Autocorrelation-Based Method for Characterization ... - ACS Publications

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Autocorrelation-Based Method for Characterization of the SelfHexagonal Lattice Mohammadreza Pourfard,† Karim Faez,*,† and S. Hadi Tabaian‡ †

Electrical Engineering Department and ‡Mining & Metallurgical Engineering Department, Amirkabir University of Technology, Tehran, Iran, 15914 ABSTRACT: Lattice characterization methods are normally used to quantify the effects of different anodization conditions on hexagonal lattices like nanoporous anodized aluminum oxides (AAOs). In this paper, a fast and accurate approach is recruited which uses the periodicity of the material’s texture to quantify the degree of ordering in the hexagonal lattices. This automatic correlation-based approach is the first method which could give the observer a spectrum judgment about the degree of lattice order and could evaluate the order of near-regular textures. This method gives the observer an objective criterion for judging the order of hexagonal lattices, and two new criteria are also proposed to define the amount of order in these structures. This may help to unveil the mechanism for the self-organization process and deposition techniques in AAO.

1. INTRODUCTION Anodic aluminum oxide (AAO) as one of the most popular self-organized hexagonal lattices has recently attracted considerable attention.1−9 A few specific features of AAOs like low cost fabrication, simplicity, controllability of pore dimensions, accessibility of high aspect ratio pores, long-range stability, and compatibility of these porous films with existing semiconductor technology have attracted researchers’ interests, making them a viable candidate for large-scale industrial processing and a proper alternative to conventional nanolithography techniques. Therefore, this specific structure can be used as a proper template for fabrication of ordered carbon nanotube arrays,10 photonic crystal,11 high density magnetic storage devices,12,13 nanowires,10,12,14,15 pattern transfer masks,16 and many other devices. An important issue for optimizing the performance of the obtained devices is the amount of order of these structures. Preparation conditions of the AAO including electrolyte type, temperature, anodization time, and applied voltage determine the ordering degree of this structure. In fact, at the outset of the anodization process, the surface of the aluminum seems like randomly scattered pores. However, as the time of anodization passes, the degree of ordering gradually increases and an ordered hexagonal lattice will form.4,17−20 In this structure, there are also different ordered regions with different orientations called domains. At first, researchers like Li et al.21 thought that the average domain size increases linearly with the anodization time, but Nielsch et al.22 have shown that this is not the case and there is an optimum anodization time to obtain the highest order and largest domain size. Any deviation from this optimum time value will reduce maximum domain size and consequently the ordering degree. © 2013 American Chemical Society

To determine the domains’ borders, researchers normally use subjective visual judgment, but there is an essential need to make this process objective and automatic. For this goal, some image processing techniques have recently been proposed.1−7,9,23,24 These techniques can be categorized into two major types:5 Transformation-domain methods and spatial-domain methods. The Fourier method is an example of the first type.1,2 The triangular3,23,24 and summation5 method are an example of the second type. All of these methods have their own pros and cons. Rahimi et al.1,2,25,26 have used a 2D Fourier method for qualitative identification of the lattice type. Their methods take the angular FFT of the image and then the image is normalized to the largest pixel value. A threshold is exerted on the Fourier image, and the low-value pixel will be eliminated. At the next step, a histogram of the angle of Fourier line in the image will be constructed. The sharpness of this histogram will be considered as an ordering criterion. In fact, more ordered image texture will have a more condensed histogram, and a few peaks will be created around some angles which have a 60° difference. The complexity of this method is not too much, but it would not have been able to discriminate between some medium-order textures because the Fourier of a few different images could have the same shape and consequently the same histogram format. This problem has caused the Fourier method to only be able to define the order of the AAO images which are disordered or highly ordered and has a major problem for medium-order textures. Received: February 11, 2013 Revised: July 14, 2013 Published: July 23, 2013 17225

dx.doi.org/10.1021/jp401504q | J. Phys. Chem. C 2013, 117, 17225−17236

The Journal of Physical Chemistry C

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Figure 1. (a) Scanning electron microscopy (SEM) image obtained from a long-range ordered anodic porous alumina formed in sulfuric acid at 25 V. (b) Domains of this figure shown manually. Previous methods consider the largest domain size to the whole image as an ordering criterion. It is observable that the small movement of the quadratic box has a great influence on the image order based on their criteria. In the black box (1), the whole image is detected as ordered while in the red box (2) nearly half of the image would be the largest domain size.

