J. CHEM. SOC. FARADAY TRANS., 1994, 90(13), 1923-1929 Studies of Silver Electronucleation onto Carbon Microelectrodes J. P. Sousa* Departamento de Engenharia Quimica , FEUP/Universidade do Porto, Rua dos Bragas, 4099 Porto Codex, Portugal S. Pons Department of Chemistry, University of Utah, Salt Lake City, UT 84112,USA M. Fleischmann Department of Chemistry, University of Southampton, Southampton, UK SO9 5NH It is shown that it is possible to use microelectrodes to develop a number of strategies for the measurement of silver electronucleation kinetics at the molecular level. The direct measurement of individual events and there-fore stochastic deterministic growth of the centres allows determination of an accurate value of the arrival time of the first nucleus.The distribution of these arrival times follows the simplest model of a pure birth process with identical rate constants for all the aggregation steps. The electrodeposition and electrocrystallisation of several metals have been widely studied and the mechanisms inten- sively investigated. 1-3 The early development of the theory of crystal growth stemmed from the ideas of Gibbs, the main contributors being Erdey-Gruz, Volmer, Becker and D~ring.~ The model arrived at was that crystals would grow until the object (substrate surface) was bounded by perfect low-index faces so that the crystal minimises its surface energy. Further growth would then require two-dimensional nucleation of layer planes and the formation of each nucleus would lead to the deposition of unit lattice repeat or, possibly, a subunit of lattice. On the basis of this model, crystal growth takes place at the edges of layer planes.In the 1950s a new model was proposed by Burton et according to which an emerging screw dislocation generates a self-perpetuating step, i.e. crystal growth leads to the winding up of these steps into spiral growth forms (which have been frequently observed by electron microscopy). The edges are 'rough' and lattice formation occurs at these self-perpetuating edges. The electrocrystallization of silver was investigated repeat- edly during the 1960s using a variety of electrochemical tech- niques (ac impedance, galvanostatic steps and double potential step methods).6-' In these investigations no special steps were taken to ensure the formation of smooth surfaces, i.e.these measurements can be related essentially to those in the present study. A general conclusion drawn from those investigations was that there is a relatively high concentra- tion of adatoms at the substrate surface and that the exchange current associated with the formation of these adatoms is comparable to that of the Hg2+/Hg reaction, which is one of the fastest electrochemical processes known (ky = 1-10 cm s-').'' During the past decade, special attention has been devoted to the study of the initial stages leading to the formation of metallic centres. However, several difficulties in characterising the nucleation process have been reported.'2*' The nucleation of silver on conventional carbon electrodes (diameter in the cm range) under potentiostatic conditions has been studied by Hills and co-~orkers.~~-~~ It was found that the growth of metallic crystallites is a three-dimensional process and is diffusion controlled.Attempts have been made to establish a relationship between the applied overpotential and the kinetics of the process. The main difficulty in estab- lishing such a relation was the overlap of the metallic centres undergoing deposition. Therefore, the only kinetic parameter obtainable was the number density of nuclei generated during a certain potentiostatic pulse by an indirect process of rela- ting the observed current transient to the area of the cluster formed.The rate constants and the kinetics of formation for the growth of single, three-dimensional centres can be estimated by using small devices such as microelectrodes (diameter in the pm range) as substrates. The importance of microelec- trodes is widely recognised and interest in their application to several areas of research has increased dramatically over the past decade.17-22 These devices have made an impact in the electrocrystallization domain, both fundamental and applied, unequalled by almost any other technique in modern times. Microelectrode properties contributing to improvements in the quality of the experimental data in the electrodeposition field include:23 (i) the ability to define the formation of an even smaller microelectrode (area in the A range) on the sub- strate surface; (ii) high sensitivity, which produces a high signal when a single molecular event takes place at the sub- strate surface; (iii) the individual molecular events are sto- chastic and therefore constitute a deterministic process.The above properties, among others, allow study of the nucleation and growth of an extremely small crystal, i.e. a single