Deformation of anodic aluminum oxide nano-honeycombs during nanoindentation

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Acta Materialia 57 (2009) 2710–2720 www.elsevier.com/locate/actamat

Deformation of anodic aluminum oxide nano-honeycombs during nanoindentation K.Y. Ng, Y. Lin, A.H.W. Ngan * Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China Received 14 August 2008; received in revised form 18 February 2009; accepted 19 February 2009 Available online 26 March 2009

Abstract Anodic aluminum oxide with a nano-honeycomb structure is subjected to nanoindentation along the axial direction of the honeycomb. The load–displacement behavior is discontinuous with periodic strain excursions. Top-view and cross-sectional microscopic examination reveals a localized mode of deformation with very clear-cut elastoplastic boundaries. A crack system that is self-similar with respect to the indent size is also found, and this is thought to correspond to the discontinuous load–displacement behavior. A simple column model is proposed to explain certain features of the deformation microstructure. Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Anodic aluminum oxide; Nanoindentation; Focused-ion-beam; Plastic deformation; Cracks

1. Introduction

2. Materials and methods

Anodic porous alumina (AAO) is a self-ordered nanostructured material formed by electrochemical oxidation of metallic aluminum. It has a unique nanoporous structure consisting of vertically aligned pore channels which are closely packed together forming a hexagonal honeycomb structure as shown in Fig. 1. Since its discovery [1–3], attention has been drawn to the feasibility of utilizing it as functional devices [4–7] and host templates for the fabrication of other nanostructures [8–13]. However, the mechanical behavior of AAO has been much less studied, despite the fact that the mechanical properties of macroscopic honeycomb structures have been a long-term interest of the mechanics and materials science community [14–24]. In this paper, we report for the first time a systematic study of the deformed microstructure of AAO honeycombs by nanoindentation.

Aluminum discs 1 in. in diameter were cut out from a 300 lm thick pure (99.999%) Al sheet by electrical discharge machining. The prepared discs were annealed in a vacuum better than 107 torr at 550 °C for 2 days to allow grain growth to occur. The surfaces of these Al discs were then mechanically polished and subsequently electropolished in 20% perchloric acid in methanol to achieve a surface smoothness of about 1 lm, which is suitable for the subsequent anodization steps, and at the same time to remove any residual stresses on the surface. Nanoporous AAO structures were then prepared by anodic oxidation of the pre-treated Al discs in a 0.3 M oxalic acid solution at 40 V DC in a constant-temperature environment of 17 ± 0.1 °C, achieved using an electronic feedback-controlled waterbath. Since the structure of the resultant AAO film is very sensitive to the anodization environment, to obtain a highly ordered AAO nano-honeycomb structure, we employed the two-step anodization method first suggested by Masuda et al. [1–3]. The first anodization step involved forming an initial layer of AAO on the Al substrate by performing anodic oxidation in the above conditions for 6–10 h. This initial layer was then removed by a

*

Corresponding author. Tel.: +852 2859 7900; fax: +852 2858 5415. E-mail address: [email protected] (A.H.W. Ngan).

1359-6454/$36.00 Ó 2009 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2009.02.025

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results are presented below. After the indentation tests, the fine structures of the resultant indents were investigated by scanning electron microscopy (SEM, LEO 1530 FEG microscope). Cross-sectional subsurface microstructure underneath an indent was investigated by focused ion beam (FIB) milling in an FEI Quanta 3D FIB system. Five repeats were carried out for each of the loading rates of 0.5, 5 and 50 mN min1 in the plan-view analysis, and three repeats were done for each of the cross-sectional analyses. The results were highly reproducible. 3. Results

Fig. 1. Schematic showing the structure of the AAO.

