1982_theoretical and experimental studies of multiple nucleation

8/12/2019 1982_Theoretical and Experimental Studies of Multiple Nucleation

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THEORETICAL AND EXPERIMENTAL STUDIES OFMULTIPLE NUCLEATIONBENJ MIN SCH RIFKER

Depat-tamento de Quimica, Universidad Simdn Bohvar, Apartado 80659, Caracas 1080, Venezuelaand

GRAHAM HILLSUniversity of Strathclyde, McCance Building, 16 Richmond Street, Glasgow Gl lXQ, U.K.

Recei ved 16 November 1982)Abstract-The theory of thepotentiostatic current transient for three-dimensional multiple nucleation withdiffusion controlled growth isdiscussed. Reliable values of nuclear number densities and nucleation rates areobtained from the analysis of the current maximum, and good agreement is obtained with experimental datafor nucleation in several electrochemical systems. The termination of the nucleation process by the expansionof diffusion fields is considered, as well as the deviations from randomness observed in the distribution ofnuclei on the electrode surface.

INTRODUCTIONThe early stages of electrochemical phase transform-ations are usually associated with two- or three-dimensional nucleation processes, the rate of which,and hence the number of nuclei SO formed, beingstrongly dependent on the overpotential. It is thereforeimportant to establish the exact relationship betweenthe overpotential and the kinetics of nucleation, eitherby the direct microscopic observation of the electrodeCl, 21, or by the indirect procedure of relating thecurrent to the area of electroactive surface and thenceto the total number of nucleiC3.41. In many cases ofelectrodeposition reactions, notably in the depositionof metals from molten salts[5, 61 or aqueous sohitions[7,8], the charge-transfer step is found to be fast andthe rates of growth of mature nuclei are well describedin terms of control by mass-transfer of electro-depositing ions to the growing centres.Earlier descriptions of the growth of nuclei underdiffusion control assumed that the transfer of ions toindividual nuclei was by one-dimensional, ie linear,diffusion[4]. However, it was later realized that, be-cause of the small size of the nuclei, their growth wouldbetter be described in terms of localized sphericaldiffusion[5]. The current of growth of a single hemis-pherical nucleus is then

3 2M1 2110I zFx(2Dr),.I = 1:2 (1)Pwhere D is the diffusion coeficient, L he bulk concen-tration and r the motar charge of the electro-depositing species, M and p are the molecular weightand the density of the deposited material, respectively.That this description of the growth of ind i t i idua l nucleiis essentially correct was subsequently verified bycomputer simulation[9] and by monitoring thegrowth current of single nuclei of various metals onmicroelectrodes[S, 10, 1 I].

In experimental studies of electrochemical nu-cleation, however, single nuclei are the exception ratherthan the rule, and consideration of the mutual inter-play of nucleation and growth processes that occurduring multiple nucleation become important. Thuspotentiostatic transients of, say, the three-dimensionalnucleation of mercury on a macroscopic electrode ofvitreous carbon[ 121 take the form shown in Fig. 1. Atthe beginning of each transient there is inevitably acharging current that decays during the process ofnucleation and growth. The succeeding part reflects therise In current as the electroactivc area increases. eitheras each independent nucleus grows in size and/or thenumber of nuclei also increases. During this stage ofthe growth of the deposit, the nuclei develop drffusionzones around themselves and as these zones overlapthe hemispherical mass-transfer gives way to linearmass-transfer to an effectively planar surface. Thecurrent then falls and the transient approaches thatcorresponding to linear diffusion to the total area ofthe electrode SUrFdCe, as indicated by the broken line inFig. 1.

In previous treatments of multlple nuclealion[5, 9,131, the assumption has invariably been made thatnuclei grow indrpen&nrl~~ of each other, and that wellbefore the current maximum the total current could beidentified with the sum of individual currents. ir

where Ii r is the current at an individual nucleus i ofager and Nis the total number of nuclei. Furthermore, ifthe initial nucleation is effectively instantaneous, all thenuclei would be of the same age and grow at the samerate, so that (2) could be simplified to[.5]

879


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88 BENJ MIN SCH RIFKER ND GR EI M HILLS

A1 LFig. 1. Potentiostatlc current transients for the electrodeposition of mercury onto vitreous carbon from0.01 M Hg, (NOX), in aqueous 1 M KNIUJ and at the overpotentlals Indicated in mV. Electrode area= 0.32 cm.

