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Chemometrics and Intelligent Laboratory Systems journal homepage: www.elsevier.com/locate/chemolab

Pore diameter of nanoporous anodic alumina: Experimental study and application of ANFIS and MLR Hamed Akbarpour a,⁎, Mahdi Mohajeri b, Masoumeh Akbarpour c a b c

Department of Civil and Environmental Engineering, The Pennsylvania State University, State College, USA Materials Science and Engineering, Texas A&M University, College Station, USA Department of Civil Engineering, K.N. Toosi University of Technology, Tehran, Iran

a r t i c l e

i n f o

Article history: Received 23 October 2015 Received in revised form 25 February 2016 Accepted 27 February 2016 Available online 5 March 2016 Keywords: Pore diameter Nanoporous anodic alumina Anodization TGA ANFIS MLR

a b s t r a c t This paper presents numerical and experimental results to show the effect of nonlinear models has on the prediction of pore diameter of nanoporous anodic alumina (NPAA). For this purpose, a database of pore diameters was collected from the literature to developing two models: (1) an adaptive neuro-fuzzy inference system (ANFIS) model and (2) a multiple linear regression (MLR) model. Inputs for these two models compromised of the concentration, the temperature and the voltage whereas pore diameter in a NPAA membrane was the target output. To illustrate the accuracy of two proposed models, a comparison between outputs obtained from the models, test results and two empirical formulas has been made. The comparison reveals that the models have a good capability of predicting pore diameters with an acceptable error with respect to empirical formulas. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Fabrication of nanostructured aluminum oxide membranes using two-step anodization of pure aluminum has strikingly attracted attentions because of its numerous applications from electro-optical [1] and mechanical engineering [2] to the bioengineering [3] and human life science [4,5]. Researchers have focused on special properties of NPAA membranes since they can be used in so many ﬁelds: gas separation [6], synthesis of magnetic materials [7], Nano-ﬂuid [8] and hemodialysis [9]. They described the effect of the geometry and arrangement of nanopores in a NPAA membrane has on a ﬁnal product. In addition to geometry and distribution of pores, anodizing conditions should be taken into account to control dimensions of nano-pores precisely [10–15]. It is understood that any changes in anodizing conditions, voltage, electrolyte temperature, anodizing duration and acidic electrolyte concentration would result in a change in pores diameter in a NPAA membrane from a few to hundreds of nanometer [16–18]. O'Sullivan and Woods [10] presented a relationship between the pore diameter and anodizing voltage with a correlation constant of 1.29 nm·V−1. Although the relationship was developed for a phosphoric acid electrolyte solution with a limited applied anodization (80–120), but it has been broadly conﬁrmed as a useful guideline by subsequent studies [19]. ⁎ Corresponding author. E-mail address: [email protected] (H. Akbarpour).

http://dx.doi.org/10.1016/j.chemolab.2016.02.012 0169-7439/© 2016 Elsevier B.V. All rights reserved.

Palibroda et al. [20,21] reported the dependence of the pore diameter on anodizing potential. They suggested a linear relationship for anodization under potentiostatic conditions. The empirical equations are given, respectively, as follows: DP ¼ 1:29 U

ð1Þ

DP ¼ 4:986 þ 0:709 U

ð2Þ

where DP and U are pore diameter (in nanometer) and anodizing voltage (in V), respectively. Sulka et al. [11] reported the dependence of pore diameter on temperature of sulfuric acid electrolyte solution. Parkhutik et al. [22] presented that pore diameter could be affected by electrolyte concentration. In this study, membranes were prepared by rapid detachment process in a two-step anodization and pores diameters were measured by ﬁeld emission scanning electron microscope (FESEM). Thermal stability and crystalline structure were investigated by X-ray Diffraction (XRD) and Thermo-gravimetric analysis (TGA). In addition, an artiﬁcial intelligence tool was used to ﬁnd patterns existed in database by learning and inferring from available data. In this regard, ANFIS and MLR methods both were used to develop predictive models. A total of 32 test results were collected from the literature and experiments done by the authors. Three factors affect pore diameter in a NPAA membrane which were considered as input variables, as follows: (1) concentration (C),

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(2) temperature (T) and (3) applied voltage (U). Finally, results obtained from the models and two empirical formulas were compared to target values. The results indicate that the models estimate pore diameter in a NPAA membrane more accurate than empirical formulas.

contains 32 test results for training, validation and testing the models. Each set of records constitutes of three parameters affecting on the pore diameter, including: (1) concentration, (2) voltage and (3) temperature. The distribution of them is shown in Fig. 2.

