Some theoretical aspects of pulse electrolysis

Surface Technology, 10 (1980) 81 - 104 © Elsevier Sequoia S.A., Lausanne - - Printed in the Netherlands

81

SOME T H E O R E T I C A L ASPECTS OF PULSE ELECTROLYSIS*

N. IBL

Laboratory of Industrial and Engineering Chemistry, Swiss Federal Institute of Technology, Zurich (Switzerland) (Received August 29, 1979)

Summary Some general aspects of pulse plating are treated, in particular mass transport and capacitance effects. A simple model which allows the visualization of the phenomena involved and which facilitates their estimation is used to describe the mass transport. The paper also deals with the factors which limit the useful range of conditions in pulse plating, i.e. the pulse generator, capacitance effects and mass transport effects. Mass transport effects are limiting in two ways: (a) depletion near the cathode actually limits the rate of metal deposition; (b) the depletion also causes p o w d e r formation. The current density distribution in pulse plating is discussed briefly in relation to the latter phenomenon.

1. Introduction Metal deposition by pulse electrolysis has received much attention in recent years. A symposium devoted to the subject [1], held in Boston in April 1979, had over 200 participants. There are many ways of performing non-d.c, electrolysis [2 - 5]. In this paper we shall focus our attention on electrolysis with galvanostatic square pulses separated by intervals with zero current in the external circuit (open circuit) (Fig. 1). Let us start with a few remarks a b o u t the three characteristic parameters of such pulse plating, i.e. the height of the pulse (pulse current density jp), the ON time T and the O F F time T'. (In principle most of the following discussion applies to any kind of pulse electrolysis with square pulses but in general we shall refer more specifically to the case of metal deposition.) Generally a specific feature of electrolysis compared with other methods of producing a metal or a c o m p o u n d is that we can prescribe the

*Paper presented in somewhat abridged f o r m at the International A m e r i c a n Elect r o p l a t e r s ' Society Symposium on Pulse Plating held in Boston, April 19 - 20, 1979.

82 B T

¢J

T'

Jp

Jm Time Fig. 1. Electrolysis with galvanostatic p u l s e s ; j m = jpT/(T + T') is the average current density.

driving force (i.e. the free energy) of the process by applying a given potential to the electrode or we can prescribe a certain reaction rate by applying a given current. The values of either variable can be selected as desired over a broad range. The progress of modern electronics allows us to make a much more far-reaching use of this advantage of the electrolytic m e t h o d by applying a current or a potential which is some function of time. This permits a great increase in the number of possible ways of varying the conditions. Indeed, in pulse electrolysis, instead of one parameter three parameters -- the pulse current density, the ON time and the OFF time -- can be varied independently over a very broad range. This results in two important features of pulse electrolysis. (1) Firstly, very high instantaneous current densities (see Section 4 and ref. 2) and hence very negative potentials can be attained. The high overpotential causes a shift in the ratio of the rates of reactions with different kinetics. This effect is illustrated in Fig. 2 which shows the partial currentpotential (j-E) curves for an electrode with two possible reactions (e.g. metal deposition a n d hydrogen evolution or the deposition of two metals as in alloy plating). At a given value of the electrode potential E the ratio of the current efficiencies of the two reactions is given by the ratio Jl/J2 of the two partial currents. If the slopes of the two partial j - E curves are different the ratio Jl/J2 can change markedly when the total current It = ]1 + ]2 is substantially increased, as is the case for the pulse current density jp which is often much larger than the current density normally applied in d.c. electrolysis. For this and other reasons the current efficiency in pulse plating can be very different from t h a t in d.c. plating. The high overpotential associated with the high pulse current density also greatly influences the nucleation rate because a high energy is available for the formation of new nuclei. (2) The second characteristic feature of pulse electrolysis is the influence of the O F F time during which important adsorption and desorption phenomena as well as recrystallization of the deposit occur. Because of the

83

t

p~

Voltage E Fig. 2. The change in current efficiency for c o m p e t i n g reactions with different kinetics produced by increasing the overall current density Jt = Jl + J2 where J l and J2 are t h e partial current densities for reactions 1 and 2 respectively and 01 = Jl efficiency for reaction 1.

~Jr is the

current

great sensitivity of electrocrystallization to adsorption the properties of the deposit can depend to a large extent on the pulse conditions, especially if these are varied over a broad range. However, although the range of possible variations of the parameters is quite large it is not unlimited. In the following we shall discuss the useful range of pulse conditions and at the same time we shall elaborate on some general theoretical aspects o f pulse electrolysis. At one end of the spectrum are high values of T and T', i.e. conditions close to those for d.c. electrolysis. In the following we shall deal more specifically with the other end of the spectrum, i.e. with high instantaneous current densities and short pulse times. The main limiting factors are then (a) the pulse generator, (b) the capacitive charging of the double layer and (c) mass transport effects.

2. The pulse generator The limitations due to the pulse generator are summarized in Table 1. In column 2 are listed the performances of the generators that are, to our knowledge, currently available commercially. These generators have been designed for pulse plating. The shortest pulse length is about 50 ps and the highest current up to 2000 A; the pulse current density attainable depends of course on the size of the electrode. Column 3 shows the characteristics of a generator which allows pulse lengths down to less than 1 Us but which is less suitable for pulse plating because it permits a very high voltage but only a low pulse current. We use it in conjunction with a transformer which de-

84 TABLE 1 Characteristics of commercially available pulse generators Characteristic

Generators designed for electroplating

Special pulse generators

Pulse amplitude Pulse voltage Pulse length Pulse frequency Duty cycleb

Up to 200 A; 2000 A Up to 20 V From 50 ps; 100 ps 10 Hz - 1 kHz (possibly 10 kHz) 0.01 0.9

Up to 10 A (100 A)a Up to 2000 V (200 V) a From 0.1 ps (1 ps)a 1 Hz - 1 MHz 0.015

-

aThe values in parentheses are for generators used together with a transformer. bWhe duty cycle equals the pulse length divided by the period.

creases-the voltage but increases the attainable pulse current to 100 A. Generators th at are m o r e powerful than those described here can be built; we k n o w of one at least but that has been c on st ruct ed by its users. However, there is little d o u b t that if pulse plating becomes m ore widely e m p l o y e d cheaper and mo r e powerful generators will com e on the market.