However, detailed experimental studies of multiplenucleation combining microscopic observation of theelectrode surface with current measurements[ 14,151indicate that the rate of growth of N crystallites issignificantly lower than that predicted by (3), ir it is lessthan the product of N and the known rate ofgrowth ofa single isolated crystallite. These results thereforesuggest that the growth of mature nuclei on a poly-nucleated electrode cannot be considered to occurindependently of each other, even well before thecurrent maximum of potentiostatic transients. In ad-dition, the possibility of an effective decrease of thenucleation rate caused by irreversible processes nearthe boundary of a growing nucleus can and must beconsideredC16 For the experiments shown in Fig. 1, aplot of I us t lJ/2 for instantaneous nucleation, Fig. 2a,shows good linearity. However, the currents regress toa positive intercept on the timeaxis and this intercept 1spotential dependent. Further analysis of the earlystages of deposition, eg by means of log I L:S og t plotsas in Fig. 2b, show that there is a transition from anI-c~~ relationship to that of lPtliz, and therefore thata progressive regime is to be detected in the early stagesof the transients. This part of- the transient is de-scribed[5] by the equation

I(t) = 2zFAN,n (~Dc) M ~~____3P 1/t (4)where A is the steady slate nucleation rateconstant persite and N, the number density of active sites. Afterthe period of progressive nucleation, further nu-cleation is arrested and instantaneous behaviourthen follows, with a certain delay, which corresponds

to that period in which nuclei were still being producedat a rate comparable with their growth. This picturenow seems to be a general feature of nucleation in anumber of systems where nuclear growth is mass-transfer controlled, and the question then arises as towhy the nucleation process terminates with a largeproportion of the surface still un-nucleated, i un-affected by the persistent supersaturation. It is ap-parent, then, that the final nuclear number densityresponsible for the instantaneous regime of thecurrent transients does not correspond to the totalnumber density of active sites, but is a consequence ofthe irreversible processes that occur in the nucleargrowth process. That being so, it will be shown that thesaturation nuclear number density, N,, contains infor-mation on both nucleation and growth rates, incontrast to the trivial limit ofthe final number of nucleicontrolled by the exhaustion of active sites. Thedistribution of mature nuclei on the electrode surfacewill also be considered, as recent results indicate thattheir distribution is not random [17, iSI, as would beexpected if nuclear growth and nucleation were mutu-ally independent.

THE CURRENT TRANSIENTThe physical model that accounts for the variousfeatures of the current transient for three-dimensionalnucleation with diffusion controlled growth has been

known for some time[5]. Nevertheless, expressions forboth rising and falling parts of the transients, as well asthe current maximum, have not been obtained until


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Theoretical and experimental studies of multiple nucleation SSl

Fig. 2. (a) Linear dependence between current and tI* for the middle, rising sections of the transients in Fig.1. (b) Log-log plots of the early partsof potentiostatic transientsunder conditions similar to those obtainedin Fig. 1 but at lower overpotentials. The slopes of the straight line portions are l/2 and 3/2.recently. The main difficulty in the mathematicalanalysis is that although nucleation occurs on the planeof the electrode surface the growth of nuclei and thediffusion of depositing species extend into the bulk ofthe electrolyte, thus generating a 24 dimensionalproblem[19], intractable in terms of the Avramitheorem[20]. Up till now two approaches have beenfollowed. In one of them[21], overlap between neigh-bouring centres at a height h from the substrate isconsidered by means of a thin layer of width dh. Thetrue area of the 2-D projection of this layer on theplane of the substrate is then obtained by applicationof the Avrami theorem fot 2-D overlap. The truevolume of overlapping hemispherical centres is thenreconstructed through an integration with respect to hand the current transient follows from application ofFaradays law. The other approach[16], which will befollowed here, considers the equivalent area of planesurface towards which diffuses, by linear diffusion, thesame amount of material that would be transferred,through spherical diffusion, to a hemispherical grow-ing centre. The overlap of diffusion fields of individualnuclei is thus reduced to a 2-D problem, for which theapplication of Avrami theorem is exact. The currenttransient is obtained by considering the linear dif-fusion of electrodepositing species to the true area ofprojected diffusion fields.Thus we can consider a set of hemispherical nucleirandomly distributed on the electrode surface andgrowing under diffusion control, as shown in plan viewin Fig. 3. A hemispherical diffusion zone radiatingfrom each centre grows at a radial velocity such that itsradius, 8, is described as a function of time by the

