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Quantitative three-dimensional characterization of pearlite spheroidization
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Acta Materialia 58 (2010) 4849–4858 www.elsevier.com/locate/actamat
Quantitative three-dimensional characterization of pearlite spheroidization Yuan-Tsung Wang 1, Yoshitaka Adachi *, Kiyomi Nakajima, Yoshimasa Sugimoto National Institute for Materials Science, Innovative Materials Engineering Laboratory, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan Received 22 March 2010; received in revised form 5 May 2010; accepted 10 May 2010 Available online 8 June 2010
Abstract We investigated the pearlite spheroidization of a 0.8 mass% C–Fe steel under 700 °C static annealing conditions using a combination of computer-aided three-dimensional (3-D) tomography and electron back-scattered diffraction. The holes present in naturally grown cementite lamellae cause shape instability and induce shape evolution of the lamellar structure during spheroidization. 3-D visualization demonstrated that the intrinsic holes play an important role in the initiation and development of pearlite spheroidization. The hole coalescence and expansion causes the break-up up of large cementite lamellae into several long narrow ribbons. Furthermore, the growth mechanism of inter-hole coalescence is related to the ratio of half the inter-hole distance on a cementite lamella to the thickness of that lamella. The driving force for hole growth is either the difference in surface energy or the curvature between the hole edges and the adjacent flat surface of the lamella. The morphologies of cementite ribbons depend on the hole expansion position on cementite lamella, and can change their shape to cylinders or small spheres by Rayleigh’s perturbation process after prolonged spheroidization. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hole; Cementite lamella; Surface energy; Curvature; 3-D tomography
1. Introduction Several two-phase materials, such as eutectoid pearlite and eutectic alloys, possess a lamellar structure. It is frequently observed that such a structure tends to disintegrate into small spheres during annealing. Many investigations and models for the mechanism of spheroidization of lamellar structures have been proposed to explain certain aspects of spheroidization using two-dimensional (2-D) observations [1–8], but many details of the phenomena are still unclear. The thermal groove model is generally accepted as the main mechanism for the spheroidization of deformed pearlite, since many sub-grooves are present on grain boundaries [5,6]. However, the main mechanism for the spheroidization of undeformed pearlite under static annealing is still ambiguous. For instance, some reports have adopted Mullin’s modified perturbation model to explain the instability of *
1
Corresponding author. Tel.: +81 29 859 2159; fax: +81 29 859 2159. E-mail address: [email protected] (Y. Adachi). Y.T. Wang is on leave from China Steel Corporation, Taiwan.
lamellar structures [7]. Others found that the initiation and development of the break-up of cementite lamellae were associated with lamellar faults in cementite [4,5,8]. It has been reported that holes are always present in naturally grown cementite lamellae [9,10] and may be the most important morphological feature affecting the spheroidization of undeformed pearlite [2]. However, these features were not considered in most previous studies, probably because the existence of these faults could not be easily identified or visualized in 2-D polished sections. Recently, 3-D visualization technology has been successfully applied in microstructural analysis, and has provided information on many important aspects of microstructural characterization, such as the particle shape and size distribution [11,12]. To understand the contribution of microstructural holes to the initiation of the break-up of cementite lamellae during spheroidization and to model spheroidization of undeformed pearlite under static annealing conditions accurately, the true 3-D microstructural characteristics of cementite lamellae at different annealing
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.05.023
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times must be examined. The objective of this study is to investigate the influence of the inter-hole distance, location and lamellar thickness on hole growth and the shape evolution of cementite lamellae during static annealing, and to clarify the elements contributing to the mechanism of complete pearlite spheroidization at different stages using computer-aided 3-D visualization. 2. Experimental procedure The chemical composition of the steel used in this study was 0.8 mass% C–Fe. Hot-rolled steel specimens were reaustenitized at 1200 °C for 30 min, followed by furnace cooling to room temperature at a cooling rate of 1 °C s1. The samples were subsequently isothermally annealed at 700 °C for 0, 10, 60 and 360 min, followed by furnace cooling at 1 °C s–1 down to room temperature. The microstructure was thoroughly examined by optical microscopy, scanning electron microscopy (SEM), transmission electron microscopy (TEM) and field emission gun SEM (FESEM) using a JSM7000F microscope equipped with an electron backscattered diffraction (EBSD) device (TexSEM Laboratory). A mechanically polished specimen was etched using 3 vol.% nital. For EBSD examination, the specimens were mechanochemically polished with colloidal silica to obtain a smooth and damage-free surface. The aspect ratio of cementite lamellae was measured by observing the SEM images of the as-prepared specimens annealed at different times, for more than 300 particles per specimen. For each cementite lamella, the longest and thickest positions of the platelet are defined as its length (l) and thickness (b), respectively. The values for each condition were obtained from several pieces of specimen and the average aspect ratio (l/b) was
Fig. 2. Effects of annealing time on the length, thickness and aspect ratio of cementite lamellae for pearlitic structures annealed at 700 °C.
determined. Thin TEM foils were fabricated by the conventional twin-jet method, using a solution of 1 vol.% perchloric acid and methanol at 50 mA, at room temperature. For 3-D tomography, a SII-Zeiss XVision200DB doublebeam scanning electron microscope was used, in which serial sectioning by focused ion beam (FIB) and SEM observation were alternately and repeatedly conducted in an automatic operating mode. The sectioning interval was set at 100 nm, and there were about 92 total sections for each condition. Before FIB–SEM operation, 5 3 3 mm3 specimens were mechanically polished on the two sections perpendicular to each other, to achieve a sharp edge. The 3-D microstructures were reconstructed by stacking and alignment of the serial sectioning images and the appropriate crystallographic ori-
Fig. 1. SEM micrographs of pearlite (a) before and after spheroidization at 700 °C for (b) 10 min, (c) 60 min and (d) 360 min.