2. Experimental study

4. ANFIS

Aluminum foil (99.9%, 0.2 mm thickness) is utilized as a working substrate and is precut into two coupons (0.5 cm × 2 cm) with annealing in air at 450 °C and for 3 h. The native oxide layer is removed by sinking coupons in the chemical polish solution (1 M NaOH) for 3 min. The samples are then immersed in ethanol and deionized water. Two prepared coupons are employed as working and counter electrode which are located at 3 cm interval. Conventionally, two-step self-organized procedure is applied to a NPAA ﬁlm fabrication with electrolyte of oxalic acid (0.3 M) at potentiostatic conditions (40 V, 27 °C in ﬁrst step and 40 V, 27 °C in second step). An electrochemical cell which is equipped with a stirrer and a circulator is used to keep the electrolyte temperature constant during anodizing process. In detachment step, one step pulse is input at potentiostatic condition (55 V) in 70% HClO4 + CH3CH2OH (v:v = 1:1) solution [23]. Fig. 1 shows the procedure for preparing a NPAA membrane. The morphology of the sample is investigated by a ﬁeld emission scanning electron microscope (Hitachi S-4160, Japan). Pore diameter is estimated via Image J 1.37v software. X-ray diffraction (XRD) is performed using Philips Analytical PC-APD diffract meter. Thermal analysis of samples is done by Netzch STA/409 PC thermal analysis system. TGA and DSC have been performed on simultaneous thermal analyzer apparatus of Rheometric Scientiﬁc Company (STA1500+ Model, England) with a heating rate 15 °C·min−1 under dry air ﬂow.

ANFIS is a neuro-fuzzy system which can be used to model different types of systems [32–35]. ANFIS uses fuzzy sets and a linguistic model with some if–then rules. The most important strength of ANFIS is that it could provide approximations of a parameter with acceptable errors [36]. ANFIS contains three main parts: a fuzzy rule base, a set of data and a reasoning procedure. Data would deﬁne membership functions while the reasoning procedure performs an inference process [37,38]. In the rule base, ANFIS uses a ﬁrst-order Sugeno-type fuzzy inference system. This system has two inputs (x and y) and an output f. A ﬁrst-order Sugeno-type fuzzy system is described in the following part: Rule 1: If x is A1 and y is B2, then f1 = p1x + q1 y + r1 Rule 2: If x is A2 and y is B2, then f2 = p2x + q2 y + r2 where x and y are the input features, p1 , p2 , q1 ,q2 , r1 , r2 are linear coefﬁcients in the consequent part, A1 , A2 , B1 , B2 are nonlinear coefﬁcients and f(x, y) is a ﬁrst-order polynomial [39]. The inference methodology

3. Data gathering The aim of this research is to conduct an experimental study as well as developing ANFIS and MLR models to predict the pore diameter of a NPAA membrane. For this purpose, a database is collected from the literature and the study done by the authors [12,24–31]. The database

Fig. 1. Preparation process of a NPAA membrane.

Fig. 2. Distribution of samples in terms of anodizing parameters.

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where μ Ai and μ Bj are the membership functions. They could have different shapes as a triangular–trapezoidal shape, a Gaussian shape, or a bell shape [40]. Layer 2 (Multiplication): This layer has a number of nodes labeled Π, in which, incoming signals have been multiplied by each other. The output would be: ωi ¼ μ Ai ðxÞ μ Bi ðyÞ

i ¼ 1; 2:

ð5Þ

Each output represents the strength of the rule [41]. Layer 3 (Normalization): In this layer, every strength has been normalized with respect to the sum of strengths. ω¼

Fig. 3. Inference methodology of ANFIS.