centre. The essentially deterministic growth of the first nucleus (defined as a set of metal atoms large enough that they are thermodynamically stable at a given potential) allows one to determine an accurate value of the arrival times. A large collection of these arrival times for each over- potential gives a distribution of first passage times.These dis- tributions are determined by the kinetics of the formation of the first nucleus and therefore can be conveniently exploited in order to determine the kinetic parameters that characterise the metallic nucleation process. In the early 1970s, nucleation started to be regarded as a random phen~rnenon;~~ the random birth of nuclei within a given time interval was assumed to follow a simple Poisson distribution law: P, = N" exp(-N) m! where N is the average number of nuclei to be expected in the average time interval. When using small devices such as microelectrodes in metal- lic electrodeposition studies one can observe from analysis of the ensembles of the experimental crystal growth transients that the process of forming a single nucleus can be described by the Chapman-Kolmogorov relations : The difficulties in obtaining general solutions to such sets of equations are well known.Both Aj and pj are complicated functions of the nucleus size. One cannot therefore derive any sensible master equation. Algebraic solutions can be attained only for values of K < 3.25 Recently, Fleischmann et have proposed a model based on the facts that under severe experimental conditions (high overpotentials and highly inert substrate surfaces) the growth rates were much higher than the dissolution rates and that the birth rates were equal (Ao = A1 = = Ij) through- out the nucleation process, i.e. nucleation occurs by a pure birthprocess. The probability of forming a cluster of size j is governed by the system of equations: with initial conditions P0--1; Pj+l=0; t=O (4) The expressions comprising eqn.(3) can be integrated one at a time to give the general Poisson distribution law (At)jPj(t)= -exp(-It)I! The probability of forming a cluster of size greater than K (the size of the critical cluster), in a time interval 0 to t is then given by m (It)j1Pj(t) = 1 -C 7exp(-IIt) K+l o J! and this is illustrated in Fig. 1 as plots of PXt) us. the dimensionless variable T(=At) for K = 1 to 10, while in Fig. 2 the natural logarithms of these parameters are plotted. It can be anticipated that the first major detectable devi- ation from this behaviour (pure birth process) would be the influence of low rates of death (which occurs accordingly with the classical and atomistic approach for the nucleation process) in a general birth and death process.J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.oo 0.80 0.60 P 0.40 0.20 0.00 T Fig. 1 Model of the pure birth process for different K values, using T = At. Plots of Cz+ Pj(T)US. T.k = 1 (a),.. . ,10 (j). (7) Closed-form solutions of eqn. (7), can be derived only if K < 3. However, by solving the Laplace transforms of these equations one can arrive at the general expression m * (1”t)j1pj(t) = 1-C Iexp(-k) + K(AK+ l)ptK(+2) exp(-At)K+ 1 j=o 1-(K + 2)! Eqn. (8) has been used to construct Fig. 3-5. 0.00 -2.00 -4.00 k c--6.00 -8.00 -1 0.00 -3.00 -2.00 -1.00 0.00 1.00 2.00 In T Fig.2 Plots of In Cz+l Pj(T)us. In T for the pure birth model; (a)-(j) as Fig. 1 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T Fig. 3 Plots of cr (T)0s. T for the birth and death model for dif- ferent values of /3, where /3 = p/2 and A. = A, = . ..ik= A. Arrow indicates p increasing from 0.1 to 1.0 in increments of 0.1. /3 = 5 (a), 10 (b),50 (c),100 (d)and 500 (e). As can be seen, the inclusion of death rates accounts for a slight broadening of the theoretical plots only for small values of K. This leads to the conclusion that, under forced experimental conditions, p assumes very low values which can be ignored and the nucleation process can be regarded as being a pure birth process.The pure birth model is here applied to study the kinetics of the nucleation and growth of single centres of silver on highly inert substrate surfaces (carbon fibres) under poten- tiostatic conditions. These substrates are characterised as having a very low energy of adhesion between the substrate and the new phase. This leads to a low rate of nucleation and hence causes the deposition of only one or, at most, a few centres, which is followed by their subsequent growth. A sta-tistical treatment of the arrival time ensembles is presented, showing that silver nucleation can be regarded as a random process, i.e. obeying the Poisson distribution law. 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T Fig. 4 Plots of cg+(T) us.T showing broadening due to the inclusion of small values of fl (fl = p/A) [eqn. (8)]. Arrow indicates increasing fl, as in Fig. 3. k = 1 (a), 5 (b),10 (c). 