3. Capacitance effects The electrical double layer at the e l e c t r o d e - s o l u t i o n interface is the interfacial region where there is a space charge (i.e. where the electroneutrality co n d ition t hat holds for the bulk solution is n o t fulfilled). It can be a p p r o x i m a t e d as a plate capacitor with an interplate distance of a few ~lngstrSms (hence the high capacitance). This distance is several orders of magnitude smaller than the thickness of the diffusion layers discussed in Section 4. ( F o r f u r t her details the reader is referred to the t e x t b o o k s on electrochemistry by Milazzo [6] and Bockris and Reddy [7] .) The double layer has a high capacitance and charge must be provided in order to raise its potential to the value required for metal deposition at a rate corresponding to the applied pulse current which is supplied by the generator (Fig. 3). The applied pulse cur r ent is represented by a thin line in Fig. 3. At the very beginning of the pulse the cathode potential has the equilibrium value and no metal is deposited. T he whole current is used to charge the double layer. However, as the double layer is being charged the potential becomes m o r e negative and t he part of the current JF corresponding to metal deposition (the faradaic cur r ent which is represented by a bold line in Fig. 3) thus increases with time and eventually reaches the value of the total pulse current. The charging of the double layer is then terminated. Depending on the conditions the charging time is longer or shorter than the pulse length. In Fig. 3(a) the charging time is negligible com pared with the ON time and the cur r ent of metal deposition coincides virtually everywhere

85

0 (a)

JP

--Jm

t~ ~

faradaiccurrentJF ~ . __/____ ~ t : t a [ currentJt

(b)

(c)

(d)

Fig. 3. The influence of the capacitance of the electrical double layer: (a) negligible effect; (b) medium effect; (c), (d) strong effects.

with the overall current provided by the generator. In Fig. 3(b) the charging time is comparable with but smaller than the pulse length. In Fig. 3(c) the charging time is longer than the pulse length and the faradaic current never reaches the value of the total current. Let us now consider the situation after the end of a pulse. The faradaic current does not drop immediately to zero because the discharging of the double layer (and thus the decay of the potential) takes some time. In Fig. 3(c) the discharge time is shorter than the O F F time whereas in Fig. 3(d) it is longer. In the latter case the faradaic current never drops to zero because the double layer is never completely discharged. The current of metal deposition is then more or less strongly damped compared with the total current and only slightly fluctuates around the average current density ]m : d.c. conditions are then being approached. This limits the choice of the pulse conditions because it is clearly pointless to carry out a pulse electrolysis in such a way that the current is more or less direct. In a few experiments with copper and cadmium we verified t h a t deposits are then obtained with properties that are virtually the same as those observed in d.c. conditions although the current density and length of the pulse were such that we might have expected to observe the effects of pulsing. A n o t h e r double-layer effect that we might expect a priori is a low current efficiency for metal deposition in the cases shown in Figs. 3(b) - 3(d) since only part of the pulse current is then used for the discharge of the metal cations. In fact this is not so. During the O F F time the total current flowing in the external circuit is zero and therefore the total current flowing

86

10 for T :~ t~ neg[igibte charging effect 3

m

u

for T<< t~

0.3

substantial charging effect 0.1

0.03 J

0.1

0.3

,

i

i l l ,

I

,

i

l I I I ,11 I ~ l I l l

,,ii

3

10 ip

30

100

|

300

[A/cm2]

Fig. 4. The charging t i m e o f the double layer v s. the pulse current density in 0.75 M CuSO 4 + 1 M H 2 S O 4 : C = 5 0 p F c m - 2 , j o - - 5 $ 10 - 3 A c r e - 2 .

through the electrode is also zero; however, the capacitive current and the faradaic current are not zero. The double layer is discharged and this discharge current is exactly compensated by a faradaic current. The charges used for the charging of the double layer during the pulse are eventually used in depositing metal but during the OFF time. There is a partial shift of the metal deposition from the ON time to the OFF time but without loss of current efficiency. Thus this effect does not limit the useful pulse conditions; t h e y are limited only by the approach to d.c. conditions. This view is confirmed by experimental results which we shall discuss later (see Table 2). First, however, let us sketch the principle of the evaluation of the magnitude of capacitance effects. A first approximation is obtained as follows. From the current-potential relation (eqn. (2)) we calculate the overpotential for a current density equal to the total pulse current density jp. The next step is to compute the time necessary to raise the electrode potential to the value 7, assuming the whole current to be capacitive. This gives a lower limit for the charging time because as the time increases an increasing part of the total current is used to deposit metal. Figure 4 shows the charging time t* estimated in this manner for the deposition of copper from an acidified CuSO4 solution (C = 50 pF cm -2, J0 = 5 mA cm-2): t c decreases with increasing jp. For pulse lengths substantially longer than the times given by the line in Fig. 4 charging effects are negligible but for pulse lengths smaller than

¢000

~--I

I T"I X

O01.~

I

0 X

~.~ 0

x x

Q

X X X

C,

88 t* they become increasingly important. The double-layer capacitance of 50 p F cm-2 used in the calculation is a typical value. We have also carried out a more refined calculation which is presented in greater detail elsewhere [8]. We start by writing the total current Jt as equal to the sum of the capacitive current j¢ and the faradaic current JF : (1)

Jt = Jc +iF The faradaic current is given by the usual kinetic exponential expression

JF =jo[exptaZF'~t\--~/-

exp{

( 1 - - a ) z F ~ }}

(2)

and the capacitive current obeys the relation

Jc = dQ/dt = CdT?/dt

(3)

