Study of the initial stages of silver electrocrystallisation from silver thiosulphate complexes

JOUI~AI,Of

N~ ELSEVIER

7

lounlal of Electn~analytical Chelni.~lry 4:~3 ( I tY)7i 141 - 151

Study of the initial stages of silver electrocrystallisation from silver thiosulphate complexes Part I: Modelling of the silver nuclei formation during the induction period W. Simons

, D. OOmllS. en, A . H t i b i n

t"rl~e thlh t,r<~iti,it llrill~iq, th'/iorlnletll of Metallurgy !;'h,,cm.'hemi,~trv mid Matcrinl,~
Abstract

Chronoampemmeiry was used to investigate the initial non-slaiionary stages during the elecm)crystallisation of ~ilver thiosulphate complexes on a polycrystalline silver rotating disc electrode. At the beginning o1' the experiment a large current i~lik was always observed, followed by a current decrease after several seconds. A new electrochcnlical reaclion model was i,troduced..,,uggesiiiig the fiwination o1' 2D silver clusters as a reaction step which precedes the actual moment of 3D nucleation. By doing so, the nucleation phenomena observed during the so called induclion period do ilot have Io he considered lls being a slochastic proce~ c,,m~ulning no current, its is otten done in the literature. The short lime transients are siiliulated using a niunerical conlpullilional method, ~olving the biisic nla,~siranspoll eqnaliOllS witl~ the bOulldary conditions as il funclion ill" the newly proposed nlodel. Salisl'ying fitting l'esulls tire otllained, lhu~ allowing tllie Io d0lliici: vlihle~ for the diflTrenl electrochemical parameters involved, © 1997 Elsevier Science S.A. Keyll'ord,~: Silvo' Ihio~ulldiiile ¢oniplcxes: l{Icclrocry~tiilli.~litiiul; Induction lleriod; (,hrllnoanil~erinii¢lry

1. Introduction

i.I. Reduction o.f silt,er lhiosulphate c~mlph~ws The reduction of silver Ihiosulphate complexes has important applications in the photographic industry, especially in silver halide systems where a silver image is formed during the development step [I]. An amelioration of the quality of the silver image obtained has been sought for many years through a pmlbund electrochemical study of the basic reduction reaction. In spite of this, a quantitative evaluation of the electrocrystallisalion process which is involved in the formation of a silver deposit, is still incomplete. Vandeputte [2-6] investigated the reduction reaction through the combined use of different elecm)chenfical measurement techniques, Alkaline nitrate + thiosulphaie solutions were used, I'or which it was shown [7] thai

" Corresponding author. e-mail:[email protected].

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+ 32-2-6293535:

0022-0728/97/'$17.00 © 1997 Elsevier Science S.A. All rights reserved. PH S 0 0 2 2 - 0 7 2 8 ( 9 7 ) 0 0 2 / t~- 3

Ag(S~O~)~ was the only relevant complex present in the bulk of the electrolyte. Working on a I'ully covered, preo plated polycrystalline silver rotating disc electrode (RDE). the further growth of the silver crystallite.', was in a stationary regime, and the following mechanisin was pro° posed: /~0h,,

AgL~ ~ AgL ,~ L

(la)

AgL + e

(lb)

~ Ag"+ L

And so: AgL~+e

~k, Ag ° + 2 L

(Ic)

k ..... I

For reasons of clarity, the notation L is u~ed for !h¢ S 2 0 ~ . and Agl, and AgL 2 fi)r the compl~:~c~ Ag(S~Oj) and Ag(S~Oj)~ . re,~pectively. The charge~ of° the ionic components are omitted for simplicity. In thi,~ 2-step CE mechanism, k,i~.~, and k,.~ represent the rate

ligandion

142

W. Simons et aL / Journal of Eh'ctr¢mnalytical Chemisto" 433119971 141- t51

c o n s ~ t s of the preceding homogeneous decomplexation and association reactions, and k~" and k 7 the rate constants for the heterogeneous reduction of AgL and the oxidation of Ag". Reaction (l c) is the overall electrochemical reduction reaction. in Ref. [5] the values of k,~,. k,.,,, k." and k'~ are given as a function of the NaNO3 concentration, added as a s n o r t i n g electrolyte, it was tbund [3-5] that for a low content of ;,~xtium nitrate (i.e., 100 tool m-~). besides the electron transfer step (lb) the chemical decomplexafion step (la) could also be detected. At higher NaNO~ concentrations a remarkable increase of the value of the decomplex~tion rate constant and an overall acceleration of the reduction rate were ob~rved. Althouggh it is sure that the preceding chemical step was ~till present [4,5] it could not detected under these conditions. In that case° the global reaction (Ic)conveniently descried the reaction mechao nism, indicating that the AgL~ instead of the AgL complex was active in the ~pplicd l~)tential region. In spite of all the previously obtained results, it is hitherto still not undersr~l at which rate silver is nude° ated on the surl;ace and at which rate the growing crystal° lites cover the initial substrate. Yet. these nonostationary nucleation and growth pro,2esses determine to a large extent the developing ntte. the morphology and final quality of the silver image formed. The quantitative chatacterio ~tion of the non-stationary electrodeposition kinetics of silver will the~Ibre ~ the subject of this study. !,2,