Fig. 3. Schematic plan view of hemispherical nuclei (D)randomly distributed on the electrode surface. The surround-ing circles represent theirdiffusion zones.current will be given by (3) but as the radii of diffusionzones grow and overlap, replacement of material in theplanes close to the electrode surface is hindered, andthe only source of depositing species is that reachingthe electrode perpendicularly, as in Fig. 4. The cor-responding diffusive flux and growth current can thenbe expressed in terms of semi-infinite linear diffusionto that fraction of the electrode surface area within thecircular perimeter of the growing diffusion zones. Theplanar area of a single diffusion zone is given by

S(t) = nS2 (t) = nkDt. (6)If immediately following t = 0, N centres were instan-taneously nucleated per unit area, then,at a later time t,relation:

s(t) = (kDr), O,, = NnkDt, (7)(5) where 8.. is the fraction of the area covered bvwhere k is a numerical constant determined by the diffusion%nes without taking overlap into account. ifconditions of the experiment. At short times, the the N centres are randomly distributed on the elec-


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882 BENJ MIN SCH RIFKER NDGR H M HILLS

Fig. 4. Schematic representatmn of the growth of diffusionzones and their eventual overlap. The arrows indicate thedirectIons of the diffusional field durmg the growth of the

nuclei.

trode surface, the actual fraction of area covered, 8, canbe related to H,, by the Avrami theorem, ie0 = I -exp(-OU,,), (8)

so that from (7)B = I -exp(-NN?rkDt) (9)

and the radial flux density through the boundaries ofthe diffusion zones will be given by the equivalentplanar diffusive flux to an electrode of fractional area 0.Conservation of mass requires that the amount ofmaterial entering the diffusion zones is equal to theamount being incorporated into the growing nucleiand the current density to the whole electrode surfaceis therefore

I= zFDct)7L1,2t1 2 =s [l ~~exp(~NnkDt)].

At very short times, NnkDr * 1 and1 -exp( - NJrkDt) C= NrkDt. (11)

Thus, in the limit Nt + 0, (10) becomesI Nr - 0 = .FD32C/2 Nkr2 12)

and this current must be identical to that flowing to Nisolated hemispherical nuclei, i as described by (3).Equating these two expressions defines k ask = @lrcM/p). (13)

If, on the other hand, nucleation is progressive, thenN(r) = AN,l and

I c),, = ANnkDtdt = AN,rckDt/2. (14)0It follows that

I= s[l -exp(~AN,~kDt/2)]. (15)and k is again evaluated by taking the limit AiV,t + 0and comparing it with (4). For progressive nucleation,k is then defined by the relation

k = ; (&rcM/~)~. (16)Thus (10) and (15) describe the current transient forinstantaneous and progressive nucleation. In each case,the current passes through a maximum and thenapproaches the limiting diffusion current to a planaretectrode. The current, I,, and the time t,, cor-responding to the maximum can be evaluated[16] byequating the first derivative of the I-t relation to zero.The resulting expressions for these quantities for bothinstantaneous and progressive nucleation are given inTable 1. The product Ikr,,, does not contain thequantities k, k, AN, or N and is therefore a convenientdiagnostic criterion of the particular form of nucle-ation occurring. Alternatively, the transients can bepresented in non-dimensional form by plottingI/li vs t/t, and compared with theory through (20)and (24) for instantaneous and progressive nucleation,respectively, which also appear in Table 1. The theor-etical non-dimensional plots arising from these equa-tions are shown in Fig. 5.

