ON THE MECHANISM OF ELECTROLYTIC DEPOSITION AND DISSOLUTION OF SILVER1

INTRODUCTION

The mechanism of crystal growth has beeu an object of study for many years. Be- ginning with Gibbs ( I ) , major contributions have been made by Voliner (2), ICossel (3), Strailski (4), Becker and Doring (5), Frenkel (6), and Frank (7). From this worlc it was concluded that the first step in crystal growth is "adsorption" on the surface, whereafter "adatoms" (2) diffuse to steps and along steps to Icinlcs, where they are incorporated into the lattice, after which repetition of these steps ("repeatable step") brings about the incorporation of a kink atom into the lattice (cf. Fig. 1 and Conway and Boclcris (21)).

FIG. 1. Schematic view of metal-solution interface depicting ( a ) atom a t edge, (b) kink, (c) atom in edge, ( d ) adion.

I

After all the stepped surfaces have been used up, two-dimensional nucleation was expected in the earlier theory (1, 2, 3, 4, 5, 6) to occur on surfaces of low index, which implied that some degree of supersaturation was necessary for the continuation of the growth process. Experiments show, however, that usually crystals can grow indefinitely without a large degree of supersaturation of the surface. This discrepancy arises because the model for crystal growth, sketched above, applies to the growth of a perfect crystal, but most real crystals contain imperfections which provide the steps necessary for growth.

Frank et a2. (7) have studied the influence of imperfections, mainly screw dislocations, on the growth mechanism. Screw dislocations were found to provide a practically in- exhaustible source of steps which turn a t higher supersaturations into spirals, forming growth pyramids on the surface.

The equations deduced by Frank (7, 8) can also be used for a discussion of electrolytic crystal growth if the parameters are appropriately changed, but i t is necessary to keep in mind that the energetic properties of a metal surface in contact with an electrolytic solution differ essentially from those of a surface in contact with its vapor. Thus, in a solution there exists a t the metal surface an electrical double layer, with a field strength of about lo7 v cm-l; ions from the solution phase are adsorbed on the surface, etc. (13).

l&Ianuscript received J u l y 83, 1968. Contribution frotit the John Harrison Laboratories, Universily of Pennsylvania, Philadelphia, U.S.A.

T h i s paper was presented at the Synsposiuns on Charge Transfer Processes held at the Universzty of Toronto, Toronto, Ontario, Septenzber 4 and 5, 1958.

Can. J. Chem. Vol. 37 (1959)

190

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MEI-IL XSD BOCI.;IZIS: ELISCTROLYTIC DEPOSITION 191

For electrolytic crystal gl-owth an impoi-tant difference to growth of crystals from the gas phase is also that an activation energy has to be overcome to make it possible for the metal ions to cross the electrical double layer and to strip off the solvation sheath (21).

I t is the electric fielcl a t the metal-solution interface which gives rise to two possible extreme iclealizatioils for rnoclels of electrolytic crystal grolvth (9).

In model A nearly all the elcctrocle surface is active for the process of transferring ions from the solutioil to the electrode (transfer reaction). The particles on the surface carry probably a partial charge ancl are therefore best termed adions (21). The consecutive process to that of transfer is surface cliffusion of adions to the growth positions on the surface, the final step being illcorporation of adions into the lattice. In addition, a t higher concentration of aclions, nucleation on the surface may occur.

In model I3 the ions from the solutioil tra~lsler directly to growing poiilts on the surface (active centers).

Little informative work has been hitherto possible on electrolytic crystal growth, mainly because of experimental clifficulties. Metal surfaces readily adsorb impurities, possibly a t the active centers importallt for the crystal growth process. Because of this sensitivity to impurities, one of the main experimental difficulties in such work is lteeping a small metal area (of the order of 0.01-1 sq. cm in area) in contact with solutioil but free of adsorbed impurities. A further intrinsic difficulty is that of carrying out significant measurements \vllilst the electrode surface remains essentially constant, i.e., observations of the rate of the reaction of a given potential must be made rapidly.

