Potentiostatic-Galvanostatic-Potentiostatic Study of the Deposition of Mercury on Graphite BY GAMINE GRAHAM MONTENEGRO A. GUNAWARDENA J. HILLSAND IRENE Chemistry Department The University Southampton SO9 5NH Received 23rd August 1977 A potentiostatic-galvanostatic-potentiostaticpulse train has been used to investigate the electro- chemical nucleation of mercury on graphite. The second potentiostatic pulse was used to evaluate the nuclear number density along the galvanostatic transient. The maximum in the galvanostatic- potential-time transient is seen to be related to the faradaic charge transfer reaction rate and can be made the basis of the determination of exchange currents. The kinetics of electrochemical nucleation are invariably investigated by the analy- sis of the time transients observed using single or double potentiostatic or galvanostatic pulse techniques.l-' The latter stages of such transients reflect the growth of estab- lished nuclei their geometry and the nature of the rate-controlling faradaic process.In the simple metal deposition process considered here namely the electrodeposition from aqueous solution of mercury onto vitreous carbon hemispherical nuclei are formed the growth of which is eventually entirely mass-transfer controlled. At the beginning of the transients however the faradaic current is controlled by the kinetics of the charge transfer reaction leading to the formation of clusters and nuclei. If the transients are appropriately recorded i.e. using equipment with suffi- ciently short response times the fine structure of the transient can be delineated and in principle at least deconvoluted into the various and varying contributions to the current or potential including that of the charge-transfer process whereby clusters of metal atoms are built up on the surface to a critical size formally defining the birth of stable nuclei under the prevailing conditions.There are basically two models of electrochemical nucleation onto a solid substrate. The first envisages favoured sites which trap and concentrate metal atoms by the direct deposition of single atoms and/or surface diffusion of neighbouring ad-atoms. The second model is of a more general character and envisages an initial build-up of a two-dimensional array of directly deposited ad-atoms which by surface diffusion and/ or by direct discharge onto ad-atoms and other clusters form a statistical distribution of differently sized clusters.Again a fraction of the clusters reach their critical size and will then grow irreversibly and spontaneously to macroscopic proportions. The two models are not very different. Short of a complete molecular description of the clustering process (and such calculations are in progress) it is necessary to assume (i) that at each stage during cluster formation there is instantaneous equilibrium between ad-atoms and clusters and (ii) that the properties of clusters can be described in macroscopic terms. The rate of their formation and disappearance can then be described in terms of the kinetics of charge transfer (a) to form isolated ad-atoms localised at the electrode surface and (b) to add metal atoms to existing two-dimen- sional or three-dimensional groups or clusters already adsorbed on the surface.Such varied processes are likely to be characterised by a range of transfer rate constants or C. A. GUNAWARDENA G. J. HILLS AND 1. MONTENEGRO exchange currents depending on the size of the cluster to be formed and for large clusters approximating to the standard equilibrium rate constant for the bulk metal =+= metal ion reaction. Whether these clustering rate constants can properly be interrelated through the Gibbs-Kelvin correction for excess free energy and other macroscopic relationships has yet to be established. Even if this could be done there remains the unknown rate constant for ad-atom formation.The evaluation of all these parameters is a lengthy and complex procedure not yet completed for any one system. Here we report some attempts to define certain parts of the problem and in particular to use the recently established relationship between overpotential and nuclear density to evaluate the nuclear current density and hence the apparent charge-transfer rate constant for the metal +metal ion deposition reaction. The experimental basis for the work is the so-called PGP pulse train in ref. (4) in which successions of potentiostatic galvanostatic and potentiostatic pulses are used respectively (a) to stabilise the electrode surface (b) to initiate a forced growth of nuclei and to grow the established nuclei to a countable and even visible size.The pulse train and corresponding responses are shown diagrammatically in fig. I n I 1 1 t FIG.1.-Schematic representation of the PGP method. and in fig. 2 is shown a set of the middle galvanostatic transients recorded for a range of current densities. The first potentiostatic pulse serves only to maintain the electrode surface at a convenient anodic potential i.e. in a controlled state of readiness for the subsequent electrodeposition process. By means of a mercury wetted relay this is followed in <10 ,us by the galvanostatic pulse which throughout its existence will be a convolution of (i) the faradaic process attending ad-atom formation cluster formation and nuclear growth and (ii) the non-faradaic process of charging the interfacial capacity.Although it is not yet proved that they can be totally separated there is no reason to suppose that such a separation involves great uncertainty. It may be noted that throughout the process of nucleation the electrode substrate surface remains essentially graphite. The electroactive area is relatively small and will therefore not contribute significantly to the " common " capacity'O except through the pseudo-capacitance which is part of PO TEN TI0 S TA TI C-GA L V A N 0STAT I C-P OT EN TI0STAT I C S T UDY 0 1 2 10 tls FIG.2.