An overview of recrystallization and grain growth - Semantic Scholar

Materials Science Forum Vols. 558-559 (2007) pp. 33-42 online at http://www.scientific.net © (2007) Trans Tech Publications, Switzerland

An Overview of Accomplishments and Challenges in Recrystallization and Grain Growth A. D. Rollett1, A. P. Brahme1,2, and C. G. Roberts1 1

Dept. Materials Science and Engineering Carnegie Mellon University 5000 Forbes Avenue Pittsburgh, PA 15213, USA

2

Max-Planck-Institut fuer Eisenforschung Max-Planck-Str. 1 40237 Duesseldorf, Germany

[email protected], [email protected], [email protected] Keywords: recrystallization, grain growth, simulation, modeling, grain boundary character distribution, stagnation Abstract. The study of microstructural evolution in polycrystalline materials has been active for many decades so it is interesting to illustrate the progress that has been made and to point out some remaining challenges. Grain boundaries are important because their long-range motion controls evolution in many cases. We have some understanding of the essential features of grain boundary properties over the five macroscopic degrees of freedom. Excess free energy, for example, is dominated by the two surfaces that comprise the boundary although the twist component also has a non-negligible influence. Mobility is less well defined although there are some clear trends for certain classes of materials such as fcc metals. Computer simulation has made a critical contribution by showing, for example, that mobility exhibits an intrinsic crystallographic anisotropy even in the absence of impurities. At the mesoscopic level, we now have rigorous relationships between geometry and growth rates for individual grains in three dimensions. We are in the process of validating computer models of grain growth against 3D non-destructive measurements. Quantitative modeling of recrystallization that includes texture development has been accomplished in several groups. Other properties such as corrosion resistance are being related quantitatively to microstructure. There remain, however, numerous challenges. Despite decades of study, we still do not have complete cause-and-effect descriptions of most cases of abnormal grain growth. The response of nanostructured materials to annealing can lead to either unexpected resistance to coarsening, or, coarsening at unexpectedly low temperatures. General process models for recrystallization that can be applied to industrial alloys remain elusive although significant progress has been made for the specific case of aluminum alloy processing. Thin films often exhibit stagnation of grain growth that we do not fully understand, as well as abnormal grain growth. Grain boundaries respond to driving forces in more complicated ways than we understood. Clearly many exciting challenges remain in grain growth and recrystallization. Grain Boundary Properties: Energy It is not difficult to appreciate that the energy of grain boundaries varies with type even in the most symmetric crystal classes. Dihedral angles, a direct measure of anisotropy of interfacial energy, vary widely from the 120° value expected for isotropy; even in strongly fiber textured thin films of aluminum, the range is from 90-180°. In fact, the dihedral angles can be used to extract the variation in grain boundary over the whole range of five (macroscopic) degrees of freedom required to describe the crystallographic type [1]. Experiments in which a liquid is allowed to penetrate down grain boundaries show that certain boundary types are much more likely to be penetrated than

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Recrystallization and Grain Growth III

