chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147

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Chemical Engineering Research and Design

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Design of experiments for statistical modeling andmulti-response optimization of nickel electroplating process

Maria Poroch-Seritana, Sonia Gutta, Gheorghe Gutta, Igor Cretescub,∗,Corneliu Cojocarub,∗∗, Traian Severina

a “Stefan cel Mare” University of Suceava, Faculty of Food Engineering, 9 Universitatii street, 720225, Suceava, Romaniab “Gheorghe Asachi” Technical University of Iasi, Department of Environmental Engineering and Management, 71 Mangeron Blvd.,700050, Iasi, Romania

a b s t r a c t

The central composite experimental design and response surface methodology have been employed for statistical

modeling and analysis of the results dealing with nickel electroplating process. The empirical models developed in

terms of design variables (current density J (A/dm2), temperature T (◦C) and pH) have been found statistically adequate

to describe the process responses, i.e. cathode efficiency Y (%), coating thickness U (�m), brightness V (%) and hardness

W (HV). The graphical representations consisted of 2D contour plots and 3D surface plots have been used for exploring

and analysis of response surfaces in order to identify the main, quadratic and interaction effects. The multi-response

optimization of nickel electroplating process has been carried out by means of desirability function approach. To

this end, a genetic algorithm has been used for mathematical optimization of the multi-response problem. The

optimization algorithm has conducted to a set of equivalent solutions named Pareto optimal set. The confirmation

runs have been employed in order to make a decision about the optimal solution approved by experiment. Thus,

the optimum conditions of nickel electroplating has been defined in this work as J* = 5.35 (A/dm2), T* = 33.44 (◦C) and

pH* = 6.22 and respectively the responses confirmed by experiment were Y = 79.12 ± 0.18 (%), U = 52.77 ± 0.48 (�m),

V = 26.12 ± 0.45 (%) and W = 371.6 ± 1.77 (HV). In such conditions the quality of nickel electroplating deposit was the

best one in accordance with experimental results.

© 2010 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Electroplating; Experimental design; Response surface method; Desirability function; Genetic algorithm

investigations dealt with conventional methodology of exper-

1. Introduction

Nickel electroplating is used extensively for decorative, engi-neering and electroforming purposes. Decorative applicationsaccount for about 80% of nickel consumption in plating while20% is consumed for engineering and electroforming aims (DiBari, 2000).

The environmental aspects related to electroplating indus-try such as waste minimization, toxicity of electroplatingmetals and reduction of both chemicals and water con-sumption have been deeply approached by different authors(Abou-Elela et al., 1998; Babu et al., 2009; Fang and Chan, 1997;

Kuntay et al., 2006).

∗ Corresponding author. Tel.: +40 741914342; fax: +40 32 271311.∗∗ Corresponding author. Tel.: +40 742176747.

E-mail addresses: icre@ch.tuiasi.ro (I. Cretescu), cojocaru c@yahoo.cReceived 24 November 2009; Received in revised form 11 May 2010; Ac

0263-8762/$ – see front matter © 2010 The Institution of Chemical Engidoi:10.1016/j.cherd.2010.05.010

In the last decades, many scientific and technical studieshave been carried out to account for the effects of factorson electroplating performance and to address the enhance-ment of nickel electroplating process (Abd El Wahaab etal., 1986; Dolgikh et al., 2009; Njau et al., 1998; Tsuru etal., 2002; Hoffmann et al., 2008; Gupta et al., 2002; Yoshidaet al., 2003, 2004; Abdel-Hamid, 1998; Wang et al., 2008;Shpanko et al., 2004; Badarulzaman et al., 2009; Balakai etal., 2009; Sotskaya and Dolgikh, 2008; Mohanty et al., 2001;Ciszewski et al., 2004; Oliveira et al., 2006; Orinakova et al.,2006). However, it is worth to mention that most of such

om (C. Cojocaru).cepted 28 May 2010

imentation in which one factor is varied while others are fixedat constant levels. The classical methods of experimentation

neers. Published by Elsevier B.V. All rights reserved.

chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147 137

Nomenclature

ai1, ai2, coefficients for desirability functionb0, bi, bii, bij regression coefficients for polynomial equa-

tionsb vector of regression coefficientsd individual desirability functionD overall desirability functionD� black intensity of sampleF ratio of variances (Fischer test)g population size for GA (integer number)I current intensityi and j subscripts (integer variables)J current densityk iteration indicating the current population in

GAL number of significant regression coefficientsm integer number (m < g)mp experimental amount of nickel depositionmt theoretical amount of nickel depositionn number of variablesn0 number of experiments in center pointN number of experimental runs in CCDp level of significanceR reflection (relative percent)R� intensity of standard reference substances number of responsesS2

