- Email: [email protected]
Scanning probe microscopy study of grain boundary migration in NiAl
Acta Materialia 52 (2004) 4953–4959 www.actamat-journals.com
Scanning probe microscopy study of grain boundary migration in NiAl Eugen Rabkin, Yaron Amouyal *, Leonid Klinger Department of Materials Engineering, The Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel Received 23 March 2004; received in revised form 30 May 2004; accepted 2 June 2004 Available online 27 August 2004
Abstract A post-mortem scanning probe microscopy (SPM) study of grain boundary migration in Ni-rich NiAl at 1400 C is presented. The migration of grain boundaries during annealing is quantified using the SPM measurements of surface topography of the regions swept by migrating grain boundaries. It is shown that the quantitative conclusions about the dynamics of grain boundary motion can be drawn from the study of surface topography in the vicinity of both individual migrated boundaries and migrated triple junctions. In the case of individual boundaries, the curvature of the blunted root of the grain boundary groove formed at original boundary position provides the information about the beginning of migration process. In the case of triple junctions that moved along one of three boundaries forming the junction, the variable width of the grain boundary groove allows to recover the dynamics of migration process. Using the Mullins model of grain boundary grooving and its modifications we estimated that the grain boundary migration rate is 0.52 0.09 lm/s. This is much higher than the average migration rate obtained by dividing the migrated distance by total annealing time. It is concluded that in the near-surface region of NiAl the grain boundaries migrate in jerky, spasmodic fashion. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain boundaries; Grain boundary migration; Scanning probe microscopy
1. Introduction Grain growth can occur in many polycrystalline materials when heated to sufficiently high temperatures. Because the material properties depend strongly on the microstructure, grain growth has been extensively studied for both fundamental and technological reasons. It is now well established that the mechanism of grain growth is a migration of individual grain boundaries (GBs) under the action of capillary driving force. Most of the data on grain growth in metallic materials were obtained by comparing the microstructures before and after annealing (post-mortem studies). This approach reduces the rich temporal behavior of ensemble of interacting GBs to the ‘‘snapshots’’ of the microstructure *
Corresponding author. Tel.: +972-482-938-75; fax: +972-482-956-
77. E-mail addresses: [email protected] (E. Rabkin), [email protected] (Y. Amouyal).
taken at few consecutive moments of time. In-situ observations of grain growth can provide much more accurate picture of the process, however, such studies are rare, especially at high temperatures. The in-situ studies of grain growth were conducted using the transmission electron microscopy (TEM), scanning electron microscopy (SEM) and X-ray tomography [1–3]. All these insitu methods suffer from various serious drawbacks. For example, TEM studies can be conducted only on thin samples that are transparent for the electron beam. At high temperatures the same capillary forces that cause grain growth can also change the shape of these thin samples and lead to their eventual disintegration [1]. SEM and X-ray tomography are limited in their time and spatial resolution, respectively. In a recent study of Roost et al. [4] the high potential of scanning tunnelling microscopy (STM) for in-situ investigation of grain growth was demonstrated. It was shown that some GBs that initially migrated at a high speed stopped suddenly at certain positions and remained immobile for most of
1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.06.027
4954
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
the annealing time. This phenomenon was explained in terms of orientation pinning. In his seminal work [5] Mullins demonstrated that surface topography produced by migrating GBs provides important information on modalities of GB motion. It was shown that in many cases the migrating GB leaves behind a family of densely spaced lines (ghost lines) indicative of jerky, spasmodic character of GB motion. It was also noted that after the GB leaves its original groove the latter starts to heal and immediately forms a zero slope at the groove root. However, the healing time of the abandoned groove is much longer than its formation time and the grooves formed in minutes can persist for hours of annealing time after abandonment. Computer simulations of GB migration in the ideal, defect- and impurity-free lattice also indicate that the GB motion is jerky [6,7]. For instance, in a curvaturedriven GB migration the moving GB emits vacancies to the bulk as a result of its area reduction. These vacancies generate a dragging force that stops the GB until the local curvature increases to the point of a breakaway. These, as well as other [8–11] observations imply that the nature of GB jerky motion is intrinsic. Rabkin et al. [12] demonstrated that analysis of ghost line morphology of the quenched sample in ambient scanning probe microscope (SPM) allows a quantification of the jerky motion. In this work, further methods of the quantification of GB motion using the topographies observed in ambient SPM will be presented.
2. Experimental procedure Coarse grained Ni-rich NiAl polycrystals with the grains of a several hundreds of microns in size have been investigated. The samples were supplied as ingots that were obtained by melting under vacuum in cold-crucible device (silver crucible, chilled with water under pressure). Three-four remeltings were performed in order to get sufficient homogeneity of the chemical composition. Samples of about 1–2 cm in diameter and 4 mm in thickness were cut from the ingot, ground and polished using SiC papers and diamond pastes down to 0.25 lm particle size. The chemical composition of the samples was determined by energy dispersive X-ray spectroscopy (EDS) in Philips XL30 SEM using accelerating voltage of 20 keV. The chemical compositions of these Ni-rich NiAl specimens were 45.6 0.2 at.% Al (Sample I) and 43.0 0.2 at.% Al (sample II). All annealings were conducted in an ultra-high vacuum (UHV) furnace with the total vacuum of 107 Torr at 1400 C. Sample I was annealed for 30 min and Sample II for 60 min. After annealing, the SPM measurements were performed with the AutoProbe CP SPM (Park Scientific, USA) operated in the contact mode. SPM scans were done with the CSC11/50 Ultrasharp Si tips, coated with W2 C. Their
nominal radius of curvature was 50 nm. The SPM images contained 256 256 pixels and were taken in the region of the GB grooves, with the scanning direction perpendicular to the groove. Electron back-scatter diffraction (EBSD) analysis was utilized as an additional tool for the determination of crystallographic orientation (LINK OPAL System of Oxford Instruments, mounted on the high-resolution SEM LEO982 Gemini, Zeiss – Leica).
