Quantitative Evaluation of Self-ordering in Anodic Porous Alumina

Abstract

An important requirement for anodic porous alumina to be used as a template in synthesis of other nanomaterials, is that the in-plane arrangement of the nanopore channels should be self-ordered, because the self-ordered pattern corresponds to the growth of straight pore channels from top to bottom.

1 Introduction

An important requirement for anodic porous alumina to be used as a template in synthesis of other nanomaterials, is that the in-plane arrangement of the nanopore channels should be self-ordered, because the self-ordered pattern corresponds to the growth of straight pore channels from top to bottom [1, 2]. If the pattern is not self-ordered, the corresponding pore channels will frequently split or terminate during growth, and this kind of structure cannot be used as template for long nanowires or nanopillars [3–5].

It has been found that the self-ordering quality of anodic porous alumina can be influenced by various anodization conditions, such as anodization voltage, acid concentration, temperature, anodization time, and substrate aluminum grain orientation [6–11]. However, by direct observation of the scanning electron microscopy (SEM)-captured porous patterns formed under different anodization conditions, it is hard to examine the ordering change, and so it is not possible to find the optimal anodization conditions for the best self-ordered anodic porous alumina formation. Thus, in order to find the optimum conditions, it is necessary to develop a quantitative method to evaluate the self-ordering qualities of anodic porous alumina fabricated under various anodization conditions.

In this chapter [12], to quantitatively evaluate the ordering of in-plane porous patterns, a new technique called angular orientation distribution (AOD), further developed from Hillebrand et al. [13] and Mátéfi-Tempfli et al. [14] is presented, in which triangles formed by three neighboring pore centers are colored according to their angular orientation. The self-ordering results computed by this technique on a series of porous patterns in anodic porous alumina fabricated under a wide range of oxalic acid (H₂C₂O₄) electrolyte concentrations under different temperatures are compared with those results obtained by conventional ordering evaluation methods, such as the radial distribution function (RDF) [15] and angle distribution function (ADF) [13]. We will show that the present AOD method is more sensitive in quantifying the ordering of porous patterns. In this chapter, the anodic porous alumina samples are mainly used for judging the effectiveness of the present AOD method on evaluation of self-ordering qualities, and a systematic search of the optimal self-ordering conditions based on this powerful evaluation tool by considering all of the key influencing factors on self-ordering will be shown in detail in Chap. 7.

2 Quantitative Evaluation Methods for Porous Patterns in Anodic Porous Alumina

In order to quantitatively evaluate the ordering quality of different porous alumina patterns, the coordinates of the pore centers in each pattern were captured by the ImagJ software [16] as previously used by Hillebrand et al. [13]. Three analysis methods were used to quantify the pore ordering on the patterns.

2.1 Radial Distribution Function (RDF)

The two-dimensional (2-D) RDF was defined by Eq. (3.8) in Chap. 3, which is RDF = S _pattern/[2πrN(dn(r)/dr)], where S _pattern is the pattern’s area, r is the distance between the centers of any two pores in the pattern, N is the total number of pore pairs, and n(r) is the number of pore pairs in which the pores are separated by a distance ≤ r. The RDF gives the probability density of finding a neighbor pore distanced r away from any given pore in the pattern.

2.2 Angle Distribution Function (ADF)

In a porous pattern, three nearest neighboring pores form one triangle, and so the whole pattern can be represented as a mesh of triangles with the pore centers as the mesh nodes. In order to avoid unwanted side effects, the nearest neighbors of a given pore are found in a region around that pore center which is less than 1.8 times of the first RDF peak position, and each angle of a triangle should be in the range of 30°–90° or else that triangle is not formed. Sometimes, in a disordered pattern, edges from two different triangles may intersect each other, and in this case, the triangle whose three edges have a smaller standard deviation $\left( {{\text{Dev}}\_d} \right)$ will be chosen. For a triangle, $\left( {{\text{Dev}}\_d} \right)$ is defined as [13, 14]

$$ {\text{Dev}}\_d = \frac{1}{{\bar{d}}}\sqrt {\frac{1}{3}\sum\limits_{i = 1}^{3} {(d_{i} - \bar{d})^{2} } }, $$

(6.1)

where d _i (i = 1, 2, 3) are the edge lengths of that triangle, and $\bar{d}$ is the mean of d _i . The angles of all the formed triangles are statistically evaluated to give the ADF, which represents the probability of finding a particular angle value in the pattern [17].