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entation assigned to the section. AVS and MAVI express software packages were used for all image processing, visualization, microstructural and quantitative analysis. 3. Results 3.1. Two-dimensional observation Although the main purpose of this study is to perform 3D visualizations of pearlite spheroidization with computeraided tomography, it is worth emphasizing some subtle differences between 2-D and 3-D images, which have an important bearing on the shape stability of cementite
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lamellae. The microstructures of 0.8 mass% C–Fe steel observed as hot-rolled and after annealing at 700 °C for different times are shown in Fig. 1. In the hot-rolled condition, the microstructure exhibits the typical lamellar structure of full pearlite, as shown in Fig. 1a. After annealing, the longitudinal cementite lamellae gradually evolve into smaller ones and change shape with the increasing annealing time (Fig. 1b and c). They form a well-spheroidized structure after spheroidization heat treatment at 700 °C for 360 min, and some particles start to coarsen by Ostwald ripening [1,13], as shown in Fig. 1d. On the other hand, the aspect ratio result, shown in Fig. 2, indicates a high aspect ratio for the initial structure of cementite lamellae,
Fig. 3. (a–e) 2-D serial section images of full pearlite for each 100 nm sectioning interval and (f) 3-D reconstructed image of cementite, respectively. Yellow arrows in serial section images show that the continuous cementite lamellae in a colony are separated from each other in other sections, but red arrows show the cementite lamellae are discontinuous in one section, but are continued in other sections. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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gradually decreasing with increasing annealing time until spheroidization is complete. This indicates that the length decreases with increasing annealing time, but the thickness shows an inverse trend. It has been reported that the fraction of the volume disappearing due to shape variation during spheroidization is redistribution evenly over the remaining plate, thickening the lamellae slightly [4]. 3.2. Three-dimensional observation 3.2.1. Full pearlite microstructure To understand the morphology variation of the same cementite lamella at different depths in 2-D polished sections, serial sectioning images of the full pearlite specimen for each 100 nm sectioning interval were compared, as shown in Fig. 3a–e. Clearly, the cementite lamella changes its morphology not only in terms of lamella length and thickness in 2-D images, but also at different depths of the 3-D microstructures. It is worth mentioning that the continuous cementite lamellae in a colony are likely to be separated from each other in other sections (shown as yellow arrows in Fig. 3). Some cementite lamellae that are discontinuous in one section, however, are continuous in other sections (shown as red arrows in Fig. 3). The serial sectioning images were reconstructed in 3-D using the AVS express software packages. The 3-D visualization images show that these discontinuous parts were holes present in naturally grown lamellar microstructures (Fig. 3f). These features are formed due to dislocations in ferrite which appear to interrupt further growth of cementite according to TEM observation (Fig. 4). Similar results were observed and discussed by Darken and Fisher [14] and Bramfitt and Marder [15]. 3.2.2. Partial spheroidization microstructure Figs. 5 and 6 show the 3-D topography and shape evolution of cementite lamellae, respectively, during spheroidization at 700 °C for 60 min. The 2-D morphologies of the cementite lamellae differ significantly from different viewpoints of the 3-D image, as shown in Fig. 5a. Although the same cementite lamella is located on both sides, it changes size and shape with different depth positions in the 2-D images (Fig. 5b). This also shows a variety of morphological features in cementite lamellae from partial spheroidization cementite. Further detail was identified in the 3-D images of each individual cementite lamella in Fig. 5a, indicating that large cementite lamella can breakup into long narrow ribbons (Fig. 6a–d) owing to hole growth in naturally grown lamellar microstructures. Fig. 6a illustrates the hole growth and expansion during the initial stage of spheroidization. Some holes coalesced and caused large cementite lamella to break-up into several long narrow ribbons (Fig. 6b), and some further disintegrated ribbon cementite into cylindrical shapes (Fig. 6c) or broke down into small particles (Fig. 6d). The relationship between the positions of cementite lamellae (as shown in Fig. 6a–d) and the corresponding ferrite orientation in
Fig. 4. TEM micrograph of full pearlite showing the interrupting effect of dislocations on the growth of cementite.
polished section #46 of partially spheroidized pearlitic microstructure is also shown in Fig. 6e. Inspection of Figs. 5 and 6 reveals that the 3-D topography and shape evolution of cementite lamellae deviate significantly from what can be extracted from 2-D information, as shown in Figs. 1 and 3. This implies that the true 3-D microstructural characteristics of cementite lamellae at different spheroidization stages must be understood fully to model its spheroidization behavior and mechanism accurately. Therefore, in this study we discuss a correlation between the inter-hole distances, present in naturally grown full pearlite lamellar, and its lamellar thickness on the hole coalescence and expansion. In addition, we investigate the geometric evolution of 3-D cementite lamellae during static spheroidization at different stages. 4. Discussion 4.1. Models for hole growth Fig. 7 shows an idealized cementite lamella with holes and embedded in a surrounding matrix of ferrite. To simplify the hole growth model, we considered isotropic and circular holes; the initial radius of the hole is assumed to be R (Fig. 7a). The horizontal and vertical distances between the centers of the holes are denoted as 2a and the original lamella thickness as b. The dimension of a structural unit is given by 2a 2a b. In our models we assumed that the surface energy of the lamellae is isotropic and the isotropic growth of the unit hole in a cementite
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Fig. 5. (a) The 3-D topography and (b) cross-sectional profile of cementite lamellae.