Theories on the o,s~?l ~!t" ch~clrocO,,~latlisalio,~l ; ~,,~o

further 2D growth is very rarely encountered in teal crystal growth, because real crystals are never perfect nor intact. In addition, the presence of adsorbed species on the substrate surface can hinder a 2D growth phenomenon [ 16,17]. This means that further growth is then possible only after unstable 2D clusters are transfomled into stable 3D nuclei. A considerable amount of literature exists about how to describe the nucleation and growth processes mathematically [18-20]. On a typical ! - t transient, separate time domains can be distinguished, in each of them one particular a s s e t is so dominant that an analytical solution becomes possible. Qualitatively. the overall picture is as follows. Immediately after a I~tential step is applied, a current spike occurs due to the charging of the double-layer, preceding a so called induction period where n,:~ relevant reaction is measured. This is assumed to con'espt~nd to the time ~"o during which supercritical 3D nuclei a:~'e being I'onned. Due to their growth, in a fi~llowing ntage an increasing current is observed. Further nucleation then proceeds at a constant rate until it becomes limited, either by the available surface a~a or the available number of nucleation sites, An existing supercritical nucleus will grow at a rate that is mostly controlled by diffusion of the electroactive species being reduced. After some electrolyo sis time, overlap of growing crystals occurs, or in the case of mass transl'~n control, diffusion zones around the crystals ~ g i n to interfere, Finally, the whole surface will be covered with growing cryslallites and the reaction rate reaches its maximum, in the special case of a RDE where a continuous transFa~rl of electroactive species towards the electrode surface is established, further growth proceeds at a conslant tale,

Chron~mperomett3, is often h~und to be one of the most app~priate techniques to cxamtnc ", ' ,, electrocrystallisao tion ~ : t i o n s , This technique allows the control of the driving three of the nucleation princess in a convenient way thn~gh u ~ of the electl~'vde p~ential as a perturbation, It has the ~dditional advantage that the consequential current is a direct n',easu~ of the rate of phase formation ([8-10] and vet's, c i ~ thevein), Furthermore, t ~ measured current tes*pon~ contains characteristic information about the nucleation and c~stal growth phenomena, Relative to a t ~ i g n substtate, metal ions or metal ~ l } l e x e s in ~'~lution are pveferentially ~ u c e d on their own me~al phase [1(~ 1411, Only thn,'~ dimensional nuclei with a critical size will then start to grow on a foreign f ~ n o J o n l y by a~lying a polential lha! is tk~r mo~ ~gativ¢ than the ~uilibfium I~ential, ~ a u s e the prod ~,'¢<~ ~ m a n d s a higher s u ~ t u r a t i o n due to the large surface free enemy ot" the clusters [ It)], When clusters are t'm~'~ed on a like metal substrate, as in ~ case of silver complexes t~xtucc-xi on a polycrystalline silver e~lrod¢, they are expected [15] to be two d i ~ i c m a l and epitaxially oriented to the substrate. Among many others, Budevski [15] however claims that

The reaction stage where an increasing current is observed will be analysed mathematically in part II of this paper [21], giving a quantitative description {+f the nucleation rate for the reduction of silver complexes at a polycrys~lline electrode, This article deals wimhthe time period immediately after a potential perturbation is applied. It is common practice in electrocrystallis~tion experiments to consider that the reduction reaction begins only after an induction period. Even in cases where measurements showed an early electrical current, interpretation of the transients was mostly given for the growth part, thus ignoring eventual electrochemical nucleation phenomena happening during the first few seconds of electrolysis. H~evt-r. Ft~:i,~chm~tnn ~md Thirsk [22] ~.md Gunaw~,.rden~t el ~d. [23]. for example, already obser~,cd that the moment r~ at which a minimum rate was reached, largely exc~ded the charging time for the double-layer. Olher workers [24-27] suggested the existence of discharged species like adatoms or small clusters, being present on the surface before what is usually called 'the actual moment r o for first nucleation'. As stated by Sluyters et al. [25,28], the induction period should indeed not be discarded by simply subtracting it

143

W. Simons et al./ Jouma! of Electroamdytical Chemistr)" 433 t i997~ 141-15 !

from the time axis. On the contrary, it could render valuable information on the nucleation process(es) affecting the second stage of the electrolysis. In recent work done by Li [29,30] and Sanchez Cruz [3 I] on the electrodeposition of gold and zinc respectively, the transients show a remarkable current peak at the beginning of the experiment, decaying for a few seconds and certainly not zero or negligible during that time. The electric charge consumed during this period is always higher than required to deposit a monolayer of metal atoms. Therefore to these workers it was clear that in such cases an 'electrochemical reaction' took place tbr 0 < t < 7'1|.

1,3. Aim c!l" the present work

toLab PGSTATI0-Ecochemie) and perfomled in a double wailed glass cell of 200 mi, thermostated at 25.0 + 0.50(?. The rotating disk electrodes consisted of a 2 mm diameter polycrystalline silver rod (Johnson and Matthey, 99.99%), embedded in an insulating mantle of polyvinylidenefluoride (PVDF) with an outer diameter of 25 mm. The reference electrode was a calomel electrode (Tacussel) with saturated KCI, used with a salt bridge containing a 2000 mol m-.a NaNO 3 solution. As a counter electrode, a platinum grid with a large area was used. For all experiments, a rotation speed of 1000 rpm was used. Before and during the experiment the electrolyte was deexygenated as described elsewhere [2].

3, Results and discussion

The polentiostatic reduction o1' silver thiosulphate corn° plexes on a polycrystalline silver electrode shows an initial transient hehaviour comparable to the one described in [29~31]. This paper therefore fi~cuses on the description of the initial non-stationary electrocrystallisation phenomena for the silver deposition. Contrary to the literature, where formation of successful nuclei is often described as a stochastic process consuming no current, here an electrochemical model is put forward. A possible quantitative determination of its reaction kinetics is done through the simulation of different current transients, monitored durh,;2 the first second of experiment.

Since fiw this study there was no benefit in detecting the chemical decomplexation step by working with a low nitrate content, as discussed in [3-5], the overall ionic strength was made suMciently high (6IX) reel in ~ ). The pH of the solution was adjusted to a value of 12 by adding NaOH, in order to approach pho~ographic development conditions [I]. The equilibrium p,,tential E ° of this solution, measured on a polycrystatline silver electrode at 298 K, was - 2 0 0 5: I mV versus SCE. All measured potem tials are given in millivolts with respect to the saturated calomel electrode.

