Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.issp.ac.ru/libm/papers/1999-21.pdf Äàòà èçìåíåíèÿ: Sun Jun 25 20:23:04 2006 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 04:29:02 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: molecular cloud

PII:

Acta mater. Vol. 47, No. 14, pp. 3901±3914, 1999 # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Printed in Great Britain S1359-6454(99)00240-2 1359-6454/99 $20.00 + 0.00

MISORIENTATION DEPENDENCE OF INTRINSIC GRAIN BOUNDARY MOBILITY: SIMULATION AND EXPERIMENT
M. UPMANYU1,2, D. J. SROLOVITZ2,3{, L. S. SHVINDLERMAN G. GOTTSTEIN 5
1

4,5

and

Department of Materials Science and Engineering, University of Michigan, Ann Arbor, MI 481092136, U.S.A., 2Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, U.S.A., 3Princeton Materials Institute, Princeton University, Princeton, NJ 08544, U.S.A., 4Institut fur Metallkunde und Metallphysik, RWTH Aachen, D-52056 Aachen, õ Germany and 5Insitute of Solid State Physics, Russian Academy of Sciences, Chernogolovka, Moscow distr. 142432, Russia (Received 2 February 1999; accepted 25 June 1999)

AbstractïBoth experimental and atomistic simulation measurements of grain boundary mobility were made as a function of temperature and boundary misorientation using the same geometry that ensures steady-state, curvature-driven boundary migration. Molecular dynamics simulations are performed using Lennard±Jones potentials on a triangular lattice. These simulations represent the ®rst systematic study of the dependence of intrinsic grain boundary mobility on misorientation. The experiments focus on high purity Al, with h111i tilt boundaries, which are isomorphic to those examined in the simulations. Excellent agreement between simulations and experiments was obtained in almost all aspects of these studies. The boundary velocity is found to be a linear function of the curvature and the mobility is observed to be an Arrhenius function of temperature, as expected. The activation energies for boundary migration varies with misorientation by more than 40% in the simulations and 50% in the experiments. In both the simulations and experiments, the activation energies and the logarithm of the pre-exponential factor in the mobility exhibited very similar variations with misorientation, including the presence of distinct cusps at low S misorientations. The activation energy for boundary migration is a logarithmic function of the pre-exponential factor in the mobility, within a small misorientation range around low S misorientations. # 1999 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved. Keywords: Grain boundaries; Computer simulation; Kinetics

1. INTRODUCTION

Most analyses of processes involving the motion of grain boundaries explicitly assume that grain boundary velocity v is directly proportional to the driving force F: v MF 1

system accompanying a reduction in grain boundary area, the rate of grain boundary migration is v Mgk 3

where the constant of proportionality M is the intrinsic grain boundary mobility (see, e.g. Ref. [1]). If we assume that boundary mobility is controlled by a single activated process, we should expect that the temperature dependence of the boundary mobility may be described using an Arrhenius relation M M0 e
þQakB T

2

where Q is the activation energy (enthalpy) for migration, kB is Boltzmann's constant and M0 is a (nearly) temperature independent pre-exponential factor [2]. In the case of curvature-driven grain boundary migration, where the driving force is associated with the decrease in the free energy of the
{To whom all correspondence should be addressed.

where g is the grain boundary energy (per unit area) and k is the grain boundary curvature. While the curvature of the grain boundary is a purely geometrical variable, the mobility M and the grain boundary energy g are intrinsic material parameters. The grain boundary mobility is known to vary by several orders of magnitude with variation in grain boundary crystallography within a single material [3±6], while the grain boundary energy rarely varies by more than a factor of two (neglecting twins and very low angle boundaries). Therefore, the key parameter determining the variation of grain boundary velocity within a given material is the variation in grain boundary mobility with boundary crystallography. Consequently, this variation in boundary mobility plays a key role in determining the evolution of crystallographic texture and grain boundary distribution during materials processing. Over the past several decades, many experiments have been performed and theories developed to

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clarify our understanding of grain boundary mobility. Initially, grain growth experiments were conducted in polycrystalline samples to extract an " average grain boundary mobility M from the relationship between grain size and time [7, 8] " "2 " D t2 þ D 0 AMt 4
Fig. 1. The geometry used in both the simulations (twodimensional) and experimental U-shaped half-loop grain boundary migration study. The half-loop width is w, the surface area of the half-loop grain is A and the velocity of the grain boundary at the base of the U is v ( þA aw).

