MEASURING RELATIVE GRAIN BOUNDARY ENERGIES AND MOBILITIES IN AN ALUMINUM FOIL FROM TRIPLE JUNCTION GEOMETRY

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1 v MEASURING RELATIVE GRAIN BOUNDARY ENERGIES AND MOBILITIES IN AN ALUMINUM FOIL FROM TRIPLE JUNCTION GEOMETRY C.-C. Yang, A. D. Rollett and W. W. Mullins Carnegie Mellon University, Materials Science and Engineering Department 5000 Forbes Avenue, Pittsburgh, Pennsylvania , U.S.A. ABSTRACT Determining grain boundary energies and mobilities as a function of their crystallographic parameters is essential to a quantitative understanding of microstructural evolution. In an effort to extract the grain boundary energies and mobilities based on their crystallographic types, new techniques for image processing and curve fitting were developed and applied to a columnar microstructure in a thin aluminum foil. Relative grain boundary energies and mobilities have been extracted from a large set of triple unctions, and mapped as a function of their crystallographic character. 1. INTRODUCTION Relative grain boundary energies and mobilities in Al foil have been extracted from the geometry and crystallography of triple unctions. This approach is based on the assumption of local equilibrium of grain boundary energies at triple unctions described by Herring (1951), 3 ˆ σ σ ˆ b + n = 0 1 (1) φ = where the quantities in this equation are depicted in Fig. 1. ˆb 1 ˆn 1 χ χ 3 σ 1 σ σ 3 φ 3 χ 1 ˆb ˆn ˆb 3 ˆn 3 Fig. 1. Definition of geometric parameters associated with boundaries adoining a triple unction. Each boundary is associated with an interfacial energy σ, a unit normal nˆ, a unit direction bˆ, an inclination φ, and a dihedral angle χ. Only one inclination angle (φ 3 ) is labeled for clarity. *It is a pleasure to dedicate this paper to Dr. Niels Hansen in honor of his lifelong work on microstructure.

2 Yang, Rollett, and Mullins Previous work, Miller and Williams (1967), suggests that only a small fraction of boundaries in Al are subect to appreciable torque. Therefore, as a first approximation, we assume that the σ grain boundary energy is independent of boundary orientation, the torque terms ( ) nˆ φ in (1) can be neglected, and Herring s equations reduce to Young s relations, σ 1 sin χ 1 = σ sin χ = σ 3 sin χ 3 () Further, a geometrical relationship between boundary migration rates has been described by Adams et al. (1998) based on the assumption that boundaries migrate only due to their capillarity and that other driving forces are negligible. Then the relation M [sin( ) i ] = 0 i i i σ i κ χ (3) holds, where κ i is the sum of two principal curvatures at triple unction associated with grain boundary i. The assumptions underlying equation (3) are that the boundary migration rate, v, is linearly related to the driving force f (=σκ) by a proportionality factor M, representing mobility, and that M and the energy are independent of the boundary inclination. Thus each triple unction geometry yields two energy related equations and one mobility related equation. After characterizing a sufficient number of triple unction geometries, relative boundary energies and mobilities can be extracted through a statistical/multiscale analysis, Kinderlehrer et al. (1999), of equations () and (3). The number of unctions to be characterized depends on the resolution desired for the properties.. EXPERIMENTAL APPROACH An Al foil sample of purity % was annealed at 550 o C for 9 hours in a N environment. This resulted in a columnar grain structure as observed by optical microscopy, and shown in Fig.. The texture was dominated by a strong cube component, {100}<001>, with occasional randomly oriented grains as shown in Fig µm Fig.. Cross section of the annealed sample (multiple layers) showing columnar structure. Fig. 3. {111}, {110} and {111} pole figures calculated from electron back scatter diffraction measurements, showing strong cube texture.