2.3 Angular Orientation Distribution (AOD)

In the following analysis involving Figs. 6.1 and 6.2, we will see that RDF and ADF are helpful when the differences in ordering qualities of the porous patterns are large; however, they are not sensitive enough to distinguish between patterns with close but different ordering qualities. A more sensitive method based on the orientation of the triangles formed in the ADF representation is proposed here. For any given triangle in the ADF representation described in Sect. 6.2.2 above, as shown in the examples below the color bar of Fig. 6.3, its orientation θ with the SEM image horizontal boarder as the reference direction can be calculated as

$$ \theta = \frac{1}{3}\left[ {{\mathbf{atan}}\left( {\frac{{y_{1} - y_{2} }}{{x_{1} - x_{2} }}} \right) + {\mathbf{atan}}\left( {\frac{{y_{2} - y_{3} }}{{x_{2} - x_{3} }}} \right) + {\mathbf{atan}}\left( {\frac{{y_{3} - y_{1} }}{{x_{3} - x_{1} }}} \right)} \right]\frac{{180^{\text{o}} }}{\pi }, $$

(6.2)

where (x _i , y _i ), i = 1, 2, 3, are the coordinates of three vertices of the triangle. Most of the triangles in a mildly disordered pattern usually have θ calculated from Eq. (6.2) falling in the range [−30°, 30°], but if θ does exceed this range, it is reduced to within [−30°, 30°] by adding or subtracting 60°, because the mesh structure is a hexagonal arrangement of the triangles. The triangles are then colored according to their θ values with the wrapping scale shown at the top of Fig. 6.3, in which −30° and 30° correspond to the same (red) color.

Fig. 6.1

SEM micrographs of (a, c, e, g) the top patterns of anodic porous alumina obtained by a second anodization step using Al substrates pre-pattern obtained in the first anodization step; (b, d, f, h) the bottom patterns of the aluminum substrate after selectively removing the anodic porous alumina on top. Two-step anodization was conducted in (a, b) 0.3 M; (c, d) 0.6 M; (e, f) 0.8 M; (g, h) 0.95 M H₂C₂O₄ at 40 V at 20 °C, with first step for 1 h and second step for 0.5 h. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

Fig. 6.2

a–d Radial distribution function for patterns shown in Fig. 6.1a–h, respectively. e, f Angle distribution function for patterns shown in Fig. 6.1a–h, respectively. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

Fig. 6.3

Color-coded patterns with pore centers as triangle mesh nodes, and triangles are colored according to their orientation θ. a, b corresponds to the pore structure in Fig. 6.1b; c, d corresponds to Fig. 6.1f. Both ${\text{Tol}}\_d$ and ${\text{Tol}}\_\phi$ are equal to 1 in (a, c), and 0.1 in (b, d) (scale bar = 1 µm). Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

In order to distinguish ordered zones in the pattern, which are ordering domains in which the triangles are similarly oriented, and then to calculate the average ordered zone size (AOZS), as is similar to Hillebrand et al. [13] and Mátéfi-Tempfli et al. [14], tolerance deviations of ${\text{Tol}}\_d$ and ${\text{Tol}}\_\phi$ are introduced. For a given triangle, if its ${\text{Dev}}\_d < {\text{Tol}}\_d$ and ${\text{Dev}}\_\phi < {\text{Tol}}\_\phi$, this triangle will be regarded as within the ordered zones (ordering domains), where

$$ {\text{Dev}}\_\phi = \frac{1}{{\overline{\phi } }}\sqrt {\frac{1}{3}\sum\limits_{i = 1}^{3} {(\phi_{i} - \overline{\phi } )^{2} } }, $$

(6.3)

is the standard deviation of the three angles $\phi_{i}$ (i = 1, 2, 3) in that given triangle, and $\bar{\phi }$ is the mean of $\phi_{i}$. Because in real porous patterns, neighboring triangles within the same ordered zone may not have exactly the same orientation θ, here we introduce a third tolerance parameter ${\text{Tol}}\_\theta$, which means that if the difference in orientation θ between two neighboring triangles $\Delta \theta < {\text{Tol}}\_\theta$, these two triangles are regarded as within the same ordered zone (ordering domain). In the following, the use of the three tolerance conditions involving ${\text{Tol}}\_d$, ${\text{Tol}}\_\phi$, and ${\text{Tol}}\_\theta$ will be illustrated by examples. The calculation program for numerical realization of the above evaluation methods is shown in Appendix II of the present thesis.

