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Prediction of grain size of nanocrystalline nickel coatings using adaptive neuro-fuzzy inference system
Solid State Sciences 13 (2011) 163e167
Contents lists available at ScienceDirect
Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie
Prediction of grain size of nanocrystalline nickel coatings using adaptive neuro-fuzzy inference system M. Hayati a, b, *, A.M. Rashidi b, A. Rezaei c a
Computational Intelligence Research Center, Razi University, Tagh-E-Bostan, Kermanshah 67149, Iran Faculty of Engineering, Razi University, Tagh-E-Bostan, Kermanshah 67149, Iran c Electrical Engineering Department, Kermanshah University of Technology, Kermanshah, Iran b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 September 2010 Received in revised form 1 November 2010 Accepted 4 November 2010 Available online 11 November 2010
This paper presents application of adaptive neuro-fuzzy inference system (ANFIS) for prediction of the grain size of nanocrystalline nickel coatings as a function of current density, saccharin concentration and bath temperature. For developing ANFIS model, the current density, saccharin concentration and bath temperature are taken as input, and the resulting grain size of the nanocrystalline coating as the output of the model. In order to provide a consistent set of experimental data, the nanocrystalline nickel coatings have been deposited from Watts-type bath using direct current electroplating within a large range of process parameters i.e., current density, saccharin concentration and bath temperature. Variation of the grain size because of the electroplating parameters has been modeled using ANFIS, and the experimental results and theoretical approaches have been compared to each other as well. Also, we have compared the proposed ANFIS model with artificial neural network (ANN) approach. The results have shown that the ANFIS model is more accurate and reliable compared to the ANN approach. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Adaptive neuro-fuzzy inference system Grain size Nanocrystalline nickel coating
1. Introduction Nanocrystalline (NC) materials are novel polycrystalline bulk materials with grain sizes less than 100 nm [1]. The research trends are attributed to the expectations that the modification of the crystallite size to nanoscales which, lead to the unique properties is not obtained for the conventional crystalline metals. For example, the hardness of the direct current (DC) the electrodeposited nickel with average grain size of 14 nm is about 6.4 GPa, whereas for the conventional nickel with average grain size of 15 mm, it is about 1.2 GPa [2]. The superior properties of NC materials have led to great potential for a number of technological applications such as soft magnets, catalysis, hydrogen storage and purification, corrosion and wear resistant coatings, electrical connectors and alkaline fuel cell electrodes [3e5]. Nanocrystalline materials have been synthesized by several methods [6]. Among them the electroplating is a powerful method for the fabrication of many highly precise products with controlled shapes and sizes [7], and consequently plays an important role in today’s applied NC coating [8]. It has been known [9e11] that the properties of electrodeposits depend on their microstructure, which can be substantially influenced by the deposition conditions.
* Corresponding author. Faculty of Engineering, Razi University, Tagh-E-Bostan, Kermanshah 67149, Iran. Tel.: þ98(831) 4274535; fax: þ98(831) 4274542. E-mail address: [email protected] (M. Hayati). 1293-2558/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2010.11.007
Since, the performance of electrodeposition for application of the nanocrystalline coatings is actually related to the electroplating conditions and under chosen suitable electrodeposition parameters, only the electroplating yields grain sizes in the nanometer range [12e14]. Although, the prediction of the grain size of nanocrystalline coatings using ANN has been done [15], but obtaining an accurate model has remained as a challenge. This paper presents the applicability of ANFIS as an accurate and reliable model for the prediction of the grain size of nanocrystalline nickel coatings. Finally, the results are compared with ANN model [15]. 2. Experimental procedure Nanocrystalline nickel coatings have been deposited on copper substrate from a typical Watts bath containing 0e10 g/l sodium saccharin (C7H4NO3S.Na) as a grain refiner agent at current density of 10e300 mA/cm2 and bath temperature of 45e70 C, by DC electrodeposition. The details of the deposition set-up, bath composition and the other conditions can be found in [15]. The grain size of the deposits has been calculated from the full width at half maximum (FWHM) intensity of X-ray diffraction (XRD) as described in [16]. The XRD examination has been carried out using a Philips X’Pert-Pro instrument operated at 40 kV and 30 mA with CoKa radiation (l ¼ 1.789 A ) at a scan rate of 0.05 s1 and 0.02 step size. An annealed nickel (coarse grained) sample with an
164
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167
Fig. 1. The inference method of Sugeno model.
average grain size of 30 mm has been also used for the correction of the instrumental line broadening.