3. Results After annealing, both samples exhibited a typical pattern of Mullins grooves together with series of ghost lines. Fig. 1 presents a high resolution SEM and SPM images of such Mullins groove and ghost lines pair in the Sample I. While the groove at final GB position exhibits sharp root, the abandoned groove at original GB position (ghost) is blunted. In order to verify that the blunted groove is really a ghost and does not correspond to any GB, EBSD patterns (EBSPs) were taken from the three regions defined by these two lines. These EBSPs shown in Fig. 1 indicate that two crystallographically identical regions straddle the ghost line, while the actual change of orientation occurs across the narrow sharp groove. In addition to the groove/ghost line pairs similar to those observed in Sample I, several migrating triple junctions (TJs) were observed in Sample II. One such TJ that migrated along one of its GBs is shown in Fig. 2. In this configuration two GBs migrate while the third one is immobile but increases its length. As a result, the width of the third GB groove decreases along the direction of TJ migration. Though several ghost lines left by migrating GBs A and B can be distinguished in Fig. 2, the original positions of these GBs cannot be determined unequivocally. One of the possible locations where the GB B spent some time at the beginning of annealing process is marked in Fig. 2 by a bold broken line. After migrating a certain distance the GBs A and B were stopped at the intermediate positions which are clearly marked in Fig. 2 by the corresponding ghost lines. After spending some time there both GBs migrated again until being stopped at their final positions. Fig. 2 also presents the SPM topography image of migrated TJ composed of five individual SPM images of 6 6 lm2 in size. The topography line profiles taken at different locations along the third GB (marked by C in Fig. 2) are shown in Fig. 3. One can see that the GB groove width decreases along the direction of TJ migration.
4. Discussion The migration of GBs during annealing can be associated with the following three processes:
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
4955
Fig. 1. A high-resolution SEM image of ghost line (left) parallel to a real GB groove (right). The corresponding SPM topography profiles are shown above the micrograph. Two identical EBSPs were acquired from both sides of the ghost line in its close vicinity indicating that the ghost line is not associated with any GB. The change of grain orientation occurs across the new GB groove.
• Formation of Mullins-type groove at the original GB position. • GB migration from its original position to a new one simultaneously with the healing of abandoned groove. • Formation of another groove at the final GB position with continuing healing of the old groove. At the beginning of the thermal annealing Mullins’ thermal grooves are formed at every GB-free surface intersection. During annealing, the driving force for migration can exceed the pinning force for some of the GBs. These GBs leave their original positions, which leads to the violation of mechanical equilibrium condition at the roots of abandoned GB grooves. Since the surface curvature at the root is formally infinite, the blunting of the abandoned root by the surface diffusion mechanism starts immediately after the GB has left. While the GB arrives at its final position and a new GB groove starts to grow, the decay of the old groove continues. This process is illustrated schematically in Fig. 4. 4.1. Analysis of GB migration It has been shown that the temporal evolution of the Mullins groove depends on the dominating mode of the
Fig. 2. An optical microscopy image of a TJ moving from the right to the left along the GB C, leaving behind a series of ghost lines. The original position of the GB B is marked by a bold broken line. A series of 5 SPM images demonstrating graduate reduction of the GB groove width along the direction of migration is presented at the bottom. The discontinuities that appear at the right three images originate from a routine process of flattening.
mass transfer. We assume that in the present experimental conditions surface diffusion is the predominating mechanism of GB grooving. This is based on results of
4956
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
Fig. 4. The GB migration process is divided into three main stages: formation of original GB groove (a), the decay of this groove during GB migration (b) and the formation of a new groove with continuing decay of the abandoned groove (c).
Fig. 3. The 4 topography line profiles taken at different locations (1–4) along the third GB C (marked in Fig. 2) indicate a decrease in the groove width along the direction of TJ migration.
previous studies, which demonstrated that in typical metals the GB grooves grow by surface diffusion if the groove width is significantly lower than 10 lm [13]. In this work, the lateral dimensions of the grooves were well below 10 lm. There are two ways to estimate the migration time of the GB: (i) by subtracting the formation times of the newly-formed and abandoned grooves from the total annealing time and (ii) by subtracting the formation time of the new groove from the time of decay of the old groove (Fig. 4). Mullins derived a general equation describing shape evolution of abandoned GB groove [5], which can be used for estimating the formation time of the old groove in method (i). However, only symmetrical GB grooves were considered in [5]. In our case the shape of the GB groove at the moment of GB depinning is not exactly known. Generally, the thermal grooves formed at random GBs in polycrystal are asymmetric because of the anisotropy in surface diffusivities and surface energies, and because of the nearGB lattice rotations [14]. This asymmetry makes the use of a theory developed in [5] problematic. There-
fore, we used the method (ii) for estimating the migration time of the GB. The advantage of this method is in the fact that initially sharp root of the abandoned groove can be treated locally as an ideal wedge that is not affected by the global asymmetry of the groove. The decay time of the old groove can be estimated from the rounding of the groove root while the formation time of the new groove can be determined from its width. In order to determine the decay time, the kinetics of root rounding should be described in terms of Mullins theory [15]. Considering only the old groove root (since most of the surface topography evolution takes place there, see the SPM image in Fig. 1), the initial condition can be approximated by f ðx; 0Þ ¼ mjxj, where m is the initial slope of the surface at the groove root. The initial and boundary conditions for groove decay can be formulated in complete analogy with the initial and boundary conditions of the original Mullins problem: ðiÞ f ðx; 0Þ ¼ mx; ðiiÞ f 0 ð0; tÞ ¼ 0;
ð1Þ
000
ðiiiÞ f ð0; tÞ ¼ 0; where prime, double prime, etc. denote the first and higher x-derivatives. The present problem defined by Eqs. (1) and the classical Mullins problem [15] are compared in Fig. 5. The following equation describes the kinetics of surface topography evolution by surface diffusion in the approximation of small surface slopes both in the present and in Mullins’ cases:
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
Fig. 5. A schematic description of the initial conditions and the final groove profile in original Mullins problem [15] and in the present case.