3 Experimental Method

Anodic porous alumina samples, which are used to evaluate the effects of the above three ordering qualification methods, are fabricated as follows. Before anodization, polycrystalline 99.99 % pure Al foils were pretreated as in Sect. 4.2.1, and then anodized in the electrochemical cell setup as in Sect. 4.2.2. The anodization was conducted under a constant voltage of 40 V in H₂C₂O₄ of different concentration under 20, 10, and 2 °C. After anodization, in order to observe the pore arrangements at the pore bottom, oxide–metal interface region, the samples were put into a mixed solution of H₂CrO₄, H₃PO₄, and H₂O with composition 1.8:6:92.2 by weight at 60 °C, to selectively dissolve the anodic oxide away. Because of the scalloped shape of the oxide barrier layer at the pore bottom, each pore will leave behind a small dimple on the Al substrate, and so microscopic examination of the dimpled Al substrate after the selective dissolution of the alumina on top would directly reveal the arrangement of the pores at the end of the anodization. SEM examination was carried out in a Hitachi S-4800 field emission microscope, and a LEO 1530 field emission microscope. In the anodization experiments under 20 and 10 °C, two-step anodization was conducted, in which the second anodization step started from the pre-textured aluminum substrate after selectively removing the formed porous alumina in the first anodization step, and the first and second steps were carried out at the same anodization conditions. In the anodization under 2 °C, only one-step anodization was conducted, but actually for the long anodization time used, the pore arrangement has already reached a steady state corresponding to that anodization condition.

4 Effects of the Quantitative Evaluation Methods

In Fig. 6.1a–h, two-step anodization experiments were conducted in 0.3, 0.6, 0.8 M, and 0.95 M H₂C₂O_4, respectively, while other anodization conditions, namely, voltage of 40 V, temperature of 20 °C, and duration of 1 h for the first step and 0.5 h for the second step, were kept as the same. Figure 6.1a, c, e, g are SEM micrographs of the top views of the anodic porous alumina obtained by a second anodization step using Al substrates pre-textured in a first anodization step, and Fig. 6.1b, d, f, h are SEM micrographs of the dimpled Al substrate after the selective dissolution of the oxide obtained in the second anodization step, revealing the ordering of the pore channels at their bottom at the end of the second anodization step. From direct observation of these SEM images, a trend of the ordering tendency of the pore channels with changing acid concentration is not obvious. For example, at a first glance, one may find that the ordering quality in Fig. 6.1a is lower than that in Fig. 6.1c, but it is hard to distinguish the difference amongst Fig. 6.1c, e, g or Fig. 6.1d, f, h. Thus, the quantitative pattern characterization methods introduced in Sect. 6.2 were employed.

In Fig. 6.2, the RDF and ADF are plotted for each pattern shown in Fig. 6.1. By comparing the RDF in Fig. 6.2a–d, on increasing acid concentration, ordering first increases from 0.3 to 0.6 M, and then decreases from 0.8 to 0.95 M, but the difference in ordering between 0.6 and 0.8 M is not clear. In addition, the ordering difference between the top and bottom patterns for each anodization condition is small, except that at 0.6 M the first peak in the bottom pattern is stronger than that in the top pattern. However, this does not change the ordering tendency on increasing acid concentration, which is clearly verified by the ADF shown in Fig. 6.2e, f. One can see that the peak position of the ADF for all patterns are located at around 60°, and the higher the peak intensity and the narrower the full width at half maximum (FWHM), the closer is the pattern to the perfect hexagonal case. From both the top patterns in Fig. 6.2e and bottom patterns in Fig. 6.2f, the best ordered pattern is formed under 0.8 M, while the worst ordered pattern is formed under 0.3 M. The ordering tendency with acid concentration is first increasing and then decreasing. Although RDF and ADF are useful in indicating the ordering tendency, they are not sensitive enough to distinguish between two patterns with similar pore center and angle distributions, e.g., two patterns formed under 0.6 and 0.8 M in Fig. 6.2f. The method involving the AOD in Sect. 6.2.3 is more useful in overcoming above difficulties.