Layer 1: Every node in this layer contains membership functions described by generalized Gaussian function:
3. Modeling approach ANFIS [17,18] is an adaptive network, which permits the application of neural network topology together with fuzzy logic. The goal of ANFIS is to find a model which will correctly simulate the inputs with the outputs. An ANFIS consisting of a set of TSK-type fuzzy IF-THEN rules is used to map the system inputs to its outputs. At the computational level, ANFIS can be regarded as a flexible mathematical structure that can approximate a large class of complex nonlinear systems to a desirable degree of accuracy. For simplicity, we assume that the fuzzy inference system has two inputs (x, y) and one output (f). For the first order Sugeno fuzzy model, a typical rule set with fuzzy based IF-THEN rules can be expressed as follows: Rule1:If x is A1 and y is B1, then f1 ¼ p1 x þ q1 y þ r1 . Rule2: If x is A2 and y is B2, then f2 ¼ p2 x þ q2 y þ r2 . pi, qi and ri are linear output parameters (consequent parameters) where i ¼ 1, 2. Fig. 1 illustrates the reasoning mechanism for this Sugeno model, the corresponding equivalent ANFIS architecture is shown in Fig. 2. Layer 1
Layer 2
Layer 3
Layer 4
mA ðxÞ ¼ exp
0:5ðx cÞ2
!
s2
where c, s are referred to premise parameters. Layer 2: Each node in this layer is a fixed node and calculates the firing strength of a rule via multiplication. Layer 3: Every node of this layer calculates the weight, which is normalized. The outputs of this layer are called normalized firing strengths. Layer 4: The output of this layer is compressed of a linear combination of the inputs multiplied by the normalized firing strength w. Layer 5: This layer is the simple summation of the outputs of layer 4. The adjustment of the modifiable parameters is a two-step process. First, the consequent parameters are identified by the least
Layer 5
x y
A1 w1
x
The layers shown in Fig. 2 are defined as follows [19,20]:
w1
w1 f1
A2
f B1 y
w2
w2
w2 f2
B2 x y Fig. 2. ANFIS architecture based on TakagieSugeno.
Fig. 3. A simplified overview of ANFIS model.
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167 Table 1 Optimal architecture and specification of proposed ANFIS model.
450 Sugeno 3/1 6 for all inputs 6 Gaussian Linear 6 100
square estimation, then the premise parameters are updated by the gradient descent. The ANFIS is a strong tool for the prediction and simulation in engineering applications. In this study, the variation of grain size with electroplating parameters has been modeled by the application of ANFIS. The inputs parameters are bath temperature, saccharin concentration and current density, the output parameter is grain
400 Experimentally Measured Grain Size (nm)
Type Inputs/Outputs No. of input membership functions No. of output membership functions Input membership function type Output membership function type No. of fuzzy rules No. of epochs
165
350 Correlation Factor = 0.999999 300 250 200 150 100 50 0
0
50
100
150 200 250 300 Predicted Grain Size (nm)
350
400
450
Fig. 5. Comparison of experimental and predicted (ANFIS model) results for training data for grain size at different current densities, saccharin concentrations and bath temperatures.
size of nanocrystalline nickel coatings. A simplified overview of the proposed ANFIS model is shown in Fig. 3. To build the ANFIS model, 16 data (about 70% of whole data) have been used for training and 7 data (about 30% of whole data) were used for testing the ANFIS model. The final ANFIS architecture used in this study is described in Table 1. 4. Results and discussion Fig. 4a depicts the typical X-ray diffraction patterns of an annealed nickel (coarse grained) sample and nanocrystalline deposits produced at various current densities. It can be observed that the crystal structure of the annealed and nanocrystalline samples is pure fcc nickel and no characteristic peaks of the other phases have been recorded. As seen, in spite of the intensity of Xray scattered from (111) crystallography plans, the diffraction intensity of (200) texture of nickel deposit has been sharply decreased by increasing the current density, indicating that the
70
Experimentally Measured Grain Size (nm)
65 60
50 45 40 35 30 25 20 20
Fig. 4. Comparison between (a) XRD patterns and (b) the peak broadening of profiles of X-ray scattered from (111) and (200) plans of annealed (reference) sample and nickel coatings electrodeposited at current density of 10 and 150 mA/cm2.