oy 0000 ¼ B y ; ð2Þ ot where B is Mullins coefficient. Using the superposition principle for linear differential equations, the solution of the present problem, f ðx; tÞ, can be expressed through the original Mullins solution, yðx; tÞ: f ðx; tÞ ¼ mx yðx; tÞ, or: " # x 1=4 f ðx; tÞ ¼ mx mðBtÞ Z ; ð3Þ ðBtÞ1=4 where ZðuÞ is the Mullins function [15]. It can be easily confirmed that the expression (3) satisfies both Eq. (2) and the initial and boundary conditions (1) and is, therefore, the solution of the present problem. Since the mathematical treatment of the problem of groove decay is confined to the groove root region, reducing the expression (3) to the 2nd order polynomial in x is a good approximation. The substitution of the explicit coefficients of Mullins function in Eq. (3) yields: m 1=4 f ðx; tÞ 0:780 mðBtÞ þ 0:289 x2 : ð4Þ 1=4 ðBtÞ
4957
Fig. 6. An SPM line profile of the abandoned GB groove. The coefficient s was obtained from polynomial regression at the blunted root.
from the groove width (Wn ) according to Mullins theory [15]: 4 1 Wn : ð6Þ tnew ¼ B 4:6 The procedure of determining Wn from the line profile taken perpendicular to the new groove is illustrated in Fig. 7. It follows from the definitions of tdecay and tnew that the GB migration time is tmigration ¼ tdecay tnew . We analyzed 6 pairs of Mullins groove and ghost lines in the Sample I for which the new GB grooves were nearly symmetric and, hence, the Eq. (6) based on original Mullins theory was applicable. For all of them tdecay tnew was found. The meaning of this finding is that most of the annealing time the GBs are immobile, either at their initial or final positions. The process of GB migration is very fast. This conclusion is in
Comparing the coefficient at x2 of the parabolic least square fit of experimentally measured topography of abandoned groove at the root region (denoted by s) with the parabolic coefficient in the RHS of Eq. (4) yields the time of decay: 4 1 0:289m tdecay ¼ : ð5Þ B s Determining the coefficient s (which represents the ghost curvature at the blunted root) from the ghost profile is illustrated in Fig. 6. The slope m that is needed for calculating tdecay was determined from the line profiles taken perpendicular to the new groove formed by the same GB that formed the original groove (see Fig. 7). The formation time of the new groove was calculated
Fig. 7. An SPM line profile of the new GB groove demonstrating the measurement of groove width Wn .
4958
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
agreement with the results of some previous studies [5,6,16,17]. 4.2. Analysis of TJ migration In the case of ideal symmetric GB grooves (Mullins’ grooves) the dependence of groove width on the time of growth is given by Eq. (6). However, the grooves formed by the immobile GB C of migrated TJ are highly asymmetric and, in addition, the difference in levels of the adjacent grains can be clearly seen in the SPM line profiles shown in Fig. 3. This difference decreases in the direction from original to final TJ position which is an indication of continuous sliding of two grains along the GB C during annealing. In [14] it was suggested that the physical reason for this sliding is the near-GB lattice rotation. It was shown that in certain cases the GB grooving with simultaneous GB sliding results in the time-independent groove shape and that the groove width, W , can be expressed in the general form W ¼ aðBtÞ1=4 ;
ð7Þ
where a is a constant that depends on dihedral angle at the root of the groove and on the rate of GB sliding. Though the linear dimensions of the grooves shown in Fig. 3 decrease along the direction of TJ migration, there are little changes in the overall shape of the grooves. This means that the assumptions of [14] are valid and Eq. (7) represents a good approximation. The width of the groove, W0 , that was growing during the whole annealing time t0 (position 1 at the GB C in Fig. 2) is 1=4
W0 ¼ aðBt0 Þ
:
ð8Þ
From Eqs. (7) and (8) the following expression for the growth time t of the GB groove formed by the GB C between the original and final positions of the TJ can be obtained: 4 W t ¼ t0 : ð9Þ W0 The analysis of 25 25 lm2 SPM image of the GB groove taken at position 1 (see Fig. 2) yielded W0 ¼ 6:9 0:05 lm. The SPM image of identical dimensions taken at position 2 in the vicinity of the intermediate positions of the TJ never revealed any systematic dependence of the groove width on position along the GB C. This means that the TJ migrated with a very high velocity in the scanned region. The average width of the GB groove in this region was W ¼ 5:9 0:05 lm. From Eq. (9) one obtains t 0:5t0 . This means that half of the annealing time the TJ spent moving between its original and intermediate positions, with several short stops and one longer stop at the location marked in Fig. 2 by a bold broken line. In the region between the intermediate and final positions of
the TJ we analyzed 730 line profiles that were sufficiently smooth for measuring the width W of the groove formed by GB C. For each profile the distance from the final TJ position was measured and the time t was calculated using the Eq. (9). Fig. 8 shows the dependence of the groove growth time t on the distance from the final TJ position. Relatively high scatter of the data is caused by the fourth power in Eq. (9). The least square fit of the data in Fig. 8 yielded v ¼ 0:6 0:1 lm/s for the TJ velocity. In fact, the real normal migration velocity of GBs A and B is: v0 v cos 30 ¼ 0:52 0:09 lm/s, since the angle between GBs A and C is approximately 60. Fig. 8 shows that the total TJ migration time between the intermediate and final positions was about 70 s. The following overall scenario of TJ migration emerges from our SPM analysis: the first 30 min of the annealing the TJ was moving toward its intermediate position which is clearly marked in Fig. 2 by the ghost lines, possibly making several short stops on the way. Another 29 min of the annealing the TJ was pinned at this intermediate position and then moved with the average migration rate of 0.6 lm/s to its final destination. Again, as in the previous case considered in Section 4.1 a considerable fraction of the annealing time the GBs were immobile in spite of the fact that total migration distance of the TJ is about 120 lm. The results of our study are in a good agreement with the results of direct in-situ TEM measurements of GB or interphase boundary migration in alloys. All results give the same order of magnitude for instantaneous GB velocity [16–18], irrespectively to the driving force for GB migration. The advantage of our approach is that it allows the reconstruction of dynamic events that occurred during high temperature heat treatment from the post-mortem SPM analyses of surface topography. Finally it should be noted that in our study we never considered the origin of the driving force for GB mi-
Fig. 8. The dependence of the time of formation of the GB groove on its distance from the final TJ position calculated according to Eq. (9). The TJ velocity was calculated using the linear regression.