plate is possible by reduction of the surface energy upon annealing. This is certainly a simplification, since in reality the surface energy between the ferrite and cementite phases is anisotropy and it varies with the orientation [2,16]. Surface energy anisotropy has already been suggested as the cause of micro-faceting at the ferrite–cementite lamellar interface [17]. An isotropic surface energy assumed in our models is to simplify the mathematical treatment of the growth models. To treat these in the simplest manner, we rely on the concept of the model developed by Werner [4]. If we consider the orientation relationship between the ferrite and cementite, it has been reported that the effect of anisotropy of the ferrite–cementite phase boundary energy may assumed to be the order of 10% [4], and therefore if taking this into account will not alter markedly the results of the model calculations presented in this study. For lamellar pearlite structure, the surface energy reduction when the considered unit hole is grows indicates a reduction in the ferrite–cementite phase boundary area [4]. In addition, the volume of the cementite lamella does not change during hole expansion from R to R0 (Fig. 7b). A similar mechanism for hole growth and expansion in cementite lamellae for a constant volume has been proposed in Ref. [4]. The fraction of the volume disappearing due to the growth is considered to be redistributed evenly over the flat surface adjacent to the hole, thus slightly thickening the round hole edges. Two variables, k and n, are introduced to illustrate the relationship between the ratio of half the hole spacing to the initial lamella thickness and the radius of the hole, respectively; this can be expressed as in Eqs. (1) and (2): a k¼ P0 ð1Þ b a n¼ P1 ð2Þ R In the present study, it is assumed that the volume of the cementite lamella is constant during hole expansion; there-
fore, the variation of the hole radius (R0 ) and lamella thickness (b0 ) can be described by Eqs. (3) and (4): R0 ¼ mR ¼
mkb n
where 1 6 m 6 n, 4n2 p b0 ¼ b 4n2 m2 p
ð3Þ
ð4Þ
Since the hole expansion will reduce the surface energy by reducing the surface area of ferrite–cementite phase during static annealing, the area decrease of a unit hole’s surface on a lamella can be expressed by Eq. (5): 2pk 2 b2 2 ðm 1Þ ð5Þ n2 However, the hole boundary area increases when the hole expands; therefore, the increase in the hole boundary area is given by: 2pkb2 4n2 p 0 0 DF 2 ¼ 2pðR b RbÞ ¼ m 2 1 ð6Þ 4n m2 p n
2DF 1 ¼ 2pðR02 R2 Þ ¼
The isotropic hole growth should be possible [4], if: 2DF 1 P DF 2
ð7Þ
Then, the inequality can be expressed as in Eq. (8): kðm þ 1Þð4n2 m2 pÞ P nð4n2 þ mpÞ
ð8Þ
Taking the limit m ? 1, when infinitely small variations occur in the hole’s diameter during expansion, the inequality can be obtained as in Eq. (9): 4n3 8kn2 þ pn þ 2pk 6 0
ð9Þ
Fig. 8a shows the critical solutions of the growth models for circular holes in Eq. (9); the isotropic growth of circular holes is possible under the right side region bound by the curve of the red line (curve (i)). To examine the influence of the inter-hole distance with respect to the lamella thickness
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and the radius of the hole on hole expansion and interconnection, the relationship between the two variables k and n shown in Eqs. (1) and (2) is also considered. According to the growth model for circular holes derived by Werner [4], the relationship between k and n can be expressed by Eq. (10) under a constant lamella volume condition: p a p kðnÞ ¼ k 0 1 2 ¼ 1 2 ð10Þ 4n b0 4n where k0 is the ratio of half the distance between two infinitesimally small holes to that of the lamella thickness b0.
In this study, the k and n values were measured from true 3-D images of cementite lamellae (Figs. 3f and 8b) and the measurement results labeled A in Fig. 8b are also plotted in Fig. 8a. Some data are located within the red line, but some are outside of it; the data inside the red line indicates that the inter-hole distance is greater than the lamella thickness. The data outside the red line indicates that the inter-hole distance is very small or the lamella is very thick. As reported by Werner [4], this can be divided into three different regions, which are labeled I–III and marked with dashed curves in Fig. 8a.
Fig. 6. (a–d) Shape evolution of cementite lamellae during the spheroidization process at 700 °C for 60 min and (e) the relationship between the positions of cementite lamellae and the corresponding ferrite orientation in polished section #46 of partially spheroidized pearlitic microstructure. The square multicolor region, except the black one (cementite lamellae), is the corresponding ferrite orientation image in section #46.
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(a)
(b) 2a
R
R = mR
2a
Structural unit b
b
Fig. 7. An idealized cementite lamella pierced by holes (a) before and (b) after hole growth at constant volume condition. The horizontal and vertical distance between the centers of the holes is 2a and the original thickness of a lamella is b. The dimensions of a unit hole are 2a 2a b, and the radius of the circular hole before and after growth are R and R0 .
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Region (I): for k0 > 20 (e.g. the inter-hole distance is at least 40 times the lamella thickness, i.e. the lamella is very thin); this region below the solutions of Eq. (9) lies entirely under the right side of the red curve corresponding to favorable hole growth. This indicates that once the holes grow, they continue to expand until they coalesce at n = 1, as schematically illustrated in Fig. 9a. The driving force for isotropic hole growth is obtained from the surface energy reduction upon annealing. Region (II): for 1.8 < k0 < 20, the surface-energy-driven holes grow in the initial stage, but stop before they can coalesce (Fig. 9b). Because the morphology between the edges of the two holes resembles the neck of bridge when the holes are very close, it is unstable. At this time, the other mechanism of Rayleigh’s perturbation theory [3] dominates and continues the break-up of bridges until the coalescence of holes, as shown by the blue curve (curve (ii)) in Fig. 8a. The curve equation of the blue line follows from Rayleigh’s perturbation theory and is expressed by Eq. (11) [4]. The capillary forces that drive this shape transformation derive from the curvature difference between the hole edges and the adjacent flat surface of the lamella; the curvature difference induces diffusion processes [3,18]: n¼
2k 2k 1
ð11Þ
Region (III): for k0 < 1.8, when the holes are very close, they cannot grow and expand; this always will increase the energy [4]. Therefore, Rayleigh’s perturbation theory dominates the entire break-up of bridges until coalescence of holes, as illustrated in Fig. 9c. 4.2. Shape evolution of cementite lamellae
Fig. 8. (a) Critical solutions of the growth models for circular holes assuming constant volume of lamella in Eq. (9). Curve (i) show the critical solutions for isotropic growth of circular holes and curve (ii) follows from the Rayleigh’s perturbation theory. Regions I–III show different conditions for inter-hole coalescence. (b) One case of the measurement results for hole diameter, inter-hole distance and lamellar thickness is shown as labeled A.
As discussed above, during the initial spheroidization, large cementite lamellae can break-up into several long narrow ribbons through the following routes: growth of holes, break-up of bridges adjacent to the holes, and a combination of both effects in the lamellae. A snapshot of the true 3-D morphology of the cementite lamellae (Fig. 10a and b) shows mass transfers from the edges of the holes to the flat interface, resulting in the dissolution of the hole edges and thickening of the adjacent flat surface of the ribbon-like bulges. These results are consistent with the initial assumption that the volume of the cementite lamella remains constant during hole expansion. In general, these long, narrow ribbons can be classified into three types of shaped-instability morphologies that depend on the position of hole expansion on cementite lamella, as shown in Fig. 10. First, when the coalescence holes are located away from the edges of the lamella (Fig. 10a and b), the formation of two types of ribbon morphologies, with symmetrical and unsymmetrical bulges at the ends of the ribbon edges, will be discussed. The progression in shape instability during static annealing is schematically illustrated in Fig. 11, using a finite cylinder with a semicylindrical edge. When the narrow ribbon has symmetrical
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Increasing time
(a) 2a b
(b)
(c)
Fig. 9. An illustration of the progress of inter-hole coalescence at different conditions: (a) k0 > 20, (b) 1.8 < k0 < 20 and (c) k0 < 1.8. The red arrows in figures indicate the direction of solute fluxes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Top View
(a)
(b)
Front view
Top View
(c)
(c) Front view
Fig. 10. Three types of ribbon morphologies obtained through the hole coalescence on cementite lamella. The hole located close to (a and b) the center and (c) the edge of the lamella. (a) and (b) have symmetrical and unsymmetrical bulges formed at the ends of ribbon edges, respectively.
