Computational Materials Science 79 (2013) 75–81

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Computational Materials Science

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Investigation on the synthesis conditions at the interpore distanceof nanoporous anodic aluminum oxide: A comparison of experimentalstudy, artificial neural network, and multiple linear regression

0927-0256/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2013.05.048

⇑ Corresponding author. Tel.: +98 21 64542220.E-mail address: hakbarpour@aut.ac.ir (H. Akbarpour).

Hamed Akbarpour a,⇑, Mahdi Mohajeri b,c,d, Momene Moradi e

a Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iranb Faculty of Engineering, Tarbiat Modares University, Tehran, Iranc Department of Mining and Metallurgical Engineering, Amirkabir University of Technology, Tehran, Irand Nanotechnology Division, Research Institute of Petroleum Industry, Tehran, Irane Department of Chemical Engineering, Tarbiat Modares University, Tehran, Iran

a r t i c l e i n f o

Article history:Received 14 March 2013Received in revised form 23 April 2013Accepted 27 May 2013

Keywords:Artificial neural networkMultiple linear regressionInterpore distanceNanoporous anodic aluminum oxide

a b s t r a c t

Using nanoporous anodic aluminum oxide thin layer becomes more popular in recent years due to itscapability to be a membrane in some engineering applications. The main purpose of this paper is to inves-tigate the synthesis conditions at the interpore distance of nanoporous anodic aluminum oxide throughan experimental study, an artificial neural network (ANN), and a multiple linear regression (MLR) model.A total of 33 experimental data used to establish both models. The models have three inputs including theconcentration of electrolyte, temperature, and applied voltage. The interpore distance of nanoporous ano-dic aluminum oxide is considered as output in the models. The results of the models are compared withthe results of experimental study and an empirical formula proposed by Nielsch. The results reveal thatthe proposed models have good prediction capability with acceptable errors. However, in this research,the proposed ANN model is accurate than the MLR analysis and both of them are better than empiricalformula. The proposed models can also predict the results of experimental study successfully.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

Nanoporous anodic aluminum oxide (AAO) had attracted inten-sive interest due to its potential to use as a membrane in someapplications such as the gas separation [1–4], drug delivery [5,6],and bone fixation [7]. Anodizing aluminum in an acidic electrolyteresulted in a thin layer of compact aluminum oxide, following byan ordered array of nanopores [8–10]. The synthesis process ledto mechanically robust and thermally durable [3] and inert, wideand variety of pore size distribution, which showed unique capa-bility for biomolecule separation process [11] and hemodialysis[12–15].

The geometry of aluminum oxide layer executed important rolefor a separation process which could lead to maximize permeationand flux across nanoporous anodic aluminum oxide membrane.For example, as a hemodialysis membrane, a pore size was desir-able which had capability to clear urea, creatinine, vancomycinand inulin as a waste product with small and middle of molecularweight while maintaining large molecular weight solutes(albumin) [16–21].

The geometry of nanoporous anodic aluminum oxide hadgrown through anodization was represented schematically inFig. 1.

Many equations were suggested for correlation between thegeometry of aluminum oxide layer and conditions of anodized alu-minum [22–25]. According to Nielsch et al. [22], the interpore dis-tance (Dc) in nanometer (nm) was linearly proportional to theapplied voltage (U) in volts (V) of the steady-state growth of oxidelayer as follows:

Dc ¼ kcU ðkc � 2:5 nm V�1Þ ð1Þ

where kc is a proportional constant in nm V�1.It was assumed that the pore diameter was a function of applied

voltage. The other efforts showed that anodizing temperature andconcentration of electrolyte cause changing the interpore distance[23–25]. In this regard, three important effecting parameters onthe interpore distance of nanoporous aluminum oxide were identi-fied as following:

� Concentration of electrolyte (C) in mol/dm3.� Anodizing temperature (T) in K.� Applied voltage (U) in V.

Fig. 1. The schematic geometry of nanoporous anodic aluminum oxide layer and aluminum substrate.

76 H. Akbarpour et al. / Computational Materials Science 79 (2013) 75–81

In general, these parameters had a considerable effect on theinterpore distance of nanoporous aluminum oxide, but the contri-bution of each parameter separately was studied a little. Nowa-days, computer-based methods such as artificial neural networks(ANNs) and multiple linear regression analysis have attracted someattentions to be replaced with high-cost experimental studies. Theabove-mentioned techniques were used in different fields of engi-neering applications such as civil engineering [26–29], chemicalengineering [30,31] and material science [32–35].

The artificial neural networks (ANNs) are known as the impor-tant and successful simulating tools of input–output datasets. Themost fundamental part of such networks is the training process.The ANN model is trained with some relevant experimental resultsand then can predict the output with an accepted error.