2. Experimental

The rate of the reduction reaction, i.e., the cathodic current, was monitored for different potential steps. Fig. I shows the long time transients obtained. The measure°

2. !. Composition of the solutions

3. i. E~perimental i-t transients

;1~

A detailed description l'or the preparation of silver thiosulphate solutions has been given elsewilere [2,3,32,33]. In this study, a 25 reel m ~~ Na~S~O~ (p.a. Merck) ~olution was investigated. The concentration of AgNO; (p.a. Agfa-Gevaert) was I molm ~:~, and the ionic strength was brought to 600 reel m 3 by adding 520 reel m N ~ O ~ (p.a. UCB) as a supporting electrolyte.

0

~11[} . . . . . . . . . .

~ .............................

4(ll

~Jl

-. . . . . . . . . . . . . . . . . . . . . . . . . . t . . . . . . . . . . . . . . . . . . . . . . . . . .

I~J)

ItS)

,,. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . -~

U

o15

-25

2.2. Electrode pretreamtent Before each experiment, the electrode surface was mechanically polished 3 times on a rotating table (Struers DPIO), successively with a 6. i and 0.25 /x~ diamond paste (Strucrs). Afterwards, the sarface was rinsed ultrasonically with deionized water.

-35

~55

f

2.3. Experimental set-up I0" I,~,,hl A

For all the chronoamperometric experiments, the single potential steps were applied from equilibrium E ° to different values in the cathodic region. The transients were recorded with a high resolution/stability potentiostat (Au-

Jq~, l, Overview of the eurre|)! Irfl|l~i~J|l~ on a poli~;t|~d j~)lycry~t~JlJ,e ~iiver RDE for 0 djl'Jcrcn! polential~ (a ~ f) E/mV; (a) ,~=~2250 (b) 250. (¢) -275. (d) -3(X). (c) -325. and (f) -350; Iile fir~! c~leerio mental point is at 0.5 s.

144

W. Simons et aL / Jnurmd of Eh,ctroanalytical Chemistry 433 (1997) 14t- 151 tls

0

0.2

0.4

0.6

0.8

0

-20 ,40

-

~

~

i -

4

.....

c d

f

-60 40 -100 ,120

tO~ I , , , . , / a

Fig, 2 Similar Iran.~i~nt~ ~ ill Iq~, I, hut intcrnlptcd m t ,~ I ~; the firsl cxpcrimcmal i~tm i~ at 0,01 ~,

ments were stopped at the moment a maximum current was ~ached. As can be seen. the reaction current rises after a minimum has ~ e n reached. The higher the cathodic over~tentinl applied, the higher is the initial ~ a k value of the current, the faster its decay ~ shifting the minima of the curves to shorter times ~ and the steeper the following increase. The rising ~ r t of the experimental curves is due to the grow~ of silver nuclei, where charge transfer takes place directly at the growth sites and newly formed silver atoms are immediately incorporated in the crystal lattice [15.1934]. The increase of the current, corresponding to an increase of the reaction surface, lasts until a maximum value is reached. At that time the whole sud~tce is covered with growing silver crystallites [21], and no new active growth sites are tbmled. In this case Vandeputtc's mecha° nism is valid [3]. As already mentioned in the introduction. a further eva!uation of this se~a~nd stage will be presented i~ part II of this arlicle, Fig. 2 gives the results of the ex~riments, perforated under the same condititms, but interrupted at I s to be sure that the minima of the curves were not yet reached and that no 3D growth would already have taken place. Notice the e n l a ~ current scale compared to Fig. I, due to the fact that now a high peak is monitored at the beginning o" the experiment, Very good reproducibility was obtained: ~ m ~ s o n of the ~ort time transients on different electrodes ~owed some small differences, but these never exceeded the 10% margin at the end of the experiment (t~ I s). All the measurt~:l curves of Fig, 2 showed an initial current clearly different from ~ero and decreasing rapidly fr~m~ a peak value during the first moments of the experiment, it is clear that the. decay was ttx:~ long to be simply a ~ r i ~ to the charging of the double layer, The measured current mua therefore be related to a Faradaic reaction (see also [29-31]), According to Vandeputte [2], the only possible reaction that can take place in this potential zone

is the reduction of silver thiosulphate complexes, tbnning silver atoms on the surface. When, however, a response to a potential step is calculated using Vandeputte's values of the parameters corresponding to reaction (lc) in a stationary regime [5], a transient as given in Fig. 3 is obtained. It has to be remarked that in Rel\ [5] the values of the rate constants were obtained tbr solutions with a slightly higher sodium nitrate concentration (0.6 M). A comparison was made with the measured transient of Fig. 2. lbr E = - 3 0 0 mV. The simulated current was about 4 times higher than the measured one and decreased in about 0.2 s to a stationing' value, whereas the measured CmTent continuously diminished during die full-time experiment. The rapid current fall Ibr the calculated current was due totally to the Ibrmation of a diffusion layer. The concentnltion of the electroactive species, being consumed at the surlhce, was then lowered to a minimum value. It is clear that the simulation of potential step responses, using Vandeputte's reaction model and rate constant values [5]. cannot give satislhctory agreement with the short time transients measured on a polished silver electrode, even if com?cnsation were made Ior the values of k~ and k.~ ~ due to th,'~r de~ndence on the sodium nitrate concentration, in the tbllowing paragraph, an explanation for this discrepancy is stated, and a new reaction model is proposed. 3.2. Pr~vn~sed reaczhm mo~h,I ./'orthe initial stage n m'h,alion

o./"

Considering the measured transients of this work and the experimental observations made by Li [29.30] and Sanchez Cruz [31]. an electrochemical reaction is needed. It is suPi:msed [24] that the lbrmation of a successful 3D nucleus pr~'eeds from the aggregation of one or more 2D

0 02 0,4 0,6 O~ ........................................................ ~

0~8

I

-4 -6

-10 -12

~-

Fig. 3. Comparison of the measured ( ) short time response (E=-300 mV) with a simulated current ( - I ! - ) . using Vandeputte's values [5] of k I and k_ i conresl~nding to reaction ( I c ) in a stationary regime.