0 error mean squareS2

res residual mean squareS� intensity of sampleT temperaturet time of electrolysisU average thickness of metallic layer (experimen-

tal value)U average thickness of metallic layer (predicted

value)V brightness of metallic coating (experimental

value)V brightness of metallic coating (predicted value)W hardness of metallic coating (experimental

value)W hardness of metallic coating (predicted value)X matrix of experimental designx vector of variables (coded values)

x(k)cj

chromosome (vector of variables in GA)

x(k)pj

parent chromosome (vector of variables in GA)y response (general term)y0i response recorded in the center pointy0 average value of responsey predicted values of responsey response vectorY cathode efficiency (experimental value);Y cathode efficiency (predicted value by regres-

sion model);z actual value of design variablez0 center point of design variable (actual value)�z interval of variation for design variable* superscript indicating optimal value

Greek letters˛ axial point or “star” point in CCD;� wavelength

�1, �2 degree of freedoms˝ valid region (initial region of experimentation)

usually ignore the interaction effects between variables (fac-tors) and lead to the counterfeit optimal conditions. In orderto overcome this issue the response surface methodology(RSM) may be used. This methodology (RSM) is a collec-tion of mathematical and statistical techniques useful fordeveloping, improving, and optimizing processes that canbe used to assess the relative significance of several inde-pendent variables (factors) even in the presence of complexinteractions. According to RSM strategy, the fitting of exper-imental data to a proper response surface model must beperformed based on statistical design of experiments. After-wards, the developed mathematical model is used to establishthe optimum operational conditions of the process underinvestigation.

Few studies related to implementation of experimentaldesign for nickel plating process have been reported in litera-ture (Oraon et al., 2006; Hu and Bai, 2001a, 2001b).

Oraon et al. (2006) have applied the response sur-face method for predicting of electroless nickel platingprocess. In this respect, authors have used a centralcomposite design (CCD) for experimentation. It has beenobserved that reducing agent (NaBH4), source of metal(NiCl2·6H2O) and temperature significantly affect the deposi-tion.

The authors (Hu and Bai, 2001a) have studied the com-position control of electroplated nickel–phosphorus deposits.To this end, the influences of electroplating variables, suchas temperature, current density, pH, NaH2PO2·H2O concentra-tion and agitation rate, on the phosphorus content of Ni–Pdeposits electroplated from a Watts nickel bath modified withNaH2PO2·H2O were systematically compared using fractionalfactorial design (FFD).

In addition, the same authors (Hu and Bai, 2001b) haveoptimized hydrogen evolving activity on nickel–phosphorusdeposits using experimental strategies. In this regard, theeffects of electroplating variables on the hydrogen evolvingactivity of Ni–P deposits were systematically examined usingfractional factorial design (FFD), path of steepest ascent, andcentral composite design (CCD) coupled with the response sur-face method (RSM). The FFD study indicated that the main andinteractive effects of temperature, pH, and NaH2PO2·H2O con-centration are the key preparation factors influencing the Ni–Pcathode.

The implementation of statistical experimental designtechniques for electroplating process assisted by develop-ing of empirical models may conduct to an enhancedelectroplating performance and to reduction of overallcosts of experimentation. To the best of our knowledge,there is a lack of reports in the literature regardingmulti-response optimization of nickel electroplating process.Therefore, this work deals with statistical modeling andmulti-response optimization of nickel electroplating processin order to improve the cathode efficiency as well as to

obtain the best characteristics of the electroplated coversin terms of thickness, brightness and hardness of metalliclayer.

138 chemical engineering research and

Table 1 – Chemical composition of Watts-type bath usedfor nickel electroplating.

Chemical composition Concentration (g/L)

Nickel chloride, NiCl2·6H2O 40.5Nickel sulfate, NiSO4·6H2O 292.5

Boric acid, H3BO3 31.5

2. Experimental

Electroplating experiments were carried out using a Watts-type bath. Details of the bath composition employed aregiven in Table 1. All chemicals used in electroplating exper-iments were of analytical reagent grade. Solution of sodiumhydroxide (0.01N) was used for pH adjustment. The homemade electrochemical cell was employed for nickel electro-plating experiments. The dimensions of cell are as follows:13.9 cm (length) × 12.5 cm (width) × 10 cm (height), correspond-ing to a volume of 1.75 L. In addition, the cell was equippedwith a cathode (made of 99.98% purity copper) and ananode (made of 99.7% purity nickel). The cathode dimen-sions are of 8.0 cm (height) × 9.8 cm (length) × 0.1 cm (width)while the dimensions of anode were of 10.0 cm (height) × 4.0 cm(length) × 0.3 cm (width). The immersion surface of cathodewas of 0.6 dm2 and respectively the anodic immersion sur-face was of 0.25 dm2. The above-described cell was connectedin galvanostatic regime to the GWINSTEK GPR-1810HD powersupply, having a digital control of current and voltage.The temperature of electrolyte solution was kept constantusing the Lauda E100 thermostatic bath. Before each elec-troplating experiment the cathode surface was preparedaccordingly.