-Early stages of the overpotential transients during single galvanostatic pulsing for deposition of mercy onto viteous carbon. Hgg+ concentration = 5 x lo-’ mol CM-~, area of electrode = 0.32 cm’.The values of each curve refer to 1041/A. the faradaic process under investigation. At the beginning of the transient all of the current is non-faradaic in character and is given by I=.@) . t+O The double layer capacitance C,can therefore be determined from the initial slope of the overpotential-time transient n =f(t). In fact the slope was linear over 20 mV or so and gave a value for the integral capacity which was not only of expected magnitude but also sensibly independent of Z (cf. table 1). Using this value the non-faradaic current throughout the transient was evaluated from a computer fit of the variation of the overpotential with time. A typical deconvolution is shown TABLE INTEGRAL CAPACITY AT DIFFERENT GALVANOSTATIC CURRENT DENSITIES 1 .-CALCULATED 103 x I 106 x c A cm-2 farad.cm-2 7.03 57.7 9.38 57.8 11.22 56.3 24.11 53.1 18.63 53.4 in fig. 3 and for each total current density the faradaic component was evaluated as a function of time (and subsequently used to determine the average nuclear size in terms of the faradaic charge passed in a particular elapsed time). It is interesting to note that throughout the region of interest the galvanostatic method is not even approxi- mately a constant current procedure so far as the faradaic process is concerned. It may also be remarked that even if the capacity were not accurately defined by this method the charging current at the potential maximum will certainly be zero.The faradaic current at this point is therefore well established. The faradaic current flowing to the electrode is of course flowing to a large num- G. A. GUNAWARDENA G. J. HILLS AND 1. MONTENEGRO I I 2 4 103tis FIG. 3.-Deconvolution of total applied constant into charging and faradaic components. ber of small nuclei and for the faradaic current density to be determined the number density of nuclei must be known. In principle the nuclei can be grown under galvano- static conditions to visibly countable size but in practice to determine the nuclear density in this way is not so easy. First the visual method is tedious. Secondly it is subject to error because of the inhomogeneous distribution of nuclei on a sheathed and bounded electrode surface of the type used here.Thirdly under galvanostatic conditions it is easy to flood the surface with nuclei which further aggravates the preceding two difficulties. Here we preferred to determine the nuclear density by means of a succeeding potentiostatic pulse of appropriate magnitude which was switched in again by means of a mercury wetted relay. The relation between the potentiostatic current- time transient and the nuclear density has recently been well established.ll Thus the middle sections of a typical set of potentiostatic transients (fig. 4) show an exact Iversus t* relationship (fig. 5) corresponding to mass transfer controlled growth the slope of which is directly proportional to the number of nuclei. The precise description of the faradaic current to a growing microscopic hemisphere is not easily obtained.In terms of spherical diffusion to an electrode of stationary radius but of growing area (a slight contradiction in terms) a reasonably good value of the nuclear density N,can be found from the equation1 -Z~F(~DC)~~~M% *N rfaradaic -pk 3 where Mis the inolecular weight and p the density of the electrodeposited metal and where the other terms have their usual meaning. P OTEN TI 0S T A TI C-G A L V AN 0S T' A TI C-P 0TEN TI 0ST A TI C STUDY _................... ,.a. ...-.... .I.. -__.... ........ ......_.... ................. L.",.. I 1 I 1 2 tls FIG.4.-Family of potentiostatic single pulse transients for the deposition of mercury onto vitreous carbon Hg$+ concentration = 1 x mol crnp3,area of electrode = 0.32 cmz.I 2 4 6 8 10 11 12 14 16 10t2/s2 FJG.5.-Linear dependence between current and t* for the middle rising sections of the transients in fig. 5. q values are shown on curves in mV. G. A. GUNAWARDENA G. J. HILLS AND I. MONTENEGRO A more exact calculation of the current density to a growing hemisphere can be made by using numerical methods and computer simulation. The use of implicit numerical procedures for this purpose is described elsewhere in a series of papers con- cerned with the application of numerical analysis to electrochemical problems in general.lZ They all involve Crank-Nicolson or other such algorithms and matrices rather than iterative algebraic procedures.Such calculations allow the current to a single nucleus at any instant of time to be accurately expressed simply in terms of ionic concentration and diffusion coefficient. From the observed total current or its linear dependence on t3 the corresponding nuclear density can be derived and was found to be only slightly higher than that predicted by eqn (2) in generally good agreement with values established by visual counting. Suffice to say here that all of