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
gration and the physical mechanisms of the jerky motion. As for the driving force, it may come from the grain growth inside the sample that cannot be observed at the specimen surface. As for the pinning/depinning of GBs, it is unlikely that impurity drag plays any role in it since the annealing temperature of 1400 C employed in the present study is pretty close to the melting point of NiAl and, therefore, little or no GB segregation of impurities should be expected. Most probably, the jerky nature of GB motion observed in the present study is associated with the fine details of the interaction between the GB and its own thermal groove [5,19]. Shvindlerman and co-workers have shown that the period of time between two consecutive detachments of migrating GB from its own thermal groove increases with the decrease of migration driving force [19]. This may explain why the distance traveled by migrating GBs between their original and final positions was so long (see Fig. 1), since the capillary driving force for GB migration in coarse grain polycrystals studied in the present work is very low. We would like to emphasize that the migration of near-surface GBs should not be always jerky. For example, under experimental conditions at which no GB grooves are formed at the surface of the sample the GB migration in the near-surface region can be continuous. This is probably the case for migration of individual GBs in high-purity Al studied by Gottstein, Shvindlerman and co-workers [20]. Thin alumina film always present on the surface of solid Al effectively suppresses surface or interfacial diffusion of Al atoms and prevents formation of GB grooves. Indeed, alumina film grows epitaxially on the surface of Al, which means that the corresponding Al/alumina interface is coherent or semicoherent. The self-diffusion along such low-energy coherent interfaces is usually very slow [21], which should prevent any capillary-driven shape changes at the surface of the sample, including GB grooving.
5. Conclusions We performed an SPM study of the surface topography in the near-GB regions of Ni-rich NiAl annealed at 1400 C. 1. The Mullins model of GB grooving by surface diffusion was modified to describe the blunting of the root of abandoned GB groove (ghost line). The time of the groove decay was determined by comparison of the curvature of the blunted root determined from the SPM image with the prediction of the model. For all the ghost/groove pairs studied the time of decay of the abandoned groove was approximately equal to the formation time of the new groove at the final GB position. It was concluded that most of the annealing time the GBs were immobile and
4959
that the migration rate between the original and final GB positions was very high. 2. The dynamics of TJ migration was investigated by analyzing the dependence of topography of the GB groove formed at the only immobile GB of the junction on the distance from final TJ position. As in the previous case it was concluded that a considerable fraction of the annealing time the GBs of the TJ were immobile. The TJ migration velocity between its intermediate and final positions was found to be 0.6 0.1 lm/s, which is in a good agreement with the results of some in-situ studies. Both examples demonstrate a high potential of postmortem SPM studies for determining of dynamic behavior of interfaces during high temperature annealings.
Acknowledgements This research was supported by the Binational US– Israel Science Foundation under the Grant No. 2000066. Dr. A. Fraczkiewicz from Ecole des Mines de Saint Etienne is heartily acknowledged for supplying us with the NiAl samples. Helpful discussions with Prof. Y. Mishin from the George Mason University are heartily appreciated. References [1] Dannenberg R, Stach EA, Groza JR, Dresser BJ. Thin Solid Films 2000;370:54. [2] Huang Y, Humphreys FJ. Acta Mater 1999;47:2259. [3] Larson BC, Wenge Yang, Ice GE, Budai JD, Tischler JZ. Nature 2002;415:887. [4] Roost MJ, Quist DA, Frenken JWM. Phys Rev Lett 2003;91(26101). [5] Mullins WW. Acta Metall 1958;6:414. [6] Sch€ onfelder B, Wolf D, Phillpot SR, Furtkamp M. Interface Sci 1997;5:245. [7] Upmanyu M, Srolovitz DJ, Shvindlerman LS, Gottstein G. Interface Sci 1998;6:287. [8] Li Chongmo, Hillert M. Acta Metall 1982;30:1133. [9] Beers AM, Mittemeijer EJ. Thin Solid Films 1978;48:367. [10] Mittemeijer EJ, Beers AM. Thin Solid Films 1980;65:125. [11] Babcock SE, Balluffi RW. Acta Metall 1989;37:2367. [12] Rabkin E, Semenov V, Izyumova T. Scripta Mater 2000;42:359. [13] Mullins WW, Shewmon PG. Acta Metall 1959;7:163. [14] Rabkin E, Klinger L, Izyumova T, Berner A, Semenov V. Acta Mater 2001;49:1429. [15] Mullins WW. J Appl Phys 1957;28:333. [16] Abdou S, Solorzano G, El-Boragy M, Gust W, Predel B. Scripta Mater 1996;34:1431. [17] B€ ogel A, Gust W. Z Metallkd 1988;79:296. [18] Grovenor C. Acta Metall 1985;33:579. [19] Aristov VYu, Fradkov VE, Shvindlerman LS. Phys Metall Metalloved 1979;45:83. [20] Gottstein G, Shvindlerman LS. Grain boundary migration in metals. Boca Raton, FL: CRC Press; 1999. [21] Kaur I, Mishin Y, Gust W. Fundamentals of grain and interphase boundary diffusion. Chichester: Wiley; 1995.