The performance of nickel electroplating process was esti-mated in terms of four responses, namely, cathode efficiency,Y (%); average thickness of metallic layer, U (�m); brightnessof metallic coating, V (%) and hardness of metallic coating, W(HV).

The cathode efficiency, Y (%), was determined by using Eq.(1):

Y = mp

mt× 100 = mp

3.041 × 10−4 × I × t(1)

where mp denotes the experimental amount of nickel depo-sition (g); mt the theoretical amount of nickel deposition (g);I the current intensity which cross the electrical circuit (A); ttime of electrolysis (s); 3.041 × 10−4 (g/C) is the electrochemicalequivalent of nickel (Paunovic and Schlesinger, 2006; Popov etal., 2002).

The thickness of metallic layer was measured by means of aPosiTector 6000 – DeFesko Analyzer, based on non-destructivephysical method. To this end, seven local values of cathodethickness were measured in order to determine the averagethickness of metallic layer, U (�m).

The brightness of metallic cover V (%) was evaluated usingthe HR 4000 CG-UV-NIR spectrometer (Ocean Optics Inc.,Dunedin, FL) based on the reflection property of the deposedmetallic layer. In this respect, the scanning with 0.025 nmresolution has been carried out for the entire wavelength spec-trum ranged from 200 up to 1100 nm. A tungsten halogen lightsource (UV-VIS-NIR Light Source DH-2000, Mikropack) was

used for this study. The experimental data were displayed andstored using the operating software OOI Base32 from OceanOptics. The light from Light Source DH-2000 was conveyed to

design 8 9 ( 2 0 1 1 ) 136–147

the sample through an array of optical fibers (QR400-7-UV/BX,Ocean Optics Inc., Dunedin, FL).

The reflection was quantified as a relative percent (R)from the reflection of standard reference substance STAN-SSH(STAN-SSH High-reflectivity Specular Reflectance Standard,Ocean Optics Inc.), according to Eq. (2):

R = S� − D�

R� − D�× 100% (2)

where S� means the intensity of sample at wavelength �; D�

the black intensity of sample at wavelength �; R� is the inten-sity of standard reference substance at wavelength �.

The reflections were measured at the wavelengths, wherethe visual sensibility is maximal one, i.e. 420 nm (blue),534 nm (green) and 564 nm (yellow), and by considering threepoints from the electrode surface. The final reported value ofbrightness V (%) represents the average value of reflectionsdetermined based on nine measurements as described previ-ously.

The hardness of metallic coating, W (HV), was measuredusing the Shimadzu, HMV – 2T, micro-hardness tester. Theloading weight was of 490.3 mN and the load duration of 15 s.

3. Results and discussion

3.1. Statistical modeling of electroplating process

The statistical method of experimentation was employed formodeling and optimization of the electroplating process. Theapplying of statistical method is useful to understand theinteraction effects between factors and to reduce the totalnumber of experimental runs by saving time and adjacentcosts.

The most significant design variables (key factors) thatinfluence the performance of nickel electroplating processdeals with current density, J (A/dm2); temperature, T (◦C); andpH value of the solution. The significance of these factors aswell as their operating range was deduced based on the pre-liminary experimental tests (not shown in the manuscript).The importance of each selected factor is discussed briefly inthe following.

The current density, J (A/dm2), is a key factor for two mainreasons. Firstly, the current density controls the rate of depo-sition (Rose and Whittington, 2002). Secondly, the currentdensity must be controlled within the correct operating rangein order to obtain sound deposits having uniform appearanceand free from burning or treeing. For optimum productivityit will be desirable to operate with the desired current den-sity and the available current. In practice, the size, shape orweight of the work may limit the surface area that can beloaded. In such cases, the current may need to be reducedto achieve the desired current density. The control of currentdensity is particularly critical where work is being plated toa specified thickness and the equipment operates on a fixedtime cycle. Thus, in electroplating, current density and its dis-tribution play a centrally important role in determining thequality of the final deposit (Paunovic and Schlesinger, 2006).

Changes in temperature can affect the performance ofnickel plating and other process solutions. Specifically, tem-

perature can influence the brightness range, throwing power,ductility, hardness, internal stress and burning characteristics(Di Bari, 2000; Rose and Whittington, 2002).

chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147 139

Table 2 – Correspondence between actual and coded values of design variables.

Design variables Symbol Actual values of coded levels Step of variations

−˛ −1 0 +1 +˛

J, current density (A/dm2) x1 1.33 1.66 3.33 5 5.36 1.66T, temperature (◦C) x2 11.77 15 30 45 48.22 15Solution pH x3 3.78 4 5 6 6.22 1

topoaW

wK

x

wcvvo(tc

(iten˛

ol

The solution pH is a measure of the hydrogen ion concen-ration, or more simply the acidity, of a solution. In the casef nickel solutions the pH has an important influence on batherformance. The pH can affect the bright plating range, cath-de efficiency, effects of impurities, throwing power, stresss well as the physical properties of the deposit (Rose andhittington, 2002).For statistical calculations the actual values of variables

ere scaled-up (coded) according to Eq. (3) (Akhnazarova andafarov, 1982; Myers and Montgomery, 2002):

i = zi − z0i

�zi∀i = 1, n (3)

here z denotes the actual value of design variable; z0 theenter point of design variable (actual value); �z the interval ofariation; x the coded level of design variable (dimensionlessalue) and n is the number of variables. Basically, the extentf each variable involves three different coded levels from low−1) to medium (0) and to high (+1). In addition, depending onhe type of experimental design, the axial levels (±˛) can beonsidered.

