-stationary nucleation 3. STOCHASTIC APPROACH TO NUCLEATION 3.1 Temporal distribution of clusters 3.2 Spatial distribution of clusters 4. ELECTROCHEMICAL CRYSTAL GROWTH 4.1 Growth of an individual crystal face 4.2 Growth of a hemispherical liquid drop

ix

1 1 11 46 68 83 83 89 106 128 165 165 177 189 189 203

viii 5. MASS ELECTROCRYSTALLIZATION

219

219 5.1 General concepts 5.2 Progressive and instantaneous nucleation without overlap 220 228 5.3 Progressive and instantaneous nucleation with overlap

Author Index

255

Subject Index

263

Preface

“Electrocrystallization is a particular case of a first order phase transition” and “Electrocrystallization is a particular case of electrochemical kinetics” are two statements that I have heard and read many times. I do not like them for a simple reason: it is annoying to see that the subject to which you have devoted more than 30 years of your life may be considered as a “particular case”. Therefore, I decided to write this book in which Electrocrystallization is the main subject. To become competent in the field of Electrocrystallization one should possess knowledge of Electrochemistry, Nucleation and Crystal Growth, which means knowledge of Physical Chemistry, Physics and Mathematics. That is certainly difficult and in most cases those who study Electrocrystallization are either more electrochemists, or more physical chemists, or more physicists, very often depending on whom has been their teacher. Of course, there are scientists who consider themselves equally good in all those fields. Very frequently they are, unfortunately, equally bad. The difference is essential but strange enough, it is sometimes not easy to realize the truth immediately. Being a pupil of the Bulgarian Crystal Growth School I feel myself closer to Nucleation and Crystal Growth than to Electrochemistry and this has left an imprint on my book. For instance, I did not always follow the Electrochemical Convention. Just a simple example: the cathodic current and the cathodic overpotential are defined as positive quantities. I do apologize to my readers for that but frankly speaking my sense of guilt is not too strong because many others do that also. It was not my aim to include in this book everything that may concern the electrochemical phase formation. Instead, I tried to write a comprehensible introduction to the fundamentals of the initial stages of

x Electrocrystallization, which are dominated by nucleation and growth of the first clusters of the new phase. For the sake of simplicity, I have considered the electrochemical formation and growth of single component, solid and liquid phases. The electrochemical deposition of alloys, the formation of gas bubbles, the growth of compact galvanic coatings, as well as metal electrodissolution are subjects beyond the scope of this book. To put it briefly, the book does not contain all that one should know of Electrocrystallization to ensure a good success in this extremely interesting field. However, I would like to believe that it does contain a lot of what one could not be without. I tried to offer a readable exposition of the topic, in simple terms, providing a detailed theoretical description of the phenomena involved. In some cases I have described also the most relevant aspects of the experimental studies of electrochemical nucleation and growth, including some important methods for acquiring and analyzing experimental results. I think that the book is suitable not only for specialists in Electrocrystallization but also for graduate and PhD students, as well as for scientists from diverse but related fields like pure and applied electrochemistry, electrocatalysis, corrosion, electrochemical adsorption, materials science etc. Those who are interested in further reading could find additional information on general Nucleation and Crystal Growth and on Electrocrystallization in many excellent books and review articles. The Reference List attached to this Preface contains only a small part of what is worth reading. It should be mentioned here that I have endeavored to cite everybody who has made significant contributions to the field of Electrocrystallization. However, I feel sure that many excellent papers have been missed and I apologize to those colleagues who will not find their names in my References. It is a matter of lacking information and/or poor memory, not of bad intention. I should add that I always tried to cite the authors’ original publications, even when they have been published a long time ago. I realize that it would be more convenient for the reader to find the information needed in some recent review of another author, in which those publications have been mentioned and even discussed. This is, however, my way to pay tribute to our predecessors. Initially, I thought that it would not be difficult to describe the fundamentals of Electrocrystallization and that I would be able to write the book among other things. However, it turned out that “What is important and what to miss?”, “What to say in details and what just to mention?” are not at all easy questions to answer. Gradually, the book occupied all my time including the weekends and even my summer vacation and this aroused the merited rebuke of my family: my 9 years old son, my 19 years old daughter

xi and my dear wife. However, they succeeded to stand by me and I thank them cordially for that. At this place somebody may say: “Yes, but he did the job and he did it by the time fixed!” Not at all! I missed all deadlines. Fortunately, the Publishers did not press me and I thank them very much for their understanding. I should say that I learned a good lesson from all this. On another occasion I will first write the book and then fix a deadline, not vice versa as I did this time. Well, I have one excuse for being late. This book had to be written in a co-authorship with my colleague Prof. Benjamin Scharifker from the University “Simon Bolivar”, Caracas, Venezuela. Due to unpredictable circumstances, however, Ben was able to participate in the preparation of the first 30 pages only and I had to do the rest alone. I thank him for his gratuitous help. There are many other people to whom I should express my sincere gratitude and not because they helped me directly in the writing of this book but because of the fruitful scientific contacts that we have had during my work in the field of Electrocrystallization. For understandable reasons I cannot mention the names of all of them. However, I should not miss my younger co-workers Dr. E.Michailova, Dr. V.Tsakova and Miss I. Lesigiarska; my colleagues Prof. S.Stoyanov, Prof. I.Markov, Prof. D.Kaschchiev, Prof. D.Stoychev, Prof. I.Krastev, Prof. Chr.Nanev, Prof. M.Paunov and Prof. I.Gutzov from the Institute of Physical Chemistry of the Bulgarian Academy of Sciences (IPC, BAS); my colleagues Prof. V.Bostanov, Prof. G.Staikov, Prof A.Popov and Prof. T.Vitanov from the Central Laboratory of Electrochemical Power Sources, (CLEPS, BAS); Dr. T.Chierchie (University of Bahia Blanca, Argentina), Prof. M.SluytersRehbach and Prof. J.H.Sluyters (University of Utrecht, The Netherlands); Prof. Cl.Buess-Herman (Free University of Brussels, Belgium), Prof. L.Heerman (Catholic University of Leuven, Belgium), Prof. G.Kokkinidis and Prof. A.Papoutsis (University of Thessaloniki, Greece), Prof. K.Juettner (University of Karlsruhe, Germany) and Prof. A.Danilov (Institute of Physical Chemistry, Russian Academy of Sciences). I owe a debt of gratitude to my elder colleagues from whom I learned a lot: Prof. E.Budevski (CLEPS, BAS), Prof. Sir G.Hills (University of Southampton, UK), Prof. R.Lacmann (University of Braunschweig, Germany), Prof. W.J.Lorenz (University of Karlsruhe, Germany), the late Prof. J.Malinowski (CLAPHOP, BAS) and the late Prof. S.Toschev (IPC, BAS). My most sincere gratitude is due to my teacher Academician R.Kaischew, the founder (together with the late Prof. I.Stranski) of the Bulgarian School of Crystal Growth. Without these two outstanding scientists this School would not exist.

xii Finally, I should not forget to thank also the Alexander von Humboldt Foundation, Germany for the generous equipment support offered to my research group. Some experimental results obtained with this equipment are included in this book. Last not least I thank my co-worker Mrs. L.Dragomirova for her precise technical assistance during the preparation of this book.

Alexander Milchev

27 February 2002 Sofia

REFERENCES (The asterisks mark the books and reviews in which Electrocrystallization is not or is only partly concerned.) 1.* J.W.Gibbs, Collected Works, New Haven, 1928, Vol. 1.

2.* M. Volmer, Kinetik der Phasenbildung, Teodor Steinkopf Verlag, Leipzig, Dresden, 1939. 3.* W.K. Burton, N.Cabrera and F.C.Frank, Phil.Trans.Roy.Soc., A243( 1951)299. 4.* Ja.I.Frenkel, Kinetic Theory of Liquids, Dover, New York, 1955. 5.* K.J.Vetter, Elektrochemische Kinetik, Springer-Verlag, 1961. 6. M.Fleischmann and H.R.Thirsk, in Advances in Electrochemistry and Electrochemical Engineering, Ed. P.Delahay and C.Tobias, Interscience, New York, 1962. 7.* J.P.Hirth and G.M.Pound, Condensation and Evaporation, Pergamon, Oxford, 1963. 8. J.O’M.Bockris and G.A.Razumney, Fundamental Asdpects of Electrocrystallization, Plenum Press, New York, 1967 9.* A.C.Zettlemoyer (Editor), Nucleation, Dekker, New York, 1969. 10.* H.R.Thirsk and J.A.Harrison, A Giude to the Study of Electrode Kinetics, Academic Press, London, 1972 11.* S. Toschev, in Crystal Growth: An Introduction, Ed. P. Hartmann, North-Holland Publ. Co, 1973, p.1. 12.* B.B.Damaskin and O.A.Petrii, Elektrochimicheskaya Kinetika, Visshaya Shkola, Moskow, 1975 (in Russian). 13. A.N.Baraboshkin, Electrocrystallization of Metals from Molten Salts (in Russian), Moskow, Nauka, 1976.

xiii 14.* A.C.Zettlemoyer (Editor), Nucleation Phenomena, Adv.Colloid.Interface Sci., Vol.7, 1977. 15.* S.Stoyanov, in Current Topics in Material Sciences, Ed. E.Kaldis, North - Holland Publ. Co, Vol. 3, 1978. 16.* A.C.Zettlemoyer (Editor), Second Special Issue on Nucleation Phenomena, Adv.Colloid. Interface Sci., Vol.10, 1979 17.* R.Kern, G.Le Lay and J.J.Metois, in Current Topics in Material Sciences, Ed. E. Kaldis, North - Holland Publ. Co, 1979, Vol. 3, p.131. 18.* R. Kaischew, Selected Works, Bulg. Acad. Sci., 1980 (in Bulgarian). 19.* S.Stoyanov and D.Kashchiev, in Current Topics in Material Sciences, Ed. E.Kaldis, North - Holland Publ. Co, 1981, Vol. 7, p.69. 20. E.Budevski, in Comprehensive Treatise of Electrochemistry, Vol.7, B.E. Conway, J.O'M. Bockris, S.U.M. Khan, R.E.White (Eds.), Plenum Press, NY, 1983, 399. 21.* A.A. Chernov, Modern Crystallography, Vol.3, Springer, Berlin, 1984. 22. A.I.Danilov and Yu.M.Polukarov, Uspekhi.Khimii, 56(1987)1082. Century, 23. B.R.Scharifker, in Electrochemistry in Transition from the 21th to the O.J.Murphy, S.Srinivasan and B.Conway (Eds.), Plenum Press, New York, 1992. 24.* I.Markov, Crystal Growth for Beginners, World Scientific, Singapore, 1995. 25.* I.Gutzow and J.Schmelzer, The Vitreous State, Springer, Berlin, 1995. 26. E.Budevski, G.Staikov and W.J.Lorenz, Electrochemical Phase Formation - An Introduction to the Initial Stages of Metal Deposition, VCH, Weinheim, 1996. 27. Ju.D.Gamburg, Electrochemical Crystallization of Metals and Alloys, Janus-K, Moskow, 1997 28. G.Staikov, W.J.Lorenz and E.Budevski, in Imaging of Surfaces and Interfaces (Frontiers of Electrochemistry, Vol.5), 1999. 29.* D.Kashchiev, Nucleation: Basic Theory with Applications, Butterworth Heinemann, Oxford, 2000. 30.* B.Mutaftschiev, The Atomistic Nature ofCrystal Growth, Springer, Berlin, 2001.


Chapter 1 THERMODYNAMICS OF ELECTROCHEMICAL NUCLEATION

1.1

General concepts

1.1.1

Electrochemical potential and the state of thermodynamic equilibrium

Electrochemical nucleation takes place at the boundary between two phases. One of them is an electronic conductor, frequently a metal. The other phase is an electrolyte solution. The charge carriers in this phase are ions and under certain circumstances the ions in the region close to the electronic conductor may interchange electrons with it. It is, namely, during the interfacial charge transfer that electrochemical nucleation occurs. When considering the thermodynamics of nucleation and the mechanism of the elementary acts of single ions attachment and detachment to and from a growing cluster or a crystal surface, it proves convenient to work not with molar but with molecular quantities. For that reason the molecular unit to be used in Chapters 1 and 2 of this book will be not a mol but a particle - atom, ion or molecule - and all physical quantities like chemical and electrochemical potential, electric charge etc. will be referred to this unit. In what follows we consider the conditions that characterize the thermodynamic equilibrium in chemical and electrochemical systems. Chemical equilibrium between bulk phases is expressed by a minimum of the Gibbs free energy G of the whole system comprising the different phases. Thus the change of G due to the reversible exchange of infinitesimal amounts of matter between phases at constant temperature T and pressure P will be given by:

2

where

Chapter 1

is the number of building units of component i in phase

and

is the chemical potential of a building unit. Therefore equation (1.1) can be rewritten as:

It turns out then that at equilibrium the Gibbs free energy change of extracting an infinitesimal amount of i from one phase is exactly compensated by the Gibbs free energy change of incorporating that amount of i into another phase. Consider now a closed system of fixed content for which

Solving the set of equations (1.3) and (1.4) by means of the method of Lagrange one obtains that:

These equalities tell us that at constant pressure and temperature the condition for a thermodynamic equilibrium in a closed system is expressed through the equality of the chemical potentials of components i in the different phases ...comprising the system.

1.THERMODYNAMICS OF ELECTROCHEMICAL NUCLEATION

3

When the chemical species are electrically charged, the system is called an electrochemical system and the energy change of transferring charged species between phases includes also an electrical work. The phases generally have nonzero net charges and electric potential differences exist between them. Then the electrical work of interfacial charge transfer is the product of the ionic charge transferred and the difference between electrical potentials, … in the bulk of the phases involved. The potential in phase is called the inner potential or Galvani potential of phase and is the electric potential in the bulk of the phase. We shall denote the Gibbs free energy of an electrochemical system by and analogously to equation (1.2) shall define the electrochemical potential of building units of component i in phase by:

Then the thermodynamic equilibrium in a closed electrochemical system of fixed content, which is as a whole electrically neutral, is expressed through the equalities:

The electrochemical potential was first introduced by Gibbs in 1899 [1.1] but later reintroduced by Guggenheim in 1929 [1.2]. The Gibbs free energy changes accompanying the transfer of electrically charged matter between phases can be expressed as the sum of two contributions to the electrochemical potential, one expressing a chemical and the other an electrical term. These contributions, however, are actually inseparable, since the transfer of particles of a charged species involves both the transfer of that amount of matter as well as a charge of Coulombs between phases. As discussed by Guggenheim [1.2], for any two phases and of identical chemical composition and any ionic species i we may write:

4

Chapter 1

where is the valence, or charge number, of species i, e is the elementary electric charge and is the electrical potential difference between the two phases. Any splitting of into a chemical part and an electrical part is in general arbitrary and without physical significance.

In spite of that, in 1935 Lange and Nagel [1.3] suggested to present the electrochemical potential as:

that turned out to be a rather convenient formula since considerations of electrochemical nucleation and crystal growth phenomena invariably involve differences between electrochemical potentials and the splitting of into a chemical and an electrical part does not lead to any mistakes. Equation (1.7) shows once again that the electrochemical potential is influenced by the electrical potential of phase and by its chemical


composition. This can be stated explicitly in terms of the activity solute in

and its standard state chemical potential

5

of the

at

where k is the Boltzman constant. For a phase consisting only of substance i, such as, e.g., a pure bulk metal, For the purpose of describing further the conditions for thermodynamic equilibrium between phases, we shall consider the following electrochemical system: a solution of metal ions of charge with activity , a bulk crystal of the same metal (M) and an ideally polarizable inert electrode of a different chemical nature that is not able to exchange ions with the electrolyte solution. This inert electrode provides an electrically conducting foreign substrate where metal ions can discharge and transform into adsorbed metal atoms or adatoms of M. Using an external source, the Galvani potential of the inert foreign substrate can be made equal to the Galvani potential of the bulk crystal of M. Under these conditions the activity of the adsorbed metal phase is The state of thermodynamic equilibrium in such a system involving charged species can be described through the equality of the electrochemical potentials of in solution, of M in the crystal lattice, and of adatoms on the foreign substrate,

Making use of (1.7') these quantities are expressed as:

In equations (1.9)-(1.11) and are the chemical potentials of the metal ions in the electrolyte, the metal atoms in the bulk crystal and the

6

Chapter 1

metal adatoms, respectively; and are the standard state chemical potentials, as described above for the bulk phases and as it will be discussed further below for the adsorbate. The condition for equilibrium between the electrolyte and the bulk metal phase is obtained equating and

This is the Nernst equation, in which is the equilibrium potential difference between the bulk metal M and a solution of with activity The ratio defines the standard potential of the redox equilibrium between and M at ionic activity Equation (1.12) describes the equilibrium between the "infinitely large" solid M and a solution of its ions. Therefore the equilibrium potential does not depend on the surface properties of the bulk metal electrode, such as the number of kink sites, surface roughness, etc. If, however, the bulk metal crystal is internally strained, then its chemical potential will be in general larger than resulting in an experimentally measured electrode potential different from In a similar fashion, the condition for equilibrium between the electrolyte and the adsorbed metal phase may be obtained equating and

Combination of equations (1.12) and (1.13) yields a simple expression for the equilibrium activity of adatoms,

1.1.2

Electrochemical supersaturation

Equations (1.12-1.14) describe the state of thermodynamic equilibrium in an electrochemical system comprising metal ions in solution, metal adatoms on an inert electrode and metal atoms M in a large crystal of M. Under these conditions the system is stable, that is, the formation, growth or dissolution of a phase cannot occur. Favorable conditions for a first order phase transition, i.e., for metal ions discharging either on the crystal of M or on the foreign substrate, occur when the solution is supersaturated. This


7

means that the electrochemical potential of the electrochemically active species in the parent phase (the electrolyte solution) is larger than the electrochemical potential of the bulk metal crystal, That being the case, the difference defines the electrochemical supersaturation, which is the thermodynamic driving force for the phase transition [1.4]. If, under certain circumstances, the opposite inequality, is fulfilled then the quantity defines an electrochemical undersaturation. The latter is the thermodynamic driving force for the electrodissolution of the metal, which is another case of a first order phase transition. We shall now consider two possibilities for meeting the supersaturation condition The first one is to keep the electrochemical potential of metal ions equal to its equilibrium value

but to polarize the

metal crystal (M) to reduce its electrochemical potential This can be done by imposing on it a potential more negative than the equilibrium one Then at constant T and of the metal ions, the supersaturation is given by:

and we define the difference as the electrochemical overpotential (or electrochemical overvoltage) This definition implies that the overpotential of a cathodic reaction, such as the reduction of to M as occurs during the electrocrystallization of M, is considered a positive quantity:

Under these conditions, in the presence of an overpotential, metal ions may discharge on the surface of the crystal of bulk M. This type of electrodeposition is called electrochemical crystal growth or electrocrystallization on a like substrate and will be considered in Chapter 4. Polarization to a potential may be also imposed on a foreign substrate. If this is the case then it is possible to form metal nuclei on its surface. A nucleus is a cluster of metal atoms that carries the physical properties of the new crystalline phase and it is the nucleus formation what we consider as a precursor of the overall electrocrystallization phenomenon.

8

Chapter 1

Again, the supersaturation is and holds for the electrochemical overpotential The second way to fulfil the inequality and to supersaturate the electrochemical system is to keep the metal crystal (M) at the Galvani potential but to make the electrochemical potential of metal ions in solution higher than the equilibrium one This can be accomplished by raising the activity of the metal ions to a value higher than the equilibrium activity In that case the supersaturation is given by:

whereas:

holds good for the electrochemical overpotential As before, under such conditions the metal ions may either deposit on the bulk crystal or may form nuclei of the new phase on the inert foreign substrate, depending on which of them is the working electrode for the metal electrodeposition. Finally, since an increased adatoms activity corresponds to the electrode potential the supersaturation and the electrochemical overpotential can be presented as:

We should emphasize that equations (1.15-1.20) suggest purely thermodynamic definitions of and In particular, the electrochemical overpotential described above differs essentially from the electrode kinetic quantity defining the kinetic overpotential for charge transfer across an interface with flow of a current i through the electrochemical system. The two quantities, and can be considered as measures for the deviation from the state of stable thermodynamic equilibrium, but the mere fact that the parent phase is supersaturated does not mean that a phase


9

transition should necessarily occur. The rate of this process depends on its mechanism and can be determined only by means of kinetic considerations. This subject will be considered in Chapter 2. Description of the thermodynamic state of the adsorbate needs further consideration. Foremost it is necessary to stress that equation (1.19) represents a general expression for the supersaturation dependence of the activity of adatoms on the inert foreign substrate. In order to derive an explicit formula for the surface concentration of adatoms, it is necessary to find out the interrelation between the activity the number of single adsorbed atoms, and the number of adsorption sites on the foreign substrate (see e.g. [1.5-1.7] and the references cited therein). The result will depend on how the free energy of the adsorbed metal phase is calculated, and here we present three examples for the functional relationship Assuming the number of adsorption sites to be much greater than the number of adatoms, is given by:

Combining equations (1.14) and (1.19-1.21) for

known as Henry’s isotherm. Assuming then the activity of adatoms is given by:

it results:

to be commensurable with

and Langmuir’s isotherm [1.8] is obtained from equations (1.14), (1.20) and (1.23):

Taking into account the lateral interaction between the adsorbed atoms then:

and Frumkin’s isotherm [1.9] follows from equations (1.14), (1.20) and (1.25):

10

Chapter 1

Here g is a non-dimensional constant accounting for the interaction energy between nearest neighbor adatoms and is positive for attraction and negative for repulsion. It can be obtained in an explicit form in the framework of the mean field approximation of Bragg-Williams [1.10], Note that for equations (1.22), (1.24) and (1.26) transform into expressions for the corresponding equilibrium adatoms concentration In Chapter 1.1.1 we defined the chemical and electrochemical potentials as molecular quantities. However, sometimes it proves convenient to work with the corresponding molar quantities. In that case the building unit of a given phase is selected to be not a particle but a mol. Bearing in mind the simple relation between the number m of moles and the number n of particles, where is the Avogadro number, the molar electrochemical potential is defined as:

Correspondingly, the Nernst equation (1.12) transforms into:

where and are the corresponding molar chemical potentials, is the molar gas constant and is the Faraday constant. The supersaturation becomes:

or

etc. Throughout this book we shall use the molar quantities when considering the growth of stable clusters (Chapter 4) and the theory of mass electrocrystallization (Chapter 5).


1.2

Nucleation work

1.2.1

General formulation

11

We have already stated in Chapter 1.1.2 that to supersaturate the parent phase may not be enough to initiate the phase transition. Why is the nucleus formation not an instantaneous process? In order to answer this question we shall consider the formation of an n-atomic nucleus of the new phase from n single metal ions of the electrolyte solution. The supersaturation is given by equation (1.15) and the nucleus is formed on the surface of an inert foreign substrate.

To immerse the working electrode in the electrolyte solution means to create a new dividing surface in the bulk of the liquid phase. The work done in this process accounts for the energy excess due to the creation of the new interface and takes into consideration the energy contribution of all physical processes that may take place at the solution-substrate phase

12

Chapter 1

boundary: the formation of an electric double layer, the adsorption of atoms, ions, water and organic molecules, etc. Therefore, in the initial state, before the formation of a nucleus, (Fig. 1.1a), the Gibbs free energy of the whole system is:

where is the Gibbs free energy of the electrolyte solution, N is the initial total number of metal ions and takes into consideration the energy contribution of the foreign substrate, including the interaction energy with the electrolyte solution. Clearly, is a function of the electrode potential E. For sufficiently large, finite working electrodes, can be split into a volume, and a surface, term:

being the Gibbs free energy of the foreign substrate reflecting its bulk properties that remained unaffected by the contact with the liquid phase. For small, say, nanoelectrodes, however, such splitting would be wrong and without physical significance. In the final state when the n-atomic metal nucleus is already formed (Fig. 1.1b) the Gibbs free energy of the whole system changes to:

where (N – n) is the reduced number of ions in the electrolyte after the nucleus formation. Here we assume that N is always much bigger than n and that the transfer of n ions from the bulk of the electrolyte to the electrode surface does not change the electrochemical potential The term in equation (1.30) accounts for the energy contribution of the foreign substrate, as changed by the presence of the n-atomic nucleus thereon. Thus it is the difference that gives the Gibbs free energy of the n-atomic cluster on the electrode surface. It depends on the cluster structure, which may not necessarily coincide with the structure of the bulk new phase. According to the classical thermodynamics, the difference gives the energy barrier for the formation of the nucleus consisting of n atoms and is called nucleation work. Thus from equations (1.28)-(1.30) for one obtains:


13

Equation (1.31) tells us that what we call nucleation work is the difference between the Gibbs free energy of n atoms, when they form an individual n-atomic cluster of the new phase on the foreign substrate, and the Gibbs free energy of the same number of particles, but when they belong to the bulk, supersaturated parent phase – the electrolyte solution of metal ions. In order to introduce the supersaturation (equation 1.15) in equation (1.31), the expression for the nucleation work . rewritten in the form:

must be

where

is another important quantity in the nucleation theory. It represents the difference in the Gibbs free energies of n atoms when they form an individual n-atomic cluster on the foreign substrate, and when they are part of the bulk new phase, the latter being set under the nucleation conditions – temperature T, metal ions activity

and electrode potential

It is that accounts for the energy contribution of the nucleus-substrate and the nucleus-solution interface boundaries and has the physical meaning of the concept surface free energy defined in this general form by Stranski in 1936 [1.12, 1.13]. The physical meaning of equation (1.32) is simple. The positive term in it, appears because some work has to be done to create the new interfaces. The negative term, is exactly the work gained in the transfer of n particles from the state of higher to the state of lower electrochemical potential since under the nucleation conditions Thus the difference

is the net work done

for the nucleus formation. For small clusters the inequalities and hold, and this is the purely thermodynamic reason for the nucleus formation to be connected with the overcoming of an energy barrier and to be not an instantaneous process. For large clusters the volume term exceeds the surface one becomes smaller than zero

14

Chapter 1

and the phase transition leads eventually to a total decrease of the Gibbs free energy of the whole system. The general expressions for the nucleation work equations (1.31) and (1.32), are valid for condensed, liquid and solid, phases when the total volume of the supersaturated system does not change essentially if an natomic nucleus forms. In this form they apply without any restrictions even to clusters consisting of a very small number of particles, including the limiting case n = 1. Similar formulae for were derived by Kaischew [1.14, 1.15] who considered the homogeneous and heterogeneous formation of crystalline nuclei from vapours and melts.

The above thermodynamic derivation of the nucleation work raises the important question for the meaning of the concept electrochemical potential of the atoms of a small cluster. In section 1.1.1 we defined the electrochemical potential of a single particle (atom, ion, molecule) but when the latter belonged to a sufficiently large ensemble and the Gibbs free energy could be considered as a function of the continuous variable For small clusters consisting of a low number of atoms the Gibbs free energy is a discrete quantity and the electrochemical potential of a cluster atom cannot be defined by means of equation (1.5). The problem is solved by introducing the following atomistic definition of the concept electrochemical potential [1.4, 1.11, 1.15]: if the Gibbs free energies of an n–1-atomic and of an n-atomic clusters are and respectively, the difference gives the electrochemical potential of the n-th atom in the n-atomic cluster. As an example, Figure 1.2 shows two clusters of the new phase – the 3- and the 4atomic ones. In this case the difference gives the electrochemical potential of the fourth atom in the 4–atomic cluster. The above considerations have been performed expressing the supersaturation by the difference (equation 1.15). The same general formula for the nucleation work

(equation 1.32)


would be obtained if the supersaturation

15

was expressed by the

difference (equation 1.17). Finally, equation (1.32) shows that given the supersaturation the work of formation of any n-atomic cluster of the new phase can be found if is calculated taking into account the supersaturation dependence of the nucleus size n. In what follows we shall show how this is done in the framework of the classical nucleation theory. 1.2.2

Classical nucleation theory

The general expression for the nucleation work, equation (1.32), says that has a minimal value for clusters formed with a minimal excess energy In terms of the classical nucleation theory developed in the pioneering works of Gibbs [1.1], Volmer [1.11], Volmer and Weber [1.16], Kossel[1.17], Stranski [1.12, 1.13, 1.18, 1.19], Farkas [1.20], Stranski and

Kaischew [1.21-1.23] and Becker and Döring [1.24], such clusters are supposed to satisfy the requirement for an equilibrium form: they must achieve a minimal total surface free energy for a constant volume V .This

16

Chapter 1

outlines the region of validity of the classical theory of nucleation – it applies to sufficiently large clusters for which macroscopic concepts like surface and volume do have physical significance. That being the case, the size of the nucleus n can be considered as a continuous variable. 1.2.2.1 The concept specific free surface energy Before deriving explicit expressions for the nucleation work it is necessary to define the concept specific free surface energy [1.25]. For that purpose let us consider two finite solid bulk phases with equal size, and (Figure 1.3a) immersed in a solution of metal ions which represents a third, liquid phase. The pressure P and the temperature T are kept constant and the two solid phases are polarised to the same potential E. Cleaving, reversibly and isothermally, the into two halves, A and B, results in the creation of two new dividing surfaces in the bulk of the electrolyte solution, each of them having a surface area S. If is the work done in this process, the work referred to unit surface area, is defined as the specific free surface energy, of the interface boundary. Analogously, if is the work done to cleave into two halves A and B, the quantity is defined as the specific free surface energy, of the interface boundary. Note that the works and should be expressed, though formally, as and respectively, where and are the works done in cleaving and into two halves in vacuum and and are the works gained due to the solvation of the new interfaces in the electrolyte solution, the appearance of the electric double layer, etc. It is the potential dependence of and that makes the specific free surface energies potential-dependent quantities. Since and are the specific free surface energies of and in vacuum, the conditions and are always fulfilled. Indeed, of silver measured in vacuum is [1.26], whereas a value of is obtained for of silver clusters in buffer solutions with different Redox potentials [1.27, 1.28]. Suppose now that is a substrate and is a crystal that has to be deposited thereon. In doing this, two new identical interface boundaries each of them having a surface area S, are created (Figure 1.3b). The net work done in this process is:

where the sum halves A and B and

is the work done in cleaving and into is the work gained in creating the two contact


17

18

interface boundaries area, one obtains:

Chapter 1

Referring the work

to unit surface

which can be rewritten in the form known as a relation of Dupré [1.29]:

Here,

is defined as the specific free interfacial energy, and is defined as the specific free adhesion energy, the latter giving the work per unit surface area that has to be done to separate the crystal from the foreign substrate. Apparently, in the case of identical phases 1 and 2, and As expected, the last relation says that the work done in splitting two identical phases in parts equals exactly the work gained in rebuilding the phases to their initial state. 1.2.2.2 Equilibrium forms 1.2.2.2.1 Liquid droplets on a foreign substrate A liquid droplet formed on a flat solid foreign substrate has the “capshaped” form of a spherical segment (Figure 1.4) with a volume:

where R is the radius of the homogeneously formed sphere and is the so called wetting angle. In order to find the total surface free energy of the three phase electrochemical system droplet–solution–foreign substrate we shall proceed in the following way. In the initial state, before the droplet appearance, the surface free energy of the substrate/solution interface is:

where is the substrate total surface area. In the final state when the liquid droplet is already formed, the surface free energy becomes:

Here and are the droplet/solution and the droplet/substrate contact surface areas, respectively. Thus for the difference giving the total surface free energy one obtains:


19

where, to simplify notations, and stand for the specific free surface energies and respectively. Bearing in mind the condition for an equilibrium form, dV = 0, after differentiation of equations (1.36) and (1.39) one obtains:

20

Chapter 1

Equating (1.36') and (1.39') yields the simple relation:

known as the Young’s rule. Introducing the adhesion energy (equation (1.35)), one obtains:

according to

which is a combination of the Young’s rule and the rule of Dupré. Equations (1.40) and (1.40') show that the wetting angle of a cap shaped liquid droplet on a flat foreign substrate is exactly determined by the values of the specific free surface energies and If the foreign substrate plays the role of an working electrode and the specific free surface energies are potential dependent quantities, the wetting angle is a function of the electrode potential, too. For and which means that the droplet does not feel the presence of the foreign substrate. This is the case of complete

non-wetting for which and the equilibrium form is a sphere (Figure 1.5a). For the droplet transforms into a monolayer disk (Figure 1.5b), which is the case of complete wetting. For the equilibrium form is a hemisphere (Figure 1.5c), etc. Equations (1.40) and (1.40') determine the equilibrium form of sufficiently large liquid droplets. With decreasing the droplet size it is necessary to take into consideration the contribution of the specific free line


21

energy (or line tension) to the energy balance [1.30-1.34]. In this case the expression for the total surface free energy transforms into:

where is the length of the three phase contact line (Fig. 1.4a). Applying the same procedure dV = 0), the condition for the equilibrium form becomes:

which tells us that for small droplets the wetting angle curvature radius R. For and (1.42) can be rewritten in the form:

is a function of the so that equation

This allows us to find the relationship given the ratio and the macroscopic wetting angle Since the line tension could be either zero or a positive, or a negative quantity [1.1, 1.34-1.37] and its order of magnitude is likely to be [1.35, 1.37], we shall select for an illustrative calculation of the relationship. This should be a reasonable value for mercury droplets deposited on a platinum working electrode from a mercury nitrate solution (Figure 1.6.) since in that case [1.38, 1.39] and [1.40].

