Surface Technology, 7 (1978) 493 - 503 493 © Elsevier Sequoia S.A., Lausanne -- Printed in the Netherlands

OPTIMIZATION OF THE COMPOSITION OF THE SULPHATE BATH FOR CADMIUM PLATING

PIOTR TOMASSI, JERZY A. WEBER and TADEUSZ ZAK

Instytut Mechaniki Precyzyjnej, Warsaw (Poland)

(Received May 15, 1978)

Summary

The throwing power of the cadmium sulphate bath for cadmium plating has been investigated taking into consideration measurements in a Hull cell. Seventeen statistically designed experiments enabled the opt imum bath composit ion to be found. The influence of the molecular weight of the surfactant on the microstructure of cadmium deposits was also investigated.

1. In t roduct ion

For cadmium plating the sulphate bath is the most advantageous non- cyanide cadmium electrolyte owing to the low cost of the chemicals and the facility for waste t reatment. However, the weakest point of this type of bath is its poor throwing power.

Our first efforts were directed to increasing, as far as possible, the throwing power of the sulphate electrolyte for cadmium plating.

An extensive series of experiments was conducted to examine the influence of various additives on the throwing power of the bath. It was found that the wetting agent Arkopal greatly improves the throwing power of the sulphate bath. Arkopal is the trade name of the reaction product of an alkylphenol with ethylene oxide. Its chemical structure is as follows

CgH 19 - ~ O - - [--CH2CH20--] n--H

Products of molecular weight from 660 to 1540 are available. We decided to employ a statistical design in our experiments to find the

bath composition yielding the best throwing power of the electrolyte. In experiments to find the optimum of a multivariable function a

statistical design is often applied when a mathematical model of the function is unknown and a dependent variable results from the experiment. The aim of the present study was to check the efficiency of the statistical design employed in our experiments to investigate new plating baths.

494

d !urn

20

16

12

8

4

. . . . . ~ ' 0 ' ' 0 0 ,2 0~1- 0 6 ,8 log z

Fig. 1. Coat ing th ickness d i s t r i bu t ion on the Hull cell c a thode for e lec t ro ly te 5.

2. Exper imen ta l

2.1. Throwing power measurements We had first to choose a cor rec t and accurate m e t h o d for measuring

th rowing power . The classical m e t h o d with a Har ing-Blum cell is character- ized by p o o r reproduc ib i l i ty and there is l i t t le re fe rence to the results observed in industrial plants. While the slope o f the polar iza t ion curve can be t aken as a measure o f the th rowing power o f the e lec t ro ly te , a disadvantage is t ha t the er ror in d i f fe ren t ia t ion is usually high; in this case the inf luence o f cu r r en t dens i ty on cur ren t e f f ic iency must also be t aken in to account .

In ou r work we dec ided to take the coat ing thickness d is t r ibut ion on the ca thode o f a Hull cell as a measure o f the th rowing power o f the electro- lyte . The m or e un i fo rm the thickness dis t r ibut ion, the be t t e r the th rowing power o f a specific e lec t ro ly te . A similar m e t h o d for evaluat ing th rowing power was p roposed by Mohler [ 1 ] .

The thickness d is t r ibut ion on the Hull cell ca thode is o f t en l inear on a semilogari thmic scale. The thickness d is t r ibut ion for e lec t ro ly te 5 is shown in Fig. 1. The d is t r ibut ion is pract ical ly l inear and can be descr ibed b y the fol lowing equa t ion

d = a - - c l o g z (1)

where d is the local coat ing thickness (~m), z the distance f rom the edge o f the ca thode (cm) and a and c are constants .

The slope o f the line in Fig. 1 is equal to the cons tan t c in eqn. (1). This pa rame te r depends on the un i fo rmi ty of the thickness d is t r ibut ion and for ou r work was t aken as a measure o f the th rowing power o f the electro-

ly te . The exper imen t s wi th the Hull cell were always c o n d u c t e d in the same

way, at a cell vo lume o f 0 .25 dm 3, cu r ren t 1 A and dura t ion o f electrolysis 15 min. The ca thode was o f o rd inary steel and its wid th was 10 cm. The cadmium coat ing thickness was measured using an e lec t romagnet ic m e t h o d at nine poin ts a dis tance o f 1 cm f rom each other . F o r a given e lec t ro ly te , f ou r expe r imen t s were p e r f o r m e d wi th the Hull cell and average local thick-

4 9 5

nesses were calculated. The values of the parameter c were calculated using the least-squares method.

