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Phase formation — Stability and nucleation kinetics of small clusters
Surface Science 156 (1985) 36-43 worth-Holland, Amsterdam
36
PHASE FORMATION - STABILITY AND NUCLEATION KINETICS OF SMALL CLUSTERS A. MILCHEV
* and J. MALINOWSKI
Central Laboratory
of Photographic
Received
1984
6 August
Processes, Bulgarian Academy
of Sciences. 1040 Sofia, Bulgaria
The properties of small particles and clusters have proved significant for a number of physical and chemical phenomena such as catalysis, adsorption, electrochemical deposition, etc. The formation of nuclei in a supersaturated phase is the basic process described by the theory of phase fo~ation. It allows the evaluation of the nucleation work, the size of the critical nuclei and its dependence on supersaturation. The energetics of clusters and therefore any phenomenon dependent on their energetic state can be elucidated by studying the phase formation process. The theory of phase formation so far developed relies on two different approaches. The classical theory is based on thermodynamic considerations and its consequences are restricted to clusters consisting of a significant number of atoms, allowing the use of thermodynamic magnitudes and functions. In many cases, however, the critical nuclei have been found to contain only a few atoms. For such cases the so-called atomistic approach has been developed. It has been shown that there is a full continuity in the results deduced by both approaches. Published experimental work is reviewed, making use of either of them.
1. fntroduction It has long been realized that microclusters of atoms and molecules have properties quite different than bulk materials. It is trivial to point out here the reasons for which clusters have attracted considerable attention in recent years. The availability of powerful computers has made possible extensive work aimed to develop discrete models of microclusters. Monte Carlo simulations, based on quantum mechanics and molecular dynamics, tend to reduce the assumptions needed to describe cluster structure and energetics. This paper, however, is not going to discuss the achievements in this field. It would rather give an overview of some work of the Sofia School of Phase Formation founded by the late Prof. Dr. I.N. Stranski and headed by Prof. Dr. R. Kaischev.
* Institute
of Physical
Chemistry,
Bulgarian
Academy
of Sciences,
0039~6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
1040 Sofia, Bulgaria.
B.V.
A. Miichev, .I. Malinowski / Phase formation
31
Nowadays quite complicated and sophisticated methods and devices are being generally used for cluster studies. In spite of this, it can be shown that some impo~~t characteristics of small clusters can still be obtained by rather simple experiments based on electrochemical phenomena. Nucleation and growth of clusters in a supersaturated ambient phase leads eventually to a decrease of the thermodynamic potential of the whole system. However, the isolated act of formation of an n-atomic cluster from n single atoms is connected with an initial increase of the thermodynamic potential due to the creation of a new interface [l]. The work A, for reversible and isothermic formation of an n-atomic nucleus is given by the equation A,=
-nAp+t#+z),
0)
where Ap is the supersat~ation and #Pfn) accounts for the energy contribution of the “cluster-ambient phase” boundary. Most generally, cp(n) can be expressed as a difference between the thermodynamic potential of the cluster and the thermodynamic potential of the equivalent number of atoms in the infinitely large new phase. Eq. (1) presents the nucleation work in a very general form and may be used without any restrictions about the structure and the size of the cluster. However, #(n) is generally not known and cannot be calculated without an assumption of some model. It is the task of any theory therefore to calculate $(n) and hence to determine the work of nucleation A,t.
2. The elassiesl approach In the classical theory [l] the size of the cluster is considered as a continuous variable and the quantity +(n) is expressed as a product of both the surface area S and the specific surface energy u of the phase boundary +(,, = SO = aan2/3. It is easily shown in this case that the nucleation work displays a maximum at some critical size n, (fig. 1) determined by the Gibbs-Thomson equation nk = ($aa/dp)3. Due to its excess surface free energy the critical nucleus is in a metastable equilibrium with the supersaturated ambient phase. Any deviation of the cluster size from the critical value nk leads either to its spontaneous growth or to its spontaneous decay. This unique property of the critical nuclei has been used to study the stability of small silver clusters and to determine experimentally their specific surface energy [2,3].