1925 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T 0.00 3.00 6.00 9.00 12.00 15;OO T Fig. 5 Plots of IF+(T)vs. T for the birth and death model [eqn. (8)]; using A, fl = 0.1, B, fl = 1.0.(a)+) as for Fig. 1. Experimental The measurements were carried out using the potentiostatic technique for the cathodic deposition of silver from an aqueous solution of composition 0.1mol dm-3 KN0,-5.0 mmol dm- AgNO, . All chemicals were purchased from J. T.Baker Chemical Co. and the water used to prepare the solutions was triply distilled from a Corning MP-1system. The working electrode, Fig. qa), consisted of a carbon fibre microdisk of 5 pm diameter sealed into glass, as has been reported in the literature.18*’ 1,27*28These electrodes were treated with 1 :1 HNO,-H,O solution and mechanically pol- ished with fine-grade alumina powder (1.0to 0.05 pm) after several measurement cycles. Between each potentiostatic measurement the working electrode was cleaned by anodic stripping (g = 180mV). As a secondary electrode, Fig. qb),a silver wire (99.9% purity), of dimensions much larger than the working electrode, was used. All of the potentiostatic mea- surements were carried out using a two-electrode system on a single-compartment cell.A waveform generator, Hi-Tek PPR1,was employed to apply the chosen overpotentials to the investigated system. The currents were measured using a Keithley Model 617pro-grammable electrometer and recorded on a Houston Instru- ments Model 200 X-Y recorder. Low-noise coaxial cables were used to make the electrical connections and the electro- 1926 (a) copper /'' lead epoxyc./A 2F epoxy g-resin resin seal seal ~'glass 1 ,glass tube tube i . ~ lead seal carbon microelectrode Fig. 6 Side view of the electrodes: (a) carbon disc microelectrode (WE); (b)silver wire electrode (AE) chemical cell was placed inside a Faraday cage to avoid external interferences.Dry nitrogen was bubbled through the electrochemical solution for 20 min before each experiment and for each set of potentiostatic measurements a fresh solution was prepared. All measurements were carried out at laboratory pressure and temperature. Results and Discussion The linear sweep voltammograms, such as that shown in Fig. 7, display the characteristic features of nucleation and phase growth, namely the large peak separation and crossover on the cathodic branch." The shapes of these voltammograms are a function of the induction times, chosen limits of poten- tial sweep, sweep rates and surface reactions. For the case of metal electrodeposition on microelectrodes the current-potential (Aq = vt) transients present character- istic features: on the forward sweep, the current rises owing to the formation of a metallic centre at the substrate surface and this rise continues until the cathodic potential limit is reached.On the return sweep, these transients can assume a variety of responses depending on the complexity of the J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 / I/\ /I I ,f i,' 11 200 -2 00 overpotential/mV Fig. 7 Linear sweep voltammogram for the deposition of silver onto a carbon microdisc electrode from 5.0 mmol dm-, AgNO, in 0.1 mol dmP3 KNO, aqueous solution; v = 150 mV s-'; I = 165 pA cm-system undergoing investigation.' 3,2 ',23 A commonly observed pattern is that the current continues to increase for the whole of the cathodic region of the return sweep owing to the continued expansion of the growth centre(s).The different shapes observed within the anodic branch can be regarded as being due to surface passivation processes, i.e. film formation. Metallic silver centres were occasionally formed on the carbon fibre surface following the cathodic reduction of Ag', according to the following reaction : Ag+ + 1 e- -+Ago (1) A large number of current-time transients were recorded over a large overpotential range. At least 200 measurements were made for each value of overpotential applied to the system (-200, -190, -180 and -170 mV) in order to obtain a statistically accurate distribution of the induction times as shown in Table 1for the highest overpotential used.The data presented in this table show that the Poisson dis- tribution is indeed appropriate to describe the nucleation phenomena. Considering that under the experimental poten- tiostatic conditions, the observed induction times are distrib- uted randomly (Fig. 8), one has to assume that the measurements fluctuate from observation to observation, i.e. statistical fluctuations are observed. Therefore, the method of Table 1 Distribution of experimental induction times for silver electronucleation onto a carbon microdisc electrode at q = -200 mV 0.00-0.25 0.26-0.50 0.51-0.75 0.76-1.10 1.10-1.25 1.26-1.50 1.51-1.75 1.76-2.00 2.10-2.25 2.26-2.50 xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxx xxx xx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx 52 110 35 8 5 210 J.CHEM. SOC. FARADAY TRANS., 1994, VOL. 90 Table 2 Experimental nucleation times and probabilities -200 mV -190 mV In t In P(t) In t In P(t) -1.386 -3.057 -0.693 -1.820 -0.693 -1.398 O.OO0 -0.814 -0.278 -0.629 0.405 -0.454 O.OO0 -0.260 0.693 -0.283 0.223 -0.1 16 0.9 16 -0.171 0.405 -0.064 1.098 -0.115 0.559 -0.039 1.252 -0.056 0.693 -0.024 1.386 -0.020 0.810 -0.010 1.504 -0.005 0.9 16 O.OO0 1.609 O.OO0 maximum likelihood must be used in order to determine the best statistical mean of the di~tribution.