phosphochromic acid solution (1.5 wt.% H2CrO4 in 6 wt.% H3PO4), followed by performing a second anodization step on the now anodic-oxide free Al substrates under the same anodizing conditions for 10 h to form a better AAO layer. The structure of the prepared anodic-oxide films was revealed by X-ray diffraction (XRD) in a Siemens Bruker AXS D8 Advance X-ray diffractometer, as well as by electron diffraction in a Tecnai 20 transmission electron microscope. In order to prepare the transmission electron microscopy (TEM) sample, two AAO films were glued face-to-face together and the assembly was then glued onto a copper disc with a 300 lm hole for the subsequent ion milling process. The assembly was then polished to the micron level, and final perforation was achieved by argon ion beam milling at an angle of 7° with respect to the surface. Nanoindentation experiments were then conducted using a CSEM Nanohardness Tester for tests requiring higher loads (i.e. >10 mN), or a Hysitron triboscope nanoindenter for tests requiring lower loads (i.e <10 mN). Both machines are load-controlled machines without a feedback loop to control the indenter displacement, so that the test conditions are ‘‘quasi-static”. Berkovich tips were used in both machines. Unless stated otherwise, the loading profile used involved loading the sample at a constant rate until reaching the preset maximum load. Several values of the maximum load, namely 80, 60, 40, 20, 10, 6, 3, 1, 0.5, 0.1 mN, were used at a fixed loading rate of 1.5 mN min1. In addition, different loading rates of 0.5, 5 and 50 mN min1 up to the same maximum load of 120 mN were used. After the load ramp, the load was programmed to maintain at the peak value for 1 min, and was then reduced to 10% of the preset maximum load at an unloading rate of 40 mN min1. The load was held at the 10% value for another 1 min in order to estimate the thermal drifting rate during the testing period, followed by complete unload. A few other loading profiles were used in some specific experiments and these are described as the

Fig. 2a shows the top-view SEM images of the as-prepared AAO film. It can be seen that closely packed nanopores are well aligned within randomly oriented domains a few microns wide. Closer inspection of the pores at a higher magnification (inset diagram in Fig. 2a) reveals that they are round in shape with a diameter of about 70 nm, and are packed together into a close-packed honeycomb structure with a planar density of typically about 5  1013 m2. These planar pores are parallel aligned tubular channels which, after 10 h of anodization in the second step, extend

Fig. 2. (a) Top-view and cross-sectional (bottom Al substrate removed) (b) SEM images showing the morphology of the as-prepared AAO film after 10 h duration of second step anodization in 0.3 M oxalic acid.

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Fig. 3. Typical load–displacement curve at different loading rates.

from the film surface to the Al substrate in the normal direction, as indicated in the cross-sectional SEM image in Fig. 2b. The AAO film used in our study was measured to be about 100 lm in thickness which is more than 10 times the nanoindentation depth in all our testing schemes. XRD results of the fabricated AAO film (not shown) indicate no distinctive peak belonging to crystalline aluminum oxide, implying that the structure of the AAO was amorphous in nature. Electron diffraction in the transmission electron microscope also confirmed the AAO structure to be amorphous. Fig. 3 shows some typical indentation curves at different loading rates. It can be seen that the deformation of the AAO sample is not smooth but is rather a discontinuous process decorated with rather large displacement steps. At applied loads <10 mN, these steps are not obvious as shown by the inset diagram in Fig. 3, which shows the result at the loading rate of 0.5 mN min1. However, at higher loads at small loading rates, the steps become clearly distinguishable from the background noise, and their occurrence is rather periodic. The mean step sizes (±SD) are 100 ± 27 and 83 ± 12 nm at 0.5 and 5 mN min1 respectively, i.e. the steps at 5 mN min1 were smaller in size than those at 0.5 mN min1, although their occurrence frequency was similar in both cases. At the highest loading rate of 50 mN min1, the individual steps were no longer

clearly definable so they were either very small or no longer occurred. It should be noted that the load–displacement data in Fig. 3 are presented as discrete symbol points corresponding to the real-time data acquisition rate of 10 points per second during the experiments, i.e. no line is drawn to link up the data points, and the apparent continuous nature of the plots is due to the fact that the individual symbols for the data points overlap. Thus, the apparent absence of excursions at the highest loading rate of 50 mN min1 is a true behavior, rather than an artifact due to the data acquisition rate being slower than the excursion occurrence frequency. Table 1 shows the hardness of the AAO film measured using the three loading rates with a peak load of 120 mN. The Oliver–Pharr method in the standard depth-sensing indentation literature [25] assumes the specimen to be a solid continuum, which is not the case for the present nanochannel-structured AAO. Hardness, on the other hand, is defined as the indentation load divided by the (true) contact area of the indent, and so for the present material, the actual solid area of the indent as measured from the SEM image of the indent, not counting the areas of the pores, should be used. The solid area ratio of the present AAO samples was estimated from the SEM images to be 55.4%, and so Table 1 shows that the apparent hardness referred to the gross indent area (i.e. the indent area counting both the pores and the solid) is 55.4% of the true hardness referred to the true solid area. The true hardness is in fact independent of the loading rate as shown. Also shown in Table 1 are the hardness values calculated from the Oliver–Pharr method (using the nanoindenter’s software). It can be seen that the data fall between the two estimates of the hardness measured by SEM imaging. Alternatively, the apparent contact areas under full load calculated from the Oliver–Pharr hardness values (i.e. area = load/hardness, also shown in Table 1) are smaller than the gross areas of the residual indents (i.e. the areas including both solid and pores) as measured by SEM. This is understandable as the Oliver–Pharr hardness is a measure of the solid’s resistance to indentation plasticity, and so the effective indent area calculated from it should reflect the solid’s presence rather than the gross indent area including also the pores. Nevertheless, the Oliver–Pharr hardness values are somewhat closer to the apparent hardness referred to