bulges at two ends of the ribbon edges (Figs. 10 and 11a), the capillary forces driving this shape transformation derive from the curvature difference between the ribbon
edge and the adjacent flat surface of the ribbon [7,19]. This transport causes the ribbon edges to recede and produces a build-up of material on the flat surface immediately
Y.-T. Wang et al. / Acta Materialia 58 (2010) 4849–4858 Increasing time
(a)
(b)
(c) Fig. 11. Schematic of the progression in shape instabilities of three types of ribbons into cylinderization or further spheroidization during static annealing processes.
adjacent to the edge. As long as there is a curvature difference between the ribbon edge and the adjacent flat surface, the mass transport from the ribbon edge is sustained. When edge diffusion fields overlap, the final equilibrium cylindrical shape is approached. Following cylinderization, the cylinder eventually decomposes into a row of spheres, via Rayleigh’s perturbation process, after long period of spheroidization [2,20]. Furthermore, reconstructions of the 3-D cementite microstructures and calculations of the surface area of cementite phases per unit volume for three different annealing times by MAVI software (Fig. 12) show that the surface area decreases with increasing annealing time, which is in agreement with the tendency to reduce the surface energy as presented above. On the other hand, when the long, narrow ribbon has unsymmetrical bulges on two ends of the ribbon edges (Figs. 10 and 11b), it is more susceptible to shape instability than the ribbon with symmetrical bulges (Fig. 11a), and the
Fig. 12. Effects of annealing time on the surface area of cementite phases per unit volume for pearlitic structures annealed at 700 °C.
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curvature edge would be unstable with respect to the flat surface of the ribbon along its length. This shaped instability is enhanced by increasing the aspect ratio of the lamella and the radius of curvature of the bulge; it has been reported that lamellae with large aspect ratios and bulges with a lower curvature radius require longer to evolve into cylinders [21,22]. Therefore, a prominent curvature difference and curvature-induced diffusion forms between the protruding bulge edges and the nearest flat surface of the ribbon, which will accelerate the development of a stable longitudinal perturbation. Then, edge spheroidization proceeds in the bulged side of the ribbon, but may interrupt the cylinderization process on the other sides of the ribbon [23]. The bulges can be considered partial cylinders along the bulge edge and their decomposition into a row of spheres is similar to Rayleigh’s perturbation of an isolated cylinder [7,23]. Finally, if the holes are located near the edges of the lamella (Figs. 10 and 11c), the holes not only expand within the lamella during spheroidization, but also gradually thin and disintegrate the long, narrow ribbons into cylindrical shapes (Fig. 6c) or further break them down into small particles (Fig. 6d). Because a long rod shape in the neck of a lamellar ribbon is intrinsically unstable with respect to the platelet shape in the other part of ribbon, it accelerates the spheroidization of pearlite and eventually breaks it up into a ring of spheres, as shown in Fig. 11c. Therefore, two mechanisms (decreasing of the surface energy for hole growth, and the capillary force for ribbon cylinderization) compete with each other; however, no matter which mechanism dominates, the tendency is to decrease the total energy of the cementite lamella during annealing treatment. 5. Summary In this study, we examined the 3-D morphologies of cementite lamella in a eutectoid pearlitic steel during spheroidization by a combination of SEM, using a double-beam field emission gun, and EBSD. The influence of the inter-hole distance with respect to the lamella thickness on hole coalescence and expansion was discussed. The effects of shape and energy instability on further morphological evolution of the 3-D lamellar structure during spheroidization were also discussed. The results imply that the intrinsic holes present in cementite lamellae play an important role in the initiation and development of pearlite spheroidization. The hole coalescence and expansion cause the break-up of large cementite lamellae into several long, narrow ribbons, decreasing the difference in surface energy, or the curvature between the hole edges and the adjacent flat surface of the lamella. Furthermore, the growth mechanism of the inter-hole coalescence is related to the ratio of half the hole distance on a lamella to the thickness of a cementite lamella. These cementite ribbons can change their shape to cylinders or further small particles due to the curvature difference between the ribbon edge and the adjacent flat surface of the ribbon
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after long periods of spheroidization. Compared with the 3-D images of the true sizes and shapes of cementite lamellae during spheroidization, the morphology variations of cementite lamellae and imperfections present at different depths in the pearlite structure cannot easily be identified by 2-D polished sections. It is demonstrated that, although some of the statistically significant information about the microstructure of the material can be extrapolated based on 2-D observations, it is only through comprehensive 3-D analysis that the local structure and crystallography for material characterization can be understood completely. Acknowledgements This research was partially supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 20560696, 2008. One of the authors (Y.T.W.) gratefully acknowledges the financial support of the China Steel Corporation of Taiwan, ROC. All authors express their gratitude to the National Institute for Materials Science, Japan for supporting this study. We acknowledge Professor Peter Voorhees, Northwestern University for fruitful discussions on the topological view point.
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Tian YL, Kraft RW. Met Trans A 1987;18:1359. Tian YL, Kraft RW. Met Trans A 1987;18:1403. Rayleigh Lord. London Math Soc 1978;12:113. Werner E. Mat Sci Eng A 1991;132:213. Graham LD, Kraft RW. Trans AIME 1966;236:94. Mullins WW. J Appl Phys 1959;30:77. Nichola FA, Mullins WW. Trans AIME 1965;233:1840. Chojnowski EA, McG Tegart WJ. Met Sci J 1968;2:14. Adachi Y, Morooka S, Nakajima K, Sugimoto Y. Acta Mat 2008;56:5995. Zhang MX, Kelly PM. Mater Charact 2009;60:545. Alkemper J, Voorhees PW. J Microsc 2001;201:388. Kral MV, Spanos G. Acta Mat 1999;47:711. DeHoff RT. Acta Metall 1984;43:43. Darken LS, Fischer RM. Some observations on the growth of pearlite. In: Zackay VF, Aaronson HI, editors. Decomposition of austenite by diffusion processes. New York: Interscience Publishers, AIME; 1962. p. 249–65. Bramfitt BL, Marder AR. Metallography 1973;6:483. Hackney SA, Shiflet GJ. Script Metall 1986;20:389. Frank FC, Puttick KE. Acta Metall 1956;4:206. Nichols FA, Mullins WW. J Appl Phys 1965;36:1826. Nichols FA. J Mater Sci 1976;11:1077. Werner E. Acta Metall 1989;37:2047. Lee JK, Courtney TH. Met Trans A 1989;20:1385. Malzahn Kampe JC, Courtney TH, Leng Y. Acta Metall 1989;37:1735. Courtney TH, Malzahn Kampe JC. Acta Metall 1989;37:1747.