" " where D t is the average grain size at time t, D 0 is " the initial grain size and M is the average grain boundary mobility. However, since this process involved a statistical averaging over grain boundaries whose structure continuously changed during grain size evolution, the mobility data extracted from such experiments did not yield reliable information on the driving force or the misorientation dependence of the grain boundary mobility. The ®rst experiments on individual grain boundaries were conducted by Aust and Rutter [3±5], who studied the eect of grain boundary misorientation and inclination on grain boundary migration in lead and lead ±tin alloys. In these early experiments, grain boundary migration was induced by a wellcharacterized sub-grain (striation) structure in one of the grains meeting a ¯at boundary. These authors observed a non-monotonic variation of the activation energy for grain boundary migration with misorientation, with well-de®ned minima at several low sigma (coincidence site lattice) grain boundaries. Rath and Hu [9] used wedge-shaped bicrystals to experimentally extract grain boundary mobilities over a wide range of (curvature) driving forces in zone-re®ned aluminum. In these studies, the grain boundary curvature and velocity increased as the boundary migrated towards the apex of the wedge. Their non-steady-state data suggested a power law velocity±driving force relation: v MF m , where the exponent m was in the range 1X5m12X3 and was a function of the boundary misorientation. Sun and Bauer used the ``reverse capillarity'' technique in high purity NaCl bicrystals to extract grain boundary mobilities [10, 11]. This experimental geometry resulted in a steadily decreasing driving force during the experiment. Sun and Bauer reported a linear relation between the velocity and the driving force over a limited range of driving forces and an Arrhenius-like temperature dependence of the mobility, albeit with a temperature dependent activation energy. In retrospect, these techniques clearly involve the motion of the grain boundary/sample edge contact lines, which can induce signi®cant drag on grain boundary migration (either intrinsically or via groove dragging). Constant integral curvature (hence constant driving force) migration experiments on single grain boundaries designed to yield steady-state grain boundary motion were ®rst performed by Shvindlerman and co-workers [6, 12] in various high purity metals (including Al, Sn and Zn) using a half-loop grain bicrystal geometry (see Fig. 1), as

well as a quarter-loop geometry. The quarter-loop geometry was ®rst introduced by Dunn and coworkers [13] and employed by Aust et al. [14]. As the tip of the half-loop migrates, the sides of the grain boundary remain nearly parallel. Therefore, the integral curvature of the grain boundary remains constant throughout the migration process, leading to constant curvature/constant velocity boundary migration. Since our earlier experiments yielded a linear relationship between driving force and boundary velocity, we conclude that thermal groove drag eects are minimal [15]. The existence of a strong variation of impurity drag with the heat of impurity segregation, impurity concentration and boundary velocity makes it clear that there are (at best) few reliable experimental data on intrinsic grain boundary mobility. Determining the intrinsic boundary mobility and its dependence on boundary crystallography is key to the development of a predictive understanding of grain boundary motion. Once this is put on a ®rm foundation, it will be much easier to account for such complicated extrinsic eects such as those associated with impurities, defect absorption/generation, and contact lines. This realization has led several researchers to examine grain boundary migration from the perspective of atomistic computer simulations. These include the early work of Jhan and Bristowe [16] and Sutton [17], that did not lead to any predictions of boundary mobility. More recently, both Shonfelder et al. [18] and Upmanyu õ et al. [19] performed molecular dynamics simulations of the migration of grain boundaries with just a few misorientations and showed that the boundary velocity was indeed proportional to the driving force and that the temperature dependence of the boundary mobility was Arrhenius. However, these studies did not systematically examine the variation of grain boundary mobility with boundary crystallography. The goal of the present paper is to systematically examine the variation of the intrinsic grain boundary mobility with grain boundary misorientation and temperature. In order to extract the essential el-


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ements of this dependence, special attention is focussed on the mobilities of boundaries with misorientations at or near several low sigma (high coincidence) misorientations and comparisons of these to general boundaries. The main approach employed in this study is molecular dynamics computer simulations with the half-loop bicrystal geometry employed with considerable success in the experiments of Shvindlerman and co-workers [6, 12, 20, 22]. The general trends in these results are directly compared with the experimental grain boundary mobility data. The plan of the present paper is to review the simulation and experimental approaches employed, demonstrate that the steadystate mobilities can be determined, and examine the temperature and misorientation dependence of boundary mobility. Comparisons between the simulation and experimental data are made throughout.
2. SIMULATION AND EXPERIMENTAL APPROACHES