ω1 ω1 þ ω2

i ¼ 1; 2

ð6Þ

where ω is a normalized strength [41]. Layer 4 (Defuzziﬁcation): This layer performs the defuzziﬁcation task through fuzzy rule procedures. Every node i is a square node with a speciﬁc node function. Inputs and outputs are ﬁnally correlated to each other by: O4i ¼ ωi f i ¼ ωi ðpi x þ qi y þ r i Þ

i ¼ 1; 2

ð7Þ

where pi, qi, and ri are consequent coefﬁcients of fi [41]. Layer 5 (Summation): The total output is the sum of all input signals as [41]: X

Fig. 4. Schematic structure of ANFIS.

of a ﬁrst-order Sugeno-type fuzzy inference system is illustrated in Fig. 3. A typical ANFIS model includes ﬁve different layers that everyone performs a distinguish action. A schematic of ANFIS architecture is illustrated in Fig. 4. The functions of each layer are explained brieﬂy, as follow: Layer 1 (Fuzziﬁcation): This layer converts given inputs into a fuzzy set by membership functions. Each node i in the ﬁrst layer is an adaptive node with the node function of: O1Ai ¼ μ Ai ðxÞ

i ¼ 1; 2

ð3Þ

O1B j ¼ μ B j ðyÞ

j ¼ 1; 2

ð4Þ

O5i ¼

X

ωi f i

i ωi f i ¼ X

i

ωi

i ¼ 1; 2:

ð8Þ

i

4.1. ANFIS parameters ANFIS model consists of three inputs (C, T and U) and one target output (DP). DP, C, T and U are pore diameter, concentration, temperature and voltage, respectively. In this model, a Gaussian membership function is used. The model has been trained and tested by 20 and 6 data records. In addition, up to 800 epochs is speciﬁed in the training phase to get the minimum error. The rule base would be as follows [42]: Rule 1: If C is A1, T is B1, U and is C1, then f1 =p1C+ q1T + r1U+ s1. Rule 2: If C is A2, T is B2, U and is C2, then f2 =p2C+ q2T + r2U+ s2.

Fig. 5. An ANFIS model based on Sugeno-type fuzzy rule.

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The proposed ANFIS model is illustrated in Fig. 5. The corresponding functions of each layer are given here, as follows: Layer 1: Every node i has a function of: O1i ¼ μ Ai ðC Þ

i ¼ 1; 2

ð9Þ

O1i ¼ μ Bi ðT Þ

i ¼ 1; 2

ð10Þ

O1i ¼ μ C i ðU Þ

i ¼ 1; 2

ð11Þ

where Ai, Bi and Ci are the linguistic labels associated with the node functions. Layer 2: The outputs of this layer are presented, as follows: ωi ¼ μ Ai ðC Þ μ Bi ðT Þ μ C i ðU Þ

i ¼ 1; 2:

ð12Þ

Layer 3: The ratio of ith rule's strength to the total sum of weights is as: ωi ¼

ωi ω1 þ ω2

i ¼ 1; 2

ð13Þ

where ωi is the normalized strength. Layer 4: The correlation between inputs and outputs is: O4i ¼ ωi f i ¼ ωi ðpi C þ qi T þ ri U þ si Þ

i ¼ 1; 2:

ð14Þ

Layer 5: This layer will compute the sum of all signals. It is also noted that in Fig. 5, inputmf and outputmf are two sets of input and output characteristics which map inputs to the rule phase and from this phase to the ﬁnal output. 5. MLR MLR is a widely used statistical method for showing the dependency between two or more variables by ﬁtting a linear equation to the observed data [43,44]. The relationship between independent variables and the dependent variable is: Y ¼ β0 þ β1 X 1 þ β2 X 2 þ ::: þ βn X n þ ε

ð15Þ

where Y is the dependent variable; β0 is a constant; βi is a slope associated with Xi, where Xi is an independent variable and ﬁnally ε is an error [45]. In this paper, MLR is chosen to develop a relationship between the pore diameter and three parameters affecting on it. The pore diameter is the dependent variable while other parameters are independent. 6. Error estimation Five statistical parameters would determine the errors between the target values and the models' outputs. These parameters would also show the models' strength. MAE ¼

N 1X jo −t j N i¼1 i i

MAPE ¼

N 1X joi −t i j 100 N i¼1 t i

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u N u1 X ðo −t Þ2 RMSE ¼ t N i¼1 i i

ð16Þ

Fig. 6. FESEM images of the NPAA membrane formed in 0.3 M oxalic acid electrolyte at 40 V: (a) top view after detachment in a solution of 70% HClO4 + CH3CH2OH (v:v = 1:1) at 55 V, (b) bottom view and (c) cross-section.