Because of these experimental difficulties only qualitative discussions of electrolytic crystal growth have been possible until nolv (9, 10). But experiences gained in the preparation of electrodes for studies of the hydrogen evolution reaction (11) and in the development of improved electi-onic measuring techniques (12) seemed to malie worth while an experin~ental study of the kinetics of electrolytic metal cleposition and dis- solution.

At silver electrocles, Gerischer et al. (14) observed a deviation from the current- potential relationship to be expected if a rate-determining transfer reaction occurs (a similar observation was recorded by Mattson (15) for copper deposition on copper). These workers found that the exchange current for the system Ag/Ag+ is indepenclent of the conce~ltratioil of silver ions in the solutions \vllich appeared to indicate a rate- determining crystallization process for the over-all metal deposition.* Kaischew and co-workers (16) showed by microscopic studies that growth spirals are formed on the silver surface during deposition, thus supporting the applicability of Frank's theory of crystal growth by means of dislocatioil to electrolytic crystal growth.

I t is the aim of the present work to find out: (1) which is the general path (i.e. that of model A or q model B) by which the ions pass from their initial state of hydration in the ,solution to their final state in the lattice and vice versa, and (2) which step in this path is rate-determining under given sets of conditions.

METHOD OF INVESTIGATION: T H E GALVANOSTATIC METHOD

1. Introduction In the study of the mechanism of crystal growth from the gas phase, the rate of crystal

growth is studied as a function of the degree of supersaturation of the vapor. The corre- sponding measurement here is that of the rate of growth as a function of potential.

*Gerischer has also recently carried ozct, independent ly of the present workers, potentiostatic (24 ) and galvano- static (26 ) n~easzcre~tzents on szlver dzssolutzon, and conzes to conclzcsions sivzzlar to those described here (cf. Mehl and Bockris (28)). Different io valzres recorded i n the work of thzs azctlzor, couzpared zvzth those recorded Izere, depend tcpo~z the qlcite dzfferent i~ie tkods of preparation of the electrode stcrfc~ces tcsed.

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192 CANADIAN JOURNAL O F CI-IEMISTRY. VOL. 37. 1959

If one tries to obtain this i-q relationship under conditions of long-time (minutes) polari- zation, changes of the electrode surface with time due to the deposition or dissolution process itself, or adsorption of residual impurities, even from highly purified solutions, may make interpretation difficult. Consequently, short-time (milliseconds) polarization is preferable, and the galvailostatic (constant current) method (17) was chosen for the present investigation. Another advantage of this method compared with long-time measurements is that it enables additional evidence concerning the deposition mechanism to be obtained from the potential-time function.

The electrodes are in contact with the solution for minutes (between the time the protecting bulb around the electrodes is broken and that a t which the sweep is taken). Therefore, extreme purificatioil of solutions was carried out.

2. Double Layer Chargi~zg and Transjer If a constant current density, i, is switched onto an electrode, this current divides

itself into two components. The first one, ic, is used for charging the electrical double layer a t the electrode-solution interface; the second part, iF, brings about transfer of ions across the double layer from the solution to the electrode surface. The current density, ic, rapidly decreases with time, during the galvanostatic charging process, which means that i F increases with time, because

is constant. If iF is controlled exclusively by the transfer reaction, then:*

where C = the double layer capacity; io = the exchange current density for the transfer controlled reaction; 0 < f l < 1 ; X, T, and F have their usual significance; and z is the valency of the cation in solution.

A general solution of this equation with the assun~ption that the double layer capacity is independent of potential was given by Rojter, Juza, and Polujan (17). The q-t relationship thus obtained is complicated and therefore difficult to verify by experiment. For small overpotentials, C Y ~ zF/IiT < 1, and the solution of equation [2] is, again assuming C independent of potential,

with q, = steady state overpotential,

[41 T = (II T/Z FG) C.

Equation [3] is only an approximation as the double layer capacity changes during the charging process with time so that an exact separation of iF and ic is not possible (18). This means that for the analysis of electrode reactions with help of the q(t) functions, only such reactions are detectable which have a relaxation time of 7 > 71. 3. Electrolytic Iiesistance o j Solz~tion

The ohmic potential drop between reference electrode and measurement electrode can easily be determined because it appears in the 77(t) function as a jump parallel to the potential axis (see Fig. 5 ) .