the succeeding potentiostatic transients exhibited the required I versus t* relationship from the slopes of which Nwas found from the relationship I= 1cND3/2c3/2t* (3) where k is a constant equal for mercury to 9.82 x lo6C cm* mol-3 established by the computer simulation of the growth of a single isolated nucleus for known experimental values of the concentration and diffusion coefficients of Hgg+ in aqueous KN03 soIution.The number of established nuclei can therefore be calculated from any rising potentiostatic transient including those recorded by interrupting the galvanostatic transient and fast switching to the potentiostatic mode. For comparison purposes they can be recorded at a fixed lower overpotential or better still at the overpotential at the point of interruption. Either way the number density of stable nuclei was found to increase progressively up to the potential maximum and then to remain constant. This is evident in fig. 6 which shows the slopes of a number of current-time 130 102t/s FIG.6.-Nuclear number density as a function of time along the galvanostatic transient.q2 values are shown on curves in mV. transients and hence the nuclear number density at times up to and beyond the gal- vanostatic potential maximum. The number of nuclei evidently reaches a limiting value from which a number of conclusions can be drawn. (1) The galvanostatic potential maximum corresponds closely to a potential of nuclear arrest i.e. no further nuclei are formed beyond it. (2) Above a critical potential and up to the maximum nuclei are formed continu- POT E N TI 0S TA TI C-G A L VAN 0STAT I C-POT E N TI 0STATIC S T U D Y ously and most if not all of these continue to grow thereafter simply because there was never any indication of any relaxation of the number density of nuclei.The slope of the potentiostatic transients following the potential maximum was never less than the limiting value. (3) At the galvanostatic maximum there will be a distribution of nuclear sizes. The range of this distribution is not easy to assess and will require for its delineation a more detailed application of the PGP method to the rising part of the maximum. However there will be an average nucleus size and it is to the determination of this we next turn our attention. It was found simply from a knowledge of the faradaic charge passed up to the time of the potential maximum i.e. from the relation where fmax is the average nucleus size at the potential maximum. From the average nuclear area 27rt?2max the faradaic current density was calculated for each value of the overpotential maximum; these values are given in table 2 to-TABLE2.-cHARACTERISTICS OF THE MAXIMA IN GALVANOSTATIC POTENTIAL AGAINST TIME TRANSIENTSOBSERVED AT A VITREOUS CARBON ELECTRODE OF AREA 0.32 cm2IN A 50 mmol dm-3 AQUEOUS SOLUTION OF Hg2(N03)2IN mol dm-3 KN03.nuclear current 209 2.25 2.31 4.31 3.40 2.8 206.2 0.72 215 2.57 2.12 4.53 3.25 2.9 212.1 0.86 226 3.00 2.02 5.65 2.97 3.2 222.9 0.96 232 3.60 1.90 6.18 2.82 3.3 228.7 1.17 243 4.50 1.so 8.30 2.51 3.7 239.3 1.37 262 6.77 1.53 11.6 2.13 4.4 257.6 2.05 277 9.00 1.46 16.0 1.88 4.9 272.1 2.59 287 11.3 1.36 18.6 1.71 5.3 281.7 3.30 296 12.9 1.28 20.9 1.65 5.6 290.4 3.61 306 15.0 1.25 22.7 1.59 5.8 300.2 4.16 319 18.0 1.12 27.0 1.45 6.4 3 10.6 5.06 gether with the corresponding values of N and fmax.The table also includes the excess interfacial energy term 3aM/Jmaxp, the Gibbs-Kelvin term which is part of the nuclea- tion overpotential and which must be subtracted from the total overpotential before the correct dependence of faradaic current density on reaction overpotential can be delineated. The interfacial free energy term 0,is a complex function of the three interfacial tensions between the metal the substrate and the solution. In this case it will be close to the surface tension of mercury. is given in fig 7. ~t is linear over its entire length and although it might be argued that the range of overpotentials is not large its conformity to the simplest irreversible form of the Tafel relation is striking.The slope of the line leads to a value of the transfer coefficient u = 0.24 G. A. GUNAWARDENA G. J. HILLS AND I. MONTENEGRO which is in good agreement with previously published ~a1ues.I~ The intercept how- ever leads to a value for the exchange current density of 2 x A cm2. Since the solution under study was a 50 mmol dm-3 solution of Hg2(N03)2 (in 1 mol dm-3 KN03),this corresponds to a standard rate constant of -2 x cm s-l. Bindra et a1.I3have recently published a table of the best of such values determined by a wide variety of methods; the value obtained here is well below the value of 2.0 x lo-' cm s-l obtained by them the value of 2.1 x low2cm s-I obtained by Weir and Enke14 250 300 350 OlmV FIG.7.-Logarithmic behaviour between nuclear current density and the corrected overpotential at the maximum.and the value of 1.9 & 0.2 x cm s-l obtained by Matsuda Oda and Delahay." Although the present procedure therefore seems to be another quasi-stationary method for the determination of metal-deposition exchange currents it is evidently in need of refinement. The principal criticism of the method is that it is a high-over- potential method and that in " driving " this already fast charge-transfer reaction even faster there is likely to be interference from mass transfer effects even though the nuclei represent individual hemispherical micro-electrodes of very small radii. It is well known that the rate of mass transfer to such small electrodes is very high. Thus under potentiostatic conditions the diffusive flux to a stationary hemisphere of radius r is equivalent to a current density of I= zFD"c ZFDC + 7.