~'-~' Also, all of the measurements are affected by different uncertainties.Thus, for the present case, the mean is given by (9) The variance is equal to the mean, but the mean is affected by an uncertainty estimated according to Eqn. (9) and (10) have been applied to determine the sta- tistical parameters that characterise the data presented in Table 1. The results obtained were: p = 79.97 counts s-', CJ = 8.94 counts s-and o,, = 0.22 counts s-'. Once the statistical parameters that characterise the experi- mental data (p, CJ and CJ,,)have been calculated, one can analyse the parent distribution, which describes the experi- mental data in terms of its shape. This can be carried out by performing the x2 test of the distribution.For the Poisson distribution, x2 is defined according to the following equation where NP(xj)for the Poisson distribution is simply the mean. However, when dealing with binned data the method of least squares3' provides a more convenient expression for x2, without significant loss of information. For a very large ensemble, the N events can be sorted into bins of width W, where each bin contains nj events. The total x2, summed over i /-/ time/s Experimental current-time transients of silver growth onto a carbon microdisc electrode from 5.0 mmol dm-3 AgNO, in 0.1 mol dmP3 KNO, aqueous solution: q = -190 mV; t = 0.5 cm s-'; I = 25 pA cm-' -' -180 mV -170 mV In t In P(t) In t In P(t) O.Oo0 -2.145 O.Oo0 -4.074 0.693 -1.136 0.693 -1.966 1.098 -0.614 1.098 -1.366 1.386 -0.384 1.386 -0.823 1.609 -0.248 1.609 -0.623 1.791 -0.187 1.791 -0.462 1.945 -0.147 1.945 -0.360 2.079 -0.116 2.079 -0.283 2.179 -0.092 2.179 -0.191 2.303 -0.067 2.302 -0.141 2.397 -0.054 2.397 -0.108 2.484 -0.03 1 2.484 -0.080 all bins, is given by the following equation: j=1 Jj whereL represents the ideal number of expected events for each bin and is given by fj = NWP(xj;p) (13) The results obtained for the analysis of Table 1 using both eqn.(12) and (13) were x = 1.97, v = 4 and P = 0.75. These values indicate that the determined probability (P)is >0.5 which leads to the conclusion that the Poisson distribution fits reasonably the random sets of experimental data.The fea- sibility of performing such an analysis to gather information on the nature of the statistical distribution of arrival times when dealing with metal nucleation using high overpotentials and small devices, such as substrate surfaces, is demonstrated. The time dependences of the distribution Pj(t) are obtained directly from Nt/NtOtal,where N, is the number of nuclei formed within a period of time, t, for any given ensem- ble of first passage times and Ntota,represents the total number of events for each overpotential. The data obtained for the system under investigation are given in Table 2. Through an iteration method and a fit with the theoretical data for the pure birth model, it appears possible to deter- mine the most appropriate values of the rates of birth, i,for each overpotential, and thus obtain an experimental value of T.This procedure also allows the experimental sizes of the initial clusters to be obtained. The data obtained for the silver studies herein reported are given in Table 3 for the four overpotentials for which the experimental investigation was carried out. In Fig. 9 the experimental values of the dimensionless vari- able T are plotted against I;+ Pj(T).From Fig. 9 we can see that there is excellent agreement between the theoretical plots and experimental data when using the determined A and K values (Table 3). The applicability of such a simple model to describe the nucleation of silver might appear to be a matter of coin- cidence, essentially because A is assumed to be constant for successive steps in the process and there is no dissolution of Table 3 Experimental values for 1,K and ky overpotential/mv critical cluster size, K birth rate /s-' k,mo1 cmW2 s-' ~~~~~~ ~ -200 1 3.64 9.03 -190 3 2.21 5.52 -180 5 0.94 2.36 -170 6 0.71 1.78 1.oo 0.80 0.60 P 0.40 0.20 0.00 0.00 3.00 6.00 9.00 12.00 15.00 T Fig.9 Fit of the experimental data obtained at q = -200, -190, -180 and -170 mV (using the values of I and K + 1 listed in Table 3) to Egtl Pj(T) us. T plots calculated for the pure birth model. (a)+) as for Fig. 1. the centres formed. Nevertheless, the most likely effect of the increase in the surface energy of the subcritical nuclei with cluster size will be a