On combining eqns. (1) - (3) and integrating we obtain the charging time tc and the discharging time td as functions of the relevant variables. The calculation has to be performed numerically with a computer [8]. The charging and discharging times calculated in this manner are shown in Table 2 (columns 5 and 8) for various values of jp (column 3). The charging and discharging times to, t* and td (columns 5, 6 and 8) were calculated for a given jp w i t h o u t taking T and T' into account (i.e. it is assumed that T and T' are long enough to allow a complete charge and discharge of the double layer). As can be inferred from Fig. 3(b), there is the complication that Jc drops asymptotically to zero at the end of the complete charge or discharge. Therefore in principle the time needed to raise the potential of the cathode from the value corresponding to JF = 0 to that corresponding to JF = Jp is infinite. Similarly the time needed for the decrease in the potential from the value at JF = ]p to that at JF = 0 is alSO strictly infinite. An arbitrary definition is thus necessary. In Table 2 t* is the charging time calculated as in Fig. 4 (see earlier), i.e. it is assumed that Jc = Jt = Jp during the whole charging. In contrast, tc is defined as the time that elapses while JF increases from zero to JF = 0.99jp (or equivalently while Jc decreases from Jc = Jt = Jp to Jc = 0.01]p ). Similarly td is the time needed for JF to decrease from J F ---Jp = Jt to JF = 0.0lip during the OFF time. For tc/T or t~/T > 1 and td/T' > 1 the quantities to, t~ and td are virtual values because there is not enough time during the pulse to charge the double layer completely or during the O F F time to discharge it c o m p l e t e l y ; j F does n o t rise then to j , but only to some smaller value (iF)max < Jp and during the OFF time it does not drop to zero but only to (JF)m~ > 0 (see Fig. 3(d)). A measure of the degree of damping of the pulsed current is the quantity A defined as A = (jp T - -

T

f JF dt)/jm T'

0

(4)

If there is no damping (Fig. 3(a)) JF = Jp and A = 0. In the case of complete damping JF = Jm all the time and A = 1 because of the definition of the average current density:

89

jm(T' + T) =jpT

(5)

The values of A for various conditions for the electrolysis of acidic CuSO4 solutions are shown in column 9 of Table 2. A further interesting quantity is the ratio T

f jFdt/jp T = 0t

0

(6)

which represents the fraction of the total a m o u n t of electricity used to deposit metal during the pulses, i.e. the current efficiency for metal deposition, assuming that metal is deposited only during the pulses. The calculated values of 0t are shown in Table 2 (column 10) where they are compared with the experimental current efficiencies of metal deposition (column 11); as expected 0t for large tc/T and tc/T' (or for large A) is very small and the experimental current efficiency 0e is much larger. This confirms the view expressed earlier in this section. The calculation of the capacitance effects shown in Table 2 is reported in more detail in a separate paper [8].

4. Mass transport effects

4.1. Model of a duplex diffusion layer 4.1.1. Qualitative considerations The limitations on the useful pulse conditions due to mass transport effects are caused by the depletion of cations in the diffusion layer. However, for short pulses we have to distinguish between the depletion in two distinct diffusion layers. Let us first discuss this concept briefly. Cheh has treated the theory of mass transport in pulse electrolysis in several papers [2, 9, 1 0 ] . The justification for dealing with this subject here is that a different approach is used: a simplified view is presented which shows the main effects qualitatively and allows them to be estimated simply. Figure 5 shows the concentration as a function of the distance x from the electrode, the concentration profiles (full lines) being represented by straight lines. This is an extension of the Nernst approximation of a linear concentration profile in a d.c. diffusion layer to the case of pulse electrolysis [6, 11]. In the immediate vicinity of the cathode the concentration pulsates with the frequency of the pulsating current: it decreases during the pulses and relaxes in the interval between them. We thus have a pulsating diffusion layer close to the cathode. If the duration of the pulse is short the diffusion layer does n o t have time to extend very far into the solution and it does not reach the region where convection takes over the mass transport [11]. Therefore the metal deposited during the pulse must be transported from the bulk of the solution towards the pulsating diffusion layer by diffusion, which means that a concentration gradient also builds up into the bulk of the electrolyte. The extent of this diffusion layer corresponds essentially to the extent of the layer that would be obtained under the same hydrodynamic conditions in

9O u

~N

8p

"5

8s M

8

)

c~

D

Distance x f r o m

the cathode

Fig. 5. C o n c e n t r a t i o n profiles o f the t w o diffusion layers in pulse electrolysis at the end o f a pulse. The b r o k e n lines (-- - - --) s h o w t h e r e c o v e r y o f the c o n c e n t r a t i o n in the pulsating diffusion layer during the O F F time. (T < t 1 < t 2 < 0.)

d.c. electrolysis. Through this outer diffusion layer cations are also supplied towards the cathode during the O F F time and it is this supply which allows the relaxation of the pulsating diffusion layer during the O F F time. The outer diffusion layer is essentially stationary. According to Fick's law the flux density N of diffusion ( i . e . the number of moles diffusing per unit time through a cross section of unit area) is proportional to the concentration gradient: (7)

N = Ddc/dx

(the symbols used are defined in the Nomenclature). During a pulse the concentration profile in the pulsating diffusion layer is approximated in our model by a straight line. Equation (7) thus takes the form gp

= D(c' e --

Ce)/5,

(8)

and for the pulse current density we can write jp = zFNp

= zFD(c'e

--

Ce)/Sp

(9)

where Np is the flux density of the cations in the pulsating layer during a pulse. (Equation (9) is valid only if the current efficiency for metal deposition is 100%. This is the situation assumed throughout Section 4 unless otherwise indicated. It is also assumed that the capacitance effects discussed in Section 3 are negligible.) The flux density of the cations is equal to the number of moles of metal deposited per unit time per unit electrode area during the ON time. The pulse current density jp is proportional to the slope of the concentration profile during the pulse. During the O F F time (when no current flows through the cathode) the concentration gradient at the interface must be zero because of eqns. (7) and (9), i.e. the concentration profile terminates with a horizontal branch at the interface, as shown by the broken lines in Fig. 5.