A. Milchev, J. Malinowskr
38
Fig. 1. The nucleation work A,, displays a maximum increase of the supersaturation Ap lowers n L.
/ Phase formation
for the cluster with the critical
size n k. The
3. Stability of small metal clusters The equilibrium the bulk electrode
potential E, of a small electrode with radius r differs from E, by a term determined by the Gibbs-Thomson equation
E, = E, - 2aV/zFr. In such a case the actual overpotential on its dimension AE,=AE(l
A E, acting on a small particle
depends
-r,/r).
It is readily seen that for particles just equal to the critical size rk the supersaturation is zero and they are in metastable equilibrium. For particles smaller than rk the supersaturation is negative and they would decay, while for r > rk the particles would grow spontaneously. This dependence has been found experimentally for Ag clusters dipped in buffers with different Red-Ox potentials [2,3]. The data for the critical radius obtained at different overpotentials are presented in fig. 2 as a Gibbs-Thomson plot of AE versus l/r,, the slope giving directly the value of u - the specific surface free energy of the “silver/solution” interface. Reliable data of u are known only relative to vacuum and this gives a value between 1100 and 1300 erg cme2. Therefore the value of 920 erg cmp2 obtained here should be considered very reasonable. In another set of experiments the stability of small silver clusters has been studied in presence of surfactant-gelatine [2]. It is readily seen that gelatine lowers the slope of the curve, corresponding to a reasonable lowering of u. Recently Plieth [4] has shown that some other physical parameters expected to depend on the cluster size also follow the requirement of the Gibbs-Thom-
A. Milchev, J. Malinowski
/ Phase formation
Fig. 2. Gibbs-Thomson plot of the equilibrium potential AE of small silver clusters size l/r,. The lower curve is obtained in the presence of gelatine [2,3].
39
versus their
son equation. Hills et al. [5] have found a similar dependence of the electrode potential of Ag, Cu and Hg small electrodes. The experimental results described so far prove convincingly the validity of the Gibbs-Thomson equation for the equilibrium potential of small metal clusters. Such studies yield values of u, which may be used to characterize the excess energy of the cluster/ambient interface. Of course it should be reminded that the critical nuclei in these experiments consisted always of more than 10’ atoms, a value obviously large enough to justify the application of the continuum thermodynamic approach. The kinetic studies of electrochemical phase formation [6], however, show that in most cases the critical nuclei contain not more than 10 atoms and these results can be interpreted only on the basis of discrete atomic models [7]. In this case the size of the clusters cannot be considered as a continuous variable and the excess free energy of the “nucleus-ambient phase” boundary cannot be expressed as a simple product of the surface area S and the specific surface energy u, both these quantities having no physical meaning at small dimensions.
4. The atomistic approach For cases when the critical nuclei represent clusters of atomic sizes, Gccn,has to be presented in a more general form, namely as a difference in the energy of n atoms when they are part of the infinitely large new phase in one case, and
A. Milcheu, J. Malinowski
40
when they form a separate n
/ Phase formation
two- or three-dimensional
cluster
in the other [8]
In eq. (2) vi/2 is the separation work from the half crystal position equivalent in vacuum to the heat of evaporation and the sum represents the dissociation energy of the n-atomic cluster, QJ~being the separation work of the i th atom. The sum in eq. (2) cannot be calculated without the assumption of some model. An approximate but very illustrative method of studying the behaviour of microclusters formed on a foreign substrate is based on the consideration of the interaction only between first next-neighbour atoms [9]. It is easily shown in this case that when the bonding energy between the atoms and the substrate is weaker than the bonding energy between like atoms, clusters with a five-fold symmetry axis are formed. The method allows the calculation of the nucleation work in a most simple way and here it is used to demonstrate the A, versus n relationship at high supersaturations, see fig. 3. Due to the strongly expressed discrete character of the cluster-size alteration at small dimensions, the nucleation work is not a monotonous function but
11. 10. 9. 0, 7. 6, 5,
0
2
4
6
6
10
12
14
16
16 n-
Fig. 3. The nucleation work A, as dependent supersaturation increases from a to g.
on cluster size n according
to atomistic
theory.