an ordering criterion. This statistical method is simple and fast and can show the dominant orientations of the material’s texture. However, this method has a problem which has caused the judgment of this method to be imprecise in some cases. This method decreases the dimensionality of the image from 2 to 1 and acts on a one-dimensional variance signal. This dimension reduction causes some important data loss, and the accuracy of this algorithm decreases in some special cases, e.g. two reverse rotations of two domains can neutralize themselves in projection to an axis while they have influence on the image order. At the same time, Pichler et al.6 have introduced the local autocorrelation function (ACF) by fitting two Gaussians into this function. They have also defined short-range and longrange orders in colloidal nanostructure (NC) and epitaxial quantum dots and have used their local autocorrelation function to calculate these orders. They have also defined an anisotropy parameter based on this autocorrelation to evaluate the amount of defects in the above-mentioned materials. However, their order parameters rely on a purely empirical fitting procedure of autocorrelation, and it has not been able to evaluate the medium-order texture well.7 In another work done recently, Khushin et al.9 have exploited the 6-fold symmetry of a hexagonal lattice by using the “opposite partner sphere” concept. They have introduced the regularity measure as a new ordering criterion for the AAO image. This method has defined a local regularity measure and counts the number of pores which have opposite partners in the radius of r to the total number of pores in that circle. Then, the regularity measure calculates the average of local regularity measure in the whole image. The last method which has been proposed in this field is Borba et al.’s7 method. It has defined positional and orientational order parameters for quantifying the order of a nanoporous aluminum array (NNA). They have categorized the defects of AAOs into dislocation and isolated disclination. They have shown that dislocation affects positional order and isolated disclination affects orientational order. After they created Voronoi cells in the pore’s center, they used the radial distribution function (RDF)23 for quantifying the positional order and hexatic parameter for quantifying the orientational order. Their local order parameter (LOP), which is motivated by the idea of 2D melting, its correlation, and the distribution of the neighbors’ number have been proposed as three ordering criteria. While their idea works well on extreme cases (highly

The necessity to judge about the medium-order image has impelled the researcher to seek a more powerful method. Therefore, Mátéfi-Tempfli et al.23 have introduced their method as a simple triangular-based technique. They have observed that more ordered AAO images will have a larger domain size. Then their method tries to find the largest domain size. For this reason, they create a triangular mesh on the pore’s center of the image. Then the region of the image which has nearly equilateral triangles will be considered as the ordered area, and the other region will be considered as the disordered one. Mátéfi-Tempfli et al.23 have introduced a criterion for the degree of ordering which is the proportion of the total ordered areas to the whole image area. In their method, there was no obligation for connectivity of the ordered areas and they considered every equilateral triangle of pores in their ordering criterion. Hillebrand et al.24 have improved the idea and proposed that the connected ordered areas (domain) can be a much better order characterization. In fact, they have claimed that a wellstructured AAO image will have larger domains than a disordered structure. Hillebrand et al.24 have introduced another two ordering criteria. Their criteria are the number of domains and the proportion of the largest domain size to the whole image area. Both Mátéfi-Tempfli et al.23 and Hillebrand et al.’s24 methods do not use the orientation of the AAO image. Abdollahifard et al.3 have shown that the ordering of an AAO image has a direct relation with orientation of domains, and a pure connectivity of the image domains without considering their orientation is not a very precise criterion. Therefore, Abdollahifard et al.3 have only connected domains which have the same orientation, and they have considered the largest domain size to the whole image as an ordering criterion. Our method is highly robust against ordering tolerance parameters, but this method is a complex one and still depends on an ordering tolerance parameter. After that, our summation method5 has been proposed. This method is based on the fact that the highly ordered hexagonal lattices (aluminum oxide,1−3,5,23−25 titanium oxide27,28) have periodic structure and the 1D signal of image projection in the two vertical and hexagonal lattices is also periodic. As the image goes far from ordering, the variance of these two signals decreases. So this method has considered the variance of summation of image projection in the x- and y-axis. Then, by rotating the image 60°, the maximum variance is considered as 17226