others [2]. The resistant boundaries are low energy and the penetrable boundaries are high energy types. The received wisdom on grain boundary energy has been that grain boundaries have disordered structures with high energy except for boundaries with misorientations close to coincidence relationships (CSLs). Near-CSL boundaries were assumed to be special, i.e. low energy, because of better atomic fit. However, theory and simulation [3, 4], combined with recent experimental results [1] suggest that special boundaries are those that combine low surface energies, placing the emphasis on boundary normal as opposed to lattice misorientation. Moreover there is a strong inverse correlation evident between grain boundary energy and the relative frequency of boundaries of different types [5, 6]. To choose a simple example in fcc metals, the Σ3 and Σ5 misorientation relationships might be expected to exhibit good atomic fit and low energies: this, however, is not borne out by the experimental evidence and, even for the Σ3, only the pure twist boundary type, with the normal parallel to the misorientation axis to give a coherent twin, gives a low energy configuration. This has been followed up by simulation of grain growth, which has demonstrated that anisotropies on the order of a few percent are sufficient to induce a corresponding anisotropy in the grain boundary population [7]. Some experimental support for an evolution in GBCD that mirrors the energy anisotropy has been reported by Piazolo et al. for grain growth in an aluminum foil with a columnar grain structure [8]. Both an increase in the number of low angle boundaries, and an increase in the number of low-index planes in boundaries was reported. This correspondence offers the tantalizing possibility that it may be possible to infer the anisotropy of grain boundary energy from the anisotropy of the distribution of boundary types, i.e. the grain boundary character distribution (GBCD). Such an approach is likely to require near random texture and sufficient (normal) grain growth, however, to ensure that the GBCD is a product solely of anisotropy in the energy function. Other applications in which the anisotropy of grain boundary energy may play a role include intergranular corrosion which has received considerable attention with respect to the importance of the way in which boundaries are connected together in networks [9, 10]. Grain Boundary Properties: Mobility We define mobility as the (linear) material property that multiplies a driving force for boundary motion to give its velocity [11]. Most of the knowledge on grain boundary mobility has been determined only for cubic metals, and, amongst that class, nearly all for (fcc) aluminum of various compositions. The general characteristics of mobility are that it is strongly dependent on temperature, grain boundary character and purity. The temperature dependence can be understood in terms of thermal activation of atom transfer across the grain boundary [12]. Although some attempts have been made to develop this approach and account for grain boundary character, the results have not been to explain the essential features of the anisotropy of mobility [13, 14]. The experimental results for Al show that the enthalpy of mobility varies strongly with boundary type over a range that is difficult to reconcile with the existence of a single type of atomic transfer mechanism. Some insight into the complexity of boundary motion has been found in detailed analysis of the complex individual atomic motions that occur during migration in molecular dynamics simulations [15]. Overall, for pure aluminum and grain boundary character characterized by low-index misorientation axes, tilt boundaries move more rapidly than twist boundaries and the ranking goes as 111>100>110. The relative activation energies, Q, are in the opposite order with Q111 < Q100 < Q110 [16]. Given the need for constitutive descriptions of grain boundary properties for use in modeling microstructural evolution, one interesting possibility is to attempt to generate a master curve for various different materials. The discussion is once again confined to fcc metals. One approach is to scale both the pre-factor and the activation energy for mobility as a function of the solute content but assume that the variation of mobility with crystallographic type is independent of solute level. As an example, activation energies for several different experimental and simulation results by

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Aristov, Molodov, Upmanyu, Zhang, Taheri and Winning [17, 18, 19, 20, 21] are shown in fig. 1, where each data set has been scaled (linearly) to minimize the differences, using Molodov’s results as the reference set. Essentially all the results are for tilt boundaries which means that certain misorientations correspond to coincident lattice site structures for angles at 27.8°, 38.2° 46.8° and 60°. Notwithstanding the significant degree of scatter in the results, there appear to be a series of maxima in activation energy, interspersed between minima at or near CSL locations.

Figure 1. Plot of activation energies scaled to bring them into coincidence as far as possible. Note the presence of minima at the CSL boundary types noted on the figure.

Grain Boundary Motion The equation of motion that applies to grain boundary motion is as follows, where v is the velocity (normal to the interface), M the mobility and P the driving force: v=MP

(1)

For boundaries moving as a consequence of capillarity [22], the equation can be expanded to show the dependence on the boundary normal, via the inclination: v = M{(γ + γθθ )κ1 + (γ + γ φφ )κ 2}

(2)

Here, the principal curvatures are denoted by κi and the corresponding second derivatives of the grain boundary energy with respect to inclination are denoted by γθθ and γφφ. Some care is needed here because some authors refer to the sum of the two curvatures as the “mean curvature” even though there is no factor of two as one might expect for an arithmetic average. This somewhat complex expression has the undesirable feature of not being a coordinate-free equation. The capillarity vector, introduced by Hoffman and Cahn [23], can be used to provide a coordinate free equivalent expression [24]: r r v = M ∇ Sξ = M ⋅ (∇ − nˆ ⋅ ∇)ξ (3)

All this seems very neat and orderly in terms of boundaries moving towards their centers of curvature except that counterintuitive behavior can occur in real life. For example, applying a shear force to boundaries results in boundary motion even in the high angle regime [25, 26], suggesting that boundaries retain a dislocation-like character even when the individual lattice dislocations cannot be discerned. Grain boundary sliding is well known but the interaction between shear stresses and normal motion is significant, as documented in recent simulation work [27]: applying a shear stress to a boundary that might normally be expected to generate sliding, can also result in