Correlation Factor = 0.999615
55
25
30
35
40 45 50 55 Predicted Grain Size (nm)
60
65
70
Fig. 6. Comparison of experimental and predicted (ANFIS model) results for testing data for grain size at different current densities, saccharin concentrations and bath temperatures.
166
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167
Table 2 Relative errors for training and testing results of the proposed ANFIS model.
450 Experimental ANFIS Model ANN Model
400
RE%
Train
Test
Min Max Mean
6.23E-07 0.001473 0.00051
0.001473 1.99 0.93
350
preferred orientation (textures) of the nanocrystalline Ni has been markedly influenced by the deposition parameters. However, this is out of the scope of the present work. In order to have a better view of the distinction between the peak broadening of the samples, the (111) and (200) peaks have been only represented in Fig. 4b. The peak broadening of electrodeposited coatings with respect to the reference sample is evident in this figure indicating the grain refining to nanoscale size, because according to line-broadening theory [21,22], the peak width difference between the reference sample and investigated specimens become readily observable when the grain sizes are in the nanometer ranges. The training and testing results of the proposed ANFIS model are shown in Figs. 5 and 6 and Table 2, wherein mean relative error (MRE) is given by:
N 1 X Xi ðExpÞ Xi ðPredÞ MRE ¼ N i¼1 Xi ðExpÞ where ‘X(Exp)’ and ‘X(Pred)’ stand for experimental and predicted (ANFIS) values, respectively and N is the number of data. The predicted values of the grain sizes using proposed ANFIS model in comparison with the experimental data and the best ANN configuration obtained in [15] as a function of current density, saccharin concentration and bath temperature are shown in Figs. 7e9, respectively. As it can be seen, there is a remarkable agreement between the experimental and predicted values using ANFIS. Also, it is clear that the ANFIS model is better or rather more accurate than the ANN model presented in [15], which implies that the proposed ANFIS model can be used as an efficient tool to predict the grain size of nanocrystalline nickel coatings. In this paper we have proposed the ANFIS model as an improved approach over ANN model in [15]. The ANFIS model could
Grain Size (nm)
300 250 200 o
Bath Temperature = 55 C 2 Current Density = 100 mA/cm
150 100 50 0
0
1
2
3
4 5 6 Saccharin Concentration (g/l)
7
8
9
10
Fig. 8. Comparison between the experimental and predicted grain size of nanocrystalline nickel coating using ANFIS and ANN models as a function of saccharin concentration.
significantly reduce the overall (train and test) MRE% to less than 0.5% in comparison with ANN model that the overall (train and test) MRE% were less than 2%. Another aspect of the superiority of ANFIS model in comparison with ANN model is the lower number of epochs which is needed to reach convergence. Therefore, the training time for the ANFIS model is definitely less than the required time for designing similar model using pure neural network. It means that the ANFIS model is better than ANN for redeveloping the model and increasing the input parameters. It may be noted that for an ANN model, we have to perform a trial and error process to develop the optimal network architecture, while the ANFIS model does not require such a procedure. Because one of the advantages of ANFIS as opposed to ANN is that the ANFIS is more transparent and we can obtain inputeoutput relationship from membership functions and IF-THEN rules.
32 200 Experimental ANFIS Model ANN Model
160
30
140
29
Grain Size (nm)
Grain Size (nm)
180
120 100 o
Bath Temperature = 55 C Saccharin Concentration = 5 g/l
80
Experimental ANFIS Model ANN Model
31
2
28
Current Density = 100 mA/cm Saccharin Concentration = 5 g/l
27
26 60
25
40 20
0
50
100
150
200
250
300
Current Density (mA/cm 2 )
Fig. 7. Comparison between the experimental and predicted grain size of nanocrystalline nickel coatings using ANFIS and ANN models as a function of current density.
24 45
50
55
60
65
o
Bath Temperature ( C )
Fig. 9. Comparison between the experimental and predicted grain size of nanocrystalline nickel coatings using ANFIS and ANN models as a function of bath temperature.
