Computer simulation of the elastically driven migration of a flat grain boundary

Transcription

1 Acta Materialia 52 (2004) Computer simulation of the elastically driven migration of a flat grain boundary H. Zhang *, M.I. Mendelev, D.J. Srolovitz Department of Mechanical and Aerospace Engineering, Princeton Materials Institute, Princeton University, Bowen Hall, 70 Prospect Avenue, Princeton, NJ 08540, USA Received 23 December 2003; received in revised form 2 February 2004; accepted 4 February 2004 Available online 5 March 2004 Abstract This paper describes a strategy for the measurement of boundary mobility using molecular dynamics (MD) simulations and its application to the migration of nominally flat h001i tilt grain boundaries in nickel, as described using an EAM potential. Determination of the driving force for boundary migration requires proper accounting for non-linear elastic effects for strains of the magnitude needed for MD simulation of stress-driven boundary migration. The grain boundary velocity was found to be a nonlinear function of driving force, especially at low temperature. However, extrapolation of the data to small driving force allows for the determination of the mobility at all temperatures. The activation energy for grain boundary migration was found to be 0.26 ev. This demonstrates that boundary migration in pure metals is not athermal, but the activation energy is much smaller than expected based upon experimental measurements. This discrepancy is similar to that found in earlier simulation measurements. Ó 2004 Published by Elsevier Ltd on behalf of Acta Materialia Inc. Keywords: Grain boundaries; Molecular dynamics; Nickel; Mobility 1. Introduction The vast majority of applications of metals in modern technology use these materials in their polycrystalline form. The properties of such polycrystalline materials are, not surprisingly, strongly influenced by the distribution of grain orientations and the density and nature of the grain boundaries within the material. For example, corrosion, creep, yield and fracture properties can all be modified by manipulating grain size, texture and/ or grain boundary character. Most types of thermal mechanical processing (e.g., annealing, hot rolling, extrusion, forging, super plastic deformation, recrystallization and grain growth) modify the grain structure and the distribution of grain boundaries whether intentionally or not. One key to predicting the effects of these types of processing methods on materials properties is to understand how grain boundaries move. This remains * Corresponding author. Tel.: ; fax: address: hzhang@princeton.edu (H. Zhang). one of the longest standing, yet least understood problems in physical metallurgy. In the present paper, we develop a widely applicable method for studying the motion of individual grain boundaries as a function of temperature and grain boundary bicrystallography. In most studies of grain boundary migration, it is assumed that the grain boundary velocity m is proportional to the driving force P: m ¼ MP: ð1þ The theoretical justification for this assumption appears to be sound provided that the driving force is sufficiently small [1]. The coefficient of proportionality M is called the grain boundary mobility. In general, however, there is no reason to assume that the velocity driving force relation is linear. Hence, the mobility can be uniquely defined only in the small driving force limit: M ¼ om op : ð2þ P¼0 This mobility is usually considered to be an intrinsic boundary property, which does not depend on the type of driving force. On the other hand, the mobility should /$30.00 Ó 2004 Published by Elsevier Ltd on behalf of Acta Materialia Inc. doi: /j.actamat

2 2570 H. Zhang et al. / Acta Materialia 52 (2004) be expected to be a function of temperature, the five parameters that describe the grain boundary bicrystallography and the boundary composition. Because the grain boundary bicrystallography parameter space is five-dimensional (three Euler angles define the relative misorientation of the two grains and two parameters describe the inclination of the boundary plane), most experimental and simulation studies of boundaries have focused on a restricted set of boundary types (e.g., pure twist or symmetric tilt boundaries). The difficulty in obtaining reliable boundary mobilities in pure materials makes it difficult to develop a systematic understanding of how impurities or solute modify boundary mobility since most theoretical