~ 713t+ (5) Using rounded values of the coefficients (D= cm2s-l c =5 x mol ~m-~) it follows that when t = s and Y = lo-' cm then the first term is negligibly small and the second equal to 10 A cm2.This is a minimum value because of course the nuclei are not stationary but growing. This is a well known " moving-boundary '' or Stefan problem which has not been solved analytically12 but which was solved numerically in order to evaluate the constant in eqn (3). It need only be noted here that depending on the starting radius the mass transfer controlled faradaic current to a growing hemispherical nucleus can be an order of magnitude greater than that pre- dicted by eqn (5) i.e.-100 A cmw2. It was a comparison between this value and those POT E N TI 0STAT I C-GA L V A N 0STATI C-POT E N T 10s TA TI C S T U D Y in table 2 which supported the initial belief that the potential at the galvanostatic maximum was charge-transfer controlled. On the other hand a closer examination of the early stages of separate potentiostatic experiments carried out on the same system reveals a rapid transition to mass transfer control. This fig. 8 shows a log-log plot of the current-time dependence. There is an evident break from a t9 to a t3 time-dependence i.e. from conditions of progressive nucleation to those of 0.d c 0 0 d 0. 0.5 1.0 log t FIG.8.-Log-log plot of the early part of the potentiostatic current-time transient under conditions similar to those obtaining in fig.4. instantaneous nucleation. For an overpotential of -200 mV this is seen to occur at t z 50 ms and at an average nucleus radius of cm i.e. close to the values ob-taining at the galvanostatic maximum. It is therefore evident that the potential-time profile at the beginning of the galvano- static transient is the result of the build up of both activation and concentration overpotentials and their subsequent relaxation as the electro-active area grows suffi- cient to relax the current density. At a constant current density I this area is given by A = 2N,~nr'f (6) G. A. GUNAWARDENA G. J. HILLS AND I. MONTENEGRO 99 where ft is the average nuclear radius at time t i.e.3Mli *. = (w) (7) As the area increases so the current density decreases monotonically with time. However if at the same time the faradaic current increases from zero in the non- linear manner shown in fig. 3 the overpotential must pass through a maximum deter- mined largely by the applied current density and the exchange current density for the metal ---L metal ion reaction. The response of an electrode reaction to a non-linear current-time waveform has been considered by Rao and Rangarajan16 in the context of electrode processes in general. Even for a " homogeneous '' (non-nucleating) reaction a potential-time maximum can be observed and in principle at least made the basis for the deter- mination of the exchange current and the energy transfer coefficient.The involve- ment of the kinetics of nucleation and of the non-faradaic current were not con- sidered and it seems probable that the extent of their involvement and hence the description of the galvanostatic transient will require more extensive experimental examination e.g. of the density and of the size distribution of nuclei along the whole transient. Kashchiev has also considered the galvanostatic potential-time maximum and in the present context i.e. in the study of electrochemical nucleation under galvano- static conditions. His description of the maximum is in broad agreement with that presented here with however a few exceptions. Thus he identifies a critical over- potential yc below which nucleation will not take place and he allows nuclei to form along all of the crest of the transient above rc.The evidence here is that it is not so for the nucleation process is seen to terminate at the maximum not after simply because the onset of mass-transfer control has already begun to attenuate the activa- tion overpotential or " phase boundary transition control " (PBTC) overpotential as it is referred to in Kashchiev's comprehensive treatment.lo It is true that he sought only to describe the effects of the separate effects of ohmic volume diffusion (mass transfer) or PBTC control and that he recognised the likelihood of mixed control. On the other hand the equations describing even the separate effects are complex and in the main not subject to analytical solutions.It might therefore be better at this stage not to attempt a single synthesis of the combined effects of capacitance mass-transfer and charge transfer in this way but rather to continue with the experimental delineation of the separable parts of the problem e.g. the direct determination of the nuclear density the direct determination of dN/dt along the transient and the direct determination of the activation and mass transfer components of the overpotential. The aim of the present work is to con- tinue with the application of multiple pulse methods to describe the complex features of nucleation transients. The authors are grateful to Dr. L. M. Peter for stimulating discussions. G. G. thanks the British Council and I. M. thanks the Instituto Nacional de Investigagao Cientifica (Portugal) for scholarships.G. J. Hills J. Thompson and D. J. 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