22

Chapter 1

Figure 1.7 shows that “cap-shaped” liquid droplets cannot exist below certain minimal radius and the corresponding limiting wetting angle is

Making use of equation (1.36) for one obtains which means that mercury droplets containing less than atoms cannot maintain the spherical form on a platinum surface. The evaluation of is made bearing in mind that being the volume of a single mercury atom. For cm and the corresponding limiting wetting angle is This is, however, a physically unrealistic result since appears to be smaller than the diameter of the mercury atom Another peculiar effect caused by the line tension is that the droplets can have two different wetting angles for each curvature radius R. For the two wetting angles are bigger than the macroscopic one since the line tension tends to reduce the length L of the three phase contact line. For the two wetting angles are smaller than the macroscopic one since for a negative line tension the length L increases thus diminishing the total surface free energy Concluding, we should emphasise that equations (1.40) and (1.42) were derived neglecting the dependence of and on the radius R [1.41-1.47].


23

Since this dependence becomes significant for small R all conclusions drawn on the basis of equations (1.40) and (1.42) should be accepted with caution if the droplets contain low number of atoms. Certainly, it is difficult to say what does “low number of atoms” mean, particularly for liquid droplets formed on a foreign substrate. In order to give an idea of the order of magnitude, we shall use the Tolman formula [1.41] for the relationship of a homogeneous sphere:

Here refers to the flat interface boundary and is evaluated to 1 to 2 atomic diameters [1.43]. Assuming that the dependence could be neglected for radii for which for mercury one obtains The volume of such a spherical droplet is which corresponds to 1.2.2.2.2 Three-dimensional crystals on a foreign substrate A crystal is characterized by a periodic arrangement of its building units – ions, atoms or molecules. There are 14 possible different arrangements of the crystal building particles in the three-dimensional space. They are known as spatial lattices of Bravais [1.48, 1.49] (Figure 1.8) and describe the crystals’ structure by means of three crystallographic axes a, b and c, and the angles and between them. Three co-ordinates, known as lattice constants fix the location of each lattice point: along the a-axis, along the b-axis and along the c-axis. Values of the lattice constants of some face-centred cubic (fcc) metal crystals are given in Table 1.1.

The crystals are surrounded by faces, edges and apexes the number of which fulfils the relation of Euler-Deckart: Faces + Apexes = Edges + 2 The position of each crystal face ABC (Figure 1.9a) can be defined by the intercepts OA, OB and OC and the angles and between the normal OM to the face ABC and the axes a, b and c. The definition ratio is:

24

Chapter 1


25

where the intercepts OA, OB and OC refer to the corresponding lattice constants and as

m, n and p being integers. Thus the ratios 1/m : 1/n : 1/p = h : k : l define the Miller indices h, k and l used in crystallography to characterise the orientation of a crystal face. Figure 1.9b shows a crystal with a simple cubic lattice and (100), (111) and (110) faces. If a crystal contacts a foreign substrate the total free surface energy of the system crystal–solution–foreign substrate is [1.50]:

Here and are the specific free surface energies and the surface areas of the crystal faces which contact only the solution; and are the specific free interfacial energies and the surface areas of the faces which contact the foreign substrate and is the substrate specific free surface energy. Substituting the difference for according to equation (1.35) one obtains:

where is the specific free adhesion energy and is the specific free surface energy that the face k would have if the substrate was absent. Since a closed crystal polyhedron can be considered as composed of pyramids with a common apex P (point of Wulff) inside the crystal, its volume V is given by:

where and are the distances between the Wulff ’s point and the j and k faces. Making use of equations (1.44') and (1.45) the condition for the equilibrium form dV = 0 leads to the Wulff’s rule (Gibbs-CurieWulff theorem [1.51-1.53]) generalized by Kaischew [1.50] to account for the crystal-substrate interaction:

26

Chapter 1

or

for the crystal faces, which contact only the parent phase, and

or

for the crystal faces, which contact the foreign substrate (Figure 1.10), where is a material constant. Combining equations (1.46) and (1.47) one obtains another expression of the same theorem:

Substituting

for

and

for

equation (1.48) transforms into:

which shows that the distances from the Wulff’s point to the faces that contact the foreign substrate are exactly times shorter than the distances from the Wulff’s point to the same faces but when the substrate is missing [1.50] (Figure 1.10a). The height of the crystal is For which is the case of complete non-wetting (Figure 1.10b), and the height h of the crystal equals For values of approaching tends to and the three-dimensional crystal degenerates into a twodimensional monolayer (Figure 1.10c). This is the limiting case of complete wetting when the height h equals the atomic diameter. The Wulff’s point P lies inside the crystal if (Figure 1.10a, b and d) and lies outside the crystal if (Figure 1.10c). In the last case the height of the crystal is


27

28

Chapter 1

Very often the theoretical models of electrochemical phase formation consider the nucleation and growth of three-dimensional spherical clusters. The derived theoretical formulae are used also for interpretation of experimental data obtained with crystalline phases and this brings up the question to what extent does the concept of wetting angle have any meaning with solid substances. The answer lies in the combination of equations (1.40') and (1.48') giving the connection between the wetting angle, the specific free surface and adhesion energies and the distances from the Wulff’s point to the crystal faces:

The relation (1.49) reveals that is the wetting angle of the spherical segment with a curvature radius R inscribed in the crystal polyhedron (Figure 1.10d, ). Note, however, that this is valid only for simple crystallographic forms. Besides, one should bear in mind that to apply the spherical approximation to a crystalline cluster means to replace the specific free surface and adhesion energies of the different crystallographic faces for single, averaged values and Finally, the Wulff’s rule enables us to construct the equilibrium form of a crystal given the values of the specific free surface and adhesion energies. For that purpose perpendiculars with lengths proportional to the corresponding specific free energies are drawn from the Wulff’s point to all possible crystallographic planes. The most inner from all obtained closed polyhedrons represents the equilibrium form (Figure 1.11).


29

1.2.2.2.3 Two-dimensional crystals on a like substrate The conditions hold good for a like substrate when two-dimensional crystals form on the top of a three-dimensional one having the same chemical composition. In this case the condition for the equilibrium form reads: where

is the total free edge energy of the two-dimensional crystal polygon. The latter consists of triangles with a common apex in the Wulff’s point and its surface area is:

In equations (1.50) and (1.51) are the specific free edge energies, are the lengths of the edges and are the distances from the Wulff’s point inside the polygon to its j-th side. Note that the quantity is defined as the work done to create a unit edge length. It does not coincide with the line tension and can never be negative. In the case of two-dimensional crystals the condition for the equilibrium form leads to [1.54]:

which is another form of the Wulff’s rule,

being a constant.

1.2.2.3 Growth forms The Young’s and Wulff’s rules determine the form of droplets and crystals that are in equilibrium with the ambient phase. However, what we can and do observe in growth experiments are forms of growth (Figures 1.6 and 1.12), which may depend strongly on the particular experimental conditions. While liquid droplets grow, generally preserving the form of a spherical segment and it is the wetting angle that may change at the growth overpotential, the different crystallographic faces could have quite a different growth velocity and the growth forms differ significantly from the equilibrium ones. A most general rule says that faces having a higher growth rate disappear from the growing crystal and the growth forms contain simple, closely packed crystallographic faces that spread with a lower velocity. Thus the equilibrium polyhedron from Figure 1.9b should

30

Chapter 1


31

transform into a simple cube due to the anisotropy of the growth rate Organic and inorganic additives to the electrolyte solution may also change the form of a growing crystal. Due to their specific, potential dependent adsorption on the different crystallographic faces such substances may affect the growth rate in a different manner depending on the adsorption capacity of the particular crystallographic face. The situation is additionally complicated at the advanced stage of the growth process when the mass transport dominates the growth kinetics. At that stage diffusion limitations may cause a non-uniform concentration distribution around the growing crystal thus compensating the anisotropy of the growth rate and leading to skeletal and dendritic modes of growth [1.571.61]. In the case of electrocrystallization such a morphological instability was experimentally observed at relatively high densities of the growth current of silver and cadmium single crystals (Figure 1.13) [1.62-1.64]. Without going into more details, we should stress that to construct a growth form is a difficult task which requires profound knowledge of the crystal structure, of the mechanism of growth, and last not least, of the values of the specific free surface and adhesion energies. Unfortunately, the latter are not always known with a sufficient accuracy. 1.2.2.4

The concept critical nucleus and the work for critical nucleus formation The general formula for the nucleation work (equation (1.32)) provides the possibility to obtain explicit expressions for this quantity if the surface free energy is evaluated accounting for the supersaturation dependence of the nucleus size To illustrate the thermodynamic method developed by Gibbs [1.2] and Volmer [1.11,1.16] we shall calculate the work of formation of two-dimensional (2D) and threedimensional (3D) liquid and crystalline nuclei on flat foreign substrates. For the sake of simplicity in these calculations we shall consider the idealized case of a “structureless” substrate thus neglecting any lattice mismatch between the electrode surface and the nuclei of the new phase. 1.2.2.4.1 Liquid nuclei on a foreign substrate In this case the surface free energy of an n-atomic liquid cluster is given by equation (1.39), which can be rewritten in the form:

32

Chapter 1

if the term is replaced by according to the Young’s rule. In order to present the surface areas and as functions of n it is enough to recall the simple relation which combined with equation (1.36) yields:

Substituting the last expression in equation (1.53) after simple algebra one obtains:

and the formula for the nucleation work

Here

becomes:


33

and the wetting angle function equals exactly the ratio of the volume V of a spherical segment (equation (1.36)) and the volume of a homogeneous sphere with the same radius R., i.e.:

The shape of the function is shown in Figure 1.14. A simple inspection of equation (1.55) shows that the function displays a maximum at a certain critical value of the nucleus size (Figure 1.15). Hence the first definition of the concept critical nucleus follows: (1) Cluster of the new phase formed in a supersaturated system with a maximal work

34

Chapter 1

We should remind, however, that the critical nucleus is defined as a cluster of the new phase having the equilibrium form. The formation of any cluster of size that does not obey the condition dV = 0 requires higher nucleation work. In the classical theory the number of atoms n is considered as a continuous variable, is a differentiable function and the condition for extremum

applied to equation (1.32) yields:

The last formula is a general expression of the Gibbs - Thomson equation giving the interrelation between the size of the critical nucleus and the supersaturation It must be pointed out that equation (1.56) can also be derived without calculating the nucleation work [1.65-1.67]. In order to demonstrate this we consider an electrolyte solution containing metal ions with electrochemical potential and an n-atomic cluster of the new metal phase formed on an working electrode with a Galvani potential The temperature T and the total number of particles in the whole system are kept constant. Assuming that the cluster is a sufficiently large spherical segment its thermodynamic potential can be presented as a sum of a volume term and a surface term and the thermodynamic potential of the three phase electrochemical system electrode – electrolyte – n-atomic cluster is expressed as:

Here is the Gibbs free energy of the foreign substrate (equation (1.29)) and the term gives the substrate free surface energy accounting for the fact that part of the substrate, is occupied by the nucleus of the new phase. What is, then, the size of the cluster that remains in equilibrium with the ambient phase? For the equilibrium state is characterized by a minimum of the thermodynamic potential at a constant composition the conditions dN=0 yield:


Substituting

35

for (-dn) in equation (1.58), the latter transforms into:

Since the equilibrium is established at a cluster size

and since

it follows that:

which coincides with the Gibbs-Thomson equation (1.56) for second definition of the concept critical nucleus can be given:

Hence a

(2) Cluster of the new phase that is in unstable equilibrium with the supersaturated parent phase. We speak about unstable equilibrium because attaching more atoms from the parent phase the critical nucleus turns into a stable cluster and grows irreversibly. On the contrary, detachment of atoms from the critical nucleus leads to its irreversible decay. In the classical nucleation theory the definitions (1) and (2) are fully identical. However, we shall show in Chapter 1.4 that the first definition is more general. Combining equations (1.56) and (1.53') one obtains:

which represents the supersaturation dependence of the critical nucleus size The latter is schematically shown in Figure 1.16. Correspondingly, the work for critical nucleus formation is [1.11, 1.68]:

36

Chapter 1

Thus equations (1.53'), (1.62) and (1.63) reveal the fundamental relations between and

Making use of equations (1.53') and (1.64) and presenting

and

as:

and

the substitution of equations (1.64') and (1.64") in equation (1.55) yields [1.50]:


37

or

where V and are the volumes of the n- and the clusters, respectively. The above formulae allow us to calculate the work of formation of any n-atomic cluster of the new phase given the size of the critical nucleus and the work of critical nucleus formation. We should emphasize that equations (1.56), (1.64), (1.65) and (1.65') are general expressions valid for all kinds of three-dimensional clusters with an equilibrium form and can be applied to the case of crystalline nuclei, too. Using the R(n) relationship (equation (1.54)) and expressing in equation (1.62) through one obtains:

which is another form of the Gibbs-Thomson equation. It gives the interrelation between the supersaturation and the radius of the homogeneously formed sphere that would be a critical nucleus in the case of a complete non-wetting when and the nucleation work is:

Another useful expression of the Gibbs–Thomson equation can be obtained if the supersaturation is presented through the difference between the equilibrium potential of the bulk metal phase and the actual potential E of the working electrode where the critical nucleus is formed (equation (1.15)). In that case equation (1.66) transforms into [1.69]:

38

Chapter 1

where is the electrochemical overpotential. Since the critical nucleus is defined as a cluster in equilibrium with the ambient phase, we can consider it as a spherical electrode with an equilibrium potential more negative than . In the same fashion it is possible to ascribe an equilibrium potential E(R) to every cluster of the new phase with a curvature radius R and to formulate the general relation:

Thus the difference defines the effective overpotential and correspondingly, the effective supersaturation that act on the clusters having sizes different from the size of the critical nucleus determined by the externally applied overpotential Combining equation (1.66') and (1.68) one obtains [1.70]:

or

which shows that given the applied overpotential the fate of any n-atomic cluster of the new phase is determined by the sign of its effective overpotential Thus clusters having exceeded the critical size, will grow since for and the effective supersaturation is Subcritical clusters having sizes will dissolve since for them the system is effectively undersaturated, i.e. and Finally, clusters having the critical size will stand in equilibrium with the ambient phase since for them and are both equal to zero. Thus far we derived analytical expressions for the nucleation work and for the size of the critical nucleus neglecting the line tension effect. The latter can be taken into consideration if the general equation (1.32) is combined


39

with equation (1.41) for making use of the R(n) relationship (1.54). Here we shall present only the final results for and as obtained by Gretz[1.31]:

Note, however, that the above formulae do not reveal the explicit dependencies of and on the supersaturation since the critical wetting angle is a function of that is implicitly included in equations (1.70) and (1.71). Equation (1.70) tells us that the line tension may increase or decrease the nucleation work depending on the sign of What is more, for negative values of the line tension the condition applied to equation (1.70) allows definition of a limiting supersaturation:

at which the nucleus formation proceeds without any thermodynamic barrier [1.71]. At the same time the size of the critical nucleus:

may differ significantly from zero. We should remind, however, that positive or negative the line tension effects become important for relatively small clusters. As we mentioned in Chapter 1.2.2.2.1 in that case it is necessary to take into account the and the relationships [1.40-1.47] in the derivation of equations (1.70) – (1.73). Certainly this would be a difficult problem and there is no doubt that such a general theory, if developed, would predict different formulae for and In the

40

Chapter 1

electrochemical case the situation should be additionally complicated since and are potential dependent quantities. 1.2.2.4.2 Three-dimensional crystalline nuclei on a foreign substrate If crystalline clusters are formed on the electrode surface the nucleation work and the size of the critical nucleus are calculated in an analogous way combining equations (1.32), (1.45), (1.46) and (1.56) [1.50, 1.72]. The obtained results are:

where is a material constant that can be found in each particular case using the generalized Gibbs-Curie-Wullf theorem (equations (1.46) and (1.47)). Thus for crystalline nuclei with a face centered cubic lattice:

when the nuclei contact the substrate with the cubic, (100), face and

when the nuclei contact the substrate with the octahedral, (111), face [1.73] etc. Another form of the Gibbs-Thomson equation following directly from the Wullf’s rule is [1.50, 1.72]:

which reveals the meaning of the constant For the critical nucleus

in equations (1.46) and (1.47).


41

1.2.2.4.3 Two-dimensional nuclei on a foreign substrate For high supersaturations and/or strong adhesion between the substrate and the new phase the classical nucleation theory predicts the possibility to form two-dimensional nuclei on a foreign substrate. Brandes [1.74] was the first who in 1927 considered this case of phase formation and found that the nucleation work equals one half of the total edge energy of the critical nucleus. Here we shall derive explicit expressions for the nucleation work of an disk-shaped two-dimensional nucleus (Figure 1.17) with a monoatomic height formed on a foreign substrate with a surface area For the purpose we shall use the general equation (1.32) which requires the determination of the surface free energy excess . In

order to find this quantity we shall use the approach described in Chapter 1.2.2.2. Thus before the nucleus formation the surface free energy is:

whereas after the nucleus formation it changes to:

Here l is the radius of the two-dimensional disk and the term accounts for its peripheral edge energy. Denoting by s the surface area occupied by a single atom in the diskshaped cluster for the number of atoms one obtains and for the total surface free energy it results:

42

Chapter 1

where according to the relation of Dupré (equation (1.35)). Therefore the nucleation work becomes:

Making use of equation (1.56) and (1.81) the Gibbs-Thomson equation is expressed as:

Substituting equation (1.83) in equation (1.82) for critical nucleus formation is obtained in the form:

the work for

or

The Gibbs-Thomson equation (1.83) can be expressed also through the critical radius of the two-dimensional disk if is replaced by In that case one obtains:

Substitution of

from equation (1.86) into equation (1.84) yields:

which equals exactly one half of the total peripheral edge energy the critical nucleus as obtained by Brandes [1.74]. If two-dimensional nuclei with a crystalline structure are formed on foreign substrate the nucleation work and the size of the critical nucleus derived in an analogous way and the formulae for and given by:

of the are are


43

where and are constants depending on the orientation of the twodimensional nuclei. Thus for crystals with a face centered cubic (fcc) lattice oriented with the (100) face parallel to the substrate

whereas in the case of (111) orientation it holds:

In the last expressions is the surface area occupied by a single atom in the crystal lattice, is the volume of one atom and and are the specific free edge energies of the cubic and the octahedral edges of the two-dimensional fcc crystal. If only interactions between first neighboring atoms are taken into consideration the specific free edge energies refer to the corresponding specific free surface energies as and [1.73]. Bearing in mind that for fcc crystals [1.75, 1.76], it appears that Finally, we should note that the expressions derived for and apply to two-dimensional nucleation on a like substrate, too, taking into account that in that case equals zero.

and the constant

44

Chapter 1

1.2.2.5 Modes of nucleation and growth on a foreign substrate An important point in the nucleation theory and experiment concerns the mode of phase formation – two- or three-dimensional nucleation and growth – on a foreign substrate. The problem has been considered from a thermodynamic point of view [1.72, 1.73, 1.77-1.81], by comparing the works of two- and three-dimensional nucleation and analyzing their dependence on the supersaturation and on the specific free surface and adhesion energies. The thermodynamic criteria derived for the case of structureless substrates [1.73] tell us that three-dimensional nuclei will be formed in the case of incomplete wetting, and always under supersaturations In

that case the condition

is fulfilled for supersaturations

satisfying the condition:

Two-dimensional nuclei will be formed if

and:

This inequality implies that for strong attractive forces between the nuclei and the foreign substrate when the adhesion energy (complete wetting) 2D nuclei are formed at any supersaturation In that case two-dimensional nuclei can also be formed at equilibrium, and even at undersaturations if In electrochemistry the last case is known as underpotential metal deposition (UPD) but this type of processes are beyond the scope of this book. For crystalline clusters equations (1.94) and (1.95) transform into:

where is a numerical constant depending on the orientation of the crystals on the foreign substrate. Thus in the case of fcc crystals contacting the surface with their cubic, (100), face and If the fcc crystals contact the surface with their octahedral, (111), face and


The transition supersaturation

45

at which 3D nuclei transform into

2D ones with a mono-atomic height is determined by the equality and is:

or

The situation is more complex if crystalline nuclei are formed not on “structureless” but on foreign crystalline substrates. In that case it is necessary to take into consideration both the orientation effect of the substrate and the crystallographic lattice mismatch at the nucleus-substrate interface boundary. Such phenomena are known as epitaxial crystal growth and for more information on this important subject the readers are referred to [1.72, 1.82-1.86]. Here we shall present only the widely used classification of the different modes of epitaxial crystal growth proposed by Bauer [1.77] in 1958 (Figure 1.18). In the case of weak adhesion between the substrate and the depositing crystals only three-dimensional nuclei are formed on the foreign substrate as predicted by Volmer and Weber [1.16]. Depending on the lattice mismatch the 3D nuclei could be more or less internally strained. This mode of epitaxial crystal growth is known as Volmer-Weber mechanism (Figure 1.18a).

In the case of strong adhesion between the substrate and the depositing clusters and relatively small crystallographic lattice mismatch twodimensional adsorption or phase formation may take place already at

46

Chapter 1

equilibrium and even at undersaturations (UPD). Two-dimensional nuclei are formed also at supersaturations and the growth follows a layer - by layer mechanism as predicted by Frank and van der Merwe [1.87, 1.88], This mode of epitaxial crystal growth is known as Frank - van der Merwe mechanism (Figure 1.18b). In the case of a strong adhesion between the substrate and the depositing clusters but significant crystallographic lattice mismatch the layer-by-layer growth takes place only during the deposition of the first few monolayers. After that, the accumulated internal strain energy due to the strong lattice mismatch compensates the attractive forces with the substrate and internally strained 3D nuclei form on the top of the 2D monolayers. This mode of epitaxial crystal growth is known as a Stranski-Krastanow mechanism (Figure 1.18c) named after I.N.Stranski and L.Krastanow who considered such type of nucleation and crystal growth phenomena already in 1938 [1.89]. Concluding, we should emphasize that the above criteria for the mode of nucleation on foreign substrates (equations (1.94) - (1.96)) were formulated on the basis of macroscopic thermodynamic considerations. However, the phase formation phenomena depend on the kinetics of nucleation and growth, too, and for that reason the conclusions of the above purely thermodynamic analysis may not apply to real physical systems.

1.3

Theory of mean separation works

1.3.1

The concept half-crystal position

The theory of phase formation developed in the works of Gibbs [1.1] and Volmer [1.11] proposes a comprehensive thermodynamic description of the nucleation phenomena. However, it does not provide any information on the mechanism of the physical processes taking place on the crystal surface. Kossel [1.17] and Stranski [1.18,1.19] were the first who in 1927 realized, independently of each other, the necessity of a close consideration of the elementary acts of attachment and detachment of single particles (atoms, ions or molecules) to and from the crystal surface. Undoubtedly, the most important contribution of Kossel and Stranski to the theory of nucleation and crystal growth is the definition of the concept half-crystal position. In an infinitely large crystal, the atom in this, often called kink site position, (Figure 1.19b), is bonded with a semi-infinite crystal block, with a semi-infinite crystal lattice plane and with a semiinfinite crystal row. Attaching or detaching one atom to and from the halfcrystal position a new half-crystal position is created and this is what makes


47

this position a repeatable step in the successive building or disintegration of the bulk crystals.

Another important finding of Kossel and Stranski is that all atoms of the infinitely large crystal are bonded as strongly as the atom in the half-crystal position. In order to prove this we shall first elucidate the concept neighboring atoms in a crystal body. The easiest way is to consider the crystal with a simple cubic lattice, often called Kossel’s crystal (Figure 1.19a), and to imagine that its atoms are spheres inscribed in small cubes with an edge equal to the atomic diameter d. Then the meaning of the first, second and third neighboring atoms becomes clear from Figure 1.20.

As seen, in this particular case the distance between first neighbors equals d, the distance between second neighbors equals and the

48

Chapter 1

distance between third neighbors equals Note that the distances between the centers of the neighboring atoms.

and

are

Consider now a three-dimensional cubic crystal consisting of building units (Figure 1.2la). For the sake of simplicity, at this stage we shall assume that the crystal is formed in a vapor phase. The specific influence of the electrolyte solution will be considered afterwards. In order to calculate the total bond energy of the crystal lattice we shall disintegrate the crystal in the following way: (i) The dark cube containing atoms (Figure 1.2la) is disintegrated to single particles. To simplify the calculations, we consider only the interactions between first neighbors, denoting the bond energy between them by Since each of the atoms is separated from three first neighbors the work done is:

(ii) The three quadratic crystal planes (the dark cubes in Figure 1.21b) containing atoms each are disintegrated to single particles, the work done in this process being:


49

since, in this case, each atom is separated from two first neighbors. (iii) The three crystal rows (the dark cubes in Figure 1.21c) containing (n1) atoms each are disintegrated, the work done in this process being:

since each atom is separated only from one first neighbor. Thus the work that represents the total bond energy of the crystal lattice is:

Correspondingly, the bond energy per atom is:

Therefore, if the crystal is considered infinitely large, and which is the value of the separation work from the half-crystal position. Indeed, the atom in this position of the simple cubic lattice possesses three first neighbors, as seen also from Figure 1.19b. The separation work becomes or if the bond energies between second and third neighbors are taken into consideration, etc. Note that the value of does not depend on the mode of the crystal disintegration. The same results for this quantity will be obtained if the crystal is disintegrated separating the single atoms at random. The term in equation (1.98) accounts for the contribution of the edge and the corner particles to the total bond energy balance of the crystal. It can be ignored for high values of n. This allows us to evaluate, though approximately, the size of the crystal that could be considered “infinitely” large. The mathematical condition is 1/n 100 and atoms. Therefore a crystal with a linear size of ~30 nm should be considered sufficiently large. The separation work of the atoms from the half-crystal position is a bulk energy characteristic of the infinitely large crystal and determines its equilibrium with the ambient phase. Under equilibrium conditions, the probabilities of attachment and detachment of atoms to and from the half-crystal position are both equal to 1/2. In the same time the inequality holds good for the probabilities of attachment and detachment of an atom from the ith site of the crystal surface if that atom is bonded less strongly than the atom in the half-crystal position, i.e. if

50

Chapter 1

Such atom should, soon or later, leave the crystal lattice and this is what determines the equilibrium form of the bulk crystals.

Note that deriving the equilibrium form one would obtain different results if attraction forces between first, between first and second, between first, second and third etc. neighbors are taken into consideration. For example accounting only for the first neighbors interactions the equilibrium form of a crystal with a simple cubic lattice is the cub containing only (100) faces. If, however, interactions between second neighbors are taken into consideration, too, it appears that the corner atoms should leave the crystal lattice since their separation work is smaller than the separation work from the half-crystal position The detachment of the eight corner atoms will be followed by the detachment of the atoms in the 12 edges of the cubic crystal, too, since their separation work is i.e. smaller than Thus the equilibrium form transforms from a simple cub to a polyhedron containing (100), (111) and (110) faces (Figure 1.9b). Accounting for the third neighbors interactions leads to the appearance of the (211) face in the equilibrium form. More distant neighbors, however, do not influence the equilibrium form since the attraction forces decrease steeply with increasing the atomic distance. Based on such considerations Stranski and Kaischew [1.90] predicted the equilibrium form of large homopolar crystals (Table 1.3) in an excellent agreement with the experimental findings [1.91]. 1.3.2

The concept mean separation work

In 1934 Stranski and Kaischew made another important step in the development of the nucleation theory. In three papers [1.21-1.23] they proposed the theory of mean separation works, revealing for the first time


51

the interrelation between the thermodynamic and the molecular-kinetic approach to the phase formation phenomena.