Many experimental design methods have backgrounds that are theoreti- cally advanced, e.g. random methods, elimination methods, simplex methods, stochastic approximation methods and others. Reviews of statistically designed experiments can be found in the literature [2 - 5]. The basic advantage of using these methods of optimization is the reduction in the number of experiments necessary to determine the optimum of a multi- variable function. A statistical experimental design can even be used when the phenomenon examined cannot be described by existing theory.

In the first part of the present study we used the gradient method known as the steepest descent method [2]. In this method, the first few experiments are performed to evaluate the derivative of the optimized func- tion. The partial derivatives form a vector, which indicates the direction of steepest descent of the function examined. This direction determines the conditions for the succeeding experiments, conducted to find the minimum of the optimized function.

The general form of the function examined can be expressed as

Y = f(xl, x2 . . . . . x~) (2)

where y is the optimized dependent variable, resulting from experiment and Xl, x2 . . . . . xk are independent variables, the controllable factors.

This function is usually presented as a regression equation

k k k y = b o + Z 1 b ix i + Y_, b i i x i x i + bii x 2 + = i < j i=zl . . . (3)

where bo, bi, bi.j, bii are regression coefficients. Within a limited region the optimized function can usually be des-

cribed by the linear part of the regression equation

y = b o + b l x I + b 2 x 2 + ... + bhx k (4)

The regression coefficients bl, b2, ..., bk can be used to estimate the partial derivatives d y / d x i. The values of the regression coefficients are measures of the influence of the definite variable xi on the quanti ty y.

TABLE 1

I n d e p e n d e n t var iab les c h o s e n f o r o p t i m i z a t i o n o f c a d i u m b a t h c o m p o s i t i o n

No. Var iab le S y m b o l Region Interval N u m b e r e x a m i n e d o f p o i n t s

1 C a d m i u m c o n c e n t r a t i o n x I 0 . 0 5 - 0 . 5 M 0 . 0 5 10 2 M o l e c u l a r w e i g h t o f A r k o p a l x 2 6 6 0 - 1 5 4 0 4 4 2 0 3 A r k o p a l c o n c e n t r a t i o n x 3 2 - 30 g d m - 3 2 15 4 S u l p h u r i c ac id c o n c e n t r a t i o n x 4 0 .1 - 1 .0 M 0 .1 10 5 T e m p e r a t u r e x 5 15 - 35 °C 2 10

496

o,1

E~

E~

0

o~

..= " 0

0~

c~

0,10,1

C ~

O C q Cg 0 0 ~

CO 0,1

0 0

• o ,

LO

0'~ CO LO

0 ,10~ ¢q CO Cq 0"~ Cq CO

0

C.O O O~

LO LO LO 0 ,10 ,1C~

o C q - ~

¢q oO ~4

¢ ~ L O 0

TABLE 3

Coating thickness distribution on the Hull cell cathode

497

Experiment no. Coating thickness (pro) at definite distance z (cm) from the cathode edge

z =1 z =2 z =3 z - - 4 z - - 5 z - - 6 z - - 7 z =8 z - - 9

1 22.0 15.8 12.2 9.3 7.8 6.4 3.1 1.4 0.6 2 22.7 17.0 14.0 11.2 7.2 4.8 2.9 4.7 4.3 3 21.1 16.9 13.0 9.8 8.7 5.6 3.8 2.8 1.7 4 22.3 16.3 14.0 11.2 8.8 7.2 4.9 2.8 2.0 5 21.7 16.7 13.3 10.3 8.2 7.5 5.7 3.1 1.9 6 26.6 20.3 16.9 13.8 9.6 7.8 5.4 4.4 2.2 7 23.7 18.9 15.7 11.8 8.8 9.4 5.3 4.0 2.7 8 24.4 17.9 16.5 12.5 9.8 7.5 4.7 3.0 1.8

9 20.7 14.9 11.0 9.7 7.4 5.5 4.0 2.0 2.1 10 20.4 13.9 13.5 9.4 7.1 5.7 4.1 2.8 1.7 11 11.8 16.2 12.7 10.8 7.1 5.2 4.2 3.2 2.0