Acta Materialia 58 (2010) 4849–4858 www.elsevier.com/locate/actamat
Quantitative three-dimensional characterization of pearlite spheroidization Yuan-Tsung Wang 1, Yoshitaka Adachi *, Kiyomi Nakajima, Yoshimasa Sugimoto National Institute for Materials Science, Innovative Materials Engineering Laboratory, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan Received 22 March 2010; received in revised form 5 May 2010; accepted 10 May 2010 Available online 8 June 2010
Abstract We investigated the pearlite spheroidization of a 0.8 mass% C–Fe steel under 700 °C static annealing conditions using a combination of computer-aided three-dimensional (3-D) tomography and electron back-scattered diffraction. The holes present in naturally grown cementite lamellae cause shape instability and induce shape evolution of the lamellar structure during spheroidization. 3-D visualization demonstrated that the intrinsic holes play an important role in the initiation and development of pearlite spheroidization. The hole coalescence and expansion causes the break-up up of large cementite lamellae into several long narrow ribbons. Furthermore, the growth mechanism of inter-hole coalescence is related to the ratio of half the inter-hole distance on a cementite lamella to the thickness of that lamella. The driving force for hole growth is either the difference in surface energy or the curvature between the hole edges and the adjacent flat surface of the lamella. The morphologies of cementite ribbons depend on the hole expansion position on cementite lamella, and can change their shape to cylinders or small spheres by Rayleigh’s perturbation process after prolonged spheroidization. Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Hole; Cementite lamella; Surface energy; Curvature; 3-D tomography
1. Introduction Several two-phase materials, such as eutectoid pearlite and eutectic alloys, possess a lamellar structure. It is frequently observed that such a structure tends to disintegrate into small spheres during annealing. Many investigations and models for the mechanism of spheroidization of lamellar structures have been proposed to explain certain aspects of spheroidization using two-dimensional (2-D) observations [1–8], but many details of the phenomena are still unclear. The thermal groove model is generally accepted as the main mechanism for the spheroidization of deformed pearlite, since many sub-grooves are present on grain boundaries [5,6]. However, the main mechanism for the spheroidization of undeformed pearlite under static annealing is still ambiguous. For instance, some reports have adopted Mullin’s modified perturbation model to explain the instability of *
1
Corresponding author. Tel.: +81 29 859 2159; fax: +81 29 859 2159. E-mail address: [email protected] (Y. Adachi). Y.T. Wang is on leave from China Steel Corporation, Taiwan.
lamellar structures [7]. Others found that the initiation and development of the break-up of cementite lamellae were associated with lamellar faults in cementite [4,5,8]. It has been reported that holes are always present in naturally grown cementite lamellae [9,10] and may be the most important morphological feature affecting the spheroidization of undeformed pearlite [2]. However, these features were not considered in most previous studies, probably because the existence of these faults could not be easily identified or visualized in 2-D polished sections. Recently, 3-D visualization technology has been successfully applied in microstructural analysis, and has provided information on many important aspects of microstructural characterization, such as the particle shape and size distribution [11,12]. To understand the contribution of microstructural holes to the initiation of the break-up of cementite lamellae during spheroidization and to model spheroidization of undeformed pearlite under static annealing conditions accurately, the true 3-D microstructural characteristics of cementite lamellae at different annealing
1359-6454/$36.00 Ó 2010 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2010.05.023
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times must be examined. The objective of this study is to investigate the influence of the inter-hole distance, location and lamellar thickness on hole growth and the shape evolution of cementite lamellae during static annealing, and to clarify the elements contributing to the mechanism of complete pearlite spheroidization at different stages using computer-aided 3-D visualization. 2. Experimental procedure The chemical composition of the steel used in this study was 0.8 mass% C–Fe. Hot-rolled steel specimens were reaustenitized at 1200 °C for 30 min, followed by furnace cooling to room temperature at a cooling rate of 1 °C s1. The samples were subsequently isothermally annealed at 700 °C for 0, 10, 60 and 360 min, followed by furnace cooling at 1 °C s–1 down to room temperature. The microstructure was thoroughly examined by optical microscopy, scanning electron microscopy (SEM), transmission electron microscopy (TEM) and field emission gun SEM (FESEM) using a JSM7000F microscope equipped with an electron backscattered diffraction (EBSD) device (TexSEM Laboratory). A mechanically polished specimen was etched using 3 vol.% nital. For EBSD examination, the specimens were mechanochemically polished with colloidal silica to obtain a smooth and damage-free surface. The aspect ratio of cementite lamellae was measured by observing the SEM images of the as-prepared specimens annealed at different times, for more than 300 particles per specimen. For each cementite lamella, the longest and thickest positions of the platelet are defined as its length (l) and thickness (b), respectively. The values for each condition were obtained from several pieces of specimen and the average aspect ratio (l/b) was
Fig. 2. Effects of annealing time on the length, thickness and aspect ratio of cementite lamellae for pearlitic structures annealed at 700 °C.
determined. Thin TEM foils were fabricated by the conventional twin-jet method, using a solution of 1 vol.% perchloric acid and methanol at 50 mA, at room temperature. For 3-D tomography, a SII-Zeiss XVision200DB doublebeam scanning electron microscope was used, in which serial sectioning by focused ion beam (FIB) and SEM observation were alternately and repeatedly conducted in an automatic operating mode. The sectioning interval was set at 100 nm, and there were about 92 total sections for each condition. Before FIB–SEM operation, 5 3 3 mm3 specimens were mechanically polished on the two sections perpendicular to each other, to achieve a sharp edge. The 3-D microstructures were reconstructed by stacking and alignment of the serial sectioning images and the appropriate crystallographic ori-
Fig. 1. SEM micrographs of pearlite (a) before and after spheroidization at 700 °C for (b) 10 min, (c) 60 min and (d) 360 min.