not pinned by an impurity cloud. Unfortunately, it is not possible to conclude that there was no eect of impurity drag even in the free motion regime of grain boundary displacement. Assuming that this motion was indeed free migration of the grain boundary, the reduced mobility of the grain boundary M* can be calculated from the extracted migration rate as k w vw ö þ 5 M Mg Mg w Mgk 2 2 2 where we relate the curvature to the width of the U-shaped half-loop w: namely k 2aw. Simulations in this study are performed in two dimensions using the molecular dynamics simulation method and simple, empirical (Lennard± Jones) pair potentials. We employ the same Ushaped, half-loop bicrystal geometry used by Shvindlerman and co-workers and constrain the sample to lie entirely in the X±Y plane (see Fig. 1), as described in Ref. [6]. Three edges of the simulation cell are left free, so as to decrease the stresses produced during grain boundary migration. Three atomic planes are frozen on the fourth edge to maintain the desired grain misorientations. The simulations were all performed at constant temperature. The underlying crystal structure is that of a triangular lattice with a nearest neighbor spacing r0. For more details, see Ref. [19]. Times are reported in units of t, energies in units of the Lennard±Jones potential well depth, distance in units of the equilibrium atom separation r0, area in units of the perfect crystal area per atom a0 and times in units of t Mat r2 ae1a2 , where Mat is the atomic mass. The 0 simulations were performed with bicrystal misorientations corresponding to various general (e.g. S I ±308), singular (S 7±38X228 and 13 ± 32X238) and vicinal (near S 7 and 13) boundaries. The present study focuses on high angle boundaries, the misorientations examined here are limited to the range 308 ` y ` 408 (the entire range of unique boundary misorientations is 308 ` y ` 608 or, equivalently, 08 ` y ` 308). The rate of change of the area of the half-loop grain, is simply A vw (see Fig. 1), assuming a steady-state half-loop shape. If the migration is curvature driven, we may write A as vw þMgkw þMg 2 w þ2MgX 6 A w Hence, we extract the reduced mobilities from the rate of change of the area of the half-loop grain Mö Mg þ A X 2 7

The present simulations are designed to study steady-state, curvature-driven grain boundary migration. Grain boundary curvature-driven boundary migration was chosen for the present study because it is ubiquitous in material processing and other driving forces commonly entail modi®cation of boundary structure (e.g. as in recrystallization). In addition, the most unambiguous experimental measurements of grain boundary mobility were performed using curvature-driven growth [6, 12, 20, 22]. In order to measure intrinsic, steady-state grain boundary migration, the simulation geometry must be chosen such that the driving force does not change during the migration process and such that there are no impurities or triple junctions to produce drag. Shvindlerman and co-workers [6, 12] introduced the U-shaped half-loop bicrystal geometry shown in Fig. 1 in order to study steady-state, curvature-driven boundary migration. This geometry is inherently two-dimensional in that the grain boundary traverses the entire thickness of the thin samples and remains (nearly) perpendicular to the surface during grain boundary migration. The curved section of the grain boundary has been observed to migrate with a self-similar shape [6], enabling the quanti®cation of the grain boundary migration rate with the driving force (the curvature). In the experiments, the position of the grain boundary (wI5 á 10þ4 m) was followed using a continuous X-ray tracking method which monitors the change in orientation of the crystals meeting at the grain boundary [20]. The position of the grain boundary was plotted as a function of time and the grain boundary migration rate is extracted from the time derivative of the grain boundary displacement±time curve. Care was taken to ensure that the migration of the grain boundary is observed in an ``unloaded'' regime, i.e. when the grain boundary is

This equation implies that the reduced mobility is simply minus a half of the rate of change of the area of the half-loop grain and is independent of half-loop width. In order to determine the instan-


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taneous area of the half-loop grain, each atom is assigned to one of the two grains based upon the angles between it p its nearest neighbors (i.e. and atoms nearer than 2r0 ) relative to a ®xed direction in the laboratory frame. The assignment to each grain is made based on whether the average nearest neighbor angles (modulo 2p/6) are closer to those of the lattices interior to or exterior to the half-loop boundary. Unless otherwise noted, all data are averaged over at least three simulation runs.

The temporal evolution of the atomic con®guration of the half-loop computational cell is shown in Fig. 2 at T 0X175eak, w 25r0 and y 28X58. Following an initial, short time transient, the asconstructed boundary equilibrates and establishes a pro®le [Fig. 2(a)] which remains self-similar throughout the migration process [cf. Figs 2(a) and (b)]. The evolution of the area of this half-loop A with time t is shown in Fig. 3. The main trend in

Fig. 2. (continued opposite)


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this ®gure is for the area of the half-loop grain to decrease linearly with time. The undulations in this curve are primarily normal thermal ¯uctuations associated with the atomic nature of the boundary migration process. The linearity of A vs t demonstrates the steady-state nature of the grain boundary migration during half-loop retraction. The slope

of this linear plot yields the rate of change of area of the U-shaped half-loop, A . In some cases, the steady-state retraction of the half-loop is interrupted by periods in which the boundary migration is signi®cantly slowed. This is commonly due to the formation and emission of vacancies from the grain boundary as the system tries to eliminate the excess

Fig. 2. Two atomic resolution images of the simulated retracting half-loop at dierent times: (a) t 50t; (b) t 860t. In this case, T 0X175eak, w 25r0 and y 28X58. Note the similarity between the half-loop shapes.