N X

ðoi −t i Þ2

R2 ¼ 1− i¼1N X

ð19Þ ðoi Þ2

i¼1

ð17Þ

ð18Þ

N X ðoi −oi Þ t i −t i i¼1 ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ COR ¼ v u N N X uX 2 t ðoi −oi Þ2 t i −t i i¼1

i¼1

ð20Þ

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Fig. 7. TGA-DTA curve for the nanoporous alumina membrane.

where ti is the target, oi is the output predicted by the models, N is the number of samples in each set, oi is the mean value of the predictions and t i is the mean value of the observations.

the order of nano-pores and on the other hand using the pore sizes for the proposed numerical study. 7.1. Morphology

7. Results and discussions This paper presents the results of an experimental study and an application of nonlinear models, ANFIS and MLR, which are used to predict the pore diameter of NPAA membranes. Three parameters of concentration, temperature and voltage have been into account to simulate the behavior of such membranes. A comparison has been made between the results obtained from the models, test data and two empirical formulas to show the accuracy of the models. In experimental study, the morphology of NPAA membranes is investigated by FESEM to obtain

Fig. 6 shows FESEM images of the NPAA membrane after the second anodizing process and also after detaching that away from the aluminum substrate. Fig. 6(a) indicates that the pore diameter of 55 nm is in good common with the modeling results. Different conditions for the anodizing process are given in the literature, to achieve a free-standing nanoporous alumina membrane and also to gain different etchants for selectively dissolution of the aluminum substrate [46–48]. The chemical dissolution method causes obtaining a nanoporous alumina membrane with close-end pores. Thus, pores-opening procedure in a phosphoric acid solution should be applied to remove the barrier layer [49]. In this study, the electrochemical method is used to detach the nanoporous alumina layer from the aluminum substrate. The detachment procedure commences the pores opening process, synchronically. Fig. 6 shows the detachment process in a 70% HClO4 + CH3CH2OH (v:v = 1:1) solution by a voltage of 55 V, which results in a freestanding highly ordered NPAA membrane. Fig. 6(b) shows the bottom view of nano-pores after detaching at 3 s, which indicates that the

Table 1 Statistical values of the pore diameter of the proposed models and empirical formulas. Data set

Model type

MAE

MAPE

RMSE

R2

COR

Training

ANFIS MLR O'Sullivan Eq. [10] Palibroda Eq. [21] ANFIS MLR O'Sullivan Eq. [10] Palibroda Eq. [21] ANFIS MLR O'Sullivan Eq. [10] Palibroda Eq. [21]

0.8250 5.6649 6.3088 16.5965 2.2833 2.3296 2.4000 11.5073 2.8500 3.6166 3.2667 11.4107

1.4789 13.1253 13.8090 32.7629 7.1692 8.1683 8.3491 30.4207 8.1826 9.5476 9.1297 28.1523

1.6042 7.2679 8.5577 19.8116 2.7698 3.3040 2.6255 13.6171 3.2003 4.4844 3.6814 13.7241

0.9989 0.9777 0.9689 0.5979 0.9949 0.9935 0.9955 0.7245 0.9935 0.9892 0.9930 0.7701

0.9969 0.9360 0.9201 0.9201 0.9879 0.9860 0.9929 0.9929 0.9931 0.9739 0.9892 0.9892

Validating

Testing Fig. 8. The X-ray diffraction spectra of the NPAA membrane annealed at different temperatures.