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statistical characteristics of their images. One of these three criteria is the slope of the line between the first and second peaks of the autocorrelation image in the xz plane. The other one is the number of peaks of the autocorrelation image above a few predefined thresholds. Our automatic method with its two new ordering criteria could help the observer have objective criteria to quantify the degree of order in AAOs and other hexagonal lattices, and it could open a window to better understand and model the formation process of these structures.

ordered and disordered AAO’s texture), it has not been able to discriminate well between medium-order textures, and it also has high computational cost. The important point about all these methods is that it is essential for all these methods to have a fuzzy (spectrum) judgment about the order of AAOs. In other words, it is necessary that these methods be able to discriminate between medium-order AAO images and have a precise judgment about the order of every texture. This is because of the fact that most of the AAO’s texture has a near-regular structure. They must also be able to compare every two textures with each other in the ordering degree. However, in the past, only a few methods would be able to do this. The triangular method23,24 has done this job to some extent. As they consider the largest domain size for ordering, two different textures with different domain structures but the same domain size can be judged the same. The complexity of triangular methods is also high, and it is necessary to create a mesh between pore centers to calculate their equilaterality. The triangular method with orientation is more complex, and it requires unsupervised segmentation.3 The ordering criterion is also very heuristic, and it is highly dependent on the place of the material which the SEM image has been taken. In other words, if the frame of the SEM image moves a few pores to the right or left, it would be possible to have another domain size and another ordering criterion (Figure 1b). Thus, there is a demand for a method which will be less sensitive to SEM image movement and has more robust results to the place of the SEM image. In this paper, a new, fast, simple, and accurate method is proposed which does not include most of the previous methods. It has used some ideas in texture recognition and analysis29 for characterizing the texture of AAOs. This method (the global autocorrelation method) is based on the statistical characteristic of the image which is an autocorrelation of a texture (correlation of a texture with itself). The distinguishing features of this method are as follows: • This method is a very fast method with a speed comparable to our previous method.5 • This method has used global autocorrelation29 for judging the periodicity of a texture instead of the previous heuristic or local approach, so it is the most accurate method for detection of ordering in hexagonal lattices like AAO. • This method has a fuzzy judgment about the image order. The spectrum judgment makes the method able to discriminate between the medium-order hexagonal images, and it works on every possible texture. • This method uses the internal information of the image texture in a single view, and it uses the data of the neighboring pixels of every pixel.29 • This method works on the 2D signal and does not neglect any useful information. • By using the autocorrelation method, this method is robust to small defects and it is only sensitive to the structure variation. This method also has some special innovations. Initially, this method is the first method which has recruited classical texture analysis techniques29 instead of heuristic ones. Second, this method has also introduced two new ordering criteria for hexagonal texture (e.g., AAOs) based on the

2. GLOBAL AUTOCORRELATION METHOD The global autocorrelation method which has been used in this paper is based on the fact that a totally ordered AAO image has

Figure 2. (a) Totally ordered hexagonal lattice. (b) Hexagonal texton of the texture of part a (large texton). (c) Another texton of the image, which has encompassed three pores in a triangular structure (small texton). By repeating these textons, the texture will be built.