This paper presents an artificial neural network (ANN) and amultiple linear regression (MLR) model accompanished with anexperimental study to investigate the synthesis conditions at theinterpore distance of nanoporous anodic aluminum oxide. In thisregard, a total of 33 experimental records are used with three fac-tors each. The results reveal that output in the models have anagreement with experimental records. In addition, all results arecompared with output of formula proposed by Nielsch [22]. It isclear that ANN is more accurate than formula proposed by Nielsch[22] and MLR model.

Table 1The range of parameters in collected database.

Variables Minimum Mean Maximum Standard deviation

C (M) 0.1 0.5309 2.4 0.4669T (K) 273 280.576 300 8.4317U (V) 12.5 45.5303 195 43.6791

2. Experimental study

A high purity aluminum foil (99.99% Alfa Aesar) was used as theprimary material. At first, the aluminum substrate was degreasedby Acetone, and then the native oxide layer upon the aluminumfoil was removed through sinking the sample in a caustic solution(3 mol/dm3 (M)) for 30 s. A self-organized two-step anodizationwas used to fabricate the thin film layer of anodic aluminum oxide.The simple electrochemical cell was equipped with a magnetic stir-rer under the constant temperature by the powerful circulator sys-tem (Thermo Haake, DC 10-K15) and a platinum electrode wasemployed as a counter electrode. The distance between the anodeand cathode was around 3 cm. The first step of anodization wasconducted on an oxalic acid electrolyte (0.3 M) at a constant poten-tial of 40 V. This process lasted 2 hours (h) while anodizing tem-perature was constant at 27 �C. The chemical etching procedurewas used to remove thin anodic aluminum oxide with the presenceof 6 and 1.8 wt% of H3PO4 and H2Cr2O4 at 75 �C for 3 h, respec-tively. After then, the second anodization was performed in thesame experimental condition and this step lasted 3 h. All solutions

were prepared from reagent grade chemicals and deionized water.Prior to each step, the aluminum sample was ultrasonicallycleaned using Acetone for 15 minute (min) following with a5 min-ultrasonic cleaning by deionized water.

The geometrical characterization of nanoporous anodicaluminum oxide was done by a Field Emission Electron ScanningMicroscope (FE-SEM, Hitachi SE-4160) after sputtering a 15-nmthickness of Au layer.

3. Data collection

The objective of this research is to develop an artificial neuralnetwork (ANN) and a multiple linear regression analysis (MLR)for predicting the interpore distance of nanoporous anodic alumi-num oxide. The first step is data gathering for the training, validat-ing, and testing process of the ANN model. All data are collectedfrom the literature [22,36–48] and self-carried out experimentalstudy.

In total, 33 records are collected from the literature. Of these;19, 7, and 7 records are used for training, validating, and testingof the model, respectively. The selection process is done randomlyto reduce hand-selection errors. Table 1 summarizes input andoutput ranges for the ANN model.

4. Artificial neural network

Artificial neural network modeling as a computer-based meth-odology is developed in an attempt to imitate the obtained knowl-edge and skills of the human brain. It offers considerable support interms of organizing, classifying, and summarizing data. It alsohelps to distinguish among input data and gives a high degree ofprediction accuracy because of requiring a few assumptions. Thesecharacteristics make neural network approach an important toolfor forecasting some complex problems. Here, a brief descriptionof ANN is presented to help readers [49].

H. Akbarpour et al. / Computational Materials Science 79 (2013) 75–81 77

4.1. Description of ANN

An artificial neural network is typically composed of severallayers of nodes. The first layer is an input layer where externalinformation is received. All the input nodes are set in one inputlayer, and the number of input nodes corresponds to the numberof independent effecting variables. The highest layer is an outputlayer where the problem solution is obtained. The number of out-put nodes is related to the problem. These two layers are separatedwith a number of hidden layers in which nodes are connected toeach other with weighted interconnection from a lower layer toa higher layer. Also, there is a bias which is connected to the hid-den and output layers with weighted interconnections [26,50,51].

The neural network training is an unconstrained nonlinear min-imization problem in which weights of a model are iterativelymodified to minimize the overall mean or total squared error be-tween the target and actual output for all output nodes over all in-put patterns. The existence of many different optimizationmethods provides various choices for neural network training.There is no algorithm currently available to guarantee the globaloptimal solution for a general nonlinear optimization problem ina reasonable amount of time [51,52].