Table 1 Comparison of hardness measured by direct SEM imaging of the true indent area and by the Oliver–Pharr method. Applied max load

120 mN

Applied loading rate (mN min1)

0.5

5

50

Apparent hardness (MPa) (measured by Oliver–Pharr method) Apparent contact area (load/ apparent hardness) Apparent hardness (MPa) (measured from the gross area of the residual indent as imaged by SEM)

162 ± 9 (743 ± 41 lm2) 120 ± 5 (998 ± 41 lm2) 217 ± 9 (553 ± 23 lm2) 0.584 ± 0.005

146 ± 14 (829 ± 79 lm2) 118 ± 3 (1013 ± 30 lm2) 214 ± 6 (561 ± 16 lm2) 0.492 ± 0.010

138 ± 6 (871 ± 38 lm2) 124 ± 4 (964 ± 32 lm2) 225 ± 7 (534 ± 17 lm2) 0.404 ± 0.025

True hardness (MPa), (measured from the true solid area of the residual indent, with solid area ratio of 55.4%, as imaged by SEM) Unloading stiffness (mN nm1)

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the gross area of the indent, rather than to the true hardness referred to the solid area of the indent. This indicates that although the Oliver–Pharr method is problematic in the present nanoporous material, it fortuitously can produce a hardness estimate which is quite close to the apparent hardness, ignoring the fact that the structure is indeed nanoporous. It should be noted that the Oliver–Pharr results reported in Table 1 are free of influence from creep. Creep during the unloading process is known to give rise to overestimation of the unloading stiffness using the Oliver– Pharr method [26]. However, as the holding at the peak load before unloading was long enough (1 min), the typical creep factors [26] for the three loading rates of 0.5, 5 and 50 mN min1 were 0.02 ± 0.01, 0.06 ± 0.01 and 0.09 ± 0.03, respectively. At such small creep factors, the effects of creep on the Oliver–Pharr estimates are negligible [26]. Fig. 4a–d shows top-view, high-resolution SEM images of the indents after the indentation experiments. In general, the edges of the indent are observed to be very sharp and straight, without any noticeable evidence of pile-up or sink-in which would produce curved indent boundaries. Another feature, shown in Fig. 4a, is that two systems of cracks can usually be observed: one system has three straight cracks running along the three median lines of

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the triangular indent, meeting at the indent center, and the other system has three almost bilinear cracks, one in each of the three sloping faces of the indent. For the sake of convenience, these two systems are referred to hereafter as the median system and the bilinear system, respectively. An interesting aspect of the bilinear system is that, as shown in Fig. 4b, the bilinear cracks always run close to the angular bisector lines of the two 30° apexes of each sloping face of the indent, irrespective of the maximum indent load or size. In other words, the bilinear crack system is always self-similar. Fig 4c is a high-magnification SEM micrograph of a portion of a bilinear crack in a Berkovich indent. On the side of the crack that is closer to the indent center, rod-like structures, which appear to be the longitudinal fragments of the fractured walls of the original honeycomb nanostructure, can be seen to lie flat on the indent surface, whereas on the other side of the crack, the honeycomb structure is much less perturbed with the honeycomb walls apparently still partially upright. Furthermore, in the zone closer to the indent center, the collapsed rod-like fragments can be seen to have fallen in a direction pointing towards the indent center. Such a feature on either side of the crack extends to the ends of the bilinear cracks at the indent corners, as shown in the inset in Fig. 4a. The insets in Fig. 4d show the enlarged views of

Fig. 4. (a) Top-view SEM images of a typical Berkovich indent revealing two systems of cracks. Maximum load 120 mN, loading rate 1.5 mN min1 and unloading rate 40 mN min1. (b) Comparison of Berkovich indents made at increasing peak loads with a loading rate of 1.5 mN min1, showing the selfsimilar nature of the bilinear crack system. (c) High-magnification SEM micrograph of a portion of a bilinear crack in a Berkovich indent. Maximum load 120 mN, loading rate 0.5 mN min1 and unloading rate 40 mN min1. (d) Magnified views of the cracked area at loading rates of 0.5, 5 and 50 mN min1, respectively, to a maximum load of 120 mN.