a hexagonal structure and its texture is repeated every few pixels. This fact can be rephrased in image processing as a periodic texture. For defining the amount of ordering in textures, a number of algorithms have been proposed,29 for which one of the fastest and most powerful methods has been proposed in this paper. 2.1. Texton Characterization. For defining a texture in image processing, its textons will occasionally be considered. Texton is a building block of a texture which will shape a new texture by its repetition.29 They are mostly used for characterizing periodic textures. An example of the totally ordered image which has a periodic texture is shown in Figure 2. It is also worthy to express that a periodic texture does not always have a unique texton. Two different textons of a synthesized, totally ordered AAO are shown in Figure 2. An AAO texture would be ordered if its larger textons have a hexagonal structure (like Figure 2b). They are also ordered if their small texton (like Figure 2c) has an equilateral triangular shape. However, defining a texton for a medium-order texture would be hard, and an improper texton will decrease the accuracy of the algorithm. 2.2. Autocorrelation Characterization. Another observation can direct us to a more precise method. If a periodic texture is moved onto itself pixel by pixel, and then the image of the textures are multiplied with each other, the resulting image will have peaks only in the place of repeating textons. In the AAO image, this resulting picture will have peaks only in pore places. Thus, every movement from a place of a pore will decrease the peak of multiplication image until another pore is observed. In fact, a periodic texture will have the same structure 17227



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Figure 3. (a) Totally ordered synthesis image. (b) Autocorrelation of part a. (c) Contour of part a. It is observable that the autocorrelation and contour of the image are repeated by the same distance and they also have hexagonal structures.

the autocorrelation has a periodic structure and it is also hexagonal. As it is expected, the contour center’s distance is the same as the pore’s distance. This is due to the fact that the autocorrelation image will have a peak when the x(m + k, n + l) signal comes exactly to the pore center of signal x(m, n). Thus, k and l are the same as the pore’s distances. Autocorrelation is one of the well-known tools for defining a texture29 in image processing. As the image goes away from periodicity (ordering), the autocorrelation image also moves away from periodicity. In fact, autocorrelation shows the periodicity of the image in a more precise and user-friendly manner. In other words, in this method, it is not important how complicated the texture is. This method transforms the image into autocorrelation space, and in that space, there are a few 2D (two-dimensional) mountain-like signals of which their peaks give useful information. 2.2.2. Medium-Ordered Texture. In a totally ordered texture, these mountain peaks are the same. However, as the signals go far from the periodicity, the peaks amplitude of the signal decreases. So if the slopes of two neighboring peaks are considered, these slopes would decrease when the distances from the origin increases.29 This fact is shown in Figures 4 and 5. Clearly, as the movement increases, the correlation decreases because every

after every texton. It is the same as a periodic signal. A periodic signal will have the same structure in every period of itself and by passing a period, a signal will be the same as its initial place. This is the fundamental idea of the autocorrelation method. In mathematical language, the formula of the autocorrelation function is as follows (Formula 1): Definition. Image. A digital image x is a 2D signal which consists of a few pixels. The positions of these pixels are described by the position coordinates m and n. Every pixel has a brightness x(m, n) which is defined by an 8-bit gray-level value between 0 (black color) and 255 (white color). Then, every gray-level will be normalized between [0, 1]. rxx is the autocorrelation of the image (Formula 1) rxx(k , l) = E[x(m , n)x(m + k , n + l)] =

∑ ∑ ∑ ∑ E[x(m − i , n − p)x i

j

p

q

(m + k − j , n + l − q)]

(1)

and it is also a 2D signal with the same dimension of original image, so it can be shown as an image which is called the autocorrelation image. 2.2.1. Totally Ordered Texture. An example of this event is shown in the synthetic image of Figure 3. It is observable that 17228



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Figure 4. (a) Highly ordered AAO image. (b) Obtained by using histogram equalization and adaptive thresholding on part a. Moving this image onto itself creates an autocorrelation signal which has been shown in contour mode in part c and in 3D mode in part d. Three large peaks and red contour are also observable around the main peak which coincides with the triangular structure of the image pores.

