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Remnant grooves on alumina surfaces
Surface Science 573 (2004) 391–402 www.elsevier.com/locate/susc
Remnant grooves on alumina surfaces N.E. Munoz, S.R. Gilliss, C.B. Carter
*
Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE, Minneapolis, MN 55455-01432, USA Received 17 May 2004; accepted for publication 6 October 2004 Available online 19 October 2004
Abstract Thermal grooving of grain boundaries in alumina has been studied using a combination of visible-light microscopy (VLM) and atomic-force microscopy (AFM). The partial angles of grooves that develop during heat treatments at 1650 C were estimated by AFM. The observation of remnant grooves is key to determining the magnitude of the movement of a grain boundary. Imaging using VLM and AFM enabled the smoothing of remnant grooves to be observed through a progression of heat treatments, which enables the anisotropic nature of the surface diffusivity of alumina to be monitored. 2004 Elsevier B.V. All rights reserved. Keywords: Grain boundaries; Diffusion and migration; Surface diffusion; Growth; Aluminum oxide; Atomic force microscopy
1. Introduction The scaling down of electronic devices is limited, in part, by the ability to obtain smooth surfaces. Surface roughness is not only important for mechanical reasons but it can also affect the chemical stability of a component. Similarly, biomaterials that must be able to withstand the hostile environment inside the body may benefit from a low surface roughness in order to minimize the sur*
Corresponding author. Tel.: +1 612 625 8805; fax: +1 612 626 7246. E-mail address: [email protected] (C.B. Carter).
face area that can be chemically attacked [1,2]. Surface changes during post-processing heat treatments can alter the topography of a surface, which can have a significant impact on material performance. Two such thermally activated phenomena are surface faceting and grain-boundary grooving [3,4]. Crystal surfaces facet to expose lower energy surfaces, which more than compensates for the increase in surface area. The facet structures that develop depend on the material and crystallographic orientation of the surface [3,5]. Grain-boundary grooving modifies the topography of a surface through the formation of channels along the triple junctions where the grain boundaries intersect the
0039-6028/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2004.10.006
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surface. Understanding the mechanisms through which these phenomena occur is important to developing methods for processing ceramics with a specific microstructure. This paper presents a method for studying the evolution of the surface microstructure and grain-boundary grooves. The development of surface features through a progression of heat treatments was studied using a non-destructive technique involving visible-light microscopy (VLM) and atomic-force microscopy (AFM). Brinson and Moore [6] carried out a related study involving heating of zinc specimens in situ on the stage of a VLM in order to observe the nucleation and growth of new crystals. In the present study, by using AFM in conjunction with VLM, the same regions and grooves could be imaged and measured in 3D through a progression of heat treatments [7,8]. The monitoring of remnant grooves enables stationary boundaries to be distinguished from migrating grain boundaries. These remnant grooves also provide a means by which the surface diffusivity of varying grain orientations can be monitored.
2.2. Grooving by surface diffusion The evolution of a thermal-grain-boundary groove occurs by mass transport through a combination of surface diffusion, evaporation–condensation, and volume diffusion [9,10]. MullinsÕ theory of thermal grooving by surface diffusion describes the grooving process at temperatures well below the melting point in many ceramics and metals. The theory predicts two surface maxima, one on each side of the groove. These ridges have been observed experimentally in W [11,12], Fe [13], Pd [14], NiAl [15], Al2O3 [16–18], ZrO2 [19], SrTiO3 [20], and CeO2 [21]. MullinsÕ theory describes the development of a thermal-grain-boundary groove by surface diffusion and allows surface diffusion coefficients to be deduced from the geometries of grain-boundary grooves: x ¼ 4:6ðbtÞ1=4 d ¼ 0:973mðbtÞ
ð2Þ 1=4
ð3Þ
2. Background
The groove width (the distance between the surface maxima), x, and the depth of the groove (vertical distance from the surface maxima to the base), d, are each a function of time, and b is given by
2.1. The energetics of grooving
b¼
Thermal grooving occurs when a grain boundary intersects the surface. In the classical view, the topology of the three interfaces in the vicinity of this triple junction is determined by the balance of the interfacial tensions [4]. The three interfaces intersecting at the groove are the grain boundary and the two solid–vapor interfaces. Herring [4] originally applied the concept of interfacial tension to describe this equilibrium geometry:
In Eqs. (2)–(4), t is the duration of heat treatment, m is the slope of the surface at the triple junction (the base of the groove), dDs is the product of the thickness of the layer, d, in which surface diffusion occurs and the surface diffusion coefficient, Ds, respectively; X is the atomic volume of the diffusion species; cs is the free energy of the solid–gas interface; k is BoltzmannÕs constant and T is the absolute temperature. The surface diffusion coefficient of the material can thus be calculated from MullinsÕ theory using Eqs. (2) and (3). The evolution of groove geometries described by MullinsÕ theory [9] assumes that surface diffusion is driven only by capillary forces; i.e., that the grain boundary plays no role in the transport of material. Furthermore, the model assumes that the surface energy is isotropic and the grain boundary is normal to the original surface. The
3 X oc c i t i þ ni i ¼ 0 ohi i¼1
ð1Þ
The interfacial tensions, ci, of the respective surfaces are linked to the terms describing their orientation dependence, oci/ohi (the torque terms for each surface); the vector ni is defined as ti · l where l is a vector along the triple junction.
dDs Xcs kT
ð4Þ
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implications are that the surface diffusivity must be isotropic, the groove must be symmetric and the ridges on each side of the groove must be of equal height. The surface ridges are indicated in an AFM profile of a grain-boundary groove in Al2O3 in Fig. 1(a). If surface diffusion is the only grooving mechanism, then in this cross-sectional view the area below the original surface in the groove should equal the area above the original surface; i.e., the area of the ridges. Eq. (1) only describes the equilibrium condition at point G in Fig. 1(a); the ridges may be in a condition of local non-equilibrium. Consequently, the orientations specified by Eq. (1) are those at point G in Fig. 1(a), which is referred to as the groove base. If the ridge material continued to diffuse away from the groove or if material evaporated during groove formation, then a notch profile, more similar to that shown in Fig. 1(b) for a chemically etched polycrystalline material [22], is expected. The AFM profile in Fig. 1(b) is of a notch at a grain boundary in Al2O3 that has been etched in hot phosphoric acid.