12 15.8 13.9 10.5 7.1 5.1 4.1 2.5 1.5 0.9 13 20.0 15.6 11.5 9.2 6.6 4.2 3.1 1.4 0.8 14 14.1 13.7 10.9 8.6 7.1 4.5 2.6 1.6 0.9 15 8.3 13.5 11.2 7.9 6.4 5.1 3.4 2.7 1.4 16 12.0 15.7 12.4 8.4 7.2 5.5 3.6 3.0 1.3 17 16.6 14.0 10.3 8.3 6.2 4.2 2.5 2.3 1.2 18 14.5 13.4 10.7 8.2 6.3 4.8 3.2 1.7 0.7 19 19.8 14.1 9.9 8.3 6.8 4.9 2.7 1.4 0.7

In our inves t iga t ions the o p t i m i z e d var iable y is equal to t he p a r a m e t e r c f rom eqn. (1). Our i n t e n t i o n was to d e t e r m i n e the b a t h c o m p o s i t i o n for

w h i c h p a r a m e t e r c r eached a m i n i m u m ( m a x i m u m t h r o w i n g power ) . We assumed t h a t p a r a m e t e r c d e p e n d s on five var iables , l i s ted in Tab le 1.

The ba th c o m p o n e n t s were as fo l lows: c a d m i u m su lpha te , sod ium su lpha te , su lphur ic acid and A r k o p a l . In all the e x p e r i m e n t s the sod ium su lpha t e c o n c e n t r a t i o n was k e p t c o n s t a n t a t 40 g d m -3.

To d e t e r m i n e the regress ion coe f f i c i en t s x l , x2 . . . . , x5 we p e r f o r m e d a series o f e ight e x p e r i m e n t s des igned to use a f r ac t iona l r ep l i ca t ion scheme t y p e 2 ~-2 t a k e n f rom the l i t e r a tu re [ 5 ] . This t y p e o f e x p e r i m e n t p l ann ing has a n u m b e r o f advantages : (1) the ca lcu la t ions are very s imple ; (2) the regress ion coe f f i c i en t s are eva lua ted i n d e p e n d e n t l y ; (3) the var iance o f regress ion coe f f i c i en t s is min ima l .

The c o n d i t i o n s fo r t he e x p e r i m e n t s p e r f o r m e d and resul ts o b t a i n e d are given in T a b l e 2. Every f a c t o r exis ts on two levels. The s ta r t ing p o i n t and f a c t o r in tervals were chosen a rb i t ra r i ly . Each e x p e r i m e n t cons i s t ed o f the fo l lowing o p e r a t i o n s : (1) the p r e p a r a t i o n o f a c a d m i u m b a t h o f de f in i t e c o m p o s i t i o n ; (2) e l ec t ro lys i s in t he Hul l cell , r e p e a t e d fou r t imes ; (3) c o a t i n g th ickness m e a s u r e m e n t s ; (4) the eva lua t ion of p a r a m e t e r c using the least~ squares m e t h o d .

498

The results o f the coa t ing th ickness m e a s u r e m e n t s are given in Tab le 3. The regression coef f ic ien ts were ca lcula ted f rom the equa t ion

1 N bi N u~ =1 $iuCu (5 )

i = 0 , 1 , . . . , k (k =5) u = l , 2 , . . . , N ( N = 8 )

where i is the index of the regression coef f ic ien t , u the ordinal n u m b e r of the e x p e r i m e n t , x;u the coef f i c ien t equal to +1 or - -1 , depend ing w h e t h e r the f a c to r xi is on the higher or lower level, and Cu is the resul t o f e x p e r i m e n t n u m b e r u.

We ob t a ined the m a t h e m a t i c a l m o d e l o f the op t imized func t ion as the regression equa t ion

c = b 0 + b l x 1 + b2x 2 + b3x 3 + b4x4 + bsx 5 (6)

The values o f the regression coef f ic ien ts are given in Table 2. The c o n f o r m i t y o f eqn. (6) wi th the o p t i m i z e d func t ion was tes ted

using the stat is t ical func t ion F. The value o f F was ca lcula ted f rom the e q u a t i o n

F = SR/ fRS2(c) (7)

where

N k Sa = Z C 2 - - N Z b 2

u=l i=O

fR = N - k - - 1 and here fR = 2 ; $ 2 ( c ) is the var iance of p a r a m e t e r c, $2(c) = 0.04.

T h e ca lcula ted value F = 2 .56 is a measure of the devia t ion of the m a t h e m a t i c a l m o d e l (eqn. (6)) f r o m the real shape o f the o p t i m i z e d func- t ion . I t was c o m p a r e d wi th the F d i s t r ibu t ion value, when the n u m b e r s of degrees of f r e e d o m were V 1 = fR, V2 -- N with signif icance level 0.05. Tables o f F d i s t r ibu t ion show t h a t F0.05 reached the value 4.46. Consider ing tha t F < Fo.o5 , the m a t h e m a t i c a l m o d e l appears to be in c o n f o r m i t y wi th the o p t i m i z e d func t ion in the region examined .