*I t i s assu~ned that the n~axinzz~tn supply of ions from the bulk of the solzition to the double layer (by diffusion and electrolytic transport) far exceeds that required for deposztzon.

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MEHL A N D BOCKRIS: ELECTROLYTIC DEPOSITION 193

Q. Transfer and Surface Dz fus ion of Ions In the following considerations, constant Faradaic current (t >> 71) is assumed. If

the surface diffusion coefficient for adions on the metal surface is sufficiently small, so that this process is important for the rate of the over-all process, the surface concen- tration of adions increases during polarization to a value above that of the surface concentration corresponding to equilibrium, and becomes less than this value during dissolution. In this section the discussion will be limited to the consideration of low overpotential, i.e., a small degree of supersaturation.

Assuming that the adions cause only a small reduction of the free surface area for the transfer reaction and neglecting concentration polarization in the solution, the potential- time function is:

[51 (iF)' = io[e (cl( 1 ) / cO) e(1-8)vr zFIRT I -811 zFIRT-

where c0 is the concentration of adions a t equilibrium and c,(,, is the concentration of adions a t a time t when the overpotential has reached a value of v. For v < RT/a zF and neglecting second-order terms, one obtains, by expansion of [5] and rearrangement,

where AC,(l) = C q ( 1 ) -60 .

Hence, v depends in this case both on i o for the transfer reaction and the "concentration overpotential due to surface diffusion".

I t is necessary to express c v ( l ) as a functioil of i and t . Now:

where v is the average of the surface diffusion flux of the adions during their passage from the point a t which they are transferred to the inetal surface and that a t which they meet a crystal building point.

Surface diffusion on metal crystals has been dealt with thoroughly by Frank et al. ( 7 ) and applied to electrolytic deposition by Vermilyea (10). The present results are consistent with the simple assumption (for I r l l < 5 nlv) that the surface diffusion flux is pro- portional to the surface supersaturation, [ ( L ~ ( ~ ) / L o ) - 11. Hence:

assuming that, near to ltinlts, the adion concentration rcmains a t equilibrium value, GO, whilst on crystal planes, where the metal ions are regarded as first being transferred from the solution to the metal surface, the mean surface concentration of adions is c , ( ~ ) , Substituting [8] into [7] and integrating yielcls

Thus, the steady state value of the increased concentration of adions over that expected for pure transfer control is given ( t = m ) by :

[ I l l Ac/co = ~F/zFvO. With [9] and [6] the v ( t ) function a t small overpotentials, with both surface diffusion

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104 CANADIAN JOURNAL O F CHEMISTRY. VOL. 37. 1959

and transfer of ions across the double layer as influencing the value of q,, and for time t > 71, becomes:

For times t, for which 71 < t < 7 2 and with 7 2 = CO/VO, from [12]

and

6. Current-Potential Relations in the Steady State (a) Transfer is rate-determ,ining.-The i-q function will then be controlled:by:

[I41 . -Bq z F I R T - e(l-B)rl F z / R 1 9 i, = zo(e ).

The adion coilcentration is a t its equilibrium value and therefore surface coverage can be neglected (6).

(b) Transfer and surface dzyusion control the role of the over-all reaction.-With [12] and [ l l ] we get

(c) Surface dzfusion is rate-determi7zing.-For

there follows,

[I61

Equations [Is] and [16] have been derived under the assumption that vo is overpotential- independent. Theoretical work by Frank et al. (7) shows, however, that this assumption is incorrect; vo increases with increasingly negative overpotential (cf. Discussion).

EXPERIMENTAL

I . Galvanostatic Arrangement (fig. 2) The potential source consisted of dry batteries, e.m.f. 90-200 v. Precision resistors

(I.R.C.) are represented by R and were arranged in 10 steps, from 1 kilohm to 6.2 meg. The letter 0 represents a Tektronix oscilloscope, type 531, with type 53/54D plug-in preamplifier. The potential-time curves were photographed with an Exakta camera having a 50-mm lens. Projection of the 35-mm film was made with an enlargement of about ten times to obtain numerical q-t curves. The switch S contained a Western Electric relay, type 175, having a rise time of about lo-' ~ e c ~ ~ .