The operating region and the levels of the design variableskey factors) are given in actual and coded values as it is shownn Table 2. A central composite design (CCD) of orthogonalype was employed in this study to perform the electroplatingxperiments in a systematically manner by varying simulta-eously all the factors (Table 3). The axial level (“star point”)

has been computed from the condition for a CCD to be an

rthogonal design and may be written in this case as fol-ows (Akhnazarova and Kafarov, 1982; Khuri and Cornell, 1996;

Table 3 – Central composite orthogonal design applied for elect

N Design variables

Current density Temperature Solut

J (A/dm2) x1 T (◦C) x2 pH

1 5 +1 45 +1 62 1.66 −1 45 +1 63 5 +1 15 −1 64 1.66 −1 15 −1 65 5 +1 45 +1 46 1.66 −1 45 +1 47 5 +1 15 −1 48 1.66 −1 15 −1 49 5.36 +˛ 30 0 5

10 1.33 −˛ 30 0 511 3.33 0 48.22 +˛ 512 3.33 0 11.77 −˛ 513 3.33 0 30 0 6.2214 3.33 0 30 0 3.7815 3.33 0 30 0 516 3.33 0 30 0 5

Myers and Montgomery, 2002):

˛ ={√

2n(2n + 2n + 1) − 2n

2

}1/2

(4)

According to the experimental design a total number of 16experimental runs were carried out. The CCD consists of threedistinct regions: (i) full factorial design in which the factor lev-els are coded to the usual low (−1) and high (+1) values; (ii)axial points localized on the axis of each variable at a distance˛ from the designed center; and (iii) center points that can bereplicated to provide an estimation of the experimental errorvariance. Four responses have been determined experimen-tally in accordance with designed runs in order to ascertain theperformance of the electroplating process. These responsesare related to cathode efficiency, Y (%); average thickness ofmetallic layer, U (�m); the brightness, V (%) and the hardness,W (HV) of metallic cover.

Commonly, a quadratic response surface model with crossterms can be constructed to fit the experimental data obtainedin accordance with CCD. The response surface model (RS-model), known also as regression or empirical equation,represents a polynomial approximation of experimental dataand is stated by the following relationship:

y = b0 +n∑

i=1

bixi +n∑

i=1

biix2i +

n∑i<j

bijxixj (5)

where y denotes the predicted response (e.g. predicted cath-ode efficiency), xi the coded levels of the design variables, andb0, bi, bii, bij are the regression coefficients (offset term, main,

roplating experiments.

Responses (experimental values)

ion pH Y (%) U (�m) V (%) W (HV)

x3

+1 100 47.85 24.71 346.00+1 99.48 35.86 1.99 289.20+1 85.56 33.43 17.91 218.10+1 99.10 42.57 21.76 336.00−1 99.27 30.29 2.50 228.10−1 99.70 38.86 3.96 237.90−1 97.05 38.00 9.36 214.00−1 98.67 41.43 16.38 273.000 100 43.57 37.13 343.700 98.18 35.86 3.49 271.700 97.07 33.57 4.15 237.100 99.73 38.29 11.00 211.50+˛ 96.99 58.29 15.39 502.90−˛ 100 34.14 25.19 318.400 99.10 33.71 3.27 248.000 98.82 34.30 4.20 252.5

140 chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147

atic

Fig. 1 – Comparison between experimental data and mathemV (%) of metallic layer.

quadratic and interaction or cross effects). The least squareestimations of the regression coefficients have been computedby means of ordinary least squares (OLS) method and can bewritten as follows (Akhnazarova and Kafarov, 1982; Myers andMontgomery, 2002; Bezerra et al., 2008):

b =(XTX)−1

XTy (6)

where b is a vector of regression coefficients, X the designmatrix of the coded levels of input variables, and y is a col-umn vector of response determined experimentally accordingto the arrangement points into CCD.