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the cracks in the indents obtained from different loading rates. It can be seen that as the loading rate increases, the top-view opening of the bilinear cracks becomes significantly smaller, whereas that of the median cracks remains more or less the same—the median crack width was measured to be 0.15 ± 0.12, 0.18 ± 0.10 and 0.16 ± 0.10 lm at the three loading rates of 0.5, 5 and 50 mN min1, respectively, and so there is no systematic trend with the loading rate. Recalling that the load–displacement curve is less discontinuous at high loading rates (Fig. 3), the displacement steps in the load–displacement behavior are therefore likely to be associated with the bilinear cracks. This will be explored further below. To further study the deformed microstructure of the AAO, cross-sectional SEM examination of the indents was performed. To do this, a trench was milled out by FIB so that one edge of it ran across the indent center, and the exposed cross-section was examined by SEM. The exposed cross-section was chosen to be either perpendicular to an edge of the indent, or parallel to it. However, both orientations of the cross-section were found to reveal more or less the same information about the cracks on the SEM level, and so the choice does not really matter. Fig. 5a is a typical SEM micrograph showing the cross-sectional morphology of an indent produced by a maximum load of 120 mN with a loading rate of 0.5 mN min1. The top layer of white contrast 0.3–0.5 lm thick was a layer of tungsten applied before the ion milling process to protect the material underneath from being damaged by the milling. It can be seen that the median crack is vertical and straight, and extends deep into subsurface along the indent axis. On the other hand, the bilinear crack is actually an oblique crack making an angle of about 70° with respect to the indent’s vertical axis, and it extends only to a finite subsurface depth of a few micrometers, terminating at what appears to be a very sharp elastoplastic boundary. The two inset diagrams in Fig. 5a show the deformed zone and its boundary in a clearer manner. It can be seen that below the elastoplastic boundary, the honeycomb walls are still intact and remain vertical, while above this boundary, the honeycomb structure is severely deformed. In the deformed zone above the sharp boundary, a line-like contrast is still visible, and this probably corresponds to the remains or fragments of the fractured honeycomb walls. In the deformed zone closer to the indent center, the line-like features are rather broken and perturbed, but they tend to align along the sloping surface of the indent, corresponding well to the plan-view observation in Fig. 4c. Because the honeycomb-wall fragments now lie flat, intense shearing and height reduction must have occurred. Close examination of the cross-sectional morphology reveals that such a zone extends horizontally from the indent center to the position of the bilinear crack, and interestingly, this zone leaves the surface and becomes a subsurface layer on the other side of the bilinear crack. On the side of the bilinear crack away from the indent center, the line-like features in the cross-sectional view adopt a curved morphology, so

that in an immediate subsurface layer they are still upright with an appearance not very dissimilar to the undeformed zone underneath the elastoplastic boundary (see top-right inset of Fig. 5a), but bending with increasing depth to join the shear zone which extends from the indent center. The observed deformed microstructure is schematically illustrated in Fig. 5b. The fact that the immediate subsurface line-like features on the side of the bilinear crack far from the indent center are still upright corresponds well to the plan-view observation in Fig. 4c. Fig. 4d shows that, viewed from above, the opening of the bilinear cracks becomes smaller at higher loading rates. The same can also be observed from the cross-sectional morphology, as shown in Fig. 5c. Evidently, the width of the bilinear crack in the cross-sectional view becomes smaller as the loading rate increases. On increasing the loading rate, the opening of the bilinear cracks diminishes, and the load–displacement behavior becomes less discontinuous. As mentioned above, these two observations suggest that the bilinear cracks are formed during the loading process, rather than during unloading as a result of, for example, stress relaxation. Two further pieces of evidence support this conclusion. First, additional nanoindentation tests were done using two special loading profiles with very different unloading processes. In the first profile, the sample was first loaded to the maximum load at 120 mN at 5 mN min1 and was then sequentially stepwise unloaded by one-fourth of the maximum load after each 25 min of load-hold period, until it is fully unloaded. In the second profile, the sample was loaded to the same maximum load at 120 mN at 5 mN min1, and was then unloaded at a slow unloading rate of 0.5 mN min1. To save space, the corresponding data are not shown, but no observable difference in the top-view and cross-sectional morphologies of the resultant indents can be noticed, compared to the indents made using the original loading scheme at the same maximum load and loading rate, and an unloading rate of 40 mN min1. This suggests that the formation of the crack structure is insensitive to the unloading process. Secondly, if the bilinear cracks are formed during the loading process, and given that they are self-similar with respect to the indent size (Fig. 4b), they then need to move in a perpendicular direction with increasing load. This is possible by considering the following sequence of events. During a loading process, when the load is still small, a small crack pattern may be formed, which may correspond to the generation of a step in the load–displacement response. When the load increases to a higher level, a new crack pattern maintaining the self-similarity with the now enlarged indent needs to be formed in new positions that are further away from the indent center, and a new step may also be formed in the load–displacement response. At the new increased load, in addition to the newly formed crack pattern, traces of the old crack pattern should still be present in the deformed microstructure. Close examination of the cross-sectional morphology of the indents reveals some