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167
5. Conclusions In this paper, we have presented the applicability of adaptive neuro-fuzzy inference system for the modeling and prediction of the grain size of nanocrystalline coatings produced at various conditions. The comparison between the experimental and the predicted values of the proposed ANFIS model shows that there is an excellent agreement between the predicted grain size and the experimental results with least error. This means that the proposed model can simulate the grain size of nanocrystalline coatings produced at various conditions in very short time with good accuracy. The results obtained with the ANFIS model is also compared to an ANN model and it has shown that ANFIS is more reliable for modeling of such nonlinear systems and has more accuracy and flexibility in comparison to ANN model. References [1] C. Suryanarayana, C.C. Koch, Hyperfine Interactions 130 (2000) 5. [2] C.A. Schuh, T.G. Nieh, T. Yamasaki, Scripta Materialia 46 (2002) 735.
167
[3] U. Erb, A.M. EI-Sherik, G. Palumbo, K.T. Aust, Nanostructured Materials 2 (1993) 383. [4] D. Clark, D. Wood, U. Erb, Nanostructured Materials 9 (1997) 755. [5] A. Robertson, U. Erb, G. Palumbo, Nanostructured Materials 12 (1999) 1035. [6] S.C. Tjong, H. Chen, Materials Science and Engineering R45 (2004) 1. [7] D. Bera, S.C. Kuiry, S. Seal, Journal of the Minerals 56 (2004) 49. [8] H. Natter, R. Hempelmann, Journal of Physical Chemistry 222 (2008) 319. [9] W.H. Safranek, Journal of Vacuum Science and Technology 11 (1974) 765. [10] C. Kollia, N. Spyrellis, Surface and Coatings Technology 57 (1993) 71. [11] Y. Xuetao, W. Yu, S. Dongbai, Y. Hongying, Surface and Coatings Technology 202 (2008) 1895. [12] A.M. El-Sherik, U. Erb, Journal of Matierals Science 30 (1995) 5743. } cs, K. Varga-Josepovits, [13] I. Bakony, E. Tóth-Kádár, L. Pogány, Á Cziráki, I. Gero B. Arnold, K. Wetig, Surface and Coatings Technology 78 (1996) 124. [14] D. Pin-Qiang, Y. Hui, L. Qiang, Transactions on Material and Heat Treatment 25 (2004) 1283. [15] A.M. Rashidi, A.R. Eivani, A. Amadeh, Computational Materials Science 45 (2009) 499. [16] A.M. Rashidi, A. Amadeh, Surface and Coatings Technology 202 (2008) 3772. [17] J.-S.R. Jang, C.-T. Sun, Prentice Hall 19 (1997) 514. [18] T. Takagi, M. Sugeno, IEEE Transactions on Systems 15 (1985) 116. [19] S. Chiu, Journal of Intelligent and Fuzzy Systems 2 (1994) 762. [20] J.-S.R. Jang, C.-T. Sun, Proceedings of the IEEE 83 (1995) 378. [21] P. Scherrer, Göttinger Nachrichten 2 (1918) 98. [22] E. Warn, Journal of Applied Physics 12 (1941) 379.
Contents lists available at ScienceDirect
Solid State Sciences journal homepage: www.elsevier.com/locate/ssscie
Prediction of grain size of nanocrystalline nickel coatings using adaptive neuro-fuzzy inference system M. Hayati a, b, *, A.M. Rashidi b, A. Rezaei c a
Computational Intelligence Research Center, Razi University, Tagh-E-Bostan, Kermanshah 67149, Iran Faculty of Engineering, Razi University, Tagh-E-Bostan, Kermanshah 67149, Iran c Electrical Engineering Department, Kermanshah University of Technology, Kermanshah, Iran b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 11 September 2010 Received in revised form 1 November 2010 Accepted 4 November 2010 Available online 11 November 2010
This paper presents application of adaptive neuro-fuzzy inference system (ANFIS) for prediction of the grain size of nanocrystalline nickel coatings as a function of current density, saccharin concentration and bath temperature. For developing ANFIS model, the current density, saccharin concentration and bath temperature are taken as input, and the resulting grain size of the nanocrystalline coating as the output of the model. In order to provide a consistent set of experimental data, the nanocrystalline nickel coatings have been deposited from Watts-type bath using direct current electroplating within a large range of process parameters i.e., current density, saccharin concentration and bath temperature. Variation of the grain size because of the electroplating parameters has been modeled using ANFIS, and the experimental results and theoretical approaches have been compared to each other as well. Also, we have compared the proposed ANFIS model with artificial neural network (ANN) approach. The results have shown that the ANFIS model is more accurate and reliable compared to the ANN approach. Ó 2010 Elsevier Masson SAS. All rights reserved.