descriptions assume that the mobility of the boundary in the pure system is known. Because boundary mobility is key to any quantitative description of a wide range of types of microstructure evolution, we should expect the literature to be replete with measurements. Unfortunately, this is not the case in fact there have been only a relatively small number of explicit experimental determinations of the mobilities of individual boundaries (for a review, see [1]). This may be attributed, in part, to the difficulty in performing such experiments. They require the production of bicrystals with accurately determined crystallography and with particular boundary shapes. The measurements should also be made in pure materials. This is particularly difficult given the fact that even extremely small concentrations of impurities can produce large impurity concentrations on the grain boundary as it sweeps through the material. For example, the analysis presented in [2] clearly shows that impurity concentrations as 1 ppm can dramatically change the boundary mobility in comparison with that in a truly pure material. Therefore, it is doubtful whether intrinsic, or impurity free, grain boundary mobilities can be extracted from experiment. Therefore, atomistic simulations provide the most realistic hope of obtaining this basic materials information. (We note that it is much easier to perform simulations of boundary motion in perfect pure materials than in impure materials.) Molecular dynamics (MD) is the atomistic method of choice for observing dynamical effects, such as grain boundary migration. Two distinct approaches have been developed to study grain boundary motion in pure materials with molecular dynamics. The first used an applied strain to drive boundary migration [3]. The authors studied the migration of a flat R29 twist boundary in a system described by a Lennard Jones potential with parameters chosen to describe copper. Although pair potentials are known to give erroneous results for such simple properties as the difference between the elastic constants C 12 and C 44 and the vacancy formation energy, the authors assumed that such a potential can reproduce the main features of grain boundary migration. This is not completely unreasonable given a body of experience that shows that pair potentials can reproduce many qualitative features of materials behavior. The driving force for boundary migration was the difference between the stored elastic energies in the two grains at fixed strain. Since the authors used periodic conditions, they had to use a simulation cell large enough to accommodate the motion of two boundaries and had to modify the equations of motion of the atoms [4] to keep the driving force fixed. During the simulation, each boundary traveled approximately 10 lattice parameters (35 A). The main conclusions of this study were that: (i) the velocity driving force relation is linear (up to applied strain of 4%) and (ii) the grain boundary mobility is an Arrhenius function of temperature with an activation energy of 0.23 ev/atom. The last result is in clear contradiction with experimental data. For example, the activation energy of the grain boundary migration in Al is about 1.5 ev/atom [2]. Such a difference appears to be too large to attribute to inadequacies of the interatomic potential. A second type of molecular dynamics simulation of (tilt) grain boundary migration relied on capillarity (curvature) rather than elasticity to drive boundary migration [5,6]. Using this approach, the authors extracted a composite parameter, the reduced mobility, which is the product of the mobility and the grain boundary stiffness, M ¼ Mðc þ c 00 Þ: ð3þ The stiffness is the sum of the grain boundary free energy c and its second derivative with respect to boundary inclination. The obvious advantage of this simulation method is that it is the reduced mobility, rather than the mobility itself, that is most measured in experiments [1]. Comparisons of the simulation results with experimental data show that the simulations are able to reproduce the dependence of the reduced mobility on misorientations. In addition, the simulations were able to reproduce the experimentally observed trends in the variation of activation energy with misorientation. However, the simulations showed that the activation energy for boundary migration was again an order of magnitude smaller than the experimental value. This general observation can be explained on the basis of the effects of impurities [7,8]. This approach has the disadvantage that it is difficult to deconvolute the boundary mobility from the measured reduced mobility. Additionally, since this approach, by its very nature, focuses on curved boundaries it is not obvious how it can be used to extract boundary mobility as a function of all five grain boundary bicrystallography parameters (especially inclination). In the present report, we present a new approach for extracting grain boundary mobilities from atomistic simulations. In this approach, grain boundary migration is driven by the application of a strain. The driving force is proportional to the difference in the elastic energy