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motion normal to the boundary plane. This phenomenon is readily explicable in terms of the interface defect structure. Models for Boundary Motion, Polycrystals There exists a plethora of methods for simulating grain boundary motion and microstructural evolution in polycrystals. No one method has obvious advantages over all others. In approximate order of application to the problem of grain growth, we describe the Monte Carlo model, the vertex model, the phase field model, and the moving finite element model. All such models use a discretization of the grain structure based on either the interfaces themselves (vertex and finite element) or on a volumetric basis such that the boundaries are dividing planes between different orientations (grains). The Monte Carlo (MC) model is of the latter type, and represents the microstructure on a regular grid such that each gridpoint represents an element of volume or voxel. Each voxel is associated with an orientation such that a grain boundary exists between any pair of dissimilar orientations. Grain boundary properties, such as energy and mobility, can then be associated with the misorientation. In its simplest form, the model is advanced one change of a voxel at a time, based on the change in the system energy. Energy decreasing moves are favored for obvious reasons and the model can be shown to reproduce motion by curvature [28]. It is straightforward to implement misorientation-dependent boundary properties in this model but dependence on the boundary normal is numerically awkward. The phase field (PF) model also discretizes microstructure on a regular grid which again is typically a simple cubic grid in 3D [29]. In this model, each orientation is associated with a field variable but in contrast to the MC model, each voxel can, in principle, have a finite fraction of any of the orientations. An energy functional specifies minima in energy to induce ordering in the system (domains of like-oriented voxels) together with a gradient penalty to control the width of interfaces (grain boundaries). The system is advanced in time by allowing the value of each field variable to change in response to the gradient of the energy functional with respect to that variable. For numerical efficiency in grain growth simulations, which are effectively coarse domain structures, the number of field variables evaluated at any given point is restricted to the locally active set [30]. For a review of the general phase field method, see Uehara and Sekerka [31]. Since boundary normals can be computed from gradients in the field variables, implementing normaldependent properties is straightforward in the PF model [32]. The vertex, differential equation and moving finite element models are examples of sharp interface models in which the discretization is based on the network of interfaces. For all of these models, implementing boundary properties as a function of all five degrees of freedom is straightforward. The simplest of these, i.e. the vertex model, places a node at each vertex and also distributes nodes along the edges (triple lines in 3D). The forces on each node are computed by a balance of forces and each node is moved at a rate that depends on the net force and the local mobility [33]. The differential equation model [34] is different from the vertex model in a subtle but important way. This also distributes nodes along the interfaces and vertices so the discretization is essentially identical but the time advance uses the Mullins equation to move the nodes; interleaved with these are time steps that update the local equilibrium at triple points (2D) or lines (3D). The moving finite element model, also known as Gradient Weighted Finite Element (GWFEM), uses a fully volumetric discretization with a tetrahedral mesh [35]. Forces on interface nodes are computed so as to simulate motion-by-curvature and time steps advance the model using a similar scheme to that described for the vertex model but implemented within a finite element scheme. The nodes in the interior of grains are also updated but with numerical factors to ensure that they do not significantly affect the motion of the interfaces.


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Validation of Grain Growth Models Validation of computer models is an essential part of building confidence in a simulation technique. Validation is distinct from verification, which means checking that a computer code performs as intended. So, for example, a diffusion code can be verified that it solves the relevant time dependent diffusion equation. In the case of grain growth, the weakest form of validation is the comparison of the average coarsening kinetics to see that the mean area is proportional to time. A more severe test is to compare microstructures in detail by comparing computed microstructures against experimental measurements pixel by pixel. A study of 2D subgrain microstructures concluded, as one might expect that it is important to include the anisotropy of grain boundary properties [36]. More recently, a similar comparison has been attempted in 3D using a data set provided by Budai et al. at Oak Ridge National Laboratory. A sample of commercial purity, hot rolled aluminum was examined using a 3D synchrotron microscopy method [37]. The sample was annealed for one hour at successively higher temperatures to explore its grain growth. A snapshot at 250°C was used as input to a Monte Carlo model of grain growth [38] that incorporates anisotropic grain boundary properties. Since the microstructure was measured on a regular grid, the information could be read into the simulation code without any interpretation being required. The results of each time step were compared to a subsequent experimental snapshot from the next annealing step at 350°C. The error signal is based on comparing corresponding voxels in the model and in the experiments: perfect agreement corresponds to a zero error. The hoped-for result is a perfect agreement at a simulation time that corresponds to the experimentally measured microstructural evolution. The results indicate, however, that although a minimum in the error is observed at a simulation time for which the number of grains present is close to that in the experimental microstructure, the agreement is far from perfect. This is easily understood because the experiments showed that new orientations appeared in the microstructure at later times that were not present in the initial microstructure.