Keywords: Adaptive neuro-fuzzy inference system Grain size Nanocrystalline nickel coating
1. Introduction Nanocrystalline (NC) materials are novel polycrystalline bulk materials with grain sizes less than 100 nm [1]. The research trends are attributed to the expectations that the modification of the crystallite size to nanoscales which, lead to the unique properties is not obtained for the conventional crystalline metals. For example, the hardness of the direct current (DC) the electrodeposited nickel with average grain size of 14 nm is about 6.4 GPa, whereas for the conventional nickel with average grain size of 15 mm, it is about 1.2 GPa [2]. The superior properties of NC materials have led to great potential for a number of technological applications such as soft magnets, catalysis, hydrogen storage and purification, corrosion and wear resistant coatings, electrical connectors and alkaline fuel cell electrodes [3e5]. Nanocrystalline materials have been synthesized by several methods [6]. Among them the electroplating is a powerful method for the fabrication of many highly precise products with controlled shapes and sizes [7], and consequently plays an important role in today’s applied NC coating [8]. It has been known [9e11] that the properties of electrodeposits depend on their microstructure, which can be substantially influenced by the deposition conditions.
* Corresponding author. Faculty of Engineering, Razi University, Tagh-E-Bostan, Kermanshah 67149, Iran. Tel.: þ98(831) 4274535; fax: þ98(831) 4274542. E-mail address: [email protected] (M. Hayati). 1293-2558/$ e see front matter Ó 2010 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.solidstatesciences.2010.11.007
Since, the performance of electrodeposition for application of the nanocrystalline coatings is actually related to the electroplating conditions and under chosen suitable electrodeposition parameters, only the electroplating yields grain sizes in the nanometer range [12e14]. Although, the prediction of the grain size of nanocrystalline coatings using ANN has been done [15], but obtaining an accurate model has remained as a challenge. This paper presents the applicability of ANFIS as an accurate and reliable model for the prediction of the grain size of nanocrystalline nickel coatings. Finally, the results are compared with ANN model [15]. 2. Experimental procedure Nanocrystalline nickel coatings have been deposited on copper substrate from a typical Watts bath containing 0e10 g/l sodium saccharin (C7H4NO3S.Na) as a grain refiner agent at current density of 10e300 mA/cm2 and bath temperature of 45e70 C, by DC electrodeposition. The details of the deposition set-up, bath composition and the other conditions can be found in [15]. The grain size of the deposits has been calculated from the full width at half maximum (FWHM) intensity of X-ray diffraction (XRD) as described in [16]. The XRD examination has been carried out using a Philips X’Pert-Pro instrument operated at 40 kV and 30 mA with CoKa radiation (l ¼ 1.789 A ) at a scan rate of 0.05 s1 and 0.02 step size. An annealed nickel (coarse grained) sample with an
164
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167
Fig. 1. The inference method of Sugeno model.
average grain size of 30 mm has been also used for the correction of the instrumental line broadening.
Layer 1: Every node in this layer contains membership functions described by generalized Gaussian function:
3. Modeling approach ANFIS [17,18] is an adaptive network, which permits the application of neural network topology together with fuzzy logic. The goal of ANFIS is to find a model which will correctly simulate the inputs with the outputs. An ANFIS consisting of a set of TSK-type fuzzy IF-THEN rules is used to map the system inputs to its outputs. At the computational level, ANFIS can be regarded as a flexible mathematical structure that can approximate a large class of complex nonlinear systems to a desirable degree of accuracy. For simplicity, we assume that the fuzzy inference system has two inputs (x, y) and one output (f). For the first order Sugeno fuzzy model, a typical rule set with fuzzy based IF-THEN rules can be expressed as follows: Rule1:If x is A1 and y is B1, then f1 ¼ p1 x þ q1 y þ r1 . Rule2: If x is A2 and y is B2, then f2 ¼ p2 x þ q2 y þ r2 . pi, qi and ri are linear output parameters (consequent parameters) where i ¼ 1, 2. Fig. 1 illustrates the reasoning mechanism for this Sugeno model, the corresponding equivalent ANFIS architecture is shown in Fig. 2. Layer 1
Layer 2
Layer 3
Layer 4
mA ðxÞ ¼ exp
0:5ðx cÞ2
!