3 H. Zhang et al. / Acta Materialia 52 (2004) stored in the two grains bounding a single grain boundary. The present method is readily applicable to a wide range of grain boundary types, including asymmetric boundaries. In comparison with capillarity-based methods, this approach allows direct extraction of the grain boundary mobility rather than the reduced mobility. In this report, we first describe the bicrystal geometry employed in the simulations and emphasize the importance of choice of simulation geometry. Next, we discuss the accurate determination of the driving force for boundary migration. We then apply this method to the determination of velocity driving force relation in pure Ni as a function of temperature. We find that the grain boundary mobility is an Arrhenius function of temperature with an activation energy that is much smaller than that found experimentally (in agreement with some earlier simulations). Finally, we discuss the origin of the non-linearities in the driving force vs. strain and the boundary velocity vs. driving force. 2. Simulation approach In order to study migration of flat grain boundaries, we must first choose both a method of applying a driving force and a corresponding bicrystal/simulation cell geometry. In order to ensure that the observed boundary migration corresponds to steady-state motion, we prefer a simulation method in which the driving force does not change with time during the simulation. At the same time, we prefer performing the molecular dynamics simulations within the NVT ensemble rather than applying a driving force through a non-equilibrium molecular dynamics approach (NEMD) in which the dynamics of the system are artificially modified (see, e.g. [4]). Additionally, in order to avoid grain boundary sliding, it is advantageous to elastically load the system such that there is no resolved shear across the boundary plane. In order to satisfy all of these requirements, we propose following simulation geometry. A periodic simulation cell containing a single grain boundary and two parallel free surfaces is constructed, as shown in Fig. 1. We then apply a biaxial strain, e xx ¼ e yy ¼ e, using the coordinate system described in the figure. The surfaces with normal z are kept free such that the bicrystal can expand or contract during the simulation without inducing a stress with a component in the z-direction, i.e., r zz ¼ r xz ¼ r yz ¼ 0. This implies that there is no shear across the grain boundary. In order to create an elastic driving force for boundary migration, the elastic energy stored in the two grains as a result of the application of the strain must be different. Note that in order for the driving force to be non-zero, the crystal must be elastically anisotropic. This is seldom a problem since isotropy is rare in crystalline solids. In order for the elastic Fig. 1. Schematic diagram of the simulation cell. The boundary normal is the z-direction, which coincides with the [0 0 1]-direction in the lower grain. The upper grain is rotated about [1 0 0]-axis by an angle h. The model is periodic in the x and y directions and the top and bottom surfaces are free. energy to be different in the two grains, the grain boundary must be neither a plane of mirror symmetry (nor a glide plane) nor a pure twist boundary. In the present simulations, we choose a tilt axis along the [0 1 0] and the boundary plane to be (0 0 1), all in the reference frame of the lower crystal, as shown in Fig. 1. The tilt angle, h, is also defined in this figure. This boundary is asymmetric. 3. Driving force for boundary migration The driving force for grain boundary migration in the present simulation method is the difference between the stored elastic energy densities in the two bounding grains. The stored elastic energy density is F ¼ 1r 2 ije ij : ð4þ Let the x, y and z axes of coordinates coincide with the crystallographic directions [1 0 0], [0 1 0] and [001] of the lower grain, respectively (see Fig. 1). For the case of a linear elastic cubic crystal (e.g., Ni), the stress components in the bottom grain are: r xx ¼ C 11 e xx þ C 12 e yy þ C 12 e zz ; r yy ¼ C 12 e xx þ C 11 e yy þ C 12 e zz ; ð5þ r zz ¼ C 12 e xx þ C 12 e yy þ C 11 e zz : Note that all shear stress stresses are zero, since the applied strain is purely balanced biaxial e xx ¼ e yy ¼ e and the simulation cell axes of aligned with the cubic axes of this grain. e zz is found from the constraint of zero