COMPARISON WITH EXPERIMENTA family of potentiostatic transients for the electro-chemical nucleation of lead onto a single crystal zincoxide electrode[22] from a 50 x 10m3M solution ofPb(NO& in aqueous 1 M KNO, is shown in Fig. 6. Areduced variable plot of the data is shown in Fig. 7,from which it is evident that the nucleation of lead onZnO follows closely the response predicted for instan-taneous nucleation, ie following the onset of thepotential step that nucleation is fast and on a relatively

small number of active sites that are exhausted at anearly stage of the process,

Table 1. Expressions resulting from the analysis of the current maxima for instantaneous and progressivenucleationInstantaneous nucleation Progressive nucleation

1.2564tin - NnkD (17) f, = y3k3D)ll (21)I, = 0.6382 zFDc (kN)12 (18) I, = 0.4615 rFD3 4c(kAN,) 4 (22)I&t, = 0.1629 (zFc)~L (19) tl,,t, = 0.2598 (zF#D (23)I2 ?{I eexp[ 12564(r/t,)]}* (20) 1=-=I:, m P- = F ( 1 - exp [ - 2.3367 (t&J2 1) 2 (24)m m


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Theoretical and experimental studies of multiple nucleation 883

1 3 aFig. 5. Non-dimensional lz/l& vs t/r,plots for (a] instantaneous nucleation and fb) progressive nucleation.

IO6 I/A

Fig. 6. Potentiostatic transients for thedepositionofPbon ZnO from 50 x 10e3 MsolutionofPb(NO,), inaqueous 1 M KNO at the potentials indicated, in mV us XP. Electrode area = 0.033 cm*.

The treatment of the potentiostatic transient dis-cussed above requires that the product I$, be in-dependent of the nucleation and growth rates. Thus ata given bulk concentration of electrodepositingspecies K&r,,,should not vary with the overpotential forsufficiently high step potentials, for which the surfaceconcentraton of lead ionsis effectively zero That this isso is shown in Table 2, where the values of that productremain constant, within experimental error, over a

wide range of overpotentials. From the mean value,(64.1+ 3.1) x 10m6 A cmW4, and the known concen-tration of PbzC ions in solution, the diffusion coef-ficient is obtained, by means of (19), as D = (0.42& 0.02) x 10-5cm2s-. This value is in good agree-ment with that obtained from an I us t-l piot of thedecaying portion of high overpotential transients bymeans of the Cottrell equation, ie D =0.46x 10-5cm~s-i. Once it is established that the nu-


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884 BENJ MIN SCH RIFKER ND GRAHAMHILLS

,1 z 3 4Fig. 7. Non-dimensional I/I~ CT /t,plot ofthe data in Fig. 6 for 490 (o), SO0 (o), 510 (A),and 520 mV (A).The continuous lines correspond to (20) and (24) for instantaneous and progressive nucleation, respectively.

Table 2. Analysis of the current maxima for deposition of Pb on 200

I@VI

485 57490 62495 67500 72505 77510 82515 87520 92525 97

IOJI,(Acm-*)--

9.2?10.311.712.914.516.117.920.021.2

lWN(cm )

0.75 64.4 84.40.6 1 64.7 1050.50 68.4 1350.40 66.6 1640.31 65.2 2070.22 57.0 2560.20 64.1 3160.16 640 3950.14 62.9 443cleation is instantaneous, the nuclear number densiticsat different overpotentials can be calculated from thecurrent maxima by means of (IS), and the values thusobtained also appear in Table 2.Three-dimensional nucleation with diffusion con-trolled growth, a common phenomenon in the elec-trodeposition of metals, also obtains in some processesofanodic film formation, especially when the deposit isa good electronic conductor. Copper sulphide is one ofthese1231 and thus potentiostatic transients for itselectrodeposition[24] from a 10 Y 10m3M solution ofNaZS in aqueous 1 M NaHCO, take the form shownin Fig. 8. The most significant feature of them is thecurrent maxima followed by decaying portions whichin all cases converge to the limiting current cor-responding to linear mass-transfer of sulphidc ions tothe copper substrate. A large decaying current is alsoobserved at the beginning of each transient, due todouble layer charging and possibly to adsorp-tion-desorption processes[25], whtch obscure thenuclear growth current at short times. Nevertheless, inFig. 9, non-dimensional plots of some of the transients