2. Treatment of Solution and Electrodes The solution was made from A.R. HC104, which was distilled three times in vacuo, the

last distillation being into a burette leading directly to the cell. The water was distilled

*Eqzration [15] i s valid only ultdcr conditiows sz~ch that the surface coverage with adiolzs i s szrficientlylozo to yield negligible lzindrance to the transfer of ions from the solution. I n fact, i f uo i s suificiently low a surface coverage could build u p and the first term i n [5] shozrld be preceded by a tervz of the type 1 - 0 /1 -go, where O,, i s the surface coverage at overpotential 7. A limiting current cotrld t l ~ e n ensue at szrificiently h$h overpotelztials. 13ut this case i s ulzlikely because of nztcleatio?z and a possible increase of vo with i?zcreasilzg overpotential.

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MEHL AND BOCKRIS: ELECTROLYTIC DEPOSITION

FIG. 2. Galvanostatic arrangement: I B: battery, 90 v ; C: measuring cell; 0: oscilloscope; S: switch; and R: resistances.

several times and had a final conductance of about 10-7 mho cm-l. The AgC104 ("pure"

1 grade) was recrystallized twice outside the system and twice inside and then introduced into a special cell in which i t was subjected to pre-electrolysis for about twenty-four hours with alternating current density of about 3x10-I amp ~ m - ~ . The solution was vigorously stirred with carefully purified nitrogen gas, the cathode was of silver, the anode ~ of platinum (vigorous stirring prevented silver needles from growing through the closed

I tap between cathode and anode compartments).*

I

I 1 The electrodes were prepared by heating a thin (spectroscopically pure) silver wire (0.5 mm diameter) in an atmosphere of purified hydrogen in a quartz tube. A spherical shape was attained by partial melting. Finally, the electrode was sealed in a glass bulb containing only hydrogen, or helium, and the bulb was broken in the solution shortly before measurement. A detailed description of this procedure has been published (19). This method of preparing electrodes is similar in nature to "thermal etching", which has been studied experimentally and theoretically by several authors (20). During the heating process a rearrangement of surface atoms of the polycrystalline surface occurs, to produce facets with simple crystallographic indices; grain boundaries occur as grooves. As the electrodes are being cooled down in a hydrogen stream (high heat conductivity), a relatively high thermal stress would appear in the metal, producing a high concen- tration of dislocations.

Electron micrographs were taken of the electrodes prepared in this way. They show etch pits with a density of about 107 per sq. cm (Fig. 3). According to Hirth and Pound (26, 27), these may be interpreted as due to dislocations with a screw component normal ,

I to the surface. Occasionally electrodes indicate surface films of unknown origin (possibly connected with accidental contact between the hot metal and the glass tube during electrode preparation) and results from such electrodes were not further examined.

Other experimental conditions, particularly with regard to purification of gases, etc., were as already described (29).

*The exterior of the pre-electrolysis cell was coated black to exclzrde light effects. The measzrrernents were carried out under conditions of minimum illun~ination.

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CANADIAN JOURNAL OF CHEMISTRY. VOL. 37, 1959

FIG. 3. Electromicrograpl~s of polycrystalline thermally etched silver surfaces.* (a) Etch pits due to dislocation sites

hlagnification 4000 X Magclificatio11 33000 X Magnification 78000 X

*For footnote see following page.

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MEHL A N D BOCKRIS: ELECTROLYTIC DEPOSITION

FIG. 3. Electromicrographs of polycrystalline thermally etched silver surfaces." (b) Grain boundary with precipitate

Magnification 13000 X Magnification 33000 X

*SiO replicas tlze conlact side of which was shadowed wi th tz~fzgs!en oxide. T h e work was carried 0211 by H. G. Wilsdorf to w k o n ~ the azitlzors' tlza?zks are due.

3. Procedure After one of the four electrodes in the cell had been lowered by means of a sliding

fit joint into the solution, the glass bulb surrounding it was brolten and the electrode adjusted until it was inside a platinum gauze basket which acted as counterelectrode. The outside resistance R (Fig. 2) was set a t a value appropriate for the given current density, and a current impulse given to the electrode. By means of the switch S (22) (Fig. 2) the impulse was restricted to an amount of electricity not more than coulombs ern+ (i.e., the amount to cover about 1 geometric cmy. The resulting potential- time curve was photographed.