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2.3. Thermal-grain-boundary groove angles The measurement of grain-boundary groove angles is perhaps the most widely used method for determining the relative interfacial energies of a material. This type of analysis involves polishing the surface of a specimen in order to obtain an initially flat surface, then heat treating until grooves form. The calculation of relative interfacial energies from Eq. (1) assumes that the groove has reached its equilibrium shape. For liquids, the surface energy is equal to the surface tension. However, this condition is not necessarily the case for crystalline materials except at equilibrium [23]. Since Eq. (1) describes an equilibrated geometry, the interfacial tension terms are then taken to be equivalent to interfacial energies. In addition to assuming an equilibrated system, relative interfacial-energy analyses simplify the calculations associated with Eq. (1) by making the following assumptions [24–28]: (i) surface energies of the material are isotropic; (ii) the grain boundary is stationary and normal to the original surface; (iii) the grain-boundary energy is independent of the tangent plane; (iv) the torque terms, oci/ohi are negligible. Under assumptions (i) through (iv), Eq. (1) simplifies to the following relationship, which is known as YoungÕs equation: cgb ¼ 2cs cos hs
Fig. 1. AFM profiles measured at alumina surfaces. (a) Typical geometry of a thermal grain-boundary groove formed by diffusion. (b) Profile measured at a grain boundary that has been etched with hot phosphoric acid. If the ridges in (a) continued to diffuse away from the groove this type of profile is expected.
ð5Þ
Here cgb and cs are the interfacial energies of the grain boundary and the surface at the base of the groove, respectively, and hs is the partial angle of each grain. The sum of the partial angles is the dihedral angle of the groove. Fig. 2(a) is a schematic profile of a groove in which assumptions (i) through (iv) apply. Under these conditions a symmetric groove profile will form such that the partial angles of both grains are equal. The ratio of grain-boundary energy to surface energy, cgb/ cs, can be calculated by measuring the angle hs. Since surface energies are not necessarily equal, assumption i need not hold so that Eq. (1) simplifies to:
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rium for the grain-boundary configuration shown in Fig. 2(c) are
Fig. 2. Schematic groove profile at a grain boundary that is normal to the original surface in a material with (a) isotropic surface energies and (b) anisotropic surface energies. (c) Schematic groove profile that developed at a boundary that is inclined towards the original surface such that hgb 5 0.
cgb ¼ c1 cos h1 þ c2 cos h2
ð6Þ
c1 sin h1 ¼ c2 sin h2
ð7Þ
where h1 and h2 are the partial angles of the respective grain. For example, h1 is the partial angle of grain 1 in Fig. 2(b); c1 and c2 are the respective surface energies of each grain at the groove base. Eq. (6) satisfies the equilibrium condition in the direction normal to the original surface. Eq. (7) satisfies the equilibrium condition in the plane of the original surface and normal to the line of the grain boundary at the surface. At a polished surface the grain boundaries do not necessarily intersect the surface at right angles, as illustrated in Fig. 2(c). Now, only assumptions (iii) and (iv) hold so that the equations of equilib-
cgb cos hgb ¼ c1 cos h1 þ c2 cos h2
ð8Þ
cgb sin hgb ¼ c1 sin h1 c2 sin h2
ð9Þ
where hgb is the angle of inclination of the grain boundary with respect to the normal of the original surface. Eqs. (8) and (9) satisfy equilibrium requirements in the direction normal to the original surface and in the direction in the plane of the original surface and normal to the grainboundary line, respectively. Previous grooving studies have shown that the grain-boundary energy depends on the misorientation of the two grains [28,29]. For example, Greenough and King [29] fabricated sets of silver bicrystals with gradually increasing grainboundary misorientations. Their study found that bicrystals with a greater misorientation resulted in grain-boundary grooves with a smaller dihedral angle, indicating that the grain-boundary energy increases with misorientation [29]. Saylor and Rohrer [28] determined surface orientations of polycrystalline Al2O3 and MgO using electron backscattered diffraction (EBSD) patterns in the scanning electron microscope (SEM) [30] and correlated the results with the measured geometries of grain-boundary grooves. This study found that smaller misorientations or near-coincident lattices have grain boundaries of a lower energy and shallower grain-boundary grooves [28].
3. Experimental LucaloxTM, which is a high-purity polycrystalline alumina with up to 0.5 wt.% magnesia [31], was used in this study. LucaloxTM was obtained 1 in the form of a tube with an outer diameter of 8.8 mm and thickness of 1 mm. Samples approximately 2-mm square were cut from tubes using a diamond saw. The convex surface was polished flat to a 0.5-lm finish using diamond lapping film. Samples were enclosed in an alumina crucible,
1
General Electric Co., Cleveland, OH.
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which was surrounded by high-purity Al2O3 powder inside a box furnace and heated at a rate of 20 C/min to a temperature of 1650 C in air. The temperature was held at 1650 C for 30 min and the lowered at a rate of approximately 150 C/min to 1300 C and 40 C/min to room temperature. The samples underwent a total of four similar heat treatments. A VLM (Olympus BH2-UMA) map was made from the central region of the sample and the same regions were re-imaged and remeasured using AFM after each heat treatment. AFM images were recorded in contact mode and in air using Digital Instruments NanoScope III MultiMode AFM equipped with a VLM for tip positioning. Si3N4 cantilevers (Veeco NP-20) with a manufacturer-specified spring constant and tip radius of 0.32 and 20–40 nm, respectively, were used for AFM imaging. In order to avoid uncertainties in the AFM measurements due to the inherent variability of probe-tip size, a groove from a standard specimen (same material and similar processing as sample specimens) was measured with each new tip. The groove from such a ‘‘standard specimen’’ was chosen because the value of dihedral angle and groove depth remained constant along the length of the grain boundary. Only AFM tips that gave consistent measurements of the standard-groove geometry were used in this study. AFM imaging consisted of scanning macroareas as large as 70 lm · 70 lm to observe the evolution of the microstructure. The imaged areas were marked on the VLM maps to facilitate a return to the same area after each 30-min heat treatment. AFM scans having dimensions 10 lm · 10 lm and 5 lm · 5 lm with a resolution of 512 · 512 pixels were made of grain-boundary grooves within these macroregions. Since the inherent AFM error in measuring grain-boundary grooves changes with the scanning direction [32,33], the same scanning direction was used for each particular groove. The groove profiles and approximate partial angles were measured from the 5 lm · 5 lm scans. An example of the sequence of VLM and AFM imaging from the same regions through a progression of two 30-min heat treatments is shown in
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Fig. 3. The first VLM image was recorded after a heat treatment at 1650 C. The region enclosed by the square corresponds to the AFM image shown in Fig. 3(b). The dark lines in the AFM image are grain-boundary grooves and indicate where the grain boundaries intersect the surface. Fig. 3(c) shows corresponding images of the same area after re-heating the sample for a second 30min period. AFM measurements of partial angles and groove depths are reported as an average of five measurements made along the length of the groove. Partial angles are measured from the slope of the surface at the base of the groove. The groove depth is measured as the vertical distance on an AFM groove profile from the base of the groove to the original surface. The original surface is assumed to be the surface at least 3 lm away from the groove base. If the surface is faceted, then the original surface is assumed to be at the location midway between the top and bottom of the largest observed facet. Although the reported AFM images do not always show the entire grain of interest, the number of sides of a grain at the surface has been determined from the VLM map of the sample. All AFM results presented here are height-mode images.