decrease of the I values rather than an increase of the dissolution rates, p, for such high over-potentials. At these potentials the electrical work term, neq, is Imucn larger man any conceivame sunace-energy term, so that one would expect the condition 2 % p to be satisfied.Therefore, the model of the pure birth will apply as a limiting case at high overpotentials. As for the data presented in Table 3, it is clear that there is a relationship between the applied overpotential and the kinetic parameters. In the table the rate constants for silver deposition onto unit area of surface (1.0 cm’) derived from the rates of birth are presented. The molar volume of Ag is 10.28 cm3 giving a volume per atom of 1.71 x cm3 (considering N = 6.023 x atoms mol- ’).Taking into consideration the face-centred cubic (fcc) structure of silver, then the area of any face of the cube (assuming that any edge has a length of 2.58 x lo-* cm) is 6.64 x cm2. This can be considered as the area onto which deposition of silver takes place giving the rates 5.42 x 1015, 3.31 x lo1’, 1.41 x lo’’ and 1.07 x lOI5 atoms cm-’. Division by N gives the rates of deposition in mol cmP2s-l listed in Table 3. These are much lower than the rates one would calculate from literature values of k; appro-priate to the experimental conditions (5.0 mmol dmP3 Ag+) and overpotentials (the literature values are 4.34 x 3.72 x 3.19 x lop6 and 2.74 x mol cm-2 sP1).lJ2 Needless to say, the rates of deposition to form the nuclei must be much smaller than the rates of deposition on the growth centres, otherwise the phenomenon of nucleation would not be observed.On the other hand, 2 changes very rapidly with potential. This variation is faster than can be accounted for by any simple electrochemical rate law,’ 1,23,25926 even faster than which might be expected to hold for a rate-determining trans- formation of an adatom of silver. However, such a mecha- nism could explain the lack of dependence of the rates of birth on the cluster size, since the rates of these steps could then not depend on the number of lattice-forming sites for sufficiently small clusters. J. CHEM. SOC. FARADAY TRANS., 1994, VOL.90 The simplicity of the results obtained here, namely the applicability of a simple pure birth model and the marked increase of the rates of nucleation with overpotential, may be contrasted with the predictions based on the classical theory of crystal gr~wth.‘~,~~ However, the results attained upon using microelectrodes show a marked change in the sizes of the critical clusters with overpotential, which does not in any way fit the predictions based on the classical theory of crystal growth. With regard to the sizes of the clusters, Table 3 also reveals a progressive decrease in the critical cluster size dimensions with increasing overpotential, as described in the literature for other system^.'^,^^,^^ The sizes of these critical clusters can be interpreted in terms of a face-centred cubic (fcc) struc- ture of silver.21 Therefore, one can expect the deposition of a stable single entity (K = 1) at the origin of a plane, entities of size three and six (K = 3, 6) for the formation of sections of the (111) plane and an entity of size five (K = 5) for an element of the (100) plane of the unit cell.A similar behaviour has been observed for the nucleation of other systems on carbon rnicroele~trodes.~~~~~*~~In the present study, tran- sitions between the various cluster sizes (or types) appear to take place in a very narrow range of potential. This indicates the necessity for further research in this area in order to establish whether such transitions can be detected experimen- tally and whether these transitions are theoretically feasible.However, considering that we are dealing with an elementary system under special experimental conditions, it is reasonable to expect to observe the formation of very small critical clus- ters. Conclusions It has been shown that the use of microelectrodes enables the study of silver nucleation as well as determination of the kinetic parameters controlling the crystal growth. The main reasons why microelectrodes are so useful compared with conventional electrodes for these kind of studies are (i) they restrict nucleation to single centres, and (ii) they allow direct measurement of the initial stages of growth, leading to mea- surement of the arrival time of the first nucleus.The distribu- tion of these arrival times is, in turn, defined by the kinetics of formation of the critical nuclei, i.e. by the kinetics at the molecular level. Silver nucleation under limiting conditions is satisfactorily explained by the pure birth model which, in turn, is a limiting case of the depositiondissolution model. Under these condi- tions the sizes of the nucleus can be explained in terms of the known structure of silver. 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