Such photographs were made a t some ten current densities (over the range 3 X t o 8x10-I amp/cm2) for each electrode and in each run a maximum of four electrodes were examined. AiIeasurements were made a t a variety of concentrations between and lo-' moles I.-' so that concentration polarization could be avoided. At the end of a series of measurements with increasing current density, some sweeps a t lowest current densities were repeated and results, closely similar t o those taken earlier, found.

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198 CANADIAN JOUKNAL OF CI-IE1LIISTKY. VOL. 37. 1969

FIG. 4. (Top) Chargi~~g curve a t low current density: (AgCIO,) : 2x10-1 mole/]. ; i = 2 ma cmLZ; units : t -axis, 2 msec, 7 -axis, 1 mv. FIG. 5. (Center) Charging of electrical double layer: i = 2.5 ma/cmz; t = sec; q = 2 mv. FIG. 6. (Bottom) Superpolarization: i = 500 ma/cmZ; units: t = 5X10-3 sec; q : 30 mv.

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MIZHL AND BOCKRIS: ELISCTROLYTIC DEPOSITION 109

4. Results Figure 4 shows a typical potential-time curve for the current density 2 ma cm-2 a t a

concentratioil of 0.2 N AgC104 in 1 N HC104 for cathodic polarization. Similar results were obtaiiled for anodic polarization. In Fig. 5, a typical 7-t relationship for short times is shown. Figure 6 shows a typical 9-t curve measured in unpurified solution, which exhibits the phenomenon of superpolarization.

Figure 7 shows a typical plot of the log i7, relation for cathodic and anodic polari- zation, and Fig. 8 a typical linear relatioil of i to 9 , a t low current densities.

FIG. 7. Log i-7 plot for deposition (a), dissolution (6). cagclo, = 2 X LO-' rnole/l.

(c) liepresents (- - - -) slow trar~sfer reaction. (d) Represents (- - - -) si~nple diffusion theory.

DISCUSSION

1. Potential-Time Curves At very short times the current flowiiig during build-up of potential is used almost

exclusively for the charging of the double layer, the capacity of which is therefore given (in the linear gcLl-t region) by:

This linear 7-t relationship can be expected for times limiting low value of t . With a double-layer capacity of about 20 ,u F cm-?, RT/zFio) = 2 ohm cm2 and allowing a roughness factor 2, the linear relationship call be expected for times of about t < second, when 9 < R T / z F . In fact, a linear relationship was observed for more than lop3 second with a slope which accordiilg to [17] would give an impossible value of 500

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CANADIAN JOURNAL OF CHEMISTRY. VOL. 37. 1959

M A CMT2

FIG. 8. i-9 plot f o r l o w curren t densities. (AgC104) = 2.10-I mo le ].-I.

p F/cm2 for the double-layer capacity. Similarly, if the transfer step alone were rate- determining, equation [3] shows that the steady state overpotential should be reached a t a time of about

for 7 < RT/zF. The actually observed values for this time are more than ten times larger than those predicted.

This slow rise time may be interpreted in terms of the build-up of an adsorption layer of adions, as shown in the next section.*

2. Potential-Time Relation and the Role of Surface Dzfusion

From [13] :

Also, from [12] :

and

*A slow reaction in the solzltion cannot be resporssible for th is behavior because dur ing deposition of metals u p o n amalganss, behavior nzay be satisfactorily interpreled in terms of rate-controlling transfer (23).

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MEHL A N D BOCKRIS: ELECTROLYTIC DIZPOSITION

Further, from [12] :

Note that in the above equations the double-layer charging has been neglected; they are therefore valid only for times much greater than the rise time of the transfer reaction, i.e., in the systems reported on here, about lop4 seconds (also, i t has been assumed that 1771 < 5 mv).