91

In contrast with Np, the flux density Ns through the outer stationary diffusion layer does n o t drop during the O F F time b u t continues in the same way during the O F F time and the ON time. Ns is therefore proportional to the average current density Jm and for a linear concentration profile we can write Jm = zFNs

= zFD

(Co - -

c'e)/6~

(10)

Summing up, we can say that the current density jp during a pulse is proportional to the concentration gradient in the pulsating layer during the pulse whereas the average current density is proportional to the concentration gradient in the stationary outer diffusion layer. The first gradient can be made very large by applying very short pulses, the thickness 5 p of the pulsating diffusion layer being then very small (see Section 4.1.2, eqn. (12) or (14)). This explains why extremely high instantaneous current densities can be used in pulse electrolysis. We have worked with jp up to 250 A cm- 2. However, the slope of the concentration profile in the stationary layer is then much smaller than that in the pulsating layer and Jm < ]p. Indeed, from the geometry of the model (see Figs. 5 and 9) it appears that the gradient in the stationary layer cannot be made larger than the gradient that occurs at the limiting current in d.c. conditions. We shall come back to this point later. An interesting question is that of the concentration which prevails at the interface between the cathode and the solution in pulse electrolysis. One is t e m p t e d to say that this concentration is higher than that in d.c. electrolysis. In fact this depends on h o w the comparison between d.c. and pulse electrolysis is made. At present there is no consensus in the literature a b o u t the best way of making this comparison. Sometimes one compares the results obtained with a pulsed current (p.c.) at a]p equal to the current density for d.c. In this case the interfacial concentration of metal cations is indeed higher with p.c. than with d.c., as has been shown by Cheh [9]. However, this is not necessarily the case if the comparison is made for a current density in d.c. conditions equal to the average current density in p.c. conditions. Indeed the slope of the concentration line for d.c. is then the same as the slope of the line NM for p.c. (see Fig. 5). Thus the interfacial concentration in the d.c. electrolysis is Ce~. For p.c. the interfacial concentration oscillates around Ce' and can be on the average much smaller than c~'. This is particularly true if the conditions (high ]p and small T) are such that the interfacial concentration drops more or less to zero at the end of a pulse (the situation in Fig. 9(a); see also Section 4.2.1). Here a few remarks a b o u t the nature of the approximation made in the above model are appropriate. For short pulses* it is certainly possible to dis-

* S h o r t pulses m e a n that ~p ffi (2DT) 1/2 is s u f f i c i e n t l y small, s a y ~N/~p > 3 . U n d e r the m o s t c o m m o n h y d r o d y n a m i c c o n d i t i o n s ~ N is i n the range 0 . 0 1 - 0 . 3 r a m . F o r D = 1 0 - 5 c m 2 s - 1 ( w h i c h is a c o m m o n value for salts such a s C u S O 4 ) ~ p is in the r a n g e 0.003

- 0.01 mm

for T = 5 - 100 ms.

92 tinguish roughly between a diffusion layer that pulsates and, further towards the bulk, a diffusion layer that is stationary. The approximation of a linear concentration profile made here means that the transition between the two layers is assumed to be sharp: it takes place at point N. In fact the transition is continuous and the damping of the concentration oscillations increases progressively with increasing distance x from the cathode. The simplification made by assuming the transition to be sharp is similar to that underlying the well-known concept of a Nernst diffusion layer which was introduced at the beginning of this century [ 11, 12]. Convection usually participates in mass transport towards an interface. However, in the immediate vicinity of a solid interface the h y d r o d y n a m i c flow and therefore mass transport by convection are negligible because of the frictional forces at the interface. Thus we may distinguish roughly between a zone near the cathode where mass transport by diffusion is predominant and a region located further towards the bulk solution where mass transport takes place essentially by convection and where a uniform concentration is ensured by convective mixing. (In the case of ionic species mass transport by migration may also occur in both regions to a greater or lesser extent but this does not m o d i f y our argument essentially.) The approximation of the concentration profile by a straight line up to the point where the bulk concentration is reached implies that there is a sharp transition at that point between the stirred and the unstirred regions (in Fig. 5 this transition is assumed to take place at point M). (For a more detailed discussion of the Nernst concept of an unstirred diffusion layer the reader is referred to the literature [7, 11, 12] .) In fact the transition is progressive; the concentration profile is a straight line only quite close to the electrode and becomes curved with increasing x. Nevertheless, the Nernst model has proved useful over the years. Similarly the simplification of a duplex diffusion layer with a sharp transition in N between the two layers is helpful for the visualization of the main phenomena involved in pulse electrolysis. Furthermore, a simple quantitative estimate of the main quantities of practical interest is also possible, as will be seen in the next section.

4.1.2. A n estimate o f quantities o f practical interest and qualitative considerations on the build-up o f a quasi-stationary state Let us consider first the case of a single pulse, i.e. a pulse at the very beginning of electrolysis. Before the start of electrolysis the concentration is Co everywhere. After the pulse current has been switched on the concentration near the cathode drops and a diffusion layer builds up (Fig. 6). We again approximate the concentration profile by a straight line; its slope is proportional to jp according to eqn. (9) and thus remains constant during a galvanostatic pulse. Therefore the increase in depletion with increasing time t results in a parallel displacement of the concentration line toward lower concentrations. The full line shows the concentration profile at the end of the pulse (t = T). The thickness of the diffusion layer increases with time and reaches the value 5 u at the end of the pulse. We first calculate 5p. The number of moles of metal deposited by the current per unit electrode area during the

~~ 0 ¢e

~t~. 8p

u

.~

93

"

"N

B

..'"

0..°'*'°°°° gO

Distance x from cathode

Fig. 6. C o n c e n t r a t i o n profiles o f t h e d i f f u s i o n layer at various t i m e s ( t 1 < t 2 < T) d u r i n g a single pulse.

pulse is j , T / z F . This is equal to the number of moles of cations removed from the solution which is given by the hatched area in Fig. 6, i.e. by (Co -Ce)6p/2 where ce is the interfacial concentration reached at the end of the pulse, jp is related to the slope (Co -- Ce)/6, of the concentration line through eqn. (9) (with Co instead of c~'). We can thus write (Co -- ce)6p _ 2

jp T zF

_ D(co

-- ce)T

(11)

6p

or

59 ---- (2DT)I/2

(12)

It is remarkable that the thickness of the diffusion layer is independent of the concentration co and of the pulse current density jp. It depends only on the diffusion coefficient D of the cations and on the pulse length T, being proportional to the square root of the product of these two quantities. Equation (11) also allows us to calculate the interfacial concentration Ce reached at the end of the pulse. When the pulse length is equal to the so-called transition time T the interracial concentration drops precisely to zero at the end of the pulse (line OB in Fig. 6). We obtain T by putting Ce = 0 in eqn. (11) and rearranging: T = (zF)2c2D/2j2p

(13)

A pulse with T = T will be called a limiting current pulse because jp then becomes the limiting current density at the very end of the pulse. The above approximate approach can be regarded as a particular case of a procedure called recently the "simplicissima m e t h o d " in which the relations involved are quite generally linearized and which gives a fair approximarion in a variety of cases* [13].