14.5 clusters of molecular dimensions and that it is likely to be a

stochastsc process. A mathematical description of the accumulaL)n process of adatoms during the so called induction period. building up a collection of stable nuclei, has alrecldy been given by Sluyters [25,28]. Nevertheless. no current is accorded to this process since discharged adatoms are involved. With respect to the nucleation of silver deposits on a polycrystalline silver electrode, 2D silver clusters, epituxially oriented toward the electrode surface, are formed shortly after a potential step is applied. Further 2D growth of those clusters is hampered by the polycrystaliine nature of the silver electrode [ 151 and by the presence of adsorbed ti~iosulpi~atc and nitrate aniljns [4]. If the applied overpatcntial is high enough. however, they can be trmformcd iilto 31~nuclei, giving rise to further 38 growth [29-3 I ,35], The formntioii of those nuci6i from single atoms or lkom tht! II

I of

ill1 cl

owtlr site is born,

SCVGISIII cxistin

clusters, but still bofo~~c cluircs I\ ccrtuin mount

al’ energy and takes a ccrtuin time. It con, therefore, be seen as uu ~ddilil~uili rclardution process, thus giving the opportunity

to the reduced silver MOW to ilccunIu!iite :lt the surr‘ilce side ol‘ the double layer region. It is, therefore, assumed that this process can influence the net rate of the reduction reaction. A way to describe it, is to add a reeectionstep to the c~d*rall reaction mechanism proposed by Vandeputte [3]. First, silver iltorns Ag *’ i\re gained on the surface through the reduction of silver con~picxcs (cfr. ICI, which in a second step are transformed into subcritical, critical and finally into supercritical silver nuclei Ago, the growth of which can experimentally be detected. The following eicctrochcmicai reaction mechanism is thus proposed concernin the initial stages of the deposition of siivcr on a poiycrysliliiinc hiiver cicctrodc:

AgL, + e k -;

(2n)

Ago

The following sections are dedicated to the description of the basic equations necessary to calculate the current transients during the induction period, according to the model introduced. The experimental data will be simulated numerically, the validity of the proposed model wiH be checked, and values will be deduced for the different parameters involved.

The quantity of interest, i.e., the net cathodic electrode current at any instant, is given by [%I:

c cicctrodcsurface arca A and the diffusi(~~~ $I*1 of the clcclroactive specks, it is possible to culct&ttc the &ITW~ if the time dependent axial gradient L 2 concentration iIt the cl~~t113d~SUII’WC ci]II IX

d. Therefore, the cxprcssion ~fcscribiil~ the mass towards the KBE surface, where it is cicctrociicniicaiiy uftcr a potcntiai pe~urbati~~~~ has been applied, must be solved mathematically.

tiorks

In accordance with [37], the whole surface is considered to be homogeneously active. As stated before, after a potential step is applied and the double lnycr is charged, the silver complexes arc reduced to silver atoms A progressively giving siivcr clusters. Since these (Ire gcnerally accepted to he very small, and in the cusc of a RDE the surface is uniformly accessible, it is reasonabis to m~pf a planar diffusion mechanism takin s of nucleation [ 12- 14,24-27%. tin e assumptions into account, and 3 as ts due to a sufficient quantity of eiect~oiytc, the one-dimensional convective difng mss transport of the active can be written

as [XQ

The constants k, and k _ . represent the potential dcpcndent rate constants

for the reduction of silver complexes and the oxidation of discharged silver atoms respectively. The rate constant k, is the speed by which stable nuclei arc gained through the consecutive nucleation steps. It should be remarked here that the reaction model holds for the entire electrolysis period. During the first moments of reaction, step (2b) plays a rate determining roic, meaning at the least that the rate constant k, is finite. From the moment successful 3B nuclei arc formed they will start to grow, indicating that at these sites the reduced Ag * atoms are immediately incorporated in the crystal lattice. Consequently, for the reaction occurring at thcsc particular places, k, must be taken as infinite.

For the proposed model, the appropriate initial and boundary conditions are: tsOandO,
(2.3)

I~

W. Simons ez (d./,hmrnal nj Eh'ctrntnudytical Chemistry 433 t 1997) 14 ! - 151

The b o u n d a r y Eq. (2.3) expresses the axial flux of AgL, on the electrode surface as a function of heterogeneous reactions taking place thereon. DA~L: represents the diffusion coeft'icient of the AgL., complex, z the axial distance from the electrode surface and (5 the diffusion layer thickness, being equal to 1.61 ~A~n)/~i.W , .....|/"Z'~t~' under steady state conditions [39]. The term ,Oz" in Eq. (2) ~ f i b e s the convective flow clo.~ to the electrode surface according to C~hran's asymptotic solution for the velocity vector ( O - 0.51 to++/"v - |/" ). Because radial concentration gradients are small, radial convection and diffusion terms can be omitted to a good approximation. Since, in addition, ~aetion (2a) occurs only at the electrode/electrolyte interface, any terms t'or homogeneous reactions consuming or producing AgL~ outside the electrochemical double layer are not present in Eq. (2). Regarding the excess of ligand L. i.e.. c~. ~ CA~~. the concentration of L is assumed to I~ constant o v e r the whole region oi' the dil't'usion layer (<'t ~ <'t'i). T h e r e t b r e the transport of ligand L+ described by a convective difl'ustun exp~ssion analogue to Eq. (2). may be ignored. The difficulty which is encountered in solving the set of Eq, (2) to Eq+ (2.3). is that boundary condition (2.3) contains the unknown time dependent concentration of the intermediate species Ag + on the elecmxle surface. There° tore an analogous expression lbr the mass conservation of Ag" has to be written. The corresponding partial differen+ tial equation (Eq. (3)) contains no diffusion tenn. since Ag" is tbrmed only at the surface side of the electrochem° teal double layer and its concentration in solution is con+ side~d to be zero at all times (see Eq. (3.2)). However, and in contrast with Eq. (2), three terms describing the heterogeneou~ reactions regarding Ag + have to be taken into account: Ot'A~ "

a,

dependent concentration profile of the electroactive species AgL2 and the electrode current derived from it (Eq. (!)). This mathematical problem can be solved numerically using an implicit finite difference formalism, as it has already been proved to be an efficient strategy in the numerical solution of mass flow equations [40-46]. In order to optimize the numerical algorithms for accuracy and stability, the equations can be translbrmed into dimensionless forms. For a RDE, it is convenient to relate the axial distance - to its dimensionless form - through division by the diffusion layer thickness/~ [42,44]. Dimensionless concenmltiOllS are obtained by using the bulk concentratioll of AgL, as a rel~rence. The foih)wing dimensionless variables are obtained: +~ :.~,