Figure 2. Plot of number of grains in the simulation (right axis) and the voxel difference signal (left axis) between the simulated microstructures and the measured microstructure [39]. The dotted horizontal line indicates the number of grains, 340, present in the measured microstructure.

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(a)

I

(b) Fig. 3. Snapshots of individual layers in the 3D data set taken from a commercial purity aluminum. The earlier time, (a), shows a large number of grains, whereas the later time, (b), shows that some large grains have grown into the measured volume that were not initially present; one such invading grain is labeled “I”. Problems in Abnormal Grain Growth One of the perennially fascinating problems in materials science is that of abnormal grain growth. There are many observations of the phenomenon, from the classical production of oriented electrical steel in Fe-Si alloys [40] to the unwanted coarsening that can occur near a solvus temperature, i.e. where precipitates are dissolving [41]. Theory shows that abnormal grain growth can be a consequence of anisotropic grain boundary properties [42, 43] such as high mobility of the boundary of an abnormal grain or low energy. There have been many attempts to link this to special grain boundary properties, e.g. [44], especially in the technologically important case of Fe3Si. However, although grain boundaries are undoubtedly anisotropic, the link to abnormal grain growth appears to be weak [45, 46]. In thin films, it is possible for surface energy to be a dominant driving force. Additionally, elastic energy has been postulated to cause abnormal grain growth in thin films of silver [47]. A recent detailed examination of abnormal grain growth in a Fe-1Si alloy has, however, shown that it is in fact low levels of stored energy arising from temper rolling that drive the process [48]. Although this illustrates the importance of heterogeneous deformation in providing a driving force, it by no means solves the general problem. As mentioned above, abnormal grain growth is often associated with the presence of second phase particles. This phenomenon is an essential part of microstructure control in polycrystals, from structural metals [49] to superconductors [50], to ceramics [51]. Although the pinning effect on boundaries has been much studied and significant progress has been made recently [52], it is unclear what role, if any, boundary anisotropy plays in the interaction with particles. Recently we have measured the correlation between particles and boundaries in a nickel-base alloy. Waspaloy was processed by vacuum-induction melting (VIM) and vacuum-arc remelted (VAR). It was then homogenized and subjected to a series of supersolvus and subsolvus hot working operations to produce a wrought (recrystallized) microstructure with negligible texture. A sample was analyzed by conventional scanning electron microscopy and EBSD for grain boundary analysis, characterized here by the MDF. In contrast to the assumptions of Zener-Smith [53], the fraction of particles on boundaries, 0.61, was found to be significantly higher than expected from a random intersection of boundaries with particles, 0.19 [54]. Moreover the character of boundaries on which particles are found, termed the particle associated misorientation distribution (PMDF), is significantly different from the measured misorientation distribution (MDF), see fig. 4, although the PMDF appears to converge with the MDF after grain growth has taken place. A detailed description will be published elsewhere.


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Figure 4. Plots of the measured misorientation distribution (MDF) for a sample of wrought Waspaloy, contrasted with the so-called Mackenzie distribution for a random cubic material. The particle-associated MDF (PMDF) differs markedly from the MDF, showing that the interaction between particles and boundaries cannot be assumed to be random.

Stagnation in Grain Growth An interesting problem in grain growth is that of stagnation of coarsening in thin films. Barmak et al. have measured grain size and topology in high purity aluminum thin films of various thicknesses as a function of time. The results show unambiguously that growth ceases at a grain size that is comparable to the film thickness [55]. The purity is such that the films are single phase and do not have enough solute for solute drag to be effective [55]. Most other possible mechanisms can also be excluded; the fiber texture means, for example, that the surfaces are all {111} facets and therefore all have the same (minimum) surface energy. Such problems appear to offer rich sources of challenges for research on grain growth. One possibility is that triple junction drag plays a significant role in constraining grain growth [56]. Another is that stress could play a role in constraining the coarsening process [57]. The best-known source of stagnation in grain growth (and, indirectly, abnormal grain growth) is, however, pinning by second phase particles [53, 58] as discussed above. Recrystallization Recrystallization has received a substantial amount of attention in recent years. The basic equation governing the migration of grain boundaries is the same as equation (3) except for the additional term to account for the volumetric driving force, ∆F, contributed by the elimination of dislocations as the boundary of a recrystallizing grain sweeps through a deformed matrix: r v = M ∇ Sξ + ∆F (4)