Analyzing the specific properties of the small phases these authors came to the conclusion that the equilibrium of a finite crystal with its surrounding is determined by the mean works of separation of the atoms from the top layers of the crystal faces (hkl) that contact the supersaturated ambient phase. In order to reveal the meaning of the quantity let us consider again the Kossel’s cubic crystal, this time formed on a foreign substrate (Figure 1.22). If only attraction forces between first neighbors are taken into consideration the equilibrium form of such a crystal is a parallelepiped consisting of building elements. In order to calculate the mean separation work of the atoms constituting the top layer of the upper crystal face we shall disassemble it in the following way: (i) The quadratic crystal plane containing atoms (the dark cubes in Figure 1.22a) is disintegrated, the work done in this process being:

(ii) The two crystal rows containing (n-1) atoms each (the dark cubes in Figure 1.22b) are disintegrated, the work done in this process being:

52

Chapter 1

(iii) The single atom (the dark cube in Figure 1.22c) is separated from the crystal, the work done being:

Thus the work giving the total bond energy of the atoms in the top layer of the uppermost crystal face is:

Correspondingly, for the mean separation work

it results:


As expected, for infinitely large crystals work tends to the separation work

53

and the mean separation from the half-crystal

position. Clearly, an infinitely large crystal does not feel the presence of the foreign substrate and the quantity remains unchanged. Using the same procedure, let us now calculate the mean separation work of the atoms constituting the top layer of the front face of the cubic crystal (Figure 1.23). Denoting by the bond energy between a crystal atom and its first neighbor from the foreign substrate the works (Figure 1.23a), (Figure 1.23b), (Figure 1.23c) and are obtained as:

54

Chapter 1

and

respectively. Thus for the mean separation work

one obtains:

A crucial point in the Stranski - Kaischew theory is the condition for the equality of the mean separation works of the atoms from all crystallographic faces of an equilibrium finite crystal. In the case under consideration the equality leads to:

which is nothing other than a different expression of the condition for the equilibrium form of a crystal formed on a foreign substrate. Indeed, since the specific free surface and adhesion energies can be defined as and respectively [1.92], (see also Chapter 1.2.2.1) equation (1.103) is expressed as:

In the same time the simple relations nd = 2h and n'd = h+h* (cf. Figures 1.23a and 1.24) allow us to substitute the ratio n'/n for (h+h*)/2h, which transforms equation (1.104) into:


55

The last formula coincides with equation (1.48') following from the purely thermodynamic condition for the equilibrium form, dV = 0. The theory of mean separation works allows us to draw some important conclusions on the equilibrium properties of the two-dimensional crystals formed on the top of three-dimensional ones having the same chemical composition (Figure 1.25).

In this case the material exchange between the ambient phase and the 2D crystal is realized through the attachment and detachment of atoms to and from its outer peripheral rows. Correspondingly, the mean works of separation of the edge atoms determine the equilibrium between a finite 2D crystal and its surrounding. Consider now a quadratic two-dimensional crystal consisting of atoms formed on the upper face of the three-dimensional cubic crystal consisting of

56

Chapter 1

atoms (Figure 1.25). In order to determine the mean separation work the front edge of the 2D crystal will be disassembled as follows: (i) The (m - 1) atoms (the dark cubes in Figure 1.25a) are separated from the crystal, the work done being:

(ii) The single atom (the dark cube in Figure 1.24b) is separated from the crystal, the work done being:

Thus the work giving the total bond energy of the atoms in the front row of the 2D crystal is and for the mean separation work it results:

The separation works characterize the energy state of the atoms in the crystal lattice and are closely related to the corresponding chemical potentials In the frameworks of the Einstein’s solid state model considering the crystal’s atoms as independent linear harmonic oscillators [1.5,1.93], the chemical potential of the ith atom of a given crystallographic face is presented as [1.4,1.11,1.94]:

where h is the Planck constant and is the atom’s vibrational frequency. Correspondingly, the mean chemical potential of the atoms in the top layer of the same face is:

where is the average geometric value of the vibrational frequencies defined as:


57

M being the number of atoms in the top layer. Finally, the chemical potential of the atom in the half-crystal position is:

where stands for the atom’s vibrational frequency. Bearing in mind the relatively small difference between the frequencies and from equations (1.106) and (1.108) it follows that:

As known, equations (1.106)-(1.108) for the chemical potentials are good approximations for the classical, high temperature region where [1.4, 1.5, 1.11, 1.93, 1.94]. The considerations performed thus far refer to crystals formed in the bulk of a vapor phase. In the case of electrochemical phase formation one should bear in mind that the crystals contact an ionic electrolyte solution and, also, that an electrical potential difference exists between the crystal and the liquid phase. In order to define the separation works in this more complex case we shall follow the approach proposed by Kaischew in 1946 [1.69]. For the purpose, the transfer of an atom from the ith site of the crystal surface to the bulk of the electrolyte solution will be schematically carried out in five steps: (i). The atom is separated from the ith site not allowing any changes to occur on the phase boundary. The work done in this process equals the separation work of the atom in vacuum. (ii). The atom is ionized doing the ionization work J. (iii). The separated electrons are returned to the crystal, the work J' gained being equal to the separation work of an electron multiplied by the ionic valence. (iv). The ion is solvated in the electrolyte thus gaining the solvation work (v). The changes in the phase boundary that should spontaneously occur because of the separation of the atom from the ith site of the crystal surface are reversibly carried out. The work gained in this process depends on the potential difference E between the crystal and the electrolyte solution. Bearing in mind this scheme the separation work of the atom from the ith site of the crystal surface is obtained as:

58

Chapter 1

Concerning the separation work by:

from the half-crystal position, it is given

since in this case the quantity equals zero. Indeed, when separating one atom from the half-crystal position (Figure 1.19b) a new half-crystal position is created and the structure of the phase boundary does not change. Note that equations (1.110) and (1.111) give the “chemical” separation works and in this form do not contain the electrical energy term 1.3.3

The crystal-solution equilibrium and the nucleation work

Let us now comment upon the crystal-solution equilibrium in terms of the kinetic theory of the mean separation works. Since this theory considers the elementary processes of material exchange across the phase boundary it is first necessary to define the frequencies of attachment, and detachment, of single particles to and from the ith site of the crystal surface. In the case of electrocrystallization on a like surface the quantities and are expressed as [1.95-1.98]:

where c is the ionic concentration, and are the energy barriers to transfer of an ion from the electrolyte to the ith site of the crystal surface and back (from the ith site to the electrolyte), E being the electrode potential difference between the crystal and the solution. The meaning of and in equations (1.112) and (1.113) is revealed by the theory of the absolute reaction rates (see e.g. [1.99]-[1.102]), according to which the two quantities depend on the ratio kT/h. In the latter h is the Plank’s constant and is the transmission coefficient, which determines the probability of transfer of an ion/atom from its initial to its final state. Here we shall clarify the physical significance of and in a slightly different way [1.97]. The frequency of attachment can be expressed as a product of the probability of finding one ion in a given site in the electrolyte solution and the frequency surface. If electrolyte and

of transfer of this ion to the ith site of the crystal is the elementary volume occupied by one ion in the is the number of ions per cubic centimeter then


59

where and are the corresponding molar quantities, being the Avogadro number. In the same time, the quantity is given by where is the ion’s vibrational frequency. Thus it appears that whereas coincides with the vibrational frequency of the atom in the ith site of the crystal surface. Splitting, though formally, the energy barriers and into “chemical” and “electrical” parts,

for

and

one obtains:

where is the cathodic ions’ transfer coefficient. According to Kaischew [1.95] (see also [1.96-1.98]) the difference gives the work of separation of one atom from the ith site of the crystal surface. As we have already seen (cf. Equation (1.110)) is a potential dependent quantity, which shows once again that splitting the energy barriers and into purely “chemical” and purely “electrical” parts is only formal. However, the dependence can be neglected when working within relatively short potential intervals. In the same fashion for the frequencies of attachment, and detachment, of single particles to and from the half-crystal position it results:

where the difference gives the separation work from the halfcrystal position. (Strictly speaking should have different values for the ith

60

Chapter 1

site and for the half-crystal position due to the different local structure of the electrical double layer.) In the case of a stable equilibrium between the electrolyte and an infinitely large crystal, the potential difference is and Then from equations (1.116) and (1.117) it follows that:

where As discussed by Kaischew [1.95] the last expression turns into the equation of Nernst (1.12) if at the standard state ionic concentration the potential difference is In that case equation (1.118) can be rewritten in the form:

where Here we consider dilute solutions for which the activity of the metal ions coincides with their concentration. Concerning the ratio of the frequencies of attachment, and detachment, of an atom to and from the ith site of the crystal surface at the equilibrium potential difference from equations (1.114) and (1.115) one obtains:

where Substituting c from equation (1.118) in equation (1.120) the latter transforms into [1.11,1.95]:

In this expression the quantities and stand for the electrochemical potentials of the atoms in the half-crystal position and in the ith site of the crystal surface, respectively. Clearly, the difference equals the difference in the corresponding chemical potentials since the ith atom and the atom in the half-crystal position belong to the same bulk crystal with a Galvani potential Note that for a metal crystal coincides with the molecular electrochemical potential

defined in Chapter 1.1.1.


61

Equation (1.121) shows that if i.e. if then and the atom in the ith site will leave the crystal surface. On the contrary, if i.e. if then and the ith atom will remain in the crystal lattice and will participate in the equilibrium form of the bulk crystal. Consider now the frequencies and when the electrochemical system is deviated from the equilibrium state by applying a cathodic overpotential to the bulk crystal. In this case the potential difference between the crystal and the solution is and from equations (1.114-1.117) one obtains:

Correspondingly for the ratios

and

it results:

where is the electrochemical supersaturation. Equations (1.122) and (1.123) allow us to draw the following important conclusions. Since it appears that the inequality is fulfilled for any value of the cathodic overpotential Therefore the bulk crystal will grow through a cathodic ions discharge on the kink sites. The inequality is fulfilled for the ith sites of the crystal surface for which and For these sites the attachment probability is higher than the detachment one at any cathodic overpotential too. The situation is different for the sites of the crystal surface where the atoms are less strongly bound than at the half-crystal position, i.e.

62

Chapter 1

and Such sites will be occupied only at sufficiently high supersaturations for which the condition holds and the inequality is again fulfilled. The attachment and detachment frequencies are closely related to the quantity electric current density i. Thus the net current to the surface of a macro-crystal with kink sites is:

or

where

stands for the exchange current density. In the same fashion one can define the local current density to the ith sites of the crystal surface. An important achievement of the theory of the mean separation works is the detailed kinetic description of the equilibrium between a small crystal and the ambient phase. In that case the equilibrium state is attained only at a supersaturation and is expressed through the equality of the probabilities of attachment and detachment of atoms to and from the different crystallographic faces. In order to demonstrate this let us consider again the cubic crystal formed on a foreign substrate (Fig.l.22a). Since the equilibrium state does not depend on the mechanism of its attainment we can assume that the equilibrium of the upper crystal face is expressed through the equality of the probabilities and for the simultaneous attachment and detachment of atoms to and from the top layer. Considering the frequencies as independent quantities for the ratio it results:

and

and therefore the condition for equilibrium of the upper face of the cubic crystal is reduced to:


63

Taking the logarithm of this expression yields:

where

is the mean electrochemical potential of the atoms constituting

the top layer of the upper face of the cubic crystal. Since is a constant characterizing the specific bulk properties of the crystal lattice equation (1.125) tells us that it is the mean separation work that depends essentially on the electrochemical supersaturation The latter being fixed, the equilibrium between the small crystal and the electrolyte solution is expressed through the equality of the mean separation works of the atoms from all crystallographic faces participating in the equilibrium form, as we already mentioned above. The profound physical significance of equation (1.125) was revealed by Kaischew in the following way [1.69]. Making use of equations (1.110) and (1.111) and bearing in mind that for the model cubic crystal from Figure 1.22a coincides with (equation (1.100)), the formula (1.125) is rewritten as:

where

The detailed calculation of the sum in equation (1.127) is unfeasible task since the values of the elementary works are not known. However, it is possible to evaluate this sum accounting for the total change of the phase boundary that takes place when the top layer of the upper crystal face

64

Chapter 1

is disintegrated. As a result of this process the total surface area of the crystal-solution interface is diminished by where d is the atomic diameter (cf. Figures 1.20a and 1.22a). Therefore if the work of solvation of unit surface area and the formation of the electrical double layer thereon is denoted by it follows that:

and

Thus equation (1.126) becomes:

As we already know, the work done in creating unit surface area in the vapor phase equals the specific surface free energy at the crystal/vapor interface boundary [1.21]. Therefore, since the work gained in solvating unit surface area and creating the electrical double layer thereon in the electrolyte is then the specific free surface energy at the crystal/solution interface is defined as:

Substituting equation (1.130) into equation (1.129) yields:

which is equivalent to the Gibbs-Thomson equation where is the volume of the single atom and nd= 2h is the upper edge length of the cubic critical nucleus consisting of atoms (cf. Figure 1.24 and equation (1.78)). Therefore in equation (1.131) and This analysis convincingly shows that the formula (1.125) derived by means of kinetic considerations coincides with the thermodynamic GibbsThomson equation (1.78). The beauty of this result is that it demonstrates in an unambiguous way the complete agreement between the purely thermodynamic and the purely kinetic approach to the equilibrium state of the small phases. Nowadays this may seem clear and even self-evident.


65

However, this was not the case in the thirties when a debate on the StranskiKaischew kinetic theory arose between Kossel and Volmer [1.103, 1.104]. The readers who are interested in this subject are referred to the thrilling historical review of Kaischew [1.105], one of the famous old masters in the field of nucleation and crystal growth phenomena. In the case of two-dimensional nuclei formed on the top of threedimensional ones the Gibbs-Thomson equation is presented as [1.66]:

where and are the mean electrochemical potential and the mean separation work of the atoms constituting the edge of the quadratic twodimensional nucleus, respectively (Figure 1.25). Making use of equations (1.110) and (1.111) and bearing in mind that for the 2D crystal in Figure 1.25a coincides with (equation (1.105)) the formula (1.132) transforms into:

where

In order to evaluate the sum in equation (1.134) we shall apply the approach of Kaischew [1.69] bearing in mind that when the front edge of the 2D crystal (Figure 1.25) is disintegrated the total periphery is reduced by 2d. Denoting the work of solvation of unit length and the formation of the electrical double layer thereon by it follows that:

and therefore

Thus equation (1.133) becomes:

66

Chapter 1

Since the work done in creating unit edge length in the vapor phase equals the specific free edge energy at the crystal/vapor interface boundary [1.21] and since the work gained in solvating the unit edge length and creating the electrical double layer thereon in the electrolyte is then the specific free edge energy at the crystal/solution interface is defined as:

Substituting equation (1.137) into equation (1.136) one obtains:

The last expression is equivalent to the Gibbs-Thomson equation for the two-dimensional crystal in which l = md is the edge length of the quadratic critical nucleus consisting of atoms, being the surface area occupied by a single atom. Therefore and We should emphasize that in the concrete case of a quadratic 2D crystal formed on the top of a cubic 3D one the work gained in creating the double layer and solvating the elementary edge length 2d is equivalent to the work gained in creating the double layer and solvating the elementary surface area Therefore it appears that Taking this into consideration and bearing in mind that the 3D and the 2D crystals are in equilibrium with the same ambient phase and at the same supersaturation from equations (1.131) and (1.138) one obtains that This means that the edge of the quadratic 2D critical nucleus is exactly twice shorter than the edge of the critical 3D cubic crystal carrier. (The determination of the quantity in the two-dimensional case is performed for the first time in this book). Concluding this Chapter we shall show how should one present the general formulae for the nucleation work and for the surface free energy (equations (1.32) and (1.33)) in terms of the theory of the mean separation works. For the purpose, let us assume that the cubic crystal consisting of atoms (Figure 1.22a) is the critical nucleus at the

1.THERMODYNAMICS OF ELECTROCHEMICAL NUCLEATION supersaturation

67

(equation (1.125)) and that the total

surface free energy of the nucleus-substrate and the nucleus-solution interface boundaries is:

Here is the electrochemical potential of the ith atom in the nucleus and is the electrochemical potential of the atom in the halfcrystal position. Since the differences between the molecular electrochemical potentials are approximately equal to the differences between the corresponding separation works taken with an opposite sign equation (1.139) becomes:

The meaning of equation (1.139') derived by Stranski [1.12, 1.13] in 1936 can be revealed in the following way. If atoms are separated from the half-crystal position (Figure 1.19b) the work done is equal to If, afterwards, these atoms are used to build up an nucleus on the foreign substrate (Figure 1.22a) the work gained is equal to the sum of the individual separation works

Apparently, the difference

reflects the difference in the energy state of atoms when they belong to the infinitely large crystal and when they form an individual cluster on the foreign substrate. Thus represents the energy of the unsaturated bonds of the cluster’s atoms and have the physical significance of the concept free surface energy of the cluster, accounting for the interaction with the foreign substrate. Under these conditions the nucleation work becomes:

68

Chapter 1

where, as before,

and

are the mean electrochemical potential and

the mean separation work of the atoms in the top layer of the upper crystal face containing particles, respectively. The same expression for the nucleation work is obtained if the critical nucleus is build up atom by atom, calculating the energy change at any elementary step [1.22,1.65]. Concerning the total edge free energy and the work of formation of a 2D critical nucleus consisting of atoms, these two quantities are given by:

and

It is not difficult to show that equations (1.140) and (1.142) turn into the corresponding expressions for the 3D and the 2D nucleation works:

derived in the classical thermodynamic theory of Gibbs and Volmer. In this Chapter we have demonstrated the Stranski-Kaishew kinetic theoretical method considering the simple Kossel’s cubic crystal. The same general physical results can be obtained with any other crystallographic lattice but at the expense of more complex algebraic calculations.

1.4

Atomistic considerations

The Stranski – Kaischew theory of the mean separation works [1.21-1.23] provides the first consistent atomistic description of the elementary processes taking place on the crystal surface. Considering the nucleation


69

phenomena, however, this theory operates with average physical quantities and can be applied only to sufficiently large clusters for which statistical definitions do have physical significance. In this respect, the kinetic theory of the mean separation works is a macroscopic theory as the thermodynamic theory of Gibbs and Volmer. The present Chapter comments upon some specific properties of the very small clusters - those consisting of several atoms [1.106,1.107]. The size n of such nuclei cannot be considered as a continuous variable and one cannot derive a simple analytical relation between and using the classical approach described in Chapter 1.2.2.4. Certainly, it would be trivial to point out here the reasons for which small clusters have always attracted considerable attention. Their unusual behavior has proved significant for a number of physical phenomena such as catalysis, adsorption, photography, electrochemical deposition of metals, semiconductors and alloys etc. Several authors already in the seventies obtained valuable information on the structure, the energetics and the thermodynamic properties of the microclusters [1.108-1.113]. This Chapter, however, is not going to discuss the basic achievements in this field. Here we shall provide information on the small clusters’ behavior obtained by means of illustrative model considerations. For the purpose we shall firstly calculate the nucleation work of 1- to 19-atomic clusters formed on a structureless foreign substrate [1.107] (see also [1.67] and [1.114]). Using the approach proposed by Stoyanov [1.107] we consider nearest neighbor interactions assuming that the bond energy between a cluster atom and its first neighbour from the foreign substrate is lower than the bond energy between the first neighboring atoms in the cluster itself. Under these conditions the clusters attain structures like those schematically shown in Figure 1.26. The clusters having exceeded the size of 6 atoms posses a pentagonal garland which is often considered as a necessary step in the building of the macro-crystals.

70

Chapter 1

For small clusters the size n is a discrete variable and we shall calculate the nucleation work using the general formula (1.32) rewritten in the form:

following from the equality of the differences between the electrochemical potentials and the separation works. Bearing in mind equations (1.110) and (1.111) the last expression transforms into:

where

Since the values of the quantities are not known we shall calculate the vs. n relationship according to equation (1.145). This means to neglect the potential dependence of the free energy excess which is a frequently used approximation. Besides, we assume that and that the bulk new phase is a crystal with a face centered cubic lattice for which The inequality is a necessary condition for clusters with a five-fold symmetry axis formed on a structureless foreign substrate [1.107]. For the sake of convenience the supersaturation will be measured in units. The results obtained for are shown in Figures 1.27 and 1.28 and can be explained in the following way. Due to the strongly expressed discrete character of the cluster size alteration at small dimensions, the nucleation work is not a monotonous function of n (cf. Figure 1.15) but displays several maxima and minima. The highest maximum of each line corresponds to the work for critical nucleus formation at the corresponding supersaturation. Thus at (Figure 1.27, line b) the critical nucleus is the cluster consisting of 8 atoms. With increasing the


71

supersaturation to (Figure 1.27, line c) the work for nucleus formation decreases but the highest maximum corresponds again to the 8atomic cluster. The work of formation of this cluster is maximal up to the supersaturation (Figure 1.27, line d) at which the vs. n relationship displays two equally high maxima, at n = 8 atoms and at n = 6 atoms. Thus the supersaturation marks the end of the interval within which the 8-atomic cluster is the critical nucleus. The beginning of this interval is the supersaturation (Figure 1.27, line a) at which four different clusters, those consisting of 8, 9, 10 and 11 atoms are formed

with the same nucleation work. However, the 11-atomic one is the critical nucleus at the supersaturation since the nucleation work diminishes only for n > 11.

72

Chapter 1

Increasing the supersaturation above the value causes a change of the critical nucleus size. Thus the 6-atomic cluster plays the role of a critical nucleus at supersaturations Indeed, the highest maximum of lines e and f corresponds to the 6-atomic cluster. This is the case up to the supersaturation (line g, Figure 1.27 and line a Figure 1.28) when again four clusters, those consisting of 3, 4, 5 and 6 atoms are formed with the same nucleation work. The 6-atomic cluster is the critical nucleus at this particular supersaturation, too.

Increasing the supersaturation above the value changes the size of the critical nucleus from 6 to 3 atoms and the 3-atomic cluster remains the critical nucleus within the supersaturation interval Lines b and c in Figure 1.28 correspond to supersaturations


73

and respectively, whereas the supersaturation (line d) marks the end of the interval within which the 3-atomic cluster is the critical nucleus. As seen from Figure 1.28 the further increase of the supersaturation leads to critical nuclei consisting of 2 and 1 atom correspondingly within the intervals and In particular, lines e, f, g, h and i in Figure 1.28 are obtained at and respectively, the supersaturation (line g, Figure 1.28) being the one at which the size of the critical nucleus changes from 2 to 1 atoms. Certainly, it is not difficult to evaluate the limiting supersaturation above which the nucleation takes plays without any thermodynamic barrier. Since according to equation (1.145) the work of formation of the 1atomic cluster is:

the condition at yields Above the supersaturation the size of the critical nucleus equals zero atoms, the nucleation work equals zero, too, and the formation of any cluster of size n > 0 is connected with a gain of thermodynamic work, i.e. the energy barrier has a negative value. The above considerations unambiguously show that each critical nucleus is associated with a supersaturation interval but not with a fixed supersaturation as predicted by the Gibbs-Thomson equation (1.62, 1.75). This result has a clear physical significance. Since the supersaturation can be varied in a continuous manner i.e. by infinitesimally small amounts, whereas the cluster size n cannot change by less than one atom, it appears that there will be no different critical nuclei for each supersaturation that can be applied to a real physical system. Therefore the vs. relationship should be in steps, a different supersaturation region corresponding to each critical nucleus (Figure 1.29). This also means that we cannot define a metastable equilibrium between the bulk parent phase and a very small critical nucleus. The reason is that varying the supersaturation the Gibbs free energy of the parent phase varies in the whole interval, whereas the Gibbs free energy of the small critical nucleus could either remain constant or could change in a discrete manner. Apparently, this invalidates the second definition of the concept critical nucleus given in the classical nucleation theory (Chapter 1.2.2.4). Thus in terms of the more general, atomistic thermodynamic treatment [1.67, 1.96-1.98, 1.106, 1.107, 1.114] the critical nucleus at a given supersaturation should be defined as the largest cluster

74

Chapter 1

formed with maximal nucleation work. The specific value of depends on the interatomic forces and the critical nuclei in two neighboring supersaturation intervals should not necessarily differ by only one atom. For instance in the case under consideration the 7- and the 13-atomic clusters can never be critical nuclei since due to their very compact structure no supersaturation exists at which and have maximal values (see Figures 1.27 and 1.28).

Having revealed the discrete character of the relationship it is important to derive a general expression for the length L of the supersaturation interval within which a given cluster plays the role of a critical nucleus [1.115,1.116]. For the purpose we consider again Figure 1.28. As we have already seen the supersaturation is the highest supersaturation at which the 6-atomic cluster is the critical nucleus (Figure 1.28, line a). Above it, the size of the nucleus changes to atoms (Figure 1.28, lines b, c and d). What we know, too, is that at the


75

supersaturation the 6- and the 3-atomic clusters are formed with the same nucleation work, i.e. whereas at the supersaturation (Figure 1.28, line d), Denoting the limiting supersaturations and by and respectively, for the length of the supersaturation interval within which the critical nucleus consists of 3 atoms it results Making use of equation (1.145) and bearing in mind the equalities and one obtains also:

The separation works in equation (1.146) depend on the specific structure of the 2-, 3- and the 6-atomic clusters shown in Figure 1.26 and are given by and Thus it follows that for the 3-atomic cluster where as before Using the same approach let us now derive an expression for the quantity in the general case when the clusters consisting of and atoms are the critical nuclei in three neighboring supersaturation intervals and We assume that and correspondingly (Figure 1.30). In this case the supersaturations and delimit the interval within which the cluster is the critical nucleus and for these supersaturations and Making use of the general formula for the nucleation work (equation (1.32)),

for the supersaturations expressions:

and

one obtains the general

76

Chapter 1

In equations (1.147) and (1.148) the free energy excess is given by [1.12,1.13]:

and accounts for the potential dependence of the separation works according to equation (1.110). Thus for and it results:

Correspondingly, for the length supersaturation interval within which the critical nucleus one obtains:

of the cluster plays the role of a

Equations (1.150)-(1.152) allow us to obtain useful information for the separation works and for the bonding energy of the atoms in the small


77

clusters if the limiting supersaturations and are determined from experimental studies of the nucleation kinetics carried out within sufficiently wide supersaturation intervals [1.67, 1.115-1.117]. At the end of this Chapter it is worth commenting upon the correlation between the atomistic thermodynamic treatment of the relationship (Figure 1.29) and the continuity, classical approach leading to the GibbsThomson equation (1.62, 1.75, 1.136), (Figure 1.16). For the sake of simplicity let us consider quadratic two-dimensional critical nuclei formed on the (100) face of a cubic crystal (Figure 1.25a). Using the general expressions (1.150)-(1.152) it is not difficult to calculate the supersaturations and and the length of the intervals within which clusters containing from to atoms in the edge are critical nuclei. Since the classical nucleation theory considers only completely built two-dimensional clusters, the edge lengths of the neighboring critical nuclei will differ by one atom and therefore we can set and Considering the case of interactions between first neighbors and neglecting the potential dependent quantities for the sum of the separation works of an quadratic 2D cluster one obtains:

Substituting i for and respectively, for the sums of the separation works in equations (1.150) and (1.151) it results:

78

Chapter 1

Thus for the supersaturations and delimiting the interval within which the quadratic cluster plays the role of a critical nucleus from equations (1.150) and (1.151) one obtains:

For instance, if atoms, and the length of the interval within which the 9-atomic 2D cluster is the critical nucleus equals (4/35) In the same time the classical Gibbs-Thomson equation (1.136) taken at results in the single value


79

The stepwise relationship calculated by means of equations (1.157) and (1.158) is shown in Figure 1.31. For comparison, the continuous line represents the macroscopic Gibbs-Thomson equation (1.136). As seen, for large clusters the supersaturation intervals are short and the fluent curve describes well the real size alteration. In particular, when equations (1.157) and (1.158) reduce to which coincides with the Gibbs-Thomson equation (1.136) if Correspondingly, Increasing the supersaturation, the intervals become wider and the Gibbs-Thomson equation is no longer a good approximation. Concluding, we should emphasize that equations (1.147) and (1.148) allow us to obtain useful information on the free energy excess of small clusters. This can be achieved without the implication of any assumptions concerning the cluster structure and the interatomic interactions if the limiting supersaturations and are determined from experimental studies of the nucleation kinetics performed within sufficiently wide supersaturation intervals [1.67, 1.115-1.117]. Making use of equations (1.149-1.151) and assuming some model of the cluster structure, information can also be obtained on the separation works and on the bond energies of the atoms in the critical nuclei.

REFERENCES 1.1 J.W. Gibbs, Collected Works, New Haven, 1928, Vol. 1. 1.2 E.A. Guggenheim, J.Chem.Phys., 33(1929)842 (see also Thermodynamics, North Holland Publishing Company, Thirdedition, 1957). 1.3 E. Lange and K. Nagel, Z. Elektrochem., 41 (1935) 575.. 1.4 R. Kaischew, Bulg. Chem. Commun., 26 (1993) 143. 1.5 T.L. Hill, Statistical Thermodynamics, Addison-Wesley, Reading, MA, 1960. 1.6 E.A.Guggenheim and R.H.Fowler, Statistical Thermodynamics, Cambridge Univ. Press, Cambridge 1965. 1.7 J.S. Dash, Films on Solid Surfaces, Academic Press, New York, 1975. 1.8 I. Langmuir, J. Am. Chem. Soc., 40 (1918) 1361. 1.9 A.N. Frumkin, Z.Phys.Chem., 116(1925) 466. 1.10 W.L. Bragg and E.J. Williams, Proc. Roy. Soc., London, 145A (1934) 699. 1.11 M. Volmer, Kinetik der Phasenbildung, Teodor Steinkopf Verlag, Leipzig, Dresden, 1939. 1.12 I.N. Stranski, Ber. Wien. Acad., 145 (1936) 840. 1.13 I.N. Stranski, Ann. Univ. Sofia, 30 (1936) 367. 1.14 R. Kaischew, Bull. Acad. Bulg. Sci. 2 (1950) 191. 1.15 R. Kaischew, Selected works, Bulg. Acad. Sci., 1980 (in Bulgarian). 1.16 M. Volmer and A. Weber, Z. Physik. Chem., 119 (1926) 277. 1.17 W. Kossel, Nachr .Ges. Wiss. Goettingen, Math.-physik. Klasse, Band 135,1927. 1.18 I.N. Stranski, Ann. Univ. Sofia, 24 (1927) 297. 1.19 I.N. Stranski, Z. Physik. Chem., 136 (1928) 259.