* T h e a u t h o r was n o t a w a r e o f t h i s w h e n h e was w r i t i n g the present article a n d is grateful t o Professor Le G o f f a n d Dr. S t o r c k for d r a w i n g his a t t e n t i o n t o ref. 13.

94 The exact relations derived by integrating Fick's differential equation are [6, 7, 14] 5p = 2(DT/Tr) 1/2

(14)

.~ = zr(zF) 2 c 2 D / 4 j ~

(15)

which show the same dependences on the relevant variables as the approximate eqns. (12) and (13) obtained very simply with the assumption of a linear concentration profile. Only the numerical coefficients are somewhat different. After the end of the first pulse the concentration recovers during the O F F time but the initial concentration is not fully restored because at the end of the first pulse there is no concentration gradient to the right of point N and hence no supply of cations by diffusion from the bulk towards the diffusion layer established during the pulse. Therefore during the O F F time the solution depletes to the right of N and the next pulse starts from a concentration lower than Co. For some time from pulse to pulse the concentration at N decreases and the depleted zone to the right of N extends itself further into the bulk of the solution. This corresponds to the build-up of the outer diffusion layer with the concentration profile MN in Fig. 5. The buildup of this layer lasts for a period of a few seconds or less. (This time can be estimated from eqn. (12) which also describes roughly the unsteady state growth of the outer diffusion layer: with 5N = 0.01 - 0.1 m m and D = 10 -5 cm 2 s-1 we obtain 0.05 - 5 s. In natural convection where 5N Can be, say, 0.3 m m this time will be 45 s.) The build-up is terminated when the depleted zone has extended itself so far that it reaches into the region where convection (which increases with increasing x because of the diminishing influence of frictional forces at the wall) prevents its further growth by equalizing the concentration through the h y d r o d y n a m i c movement of the liquid. A stationary or quasi-stationary state is then established. This is the situation which we discussed in Section 4.1.1 and which is represented in Fig. 5. Near the cathode the concentration oscillates, but between the same values, the same cycle being repeated continually. At the end of each pulse the same concentration profile (line ON in Fig. 5) is established. In contrast, the outer diffusion layer (line MN in Fig. 5) is truly stationary. As has been pointed out already, the boundary between the two layers is n o t as sharp as suggested by the point N in Fig. 5. Nevertheless, 8 p gives a good measure of the depth of penetration of the concentration oscillations into the solution. Note that in the quasi-stationary state 5p is still given by eqn. (12) since it does not depend on concentration and it is thus immaterial whether the pulse starts from the concentration c o or c~, which is the lower concentration t h a t is established at N in the steady state. However, this is strictly true only for T/O ~ 1; otherwise the mass balance on which eqn. (12) is based has to be somewhat modified to take into account the number of cations that enter

95

from the bulk into the growing diffusion layer during the pulse. Equation (12) has then to be replaced by

6p = {2DT(1 -- T/O)} 1/2

(12')

The value of Ce is obtained from the following consideration. In the stationary (or quasi-stationary) state the n u m b e r of cations diffusing from the bulk because of the slope of the concentration line MN during a whole period (0 = T + T ~) must be equal to the number diffusing to the cathode because of the slope of the concentration line ON during the ON time T (this number corresponds to the quantity of metal deposited during a pulse). During the initial unsteady state at the beginning of the pulse electrolysis the concentration at N drops until this condition is fulfilled. We thus obtain c" by making a mass balance. A more detailed account of the calculation is given in ref. 15. The result is

c~ = Co

zFD

0 ~N --

2DT

--

(16)

Ce' is the value which must be inserted into eqn. (13) instead of Co in calculating r for the quasi-stationary state. Another quantity of interest is the average limiting rate* n~g of metal deposition which is the time average of the a m o u n t of metal deposited per unit time when limiting current pulses are used (i.e. T = r, Ce ----0 at the end of a pulse). The corresponding average current density is/rag = zFn~g (see also eqn. (5)). The ratio of this quantity to the limiting current density for d.c. under the same hydrodynamic conditions is*

jmg_ n~ng_ [ {2DT(1--T/O)} 1/2 (O jgg

n;g

6N

~

]--1

) --1

+ 1

(17)

On multiplying the numerator and the denominator of the fraction on the right-hand side by (0/T) 1/2 we obtain

Jmg - {{2XY(I--y)}I/2(I --1) +I] -I

(17')

Jgg where X = DO/5N 2 and Y = T/O.By introducing the dimensionless variables X and Y we reduce to two the number of variables on which the ratio Jmg/Jgg= n*g/n*, depends. This ratio is plotted in Fig. 7 as a function of X and Y. By using the definitions of these two quantities it is possible to calculate from

*The rates nmg and ngg are equal to the flux densities in the solution. We have denoted the flux densities in the solution by N to be in agreement with the IUPAC nomenclature recommendations [ 1 6 ] but have retained the symbol n for the rates to be in agreement with our earlier paper [ 15 ]. tEquations (16) and (17) are eqns. (11) and (16) of ref. 15 slightly corrected according to footnote 5 on page 55 o f ref. 15.

96

0.8

~\-~

.9

-

~

-

,'o !