Fig. 4. The size of the critical nucleus versus the supersaturation Ap. The Gibbs-Thomson equation (the smooth curve) is a good approximation only at low Ap (high n k).
The
A. Milchev, J. Malinowski
/ Phase formation
41
displays several maxima and minima, the highest maximum corresponding to the work for critical nucleus formation. Thus at supersaturation Apa, the critical nucleus is a cluster consisting of 8 atoms. Increasing Ap, the nucleation work diminishes but the highest maximum corresponds again to the 8-atomic cluster. Further increase of Ap leads to a change of the critical nucleus size. Thus at super saturations higher than ApL,, a cluster consisting of 6 atoms is formed with a maximal work. Therefore the nk versus A/J relationship (fig. 4) will be a stepwise one - not a fixed supersaturation but a supersaturation interval will correspond to each critical nucleus. At low supersaturations the intervals are short and the Gibbs-Thomson equation (the fluent curve in fig. 4) describes well the real size alteration. Increasing the supersaturation, the intervals become wider and the Gibbs-Thomson equation is no longer a good approximation. The above considerations lead to the following important conclusion: at high supersaturations the critical nucleus is not an equilibrium cluster. Actually its chemical potential remains constant in a given supersaturation interval while the chemical potential of the ambient phase changes continuously when changing the supersaturation. Of course all these considerations concern only the thermodynamics of formation of clusters of the new phase. The rate of this process essentially depends on its mechanism and therefore this important quantity can be determined only in the frameworks of a kinetic treatment.
5. Nucleation kinetics
The classical and the atomistic theory of phase formation describe the same physical process at low and high supersaturation respectively. Following the classical work of Becker and Dijring [12], it has been shown that in the two cases there is a full continuity in the general expression for the stationary rate of nucleation [9-111
where Z, is the number of sites on the substrate where nucleation proceeds with a measurable rate and IV, is the frequency of attachment of single atoms to the critical nucleus. The only difference between the continuum and the discrete approach to the phase formation kinetics appears in the supersaturation dependence of the nucleation work A,x. This quantity is a continuous function in the first case and a stepwise, discrete function in the second. Taking this into account, the supersaturation dependence of the nucleation rate comes out as follows In I,, a l/Ap*,
A. Milchev, J. Mahnowski
42
according
to the classical
/ Phase formatum
theory, and
In I,, a AP, according to the atomistic theory. Data for the steady-state nucleation rate obtained when silver has been electrodeposited on glassy carbon from a silver nitrate solution are shown in fig. 5 [13,14]. The interpretation of these results on the basis of the atomistic theory shows that in the interval 25-51 mV the critical nucleus consists of 4 atoms. In the interval 51-160 mV the single atom plays the role of a critical nucleus and at overpotentials higher than 160 mV the number of atoms in the nucleus equals zero. In terms of the atomistic model the results nk = 0 means that the single atom adsorbed on the substrate forms a stable cluster which grows spontaneously at the corresponding supersaturation. In this case the work for nucleus formation is equal to zero and only the kinetics determines the rate of the deposition process. As is seen, the overpotentials 51 and 160 mV mark points of supersaturation at which critical nuclei with different size coexist. Returning to the theory [14] this enables one to calculate the excess free energy c$(,,, connected with the formation of clusters with the corresponding number of atoms. Therefore, based on the atomistic theory, studies of electrochemical nucleation kinetics allow the evaluation of the excess energy of the clusters. This is achieved without the implication of any assumptions concerning the cluster structure and energetic interactions between the atoms of the cluster itself or with the substrate. The values of $(,,, thus obtained characterize fully the energetics of the clusters under study. Further, making use of eq. (2) using however some model of the cluster structure, information can be also obtained for the bonding energy of the atoms in the cluster.