Accordiilg to [I91 qr=o call be got by extrapolating a plot of log (71r-778t) against t to t = 0 (Fig. 9).* The value for io obtained this way [22] can be compared with values obtained by the independent method of extrapolating the linear part of 77-log i relation to = 0. The values agree t o about f 30%. If one takes into consideration that the accuracy with urhich the potential can be determined is ~ 5 % and the overpotential due to transfer (at 1771 Q 5 mv) is about l0yo of the total overvoltage, the agreement betlveen the io values obtained in these two independent ways is satisfactory. This agreement confirms the interpretation of the slow rise time of the q-t curve on the basis of slow surface diffusion of a d ' ions.

t (rn s e c )---+

FIG. 9. Separation of total polarization illto difiusion and transfer polarization by extrapolatio~l of galvanostatic q ( t ) curves (for parameters, see Fig. -1).

Table I gives values of i o obtained by the extrapolation of the Tafel 1ine.t With the known values of i o , [22] allows the calculatio~l of vo, values of which are shown in Table I

3. Concentration oJ Adions at the Reversible Potential From [19] we get:

*Tlzc qr-0 referred to i s a quantity si~sceptible onby to calczllat.ion rlpon a model szlclz as that szlggested. I t signijies tlze potential of the electrode dzlring cl~argiug with respect to the corresponding electrode, under conditions szcck tlzat Ihe do~~ble- layer charging i s tcrntinntcd, but before the sllrface diffusion has lzad t ime to affect tlze concentration of adio~zs over that corrcspo~zdin~g to Ihc reversible potential. Correspo~tdingly, qt30 depends only ztpott in, and not ztpolt 510.

t Tlzc i n valztesfor deposition and d.issolrlkion agree well; values o f 0 are alnzost the sanze ( e0.60) for each process.

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202 CANADIAN JOURNAL OF CHEMISTRY. VOL. 37, 1959

From [23] (with vo from [21]) and from [13a] directly, the concentration of adions a t the reversible potential may be calculated. Both methods lead to the same results. The values obtained this way are presented in Table I.

TABLE I

AgCI04 io z Fvo co mole/liter amp cm2 amp/cm2 mole/apparent cm2

2.0X10-' 0.10 1.2x10-2 8 X lo-"' 1.3X10-' 0.08 l.0X10-2 2 x 10- l.OXIO-l 0 .07 1.4X10-2 4X

4. Current-Potential Relation With the aid of values of vo and i o calculated as described above, it is possible to

evaluate the theoretical q-i and q - log i relations for the steady state. These are compared with experiment in Fig. 7.

6. AJeclmnism of Deposition and Dissolution As Fig. 7 shows, for silver deposition and dissolution for electrodes prepared by thermal

etching, surface diffusion of adions is the principal rate-determining step a t low over- potentials, and a t high overpotentials the transfer of ions from the solution to the electrode is the rate-determining step. As vo does not depend upon the concentration of silver ions in solution (see Table I), the concentration dependence of i o is given by:

I t can be pred'icted from the values for io and vo (Table I) that the above statement concerning rate control will be valid down to Ag+ concentrations of about to mole l.-l. At lower concentrations than these, the transfer step will be rate-controlling a t all current densities (for the type of silver surfaces used in the present study).

The path determined for this fundamental example of simple metal deposition corre- sponds therefore more closely to model A rather than model B (see above).

The transition from rate-determining surface diffusion to rate-determining transfer is due to the decrease of the average distance between growing lines (and hence increase of vo) with increasing overpotential. The dependence of vo upon potential has not yet been subject to a satisfactory theoretical treatment. Provisionally, the following empirical equation may be suggested: vo(q) = vo e-n ZFIRT, whereupon the complete course of the i-q relation can thus be described by the equation:

For higher overpotentials (24) gives:

and for low cathodic overvoltages:

[261 i = Z ~ v o e-n zPIR2'(e-zF nIRT-

1).

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MEI-IL AND BOCKRIS: ELECTROLYTIC DEPOSITION 203

Correspondingly, for crystal growth from the vapor phase, a linear dependence of the rate of growth upon the degree of supersaturation has been found a t higher super- saturations (linear law), and a parabolic dependence upon supersaturation a t low super- saturations (cf. Franlc et al. (7)).