The most popularly training method used for multi-layeredfeed-forward networks is the back-propagation (BP) algorithmwhich is essentially a gradient steepest descent method. For thegradient descent algorithm, a step size, which is called the learningrate in ANN literature, must be specified. The learning rate is cru-cial for back-propagation learning algorithm since it determinesthe magnitude of weight changes. The BP algorithm contains twosteps that the transfer function is propagated from the input tooutput layer firstly and then the weights and biases are modifiedto spread the difference between actual and desired outputs back-ward. The weighted sum of inputs is determined using followingequation [50,51]:

Hj ¼XN

i¼1

WijXi þ biasj ð2Þ

where Hj the activity level generated at jth hidden neuron; Wij

weights on connections to the hidden layer of neurons; Xi input va-lue transmitted from the xth input neuron; and biasj is bias at jthhidden neuron.

The activation function illustrates the relationship between in-puts and outputs of a node and a network. In fact, it demonstrates adegree of nonlinearity that is applicable for the most ANN model-ing approaches. A sigmoid function is selected for calculating out-put of jth neuron as follows:

f ðHjÞ ¼1

1þ e�ðHjÞð3Þ

The training of network is an iterative procedure through whichthe connection weights are determined using input and outputsets. Generally, artificial neural network is trained to learn the in-ner relationships between the inputs and outputs by repeatedexposure to examples of the input–output data patterns. A success-fully trained ANN can generalize beyond the training data and pro-duces approximately correct results for new cases or data patterns[53,54].

Fig. 2. The schematic architecture of the proposed ANN model.

4.2. Network selection

There is no general rule to determine the number of hidden lay-ers and their neurons. It is a try and error process to identify theoptimum and general network. The number of neurons should besufficient to predict the desired parameter correctly.

Most researchers have conducted many studies to determinethe adequate neurons number, and have presented different algo-rithms. The upper bound for the number of neurons in the hiddenlayer is limited to twice the number of inputs plus one [27].

In this paper, the ANN toolbox in MATLAB is used to computethe predicted outputs. In this regard, a three feed-forward networktrained by the Levenberg–Marquardt back-propagation (LMBP)algorithm is selected. LMBP is the fastest back-propagation algo-rithm and is highly recommended but requires more memory thanother algorithms [55]. A nonlinear hyperbolic sigmoid and a linearactivation functions are used in the hidden and output layers,respectively.

4.3. Neural network parameters

The proposed ANN model contains three layers including input,hidden, and output layers. Inputs are defined by input neuronswhere their numbers equal to the input variables (i.e., C, T, andU). It is noted that there is no general rule to choose the hiddenlayer size and it is dependent on the training pattern. After tryingdifferent neural networks, the number of hidden neurons is se-lected as four setting in one hidden layer. The output layer also in-cludes one neuron (i.e., Dc). The schematic architecture of themodel is shown in Fig. 2.

All received input data are transferred to the hidden layer with-out any calculation. The useful features are derived in the hiddenlayer and all weights of the model are generated. Totally, the pre-dictions of the model are produced by the neuron of the outputlayer. The input layer weights (ILWs), input layer biases (ILBs), hid-den layer weights (HLWs), and hidden layer biases (HLB) of theoptimum proposed ANN model are given in the followingequations:

ILW ¼�1:0286 1:05347 �0:022 2:21037�0:3745 3:93645 �0:0195 �4:4202�1:7784 �0:9029 �0:7144 3:06779

�������������� ð4Þ

ILBT ¼ j2:17733 �1:3715 �0:0322 �0:8447 j ð5Þ

Table 2The interpore distance statistical values of the proposed ANN model and Nielschformula [22].

78 H. Akbarpour et al. / Computational Materials Science 79 (2013) 75–81

HLW ¼ j0:085672 �0:0326 �1:7086 �0:0342 j ð6Þ

HLB ¼ j � 0:8459j ð7Þ

Data set Modeltype

MAE MAPE RMSE R2 COR

Training ANN 2.84123 3.52544 4.29432 0.99942 0.99939MLR 4.47422 6.21282 6.39296 0.99873 0.99937NielschEq. [22]

5.44737 5.38101 8.18598 0.99780 0.99871

Validating ANN 3.87067 5.94464 4.27088 0.99754 0.99521MLR 3.21443 5.13334 3.81073 0.99799 0.99906NielschEq. [22]

2.70714 3.84038 3.9206 0.99782 0.99717

Testing ANN 4.27660 6.71049 5.64998 0.99893 0.99958MLR 4.20652 7.00891 6.58185 0.99853 0.99939NielschEq. [22]

6.41429 7.42994 9.53314 0.99671 0.99919

5. Multiple linear regression

Multiple linear regression analysis (MLR) is a statistical ap-proach which used to determine a relationship between two ormore independent variables. In fact, the dependent variable is alinear function of more than one independent variable. The generalMLR formula is given by:

Y ¼ aþ b1X1 þ b2X2 þ � � � þ bnXn � e ð8Þ

where Y is dependent variable; a is Y-intercept; b1, b2, and bn areslopes associated with X1, X2, and Xn, respectively; X1, X2, and Xn

are independent variables; and e is the error [27].In this paper, an attempt is made to develop a relationship be-

tween Dc and C, T, U where they are considered as the independentvariables and Dc is selected as the dependent variable.