in Fig. 8 indicated that nucleation of Cu,S on Cu isprogressive. A more detailed description of the phaseformation processes attending the electrocrystallizationof copper sulphide on copper will be presentedetsewhere[24].As a final evidence of the convenience of non-dimensional plots of 12/1i vs c/t,,, for the diagnosticevaluation of potentiostatic transients for 3-D nu-cleatlon with diffusion controlled growth, Fig. 10shows data obtained from several electrochemicalsystems. It is verified that, in all cases, one or other ofthe two extreme forms of nucleation considered herepredominate.

S TUR TION NUCLE R NUMBERDENSITIES ND THE SP TI L

DISTRIBUTION OF NUCLEI

It has been observed that an exclusion zone forfurther nucleation always develops around an alreadynucleated centre. The experimental study of this effect,


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Fig. 8. Potentiostatic current transients for the electrodeposition of Cu,S on Cu from a 0.01 M solution ofNa,S in aqueous 1 M NaHUJ3 and at the overpotentials indicated, in mV. Electrode area E 0.26 cm.

14

O.

I 1.0 2.0 3.0Fig. 9. Non-dimensional 12/I$,vs t/r,plot ofthe datainFig. 8 for67 (V)and 70 mV (0). Thecontinuous linescorrespond to instantaneous (upper curve) and progressive (lower curve) nucleation.

within electrochemical systems, was initiated by will remain and will reduce the probability of nu-Markov, Boynov and Toschev[28], who concluded cleation in the vicinity of each nucleus, as in thethat it was due to a local deformation of the electric analogous case of surface diffusion in deposition fromfield around the growing nuclei. The study was then vapours[30, 3 11. Nucleation will therefore be confinedextended to more dilute solutions[29] from which to those active centres that have not been includedsimilar conclusions were drawn. Whilst the defor- within an exclusion zone. On this basis, Markov andmation of the electric field may be a factor, it will be Kashchiev[32] proposed a formalism for the deri-considerably reduced by the presence of a supporting vation of the dependence of the number of nuclei onelectrolyte. Even so the concentration depletion time and more recently Markov[33] obtained a sol-around centres growing under mass-transfer control ution of the problem in terms of the rate of nucleation,


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886 BENJ MIN SCH RIFKER AND G R H M HILLS

Fig. 10. A comparison of the theoretical non-dimensional plots for instantaneous and progressivenucleation with experimental data of the deposition of Hg on C (Fig. I) (o), Ph on C[lOJ (0). Ag on C[27] (A),and Cu,O on Cu[26] (V).

AN, 9 and a power law for the rate of growth of theexclusion zone, which was experimentally determinedin the absence of supporting electrolyte.A closed form solution for the saturation nuclearnumber density can be obtained from the treatmentleading to the description of the current transient forprogressive nucleation, as new nuclei will only developon that fraction of the electrode surface uncovered bydiffusion zones.Because of the random nature of the nucleationprocess, the probability that any randomly selectednucleation site had not been crossed by the perimeterof a diffusion zone is, at any time, (1 - 3). From (8) and(14), that probability can be expressed as

The nuclear number densities at time t can be found byintegratmg the instantaneous probabilities, (23, up totime t, ie

= AN, AN, nkDr22 dr. (26)The saturation nuclear number density, NE, is found bysetting L + cc, such thatN,= lim N(t) = AN,I-e di.(27)

It may be noted that 1, exp( - a2r) dt = (1/2a) r (l/2)= (1/2a)rr;2, for a = (AN,_lckD /2) > 0, where r isthe Gamma function. Therefore the expression