pixel normally has more correlation with its closer neighbors rather than its farther neighbors. In Figures 4 and 5, it is observable that as the distances increase, the amplitude of the peaks of the signal (mountains) in autocorrelation space decreases. It is also worthy to mention that by this movement some pixels go out of the image border and some new ones come into the image. Thus, it is necessary to fulfill these new pixels with a new value. In our experiment, we have considered these

new black pixels (zero value) to have no influence on the autocorrelation signal. A comparison of Figures 4 and 5 provides valuable data. Figure 4 is visually more ordered than Figure 5. In Figure 4, it is observable that the highly ordered texture will have the structure with a few large peaks. This has shown itself in contour showing this figure through more condensed contours (especially red contours). This is while the lower 17229



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Figure 5. (a) Low-ordered AAO image. (b) Obtained by using histogram equalization and adaptive thresholding on part a. Moving this image onto itself creates an autocorrelation signal which has been shown in contour mode in part c and in 3D mode in part d. Here, there is no large peak around the main peak which shows that the structure is not highly ordered.

This idea has been adapted from texture analysis methods.29 A more ordered texture has a sharp decrease in the slope of the peaks of the autocorrelation image and vice versa. It is also worthy to express that sometimes capturing conditions of the AAO images are not the same. This causes background irregularity in an image. In that condition, two

ordered texture of Figure 5 has shorter peaks. It also has less condensed contours. In other words, the more ordered texture has larger peaks so the slope of the line between every neighboring peak is lower while the less ordered texture has shorter peaks and the slope of the line between every neighboring peak is higher. 17230



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Figure 6. Projection of autocorrelation of Figure 5 in the xz plane. The slope of this signal between its first and second peak can be considered as an ordering criterion.

Table 1. Each of A0 to A9 Considered As a Threshold for Defining the Order of Hexagonal Texture number of local maximum above zero number of local maximum above the 90% of the largest peak. number of local maximum above the 80% of the largest peak. number of local maximum above the 70% of the largest peak. number of local maximum above the 60% of the largest peak. number of local maximum above the 10% of the largest peak.

A9 (more division if it is needed) A8 A7 A6

Figure 7. Two different AAO images. Both of them have the same ordering criterion based on the largest domain size3,5,23,24 while they characterize different textures. Thus, the largest domain size could not be a precise method for texture characterization.

A5 A0

famous image processing techniques are used which are histogram equalization and adaptive thresholding.30 These methods had also been used in our previous paper and increased its capability.5 The first method equalizes the gray-

level distribution of the image and increases the image contrast, and the second method eliminates the background dissimilarity. Therefore, a binary image which shows the pores with black color more distinctively will remain (Figures 4b and 5b). Then,

Table 2. Algorithm for Comparing a Few Textures’ Ordering

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Figure 8. (a) SEM image obtained from a long-range ordered anodic porous alumina formed in sulfuric acid at 25 V. (b) Result of the exertion of our algorithm on part a. The preparation condition of part c is as follows: 0.4 M H2C2O4, 30 V at 17 °C, Al (99.95%). (d) Result of the exertion of our algorithm on part b.

the autocorrelation method will be exerted on this modified image. This causes the robustness of the algorithm against the capturing conditions and minor defects. 2.3. First Ordering Criterion. Now, this idea can be used for defining the ordering criterion. Visually, it is observable that the more ordered texture has lots of large peaks, so they are like a rough surface. Low ordered texture has shorter peaks, so they have a smoother autocorrelation texture in the image surface. Thus, if the slope of the peaks of the autocorrelation signal is considered, then this value can show the periodicity of the image. Consequently, as the periodicity has a direct relation with ordering, this slope can be used as a criterion for image ordering. A sample of this signal in x axis is shown in Figure 6. But the question is which two peaks should be considered? Two possible answers can define two criteria. Normally this slope is considered between the first and second largest peaks (formula 2). But it can also be defined as a slope between the two largest peaks in the xy or xz plane for more simplicity (formula 3).

ordering criterion = slope of the line between the first and second largest peak of the projection of autocorrelation image in the xz or xy plane

(3)