0.2--

I lO-6

10 -~

.h 10-4

10 -3 X { :

10 "2 D8/6

~'~

1

10

100

z )

Fig. 7.Jmg/jgg vs. X for various values of Y: curves A, obtained by Cheh's method (eqn. (16) in ref. 9); curves B, calculated from eqn. (17') by J. C. Puippe).

the plot the ratio Jmg/Jgg for any set of D, T, 0 and 5 N. Typical figures for plating in aqueous solutions are D = 10 -5 cm 2 s-1 and 5N ranging from about 0.01 to 0.001 cm for moderately to strongly agitated baths. Curves B were calculated using eqn. (17') and curves A on the basis of the more sophisticated treatment given by Cheh [9]. The two sets of curves show the same trend and are close. The inaccuracy of the values calculated using the simplified model (eqn. (17')) is probably of the same order as the inaccuracy due to the uncertainties which usually prevail with respect to the diffusion coefficient, the transport number and the current efficiency. (Curves A and B and eqns. (16) and (17) have been derived for the case of 100% current efficiency for metal deposition and a great excess of neutral electrolyte (compared with the cations of the deposited metal). Quite often these conditions are not fulfilled, at least not exactly. The correction which then has to be made for the current efficiency and for the migration of the metal cations under the influence of the electric field (transport number) cannot usually be achieved accurately. Furthermore, the solutions used in plating are usually concentrated with respect to the metal ions; their diffusion coefficient may thus vary appreciably over the thickness of the diffusion layer and is usually not accurately known.) Whenever they are available, more exact calculations such as those made by Cheh [2, 9, 10] are to be preferred but we expect t h a t the simplified m e t h o d presented here can still be used to advantage in situations where more exact calculations have not yet been carried out or are very difficult to make. In addition, the simplified m e t h o d does at least provide an essentially correct physical insight into the phenomena involved. 4.2. L i m i t a t i o n s d u e to mass transport

Mass transport effects limit the useful range of conditions in pulse electrolysis (a) by actually limiting the rate of metal deposition and (b)

97 6N 5p

o

6s M

2

8

0 Distance x from cathode

Fig. 8. C o n c e n t r a t i o n profiles o f t h e d i f f u s i o n layer at various t i m e s ( t 1 < t 2 < T < t 3 < T) d u r i n g a pulse l o n g e r t h a n t h e t r a n s i t i o n t i m e ( T > T).

by causing powder formation. Furthermore we shall distinguish two cases: (1) depletion mainly in the pulsating diffusion layer; (2) strong depletion in both diffusion layers.

4.2.1. The limitation o f the rate o f metal deposition Let us first compare experiments made at the same T and T' but with different jp. If the pulse current density ]p is increased the average current density increases and the interfacial concentration Ce that is reached at the end of the pulse decreases, as can be inferred from eqn. (11). Further, the transition time T decreases according to eqn. (13). At a certain value jpg of the pulse current density the interfacial concentration just drops to zero at the end of the pulse. The transition time r has then decreased to the value of the pulse length T and we thus obtain the value of jpg by setting T = T in eqn. (13). In Section 4.1.2 we called such pulses limiting current pulses. The corresponding average current density Jmg is plotted in Fig. 7 and can be calculated from eqn. {17') or from the correlations given by Cheh [9]. It is tempting to regard ]pg and ]mg as the m a x i m u m values of the instantaneous and the average current densities for a given T and T' but in fact this is not the case. jpg and Jmg are only the m a x i m u m instantaneous and average current densities at which the current efficiency for metal deposition is still 100%. Indeed, if we select for a given T and T' a pulse current density higher than ]pg all that happens is that the interfacial concentration drops to zero before the end of the pulse (since now r < T) but at least for times t < r the -current density of metal deposition is greater than jpg. The situation in the diffusion layer is illustrated in Fig. 8 which shows the concentration profiles at various times t for an electrolysis with T ~ T. For t < r the concentration profiles remain parallel since the rate of metal deposition and thus the slope of the concentration lines remain constant. However, at t = T the metal deposition switches from deposition at a constant rate to deposition with a

98

M u

J-"~'~Slope

= Jm

"E

g

(_.)

Slope = jp ,/

0

(a)

=

0

Distance x from cathode

= Distance x from cathode

(b)

Fig. 9. Depletion close to the cathode (a) in the pulsating diffusion layer only and (b) in both

diffusion

layers.

constant interfacial concentration since for t > r the interfacial concentration cannot drop further. Nevertheless, the metal continues to be deposited and metal ions continue to be removed from the solution so that the zone with depleted concentration in the immediate vicinity of the cathode grows. Because the beginning of the concentration line (at x = 0) is fixed this can take place only through a decrease in the slope of the concentration line (see Fig. 8). Therefore the rate of metal deposition drops and if the overall current is kept constant an increasing part of it must now be used for hydrogen evolution. The current efficiency is therefore no longer 100%. However, in spite of the decrease in the rate of metal deposition during the interval from r to T the slope of the line MN and hence the average rate of metal deposition may be somewhat greater than for electrolysis with a pulse current density equal to jpg [15]. Even if the plating is carried out with limiting current pulses (]p = ]pg ) the average current density may be small, depending on the circumstances. If the ratio T/O is small the average current density may be much smaller than the limiting current for d.c. under the same h y d r o d y n a m i c conditions. For instance it is found from Fig. 7 that for Y = T/O = 0.001 and for X = D O / S N2 = 0.1 the ratio ]mg/Jg~ is only about 0.06. However, the limiting pulse current density for the same T and 0 is then about 60 times larger than the limiting current for d.c. We then have the situation shown in Fig. 9(a): strong depletion in the pulsating diffusion layer but almost no depletion in the outer stationary diffusion layer. In general the depletion in the outer diffusion layer increases with increasing average current density. If the latter approaches the limiting current density for d.c. the depletion in the outer layer is strong. We then have the situation shown in Fig. 9(b). From the geometry of the figure it is apparent t h a t the slope of the line NM and therefore the average rate of metal deposition in pulse plating cannot be larger than at the limiting current in d.c. plating. However, this statement implies that 5N in Figs. 9 and 5 is the same as in the corresponding d.c. experiments. This is generally true inasmuch as the h y d r o d y n a m i c conditions are the same in both cases. However, they may be modified to some extent by the pulsed current itself. For instance in

99

6

\1/

\11 (a) A /AN /

A /A\ \N//J"

/

"~

/

(b) Fig. 10. Mass transport to a surface profile: the broken line shows the boundary of the diffusion layer, h is the characteristic height of the profile and ~ is the thickness of the diffusion layer; (a) ~ ~ h, microprofile, powder formation; (b) ~ < h, macroprofile, no powder formation.

natural convection caused by differences in density in the solution the pulsation of the concentration near the cathode interferes with the b u o y a n c y force which causes the hydrodynamic movement and ~N m a y be modified. Another example is hydrogen evolution. In rhenium deposition we have observed under certain circumstances a current efficiency for metal discharge which is five times higher in pulse plating than in d.c. plating under corresponding conditions [ 17]. Since hydrogen evolution is a very effective way of stirring, 8N m a y in such a case be substantially different for pulse plating and d.c. plating.