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entation assigned to the section. AVS and MAVI express software packages were used for all image processing, visualization, microstructural and quantitative analysis. 3. Results 3.1. Two-dimensional observation Although the main purpose of this study is to perform 3D visualizations of pearlite spheroidization with computeraided tomography, it is worth emphasizing some subtle differences between 2-D and 3-D images, which have an important bearing on the shape stability of cementite
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lamellae. The microstructures of 0.8 mass% C–Fe steel observed as hot-rolled and after annealing at 700 °C for different times are shown in Fig. 1. In the hot-rolled condition, the microstructure exhibits the typical lamellar structure of full pearlite, as shown in Fig. 1a. After annealing, the longitudinal cementite lamellae gradually evolve into smaller ones and change shape with the increasing annealing time (Fig. 1b and c). They form a well-spheroidized structure after spheroidization heat treatment at 700 °C for 360 min, and some particles start to coarsen by Ostwald ripening [1,13], as shown in Fig. 1d. On the other hand, the aspect ratio result, shown in Fig. 2, indicates a high aspect ratio for the initial structure of cementite lamellae,
Fig. 3. (a–e) 2-D serial section images of full pearlite for each 100 nm sectioning interval and (f) 3-D reconstructed image of cementite, respectively. Yellow arrows in serial section images show that the continuous cementite lamellae in a colony are separated from each other in other sections, but red arrows show the cementite lamellae are discontinuous in one section, but are continued in other sections. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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gradually decreasing with increasing annealing time until spheroidization is complete. This indicates that the length decreases with increasing annealing time, but the thickness shows an inverse trend. It has been reported that the fraction of the volume disappearing due to shape variation during spheroidization is redistribution evenly over the remaining plate, thickening the lamellae slightly [4]. 3.2. Three-dimensional observation 3.2.1. Full pearlite microstructure To understand the morphology variation of the same cementite lamella at different depths in 2-D polished sections, serial sectioning images of the full pearlite specimen for each 100 nm sectioning interval were compared, as shown in Fig. 3a–e. Clearly, the cementite lamella changes its morphology not only in terms of lamella length and thickness in 2-D images, but also at different depths of the 3-D microstructures. It is worth mentioning that the continuous cementite lamellae in a colony are likely to be separated from each other in other sections (shown as yellow arrows in Fig. 3). Some cementite lamellae that are discontinuous in one section, however, are continuous in other sections (shown as red arrows in Fig. 3). The serial sectioning images were reconstructed in 3-D using the AVS express software packages. The 3-D visualization images show that these discontinuous parts were holes present in naturally grown lamellar microstructures (Fig. 3f). These features are formed due to dislocations in ferrite which appear to interrupt further growth of cementite according to TEM observation (Fig. 4). Similar results were observed and discussed by Darken and Fisher [14] and Bramfitt and Marder [15]. 3.2.2. Partial spheroidization microstructure Figs. 5 and 6 show the 3-D topography and shape evolution of cementite lamellae, respectively, during spheroidization at 700 °C for 60 min. The 2-D morphologies of the cementite lamellae differ significantly from different viewpoints of the 3-D image, as shown in Fig. 5a. Although the same cementite lamella is located on both sides, it changes size and shape with different depth positions in the 2-D images (Fig. 5b). This also shows a variety of morphological features in cementite lamellae from partial spheroidization cementite. Further detail was identified in the 3-D images of each individual cementite lamella in Fig. 5a, indicating that large cementite lamella can breakup into long narrow ribbons (Fig. 6a–d) owing to hole growth in naturally grown lamellar microstructures. Fig. 6a illustrates the hole growth and expansion during the initial stage of spheroidization. Some holes coalesced and caused large cementite lamella to break-up into several long narrow ribbons (Fig. 6b), and some further disintegrated ribbon cementite into cylindrical shapes (Fig. 6c) or broke down into small particles (Fig. 6d). The relationship between the positions of cementite lamellae (as shown in Fig. 6a–d) and the corresponding ferrite orientation in
Fig. 4. TEM micrograph of full pearlite showing the interrupting effect of dislocations on the growth of cementite.
polished section #46 of partially spheroidized pearlitic microstructure is also shown in Fig. 6e. Inspection of Figs. 5 and 6 reveals that the 3-D topography and shape evolution of cementite lamellae deviate significantly from what can be extracted from 2-D information, as shown in Figs. 1 and 3. This implies that the true 3-D microstructural characteristics of cementite lamellae at different spheroidization stages must be understood fully to model its spheroidization behavior and mechanism accurately. Therefore, in this study we discuss a correlation between the inter-hole distances, present in naturally grown full pearlite lamellar, and its lamellar thickness on the hole coalescence and expansion. In addition, we investigate the geometric evolution of 3-D cementite lamellae during static spheroidization at different stages. 4. Discussion 4.1. Models for hole growth Fig. 7 shows an idealized cementite lamella with holes and embedded in a surrounding matrix of ferrite. To simplify the hole growth model, we considered isotropic and circular holes; the initial radius of the hole is assumed to be R (Fig. 7a). The horizontal and vertical distances between the centers of the holes are denoted as 2a and the original lamella thickness as b. The dimension of a structural unit is given by 2a 2a b. In our models we assumed that the surface energy of the lamellae is isotropic and the isotropic growth of the unit hole in a cementite
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Fig. 5. (a) The 3-D topography and (b) cross-sectional profile of cementite lamellae.