4 2572 H. Zhang et al. / Acta Materialia 52 (2004) traction in the direction normal to the boundary. This gives the free energy in the lower grain F as F ¼ ðc 11 C 12 ÞðC 11 þ 2C 12 Þ e 2 : ð6þ C 11 The elastic energy in the upper grain (rotated by h with respect to the lower grain about the [0 1 0]-axis) can be found using HookeÕs law, with the appropriate rotation, to be F þ ¼ ðc 11 C 12 ÞðC 11 þ 2C 12 Þ½8ðC 11 C 12 þ C a Þ C a ð1 cosð4hþþš e 2 ; 2½4C 11 ðc 11 C 12 þ C a Þ ðc 11 þ C 12 ÞC a ð1 cosð4hþþš ð7þ where C a ¼ 2C 44 C 11 þ C 12 is a measure of the anisotropy in they system (C a ¼ 0 for an isotropic system). The driving force for boundary migration is the difference in the stored elastic energy densities of the upper and lower grains, P ¼ F þ F ¼ ðc 11 C 12 ÞðC 11 þ 2C 12 Þ 2 C a sin 2 ð2hþ C 11 ½4C 11 ðc 11 C 12 þ C a Þ ðc 11 þ C 12 ÞC a ð1 cosð4hþþš e2 : ð8þ As expected, the driving force is proportional to the square of the applied strain and, hence, is the same in compression and tension. The direction (sign) of the driving force is determined only by the elastic constants and the misorientation. Note that if the system is isotropic (C a ¼ 0), the driving force is exactly zero for all misorientations h. The accuracy of this approach for determining the driving force for boundary migration rests on the linearity of the stress strain relation. At small strains, linear elasticity must be valid. However, if the strain is large, non-linear effects may be important. An alternative approach to determine the driving force for boundary migration is to measure the stress as a function of strain in each grain directly from an MD simulation and determine the strain energy density within each grain by integrating the stress strain curve as the strain) and the elastic constants C 11 ; C 12 and C 44 for use in Eq. (8) over the appropriate temperature 800 K 6 T (K) K. The lattice parameter as a function of temperature is shown in Fig. 2, from which we determine the linear coefficient of thermal expansion to be K 1 for this potential. Although this is approximately four times larger than the experimental value K 1 [10], this is not of concern for our determination of the driving force for boundary migration. In order to get a sense of the temperature range employed, it is useful to compare it with the melting point for the potential employed in the MD simulations. The melting point was determined using the two phase co-existence method [11]. The melting point was found to be K for this interatomic potential, while the experimentally determined value for Ni is 1728 K. Fig. 3 shows the stress strain relation for perfect crystals of orientations corresponding to the upper and lower grains in Fig. 1 with h ¼ 36:8 (i.e., a so-called R5 misorientation) at 800 and 1400 K. These simulations were performed for the strain state described. This figure clearly demonstrates that deviations from a linear stress strain relation are readily apparent for strains as small as 1%. The deviations from linearity are larger in Fig. 2. The lattice parameter as a function of absolute temperature. F ¼ Z e 0 ðr xx þ r yy Þde 0 ; ð9þ where we recall that e xx ¼ e yy ¼ e and r iz ¼ 0 in this simulation geometry. This approach requires no assumption about the linearity of the stress strain relation. In order to test these two approaches, we performed a series of MD simulations to measure the stress strain relation. The simulations were performed using the embedded atom potential for Ni described in [9]. In order to perform these simulations, we must determine the equilibrium lattice parameter (in order to determine Fig. 3. The in-plane stress, r xx þ r yy, vs. the applied strain, e (see text) of the upper (+) and lower ()) grains in Fig. 1. The continuous curves correspond to the linear elastic predictions of Eq. (10) and the symbols were determined directly from the simulation.