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Fig. 3. Variation of the simulated area of the half-loop grain shown in Fig. 2 (T 0X175eak, w 25r and y 28X58) with time.

0

grain boundary volume associated with a decrease in the total length (area) of the grain boundary (see Ref. [21] for details). In such cases, the A data are taken at times following such vacancy emission events, as observed in images of the atomic con®gurations. Self-similar migration of U-shaped half-loops and quarter-loops has also been directly observed in the experiments, where X-ray interface tracking was used to identify the position of the grain boundary at various instants of time [20, 22]. Examples of the grain boundary displacement±time graph (half-loop area is proportional to displacement times width) obtained using the continuous X-ray interface tracking technique [20] in high purity Al are shown in Fig. 4 for h111i tilt grain boundaries for a misorientation of y 40X58 at temperatures of T 777 and 831 K. Just as in the simulations, the grain boundaries in the experiments migrate at constant rates. The small ¯uctuations observed in the experimental data in Fig. 4 are associated with the method in which the boundary position is detected. The migration rate v, extracted from the data taken at these two temperatures (777 and 831 K) in Fig. 4 are v 4X1 and 51.7 mm/s, respectively. The dependence of the grain boundary velocity on the driving force for boundary migration is determined by performing a series of simulations at a ®xed temperature (T 0X125eak) for dierent half-loop widths (i.e. dierent curvatures). Since A ö is proportional to the reduced mobility M Mg [equation (7)], these data can be used to determine

the curvature dependence of the boundary mobility. Figure 5 shows the boundary velocity v ( A aw) and the reduced mobility as a function of the grain boundary curvature k ( 2aw). At small curvatures (large half-loop widths), the grain boundary velocity is a linear function of the grain boundary curvature, as expected for curvature-driven growth [equation (6)]. Consequently, the reduced mobility M* is independent of curvature [see equations (6) and (7)] in this low curvature regime. These results con®rm the assumption that the velocity is indeed proportional to the grain boundary curvature [equation (3)] and that the boundary mobility is independent of the magnitude of the driving force. In the present simu lations, we ®nd that A I þ0X26r0 2 at, which corresponds to a reduced mobility of Mö Mg 0X13r0 2 at. When the grain boundary curvature becomes very large (i.e. w becomes small), the grain boundary velocity shows strong deviations from a linear dependence on boundary curvature (Fig. 5) and the grain boundary velocity exceeds our expectations based upon equation (6). Deviations from pure curvature-driven boundary migration are only signi®cant for a half-loop width below 20r0. This is presumably attributable to elastic interactions between dierent sections of the grain boundary, non-dissipative types of boundary motion at high velocity and/or discreteness eects associated with the atomic scale grain boundary pro®le [19]. These data suggest that it is possible to extract meaningful grain boundary mobilities from these types of simu-


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Fig. 4. Displacement of the top of the quarter-loop grain with time for two experiments performed in high purity Al at 777 and 831 K with a misorientation of 38.28.

Fig. 5. Simulation results showing both the reduced mobility (left side) and boundary velocity (right side) vs curvature. For curvatures less than approximately 0.1/r0, the boundary velocity is a linear function of curvature and the reduced mobility is independent of curvature. Substantial deviations from this behavior are observed at large curvatures. The reduced mobility curve is shifted upwards by 0.05a0/t.


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lations provided that the half-loop width is suciently large (i.e. above approximately 20r0). The derivative of the experimental displacement± time graphs (e.g. Fig. 4) with respect to time provides an experimental measure of the velocity v of the grain boundary migration. Figure 6 presents experimental data on the variation of the velocity of grain boundary migration with driving force k/g for U-shaped half-loops [18] (normalized by the interface energy) for h111i tilt grain boundaries in high purity Al at a constant temperature. Examination of this plot shows that the grain boundary velocity is a linear function of the driving force for suciently high driving forces, in agreement with the atomistic simulation results. At small grain boundary curvatures, Fig. 6 shows that the grain boundary velocity is slower than expected. This is presumably attributable to impurity drag eects at low velocities. The data in this section clearly demonstrate that experiment and atomistic simulation show excellent correspondence with each other and that steadystate grain boundary migration is achieved using the U-shaped half-loop bicrystal geometry (provided that the half-loop width is suciently large and impurity drag eects are negligible). Both experiment and simulation clearly demonstrate, that grain boundary velocity is proportional to grain boundary curvature.