For each SEM figure shown in Fig. 6.1, its triangularly meshed pattern is color coded according to the orientation θ obtained by Eq. (6.2), and those truncated pores at the boundaries of the SEM micrograph with centers falling outside the frame region are excluded. As an example, Fig. 6.3a–d shows the color-coded patterns at different tolerance levels corresponding to the porous structures shown in Fig. 6.1b, f, respectively. In Fig. 6.3a, c, both the tolerance factors ${\text{Tol}}\_d$ and ${\text{Tol}}\_\phi$, as stated in Sect. 6.2, were set equal to 1, which means all the triangles in the mesh pattern are included. From these figures, ordered zones in the pattern are clearly distinguishable by their different colors due to their orientations. Sudden change of color happens across ordered zone boundaries, or at single-site defects within an ordered zone, and the ordered zone sizes in Fig. 6.3a are obviously smaller than those in Fig. 6.3c. In Fig. 6.3b, d, both ${\text{Tol}}\_d$ and ${\rm Tol}\_\phi$ were set at a much smaller value at 0.1, and by this a lot of the triangles much different from the equilateral triangle are excluded, and these are colored white in the figures. At such a low acceptance level, the major part of the porous structure in Fig. 6.3b is concluded amorphous (the white regions), whereas in the much better ordered structure in Fig. 6.3d, only those highly ordered regions are concluded as ordered zones which are colored.

Under above acceptance level of Dev_d < Tol_d = 0.1 and Dev_ϕ < Tol_ϕ = 0.1, which was chosen by referring Refs. [13, 14], ordered zones (colored) are revealed and their sizes can be conveniently calculated. Even within one ordered zone the orientation values θ of the triangles are not exactly the same, and so if the difference Δθ of two neighboring triangles is less than Tol_θ, these two triangles are accepted into the same ordered zone. After checking many possible values in the range [1°, 10°], Tol_θ was chosen to be 3°, since too small a value will divide up one ordered zone into several tiny ones, while too large a value will combine two different ordered zones together. It is worth noting here that some ambiguity does exist in defining where are the ordered zones in a given pore pattern, especially when ordering is low. The demarcation of ordered zones is controlled by the tolerance factors and the difficulty to unambiguously select these was discussed in Ref. [13]. The choice of Tol_d = 0.1, Tol_ϕ = 0.1, and Tol_θ = 3° here is judged to be appropriate after considering the effects of other values, and in order to be consistent, these tolerance values are used the same in the remaining of this chapter.

Figure 6.4a shows the number of triangles in each of the 20 largest ordered zones in the patterns shown in Fig. 6.1a, c, e, g, respectively. It is clear that the pattern formed under 0.8 M has the largest ordered zones than the other three patterns, and its largest ordered zone consists of 436 triangles, and there were 92 ordered zones in total (only the largest 20 are shown in Fig. 6.4a). It is noted above in Fig. 6.2e that the ADF fails to distinguish between the ordering under 0.6 and 0.95 M acid concentration, but the angular orientation distribution (AOD) method in Fig. 6.4a can reveal a noticeable difference: the blue histogram for the 0.95 M top pattern clearly exhibits better ordering than the red histogram for the 0.6 M top pattern. For a given porous pattern, it is desirable to have a single measure of the average ordered zone size which could also be an indicator for the ordering, but evaluation of this quantity may be distorted if very small ordered zones, such as those containing fewer than six triangles, are not excluded. In fact, regions containing fewer than six triangles are really merely describable as short-range ordered with at most only the nearest neighbors identifiable, and hence they should not be qualified as ordered zones. For this reason, these regions with fewer than six triangles within are excluded in the average ordered zone size calculation, and Fig. 6.4b shows the average ordered zone sizes of all of the patterns in Fig. 6.1 plotted against the acid concentration. We noted above in Fig. 6.2b, c, f that the RDF and ADF cannot effectively distinguish between the ordering quality of the 0.6 and 0.8 M bottom patterns, but in Fig. 6.4b, the AOD method clearly shows that the average ordered zone size is significantly different at about 0.11 and 0.17 µm² at 0.6 and 0.8 M, respectively. Figure 6.4b shows that the dependence of the ordering tendency of the porous structures on the acid concentration is more obvious than what can be revealed by the ADF and RDF in Fig. 6.2. From Fig. 6.4b, the ordering first increases with acid concentration from 0.3 to 0.8 M, and then decreases from 0.8 to 0.95 M. The AOD results in Fig. 6.4b also indicate that at the acid concentration of 0.6 M, ordering in the bottom pattern is better than that in the top pattern. This is in accordance with the sharper ADF of the bottom pattern (red curve in Fig. 6.2f) than that in the top pattern (red curve in Fig. 6.2e), as well as the higher first peak in the RDF of the bottom pattern than the top pattern in Fig. 6.2b. While an improvement of ordering as the oxide layer grows could be understood as a general trend, there is no particular reason to explain why this is more obvious for the 0.6 M concentration. The important point here, however, is that the three analysis methods of RDF, ADF, and AOD are consistent in this respect.