Scanning probe microscopy study of grain boundary migration in NiAl Eugen Rabkin, Yaron Amouyal *, Leonid Klinger Department of Materials Engineering, The Technion – Israel Institute of Technology, Technion City, Haifa 32000, Israel Received 23 March 2004; received in revised form 30 May 2004; accepted 2 June 2004 Available online 27 August 2004
Abstract A post-mortem scanning probe microscopy (SPM) study of grain boundary migration in Ni-rich NiAl at 1400 C is presented. The migration of grain boundaries during annealing is quantified using the SPM measurements of surface topography of the regions swept by migrating grain boundaries. It is shown that the quantitative conclusions about the dynamics of grain boundary motion can be drawn from the study of surface topography in the vicinity of both individual migrated boundaries and migrated triple junctions. In the case of individual boundaries, the curvature of the blunted root of the grain boundary groove formed at original boundary position provides the information about the beginning of migration process. In the case of triple junctions that moved along one of three boundaries forming the junction, the variable width of the grain boundary groove allows to recover the dynamics of migration process. Using the Mullins model of grain boundary grooving and its modifications we estimated that the grain boundary migration rate is 0.52 0.09 lm/s. This is much higher than the average migration rate obtained by dividing the migrated distance by total annealing time. It is concluded that in the near-surface region of NiAl the grain boundaries migrate in jerky, spasmodic fashion. 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Grain boundaries; Grain boundary migration; Scanning probe microscopy
1. Introduction Grain growth can occur in many polycrystalline materials when heated to sufficiently high temperatures. Because the material properties depend strongly on the microstructure, grain growth has been extensively studied for both fundamental and technological reasons. It is now well established that the mechanism of grain growth is a migration of individual grain boundaries (GBs) under the action of capillary driving force. Most of the data on grain growth in metallic materials were obtained by comparing the microstructures before and after annealing (post-mortem studies). This approach reduces the rich temporal behavior of ensemble of interacting GBs to the ‘‘snapshots’’ of the microstructure *
Corresponding author. Tel.: +972-482-938-75; fax: +972-482-956-
77. E-mail addresses: [email protected] (E. Rabkin), [email protected] (Y. Amouyal).
taken at few consecutive moments of time. In-situ observations of grain growth can provide much more accurate picture of the process, however, such studies are rare, especially at high temperatures. The in-situ studies of grain growth were conducted using the transmission electron microscopy (TEM), scanning electron microscopy (SEM) and X-ray tomography [1–3]. All these insitu methods suffer from various serious drawbacks. For example, TEM studies can be conducted only on thin samples that are transparent for the electron beam. At high temperatures the same capillary forces that cause grain growth can also change the shape of these thin samples and lead to their eventual disintegration [1]. SEM and X-ray tomography are limited in their time and spatial resolution, respectively. In a recent study of Roost et al. [4] the high potential of scanning tunnelling microscopy (STM) for in-situ investigation of grain growth was demonstrated. It was shown that some GBs that initially migrated at a high speed stopped suddenly at certain positions and remained immobile for most of
1359-6454/$30.00 2004 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actamat.2004.06.027
4954
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
the annealing time. This phenomenon was explained in terms of orientation pinning. In his seminal work [5] Mullins demonstrated that surface topography produced by migrating GBs provides important information on modalities of GB motion. It was shown that in many cases the migrating GB leaves behind a family of densely spaced lines (ghost lines) indicative of jerky, spasmodic character of GB motion. It was also noted that after the GB leaves its original groove the latter starts to heal and immediately forms a zero slope at the groove root. However, the healing time of the abandoned groove is much longer than its formation time and the grooves formed in minutes can persist for hours of annealing time after abandonment. Computer simulations of GB migration in the ideal, defect- and impurity-free lattice also indicate that the GB motion is jerky [6,7]. For instance, in a curvaturedriven GB migration the moving GB emits vacancies to the bulk as a result of its area reduction. These vacancies generate a dragging force that stops the GB until the local curvature increases to the point of a breakaway. These, as well as other [8–11] observations imply that the nature of GB jerky motion is intrinsic. Rabkin et al. [12] demonstrated that analysis of ghost line morphology of the quenched sample in ambient scanning probe microscope (SPM) allows a quantification of the jerky motion. In this work, further methods of the quantification of GB motion using the topographies observed in ambient SPM will be presented.
2. Experimental procedure Coarse grained Ni-rich NiAl polycrystals with the grains of a several hundreds of microns in size have been investigated. The samples were supplied as ingots that were obtained by melting under vacuum in cold-crucible device (silver crucible, chilled with water under pressure). Three-four remeltings were performed in order to get sufficient homogeneity of the chemical composition. Samples of about 1–2 cm in diameter and 4 mm in thickness were cut from the ingot, ground and polished using SiC papers and diamond pastes down to 0.25 lm particle size. The chemical composition of the samples was determined by energy dispersive X-ray spectroscopy (EDS) in Philips XL30 SEM using accelerating voltage of 20 keV. The chemical compositions of these Ni-rich NiAl specimens were 45.6 0.2 at.% Al (Sample I) and 43.0 0.2 at.% Al (sample II). All annealings were conducted in an ultra-high vacuum (UHV) furnace with the total vacuum of 107 Torr at 1400 C. Sample I was annealed for 30 min and Sample II for 60 min. After annealing, the SPM measurements were performed with the AutoProbe CP SPM (Park Scientific, USA) operated in the contact mode. SPM scans were done with the CSC11/50 Ultrasharp Si tips, coated with W2 C. Their
nominal radius of curvature was 50 nm. The SPM images contained 256 256 pixels and were taken in the region of the GB grooves, with the scanning direction perpendicular to the groove. Electron back-scatter diffraction (EBSD) analysis was utilized as an additional tool for the determination of crystallographic orientation (LINK OPAL System of Oxford Instruments, mounted on the high-resolution SEM LEO982 Gemini, Zeiss – Leica).