(

)

In most examples of recrystallization the magnitude of the energy stored in the dislocations is substantially larger than that associated with curvature. In addition to the perennially important relevance to industrial production of metallic alloys, there are new technological areas such as damascene technology copper interconnects in microelectronic devices [59]. One area of strong interest has been in attempting to identify the critical features of the deformed structure that give rise to new, recrystallized grains. This is motivated by the many observations that recrystallization textures are largely determined by the early stages of the process. “Early stages” is equivalent to “nucleation”, for which, remarkably enough, there is still no generally agreed upon definition [60]. Several possible mechanisms exist, all of which have experimental evidence to support them, such as strain induced boundary migration (SIBM), abnormal subgrain coarsening caused by long range orientation gradients, abnormal subgrain coarsening from misoriented grains [61], twinning etc. There is an extensive literature on nucleation and growth during recrystallization, e.g. [62]. Current opinion generally holds that both factors are important but that texture is often determined early in the recrystallization process, commonly referred to as oriented nucleation. For direct simulation of recrystallization, 3D digital microstructures are required to define the initial state of the material. Notwithstanding the substantial recent advances in non-destructive characterization with synchrotron radiation [63, 64], the method is unlikely to become a widely

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available characterization tool in the near future. It is also easier to index well-annealed structures than deformed ones. Automated serial sectioning based on EBSD has also made substantial strides recently [65], but this is a destructive technique. The alternative is to employ statistical reconstruction, commonly used for modeling of two-phase materials [66], but taking account of grain shape and orientation [67]. This has been applied to the generation of representative microstructures in a commercial purity aluminum where a particular problem is the highly elongated grain shapes found in this hot rolled material [68], fig. 5. Such digital descriptions of microstructures form the basis for simulation of evolution [69] and permit quantitative evaluation of texture change in addition to grain size and shape. They are also useful for a wide range of simulation applications in which descriptions of microstructure are required, e.g. modulus, fatigue cracking [70, 71].

Figure 5. 3D digital microstructure to represent a hot rolled commercial purity aluminum. The grain size is approximately 20 x 80 x 1000 µm.

Another, high level, approach to the modeling of recrystallization is to embed it in a process model. This has been extensively developed for through-process modeling of aluminum alloy processing by the Aachen group [72, 73]. Here the focus has been on incorporating physically based submodels that permit a range of alloys to be described by the model. The approach also allows a connection to be made to up-stream models of the solidification processing that precedes the rolling and recrystallization steps. Notwithstanding the success of the various simulation methods mentioned here, it remains a considerable challenge to predict the kinetics and texture development during recrystallization for arbitrary alloys subjected to arbitrary plastic deformations. This stands in interesting contrast to plastic deformation itself, where predicting mechanical response and texture development is relatively well understood. The remaining challenges in “nucleation” of recrystallization alluded to above do, however, make the point that we need to understand substantially better the variability or heterogeneity of the deformed state. In over-simplified terms, mechanical properties are generally well described by average values over a microstructure, whereas the recrystallization process is sensitive to extreme values of distributions of misorientation, orientation gradients, stored energy and other factors. Recent studies of the heterogeneity of deformation are helping to define the deformed state [74, 75, 76, 77, 78] although fully quantitative descriptions of the evolution of deformation microstructures are not yet available and represent a significant challenge. Modeling such heterogeneities has been pursued on several fronts, including models that link heterogeneous dislocation storage to interactions between grains [79] and efforts to introduce natural length scales into crystal plasticity [80, 81].