Fig. 6.4

a Number of triangles within the 20 largest ordered zones in the patterns shown in Fig. 6.1a, c, e, g. b Average ordered zone size against acid concentration for the patterns in Fig. 6.1. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

In order to investigate the effects of acid concentration at other temperatures, two groups of anodization experiments were done at 2 and 10 °C. As shown in Fig. 6.5a–d, one-step anodization was conducted under acid concentrations of 0.05, 0.3, 0.4, and 0.45 M, respectively, at 40 V at 2 °C for about 20 h. Here, we focus on the pore bottom region revealed by the aluminum substrates after selectively removing the anodic oxide. From the RDF shown in Fig. 6.6a–d, on increasing acid concentration the pattern’s ordering quality first increases from 0.05 to 0.4 M, and then decreases from 0.4 to 0.45 M. This trend is similar to that at 20 °C shown in Figs. 6.1 and 6.4b, although the optimal acid concentration for the best ordering pattern is 0.4 M at 2 °C and 0.8 M at 20 °C. The trend noted from the RDF is also verified by the ADF as shown in Fig. 6.6e—the ADF curve under 0.4 M has the narrowest FWHM.

Fig. 6.5

SEM micrographs of bottom patterns on the aluminum substrate after selectively removing the anodic porous alumina on top. One-step anodization was conducted in a 0.05 M; b 0.3 M; c 0.4 M; d 0.45 M H₂C₂O₄ at 40 V at 2 °C for about 20 h. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

Fig. 6.6

a–d Radial distribution functions and e angle distribution functions for patterns shown in Fig. 6.5a–d, respectively. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

Under the same tolerance conditions of ${\text{Tol}}\_d$, ${\text{Tol}}\_\phi$, and ${\text{Tol}}\_\theta$ as in Fig. 6.4, the number of triangles in the first 20 largest ordered zones and the average ordered zone size in each pattern shown in Fig. 6.5 are plotted in Fig. 6.7a, b, respectively. As shown in Fig. 6.7a, the number of triangles is very sensitive to the ordering quality of the porous pattern. For example, if one compares the ordering quality between Fig. 6.5b, c by direct observation, or by RDF in Fig. 6.6b, c, the difference is far less than what the red histogram at 0.3 M and the green histogram at 0.4 M in Fig. 6.7a shows. The trend of the ordering tendency with acid concentration in both Fig. 6.7a, b is the same as that shown by the ADF Fig. 6.6e, namely, the ordering increases first from 0.05 to 0.4 M, and then decreases from 0.4 to 0.45 M. In addition, two-step anodization was conduction under 0.1, 0.3, and 0.6 M H₂C₂O₄, respectively, under 10 °C for 2 h in the first step and a further 1 h in the second step. Figure 6.8a–c shows the bottom patterns on the aluminum substrate after selectively removing the anodic porous alumina. From Fig. 6.8d, one can easily see that the optimal acid concentration which gives rise to best ordering is 0.6 M.