5 H. Zhang et al. / Acta Materialia 52 (2004) Fig. 4. The driving force for grain boundary migration as a function of the strain squared (e 2 ) at 800 K for both tension and compression. The solid lines are the simulation results and the dashed line corresponds to the linear elastic predictions of Eq. (8). compression than in tension. This is a result of the anharmonicity of the lattice which is stiffer in compression than in tension. We can describe the dependence of the stress on the strain in terms of a power series as r xx þ r yy ¼ k 1 e þ k 2 e 2 þ; ð10þ where the k i are constants which can be determined from fitting to data as in Fig. 3. The elastic driving force for boundary migration can then be found by inserting Eq. (10) into Eq. (9) for the top and bottom grains as P ¼ F þ F ¼ 1 2 ðkþ 1 k 1 Þe2 þ 1 3 ðkþ 2 k 2 Þe3 þ ð11þ Fig. 4 shows a comparison between the driving force calculated within the linear elasticity framework Eq. (8) and that determined from fitting the simulation data to Eq. (11). At the largest strain used in the present study (i.e., jej ¼0:02), the maximum difference between the linear elastic predictions (using elastic constants determined for the same interatomic potential) and the simulation data (together with Eq. (11) up to second order) is 13%. This demonstrates that elastic non-linearities can have an important effect on the determination of the driving force. It is interesting to note that the driving force for the compressive applied strain case can be significantly smaller than in the case of a tensile applied strain of the same magnitude. This difference is associated with the difference between the non-linear elastic behaviors of the differently oriented grains. The sign of the deviation from linear P e 2 behavior in Fig. 4 depends on the sign of the applied strain. Again, this is a result of higher-order terms in Eqs. (10) and (11). unstrained simulation cell dimensions were 10a 0 along the x and y directions, and 30 40a 0 in the z-direction, where a 0 is the equilibrium lattice parameter. The grain boundary position was initially located 10a 0 above the lower (normal z) free surface. This far enough above the lower free surface such that the boundary and surface do not interfere with one another and far enough below the upper free surface that the boundary could migrate a sufficient distance to ensure reliable determination of the boundary velocity without the boundary interfering with the upper surface. The MD simulations were performed using the velocity Verlet algorithm to integrate the equations of motion for the atoms. Two methods were employed to maintain constant temperature, velocity rescaling [12] and Holian Hoover [13], in order to ensure that the algorithm employed had no significant effect on the boundary mobility. The typical MD step was 1 fs. Depending on temperature and driving force, the total duration of the simulations was between 1 and 10 ns. The position of the boundary was determined based upon the atomic scale energy and/or atomic coordination numbers. Initial simulations were performed in the absence of an applied strain. Fig. 5 shows the mean boundary position vs. time. The mean boundary position fluctuates about its initial value, implying that the mean velocity is zero over sufficiently long time intervals. This further shows that any interactions between the boundary and the free surfaces are negligible and there are no systematic contributions to the driving force in the absence of an applied strain. As expected, the amplitude of the fluctuations is small at low temperature and increases with increasing temperature. The amplitude of the fluctuations determines the statistical noise in the determination of the boundary mobility in the presence of an applied strain. At the lowest temperature studied, the thermal noise is insignificant provided that the simulation is performed over a sufficient time to allow the boundary to migrate a distance of 20 A. However, at 1400 K the thermal noise was large enough to yield an 4. Simulation of grain boundary migration The simulation cell dimensions were chosen in accordance with the equilibrium lattice parameter at the simulation temperature and the applied strain e. The Fig. 5. The mean grain boundary position vs. time at several temperatures and no applied strain.

6 2574 H. Zhang et al. / Acta Materialia 52 (2004) Fig. 6. The mean grain boundary position vs. time at 1200 K and an applied strain of e ¼ 0:02. accuracy of only 15% in the boundary velocity in simulation in which the boundary moved by 80 A. To improve the accuracy of the velocity determination, we performed up to four simulations under identical conditions (at higher T and lower P, more simulations were required than in the opposite limit). Fig. 6 shows a typical boundary position vs. time from a single MD simulation (this case corresponds to 1200 K and e ¼ 2%). Using simulation data such as this, we determined the grain boundary velocity as a function of applied strain (e ¼ 1%, 1.5% and 2%) at T ¼ 800, 1000, 1200 and 1400 K, as shown in Figs. 7(a), (b), (c) and (d), respectively. While at the highest temperature, the velocity is a linear function of the driving force for strains from )2% to +2%, this is not the case at lower temperatures. At lower temperatures, the velocity driving force relation is non-linear with higher velocities observed in tension than in compression. In fact, at the lowest temperature and highest driving force the boundary velocity in the tensile case is at least five times larger than in the compressive case (see Fig. 7(a)). While the origin of this difference is not the focus of this paper, we expect that it can be attributed to a positive activation strain just as the activation volume is commonly positive for a range of different activated kinetic processes (i.e., the effective activation barrier increases with increasing compression) [14]. 5. Grain boundary mobility and activation energy The main goal of the present simulations is to establish a reliable method to obtain grain boundary mobility. At high temperature, the velocity is a nearly Fig. 7. Mean grain boundary velocity vs. driving force for tensile and compressive applied strains at (a) 800 K, (b) 1000 K, (c) 1200 K and (d) 1400 K. The data points represent simulation results and the lines are simply guides for the eye.