6. Model assessment

Five statistical parameters are used to compare the predictedresults with the desired experimental results. These norms calcu-late the error between the target and output in the models. In fact,these parameters show the capability of models as follows:

MAE ¼ 1N

XN

i¼1

jOi � tij ð9Þ

Table 3Comparision of the experimental and predicted results.

Samplesno.

DcExp DcANN DcMLR DcNielschDcExDcAN

1 501 500.69 503.35 487.5 1.02 64 65.37 64.11 62.5 0.93 208 206.91 205.71 200 1.04 103 98.87 101.41 100 1.05 103 98.81 100.91 100 1.06 109 101.50 101.11 100 1.07 92 95.35 101.25 100 0.98 100 101.50 101.11 100 0.99 104 101.65 101.23 100 1.0

10 420 419.95 415.18 400 1.011 60 64.99 61.96 62.5 0.912 420 419.95 415.18 400 1.013 40.5 39.04 39.69 37.5 1.014 100 100.16 101.17 100 0.915 150 154.85 153.19 150 0.916 200 201.08 204.82 200 0.917 100 98.87 101.41 100 1.018 104.2 98.81 100.91 100 1.019 61 63.83 61.88 62.5 0.920 32.7 34.08 29.21 31.25 0.921 36.4 34.08 29.21 31.25 1.022 45.4 34.08 29.21 31.25 1.323 31.9 33.86 29.84 31.25 0.924 38.8 38.47 36.34 37.5 1.025 31.8 32.57 30.49 31.25 0.926 32.6 29.16 31.47 31.25 1.127 29.6 28.81 35.19 31.25 1.028 65.1 69.18 74.69 75 0.929 80.5 90.94 100.70 100 0.830 60 58.74 61.69 62.5 1.031 97 90.94 100.70 100 1.032 105 93.36 100.73 100 1.133 100 93.18 101.44 100 1.0

MAPE ¼ 1N

XN

i¼1

jOi � tijti

� 100 ð10Þ

RMSE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

i¼1ðOi � tiÞ2

rð11Þ

R2 ¼ 1�PN

i¼1ðOi � tiÞ2PNi¼1ðOiÞ2

!ð12Þ

p

N

DcExpDcMLR

DcExpDcNielsch

References

01 0.995 1.028 [22]79 0.998 1.024 [36]05 1.011 1.040 [37]42 1.016 1.030 [38]42 1.021 1.030 [39]74 1.078 1.090 [40]65 0.909 0.920 [41]85 0.989 1.000 [42]23 1.027 1.04000 1.012 1.050 [43]23 0.968 0.96000 1.012 1.05037 1.020 1.080 [44]98 0.988 1.00069 0.979 1.00095 0.976 1.00011 0.986 1.000 [45]55 1.033 1.042 [46]56 0.986 0.97659 1.120 1.046 [47]68 1.246 1.16532 1.555 1.45342 1.069 1.02109 1.068 1.03576 1.043 1.01818 1.036 1.04327 0.841 0.94741 0.872 0.86885 0.799 0.80521 0.973 0.96067 0.963 0.97025 1.042 1.050 [48]73 0.986 1.000 Experimental study by

authors

Fig. 3. Comparison of ANN, MLR, and Nielsch equation results and experimental results for (a) training, (b) validating, and (c) testing.

H. Akbarpour et al. / Computational Materials Science 79 (2013) 75–81 79

Fig. 4. FE-SEM picture of the established nanoporous anodic aluminum oxide.

80 H. Akbarpour et al. / Computational Materials Science 79 (2013) 75–81

COR ¼PN

i¼1ðOi � OiÞðti � �tiÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPNi¼1ðOi � OiÞ

2PNi¼1ðti � �tiÞ2

q ð13Þ

where ti is the desired interpore distance, Oi is the predicted output,N is the total number of data records in each set of data, O is themean value of predictions, and �t is the mean value of observations.

7. Results and discussion

In this study, an experimental study, an artificial neural net-work and a multiple linear regression analysis are carried out toinvestigate the synthesis conditions at the interpore distance ofnanoporous aluminum oxide. Three effecting factors includingthe concentration, temperature, and the applied voltage areconsidered.