Based on experimental design results (Table 3) the regres-sion models have been constructed by means of OLS-methodin order to determine the functional relationship for approx-imation and prediction of responses. Thus, the second-orderRS-models with coded variables obtained for Ni electroplatingprocess are as follows:

Y = 97.99 − 1.174x1 + 1.355x2 − 1.297x3 + 1.906x1x2

− 1.371x1x3 + 1.446x2x3 (7)

U = 38.42 + 3.695x3 − 3.055x22 + 3.918x2

3

+ 1.999x1x2 + 2.249x2x3 (8)

V = 17.2 + 4.68x1 − 3.704x2 − 5.462x22 + 4.016x1x2 + 3.419x1x3

(9)

Fig. 2 – Comparison between experimental data and mathematicof metallic layer.

al models related to cathode efficiency Y (%) and brightness

W = 299.2 + 42.04x3 − 72.85x22 + 53.49x2

3 + 27.99x1x2 (10)

subjected to: xi ∈ ˝; ˝ = {xi|−˛ ≤ xi ≤ +˛}; ∀i = 1, 3Note that, the term ˝ denotes the valid region (region of

experimentation) where the regression models are valid over.The significance of regression coefficients was tested using thestatistical Student’s t-test. Thus, in Eqs. (7)–(10) only the signif-icant terms were retained. Figs. 1 and 2 show the comparisonbetween experimental data and predicted data provided byregression models.

For testing the goodness-of-fit of regression equations, thestatistical Fisher F-test was employed considering the signif-icance level of p = 0.05. In this regard, one must compute theerror mean square and residual mean square. The error meansquare (S2

0) that has been calculated by using the repeatedobservations (Akhnazarova and Kafarov, 1982; Gavrilescu andTudose, 1999):

S20 = 1

n0 − 1

n0∑i=1

(y0i − y0)2 (11)

where n0 is the number of experiments in center point (repro-ducibility), y0i denotes the values of response recorded in thecenter point and y0 is the average value of y0i. The residualmean square (S2

res) has been computed as (Akhnazarova andKafarov, 1982; Gavrilescu and Tudose, 1999):

S2res = 1

N − L

N∑j=1

(yj − yj)2 (12)

al models related to hardness W (HV) and thickness U (�m)

chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147 141

Table 4 – Statistical test for evaluation of model adequacy.

Model Significantlevel

Degrees offreedom

Fischer test(calculate value)

Fischer test(tabulate value)

Modeladequacy

Y (x1, x2, x3) p = 0.05 �1 = N − L = 8 �2 = N0 − 1 = 1 Fc = S2conc.

S20

= 215.7 Ftab(p,�1,�2) = 238.9 Fc < Ftab(p,�1,�2)

U (x1, x2, x3) p = 0.05 �1 = N − L = 9 �2 = N0 − 1 = 1 Fc = S2conc.

S20

= 236.1 Ftab(p,�1,�2) = 240.5 Fc < Ftab(p,�1,�2)

V (x1, x2, x3) p = 0.05 �1 = N − L = 9 �2 = N0 − 1 = 1 Fc = S2conc.

S20

= 214.9 Ftab(p,�1,�2) = 240.5 Fc < Ftab(p,�1,�2)

1S2

witpNioT

F

rlom

cTs

Y

U

V

W

wrvi

as

et

W (x1, x2, x3) p = 0.05 �1 = N − L = 10�2 = N0 − 1 =

here N is the number of observations (experimental runs), Ls the number of significant coefficients in the regression equa-ion, yj is the response (experimental value) and yj denotes theredicted values of response according to regression equation.ote that, the regression model is an adequate fit to the exper-

mental data if the F-ratio is smaller than the tabulated valuene Ftab(p,�1,�2) (Akhnazarova and Kafarov, 1982; Gavrilescu andudose, 1999):

= S2res

S20

< Ftab(p,�1,�2) (13)

The results of F-ratio test are focused in Table 4 for allegression models. According to these results, the F-ratio isower than tabulated value in all cases revealing the adequacyf the regression equations from statistical standpoint, i.e. theodels can be used for the predictions.It is of real interest passing from regression equations with

oded variables to mathematical models with actual variables.o this end, the final empirical models were obtained by sub-titution technique and are presented in Eqs. (14)–(17):

ˆ = 112.5 + 1.121J − 0.6457T − 1.448pH + 0.07622J T

− 0.8224J pH + 0.0964T pH (14)

ˆ = 136.2 − 2.398J − 0.2015T − 39.98pH − 0.01358T2

+ 3.918pH2 + 0.07994J T + 0.1499T pH (15)

ˆ = 43.64 − 12.27J + 0.6743T − 6.836pH − 0.02428T2

+ 0.1606J T + 2.051J pH (16)

ˆ = 1247 − 33.58J + 15.7T − 492.9pH − 0.3238T2

+ 53.49pH2 + 1.119J T (17)

here J, T and pH are the real values of independent variableselated to current density (A/dm2), temperature (◦C), and pHalue of solution, respectively. The valid region in actual spaces defined by the following set of constraints:

1.33 ≤ J ≤ 5.36 (current density, A/dm2);11.77 ≤ T ≤ 48.22 (temperature, ◦C);3.78 ≤ pH ≤ 6.22 (pH of solution).

Eqs. (14)–(17) have been used for simulation (predictions)nd graphical representations of response surfaces as it ishown in Figs. 3–10.

Figs. 3 and 4 show the response surface plot of the cathodefficiency Y (%) as a function of design variables. In addi-ion to main effects, the interaction effects between factors

Fc = conc.