4. Results 4.1. Factors affecting groove measurements The measurements of groove geometries were found to be sensitive to two factors: AFM-tip geometry and surface faceting. When measuring the standard groove using two different AFM tips, differences in groove depths and partial angles up to 55 nm and 20, respectively, were observed. A large AFM-tip radius will cause an underestimate of groove depth. The surface faceting of a grain often continued down the sides of a groove. When an AFM tip with a large radius was used to measure a faceted groove, the surface steps that extended to the base of the groove could greatly affect the measured depth and partial angles of the faceted groove. Fig. 4 shows the effect that an AFM tip with a large radius can have on the AFM measurements of a faceted groove.
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Fig. 3. Sequence of VLM and AFM imaging used to return to the same regions after successive heat treatments. The AFM image of the boxed region is shown in (b). The highlighted groove seen in (b) can be seen in the 5 lm · 5 lm AFM image in (c). The profile of this groove is shown in (d). After 60 min at 1650 C the same area as (a) was located by VLM (e). The corresponding AFM image of this region is (f).
The grain-boundary groove from an Al2O3 specimen that underwent a total heat treatment for 120 min at 1650 C shown in the AFM image in Fig. 4(a) separates a faceted grain (left side) from a non-faceted grain (right side). Fig. 4(b) is an AFM profile along what was measured to be the base of the groove. The facet structure from the left grain extends down the sidewall to the base of the groove. The largest surface step at the measured groove base is 40 nm. Clearly the measured depth of the groove can vary up to 40 nm depending on where the measurement is made; this is a
real variation not a measurement uncertainty. The AFM groove profiles in Fig. 4(c) and (d) show a similar effect on the measured dihedral angle of a faceted groove. The groove profiles in Fig. 4(c) and (d) were made along lines l1 and l2, respectively, in Fig. 4(a). Along l1 the dihedral angle is 140. Along l2 the dihedral angle is 100. Differences this large were only observed at faceted grooves that were measured using AFM tips that gave inconsistent measurements of the standard groove. Typical variations in depth and dihedral angle along a faceted groove were 10 nm and 20, respectively.
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Fig. 4. (a) 5 lm · 5 lm AFM image of a groove in alumina heat treated for 120 min at 1650 C. The left grain is faceted. (b) AFM profile measured along the base of the groove. The largest surface step is 40 nm. (c) AFM profile measured along line l1 in (a). The dihedral angle at this location is 140. (d) AFM profile measured along line l2 in (a). The dihedral angle at this location is 100.
4.2. Remnant grooves Remnant grain-boundary grooves were observed at the location where a migrating boundary had previously intersected the surface. The AFM images in Fig. 5(a)–(d) are of the same area after each heat treatment and show that the boundary at the dotted line migrated during each heat treatment. The remnant grain-boundary groove appears as a shadow in the AFM images and marks the location where the boundary previously intersected the surface. The AFM profiles measured along the dotted line are shown below the respective AFM image. The remnant groove corresponding to the position of the boundary at the end of each heat treatment is numbered in the profile in Fig. 5(d). For example, position ‘‘1’’ corresponds to the position of the boundary after the first heat treatment, which is shown in Fig. 5(a). The direction of grain-boundary migration can therefore be inferred from the location of the remnant grain-boundary groove relative to the new position of the boundary. A rounded bottom, unlike the cusp of grain-boundary grooves is characteristic of a remnant groove. The groove depth and the height of the surface ridges decrease over the course of heat treatments. This process, which will be referred to as smoothing of remnant grooves, is slower than the formation of new grooves. As can be seen by comparing the groove profiles in Fig. 5(a) and (b), the new groove that formed is
approximately the same depth of the groove that formed at the previous boundary position while the remnant groove has not completely healed. In fact, the geometry of the remnant groove at position ‘‘1’’ did not significantly change upon further heat treatments after 60 min. 4.3. Healing of remnant grooves The rate of smoothing of remnant grooves was found to depend on the surface orientation, as can be seen in Fig. 6. The same area is shown in Fig. 6(a)–(d) after each heat treatment in the sequence. Grains A–D are labeled in Fig. 6(a). Grain A has five sides and grain B has four sides after the first heat treatment (Fig. 6(a)). A remnant groove on grain C near boundary A/C in Fig. 6(a) indicates that this boundary migrated into grain A during the first heat treatment. Throughout the remaining heat treatments grain A exhibits more extensive faceting compared with grain B as boundary A/C migrates towards the top of the image as grain C grows, and boundary A/B moves towards the bottom of the image as grain B grows. As grains C and B grow at the expense of grain A, boundary A/D is eliminated, which is first observed after the third heat treatment in Fig. 6(c), and grain A completely disappears after the fourth heat treatment as seen in Fig. 6(d). During the progression of heat treatments, grain B grows at the expense of the surrounding grains, all of which are significantly
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Fig. 5. (a)–(d) AFM images of the same region after a total heat treatment for 30, 60, 90, and 120 min, respectively at 1650 C. The grain boundary at the dotted line in (a) migrated after each heat treatment. An AFM profile made along the dotted line is shown below the respective AFM image. The position of the boundary after each heat treatment is indicated in the profile in (d).
larger than grain B at the surface. Examination of the AFM image in Fig. 6(d) indicates that remnant groove A/D within grain B has smoothed more compared with the portion of remnant groove A/
D that remains within grain C. Remnant groove A/D is appears to be parallel to the facet structure of grain C. An AFM profile measured perpendicular to the portion of remnant groove A/D within
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Fig. 6. (a)–(d) AFM images of the same region after a total heat treatment for 30, 60, 90, and 120 min, respectively. (e) AFM profile of remnant groove A/D within grain C. The arrows indicate that the walls of the remnant groove are nearly parallel to the facet orientations on grain C.
grain C in Fig. 6(e) shows that the sidewalls of the groove are nearly parallel to the surfaces of the facets of grain C, as indicated by the arrows. It should also be noted that the faceting of grain C is more pronounced in the region which boundary A/C migrated through, which corresponds to the region to the right of the remnant-groove profile in Fig. 6(e).