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Fig. 5. (a) Cross-section SEM images of a typical indent at 120 mN at 0.5 mN min1. The insets show an enlarged portion of the crack and the region close to the indent edge. (b) Schematic of the observed deformation microstructure. Enlarged cross-section images of the bilinear crack made at different loading rates to the same maximum load of 120 mN at c1: 0.5 mN min1 c2: 5 mN min1 and c3: 50 mN min1.

possible traces of these old cracks—for example, a few can be seen on the right side of the current crack in the lower inset in Fig. 5a. To increase the visibility of these old cracks, their opening should be increased, and it was thought that a short period of holding at some intermediate load values might achieve this. Thus, further nanoindentation experiments with a multicycle loading profile were conducted. In these experiments, the sample was first loaded to 30 mN at a rate of 1.5 mN min1 followed by a load-hold period and then complete unloading. After that, the loading cycle was repeated at higher peak loads of 60, 90 and 120 mN, respectively. The resultant cross-sectional morphology is shown in Fig. 6a, in which traces of three

old cracks, in addition to the largest crack which should correspond to the final load cycle, are clearly visible. The four cracks correspond to the four peak loads at 30, 60, 90 and 120 mN, and a plot of these loads vs. the square of the crack distances from the indent center (Fig. 6b) is approximately linear, in agreement with the self-similar nature of the bilinear crack pattern. Five repeats were carried out for each of the loading rates of 0.5, 5 and 50 mN min1 in the plan-view analysis, and three repeats were done for each of the cross-sectional analyses. The deformed microstructure was always similar, and the trend of the reduction in the bilinear crack opening with increasing loading rate was always consistent. The

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4. Discussion 4.1. Deformation microstructure

Fig. 6. (a) Cross-section of an indent produced after a multicycle load schedule revealing the major crack and the traces of three previous cracks. (b) Load vs. square of crack distance from indent center, for the four cracks in (a).

bilinear crack, the median crack and the deformed microstructure on both sides of the bilinear crack were always similar (about 50 indents were studied in this part). The discontinuous behavior was also very reproducible in most cases. The deformed microstructure of indents made at ultralow loads (100 lN–3 mN) was also studied, but the results are not shown here due to the lack of space. At such small deformations, no crack pattern or clear collapse/tilting of the column walls can be seen—the plan-view morphology of the indent appears like that of the more mildly deformed region on the left side of the crack in Fig. 4c. Cross-sectional examination again shows no cracking or collapse of the column walls. The indent morphology at such small loads is no longer self-similar to that at larger loads described above. Finally, the steps in the load–displacement curve in Fig. 3 are unlikely to relate to the median crack system since the opening of the latter is not sensitive to loading rate, as shown in Fig. 4d, though the occurrence of the jumps is.