plate is possible by reduction of the surface energy upon annealing. This is certainly a simplification, since in reality the surface energy between the ferrite and cementite phases is anisotropy and it varies with the orientation [2,16]. Surface energy anisotropy has already been suggested as the cause of micro-faceting at the ferrite–cementite lamellar interface [17]. An isotropic surface energy assumed in our models is to simplify the mathematical treatment of the growth models. To treat these in the simplest manner, we rely on the concept of the model developed by Werner [4]. If we consider the orientation relationship between the ferrite and cementite, it has been reported that the effect of anisotropy of the ferrite–cementite phase boundary energy may assumed to be the order of 10% [4], and therefore if taking this into account will not alter markedly the results of the model calculations presented in this study. For lamellar pearlite structure, the surface energy reduction when the considered unit hole is grows indicates a reduction in the ferrite–cementite phase boundary area [4]. In addition, the volume of the cementite lamella does not change during hole expansion from R to R0 (Fig. 7b). A similar mechanism for hole growth and expansion in cementite lamellae for a constant volume has been proposed in Ref. [4]. The fraction of the volume disappearing due to the growth is considered to be redistributed evenly over the flat surface adjacent to the hole, thus slightly thickening the round hole edges. Two variables, k and n, are introduced to illustrate the relationship between the ratio of half the hole spacing to the initial lamella thickness and the radius of the hole, respectively; this can be expressed as in Eqs. (1) and (2): a k¼ P0 ð1Þ b a n¼ P1 ð2Þ R In the present study, it is assumed that the volume of the cementite lamella is constant during hole expansion; there-
fore, the variation of the hole radius (R0 ) and lamella thickness (b0 ) can be described by Eqs. (3) and (4): R0 ¼ mR ¼
mkb n
where 1 6 m 6 n, 4n2 p b0 ¼ b 4n2 m2 p
ð3Þ
ð4Þ
Since the hole expansion will reduce the surface energy by reducing the surface area of ferrite–cementite phase during static annealing, the area decrease of a unit hole’s surface on a lamella can be expressed by Eq. (5): 2pk 2 b2 2 ðm 1Þ ð5Þ n2 However, the hole boundary area increases when the hole expands; therefore, the increase in the hole boundary area is given by: 2pkb2 4n2 p 0 0 DF 2 ¼ 2pðR b RbÞ ¼ m 2 1 ð6Þ 4n m2 p n
2DF 1 ¼ 2pðR02 R2 Þ ¼
The isotropic hole growth should be possible [4], if: 2DF 1 P DF 2
ð7Þ
Then, the inequality can be expressed as in Eq. (8): kðm þ 1Þð4n2 m2 pÞ P nð4n2 þ mpÞ
ð8Þ
Taking the limit m ? 1, when infinitely small variations occur in the hole’s diameter during expansion, the inequality can be obtained as in Eq. (9): 4n3 8kn2 þ pn þ 2pk 6 0
ð9Þ
Fig. 8a shows the critical solutions of the growth models for circular holes in Eq. (9); the isotropic growth of circular holes is possible under the right side region bound by the curve of the red line (curve (i)). To examine the influence of the inter-hole distance with respect to the lamella thickness
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and the radius of the hole on hole expansion and interconnection, the relationship between the two variables k and n shown in Eqs. (1) and (2) is also considered. According to the growth model for circular holes derived by Werner [4], the relationship between k and n can be expressed by Eq. (10) under a constant lamella volume condition: p a p kðnÞ ¼ k 0 1 2 ¼ 1 2 ð10Þ 4n b0 4n where k0 is the ratio of half the distance between two infinitesimally small holes to that of the lamella thickness b0.
In this study, the k and n values were measured from true 3-D images of cementite lamellae (Figs. 3f and 8b) and the measurement results labeled A in Fig. 8b are also plotted in Fig. 8a. Some data are located within the red line, but some are outside of it; the data inside the red line indicates that the inter-hole distance is greater than the lamella thickness. The data outside the red line indicates that the inter-hole distance is very small or the lamella is very thick. As reported by Werner [4], this can be divided into three different regions, which are labeled I–III and marked with dashed curves in Fig. 8a.
Fig. 6. (a–d) Shape evolution of cementite lamellae during the spheroidization process at 700 °C for 60 min and (e) the relationship between the positions of cementite lamellae and the corresponding ferrite orientation in polished section #46 of partially spheroidized pearlitic microstructure. The square multicolor region, except the black one (cementite lamellae), is the corresponding ferrite orientation image in section #46.
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(a)
(b) 2a
R
R = mR
2a
Structural unit b
b
Fig. 7. An idealized cementite lamella pierced by holes (a) before and (b) after hole growth at constant volume condition. The horizontal and vertical distance between the centers of the holes is 2a and the original thickness of a lamella is b. The dimensions of a unit hole are 2a 2a b, and the radius of the circular hole before and after growth are R and R0 .
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Region (I): for k0 > 20 (e.g. the inter-hole distance is at least 40 times the lamella thickness, i.e. the lamella is very thin); this region below the solutions of Eq. (9) lies entirely under the right side of the red curve corresponding to favorable hole growth. This indicates that once the holes grow, they continue to expand until they coalesce at n = 1, as schematically illustrated in Fig. 9a. The driving force for isotropic hole growth is obtained from the surface energy reduction upon annealing. Region (II): for 1.8 < k0 < 20, the surface-energy-driven holes grow in the initial stage, but stop before they can coalesce (Fig. 9b). Because the morphology between the edges of the two holes resembles the neck of bridge when the holes are very close, it is unstable. At this time, the other mechanism of Rayleigh’s perturbation theory [3] dominates and continues the break-up of bridges until the coalescence of holes, as shown by the blue curve (curve (ii)) in Fig. 8a. The curve equation of the blue line follows from Rayleigh’s perturbation theory and is expressed by Eq. (11) [4]. The capillary forces that drive this shape transformation derive from the curvature difference between the hole edges and the adjacent flat surface of the lamella; the curvature difference induces diffusion processes [3,18]: n¼
2k 2k 1
ð11Þ
Region (III): for k0 < 1.8, when the holes are very close, they cannot grow and expand; this always will increase the energy [4]. Therefore, Rayleigh’s perturbation theory dominates the entire break-up of bridges until coalescence of holes, as illustrated in Fig. 9c. 4.2. Shape evolution of cementite lamellae
Fig. 8. (a) Critical solutions of the growth models for circular holes assuming constant volume of lamella in Eq. (9). Curve (i) show the critical solutions for isotropic growth of circular holes and curve (ii) follows from the Rayleigh’s perturbation theory. Regions I–III show different conditions for inter-hole coalescence. (b) One case of the measurement results for hole diameter, inter-hole distance and lamellar thickness is shown as labeled A.