5. Discussion 5.1. Uncertainties in AFM measurements Uncertainties associated with using AFM to measure concave features, mean the numerical values reported for the partial angles can only be estimates of the actual values. However, the reported
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values do show a clear difference between the symmetry of the groove profiles that were measured at stationary and migrating boundaries. The uncertainty is a consequence of the finite radius of AFM tips. When scanning across the surface, the AFM tip may not reach the base of the grainboundary grooves, which would result in an underestimate of the actual groove depth. The AFM tips used in this study had a manufacturerspecified radius ranging between 20 and 40 nm. Checking the performance of each AFM tip on a ‘‘standard groove’’ can reduce inconsistencies in the uncertainties in AFM measurements that result from an inherent variable tip geometry. The correlation of groove geometries through a progression of heat treatments then becomes more accurate by using tips that give consistent measurements of the standard groove. Detailed discussions on how to account for the uncertainties associated with measuring grain-boundary grooves using AFM are given elsewhere [28,34]. 5.2. Remnant grooves Remnant grooves have been observed in many previous studies [6,35–39]. Mullins suggested that the presence of remnant grooves implies that the grain-boundary grooves had been pinned [35]. A migrating grain boundary will be pinned by a groove if hgb < hcrit where hcrit is the complementary angle of the partial angle in the direction in which the boundary is migrating [35]. As proposed by Mullins, boundary movement is impeded because when hgb < hcrit the boundary must increase in length [35]. Consequently, the subsurface portion of a migrating grain boundary will migrate while the boundary is pinned at the groove until hgb > hcrit [35]. Mullins argued that when this condition has been reached the migrating grain boundary escapes the groove and moves along the surface fast enough preventing thermal grooving until the boundary is nearly perpendicular to the surface, at which point the speed of the boundary decreases. When the migrating boundary at the surface slows down sufficiently, a grain-boundary groove develops at the new surface position of the boundary, thereby pinning the grain boundary once again until the process repeats.
The sequential pinning of a grain boundary by its grain-boundary groove can be seen in the series of images in Fig. 5. The study of grain-boundary migration at the surface by examination of remnant grooves is analogous to classic studies on the movement of dislocations in silicon by Dash [40] and lithium fluoride by Gilman and Johnston [41]. In these studies a chemical etchant is used to reveal dislocations that intersect the surface. An etch pit with a sharp bottom develops at locations where dislocations intersect the surface. If a stress is then applied to move the dislocations, when the crystal is chemically etched again, a sharp-bottom pit reveals the new position of the dislocation. The chemical etchant also attacks the old pit but now flattens the bottom of the pit. Once a grain boundary migrates away from the grain-boundary groove, Eq. (1) no longer determines the geometry of the groove. If the energy is not dependent on orientation, the forces due to surface curvature will then act to flatten the surface at the old groove [42], slowly healing the remnant groove. The healing of grain-boundary grooves appears to be much slower than the formation of new grooves. Comparison of position 1 in the profiles in Fig. 5(a) and (b) shows that the remnant groove is less deep than when the boundary was originally located in this position. However, the geometry of the remnant groove at position 1 does not change significantly during subsequent heat treatments while new grooves of equal or greater depth develop. The smoothing of remnant grooves has been treated mathematically by Mullins [35] for the isotropic case. The series of images in Fig. 6 indicates that the anisotropy of the surface energy may play a significant role in the slow process of smoothing of remnant grooves. In Fig. 6 boundary A/C migrates towards the top of the image while grain boundary A/B migrates towards the bottom of the image. Therefore, the portions of remnant groove A/D within grains C and B have had approximately the same amount of time to smooth, and might be expected to be comparable in depth. However, as can be seen in Fig. 6(d) remnant groove A/D within grain B has undergone more extensive smoothing. Anisotropy of surface diffusivities does not fully explain the difference in the smoothing
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rates observed in Fig. 6, because remnant grooves A/C and C/D within grain C have undergone more extensive smoothing. Remnant groove A/D appears to lie parallel to the faceting structure of grain C, and the AFM profile of remnant groove A/D within grain C in Fig. 6(e) shows that the sidewalls of the remnant groove are nearly parallel to the facet orientations, which is indicated by the respective arrows. Consequently, this portion of remnant groove A/C, which consists of low-energy surfaces, is not expected to smooth because the shape of the remnant groove minimizes the total surface energy [43,44]. The ability to re-visit and re-measure these remnant grooves opens the possibility to determine surface diffusivities of various surface orientations from the same specimen (cf. the technique used in Ref. [45]). Combining electron backscattered diffraction analysis, coupled with the VLM and AFM techniques described in this work enables relative diffusivities for many surface orientations of Al2O3 to be studied.
6. Summary and conclusions Issues related to tip geometry in the AFM and the morphology of the groove have been discussed. Care must be taken when interpreting AFM results because of errors which can be introduced by inconsistencies in tip geometry or due to the faceted nature of the groove. Remnant grooves, together with VLM maps, allow the movement of individual grain boundaries to be observed and the magnitude and sporadic nature of migration to be analyzed. Monitoring the smoothing of remnant grooves over a series of heat treatments serves as a reminder of the anisotropic nature of Al2O3 and how this anisotropy influences surface diffusivities.
Acknowledgments This work has been supported by the US Department of Energy through grants DE-FG0292ER45465-A004 and DE-FG02-01ER45883. N.E.M. would like to acknowledge additional support from the Microscopy Society of America
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through the Undergraduate Research Scholarship Award and the University of Minnesota, Minneapolis, through an Undergraduate Research Opportunities Program (UROP). The authors would like to thank Jeffrey K. Farrer (BYU) and Prof. Ravishankar (IISc, Bangalore) for many stimulating discussions.