80

Chapter 1

1.20 L. Farkas, Z. Phys. Chem., 125 (1927) 236. 1.21 I.N. Stranski and R. Kaischew, Z. Physik. Chem., B26 (1934) 100. 1.22 R. Kaischew and I.N. Stranski, Z. Physik. Chem., B26 (1934) 114. 1.23 I.N. Stranski and R. Kaischew, Z. Physik. Chem., B26 (1934) 132. 1.24 R. Becker and W.Döring, Ann. Phys., 24 (1935) 719. 1.25 M. Born and O. Stern, Ber. Berliner Akad., 48 (1919) 901. 1.26 R.A. Swalin, Thermodynamics of Solids, J. Wiley & Sons, Singapore, 1991, p.231. 1.27 I. Konstantinov and J. Malinowski, J. Phot. Sci., 23 (1975) 1. 1.28 I. Konstantinov and J. Malinowski, J. Phot. Sci., 23 (1975) 145. 1.29 M.Dupré, Théorie méchanique de la chaleur, Paris, 1969. 1.30 V.S. Vesselovski and V.N. Pertzov, Zh. Fis. Khim., 8 (1936) 245. 1.31 R.D. Gretz, J. Chem. Phys., 45 (1966) 3160. 1.32 P. Tarazona and G. Navascués, J. Chem. Phys.,75 (1981) 3114. 1.33 J. Mostany, J. Mozota and B.R. Scharifker, J. Electroanal.Chem.,177 (1984)25 1.34 B.V. Toschev, D. Platikanov and A. Scheludko, Langmuir, 4 (1988) 489. 1.35 T. Kolarov, A. Scheludko and D. Exerova, Trans. Faraday Soc., 64 (1968) 2864. 1.36 G. Navascués and P. Tarazona, J. Chem. Phys., 75 (1981) 2441. 1.37 D. Exerova, D. Kashchiev, D. Platikanov and B.V. Toshev, Adv. Colloid Interface Sci., 49 (1994) 303. 1.38 B. Mutaftschiev and R. Kaischew, Commun. Inst. Phys. Chem. Bulg. Acad. Sci., 1 (1960) 43. 1.39 E. Michailova, M. Peykova, D. Stoychev and A. Milchev, J. Electroanal. Chem., 366 (1994) 195. 1.40 A. Scheludko and M. Todorova, Commun. Bulg. Acad. Sci. (Phys.), 3 (1952) 61. 1.41 R.C. Tolman, J. Chem. Phys., 17 (1949) 333. 1.42 J.G. Kirkwood and F.P. Buff, J. Chem. Phys., 17 (1949) 338. 1.43 S. Ono and S. Kondo, Handbuch der Physik, Vol. 10, Springer, Berlin, 1960. 1.44 R. Defay and I. Prigogine, Surface Tension and Adsorption, Longman, London, 1966. 1.45 S. Toschev, in Crystal Growth: An Introduction, Ed. P. Hartmann, North-Holland Publ. Co, 1973, p.1. 1.46 G.Navascués and P. Tarazona, Chem. Phys. Letters, 82 (1981) 586. 1.47 J. Schmelzer, I. Gutzow and J. Schmelzer Jr., J. Colloid. Interface Sci., 82 (1996) 83. 1.48 I. Kostov, Kristalographia, Nauka i izkustvo, Sofia, 1958. 1.49 W. Kleber, An Introduction to Crystallography, VEB Verlag Technik, Berlin, 1970. 1.50 R. Kaischew, Commun. Bulg. Acad. Sci., 2 (1951) 191. 1.51 J.W. Gibbs, Am. J. Sci. Art, 16 (1878) 454. 1.52 P. Curie, Bull. Soc. Mineralog., 8 (1885) 145. 1.53 G. Wulff, Z. Kristallogr., 34 (1901) 449. 1.54 R. Kaischew, Commun. Bulg. Acad. Sci., 4 (1954) 85. 1.55 Gmelin 's Handbuch der Anorganische Chemie, 8 Auflage, Teil B 1 , p.286, Springer, 1974. 1.56 E. Michailova and A. Milchev, J. Appl. Electrochem., 18 (1988) 614. 1.57 A.A. Chernov, Kristallografiya, 8 (1963) 87. 1.58 A.A. Chernov, Kristallografiya, 16 (1971) 844. 1.59 A.A. Chernov, J.Cryst. Growth, 24/25 (1974) 11. 1.60 A.A. Chernov, in Modern Crystallography, Vol.3, Springer, Berlin, 1984. 1.61 C.N. Nanev, Prog.Crystal Growth and Charact., 35 (1997) 1. 1.62 R. Rashkov and C.N. Nanev, Cryst. Res. Techn., 23 (1988) 805. 1.63 C.N. Nanev and R.Rashkov, J. Cryst. Growth, 121 (1992) 209.

1.THERMODYNAMICS OF ELECTROCHEMICAL NUCLEATION 1.64 R. Rashkov and C.N. Nanev, J. Cryst. Growth, 158(1996)136. 1.65 R.Kaischew, Z.Phys., 102(1936)684. 1.66 I.N.Stranski and R.Kaischew, Z.Phys.Chem. B 35(1937)427. 1.67 A.Milchev, Contemp.Phys., 32(1991)321. 1.68 J.P.Hirth and G.M.Pound, Condensation and Evaporation, Pergamon, Oxford, 1963. 1.69 R.Kaishew, Ann.l’Univ.Sofia (Chem.), 63(1946/1947)53. 1.70 D.Kashchiev and A.Milchev, Thin Solid Films, 28(1975)189. 1.71 A.Scheludko, Colloid.Surf., 1(1980)191. 1.72 R.Kern, G.Le Lay and J.J.Metois, in Current Topics in Material Sciences, Ed. E.Kaldis, North - Holland Publ. Co, 1979, Vol. 3, p.131. 1.73 I.Markov and R.Kaischew, Kristall Technik, 11(1976)685. 1.74 H.Brandes, Z.Phys.Chem., 126(1927)198. 1.75 J.K.Mackenzie, A.J.W. Moor and J.F.Nicholas, J.Phys.Chem.Solids, 23(1962)185. 1.76 J.K.Mackenzie, A.J.W. Moor and J.F.Nicholas, J.Phys.Chem.Solids, 23(1962)197. 1.77 E.Bauer, Z.Krist., 110(1958)372. 1.78 R.Lacmann, Z.Krist., 116(1961)13. 1.79 W.Kleber, Z.Phys.Chem.(NF), 53(1967)52. 1.80 S.Toshev, M.Paunov and R.Kaischew, Commun.Dept.Chem.Bulg.Acad.Sci., 1(1968)119. 1.81 S.Stoyanov and D.Kashchiev, in Current Topics in Material Sciences, Ed. E.Kaldis, North - Holland Publ. Co, 1981, Vol. 7, p.69. 1.82 H.Mayer, in Advances in Epitaxy and Endotaxy, Eds. R.Niedermayer and H.Meyer, VEB Deutscher Verlag für Grund-stoffsindustrie, Leipzig, 1971. 1.83 D.W.Pashley, in Epitaxial Growth, Part A, Academic, 1975, p.1. 1.84 J.W.Mattews, Epitaxial Growth A and B, Materials Science Series, Academic Press, New York, 1975. 1.85 G.Honjo and K. Yagi, in Current Topics in Material Sciences, Ed. E.Kaldis, North - Holland Publ. Co, 1980, Vol. 6, p. 196. 1.86 I.Markov and S.Stoyanov, Contemp.Phys., 28(1987)267. 1.87 F.C.Frank and J.H.van der Merwe, Proc.Roy.Soc., A198(1949)205. 1.88 F.C.Frank and J.H.van der Merwe, Proc.Roy.Soc., A200(1949)125. 1.89 I.N.Stranski and L.Kratsanov, Ber.Akad.Wiss.Wien., 146(1938)797. 1.90 I.N.Stranski and R.Kaischew, Z.Krist., 78(1931)25. 1.91 B.Honigmann, Gleihgewihts- und Wachstumsformen von Kristallen, Dietrich Steinkopff Verlag, Darmstadt, 1958. 1.92 R.Kaischew, in Festkörperphysik und Physik der Leuchtstoffe, Berlin, Akad. Verlag, 1958, p.133. 1.93 L.D.Landau and E.M.Lifschitz, Statistical Physics, Edition, Nauka, Moskow, 1964. 1.94

R.Kaischew, Z.phys.Chem., B48(1940)82.

81

82

Chapter 1

1.95 R.Kaischew, Ann.Univ.Sofia Fac.Physicomat, Livre 2 (Chem), 42(1945/1946) 109. 1.96 A.Milchev, S.Stoyanov and R.Kaischew, Thin Solid Films, 22(1974)255. 1.97 A.Milchev, S.Stoyanov and R.Kaischew, Thin Solid Films, 22(1974)267. 1.98 A.Milchev, S.Stoyanov and R.Kaischew, Soviet.Electrochem.,13(1977)723. 1.99 S.Glasstone, K.J.Laidler and H.Eyring, The Theory of Rate Processes, MacGraw-Hill, New York, 1941. 1.100 K.J.Vetter, Elektrochemische Kinetik, Springer-Verlag, 1961. 1.101 B.B.Damaskin and O.A.Petrii, Elektrochimicheskaya Kinetika, Visshaya Shkola, Moskow, 1975. 1.102 J.W.Christian, Transformations in Metals and Alloys, Pergamon, 1981. 1.103 W.Kossel, Ann.Phys., 21(1934)457. 1.104 M.Volmer, Ann.Phys., 23(1935)44. 1.105 R.Kaischew, J.Cryst.Growth, 51(1981)643. 1.106 A.Milchev and S.Stoyanov, J.Electroanal.Chem., 72(1976)33. 1.107 S.Stoyanov, in Current Topics in Material Sciences, Ed. E.Kaldis, North – Holland Publ. Co, 1978, Vol. 3. 1.108 S.Ino, J.Phys.Soc.Japan, 27(1969)941. 1.109 M.R.Hoare and P.Pal, Adv.Phys., 20(1971)161. 1.110 M.R.Hoare and P.Pal, J.Cryst. Growth, 17(1971)77. 1.111 A.Bonissent and B.Mutaftchiev, J.Chem.Phys., 58(1973)3727. 1.112 T.Halicioglu, J.Cryst. Growth, 29(1975)40. 1.113 C.L.Briant and J.J.Burton, J.Chem.Phys., 63(1975)2045. 1.114 A.Milchev and J.Malinovski, Surface Sci., 156(1985)36. 1.115 A.Milchev and E.Vassileva, J.Electroanal.Chem., 107(1980)337. 1.116 A.Milchev, Electrochim.Acta, 28(1983)947. 1.117 P.M.Rigano, C.Mayer and T.Chierchie, J.Electroanal.Chem., 248(1988)219.

Chapter 2 KINETICS OF ELECTROCHEMICAL NUCLEATION

2.1

General formulation

2.1.1

Non-stationary nucleation kinetics

In Chapter 1 we considered the equilibrium properties of small clusters and derived explicit thermodynamic expressions for the work of nucleus formation. We should emphasize, however, that giving us the value of the energy barrier thermodynamics do not say anything about the rate J of appearance of nuclei within the supersaturated parent phase. The reason is that this important physical quantity depends on the mechanism of nucleus formation and can be determined only by means of kinetic considerations. It is the purpose of this Chapter to present the fundamentals of the nucleation kinetics, to derive explicit expressions for the nucleation rate J and to reveal the supersaturation dependence of this quantity. In doing this we consider the nucleus formation on the assumption that the process is a set of consecutive bimolecular reactions of the type:

Here is the monomer and and are aggregates consisting of n and of n+l particles respectively. In terms of this model [2.1-2.5] the n-atomic clusters form on the substrate in two ways: (i). through attachment of single atoms to the clusters consisting of n-1 atom and (ii). through detachment of single atoms from the clusters consisting of n+1 atom. Thus the rate of birth of the n-atomic clusters in the supersaturated system is:

84

Chapter 2

whereas the death rate

of the n-atomic clusters is:

In equations (2.1) and (2.2) and are the number densities of n-1, of n and of n+1-atomic clusters at time t and and are the frequencies of attachment of single atoms to the clusters consisting of n-l and of n atoms, respectively. Accordingly and are the frequencies of detachment of single atoms from the clusters consisting of n and of n+1 atoms. Both and are considered as time independent and the rate of appearance n-atomic clusters is expressed as:

n = 1,2,3,… Equation (2.3) can also be presented in the form:

n =1,2,3,... where

and

In equations (2.4) – (2.6) and are the net fluxes of clusters through size n and through size n + 1 respectively and the rate of nucleus formation is defined as the net flux through the critical size

2.KINETICS OF ELECTROCHEMICAL NUCLEATION 2.1.2.

85

Stationary nucleation kinetics

The formation and growth of nuclei on the substrate causes a gradual exhaustion of single atoms and leads eventually to establishment of a thermodynamic equilibrium between the bulk new and the bulk parent phase. However, if the supersaturation is kept constant the simple case of a stationary (or a steady state) nucleation may be realized after a certain time. Under such conditions the equality

holds good for any cluster size and defines the stationary nucleation rate The equality (2.7) tells us that under steady state conditions the number of clusters that turn, per unit time, from size n - 1 to size n equals exactly the number of clusters that turn from size n to size n + 1. In other words, it appears that a constant clusters flux flows through each point n of the size axis, including the critical size Thus one obtains:

being the number density of sites on the substrate where the nucleus formation takes place.

86

Chapter 2

The set of algebraic equations (2.8) has been solved first by Becker and Döring [2.6] (see also [2.7-2.9]) on the simplifying assumption that the clusters having reached the size s were disintegrated to single particles and were returned to the parent phase in order to keep constant the supersaturation Thus the number was considered to be equal to zero. The simple mathematical method used by Becker and Döring consists in the following. The first equation of the set (2.8) is multiplied by the second is multiplied by the third by the nth by etc. Then the left and the right hand sides of all equations are summed up to yield, after rearrangement, the following expression for the stationary nucleation rate

We must note that equation (2.9) represents a very general formula for and can be applied to any particular case of two-dimensional or threedimensional phase formation, both on a like and on a foreign substrate, if and are expressed through appropriate physical quantities. In Chapter 1.3.3 we have already defined the frequencies of direct attachment and detachment, and of single atoms to and from the ith site of the crystal surface in the case of electrocrystallization. In equation (2.9) the quantities and represent the frequencies of direct attachment and detachment of single atoms to and from an n-atomic cluster and were defined by Milchev, Stoyanov and Kaischew in the framework of the atomistic theory of electrochemical nucleation [2.10-2.12]. Here we shall clarify the meaning of this definition as follows. We consider identical nucleation sites on the electrode surface, which means that the separation work of an atom adsorbed on a nucleation site is the same for all sites. However, in determining the frequencies and we shall account for the presence of different ith sites depending on whether the atom is attached or detached only to and from the site alone or to and from the site and an n-atomic cluster already formed thereon. Under such conditions for the frequencies of attachment and detachment of an atom to

2.KINETICS OF ELECTROCHEMICAL NUCLEATION

87

and from the ith site of the substrate we can write analogously to equations (1.114) and (1.115):

In these expressions the quantities and have the same physical significance as and in equations (1.112) and (1.113) (Chapter 1.3.3) but refer to clusters formed on a foreign substrate, and are the “chemical” parts of the energy barriers to transfer of an atom to and from the ith site of the electrode surface and the difference defines the atom’s separation work from the ith site. Having defined the quantities and it is now important to determine the frequency ratios in the denominator of equation (2.9). However, the following important circumstance should be taken into account beforehand [2.10, 2.11]. In accordance with the accepted formalism, let us assume that the ith site of the electrode surface is a cluster consisting of n-1 atoms. Thus the attachment of one atom to the ith site is in fact an attachment of one atom to a cluster consisting of n-1 atoms and leads to the formation of an n-atomic cluster. Therefore, the frequency corresponds to the frequency from the theory of Becker and Döring [2.6]. On the other hand, the detachment of one atom from the ith site means the detachment of one atom from its n-1 neighbors, which is the case when one atom is detached from a cluster consisting of n atoms. Thus the frequency corresponds to the frequency from the theory of Becker and Döring [2.6] and the ratio coincides with the ratio from the denominator of equation (2.9), i.e.:

Here is the separation work of the nth atom from the n-atomic cluster and is, rigorously speaking, a potential dependent quantity (cf. Equation (1.110) [2.13]). Substituting c for according to equation (1.118), one obtains:

88

Chapter 2

and therefore we find that:

Here

is the free energy excess of the n-atomic cluster, is its Gibbs free energy and

is the nucleation work

given by the general formula (1.32). Equation (2.14) shows that each term of the sum in the denominator of equation (2.9) depends strongly on the nucleation work of a certain cluster of the new phase. In particular, the term can be presented as where is the nucleation work of the zeroatomic nucleus. Thus the formula for the stationary nucleation rate becomes:

and in this general form applies to any process of heterogeneous phase formation. Apparently the derivation of an explicit expression for the


89

stationary nucleation rate requires the calculation of the sum in equation (2.15) and what follows describes the classical approach to this problem.

2.2.

Classical theory of stationary nucleation

Volmer and Weber [2.14], Farkas [2.15] and Kaischew and Stranski [2.16-2.18] were the first who examined the stationary nucleation kinetics and derived theoretical expressions for the stationary nucleation rate. However, in this Chapter we shall present the results of the more rigorous treatments of Becker and Döring [2.6] and of Zeldovich [2.19] and Frenkel [2.20] who laid the foundations of the contemporary classical nucleation theory (see also [2.7-2.9] and [2.21-2.24]). For the sake of simplicity we shall neglect both the line tension effects (equations (1.42) and (1.70)) and the dependence of the specific free surface energy on the size of the clusters (equation (1.43). 2.2.1.

Becker and Döring approach

In the frameworks of the classical nucleation theory the size n of the clusters is considered as a continuos variable, the nucleation work is expressed by equation (1.65) and the sum in equation (2.15) is replaced by an integral, i.e.,

After that the integral is solved on three simplifying assumptions: (i). The function (equation (1.65)) is expanded in Taylor series around the maximum at and the first three terms are taken into account. Bearing in mind that the first derivative of equals zero at (the condition for a maximum of the nucleation work) one obtains:

(ii). The limits of integration are extended to has a very sharp maximum at the vicinity of the critical size do count [2.7].

since the function and only the values in

90

Chapter 2

(iii). Being a relatively weak function of n the frequency is replaced by its value at Under these conditions the integral in equation (2.16) transforms into:

where and Finally, substituting equation (2.18) into (2.16) one obtains the well-known classical formula for the stationary nucleation rate:

Here

is known as a “non-equilibrium” factor of Zeldovich [2.19] and its physical meaning can be revealed in the following way. Let us assume that a metastable equilibrium can be established in the supersaturated parent phase. In this case the constant flux of clusters through the size axis equals zero and equations (2.8) transform into:


Solving this set of algebraic equations we obtain the number

91

of n-

atomic clusters in the form of a quasi-equilibrium distribution function

and the nucleation rate could be expressed, though formally, as the number of critical nuclei that turn into stable clusters per unit time, i.e.,

Comparing equations (2.19) and (2.23) it is not difficult to see that the only difference between and is the Zeldovich factor That being the case, it appears that this quantity should be associated with the difference between the quasi equilibrium and the actual, steady state number of critical nuclei. The considerations that follow shed additional light on this subject. 2.2.2.

Zeldovich-Frenkel approach

The Zeldovich-Frenkel approach to the stationary nucleation kinetics [2.19, 2.20] consists in the replacement of the Becker and Döring expression

by a first order differential equation. For the purpose the detachment frequency is expressed as in accordance with equations (2.21) and is substituted in equation (2.24). Thus the stationary nucleation rate becomes:

where and Following Frenkel [2.20] and assuming that for n > 10 it is justified to pass from finite differences to continuous functions equation (2.25) transforms into:

92

Chapter 2

and can be integrated within limits [n,s] to yield:

Making use of the boundary conditions:

for the stationary nucleation rate

one obtains:

Finally, substituting the equilibrium distribution function (2.22) into equation (2.28) for it results:

from equation

which is an expression practically equivalent to the Becker and Döring fromula (2.16). The small difference in the integration limits disappears when the latter are extended to so that the Zeldovich-Frenkel explicit expression for the stationary nucleation rate coincides with the Becker and Döring’s one (2.19). Certainly, it would be important to find out the connection between the quasi-equilibrium and the steady state number of clusters and for this purpose we shall combine equations (2.27) and (2.28) bearing in mind that and that In this way one obtains:


93

and it is readily seen from equation (2.30) that the stationary number of clusters is always smaller than the equilibrium one. Extending the upper limits of both integrals to and replacing the function (equation (1.65)) by the first three terms of the Taylor expansion, equation (2.30) transforms into [2.25]:

where

and

94

Chapter 2

are the error function and the complementary error function, respectively [2.26]. Finally, for erf(0) = 0 (erfc(0) = 1) and it appears that the stationary number of critical nuclei is twice smaller than the equilibrium one The ratio is schematically shown in Figure 2.1 where the curve represents the stationary solution given by equation (2.31). As seen, the difference between and is most significant within the region around the critical nucleus consisting of atoms. According to Zeldovich [2.19] is defined as the critical region within which the nucleation work differs from the maximum one by no more than kT (Figure 2.2). Within this region the quantities and are

connected with the simple relation [2.3]:

whereas without lack of accuracy we can postulate that for clusters of sizes between 1 and n' and for clusters of sizes n > n". This approximation is represented by the broken straight line in Figure 2.1. Making use of the Taylor expansion of and combining equations (2.17) and (2.20) for n = n", the condition yields Thus for the length of the critical region one obtains and [2.25, 2.27, 2.28]. The geometrical meaning of the Zeldovich factor is quite simple, too. Being determined by the second


95

derivative

is

(equation (2.20)), it appears that

associated with the curvature of the function at (Figure 2.2). Some numerical evaluations of show that it varies from to 1 at extremely high supersaturations. 2.2.3

Supersaturation dependence of the stationary nucleation rate

The supersaturation dependence seems to be the most frequently and the most extensively studied feature of the stationary nucleation rate ever since the time of Volmer [2.29]. For that reason in this Chapter we shall derive some explicit expressions for the relationship assuming that the number of the nucleation sites is not a potential dependent quantity. 2.2.3.1 Three dimensional nucleation In this particular case the work for critical nucleus formation is given by equation (1.63) if the nuclei are liquid droplets and by equation (1.74) if the nuclei are crystals formed on a foreign substrate. As seen, in both case where or

respectively, the quantity being dependent on the orientation of the crystalline critical nuclei on the foreign substrate (see e.g. equations (1.76) and (1.77)). Concerning the supersaturation dependence of the Zeldovich factor the classical relation (equation (1.64)) combined with equation (2.20) yields:

What one should do next is to find out the supersaturation dependence of the frequency in equation (2.19). This quantity depends on the mechanism of single atoms attachment to the critical nucleus and here we shall consider two possibilities: (i). direct attachment of ions from the bulk of the electrolyte solution and (ii). surface diffusion of ad-atoms on the electrode surface. 2.2.3.1.1 Mechanism of direct attachment In this case the frequency is expressed as:

96

Chapter 2

where is the surface area of the critical nucleus and is the number of ions impingements per square centimeter, per second. The supersaturation dependence of the first quantity is given by being a material constant depending on the shape of the critical nucleus. Thus for liquid spherical clusters whereas for crystalline clusters with simple crystallographic forms one could apply the relation (equation 1.49) and the constant transforms into As we mentioned before, using equation (1.49) means substituting the specific free surface and adhesion energies and of the different crystallographic faces for single, averaged values and In the classical nucleation theory the ions’ flux to the critical nucleus is set equal to the ratio where coincides with the cathodic current in the Butler-Volmer equation [2.30] and is the amount of electricity carried out by a single ion. As shown in [2.31] the quantity:

should not depend on the curvature of the critical nucleus if the ions coming from the electrolyte solution do not “know” whether they will join a small cluster or will adsorb on the flat electrode surface. Once being attached to a cluster of the new phase, however, the atoms start feeling the specific influence of their neighbors and the ions’ detachment flux depends on the curvature of the critical nucleus [2.31]. We should point out that in the case of crystalline clusters the different crystallographic faces have different exchange current densities. Therefore, to apply equation (2.37) to crystalline nuclei means to replace the actual exchange current densities for a mean quantity averaged over all available crystallographic faces. Taking all this into consideration the supersaturation dependence of the stationary nucleation rate is expressed as:

where


In the particular case of spherical liquid droplets

97

becomes:

whereas for crystalline clusters with simple crystallographic forms for it results:

We should point out that the quantity is usually considered as a constant although and are potential dependent in the electrochemical case. However, this appears to be a justified approximation when the nucleus formation takes place within relatively short overpotential intervals. Bearing in mind that equation (2.38) can be presented in a form suitable for a direct interpretation of the overpotential dependence of the stationary nucleation rate:

Here

Denoting the product

by

equation (2.40) transforms into a linear relationship:

the slope of which the nucleation work any value of

according to:

allows the determination of both and the size

of the critical nucleus at

98

Chapter 2

Note that very often experimentalists forget the term equation (2.38) and interpret the vs. instead of the

in vs.

relationship which is certainly not correct. 2.2.3.1.2 Mechanism of surface diffusion In this case the frequency is presented as a product of three

quantities: the probability of finding an adsorbed atom on a given site of the electrode surface, the frequency of transfer of an adsorbed atom to an adjacent adsorption site [2.32] and the number of atomic sites at the periphery of the three dimensional critical nucleus, d being the atomic diameter. Thus for one obtains [2.33]:

In equation (2.44) is the surface diffusion coefficient of the ad-atoms on the electrode surface and is the jump distance at surface diffusion which is considered to coincide with the diameter of a substrate atom, the product being equal to 1. The quantity in equation (2.44) is the length of the periphery of the critical nucleus and for spherical clusters is given by being the radius of the homogeneous sphere. Substituting for according to the Gibbs-Thomson equation (1.66) for one obtains:

or, making use of equation (1.49),


99

In the classical nucleation theory the number of single atoms on the foreign substrate is expressed through the quasi-equilibrium distribution function (equation (2.22)), so that

Therefore, since according to the general formulae (1.32) and (1.143) one obtains:

or

where

is the free energy excess of the single atom adsorbed

on the foreign substrate and stands for the equilibrium adatoms concentration at As seen, equation (2.47) coincides with the Henry’s adsorption isotherm (equation (1.22)) if Finally, since in equation (2.19) the nucleation work and the Zeldovich factor are thermodynamic quantities and do not depend on the mechanism of the atoms attachment to the critical nucleus, the expression for the stationary nucleation rate in the case of a surface diffusion mechanism becomes:

Here the constant

reads:

100

Chapter 2

in the case of spherical liquid droplets, and

in the case of crystalline clusters with simple crystallographic forms. Denoting the product by equation (2.48) transforms into a linear relationship:

Analogously to the case of direct attachment mechanism (equations (2.40, 2.41)) the slope of the straight line allows us to determine the nucleation work and the size of the critical nucleus. Equations (2.38) and (2.48) show that the stationary nucleation rate is a different function of the supersaturation depending on the mechanism of the atoms attachment to the critical cluster. Therefore the question arises how to choose between the two expressions when interpreting experimental data for the stationary nucleation rate? As shown in [2.34, 2.35] the problem can be solved by verifying whether depends on the concentration of the electrolyte solution at a constant overpotential Indeed, while in equation (2.48) is concentration independent the quantity (2.38) depends on the concentration as

in equation

In order to prove this we

shall remind that the exchange current density included in is associated with the crystal-solution equilibrium and is determined by the frequencies of attachment and detachment of single ions to and from the half-crystal position at the equilibrium potential Therefore where

and substituting the Nernst equation (1.119) in (2.52) one obtains:


101

Experimental studies of the stationary nucleation rate of silver on a glassy carbon substrate [2.34] do confirm the vs. relationship, the value obtained for the charge transfer coefficient being However, one must not forget that studies of the concentration dependence of the stationary nucleation rate would provide a reliable information on the actual mechanism of the nucleation process only if the experimental studies are carried out at a constant overpotential The reason is that when the bulk concentration of the electrolyte changes, the equilibrium potential also changes. Therefore, in order to keep the electrochemical overpotential constant it is necessary to polarize the working electrode to one and the same E but always in reference to the equilibrium potential of the bulk new phase in the solution with the corresponding concentration c. Concluding we should note that the mechanism of ions’ direct attachment has been confirmed by a number of experimental studies of the current of growth of both 3D metal crystals and droplets formed on a foreign substrate [2.36-2.44] and isolated faces of single crystals grown in a capillary [2.45,2.46]. In general, every case of an i vs. or an i vs 3D potentiostatic “current-time” growth law indicates that the ions are attached to the crystal (or droplet) surface, which means direct attachment mechanism. 2.2.3.2 Two dimensional nucleation For high supersaturations and/or strong adhesion of the new phase to the electrode surface two-dimensional nuclei can be formed also on a foreign substrate. In this case the expression for the stationary nucleation rate retain the general form of equation (2.19). However the size of the critical nucleus and the nucleation work are given by equations (1.83) and (1.84) or by equations (1.88) and (1.89), respectively. 2.2.3.2.1 Mechanism of direct attachment In this case the frequency is presented, analogously to equation (2.36), as:

Here the product gives the “surface area” of the edges of the 2D critical nucleus and is the number of ions impingements per square centimeter, per second. If the critical nucleus is considered as a twodimensional disc (Figure 1.17) and its radius is expressed through the Gibbs-Thomson equation (1.86) for one obtains

102

Chapter 2

where is the surface area occupied by a single atom in the disk-shaped nucleus. The quantity in equation (2.54) reads:

where now is the exchange current density referred to unit edge surface area. Finally, since in this case the supersaturation dependence of the Zeldovich factor is given by:

combination of equations (1.84), (2.19) and (2.54)-(2.56) yields:

Here

and

being the volume of one atom. Thus the overpotential dependence of the stationary nucleation rate can be presented in the form of a non-linear relationship:

where and Therefore, in this case experimental data for the stationary nucleation rate should be interpreted by means of some “best fit” method with three free


103

parameters and Note that if not disk-shaped but crystalline 2D nuclei are formed on a foreign substrate equation (2.59) attains the same form but with different values of and depending on the orientation of the crystalline nuclei on the foreign substrate (see equations (1.88)-(1.93)). 2.2.3.2.2 Mechanism of surface diffusion If 2D nuclei are formed through surface diffusion of ad-atoms on the foreign substrate what does change is only the frequency of single atoms attachment to the critical cluster. Similarly to the case of 3D nucleation (equation (2.44)) the quantity reads:

and therefore the expression for the stationary nucleation rate

becomes

where

Thus the overpotential dependence of the stationary nucleation rate is given by:

where and Finally we must point out that equations (2.57), (2.59), (2.61) and (2.63) turn into the expressions for the stationary nucleation rate on a like substrate if Of

104

Chapter 2

course, one should bear in mind that the quantities different values for a like and for a foreign substrate. 2.2.4

and

have

A confrontation with experiment

A most direct way to study the nucleation kinetics is to examine the “number of nuclei vs. time” relationship at a constant supersaturation Since the number N(t) of nuclei obtained within a time interval [0,t] is defined as:


105

where J(t) = dN(t)/dt is the nucleation rate, it appears that for a stationary nucleation process Therefore one should expect a linear “number of nuclei vs. time” relationship. Such experiments have been carried out by depositing silver and copper crystals and mercury droplets on platinum and on glassy carbon electrodes [2.5, 2.34, 2.35, 2.39, 2.40, 2.472.53] and some results are shown in Figure 2.3. As seen, a stationary state is established after a certain induction time but the nature of the non-stationary effects is considered in Chapter 2.4. Here we comment upon the overpotential dependence of the stationary nucleation rate calculated directly from the linear portions of the N(t) relationships.