The temperature dependence of the reduced grain boundary mobility is determined by performing a series of simulations over a temperature range from T 0X075eak to 0X250eak and with a half-loop width w for which classical curvature-driven growth occurs, w 25r0 . The simulations were performed for boundary misorientations corresponding to singular (low S), vicinal (near singular) and general grain boundaries. Figure 7 shows the variation of the logarithm of the reduced mobility Mg vs inverse temperature for S 13 (y 32X218), near S 13 (y 338, 31X58), S 7 (y 38X238), near S 7 (y 38X988, 37X518) and S I (y 308). These data sets are each well ®tted by a straight line (ln Mö ln Mö þ QakB T), the slope of which yields 0 the activation energy for grain boundary migration (þQ/kB) and the y-intercept (1aT 0) yields the pre-exponential factor for the intrinsic reduced mobility, for each grain boundary. The data in Table 1 show that the activation energy for these grain boundaries lies between 0.3e and 0.5e, where e is the bond strength (i.e. the well depth of the Lennard±Jones pair potential). The linearity of the data (on the axes of Fig. 7) demonstrates that the mobility is indeed an Arrhenius function of temperature [as per equation (2)], at least in the temperature range examined (QakTw1). This is

Fig. 6. Experimental measurements of grain boundary velocity vs boundary curvature for a general grain boundary in high purity Al.


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Fig. 7. Arrhenius plots of reduced mobility (logarithm of reduced mobility vs inverse temperature) for a series of simulations corresponding to misorientations at or near S 7 (®lled circles, squares and diamonds), at or near S 13 (open symbols) and S I (i.e. y 308, inverted, ®lled triangles). The S 7 series data are shifted upwards by ln(2) and the S I curve is shifted downwards by ln(1/8) for clarity.

consistent with the predictions of absolute reaction rate theory. Figure 8 shows the experimentally determined reduced mobilities for h111i tilt grain boundaries in high purity Al (y 38X28 and 40.58) vs inverse temperature in the same form as the simulation data (Fig. 7). The reduced mobilities for both boundaries are well ®tted by straight lines, yielding activation energies of 1.25 and 2.2 eV, respectively. The linearity of the two data sets in Fig. 8 demonstrate that the reduced grain boundary mobility is Arrhenius with temperature, con®rming the results obtained through computer simulations above and the ansatz in equation (3). It is not possible to make accurate direct comparisons between the simulation and experimental data (e.g. activation energies for boundary migration) because of the inaccuracies of the interatomic potentials and the dierent crystal structures in the two cases. Nonetheless, crude comparisons are possible based on scaling the parameters in the interatomic potential and similarities between the crystallography. While the experimental study focuses on h111i tilt grain boundaries in a face centered cubic material (Al), the simulations are performed on a two-dimensional triangular lattice. Since the two-dimensional triangular lattice is isostructural with {111} planes, the grain boundaries are closely (but by no means completely) equivalent. We can relate the experimental measurements of

the activation energies to the simulation results by normalizing the experimental results by the eective bond energy, e: i.e. the cohesive energy of the solid Ecoh divided by half the number of nearest neighbor atoms. Using the experimental value for the activation energy for the migration of a S 7 grain boundary in Al [22] and normalizing it by the bond energy (e), we ®nd Qae 2X2. The corresponding value from the simulations is Qae 0X30. The agreement between the two is poor, with the experimental value 7.3 times larger than that predicted by the simulations. A similar discrepancy was observed by Shonfelder et al. when comparing õ
Table 1. The activation energies and pre-exponential factors for reduced mobilities for various misorientations extracted from the performed simulations y (8 ) 40.000 39.500 38.976 38.213 (S 7) 37.517 37.000 35.567 33.992 33.000 32.204 (S 13) 32.000 31.500 31.000 30.000 (S I) Q (e ) 0.4562 0.4102 0.3652 0.3022 0.3902 0.4292 0.4682 0.4562 0.4352 0.3192 0.3582 0.3932 0.4252 0.4662 0.013 0.033 0.021 0.039 0.030 0.044 0.033 0.020 0.031 0.020 0.037 0.015 0.027 0.025 ln(M0g) 17.422.2 18.122.6 18.023.7 14.823.5 18.924.8 20.223.2 18.423.0 18.223.4 14.622.9 7.521.3 10.121.9 12.122.8 12.523.3 15.520.6


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Fig. 8. As in Fig. 7, but for experimental data on Al for misorientations of y 38X28 (S 7) and 40.58.