Another extension can be made here. In fact, the slope of the line between the second largest peak with its largest neighboring peak can also be considered. However, this item must have low impact on the ordering criterion. Thus, if a coefficient is considered for the image order, this coefficient would be lower for the slope of the line between the first and second peaks (α in formula 4) than the second and third peaks (β in formula 4). extended ordering criterion = α(slope of the line between the first and second largest peaks) + β(slope of the line between the second and third largest peaks (neighborhood of second peak))

(4)

ordering criterion = slope of the line between the first and second largest peak

This criterion has this feature that it acts on a simple 1D signal, but it has a problem that the projection decreases the

(2) 17232



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Figure 9. (a−c) AAO with pore distance of 22.0 ± 0.4, 61.9 ± 1.2, and 39.2 ± 1.3 nm, respectively. By visual judgment and experiments, a is more ordered than b and b is more ordered than c.7 Our experimental results confirm this judgment. (g and h) Autocorrelation signals before and after scaling. (i and j) Peak numbers before and after scaling.

signal dimension and excludes some information. Therefore,

peaks higher than that threshold. The method of defining this threshold will be described in the following section. 2.4. Second-Order Parameter and Comparing a Few Textures. Sometimes, an absolute value for texture order is not demanded and a relative comparison would be enough. In other words, in some experiments, by changing the input parameters, a few samples of a material are fabricated. Users

another criterion is also considered here. Representing autocorrelation in contour mode (Figures 3c or 4c) reveals more contours for higher peaks. This means that if a specific amplitude is considered as a threshold, then a more ordered texture would have more mountain-shaped signals with 17233



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Figure 10. (a−c) SEM images of two-step anodized specimens applying various striping step (STP) times of our previous work.2 Adapted from Figure 3 of ref 7. Copyright 2013 American Chemical Society. (d−f) Result of the exertion of our algorithm on them. Their peaks in 10 intervals have been shown in Table 3. It is observable that our algorithm has recognized part a as more ordered than part b and part b as more ordered than part c. This coincides with visual judgment, and the slope of the line between A9 and A8 are in this order.

10h and Table 3. In this figure, texture a is more ordered than b and b is more ordered than c. Figure 10g also confirms this fact. All the signals have peaks and valleys in the same place on the x-axis of 50 pixels (this number is arbitrary but fixed for all the figures and contains a few peaks of every figure). The difference is related to the magnitude of peaks. But looking more deeply into this criterion, it is revealed that larger pore sizes stretch the autocorrelation function and this causes the slope of the line between the first and second peaks of signal b of Figure 9g to be lower than signal a. This problem can be solved by considering the pore’s size in scaling the x-axis and resampling this axis based on that. The formula is as follow:

want to know which one is more ordered than the others and sort them based on their order. Our method (global autocorrelation method) would be able to do both the absolute and relative comparison of the texture’s order. For a relative comparison, the following algorithm will be considered. The first 10 peak height intervals are considered, and the number of peaks above 10% of the largest peaks (A0 criterion) and above 20% of the largest peaks (A1 criterion), etc., will be defined (Table 1). At first glance, it seems that the more ordered texture would have larger A0 than the low ordered one. So, at first this criterion will be considered. If two textures have the same value for this criterion, then the second criterion (A1) would be considered. This is continued until the comparison is made on the last criterion. This fact will be shown further on in Figure

target (scaled) x ‐axis = base x ‐axis ×

base pore size target pore size (5)

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The result of our algorithm has been shown in Figures 10d, e, and f, and the parameter values A0, A1, ..., A9 have been shown in Table 3. Borba et al.7 have recognized Figure 9a as more ordered than Figure 9b and Figure 9b to be more ordered than Figure 9c. Our results in Table 3 also confirm their results. We have also used our methods (the global autocorrelation method) in our previous experiment in ref 2. Figures 10a, b, and c are SEM images of two-step anodized specimens applying various STP times. The result of our algorithm has been shown in Figures 10d−f and the parameter values A0, A1, ..., A9 have been shown in Table 3.