3. Results After annealing, both samples exhibited a typical pattern of Mullins grooves together with series of ghost lines. Fig. 1 presents a high resolution SEM and SPM images of such Mullins groove and ghost lines pair in the Sample I. While the groove at final GB position exhibits sharp root, the abandoned groove at original GB position (ghost) is blunted. In order to verify that the blunted groove is really a ghost and does not correspond to any GB, EBSD patterns (EBSPs) were taken from the three regions defined by these two lines. These EBSPs shown in Fig. 1 indicate that two crystallographically identical regions straddle the ghost line, while the actual change of orientation occurs across the narrow sharp groove. In addition to the groove/ghost line pairs similar to those observed in Sample I, several migrating triple junctions (TJs) were observed in Sample II. One such TJ that migrated along one of its GBs is shown in Fig. 2. In this configuration two GBs migrate while the third one is immobile but increases its length. As a result, the width of the third GB groove decreases along the direction of TJ migration. Though several ghost lines left by migrating GBs A and B can be distinguished in Fig. 2, the original positions of these GBs cannot be determined unequivocally. One of the possible locations where the GB B spent some time at the beginning of annealing process is marked in Fig. 2 by a bold broken line. After migrating a certain distance the GBs A and B were stopped at the intermediate positions which are clearly marked in Fig. 2 by the corresponding ghost lines. After spending some time there both GBs migrated again until being stopped at their final positions. Fig. 2 also presents the SPM topography image of migrated TJ composed of five individual SPM images of 6 6 lm2 in size. The topography line profiles taken at different locations along the third GB (marked by C in Fig. 2) are shown in Fig. 3. One can see that the GB groove width decreases along the direction of TJ migration.
4. Discussion The migration of GBs during annealing can be associated with the following three processes:
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
4955
Fig. 1. A high-resolution SEM image of ghost line (left) parallel to a real GB groove (right). The corresponding SPM topography profiles are shown above the micrograph. Two identical EBSPs were acquired from both sides of the ghost line in its close vicinity indicating that the ghost line is not associated with any GB. The change of grain orientation occurs across the new GB groove.
• Formation of Mullins-type groove at the original GB position. • GB migration from its original position to a new one simultaneously with the healing of abandoned groove. • Formation of another groove at the final GB position with continuing healing of the old groove. At the beginning of the thermal annealing Mullins’ thermal grooves are formed at every GB-free surface intersection. During annealing, the driving force for migration can exceed the pinning force for some of the GBs. These GBs leave their original positions, which leads to the violation of mechanical equilibrium condition at the roots of abandoned GB grooves. Since the surface curvature at the root is formally infinite, the blunting of the abandoned root by the surface diffusion mechanism starts immediately after the GB has left. While the GB arrives at its final position and a new GB groove starts to grow, the decay of the old groove continues. This process is illustrated schematically in Fig. 4. 4.1. Analysis of GB migration It has been shown that the temporal evolution of the Mullins groove depends on the dominating mode of the
Fig. 2. An optical microscopy image of a TJ moving from the right to the left along the GB C, leaving behind a series of ghost lines. The original position of the GB B is marked by a bold broken line. A series of 5 SPM images demonstrating graduate reduction of the GB groove width along the direction of migration is presented at the bottom. The discontinuities that appear at the right three images originate from a routine process of flattening.
mass transfer. We assume that in the present experimental conditions surface diffusion is the predominating mechanism of GB grooving. This is based on results of
4956
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
Fig. 4. The GB migration process is divided into three main stages: formation of original GB groove (a), the decay of this groove during GB migration (b) and the formation of a new groove with continuing decay of the abandoned groove (c).
Fig. 3. The 4 topography line profiles taken at different locations (1–4) along the third GB C (marked in Fig. 2) indicate a decrease in the groove width along the direction of TJ migration.
previous studies, which demonstrated that in typical metals the GB grooves grow by surface diffusion if the groove width is significantly lower than 10 lm [13]. In this work, the lateral dimensions of the grooves were well below 10 lm. There are two ways to estimate the migration time of the GB: (i) by subtracting the formation times of the newly-formed and abandoned grooves from the total annealing time and (ii) by subtracting the formation time of the new groove from the time of decay of the old groove (Fig. 4). Mullins derived a general equation describing shape evolution of abandoned GB groove [5], which can be used for estimating the formation time of the old groove in method (i). However, only symmetrical GB grooves were considered in [5]. In our case the shape of the GB groove at the moment of GB depinning is not exactly known. Generally, the thermal grooves formed at random GBs in polycrystal are asymmetric because of the anisotropy in surface diffusivities and surface energies, and because of the nearGB lattice rotations [14]. This asymmetry makes the use of a theory developed in [5] problematic. There-
fore, we used the method (ii) for estimating the migration time of the GB. The advantage of this method is in the fact that initially sharp root of the abandoned groove can be treated locally as an ideal wedge that is not affected by the global asymmetry of the groove. The decay time of the old groove can be estimated from the rounding of the groove root while the formation time of the new groove can be determined from its width. In order to determine the decay time, the kinetics of root rounding should be described in terms of Mullins theory [15]. Considering only the old groove root (since most of the surface topography evolution takes place there, see the SPM image in Fig. 1), the initial condition can be approximated by f ðx; 0Þ ¼ mjxj, where m is the initial slope of the surface at the groove root. The initial and boundary conditions for groove decay can be formulated in complete analogy with the initial and boundary conditions of the original Mullins problem: ðiÞ f ðx; 0Þ ¼ mx; ðiiÞ f 0 ð0; tÞ ¼ 0;
ð1Þ
000
ðiiiÞ f ð0; tÞ ¼ 0; where prime, double prime, etc. denote the first and higher x-derivatives. The present problem defined by Eqs. (1) and the classical Mullins problem [15] are compared in Fig. 5. The following equation describes the kinetics of surface topography evolution by surface diffusion in the approximation of small surface slopes both in the present and in Mullins’ cases:
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
Fig. 5. A schematic description of the initial conditions and the final groove profile in original Mullins problem [15] and in the present case.