The F d i s t r ibu t ion was also used in tes t ing the signif icance o f regression coeff ic ients . In this case an Fi value was ca lcula ted for each regression coeffi- c ient b; f r o m the fo l lowing equa t ion

F z. = N b 2 / S 2 ( c ) (8)

T h e fo l lowing values were ob ta ined : F 1 = 125.14, F 2 = 16.94, F a = 118 .89 , F 4 = 43.43, F5 = 0.11.

These values were c o m p a r e d wi th the F d is t r ibu t ion value, w h e n the n u m b e r s o f degrees of f r e edom were V 1 = 1, V 2 = p - - 1 wi th signif icance level 0.05. The Fo.o5 value is equal to 10.1. P a r a m e t e r p co r r e sponds to the n u m b e r o f e x p e r i m e n t s c o n d u c t e d for evaluat ing var iance $2(c) (here p = 4).

TABLE 4

Experimental design applied in the simplex method

499

Experi- Cd concen- Mol. weight Arkopal H2SO 4 Tempera- Para- ment no. tration (M) of Arkopal concentra- concen- ture meter

tion tration (°C) (M) (g dm -3 ) (M)

12 0.1 880 14 0.32 25 0.266 13 0.1 880 14 0.24 25 0.274 14 0.1 880 21 0.28 25 0.237 15 0.1 880 21 0.36 25 0.200 16 0.1 880 28 0.32 25 0.211 17 0.1 880 28 0,40 25 0.249 18 0.1 880 21 0,44 25 0.237 19 0.1 880 14 0,40 25 0.274

The regression coefficients for which Fi < F0.05 can be disregarded. Here F~ = 0 means that the influence of temperature on the throwing power of the electrolytes examined is negligible. This conclusion must not be generalized.

The values of other regression coefficients indicated that the throwing power of the cadmium electrolyte could be improved by increasing the Arkopal concentration and its molecular weight and by decreasing the cadmium and sulphuric acid concentrations.

In accordance with the steepest descent method, the next experi- mental points were located in the direction determined by the vector composed of partial derivatives of the optimized function. That means practically, that independent factors xl, x2 .. . . . x5 were changed proportio- nally to regression coefficients bl, b2 . . . . , bs. The conditions for this series of experiments (nos. 9, 10 and 11) are given in Table 2.

As was foreseen, the optimized parameter c decreased markedly, especially in experiment 11. This was caused mainly by the decreasing coating thickness in the high current density region.

It was not advantageous to perform the next experiments in the same direction, because further reduction in cadmium concentration resulted in a decrease in current efficiency and deposition speed. In the following experiments the cadmium concentration was constant at 0.1 M.

Further increase in the molecular weight of Arkopal was also disadvantageous because of the poorer solubility of higher fractions of this polymer. In the next experiments the molecular weight of Arkopal was constant at 880.

Further optimization of the cadmium bath with regard to Arkopal and sulphuric acid concentration was conducted using the simplex experimental design [6] . In this method the experimental points are located in the vertices of a regular solid, the so-called simplex, in k-dimensional space. For

500

)C

1

0,20

~ 17

e18

13 ~ ~2 • 19 8

p i i J I J •

0,24. 0,28 0,~2 0,36 0,140 0,44 0 2 ~. 6 8

X 4

Fig. 2. Distribution of experimental points using the simplex method.

Fig. 3. Coating thickness distribution on the Hull cell cathode for electrolytes 1 (curve 1 ) and 15 (curve 2).

10

example , when there are two independen t variables (k = 2), exper imenta l points are loca ted in the vert ices o f an equilateral tr iangle; when k = 3 they are placed in the vert ices of a t e t r ahed ron , etc. Af ter the first series of h + 1 expe r imen t s had been pe r fo rmed , the po in t o f the wors t result was re jec ted and a new simplex was buil t using the remaining points and the po in t fo r the n e x t exper iment . Every successive ex p e r im en t caused displace- m e n t o f the s implex closer to the op t imum.

The condi t ions and results of the exper iments designed using the s implex m e t h o d are given in Table 4. Dis t r ibut ion o f exper imen ta l points is shown in Fig. 2.