three-dimensional molecular dynamics simulation of h100i ¯at twist boundary motion to experimental results [18]. This much larger experimental activation energy for boundary migration is likely attributable to impurity drag. Even though the purity of the Al in these experiments is very high impurity, the total impurity content at or near the grain boundary can still be quite substantial. If the distribution of impurities in the material are indeed modi®ed by the grain boundary, the activation energy for grain boundary migration can be as large as the sum of the bulk impurity diusivity plus the impurity± boundary binding energy. The activation energy for diusion of a typical impurity (Mn) in Al, is 2.2 eV [23] or Qae 1X7, which diers from the experimental activation energy for boundary migration by 0.5e (0.65 eV) and represents a plausible impurity± boundary binding energy. There is a large body of evidence which shows that decreasing impurity concentrations decreases the activation energy for boundary migration [3±6]. The fact that the present simulations are two-, rather than three-dimensional, suggests that the boundary structures and the nature of the activated states in the simulation and the experiments may be substantially dierent. There is no doubt that the activated state is more constrained in two dimensions, compared with three. This should lead to higher activation energies and, hence, does not resolve the discrepancy. The Lennard±Jones potentials used in the present simulations are known to be too sti in compression to describe metals. This too would be expected to

yield activation energies that are too high (in contradiction to the present results). Therefore, our preliminary conclusion is that the discrepancy between simulation and experimental determination of the intrinsic grain boundary mobility is primarily attributable to impurity drag eects in the experiments (i.e. the experiments are not measuring intrinsic boundary mobility).
5. MISORIENTATION DEPENDENCE OF MOBILITY

A series of simulations was performed for misorientations in the range of 308 ` y ` 408 (corresponding to singular, vicinal and general grain boundaries) and for temperatures in the range from T 0X075eak to 0X250eak (half-loop width w 25r0 ). These data were tabulated in the form of Fig. 7 and ®tted to obtain the activation energy for boundary migration and the pre-exponential factor for the reduced mobility Mö (Table 1). The vari0 ations of the activation energy and the pre-exponential factor with misorientation are shown in Fig. 9. Clearly neither Q nor Mö vary monotonically with 0 misorientation. The activation energy vs misorientation plot shows two pronounced minima (cusps) at misorientations corresponding to S 13 (y 32X208) and S 7 (y 38X238). The variation of the pre-exponential factor Mö for the reduced 0 mobility also shows distinct minima (cusps) at the same misorientations (S 13, y 32X208 and S 7, y 38X238). The variation of Mö with y is 0 much less smooth than the Q vs y plot, however to within the error bars, each curve shows smooth


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variation with y away from the cusps. The similarities between the variation of Q and ln(M0g) with y suggests that the pre-exponential factor and activation energy for boundary migration are positively correlated. This is discussed in the next section. The variation of the activation energy and the pre-exponential factor (for the reduced mobility) with misorientation of h111i tilt grain boundaries in high purity Al has been extracted from the experimental data and is shown in Fig. 10. As in the simulations (Fig. 9), this variation is not monotonic, with minima in both Q and Mö occurring at 0 or near y 38X28 (i.e. near S 7). No experimental data near S 13 (y 32X208) are available for comparison with the simulation results. As in the case of the simulation study, the misorientations for which maxima and minima in the activation energy vs misorientation curves are observed are replicated in the pre-exponential curves. This is experimental con®rmation of the close correlation between Q and ln(M0g) observed in the simulations. Similarly, the misorientation dependence of the grain boundary mobility in Al is much more pronounced at low temperature then at higher temperature, as in the simulations.

Fig. 10. Same as Fig. 9, but for the experimental data on Al.

Examination of the reduced mobility data in Figs 7 and 8 shows that the magnitude of the variation of the mobility with misorientation is temperature dependent. Assuming that the mobility is an Arrhenius function of temperature, we ®nd that the temperature dependence of the variation of the mobility with respect to misorientation d ln Mö d ln Mö 1 dQ 0 þ dy dy kB T d y 8

is controlled by the misorientation dependence of the activation energy alone. Therefore, the misorientation dependence of the mobility should be more pronounced at low temperatures than at high temperatures, as is borne out by Fig. 7 and Tables 2 and 3. Equation (8) also suggests that there is a speci®c temperature at which the variation of the mobility with misorientation is exactly zero (i.e. the mobility is misorientation independent). This too is discussed in the next section.
6. COMPENSATION EFFECT Fig. 9. The activation energy for boundary migration and the logarithm of the pre-exponential factor (b) of reduced mobility in the simulations as a function boundary misorientation. These data are tabulated Table 1. (a) the of in

The simulation and experimental results presented in the previous section suggest that the activation energy and the logarithm of the reduced mobility vary in the same manner with changes in boundary misorientation or, therefore, boundary


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Table 2. The reduced mobilities [in units of a2 atr0 ) from simulations performed for four dierent grain boundaries compared at two 0 dierent temperatures S 13 (y 32X208) T 0X125eak (I0X3Tm ) T 0X200eak (I0X5Tm ) 0.27420.038 0.86520.148 y 30X008 0.15520.048 0.64920.005 y 33X998 0.39220.039 1.01420.124 S 7 (y 32X208 0.68220.067 1.51520.285