I
0
a04
OD3
o,l2
0.16
a20
0.24 Q/v-
Fig. 5. The stationary nucleation rate In 1 follows the dependence predicted by the atomistic theory. The three sets of experiments activation procedures of the electrode.
on the overpotential 7 as are carried out at different
A. Milchev, J. Malinowski
/ Phase formation
43
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] 1131 [14]
M. Volmer, Kinetik der Phasenbildung (Dresden, Leipzig, 1939). I. Konstantinov and J. Malinowski, J. Phot. Sci. 23 (1975) 145. I. Konstantinov and J. Malinowski, J. Phot. Sci. 23 (1975) 1. W.J. Plieth, Electrochemie der Metale, Ed. W. Luz (Verlag Chemie, Weinheim, 1982); Surface Sci. 156 (1985) 530. G. Hills, A.K. Pour and B. Scharifker, Electrochim. Acta 28 (1983) 891. A. Milchev and S. Stoyanov, J. Electroanal. Chem. 72 (1976) 33. A. Milchev, S. Stoyanov and R. Kaischev, Thin Solid Films 22 (1974) 255. I. Stranski, Ann. Univ. Sofia, Livre 2 (Chemie) 367 (1936/1937). S. Stoyanov, in: Current Topics in Material Science, Ed. E. Kaldis (North-Holland, Amsterdam, 1978). A. Milchev, S. Stoyanov and R. Kaischev, Thin Solid Films 22 (1974) 267. A. Milchev, S. Stoyanov and R. Kaischev, Soviet Electrochem. 13 (1977) 723. R. Becker and W. Doring, Ann. Physik 24 (1935) 719. A. Milchev and E. Vassileva, J. Electroanal. Chem. 107 (1980) 337. A. Milchev, Electrochim. Acta 28 (1983) 947.
36
PHASE FORMATION - STABILITY AND NUCLEATION KINETICS OF SMALL CLUSTERS A. MILCHEV
* and J. MALINOWSKI
Central Laboratory
of Photographic
Received
1984
6 August
Processes, Bulgarian Academy
of Sciences. 1040 Sofia, Bulgaria
The properties of small particles and clusters have proved significant for a number of physical and chemical phenomena such as catalysis, adsorption, electrochemical deposition, etc. The formation of nuclei in a supersaturated phase is the basic process described by the theory of phase fo~ation. It allows the evaluation of the nucleation work, the size of the critical nuclei and its dependence on supersaturation. The energetics of clusters and therefore any phenomenon dependent on their energetic state can be elucidated by studying the phase formation process. The theory of phase formation so far developed relies on two different approaches. The classical theory is based on thermodynamic considerations and its consequences are restricted to clusters consisting of a significant number of atoms, allowing the use of thermodynamic magnitudes and functions. In many cases, however, the critical nuclei have been found to contain only a few atoms. For such cases the so-called atomistic approach has been developed. It has been shown that there is a full continuity in the results deduced by both approaches. Published experimental work is reviewed, making use of either of them.
1. fntroduction It has long been realized that microclusters of atoms and molecules have properties quite different than bulk materials. It is trivial to point out here the reasons for which clusters have attracted considerable attention in recent years. The availability of powerful computers has made possible extensive work aimed to develop discrete models of microclusters. Monte Carlo simulations, based on quantum mechanics and molecular dynamics, tend to reduce the assumptions needed to describe cluster structure and energetics. This paper, however, is not going to discuss the achievements in this field. It would rather give an overview of some work of the Sofia School of Phase Formation founded by the late Prof. Dr. I.N. Stranski and headed by Prof. Dr. R. Kaischev.
* Institute
of Physical
Chemistry,
Bulgarian
Academy
of Sciences,
0039~6028/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
1040 Sofia, Bulgaria.
B.V.