S20

= 213.0 Ftab(p,�1,�2) = 241.9 Fc < Ftab(p,�1,�2)

are considerable. Thus, at lower values of temperature theincrement of current density J (A/dm2) conducts to straightdecreasing of cathode efficiency, while for higher tempera-ture levels the growth of current density leads to slightlyincreasing of this response. Concerning the influence of tem-perature, the cathode efficiency goes up with increasing oftemperature, especially at higher levels of current density.For low levels of current density the increment of temper-ature conducts to slightly decreasing of cathode efficiency.The influence of pH is also affected by interaction effectsbetween variables. As one can see, the increasing of solu-tion pH leads to descent of cathode efficiency, especiallyat higher levels current density. For low values of currentdensity the influence of pH becomes insignificantly. Thereare no quadratic effects of the factors on cathode efficiencyresponse.

Figs. 5 and 6 illustrate the response surface plots andcontour-lines maps in case of the brightness V (%) of nickeldeposition. Owing to strong quadratic effect, an optimal valueof temperature is observed in the range of 30–36 ◦C leadingto the maximum level of the brightness. The increment ofsolution pH conducts to improving of the brightness. Due tothe main effect, the increasing of current density leads toenhancement of the brightness. The improving of brightnesswith current density is especially conspicuous at higher valuesof temperature.

Response surfaces and contour curves illustrated inFigs. 7 and 8 highlights the effects of design variables onperformance criterion W (Vickers hardness). In this case, theincreasing of solution pH leads to improving of deposit hard-ness. Because of strong quadratic effect, the optimum rangeof temperature is located in the limits of 30–36 ◦C where theVickers hardness is the highest. The effect of current densityon hardness is the lowest one if compared with the effects ofother two variables. This effect is due to interaction betweentemperature and current density. Thus at low values of tem-perature, the increment of current density leads to a minordecrease of the response, W (HV). While at high levels oftemperature the increasing of current density cause a slightimproving of hardness.

In Figs. 9 and 10 are shown the response surfaces related tothe thickness of electroplated layer, U (�m). One can observethat in this case, pH of solution has the greatest influence onthe response. More precisely, the increment of pH in the rangestudied contributes to the evident growth of the thickness U(�m). Due to quadratic effect associated with temperature fac-tor, there is a ridge-type surface with a maximum visible inthe range of 20–40 ◦C. The influence of current density is dom-

inated by the effect of interaction. Thus, for low temperature(<25 ◦C), the increment of current density leads to diminishingof thickness. By contrast, at high levels of temperature (>30 ◦C)

142 chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147

Fig. 3 – Y – response surface plot and contour-lines map depending on T (◦C) and J (A/dm2) variables, holding the thirdvariable at fixed level, pH 5.0.

endi 2

Fig. 4 – Y – response surface plot and contour-lines map depat fixed level, T = 30 ◦C.

the increasing in current density conducts to growth of thethickness.

3.2. Optimization of electroplating process

The optimization of a process described by two or moreresponses, it is helpful to carry out by using the conceptof desirability function. According to this methodology, each

response yi must be converted into the individual desirabil-ity function di that ranges from 0 (very undesirable) to 1 (verydesirable). Such transformation may be presented as follows

Fig. 5 – V – response surface plot and contour-lines map dependfixed level, J = 3.33 A/dm2.

ng on pH and J (A/dm ) variables, holding the third variable

(Akhnazarova and Kafarov, 1982):

di(yi(x)) = exp [−exp(−(ai1 + ai2yi(x)))] ∀i = 1, s (18)

where s is the number of responses involved (in our spe-cific case s = 4); the coefficients ai1 and ai2 were calculated byassigning for two values of yi the corresponding two valuesof di, preferably in the range 0.2 < di <0.8 (Akhnazarova and

Kafarov, 1982).

Afterwards, the individual desirability values are combinedinto the overall desirability function, D, which is computed as

ing on pH and T (◦C) variables, holding the third variable at

chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147 143

Fig. 6 – V – response surface plot and contour-lines map depending on J (A/dm2) and T (◦C) variables, holding the thirdvariable at fixed level, pH 5.0.