As discussed above, during the initial spheroidization, large cementite lamellae can break-up into several long narrow ribbons through the following routes: growth of holes, break-up of bridges adjacent to the holes, and a combination of both effects in the lamellae. A snapshot of the true 3-D morphology of the cementite lamellae (Fig. 10a and b) shows mass transfers from the edges of the holes to the flat interface, resulting in the dissolution of the hole edges and thickening of the adjacent flat surface of the ribbon-like bulges. These results are consistent with the initial assumption that the volume of the cementite lamella remains constant during hole expansion. In general, these long, narrow ribbons can be classified into three types of shaped-instability morphologies that depend on the position of hole expansion on cementite lamella, as shown in Fig. 10. First, when the coalescence holes are located away from the edges of the lamella (Fig. 10a and b), the formation of two types of ribbon morphologies, with symmetrical and unsymmetrical bulges at the ends of the ribbon edges, will be discussed. The progression in shape instability during static annealing is schematically illustrated in Fig. 11, using a finite cylinder with a semicylindrical edge. When the narrow ribbon has symmetrical
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Increasing time
(a) 2a b
(b)
(c)
Fig. 9. An illustration of the progress of inter-hole coalescence at different conditions: (a) k0 > 20, (b) 1.8 < k0 < 20 and (c) k0 < 1.8. The red arrows in figures indicate the direction of solute fluxes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Top View
(a)
(b)
Front view
Top View
(c)
(c) Front view
Fig. 10. Three types of ribbon morphologies obtained through the hole coalescence on cementite lamella. The hole located close to (a and b) the center and (c) the edge of the lamella. (a) and (b) have symmetrical and unsymmetrical bulges formed at the ends of ribbon edges, respectively.
bulges at two ends of the ribbon edges (Figs. 10 and 11a), the capillary forces driving this shape transformation derive from the curvature difference between the ribbon
edge and the adjacent flat surface of the ribbon [7,19]. This transport causes the ribbon edges to recede and produces a build-up of material on the flat surface immediately
Y.-T. Wang et al. / Acta Materialia 58 (2010) 4849–4858 Increasing time
(a)
(b)
(c) Fig. 11. Schematic of the progression in shape instabilities of three types of ribbons into cylinderization or further spheroidization during static annealing processes.
adjacent to the edge. As long as there is a curvature difference between the ribbon edge and the adjacent flat surface, the mass transport from the ribbon edge is sustained. When edge diffusion fields overlap, the final equilibrium cylindrical shape is approached. Following cylinderization, the cylinder eventually decomposes into a row of spheres, via Rayleigh’s perturbation process, after long period of spheroidization [2,20]. Furthermore, reconstructions of the 3-D cementite microstructures and calculations of the surface area of cementite phases per unit volume for three different annealing times by MAVI software (Fig. 12) show that the surface area decreases with increasing annealing time, which is in agreement with the tendency to reduce the surface energy as presented above. On the other hand, when the long, narrow ribbon has unsymmetrical bulges on two ends of the ribbon edges (Figs. 10 and 11b), it is more susceptible to shape instability than the ribbon with symmetrical bulges (Fig. 11a), and the
Fig. 12. Effects of annealing time on the surface area of cementite phases per unit volume for pearlitic structures annealed at 700 °C.
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curvature edge would be unstable with respect to the flat surface of the ribbon along its length. This shaped instability is enhanced by increasing the aspect ratio of the lamella and the radius of curvature of the bulge; it has been reported that lamellae with large aspect ratios and bulges with a lower curvature radius require longer to evolve into cylinders [21,22]. Therefore, a prominent curvature difference and curvature-induced diffusion forms between the protruding bulge edges and the nearest flat surface of the ribbon, which will accelerate the development of a stable longitudinal perturbation. Then, edge spheroidization proceeds in the bulged side of the ribbon, but may interrupt the cylinderization process on the other sides of the ribbon [23]. The bulges can be considered partial cylinders along the bulge edge and their decomposition into a row of spheres is similar to Rayleigh’s perturbation of an isolated cylinder [7,23]. Finally, if the holes are located near the edges of the lamella (Figs. 10 and 11c), the holes not only expand within the lamella during spheroidization, but also gradually thin and disintegrate the long, narrow ribbons into cylindrical shapes (Fig. 6c) or further break them down into small particles (Fig. 6d). Because a long rod shape in the neck of a lamellar ribbon is intrinsically unstable with respect to the platelet shape in the other part of ribbon, it accelerates the spheroidization of pearlite and eventually breaks it up into a ring of spheres, as shown in Fig. 11c. Therefore, two mechanisms (decreasing of the surface energy for hole growth, and the capillary force for ribbon cylinderization) compete with each other; however, no matter which mechanism dominates, the tendency is to decrease the total energy of the cementite lamella during annealing treatment. 5. Summary In this study, we examined the 3-D morphologies of cementite lamella in a eutectoid pearlitic steel during spheroidization by a combination of SEM, using a double-beam field emission gun, and EBSD. The influence of the inter-hole distance with respect to the lamella thickness on hole coalescence and expansion was discussed. The effects of shape and energy instability on further morphological evolution of the 3-D lamellar structure during spheroidization were also discussed. The results imply that the intrinsic holes present in cementite lamellae play an important role in the initiation and development of pearlite spheroidization. The hole coalescence and expansion cause the break-up of large cementite lamellae into several long, narrow ribbons, decreasing the difference in surface energy, or the curvature between the hole edges and the adjacent flat surface of the lamella. Furthermore, the growth mechanism of the inter-hole coalescence is related to the ratio of half the hole distance on a lamella to the thickness of a cementite lamella. These cementite ribbons can change their shape to cylinders or further small particles due to the curvature difference between the ribbon edge and the adjacent flat surface of the ribbon
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after long periods of spheroidization. Compared with the 3-D images of the true sizes and shapes of cementite lamellae during spheroidization, the morphology variations of cementite lamellae and imperfections present at different depths in the pearlite structure cannot easily be identified by 2-D polished sections. It is demonstrated that, although some of the statistically significant information about the microstructure of the material can be extrapolated based on 2-D observations, it is only through comprehensive 3-D analysis that the local structure and crystallography for material characterization can be understood completely. Acknowledgements This research was partially supported by the Japanese Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Research (C), 20560696, 2008. One of the authors (Y.T.W.) gratefully acknowledges the financial support of the China Steel Corporation of Taiwan, ROC. All authors express their gratitude to the National Institute for Materials Science, Japan for supporting this study. We acknowledge Professor Peter Voorhees, Northwestern University for fruitful discussions on the topological view point.
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