One striking feature of the deformed microstructure is a very sharp elastoplastic boundary. Closer inspection of the plan-view SEM micrographs near the indent edges (see, for example, the inset in Fig. 4a) shows that the resolution of the elastoplastic boundary in this material could be up to the pore diameter of the AAO channel, i.e. around 70 nm in this case. This is a unique feature compared to the indents in continuum materials which usually exhibit curved edges corresponding to pile-up or sink-in events. The cross-sectional micrographs in Fig. 5 also reveal the same clear-cut mode of deformation underneath the indent—within the indent boundary, plastic bending of the tubular structure is seen up to a curved subsurface elastoplastic boundary, also sharply defined, and outside the indent boundary in the direction parallel to the specimen surface, no obvious sign of residual deformation can be seen in the tubular structure. This unique deformation mode is due to the discontinuous structure of the AAO, which allows quite sharp strain gradients to occur in the structure, making propagation of deformation through the structure inefficient. The large free volumes in the porous structure also mean that as a tube wall bends or buckles, the fragments can be accommodated by the free space around it and so an adjacent wall at a free spacing away is not greatly affected. The inset in Fig. 4a shows this very clearly. In the previous section, we observed the self-similar nature of the extension of the indents under higher loads. Moreover, in a cross-sectional profile, such as those in Fig. 5, a region closer to the indent center has a more severely deformed microstructure than a region nearer to the edge of the indent, i.e. the change in microstructure from the indent’s edge to its center should show progressive development of the deformation of a given material volume along its strain path during the indentation process. As illustrated in Fig. 5a and b, a deformed region near the outermost edge of the indent, which represents the earliest stage of deformation (position A in Fig. 5b), has its tubular structure only mildly bent, because the strain is small. Further towards the indent center, but before the position of the bilinear crack (position B), where the microstructure corresponds to a bigger accumulated strain, severe collapse of the tubular structure by shear tilting has occurred locally at a subsurface depth that corresponds to a heavily compressed subsurface layer. This layer apparently serves to accommodate the larger accumulated strain as compared to the earliest stage of deformation. On moving towards the bilinear crack, corresponding to further increase in accumulated strain, the compressed layer thickens at the consumption of the mildly deformed layer above, which is still compressible by tilting of the tubular structure at its base. The bilinear crack (position C) seems to occur at the location where the mildly deformed zone just vanishes

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in length, presumably because the stress there is too high as a result of a sudden change in microstructure. The loadingrate sensitivity of the occurrence of the bilinear cracks suggests that the rupture of the AAO material to form a crack is likely to be a viscous or rate-determining process as well. On the other side of the crack towards the indent center, which corresponds to even larger accumulative strains (position D), the already tilted structure is further densified by full collapsing of the tubular walls, so that these now lie flat along the indent surface, corresponding to the experimental observation in Fig. 4c. The subsurface deformed zone is in fact rather similar to shear bands frequently observed in deformed macroscopic honeycomb structures [27], despite the fact that these latter experiments were usually carried out under plane stress conditions with the honeycomb axis perpendicular to the normal stress directions, while the current study involved stress along the axis direction. Another interesting feature of the deformation behavior of the AAOs is that their load–displacement behavior is discontinuous as shown in Fig. 3. The steps in the load–displacement curve become smaller at higher loading rates, but their occurrence frequency is rather periodic and loading-rate insensitive. The period of occurrence of these steps, as seen from Fig. 3, is of the order of 100 nm, and this is similar to the channel diameter of the AAO structure (see inset in Fig. 2a). The steps are thus likely to be related to the sequential rupture of individual cellular walls. As discussed above, increasing the loading rate decreases both the opening of the bilinear cracks (Fig. 4d), as well as the size of the steps in the load–displacement curve (Fig. 3), suggesting that the steps in the load–displacement are probably the direct consequence of the sideways propagation of the bilinear cracks as load increases. Hence, a likely scenario is that each time a new bilinear crack develops at a distance of one channel away from an already formed

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crack, local rows of cellular walls will have to rupture to accommodate the new crack, and this rupture event may correspond to a step in the load–displacement graph. However, the dependence of the crack opening (Fig. 4d) and the step size (Fig. 3) on the loading rate seems to suggest that the discreteness of the collapse of the columns depends on the strain rate. At low strain rates, the columns seem to collapse in a more synchronized fashion, leading to bigger steps in the load–displacement curve and more clear-cut opening of the bilinear crack, but at a fast strain rates, the collapse of the columns seems to be less synchronized. More evidence for a correlation between bilinear crack opening and strain rate can be seen by comparing the 15 mN indent in Fig. 4b made at 1.5 mN min1, and the 120 mN indent in Fig. 4d made at 50 mN min1. The strain rate during indentation is approximately given by e_  P_ =P , where P is the load and P_ is the loading rate [28], and these two indents had comparable final strain rates (0.1 and 0.4 min1, respectively). The bilinear crack openings measured from these indents are also comparable at 0.3 ± 0.1 and 0.4 ± 0.2 lm, respectively, despite the large difference in the indent sizes. It was mentioned above that indents were also made at very small loads (100 lN–3 mN), and no discernible crack systems were observed. Some of these were made using a lower loading rate, e.g. 0.2 mN min1 to a maximum load of 1 mN, giving a final strain rate of 0.2 min1, which is comparable the final strain rate in the large indents studied, e.g. 0.4 min1 for the 120 mN indent at 50 mN min1 in Figs. 4d and 5c. The fact that the low-load indents show no discernible cracks but the large indents do, even at comparable strain rates, indicates that the absence of cracks at very low loads is not a pure effect of strain rate. At 1 mN, the apparent hardness measured by SEM imaging of the indent area is about 165 ± 5 MPa, and this is significantly higher than the value of 113 ± 2 MPa measured at

Fig. 7. A simple column model of deformation. (a) A column of orthotropic material with length L in the pre-deformed state. (b) After compression by distance d. (c) Further compaction of the tilted zone. (d) Details of tilted zone showing relative sliding between two periodic units.