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87

Fig. 9. Comparison of MLR outputs and the desired data.

detachment process has been completely done. The surface of the membrane and the cross-section of the pore walls have been shown in Fig. 6(a) and (c), which illustrate that the detachment process would not inﬂuence the main body and uniformity of the nano-pores in the membrane. 7.2. Thermal and structural characteristic of nanoporous alumina membranes The NPAA membrane is desired to use as a template for synthesis of the carbon nanotube by chemical vapor deposition method at a high temperature [50–52]. For this purpose, it is essential to understand the thermal behavior of the nanoporous alumina membrane. Fig. 7 shows the results of a thermo-gravimetric analysis (TGA) of the nanoporous

anodic alumina membrane prepared, where three weight losses are prominent. First section is caused by moisture removal and decomposition of oxalate anions existing in the nanoporous layer [48]. The second region with a strong exothermic peak at 890 °C presents a transition from amorphous alumina to θ, δ-alumina. The last region with the endothermic peak at 1180 °C is attributed to the transition from alumina to the stable hcp α–phase (corundum). Fig. 8 shows X-ray diffraction spectra of the NPAA membrane for produced (unheated) and after annealing at 600 °C and 900 °C. The amorphous phase is obtained for the unheated part by annealing at 600 °C for 1 h, with increasing temperature by the rate of 5 °C·min−1. By increasing the annealing temperature up to 900 °C, the peaks of the gamma phase would appear in the spectra, while the membrane would be mechanically stable after heat treatment.

Fig. 10. Comparison of MLR outputs and experimental observations.

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Table 2 Comparison of experimental data with the predicted results. No.

C

U

T

DP,EXP.

DP,ANFIS

DP,MLR

DP,O 'Sullivan

DP,Palibroda

Reference

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.31 0.31 0.31 0.53 0.53 0.76 1.10 2.40 0.18 1.10 0.30 0.34 0.40 0.30 0.40 0.40

20.0 25.0 30.0 55.0 60.0 20.0 25.0 30.0 35.0 60.0 25.0 30.0 35.0 40.0 45.0 50.0 12.5 15.0 20.0 12.5 15.0 12.5 12.5 12.5 25.0 25.0 40.0 40.0 80.0 40.0 30.0 40.0

313 313 313 313 313 318 318 318 318 318 323 323 323 323 323 323 273 273 273 273 273 273 273 273 283 278 293 288 298 290 290 275

24.0 30.0 34.0 69.0 71.0 30.0 33.0 37.0 42.0 76.0 39.0 43.0 59.0 60.0 67.0 82.0 19.9 21.6 24.0 19.1 21.4 19.0 18.4 13.7 24.0 36.0 50.0 60.0 80.0 55.0 38.0 54.0

24.1 27.0 30.2 51.8 57.4 29.3 32.9 37.2 42.3 76.1 40.9 46.9 53.9 61.9 70.5 79.7 19.9 20.0 21.3 19.1 19.6 18.9 22.8 13.7 24.0 36.0 49.9 59.9 80.0 55.2 38.0 55.6

31.2 36.6 42.1 69.3 74.8 31.8 37.3 42.7 48.2 75.4 37.9 43.4 48.8 54.3 59.7 65.2 17.9 20.7 26.1 17.8 20.5 17.6 17.4 16.5 32.9 31.7 50.5 49.8 94.6 50.1 39.1 48.1

25.8 32.3 38.7 70.9 77.4 25.8 32.3 38.7 45.2 77.4 32.3 38.7 45.2 51.6 58.1 64.5 16.1 19.4 25.8 16.1 19.4 16.1 16.1 16.1 32.3 32.3 51.6 51.6 103.2 51.6 38.7 51.6

19.2 22.7 26.3 43.9 47.5 19.2 22.7 26.3 29.8 47.5 22.7 26.3 29.8 33.4 36.9 40.4 13.9 15.6 19.2 13.9 15.6 13.9 13.9 13.9 22.7 22.7 33.4 33.4 61.7 33.4 26.3 33.4

[12] [31] [24] [31] [12] [31] [24] [31] [12] [31] [12] [31] [12] [31]

[25]

[26] [27] [28] [29] [30] Authors' study

In this paper, ANFIS and MLR models are also studied. In these models, 20, 6 and 6 records are selected for the training, validating and testing phases, respectively. The statistical parameters for these three steps of the models and for two empirical formulas [10,21] are given in Table 1. MLR is also studied based on the database gathered from the literature. The equation obtained from the MLR model is presented in the following:

ANFIS is depicted in Figs. 11 and 12. In Fig. 11, the horizontal and vertical axes are the representative of samples number and the pore diameter, respectively. It shows that the ANFIS model can predict the pore diameter accurately. Fig. 12 illustrates a comparison between the ANFIS model's predictions and experimental observations. It also reveals that ANFIS has a good ability to estimate the complex problems.