N, = (ANJ2kD): (28)gives the nuclear number density observed at longtimes, ie after the entire surface had been covered bynucleation exclusion zones. The effective rate of vari-ation of the nuclear number density and its integral,together with the current transient, are shown in Fig.11 in non-dimensional form. At a time correspondingto the current maximum, the effective rate of variationof the number density of nuclei is only 10 of itsoriginal value at t = 0, and the nuclear number densityhas reached 95 /,, of its saturation value, it follows thatat that point the nucleation process has been virtuallyarrested by the nucleation exclusion zones around eachgrowing nucleus.The inhibition of the process of nucleation withinthe exclusion zones introduces a correlation in thespatial distribution of nuclei, since it becomes increas-ingly improbable to observe a nucleus developing nearan already mature nucleus, even though an active sitefor nucleation may exist within its exclusion zone.Thus the nearest neighbour distribution of nuclei, oncethe saturation nuclear number density is reached,should differ from the distribution expected for thedirect exhaustion of randomly distributed active siteson the surface.If mature nuclei were randomly distributed on thesurface, the distribution function of the distancesbetween nearest neighbours would be[ 17, 341

p(r) dr = 2xN,r exp (- xN,r) dr, (2where N, is the number density of nuclei on the surfaceand p(r)dr denotes the probability that the nearestneighbour to a nucleus occurs between r and r + dr. Inorder to compare experimentally determined nearestneighbour distributions with the theoretical functions


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Theoretical and experimental studies of multiple nucleation a87

\ Nbt,N,

Fig. 11. Non-dim&sional representation of the rate of variation of the nuclear number density,[dN/dWt,)jlAN,. the nuclear number density, [N(t/t,)]/N,, and the current transient, i*/I , as afunction of normalized time, t/t,.on the basis of (29), it is necessary to determine theprobability p (r)A r of observing the nearest neighbourwithin a finite interval (r, r + Ar). Thus

r2P(r)Ar = J 2xN,rexp(-aN,r*) dr,rl

Ar== r2-rl W Iso that

p(r)Ar = exp(-IcN&)-exp(--nN,r:). (31)Figure 12 shows a comparison of theoretical his-tograms according to (31) with two experimentallymeasured nearest neighbour distributions[ 171 for the

nucleation of silver on glassy carbon from a 6 Msolution of AgNOJ. It is seen from the figure that thenucleation predicted at the smallest distances by (31) isnot observed in the experimental histograms due toinhibition of nucleation by the exclusion zones aroundeach growing nucleus.In order to study further the effect of the diffusionfields around each nucleus on the spatial distributionof nuclei, we carried out a digital simulation of theprocess. A square was defined as the simulation area,into which nuclei were allowed to develop at randomlyselected locations, using a random number generatorto define the pairs of co-ordinates corresponding totheir positions. The time step of the simulation waschosen in such a way as to introduce one nucleus per

0.L

0.2

Ptri a r a

5 1 15 5 1

Fig. 12. Nearest neighbour distribution of nuclei. Continuous histograms: (a) deposition of silver at 0.13 V,N, = 477; (b) deposition of silver at 0.14 V, N, = 771[17]. Broken histograms: distributions according to(31).

0.L

0.2

r-l


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888 BENJAMIN SCHARIFKER AND GRAHAM HILLStime step. Thus the steady state nucleation rate, AN,,was equal to l/t,, where t,is the length of the time step.Around each growing nucleus a nucleation exclusionzone was allowed to develop, with a radius given by (5),ie by 6, = (kDr)/, where k is the numerical constantdefined by (16) and 5 is the age of the nucleus. Theradius of each exclusion zone was updated at the end ofthe time steps. lf, during a particular time step, therandomly selected pair of coordinates was locatedinside the exclusion zone of an already growingnucleus, the pair was discarded and a new time step wasbegun. In this way nucleation inside an exclusion zonewas avoided and, as a consequence, the effective rate ofestablishment of new nuclei was continuously decreas-ing as the simulation progressed. Eventually, the entirearea of the simulation would be covered by exclusionzones and the number of nuclei would thereafterremain constant. The simulation was therefore con-tinued until the probability of the total area beingcovered by exclusion zones was better than 0.99, ieuntil no new nuclei were established after 100 con-secutive time steps. At this point the nearest neighbourdistribution was determined by calculating the dis-tance separating each nucleus from any other toestablish which was its nearest neighbour, and thedistribution was then displayed as a histogram forcomparison with experiment. Figure 13 shows theresults of simulations related to two experimentallymeasured nearest neighbour distributions[17]. Herethe nucleation rate, AN,. was taken as 7.7x 103 s-l cm- for nucleation at u = 0.13 V and 20x lo - cm- at q = 0.14 V. These. values can bededuced from the work of Milchev et al.[17] eitherfrom their N-t curves or from the N, values through(28) above. The value of the diffusion coefficient ofAg + in solution used in the simulations was D = 1x 10-5cmzs- and that of k, 1.66, was calculatedfrom the concentration of silver ions in solution andthe known values of M and p for silver.A comparison of Figs 12 and 13 shows that thesimulated histograms reproduce better the experimen-