106

Chapter 2

Figure 2.4 demonstrates an excellent qualitative agreement of the classical nucleation theory (equation (2.41)) with the experimental data. However, the quantitative interpretation of the slope of the straight line leads to unexpected results - the size of the critical nucleus turns out to vary from 6 to 3 atoms in the whole overpotential interval. (Similar values for are obtained if the data are interpreted by means of equation (2.51), (2.59) and (2.63)). This finding precludes a consequent quantitative analysis in terms of the classical nucleation theory since the latter describes the critical nuclei by means of macroscopic quantities like wetting angle, surface, surface and edge energy that have no physical significance for clusters consisting of only several atoms. The attempt to evaluate the stationary nucleation rate by means of equations (2.38), (2.48), (2.57) and (2.61) leads to even more disappointing results. Thus for and T = 308K it appears that a critical silver nucleus should consist of 20 atoms whereas the stationary nucleation rate must be equal to This means that such a silver nucleus should be formed once in 26 years! In the same time what the experiment does show at the overpotential (Figures 2.3 and 2.4) is a times higher nucleation rate, The striking quantitative contradiction between the classical nucleation theory and the experimental data accompanies the studies of electrochemical nucleation since the time of Thomfor and Volmer [2.29] who were the first to obtain a surprisingly low value for the size of a critical mercury nucleus on a platinum substrate. The problem has been successfully solved by the atomistic theory of the nucleation rate [2.10-2.12, 2.33, 2.62-2.66], which answers the question: how to interpret the experimental data on electrochemical nucleation? The next Section contains a survey of these theoretical considerations.

2.3

Atomistic theory of stationary nucleation

2.3.1

Walton - Stoyanov approach

The first attempt to overcome the quantitative contradiction between the classical nucleation theory and the experimental results was made by Walton [2.56, 2.57] who derived an explicit expression for the stationary nucleation rate valid for small critical nuclei. Walton’s theory was developed for the case of heterogeneous nucleation from a vapor phase but we consider it here


107

in order to ensure a better understanding of the atomistic approach to the nucleation kinetics. Similarly to the classical theory the Walton’s treatment is based on the idea for a critical nucleus the formation of which is the rate-determining step of the phase transition. However, the definition of the critical nucleus is conformed to the specificity of the small clusters and the high supersaturations. Thus it is assumed that the critical nucleus may consist of a very low number of atoms - two, one and even zero atoms in the case of very active substrates. The critical nucleus is considered as a cluster having a probability of decay higher than or equal to the probability of further growth. Attaching one atom, however, the nucleus acquires a probability of growth

higher than the probability

of decay and

transforms into the smallest stable cluster, which grows irreversibly. Under these circumstances the nucleation rate is defined through the number of stable clusters appearing on the substrate per unit time, i.e.,

where

is the frequency of attachment of single atoms to the critical

nucleus and is expressed as:

Here

accounts for the number of possible ways of attachment of single

atoms to the critical nucleus, is the probability to find an adsorbed atom on a given adsorption site, is the ad-atom’s vibration frequency and is the activation energy for surface diffusion, which is considered to be the operating mechanism of heterogeneous nucleation from a vapor phase. The quantity in equation (2.65) gives the number of critical nuclei per unit surface area and is calculated considering the clusters as a twodimensional gas composed of different molecules, each molecule corresponding to a cluster of specific structure and size. In the framework of this approximation it is possible to obtain the most probable size distribution function of clusters of different sizes and the number of critical clusters is determined by the minimum of at as in the classical approach to the same problem:

108

Chapter 2

In equation (2.67) is the total bonding energy of the and the number of single adsorbed atoms on the substrate reads:

cluster

where is the incidence flux of atoms from the vapor phase to the substrate and is the desorption energy of a single atom. Thus for the stationary nucleation rate one obtains:

Namely equation (2.69) is used to determine the size of the critical nucleus by studying experimentally the relationship at a constant substrate temperature T. The derivation of the function shows that in the theory of Walton [2.56, 2.57] the formation of any polyatomic cluster of the new phase is considered as a process connected with the overcome of a certain energy barrier. In the same time the condition for a minimum of the quasi equilibrium distribution function at indicates that namely the formation of the critical nucleus is connected with the use of maximal energy. Thus the Walton’s atomistic definition of the critical nucleus coincides with the first definition of this concept given in the classical nucleation theory (Chapter 1.2.2.4). There is, however, an essential difference in the second definition of the critical nucleus in the classical and in the atomistic model. While the classical theory defines the critical nucleus as a cluster having a probability of decay equal to the probability of further growth and allows us to derive the Gibbs-Thomson equation by means of purely kinetic considerations (Chapter 1.3.3), the atomistic model defines the critical nucleus as a non-equilibrium cluster. Thus which means that and the following way.

We shall clarify the meaning of these definitions in


109

Suppose that the growth and decay of an n-atomic cluster of the new phase takes place through consecutive acts of single atoms attachment and detachment. In that case the probabilities and are given by [2.11]:

and for the

critical nucleus it holds:

Therefore it follows that for the critical nucleus the smallest stable cluster consisting of

whereas for

atoms one obtains

and

Bearing in mind that in the case of heterogeneous condensation from a vapor phase the quantity is given by:

the inequalities

and

stable cluster are obtained in the form:

defining the critical and the

110

Here

Chapter 2

and

give the number of possible ways of single atoms

detachment from the critical and from the smallest stable cluster respectively, and are the bonding energies of the clusters consisting of and atoms and is the vibration frequency of the atom that is to be detached from the critical nucleus. Suppose now that at the substrate temperature T (supersaturation and a constant incidence flux the inequalities and are fulfilled. This means that the 2-atomic cluster (Figure 2.5b) disintegrates before attaching a third atom and the two-atomic configuration is unstable.

In the same time, the condition means that a single atom joins the 3-atomic cluster before its disintegration and the configuration in Figure 3c, in which each atom is connected with three bonds is relatively stable at the temperature T. In terms of the atomistic model the 2-atomic cluster is defined as a critical nucleus, the 3-atomic one is defined as the smallest stable cluster and the stationary nucleation rate is:

Decreasing the substrate temperature (i.e. increasing the supersaturation) the ratio diminishes and becomes equal to 1 at the temperature Below this temperature the equality sign changes, i.e. which means that the 2-atomic nucleus transforms into a stable cluster whereas the single atom adsorbed on the substrate plays the role of a critical nucleus. Thus the stationary nucleation rate becomes:

On the contrary, increasing the substrate temperature (i.e. decreasing the supersaturation) the ratio gradually increases and becomes equal to 1

2.KINETICS OF ELECTROCHEMICAL NUCLEATION at the temperature Above this temperature cluster is a critical nucleus, i.e.

111 and the 3-atomic

These simple considerations show that the temperatures and mark the upper and the lower limit of a supersaturation interval within which the 2-atomic cluster plays the role of a critical nucleus. Of course, we should note that in a real physical system the choice of a stable configuration cannot be made by means of such elementary geometric arguments, particularly if the clusters consist of a larger number of atoms. Therefore, a real nucleation experiment may not necessarily show exactly the sequence of critical nuclei presented in Figure 2.5. The Walton’s theory for the stationary nucleation rate was extensively used for interpretation of experimental data on heterogeneous condensation from a vapor phase [2.57-2.60]. However, it was for a long time considered applicable only to this special case of phase formation. Stoyanov [2.61] was the first to realize in 1973 that the Walton’s expression for the stationary nucleation rate can be obtained directly from the general formula of Becker and Döring [2.6] if all terms in the denominator of equation (2.9) are neglected with respect to the one corresponding to the critical nucleus. Thus assuming that

from equation (2.9) one obtains

which is the Walton’s formula for the case of a zero atomic critical nucleus; assuming that Walton’s formula for the biggest one

one obtains the atom etc. Finally, if the

term of the sum is

it follows that:

which turns exactly into the formula (2.69) if the attachment and detachment frequencies and are replaced by means of equations (2.66) and (2.74). Note that the quantity is introduced in equation (2.80) to account for the presence of terms equal to and commensurable with the biggest one in the sum of Becker and Döring (equation (2.9))[2.6].

112

Chapter 2

The readers who are already acquainted with Chapter 2.1.2 should have a fairly good idea of the physical significance of equation (2.80). However, in the early seventies the exact meaning of this expression was still obscure. Though, one thing was clear: an atomistic expression for the stationary nucleation rate could be obtained in any particular case of phase formation if the frequencies and were presented as functions of the supersaturation The first result of this finding was the atomistic theory of electrochemical nucleation developed by Milchev, Stoyanov and Kaischew in 1974 [2.10-2.12] (see also [2.5, 2.62-2.65]). The next Section presents the basic theoretical results obtained by these authors. 2.3.2

Atomistic theory of electrochemical nucleation

The crucial points in the derivation of an explicit expression for the stationary nucleation rate are: (i) the definition of the frequencies of attachment and detachment of single atoms to and from the ith site of the electrode surface (equations (2.10) and 2.11)) and (ii) the elucidation of the correlation between the ratios and (equations (2.12) and 2.13), respectively) [2.10-2.12]. Having this done, the formula for is evaluated by means of equation (2.80) and the obtained general expression reads:

We should point out that in case of atomistically small nuclei it is impossible to distinguish between two- and three dimensional clusters and between solid and liquid new phases since these are macro-concepts that have no physical significance for aggregates consisting of a few building units. 2.3.2.1 Mechanism of direct attachment In this case of nucleus formation from equation (2.81) one obtains:


Here the frequency for

coincides with the frequency

and the difference

113

(equation (2.10))

gives the thermodynamic work

of critical nucleus formation according to the general formula (1.32). Concerning the overpotential dependence of the stationary nucleation rate, combination of equations (2.82) and (2.10) for results in:

where

Thus the overpotential dependence of the stationary nucleation rate can be presented by a linear relationship:

where is a constant since for small critical clusters the size remains unchanged within sufficiently wide overpotential intervals (see Chapter 1.4 and Figure 1.29). 2.3.2.2 Mechanism of surface diffusion In this case of nucleus formation the frequencies of attachment and detachment and are given by [2.66]:

Here and are frequency factors, adsorbed on the electrode surface,

is the number of single atoms is the probability to find an

114

Chapter 2

adsorbed atom on a given adsorption site,

is the activation energy for

surface diffusion, and are the energy barriers for attachment and detachment of a single atom to and from the n-atomic cluster and the difference gives the separation work of one atom from the natomic cluster. Note that sometimes the frequency

is expressed through

the surface diffusion coefficient defined as [2.10, 2.32]. Making use of equations (2.86) and (2.87) and calculating the frequency ratio in equation (2.80) for the stationary nucleation rate one obtains:

and for the overpotential dependence of

it results:

Here

and is the equilibrium ad-atoms concentration (see equations (2.47), (2.47') and (1.22)). Thus for the overpotential dependence of the stationary nucleation rate one obtains the linear relationship:

where 2.3.3.

Interrelation between the classical and the atomistic nucleation theories

The question "How are the classical and the atomistic nucleation theories interrelated?" arose simultaneously with the Walton’s theoretical model [2.56] but was successfully settled 12 years later, in the framework of


115

the atomistic theory of electrochemical nucleation [2.10-2.12, 2.622.65,2.67]. In particular, papers [2.11], [2.63] and [2.67] provide a detailed theoretical analysis of the two models concerned and reveal the straightforward connection between the classical and the atomistic theoretical considerations. Nowadays it is quite clear that it does not make sense to speak about two different theories of the nucleation rate: the classical and the atomistic one. In reality, what we do have is a general nucleation theory comprising two limiting cases. The classical model describes the nuclei by means of macroscopic physical quantities and can be used to predict the size and to evaluate the nucleation work of sufficiently large critical clusters. The atomistic model is valid in the case of high supersaturations and very active substrates when the critical nuclei are very small. Therefore the quantitative interpretation of experimental data on the stationary nucleation rate based on the atomistic theory provides valuable information on the specific properties of clusters consisting of a few building units. The difference between the two models, as far as such difference does exist, should be sought for in the supersaturation dependence of the critical nucleus size. In the classical model this dependence is determined by the Gibbs-Thomson equation which juxtaposes a different critical cluster to each supersaturation. An analytical expression is proposed also for the supersaturation dependence of the stationary nucleation rate. The atomistic model takes into account the discrete character of the clusters size alteration at small dimensions (see Chapter 1.4) and does not propose a simple analytical relation between and Beside that, it accounts for the fact that a supersaturation interval and not a fixed supersaturation corresponds to each critical cluster. This changes also the shape of the supersaturation dependence of the stationary nucleation rate which, in coordinates vs. is a broken straight line (Figure 2.6) representing the intervals of or of the overpotential within which different clusters play the role of critical nuclei. This allows us to determine the size of the critical nuclei directly from the slope of an experimental vs. relationship. Indeed, taking the logarithm of equation (2.81) one obtains:

and differentiation with respect to

yields:

116

Chapter 2

Since for small clusters the size of the critical nucleus is constant within a given supersaturation interval and the free energy excess term of the nucleation work for this interval:

is constant, too, it appears that


117

In the same time it is readily seen from equation (2.10) that for and constant

if

whereas it is not difficult to show that if

from

equation (2.86) it results:

Thus neglecting the possible supersaturation dependence of equations (2.93)-(2.96) one obtains:

and

from

in the case of a direct attachment mechanism and

in the case of surface diffusion of ad-atoms on the electrode surface. Note that equations (2.97) and (2.98) allow us to evaluate also in those cases when the stationary nucleation rate is measured as a function of some relative overpotential where is the potential of a reference electrode. In this case the actual overpotential is expressed as where is a constant and therefore the experimental slope coincides with This is, however, not the case if data for are interpreted in terms of the classical nucleation theory since the calculation of the nucleation parameters and from the slope is possible only if the equilibrium potential is known. With decreasing the supersaturation the transition regions around the limiting values and broaden since more and more terms in the sum of Becker and Döring (equation (2.9)) become commensurable with the largest one corresponding to the critical nucleus. Thus the broken straight line turns into a smooth curve indicating the range of validity of the classical

118

Chapter 2

nucleation theory. As proved by Kashchiev [2.68], equations (2.97) and (2.98) could be applied also to the case of large critical nuclei if the potential dependence of both the specific free surface energy and the wetting angle could be neglected. However, in that case is the slope of the tangent to the vs. smooth curve (Figure 2.6). We can demonstrate this in the following way. Suppose that the stationary nucleation rate is presented by the classical formula:

where dependence of the term

Then neglecting the supersaturation one obtains:

where

Bearing in mind that according to the Gibbs-Thomson equation (1.56) it turns out that as before:

and therefore:

which practically coincides with equations (2.97) and (2.98). The analysis performed thus far demonstrates the general validity of the relation for small critical nuclei, because remains


119

constant in wide supersaturation intervals, as revealed by the atomistic nucleation theory [2.10-2.12, 2.56-2.67] and for large critical nuclei because of their specific equilibrium properties described by the Gibbs-Thomson equation, as revealed by Kashchiev [2.68]. Finally, we should remind that equations (2.97) and (2.98) would give correct values for the size of the critical nucleus only if the number of active sites on the electrode surface does not depend on, or is a week function of the overpotential Otherwise one should take into consideration the relationship when determining In that case equations (2.97) and (2.98) for transform into:

120

Chapter 2

in the case of direct attachment mechanism and

in the case of a surface diffusion mechanism. Having derived theoretical expressions for the stationary nucleation rate valid for small critical nuclei it is worth interpreting again the experimental data for (Figure 2.4) this time making use of equations (2.85) and (2.91). The obtained result is presented in Figure 2.7 showing that two different clusters play the role of critical nuclei in the whole overpotential interval: the 5-atomic one for and the 2-atomic one for These results agree well with the atomistic theory developed for small critical clusters. The formal qualitative correspondence between the classical theory and the experimental data (Figure 2.4) is due to the relatively narrow overpotential interval within which this experiment has been carried out. With increasing the overpotential region the disagreement with the classical theory is readily revealed. This is demonstrated in Figure 2.8a showing experimental data on the stationary nucleation rate of silver on a glassy carbon electrode obtained in a three times wider overpotential interval - from 0.050V to 0.140V [2.35]. As seen, the data for plotted in the classical coordinates ln vs. describe rather a smooth curve then a linear relationship. Similar results have been obtained in the case of mercury electrodeposition on a glassy carbon working electrode [2.53]. The dashed straight lines in Figure 2.8a mark the regions of seeming qualitative agreement with the classical nucleation theory, which would mislead experimentalists working within short overpotential intervals, say, (0.050÷0.090)V or In the same time the data for plotted in coordinates vs. according to equations (2.85) and (2.91) describe a single straight line which means that there is only one critical nucleus in the whole overpotential interval (0.050÷0.140)V. Making use of equation (2.97) with one obtains atom. Thus it appears that the single silver atom adsorbed on the electrode surface plays the role of a critical nucleus and the 2-atomic silver cluster is stable and can grow irreversibly under the particular experimental conditions. The results of another very informative experiment (Figure 2.9 [2.69]) show that three different clusters play the role of critical nuclei in the electrochemical nucleation of copper on polycrystalline palladium. In the


121

122

Chapter 2

overpotential interval (0.040 ÷ 0.190)V these are the clusters consisting of 4, 1 and 0 atoms. The last result, indicates that the single copper atom adsorbed on the glassy carbon substrate is stable enough and can grow irreversibly at the overpotentials It is interesting to note that exactly the same values for the size of the critical nucleus (4, 1 and 0 atoms) were obtained in the case of electrochemical nucleation of silver on a glassy carbon electrode, but in a different overpotential interval – (0.025 ÷ 0.240)V [2.35].

An important point of the theory of electrochemical nucleation is the evaluation of the nucleation work of small clusters. The difficulty arises from the fact that the structure of nuclei consisting of several atoms is not known and therefore the free energy excess is also unknown quantity. In order to elucidate this point we shall dwell upon the length L of the supersaturation interval within which a given cluster plays the role of a critical nucleus.


123

As we have already seen in Chapter 1.4, if clusters consisting of and atoms are critical nuclei in three neighboring supersaturation intervals, the quantity equals where and are the limiting supersaturations determined by the equalities and (see equations (1.147), (1.148) and (1.150)-(1.152)). However, a nucleation experiment provides information on the nucleation rate and not on the nucleation work and therefore the limiting supersaturations at which the size of the critical nucleus changes are determined by the equalities and Making use of equations (2.82)-(2.84) and bearing in mind that for small critical nuclei for the supersaturations and marking the beginning and the end of the interval within which the cluster is a critical nucleus one obtains:

As seen, these expressions do not coincide with equations (1.147) and (1.148). However, since and hardly differ too much and the same is valid for

and

without much lack of accuracy one

could neglect the terms in the square brackets of equations (2.106) and (2.107). The same result is obtained if the case of surface diffusion (equations (2.88) - (2.90)) is considered. Therefore we shall use equations (1.150) and (1.151) in order to determine the excess energies and of the 1- and the 4- atomic critical nuclei [2.5, 2.35, 2.66, 2.69, 2.70]. Since

and the obtained result is and Making use of the general formula (1.32) the calculation of the

corresponding nucleation works is straightforward: Let us now derive general formulae for the free energy excess for the nucleation work

and and

of small critical nuclei allowing us to

124

Chapter 2

evaluate these two quantities from experimental data for the stationary nucleation rate. Suppose that clusters consisting of atoms are critical nuclei in neighboring supersaturation intervals and that the limiting supersaturations are correspondingly (Figure 2.10). Then from the equalities it follows that:


125

and summing the left and the right hand sides of this set of algebraic equations one obtains:

Bearing in mind that in the case of heterogeneous nucleation the smallest critical nucleus consists of zero atoms, i.e. and equation (2.109) can be rewritten in the form:

which provides the possibility to evaluate the free energy excess of each critical nucleus from the series if the sizes and the limiting supersaturations are experimentally determined. Correspondingly, the nucleation work supersaturation according to:

can be calculated for each

For example, in the case of nucleation of silver on glassy carbon [2.34, 2.35] and of copper on palladium [2.70] we have registered two critical nuclei beside the zero-atomic one – those consisting of 1 and of 4 atoms. Therefore n = 2, and

126

which coincides with the results obtained above since

Chapter 2

and

The extrapolation of equations (2.110) and (2.111) to high values of x (large critical nuclei) can be performed if the difference is considered as infinitesimal and the function is supposed to reflect the supersaturation dependence of the critical nucleus size according to the Gibbs-Thomson equation ((1.62), (1.75)). Then the sum in equation (2.110) is replaced by an integral, i.e.,

which is nothing other than an expression for the free energy excess presented through the integrated Gibbs-Thomson equation (1.56), being the number of atoms in the critical nucleus. In equation (2.114) is the number of atoms in the critical nucleus which is big enough to justify the macroscopic approach and is its free energy excess. We should emphasize that equations (2.109)-(2.111) represent general formulae that can be used for a direct calculation of and of small clusters in any case of a first order phase transition. The key point is the experimental determination of the sizes of the critical nuclei, the limiting supersaturations and particularly the supersaturation indicating the transition between the zero-atomic and the smallest nonzero-atomic nucleus Such data are available for several experimental systems and some calculated values for and are given in Table 2.1 both in joules and in kT units. The performed considerations unambiguously show that the free energy excess and the nucleation work of small critical nuclei can be determined directly from experimental data without any


127

simplifying assumptions concerning the cluster structure and the specific cluster-foreign substrate interaction. This is certainly an important result bearing in mind that small clusters do play a decisive role in the first order electrochemical phase transitions. As we know, in that case the size of the critical nuclei hardly exceeds the value of 10 atoms [2.34, 2.35, 2.47-2.53, 2.63, 2.69-2.115].

Concluding we should point out that the atomistic theory is often called a theory for high supersaturations. However, this definition is not quite precise. For instance we have already seen (Table 2.1) that a 4-atomic silver cluster is a critical nucleus in the overpotential interval (0.025 ÷ 0.051)V. Thus it mean that the supersaturation is a high supersaturation? An estimate of the size of the spherical silver cluster, which would be the critical nucleus at no interaction with the substrate shows that for atoms. Obviously, this is a value high enough to rank among the low supersaturations. Therefore in this case the experimentally determined small size of the nucleus atoms) is due to the strong interaction between the silver atoms and the glassy carbon substrate and not to the high supersaturation. Thus the atomistic theory should be called a theory for small critical nuclei and one must bear in mind that in the case of heterogeneous phase formation a small critical nucleus does not necessarily mean a high supersaturation.

128

2.4

Chapter 2

Non-stationary nucleation

An important task of the nucleation theory is to determine the nonstationary nucleation rate J(t) and to evaluate the time needed to establish a stationary state in the supersaturated system. One possibility to do this is to solve numerically the set of differential equations (2.4) and to calculate the time dependence of the number of the n-atomic clusters [2.116-2.121]. Apparently the results obtained in this way will depend strongly on the numerical values used for the attachment and detachment frequencies and Another possibility is to apply the continuity approach of Zeldovich [2.19] and Frenkel [2.20] and to obtain explicit expressions for and for J(t). What follows describes the latter theoretical method. 2.4.1

Zeldovich-Frenkel theory

What Zeldovich did in 1942 [2.19] was to consider the nucleus size n as a continues variable and to replace the set of ordinary differential equations (2.4) by a single partial differential equation:

in which

is the non-stationary nucleation rate (c.f. equation (2.26)). As Zeldovich pointed out, equation (2.115) could be considered as a diffusion equation describing the “motion” of the size n in a thermodynamic field of force. In particular, for constant and equation (2.115) transforms into the Fick’s low:

where the attachment frequency plays the role of a diffusion coefficient [2.19]. An exact solution of equation (2.115) cannot be obtained. However, the approximate determination of the functions Z(n,t) and J(t) was attempted by


129

several authors [2.122-2.134]. As discussed by Lyubov and Roithburd [2.27] (see also [2.3]) the main assumption in those theoretical considerations was that the establishment of the steady state depends on the processes taking place in the narrow region near the critical nucleus (Figure 2.2.). Outside this critical region the stationary state was considered to be almost instantaneously fixed. Based on such simplifications different approximate formulae have been derived for Z(n,t), J(t) and N(t) but nowadays the expressions obtained by Kashchiev [2.128] are considered as most precise:

In equations (2.119) and (2.120) is the non-stationary induction period (or time lag) associated with the time needed to establish a stationary size distribution of clusters in the supersaturated system. It is interesting to note that Collins [2.126] obtained the same mathematical formula for the nonstationary nucleation rate J(t) but with a different expression for the induction time Thus according to Kashchiev [2.128]:

whereas according to Collins [2.126]:

i.e., We must point out, however, that the reliability of the Kashchiev’s solution (2.122) has been numerically proved by several authors [2.119, 2.120, 2.135-2.137].

130

Chapter 2

As expected, the induction time depends essentially on the mechanism of nucleus formation and decreases with increasing the supersaturation and the attachment frequency , the latter being given by equation (2.36), (2.44), (2.54) or (2.60). Equations (2.119) and (2.120) show that for times the nucleation rate attains its steady state value and for the number of nuclei N(t) it results:

This expression allows us to determine the value of directly from the slope of the straight line portion of an experimental N(t) relationship (Figure 2.11), whereas its intercept on the time axis relates to the induction time as Substituting the last formula in equations (2.119) and (2.120) one


131

obtains the nucleation rate J(t) and the number of nuclei N(t) as functions of the non-dimensional parameter

In this way the kinetic quantities J(t) and N(t) are presented through the two experimentally accessible nucleation parameters: the slope of the linear portion of an N(t) relationship giving the stationary nucleation rate and its intercept on the time axis that is connected with the induction time This permits a direct interpretation of experimental results based on the Zeldovich non-stationary nucleation theory [2.47, 2.48, 2.74, 2.132, 2.133]. Equation (2.121) allows us to evaluate the induction time in the framework of the Zeldovich-Frenkel continuity approach. Thus for electrochemical nucleation of silver on a platinum electrode [2.5, 2.47], for and T = 308K it appears that

atoms,

and

for the induction time it results These data differ dramatically from the experimental findings, atoms and [2.5, 2.47] (c.f. Figure 2.3), and this necessitates looking for other reasons for the experimentally observed induction times. In the following we shall show that a transient nucleation rate can also be obtained if the energetic state of the electrode surface changes during the process of electrochemical phase formation. However, I suppose that beforehand the reader might be interested to know how has Ja.B.Zeldovich himself commented the induction times observed in the case of electrochemical nucleation. What follows is a short true story: I met Jakov Borisovich Zeldovich in April 1979 during his visit to the Institute of Physical Chemistry in Sofia. At that time we often had foreign visitors and in such occasions our Director, Academician R.Kaischew, used to gather the heads of research groups and to ask them to present their recent results to the prominent guests. This is how I met Professor I.N.Stranski who, during the second half of his life, worked in former West Berlin, Academician A.N.Frumkin, Sir F.C.Frank, the Nobel Prize winner D. Hodgkin and others.

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In that case I had to tell Jakov Borisovich about the non-stationary electrochemical nucleation. I explained briefly the experimental procedure and showed him the graph given in Figure 2.3. Jakov Borisovich who listened carefully looked at it and asked a single question: “What is the size of the critical nucleus?” “5 atoms” - I replied.

Visit of Ja.B.Zeldovich to the Institute of Physical Chemistry, Bulgarian Academy of Sciences in April 1979, (front row, left to right): Ja.B.Zeldovich, V.Bostanov, R.A.Kaischew; (back row, left to right): S.Stoyanov, I.Gutzov, Chr.Nanev, A.Milchev and D.Kashchiev. “Forget about the “Zeldovich” induction times”- Jakov Borisovich said. I did not venture to ask why. And I did not have time for a longer discussion - the others waited. To my shame I said to myself: “The man has forgotten his own theory”. But he was not! Later I realized what Zeldovich meant. In aqueous solutions, room temperature and “attachment - detachment” frequencies of the order of the time needed to establish a stationary size distribution of clusters consisting of several atoms could not take milliseconds. Fortunately, I did not forget the Zeldovich comment. Three years later it led me to a different explanation of the non-stationary nucleation effects. Chapter 2.4.2 is devoted to this particular subject.