The coat ing thickness d is t r ibut ion is no longer linear on a semi- logari thmic scale and the re fo re a new cr i ter ion m of th rowing power was used. Paramete r m was def ined as a relative s tandard deviat ion o f average coat ing thickness

( ~ (di __din) 2 I/2 1 /=i 1 (9)

m : . ( . - 1) I

where di is the local coat ing thickness, d m the average coat ing thickness on the Hull cell ca thode and n the n u m b e r o f thickness measuring points (n = 9).

When the th rowing power o f the e lec t ro ly te was sat isfactory, d i f ferences di - - dm were small and pa ramete r m reached a low value.

The min im um value o f pa rame te r m was observed in expe r imen t 15 and this po in t was t aken as the m a x i m u m o f th rowing power . Exper imen t s

501

(a) (b)

(c) (d) Fig. 4. (For caption see overleaf)

502

(e)

Fig. 4. (a) Structure of cadmium deposits with various molecular weights of Arkopal : (a) 660; (b) 790; (c) 880; (d) 1230, (e) 1540.

18 and 19 were performed only to control the optimized function in other directions.

Thickness distributions for electrolytes 1 and 15 are compared in Fig. 3. For electrolyte 15 this distribution is much more uniform.

The appearance of cadmium coatings obtained from all the electrolytes examined was practically the same: the deposits were smooth and silverish.

2.2. Microstructure of the cadmium coatings The influence of the molecular weight of Arkopal on the microstruc-

tures of the cadmium deposits was investigated using scanning electron microscopy. Coatings were deposited from the optimum bath described with the following composition Cd 11 g dm -3 (0.1 M), H2SOa 35 g dm -3 (0.36 M), Arkopal 21 g dm -s, Na2SO 4 40 g dm -~a (0.28 M).

Samples of Arkopal with the following molecular weights were used: 660,790, 880, 1230 and 1540. In every case the conditions of electrolysis were the same: current density 2 A dm -2, duration of electrolysis 15 min, temperature 25 °C. The cathode was of polished carbon steel. The micro- structures of the cadmium deposits are shown in Fig. 4. The hexagonal structure of the cadmium crystals can be observed. For lower molecular weights of Arkopal, a pyramidal form of crystal growth can be seen; the crystals are large and flat (Figs. 4(a), 4(b)). For higher molecular weights spiral growth appears and the crystals are more pointed (Fig. 4(c)). This is probably caused by higher adsorption of longer chain molecules. The

503

adsorbed molecules inhibit the crystal growth at the edge faces of the crystal layer and more nuclei are formed. As a result smaller irregular crystals are formed. This phenomenon can be observed in Figs. 4(d), 4(e). A similar effect of certain additives on the microstructure of copper deposits was described by Nageswar [7] .

The changes in crystal form can also be explained by changes in the orientation of adsorbed molecules of surfactant. In ref. 8 an investigation is reported of the influence of Arkopal-type compounds on the shape of polarization curves. It was found that the best quality of zinc coatings could be obtained using an additive of molecular weight 1100.

3. Conclusions

By employing a statistical experimental design it has been possible to obtain a composition with maximum throwing power for a sulphate bath for cadmium plating by conducting only 17 experiments. Systematic research for five variables with intervals as shown in Table 1 would in theory require 300 000 experiments.

Optimum bath composition as well as a mathematical model of the function examined in the form of a regression equation were obtained. New mathematical criteria for the throwing power of electrolytes were established.

The very simple example of the application of statistical experimental design presented in this paper indicates that this kind of modern technique could be used effectively to investigate new types of bath for metal finishing.

References

1 J. B. Mohler, Met. Finish., 7 (1976) 40. 2 D. J. Wilde, Optimum Seeking Methods, Prentice Hall, Englewood Cliffs, N.J., 1964. 3 D.J . Finney, The Theory of Experimental Design, The University of Chicago Press,

Chicago, 1955. 4 H. L. Johnson and L. F. Leone, Statistics and Experimental Design in Engineering

and Physical Sciences, John Wiley, New York, 1967. 5 W. G. Cochran and G. M. Lox, Experimental Designs, John Wiley, New York, 1968. 6 W. Spendley, G. R. Hext and F. R. Himsworth, Technometrics, 4 (1962) 441. 7 S. Nageswar, Electrodeposition Surf. Treat., 3 (1975) 417. 8 Blasberg-Mitteilungen, No. 26, October 1973, Solingen, F.R.G.