4.2.2. Powder formation and current distribution problems When the solution in the immediate vicinity of the cathode is strongly depleted with respect to the metal cations, very rough or p o w d e r y deposits are obtained; this is a quite general rule [4, 18 - 2 0 ] . In d.c. plating or d.c. electrometallurgy this prevents the use of a current density close to the limiting current density. In pulse plating the influence of depletion on roughness m a y be very different depending upon whether we have the situation of Fig. 9(a) or 9(b). Let us first briefly recall the mechanism of p o w d e r formation at the limiting current [4, 18 - 20] with the help of Fig. 10 which shows

100 schematically the diffusion layer over a rough surface. The asperities of average height h may be already present before the electrolysis or may develop at the start of plating because of surface heterogeneities. Well below the limiting current the current distribution is normally governed mainly by the geometry of the system, the conductivity of the bath and the overpotential due to the slowness of charge exchange at the cathode-solution interface (activation overpotential) (secondary current distribution) [21]. If the variation of overpotential with current density is sufficiently large it equalizes the current distribution and there is no tendency towards amplification of any initial roughness. At the limiting current, however, the rate of metal deposition is controlled by the mass transport of the cations towards the cathode and this mass transport is the main factor governing the current distribution (tertiary current distribution). Its effect on the structure of the electrodeposit depends on whether the characteristic height h of the roughness is smaller or larger than the thickness of the diffusion layer (microprofile (Fig. 10(a)) or macroprofile (Fig. 10(b)) respectively). For the microprofile the peaks are favoured with regard to diffusion because the free cross section for diffusion increases with increasing distance from an asperity. Therefore more metal is deposited On the peaks than in the recesses. There is thus a strong tendency towards amplification of the initial roughness and eventually powder is formed. In d.c. plating the thickness of the diffusion layer is usually about 0.01 - 0.3 mm and the amplification mechanism sets in for a r o u g h n e s s w i t h a height as small as that order of magnitude. However, in pulse plating the pulsating diffusion layer can be made much smaller by using very short pulses. Peaks with heights of the order of 0.01 - 0.3 mm are now no longer small compared with the thickness of the diffusion layer. Figure 10(b) shows this situation: the use of short pulses turns a microprofile into a macroprofile. The diffusion layer now follows the profile, the peaks and the recesses are equally accessible for diffusion and the asperities are not amplified (only much smaller roughness is amplified and then only to a very small extent). Therefore powder formation due to depletion effects does n o t lead to an upper limit on the useful range of pulse current densities: even for limiting current pulses or pulses with jp ~ jpg and T ~ T powder formation does not occur provided that the pulsating diffusion layer is sufficiently thin and the depletion in the outer stationary diffusion layer is small, i.e. provided that the situation is as in Fig. 9(a). In contrast, strong depletion in the outer stationary diffusion layer (Fig. 9(b)) causes powder formation and dendrites grow in a fashion similar to that observed in d.c. plating. This p h e n o m e n o n limits the increase in the average current density in pulse plating and prevents us from letting it approach the value which the limiting current density would have for d.c. under the same h y d r o d y n a m i c conditions. We have verified these conclusions by depositing copper, cadmium and silver with very short pulses and very high pulse current densities [22 - 26]. The experiments were carried out in a flow apparatus which ensured welldefined h y d r o d y n a m i c conditions and small values of 8N [23, 24]. The

101

values of 8p and of r were calculated along the lines indicated in Section 4.1. The roughness of the electrodeposit was measured with a mechanical profilometer. A selection of the results is presented in Table 3 together with the experimental conditions. In some cases the ratio T/r was less than unity, i.e. the interfacial concentration did not drop to zero during a pulse. In other experiments the ratio T/r was very high, i.e. the metal was deposited at the limiting current for most of the duration of a pulse. In spite of this no marked increase in roughness was observed although current densities of up to 250 A cm- 2 were used. The thickness of the pulsating diffusion layer was smaller by two to three orders of magnitude than in normal d.c. experiments. The results are therefore in agreement with theoretical expectations. In m a n y cases the roughness was smaller for large values of T/r than for small values, i.e. there was some levelling. Indeed, a tertiary current density distribution should be uniform over a macroprofile and recesses should thus be preferentially filled through geometric levelling. Therefore this result is also in agreement with the model for the influence of mass transport on roughness under limiting current conditions. An improvement of the micro throwing power has also been observed by Cheh under certain circumstances. However, an improvement in current distribution is to be expected theoretically only for a microprofile (which is turned into a macroprofile by the use of short pulses) and only when the mass transport of the cation more or less controls the current distribution. For a secondary current distribution the situation is quite different [21] : the distribution is then more uniform the larger the dimensionless Wagner number which is defined as Kd~ / dj Wa (18) L where ~ is the specific conductivity of the solution and L is a characteristic geometric length. If Tafel's law 77 = a + b l n j

(19)

holds, the slope d,~/dj of the potential-current curve is inversely proportional to j so that Wa decreases with increasing current density. We should therefore expect a poorer throwing power in pulse plating with its high instantaneous current densities than in d.c. plating*. A decrease in throwing power with increasing pulse current density has in fact been observed by Andricacos et al. [27]. It m a y well be the case, however, that the factors affecting current distribution are actually often more complicated than would correspond to the foregoing simple considerations. In some of the experiments whose results are presented in Table 3, in particular for silver and cadmium, it was observed that the influence of the mass transport of the cations is quite markedly superseded by crystallization effects. In any case transport effects are strong mainly when the depletion of the cations is considerable. Otherwise crystallization phenomena and ad*This has i n d e e d b e e n f o u n d f o r t h e d e p o s i t i o n o f c o p p e r f r o m a CuSO 4 s o l u t i o n ( e x p e r i m e n t s o f B. S t u r z e n e g g e r in o u r l a b o r a t o r y ) .