A. Miichev, .I. Malinowski / Phase formation
31
Nowadays quite complicated and sophisticated methods and devices are being generally used for cluster studies. In spite of this, it can be shown that some impo~~t characteristics of small clusters can still be obtained by rather simple experiments based on electrochemical phenomena. Nucleation and growth of clusters in a supersaturated ambient phase leads eventually to a decrease of the thermodynamic potential of the whole system. However, the isolated act of formation of an n-atomic cluster from n single atoms is connected with an initial increase of the thermodynamic potential due to the creation of a new interface [l]. The work A, for reversible and isothermic formation of an n-atomic nucleus is given by the equation A,=
-nAp+t#+z),
0)
where Ap is the supersat~ation and #Pfn) accounts for the energy contribution of the “cluster-ambient phase” boundary. Most generally, cp(n) can be expressed as a difference between the thermodynamic potential of the cluster and the thermodynamic potential of the equivalent number of atoms in the infinitely large new phase. Eq. (1) presents the nucleation work in a very general form and may be used without any restrictions about the structure and the size of the cluster. However, #(n) is generally not known and cannot be calculated without an assumption of some model. It is the task of any theory therefore to calculate $(n) and hence to determine the work of nucleation A,t.
2. The elassiesl approach In the classical theory [l] the size of the cluster is considered as a continuous variable and the quantity +(n) is expressed as a product of both the surface area S and the specific surface energy u of the phase boundary +(,, = SO = aan2/3. It is easily shown in this case that the nucleation work displays a maximum at some critical size n, (fig. 1) determined by the Gibbs-Thomson equation nk = ($aa/dp)3. Due to its excess surface free energy the critical nucleus is in a metastable equilibrium with the supersaturated ambient phase. Any deviation of the cluster size from the critical value nk leads either to its spontaneous growth or to its spontaneous decay. This unique property of the critical nuclei has been used to study the stability of small silver clusters and to determine experimentally their specific surface energy [2,3].
A. Milchev, J. Malinowskr
38
Fig. 1. The nucleation work A,, displays a maximum increase of the supersaturation Ap lowers n L.
/ Phase formation
for the cluster with the critical
size n k. The
3. Stability of small metal clusters The equilibrium the bulk electrode
potential E, of a small electrode with radius r differs from E, by a term determined by the Gibbs-Thomson equation
E, = E, - 2aV/zFr. In such a case the actual overpotential on its dimension AE,=AE(l
A E, acting on a small particle
depends
-r,/r).
It is readily seen that for particles just equal to the critical size rk the supersaturation is zero and they are in metastable equilibrium. For particles smaller than rk the supersaturation is negative and they would decay, while for r > rk the particles would grow spontaneously. This dependence has been found experimentally for Ag clusters dipped in buffers with different Red-Ox potentials [2,3]. The data for the critical radius obtained at different overpotentials are presented in fig. 2 as a Gibbs-Thomson plot of AE versus l/r,, the slope giving directly the value of u - the specific surface free energy of the “silver/solution” interface. Reliable data of u are known only relative to vacuum and this gives a value between 1100 and 1300 erg cme2. Therefore the value of 920 erg cmp2 obtained here should be considered very reasonable. In another set of experiments the stability of small silver clusters has been studied in presence of surfactant-gelatine [2]. It is readily seen that gelatine lowers the slope of the curve, corresponding to a reasonable lowering of u. Recently Plieth [4] has shown that some other physical parameters expected to depend on the cluster size also follow the requirement of the Gibbs-Thom-
A. Milchev, J. Malinowski
/ Phase formation
Fig. 2. Gibbs-Thomson plot of the equilibrium potential AE of small silver clusters size l/r,. The lower curve is obtained in the presence of gelatine [2,3].