( 4. I )

.r~,/+;t),/,~

{ .... iD,~i;p]..i, 2/,

+~+ < ' '

(4.2)

with i~- Agl+, Ag + orL

(4.3)

From now on, the underline symbol is dropped for the sake of simplicity. The set of Eq. (2.x) and Eq. (3.x) can then be cast into the following Ibrms: il('A~t+,

+)c~.t+

=----~

+ ~}( ' ,.,Xgl+ :

:

+ " ' ~

Oz"

at

;)rA~• + +')I

,,

~

k | (',,x~l+ :

(5)

Or,

= ( /+', ) + k'; )(',x.,, ,,

(6 )

'

0 and 0 ~ : ~ ~:

Z~

,',~|.+, .-+ ' +I and c..~ ~ = 0

(

7. I )

| > 0 a n d z~ I CA~,|,. +~ I

(7.2)

t > 0 and : + 0

+

°

(

=

+

' "P, =

' )s

(3)

t ~ 0 and 0 ~ ,: ~ m CA~. + 0

= k'lcA~t,

|+'A~

(7.3)

t > OandO < z ~

( 3. I )

"a~' = 0

(3,2)

In Eq. (6) and Fal. (7.3). the dimensionless rate constants are defined as follows:

t > 0 and 0 < .-. ~

('~, ~ 0

a:

Since k~, k:.. | and k, are heterogeneous rate constants and t ~ concentration of Ag" is expressed as a volumetric ~ n t r a t i o n in reel m-~, t ~ right hand side of Eq, (3), ~ s to be multiplied by a parameter S with the dimensions of m ~, for which a physical interpretation will be given later on, Coupling with the mass conservation law for AgL~ is, in this c a ~ ~ h i e v ~ , via the reaction rate term k~t'ar.| +, Solving ~ , ({3), thus implies that the surface concentration of ,~L~ has to ~ known at all times, The integration of the two partial differential Eq. (2) Eq, (3), together with the suitable initial and boundary conditions, yields the quantities of interest, being the time

( 7.4 )

( 8 I)

k'~ = l<1 t:),~t++,12 --1/.,+ 7 ", ~ S

r

I

~.~ =

=k

+

|

o, I< 2 -'.''" • t'~o( ' '~ t ) " ~

?,tqdk

|-'A~t.,--

(8.2)

(8.3)

~, z~.,~(..'t2 "-.+'' s

k~ = k t D+x~L~I2 +|

S

(8.4) ('i+c~L.,,,,,

(8.5)

3.5. Equathm d i s c r e t i z a t i o n

A solution for the mathematical problem is established using the implicit Crank-Nicolson finite difference method,

W. Simonx el aL / Jourmd ql'Electnmnah'tical Chemistry 433 (1997) 141-151

which has been shown to be stable for all ratios of the time increment At to the space increment A z [41]. In order to obtain maximal accuracy for the calculation of the AgL 2 concentration in the vicinity of the electrode surface, it is. therelbre, satisfactory to take the increments sufficiently small, without having to be concerned about the stability of the numerical algorithm. The grid used to discretize the computational (z, t)plane consists of NT time increments and N Z - I equally spaced axial increments (with Az equal to I / ( N Z - I)). The index i for the axial coordinate varies between 0 and N Z - I, with the point indicated by i = 0 being at the distance - I / 2 A z from the electrode surface and the point i = N Z - I at + I / 2 A z from z = I. The time level is indicated by index n. The discretized amdogs of the dimensionless partial dil'fet~ntial Eq. (5) and Eq. (6) can then be rearranged It) give the following set ol' ;algebraic equations:

( -o.5 + o.25i i = o.5):( a:)'),.:,,.,, I,.~ (t +

At

I

'A~t,

+ ( - 0.5 - 0.25( i - t).5 ) 2( A :J'3'J cA~'-. ,, ,.,,., -- ( 1).5 - 0.25( i - 0.5 ) 2( A" j':")ca,,, . ' '" ~

.I.II

At

t a~L"

+ ( 0 . 5 + 0.25( i - O.5 ) " ( A ' ) ' ) t "'Ag,.'"': I b r i = I to N Z - 2 a n d n = .,.,

I =

! +

'a~'

k"' I At

+

--7. ! -

-

'2

k".At ) .

."

+k';,xt(

3.6. Practical computations Using the presented space and equation discredzation, single step chronoamperometric transients were conDputed with a Borland ~ C + + Compiler installed on a IBM compatible Pentium personal computer. Convergence of Ihe calculation procedure was checked by varying the number of time levels and axial grid points, thus changing the ratios of the lime to axial incl~2ments. This did not lead to stability problems of any kind, us was to be expected t'or the Crank-Nicoison method. For one specific initial lime increment At., chosen sufficiently small for giving the approprime accuracy. shortening of the overall calculation time could be obtained by raising the time increment value with increasing time steps. By doing so. normal calculations, with the number of interval points being equal to NZ ~ 40 and lot 2000 time levels, require about 2 real time seconds.

3. Z Comparison of the cwerimentally measured aml ,'mn~

I to NT

t A,

The second part of the calculation procedure consists in solving the linear algebraic Eq. (!0) for t""+ a~.. Its value is needed to calculate Eq. (9) in the next time step t(n + 2) in the specific case for i = I. The value of t'"A~' is known from calculations in the previous time level n. The surf'ace concentration (CAgl,2): : 0 can be obtained by using a linear combination of the first order approximations of the Taylor series for three points t :Agl. ' i (i = l to 3) of the concentration profile. These values have been determined in the firs| part of the calculation procedure.