7 H. Zhang et al. / Acta Materialia 52 (2004) linear function of driving force with nearly identical coefficients in compression and tension, as seen in Fig. 7(d). Therefore, the grain boundary mobility can be directly measured as the slope of the velocity driving force plot under these conditions. However, at low temperatures, the situation is complicated by the fact that the velocity driving force relation is not linear over the range of applied strains used in the present work. While, in principle, this could be overcome by significantly reducing the magnitude of the applied strains employed, this is not practical for two reasons: the low velocities that would result would place unreasonable demand on computational resources and the accuracy of any measurement would be unduly influenced by the fluctuations in boundary position. Therefore, at low temperature, we propose using a different method to determine the mobility. This method is based upon the observation that the mobility is defined (see Eq. 2) as the derivative of the velocity with respect to driving force in the zero driving force limit. Any non-linearity in the velocity driving force relation should tend to zero and there should be no difference between the tension and compression data in this limit. This is indeed consistent with the simulation data. Fig. 8 shows that m=p data tend to a constant, independent of the sign of the strain, as the driving force goes to zero. Since the deviation from linearity has opposite signs in the compressive and tensile applied strain cases, the true mobility will lie between those obtained in compression and tension. Therefore, a reasonable estimate of the mobility can be obtained as the average of the compression and tensile m=p data at the smallest practicable strains. In principle, it would be possible to obtain an even more accurate estimate if the analytical form of the non-linear velocity driving force relation were known. However, this is outside the scope of the present investigation. Finally, we note that the velocity driving force data only showed Fig. 8. The ratio of the mean grain boundary velocity and the driving force as a function of the driving force for tensile and compressive applied strains at 800 K. The data points represent simulation results and the lines are simply guides for the eye. Fig. 9. An Arrhenius plot for the grain boundary mobilities, extracted from Fig. 7 in the small driving force limit. The activation energy for boundary migration is the slope of the best fits straight line (shown). marked deviations from linearity at the lowest temperature investigated, i.e., T ¼ 800 K. The variation of the logarithm of the grain boundary mobility with inverse temperature is shown in Fig. 9. The activation energy for grain boundary migration, found from the slope of this plot, is ev/atom (where the error estimate is based upon a 95% confidence interval). 6. Discussion and conclusions In this paper, we described a new method for obtaining grain boundary mobility from molecular dynamics simulations. The method uses a particularly simple MD procedure base upon the Verlet algorithm in an NVT ensemble with the application of a strain in the plane of the boundary and free surfaces in the direction orthogonal to the boundary. This method ensures the constancy of the driving force during boundary migration. We also presented a critical assessment of different approaches to evaluating the magnitude of the driving force. Given the large strains that must be employed to produce sufficiently fast boundary migration, reliance on linear elasticity was found to yield unreliable estimates of the driving force. The present approach is readily applicable to most grain boundaries. However, given the nature of the applied strains, this approach cannot be applied to situations in which the strain energy in the two grains is identical, e.g., symmetric tilt boundaries. In our simulations, we encountered two different types of non-linearities. The first is the deviation of the driving force from those predicted based upon linear elasticity. These deviations are considerable, as large as 13% at strains of 0.02 (at all temperatures). The second type of non-linearity observed was in the velocity driving force relation. These deviations are, however, only significant at low temperature. At T ¼ 1400 K and