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Summary A brief review of recent developments in grain growth and recrystallization has been given. Some challenges in the various areas have been pointed out, such as establishing causal relationships between grain boundary populations and anisotropic properties; a better understanding of grain boundary mobility; a comprehensive understanding of abnormal grain growth; an understanding of the early stages of recrystallization. Acknowledgements The support of the NSF-supported Materials Research Science & Engineering Center (MRSEC) at Carnegie Mellon under grant number DMR-0520425, the Alcoa Technical Center, the U.S. Air Force, and discussions with many colleagues are all gratefully acknowledged. References [1] D. M. Saylor, A. Morawiec, G. S. Rohrer: J. Am. Ceram. Soc. Vol. 85, (2002), p. 3081. [2] M. Takashima, P. Wynblatt, B. L. Adams: Int. Sci. Vol. 8, (2000), p. 351. [3] A. P. Sutton: Phil. Mag. A Vol. 63, (1991), p. 793. [4] D. Wolf: Scripta metall. Vol. 23, (1989), p. 1913. [5] D. M. Saylor, A. Morawiec, G. S. Rohrer: Acta mater. Vol. 51, (2003), p. 3663. [6] D. M. Saylor, B. S. El-Dasher, A. D. Rollett, et al.: Acta mater. Vol. 51, (2004), p. 3649. [7] J. Gruber, D. C. George, A. P. Kuprat, et al.: Scripta mater. Vol. 53, (2005), p. 351. [8] S. Piazolo, V. G. Sursaeva, D. J. Prior: in Recrystallization and grain growth, pts 1 and 2, Vol. 467-470, 2004, 935. [9] C. A. Schuh, R. W. Minich, M. Kumar: Phil. Mag. Vol. 83, (2003), p. 711. [10] E. S. McGarrity, P. M. Duxbury, E. A. Holm: Phys. Rev. E Vol. 71, (2005), p. [11] G. Gottstein, L. S. Shvindlerman: Scripta metall. Vol. 27, (1992), p. 1521. [12] D. Turnbull: Trans. Met. Soc. AIME Vol. 191, (1951), p. 661. [13] H. Gleiter: Acta metall. Vol. 17, (1969), p. 853. [14] H. Gleiter: Acta metall. Vol. 17, (1969), p. 565. [15] H. Zhang, D. J. Srolovitz: Acta mater. Vol. 54, (2006), p. 623. [16] G. Gottstein, L. S. Shvindlerman: Grain boundary migration in metals, (CRC Press, Boca Raton, FL 1999). [17] V. Y. Aristov, C. Kopetskii, L. S. Shvindlerman: Sov. Phys. Solid State Vol. 18, (1976), p. 137. [18] M. Upmanyu, D. Srolovitz, R. Smith: Int. Sci. Vol. 6, (1998), p. 41. [19] D. A. Molodov, U. Czubayko, G. Gottstein, et al.: Acta mater. Vol. 46, (1998), p. 553. [20] M. Winning, G. Gottstein, L. S. Shvindlerman: Acta mater. Vol. 50, (2002), p. 353. [21] M. L. Taheri, D. Molodov, G. Gottstein, et al.: Z. Metall. Vol. 96, (2005), p. 1166. [22] W. W. Mullins: J. Appl. Phys. Vol. 27, (1956), p. 900. [23] D. W. Hoffman, J. W. Cahn: Surf. Sci. Vol. 31, (1972), p. 368. [24] A. A. Wheeler, G. B. McFadden: Euro. J. Appl. Math. Vol. 7, (1996), p. 367. [25] M. Winning, G. Gottstein, L. S. Shvindlerman: Acta mater. Vol. 49, (2001), p. 211. [26] M. Winning: Zeitschrift Fur Metallkunde Vol. 95, (2004), p. 233. [27] J. W. Cahn, Y. Mishin, A. Suzuki: Phil. Mag. Vol. 86, (2006), p. 3965. [28] A. D. Rollett, P. Manohar: in Continuum scale simulation of engineering materials, (Eds: D. Raabe, et al.), Wiley-VCH, Weinheim, Germany 2004, 77. [29] D. Fan, L.-Q. Chen: Acta mater. Vol. 45, (1997), p. 611. [30] J. Gruber, N. Ma, Y. Wang, et al.: Modelling Simul. Mater. Sci. Eng. Vol. 14, (2006), p. 1189. [31] T. Uehara, R. F. Sekerka: J. Cryst. Growth Vol. 254, (2003), p. 251. [32] A. Kazaryan, Y. Wang, S. A. Dregia, et al.: Phys. Rev. B Vol. 61, (2000), p. 14275 [33] H. Frost, C. Thompson: J. Electronic Mater. Vol. 17, (1988), p. 447. [34] D. Kinderlehrer, I. Livshits, S. Ta'asan: SIAM J. Sci. Comp. Vol. 28, (2006), p. 1694. [35] A. Kuprat: SIAM J. Sci. Comp. Vol. 22, (2000), p. 535. [36] M. C. Demirel, A. P. Kuprat, D. C. George, et al.: Phys. Rev. Lett. Vol. 90, (2003), p. 016106. [37] W. J. Liu, G. E. Ice, B. C. Larson, et al.: Metall. Trans. A Vol. 35A, (2004), p. 1963. [38] A. D. Rollett: JOM Vol. 56, (2004), p. 63.

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