102

0

0

0

0

0

0

0 0 0

0 0 0

e~ 0

0

0

0

0

0

0

0 0 0

0 0 0

~

oo

0 e~

0

~ V4

e~

~

~

103

sorption or desorption tend to be the dominant factors. It should be emphasized that they can influence the properties and the structure of the electrodeposits considerably. In a separate paper we shall try to work out some general trends which became apparent in the course of a systematic study of pulse plating carried out in our laboratory (see also ref. 28). Nomenclature a,b C Ce

c~ Ce' Co C D E F h I Jc JF Jgg Jm Jmg

Jp Jpg Jt Jo L nmg npg

N

N, N~ Q t tc

t; td T T' Wa X

X Y Z

5N 5p 5s A ~7

constants in the Tafel equation (eqn. (19)) concentration of metal cations interfacial concentration concentration at the inner border of the stationary diffusion layer interfacial concentration for d.c. with a current density j = Jm concentration in the bulk solution capacitance of the double layer diffusion coefficient cathode potential Faraday constant (96 500 C (g equiv)- 1 ) average height of asperities current density capacitive current (during charging or discharging of the double layer) faradaic current (current that deposits the metal) limiting current density for d.c. jp T / ( T + T ' ) , average current density (see Fig. 1) average current density in plating with limiting current pulses height of pulse (pulse current density) (see Fig. 1) pulse current density of limiting current pulse (Ce = 0 at t = T) JF + Jc, total current density exchange current density characteristic length of the geometry of the system average rate of metal deposition in plating with limiting current pulses instantaneous rate of metal deposition in limiting current pulses flux density of mass transport (tool cm - 2 S- 1 ) flux density of cations in the pulsating diffusion layer during a pulse flux density of cations through the outer stationary diffusion layer charge of capacitor time charging time of the double layer (until JF = 0.99jp) charging time of the double layer (assuming jc = jp) discharge time of the double layer (until JF = 0 . 0 l i p ) length of pulse (ON time) length of interval between pulses (OFF time) (see Fig. 1) Wagner's number distance from the cathode DO /~ 2 , dimensionless variable T/O, dimensionless variable ionic charge charge transfer coefficient total thickness of the diffusion layer (thickness of the diffusion layer in d.c. electrolysis under the same hydrodynamic conditions as in pulse electrolysis) thickness of the pulsating diffusion layer at the end of the pulse thickness of the stationary diffusion layer degree of damping (or flattening) of the pulsed current overpotential

104 0 0e 0t

T + T', repetition period (inverse of frequency) experimental current efficiency for metal deposition current efficiency for metal deposition calculated on the assumption that the metal is deposited only during the pulses specific conductivity of the solution transition time

References 1 Rep. on the American Electroplaters' Society (AES) Pulse Plating Syrup., Boston, in Met. Finish., 77 (6) (1979) 77. Ch. J. Raub, Metalloberfl~'che, 33 (10) (1979) 437. W. Paatsch, Galvanotechnik, 70 (11) (1979) 1111. 2 H. Y. Cheh,J. Electrochem. Soc., 118 (1971) 1132. 3 K. Viswanathan and H. Y. Cheh, J. Electrochem. Soc., 125 (1978) 1616; 126 (1979) 398. 4 A. R. Despic and K. I. Popov, Mod. Aspects Electrochem., 7 (1972) 199. 5 K. I. Popov, D. N. Keca, S. I. Vidojkovic, B. J. Lazevaris and V. B. Milojkovic, J. Appl. Electrochem., 6 (1976) 365. 6 G. Milazzo, Electrochimie, Dunod, Paris, 1975. 7 J. O'M. Bockris and A. K. N. Reddy, Modern Electrochemistry, Vols. 1 and 2, Plenum, New York, 1970. 8 J. C1. Puippe and N. Ibl, to be published. 9 H. Y. Cheh, J. Electrochem. Soc., 118 (1971) 551. 10 K. Viswanathan, M. A. Farrell Epstein and H. Y. Cheh, J. Electrochem. Soc., 125 (1978) 1772. 11 N. Ibl and O. Dossenbach, in J. O' M. Bockris, E. Conway and E. Yeager (eds.), Treatise on Electrochemistry, Vol. III, Plenum, New York, Chap. 3. 12 G. Milazzo, Electrochimie, Dunod, Paris, 1975, p. 175. 13 J. Villermaux and P. Le Goff, to be published. 14 H . J . S . Sand, Z. Phys. Chem., 35 (1900) 641. 15 N. Ibl, Metalloberfla'che, 33 (1979) 51. 16 IUPAC Inf. Bull., Provisional Document No. 59, July 1977. 17 J. C1. Puippe, N. Ibl, H. Angerer and K. Hosokawa, to be published. 18 N. Ibl, Proc. Int. Conf. on Protection against Corrosion by Metal Finishing, Basle, 1966, Forster, Zurich, 1966, p. 48; Oberfld'che-Surf., 7 (1966) 256. 19 N. Ibl, Adv. Electrochem. Electrochem. Eng., 2 (1962) 49 - 143. 20 N. Ibl and K. Schadegg, J. Electrochem. Soc., 114 (1967) 54, 1268. 21 N. Ibl, Les Techniques de l'Ingdnieur, 12 (1976) 57. 22 N. Ibl and M. Braun, Chem.-Ing.-Techn., 45 (1973) 182. 23 M. Braun and N. Ibl, Oberfla'che-Surf., 14 (1973) 49. 24 M. Braun, Dissertation No. 5015, Eidgen6ssische Technische Hochschule, Zurich, 1973. 25 H.J. Schenk, Dissertation No 5566, Eidgen6ssische Technische Hochschule, Zurich, 1975. 26 J. C1. Puippe, Dissertation No. 6225, EidgenOssiche Technische Hochschule, Zurich, 1978. 27 P. C. Andricacos, H. Y. Cheh and H. B. Linford, Plat. Surf. Finish., 64 (July 1 9 7 7 ) 4 2 ; (September 1977) 44. 28 N. Ibl, J. CI. Puippe and H. Angerer, Surf. Technol., 6 (1978) 287.