After the resampling process, Figure 9h will be obtained. In this figure, the slope of the line between the first and second peaks of signal a of Figure 9g is lower than signal b and c. In Table 3 and Figure 9i, it is also shown that Ai is equal to b, c for i = 0:7 and, at A8, c is more ordered than b. This problem is solved in the scaled version, and the results have been shown in Figure 9j and scaled rows of Figure 9b and c of Table 3. Thus, our algorithm is as follows (Table 2). First we scale the interval of x-axis of all the textures based on one of them. Then we compare the Ai of all the textures with each other. The texture with more Ai will be more ordered because it has more peaks than the others in the higher threshold. If both textures have the same Ai for all i = 0:9, then we consider the highest threshold which they experienced as more than one peak and break that interval into ten intervals and replace these intervals with the previous ones. This shows which texture has the higher value of peak, and this is the more ordered one. Our comparing algorithm will be able to compare every two different textures, and it has absolutely different values of ordering for two different textures. This is while the previous algorithms3−7,12 will not be able to do this perfectly, and there are some cases where they have the same judgment about the texture’s ordering for two different textures (Figure 7).

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CONCLUSION In this paper, two new criteria for the characterization of AAO structures are introduced. Our method (the global autocorrelation method) uses the autocorrelation of the image, and therefore, it is based on the statistical characteristics of the image. One of these two criteria is the slope of the line between the first and second peaks of the autocorrelation image in the xz plane. The other one is the number of peak of autocorrelation image above a few predefined thresholds. These two new criteria have introduced a tool for characterizing the hexagonal lattices and can give the researcher an objective tool to evaluate the ordering of these structures. Our comparison with the other previous results shows that the global autocorrelation method is the most precise one up to now while its computational cost is very low. We expect that our characterizing order helps improve the theoretical modeling of AAO fabrication and propels them to fabricate highly ordered hexagonal lattices.

3. EXPERIMENTAL RESULTS In this part, the result of the exertion of our algorithm has been tested on a few real AAO samples. The first one (Figure 8a) is Table 3. Parameter Value of A0, A1, ..., A9 of Figures 9 and 10 Figure

A9

A8

A7

A6

A5

A4

A3

A2

A1

A0

Figure 9a Figure 9b Figure 9b (scaled version) Figure 9c Figure 9c (scaled version) Figure 10a Figure 10b Figure 10c

13 9 11

11 4 6

11 1 2

11 1 2

8 1 1

3 1 1

2 1 1

1 1 1

1 1 1

1 1 1

12 10

5 5

1 1

1 1

1 1

1 1

1 1

1 1

1 1

1 1

26 22 25

24 15 12

11 6 4

5 3 2

3 3 1

3 3 1

1 1 1

1 1 1

1 1 1

1 1 1

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We would like to thank Mohammad Javad Abdollahifard for his help on the writing of the paper and Mahdi Kafaee for his AAO images. We would also like to thank the anonymous reviewers for their comments. This research was partially financially supported by School of Computer Science and Nano of Institute for Studies in Theoretical Physics and Mathematics (IPM).

an SEM image obtained from a long-range ordered anodic porous alumina formed in sulfuric acid at 25 V, and the second one (Figure 8c) is formed on oxalic acid at 30 V. It is observable from Figure 8b that the periodicity of the autocorrelation image is very high, and for Figure 8d, the periodicity of the autocorrelation image is low. Our experiment has also been repeated for the image of Borba et al.7 They have created three different samples of the AAO structure through the anodization process. The process was carried out in two anodization steps, in a conventional twoelectrode cell using a Cu sheet as a cathode. Their anode is made of aluminum with two different degrees of purity, that is, high-purity Al bulk (99.999%) and commercial Al (99.5%). After each anodization step, their samples were dipped in a 5 wt % H3PO4 solution at 35 ± 1 °C for different times to remove the alumina formed in the first anodization step. Then, a second anodization step was performed to allow the opening of nanopores. The anodization of Figure 9a, b, and c yielded AAO with pore distance of 22.0 ± 0.4, 61.9 ± 1.2, and 39.2 ± 1.3 nm, respectively.

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REFERENCES

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