F pendfi

gK

D

sntct

Fv

ig. 7 – W – response surface plot and contour-lines map dexed level, J = 3.33 A/dm2.

eometric mean (Derringer and Suich, 1980; Akhnazarova andafarov, 1982; Khayet et al., 2008):

(x) = [d1(y1(x)) × d2(y2(x)) × · · · × ds(ys(x))]1/s (19)

If an individual desirability function is completely unde-irable, i.e. di = 0 then the value of overall desirability isullified. In case of nickel electroplating process discussed in

he present paper, the overall desirability function approachombines four responses, namely, the cathode efficiency,hickness, brightness and hardness of nickel deposition and

ig. 8 – W – response surface plot and contour-lines map dependariable at fixed level, pH 5.0.

ing on pH and T (◦C) variables, holding the third variable at

may be written in extended form as follows:

D(x1, x2, x3) = exp{−(1/4) · [exp(−(−6.731 + 0.075 · Y(x1, x2, x3)))

+ exp(−(−1.495 + 0.039 · U(x1, x2, x3)))

+ exp(−(−0.387 + 0.031 · V(x1, x2, x3)))

+ exp(−(−1.111 + 3.71 × 10−3 · W(x1, x2, x3)))]}

(20)

The formulation of multi-response optimization problemin this case is related to maximization of desirability function

ing on J (A/dm2) and T (◦C) variables, holding the third

144 chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147

Fig. 9 – U – response surface plot and contour-lines map depend ◦

fixed level, J = 3.33 A/dm2.

and may be written as:

maxx ∈ ˝

D(x)

= max{[

d1(Y(x)) × d2(U(x)) × d3(V(x)) × d4(W(x))]1/4

, x ∈ ˝

}(21)

Note, that, the solving of multi-response optimizationproblem defined by Eq. (21) involves figuring out of a solu-tion in the design space that satisfies several specifications(objectives) in the performance space of responses. Usually,such specifications are conflicting and there is no simulta-neous optimal solution for all of them. In this respect, thesolution is not unique; instead there is a set of possible solu-tions where none is best for all responses. Such set of possibleoptimal solutions in the design space is called Pareto optimalset (Blasco et al., 2008). The Pareto optimality is a conditionwhen a single response cannot be improved without damag-ing the qualities of other responses. The region defined by theresponses for all Pareto set points is called the Pareto optimalfront. In fact, the Pareto front supplies a set of solutions wherethe designer (decision-maker, experimentalist) has to look forthe best choice according to his preferences. In order to fig-ure out Pareto optimal set, a genetic algorithm (GA) has beenemployed for mathematical optimization.

Genetic algorithms (GAs) are a special class of evolutionaryalgorithms (EA) that make use of random search techniquesinspired from evolutionary processes in nature (e.g. natural

Fig. 10 – U – response surface plot and contour-lines map depenvariable at fixed level, pH 5.0.

ing on pH and T ( C) variables, holding the third variable at

selection and genetics) (Pasandideh and Niaki, 2006; Hibbert,1993; Leardi, 2007).

GA involves a series of successive steps (Renner and Ekárt,2003; Yuan and Qian, 2010; Preechakul and Kheawhom, 2009;Shopova and Vaklieva-Bancheva, 2006) that are given in theflowchart shown in Fig. 11. The algorithm starts with initial-ization of the first population. Thus, it creates a first currentpopulation {x(k)

c1 , x(k)c2 , . . . , x

(k)cg } with g number of individuals

(population size) where k denotes the iteration number also

known as generation. Each individual x(k)cj

in the population iscalled a chromosome and represents a solution to the prob-lem. Traditionally, solutions are represented in encoded formas binary strings of 0 s and 1 s, but other encodings are alsopossible including real code. Each individual in the currentpopulation is evaluated in terms of fitness value. Note that,according to GA terminology the fitness function is identicalwith the objective function or desirability function in our case.The fitness evaluation of the current population may be noted

as{

D

(x

(k)c1

), D

(x

(k)c2

), . . . , D

(x

(k)cg

)}. Afterwards, the termi-

nation criterion is verified. If it is met, the best solution isthen returned. As usually, the termination criterion allows atmost a predefined number of iterations and verifies whetheran acceptable solution was found (Renner and Ekárt, 2003).If the termination criterion is not satisfied, then, some indi-viduals are selected from the current population based on

the fitness values. Such selected individuals are named par-

ent chromosomes{(

x(k)p1

),

(x

(k)p2

), . . . ,

(x

(k)p(g−m)

)}and are used

ding on J (A/dm2) and T (◦C) variables, holding the third

chemical engineering research and design 8 9 ( 2 0 1 1 ) 136–147 145

t of t

f

a

etN2Stnotr

Gd

Fig. 11 – The flowchar

or generation of new chromosomes (offspring) that make

new population{(

x(k+1)c1

),

(x

(k+1)c2

), . . . ,

(x

(k+1)cg

)}. To this

nd, the genetic operators like reproduction, crossover and muta-ion are used to generate a new population (Pasandideh andiaki, 2006; Hibbert, 1993; Leardi, 2007; Renner and Ekárt,003; Yuan and Qian, 2010; Preechakul and Kheawhom, 2009;hopova and Vaklieva-Bancheva, 2006). Generally, implemen-ations of genetic algorithms differ in the mode of constructedew population. In this work, the population size used in GAptimization was of g = 20. In order to develop new generationshe genetic operators were used in the following proportion:eproduction (10%), mutation (20%) and crossover (70%).

The results of optimization (Pareto optimal set) given by

A are displayed in Table 5. The set of optimal solutionsetermined by mathematical calculation contains 5 points

Table 5 – Pareto optimal solutions.