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[34] D.B. Marshall, J.R. Waldrop, P.E.D. Morgan, Acta Mater. 48 (2000) 4471. [35] W.W. Mullins, Acta Metall. 6 (1958) 414. [36] L. Chongmo, M. Hillert, Acta Metall. 29 (1981) 1949. [37] G. Gottstein, L.S. Shvindlerman, Scripta Metall. Mater. 27 (1992) 1521. [38] C. Scott, M. Kaliszewski, C. Greskovich, L. Levinson, J. Am. Ceram. Soc. 85 (2002) 1275. [39] P.A. Beck, in: R.M. Brick (Ed.), Metal Interfaces, American Society of Materials, Cleveland, 1952, p. 208. [40] W.C. Dash, J. Appl. Phys. 27 (1956) 1193. [41] J.J. Gilman, W.G. Johnston, Dislocations and mechanical properties of crystals, John Wiley, New York, 1957. [42] W.D. Kingery, H.K. Bowen, D.R. Uhlmann, Introduction to Ceramics, John wiley & Sons, New York, 1960. [43] C. Herring, in: R. Gromer, C.S. Smith (Eds.), Structure and Properties of Solid Surfaces, University of Chicago Press, Chicago, 1952. [44] H.P. Bonzel, W.W. Mullins, Surf. Sci. 350 (1996) 285. [45] S.J. Bennison, M.P. Harmer, J. Am. Ceram. Soc. 73 (1990) 833.
Remnant grooves on alumina surfaces N.E. Munoz, S.R. Gilliss, C.B. Carter
*
Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave SE, Minneapolis, MN 55455-01432, USA Received 17 May 2004; accepted for publication 6 October 2004 Available online 19 October 2004
Abstract Thermal grooving of grain boundaries in alumina has been studied using a combination of visible-light microscopy (VLM) and atomic-force microscopy (AFM). The partial angles of grooves that develop during heat treatments at 1650 C were estimated by AFM. The observation of remnant grooves is key to determining the magnitude of the movement of a grain boundary. Imaging using VLM and AFM enabled the smoothing of remnant grooves to be observed through a progression of heat treatments, which enables the anisotropic nature of the surface diffusivity of alumina to be monitored. 2004 Elsevier B.V. All rights reserved. Keywords: Grain boundaries; Diffusion and migration; Surface diffusion; Growth; Aluminum oxide; Atomic force microscopy
1. Introduction The scaling down of electronic devices is limited, in part, by the ability to obtain smooth surfaces. Surface roughness is not only important for mechanical reasons but it can also affect the chemical stability of a component. Similarly, biomaterials that must be able to withstand the hostile environment inside the body may benefit from a low surface roughness in order to minimize the sur*
Corresponding author. Tel.: +1 612 625 8805; fax: +1 612 626 7246. E-mail address: [email protected] (C.B. Carter).
face area that can be chemically attacked [1,2]. Surface changes during post-processing heat treatments can alter the topography of a surface, which can have a significant impact on material performance. Two such thermally activated phenomena are surface faceting and grain-boundary grooving [3,4]. Crystal surfaces facet to expose lower energy surfaces, which more than compensates for the increase in surface area. The facet structures that develop depend on the material and crystallographic orientation of the surface [3,5]. Grain-boundary grooving modifies the topography of a surface through the formation of channels along the triple junctions where the grain boundaries intersect the
0039-6028/$ - see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.susc.2004.10.006
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surface. Understanding the mechanisms through which these phenomena occur is important to developing methods for processing ceramics with a specific microstructure. This paper presents a method for studying the evolution of the surface microstructure and grain-boundary grooves. The development of surface features through a progression of heat treatments was studied using a non-destructive technique involving visible-light microscopy (VLM) and atomic-force microscopy (AFM). Brinson and Moore [6] carried out a related study involving heating of zinc specimens in situ on the stage of a VLM in order to observe the nucleation and growth of new crystals. In the present study, by using AFM in conjunction with VLM, the same regions and grooves could be imaged and measured in 3D through a progression of heat treatments [7,8]. The monitoring of remnant grooves enables stationary boundaries to be distinguished from migrating grain boundaries. These remnant grooves also provide a means by which the surface diffusivity of varying grain orientations can be monitored.
2.2. Grooving by surface diffusion The evolution of a thermal-grain-boundary groove occurs by mass transport through a combination of surface diffusion, evaporation–condensation, and volume diffusion [9,10]. MullinsÕ theory of thermal grooving by surface diffusion describes the grooving process at temperatures well below the melting point in many ceramics and metals. The theory predicts two surface maxima, one on each side of the groove. These ridges have been observed experimentally in W [11,12], Fe [13], Pd [14], NiAl [15], Al2O3 [16–18], ZrO2 [19], SrTiO3 [20], and CeO2 [21]. MullinsÕ theory describes the development of a thermal-grain-boundary groove by surface diffusion and allows surface diffusion coefficients to be deduced from the geometries of grain-boundary grooves: x ¼ 4:6ðbtÞ1=4 d ¼ 0:973mðbtÞ
ð2Þ 1=4
ð3Þ
2. Background
The groove width (the distance between the surface maxima), x, and the depth of the groove (vertical distance from the surface maxima to the base), d, are each a function of time, and b is given by
2.1. The energetics of grooving
b¼
Thermal grooving occurs when a grain boundary intersects the surface. In the classical view, the topology of the three interfaces in the vicinity of this triple junction is determined by the balance of the interfacial tensions [4]. The three interfaces intersecting at the groove are the grain boundary and the two solid–vapor interfaces. Herring [4] originally applied the concept of interfacial tension to describe this equilibrium geometry:
In Eqs. (2)–(4), t is the duration of heat treatment, m is the slope of the surface at the triple junction (the base of the groove), dDs is the product of the thickness of the layer, d, in which surface diffusion occurs and the surface diffusion coefficient, Ds, respectively; X is the atomic volume of the diffusion species; cs is the free energy of the solid–gas interface; k is BoltzmannÕs constant and T is the absolute temperature. The surface diffusion coefficient of the material can thus be calculated from MullinsÕ theory using Eqs. (2) and (3). The evolution of groove geometries described by MullinsÕ theory [9] assumes that surface diffusion is driven only by capillary forces; i.e., that the grain boundary plays no role in the transport of material. Furthermore, the model assumes that the surface energy is isotropic and the grain boundary is normal to the original surface. The
3 X oc c i t i þ ni i ¼ 0 ohi i¼1
ð1Þ
The interfacial tensions, ci, of the respective surfaces are linked to the terms describing their orientation dependence, oci/ohi (the torque terms for each surface); the vector ni is defined as ti · l where l is a vector along the triple junction.
dDs Xcs kT
ð4Þ
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implications are that the surface diffusivity must be isotropic, the groove must be symmetric and the ridges on each side of the groove must be of equal height. The surface ridges are indicated in an AFM profile of a grain-boundary groove in Al2O3 in Fig. 1(a). If surface diffusion is the only grooving mechanism, then in this cross-sectional view the area below the original surface in the groove should equal the area above the original surface; i.e., the area of the ridges. Eq. (1) only describes the equilibrium condition at point G in Fig. 1(a); the ridges may be in a condition of local non-equilibrium. Consequently, the orientations specified by Eq. (1) are those at point G in Fig. 1(a), which is referred to as the groove base. If the ridge material continued to diffuse away from the groove or if material evaporated during groove formation, then a notch profile, more similar to that shown in Fig. 1(b) for a chemically etched polycrystalline material [22], is expected. The AFM profile in Fig. 1(b) is of a notch at a grain boundary in Al2O3 that has been etched in hot phosphoric acid.