I m W d t'lll'retl/

2

"2

"At~t.:

) :~. ,,

147

(10)

The first part of the calculation procedure consists in calculating the expressions depicted by Eq. (9) (Ibr i= I to N Z - 2). each of which contains three unknown values of .,i.n ~ I at time level n + I. as a function of the three "At~L~ known values of tAgl, .i.,, : at the previous time level n. For the specific cases of i = I and i = N Z - 2. both the left and right hand sides of Eq. (9) contain two terms with t":°At~L:(no relevant physical point) and two with t'~X~Lu z i These terms can easily be substituted by an expression containing the values of c AgL ~= ~2 and I respectively when taking the finite difference equivalents of the boundary conditions (7.3) and (7.2) into account. Thus, for i = I to i = N Z - 2 and Ibr each time step t(n + I), the original set of" algebraic equations exhibit a tridiagonai coefficient matrix which can readily be solved by the Thomas algorithm [41], gaining the time dependent concentration profile of AgL 2.

ll'tlllSietllS

As stated befi)re, the dectr()chemical model i)resented covers at least the first second of the electrodeposition reaction, it is nevertheless stated that the single reaction step (2a), being the reduction of silver thiosulphate c o m plexes gaining atomic silver, lusts from the moment the double-layer is charged. Each chronoamperometric measurement demands lbr a constant potential perturbation, the electrochemical rate constants k~ and k ~:

k, = k ' , ' e x p [ - a F E / R r ]

(JJ)

and k , = k,!, exp[(! - a ) F E / R T ]

(12)

are constant during one experiment. To fit the transients with the proposed model, it is therefore reasonable to obtain relevant values for the rate constants of reaction (2a). by fitting the measured maximum currents. Then. the process is known as being the direct incorporation of ~ilver atoms at growing hemispherical crystallites which fully cover the electrode surface. Since the composition of the silver thiosulphate solutions investigated as well as the pre-treatment of the silver electrode surfaces differed from

IV. Si,um++ et al./.hmrmd +!]Eh'ctroamtfi'ti+'td Chemi.stO" 433 ¢1997~ 141-151

1411

those used by Vandeputte [3-6], new calculations gaining appropriate values for k~ and k ~ had to be performed. Therefore, the developed numerical routine was used but with the rate of reaction (2b) taken infinitely fast. The electrochemical mechanism was thus regarded as a simple reduction/oxidation reaction. The values ot" the potential independent constants k? and k°t ( ~ e Eq. (I I) and Eq. (12)), together with the cathodic charge transt'er coefficient a and the diffusion coefficient Da++t.,' were chosen in such a way that for different overlx~tentiais the calculated steady state currents coincide with the maximum values of the measured long time transients (of, Fig. I). A quantitative criterion for the l+,,~t fit is based on the minimisation of the sum of residuals (SR) of the measured and calculated data. A comparison of measured and calculated steady state cath~lic cmxents, together with the obtained parameters, is depicted in Table !, Considering Table I, the chosen electrochemical parameters give good agreement of the maximum c u r r e n t s tbr all overpotentials+ At the worst, i.e. in the particular case of E ~ + 250 mV. the computed current value differs 4.5+J`+ l~+m the ex~rimentally determined value. The sum of the obtained residuals equals 2.77 x I0 +' A. Regarding the values of the diftk'+rent parameters, an important remark has to be made. Under comparable experimental conditions, but with a slightly higher NaNO~ concentration (600 sol m +~), the tbllowing values wer~ recently obtained by Vandepatte [5]: ++++0.81, k ~+'" ~ 44.5 x 10 -s m s ~` k_+i" ~ 53.5 × IO s m and D,x~,, +~ 6,0 × 10 TM m--'t' s ~. The constants k ~"'~ and k+ ~ denote the potential independent heterogeneous rate conslmUs Ik+r the reduction and oxidation of AgL and Ag", respectively, but they can be comlmred to k~~ and k"~ because under the given conditions the preceding chemical reaction step is found to be much t'aster, thus having practically m+ influence on 1he following electr~rchemical step, As ctm be ~ n , unexlX,Ct~ la~e divel"gen+,~s between the values o1" the ~spe+ctive rate constants arise, A slight difference can at t'i~t be a , ~ r i l ~ to the higher NaNO~ concentration, Table I Ox~erx,+wot"t~ valuesI't~"<+,/+?,/+?,, l},x+v'and I,,,~,o~ained d~amgh tke sit~+ml~i~l el" the t~axitl'a~ of the 11~easuP,..,deu~r~a! I~utxienln I,~.,,. lbr dil't+r~l al+l~lied t~+lenlials b~',(SR + the sum of II~ n~siduats+ b+'/mV

IO +' Im,,+,/

I() ++l++k /

~.s, S ( ' ~

A

A

= 2t+,8

~ 25,P,

=

2.~)

=

325

- 49.4

-`19,9

=

.~.~I

~

-

53,3

52.4

| O +'

03

m.s- t 5.4

il~t+l "

2.8

2 s- Q

lift" .x

3.6

A 2.77

SR/

since generally an overall catalytic effect is detected when raising the sodium nitrate content [5]. However, the changes noticed were too large to be due simply to a difference in electrolyte composition. The main reason tbr the observed distinction is that Vandeputte [5] studied the reduction reaction oll a silver electrode pre-plated Ibr 8 sin, for which the surface morphology was substantially changed compared to the moment at the maximum of the current transient. It is therefore likely that under those particular circunlstances, the apparent pm-ameters ~, k~*'", k,+,+~i~ and l ~ t " are different from those determined through chronoaml~rometric experiments, for which a polished electn×le was used as the basic substrate, In the fitting el +the short time transients of Fig. 2, the obtained parameters +~. k<0~, k
W. Simons et ai. / Journal c( Electroamdytit'al Chemisto" 433 ( i 997) 14 i - 151

A

tls O 0

0.2

0.4

tag*

0.6

0.8

l

~

7"-~-

. ~ . , , ~ o - - m¢ . 7 - ~ I I ~ I - - , ~.'.- -c-

:ir.~- ~

~'---'~

J

I ....