Table 3. The experimentally extracted reduced mobilities (in units of m2/s) for three dierent boundaries, compared at two dierent temperatures y 38X28 (S 7) T 633 K (I0X7Tm ) T 753 K (I0X8Tm 2X5 á 10 9X5 á 10þ
þ9 8

y 42X08 8 á 10 1X5 á 10þ7
þ10

y 40X508 2X5 á 10þ10 8X0 á 10þ8

structure. In order to quantify this relationship we plot ln(M0g) vs Q (Fig. 11). These data clearly show that a relationship of the form Q a lnM0 g b 9 exists for boundaries of similar misorientation, where a and b are constants. In particular, we ®nd that the data point for the S 7, y 38X218 boundary and those for the boundaries that are only slightly misoriented from it (i.e. its vicinal boundaries, 378y398) accurately lie on one line (one set of a and b), while the data point for the S 13, y 32X28 boundary and those for the

boundaries that are only slightly misoriented from it (31X58y338) accurately lie on a dierent straight line. The boundaries that are outside of these angular ranges are widely distributed and fall on neither line. The fact that the data for the near S 7 and near S 13 boundaries fall onto distinct lines suggest that the underlying structure of these two sets of boundaries is quite distinct. Figure 12 shows the same type of plot as in Fig. 11 [i.e.Q vs ln(M0g)] for the experimental data on S 7 and near S 7 h111i tilt grain boundaries in high purity Al. As in the simulation results, the experimental data all fall on a single straight line.

Fig. 11. Logarithm of the pre-exponential factor of the reduced mobility vs the activation energy for boundary migration for a range of misorientations for the simulation data presented in Fig. 9 (error bars indicated in Table 1). The data plotted as ®lled squares correspond to the boundaries with misorientations at or near S 7 (38.328, 38.978, 37.518, 37.008). The data plotted as ®lled diamonds correspond to the boundaries with misorientations at or near S 13 (32.208, 32.008, 31.508 and 33.008). The open circles correspond to misorientations outside these ranges.


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Fig. 12. Same as Fig. 11, but for the experimental Al data presented in Fig. 10.

These experimental and atomistic simulation data, taken in toto, clearly demonstrate that the variation of the pre-exponential factor and activation energy for grain boundary mobility amongst structurally related boundaries is well described by the relationship in Fig. 10. This same relationship was obtained earlier [20] based upon a wide range of grain boundary mobility experimental measurements (dierent materials, dierent rotation axes). The linear relation between the activation energy and the logarithm of the pre-exponential factor describing grain boundary mobility, represented by equation (9), shows that an increase in activation energy (which tends to decrease the boundary mobility) will be compensated by a concomitant increase in the pre-exponential factor (which tends to increase the boundary mobility). This implies that the mobilities of structurally related boundaries will be much closer than the dierences in activation energy would normally suggest. This is indeed the case, as the data in Figs 7 and 8 show. This eect has become known as the compensation eect, the thermodynamic and kinetic origin of which is discussed in Ref. [24]. The existence of the compensation eect [as represented by equation (9)], and the functional form of the mobility [equation (2)], implies that there exists a compensation temperature, Tc, at which the mobilities of these closely related grain boundaries are all equal. Solving equations (2) and (9) for two dierent boundaries and setting them equal implies

that Tc aakB . Extraction of the values of a from the simulation data in Fig. 11 implies that Tc aakB 0X4020X02eakB for the near S 7 boundaries and Tc aakB 0X19220X018eakB for the near S 13 boundaries. This is very close to that obtained from the crossing of the mobilities (simulation) vs inverse temperature curves in Fig. 7: from Tc 0X38eakB to 0X41eakB for the near S 7 boundaries and from Tc 0X180eakB to 0X200eakB for the near S 13 boundaries. The experimental data for the near S 7 boundaries in Al in Fig. 12 imply Tc aakB 719 K and the temperature where the two curves in Fig. 8 cross is Tc 770 K. The accuracy to which the values of Tc are obtained using the two approaches (simulation H1%, experiment H7%) is an indication of the accuracy to which the Arrhenius form of the mobility [equation (2)] and equation (9) ®t the data. Given the uncertainty in the experimental and simulation data, this is excellent agreement.
7. CONCLUSIONS

We have made extensive measurements of grain boundary mobility as a function of temperature and boundary misorientation in a geometry where boundary migration is driven by grain boundary curvature. Both molecular dynamics simulations using Lennard±Jones potentials on a triangular lattice and experiments on h111i tilt boundaries in high purity Al were performed. These simulations