The "critical overpotential", which separates the linear and the parabolic law, depends on the ratio io/zFvo and thus upon the concentration of Ag+ ions in solution (upon which io depends).

I t should be noted here that a perfect parabolic law, viz.,

i = k(e-ZF "RT- 1)2

(where k is a constant) call only be expected for a single dislocation and a function such as that reported here for distribution of dislocations on the crystal surface.

6. Equilibrium Concentration of Adions The equilibrium concentratioil of adatoms cad for the system metal-vapor can easily

be calculated. With an energy of vaporization for silver of about 65 lical, it is found that the equilibrium concentration of adions during electrochemical deposition is very much higher than the equilibrium concentration of adatoms during deposition from the vapor phase.

7. Superpolarization This phenomenon (Fig. 6) occurs a t higher overvoltages, 100 mv, and is difficult to

study experimentally because of its poor reproducibility. Superpolarization was observed almost exclusively for solutions which had not been well purified so that the phenomenon might well be one of "over-shoot" (24) due to supersaturation of the surface with adions. Vermilyea (10) has discussed this type of phenomenon in terms of dislocation theory (7).

A P P E N D I X

Certain Terms Used in Electroche7nical Crystal Growth Compared with Those Used in Crystal Growth from the Vapor Phase

Burton, Cabrera, and Franli (7) define supersaturation in the vapor by

where p is the actual vapor pressure and PO the saturation value. The supersaturation of adsorbed molecules on the surface is defined by,

where c, and co are the actual and equilibrium concentration of adsorbed molecules respectively.

The "current" from the vapor to the surface is written as

where T, is the mean life of an adsorbed atom on the surface defined by

where W' = evaporation energy from the surface to the vapor, and v = frequency factor. In the present theory, the current from the solution to the surface is written (cf. 5) as:

[5*1 -zF qIRT- i = ( e C,/Co)io ( I - B ) ~ ZPLRT.

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204 CANADIAN JOURNAL OF CI-IEMISTRY. VOL. 37. 1859

Compar i son of [3*] and [5*] yields the following analogies:

Comparison of expressions used in crystal growth from .irapor phase and by electrolytic means

Burton, Cabrera, and This paper Franlc (7) (vapor phase) (electrolytic)

Supersat~~ration in the vapor (solution) PIP0 - 1 e - : ~ ~ R T - 1 Supersat~~ration on the surface cs/co - 1 C,/CL? - 1 Mean lifetime of adsorbed molecules T S (co/io) e-(l-")' "'RT

REFERENCES

1. GIBES, J. W. Collected works. Longmans, Green & Co., Ltd., London. 1928. 2. VOLMER, M. Kinetik der Phasenbildung. Verlag von Theodor Steinlcopff, Dresden und Leipzig. 1939. 3. KOSSEL, W. 0. Nachr. Ges. Wiss. Gottingen, 135 (1927). 4. STRANSKI. T. N. Z. ~hvs ik . Chem. 136. 259 (1928). 5. BECKER, < and D ~ R ~ N ; , W. Ann. ~ h y s i k , ~ e i p z k , 24, 719 (1035). 6. FRENKEL, I. J. Phys. Chem. (U.S.S.R), 9, 392 (1945). 7. FRANK, F. C. Disc~~ssions Faraday Soc. 5, 48, 67 (1949). CADRERA, N. and BURTON, W. I<. Dis-

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16. I<AISCHEW, R., MUTAFTSCAIEW, B., and NEXOW, D. Z. physil;. Chcm. 205, 341 (1956). 17. ROJTRR, IV. A,, JUZA, W. A., and POLUJ.ZN, E. S. Acta Physicochim. U.R.S.S. 10, 389, 845 (1939). 18. BOCKRIS, J . O'M., MEAL, W., COXWAY, B. E., and yo us^, L. J. Chem. Phys. 25, 776 (1956). 19. CONWAY, R. E., BOCKRIS, J. O'hl., and MEHL, W. J . Sci. Instr. 33, 400 (1956). 20. GWATHMEY, A. T. and BENTON, A. F. J . Chem. Phys. 8, 431 (1940). CAALMERS, B., ICING, R., and

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