2.4.2

Nucleation on a changing number of active sites

The sets of equations (2.4) and (2.8) are formulated on the simplifying assumption that the number of nucleation sites is much bigger than the number of single adsorbed atoms and polyatomic clusters. While such an


133

assumption is justified in the very beginning of the nucleation process, the inequality

is certainly not fulfilled in the advanced stage of the

phase transition when most of the nucleation sites on the electrode surface are occupied by the clusters of the new phase. The situation is additionally complicated by the fact that a real solid surface is never energetically homogeneous. Thus it is well established that the nucleus formation takes place on active sites, the number density of which is much lower than the total number density of atomic sites of the substrate that is typically of the order of Erdey-Grúz and Volmer [2.140] were the first who in 1931 noticed that nuclei of the new phase were formed on some preferred sites of the electrode surface. In a subsequent work [2.141], Erdey-Grúz and Wick confirmed this observation (see also Refs. [2.7, 2.29]) and since that time the concept of active sites inevitably accompanies all theoretical and experimental studies of electrochemical nucleation on a foreign substrate. Various theoretical models have been developed to account for the influence of active sites on the nucleation kinetics. Thus Fleischmann and Thirsk [2.142] assumed that all preferred sites were equally active, whereas Kaischev and Mutaftschiev [2.73] and later Markov et al. [2.49, 2.143, 2.144] and Fletcher et al. [2.145-2.150] explored the case of active sites with a different activity regarding the process of nucleus formation. Important contributions to the same subject were made also by Baraboshkin [2.151], Abyanech and Flesichmann [2.152, 2.153], Scharifker and Mostany [2.154], Polukarov [2.155], Sluyters-Rehbach et al. [2.156, 2.157], Mirkin and Nilov [2.158], Heerman and Tarallo [2.159-2.162] and Danilov et al. [2.1632.165]. The above mentioned theoretical and experimental studies were based on the hypothesis for an originally “frozen” energetic state of the electrode surface. Thus it has been tacitly assumed in all cases that after applying the overpotential the number of active sites on the substrate was instantaneously fixed. During the process of nucleus formation the active sites could only decrease either due to a direct occupation by the nuclei of the new phase or due to an ingestion by zones of reduced concentration and overpotential arising around the growing supercritical clusters. However, the mere fact that the number of active sites changes when changing the electrode potential [2.53, 2.89, 2.100, 2.101, 2.104, 2.105, 2.173] indicates that some time should be needed to transform the electrode surface from one to another energy state. Of course, it is a different point if such time period would be long enough to be experimentally observed – that would depend on the properties of the particular experimental system. Taking this into consideration, in this Section we describe a qualitatively new physical model

134

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of the nucleation kinetics assuming that active sites may appear on and disappear from the electrode surface because of independent chemical and/or electrochemical reactions, parallel to the process of nucleus formation [2.166-2.169] (see also [2.170, 2.171]). The latter could be oxidationreduction reactions, surface transformation phenomena within pre-formed UPD layers, electrochemical adsorption and desorption of foreign atoms and/or blocking agents like anions, organic and inorganic additives, impurities etc. 2.4.2.1 General formulation We consider a solution of metal ions and an inert working electrode containing two types of active sites at the time moment developed active sites and latent active sites which can be developed only at an electrochemical overpotential At time t the total number of active sites is distributed into developed, free active sites, active sites occupied by N(t) nuclei of the new phase and undeveloped, latent active sites. Under these conditions the following electrochemical reactions take place on the working electrode: (i). The latent active sites are developed, the rate constant of this process being (ii). The initial and the newly developed, free active sites are either occupied by the nuclei of the new phase or deactivated by a backward electrochemical reaction. The rate constants of these two processes, and respectively, are considered to have the same values both for the initial and for the newly developed active sites. (iii) The active sites deactivated with a frequency turn into latent sites and can be developed again with a frequency Assuming that an active site which is occupied by a nucleus of the new phase cannot be deactivated, the overall nucleation kinetics are described by two differential equations:

The first term of the right hand side of equation (2.124) gives the rate of birth of free active sites on the electrode surface and the second term gives the death rate of free active sites. As for equation (2.125), it gives the rate of appearance of supercritical clusters on the working electrode. We should


135

emphasize that this theoretical model does not take into account “Zeldovich” type of non-stationary effects. Instead, we assume that the quantity giving the nucleation frequency per site is time independent and is expressed as (see equations (2.14) and (2.15)):

In other words,

for large and

for small critical nuclei (c.f. equations (2.19) and (2.81)). considered as time independent, too. Substitution of and equation (2.125) into equation (2.124) yields:

and

are from

where and Thus the overall nucleation kinetics are described by a single, second order differential equation which is known to have three different solutions depending on the sign of the difference or Since after rearrangement becomes:

under the initial conditions t = 0,

N(0) = 0 for N(t) one obtains:

136

where

Chapter 2

Correspondingly, the relations

and

hold good for the time dependencies of the rate J(t) = dN/dt of nucleus formation and of the number of free active sites on the electrode surface. The inspection of equations (2.128) and (2.129) shows that in the general case the quantity J(t) has a non-zero initial value, a zero final value, and displays a maximum, at time

(Figure 2.12, line 1). As expected, the number N(t) of nuclei changes with time from N(0) = 0 at t = 0 to at (Figure 2.12, line 2), whereas the number of free active sites (equation (2.130)) changes from at t = 0 to at and attains a maximum at time If there are no free active sites on the electrode surface before applying the nucleation overpotential, i.e. if P = 0 and equations (2.128)–(2.131) simplify to:


137

In another limiting case when there are no latent active sites on the electrode surface or when the latent sites cannot be developed because the nuclei form only onto the initially available active sites and equations (2.128) and (2.129) simplify to:

138

Chapter 2

Finally, if the deactivation frequency equals zero the above expressions for N(t) and J(t) turn into the ones derived by Fleischmann and Thirsk [2.174, 2.175] and by Kaischew and Mutaftschiew [2.73] who have taken into consideration only the exhaustion of active sites due to the nucleus formation thereon:

As seen, in this case the nucleation rate has a maximal value at the initial moment t = 0, i.e. and decreases to zero at The detailed direct experimental studies of the “number of nuclei versus time” relationship [2.5, 2.34, 2.35, 2.47, 2.48, 2.50, 2.52, 2.73-2.75, 2.892.91] show that after a certain induction period the slope dN/dt of the N(t) curves attains a constant, stationary value (Figure 2.3). The observed time lags range from [2.74] to [2.34, 2.35] and cannot be explained in terms of the Zeldovich non-stationary nucleation theory predicting induction times of [2.5] (Chapter 2.4.1). Reverting to the theoretical model accounting for the “site birth and death” effects it is not difficult to show that the general equations (2.128) and (2.129) predict N(t) and J(t) relationships similar to the experimental ones (Figure 2.3) if two inequalities, and are simultaneously fulfilled within a sufficiently wide time interval In that case and equation (2.128) turns into the linear relationship the slope giving the stationary nucleation rate:


and the intercept

139

on the time axis being:

In the special case of

equations (2.132) and (2.133) transform into:

As shown in Ref. [2.167] the condition for a stationary state: depends on the values of the rate constants and and can be rewritten in the form:

140

Chapter 2

Figure 2.13 shows non-dimensional plots of equations (2.128) and (2.129) calculated for and As seen, under these conditions the theoretical N(x) vs. x relationships are quite similar to the experimental ones shown in Figure 2.3. For relatively short times, when the condition is fulfilled but the exponential term in equations (2.128) and (2.129) cannot be neglected (see the initial portions of the nondimensional plots in Figure 2.13). In that case the N(t) and I(t) relationships are given by:

where is an induction time and Equation (2.136) shows that a stationary state is established after elapsing of time However, if the rate constants and have commensurable values the nucleus formation takes place under non-steady state conditions from the very beginning till the end of the phase transition. 2.4.2.2 Particular cases We consider three cases in which inequality (2.134) is fulfilled and equations (2.135) and (2.136) describe the onset of nucleus formation.


141

2.4.2.2.1 High frequency of appearance of active sites In this case equations (2.132) and (2.133) transform into:

which means that the slope dN(t)/dt of the linear portion of an experimental N(t) relationship gives directly the stationary nucleation rate, whereas the intercept on the time axis depends on the frequency of appearance of active sites on the electrode surface. In this case the induction time is:

As seen, if

coincides with

2.4.2.2.2 High nucleation frequency In this case equations (2.132) and (2.133) transform into:

which means that now it is the intercept on the time axis that contains the information on the nucleation rate constant What we believe to be a stationary nucleation rate, reflects the steady state kinetics of appearance of active sites on the electrode surface and has nothing to do with the actual process of nucleus formation. Now the induction time equals:

142

Chapter 2

As seen from equation (2.141) the intercept on the time axis may be positive, zero or negative depending on the numerical value of the ratio Thus three different types of N(t) relationships may be obtained depending on or

(Figure 2.14). Bearing in mind that in this particular case it is clear that would be positive (Figure 2.14, (1)) only if the initial number of free active sites is low enough to ensure the validity of the inequality If which means that the initial nucleation rate coincides with the stationary one then (Figure 2.14, line 2). Finally, if i.e. if then (Figure 2.14, line 3). As before, in the case the intercept is always positive and equals The time dependence of the nucleation rate J(t) = dN(t)/dt is presented in Figure 2.15 by means of non-dimensional plots


143

2.4.2.2.3 High deactivation frequency In this case equations (2.132) and (2.133) transform into:

which means that and depend on the rate constants the three electrochemical reactions. The induction time the average deactivation time of active sites:

and

of equals

144

Chapter 2

Similarly to the case of high nucleation frequency, three different types of N(t) relationships (see Figure 2.14) may be obtained also in this case of phase formation. However, the intercept on the time axis will be positive, zero or negative depending on the numerical value of the ratio or Clearly, for the intercept coincides with and is always positive. The above considerations raise a very important question: What do we measure in a progressive nucleation experiment? Is dN(t)/dt a nucleation rate, a rate of appearance of active sites or it is a combination of the tree rate constants? As we have seen, depending on the values of and the experimentally accessible physical quantities and may have completely different physical significance although what we only can and do measure in a direct nucleation experiment is the number of nuclei vs. time, N(t) vs. t, relationship. Note that for sufficiently high values of or (cases 2.4.2.2.2, equation (2.141) or 2.4.2.2.3, equation (2.144), respectively) the induction time may become too short to be experimentally detected. Therefore even in the case of experimental N(t) relationships without induction periods [2.72, 2.176] one would not be able to assert that the constant rate of appearance of stable clusters on the electrode surface would relate to the actual process of nucleus formation. Clearly, we are in exactly the same situation when data on the nucleation kinetics are obtained indirectly, by measuring the current of progressive nucleation [2.169]. These theoretical considerations show unambiguously that it is difficult to say what kind of information do we obtain from an electrochemical nucleation experiment. The reason is that the state of the electrode surface and therefore also the number of active sites may depend strongly on the applied electrochemical overpotential. Since in all nucleation experiments the electrode potential changes from one to another level (say, from the initial to the nucleation potential) it may take certain time for the electrode surface to reconstruct and to turn from one energetic state to another. Apparently this complicates the overall nucleation kinetics and it seems impossible to distinguish between the actual process of nucleus formation and the parallel electrochemical reactions that may take place on the electrode surface. One possibility to solve this problem is described in details in Refs. [2.168, 2.169] and here we comment upon the same subject. For the sake of clarity we discuss the relatively simple case of nucleation influenced by potential dependent redox electrochemical reactions that may change the state of the electrode surface in a real experiment.


145

2.4.2.3

Electrochemical nucleation influenced by parallel redox reactions Consider a solution of metal ions with a concentration c, a reference, a counter and an inert metal working electrode that does not interchange like ions with the electrolyte solution. The whole system is kept at a constant temperature T. As we know, in order to form nuclei of the new phase on the electrode surface it is necessary to polarize the working electrode to a potential E more negative than the equilibrium potential of the bulk metal M in the same electrolyte. The electrochemical reaction of the metal ions discharge is:

This is, however, not the only reaction that may take place in such an electrochemical system. Parallel to it the working electrode itself may participate in independent redox electrochemical reactions of the type:

The equilibrium potential of such processes depend only on the temperature T and on the acidity of the electrolyte solution and is given by [2.54]:

being the standard potential. Note that very often the experimentalists neglect such secondary electrochemical reactions although they could be of major importance for the properties of the electrode surface [2.85, 2.88, 2.93] (see also [2.177, 2.178] and the references cited therein). 2.4.2.3.1 The standard pulse potentiostatic technique The pulse potentiostatic technique which is most frequently used to study the nucleation kinetics consists in the following (Figure 2.16). (i). Pulse I: The working electrode is polarized to the potential more positive than the equilibrium potential in order to dissolve the nuclei formed during the previous nucleation experiment. The dissolution

146

Chapter 2

overpotential is If the potential is more positive than a metal oxide or hydroxide will form on the electrode surface according to reactions (2.147) or (2.148), the oxidation overpotential being

(ii). Pulse II: The working electrode is polarized to the initial potential to create standard conditions for the process of nucleus formation. Often the potential is set equal to the equilibrium potential so that at this stage


147

the nucleation overpotential equals zero. However, if is more negative than the oxide layer is partly reduced (reduction overpotential and the obtained metal spots scattered within the oxide coverage play the role of active sites for the subsequent process of nucleus formation. Thus, at the end of Pulse II there will be developed and latent active sites where the metal oxide is to be reduced at a more negative potential. (iii). Pulse III: The working electrode is polarized to the potential E more negative than the equilibrium potential to form nuclei of the new phase at the nucleation overpotential At this stage the reduction overpotential is increased to and the latent active sites start reducing. Thus new, free active sites appear simultaneously with the process of nucleus formation. (iv). Pulse IV: The electrode potential is changed to the difference giving the growth overpotential The latter ensures the growth of nuclei formed at Pulse III but is not high enough to form new nuclei on the electrode surface. At this stage the nuclei of the new phase grow to a visible size and can be counted. The reduction of the oxide coverage may continue at the overpotential but this process cannot affect the nucleation kinetics if is selected sufficiently small. Finally, Pulse I is again applied to clean the electrode surface and to prepare it for a new cycle. In this way, changing the duration of Pulse III at a fixed amplitude the “number of nuclei vs. time” dependence can be measured at different constant overpotentials: etc. Simultaneously it is possible to record also the current of progressive nucleation and the pure growth current during the Pulses III and IV, respectively. The results of such combined nucleation experiments are reported in [2.40, 2.89]. As seen, using the standard pulse potentiostatic technique one takes the risk of varying more than one nucleation parameter at the same time, something that an experimentalist should never do. Indeed, any change of the electrode potential E leads to changes of both the overpotential which determines the process of nucleus formation and the overpotential which determines the dynamic properties of the electrode surface. Consequently, the three rate constants and and the total number of active sites change simultaneously and this is what makes impossible to distinguish between the nucleation rate and the rate of appearance of active sites in an ordinary nucleation experiment. Apparently, the only way to overcome this complication is to fix the energetic state of the electrode surface, which means to fix the temperature T, the solution acidity pH and the electrode potential E. In what follows we

148

Chapter 2

show how this can be done by means of a modified pule potentiostatic technique. 2.4.2.3.2 The modified pulse potentiostatic technique We have already seen in Chapter 1.1.2 that the supersaturation and the nucleation overpotential (equations 1.15-1.18) can be varied in two different ways. The first and the most frequently used one is to change the

working electrode potential E at a fixed equilibrium potential The second way is to fix the electrode potential E and to vary the equilibrium potential

2.KINETICS OF ELECTROCHEMICAL NUCLEAT1ON

149

by varying either the concentration c or the temperature T of the electrolyte solution. Having fixed the solution acidity pH, the temperature T and the potentials of Pulses I-IV at the values E and respectively (Figure 2.17), the “number of nuclei versus time” dependence can be measured by varying only the duration of Pulse III. Different nucleation overpotentials etc. can be applied by increasing the concentration of metal ions in the electrolyte solution, which makes more positive the equilibrium potential As before, it is possible to record also the current of progressive nucleation and the growth current during the Pulses III and IV, respectively. Note that it is not appropriate to vary through the temperature T because the equilibrium redox potential (equation (2.149)), and therefore also are temperature depending quantities. Using the modified pulse potentiostatic technique one should bear in mind that with increasing the equilibrium potential the growth overpotential increases, too. This may cause the formation of new nuclei during Pulse IV if the overpotentials etc. exceed the critical nucleation overpotential. If this happens, the increase of may be compensated by making the potential more positive. Clearly, this would have only a temporary effect on the oxide coverage since the surface state of the working electrode will be again fixed during pulses I and II. As for the dissolution overpotential with increasing it decreases to etc. which may be not positive enough to dissolve all nuclei formed and grown during the pulses III and IV. Since in that case a correction of by making more positive may change the initial state of the oxide coverage it is desirable to select a sufficiently positive dissolution potential from the very beginning of the nucleation experiment. 2.4.2.4 A confrontation with experiment The experimental data presented in Figure 2.3 are obtained by means of the standard pulse potentiostatic technique, which does not allow us to distinguish between the already considered three particular cases. The reason is that the slope dN(t)/dt and the intercept of the experimental N(t) relationships depend on the applied electrode potential E, which affects the two electrochemical overpotentials: and However, if the modified pulse potentiostatic technique is employed it appears that different types of N(t) relationships can be obtained depending on the values of the rate constants and

(i). If the inequality is fulfilled the slope dN(t)/dt of the linear portions of the N(t) relationships gives directly the

150

Chapter 2

stationary nucleation rate (equation (2.137)) and will increase with increasing the nucleation overpotential Since E, T and

pH are fixed, the overpotential is fixed, too, and the induction time (equation (2.138)) which depends only upon


151

and remains constant. This means that experimental N(t) relationships with different slopes but the same intercept will be obtained when varying the nucleation overpotential by varying the concentration c (Figure 2.18a). (ii) If the inequality is fulfilled the slope dN(t)/dt of the linear portions of the N(t) relationships gives the stationary rate of appearance of active sites (equation (2.140)). This quantity depends only on and does not change when changing the nucleation overpotential at a fixed electrode potential E. Therefore parallel N(t) relationships with different intercepts will be obtained when varying the nucleation overpotential by varying the concentration c (Figure 2.18b). In this case it is the intercept on the time axis (equation (2.141)) that contains the information on the nucleation rate constant As we have already mentioned in Chapter 2.4.2.2.2 the intercept may be positive, zero or negative depending on whether or Therefore, since and are fixed in the modified pulse potentiostatic technique it is clear that with increasing the overpotential will incerase, too and may pass from positive through zero to negative values at the same stationary nucleation rate. We should emphasize that this would not be the case if the experiment is carried out by means of the standard pulse potentiostatic technique (Figure 2.16) since the change of the electrode potential E will change both and Therefore the rate constants and will change simultaneously, most likely in a different manner, and one cannot predict how this would affect the value of (iii). If the inequality is fulfilled the slopes dN(t)/dt of the linear portions of the N(t) relationships depend on the three rate constants (equation (2.143)) and, through will increase with increasing the nucleation overpotential Since the induction time (equation (2.144)) depends only on which is fixed the intercept from the time axis remains constant. Similarly to the case (ii) may be positive, zero or negative, now depending on the value of the ratio and three types of N(t) relationships can be obtained (Figure 2.19 a-c). However, the sign of cannot change with increasing as in case (ii) since it is function only of which is fixed in the modified pulse potentiostatic technique. As seen, the modified pulse potentiostatic technique allows us to comprehend the information obtained in a nucleation experiment already from the shape of the experimental N(t) relationships (Figures 2.18 and 2.19). In the following we comment upon the concentration and the

152

Chapter 2


153

overpotential dependence of and We consider only the atomistic nucleation theory valid for small critical nuclei (Chapter 2.3.2) since this is always the case of electrochemical nucleation on a foreign substrate. If the nuclei are formed through a direct attachment mechanism the relationship is given by equations (2.82)-(2.84) and substituting and one obtains:

or

Here

In the same fashion, for the case of a surface diffusion mechanism from equations (2.88)-(2.90) one obtains:

or

where

Equations (2.150') and (2.152') tell us that the size of the critical nucleus can be determined from the slope of the concentration dependence of the stationary nucleation rate obtained at a fixed electrode potential E:

154

Chapter 2

The above formula applies to both mechanisms of critical nucleus formation: direct attachment of ions from the electrolyte and surface diffusion of adatoms on the electrode surface. Clearly, the number of active sites remains unchanged when varying the concentration c since E, T and pH are fixed in the modified pulse potentiostatic technique. If the slope of the linear portion of the N(t) curve gives the stationary rate of appearance of active sites (equation (2.140)) the overpotential dependence of this quantity can be determined only if the actual mechanism of appearance of active sites on the electrode surface is known. Apparently this would require detailed studies of the specific redox reactions taking place on the electrode surface. Beside that, one should bear in mind that what forms during the dissolution Pulse I (Figure 2.18) may be not a thermodynamically stable stoichiometric metal oxide but a complex coverage consisting of oxygen containing surface species [2.85, 2.88, 2.93, 2.177, 2.178]. Nevertheless, assuming that the appearance of active sites could be considered as a process of formation and growth of negative nuclei, i.e. holes within the oxide coverage, an approximate formula of the type:

can be used to interpret the data for the overpotential dependence of the stationary nucleation rate. In equation (2.155) is the size of the negative critical nucleus, is the corresponding transfer coefficient, m is the number of electrons exchanged in the particular reduction reaction and the sum gives the negative nucleation overpotential Assuming that in this case the critical size remains constant in sufficiently wide overpotential intervals, too, equation (2.155) allows us to determine the value of from the slope of an experimental versus relationship even in the case when and are unknown quantities [2.83]. Concerning the intercept it depends on the kinetics of appearance and disappearance of active sites in all cases excepting 2.4.2.2.2. In the latter case is given by equation (2.141) and can be presented in the form:

where the constants and read or and Therefore the size n of the critical nucleus can be determined by fitting the experimental relationship to equation (2.156). In all other


155

cases the interpretation of requires a theoretical model for the mechanism of appearance and disappearance of active sites on the electrode surface. Concluding this Chapter we present a set of experimental N(t) relationships obtained by means of the modified pulse potentiostatic technique in the case of mercury electrodeposition on a platinum single crystal microelectrode (Figure 2.20) [2.179]. The data for N(t) refer to the whole electrode surface and the nucleation rate J(t) = dN(t)/dt has dimensions instead of As seen the intercept on the time axis does not change when varying the concentration c, which means that it is the slope of the straight lines that contains the information on the stationary nucleation rate of mercury on the platinum surface - either according to equation (2.137) or according to equation (2.143) if part of the active sites are deactivated by a fast backward electrochemical reaction. However, in both cases the data for should

describe a linear versus Inc relationship and this is shown in Figure (2.21). From the slope (equation (2.154)) for the size of the critical mercury nucleus one obtains 3 atoms which confirms the applicability of the atomistic nucleation theory to this experiment. An important point of the nucleation theory and experiment concerns the possible activity distribution of active sites on the electrode surface. The problem was considered first by Kaischew and Mutaftchiev in 1965 [2.73]

156

Chapter 2

but later the idea was developed further by Markov et al. [2.49, 2.143, 2.144] and by Fletcher et al. [2.145-2.150].

The idea of Kaischew and Mutaftschiew [2.73] is simple and clear. The activity of the active sites is considered to be determined by the bond energy (or the “wetting”) between the active sites and the nuclei formed thereon. The stronger the bond energy, the higher the activity, the lower the nucleation work and the smaller the size of the critical nucleus formed on an active site. What does it mean? If we consider a substrate with k types of active sites with different activities the total number N(t) of nuclei that will be formed within a given time interval [0,t] will be:

where is the number of nuclei formed on the kth type of active sites. Correspondingly the total nucleation rate J(t) = dN(t)/dt is:

or


where

157

is the mean nucleation rate averaged over all types

of active sites. Clearly, it is J(t) that we measure in a real nucleation experiment and interpreting its potential or concentration dependence we determine some average number of atoms in the critical nucleus. This number will be bigger if there are many active sites with a lower activity on the electrode surface but this is hardly the case of electrochemical nucleation on a foreign substrate where the size of the nucleus does not exceed 10 atoms. Of course, the activity distribution of active sites could be extended to the region of higher activities. Apparently this would result in smaller critical nuclei but the size of a critical nucleus cannot become smaller than zero. Therefore extension of activity distribution to higher activities is of no importance after a limiting activity at which (see e.g. [2.35, 2.69, 2.81, 2.101]). In that case the thermodynamic work for nucleus formation equals zero, too and the overall nucleation process will be determined by the kinetics of single ions attachment to the active sites, to the adsorbed atoms and to the n-atomic clusters, which will be all stable under such circumstances. The activity will not matter anymore, whatever it is. Therefore nucleation rate dispersion due to activity distribution of active sites although being of principal interest does not seem to play a significant role in the case of electrochemical nucleation on a foreign substrate if is sufficiently small. Thus far we considered the particular case of a metal working electrode whose surface properties change during the process of nucleus formation due to some redox, potential dependent electrochemical reactions. However, we should remind that the general theoretical model can be used to describe the nucleation kinetics also if other physical phenomena, like underpotential metal deposition, adsorption and desorption of anions and/or different organic and inorganic blocking agents determine the dynamic properties of the electrode surface. Due to such electrochemical processes small “background” currents may flow through the electrochemical system prior to and during the process of nucleus formation. Such currents are often ignored although they could reflect the energy change of the electrode surface that might be of major importance for the overall nucleation kinetics. Finally, Table 2.2 presents data for the standard oxidation-reduction potentials (equation (2.149)) of some metal electrodes, which are most frequently used as substrates for studies of electrochemical phase formation.

158

Chapter 2

Given the value of pH, Table 2.2. allows us to evaluate the equilibrium potential corresponding to the electrochemical reaction (2.147) and (2.148) using equation (2.149). Thus one may conclude whether metal oxides may be formed and reduced within the potential region of a nucleation experiment. REFERENCES 2.1

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Chapter 3 STOCHASTIC APPROACH TO NUCLEATION

3.1

Temporal distribution of clusters

In the very initial stage of a phase transition, the appearance of nuclei on the electrode surface can be considered as a flux of random independent events along the time axis [3.1-3.16]. Therefore the probability to form exactly m nuclei within a time interval [0,t] can be expressed by the Poisson distribution law:

where N(t) is the expected average number of nuclei within [0,t]. Thus the probability to form m = 0 nuclei is the probability to form exactly 1 nucleus (m = 1) is etc. Equation (3.1) allows us to derive an expression for the probability to form at least m, i.e. m or more than m, nuclei within the interval [0,t]:

Hence from equations (3.1) and (3.2) one obtains:

166

Chapter 3

Correspondingly for the probability to form the mth nucleus within the infinitesimal time interval [t, t + dt] it results:

and the average time

of expectation is defined as:

For the special case m = 1 equations (3.3)-(3.5) transform into:

In particular, equation (3.6) defines the probability of formation of at least one nucleus within the time interval [0,t] and is a very frequently studied stochastic quantity. As seen the probabilities and contain essential information on the most important kinetic characteristics of the nucleation process: the nucleation rate J(t) = dN(t)/dt and the average number of nuclei N(t) = What follows shows how this information can be derived by means of probabilistic studies of the nucleation kinetics. 3.1.1

Stationary nucleation

In the case of a stationary nucleation process integrates to:

and equation (3.8)

3. STOCHASTIC APPROACH TO NUCLEATION

167

Correspondingly, for m =1 the average expectation time becomes:

and bearing in mind the relation we conclude that for a stationary nucleation process coincides with the time needed to form on the electrode surface one nucleus on an average.

Concerning the probabilities by:

and

these two quantities are given

168

Chapter 3

and their time dependence is schematically shown in Figure 3.1. We should note that equation (3.10) represents a very well known relationship and is often employed for an experimental determination of [3.17-3.20]. However, one should not forget that the relation (3.10) is strictly valid only for a stationary nucleation process and should not be used if nonstationary effects dominate the initial stage of the phase transition. 3.1.2

Non-stationary nucleation

In this case the nucleation rate J(t) is a function of time and equations (3.6) - (3.8) must be rewritten in the form:

As seen, now the average time has no simple physical significance and cannot be used for a direct determination of the nucleation parameters. If the nucleation rate J(t) is a monotone function of time asymptotically tending to the and plots should look like those schematically shown in Figure 3.2. In this case the evaluation of the kinetic quantities and (Chapter 2.4) is based on the following considerations [3.3-3.5, 3.16]. The condition for inflection of the transient, at applied to equation (3.3) yields:

which for m = 1,

reduces to:


Therefore if we present N(t) in the general form:

where F(x) is a differentiable function of the non-dimensional variable equation (3.16') reduces to:

169

170

Chapter 3

Correspondingly, the probability of the inflection point of a probabilistic transient is presented by the non-dimensional expression:

where and After that the values of and are evaluated from an experimental transient as follows: (i) Having a theoretical formula for the function F(x) the probability is calculated for different values of by means of equation (3.19).Thus one obtains a non-dimensional plot. (ii) The coordinates of the inflection point of the experimental transient are determined. (iii) The value of corresponding to the experimental value of is found from the plot and is calculated as (iv) Given and is calculated by means of equation (3.18) and (3.19) as:

Apparently, this method of calculation requires a theoretical formula for the non-stationary number of nuclei N(t) (equation (3.17)) and in what follows we shall apply it to the data of a typical probabilistic nucleation experiment making use of equations (2.120') and (2.135') for (Chapters 2.4.1 and 2.4.2 respectively). Numerical values of the functions and calculated for different values of are given in Table 3.1. 3.1.3


What an experimentalist should necessarily do before applying the formalism developed in Chapters 3.1.1 and 3.1.2 is to verify whether the Poisson law (3.1) applies to the process of nucleus formation in the particular experimental system.


171

According to equation (3.1) the probability to form exactly m nuclei within a fixed time interval [0,t] depends on a single parameter - the average number of nuclei N(t), which coincides with the statistical dispersion of the random quantity m. Therefore if N(t) is found by a sufficiently large

172

Chapter 3

number of measurements, the theoretical probability should coincide with the experimentally determined frequencies of appearance of 0, 1, 2 etc. nuclei on the electrode surface. Here we comment briefly upon the results of such a statistical experiment [3.16].

Mercury nuclei are deposited from on a platinum single crystal electrode by means of the standard pulse potentiostatic technique (Chapter 2.4). The electrode surface is observed microscopically and the average number of nuclei formed at fixed overpotential and time is determined from 300 independent measurements. Simultaneously the probability of formation of exactly m (0, 1, 2, ...) nuclei is determined as where is the number of favorable events “formation of exactly m nuclei”on the electrode surface. The results of two independent series of statistical measurements are shown in Figure 3.3 where the open and solid circles give the experimentally determined probabilities and the broken lines link up the values of calculated by means of equation (3.1) with the experimental average


173

values of N(t). The good correspondence between the two values of proves the random character of the nucleation process. An additional evidence in this respect is the agreement between the average number of nuclei and the statistical dispersion calculated from the experimental results: for N(t) = 0.81 and for N(t) = 4.21. We should emphasize that implementation of such probabilistic measurements requires extremely reproducible experimental conditions and sufficient statistics. The author’s experience in this respect shows the 300 independent measurements should be enough although 500 are definitely better [3.4, 3.5, 3.16]. Once having proved the satisfactory agreement between equation (3.1) and the nucleation experiment the next step is to examine the time dependence of the probability of formation of at least m nuclei on the electrode surface. Varying the duration of the nucleation pulse at a constant overpotential can easily do this. For the purpose 50 independent measurements are made for each amplitude and duration of the nucleation pulse and the probability is determined as where is the number of favorable events “formation of at least m nuclei”. The same experiment allows us to obtain a second value of substituting in equation (3.3) the average number N(t) of nuclei corresponding to each duration of the nucleation pulse. Note that a more simple probabilistic experiment consists in the measuring of the differential probability to register the first nucleation event after elapsing of time t, i.e. within the infinitesimal time interval [t,t+dt] after applying the nucleation overpotential [3.10]. Integrating the obtained differential probabilistic transients one obtains again the integral relationship. This method does not require microscopic observation of the electrode surface – the first nucleation event can be registered by monitoring the current flowing through the electrochemical cell. However, it cannot be used to measure the distribution of the probabilities and Figure 3.4 shows the time dependence of the probabilities and to form at least 1, at least 2 and at least 3 mercury nuclei, respectively. The open circles correspond to (m = 1, 2, 3) whereas solid circles represent the values of obtained by means of the formulae:

174

Chapter 3

The close values of obtained in the two ways prove the sufficient statistics of this experiment. The crucial point of such probabilistic studies is the determination of the inflection point of the transients. To minimize errors, the best is to approximate the data for with a polynomial and then to find and


175

from the maximum of the differentiated polynomial plot. Having determined and and are evaluated as described in the previous Section. The obtained results are given in Table 3.2. where the data for the stationary nucleation rate refer to the whole electrode surface and not to square centimeter. We should point out that in this experiment the values of and are determined form the coordinates of a single experimental point - the inflection point of the transient and, beside that, using essentially theoretical equations for N(t) and J(t) without knowing in advance whether they do describe the non-stationary nucleation kinetics in the particular experimental system. A simple way to verify the reliability of these results would be to substitute the data for and in the theoretical expression:

and to try to reproduce the whole experimental transient at the corresponding overpotential. Beside that, with the same data for and one should be able to reproduce the probabilities (equations (3.22) and (3.23)) etc. which are experimentally accessible at the same overpotentials, too. This is shown in Figure 3.4 where the crosses and lines represent the transients obtained by means of equations (2.120') and (2.135') respectively.