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UPMANYU et al.: MISORIENTATION DEPENDENCE REFERENCES 1. Humphreys, F. J. and Hatherly, M., Recystallization and Related Annealing Phenomena. Pergamon Press, Oxford, 1996, Chap. 4. 2. Turnbull, D., Trans. Am. Inst. Min. Engrs, 1951, 191, 661. 3. Aust, K. T. and Rutter, J. W., Trans. Am. Inst. Min. Engrs, 1959, 215, 119. 4. Aust, K. T. and Rutter, J. W., Trans. Am. Inst. Min. Engrs, 1959, 215, 820. 5. Aust, K. T. and Rutter, J. W., Acta metall., 1965, 13, 181. 6. Aristov, V. Y., Fridman, Y. M. and Shvindlerman, L. S., Physics Metals Metallogr., 1973, 35, 859. 7. Gordon, P. and Vandermeer, R. A., Recrystallization, Grain Growth and Textures. Am. Soc. of Metals, Cleveland, OH, 1966. 8. Holmes, E. and Winegard, W. C., Can. J. Phys., 1959, 37, 469. 9. Rath, B. B. and Hu, H., Trans. Am. Inst. Min. Engrs, 1969, 245, 1577. 10. Sun, R. C. and Bauer, C. L., Acta metall., 1970, 18, 635. 11. Sun, R. C. and Bauer, C. L., Acta metall., 1970, 18, 639. 12. Aristov, V. Y., Fradkov, V. E. and Shvindlerman, L. S., Soviet Phys. Solid St., 1980, 22, 1055. 13. Dunn, C. G., Daniels, F. W. and Bolton, M. J., Trans. Am. Inst. Min. Engrs, 1949, 185, 785. 14. Aust, K. T., Harrison, R. J. and Maddin, R., J. Inst. Metals, 1956, 85, 57. 15. Gottstein, G. and Shvindlerman, L. S., Scripta metall. mater., 1992, 27, 1521. 16. Jhan, R. J. and Bristowe, P. D., Scripta metall. mater., 1990, 24, 1313. 17. Sutton, A. P., in Computer Simulation in Materials Science: Nano/Meso/Macroscopic Space and Time Scales, ed. H. O. Kirchner, L. P. Kubin and V. Pontikis. NATO-ASI Series E, Vol. 308, 1996, p. 163. 18. Shonfelder, B., Wolf, D., Philpot, S. R. and õ Furtkamp, M., Interface Sci., 1997, 5, 245. 19. Upmanyu, M., Smith, R. W. and Srolovitz, D. J., Interface Sci., 1998, 6, 41. 20. Gottstein, G., Molodov, D. A. and Shvindlerman, L. S., Interface Sci., 1998, 6, 41. 21. Upmanyu, M., Srolovitz, D. J., Gottstein, G. and Shvindlerman, L., Interface Sci., 1998, 6(4), 289. 22. Molodov, D. A., Czubayko, U., Gottstein, G. and Shvindlerman, L. S., Scripta metall. mater., 1995, 32, 529. 23. Brandes, E. A. (ed.), Smithells Metals Reference Book, 6th edn. Butterworths, London, 1983, pp. 13±15. 24. Gottstein, G. and Shvindlerman, L. S., Interface Sci., 1998, 6(4), 267.

represent the ®rst systematic study of the dependence of intrinsic grain boundary mobility on misorientation. In both experiments and simulations, steady-state grain boundary migration was achieved. The boundary velocity was found to be a linear function of the curvature, as expected. Plots of the logarithm of the mobility vs inverse temperature were linear; indicating that the temperature dependence of the mobility is Arrhenius. The same plots showed that the activation energies varied by more than 40% in the simulations and 50% in the experiments with changes in misorientation. The mobilities of boundaries in a small misorientation range around low S misorientations became the same at a well-de®ned temperature, Tc (that varied from one low S misorientation to another). In both the simulations and experiments, the activation energies and the logarithm of the pre-exponential factor in the mobility exhibited very similar variations with misorientation, including the presence of distinct cusps at low S misorientations. This is a manifestation of the compensation eect. The activation energy for boundary migration was shown to be a logarithmic function of the pre-exponential factor in the mobility, within a small misorientation range around low S misorientations. This explains the observation that grain boundary mobility often does not vary as drastically with misorientation as the activation energies themselves would suggest and the observed equality of boundary mobilities at a well-de®ned temperature. Aside from a discrepancy in the absolute value of the activation energy, we ®nd that overall excellent agreement between the simulation and experimental data was obtained. Such a discrepancy in the absolute values is to be expected when comparing two-dimensional simulations using oversimpli®ed interatomic potentials with real experiments.
Acknowledgements ïM.U. and D.J.S. gratefully acknowledge the ®nancial support of the Division of Materials Science of the Oce of Basic Energy Sciences of the United States Department of Energy Grant #FG0288ER45367, under whose auspices this research was performed. L.S.S. and G.G. express their gratitude to the Deutsche Forschungsgemeinschaft and to the Russian Foundation for Fundamental Research under contract N 96-02-17483 for ®nancial support for their collaboration.