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15 mN, and the 120 ± 5 MPa at 120 mN (see Table 1). The AAO structure has an internal length scale due to its pore size and so even though the indenter may be self-similar, this may no longer be the case for the deformation. However, it must be remembered that in the present constant loading-rate experiments, the strain rate is initially large when the load is small, and increases when the load becomes large. To make definite conclusions about strain-rate effects, constant strain-rate experiments, corresponding to an exponential load profile, should be performed in the future. The present results open up new perspectives for research on using depth-sensing indentation techniques to study porous materials. In the literature, depth-sensing indentation is frequently used in materials which are not continuous but are porous, e.g. trabecular bone samples [29–32]. Our present results show that the application of the Oliver–Pharr data analysis protocol is uncertain in this situation—Table 1 shows that the hardness measured by the Oliver-Pharr protocol exhibits an apparent decreasing trend as loading rate increases, but in fact the true hardness is rather constant. Also, the localized nature of deformation in these materials means that the deformation field underneath the indenter is significantly different from predictions using continuum elastoplasticity theories [33], so that new data analysis tools that yield hardness and elastic modulus need to be developed for these porous materials. The highly localized deformation mode of these porous materials also means that they can be very effective shielding materials to protect an underlying material from receiving plastic damage.

The present AAO is a highly orthotropic structure which can be expected to be much stiffer and stronger along the axial direction of the channels than along an orthogonal direction. To shed some light on the observed deformation microstructure, let us consider a simple model, shown in Fig. 7a, which depicts a column of orthotropic material with length L in the pre-deformed state. This column is now compressed by a distance d as shown in Fig. 7b. The actual indentation geometry is to have d changing in the transverse direction perpendicular to the compression direction, but the resultant strain gradient effects between columns are ignored in this simple analysis. 4.2.1. Case 1: tilting Consider that a layer with initial thickness z undergoes an instability event leading to rotation by angle h, whereas the remaining length of the column undergoes elastic compression. The strain energy associated with the compression per unit cross-sectional area of the column is given by: E½d  zð1  cos hÞ2 þ W ðh; zÞ; 2ðL  zÞ

ee ¼

d  zð1  cos hÞ Wh ¼ zE sin h Lz

ð1Þ

ð2Þ

and Ee2e þ W z ¼ 0: ð3Þ 2 In Eqs. (2) and (3), Wh = oW/oh and Wz = oW/oz, and ee is the elastic strain of the length (L  z). Since ee << 1 , an accurate enough solution for it from Eq. (3) is ee = Wz/[E(1  cos h)], and from Eq. (2), we have: Eee ð1  cos hÞ þ

h zW z tan ¼ 2 Wh

ð4Þ

and z d=L  W z =½Eð1  cos hÞ : ¼ L 1  cos h  W z =½Eð1  cos hÞ

ð5Þ

To obtain a rough estimate of the work function W, assume that tilting of the region z is restrained mainly by a constant frictional shear stress sc acting between longitudinal periodic units as shown in Fig. 7d. The distance slid by sc during tilting is s = d tan h, and so the dissipative work for one wall is (sczb)(d tan h), where b and d are the thickness and width, respectively, of the periodic unit. Since there are 1/(bd) periodic units per unit cross-section of the column, the total dissipative work per unit area is: W ¼ sc z tan h þ W 0 ;

4.2. A simple column model

U1 ¼

where the first term is the elastic energy of the column portion with length (L  z), and W is the plastic work done of the tilted region, assumed to be an increasing function of h and z. The equilibrium condition is given by oU1/oh = 0 and oU1/oz = 0, yielding, respectively