3 Measuring relative grain boundary energies and mobilities Since the columnar structure results in triple unctions that are nearly straight and perpendicular to the surfaces, the difficulty and error of measuring true dihedral angles and principal curvatures for each grain boundary by the serial sectioning technique is effectively eliminated. Crystallographic information for the grains adacent to each triple unction was obtained by using orientation imaging microscopy in a scanning electron microscope. Each triple unction was recorded on an individual scanning electron micrograph and image processing methods, Talukder et al. (1998), and curve fitting algorithms were applied in order to extract dihedral angles and curvatures. The precision of curvature measurements from the curve fitting algorithms is better than 97%, Yang et al. (000). 3.RESULTS 3.1 Misorientation angle dependence of boundary energy and mobility. 40 grain boundaries were characterized at 134 triple unctions. The boundaries were sorted into the 13 different types which are listed in Table 1, based on their misorientation angles. Misorientation between two orientations in this paper means both the axis that yields the minimum rotation angle and the value of that angle, taking crystal symmetry into account. The resolution was limited by the number of data points. The misorientation angle distribution of the boundaries is shown in Fig. 4, which shows that most are low angle grain boundaries, i.e. misorientation angle θ <15 o. Table types of boundaries sorted by misorientation angle. Boundary Type Misorientation Angle θ (degrees) < >15 The dihedral angles were used as input to the statistical/multiscale analysis to first calculate the grain boundary energy. Fig. 5 shows the variation of relative boundary energy with misorientation angle. For low angle grain boundary energies, the experimental data is in good agreement with the Read-Shockley equation; and the average relative energy for high angle grain boundaries is A similar statistical calculation was performed based on equation (3) to obtain grain boundary mobility as a function of misorientation angle, Fig. 6. The results show that the boundary mobility increases rapidly when the misorientation angle exceeds 10. Also, high angle grain boundaries are observed to be much more mobile than low angle boundaries. 60 Total Number of Boundary =40 50 Number of Boundaries Misorientation Angle(degrees) 3 Fig. 4. Frequency plotted against misorientation angle.

4 Yang, Rollett, and Mullins Relative Boundary Energy Energy Derived from Experimental Data Read-Shockley Equation Misorientation Angle (degrees) Relative Boundary Mobility Misorientation Angle (degrees) Fig. 5. Variation of relative boundary energy with misorientation angle. Fig. 6. Variation of relative boundary mobility with misorientation angle. 3. Misorientation Axis Dependence of Boundary Energy and Mobility. In a second analysis, only low angle grain boundaries, i.e. boundaries with misorientation angles less than 15 o, were included, and the boundaries were relocated to their physically equivalent positions within the fundamental zone, i.e. the standard stereographic triangle, All boundaries were then sorted into 13 different types chosen to give uniform coverage of the standard stereographic triangle. Each boundary was assigned to a particular type by finding the nearest axis from the list shown in Table. As for the previous analysis, the resolution was limited by the size of the experimental data set. The same statistical analysis for the reconstruction of boundary energies and mobilities was performed as used in the previous section. Table. 13 types of boundaries distinguished by misorientation axes. Boundary Type Misorientation Axis [uvw] Fig. 7 shows the misorientation axis dependence of the boundary energy. Comparing three low Miller index boundaries in the corners of the triangle, [001], [101], and [111], it was found that σ [001] σ [101] > σ [111]. Fig.8 shows the variation of low angle boundary mobility with the boundary misorientation axis. [111] type boundaries are much more mobile than [101] and [001] type boundaries, i.e. M 111 >M 110 >M 001. This result parallels that of Bauer and Lanxner (1986) who found that low angle [111] tilt boundaries in gold are significantly more mobile than [001] tilt boundaries. 3.3 Tilt and Twist Rotation Dependence of Boundary Energy and Mobility. The total misorientation θ/uvw can be decomposed into two sequential operations: a tilt rotation, ψ, followed by a twist rotation, ϕ, Wolf and Lutsko (1989). The tilt rotation is about an axis lying in the boundary plane, and the twist rotation is about an axis perpendicular to the plane. The same set of low angle boundaries as in the previous section were sorted into the following three types. 4