"

J ~

reel I!i .3

[

120 -

', ~(a)

i

( )

-20 i~. ~ , . r

149

f

_c_

-,ol

80

-100

4~ 0~e

-120

20

-80

60 t1

a-

b

d

~;

-140 0

1()h L,.~ I A

.......

0

0.5

I

1.5

2

~ .......................

2.5

3

tls

B 0 1),2 0A 0,6 0.8 0 ~........................ ~................................................. *......................... t,

-40

I

Fig, S, Variation of vat,, will1 t for diflL.ren! poleittials E {nIV/SCE): (a) ....... ...5, {h) ~ -~ 250, (c) ~ - 275, (d) ~ ,~, 3()1), (c) ......325, i|lld (1'):~ --,~3511,

higher tile applied ovcrpotential, the later this stationary state is reached. This phenomenon however does not inlply that the incorporation of Ag* atoms into clusters is retarded. Indeed, the net reaction rate of the second step, being k2ca~. (in reel 111~2 s t), is higher for higher overpotentials, as can be seen from Fig. 6. The final values for the lowest and highest overpotential, at 0.9 and 2.9 s, respectively, equal 4.3 x IO (' and 2.2 x l ( ) s reel i n -~

i¢i,f ~ - ' n ' ~ . . . . . .

-60 -80 - I00

ol20 i l

s

-140 !()~ I,,,,~ I A Fig. 4. C'tmlparison of the oxlK'ritllental alld traictlh|ted l - I tl'allsielit.',i ftir 1~ different potentials E; k, and S arc taken its in Table 2. Full lines lit ~ f) tlellOte tile IllellSlll°ed ¢tirvos; small Stlliarcs, circles and trim)glen tire tile simulated values. A: {it) al|d ( ~ ) I'or L' ~ - 225 IIIV; {C) alld {O) Ior E ~ - 275 raM; {e) alld ( & ) f o r E ~ - 325 tlIV. B: (b) alld ( H ) Ibr E ~ ~ 251) mV; (d) and ( O ) Ior E ~ ~ 31)1) mV; (f) and ( A ) Ior E ~ ~ 351) inV.

" I

in spite of a reduction of k 2. the second process (2b) is thus accelen|ted for increasing overpotentials mainly through art increase in tile concentratiotl of Ag ~. Regarding the variation o1' the par~tnleler S (see T:fffle 2), its physical i110aning could be considered its tile aVel'age of the surface It) vohin|e ratio of tile deposited silver cluster layer o11 the surface. An analogous problem is I0*' k2cAu. I tool m 2 ,,i"

served for increasing overpotentials. The variation of the Ca~. with time Ibr different potentials is depicted in Fig. 5. Calculations were made until a constant concentration was attained, varying from 1.3 mol m ~ lbr E = - 2 2 5 mV to 110.4 tool m ~~ for E = ~ 3 5 0 inV. At that time, the production and consumption of silver atoms ate in equilibrium and the left hand term of Eq. (3) becomes zero. The Table 2 Variation of the lifted parameters S and k2 and SR with tile atpplied potential E E / m V vs. SCE

-

-

225 250 275 300 325 350

l()7 k , / m

32.0 17.5 I0.0 6.0 3.5 2.0

s~i

l()-c,S/m

1.3 1.7 2.7 4.3 6.6 9.5

i

!

2O

15 i I

d

I0 t~ f

t,

l()c' S R / A

15.5 17.2 13.6 5.6 6.6 13.2

0

.

0

.

.

0,5

.

.

.

I

.

.

1,5

.

.

2

.

'

2,5

3

I/r,

Fig. 6. Vut'iillion of tile tlgl i'elt¢thlll r~tle k~c'A~, with ! for different IX)tentiab; F (nIV/SCE): (:1)~ ~-225, ( b ) ~ - 250. ( ¢ ) ~ - 2 7 5 . ( d ) ~ - 300. {el = - 325. and (f) = - 350.

1~

;K Simon.~ ct aL / Journal vf Elertroanalytical Chentisto' 433 (1997) 141-151

encountered in the numerical study of the characterisation of fixation processes in photographic emulsions [42,44], where the rate terms in the general convective diffusion equations contain a similar parameter known as the specific surface area of the dissolving silver halide crystals. For the electrocrystallisation experiments of this work, an incense of the driving force provides for the generation of more clusters on the naked electrode surface, seen through a rise of the local Ag * concentration. The more small cluste~ are formed, randomly spread over the electrode surface, the more area they cover without leading to a proportional increa~ of the volume of the deposited cluster layer. This means indeed that the process acts as if ~t:,~reaction surface apparently increases for higher applied overpotentials.

5. List of symbols surface area of the silver electrode = !.26 × 10 -5

ira:) ('

volumetric concentration (mol m ~)