39
versus their
son equation. Hills et al. [5] have found a similar dependence of the electrode potential of Ag, Cu and Hg small electrodes. The experimental results described so far prove convincingly the validity of the Gibbs-Thomson equation for the equilibrium potential of small metal clusters. Such studies yield values of u, which may be used to characterize the excess energy of the cluster/ambient interface. Of course it should be reminded that the critical nuclei in these experiments consisted always of more than 10’ atoms, a value obviously large enough to justify the application of the continuum thermodynamic approach. The kinetic studies of electrochemical phase formation [6], however, show that in most cases the critical nuclei contain not more than 10 atoms and these results can be interpreted only on the basis of discrete atomic models [7]. In this case the size of the clusters cannot be considered as a continuous variable and the excess free energy of the “nucleus-ambient phase” boundary cannot be expressed as a simple product of the surface area S and the specific surface energy u, both these quantities having no physical meaning at small dimensions.
4. The atomistic approach For cases when the critical nuclei represent clusters of atomic sizes, Gccn,has to be presented in a more general form, namely as a difference in the energy of n atoms when they are part of the infinitely large new phase in one case, and
A. Milcheu, J. Malinowski
40
when they form a separate n
/ Phase formation
two- or three-dimensional
cluster
in the other [8]
In eq. (2) vi/2 is the separation work from the half crystal position equivalent in vacuum to the heat of evaporation and the sum represents the dissociation energy of the n-atomic cluster, QJ~being the separation work of the i th atom. The sum in eq. (2) cannot be calculated without the assumption of some model. An approximate but very illustrative method of studying the behaviour of microclusters formed on a foreign substrate is based on the consideration of the interaction only between first next-neighbour atoms [9]. It is easily shown in this case that when the bonding energy between the atoms and the substrate is weaker than the bonding energy between like atoms, clusters with a five-fold symmetry axis are formed. The method allows the calculation of the nucleation work in a most simple way and here it is used to demonstrate the A, versus n relationship at high supersaturations, see fig. 3. Due to the strongly expressed discrete character of the cluster-size alteration at small dimensions, the nucleation work is not a monotonous function but
11. 10. 9. 0, 7. 6, 5,
0
2
4
6
6
10
12
14
16
16 n-
Fig. 3. The nucleation work A, as dependent supersaturation increases from a to g.
on cluster size n according
to atomistic
theory.
Fig. 4. The size of the critical nucleus versus the supersaturation Ap. The Gibbs-Thomson equation (the smooth curve) is a good approximation only at low Ap (high n k).
The
A. Milchev, J. Malinowski
/ Phase formation
41
displays several maxima and minima, the highest maximum corresponding to the work for critical nucleus formation. Thus at supersaturation Apa, the critical nucleus is a cluster consisting of 8 atoms. Increasing Ap, the nucleation work diminishes but the highest maximum corresponds again to the 8-atomic cluster. Further increase of Ap leads to a change of the critical nucleus size. Thus at super saturations higher than ApL,, a cluster consisting of 6 atoms is formed with a maximal work. Therefore the nk versus A/J relationship (fig. 4) will be a stepwise one - not a fixed supersaturation but a supersaturation interval will correspond to each critical nucleus. At low supersaturations the intervals are short and the Gibbs-Thomson equation (the fluent curve in fig. 4) describes well the real size alteration. Increasing the supersaturation, the intervals become wider and the Gibbs-Thomson equation is no longer a good approximation. The above considerations lead to the following important conclusion: at high supersaturations the critical nucleus is not an equilibrium cluster. Actually its chemical potential remains constant in a given supersaturation interval while the chemical potential of the ambient phase changes continuously when changing the supersaturation. Of course all these considerations concern only the thermodynamics of formation of clusters of the new phase. The rate of this process essentially depends on its mechanism and therefore this important quantity can be determined only in the frameworks of a kinetic treatment.