As mentioned earlier, 19, 7, and 7 records are used for training,validating and testing of the proposed ANN model, respectively.The statistical values for training, validating, and testing data ob-tained from the ANN, MLR, and Nielsch equation [22] are givenin Table 2.

A multiple linear regression analysis is carried out and theeffecting factors on the interpore distance are assumed as the inde-pendent variables. The interpore distance is the dependent vari-able. The general MLR equation, in this model, is given by:

Dc ¼ �11:6757þ 2:86187 � C þ 0:0274084 � T þ 2:60092 � U

ð14Þ

where Dc is interpore distance; C is concentration; T is temperature;and U is applied voltage. The statistical values of this model are gi-ven in Table 2.

The results of ANN and MLR models are compared with anempirical formula proposed by Nielsch et al. [22]. As mentionedearlier, this formula considered just the effect of applied voltage.It explained that the interpore distance is a function of potential.In this paper, the ANN and MLR models are proposed to investigatethe effect of other parameters on the interpore distance of nano-porous aluminum oxide. The results show that the ANN and MLRresults are more accurate than empirical formula proposed by Nie-lsch et al. [22]. The obtained general MLR function also explainsthat the effect of other parameters is considerable and should beaccounted for. The performance of both models and Nielsch equa-tion is summarized in Table 3.

All results reveal that both proposed models have good predic-tion capacity with acceptable errors. In this study, ANN is betterthan MLR and also their results are more accurate than the Nielschformula [22]. The performance of ANN, MLR, and Nielsch equation[22] for training, validating, and testing are compared with exper-imental results in Fig. 3. In Fig. 3, the horizontal axis shows predictresults by ANN, MLR, and Nielsch equation and the vertical axis isthe representative of experimental results. The linear least squarefit line, its equation and R2 values are shown in this figure for train-ing, validating, and testing sets. As it is shown in Fig. 3, values ob-tained from training, validating, and testing sets in ANN, MLR, andNielsch equation [22] are very closer to experimental results. Thisreveals that experimental results with predictions show a closematch.

In experimental study, the synthesis conditions of oxalic acidelectrolyte (0.3 M) are studied at the constant potential and anod-izing temperature of 40 V and 27 �C, respectively. The geometricalcharacterization of nanoporous anodic aluminum oxide is done bya Field Emission Electron Scanning Microscope (FE-SEM,Hitachi SE-4160) after sputtering a 15-nm thickness of Au layer.Fig. 4 shows FE-SEM picture of developed nanoporous aluminumoxide. The result shows that with the concentration of 0.3 M, tem-perature of 300 K and the applied voltage of 40 V, the interpore dis-

tance equals to 100 nm. The experimental result is verified withboth models and empirical formula proposed by Nielsch et al.[22]. The results of ANN, MLR and Nielsch formula [22] are 98.82,101.44, and 100 nm, respectively. It shows that artificial neuralnetwork modeling and multiple linear regression analysis can besubstituted with high-cost experimental studies.

8. Conclusions

The synthesis conditions at the interpore distance of nanoporousanodic aluminum oxide thin layer are important problems in recentyears. In this regard, some computer-based techniques such as arti-ficial neural networks (ANNs) and multiple linear regressions aregood tools for determining the results of these complicated prob-lems. The main purpose of this study is to develop artificial neuralnetwork and multiple linear regression models for predicting theinterpore distance of nanoporous anodic aluminum oxide. The mod-els’ predictions are very close to the experimental desired results fortraining, validating, and testing. The mean absolute error, meanabsolute percentage error, root mean square error, and the correla-tion coefficient are presented to show the accuracy of models. More-over, the results of models are compared with empirical formulaproposed by Nielsch et al. [22]. It is noted that the models considerthree effecting factors on the interpore distance while the empiricalformula just considers the applied voltage as the effecting parame-ter. In fact, the proposed models investigate also the effect of otherparameters including the concentration of electrolyte and tempera-ture more than the potential.

In addition, an experimental study is carried out and the resultsare verified with other models and formula proposed by Nielschet al. [22]. The results show that models can predict the desiredoutput successfully.

In this study, it is shown that ANN model is better than anothermodel and formula proposed by Nielsch et al. [22]. The results alsoreveal that ANN and MLR can predict the interpore distance of nano-porous anodic aluminum oxide successfully. The proposed modelscan save time and avoid carrying some high-cost experimentalstudies out due to their more accurate predictions with acceptableerrors.

H. Akbarpour et al. / Computational Materials Science 79 (2013) 75–81 81

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