DP ¼ −30:013−0:689263 C þ 0:126557 T þ 1:09001 U

8. Conclusion

ð21Þ

where DP is the pore diameter; C is the concentration; T is the temperature; and U is the voltage. The values of statistical parameters corresponding to the MLR model are given in Table 1. The results are compared with two proposed empirical formulas [10,21] showing that this model is better than those empirical formulas. Table 1 also shows that the results of the MLR model are close to the O'Sullivan's formula [10]. It shows that O'Sullivan's formula predicts the pore diameter of a NPAA membrane more accurate than Palibroda's formula [10]. The performance of the MLR model is shown in Figs. 9 and 10. Fig. 9 shows a comparison between the results and the target values. In this ﬁgure, the vertical axis shows the model's predictions and the horizontal axis is a representative of the samples number. It reveals that the model can predict the pore diameter accurately. Fig. 10 presents a comparison between the model's predictions and the experimental observations. In Fig. 10, the horizontal axis contains experimental data and the vertical axis is a representative of the model's results. This ﬁgure illustrates that MLR with the mean absolute error less than 6% and R2 values of 90.4% has a good prediction capability of the pore diameter. The performance of the ANFIS and MLR models and also empirical formulas are given in Table 2. The results present that the ANFIS model can estimate the pore diameter more accurate than other models and formulas. The performance of

This paper presents the results of an experimental study and applications of non-linear models to predict the pore diameter of a NPAA membrane. In the experimental study, the morphology of nanoporous anodic alumina is studied by FESEM to know the order of nano-pores which are used for the proposed models. In this regard, the electrochemical method is used to detach nanoporous alumina layer from the aluminum substrate resulting in a free-standing highly ordered membrane. The results show that the detachment process does not inﬂuence the main body and uniformity of the nano-pores of the membrane. The thermal behavior of the membrane shows that three weight losses are prominent including: moisture removal and decomposition of oxalate anions; transition from amorphous alumina to θ, δ-alumina and transition from alumina to the stable hcp α–phase (corundum). The X-ray diffraction spectra of the NPAA membrane for produced and that of after annealing at 600 °C and 900 °C show that with increasing annealing temperature to 900 °C, the gamma phase peaks would appear in spectra and the membrane is mechanically stable after heat treatment. In numerical study, the results of two proposed models (i.e. ANFIS and MLR models) are compared with two empirical formulas and targets. For developing the models, a total of 32 test results were collected from the literature and the Authors' previous studies. In these models, 20, 6, and 6 experimental records were selected to training, validation and testing them, respectively. The statistical parameters show that

H. Akbarpour et al. / Chemometrics and Intelligent Laboratory Systems 153 (2016) 82–91

Fig. 11. Comparison of ANFIS outputs with the desired data for (a) training (b) validating (c) testing.

the models are good enough for predicting the pore diameter of NPAA membranes, where ANFIS is the best and MLR is also better than empirical formulas. Nomenclature ANFIS Adaptive Neuro Fuzzy Inference System MLR Multiple linear regression Pore diameter (nm) DP U Applied voltage (V) Constant (≈1.29 nm⋅ V−1) λP BP Back-propagation algorithm Activity level generated at jth hidden neuron Hj Weights on the connections to the hidden layer of neurons Wij Input value transmitted from the xth input neuron Xi Bias at jth hidden neuron biasj

F ILW ILB HLW HLB MAE MAPE RMS R2 COR ti Oi N O t

Activation function Input layer weight Input layer bias Hidden layer weight Hidden layer bias Mean absolute error Mean absolute percentage error Root mean square error Absolute fraction of variance Correlation coefﬁcient Desired pore diameter Predicted output Total number of data records in each set of data Mean value of predictions Mean value of observations

89

90

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Fig. 12. Comparison of ANFIS outputs and experimental observations for (a) training (b) validating (c) testing.

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