5 Measuring relative grain boundary energies and mobilities 1. Tilt boundary: with boundary plane normal perpendicular to the boundary misorientation axis; i.e. ψ/(ψ+ϕ) >0.7.. Twist boundary: with boundary plane normal parallel to the boundary misorientation axis; i.e. ϕ/(ψ+ϕ) > Mixed boundary: others. [001] Relative Energy [105] [117] 0.33 [113] 0.30 [05] 0.6 [03] [15] [8411] [335] [111] 0.3 Fig. 7. Variation of relative boundary energy with the boundary misorientation axis. [101] [001] [77] [33] [105] [05] [117] [15] [113] [335] Relative Mobility [03] [101] [8411] [77] [33] [111] Fig. 8. Variation of relative boundary mobility with the boundary misorientation axis. The statistical/multiscale analysis was repeated for these three types in order to investigate the inclination dependence of the grain boundary properties. The results are shown as ratios of relative boundary energy and mobility for these three types of boundaries are shown in Table 3. The ranking of the results is such that σ twist > σ mixed > σ tilt. This result is in agreement with those of Otsuki who measured dihedral angles for grain boundaries in aluminum in contact with a liquid aluminum alloy (1990). The mobilities are ranked similarly except that tilt boundaries appear to be much less mobile than other inclination types, M twist > M mixed» M tilt. Table 3. Relative boundary energy and mobility for tilt, twist, and mixed boundaries. Low Angle Twist Boundary Mixed Boundary Tilt Boundary Relative Energy Relative Mobility

6 Yang, Rollett, and Mullins 4. SUMMARY Relative boundary energies and mobilities as a function of crystallographic type have been extracted from triple unction geometry in an Al foil. The variation in boundary energy follows the Read-Shockley equation whereas the mobility increases sharply for misorientations above about 10. For low angle grain boundaries, it is found that boundaries with [001] misorientation axes have a higher energy than both [101] and [111] type boundaries, but are much less mobile than the others. Twist low angle boundaries have both higher energy and mobility than tilt boundaries. ACKNOWLEDGMENT This work is supported by the National Science Foundation, grant number DMR The authors would like to thank Prof. Kinderlehrer, Prof. Ta'asan, Prof. Casasent and Dr. Livshits for helpful discussions, as well as to Dr. Weiland for providing the Al foil material. REFERENCES Adams, B. L., Kinderlehrer, D., Mullins, W. W., Rollett, A. D., and Ta asan, S. (1998). Extracting the Relative Grain Boundary Free Energy and Mobility Functions from the Geometry of Microstructures, Scripta Materialia 38, Bauer, C.L. and Lanxner, M. (1986). Relationships between grain boundary structure and migration kinetics, Proc. JIMIS-4, 411. Herring, C. (1951). Surface tension as a motivation for sintering, In: The physics of powder metallurgy 1951, Ed. W. E. Kingston, (McGraw-Hill Book Conpany, Inc., New York) Kinderlehrer, D., Livshits, I., Ta'asan, S., and Mason, D.E. (1999). Multiscale Reconstruction of Grain Boundary Energy from Microstructure, Proc. of the Twelfth International Conference on Textures of Materials, Montréal, Canada, Miller, W.A. and Williams, W.M. (1967). Anisotropy boundary energy and its influence on boundary morphology in sheet, Acta Met. 15, Otsuki, A. (1990), Ph.D. thesis, Kyoto Univ., Japan. Talukder, A., Casasent, D., and Ozdemir, S. (1998). Image processing for grain boundary detection in microscope images, Proc. Conference on Grain Growth in Polycrystalline Materials III, TMS, Wolf, D., and Lutsko, J. F. (1989). On the geometrical relationship between tilt and twist grain boundaries, Zeit. Kristall. 189, Yang, C.-C., Rollett, A. D., and Mullins, W.W., in preparation. 6