ð6Þ

where W0 represents a constant dissipative energy, such as the fracture energy required before tilting. With Eq. (6), Wh = oW/oh = sczsec2 h and Wz = oW/oz = sctan h, and so Eq. (4) becomes sin h cos h = tan (h/2), giving h  52. Fig. 5a reveals the residual thickness zr = zcos h of the tilted zone, so we are interested in this quantity. Being a typical fracture stress, sc should be much smaller than E, and so, from Eq. (5), zr  1.6d. Close inspection of the tilted zone between the edge of the indent and the crack in Fig. 5a and b reveals that the tubular structure within it is tilted by an angle of 50–60° with respect to the vertical, and so the present prediction of 52° is good. In this regime, the residual thickness of the tilted zone also increases towards the crack position, i.e. along the direction of increment of d, as predicted. From Fig. 5a, zr/d  1.2 ± 0.2, which is close to the value of 1.6 predicted. Considering the simplicity of the model, the discrepancy is acceptable. 4.2.2. Case 2: full compaction Case 1 above cannot explain the densely compressed zone between the crack and the indent center in Fig. 5a and b, the thickness of which decreases instead of increases

K.Y. Ng et al. / Acta Materialia 57 (2009) 2710–2720

with d, as predicted by Eq. (5). It is thought that when the tilted zone reaches a certain thickness, it becomes energetically unfavorable (see later) for it to further thicken compared to densification into a compact zone, as shown in Fig. 7c. Let zc be the thickness of this critical tilted zone. As this zone undergoes the full compaction process, the energy of the whole column in Fig. 7(c), compared to the initial state in Fig. 7a, is: 2

U2 ¼

Eðd  zc ep Þ þ W p ðep Þ þ W ð52 ; zc Þ; 2ðL  zc Þ

ð7Þ

where the first term is the elastic energy of the column with length (L  zc), Wp is the plastic energy of the thickness zc which undergoes a compaction strain of ep, and W(52°, zc) is the work dissipated in the tilted zone before compaction. Equilibrium is reached when dU2/dep = 0, giving the elastic strain of the length (L  zc) as: ee ¼

d  zc ep dW p =dep : ¼ L  zc zc E

ð8Þ

Assuming that the plastic collapse of thickness zc takes place at a constant compressive stress rc, W p ðep Þ ¼ rc ep zc ;

ð9Þ

and so from Eq. (8):   d L rc : ep ¼ þ 1  zc zc E

ð10Þ

The plastic strain is therefore predicted to grow with d, and in particular, the residual thickness of the compressed zone is: zr ¼ zc ð1  ep Þ ¼ zc þ ðL  zc Þ

rc  d: E

ignores gradient effects, and cannot predict the depth of the tilted zone below the indent surface. Moreover, the bilinear crack is assumed to exert no effects. All these detailed features will have to be dealt with by more rigorous modeling efforts in the future. 5. Conclusion Nanoindentation experiments along the nanochannel directions of anodic aluminum oxide revealed that the load–displacement behavior was discontinuous with periodic displacement excursions. Top-view and cross-sectional microscopic examination revealed that the deformation mode was highly localized and accompanied by very sharp elastoplastic boundaries. A crack system which was self-similar with respect to the indent size was also observed, and the propagation of the cracks involved is thought to be the reason for the discontinuous load–displacement behavior. A column model was proposed to analyze the strain energies associated with a tilted zone and a fully compacted zone in an elastic column. By assuming suitable work functions for the tilting and compaction, the model predicts that tilting is more energetically favorable than full compaction when the overall column compression is small, and vice versa when compression is large. When tilting is favored, this model predicts the tilted zone to thicken with respect to increase in column compression, and when compaction is favored, the model predicts that the compacted zone thickness will decrease with increasing compression. These features are in qualitative agreement with the observed deformed microstructure of the indents.

ð11Þ Acknowledgment

Eq. (11) predicts that the thickness of the fully compressed zone decreases with increasing d with slope 1, and this is in excellent agreement with the observation in Fig. 5a that the elastoplastic boundary between the crack and the indent center is almost a straight, horizontal line, implying that zr = (constant—d). The transition from tilting at 52° (Case 1) to full compaction (Case 2) happens when U2 < U1, where U1 and U2 are the minimized energies calculated from Eqs. (1) and (7), respectively. It can be shown that this transition takes place when d is larger than some critical value dc, given by: dc 5:605s2c þ 0:5r2c  : L rc E

2719

ð12Þ

Since sc and rc are both a lot smaller than E, dc is a very small fraction of L. The present experimental value of dc/ L is (2.5 lm/106 lm) = 0.02. Therefore, sc/E and rc/E are both of the order of 102, which is not unreasonable. The present model is therefore able to predict certain geometrical features of the observed deformation microstructure of AAOs. As mentioned above, this simple model

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