oy 0000 ¼ B y ; ð2Þ ot where B is Mullins coefficient. Using the superposition principle for linear differential equations, the solution of the present problem, f ðx; tÞ, can be expressed through the original Mullins solution, yðx; tÞ: f ðx; tÞ ¼ mx yðx; tÞ, or: " # x 1=4 f ðx; tÞ ¼ mx mðBtÞ Z ; ð3Þ ðBtÞ1=4 where ZðuÞ is the Mullins function [15]. It can be easily confirmed that the expression (3) satisfies both Eq. (2) and the initial and boundary conditions (1) and is, therefore, the solution of the present problem. Since the mathematical treatment of the problem of groove decay is confined to the groove root region, reducing the expression (3) to the 2nd order polynomial in x is a good approximation. The substitution of the explicit coefficients of Mullins function in Eq. (3) yields: m 1=4 f ðx; tÞ 0:780 mðBtÞ þ 0:289 x2 : ð4Þ 1=4 ðBtÞ
4957
Fig. 6. An SPM line profile of the abandoned GB groove. The coefficient s was obtained from polynomial regression at the blunted root.
from the groove width (Wn ) according to Mullins theory [15]: 4 1 Wn : ð6Þ tnew ¼ B 4:6 The procedure of determining Wn from the line profile taken perpendicular to the new groove is illustrated in Fig. 7. It follows from the definitions of tdecay and tnew that the GB migration time is tmigration ¼ tdecay tnew . We analyzed 6 pairs of Mullins groove and ghost lines in the Sample I for which the new GB grooves were nearly symmetric and, hence, the Eq. (6) based on original Mullins theory was applicable. For all of them tdecay tnew was found. The meaning of this finding is that most of the annealing time the GBs are immobile, either at their initial or final positions. The process of GB migration is very fast. This conclusion is in
Comparing the coefficient at x2 of the parabolic least square fit of experimentally measured topography of abandoned groove at the root region (denoted by s) with the parabolic coefficient in the RHS of Eq. (4) yields the time of decay: 4 1 0:289m tdecay ¼ : ð5Þ B s Determining the coefficient s (which represents the ghost curvature at the blunted root) from the ghost profile is illustrated in Fig. 6. The slope m that is needed for calculating tdecay was determined from the line profiles taken perpendicular to the new groove formed by the same GB that formed the original groove (see Fig. 7). The formation time of the new groove was calculated
Fig. 7. An SPM line profile of the new GB groove demonstrating the measurement of groove width Wn .
4958
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
agreement with the results of some previous studies [5,6,16,17]. 4.2. Analysis of TJ migration In the case of ideal symmetric GB grooves (Mullins’ grooves) the dependence of groove width on the time of growth is given by Eq. (6). However, the grooves formed by the immobile GB C of migrated TJ are highly asymmetric and, in addition, the difference in levels of the adjacent grains can be clearly seen in the SPM line profiles shown in Fig. 3. This difference decreases in the direction from original to final TJ position which is an indication of continuous sliding of two grains along the GB C during annealing. In [14] it was suggested that the physical reason for this sliding is the near-GB lattice rotation. It was shown that in certain cases the GB grooving with simultaneous GB sliding results in the time-independent groove shape and that the groove width, W , can be expressed in the general form W ¼ aðBtÞ1=4 ;
ð7Þ
where a is a constant that depends on dihedral angle at the root of the groove and on the rate of GB sliding. Though the linear dimensions of the grooves shown in Fig. 3 decrease along the direction of TJ migration, there are little changes in the overall shape of the grooves. This means that the assumptions of [14] are valid and Eq. (7) represents a good approximation. The width of the groove, W0 , that was growing during the whole annealing time t0 (position 1 at the GB C in Fig. 2) is 1=4
W0 ¼ aðBt0 Þ
:
ð8Þ
From Eqs. (7) and (8) the following expression for the growth time t of the GB groove formed by the GB C between the original and final positions of the TJ can be obtained: 4 W t ¼ t0 : ð9Þ W0 The analysis of 25 25 lm2 SPM image of the GB groove taken at position 1 (see Fig. 2) yielded W0 ¼ 6:9 0:05 lm. The SPM image of identical dimensions taken at position 2 in the vicinity of the intermediate positions of the TJ never revealed any systematic dependence of the groove width on position along the GB C. This means that the TJ migrated with a very high velocity in the scanned region. The average width of the GB groove in this region was W ¼ 5:9 0:05 lm. From Eq. (9) one obtains t 0:5t0 . This means that half of the annealing time the TJ spent moving between its original and intermediate positions, with several short stops and one longer stop at the location marked in Fig. 2 by a bold broken line. In the region between the intermediate and final positions of
the TJ we analyzed 730 line profiles that were sufficiently smooth for measuring the width W of the groove formed by GB C. For each profile the distance from the final TJ position was measured and the time t was calculated using the Eq. (9). Fig. 8 shows the dependence of the groove growth time t on the distance from the final TJ position. Relatively high scatter of the data is caused by the fourth power in Eq. (9). The least square fit of the data in Fig. 8 yielded v ¼ 0:6 0:1 lm/s for the TJ velocity. In fact, the real normal migration velocity of GBs A and B is: v0 v cos 30 ¼ 0:52 0:09 lm/s, since the angle between GBs A and C is approximately 60. Fig. 8 shows that the total TJ migration time between the intermediate and final positions was about 70 s. The following overall scenario of TJ migration emerges from our SPM analysis: the first 30 min of the annealing the TJ was moving toward its intermediate position which is clearly marked in Fig. 2 by the ghost lines, possibly making several short stops on the way. Another 29 min of the annealing the TJ was pinned at this intermediate position and then moved with the average migration rate of 0.6 lm/s to its final destination. Again, as in the previous case considered in Section 4.1 a considerable fraction of the annealing time the GBs were immobile in spite of the fact that total migration distance of the TJ is about 120 lm. The results of our study are in a good agreement with the results of direct in-situ TEM measurements of GB or interphase boundary migration in alloys. All results give the same order of magnitude for instantaneous GB velocity [16–18], irrespectively to the driving force for GB migration. The advantage of our approach is that it allows the reconstruction of dynamic events that occurred during high temperature heat treatment from the post-mortem SPM analyses of surface topography. Finally it should be noted that in our study we never considered the origin of the driving force for GB mi-
Fig. 8. The dependence of the time of formation of the GB groove on its distance from the final TJ position calculated according to Eq. (9). The TJ velocity was calculated using the linear regression.