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2.3. Thermal-grain-boundary groove angles The measurement of grain-boundary groove angles is perhaps the most widely used method for determining the relative interfacial energies of a material. This type of analysis involves polishing the surface of a specimen in order to obtain an initially flat surface, then heat treating until grooves form. The calculation of relative interfacial energies from Eq. (1) assumes that the groove has reached its equilibrium shape. For liquids, the surface energy is equal to the surface tension. However, this condition is not necessarily the case for crystalline materials except at equilibrium [23]. Since Eq. (1) describes an equilibrated geometry, the interfacial tension terms are then taken to be equivalent to interfacial energies. In addition to assuming an equilibrated system, relative interfacial-energy analyses simplify the calculations associated with Eq. (1) by making the following assumptions [24–28]: (i) surface energies of the material are isotropic; (ii) the grain boundary is stationary and normal to the original surface; (iii) the grain-boundary energy is independent of the tangent plane; (iv) the torque terms, oci/ohi are negligible. Under assumptions (i) through (iv), Eq. (1) simplifies to the following relationship, which is known as YoungÕs equation: cgb ¼ 2cs cos hs
Fig. 1. AFM profiles measured at alumina surfaces. (a) Typical geometry of a thermal grain-boundary groove formed by diffusion. (b) Profile measured at a grain boundary that has been etched with hot phosphoric acid. If the ridges in (a) continued to diffuse away from the groove this type of profile is expected.
ð5Þ
Here cgb and cs are the interfacial energies of the grain boundary and the surface at the base of the groove, respectively, and hs is the partial angle of each grain. The sum of the partial angles is the dihedral angle of the groove. Fig. 2(a) is a schematic profile of a groove in which assumptions (i) through (iv) apply. Under these conditions a symmetric groove profile will form such that the partial angles of both grains are equal. The ratio of grain-boundary energy to surface energy, cgb/ cs, can be calculated by measuring the angle hs. Since surface energies are not necessarily equal, assumption i need not hold so that Eq. (1) simplifies to:
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rium for the grain-boundary configuration shown in Fig. 2(c) are
Fig. 2. Schematic groove profile at a grain boundary that is normal to the original surface in a material with (a) isotropic surface energies and (b) anisotropic surface energies. (c) Schematic groove profile that developed at a boundary that is inclined towards the original surface such that hgb 5 0.
cgb ¼ c1 cos h1 þ c2 cos h2
ð6Þ
c1 sin h1 ¼ c2 sin h2
ð7Þ
where h1 and h2 are the partial angles of the respective grain. For example, h1 is the partial angle of grain 1 in Fig. 2(b); c1 and c2 are the respective surface energies of each grain at the groove base. Eq. (6) satisfies the equilibrium condition in the direction normal to the original surface. Eq. (7) satisfies the equilibrium condition in the plane of the original surface and normal to the line of the grain boundary at the surface. At a polished surface the grain boundaries do not necessarily intersect the surface at right angles, as illustrated in Fig. 2(c). Now, only assumptions (iii) and (iv) hold so that the equations of equilib-
cgb cos hgb ¼ c1 cos h1 þ c2 cos h2
ð8Þ
cgb sin hgb ¼ c1 sin h1 c2 sin h2
ð9Þ
where hgb is the angle of inclination of the grain boundary with respect to the normal of the original surface. Eqs. (8) and (9) satisfy equilibrium requirements in the direction normal to the original surface and in the direction in the plane of the original surface and normal to the grainboundary line, respectively. Previous grooving studies have shown that the grain-boundary energy depends on the misorientation of the two grains [28,29]. For example, Greenough and King [29] fabricated sets of silver bicrystals with gradually increasing grainboundary misorientations. Their study found that bicrystals with a greater misorientation resulted in grain-boundary grooves with a smaller dihedral angle, indicating that the grain-boundary energy increases with misorientation [29]. Saylor and Rohrer [28] determined surface orientations of polycrystalline Al2O3 and MgO using electron backscattered diffraction (EBSD) patterns in the scanning electron microscope (SEM) [30] and correlated the results with the measured geometries of grain-boundary grooves. This study found that smaller misorientations or near-coincident lattices have grain boundaries of a lower energy and shallower grain-boundary grooves [28].
3. Experimental LucaloxTM, which is a high-purity polycrystalline alumina with up to 0.5 wt.% magnesia [31], was used in this study. LucaloxTM was obtained 1 in the form of a tube with an outer diameter of 8.8 mm and thickness of 1 mm. Samples approximately 2-mm square were cut from tubes using a diamond saw. The convex surface was polished flat to a 0.5-lm finish using diamond lapping film. Samples were enclosed in an alumina crucible,
1
General Electric Co., Cleveland, OH.
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which was surrounded by high-purity Al2O3 powder inside a box furnace and heated at a rate of 20 C/min to a temperature of 1650 C in air. The temperature was held at 1650 C for 30 min and the lowered at a rate of approximately 150 C/min to 1300 C and 40 C/min to room temperature. The samples underwent a total of four similar heat treatments. A VLM (Olympus BH2-UMA) map was made from the central region of the sample and the same regions were re-imaged and remeasured using AFM after each heat treatment. AFM images were recorded in contact mode and in air using Digital Instruments NanoScope III MultiMode AFM equipped with a VLM for tip positioning. Si3N4 cantilevers (Veeco NP-20) with a manufacturer-specified spring constant and tip radius of 0.32 and 20–40 nm, respectively, were used for AFM imaging. In order to avoid uncertainties in the AFM measurements due to the inherent variability of probe-tip size, a groove from a standard specimen (same material and similar processing as sample specimens) was measured with each new tip. The groove from such a ‘‘standard specimen’’ was chosen because the value of dihedral angle and groove depth remained constant along the length of the grain boundary. Only AFM tips that gave consistent measurements of the standard-groove geometry were used in this study. AFM imaging consisted of scanning macroareas as large as 70 lm · 70 lm to observe the evolution of the microstructure. The imaged areas were marked on the VLM maps to facilitate a return to the same area after each 30-min heat treatment. AFM scans having dimensions 10 lm · 10 lm and 5 lm · 5 lm with a resolution of 512 · 512 pixels were made of grain-boundary grooves within these macroregions. Since the inherent AFM error in measuring grain-boundary grooves changes with the scanning direction [32,33], the same scanning direction was used for each particular groove. The groove profiles and approximate partial angles were measured from the 5 lm · 5 lm scans. An example of the sequence of VLM and AFM imaging from the same regions through a progression of two 30-min heat treatments is shown in
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Fig. 3. The first VLM image was recorded after a heat treatment at 1650 C. The region enclosed by the square corresponds to the AFM image shown in Fig. 3(b). The dark lines in the AFM image are grain-boundary grooves and indicate where the grain boundaries intersect the surface. Fig. 3(c) shows corresponding images of the same area after re-heating the sample for a second 30min period. AFM measurements of partial angles and groove depths are reported as an average of five measurements made along the length of the groove. Partial angles are measured from the slope of the surface at the base of the groove. The groove depth is measured as the vertical distance on an AFM groove profile from the base of the groove to the original surface. The original surface is assumed to be the surface at least 3 lm away from the groove base. If the surface is faceted, then the original surface is assumed to be at the location midway between the top and bottom of the largest observed facet. Although the reported AFM images do not always show the entire grain of interest, the number of sides of a grain at the surface has been determined from the VLM map of the sample. All AFM results presented here are height-mode images.