This simple test unambiguously shows that the theory accounting for the changing number of active sites provides a much better description of these experimental results. The data for and obtained by means of equations (2.120') and (2.135') do not differ by more than 30% but due to the strong exponential dependence of on N(t), probabilistic studies are extremely sensitive and even small deviations of the experimental N(t) relationship from the theoretical ones are easily detected. Concerning the physical nature of active sites, in this case they are likely to be holes or other structural defects in the oxide coverage of the platinum electrode where mercury nuclei contact directly the bare platinum surface [3.21-3.24], Apparently the number of such active sites should depend on the electrode potential and may change during the process of nucleus formation.

176

Chapter 3

An important result following from the theoretical equations (2.135') and (3.19) is that the probability of the inflection point of a non-stationary transient cannot exceed the value of (see the fifth column of Table 3.1). This means that the theoretical model accounting for the “site birth” effect cannot explain the results of Gunawardena et al. [3.10] and of Toshev and Markov [3.20] who have obtained probabilistic transients with in the case of mercury and cadmium electrodeposition on a platinum surface. An interpretation on the basis of the macroscopic Zeldovich-Frenkel approach is rather questionable, too, because the size of the critical nuclei in these experiments is very small. Since the formation of an oxide coverage seems unlikely under the conditions of those experiments it might be possible to associate the observed non-stationary effects with the reconstruction of preformed UPD layers after applying the nucleation overpotential [3.25, 3.26]. Such complex electrochemical processes are certainly of major importance for the nucleation kinetics but their mechanism is not fully clarified. Concluding we shall comment briefly upon the more general case of nucleation on active sites with different activities. For the purpose let us consider a substrate with k different types of active sites, the expected average number of nuclei corresponding to each type being respectively. In this case the probability to form no nuclei on the electrode surface within the time interval [0,t] will be:

where is the probability to form no nuclei on the ith type of active sites. Hence for the probability to form at least one nucleus on the electrode surface one obtains:

where

The analysis of the integral probability

in this

more complex case of phase formation would be possible if both the activity distribution function and the time dependence of the average number of nuclei are known. The situation would be additionally complicated if k is also a time dependent quantity. However, as we pointed out in Chapter 2.4.2.4, activity distribution of active sites would hardly play a decisive role in the case of electrochemical nucleation on a foreign substrate.


177

Thus far we considered the probability distribution of clusters in time. Chapter 3.2 considers the probability distribution of clusters in space.

3.2

Spatial distribution of clusters

The experimental studies of the nucleation kinetics show that after sufficiently long time the total number of nuclei formed on the substrate remains constant. A simple explanation of this effect is that the saturation nucleus density is determined by the exhaustion of active sites on the electrode surface. That being the case, the number and location of supercritical clusters at the final stage of the phase transition would reflect the number and location of active sites, which are two most important characteristics of any process of heterogeneous phase formation. However, in many cases nucleation exclusion zones of reduced concentration and/or overpotential arise around the growing stable clusters and may overlap thus interrupting the nucleus formation. This would lead to a saturation nucleus density lower than the original number density of the active sites. Beside that bigger clusters could depress the growth of smaller ones formed nearby, clusters formed on neighboring active sites may coalesce etc. All these effects will result in a number and location of visible clusters, different from the number and location of active sites. Therefore, if the process of nucleus formation is used to decorate the electrode surface and to reveal its specific energy state it is important to verify beforehand if there is a cluster-cluster correlation or clusters are randomly distributed on the electrode surface. This may be done by inspecting the statistical distribution of the distances between nearest neighbor clusters making use of a stochastic criterion based on the Poisson distribution law [3.27-3.32]. In the following we present the generalized approach to this subject reported in [3.32]. 3.2.1

A general formulation

Consider M clusters distributed in a spatial region in such a way that the probability to find m of them in the spatial element of is given by the Poisson law (3.1). In this case N denotes the expected average number of clusters in Following Hertz [3.27] is presented as:

where

is the specific linear dimension of

and

178

Chapter 3

being the gamma function [3.33]. Thus for v= 1, and which is the case of clusters randomly distributed on a line, e.g. on a mono- or a polyatomic step. For v = 2, and which corresponds to clusters randomly distributed on a plane. Finally, for v= 3, and which is the case of clusters randomly distributed in a volume, e.g. metal nuclei incorporated within the bulk of a conducting polymer matrix. As an illustration Figure 3.5 shows silver crystals evaporated onto spiral monoatomic steps (v= 1) and on atomically flat regions (v = 2) of a crystalline AgBr layer. The principles and basic achievements of the decoration method used for the purpose can be found in the review article of Krohn [3.34] (see also the references cited therein).

Reverting to the theoretical considerations we define the probability for the cluster A located in the center of the spatial element (Figure 3.6) to have its nth neighbor B at a distance between and as a product of two probabilities: the probability clusters in

and the probability

to find n-l

to find 1 cluster in the infinitesimal

3. STOCHASTIC APPROACH TO NUCLEATION spatial element

Denoting the

average cluster density in clusters in it results for the probability

179

as

for the average number N of and for m = n-1 from equation (3.1)

one obtains:

In the same fashion the probability

is expressed as:

Thus for the probability distribution function obtains:

one

180

Chapter 3

Equation (3.30) allows us to derive general expressions for two important physical quantities - the average and the most probable distance between nth neighbors. Thus:

whereas the condition for a maximum

applied to

equation (3.30) yields:

A formula useful for comparison with experimental results can be obtained by introducing the non-dimensional distance into equation (3.30). This has been done by Scharifker et al. [3.35] for the particular case v = 2, n = 1. The reduced general formula for the probability distribution function is given by:

An alternative is to reduce the probability distribution function with respect to the average distance between nth neighbors [3.32]. In this case one obtains:

where and The advantage of equations (3.33) and (3.34) is that they do not depend on the average cluster density and provide the possibility for an easier comparison between theory and experimental results. Finally, Table 3.3 collects the theoretical expressions for and for clusters randomly distributed on a line (v = 1), on a plane (v = 2) and in a volume (v= 3).


3.2.2

181


Having derived general theoretical formulae for the probability distribution function (equations (3.30), (3.33) and (3.34)) in the following we present the results of a statistical analysis of the distances between clusters formed on a flat electrode surface. Several authors have performed such experimental studies in different experimental systems: silver [3.36-3.38], lead [3.39-3.41], mercury [3.42] and gold [3.43] on glassy carbon, copper on evaporated silver [3.44] and mercury on platinum [3.31, 3.45]. Here we comment upon the data reported in Ref. [3.38]. Figure 3.7 shows a photograph of M= 864 silver crystals deposited on a glassy carbon electrode with a surface area and the histograms in Figure 3.8 (a,b,c) represent the statistical distribution of the distances between first, second and third neighbors, respectively. In order to eliminate the edge effect only crystals

182

located at a distance larger than

Chapter 3

from the electrode periphery have been

taken into consideration, being the average distance between third neighbors in the case of clusters random location on the electrode surface. Two most important statistical characteristics of a set of experimental data consisting of M terms are [3.46]: the average,

and the standard deviation,


183

They are measures of the central value and the width of a distribution curve. In the case of a continuous distribution function dP(r) the quantities and are defined as:

and making use of equation (3.30) for v = 2 and n = 1, 2, 3 for the probability distribution functions of the distances between first, second, and third, neighbors one obtains:

The resulting values for and the ratio are given in Table 3.4 [3.47] and show that in the case of a random distributions the ratio has a constant non-dimensional value, irrespective of and Therefore it can be used as a quantitative criterion for the correspondence between an experimental and a random distribution.

184

Chapter 3

The lines in Figure 3.8 (a, b, c) represent the theoretical distribution functions and calculated by means of equations (3.38) - (3.40) and the experimental average cluster density The data for and characterizing the experimental histograms in Figure 3.8 are given in the first row of Table 3.5. The comparison between the theoretical curves and the experimental histograms in Figure 3.8 (a, b, c) shows that the smallest distances appear with a probability lower than that theoretically predicted for the random distribution. This also affects the experimental ratios which differ significantly from the theoretical ones (compare first and third row of Table 3.5).

Such findings are usually considered as a proof for the existence of nucleation exclusion zones around the growing stable clusters. However, one should not forget that the only definite conclusion that can be drawn from such statistical analyses is that the clusters of the new phase are not randomly located on the electrode surface. Apparently this might be also due to coalescence effects in the advanced stage of the phase transition and/or to specific, non-random location of active sites on the surface of the working electrode. One possibility to obtain more reliable information in this respect is to develop the actual location of active sites through a repetitive deposition of small number of nuclei at short times when the influence of exclusion zones and coalescence effects can be neglected. In order to reveal what a “small number of nuclei” mean one can proceed in the following way. We can assume that the length of the interval of missing distances in the experimental histograms for first neighbors equals approximately the average diameter of a nucleation exclusion zone. Therefore if the average distance between first neighbor clusters is it is very likely that the clusters will be sufficiently far from each other. Bearing in mind that for a random distribution one obtains the condition consideration where density should be

For example, in the case under it follows that the nucleus number This means that if the average number


185

186

Chapter 3

of clusters formed on the electrode surface is approximately 50 their location would reflect the location of active sites on the glassy carbon substrate. Figures 3.9 (a) and (b) show the location of 33 and 45 silver crystals deposited at a nucleation overpotential for 0.01s, whereas Figure 3.9 (c) is a combined picture of the coordinates X,Y of the active sites where 843 silver crystals were formed by applying 14 potentiostatic pulses consecutively. In all cases the coordinates of the nucleation sites were determined from in situ photographs of the electrode surface taken after each nucleation-growth pulse train. The results from the statistical analysis of the distances between first, second and third neighboring nucleation sites are shown by histograms in Figure 3.8 (a',b',c') whereas second row of Table 3.5 collects the data for and Again, the lines in Figure 3.8 (a',b',c') demonstrate the random distribution calculated by means of equations (3.38)-(3.40). The very good agreement between theory and experiment proves that the active sites are randomly located on the mechanically polished glassy carbon surface. This makes the electrode a suitable substrate for studies of the spatial distribution of electrochemically deposited nuclei of the new phase. To the author’s knowledge purely random distribution of nucleation sites was registered also in the case of latent image formation on AgBr layers in photography [3.48] and in the case of nucleation of pits on stainless steel [3.49].

REFERENCES 3.1 E.N.Sinitzin and V.P.Skripov, Ukr.Fiz.Zhurnal, 12(1967)99. 3.2 V.P.Skripov, V.P.Koverda and G.T.Butorin, Kristallografiya, 15(1970)1219. 3.3 S.Toshev and S.Stoyanov, Commun.Dept.Chem.Bulg.Acad.Sci., 3(1970)205. 3.4 S.Toshev and A.Milchev, Commun.Dept.Chem.Bulg.Acad.Sci., 3(1970)755. 3.5 S.Toshev, A.Milchev and S.Stoyanov, J.Cryst.Growth, 13/14(1972)123. 3.6 S.Toschev, in Crystal Growth: An Introduction, Ed. P. Hartmann, North-Holland Publ. Co, 1973, p. 1. 3.7 P.Bindra, M.Fleischmann, J.W.Oldfield and D.Singleton, Faraday Discuss.Chem.Soc., 56(1973)180. 3.8 A.N.Baraboshkin, L.T.Kosikhin and V.N.Chebotin in Kinetika i Mechanism Kristalizatsii, Nauka i Technika, Minsk, 1973. 3.9 A.N.Baraboshkin, Electrocrystallization of Metals from Molten Salts (in Russian), Moskow, Nauka, 1976. 3.10 G.A.Gunawardena, G.J.Hills and B.Scharifker, J.Electroanal.Chem., 130(1981)99. 3.11 W.Obretenov, V.Bostanov and A.Popov, J.Electroanal.Chem., 132(1982)273. 3.12 S.Fletcher, Electrochim.Acta, 28(1983)917. 3.13 R.L.Deutscher and S.Fletcher, J.Electroanal.Chem. 164(1984)1. 3.14 R.Sridharan and R.de Levie, J.Electroanal.Chem., 169( 1984)59.


187

3.15 R. de Levie, in Advances in Electrochemistry and Electrochemical Engineering, Vol. 13, Ed. H.Gerischer, John Wiley &Sons, 1984. 3.16 A.Milchev and V.Tsakova, Electrochim.Acta, 30(1985)133 3.17 R.Kaischev, A.Scheludko and G.Bliznakov, Commun. Bulg.Acad.Sci., 1(1950)137. 3.18 A.Scheludko and G.Bliznakov, Commun.Bulg.Acad.Sci.(phys), 2(1951)227. 3.19 E.Budevski, V.Bostanov, Z.Stoynov, A.Kotzeva and R.Kaischew, Electrochim.Acta, 11(1966)1697. 3.20 S.Toschev and I.Markov, Electrochim.Acta, 12(1967)281. 3.21 R.Kaischew and B.Mutaftschiev, Electrochim.Acta, 10(1965)643. 3.22 A.Milchev, T.Chierchie, K.Juettner and W.J.Lorenz, Electrochim.Acta, 32(1987)1039. 3.23 A.Milchev, T.Chierchie, K.Juettner and W.J.Lorenz, Electrochim.Acta, 32(1987)1043. 3.24 A.Milchev and T.Chierchie, Electrochim.Acta, 35(1990)1873. 3.25 H.Bort, K.Juettner, W.J.Lorenz, G.Staikov and E.Budevski, Electrochim.Acta, 28 (1983)985 3.26 G.Staikov and W.J.Lorenz, Z.Phys.Chem., 208(1999)17 3.27 P. Hertz, Math.Ann., 67(1909)387. 3.28 S.Chandrasekhar, Rev.Mod.Phys., 15(1943)1. 3.29 H.Schmeisser, Thin Solid Films, 22(1974)99. 3.30 J.C.Zanghi, J.J.Metois and R.Kern, Phil.Mag., 29(1974)1213. 3.31 A.Milchev, W.Kruijt, M.Sluyters-Rehbach and J.H.Sluyters, J.Electroanal.Chem., 350 (1993)89. 3.32 A.Milchev, J.Chem.Phys., 100(1994)5160. 3.33 G. A.Korn and T.M.Korn, Mathematical Handbook for Scientists and Engineers, McGraw-Hill, New York, 1961. 3.34 M.Krohn, Vacuum, 37(1987)67. 3.35 B.R.Scharifker, J.Mostany and A.Serruya, Electrochim.Acta, 37(1992)2503. 3.36 A.Milchev, E.Vassileva and V.Kertov, J.Electroanal.Chem., 107(1980)323. 3.37 A.Serruya, B.R.Scharifker, I.Gonzalez, M.T.Oropeza, M.Palomar-Pardave, J.Appl. Electrochem., 26(1996)451. 3.38 A.Milchev, E.Michailova and I.Lesigiarska, Electrochem.Commun., 2(2000)407. 3.39 A.Serruya, J.Mostany and B.R.Scharifker, J.Chem.Soc.Faraday Trans., 89(1993)255. 3.40 J.Mostany, A.Serruya and B.R.Scharifker, J.Electroanal.Chem., 383(1995)37. 3.41 E.Garsia-Pastoriza, J.Mostany and B.R.Scharifker, J.Electroanal.Chem., 441(1998)13. 3.42 A.Serruya, J.Mostany and B.R.Scharifker, J.Electroanal.Chem., 464(1999)39. 3.43 U.Schmidt, M.Donten, J.G.Osteryoung, J.Electrochem.Soc.,144(1997)2013. 3.44 A.Milchev, Electrochim.Acta, 28(1983)947. 3.45 W.S.Kruijt, M.Sluyters-Rehbach, J.H.Sluyters and A. Milchev, 371(1994)13. 3.46 W.H.Press, B.P.Flannery, S.A.Teukolsky, W.T.Vetterling, Numerical Recipes, Cambridge University Press, Cambridge, 1992. 3.47 V.Tsakova and A.Milchev, J.Electroanal.Chem., 451(1998)211. 3.48 V.Platikanova and A.Milchev, J.Imaging Science, 30(1986)210. 3.49 R. Salvarezza, A.J.Arvia and A.Milchev, Electrochim.Acta, 15(1990)289.


Chapter 4 ELECTROCHEMICAL CRYSTAL GROWTH

4.1

Growth of an individual crystal face

A dislocation free, atomically smooth face of a three-dimensional crystal can grow if after applying the supersaturation two-dimensional nuclei form and spread out to reach the face borders [4.1-4.9]. To form a 2D cluster on an intact crystal face means to form a circular or a polygonized monoatomic step on the atomically smooth surface where single atoms from the parent phase can attach and incorporate into the crystal lattice. Thus the 2D nucleus spreads out and eventually covers the crystal face. Several particular growth mechanisms can be considered [4.104.12]: (i) If the average time of appearance of 2D nuclei is much longer than the average growth time needed for the first nucleus to cover the crystal face we speak about a mononuclear-monolayer mechanism of growth. In this case the spread of each single two-dimensional nucleus forms a monoatomic layer on the crystal face. (ii) If the opposite inequality is fulfilled than new 2D nuclei appear before the first one has covered the crystal face and we speak about a multinuclear-monolayer growth mechanism. In this case the monoatomic layer on the crystal surface form as a result of the appearance and spread of a large number of nuclei, which interact with one another during the growth. (iii) If the nucleation rate is even higher and the time even shorter, new nuclei may form on the top of the growing ones and layers of polyatomic

190

Chapter 4

height spread over the crystal surface In this case we speak about a multinuclear-multilayer growth mechanism. (iv) If the faces of a 3D crystal are not atomically smooth but exhibit a number of mono- and polyatomic steps, kink sites, edge and screw dislocations, vacancies etc. the appearance of two-dimensional nuclei becomes highly improbable. Instead, even a small deviation from the equilibrium state leads either to growth or to dissolution of the crystal face. Now we can distinguish between the growth of step trains and spiral growth due to screw dislocations. In this Chapter we consider the mononuclear-monolayer growth, the growth of step trains and the spiral growth mechanisms. The multinuclearmonolayer and the multinuclear-multilayer growth mechanisms will be considered in Chapter 5. 4.1.1

Mononuclear - monolayer growth

In order to describe the growth kinetics of a single two-dimensional cluster we combine two independent equations for the growth current The first one is the Faraday law, which gives the material balance of the electrodeposited species independently of the growth mechanism:

Here is the valence, molar volume of the deposit and

is the Faraday constant,

is the

is the volume of the two-dimensional cluster. In equation (4.2) is the top area of the 2D cluster, its height h being equal to the atomic diameter d in the case of simple crystallographic lattices. Assuming that the nucleus is a 2D crystal polygon then where l(t) is the edge length and b is a constant depending on the crystal geometry. Thus for a quadratic nucleus b = 1, for a hexagonal nucleus if the nucleus is a disk of radius l(t), Taking this into consideration equation (4.1) is rewritten in the form:

4. ELECTROCHEMICAL CRYSTAL GROWTH

191

The second expression for the current must reveal the growth mechanism and, as an illustration, we consider the case of direct attachment of ions from the bulk of the electrolyte to the periphery of the twodimensional cluster. The rate-determining step is supposed to be the ions’ transfer across the electric double layer. Then the current is given by the Buttler-Volmer equation:

Here

is the length of the nucleus periphery, the product gives the total area of the edges on which the ions are deposited and is the exchange current density referred to unit edge area. The geometric constant b' equals 4 for a quadratic 2D crystal, 6 for a hexagon and for a disk. Note that sometimes the product is replaced by the product where is the exchange current density referred to unit edge length. Thus Combination of equations (4.3) and (4.4) yields a simple differential equation for l(t),

which with initial conditions t = 0, l(0) = 0 integrates to:

Correspondingly, for the current (4.3), (4.5) and (4.6) it results:

of a single 2D crystal from equations

192

Chapter 4

For low cathodic overpotentials and/or high temperatures T the exponential functions in equations (4.6) and (4.7) are expanded in series according to and one obtains more simple overpotential dependences of l(t) and

Equations (4.6) and (4.7) describe the l(t) and the relationships if the two-dimensional crystal grows in the central part of the 3D crystal face. When the edge of a 2D crystal comes into contact with the face borders the current starts increasing with a lower velocity, attains a maximum and then falls to zero when a complete monolayer covers the crystal face. Figure 4.1 demonstrates a sequence of current oscillations due to the appearance and spread of single two-dimensional silver nuclei on an isolated, dislocation

free cubic face of a 3D silver crystal growing in a solution [4.13]. The cathodic overpotential is corresponding to each transient equals the charge

aqueous and the charge of a monolayer of


193

silver on the cubic face However, the current spikes exhibit different shapes depending on the specific location of the 2D clusters on the crystal face. These results are obtained by means of the unique capillary method devised by Kaischew et al. [4.14] and Budevski et al. [4.15-4.17] and described in details in a number of original papers [4.18-4.24] and review articles [4.25-4.31] (see also [4.32] and the references cited therein). The method provides the possibility to grow, electrochemically, a single 3D metal crystal in a round [4.16-4.19] or in a rectangular [4.20-4.22] glass capillary in such a way that only a small region of a crystal face, comes into contact with the electrolyte solution. A special procedure is applied to remove dislocations and other surface inhomogeneities and to obtain a defect free, “quasi-ideal” crystal face. The latter is employed as a working electrode for studies of 2D-nucleation and growth kinetics both under potentiostatic and under galvanostatic conditions. Studying the spontaneous appearance of two-dimensional clusters on the electrode surface one obtains direct information on the average time needed to form a 2D nucleus at a given overpotential As we have seen in Chapter 3 (equations (3.8) and (3.10)), in the case of negligible nonstationary effects the time equals the reciprocal stationary nucleation rate. This has been used by Budevski et al. [4.16, 4.17] to examine the overpotential dependence of the stationary rate of two-dimensional nucleation. The obtained results confirm the validity of the classical theory of nucleation on a like substrate and provide the possibility to determine the nucleation work, the size of the two-dimensional critical nucleus and the specific free edge energy at the nucleus-solution interface boundary. In Figure 4.2 the data for [4.16] are plotted in coordinates versus substrate):

according to equation (2.59) taken for

Here

(like

From the slope of the straight line one obtains that in the overpotential interval

0.010V - 0.015V the nucleation work (equation (1.89) for

varies from

whereas the

size

of the critical 2D nucleus varies from 48 to 21

194

Chapter 4

atoms. From the same data for the specific free edge energy with one obtains . These results are about

10% lower than those reported by Budevski et al [4.16, 4.17] who, in their interpretation, have neglected the overpotential dependence of the preexponential term in the formula for the stationary nucleation rate. 4.1.2

Growth of step trains

The incorporation of atoms into a propagating step takes place either through a surface diffusion of adatoms on the crystal surface or through a direct attachment of ions from the bulk of the electrolyte solution. In the following we consider the two mechanisms.


195

4.1.2.1 Surface diffusion mechanism Burton, Cabrera and Frank [4.33-4.36] proposed the first quantitative treatment of this mode of crystal growth in 1949 (see also [4.37-4.41]). The authors considered in details the particular case of crystallization from a vapor phase but five years later the obtained theoretical results were translated to the language of electrocrystallization by Lorenz [4.42, 4.43]. Since that time the problem was discussed by a number of authors [4.444.51] and here we follow the treatment given in the books of Vetter [4.49] and Damaskin and Petrii [4.51].

Consider two parallel monoatomic steps separated by a flat terrace (Figure 4.3). The distance between the two steps is and they advance on the crystal surface due to the adatoms surface diffusion and incorporation into the kink sites at a cathodic overpotential The direct attachment of ions from the bulk of the electrolyte solution to the kink sites is considered negligibly small. The adatoms concentration is a function of the step distance x and to find out the relationship is the first important task of the theoretical model. For the purpose we assume that the adatoms having reached the growing steps are immediately incorporated into the kink sites of

196 the crystal lattice. Thus at x = 0 and at equals the equilibrium value i.e.

Chapter 4 the adatoms concentration Sufficiently far

from the growing steps the adatoms concentration is which coincides with the Henry’s adsorption isotherm (equation (1.22)). Namely the difference between the actual, and the equilibrium, adatoms concentration is the driving force that moves the adatoms towards the growing steps. Diffusion along the step edge is considered to be of no importance. Under these conditions the mass balance of the deposited species is given by a diffusion equation accounting for the net flux of ions from the electrolyte to the crystal surface:

Here is the surface diffusion coefficient and density given by:

is the local current

where is the exchange current density characterizing the ionsadatoms exchange at the equilibrium potential of the bulk metal phase. (Remember that in this book we consider the cathodic current and the cathodic overpotential as positive quantities.) Substituting equation (4.10) into (4.9) yields:

where

and


197

are the cathodic and the anodic current components at In the case of stationary diffusion, transforms into:

equation (4.11)

and imposing the boundary conditions:

for the stationary adatoms concentration

one obtains:

In equation (4.15) the quantity is called a penetration depth of surface diffusion and gives the average distance that the adatoms cover while moving on the flat terrace,

In fact which is the well-known Einstein formula, average life time of an adatom on the crystal surface:

being the

198

Chapter 4

The concentration distribution function is schematically presented in Figure 4.3 where lines (a), (b) and (c) correspond to cathodic deposition equilibrium state and anodic dissolution respectively. Having determined the stationary distribution function let us now derive an explicit expression for the stationary cathodic current in the case of surface diffusion. For the purpose we define the average current density as:

and substitute for (4.14). Thus the expression for

according to equation becomes:

Since the first derivative of the function

(equation (4.15)) is:

it follows that

Therefore for the growth current

one obtains [4.44, 4.48-4.52]:


199

Two particular cases of equation (4.20) may be of interest: (i) At high cathodic overpotentials and low exchange current densities i.e. long average adatoms life time the penetration depth may become much bigger than the average half distance between the monoatomic steps. Since for equation (4.20) turns into the Buttler-Volmer formula:

(ii) In the opposite case, when the inequality and equation (4.19) becomes:

holds,

Finally, the current relates to the advancement rate monoatomic steps according to [4.49]:

of the

Here is the molar concentration of surface atoms in the crystal lattice and is the total step length per unit surface area, which coincides with the number of steps per unit length. Concluding we should point out that Burton, Cabrera and Frank [4.35, 4.53] (see also Lorenz [4.54, 4.55]) have solved also the problem of stationary surface diffusion towards steps with circular symmetry. The theory of the more complex case of combined surface diffusion and bulk diffusion limitations can be found in the works of Fleischmann and Thirsk [4.48], Damjanovich and Bockris [4.50] and Gilmer et al. [4.56]. Note, however, that all these theoretical considerations do not account for difficulties connected with the so-called Ehrlich-Schwoebel barrier [4.574.62]. Elastic [4.63, 4.64] and entropic [4.65] interactions between growing steps are also neglected.

200

Chapter 4

4.1.2.2 Direct attachment mechanism In this case of crystal growth ( in Figure 4.3) the current density is defined as [4.42, 4.48, 4.49, 4.66, 4.67]:

where the product gives the total surface area of the step edges on which the ions are deposited and is the corresponding exchange current density. The bulk diffusion of ions from the electrolyte to the growing steps is again neglected. Concerning the propagation rate it is given by a formula similar to equation (4.21):

The last expression has been used by Bostanov et al. [4.21, 4.22] to interpret the experimental data for the propagation rate of monoatomic layers on the cubic face of a sliver single crystal grown in a rectangular capillary. (In [4.21,4.22] the concentration is expressed through the ratio giving us the area occupied by 1 mol of surface atoms.) The comparison between equations (4.20) and (4.22) unambiguously shows that measurements of the stationary relationship cannot answer the question: surface diffusion of adatoms or direct attachment of ions determine the growth mechanism of equidistant monoatomic step trains on a flat crystal surface. The reason is that in both cases the slope can be presented by the same general formula:

where

is a generalized exchange current density given by:

in the case of a surface diffusion mechanism, and by:


201

in the case of a direct attachment mechanism. The problem has been discussed in details by Vitanov et al. [4.68] (see also the references cited therein), who have found that for silver electrodeposition the exchange current density is more than three orders of magnitude higher than The exact experimental values obtained for the cubic face of a single silver crystal were and which means that in this particular case the contribution of surface diffusion was negligibly small. The reason for this result might be that due to their complex solvation shell the adatoms move very slowly on the crystal surface and the direct attachment of ions to the kink sites appears to be much more efficient. 4.1.3

Spiral growth of crystals

A screw dislocation creates a single monoatomic step on the crystal surface (Figure 4.4a) and if, after applying a supersaturation the length

l of this step exceeds the edge length of the corresponding critical 2D nucleus, the step starts advancing. Thus a new step, perpendicular to the first one is created (Figure 4.4b) and when the length of this second step exceeds

202

Chapter 4

the critical length it starts moving, too, and a third step, parallel to the first one forms (Figure 4.4c) etc. This leads to the appearance of a continuously rotating spiral that forms a growth pyramid on the crystal surface (Figure 4.4d). Polygonized growth spirals are obtained when the steps’ propagation rate is different in the different crystallographic directions, whereas we observe circular, or almost circular, spirals (Figure 4.5a) when the propagation rate does not depend, or depends weakly, on the step orientation. Of course, in most cases the real crystals contain many screw dislocations and their growth is an extremely complex process. As an example Figure 4.5b demonstrates the surface relief of a growing cubic face of a single silver crystal with a large number of screw dislocations that create overlapping growth pyramids.