5. Nucleation kinetics
The classical and the atomistic theory of phase formation describe the same physical process at low and high supersaturation respectively. Following the classical work of Becker and Dijring [12], it has been shown that in the two cases there is a full continuity in the general expression for the stationary rate of nucleation [9-111
where Z, is the number of sites on the substrate where nucleation proceeds with a measurable rate and IV, is the frequency of attachment of single atoms to the critical nucleus. The only difference between the continuum and the discrete approach to the phase formation kinetics appears in the supersaturation dependence of the nucleation work A,x. This quantity is a continuous function in the first case and a stepwise, discrete function in the second. Taking this into account, the supersaturation dependence of the nucleation rate comes out as follows In I,, a l/Ap*,
A. Milchev, J. Mahnowski
42
according
to the classical
/ Phase formatum
theory, and
In I,, a AP, according to the atomistic theory. Data for the steady-state nucleation rate obtained when silver has been electrodeposited on glassy carbon from a silver nitrate solution are shown in fig. 5 [13,14]. The interpretation of these results on the basis of the atomistic theory shows that in the interval 25-51 mV the critical nucleus consists of 4 atoms. In the interval 51-160 mV the single atom plays the role of a critical nucleus and at overpotentials higher than 160 mV the number of atoms in the nucleus equals zero. In terms of the atomistic model the results nk = 0 means that the single atom adsorbed on the substrate forms a stable cluster which grows spontaneously at the corresponding supersaturation. In this case the work for nucleus formation is equal to zero and only the kinetics determines the rate of the deposition process. As is seen, the overpotentials 51 and 160 mV mark points of supersaturation at which critical nuclei with different size coexist. Returning to the theory [14] this enables one to calculate the excess free energy c$(,,, connected with the formation of clusters with the corresponding number of atoms. Therefore, based on the atomistic theory, studies of electrochemical nucleation kinetics allow the evaluation of the excess energy of the clusters. This is achieved without the implication of any assumptions concerning the cluster structure and energetic interactions between the atoms of the cluster itself or with the substrate. The values of $(,,, thus obtained characterize fully the energetics of the clusters under study. Further, making use of eq. (2) using however some model of the cluster structure, information can be also obtained for the bonding energy of the atoms in the cluster.
I
0
a04
OD3
o,l2
0.16
a20
0.24 Q/v-
Fig. 5. The stationary nucleation rate In 1 follows the dependence predicted by the atomistic theory. The three sets of experiments activation procedures of the electrode.
on the overpotential 7 as are carried out at different
A. Milchev, J. Malinowski
/ Phase formation
43
References [l] [2] [3] [4] [5] [6] [7] [8] [9] [lo] [ll] [12] 1131 [14]
M. Volmer, Kinetik der Phasenbildung (Dresden, Leipzig, 1939). I. Konstantinov and J. Malinowski, J. Phot. Sci. 23 (1975) 145. I. Konstantinov and J. Malinowski, J. Phot. Sci. 23 (1975) 1. W.J. Plieth, Electrochemie der Metale, Ed. W. Luz (Verlag Chemie, Weinheim, 1982); Surface Sci. 156 (1985) 530. G. Hills, A.K. Pour and B. Scharifker, Electrochim. Acta 28 (1983) 891. A. Milchev and S. Stoyanov, J. Electroanal. Chem. 72 (1976) 33. A. Milchev, S. Stoyanov and R. Kaischev, Thin Solid Films 22 (1974) 255. I. Stranski, Ann. Univ. Sofia, Livre 2 (Chemie) 367 (1936/1937). S. Stoyanov, in: Current Topics in Material Science, Ed. E. Kaldis (North-Holland, Amsterdam, 1978). A. Milchev, S. Stoyanov and R. Kaischev, Thin Solid Films 22 (1974) 267. A. Milchev, S. Stoyanov and R. Kaischev, Soviet Electrochem. 13 (1977) 723. R. Becker and W. Doring, Ann. Physik 24 (1935) 719. A. Milchev and E. Vassileva, J. Electroanal. Chem. 107 (1980) 337. A. Milchev, Electrochim. Acta 28 (1983) 947.