4. Results 4.1. Factors affecting groove measurements The measurements of groove geometries were found to be sensitive to two factors: AFM-tip geometry and surface faceting. When measuring the standard groove using two different AFM tips, differences in groove depths and partial angles up to 55 nm and 20, respectively, were observed. A large AFM-tip radius will cause an underestimate of groove depth. The surface faceting of a grain often continued down the sides of a groove. When an AFM tip with a large radius was used to measure a faceted groove, the surface steps that extended to the base of the groove could greatly affect the measured depth and partial angles of the faceted groove. Fig. 4 shows the effect that an AFM tip with a large radius can have on the AFM measurements of a faceted groove.
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Fig. 3. Sequence of VLM and AFM imaging used to return to the same regions after successive heat treatments. The AFM image of the boxed region is shown in (b). The highlighted groove seen in (b) can be seen in the 5 lm · 5 lm AFM image in (c). The profile of this groove is shown in (d). After 60 min at 1650 C the same area as (a) was located by VLM (e). The corresponding AFM image of this region is (f).
The grain-boundary groove from an Al2O3 specimen that underwent a total heat treatment for 120 min at 1650 C shown in the AFM image in Fig. 4(a) separates a faceted grain (left side) from a non-faceted grain (right side). Fig. 4(b) is an AFM profile along what was measured to be the base of the groove. The facet structure from the left grain extends down the sidewall to the base of the groove. The largest surface step at the measured groove base is 40 nm. Clearly the measured depth of the groove can vary up to 40 nm depending on where the measurement is made; this is a
real variation not a measurement uncertainty. The AFM groove profiles in Fig. 4(c) and (d) show a similar effect on the measured dihedral angle of a faceted groove. The groove profiles in Fig. 4(c) and (d) were made along lines l1 and l2, respectively, in Fig. 4(a). Along l1 the dihedral angle is 140. Along l2 the dihedral angle is 100. Differences this large were only observed at faceted grooves that were measured using AFM tips that gave inconsistent measurements of the standard groove. Typical variations in depth and dihedral angle along a faceted groove were 10 nm and 20, respectively.
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Fig. 4. (a) 5 lm · 5 lm AFM image of a groove in alumina heat treated for 120 min at 1650 C. The left grain is faceted. (b) AFM profile measured along the base of the groove. The largest surface step is 40 nm. (c) AFM profile measured along line l1 in (a). The dihedral angle at this location is 140. (d) AFM profile measured along line l2 in (a). The dihedral angle at this location is 100.
4.2. Remnant grooves Remnant grain-boundary grooves were observed at the location where a migrating boundary had previously intersected the surface. The AFM images in Fig. 5(a)–(d) are of the same area after each heat treatment and show that the boundary at the dotted line migrated during each heat treatment. The remnant grain-boundary groove appears as a shadow in the AFM images and marks the location where the boundary previously intersected the surface. The AFM profiles measured along the dotted line are shown below the respective AFM image. The remnant groove corresponding to the position of the boundary at the end of each heat treatment is numbered in the profile in Fig. 5(d). For example, position ‘‘1’’ corresponds to the position of the boundary after the first heat treatment, which is shown in Fig. 5(a). The direction of grain-boundary migration can therefore be inferred from the location of the remnant grain-boundary groove relative to the new position of the boundary. A rounded bottom, unlike the cusp of grain-boundary grooves is characteristic of a remnant groove. The groove depth and the height of the surface ridges decrease over the course of heat treatments. This process, which will be referred to as smoothing of remnant grooves, is slower than the formation of new grooves. As can be seen by comparing the groove profiles in Fig. 5(a) and (b), the new groove that formed is
approximately the same depth of the groove that formed at the previous boundary position while the remnant groove has not completely healed. In fact, the geometry of the remnant groove at position ‘‘1’’ did not significantly change upon further heat treatments after 60 min. 4.3. Healing of remnant grooves The rate of smoothing of remnant grooves was found to depend on the surface orientation, as can be seen in Fig. 6. The same area is shown in Fig. 6(a)–(d) after each heat treatment in the sequence. Grains A–D are labeled in Fig. 6(a). Grain A has five sides and grain B has four sides after the first heat treatment (Fig. 6(a)). A remnant groove on grain C near boundary A/C in Fig. 6(a) indicates that this boundary migrated into grain A during the first heat treatment. Throughout the remaining heat treatments grain A exhibits more extensive faceting compared with grain B as boundary A/C migrates towards the top of the image as grain C grows, and boundary A/B moves towards the bottom of the image as grain B grows. As grains C and B grow at the expense of grain A, boundary A/D is eliminated, which is first observed after the third heat treatment in Fig. 6(c), and grain A completely disappears after the fourth heat treatment as seen in Fig. 6(d). During the progression of heat treatments, grain B grows at the expense of the surrounding grains, all of which are significantly
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Fig. 5. (a)–(d) AFM images of the same region after a total heat treatment for 30, 60, 90, and 120 min, respectively at 1650 C. The grain boundary at the dotted line in (a) migrated after each heat treatment. An AFM profile made along the dotted line is shown below the respective AFM image. The position of the boundary after each heat treatment is indicated in the profile in (d).
larger than grain B at the surface. Examination of the AFM image in Fig. 6(d) indicates that remnant groove A/D within grain B has smoothed more compared with the portion of remnant groove A/
D that remains within grain C. Remnant groove A/D is appears to be parallel to the facet structure of grain C. An AFM profile measured perpendicular to the portion of remnant groove A/D within
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Fig. 6. (a)–(d) AFM images of the same region after a total heat treatment for 30, 60, 90, and 120 min, respectively. (e) AFM profile of remnant groove A/D within grain C. The arrows indicate that the walls of the remnant groove are nearly parallel to the facet orientations on grain C.
grain C in Fig. 6(e) shows that the sidewalls of the groove are nearly parallel to the surfaces of the facets of grain C, as indicated by the arrows. It should also be noted that the faceting of grain C is more pronounced in the region which boundary A/C migrated through, which corresponds to the region to the right of the remnant-groove profile in Fig. 6(e).