Fig. 6.7

a Number of mesh triangles in each of the 20 largest ordered zones in the structures shown in Fig. 6.5. b Average ordered zone size of the patterns in Fig. 6.5 against acid concentration. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

Fig. 6.8

a–c SEM micrographs of bottom patterns on the aluminum substrate after selectively removing the anodic porous alumina. Two-step anodization was conducted in a 0.1 M; b 0.3 M; c 0.6 M H₂C₂O₄ at 40 V at 10 °C for 2 h in the first step and 1 h in the second step. d Number of triangles in each of the 20 largest ordered zones in the patterns of (a–c). Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

It should be noted that the highest acid concentration used in the anodization at the three temperatures of 20, 2, and 10 °C in Figs. 6.1, 6.5, and 6.8, namely, 0.95, 0.45 M, and 0.6, respectively, is already close to the solubility limit of aqueous H₂C₂O₄ at each temperature. From the results in Figs. 6.4, 6.7, and 6.8d, we can conclude that the optimal H₂C₂O₄ concentration for the best ordering at 40 V is 0.8, 0.4, and 0.6 M at 20, 2, and 10 °C, respectively. Together with the earlier established result that the optimal H₂C₂O₄ concentration is 0.3 M at 0 °C [4], the relationship between the optimal acid concentration and temperature is plotted in Fig. 6.9a.

Fig. 6.9

a Optimal oxalic acid (H₂C₂O₄) concentration (square symbols) which can result in the best pore channel ordering in anodic porous alumina at 40 V, and average ordered zone size (triangle symbols) formed under the optimal acid concentration, against anodization temperature; b Oxide growth rate against anodization temperature under 0.3 M H₂C₂O₄ at 40 V; the inset figure shows the oxide growth rate against H₂C₂O₄ concentration at 40 V at 2 °C. Reprinted from Ref. [12]. Open Access, American Institute of Physics; used in accordance with Creative Commons Attribution 3.0 Unported License

From Fig. 6.9a, the optimal H₂C₂O₄ concentration (square symbols) which can result in the best self-ordering in the pore arrangement is approximately linearly increasing with temperature. This means that at different temperatures, anodization in H₂C₂O₄ under 40 V would exhibit an optimal acid concentration corresponding to best ordering which is in general not 0.3 M. Moreover, under above optimal acid concentrations (0.4 M at 2 °C, 0.6 M at 10 °C, and 0.8 M at 20 °C), the average ordered zone size (triangle symbols) is also slightly linearly increasing with temperature. In addition, one of the main limitations previously for the wide application of mild anodization is that the oxide growth rate is too slow for batch production [8, 18]. However, this difficulty can be overcome if self-ordered porous alumina is fabricated using the optimal acid concentrations at higher temperatures as shown in Fig. 6.9a. Figure 6.9b shows that the oxide growth rate, estimated as the SEM observed oxide thickness divided by the anodization time, increases approximately exponentially with temperature under a certain acid concentration, and with acid concentration under a certain temperature. Therefore, on increasing temperature along the curve for optimal conditions in Fig. 6.9a, the oxide growth rate should increase enormously, since both temperature and acid concentration are increasing, and both of them result in approximately exponentially increase of the oxide growth rate in Fig. 6.9b.

We should note that the substrate grain orientation which is an important factor on self-ordering qualities of anodic porous alumina has not been considered during the fabrication of the above alumina samples, as a result the porous patterns evaluated in this chapter may not be formed on the same oriented Al grains. Since the substrate orientations are different, even other anodization conditions were kept the same, the ordering qualities of porous patterns may be different. Therefore, the changing tendency of ordering obtained in Figs. 6.4b, 6.7b, and 6.9a may not be accurate enough when the substrate grain orientation is taken into consideration. This issue is out of the scope of the present chapter, and will be fully addressed in Chap. 7.

5 Summary

In this chapter [12], a colorization method on quantitative evaluation of ordering in porous patterns of anodic porous alumina has been developed. This method is called angular orientation distribution (AOD) method, involving the AOD of the triangles connecting each three neighboring pore centers in the porous patterns. This method is found to be much more sensitive in delineating the in-plane self-ordering qualities of anodic porous alumina than the conventional radial distribution function (RDF) and the angle distribution function (ADF) methods.

In addition, quantitative analysis of the experimental in-plane porous patterns in anodic porous alumina formed under different oxalic acid (H₂C₂O₄) concentrations and temperatures at 40 V suggested that fast fabrication of self-ordered anodic porous alumina can be realized by performing the anodization at relatively higher temperatures using the corresponding higher acid concentrations, rather than the previously well-known hard anodization method 93, which may easily result in the breakdown of anodic porous alumina structures due to the high anodization voltages. This encourages us to further explore the anodization conditions for the fast growth of highly self-ordered and mechanically stable porous alumina, as discussed in the next chapter.