s2
where c, s are referred to premise parameters. Layer 2: Each node in this layer is a fixed node and calculates the firing strength of a rule via multiplication. Layer 3: Every node of this layer calculates the weight, which is normalized. The outputs of this layer are called normalized firing strengths. Layer 4: The output of this layer is compressed of a linear combination of the inputs multiplied by the normalized firing strength w. Layer 5: This layer is the simple summation of the outputs of layer 4. The adjustment of the modifiable parameters is a two-step process. First, the consequent parameters are identified by the least
Layer 5
x y
A1 w1
x
The layers shown in Fig. 2 are defined as follows [19,20]:
w1
w1 f1
A2
f B1 y
w2
w2
w2 f2
B2 x y Fig. 2. ANFIS architecture based on TakagieSugeno.
Fig. 3. A simplified overview of ANFIS model.
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167 Table 1 Optimal architecture and specification of proposed ANFIS model.
450 Sugeno 3/1 6 for all inputs 6 Gaussian Linear 6 100
square estimation, then the premise parameters are updated by the gradient descent. The ANFIS is a strong tool for the prediction and simulation in engineering applications. In this study, the variation of grain size with electroplating parameters has been modeled by the application of ANFIS. The inputs parameters are bath temperature, saccharin concentration and current density, the output parameter is grain
400 Experimentally Measured Grain Size (nm)
Type Inputs/Outputs No. of input membership functions No. of output membership functions Input membership function type Output membership function type No. of fuzzy rules No. of epochs
165
350 Correlation Factor = 0.999999 300 250 200 150 100 50 0
0
50
100
150 200 250 300 Predicted Grain Size (nm)
350
400
450
Fig. 5. Comparison of experimental and predicted (ANFIS model) results for training data for grain size at different current densities, saccharin concentrations and bath temperatures.
size of nanocrystalline nickel coatings. A simplified overview of the proposed ANFIS model is shown in Fig. 3. To build the ANFIS model, 16 data (about 70% of whole data) have been used for training and 7 data (about 30% of whole data) were used for testing the ANFIS model. The final ANFIS architecture used in this study is described in Table 1. 4. Results and discussion Fig. 4a depicts the typical X-ray diffraction patterns of an annealed nickel (coarse grained) sample and nanocrystalline deposits produced at various current densities. It can be observed that the crystal structure of the annealed and nanocrystalline samples is pure fcc nickel and no characteristic peaks of the other phases have been recorded. As seen, in spite of the intensity of Xray scattered from (111) crystallography plans, the diffraction intensity of (200) texture of nickel deposit has been sharply decreased by increasing the current density, indicating that the
70
Experimentally Measured Grain Size (nm)
65 60
50 45 40 35 30 25 20 20
Fig. 4. Comparison between (a) XRD patterns and (b) the peak broadening of profiles of X-ray scattered from (111) and (200) plans of annealed (reference) sample and nickel coatings electrodeposited at current density of 10 and 150 mA/cm2.
Correlation Factor = 0.999615
55
25
30
35
40 45 50 55 Predicted Grain Size (nm)
60
65
70
Fig. 6. Comparison of experimental and predicted (ANFIS model) results for testing data for grain size at different current densities, saccharin concentrations and bath temperatures.
166
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167
Table 2 Relative errors for training and testing results of the proposed ANFIS model.