E. Rabkin et al. / Acta Materialia 52 (2004) 4953–4959
gration and the physical mechanisms of the jerky motion. As for the driving force, it may come from the grain growth inside the sample that cannot be observed at the specimen surface. As for the pinning/depinning of GBs, it is unlikely that impurity drag plays any role in it since the annealing temperature of 1400 C employed in the present study is pretty close to the melting point of NiAl and, therefore, little or no GB segregation of impurities should be expected. Most probably, the jerky nature of GB motion observed in the present study is associated with the fine details of the interaction between the GB and its own thermal groove [5,19]. Shvindlerman and co-workers have shown that the period of time between two consecutive detachments of migrating GB from its own thermal groove increases with the decrease of migration driving force [19]. This may explain why the distance traveled by migrating GBs between their original and final positions was so long (see Fig. 1), since the capillary driving force for GB migration in coarse grain polycrystals studied in the present work is very low. We would like to emphasize that the migration of near-surface GBs should not be always jerky. For example, under experimental conditions at which no GB grooves are formed at the surface of the sample the GB migration in the near-surface region can be continuous. This is probably the case for migration of individual GBs in high-purity Al studied by Gottstein, Shvindlerman and co-workers [20]. Thin alumina film always present on the surface of solid Al effectively suppresses surface or interfacial diffusion of Al atoms and prevents formation of GB grooves. Indeed, alumina film grows epitaxially on the surface of Al, which means that the corresponding Al/alumina interface is coherent or semicoherent. The self-diffusion along such low-energy coherent interfaces is usually very slow [21], which should prevent any capillary-driven shape changes at the surface of the sample, including GB grooving.
5. Conclusions We performed an SPM study of the surface topography in the near-GB regions of Ni-rich NiAl annealed at 1400 C. 1. The Mullins model of GB grooving by surface diffusion was modified to describe the blunting of the root of abandoned GB groove (ghost line). The time of the groove decay was determined by comparison of the curvature of the blunted root determined from the SPM image with the prediction of the model. For all the ghost/groove pairs studied the time of decay of the abandoned groove was approximately equal to the formation time of the new groove at the final GB position. It was concluded that most of the annealing time the GBs were immobile and
4959
that the migration rate between the original and final GB positions was very high. 2. The dynamics of TJ migration was investigated by analyzing the dependence of topography of the GB groove formed at the only immobile GB of the junction on the distance from final TJ position. As in the previous case it was concluded that a considerable fraction of the annealing time the GBs of the TJ were immobile. The TJ migration velocity between its intermediate and final positions was found to be 0.6 0.1 lm/s, which is in a good agreement with the results of some in-situ studies. Both examples demonstrate a high potential of postmortem SPM studies for determining of dynamic behavior of interfaces during high temperature annealings.
Acknowledgements This research was supported by the Binational US– Israel Science Foundation under the Grant No. 2000066. Dr. A. Fraczkiewicz from Ecole des Mines de Saint Etienne is heartily acknowledged for supplying us with the NiAl samples. Helpful discussions with Prof. Y. Mishin from the George Mason University are heartily appreciated. References [1] Dannenberg R, Stach EA, Groza JR, Dresser BJ. Thin Solid Films 2000;370:54. [2] Huang Y, Humphreys FJ. Acta Mater 1999;47:2259. [3] Larson BC, Wenge Yang, Ice GE, Budai JD, Tischler JZ. Nature 2002;415:887. [4] Roost MJ, Quist DA, Frenken JWM. Phys Rev Lett 2003;91(26101). [5] Mullins WW. Acta Metall 1958;6:414. [6] Sch€ onfelder B, Wolf D, Phillpot SR, Furtkamp M. Interface Sci 1997;5:245. [7] Upmanyu M, Srolovitz DJ, Shvindlerman LS, Gottstein G. Interface Sci 1998;6:287. [8] Li Chongmo, Hillert M. Acta Metall 1982;30:1133. [9] Beers AM, Mittemeijer EJ. Thin Solid Films 1978;48:367. [10] Mittemeijer EJ, Beers AM. Thin Solid Films 1980;65:125. [11] Babcock SE, Balluffi RW. Acta Metall 1989;37:2367. [12] Rabkin E, Semenov V, Izyumova T. Scripta Mater 2000;42:359. [13] Mullins WW, Shewmon PG. Acta Metall 1959;7:163. [14] Rabkin E, Klinger L, Izyumova T, Berner A, Semenov V. Acta Mater 2001;49:1429. [15] Mullins WW. J Appl Phys 1957;28:333. [16] Abdou S, Solorzano G, El-Boragy M, Gust W, Predel B. Scripta Mater 1996;34:1431. [17] B€ ogel A, Gust W. Z Metallkd 1988;79:296. [18] Grovenor C. Acta Metall 1985;33:579. [19] Aristov VYu, Fradkov VE, Shvindlerman LS. Phys Metall Metalloved 1979;45:83. [20] Gottstein G, Shvindlerman LS. Grain boundary migration in metals. Boca Raton, FL: CRC Press; 1999. [21] Kaur I, Mishin Y, Gust W. Fundamentals of grain and interphase boundary diffusion. Chichester: Wiley; 1995.