A detailed theory of the spiral growth of crystals has been developed by Burton, Cabrera, Frank, Mott and Levine [4.33-4.41] who have considered the case of spiral growth from a supersaturated vapor phase. Later the theory was adapted to electrocrystallization by Vermilyea [4.44] and Fleischmann and Thirsk [4.48] and developed further and verified experimentally by Budevski, Bostanov, Staikov, Nanev et al. [4.28, 4.69-4.75] (see also the pioneering work of Kaischew, Budevski and Malinovski [4.76]). In the simple case of an Archimedean spiral the step propagation rate in a direction normal to the spiral depends on the curvature radius r according to [4.35]:


203

Here stands for the propagation rate of a linear step and is the radius of the circular 2D critical nucleus at the applied supersaturation This equation is valid in the case of and relatively low supersaturations [4.49]. Concerning the distance between parallel circular steps, the first approximate expression obtained by Burton, Cabrera and Frank was Later the authors proposed a better approximation, namely but the formula of Cabrera and Levin [4.41]:

is nowadays considered as most precise. As shown by Budevski et al. [4.70] equation (4.28) appears to be a fairly good approximation also for polygonized spirals with 3, 4 and 6 corners. It is worth noting that the total step length per unit surface area equals the reciprocal step distance both for parallel linear steps and for steps created by screw dislocations. Therefore the expressions for the growth current density retain the same mathematical form [4.49]. For instance, in the case of a surface diffusion mechanism the current will be given by equation (4.20) in which the step distance should be replaced by i.e.:

4.2

Growth of a hemispherical liquid drop

If the faces of a three-dimensional crystal contain a large number of growth sites with a close activity with respect to the ions incorporation in the crystal lattice the 3D crystal should grow similarly to a liquid drop. This is a very frequently used approximation, particularly when kinetics of formation and growth of multiple nuclei are theoretically considered. For that reason in this Chapter we shall derive expressions for the radius R(t) and for the current of a single hemispherical cluster growing at a constant electrode potential E. The problem has been examined by many authors and on different simplifying assumptions [4.77-4.101]. Here we confine the theoretical considerations to the case of electrochemical growth under ions’ transfer, diffusion and ohmic limitations, ions’ direct attachment mechanism, and stationary diffusion. Migration [4.96] and Gibbs-Thomson effect of the curvature of the growing clusters [4.84, 4.85] will be neglected.

204

Chapter 4

Consider a three-electrode electrochemical cell comprising an electrolyte solution of metal ions with a bulk concentration c, an inert, ideally polarizable working electrode, a counter electrode and a reference electrode with a fixed potential The whole system is kept at a constant temperature T. For the sake of convenience, we select a reference electrode made of the bulk metal M whose ions are present in the solution. The only requirement that we have to meet for the purpose is the bulk metal M to have a stable and reversible equilibrium potential That being the case, and any external potential E which we apply to the working electrode polarizes it directly to the electrochemical overpotential which is a measurable quantity with a clear physical significance. Certainly we can use any other reference electrode with a fixed potential and this is what we should do if the bulk metal M has no reversible potential in the particular experimental system or if this potential is not known. However, then the external potential E will polarize the working electrode to the overpotential which relates to according to where Namely the difference is which we eliminate with the special choice of our reference electrode. Suppose now that using an external source we apply to the working electrode the constant potential E, more negative than the equilibrium potential In this case the process that proceeds first is the charging of the electric double layer. During this process the electrochemical overpotential is and changes with time according to the law:

Here is the double layer capacitance and is the ohmic resistance of the electrolyte, which depends on the geometry of the electrolytic cell. This simple formula tells us that after time the double layer of the working electrode is almost fully charged and becomes equal to From this time onwards nuclei of the new phase appear on the electrode surface under the well-defined conditions of a constant supersaturation In the following we derive analytical expressions for the radius R(t) and for the growth current of a single hemispherical cluster in five important particular cases: pure ions’ transfer control, combined ions’ transfer and diffusion control, pure diffusion control, combined ions’ transfer and ohmic control and pure ohmic control.

4. ELECTROCHEMICAL CRYSTAL GROWTH 4.2.1

205

Pure ions’ transfer control

In the very initial stage, the growth of an .R-sized cluster with a surface area is determined by the ions’ transfer across the electrical double layer at the cluster-solution interface and the growth current is expressed by the Buttler-Volmer formula:

where is the exchange current density at the bulk concentration c. (Remember that in this book the cathodic overpotential and the cathodic current are considered as positive quantities and the cathodic and the anodic charge transfer coefficients are and respectively. As an exception, here and henceforth, till the end of Chapter 4, the molar gas constant is denoted by R'.). In order to derive explicit formulae for R(t) and the current is presented through the Faraday law, which describes the mass balance independently of the growth mechanism:

Then combining equations (4.31) and (4.32) and solving the obtained simple differential equation with boundary conditions t = 0, R(0) = 0 for R(t) and it results:

4.2.2

Combined ions’ transfer and diffusion control

We observe this mode of electrochemical crystal growth at longer times when the concentration at the cluster surface is already lower than the bulk concentration c and the ions’ transfer limitations still exist. We assume that

206

Chapter 4

the mass transport is realized only by diffusion of the electroactive species since a large excess of supporting electrolyte in the basic solution quenches the migration. Fletcher [4.88,4.89] has examined in details this mode of crystal growth and here we consider only the relatively simple case of stationary diffusion when the concentration distribution around the growing cluster is given by:

Here is the distance measured from the cluster center and to use equation (4.35) means to assume that during the growth a steady state is instantaneously established at each cluster radius R. Under such conditions the current is expressed as:

where the concentration gradient at the cluster-solution interface is Thus for one obtains:

There are three unknown quantities in equation (4.37) – R(t) and and to find them we need two other independent expressions for the growth current. Apparently the first one must be again the Faraday law (equation (4.32)) whereas the second must describe the interfacial ions’ transfer reaction according to (see e.g. [4.102] and [4.103]):

Combination of equations (4.32), (4.37) and (4.38) yields a first order differential equation for R(t):


207

which is solved with initial conditions t = 0, R(0) = 0 to give:

Substituting this formula for R(t) into the law of Faraday (equation (4.32)) for the current one obtains:

Here

Equations (4.40) and (4.41) for R(t) and were obtained first by Fletcher in 1983 [4.88, 4.89]. They can be derived also by means of a different theoretical approach described in [4.96].

208

Chapter 4

Theoretical R(t) and plots calculated by means of equations (4.40) and (4.41) are shown in Figures 4.6 and 4.7 for the case of electrodeposition of silver from in the presence of a large excess of supporting electrolyte. The values of the constants used in these calculations are: T=308K, and

If under certain experimental conditions the inequalities fulfilled equations (4.40) and (4.41) approximate to:

are

These two expressions allow us to determine the diffusion coefficient and the exchange current density from the slope and from the intercept of R(t) versus or versus experimental relationships. Such data have


209

been obtained by studying the growth current of single silver crystals from aqueous silver nitrate solutions at different constant overpotentials and a large excess of potassium nitrate in the working electrolyte [4.104]. 4.2.3

Pure diffusion control

This is the case of pure ions transport limitations of the growth kinetics and is observed at long times and/or high exchange current densities when the inequality nt > 50 is fulfilled. Then equations (4.40) and (4.41) turn into the simple expressions describing the R(t) and relationships under the conditions of pure diffusion control:

210

Chapter 4

For sufficiently large cathodic overpotentials when the surface concentration equations (4.49) and (4.50) simplify to

4.2.4

Combined ions’ transfer and ohmic control

If the cluster is growing at a low rate and the electrolyte is intensively stirred the ions concentration at the crystal surface remains equal to the bulk one However, if the solution does not contain large excess of

supporting electrolyte it is necessary to account for the ohmic drop in the electrolyte solution and the actual overpotential at the cluster surface is given by [4.93]:


211

Here is the ohmic resistance of the electrolyte around the hemispherical cluster, which can be determined as follows (Figure 4.8). The resistance of the electrolyte between two imaginary hemispheres with radii x and x + dx is given by the Ohm’s law:

where is the specific solution conductivity. Hence the ohmic resistance of the electrolyte between the R-sized cluster and a hemisphere with an arbitrary radius is:

Therefore, if the distance between the cluster and the counter electrode is for the total ohmic resistance of the circuit R-sized cluster – electrolyte – counter electrode one obtains:

or

if L>>R, which is usually the case. Equation (4.55) shows that 90% of the total ohmic resistance is concentrated within a layer of a thickness of 10 cluster radii R. Indeed for which means that it is worth using a Luggin capillary to decrease the ohmic drop only if it could be fixed to the growing cluster surface at a distance shorter than 10R. Certainly this would be difficult for clusters of size In this case the general kinetic equation for the current has to be presented in the form:

212

Chapter 4

and the combination of equations (4.57) and (4.32) yields the following differential equation for the radius R(t) of the hemispherical cluster [4.93]:

Here:

An exact solution of equation (4.58) is not possible but approximate analytical expressions for R(t) and can be obtained for low and for high ohmic drops (short and long times, and/or high and low respectively) [4.93]. Thus for a low ohmic drop:


213

where

Equations (4.63) and (4.64) are good approximations for relatively short times when the inequality is fulfilled. In the case of a high ohmic drop the expressions for R(t) and read:

where

214

Chapter 4

The validity of equations (4.70) and (4.71) is restricted to relatively long times when the inequality is fulfilled. If under certain experimental conditions the inequalities hold then:

Finally, when the overpotential compensated by the ohmic drop, i.e.

at the cluster surface is entirely one obtains:

which is the case of complete ohmic control of the growth kinetics. Scheludko and Bliznakov derived the last two expressions already in 1951 [4.77]. Concluding, we should point out that the current modifies the electric field in the cluster vicinity and this leads to the appearance of zones of reduced overpotential within which In order to find the electrochemical overpotential at a distance from the growing cluster it is necessary to evaluate not the total ohmic drop but only the part of it that is due to the ohmic resistance of the electrolyte between and L [4.93]. The latter is given by:


and for the ohmic overpotential the growing cluster one obtains:

215

at a distance

from

Correspondingly, the radius of the zone of reduced overpotential within which is given by:

For sufficiently large values of L and/or low when the inequality is fulfilled, equations (4.80) and (4.81) simplify to:

REFERENCES 4.1 4.2 4.3 4.4 4.5 4.6

J.W.Gibbs, Collected works, New Haven, 1928, Vol.1. M.Volmer and A.Weber, Z.Phys.Chem., 119(1926)277. H.Brandes, Z.Phys.Chemie, 126(1927)196. M.Volmer and M.Marder, Z.Phys.Chem., 154(A)(1931)97. T.Erdey-Gruz and M.Volmer, Z.Phys.Chem. 157(1931)165. R.Kaischew and I.N.Starnski, Z.Phys.Chem., 170(A)(1934)295.

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4.7 M.Volmer, Kinetik der Phasenbildung, Teodor Steinkopf Verlag, Leipzig, Dresden, 1939. 4.8 R.Kaischew, Ann.Univ. Sofia Fac. Physicomat., Livre 2 (Chem.), 42(1945/1946)109. 4.9 R.Kaischew, Acta Phys.Hung., 8(1957)75. 4.10 A.E.Nielsen, Kinetics of precipitation, Pergamon Press, NY, 1964 4.11 A.A.Chernov and B.J.Lyubov, Rost kristallov, Vol.V, Nauka, Moskow, 1965 4.12 W.B.Hillig, Acta Met., 14(1966)1868. 4.13 V.Bostanov, W.Obretenov, G.Staikov, D.K.Roe and E.Budevski, J.Crystal Growth, 52(1981)761. 4.14 R.Kaischew, G.Bliznakow and A.Scheludko, Izv.Acad.Bulg.Sci. (Phys.), 1(1950)146. 4.15 E.Budevski and V.Bostanov, Electrochim.Acta, 9(1964)477. 4.16 E.Budevski, V.Bostanov, T.Vitanov, Z.Stoynov, A.Kotzeva and R.Kaischew, Electrochim.Acta, 11(1966)1697. 4.17 E.Budevski, V.Bostanov, T.Vitanov, Z.Stoynov, A.Kotzeva and R.Kaischew, Phys. Status Solidi, 13(1966)577. 4.18 V.Bostanov, A.Kozeva, E.Budevski, Bull.Inst.Phys.Chem., BAS, 6(1967)33. 4.19 E. Budevski, V.Bostanov, T.Vitanov, Chr.Nanev, Z.Stoynov and R.Kaischew, Commun.Dept.Chem., BAS, 11(1969)437 4.20 V.Bostanov, R.Rusinova and E.Budevski, Chem.Ing.Tech., 45(1973)179. 4.21 V.Bostanov, G.Staikov and D.K.Roe, J.Electrochem.Soc., 122(1975)1301. 4.22 V.Bostanov, J.Crystal Growth, 42(1977)194. 4.23 W.Obretenow, V.Bostanov and V.Popov, J.Electroanal.Chem., 132(1982)273. 4.24 W.Obretenov, T.Kossev, V.Bostanov and E.Budevski, J.Electroanal.Chem., 170(1984) 51. 4.25 R.Kaischew and E.Budevski, Contemp.Phys., 8(1967)489. 4.26 E.Budevski, in Progress in Surface and Membrane Science, Vol. 11, Academic Press, NY, 1976, p.71. 4.27 E.Budevski, G.Staikov and V.Bostanov, in Growth and Properties of Metal Clusters, J.Burdon (Ed.), Elsevier, Netherlands, 1980. p.85. 4.28 E.Budevski, V.Bostanov and G.Staikov, Ann.Rev.Mat.Sci., 10(1980)85. 4.29 E.Budevski, in Interfacial Aspects of Phase Transformations, B.Mutaftschiev (Ed.), D.Reidel Publishing Compnay, 1982, p.559. 4.30 E.Budevski, in Comprehensive Treatise of Electrochemistry, Vol.7, B.E.Conway, J.O'M. Bockris, S.U.M. Khan, R.E.White (Eds.), Plenum Press, NY, 1983, 399. 4.31 E.Budevski, G.Staikov and W.J.Lorenz, Electrochim.Acta, 45(2000)2559. 4.32 E.Budevski, G.Staikov and W.J.Lorenz, Electrochemical Phase Formation - An Introduction to the Initial Stages of Metal Deposition, VCH, Weinheim, 1996. 4.33 W.K. Burton, N.Cabrera and F.C.Frank, Nature, 163(1949)398. 4.34 W.K. Burton, N.Cabrera and F.C.Frank, Disc.Farad.Soc., 33(1949)40. 4.35 W.K. Burton, N.Cabrera and F.C.Frank, Phil.Trans.Roy.Soc., A243(1951)299. 4.36 W.K. Burton, N.Cabrera and F.C.Frank, in Elementarnie Processi Rosta Kristallov, Inostrannaja Literatura, Moskow, 1959 (in Russian). 4.37 F.C.Frank, Disc.Farad.Soc., 5(1949)48. 4.38 N.F.Mott, Nature, 165(1950)295. 4.39 F.C.Frank, Z.Electrochem.56(1952)429. 4.40 F.C.Frank, Adv.Phys., 1(1952)91. 4.41 N.Cabrera and M.Levine, Philos.Mag., 1(1956)450. 4.42 W.Lorenz, Z.Naturforsch., 9A(1954)716 4.43 W.Lorenz, Z.Electrochem., 58(1954)912. 4.44 D.A.Vermilyea, J.Chem.Phys., 25(1956)1254. 4.45 V.Mehl and J.O'M.Bockris, J.Chem.Phys., 27(1957)817. 4.46 V.Mehl and J.O'M.Bockris, Can.J.Chem., 37(1959)190. 4.47 A.Despic and J.O'M.Bockris, J.Chem.Phys., 32(1960)389.


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4.48 M.Fleischmann and H.R.Thirsk, Electrochim.Acta, 2(1960)22 4.49 K.J.Vetter, Elektrochemische Kinetik, Springer Verlag, Berlin, 1961 4.50 A.Damjanovich and J.O'M.Bockris, J.Electrochem.Soc., 110(1963)1035 4.51 B.B.Damaskin and O.A.Petrii, Introduction to the Electrochemical Kinetics, Moskow, Vishsaya Shkola, 1975. 4.52 W.Lorenz, Naturwiss., 40(1953)576. 4.53 N. Cabrera and W.K.Burton, Disc.Farad.Soc., 5(1949)40. 4.54 W.Lorenz, Z.phys.Chem., 202(1953)275. 4.55 W.Lorenz, Z.Electrochem., 57(1953)382. 4.56 G.H.Gilmer, R.Chez and N.Cabrera, J.Crystal Growth, 8(1971)79. 4.57 G.Ehrlich and F.G.Hudda, J.Chem.Physics, 44(1966)1039. 4.58 R.L.Schwoebel and E.J.Chipsey, J.Appl.Phys., 37(1966)3682. 4.59 R.L.Schwoebel, J.Appl.Phys., 40(1969)614. 4.60 S.C.Wang and G.Ehrlich, Phys.Rev.Letters, 67(1991)2509. 4.61 S.C.Wang and G.Ehrlich, Phys.Rev.Letters, 70(1993)41. 4.62 S.C.Wang and G.Ehrlich, Phys.Rev.Letters, 71(1993)4174. 4.63 M.Uwaha, Phys.Rev.Letters, 65(1990)1681. 4.64 A.Pimpinelli and J. Villain, Physics of Crystal Growth, Ed. C. Godrech, Cambridge University Press, 1998. 4.65 H.-C.Jeong, E.D.Williams, Surf.Sci.Rep., 34(1999)171. 4.66 M.Volmer, in Actualites Scientifiques et Industrielles, Vol.85, Herrmann Verl., Paris, 1933,p.l. 4.67 N.Mott and R.Watts-Tobin, Electrochim.Acta, 4(1961)79. 4.68 T.Vitanov, A.Popov and E.Budevski, J.Electrochem.Soc., 121(1974)207. 4.69 Chr.Nanev, J.Cryst. Growth, 23(1974)125. 4.70 E.Budevski, G.Staikov and V.Bostanov, J.Cryst. Growth, 29(1975)316. 4.71 Chr.Nanev, J.Cryst. Growth, 35(1976)113. 4.72 V.Bostanov, E.Budevski and G.Staikov, Farad.Symp.Chem.Soc., 12(1977)83. 4.73 Chr.Nanev, Kristall und Technik, 12(1977)578. 4.74 E.Budevski, V.Bostanov, G.Staikov and G.Angelov, Commun.Dept.Chem.BAS, 11 (1978)467. 4.75 G.Staikov, W.Obretenov, V.Bostanov, E.Budevski and H.Bort, Electrochim.Acta, 25 (1980)1619. 4.76 R.Kaischew, E.Budevski and J.Malinovski, Z.Phys.Chem., 205(1955)348. 4.77 A.Scheludko and G.Bliznakov, Commun.Bulg.Acad.Sci., 2(1951)239. 4.78 M.Fleischmann and H.R.Thirsk, Electrochim.Acta, 1(1959)146. 4.79 J.Newman, I&EC Fundamentals, 5(1966)525. 4.80 H.R.Thirsk and J.A.Harrison, A Guide to the Study of Electrode Kinetics, Academic Press, London and New York, 1972. 4.81 I.Markov, A.Boynov and S.Toschev, Electrochim.Acta, 18(1973)377. 4.82 G.J.Hills, D.J.Schiffrin and J.Thompson, Electrochim.Acta, 19(1974)657. 4.83 G.J.Hills, D.J.Schiffrin and J.Thompson, Electrochim.Acta, 19(1974)671. 4.84 D.Kashchiev and A.Milchev, Thin Solid Films, 28(1975)189. 4.85 D.Kashchiev and A.Milchev, Thin Solid Films, 28(1975)201. 4.86 I.Markov and S.Toschev, Electrodep.Surf.Treat., 3(1975)385. 4.87 A.N.Baraboshkin, Elektrokristallizatsia Metallov iz Rasplavlenih Solei, Nauka, 1976. 4.88 S.Fletcher, J.Chem.Soc.Farad.Trans.l, 79(1983)467. 4.89 S. Fletcher, J.Cryst. Growth, 62(1983)505. 4.90 P.A.Bobbert, M.M.Wind and J.Vlieger, Physica, 141A(1987)58. 4.91 P.A.Bobbert, M.M.Wind and J.Vlieger, Physica, 146A(1987)69. 4.92 M.D.Pritzker, J.Electroanal.Chem., 243(1988)57. 4.93 A.Milchev, J.Appl.Electrochem., 20(1990)307. 4.94 M.Tokuyama, Physica A, 169(1990)147.

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4.95 M.Tokuyama and Y.Enomoto, J.Chem.Phys., 94(1991)8234. 4.96 A.Milchev, J.EIectroanal.Chem, 312(1991)267. 4.97 W.S.Kruijt, M.Sluyters-Rehbach and J.H.Sluyters, J.Electroanal.Chem., 351(1993). 4.98 W.Schmikler and U.Stimming, J.Electroanal.Chem., 366(1994)203. 4.99 A.Milchev and R.Lacmann, Electrochim.Acta, 40(1995)1475. 4.100 E.Michailova, A.Milchev and R.Lacmann, Electrochim.Acta, 41(1996)329. 4.101 V.Tsakova and A.Milchev, Bulg.Chem.Commun., 29(1996/1997)444. 4.102 B.B.Damaskin and O.A.Petrii, Electrokhimia, Moskow, Vishaja Shkola, 1987. 4.103. A.J.Bard and R.L.Faulkner, Electrochemical methods, Second Edition, John Wiley &Sons, Inc., 2001, p.99. 4.104 E.Michailova, A.Milchev and R.Lacmann, Electrochim.Acta, 41(1996)329.

Chapter 5 MASS ELECTROCRYSTALLIZATION

5.1

General concepts

What we call mass electrocrystallization is the overall process of appearance, growth and overlap of multiple nuclei on the electrode surface. In particular, we have instantaneous nucleation when all nuclei are formed within a short time period after supersaturating the parent phase. Then the nuclei only grow and overlap which means that in reality instantaneous nucleation is the process of growth of a constant number of supercritical clusters. The task of the theory in this case is to derive an explicit expression for the growth current density On the contrary, we speak about progressive nucleation when nuclei of the new phase form, grow and overlap during the whole period of observation and the task of the theory in this case is to derive explicit expressions for the nucleation rate for the number density of nuclei and for the total current density For the purpose it is necessary to consider the nucleus formation accounting for the appearance of zones of reduced nucleation rate around the growing stable clusters. Such zones spread out and overlap, partly deactivating the surface area available for the nucleus formation. In case of no zones one must take into consideration the reduction of the free surface area due to the spreading and overlap of the growing stable clusters themselves. The theoretical treatment of these two types of complex physical processes is one of the most important problems in the theory of mass crystallization in all cases of first order phase transitions.

220

Chapter 5

The total current density of N(t) nuclei progressively formed on the electrode surface is given by the well-known convolution integral (see e.g. [5.1-5.5] and the references cited therein):

Here

is the growth current of a single cluster born at time is the actual free surface fraction available for the nucleus formation and is the nucleation rate at time t = u. If nuclei are formed instantaneously following time t = 0 then the nucleation rate is presented as where is the function of Dirak. In this case the current density is given by:

if the nuclei grow independently of each other. In case of overlap it is necessary to evaluate the actual surface area where ions are discharged and incorporated into the crystal lattice. This Chapter acquaints the reader with the fundamentals of mass electrocrystallization and presents explicit expressions for the currents and in some important particular cases.

5.2

Progressive and instantaneous nucleation without overlap

In the very initial stage of the phase transition the appearance of a nucleus on the substrate does not change the conditions for subsequent nucleus formation, the growing supercritical clusters do not interact with one another and the actual surface fraction covered by nucleation exclusion zones (or by the growing nuclei themselves) is negligibly small. Thus the current density is given by:

and this is the simplest case of progressive nucleation.

5. MASS ELECTROCRYSTALLIZATION 5.2.1

221

Stationary two-dimensional nucleation

If 2D nuclei form on the electrode surface with a stationary nucleation rate and then grow under ions transfer control the current of progressive nucleation is obtained by substituting equation (4.7) for into (5.1') and solving the integral to yield [5.1,5.2]:

Here the rate constant

reads:

The stationary nucleation rate is where is the number density of active sites on the electrode surface and is the 2D nucleation frequency (or the nucleation rate constant) per site given by the general formula (2.126). The current of instantaneous nucleation and growth of nuclei is expressed by equation (5.2). 5.2.2

Non-stationary two-dimensional nucleation

In the case of non-stationary effects due to the dynamics of active sites on the electrode surface [5.6] (Chapter 2.4.2) and when the condition (2.134) is fulfilled the nucleation rate J(t) is given by equation (2.136) and solving the integral (5.1') with equation (4.7) for one obtains:

For times

equation (5.5) simplifies to:

which can be presented as a linear relationship:

222

Chapter 5

allowing us to determine and from the slope and from the intercept of an vs. t experimental progressive current transient. The physical significance of and in this special case of phase formation was clarified in Chapter 2.4.2.2. In the advanced stage of the phase transition when all latent active sites are developed and most of the free active sites are already occupied by the nuclei of the new phase the time dependence of the nucleation rate J(t) is given by [5.1, 5.7]:

Substituting equations (4.7) and (5.8) into (5.1') and solving the integral for the current it results:

As expected, for times when all active sites are occupied by the nuclei of the new phase equation (5.9) turns into:

which gives the current of clusters. 5.2.3

independently growing two-dimensional

Stationary three-dimensional nucleation

If 3D nuclei form on the electrode surface with a stationary rate the current of progressive nucleation is obtained by presenting the current of a singlecluster as:

and solving the integral in equation (5.1') for

to yield [5.8]:

5. MASS ELECTROCRYSTALLIZATION

223

This equation holds good in three particular cases: (i). Combined ions transfer and diffusion limitations when n* = n (equation (4.43)), p* = p (equation (4.44)) and

(ii). Combined ions transfer and ohmic limitations for low ohmic drop when (equations (4.66)), (equation (4.67)) and

where the non-dimensional constant

is given by equation (4.69).

(iii). Combined ions transfer and ohmic limitations for high ohmic drop when (equations (4.75)), (equation (4.74)) and

Equation (5.11) allows us to derive explicit expressions for also in some more simple particular cases of growth of single hemispherical clusters considered in Chapter 4. Thus for short times when 2n* t < 0.1 and equation (5.11) simplifies to:

where

and

224

Chapter 5

This is the case of progressive nucleation under pure ions transfer control of the growth kinetics. For relatively long times when the inequalities are fulfilled equation (5.11) simplifies to:

or

that is more suitable for interpretation of experimental data for the current The last two expressions hold good in the particular cases: (i). n* = n, p* = p,

and

(iii).

but for times long enough to justify neglecting the unity with respect to 2n* t. As seen, equation (5.18) allows determining the stationary nucleation rate and the exchange current density from the slope and from the intercept of a linear vs. relationship. For the purpose it is necessary to have data for the diffusion coefficient (particular case (i))


225

and for the conductivity of the electrolyte solution (particular case (iii)) from independent measurements. At even longer times (n* t > 50) equation (5.17) and (5.18) turn into:

which describes the kinetics of progressive nucleation in the particular cases of pure diffusion (n* = n, p* = p) and pure ohmic control of growth of the single hemispherical clusters. Similarly to the case of two-dimensional nucleation, the current of instantaneous nucleation and growth of three-dimensional clusters is given by the simple formula (5.2) in which the current depends on the particular growth mechanism. 5.2.4

Non-stationary three-dimensional nucleation

As an illustration of this mode of progressive nucleation we consider the same time dependence of the nucleation rate as in Chapter 5.2.2 but only in the cases in which explicit expressions can be derived for the current pure ions transfer, pure diffusion and pure ohmic control of the growth kinetics. 5.2.4.1

Pure ions transfer control

If the non-stationary effects are due to the dynamics of active sites on the electrode surface (Chapter 2.4.2) and when the condition (2.134) is fulfilled, the nucleation rate J(t) is expressed by equation (2.136). Solving the integral in (5.1') with equations (4.34) and (2.136) for and J(t), respectively for the current one obtains [5.9]:

At times

the above expression turns into the asymptotic formula:

which can be presented as a linear relationship:

226

Chapter 5

This allows us to determine the product and the constant from the slope and from the intercept of an vs. t experimental progressive current transient. How to interpret the overpotential and the concentration dependence of is explained in [5.9]. In the advanced stage of the phase transition when all latent active sites are developed and most of the free active sites are already occupied by the nuclei of the new phase, the nucleation rate J(t) is given by equation (5.8), which now refer to three-dimensional nucleation. Solving the integral in equation (5.1') with for the current in this case one obtains:

Again, for long times when all active sites are occupied by the nuclei of the new phase equation (5.25) turns into:

which gives the current of clusters.

independently growing three-dimensional

5.2.4.2 Pure diffusion and pure ohmic control In order to derive theoretical expressions for accounting for the dynamics of active sites on the electrode surface we solve the integral in equation (5.1') with equation (2.136) for J(t) and with the formula:

which describes the growth current of a single hemispherical cluster in the particular cases of pure diffusion (n* = n, p* = p, equation (4.50)) and pure ohmic ( obtains:

equation (4.78)) control. Thus for

one


where

227

and the function

As shown in [5.10] the solution of

reads [5.10]:

is:

where j stands for Numerical values for Table 5.1. For large values of (long times t and/or short equation (5.29) turns into the asymptotic formula:

and

are given in tends to

and

Correspondingly:

which allows us to determine the product and the constant from the slope and from the intercept of an vs. t experimental progressive current transient. How to interpret the overpotential and the concentration dependence of

is explained in [5.9].

228

Chapter 5

In the advanced stage of the phase transition the nucleation rate J(t) is given by equation (5.8) and with (5.26) for equation (5.1') transforms into:

Here

and [5.10]:

The numerical values of and coincide with those of and (Table 5.1.) for and For long times when all active sites are occupied by the nuclei of the new phase, tends to and equation (5.32) turns into:

which gives the current of clusters.

5.3

independently growing three-dimensional

Progressive and instantaneous nucleation with overlap

The evaluation of the actual volume or the actual surface area occupied by growing clusters or by nucleation exclusion zones, respectively is a most important point in the kinetic theory of the first order phase transitions. Several authors [5.11-5.17] have considered the problem and among them the name of Avrami [5.14-5.16] seems to be the most popular one. However, the stochastic approach proposed by Kolmogoroff [5.12] is undoubtedly the most rigorous one (see e.g. the critical analysis of Belen’ky [5.18]) and here we describe this approach. In his original work [5.12] Kolmogoroff considered the formation, growth and overlap of 3D clusters within a three-dimensional space but the proposed formalism applies without restrictions to the case of 2D nucleation and


229

growth on a plane surface, too [5.18, 5.19]. For that reason here we consider this more simple case of mass crystallization as an illustration. 5.3.1

Nucleation and growth of 2D clusters on a plane substrate

Suppose that 2D circular nuclei appear on a plane substrate with a total surface area at a rate (Figure (5.1)). After that the nuclei grow at a rate so that if a 2D nucleus has appeared at the time moment u its radius l(t) at time t > u is:

Correspondingly the top surface area of a 2D circular disk is given by:

Here comes the first important assumption made in the Kolmogoroff’s model: (i). the growth rate must not depend on the size of the crystals, i.e. it could be a function of the current time moment t but not of the birth moment u. Apparently, a constant growth rate is the simplest case in which this requirement is fulfilled.

230 In order to determine the actual surface area

Chapter 5 covered by N*(t)

two-dimensional crystals, progressively formed and growing within the time interval [0,t], we consider an arbitrary point M on the substrate (Figure 5.2.) and determine the probability for this point to be ingested by at the moment t. For the purpose, first we remember that the periphery of a 2D crystal formed at the moment u

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