450 Experimental ANFIS Model ANN Model
400
RE%
Train
Test
Min Max Mean
6.23E-07 0.001473 0.00051
0.001473 1.99 0.93
350
preferred orientation (textures) of the nanocrystalline Ni has been markedly influenced by the deposition parameters. However, this is out of the scope of the present work. In order to have a better view of the distinction between the peak broadening of the samples, the (111) and (200) peaks have been only represented in Fig. 4b. The peak broadening of electrodeposited coatings with respect to the reference sample is evident in this figure indicating the grain refining to nanoscale size, because according to line-broadening theory [21,22], the peak width difference between the reference sample and investigated specimens become readily observable when the grain sizes are in the nanometer ranges. The training and testing results of the proposed ANFIS model are shown in Figs. 5 and 6 and Table 2, wherein mean relative error (MRE) is given by:
N 1 X Xi ðExpÞ Xi ðPredÞ MRE ¼ N i¼1 Xi ðExpÞ where ‘X(Exp)’ and ‘X(Pred)’ stand for experimental and predicted (ANFIS) values, respectively and N is the number of data. The predicted values of the grain sizes using proposed ANFIS model in comparison with the experimental data and the best ANN configuration obtained in [15] as a function of current density, saccharin concentration and bath temperature are shown in Figs. 7e9, respectively. As it can be seen, there is a remarkable agreement between the experimental and predicted values using ANFIS. Also, it is clear that the ANFIS model is better or rather more accurate than the ANN model presented in [15], which implies that the proposed ANFIS model can be used as an efficient tool to predict the grain size of nanocrystalline nickel coatings. In this paper we have proposed the ANFIS model as an improved approach over ANN model in [15]. The ANFIS model could
Grain Size (nm)
300 250 200 o
Bath Temperature = 55 C 2 Current Density = 100 mA/cm
150 100 50 0
0
1
2
3
4 5 6 Saccharin Concentration (g/l)
7
8
9
10
Fig. 8. Comparison between the experimental and predicted grain size of nanocrystalline nickel coating using ANFIS and ANN models as a function of saccharin concentration.
significantly reduce the overall (train and test) MRE% to less than 0.5% in comparison with ANN model that the overall (train and test) MRE% were less than 2%. Another aspect of the superiority of ANFIS model in comparison with ANN model is the lower number of epochs which is needed to reach convergence. Therefore, the training time for the ANFIS model is definitely less than the required time for designing similar model using pure neural network. It means that the ANFIS model is better than ANN for redeveloping the model and increasing the input parameters. It may be noted that for an ANN model, we have to perform a trial and error process to develop the optimal network architecture, while the ANFIS model does not require such a procedure. Because one of the advantages of ANFIS as opposed to ANN is that the ANFIS is more transparent and we can obtain inputeoutput relationship from membership functions and IF-THEN rules.
32 200 Experimental ANFIS Model ANN Model
160
30
140
29
Grain Size (nm)
Grain Size (nm)
180
120 100 o
Bath Temperature = 55 C Saccharin Concentration = 5 g/l
80
Experimental ANFIS Model ANN Model
31
2
28
Current Density = 100 mA/cm Saccharin Concentration = 5 g/l
27
26 60
25
40 20
0
50
100
150
200
250
300
Current Density (mA/cm 2 )
Fig. 7. Comparison between the experimental and predicted grain size of nanocrystalline nickel coatings using ANFIS and ANN models as a function of current density.
24 45
50
55
60
65
o
Bath Temperature ( C )
Fig. 9. Comparison between the experimental and predicted grain size of nanocrystalline nickel coatings using ANFIS and ANN models as a function of bath temperature.
M. Hayati et al. / Solid State Sciences 13 (2011) 163e167
5. Conclusions In this paper, we have presented the applicability of adaptive neuro-fuzzy inference system for the modeling and prediction of the grain size of nanocrystalline coatings produced at various conditions. The comparison between the experimental and the predicted values of the proposed ANFIS model shows that there is an excellent agreement between the predicted grain size and the experimental results with least error. This means that the proposed model can simulate the grain size of nanocrystalline coatings produced at various conditions in very short time with good accuracy. The results obtained with the ANFIS model is also compared to an ANN model and it has shown that ANFIS is more reliable for modeling of such nonlinear systems and has more accuracy and flexibility in comparison to ANN model. References [1] C. Suryanarayana, C.C. Koch, Hyperfine Interactions 130 (2000) 5. [2] C.A. Schuh, T.G. Nieh, T. Yamasaki, Scripta Materialia 46 (2002) 735.
167
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