No. Design variables (factors)

Current density Temperature pH

O1 x1 = 1.211J = 5.35 A/dm2 x2 = 0.104T = 31.56 ◦C x3 = 1.215pH = 6.22 9O2 x1 = 1.202J = 5.34 A/dm2 x2 = 0.806T = 42.09 ◦C x3 = 1.215pH = 6.22 9O3 x1 = 1.211J = 5.35 A/dm2 x2 = 0.229T = 33.44 ◦C x3 = 1.215pH = 6.22 9O4 x1 = 1.211J = 5.35 A/dm2 x2 = 0.104T = 31.56 ◦C x3 = 1.215pH = 6.22 9O5 x1 = 0.601J = 4.34 A/dm2 x2 = 0.408T = 36.12 ◦C x3 = 1.169pH = 6.17 9

he genetic algorithm.

(solutions) that are localized on Pareto front. The desirabil-ity function scores in these points (fitness values) are rangedfrom 0.509 to 0.529 pointing out that all these points are ofinterest. However, selecting a single solution as the optimalpoint for process is a matter of decision that depends on spe-cialist preferences. In this acceptation, the sensitivity analysisand experimental confirmation plays an essential role.

In our case, the emphasis was placed on the experimentalconfirmation of Pareto solutions. After experimental verifica-tion, the best experimental run is the point O3 being defined bythe following values of factors: J* = 5.35 (A/dm2), T* = 33.44 (◦C)and pH* = 6.22. In such conditions the quality of nickel electro-plating deposit (layer) was the best one being superior over the

other points from Pareto optimal set. Note that, for other solu-tions from Pareto set (i.e. O1, O2, O4 and O5) the electroplatinglayers obtained experimentally are of lower quality. More pre-

Responses (performance criteria) Desirable function

Y U V W D

3.537 49.194 27.961 431.96 0.5147.357 50.843 25.174 408.95 0.5294.215 49.712 27.879 433.17 0.5203.537 49.194 27.961 431.96 0.5146.515 49.143 20.979 416.10 0.509

146 chemical engineering research and

Table 6 – Experimental confirmation of optimal solutionO3, J = 5.35 A/dm2, T = 33.44 ◦C, pH = 6.22.

Y (%) U (�m) V (%) W (HV)

Run 1 79.09 52.71 26.10 370.0Run 2 79.31 52.33 26.57 371.3Run 3 78.95 53.28 25.68 373.5Mean value 79.12 52.77 26.12 371.6S.D. 0.18 0.48 0.45 1.77

nickel from aqueous sulfate solutions. Part I. Current

cisely, such layers were non-uniform and had a week adhesionto the substrate.

Table 6 shows the experimental values of responses deter-mined under the conditions given by point O3, i.e. J = 5.35(A/dm2), T = 33.44 (◦C) and pH = 6.22. Three replication runswere performed in these conditions and the mean values ofresponses as well as their standard deviations (S.D.) are alsoreported in Table 6. Thus, the optimal solution O3 is describedby the following mean values of responses confirmed exper-imentally (Table 6): Y = 79.12 ± 0.18 (%), U = 52.77 ± 0.48 (�m),V = 26.12 ± 0.45 (%) and W = 371.6 ± 1.77 (HV). As one candeduce, the optimal condition confirmed experimentally rep-resents a compromise between involved responses and haveconducted to a high quality properties of metallic deposition.

All computations were performed using Matlab software asa tool for scientific programming and technical calculations.

4. Conclusions

In the present work it was demonstrated the applicabil-ity of experimental design, response surface methodologyand desirability function approach for modeling and multi-response optimization of nickel electroplating process.

Based on the experimental design results, the second-order polynomial models were developed for predicting theelectroplating responses (i.e. cathode efficiency Y (%), coatingthickness U (�m), brightness V (%) and Vickers hardness W(HV)) as a functional combination of design variables, namelycurrent density J (A/dm2), temperature T (◦C) and pH of thesolution. The polynomial models were statistically validatedusing F-ratio test. The graphical response surfaces analy-sis was employed for identification and discussion of main,quadratic and interaction effects of factors on responses.

A global objective function (desirability function) wasconstructed for multi-response optimization. In order tosolve the multi-response problem a genetic algorithm wasemployed. The results of optimization offered a set of solu-tions known as Pareto optimal front. The mathematicalsolutions were verified via experimentation and the bestexperimental conditions were established. The optimal solu-tion confirmed by experiment involves the following levelsof factors: J* = 5.35 (A/dm2), T* = 33.44 (◦C) and pH* = 6.22. Thedesirable values of responses in these conditions are as fol-lows: Y = 79.12 ± 0.18 (%), U = 52.77 ± 0.48 (�m), V = 26.12 ± 0.45(%) and W = 371.6 ± 1.77 (HV). As one may conclude, the opti-mal combination of factors conducted to desirable levels ofresponses, and finally to the best characteristics of the elec-troplated cover.

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