OL

2

a

tal results than the theoretical histograms deducedfrom (31), especially at the smaller distances. Also, theprobability maximum is displaced to somewhat largerdistances as compared to the predictions of (31),reflecting the inhibitory effect of the exclusion zones. Itis also interesting to note that, although the numberdensity of active sites for nucleation in the simulationsis, in principle, infinite, the saturation values of thenuclear number densities obtained from them are closeto the experimentally oljservcd values, confirming thatexclusion zones are of major importance in the arrestof the process of nucleation under these conditions ofmass-transfer controlled growth of nuclei.In spite of the success of the simulations in re-producing qualitatively the nearest neighbour distri-bution of nuclei, there are still important quantitativedifferences between them. One possible reason ofdiscrepancy may be the absence of supporting elec-trolyte in the experiments reported by Milchev etnl.[17]. It may be thought that its presence is irrelevantat the high concentration of electrodepositing species(6 M) used, but it is a well known fact that mass-transfer is enhanced in the absence of a supportingelectrolyte, due to the migration of ions in the electricfield. Also, the large concentration gradients generatedin concentrated solutions promote the onset of naturalconvection, thus increasing even further the rate ofmass-transfer. These two eftects increase the rate ofgrowth of the exclusion zones, shifting the experimen-tal nearest neighbour distributions to longer distances.It is therefore convenient, in order to minimize theeffects of migration and convection, to determine thenearest neighbour distributions from dilute solutionsof electrodepositing species and in the presence of anexcess supporting electrolyte. This work is now inprogress and its results will be. presented elsewhe35].One further aspect, relating the simulations to thetheory represented by (+(28), deserves comment.The nuclear number density as a function of timecan be obtained from (26) which, after integration.becomes[ 361

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Fig. 13. Nearest neighbour distribution of nuclei. Continuous histograms: experimental results as inprevious figure. Broken histograms: results of simulations which produced N, values of (a) N, = 482 and (b)N, = 800 (for details see text).


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Fig. 14. Nuclear number density as a function of time. Continuous line according to (32). (o), (V): resultsfrom simulations carried out under the same conditions as in Fig. 13.

@ 1z at)N t) = N,erf- 2where erf is the error function and a= (AN, nkDj2) .I2 A non-dimensional plot of (32) isshown in Fig. 14, together with simulated results forthe evolution of the nuclear number density with timeunder different conditions. The excellent agreementbetween the simulated results and the theoretical curvethus verifies that the simulation procedure and thetheory are self-consistent.

Acknowledgement--One of us (B.S.) thanks the Conse~oNational de Investigaciones Cientificas y Tecnol6gicas(CONICIT) of Venezuela for a reasearch grant during whichpart of this work was carried out.

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(1960).4. D. J. Astley, J. A. Harrison and H. R. Thirsk, Trans.Fnradny Sot. 64, 172 (1968).5. G. J. Hills, D. J. Schiffrin and J. Thompson, Electrorhim.Acta 19, 657 (1974).6. F. Lantelme and J. Chevalet. J. electroanal. Chem. 121.311 (1981).7. F. Palmisano. E. Desimoni. L. Sabbatini and G. Torsr. J.appl. Elecrrochem. 9, 517 (1979).8. B. R. Scharifierand G. J. Hills, J. elecrroana/. Chem. 130,

81 (1981).9. C. A. Gunawardena, G. J. Hills and I. Montenegro,Electrochim. Aeta 23, 693 (1978).10. G. J. Hills, I. Montenegro and B. R. Scharifier, J. appl.Elrrtrochrm. 10, 807 (1980).

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1982_theoretical and experimental studies of multiple nucleation

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