5. Discussion 5.1. Uncertainties in AFM measurements Uncertainties associated with using AFM to measure concave features, mean the numerical values reported for the partial angles can only be estimates of the actual values. However, the reported
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values do show a clear difference between the symmetry of the groove profiles that were measured at stationary and migrating boundaries. The uncertainty is a consequence of the finite radius of AFM tips. When scanning across the surface, the AFM tip may not reach the base of the grainboundary grooves, which would result in an underestimate of the actual groove depth. The AFM tips used in this study had a manufacturerspecified radius ranging between 20 and 40 nm. Checking the performance of each AFM tip on a ‘‘standard groove’’ can reduce inconsistencies in the uncertainties in AFM measurements that result from an inherent variable tip geometry. The correlation of groove geometries through a progression of heat treatments then becomes more accurate by using tips that give consistent measurements of the standard groove. Detailed discussions on how to account for the uncertainties associated with measuring grain-boundary grooves using AFM are given elsewhere [28,34]. 5.2. Remnant grooves Remnant grooves have been observed in many previous studies [6,35–39]. Mullins suggested that the presence of remnant grooves implies that the grain-boundary grooves had been pinned [35]. A migrating grain boundary will be pinned by a groove if hgb < hcrit where hcrit is the complementary angle of the partial angle in the direction in which the boundary is migrating [35]. As proposed by Mullins, boundary movement is impeded because when hgb < hcrit the boundary must increase in length [35]. Consequently, the subsurface portion of a migrating grain boundary will migrate while the boundary is pinned at the groove until hgb > hcrit [35]. Mullins argued that when this condition has been reached the migrating grain boundary escapes the groove and moves along the surface fast enough preventing thermal grooving until the boundary is nearly perpendicular to the surface, at which point the speed of the boundary decreases. When the migrating boundary at the surface slows down sufficiently, a grain-boundary groove develops at the new surface position of the boundary, thereby pinning the grain boundary once again until the process repeats.
The sequential pinning of a grain boundary by its grain-boundary groove can be seen in the series of images in Fig. 5. The study of grain-boundary migration at the surface by examination of remnant grooves is analogous to classic studies on the movement of dislocations in silicon by Dash [40] and lithium fluoride by Gilman and Johnston [41]. In these studies a chemical etchant is used to reveal dislocations that intersect the surface. An etch pit with a sharp bottom develops at locations where dislocations intersect the surface. If a stress is then applied to move the dislocations, when the crystal is chemically etched again, a sharp-bottom pit reveals the new position of the dislocation. The chemical etchant also attacks the old pit but now flattens the bottom of the pit. Once a grain boundary migrates away from the grain-boundary groove, Eq. (1) no longer determines the geometry of the groove. If the energy is not dependent on orientation, the forces due to surface curvature will then act to flatten the surface at the old groove [42], slowly healing the remnant groove. The healing of grain-boundary grooves appears to be much slower than the formation of new grooves. Comparison of position 1 in the profiles in Fig. 5(a) and (b) shows that the remnant groove is less deep than when the boundary was originally located in this position. However, the geometry of the remnant groove at position 1 does not change significantly during subsequent heat treatments while new grooves of equal or greater depth develop. The smoothing of remnant grooves has been treated mathematically by Mullins [35] for the isotropic case. The series of images in Fig. 6 indicates that the anisotropy of the surface energy may play a significant role in the slow process of smoothing of remnant grooves. In Fig. 6 boundary A/C migrates towards the top of the image while grain boundary A/B migrates towards the bottom of the image. Therefore, the portions of remnant groove A/D within grains C and B have had approximately the same amount of time to smooth, and might be expected to be comparable in depth. However, as can be seen in Fig. 6(d) remnant groove A/D within grain B has undergone more extensive smoothing. Anisotropy of surface diffusivities does not fully explain the difference in the smoothing
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rates observed in Fig. 6, because remnant grooves A/C and C/D within grain C have undergone more extensive smoothing. Remnant groove A/D appears to lie parallel to the faceting structure of grain C, and the AFM profile of remnant groove A/D within grain C in Fig. 6(e) shows that the sidewalls of the remnant groove are nearly parallel to the facet orientations, which is indicated by the respective arrows. Consequently, this portion of remnant groove A/C, which consists of low-energy surfaces, is not expected to smooth because the shape of the remnant groove minimizes the total surface energy [43,44]. The ability to re-visit and re-measure these remnant grooves opens the possibility to determine surface diffusivities of various surface orientations from the same specimen (cf. the technique used in Ref. [45]). Combining electron backscattered diffraction analysis, coupled with the VLM and AFM techniques described in this work enables relative diffusivities for many surface orientations of Al2O3 to be studied.
6. Summary and conclusions Issues related to tip geometry in the AFM and the morphology of the groove have been discussed. Care must be taken when interpreting AFM results because of errors which can be introduced by inconsistencies in tip geometry or due to the faceted nature of the groove. Remnant grooves, together with VLM maps, allow the movement of individual grain boundaries to be observed and the magnitude and sporadic nature of migration to be analyzed. Monitoring the smoothing of remnant grooves over a series of heat treatments serves as a reminder of the anisotropic nature of Al2O3 and how this anisotropy influences surface diffusivities.
Acknowledgments This work has been supported by the US Department of Energy through grants DE-FG0292ER45465-A004 and DE-FG02-01ER45883. N.E.M. would like to acknowledge additional support from the Microscopy Society of America
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through the Undergraduate Research Scholarship Award and the University of Minnesota, Minneapolis, through an Undergraduate Research Opportunities Program (UROP). The authors would like to thank Jeffrey K. Farrer (BYU) and Prof. Ravishankar (IISc, Bangalore) for many stimulating discussions.
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