8 2576 H. Zhang et al. / Acta Materialia 52 (2004) jej ¼2%, we found no statistically meaningful deviation from linearity in the velocity driving relation but large deviations from linearity in the stress strain relation. Therefore, these deviations from linearity are not related. This is, in fact, not surprising since the stress strain relation is a property of the bulk crystals, while the velocity driving force relation depends on the properties of the grain boundary. Using this new simulation approach, we found that the activation energy for grain boundary migration in our asymmetric R5 [0 1 0] tilt boundary in Ni is 0.26 ev/ atom. Since this value is much greater than the uncertainty in the data (0.08 ev/atom), we conclude that grain boundary migration is a thermally activated processes. Experimental investigations also find that grain boundary migration is thermally activated. However, the activation energies typically obtained are significantly larger than those found here. For example, the measured values of grain boundary mobility in high purity Al is between 1.2 and 2.5 ev/atom [1]. This is significantly larger than the present results even though the melting point and cohesive energy of Al is much smaller than that of Ni (we are aware of no reliable Ni grain boundary mobility data). This discrepancy between experiment and simulation is consistent with that found using several different interatomic potentials and simulation methods [3,5,6]. This suggests that the discrepancy between the activation energies obtained from simulation and experiment are not the result of the nature of the interatomic potential or the simulation method (flat and curved boundary migration yield similar activation energies), but rather is a general result. We have performed a series of preliminary calculations of the migration of the same grain boundary in Al in order to test the generality of the present observations. As for Ni, the activation energy for grain boundary migration in Al is quite small [15]. This deviation has been attributed to the impurity drag effect, even in the experiments performed using high purity materials. In such a scenario, the large activation energies are associated with the activation energy for impurity diffusion and/or the heat of segregation. Therefore, simulations such as those presented here provide a unique opportunity to determine the intrinsic boundary mobility. The intrinsic mobility is one of the fundamental parameters needed to predict the net boundary mobility in an impure material. Although the present simulations were all performed on a single grain boundary in a single material, we expect the main conclusions will remain unchanged: grain boundary migration is thermally activated but with small activation energy in pure metals. Additional simulations with different bicrystallography in different materials are required to confirm the generality of this conclusion. Acknowledgements The authors gratefully acknowledge useful discussions with Srinivasan G. Srivilliputhur and Danxu Du and the support of the US Department of Energy, Grant No. DE-FG02-99ER References [1] Gottstein G, Shvindlerman LS. Grain boundary migration in metals: thermodynamics, kinetics, applications. Boca Raton: CRC Press; [2] Shvindlerman LS, Gottstein G, Molodov DA. Phys Status Solidi A 1997;160:419. [3] Sch onfelder B, Wolf D, Phillpot SR, Furtkamp M. Interf Sci 1997;5:245. [4] Parrinello M, Rahman A. Phys Rev Lett 1980;45:1196. [5] Upmanyu M, Smith RW, Srolovitz DJ. Interf Sci 1998;6:41. [6] Upmanyu M, Srolovitz DJ, Shvindlerman LS, Gottstein G. Acta Mater 1999;47:3901. [7] Lucke K, Detert K. Acta Metall Mater 1957;5:628. [8] Cahn JW. Acta Metall Mater 1962;10:789. [9] Voter AF, Chen SP. Mater Res Soc Symp Proc 1987;82:175. [10] Handbook of chemistry and physics. Boca Raton: CRC Press; [11] Morris JR, Wang CZ, Ho KM, Chan CT. Phys Rev B 1994;49:3109. [12] Berendsen HJC, Postma JPM, Vangunsteren WF, Dinola A, Haak JR. J Chem Phys 1984;81:3684. [13] Hoover WG, Holian BL. Phys Lett A 1996;211:253. [14] Aziz MJ, Sabin PC, Lu GQ. Phys Rev B 1991;44:9812. [15] Mendelev MI, Srolovitz DJ, Han S, Ackland GJ, Morris JR, [to be published].