DAgL 2 diffusion coefficient of the silver thiosulphate

E Eo F /¢alh

kl 4. Conclusions

The initial phase of the reduction of silver thiosulphate complexes on a polycrystalline silver RDE, exhibiting a large current peak followed by a current decrease during several seconds, can successfully ~ quantified using a newly p r o ~ d 2-step reaction model. Herein, the electrochemical formation of silver atoms, aggregating to form silver clusters, is a phenomenon which precedes the actual moment of nucleation. The basic pa~ial differential equations, describing the mass transport phenomena towards a RDE when a potential perturbation is applied, are solved numerically. The prc~nted method, based on the Crank~Nicolson implicit finite difference approach, provides for an efficient and accurate calculation of the quantities of interest, ~ i n g the time dependent concentration profile of the electroactive , p e c l e s and the current transient derived I}om it. A plausible physical interpretation is given for the different electrochemical ~ n ~ t e r s involve, it can be concluded that k,, the rate constant of the second reaction step, decreased progressively fronl 3,2 × 10 .~' to 2,0 × 10 ~:~ m s .~ with potentials d~reasing from ~ 225 to ~ 350 mV/SCE. In spi~ of ~ observed decrease of the rate constant, the net reaction rate k~ cA~. of the nucleation event increa~s with rising overl~tentials. An almost inverse potemiai depen~y was noticed for S, the surface to volume ratio of the ~posited Ag" layer. Having an el~trochemicai model which describes the initial phase of the electrodeposition of silver in a convenient way, and having an accurate and efficient numerical cMculation rtmti~, allows us to investigate quantitatively t ~ ~ p e ~ r ~ - e of the reaction ~te on the composition and type of different silver electrolytes. The eh.'ctrochemical c~micai behaviour of specific surface phenomena and way in which they alter or disturb (seen through a "~ation of k 2 and S) the formation rate of silver nuclei, will be studied. A d d i ~ l y , the developed methodology can be extrapolal~l to ocher electrochemical systems.

I

k

"~ I

k, kde¢

complex AgL2 ( m2 s - I ) electrode potential (V) equilibrium potential (V) Faraday's constant = 96485 (C s o l - ~) net cathodic electrode current (A) measured maximum of file cathodic electrode current (A) calculated m ~ximum of the cathodic electrode current (A) overall heterogeneous rate constant Ibr the reduction of silver thiosulphate complexes AgL (m s t ) potential independent heterogeneous rate c o n s t a n t for the reduction of silver thiosulphate complexes AgL 2 (m s - I ) heterogeneous rate constant for the oxidation of silver atoms (m 7 s- l mol-~-2) potential independent heterogeneous rate constant lhr the oxidation of silver atoms (m 7 s ~+ ~ s o l +~~ ) rate constant Ibr the nucleation of stable silver clusters(m s ~) decomplexation rate constant of AgL: (s I )

association rate constant of AgL and 1. (s I) heterogeneous rate constant for the reduction of AgL(m s a ) I*,0 potential independent heterogeneous rate constant for the reduction of AgL (m s~ 1) heterogeneous rate constant for the oxidation of Ag o ( m s i) k" i" potentia~ independent heterogeneous rate constant for the oxzt~tion of Ag o (m s- i) R gas constant = 8.314 (J s o l - t K- ~) t absolute time (s) or dimensionless time T temperature (K) axial distance (m) or dimensionless axial distance to the electrode surface k(

5. I. Greek symbols" o~

6 ~r TO P OJ

cathodic charge transfer coefficient thickness of the Nemst diffusion layer (m) thickness of the reaction layer (m) induction period (s) kinematic viscosity of the electrolyte (m 2 s- i) rotation speed of the electrode (tad s-I ) 0.51 ¢o3/2u- i/2 (m- i s- I)

W. Simons et al. / Joun;al of Electrtmnalyth'al Chemixtry 433 { i~7~ 141-151

5.2. Abbteriations and other notations

AgL, complex Ag(S,O0~" '-I~chemical-electrochemical reaction scheme CE index for the dimensionless axial distance z, i dimensionless iigand ion S.,O 3L time index, dimensionless I1 RDE rotating disc electrode rpm revolutions per minute average of the surface to volume ratio of the S deposited silver cluster layer (m- t) sum of residuals between measured and calculated SR data SCE saturated calomel electrode

AcknowledRements Rese~rch financed by the Flemish Institute liar the En° couragement of Scientit'ic and Technological Research in the Industry (IWT). The authors thank P. Laevers for his suggestive contribt~tion to the formulation of the basic mass transp~r~ equations.

References [I] T.II. James (Ed.). Tile Theory of the Photographic Pr{vcess. 4th ed.. oh. 16. MacMill:m. New York. I977. [2] S. Vandeputte. B. Trihollet. A. Hubin. J. Vereecken. Eleclr,~chim. Acta 33 (1994) 2729. [3] S, Vandepntte. E. Verhoom. A. Huhin. J, Vereeckcn, J, Electroanul. Chem. 397 (1995) 249. [4] D. Gonnissen. S. Vandeputte. H. Huhin, J, Vereecken. Elcctrochin|. Acta 41 (7/8)(1|196) 1051. [5] S. Vandeputte. A. Hubin. J. Vereecken. accepted liar publication in Electrochim. Acta, [61 S. Vandept|tte. PhD Thesis. Vrije Universitcit Brusscl. Brussels. 1096.

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151

[14] 1. Konstantinov. J. Malinowski. J. Photogr. Sci. 23 (I) (1975) 145. [15] E.B. Budevski. in: B.E. Conway. J.O'M. Bockris. E. Yeager. S.U.M. Khan (Eds.). Comprehensive Treatise of Electrochemistry. vol. 7. ch. 7. Plenum. New York. 1983. [16] J.O'M Bockris. G.A. Razumney. Fundamental Aspects of Electrocrystallisation, oh. 10. Plenum. New York. 1967. [ 17] H. Fischer. Electrodepos. Surface Treat.. i ( ! 972/1973) 239. [18] B. Scharifker. G. J Hills. Electrochim. Acta 28 (7) (1983) 879. [19] M. Siuyters-Rehbach. J.H.O.J. Wijenberg. E. Bosco, J.H. Siuyters, J. Electroanal. Chem. 236 (1987) I. [20] PJ. Sonneveid. W. Visscher. E. Barendrecht. Electrochim. Acta 37 (7) (1992) 1199. [21] D. Gonnissen. W. Simons. A. Hubin. submitted for publication in J. Electroanal. Chem. [22] M. Fleischmann. H.R. 'rhirsk. Electrochim. Acta I (1959) 146, [23] G.A. Gunawardena. GJ. Hills. I. Montenegro, Electric:him. Acta 23 (1978) 693.

[24] G,J, Hills. A,K, Pour. B, Scharifker. Eleclrochim, Acla 2~ (7) (!~,)~3) 891.

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